Numerical Methods Applied Id Metallurgical Processing

NUMERICAL METHODS APPLIED ID METALLURGICAL PROCESSING

JUAN HECTOR BIANCHI

A thesis submitted for the degree of Doctor of Philosophy of the University of London and for the Diploma of Membership of the Imperial College

John Percy Research Grou Department of Metallurgy Royal School of Mines, Imper i a 1 Co 11ege London

JULY 1983 In bulk forming operations the plastic deformation is very large compared with the elastic one. This fact allows the use of constitutive laws based upon both current stress-strain rate measures. A Finite Element formulation of the mechanics involved in quasi-steady state conditions was implemented, introducing the incompressibi1ity via a penalty on the volumetric strain rate.

The perfectly plastic behaviour was considered first, and the extrusion process was thoroughly analyzed. Solutions for direct and and indirect modes of operation up to very high extrusion ratios are presented. Both plane strain and axisymmetric geometries were considered. Frictional conditions at the billet-container interface were introduced in two ways. Comparison between results obtained using these lines of approach and numerical problems associated to them are examined. Pressure results are compared with previously reported solutions resulting from Slip Lines, Upper Bounds and Finite Elements. The behaviour of the internal mechanics as a function of changes of geometrical and frictional conditions are examined.

Next, a non-linear flow stress resulting from hot torsion experiments was considered. The thermal analysis involves the solution of another equation, which is coupled with the mechanical problem due to convection, heat generation and flow stress dependence on temperature. After an appropriate transformation, that equation is formulated in a "weak" form and implemented to be solved jointly with the mechanical part. Application was made to quasi-steady-state analysis of axisymmetric extrusion. The dependence of solutions with billet length, tool parameters and numerical instabilities are examined and comparison is made with experimental results.

Finally, a review of dynamic analysis research on forming problems is carried out. The difficulties involved in implementing such formulations from the state of the work of the present thesis, and their consequences in the transient situation detected experimentally in extrusion processes are discussed. TABLE OF CONTENTS Page No ABSTRACT i i i LIST OF FIGURES Vi i LIST OF TABLES x NOMENCLATURE xi 1. INTRODUCTION AND LITERATURE SURVEY 1.1 Introduction 1.2 Analysis of Metal Working Processes 1.2.1 Experimental, empirical and hybrid methods 1.2.2 Slip line field techniques 1.2.3 Upper bound solutions 1.2 .4 Fi n i te elements 1.2.4.1 Elastoplastici ty 1.2.4.2 Viscoplasticity 1.3 Summary 22

2. BASIC CONCEPTS AND FINITE ELEMENT FORMULATION 2.1 I ntroduct ion 26 2.2 Mechanical Problem 26 2.2.1 Strain 2.2.2 Stress 2.2.3 Constitutive models 2.2.3.1 Elastoplasticity 2.2.3.2 Rigid-Plastic 2.2.3.3 Viscoplasticity 2.2.4 Variational principle - viscoplasticity model with penalty formulation 2.2.5 Finite element treatment 2.2.6 Stress analysis 2.3 Thermal Problem 43 2.3.1 Basic equations 2.3.2 Finite element treatment 2.4 47 Thermomechanic Coupling 49 2.5 Solution Procedure

3. EXPERIMENTAL 3.1 I ntroduct ion 54

3.2 The Extrusion Press 54 3.2.1 Container heating 3.2.2 Bi1 let preheat 3.2.3 Extrusion data recording 3.2.4 Direct extrusion tooling 3.2.5 Indirect extrusion tooling 3.2.6 Water quench 3.3 Experimental Procedure 61 3-3-1 Direct extrusion 3.3.2 Indirect extrusion 3.4 Materials 63

1 v 4. ANALYSIS OF EXTRUSION I: PERFECTLY PLASTIC IDEALIZATION

4.1 Introduction 65 4.2 Direct Plane Strain Extrusion Through Square Dies 68 4.2.1 Computational conditions 4.2.2 Direct extrusion, fc=2 4.2.2.1 Smooth container - smooth die 4.2.2.2 Rough container - rough die 4.2.2.3 Direct extrusion, R=10 4.3 Indirect Plane Strain Extrusion Through Square Dies 83 4.3.1 Computational conditions 4.3.2 Indirect extrusion, FL=2 and FL= 10 4.4 Alternative Introduction of Friction 92 4.4.1 Computational conditions 4.4.2 Mean strain rate 4.4.3 Plane strain extrusion 4.4.4 Axisymmetric rod extrusion 4.4.4.1 Direct extrusion, low ratios 4.4.4.2 Direct and indirect extrusion, high rat ios 4.5 Conclusions 113 5. ANALYSIS OF EXTRUSION II: THERMOMECHANICAL WORKING 5.1 Introduction 117 5.2 Mechanics H7 5.2.1 Material behaviour 5.2.2 Acceleration terms 5.2.3 Applicability of the steady state approach. I: assumptions and general considerations 5.3 One Dimensional Thermal Model 127 5.4 Extrusion Thermal Model 1 ing 133 5.4.1 General considerations 5.4.2 Applicability of the thermal model 5.5 Direct Rod Extrusion R=12.4, A15456 136 5.6 Direct Rod Extrusion R=20, Al5052 152

5.7 Higher Extrusion Ratios 156 5.8 Applicability of the Steady State II: Transient Limit and Steady State Flows 170 5.9 Conclusions and Recommendations for Further Work (l) 178 5.9.1 Post-peak load steady state 5.9.2 Initial transient 6. DYNAMIC ANALYSIS OF THE EXTRUSION PROCESS 185 6.1 Introduction 6.2 El as topiastic Model 185 6.2.1 General considerations 6.2.2 Variational principle 6.2.3 Frame of reference 6.2.4 Constitutive relationship 6.2.5 Integration of the equations of motion

v 6.3 Viscoplastic Model 6.4 Conclusions and Recommendations for Further Work (II

Appendix I: COMPUTATIONAL FORMS FOR FINITE ELEMENT ANALYSIS Appendix II: DATA ADO.UISITION AND GENERATION

ACKNOWLEDGEMENTS

REFERENCES LIST OF FIGURES

Page No Chapter One Fig 1.1 Isothermal Dynamic Analysis. Qualitative behaviour of load-displacement FE reported results 2b

Chapter Two Fig 2.1 Frictionless interface b2 Fig 2.2 Sticking interface b2

Chapter Three Plate I : General layout of the extrusion press 55 Plate I I a) Di rect tooli ng 60 b) Ind i rect tooli ng

Chapter Four Fig b. 1 FE mesh for extrusion through square dies 69 Fig 4.2 Direct Extrusion - boundary conditions imposed d i rect1y on nodes 71 Fig 4.3 Numerical convergence of the FE solution plane strain extrusion R=2, smooth walls 71 Smooth container walls - velocity field, principal Fig b.b stresses, mesh deformation and strain rate isolines for plane strain direct extrusion R=2 75 Rough container walls - velocity field, principal Fig b.5 stresses, mesh deformation and strain rate isolines for plane strain direct extrusion R=2 73 Smooth container walls - velocity field, principal Fig 4.6 stresses, flow lines and strain rate isolines for plane strain direct extrusion R=10 81 Rough container walls - velocity field, principal Fig 4.7 stresses, flow lines and strain rate isolines for plane strain direct extrusion R=10 82 Fig 4.8 Indirect extrusion - boundary conditions imposed d i rect1y on nodes 84 Fig b.3 Indirect plane strain extrusion R 2 effect of bou boundary conditions at ram for Fig b.8(a) 86 Fig 4.10 Plane strain indirect extrusion R=2. Flow patterns for the two ways of operation 86 Fig 4.11 Smooth container walls - velocity field, principal stresses, mesh deformation and strain rate isolines for plane strain-Jndirect extrusion R=2 89 Fig 4.12 Rough container walls - velocity field, principal stresses, mesh deformation and strain rate isolines for plane strain indirect extrusion R=2 90

vi i Fig 4.13 Rough container walls ^velocity field, principal stresses, flow lines and strain rate isolines for for plane strain indirect extrusion R=10 91 Fig 4.14 Dependence of strain rate, strain and hydrostatic pressure with mode of deformation and frictional conditions. Plane strain extrusion. R =20. Perfectly plastic material 97 Fig 4.15 Detail of principal stresses at the exit. Plane strain indirect extrusion R =20 98 Fig 4.16 Dependence of pressures with ram displacement. Axisymmetric direct and indirect extrusion 104 Fig 4.17 Comparison of FE and Upper Bound results. Axisymmetric direct rod extrusion 105 Fig 4.18 Comparison of FE and Upper Bound results. Axisymmetric indirect rod extrusion 106 Fig 4.19 Strain rate dependence with extrusion ratio. Axisymmetric direct and indirect rod extrusion 108 Fig 4.20 Effect of frictional conditions on flow pattern. Axisymmetric extrusion R=20 107 Fig **.21 Dependence of strain rate with mode of deformation and frictional condition. Axisymmetric extrusion R=20 110 Fig 4.22 Dependence of strain with mode of deformation and frictional condition. Axisymmetric extrusion R=20 111 Fig 4.23 Dependence of hydrostatic pressure with deformation and frictional condition. Axsymmetric extrusion R=20 112

Chapter Five Fig 5-1 Hot working behaviour of two Al alloys 119 Fig 5.2 Solution for the one-dimensional thermal model 131 Fig 5-3 Mesh geometry for FE analysis of direct rod extrusion R=12.4 137 Fig 5.4 Thermal boundary conditions 137 Fig 5-5 Direct rod extrusion R=12.4. Effect of heat transfer to container and friction on load-displacement curve 141 Fig 5-6 Effect of both initial billet and container temperature on pressures. R=12.4 142 Fig 5.7 Flow lines for axisymmetric direct rod extrusion R=12.4 144

Fig 5.8 Hot deformation history along flow lines. Direct rod extrusion R=12.4 144 Fig 5.9 Dependence of the temperature rise on initial billet and container temperatures 145 Fig 5.10 Velocity field and principal stress. Direct rod extrusion R=12.4 1i,9

vi i i Fig 5. 11 Strain dependence with ram speed. Direct rod extrusion R=12.4 150

Fig 5. 12 Details of principal stresses in the deformation region and die exit corner. Direct rod extrusion R= 12.4 151

Fig 5. 13 Effect of friction factor on loads. Direct rod extrusion R=20 155

Fig 5. 14 Temperature history along flowlines under severe deformation. Indirect rod extrusion R=80 155

Fig 5. 15 Dependence of the local equivalent strain rate with ram speed 159

Fig 5. 16 Mean equivalent strain rate dependence with ram speed 164

Fig 5. 17 Comparison of local and mean measures of strain rate variations with ram speed 167

Fig 5. 18 Dependence of temperature rise with ram speed 164 Fig 5. 19 Die wall reaction dependence with ram speed 167

Fig 5. 20 Ram force dependence with ram speed 168 Fig 5. 21 Flow pattern for direct rod extrusion R=80 172 Fig 5. 22 Flow pattern for indirect rod extrusion R=80 173 Plate 1 11 Extrusion Flow patterns 175

Fig 5. 23 Minimized Upper Bound construction for the partial extrusions of Plate III 174

Append ix 1 Fig Al .1 Extrusion geometries 209

Append i x 1 1 Fig Al 1.1 Input data 221 Fig Al I .2 Material sets for border elements 216

x i LIST OF TABLES

Page No Table 3- Alloy composition in weight percent 63 Table 4. Results for Direct Extrusion, Plane Strain, R=2, Smooth Container-Smooth Die Wall 72 Table 4. Comparison of FE results for Plane Strain Direct Extrusion, R=2, frictionless walls 74

Table 4. Results for Direct Extrusion, Plane Strain, R=2, Rough Container and Die Wall 77 Table 4. V FE Results for Direct Extrusion, Plane Strain R=10 80 Table 4.V FE Results for Indirect Extrusion, Plane Strain 87 Table 4.VI FE Results for Direct and Indirect Plane Strain Extrus ion 94 Table 4.VI I Comparison of Results for Direct Rod Extrusion, low reductions 100 Table 4.VI I I FE solutions for Axisymmetric extrusion 102 Table 5-I Hot working parameters 48 Table 5®I I Comparison of Experimental and FE results. Direct Rod Extrusion R=12.4, Al5456 139 Table 5®MI FE results: In Z dependence with temperature across the rod section 147 Table 5®IV Direct Axisymmetric Rod Extrusion R=20, Al5052 153 Table 5®V Direct Axisymmetric Rod Extrusion R=30 and R=80, 157 Ai 5052 Table 5®VI Frictional forces estimations for Bl=34mm 169

x NOMENCLATURE

Cartesian coordinates Cylindrical coordinates Local coordinates Young Modulus Poisson's ratio Shear modulus Bui k modulus Strain tensor Strain rate tensor Deviatoric strain rate tensor Volumetric strain rate Equivalent strain rate Strain rate tensor invariants stress tensor Deviatoric stress tensor Hydrostatic stress tensor Deviatoric stress rate tensor invariants Hydrostatic or mean stress Friction factor Yield stress Ram pressure Frictionless estimate of the ram pressure Extrusion ratio Billet length Bi1 let diameter Tool velocity Temperature Dens i ty Conduct i vi ty Specific heat Preheating temperature Container temperature Coefficient of heat transfer Z Zenner - Hollomon parameter or temperature compensated strain rate

AH Activation energy

Matrices and vectors

Single underlying indicates the measure is a vector. Double underlying is used for matrices. Superscript T for both of them indicate transposition. A dot on top of a symbol expresses partial time-derivative.

u> u (ch 6) D i splacement u, u (Ch 6) Veloci t ies u (ch 6) Accelerat ions K Stiffness matrix H Thermal matrix f External loads Differential operators k> I N Shape functions matrix

£ Strain rate q' Deviatoric stress

D constitutive law matrix

xi i CHAPTER ONE: INTRODUCTION AND LITERATURE SURVEY 1.1 Introduction

The initial aim was the study of extrusion of aluminium alloys in the hot working regime. In addition to load evaluations and mode of deformation _dependence wi th. jfrj.ctional jcond i tions,_pred ictions of both temperature and strain rate distribution were pursued. These latter ones for setting the connection of the compensated strain rate with process parameters.

The Finite Element Method appeared as the most versatile tool for such analysis and the work was oriented towards the production of such a code. The viscoplastic model (l, 2, 3) jointly with the penalty approach (4, 5) to satisfy the incompressible behaviour of the material was selected. Such formulation takes advantage of the flow plasticity constitutive form, since in most metal forming processes the plastic strain usually outweighs the elastic strain. Besides, its current strain rate dependence of the flow stress sets directly the tool for handling constitutive equations as found in hot working. As the main concern was the steady state analysis, such flow approach is computationally more convenient than incremental elastoplasticity formulations. It is also able to eliminate mesh updating which is the more severe drawback of elastoplastic codes when sharp geometries such as the die corner are present.

A review of available technology for analysis of bulk forming processes is given further on in this chapter,with particular emphasis on the solutions obtained through finite element applications. Chapter two sets the basic background and describes the details of the for-r mulation selected for both mechanical and thermal field under steady state condi tions.

The code was first applied with the perfectly plastic idealization of the material. Its performance were checked against previous extrusion results provided by Slip Lines and Upper Bound, with a successful outcome, as summarized in chapter four. A search for a computational tool for fast display of strain rate fields revealed unexpected difficulties.In spite of that, some contouring reported in chapter four indicates the excellent agreement of the main strain rate lines with slip line predictions. A post-procesor was designed for graphical display of principal stresses and grid distortion analysisDurtng its prbducTTibn, a break-through of the strain rate fields inspection was achieved, by numerical operations on the advancing steady state flow line. The technique was also extended to the total strain and hydrostatic pressure fields, and is used to explore the differences arising from different frictional conditions in both direct and indirect extrusion.

In chapter five a hot working constitutive equation is introduced in the model and the coupling with a thermal field is produced. First, the conditions of applicability are examined. Bound dependence with the experiments according to the procedures described in chapter three are examined. Instabilities in numerical convergence set limits to the applicability of simplifications in the thermal analysis, which are discussed in details in chapter five. Internal fields, including the compensated strain rate variation across the deformation zone, are inspected. On an experimental basis,the limits of the steady state analysis are rediscussed. In chapter six, a critical examination of the main dynamics formulations is carried out. Suggestions for further work for both steady state and transient analysis are given at the end of chapters five and six.

1.2 Analysis of Metal Working Processes

One of the traditional lines of research has been the production of empirical correlations linking together overall parameters. On a more theoretical basis, the application of the flow theory of plasticity lead to the development of slip-line and Upper Bound techniques. Initially designed as graphical methods, computational versions of both have been developed. From experimental grid distortion measures, the visioplasticity method was produced, oriented towards the strain and stress analysis of the plastic deformation; faster processing using numerical techniques was only produced lately. The advances of Finite Element techniques in elasticity lead to the extension of stress-strain material description typical of elastic range to post-yielding (elastoplastici ty); another main line has been developed by application of flow plasticity (viscoplasticity). Finite Difference codes never-achieved the versatiH-ty—in-hand 1ing different geometries, and are far less widespread (16). The current state of development of Boundary Integral Techniques (38), still prevents its general applicability; one of its drawbacks comes from the formulation in terms of the outer surface being unable to handle successfully coupled internal non-linearities, which can only be avoided at the cost of large discretization, then becoming competitive with finite elements. Other numerical techniques such as the Method of Lines (6), or Operational Methods (7), have only been tested in limited simplistic conditions, and are by far less developed for handling non-linear problems, not simple geometries and updating requi rements.

With the incursion of computers as an everyday tool, it is becoming no longer necessary to deal with closed solutions only. The present trend, in common with most of other fields of engineering is the application of numerical techniques to the metal forming analysis trying to avoid as many simplifications as possible. Thus, computer modelling is becoming a useful tool of prediction and exploration of incidence of both boundary conditions and material behaviour, which has posed new requirements regarding experimental measurements. The more integrated nature of the treatment produces a stream of field variables,from where analysis of more complex situations can be designed.

1.2.1 Experimental, Empirical and Hybrid Methods

Much of the early work in forming was concerned with assessing the capability of the tool to complete successfully the desired operation by establishing expirical relationships among global parameters as presures or forces, speed, reduction and pre-heat temperatures. In particular for extrusion, such trend has been the predominant approach in the hot working region. Even now experimental and industrial

4 expertise follows the trend of being condensed either as an empirical law (8, 12), or as limit diagrams (9), relating together those overall parameters with the feasibility of obtaining a product exempt of defects. The availability of temperature rise and strain rate from upper bound predictions allowed Sheppard and Raybould~(l0), to introduce the structural dependence in limit diagrams; a drawback arises from the inability of the numerical model to give information on the inhomogeneous nature of the deformation across the extrude, which restricts microstructural predictions. Wood (11), expanded the limit diagram concept by including the effect of extruding different section geometries. Sheppard (12) improved the continuum aspect by defining the pressure limiting line as that required to initiate extrusion and not the steady state pressure as considered by previous workers.

On the other hand, in the visioplasticity approach of Thomsen et al (13)» the main concern was the internal mechanics. The procedure consisted of determining experimentally particle velocities on a meridian plane through the use of incremental deformation steps; with such data the strain distributions were calculated. Introduction of these strains into the equilibrium equations allowed the determination of stresses; the technique was applied to extrusion of lead. Kobayashi et al (14), developed a visioplasticity computer procedure which uses a flow function; the measured values of grid displacement are smoothed out and the strain and stress distributions are produced by numerical differentiation (15). The method was applied by Shabaik (16) to axisymmetric extrusion of steel; a similar research was reported by Chi Ids (l7).Altan and Kobayashi (18), working with low extrusion ratios, calculated the temperature rise from the visioplasticity solution, and checked successfully against their experimental temperature measurements. More recently Moore and co-workers (19) produced a computationally oriented visioplasticity procedure for dynamic and non-steady metal working processes, which was applied to plain strain upsetting. So far, the main drawback of visioplasticity techniques, is set by the time consuming operation of grid distortion measurements from the experiments. A simplistic model which gives mean strain rate values for direct extrusion was proposed by Feltham (20); the deformation was assumed to occur within a cone of semi-angle of 45°, and the total strain to be equal to Ln R:

t - 6 V Ln R (1.1)

where V = ram speed and = billet diameter and R the extrusion ratio.

Alexander (21) suggested that in order to account for the inhomogeinity of the deformation, the total strain should be replaced by: a+b Ln R . This relationship still asumed a semi-angle of 45°, which is rarely so. Wilcox and Wi tton (22) pointed out that the equation should contain a term for a variation in semi-angle. Castle (23) made this addition proposing:

2 y _ 6 (a+b Ln R) tan tu (1.2)

where UJ is the deformation zone semi-angle. Unfortunately all these expressions assume conical deformation zone, whereas in practice this is usually curved, but have the advantage of their simplicity.

Direct measurements of temperature are difficult, because thermocouples inserted in the billet are sheared during the extrusion and pyrometric methods are hindered by the poor emissivity of the aluminium extrude. Nevertheless measures have been made at the National Engineering Laboratory (24, 25) using thermocouples protected

6 with stainless steel sheaths. In terms of the industrial process, measurements of surface temperature by contact thermocouple such as NANMAC have been successfully conducted (26).

1.2.2. Slip Line Field Techniques

Analytical integration of the steady state equilibrium equations is only possible for certain problems with simple boundary conditions. An alternative graphical method was developed from the properties of the directions of maximum shear (27). Thus, Slip Line Fields were constructed step by step, from some known initial slip line; if the boundary conditions were such that no starting line was evident, a trial and error procedure had to be followed at each time step. Overall forces, together with some local stress evaluation were obtained. The Saint Vernant - Von Mises perfectly plastic idealization of the material restricts its application to materials bearing an approximation of their flow stress by a constant value over the deformation range; however, some extension to work-hardening materials by defining a medium shear value have been produced (28). For extrusion, in a limited range of reduction, the normalized pressure results were found to follow in general a law of the form:

= a + b Ln R (1 .3) where a and b are friction-dependent constants (29).

Symonds (30) studied the possibility of applying the flow theory of plasticity to axisymmetric problems. He concluded that the equations become elliptic, and therefore the "characteristics" lines of integration, base upon which the slip line method was developed, were no longer applicable.

Bishop (31) proposed a slip line field for extrusion which calculated the temperature rise from the work done by shear at the velocity discontinuities; this assumed no heat losses to the container, and was extremely laborious to calculate. More recently, a numerical method called Matrix Operator Technique was developed to produce Slip Lines (32, 34). Basically, the radius of curvature of a siip 1ine or a segment of it is expressed as an n - dimensional vector whose coefficients depend on some linear operators; such operators, expressed as nkn matrices are constructed for particular boundary conditions and shapes. Thus, starting from the border the radius of curvature is evaluated for inwards positions, and the whole slip line constructed by a computational procedure.

An updated survey of previous slip line field solutions for forming problems is given by Johnson (35, 36, 37).

1.2.3. Upper Bound Solutions

Within the frame of flow plasticity, a simplification aimed at load evaluations only,was produced by Johnson (39, 40). The method applies an Upper Bound theorem due to Prager (41) and introduces regions whose borders fit the high shearing lines predicted either from slip line or experimental evidence. No deformation is allowed within such regions; all changes happen at the borders, the latter ones becoming discontinuity lines. The internal rate of work is composed of two terms (35, 43). The first is provided by the frictional forces at the tool surface. The second accounts for the energy dissipation within any of those regions and can also be transformed into surface integrals expressed in terms of nominal shear and velocity discontinuity. Thus, the total internal rate of work produced by the kinematically admissible field gives an overestimation of the required external power, and from it the load is assessed.

The technique was first applied to axisymmetric extrusion by Alexander (44). Kudo (45) developed the unit region concept to optimize the shape of the partitions at the deforming region. Different triangularization arrangements of the deforming regions were proposed (47 49, 183, 184); curved boundaries were also examined (46, 47, 50). Hailing and Mitchell (49) used an optimized double

8 triangle arrangement of velocity discontinuities; suggestions were also made for work-hardening and frictional effects (49), though this does not give a true upper bound (51). Sheppard and Raybould (10) designed a computational optimization scheme which minimized the load by varying the angles defining both the deformation zone and triangularization. Tunnicliffe (54) and Tutcher (55) examined the response of single and double triangular arrangements with the same scheme, for direct and indirect extrusion.

Tanner and Johnson (56) obtained the temperature distribution from upper bound plane strain solutions; balance of heat losses and heat generation at the discontinuity lines was considered,and conduction effects were neglected. Johnson and Kudo (57) extended this work for axisymmetric extrusion and introduced terms for heat generation within the deforming regions. Sheppard and Raybould (10) also evaluated temperature rise from upper bounds; however, the differences between the billet temperature and the tool were not taken into account. Sheppard and Wood (58) improved this analysis with an Integral Profile method which globally allowed for heat losses.

Sheppard and Raybould (59) obtained strain rates along a series of streamlines from the upper bound construction; the method permitted an indication of their variation across the section and produced mean overall strain rate figures. Its disadvantage is that deformation is assumed to occur instantaneously at a number of discontinuities. Mc Shane et al (60) improved on this method by dividing the deforming region into small elements and calculating the strain and strain rate for each element, from which an average strain rate was produced.

This refinement, jointly with the temperature predictions of Ref (58), allowed (8, 55) the flow stress to be related to local hot working variables; the method is restricted to steady state situations.

1.2.4. Finite Elements

Two main approaches have been pursued: elasto-plasticity and viscoplasticity. The results produced in connection with forming processes are reviewed below.

1.2.4.1. Elastoplastici ty

Early applications of FEM to plasticity problems started immediately after its success in linear elasticity.

The initial approaches started with the total stress-total strain form of the constitutive law in the elastic range, and two main lines were followed. In the "thermal strain" (60) one, latter

generalized with two functions £Q and aQ depending on the strain level (5):

a = D.(e - e ) + a (l .4) — = — -o -o

pure elasticity at small strain is achieved by setting £0 = = 0; at higher strain levels they become "corrections" to the elastic behaviour in order to describe the plastic range. The second line, so called "initial load" technique was started by Argyris (61, 62). Either from the imposed initial strain or stress, an initial load vector is defined within each finite element. For the first imposition such initial forces are designed to suppress elastically the imposed initial strains. In the second, an equivalent set of initial forces are obtained by their equivalent kinematic ones.

From a material point of view, the elastoplastic analysis is non- linear in nature; therefore, a method of solution based on a sequence of linearized step is preferable. Thus, elastoplastic stress-strain relationships are normally expressed as incremental laws; in the numerical model,they can be adjusted in each load increment to account for the plastic deformation history. Such considerations, lead to to the so called "tangent stiffness" method proposed by Pope (63). The original version involved so much computing time that it was "uneconomical . "MocIi fTcat i6~h~s~were~proposed "separate!y" by Swedlow and Young (64) and Argyris (62). The latter one used a variation of

10 initial load techniques. Up to that point, the main concern were structural mechanical problems.

Mar.ea_l_and King_(f>5)- introduced a partial stiffness concept by taking linear increments of the equilibrium equations. They produce solutions for a thick walled cylinder problem; comparison with known analytical solutions and experimental results were reported. Although good agreement was in general obtained, the analysis of the deformation had to be interrupted at some stage because negative strains were emerging.

Yamada et al (66) modified Marcal's algorithm and used it for collapse predictions in structural problems; here again, the procedure had limited applications because of negative strain developments. However, this work influenced several workers, and a stream of papers using such approach were published. Analysis of hydrostatic extrusion was carried out by Iwata et al (67). Plane strain upsetting, including frictional effects were studied by Nagamatsu (68). The most extensive application to metal forming problems is due to Lee and Kobayashi (69~72) They analysed the problems of flat punch indentation, Brinell hardness test, upsetting, side pressing and extrusion. They followed the deformation to gain information of plastic zone developments; strain and stress distributions and geometrical changes of the workpiece were explored. Examination of residual stresses after unloading was also carried out. However, these were exploratory applications; the accuracy of the solutions, obtained, especially where large strains were present, is doubtful.

Other lines of work were developed in order to overcome the computing difficulties of the tangential stiffness method. Depending on what is the explicit form available for the constitutive law and the range of variation of cr and £ , either £ = £ (cr) or cr= a (e) forms were introduced in the analysis; these gave origin to the so- called "initial strain" (62) and "initial stress" (5) methods respectivel The last one, developed by Zienkewicz et al (5), was applied to analysis of pressure vessels, indentation and plain strain extrusion (73) using

11 isoparametric elements. This last example was produced for extrusion ratio R=2, with frictionless conical dies; the boundary conditions were applied directly to the nodes, to deal with only the non-friction walls case. By finding the response to small ram displacements, they obtained a load-displacement curve which shows rise from zero and fall to a horizontal plateau as the curve (I) in Fig 1.1. This result is particularly important because it was obtained without the explicit inclusion of acceleration terms for the early transient and explicit time integration. However, that is partially taken into account throughout the mesh updating operation. Results of p/J for both Tresca and Von Mises yield criteria are reported, but a mis- quotation of the work by Johnson (42) lead to wrong comparisons; correcting such mistake, values of p/2k of 0.82 for Tresca and 0.95 for Von Mises are obtained, against Johnson (42) slip-line result of 0.90. However, for this particular geometry and smooth die, the updating mesh operation is greatly simplified; such is not the case when either friction increases or square dies are considered.

More research using variations of the tangent stiffness method has been lately produced. Hartley (76) and co-workers have applied it to extrusion and ring upsetting. A Najafi-Sani (77) looked at the problem of clamp design in the continuous hydrostatic extrusion.

Most of the literature reviewed so far is concerned with small- strain plasticity, originally developed for dealing with the onset of yielding and early spreading of the plastic zone. Large strains introduce geometric non-linearities which require a careful examination of the measures of stress and strain used (53). The spurious component in the resulting local stress produced by rigid body motion must be disregarded. This compels the introduction of stress rate-strain rate forms of the constitutive relationship, and normally a time integration is needed; such requirement opens new capabilities in dealing with material non-linearities and acceleration terms. A detailed examination of these formulations is carried out in 6.2. Below, only the results relevant to forming problems are reviewed.

12 Hibbit et al (79) extended Marcal's work (80) to large strain and displacement, with a description in a static configuration. With a similar approach Gordon and Weinstein (81) examined plane sheet drawijig.. for .both smooth and fractional conditions.

Kitagawa et al (83) proposed a linearized incremental form of the equilibrium equations but in a convected coordinate system embedded in the body. Extension of infinitesimal procedures to large deformation

in Eulerian frame of reference were produced by Osias et al (84, 85) and Swedlow et al (86, 87).

What is now known as Updated Lagrangian formulation can be traced to the early paper of McMeeking and Rice (88), further developed mainly by Lee and co-workers (89-91), Bathe et al (92-95) and Yamada et al (96) updated schemes were also proposed and developed by Argyris et al (97~99)

Dynamics analysis of axisymmetric extrusion, rolling and sheet stretching, using the Updated Lagrangian formulation (section 6-2) was reported by Bathe, KeyAco-workers (95)- The elastoplastic material was assumed to obey a kinematic hardening rule, with a tangent/elastic modulus ratio of 0.005 and a Poisson's ratio of 0.3.

The main features of the extrusion example in (95) are: R=1.77, conical dies and ram moving at constant speed; the effect of three different frictional coefficients were explored: 0.0, 0.1 and 0.3. The load results exhibits a dependence with the ram displacement which qualitatively agree in general with the experimental observation; for all values of friction, a rise from zero load to a maximum value is achieved. In a frictionless case a steady horizontal plateau is observed afterwards (curve I, Fig l.l);when some friction is allowed,the peak achieved is higher and from there the curve moves along a short plateau and then drops at a constant slope (curve II, Fig l.l). In spite of the extremely low extrusion ratio, such result suggests that the prominence of the load peak and the change in slope after the first decay from a maximum value experimentally observed in hot working, comes from another source. Neither the integration of the equation of motion which UL goes through, nor the inclusion of elastic effects

13 of elastoplasticity are able to predict such phenomena. From these isothermal results, and from the analysis of the ones produced with the thermocoupled model, a possible explanation is suggested in the present thesis -in section 6.4. Another feature of-the-extrusion— — example in Ref (95) is the solution exhibiting distortion of the mesh at the rear of the container wall, revealing a high shearing area, when the friction coefficient is non zero; in any case, distortions in the die area were present. Local values of stresses and pressure distributions for the steady state achieved are also reported.

The rolling example in Ref (95) features a*33%% of thickness reduction, and the deformation is traced from its onset. Records of both plate thickness and deviatoric stress variations along two lines: one on the center and a peripheral one were produced for the steady state. The surface thickness exhibits both uplifting just prior to enter in contact with the rollers and elastic recovery after exiting them. The first phenomenon happens jointly with an overshoot of local stress, which involves plastic flow, followed by a brief elastic unloading and then a return to plastic flow (once plastic flow has started, any retreat from monotone loading represents elastic unloading from a work hardening state). They attribute such result to a mistmatch of roller and plate local velocities,requiring further investigation.

In a more recent publication, Bathe et al (100) applied a more refined UL incremental analysis procedure to plain strain punch indentation problem. Load and growth of plastic zone results are reported and the dependence of the solution with mesh size and number of integration points is discussed.

Transient analysis of extrusion were carried out by Lee, Mallet, McMeeking and Yang (89, 90, 101, 102), using Updated Lagrangian schemes. They considered a material with kinematic hardening, with a ratio of tangent/elastic moduli of 0.16. The tangent modulus (da/de)was about four times the yield stress; Rice (88, 103) pointed out that when

14 these parameters are of the same order,rotational effects must be included in the analysis. The constitutive matrix had twofold description, incorporating plastic effects only if the yield stress was exceeded, with a smooth derivative in the transition between the two modulus. Constant strain triangular elements, arranged on the cross-diagonal of a quadrilateral were used; such pattern have been shown to satisfy more efficiently the condition of incompressibi1ity and to avoid locking effects, by Nagtegaal and co-workers (104). The extrusion problem solved in Ref (90) was plane strain case, R lower than 2, wi th conical smooth die. The reported ram load-displacement results exhibit the shape of curve III in Fig 1.1; slight oscillations in them were detected, which were attributed to the discontinuity of the updating mesh procedure. These oscillations were reduced by disminishing either mesh size or time step; the load does not rise from zero, which suggests that quasi-static analysis has been applied,

neglecting the initial accelerations. In Refs (89, 101), the capability of dealing with friction along wall and die was incorporated. A larger ram load than in the frictionless case was obtained; it featured a steady drop with billet length after a maximum, as in curve II, Fig 1.1, which was attributed to surface forces. A-posteriori examination of the longitudinal internal stresses, confirmed a virtual steady state being achieved in spite of such decrease of the ram load. Further developments (89, 102) allowed the analysis of axisymmetric extrusion. Results are reported for the first geometry and for another one with lower reduction

(R a O.96). For the latter one, the stresses at the axial line become less compressive and even tensile, leading to a central line with residual axial stresses; this result was not observed in plane strain situations. It is suggested that such an effect is only a characteristic of the axisymmetric case, where the hydrostatic pressure partially shields the interior from the outlet; this inhibits yielding in the interior, whereas in plane strain no such shielding takes place.

Yamada et al (96) also gave a solution for plane strain extrusion, R = 1.15, through double curved conical die. The procedure is designed as an Updated Lagrangian scheme and "crossing diagonal triangles" (104) were used. Frictional conditions were incorporated through a flow stress depending on the relative displacement at the wall, with an initial

15 linear variation followed by a cut-off value equal to the yield stress afterwards. The material was considered to be el as topi astic, with a relationship work hardening/elastic moduli of 0.016 and a yield stress 2 of 20 kgr/mm , and Poisson's ratio of 0.3. The results for ram load- displacement qualitatively agree with the findings of Lee at al (90) and Bathe et al (95); Varna da (96) does report oscillations at the departure of each surface element from the die.

Derbalian, McMeeking and co-workers (105) introduced a modified Updated Lagrangian scheme which uses a fixed mesh; it was tested with plane steady radial flow through a converging channel. This example is an idealization of smooth conical die extrusion.

Thermally coupled dynamic Finite Element analysis were carried out hy Argyris et al (98, 99) using the "natural" formulation. The strain records were kept by means of Almansi strain. The analysis included irreversible thermodynamic considerations; the material referred to as aluminium had a flow stress with the yield point depending on temperature and the tangent modulus on total plastic strain and temperature. Results are reported for heading of a bolt, axisymmetric extrusion (98) and rolling (99). The extrusion example considered a 20 mm diameter billet, and smooth conical dies with an extrusion ratio of 1.56. An extremely high ram speed of 300 mm/sec was used, which produced a maximum temperature rise of 120°C from room temperature; no report of ram load or internal mechanics is given, but thermal performance is examined for both adiabatic case and allowing some transfer to the tool. Results are reported for the evolution of the heat arising at the high shearing near the edge of the die exit; an asymmetric advance of the hot front was found. The backward effect due to conduction is much slower than the forward one due to the hot solid movement. In the rolling example (99) such behaviour ahead of the peak is shown to depend on the rollers speed; at faster deformation the process becomes more adiabatic and reaches a higher plateau temperature. At lower speed the heat transfer notoriously damps such steady temperature.

The interesting outcome of both works (98, 99) is related to the time- marching scheme used to solve the equations, giving predictions of clear experimental feasibility. In the rolling example (99) roll force and torque exhibit oscillations in the steady state, due to the separation of grid elements from the contact with the rollers. Their presence in such a smooth geometry (reduction h /h° = 13/20) shows the difficulty of a procedure operating by mesh updating in handling rough surfaces.

1.2.^.2 Rigid - plasticity and viscoplasticity

Here the flow theory is applied linking total current values of stress and strain rate rather than increments. Most formulations derived the finite element equations by minimizing the total internal work (or work rate) given particular boundary conditions; this is done through a variational principle (5, 97). Neither of the variational principles used to derive the elementary equations provides for enforcing the incompressibi1ity constraint, which is a necessary assumption. Several possibilities for such enforcing have been presented, which provide the label for each different formulation:

a) Stream function

This was the most usual procedure for describing incompressible velocity fields in fluid mechanics. Goon (106) used mapping techniques to obtain solutions for metal flow problems. The FE implementation for metals was done by Godbole (107) and Zienkiewicz and Godbole (108) who treated the plastic and viscoplastic flow of metals as a special case of non-Newtonian fluids. They solved different extrusion cases and the non-steady problem of punch indentation. The procedure is based on the introduction of a stream function and its FE implementation starts from the virtual work principle and requires elements whose shape functions have continuity of nodal variable and its first derivatives. Unless velocities are entirely defined on all boundaries, the boundary conditions are difficult "to enforce.

17 b) Lagrange multiplier/velocity and pressure fields

The application of this formulation to metal flow problems can be traced to three different and apparently independent sources. In 1972, M Lung (109) outlined the method and using triangular elements presented a solution for plane strain extrusion. In the same year, C Lee and Kobayashi (110, 111) presented a similar formulation and proposed a linearization of the resulting equations by using a simple perturbation method. They reported a solution to the problem of simple upsetting (111) and also applied the method to assure compressibility and smooth errors from visioplasticity analysis of extrusion. Godbole (107) proposed an equivalent formulation; however his starting point was not an upper bound theorem as in the previous two, but an extension of his work on the analogy of viscous fluids and metal flow; the formulation was later extensively used by Jain (112).

The basis of the method lays in the introduction of two functionals; the first accounts for the total internal energy: distortional plus dilatational (the latter one made dependent on a Lagrange mu Itiplier (110, 115)). The second functional assures that the incompressibi1ity condition is satisfied. The resulting coupled system of algebraic equations has a double set of nodal unknowns: velocities and pressures, and is solved simultaneously. Both bilinear (107) and biquadratic (110, 111) elements were used.

Lung and co-workers (113, 11^, 115) applied the method to solve extrusion and upsetting problems under various frictional conditions, in both plane strain and axisymmetric situations; hardening and non-hardening materials were also considered.

Lee and Kobayashi (111) coined the label "Matrix Method" and used it to obtain new solutions in cylinder compression. Kobayashi and co-workers extensively applied the method to compression of large cylinders (116) , cold heading (117), bore expanding and flange drawing of anisotropic sheet metals (118), piercing and extrusion (119). This work was extended by Chen et al (120), who also

18 examined the extrusion and drawing solution with experimental data and a void growth model pointed towards defining workability and fracture criteria. Matsumoto et al (121) introduced means to handle friction conditions on surfaces whose direction of flow is not known a-priori ,for work hardening materials; Chen and Kobayashi (122) extended this feature to ring upsetting, comparing with upper bound and experimental results.

Gotoh and Ishire (124) introduced a local system of convected coordinates into each element so that geometrical non-linearity could be taken into consideration; the approach was used to tackle the problem of deep drawing. Dawson and Thompson (125-128) also included rigid body movement corrections and applied the procedure to steady state rolling and axisymetric extrusion, ratio R = 4. The elastic contribution to the flow stress seems to be conserved in all the deformation range (125); no reports of it switching off after yielding is given, which suggests a strict application of the elasto-viscoplasticity (2) concept; incompressibi1ity was enforced in all elastic and plastic ranges. Another attempt to produce a constitutive law which encompassed total strain in addition to current strain rate level was carried out by Dawson(l28) temperature profiles were also produced from steady state cons iderations.

Tomita and Sowerby (129) used the Lagrange multiplier and a perturbation method to asses the incidence in the process of the parameters of an exponential strain rate law. This quasi-steady- state analysis was applied to tube nosing and drawing and plane strain side-extrusion. For the latter one, the results for the first perturbation indicated a linear decrease of the forming pressure with a decrease of the exponent of the strain rate; such dependence become parabolic when the second perturbation was also considered.

19 c) Penalty function approach

In this approach the incompressibi1ity is approximately satisfied by expressing the local volumetric strain rate as the local hydrostatic pressure divided by a large number a (the penalty parameter). Thus, only one functional needs to be minimized and the requirement for also discretizing the pressure is eliminated, together with a large saving in computing. The penalty parameter a can be identified with the value (K - 2/3 G) for an isotropic elastic material, where K is the bulk modulus and G the shear modulus (75). Thus, as: K =( 2/3) G (l + 17)/(1 - 21j) a big value of a is equivalent to the well known plasticity condition of the Poisson's ratio approaching 0.50. This approach was originally established by Zienkiewicz (130) and subsequently used by Zienkiewicz and Godbole (131) for punch indentation and extrusion. Because of its simplicity of implementation it has become widely popular.

Chen and Kobayashi (123) analyzed multi-pass rod drawing and extrusion through conical dies, R = 2, of work hardening material; the study was made under steady-state assumptions, and the strain values were obtained by integration of strain rates along flow lines as in Ref (119). They used four-noded elements, and compared the Lagrange Multiplier and Penalty approach results, establishing the requirement for under integration of the dilatational energy by using only one Gauss point.

Cornfield and Johnson (132) tackled the problem of hot rolling and studied the effects of various temperature profiles. Alexander and Turner (134) solved the problem of hot dieless drawing; they achieved some success on the prediction of the advance of the deformed shape, but some oscillations in stresses were found which was attributed to the simple elements used. Price and Alexander (135, 136) presented a solution for isothermal forging using quadratic elements and an exponential strain rate law. Pacheco (137) achieved the same results when using both (u/p) and the penalty approach for frictionless plane strain extrusion R = 2; he also applied the penalty formulation to hydrostatic coextrusion and open forging of perfectly plastic and work hardening materials four noded elements, with 2 x 2 integration points for the stiffness matrix and one central point for the penalty matrix were used.

Jain (112) and Zienkiewicz et al (138) extended the applications to a wide range of steady state problems: rolling and plane strain and axisymmetric direct, indirect and pipe extrusions. They found that nine noded isoparametric elements and use of selective integration (stiffness 3x3, penalty 2x2) produced better resul than reduced integration (both stiffness and penalty 2x2) and eight noded elements. A special element was introduced to account for frictional effects of intermediate situations but its performance was not fully explored. Perfectly plastic and work hardening materials were considered. Solutions for the transient problems of stretch forming, deep drawing and axisymmetric cup forming are reported (138). The last one is interesting for its similitude with indirect extrusion. Jain (112) managed to follow its transient from the onset of deformation, obtaining a load-ram displacement curve showing a rise from zero and a plateau as in curve IV of Fig 1.1.even for the simplistic case of full slipping boundary conditions, oscillations on that loci were observed. High amount of mesh redefinitions were applied, and the problem was not solved by a time marching scheme; rather it was implicitly integrated by mesh updating driven by the equilibrium solution. Thirty five updating steps were used, each going through the expensive operation of matrix inversion. In spite of their claim of success in this transient extrusion study, its general applicability is doubtful; full sticking conditions can produce even more severe distortion of elements at the die edge due to large mismatching in adjacent velocities. The implications of mes redefinitions have not been fully explored, in spite of the mesh being the only record of the dynamics.

Zienkiewicz et al (138, 139, 140) and Zienkiewicz and Ohate (141), used the penalty formulation together with a steady state energy dissipation to solve the problem of coupled flow in plane strain

21 extrusion and hot rolling; upwinding techniques (1*42 , 1 *43 , 1 *4*4 ) were used for the shape functions of the thermal discretization.

The extrusion example considered a ratio of R = *4 and the material referred at Ti - 6A1 - 6V exhibited no strain rate sensitivity at all, only temperature dependence on its flow

stress. Hot rolling of aluminium, with a reduction = 1.3*4 was also considered.

The penalty function approach is the simplest to implement, as its most basic form only involves a simple modification of existing elastic codes. Latter investigations (1*45, 1*46) have given mathematical formality to the approach and its particular form of numerical integration.

1.3 Summary

Most of the surveyed steady state extrusion work has been produced at very low ratios. The incidence of frictional conditions on load and deformation patterns for both direct and indirect mode need to be properly determined. The viscoplastic finite element formulation with the penalty parameter appeared to be most convenient technique. Within this steady state framework, the performance of thermal solutions under the more severe deformations inherent to axisymmetric problems need to be assesed; also the incidence of flow stress exhibiting strain sens i t i vi ty.

The initial deformation needs special treatment. So far, only very low extrusion ratios have been considered, most of them with conical dies were the mesh updating is not as critical as in the squared one. The consideration of early velocity changes - accelerations-which set the steady deformation zone reveals to account for the load rise from zero. After reaching a maximum value, either a steady plateau or an almost linear decay with billet length is reported (Fig 1.1). That suggests that the prominent load peak outside that pattern observed in experimental loci comes from other source. To gain understanding of the factors which control such phenomena, analysis of the i nterna1 thermomechanics of steady state can give some useful hints, However, the obtention of the background part (Fig 1.1) of those loci requi res mesh updating. Of particular interest is one of the renewed work which make all the complex analysis in a fixed fram ^105) -

23 Curve Extrusion Ref. Geommetry Other conditions: Ratio

a R = 2 73 conical die smooth walls and die I b R < 2 95 9 6 as I.butwork hardening material

II R< 2 as lb,but rough interfaces

III R<2 90 as lb

IV R> 2 112 square die as lb

fig 1.1- Isothermal Dynamic Analysis . Qualitative behaviour of FE Load-Displacement reported results -Direct

Plane strain extrusion t Elast oplastic material.

24 CHAPTER TV/0: BASIC CONCEPTS AND FINITE ELEMENT FORMULATION 2.1 Introduction

Most metal working processes can be regarded as problems of constrained plastic flow, ie part of the material has yielded and became plastic and part remains elastic. Therefore, to expand the linear stress-strain constitutive law of the elasticity regime to the plastic one, an incremental version has to be used. This is the approach of "deformation theories", as elastoplasticity. However, as within the deformation zone the plastic deformation is generally much larger than the elastic strain, the solution is simplified by completely neglecting the latter one. That is the assumption in flow theories such as rigid plasticity and viscoplasticity. Another general feature of the main bulk operations like rolling and extrusion is the presence of only one deformation zone and continuously plastic loading; the unloading takes place after leaving such a zone, which avoids keeping elastic records. The picture changes from the metallurgical point of view,where unloading can happen when any restoration process that drops the flow stress operates. Also, alterations of stresses due to rigid body motion are present at high rates of deformation. For both of those cases,incremental forms are again required (Chapter 6).

This chapter presents the basic concepts involved in both flow and incremental theories, complemented with a thermal analysis. The description of the specific Finite Element formulation chosen for the construction of the computer program used in this thesis is also given. All explicit matrix forms which are not relevant to the text are listed in Appendix I.

2.2 Mechanical Problem

2.2.1 Strai n

In an isotropic material, the infinitesimal strain state of a volume element surrounding a point P is given by the symmetric strain tensor e. . : •J

26 . . = 1 ( 3uj + 3uj ) (2.1) IJ 2 9xi

where u. is the displacement of the point P in the i-spatial coordinate -direction. As any tensor,it can~be~decomposed intu~^rn antisymmetric and a symmetric part. It is convenient to choose the second in the diagonal form:

h , (2.2) i j = v I j

where h stands for hydrostatic, 6.. is the Kronecker delta (5.. =1 when 1 J I .1 i = j, 0 otherwise) and f.: is the volumetric dilatation or mean normal s tra i n:

Ev = 7 Eij (2-3)

The antisymmetric part is called deviatoric, and its invariants are;

1,= 0 (2.4)

I =1 e 1 . e1 . (2 5) 2 2 ' J ' J where is the radius of the Mohr circle for strains (78).

I3 = det ( e\ j) (2.6)

The property of arising from the antisymmetry of the deviatoric strain,indicates no change in volume and only changes of shape are accounted for.

The symmetry of both eq(2.i) and (2.2) forms allows to develop the calculations considering for the general tridimensional case only six distinct components. Further reduction in their number is possible in the presence of more symmetries, as the ones arising from plane strain or axisymmetric situations. Thus, both (2.1) and (2.2) operations can be formally expressed in matrix form as:

27 £=L.u (2.7)

£h = _L . m £ (2.8) 3 ~ where now the strain state is represented by a vector the matrix L_ encompasses the operation (2.1) and the vector m the (2.2). The specific forms for all of them in both plane strain and axisymmetric situations are given in Appendix |.

The strains are uniquely determined by the components of displacement. However, if the strains are solved first, the displacements can only be solved for, if . satisfies certain compatibility conditions (148).

Infinitesimal strain has little bearing on large plastic deformation. However, its rate gives the correct relationship for instantaneous strain rates upon which flow plasticity is based.

By analogy with the strains, the strain rates are obtained as:

£ = L.u_ (2.9)

h T z• = _1 . m.-e /(2.10 \) ~ 3 where the vector stands for the velocity components of the point P.

2.2.2 Stress

The stress state at P is given by a tensor a. j, i denoting the plane on which it acts and j denoting the cartesian direction along which it acts. According to the condition of equilibrium for the moments, it follows that c;. . is symmetric, which allows to process ij only six of its components.

By analogy with the strain, it can be split as:

a.. = ah. + a!. (2.11a) ' J U 'J

28 a The hydrostatic component (J?^ = m6.j eq (2.11b) sets a "false zero" of stress, with respect to which the deviatoric a. stress measure is taken. The principal invariants of the deviatoric tensor are:

J! = a!. =0 (2.11) 1- II

J^l a|. oj. (2.12a)

2 2 2 or (178) = £ {(o] - o2) + (0] - a3) + (o2 - c^) ) (2.12b)

J^ - Det (a!.) (2.13) where /T^ is the radius of the Mohr circle for stresses (73).

The general equations of motions, including acceleration effects, are gi ven by (75)

dSij ~ b? + p (^i +u. ^i) = 0 (MNm'3) (2.1M Ox. ' 91 J 9x. J J where x. are space coordinates, and wi th

c / 9uou. , ou9u. . , . b = p (_, + u I ) (2.15) 1 91 J 9x. 'j the acceleration forces.

At the border surface 5 ,the stresses must satisfy the specified surface tractions J

a. . n. = t. (2.16) 'J ' where n. is a unit vector along the outer normal to S^. . Vector forms as for stresses, t_ for tractions and b for volume forces are normally used in the FE context (see Appendix |).

29 2.2.3 Constitutive Models

2.2.3-1 Elastoplastici ty

The total strain (and its rate) can be decomposed into an elastic and a plastic part. For a linear isotropic elastic solid, the deviatoric strain is connected to the deviatoric stress by

e! . a! . (2.17) I J = _JJ V '' 2G while the normal strain:

h h , e.. a.. a (2.18) IJ = IJ = M ' 3K 3k where G and K are the shear and bulk modulus, respectively. In the mathematical theory of plasticity a basic assumption is made that the material is homogeneous with an isotropic rule of hardening. It is assumed that there exists a scalar function, called a yield function, which depends on the stress and strain and the loading history, and which characterises the yielding of the material as follows:

F < 0 no change in plastic deformation (2.19)

F = 0 change in plastic deformation (2.20)

Assuming an isotropic behaviour of the material before and during deformation, F depends only on the invariants of stress, strain and strain history (1*}9)- If the yield function is an isotropic function of stress alone, then the plasticity involved is called "isotropic stress" theory. Furthermore, if the hydrostatic stress is assumed not to influence yielding, and plasticity is controlled by the deviation from the hydrostatic state:

a f(J J } F ( j j) = 2' 3 " 1 (2-21) c

c = constant

30 f does not depend on the strain history, which only enters through the parameter c.

The two better known yield criteria, Von Mises (150) and Tresca (151) fall within this category. The simplest form is provided by Von Mises, here the condition (2.21) for plastic flow being:

F = J^ - c2 (2.22) where for uniaxial tension: C = 0.//3 and for pure shear: c - k; o^ and k are the yield stress and the shear stress respectively, connected by the equivalence:

= k (2.23)

The Von Mises yield criteria (2.22) is independent of the sign of the deviatoric stresses (all squared terms), and depends only on their magnitudes (no preferred directions). Thus, it is only applicable when no anisotropy or Bauschinger effect is present.

The plastic strain increment is related to the yield surface by by (152, 153):

de. . = X 9F* dt _ A 9F* ^2 dt _ X o! . dt (2.2k) IJ do'.. 9J1 9a'.. 2 /J' ,J ij 2 IJ 2

A reduction on the restriction of the above rule,which is necessary for many non fully compacted materia 1s^can be obtained by specifying a plastic potential Q. When Q. = F the plastic deformation is known as associated, and otherwise as non-associated.

By transforming the elastic parts (2.17) and (2.18) into incremental forms, and addition of the plastic contribution (2.24), a rate i s obta i ned as :

e. . _ 1 a.'. + 1a + A /~3 a!. (2.25) 'j "27 ,j 3k m ttv 'j

31 which is a particular case of the well known Prandtl-Reuss relations

(154, 155, 156).

The parameter A is the one which allows the transfer of material properties from simple to complex stress states. In order to do this, an expression depending only on invariants must be produced. For a situation of pure shear stress k, only tangential deformation Y is present in (2.24) and therefore (see Appendix I):

A2 = Y (2.26)

and from (2.12a)

v/jl = k = a.'. (2.27) 2 U

Thus, by (2.25):

x - 2 Al (2.28)

For large deformations, additional considerations in the transfer of material properties becomes necessary, as discussed in Chapter 6.

The general form (2.25) has been the basis of most of the elastoplastic analyses reviewed in Chapter 1. Its incremental nature enforces to follow the rise in flow stress by steps, through some implicit or explicit time updating, as discussed in Chapter 6.

2.2.3.2 Rigid-plastic

In the rigid-plastic model, elastic effects are neglected. There- fore, only the last term in (2.25) is conserved; by introduction into it of ' g i ven by (2.28) :

?..-/[, £JJ (2.29)

32 the known Saint-Venant Levy-Mises equation for associated flow plasticity is obtained, \7hile civil engineering workers (5) are more prone to use (2.29), the trend in plasticity (13, 122, 178) is to make use of effective values:

e effective strain rate: eff = ^2 (2.30)

effective deviatoric stress: ^^ = ^3 (2.31)

which produces the entirely equivalent rigid-plastic law (122) (see Append i x I):

il?ff aJ; (2.32) 'J 2 °eff '

For both (2.29) and (2.32), stress values are determined by current strain rates which noticeably reduces the computing analysis of pro- gressive deformation.

2.2.3.3 Vi scoplast ici ty

The relations hitherto discussed are considered within the framework of time- independent plasticity. Some attempts to describe the behaviour of materials including time effects assume that mechanical properties can be described with a viscoplastic constitutive equation encompassing both rheological and plastic effects (2). The viscous properties of the material introduce a time dependence of the states of stress and strain. On the other hand, the plastic properties make this states depending on the loading path. Thus, as a result of simultaneous introduction of viscous and plastic properties a dependence on both the load path and duration of the process can be produced. Two fundamental problems are present: the determination of an adequate dynamic yield criterion and the establishing of suitable constitutive equations in the plastic range. Impulsive loading experiments (159, 160) have indicated yield stress to be sensitive to strain rate gradients. Also for the plastic regime, torsion (187, 188) and tensile

33 (157, 158) experiments have indicated that the flow stress exhibits strain rate dependence. If elastic effects are negligible, a generalized form of (2.24) for strain rate sensitive materials can be expressed by (2):

e. . _ ri $ (F) 90 (2.33) ,J 9a!. 'J where now $(F) represents the test results for dynamical loading, with the same (2.19) and (2.20) properties, and ri is a parameter dependent on some state variables as time, temperature and deformation invariants. For isotropic materials, eq(2.33) can be written as (5):

e.. _ la!. (2.34) 'J " 2y ,J which can be associated to non-Newtonian fluid behaviour. A very characteristic form, known as Bingham fluid, in which a yield stress a |s exh i b i ted i s : y

A2 = - a ) (2.35)

which by (2.34) introduces a "viscosity1':

a U =SK,r '2/n ) ) + y (2.36) 2 VI o

It can be seen from (2.36) that when r\ oq the pure plasticity behaviour of (2.29) is recovered, with a "viscosity":

h =-^-2 (2.37) ?V\ 2 jthus, both viscoplastic and rigid plastic descriptions coincide, the difference being only semantic. The fact that the viscoplastic name is more commonly used comes from the potential theoretical capability of including the n parameter rather than actual solutions produced with

3'' its consideration. Experimental difficulties in separating delayed and instantaneous plastic response (157, 158) (see discussion in 5-2.1) prevent its general use. Instead, all strain rate dependence is considered to intervene in (2.36) as plastic deformation, and allowing for a variable yield stress surface. On the other hand, if a dynamic analysis is intended (Chapter 6) incremental forms are rather used:

da _ 8a de + 9c dG + .... (2.38) 9G 9G which introduce the sensitivity of the flow stress to strain, strain rate, etc. Such dependence, which is the instantaneous one (section 5.2.1), is a good approximation for small increments where the path incidence is minimal.

In both viscoplastic and rigid-plastic treatments, nearly rigid regions where strain rates almost vanish, bring about singularities in the plastic ratio V (2.37). Thus,a cut off value is normally adopted to reset the strain rate.

The complete constitutive law, encompassing both deviatoric (2.34) and hydrostatic (2.18) components, can now be expressed in matrix form:

a = a* + a = 2 VU £+ mam = + amm c (2.39) where now the bulk modulus K has been replaced by a parameter ot. For

constrained plastic flow, both (5) a = (K - 2/3G) and K are very large, which virtually guarantees that the same effect is achieved by their i nterchange.

2.2.4 Variational Principle - Viscoplastic Model with Penalty Formulation

Instead of pursuing the solution of the partial differential eq(2.l4), a variational approach is used. Thus, by analogy with fluid treatments (5), a virtual work principle is invoked in terms of virtual velocities 5u:

35 6$== J&k\o4\I/ 6e . g - y6uT. b dV - / du1.1 dS = 0 (2.40) V v

In the above, V is the volume of the deforming body, and S^. the fraction of the whole boundary where forces £ (2.16) are specified; on the remaining fraction Su,where the velocities are specified, the condition of kinematical admissibility guarantees a null variation of velocities. The total stress o is given by the constitutive form (2.39). The compatibility of the virtual strain rate variations is introduced by (2.9). The vector b_ accounts for body (b_ ) and acceleration forces:

T T b = b° - p( Bg + (V.u ) . u ) (2.41) Bt

2.2.5 Finite Element Treatment

As standard FE techniques (5) are applied here to solve (2.40), only a summary of the method is presented. The domain V is discretized into non-overlapping elements connected at nodal points. At any point within each of these elements, the velocity is obtained by interpolation from the nodal velocities expressed in an expanded vector a_:

u = f^.a (2.42) where the interpolating "shape functions" g satisfy

1 for node i ^node i (2.42a) . 0 elsewhere

Thus, at the inter-element boundary, only continuity of the velocity field is achieved (C° continuity), all derivatives being discontinuous. The strain rates (2.9) become:

e = l^. N_.a = B.a (2.43)

Introducing (2.41) to (2.43) into (2.40):

36 T 6

which must hold for all variations <5a, therefore, a system of ordinary ..differential .equations is obtained as:

M.d^ + K.a + f = 0 (2.45) dt

where the mass matrix M is given by:

M = P / NT. N dV (2.46) -A/ and the stiffness matrix K encompasses the three contributions:

(2.47)

(2.48)

T T (<3 = P J N . ( V.(N.a) )T U dV (2.49)

V and the external forces vector f:

NV dV - J NT. t dS (2.50)

ST Equation (2.45) holds for any element of the network, provided its external forces f (generally zero) are specified, and both stiffness K and mass M contributions from neighbours assembled at the nodal points. With this clarification, (2.45) can be understood either as an element equation or as one for the whole assembled system.

• In spite of all matrices and vectors in eqs (2.46) to (2.50) being expressed as integration over the whole volume V, they are never produced in such a way. Instead, they are constructed by assembling at each node the contributions computed for each element. As different size and shape of elements are normally present, the process is greatly simplified by defining a transformation which maps any of them from its "global" system of coordinates (x,y) to a fixed "local" (P,P) one; thus:

37 dV = det J_ .dC dn (2.51)

where J is the Jacobian matrix accounting for the mapping. Thus, all integrals become referred to the local (C,n) region, adopting the . general-form; . _ -

g (C,n) dc dn (2.52) V

By selecting a squared region defined by:

-1 < £ S 1 -1 < n < 1 (2.53)

the one dimensional Gauss quadrature formula can be extended in a straightforward way and now eq(2.52) computed as:

£ H. H f(C.,n.) (2.5*0 ij J J where for each orthogonal direction,the location of the Gauss integration point (eg C.) and its weight coefficient (eg H.) are tabulated in any elementary numerical analysis text (202).

In eq (2.52), g(C,n) encompasses the computation of both the integrand (eq(2.A6) to (2.50)) and the determinant eq(2.50 at locations within each element. Global derivatives enter in that determinant, which has not been specified yet. By analogy with the velocities, any inner position x of an element can be expressed as a function of the expanded nodal coordinates vector x as:

x' = NJ.x (2.55) when the same shape functions are used: N = N1, the element is called isoparametric. Now, (2.51) can be expressed as:

dV = det f JA.x ) dC dn (2.56) where J.* depends on the local derivatives of the shape function, its product with x giving the Jacobian in terms of the global derivatives.

38 1 n the system (2.45) , J^, _M, a_ and f are f i ni te and a sol ut ion is sought when a-*00 . It is apparent that if J^ is non-singular, then a + 0 as cr*30 . This illustrates a problem of over-constraint which causes the mesh to lock, resulting in useless results. The problem of over-constraint of ..elements in .the_ isochorj_c regime has been discussed by Hugges et al (203), Fried (204), Malkus et al (145), and Nagtegaal et al (104). In order to remove that condition, j<2 must be made singular so that even if the predominant term in (2.45) is K^ a. = 0, allowance is made for a f 0. It turns out that virtually all the

1 commonly used conforming elements,result in non-singular J<2 s when "exact" (order of the interpolating functions N plus one in each direction) numerical integration is used. Singularity (or nearly singularity) is introduced by using lower order of integration. When

(order of N x order of N) Gauss quadrature points are used for all matrices in eq(2.46) to (2.49), the integration is called "reduced".

Another option is given by exact integration of all of them but J<2, which is underintegrated in one order less, a form which is called "selective" integration. Malkus and Hughes (145) proposed an heuristic theory,from which it turns out that the most effective elements in applications of the type considered are the so-called "Lagrange" isoparametric elements using selective integration. Practical applications to flow problems produced by Zienkiewcz et al (138), confirmed the success of using nine node isoparametric Lagrangian quadrilateral elements, with (3 X 3) quadrature for stiffness and (2 X 2) for penalty matrices. This has been the approach chosen for the present thesis.

As both ](.| and K.^ depend on the nodal unknowns a t the system (2.45) is non-linear. For problems where accelerations can be neglected, no time integration becomes necessary. The resulting algebraic system must be solved iteratively; by introducing:

a(n) = 3 (n-1) + A>) (257)

a Newton-Raphson scheme (5) is constructed as:

K(n-1)Aa(n) „ _f . K(n-I)a(n-1) (2.58)

39 with the a priori assumption K n - K n ^ . The label (n) stands for the n-iteration. A guess is needed for the zero-order, which is introduced by setting y = 1 in (2.39).

2-^-2^-6—-Stress Analyses- .

For both stress and strain tensors, a set of orthogonal directions exists for which only normal (diagonal) components are non-zero. Both length and orientation of such principal axes provide a useful tool for inspecting the plastic flow.

The magnitude of the planar principal axes are given by (78):

ai,2 Qm ± (2.59)

On the other hand, the angle 2$ between one of these locally varying directions and the fixed x-axis has the following properties:

a (I) tan2 ^( r°x) tan2 (2.60)

e tan2 *(*r x) tan2 (2.61)

which come out by considering that in both (x,y) and (1,2) coordinate systems the diagonal hydrostatic pressure tensor is the same.

(II) tan2 ^ (a | - c' xy (2.62) «2 xi Ref (78) (J^ -

tan2 ^ (M 1) - V (2.63) 1 X • .91 (i2 -

and by the constitutive relationship (2.34):

tan2 ^ | = ^ ty (2.64)

40 Besides, for each position, another set of normal directions (a,8) can be defined as the ones at which the shear zis maximum;

ZctB = k (2.65)

and their relative orientation to the principal axes is given by:

k (2.66) tan2<|; =

which is infinite due to eq(2.12); therefore, slip-lines are always at 45° of principal axes.

Thus, it follows from eq(2.62) that if no shear ff^ is present (or its ratio compared with the normal part of in (2.12) is negligible):

^(1,x) = 0 (2.67)

and from (2.59):

o - a =a =a 1 2 x y (2.68)

On the other hand, if the shear is maximum a' - /j' xy 2

^(1,X) = 45° (2.69)

and

= a / = a m J2 (2.70)

Now, the behaviour of plastic material-tool interface can be properly Inspected. In the FE treatment introduced in the previous section, the boundary conditions can be introduced either by specifying forces (by eq(2.50)) or velocities. In the latter case, they do not enter directly ___ ...... /n\ _„.,., in surface integrals,but indirectly through the Aa = 0 condition for any iteration in (2.58), thus coupling through K's coefficients the

41 volume with its surface. For simplicity, a tool-material interface parallel to the x*-axis is examined:

y y

Fig 2.1. Frictionless Fig 2.2 Sticking Interface Interface

1) Frictionless surface:

Here the component of the force parallel to the tool is zero, which by (2.{fc) means no shear is acting. Therefore, neither rotation (2.67) nor different magnitude of axis (2.6ft) are present, and the stress situation is as in Fig 2.1. In the direction normal to the tool the velocity is fully restricted. As the formulation (2.45) uses velocities as primal variables, the force boundary condition is not automatically satisfied, but approached as the result of convergence of the numerical solution (2.58).

2) Sticking friction:

In this case it is simpler to impose full restriction on both normal and parallel to the tool velocities. The second one implies that all forces are provided tangentially to the surface, with involvement of maximum shear. Thus, the material at the

42 interface is stressed as in Fig 2.2, with principal axes rotated and elongated as soon as y in (2.63) becomes different from zero.

3) Symmetry interface:

If the flow exhibits a plane of symmetry, its normal velocity must vanish. As no relative motion of one half with respect to the other one takes place, it is also assumed that the tangential force is zero.

2.3 Thermal Problem

2.3.1 Basic Equations

The equations for the temperature field T are obtained from the balance of the rate of energy flux q across an infinitesimal volume as (133):

3 Pc 9J = q + Q (cal/sec.m ) (2.71) 3t where c is the specific heat; 0 accounts for the fraction 8 of mechanical work converted into heat:

T Q=i£ .a (2.72) J

J being the mechanical equivalent of heat:

J = 4.186 (joule/cal)

As for 8, following the experimental results for temperature rise from Farren and Taylor (82), from tensile tests of steel, copper and aluminium, values in between O.85-O.9O may be considered.

43 In a moving medium, the energy flow per unit area and unit time is given by (133, 1**7):

q =-k-VT + pc u T (2.73)

In the above, the first term in the R.H.S. accounts for conduction effects, while the second is introduced by the convection due to the movement of the solid; k is the conductivity. Thus, by introducing eqs (2.72) and (2.73) into (2.71) and noting that for incompressible flow, VT.u= e = 0: — — v

T T pc £T = V . k 7T -p c u\vr + 3. i . o_ (2.74) 3t J

Two types of boundary conditions can be considered: either temperature: T = T^ (2.75) specification on S^, or

heat flux: « • n - -k 3T. + pc u T _ q + a (T - Tn) (2.76) M_- —-^ n-T s Z 9n specification on S^.

In the above, n_ is the inner normal and the term in the R.H.S. accounts for a generalized heat transfer with a linear dependence on T; q is a specified flux,a the heat transfer coefficient and the temperature of the outer medium.

Thus, following Zienkiewicz et al (138-1**1) and Thompson et al (125, 126) a steady state solution of (2.7*0 proceeds by using a "weak" form:

T / v (oc u . V T - Vk.W - Ji eTa ) dV + /v (T - T ) dS + J A Js T

v (-k H + Pc u T - q - a (T -..T )) dS-.= 0 (2.771 9n

kb where v and v are arbitrary testing parameters. By applying the diver-

gence theorem to the second term on the L.H.S. , and choosing v = -v"

HV pc U.VTT + Vvk.VT - 8VI.T.a) dV

'V;

v(q + as(T - T2) - pc u.n T) dS = 0 (2.78)

Sq Where, at the border where temperature is specified:

/v(k 3j) dS = 0 (2.79) J 9n ST

2.3.2 Finite Element Treatment

For hot working applications, it was found in the present work that

improvements in the stability of the solution were achieved by reducing

the perturbations through the change of scale:

T = T' + T_ (2.80) D

where Tg is the preheating or initial billet temperature.

By assuming a trial function expansion of the temperature in terms

of the nodal values h at the points of the grid:

T1 = WT.h (2.81)

where W are also interpolating shape functions, adopted such as at each

node i (5):

v = W1 (2.82)

(see eq(2.42a)); the form (2.78) can be transformed into:

H . h + £ = 0 (2.83)

where:

45 H.. = k (VT. W'V + pc W1 u.V) W1 dV U /V

1 1 + fvj (as - pc u.n) W dS (2.84)

and:

T 1 g. = / -l£ .£W' dV+/ W (q + a s (T B - T 2 ) -p" c u.- n- T'B ') dS J (2.85)

The instability of numerical solutions for elliptic problems with

substantial first order derivatives,has long been recognised in the

context of finite differences (153, 154). Typical examples of this

situation are convective heat transport eq(2.74) and the Navier-Stokes

equation in fluid mechanics. Finite Difference schemes become unstable

when the local Peclet number is such that:

p _ Pc V J > 2 (2.86) 6 " ~k

where V is the maximum velocity component and T the mesh size. That

can only be overcome at the expense of an excessive reduction of T.

In Zienkiewicz et al (165), an analysis of the difficulties that arise when FEM is used to solve eq(2.74) is made, and the need of an equival-

ent to "upwind differences" (166) in the FEM is recognized. Christie et al (142) developed the technique for one-dimensional problems and

Heinrich et al (143) expanded it to two dimensional cases. The most

important feature of the numerical schemes proposed (144),relies on

the choice of the W1 functions, selected as:

, , w'(C,n) = f(N (C),N (n),a.,3i,F(J)) (2.87)

where a. and are parameters and F is a function of the mesh size, i i r The difference equation obtained when considering the one-dimensional case of the eq(2.74), for the particular case Q. = 0, leads to "optimal"

(1*13, 1A4) values of a., 3. which are functions of the Peclet (2.86).

The functions W1 become non-symmetric. " For nine'noded isoparametric elements, a mean velocity value for the Peclet is produced as (144):

46 V = Uij = ^(u1 + ujMij (2.88)

where u/, u^ are the velocity vectors at nodes and jJ^ a unit vector in the direction of the line through i and j.

In the computer program developed for the work in this thesis, the capability of the use of upwinding in the construction of the matrix H has been incorporated to avoid possible instabilities.

Thus, with the use of the modified shape functions eq(2.87), the general techniques of mapping and numerical integration described in section 2.2.5 are here applied to eq(2.84) and (2.85). So far, 3X3

Gauss quadrature was used.

2.4. Thermomechanical Coupling

In hot deformation processes, the flow stress becomes a non-linear function of strain rate and temperature (section 5.2.1). Besides, the thermal eq (2.74) is also linked to the velocity field through both convective and heat generation terms.Both resulting algebraic systems (2.

(2.83) are not independent, and can be solved either separately (138) or simultaneously (139, 1^0) as:

(n ( C) 2 - f - K :"a "'" •

. 2 ' S /

In the above, the velocities are obtained by a Newton-Raphson algorithm, while the temperatures come from a direct iteration scheme. As each element is connected with only few neighbours, the supermatrix resulting from assembling their LHS contributions in (2.89), has few non-zero off-diagonal" coefficients. Extremely large reductions in storage are achieved by keeping only the coefficients within the bandwidth bounded by the maximum separation in non-zero values from the diagonal of the

47 sparce supermatrix (banded matrix storage). Further reduction is still

possible, by only storing the non^zero coefficients within that band

(skyline or profile housekeeping). Thus, as their locations can be

a-priori determined from both element connections andnodal restrictions

indeces,very compact algorithms avoiding zero multipliers can be

designed. Two main ways of operations are possible for such algorithms.

In the first, the whole assembly of the (2.89) contributions is completed

before the solution starts (5, 205 - 208). Such design requires to keep

the assembled matrix in core, which is a big demand in central memory

space. In the second way, adopted in this work, the LHS matrices of each

element (2.89),together with appropriate indeces, are massively stored

out of core. In a later operation, such storage is screened and the

element matrices partially assembled. The solution proceeds by solving

the unknowns in a "front" where the assembly has been completed;

simultaneously, the control indices are changed, the outer unit rewound

and the sequence started again till the whole solution has been produced.

The main advantage of these "frontal" (172-175) solvers is the big

reduction in central memory requirements.

The non-linear system (2.89) is symmetric in K and non-symmetric

in H. This means that an asymmetric solver for both equations

represented in (2.89) has to be used or else one has to pursue separate

solutions taking advantage of the K structure. The first approach was

selected for this thesis by using a frontal solution package (173, 175).

It is important to estimate separately the error norm for velocities and temperatures, due to their widely different values. In this thesis,

the con ergence of iteration is judged using an Euclidean norm with

e appropriate tolerances toj '-e. requiring that iteration stops when:

(N) IIA I|| | < etol. (max ||A ^ \\ )

(2.90) where the norm is defined in the expanded space, encompassing all the nodes of the network, as:

ia = ^si-^p1'

= u1 (2 % — 6tol (1) and = u2 = T 6tol (2)

2.5. Solution Procedure

Four different stages can be identified in the complete analysis:

a) Data adquisition and generation:

All the numerical parameters, geometrical dimensions, material

properties, nodal restrictions and macro-commands are read from

an input file as fully specified in Appendix II. Automatic mesh

generation as described in there avoids the need for feeding in

the information for the whole network; only certain elements and

nodes in line require complete specification.

b) Preprocess i ng

Three operations are carried out at this stage:

1. The matrix of the nodal connections!^ , holding for each

i-element the global numbersof its j-nodes is modified as:

l'-' < node not shared (eg: external border x ^ n0 3 nodes and central ones in the 9-noded

isoparametric)

> 0 node shared among elements 2. The maximum bandwidth is determined.

kl 3. The matrix L , where k is the degree of freedom of each

1-node is constructed as:

kl L 0 degree of freedom with a restriction x imposed, and which is not solved

1 k 1 _ K1 L = Neq where Neq is the number assigned to the equation to be solved

c) Process i ng:

Step I: Construction of the element's coefficients for (2.89)

1. From the nodal velocities evaluate at each of the 3x3

Gauss point locations the strain rate, flow stress and p (2.37) ratio, and perform numerical integration to produce the element stiffness matrix( 2.47, 2.49 ) As those distributions are unknown in the first entry, it is there set p = 1.

2. From the nodal velocities evaluate at each of the 2x2 Gauss point location the volumetric strain rate (2.10) and produce the element penalty matrix (2.48).

3. The boundary condition handling as in section 2.2.6, avoids to calculate any external load vector f in (2.58), for both frictionless and sticking walls. The remaining part in the RHS of

(2.58) introduced by the perturbation scheme does not require integration, and is obtained from the velocity distribution of the previous iteration.

4. If the problem is defined as thermocoupled, (see Appendix II), during the step 1. also the temperature and velocity are evaluated at internal positions, and the matrix H (2.84) is constructed.

50 5. If the problem is defined as thermocoupled, during the step

3. the RHS vector g (2.83) is produced.

kl 6. All k-degree of freedoms in the equation index L are set as negative if the 1-node is not shared (which becomes known by screening the nodal connections I ). The LHS element matrix ^ x (2.89) is packed and zeros disregarded. Then, in a peripheral unit 1 the following are written:

The number of unknowns the element has (NX), after

elimination of the restricted(boundary condition)ones.

The diagonal position in the shadow supermatrix (NX ).

The list of the equation numbers assigned throughout L .

- The matrix coefficients of the LHS of (2.89).

7. The RHS corresponding to the whole network is assembled in core by adding the elements contribution from (2.89).

Step I I: Assembly and solution

8. The unit 1 is rewound, and by screening element by element the following sequence of operations is carried out:

LHS (2.89) data are read from unit 1, and nodal contribution

of shared nodes added up.

When a node is fully assembled, the sign of the control kl index L^ is changed for all k degrees of freedom.

All equation numbers in the L index which hold a negative

value become completely specified and ready for solution.

Then, the Gauss elimination and triangularization is

produced and the complete equation is written to a second

external unit 2.

9. Once all element has been swept, the equations are retrie/ed

from unit 2 to the program core; then back-sust i tut ion is produced

and thus the solution is attained.

51 Step III: Numerical convergence control

10. The norm (2.91) is evaluated, and the convergence is judged as a_function (2.90) of the preset tolerances. If either the criterion (2.90) is not satisfied, or the n index in (2.89) is lower than the preset number of maximum iterations Nmax, the cycle 1. to 9. is reentered.

Step IV: Field values of the achieved solution

11. All field distributions: velocity, temperature, strain rate, hydrostatic pressure, principal stresses and their angles are computed at the location of the Gauss points of each element for a-posteriori analysis, d) Post-Process i nq

Graphical display of the fields computed in 11, as detailed in section 4.1.

52 CHAPTER THREE: EXPERIMENTAL

r 3.1 Introduction

The experimental programme considered the hot extrusion of

J:wa .aluminium based alloys^ namely Al 5Q52_-and-Al~54-5&;—tors ion —

results indicated that they exhibit different behaviour. The

effect of operational and material parameters and its changes on

the resulting data,are produced to provide the frame for assessment

of the performance and limits of the numerical predictions. This

chapter will briefly outline the experimental procedures used for

this work. These are the same for both alloys unless otherwise

mentioned.

3.2 The Extrusion Press

The extrusion press used in the experimental work was a 5MN

fast action ENEFCO hydraulic press mounted vertically over an

accessible pit (depth 3 metres). The press tooling was supported

by a backing plate and the container could be raised or lowered

hydraulically. The press layout is shown in Plate I with the

operating controls, load indicating dial on the right and induction

heater on the left.

The container dimensions allowed the extrusion of a maximum

billet length of 158 mm and different size container liners were

available for the extrusion of 57 or 75 mm diameter billets. In

order to minimize the effect of friction on the load, the length

of the billet was usually 95 mm for 75 mm diameter.

An auxiliary pump enabled a fast approach by the ram prior to

the commencement of extrusion. This reduced the time for heat

loss from the billet while the ram was being lowered. When the

ram hit the pressure pad and the load exceeded 0.2 MN the pump was

automatically bypassed and the ram moved at the pre-set speed during

extrusion. The ram speed was controlled by adjusting the oil flow

from the hydraulic pumps to give a range of speeds varying from

54 Plate I . General Layout of the Extrusion Press 2 mm/sec to 13 mm/sec. Higher speeds of the order of 100 mm/sec to 600 mm/sec could be obtained by accumulator drive, that is by discharging nitrogen filled bottles to increase the rate of delivery of oil to the main ram. _

3.2.1 Container heating

In the early work the container lining was maintained at 300°C by 8 inconel heating elements positioned in the container holding casting. The addition of an inner container linen with 12 elements gave a combined power rating of 16 kilowatts, and allowed a maximum temperature of 530°C. To prevent overheating, thermocouples were braised to each element and attached to two Eurotherm temperature controllers, one for each set. The temperature profile of the linen, obatined by attaching to its surface thermocouples in different locations, was found to vary within + 5°C of the set temperature.

For both direct and indirect extrusion the container temperature was set 50°C below the billet temperature.

3.2.2 B?1 let Preheat

For direct extrusion the billets were preheated in a Banyard

Metal heat induction furnace attached to the press. The furnace operates at the mains frequency; the geometry enables 75 mm diameter billets, of lengths upto 150 mm, to be heated upto temperatures of

600°C at a rate of 25°C/min. The billet temperature was continuously monitored by a thermocouple placed in a 15 mm deep single hole axially drilled; the same device also drives the temperature controller of the furnace. The induction heater is suspended from two rails and can move over the container by means of a pneumatic system commanded from the main panel. When the heater is aligned with the container bore, the billet is transferred manually by removing a rod holding the billet within the coil; the mobile heater is set back to its initial position. To eliminate any possibility of temperature gradients within the billet, it was allowed to soak for twenty minutes prior to extrusion.

The indirect extrusion tooling prevented the use of the induction

heater. In order to keep conditions for direct and indirect extrusion

as similar as possible, the billets were preheated in an air circulating

furnace adjacent to the press and transferred manually to the container.

With the furnace near the set temperature, a heating rate of ~35°C/min

can be achieved.

As for direct extrusion, billets were soaked for twenty minutes.

3.2.3 Extrusion Data Recording

The data for all extrusions were recorded on a Datalab DL2800

transient recorder,consisting of a control module and four memory

modules. This recorder,which was designed to capture single-shot

and low repetition events and present them for continuous oscilloscope

display, is also ideal for high speed data acquisition. The recorder

can sample at very fast rates, store the data, and then output it at

an acceptable rate in analogue or digital form.

The control module contains the timebase, delay control and

output circuitry and is independent of the number of memory modules

used. It permits recording cycles in single, delayed or continuous

modes with manual or automatic triggers. The sampling rate varies

from 0.5jjs to 200 ms per sample and records each channel simultaneously.

The data for each channel is stored in the corresponding memory

modules as 4096 x 10-bit words. A memory module consists of a

pre-amplifier, a track and hold circuit, an analogue-to-digital

converter and a digital memory. The pre-amplifier has switchable and

continuously variable gain to allow for a wider range of input signals,

and a d.c. offset control to bring biased inputs within the range of

the instrument. For the recording of the extrusion data the transient recorder was operated in the single mode with a manual trigger. The sample

rate was generally 2 ms or 5 ms which gave recording times of 8 or 20

seconds-respectivety— The-+oad-was-measured by-aMayes—loadcel 1

located directly above the ram, the output of which increased from

1.5 mV to 500 mV as the actual load and the indicated load rose from

0-500 Tons. These figures were confirmed by a calibration done by the

National Physical Laboratory in March 1980. The output was recorded

directly by the transient recorder with an input setting of 2 V full

scale and an offset value of 5.0 (centre of the scale).

The ram displacement was measured by a rectilinear potentiometer with a 0.6 m stroke, fixed between the ram and the base of the press.

The output from the displacement transducer varied by 1.75 V per 100 mm

movement of the ram and was recorded by the transient recorder with an

input setting of 10 V full scale. The offset values used were 2.0 for

direct extrusion and 3.5 for indirect extrusion.

The hydraulic pressure of the press was measured with a pressure

transducer situated at the inlet to the main cylinder. The output

from this transducer increased from 0-30 mV as the hydraulic pressure

increased from 0-3500 p.s.i. (0-24 MN/m ). The diameter of the 2 main cylinder was 22£" which gave pressure recording ranges of 0-1300 MN/m 2 for a 75 mm ram and 0-2265 MN/m for a 57 mm ram. This output was

recorded by the transient with an input setting of 0.1 V full scale

and an offset value of 5*0.

The first 4000 words of each memory channel were individually

displayed in analogue form on an oscilloscope. For a more permanent

record each channel was plotted on an X-Y recorder which was linked

to the transient. 3.2.4. Direct Extrusion Tooling

Plate Ila shows the direct extrusion tooling for 75 mm_diameter billets. The ram, container and die assembly are in the background with 95 x 75 mm dia. billets, pressure pad, scraper pad and dies in the foreground. Also shown are 72 x 57 mm dia. billets and ram.

The ram, container liner, dies, pressure and scraper pads were all machined from KEA 5% Cr-V steel.

Changing the die was effected by taking off the retaining ring, lifting out the die assembly and removing the die backer and die.

The dies used for direct extrusion were for rod shaped extrudes with reduction ratios of 10, 20, 30, 40, 50, 80, and 100:1, the dimensions of which have been described elsewhere (23).

3.2.5. Indirect Extrusion Tooling

The indirect tooling is illustrated in Plate lib. The ram and pressure pad have been replaced by a dummy block with a tapered notch for the removal of the discard after extrusion. This is shown in the left foreground in the discard removal position. The die assembly is located at the top of a mandrel, the dimensions of which allow the container to pass over it. The mandrel and die assembly are shown in the centre of the plate, slightly out of position. A spare die holder, die backer and die are shown in the foreground with two 95 x 75mm dia. billets. The horseshoe piece on the right is to facilitate the removal of the discard from the container. All components were machined from KEA 145 5% Cr~V.steel ,

Changing the die is effected by lifting the container, unscrewing the die holder and removing the die backer and die. The dies used for indirect extrusion were also for rod shaped extrudes with reduction ratios of 20, 30, 40, 50, 60, 70, and 80:1. Plate II. a) Direct Tooling b) Indirect Tooling The dimensional arrangement of the tooling for indirect extrusion

is such that only 75 mm dia. extrusions are possible with a minimum

reduction ratio of 20:1 and a maximum billet length of 120 mm.

3.2.6. Water Quench

In order that the structure of the material should reflect that due to the hot working process the extruded product was passed through a water quench immediately upon exit from the die assembly.

The quench consisted of two concentric cylinders, the inner one containing holes of 1.2 mm dia. drilled radially, 532 of which were contained in the initial 26 cm, the remainder in groups of 28 holes placed at 10 cm intervals along the 1.37 m length. Water was delivered at approximately 60 gallons/minute by a centrifugal pump.

3.3. Experimental Procedure

During all extrusions a standard procedure was adopted to minimize sources of error. The container was maintained at the temperature required for at least one hour before extrusion to allow for the die and bolster to attain a steady temperature. For indirect extrusion, the pressure pad or the dummy block used were preheated on the container for at least 20 minutes prior to extrusion. No lubrication was used

in either direct or indirect extrusion.

3.3.1. Direct Extrusion

At the start of the extrusion cycle the container was hydraulical1y lowered onto the die assembly to obtain a pressure seal and two semi- circular rings were placed on top of the container to prevent any damage to the main ram. Upon transferring the preheated billet to the container the pressure pad was dropped onto the billet and the water quench, record instruments and the fast ram approach were then initiated. Extrusion

61 was continued at the predetermined speed until the main ram touched

the rings whereupon the pumps were reversed and the ram raised. In

order to assess the heat losses from the billet the extrusion cycle was timed from then the furnace door was opened until the ram hit

the~pressure pad.

The container was then raised and the extrude cut with a hacksaw,

before being punched through the die into the pit below. The discard

(average length of 12.5 mm) was removed by placing three circular

rings between the bottom of the container and the die assembly, and

using the ram to push a tight fitting scraper pad through the container.

This also had the effect of cleaning the liner to produce similar

starting conditions for each extrusion. The overall procedure is the

same for the induction heated and air furnace heated billets.

3.3.2. Indirect Extrusion

The procedure for indirect extrusion was essentially the same

except that the 75 mm ram was removed from the main ram and immediately

prior to extrusion the container was raised such that the die assembly

at the top of the mandrel was positioned in the bottom 50 mm of the

container. Upon transferring the preheated billet the dummy block was placed in the container and the extrusion cycle was initiated as

for direct extrusion. When the main ram hit the dummy block both the

billet and container were pushed down onto the mandrel and moved

simultaneously at the predetermined speed during extrusion.

At the end of extrusion, after the main ram had been raised,

the container was hydraulically lowered to break the container/billet

linkage. The container was then raised so that the dummy block lifted

the discard off the die face, enabling the extrude to be cut. Following

this the container was again lowered and the discard, the length of which was approximately 14 mm including a diametrical allowance of

1.8 mm for the notch, was separated from the dummy block by punching

it at the tapered notch. The ease of this operation was improved by

lubricating the face of the dummy block with DAG 580 prior to extrusion.

62 The container was then raised and lowered again to clean the container

1i ner.

It should be noted that_the removal of the discard relies entirely on the hydraulic force exerted by the container. Since this is only

0.5 MN for the downward strokeand 0.75 MN for the upward stroke, large discards of the stronger alloys cannot be removed and therefore

'stickers' must be avoided.

3.4. Materials

The compositions quoted for the two alloys used are given in

Table 3-I-

Al loy Al Mg Cr Cu Mn Si Fe Ti Zn B

5052 Rem 2.55 0.24 0.003 0.003 0.07 0.17 0.019 0.03 13ppm

5456 Rem 4.88 0.08 0.08 0.72 0.11 0.23 0.03 0.14

Table 3«' Alloy composition in weight percent

The as cast logs were homogenized at 500°C for 24 hours, furnace cooled and machined to the billet requirements.

63 CHAPTER FOUR: ANALYSIS OF EXTRUSION I :

PERFECTLY PLASTIC IDEALIZATION 4.1 Introduction

The Finite Element procedure detailed in section 2.5 is applied below to analyze the steady state of the extrusion process. In this chapter, a perfectly plastic behaviour of the material is assumed. The network for the use of the numerical technique is made of nine-noded isoparametric elements.

Friction between billet and container is introduced in two different ways. In the first, boundary conditions are imposed directly to the border nodes. Thus, only the extreme cases of either full slipping or full sticking are simulated. The procedure is applied to direct and indirect extrusion through square dies, plane strain R = 2 and R - 10.

In the second way, a boundary element whose properties vary with friction is introduced; the results are compared with those obtained with the nodal specification approach and then applied to axisymmetric extrusion, ratios up to R = 150.

The load results obtained for Plane Strain problems are compared to Slip-Line Field solutions. For the axisymmetric case, comparisons are made with Upper Bound results.

In the oresent FE formulation, the normal component specified at the ram-billet border is the velocity (Fig 4.2); hence, the pressures should be computed.Those borders are not crossed by plastic flow, and the external and internal forces facing each other are in equilibrium. Thus, the external tool load there can be evaluated as equal to minus the internal forces. The plane networks of Fig 4.1 have implicit transverse

directions, which are either the width for plane strain or a circumference for axisymmetric geometries. If V* is one of these plane

"volumes", defined by the x-direction running parallel to the axis of the billet and the normal along the y-direction, the external force

P* per unit of z-length is given by: x

65 where cr stands for the total stresses. The total P load, encompassing the length of either the whole strip (plane strain) or whole ring (axisymmetric) is given by:

P = 2 L P* (4.1) x z where the factor 2 enters when considering the other half of the billet, and:

Plane strain: L = W width of the strip z K

Axisymm: L^ = 71 "width" of the circumference

Pressures p are produced by dividing (4.1) by the associated areas

Plane Strain Axisymmetric

2 Ram AR = 2 W Y1 AR = -nY ]

(4.2)

2 Die wall ADW = 2 W (Y} - Y2) AR = tt(Y^ - Y^)

In all the normalized pressure results (p/2k) presented further on, k stands for the maximum shear (2.23).

Derivative-connected fields are evaluated at the Gauss integration points of each element, their distribution being determined by the FE network. The isoparametric elements used do not allow the consideration of derivative continuity across their boundaries, hence mismatching in the related fields is expected. The standard technique used in connection with FE work, viz. extrapolating to the nodal positions using the same shape functions, averaging at confluent nodes and then contouring locally, failed to produce good results for strain rates. A refined alternative to that, using for the strain rates a method similar to conjugate stresses (170) was avoided,due to the increase in computing time involved in the solution of an additional system of equations.

The need for more data points was evident. Then, two similar techniques were explored: Bivariate Interpolation and Smooth Surface Fitting (167),

66 the latter one giving the best results. The algorithm operates by triangularizing the data region, and after identifying the neighbours to each data point to be considered, estimates the partial derivatives and interpolates to obtain the unknown values, at the inner positions.

A parameter 9 was selected for the neighbours, and regular grids of

80 X 80 points in each direction were produced. The newly created grid was next interfaced with a contour plotting algorithm (168) and strain rates were examined. At the exit corner, a high peak was detected, surrounded by spurious values. This result was attributed to the inability of the interpolation in the regularizing step to account for the transition from adjacent elements. A new step was then interfaced, which fits Minimal Weighted Least-Squares Bicubic Splines (169) to the

FE results,and thenproducing the data expansion described above. Some of the plots produced for plane strain situations show quite a good behaviour, but the scheme requires a laborious adjustment of parameters in each step, which are different for each particular problem.

A faster post-processing technique was then developed, recording the variables of interest along the movement of the material. The technique operates as follows: a starting point is located at any location in the mesh, and its position is updated in time with a velocity resulting from Bilinear interpolation of the components of the 4 surrounding points. Simultaneously, the internal fields (strain rates, hydrostatic pressure, temperature, etc) are processed in the same manner and the history along the thus generated flow line recorded for ulterior display. The accumulated plastic strain was obtained by adding at each time step the increment produced by the strain rate.

Results are presented for plane strain and axisymmetric geometries, direct and indirect mode of deformation and different frictional conditions. Updating of a strip made of squares, in contact with the ram, was used to explore the distortion which the material undergoes

in steady state for different conditions of deformation.

The availability of the strain rates during the calculations at element level, was used to estimate mean equivalent values. Comparison

is made with Feltham (20) predictions for different extrusion ratios, and with profiles produced through histoiine recording. Total principal stresses were plotted using full lines for compressive and dotted ones for tensile state, respectively. An immediate correlation between graphical display and the stress state follows from the discussion in section 2.2.6; thus, for the geometric sets of Fig 4.1

(a) Directions of principal axes of both total and deviatoric stresses

and strain rates are the same.

(b) When the lengths of principal axes are equal, no shear z is

present.

(d) Principal axes tend to align with the global directions (ie x, y)

when the tangential rate of deformation y is zero or the

normal ones E^, E^ are very high (A 1.17)- The associated slip

lines tend to depart at 45° from the plane axes.

"(e) Principal axes tend to align at 45° from the global directions

when y increase or e^, e^ are very small. The associated slip

lines tend to depart either at 0° or at 90° from the global axes.

4.2 Direct Plane Strain Extrusion Through Square Dies

4.2.1 Computational Conditions

The geometry, dimensions and distribution of elements for the different grids used are shown in Fig 4.1. Frictional effects were

introduced by restricting the degrees of freedom of border nodes, as detailed in Fig 4.2. While the choice for the normal boundary condition at the container wall was obvious, the tangenti«il one depends on the

nature of the interface with the billet. If this is fully lubricated,

no tangential forces operate and the container wall appears as a smooth

surface, allowing full slipping. On the other extreme, if the material

sticks totally to the container, the tangential velocity becomes zero

(section 2.2.6). Perfect axial symmetry was assumed for the central

surface, which implies normal velocity zero; in addition to that, both

68 fig 4.1-FE meshes used for extrusion through square dies,

(i) T T~ 4

4k r

B < L{ A6 A3 'I A4

( II ) ( V ) ,A3 4 ft o AX to •a • A6 Mesh N- of N* of' Scale nodes elements

AX f •X, 1 187 39 1 : 4.75 ( III ) ( VI )

4 4 II 199 42 : it -T

4 III 327 72 I i "

IV 327 72 ' a t& AX AS i V 339 75 ! a

VI 199 42 ! a

VII 455 102 1 —L 1:1.62 halves were supposed to move with no relative motion with respect to each other and no friction acting on the interface. A constant flow -2 stress of = 173.2 MNm was used. In order to avoid singularities

in the pseudoviscosity, arising from strain rates values close to zero -5 at ngjd regions, a cut off value of 10 j^as selected.

4.2.2 Direct Extrusion. R = 2

4.2.2.1 Smooth container - smooth die

In spite of its purely theoretical interest, the existence of known

slip line solutions (27, 171) made this problem a natural choice for comparison with numerical results.

For the particular case of the penalty formulation, using nine node

isoparametric elements,0nate (141) found variations in the solutions obtained with different meshes. In order to explore that dependence,

a set of examples were run, an account of which is given in Table 4.1.

All results show good agreement with the Slip Line Field value (27,

171) of p/2k = 1.29. The solution for DPS2-1 is very close to it, but

the velocity field shows a deficiency of -3.25% at the exit. Reduction

of the dimensions Al to A4 (Fig 4.1) allowed an improvement in velocity,

but pressure rose (DPS2-2). If the coordinates of nodes C and D are

made to be coincident, that pressure drops(DPS2~3)with a 1.9% overestimation

of exit velocity. Reduction of the thickness A5 of the layer in contact

with the container improved both pressure and velocity results, (DPS2-4) with

an even better performance when nodes C and D were made to coincide (DPS2-5).

The associated force at B for smaller A5 is closer to zero than that

of previous examples, which makes a better agreement with the boundary

condition of no tangential force. The effect of changes in the penalty

parameter ot on the solution are shown in Fig 4.3. The solution obtained o for a = 10 gives a better fulfillment of the condition of compressibi-

lity, as can be seen by the proximity of the resulting exit velocity

and the exact value provided by balance of mass; also the associated

pressure shows a faster approaching to the slip line result, from

early iterations. =0,0^ = 0 ) (u v =0.1^=0) Y / / / ' / / Cqptajn^r LjL

o o it n \ \ > q o o II o N II II it billet II It

( L Z fw= 0) ^('lf_fy= 0)

(r) \ x "iZ) ( fx =0,uyzO) (fx =0,uy=0)

a) full slipping b) full sticking

fig 4.2-Direct Extrusion Boundary conditions imposed directly on nodes

O 0 2 3*56 2 3*56 number of iterations number of iterations fig 4.3-Numerical Convergence of the FE solution Plane Strain Extrusion R = 2, smooth walls.-

71 2 [lO" m] Addi tiona1 Reaction Average Code Mesh Number Penalty Ram force Die wa11 2. of 2 condi tions (A-B) force (D-E) 2k at node u at *[nMm" X (MN] B (MNj Al A 2 i terat ion to fig-4-1 M A3 AA A 5 i section x~F (l0 m/sec]

5 DPS2-1 1 0.75 0.75 1.00 0.90 1.70 7 ,0 y 5 y 0.2670 0.2636 1.335 -2.67E-3 1.935 c D

5 DPS2-2 1 0.01 0.01 0.01 0.01 2.50 7 10 - 0.2785 0.2776 1.392: +0.0121 1.981

8 DPS2-3 1 0.01 0.01 0.01 0.01 2.50 10 y y 0.2759 0.277^ 1.376 +0.0129 2.038 7 c = D

8 DPS2-4 1 1 0.10 2.^5 0.10 0.10 0.10 6 10 - 0.2739 0.2751 1.369 -8. E-5 2.028

DPS2-5 1 1 0.10 2.U5 0.10 0.10 0.10 6 108 y 3 y 0.2725 1.362 -8. E-5 2.026 c D 0.2733

Table Results for Direct Extrusion, Plane Strain, R = 2,

Smooth Container-Smooth Die Wall. Effect of changes in

geometry and penalty on the FE solution

Mote: Forces and pressures are expressed oer unit of width length An interesting feature is the closeness of both ram force and die

wall reaction values. This means that not only the force at B, but

along all the bi 11et-?container interface is very small as should happen

to satisfy the hypothesis of frictionless walls (see section 2.2.6).

A comparison with previous FE work using penalty and u/p formu-

lations is shown in Table 4.11. Pacheco (137) used a banded matrix

housekeeping in which the whole system (2-58) is solved simultaneously.

On the other hand, in the present work a frontal out of core solver

(175) was interfaced; this one requires much less central memory.

Thus, in Systems as the one operating at Imperial College, where the

hierarchical priorities are mainly defined by memory requirements, a

drastic reduction in the running turnaround is achieved.

The strong transition in the u velocity from the zero value at D

to the higher ones of the extrude,requires Al to A4 to be small in order to produce good pressure and velocity results. Nevertheless,

this condition is a disadvantage when strain rate contouring has to be produced. The sharp behaviour of strain rate at the exit corner, and the closeness of the associated data points (Gauss points of the element), makes the necessary step of regularization very difficult;

interpolation produces spurious results, which can only be overcome at the expense of a very big reduction in the size of the sides of the regular grid element. An alternative to that is cutting off the peak, which allows the more important behaviour in the neighbourhood to be explored.

A detailed description of the internal mechanics is shown in

Fig 4.4. The appearance of a dead metal zone is clear,where displacements, velocities and strain rates vanish. The maxima of strain rates lay on two directions M^ and at«*45° from the frontal wall. Compres- sibility is satisfied, the volumetric strain rate being zero almost elsewhere but in the vicinity of the die corner. For plane strain conditions, the second strain rate invariant eq(2.5) becomes l2 = (e + Y ) (eq At-17)- As the velocity field indicates, along the M.j and W^ regions the rate of change of a velocity component in its transverse direction (viz. 9u /9y) is larger than the change in its X own direction, viz. ( ^u /^x). This makes the tangential deformation Y X

73 Author N° N° Type of Addi tional p/2k u^ at x = F element condi tion elements nodes [10~2 m/sec]

Pacheco (137) 130 156 bili near 1.52 2.16 u/p 4 nodes

Pacheco (137) 130 156 bili near y E y 1.40 2.04 c D u/p 4 nodes

130 156 bili near 1.40 2.054 Pacheco (137) yc E yD Penalty 4 nodes

i soparam. 1.45 2.094 Onate (141) 39 187 yc E yD Penalty 9 nodes

42 i soparam. 1.362 2.026 DPS2-5 199 yc E yD 9 nodes

Hill (176) p _ /3 (0.58 + 1.44 In II ) = 1.366

2k " 2 Y2

Hill (27, 171) Slip' p_ = 1.29 2k Line Field Solution

Table 4.11 Comparison of FE results for plane strain direct extrusion,

R = 2, frictionless walls

74 4 0 4

* t t ±t it tt / / + / * / / / / / tttt y X X / /t * X H H # / / / / / /i > • ii i • / t t / /

t f f 1 1 II • t

jjg 4>4-Smooth Container Walls. Velocity Field .Principal Stresses .Mesh Deformation and Strain Rate IsolTnes for Plane Strain direct extrusion R=2 X perfectly plastic material VR=tQ~2 (m/sec) much larger than the normal e , Therefore, the associated stress state

eq(2.34) is almost pure shear, and M^ and M^ are slip lines. Thus, in

this case the result is the outcome of the method used,rather than an

a-priori assumption as in slip line techniques.

An examination of the plot of total stresses also demonstrates the

shear nature of the flow, with marked changes in the lengths of prin-

cipal axes. The stress states along the line have a larger hydro-

static component than the ones along M^, which is due to the

release of the compressive state in the area facing the exit. At the

billet-container interface, equiaxed total principal stresses with no orientation changes with respect to the x-y position, clearly indicate

that the non-shearing assumed (Fig 4.2) is actually recovered in the

solution (see section 2.2.6). This is not an obvious outcome due to

the formulation being based uponvelocities as primal variables.

The distortion of an originally straight line normal to the

axis as it moves under steady state conditions,comes out with dx/DD = 0.303 at the exit. Hill (171) slip-line estimation was D dx/Dg = 1.257, and he reported a kink in the deformed grid near the central axis which is not found in the present solution.

4.2.2.2 Rough container - rough die wall

The boundary conditions assumed are shown in Fig 4.2.b and

Table 4.1 I I gives the results obtained in connection with the geometrical layouts of Fig 4.1. Table 4.1II clearly indicates a dependence of the ram force and hence of the p/2k value with the billet length, a feature not provided in the solution proposed by Johnson (177). On

the other hand, the reactions at the die wall are almost constant,

their values for different lengths not differing by more than 1.5%.

The boundary condition of full restriction on the velocities of Node

D (Fig 4.1),produces a better velocity profile at the exit than when slipping in the y-direction is allowed as it happens in the frictionless case

A further improvement was found when reducing the thickness A2. As in

the case of the smooth walls,a drop in the ram load was noticed when nodes

C and D were made to collapse.

76 2 Code Mesh (l0" m] Number Penalt / Alterat. Ram Force Die Wall P Reaction Average u^ of ocfMN from (A-B) Force at node at section Fig <4.1 [MN] (D-E) B MN A1 A 2 A 3 A'4 A5 i ter. 7- x=F m [MN] [10-2m] [l0~2 m/sec

8 DPS2-7 0.10 2.<45 0.10 0.10 0.10 6 ,o = 11.9 0.3178 0.2358 1.589 -<4.

DPS2-8 1 1 0.10 2.<45 n. 10 0.10 0.10 6 108 = 11.9 0.3161 0.23<46 1.580 -<4.<48E-3 2.022 X1

yr - yD

8 DPS2-9 • 1, 0.10 0.20 0.10 0.10 0.10 6 10 X = 21.9 0 . <4 0 0<4 0.2373 2.002 •1<4.<47E-3 2.009 1

6 DPS2-10 1,1 0.10 0.20 0.10 0.10 0.10 6 10 = 21.9 0.3082 0.2381 1 .5<41 -7.63E-3 2.0014 X1

p = 0.63 + 1.28 In L = 1.517 Johnson (177) Y2

Table <4.111 Results for Direct Extrusion, Plane Strain, R = 2,

Rough Container and Die Wall. Effect of changes in

geometry and penalty on the FE solution

Note: Forces and pressures are expressed per unit of width length §900 0 t 0

fig 4.5-Rough Container Walls. Velocity Field .Principal Stresses .Mesh Deformation and Strain Rate Isollnes for x Plane Strain direct extrusion R=2 2 perfectly plastic material , VR =1(f ( m/sec) A detailed analysis of the internal mechanics is shown in Fig A.5.

Both velocity and displacement field (inherent to grid distortion) show a dead metal zone larger and more extended backwards than for the frictionless case. The front of a vertical line leaving the ram appears at the exit of the steady state with a more pronounced defpr- mation profile than the fully lubricated case, with (dx/Dg)= 1.1A5.

The maximum strain rate direction M2 and hence its slip line now moves backwards, with semiangle bigger than 45°, as preassumed in slip line constructions (Fig 10 in (35))- The principal stresses at the container walls show the effect of the condition of sticking imposed through the velocities; principal axes lie at A5° to the tool interface.

Nevertheless, the stresses show an abnormal behaviour there, some components appearing extremely large. This fact apparently was associated with the selective integration used indescriminately for both volume and border elements and with the sharp velocity transition at the ram- container corner (Fig A.2b). When full integration was introduced, in connection with the boundary elements next described in section A.A, the anomaly was removed.

A.2.3 Direct Extrusion, R = 10

These examples were selected in order to assess the performance of the model at a higher extrusion ratio. Table A.IV summarizes the results obtained from mesh IV of Fig A.1 for both boundary conditions of

Fig A.2.

The solution for the frictionless case (DPS10-1) shows good agreement with previous results. Close values of load and die wall forces are found, indicating the convergence of the solution to the boundary conditions of Fig A.2 a. For the rough container assumption, the ram load increases noticeably with billet length, while the die wall reaction keeps within a bound of 1.2% bigger than for the frictionless case.

The incidence of frictional conditions in the flow are now more pronounced than for R = 2. Under full lubrication, the dead metal zone almost vanishes, remaining only in the very far corner of the die wall (Fig A.6). On the other hand, if full sticking of the billet to Code No of Penalty Billet Ram Die wa11 p/2k Average i ter. 1ength force force u at x== F (MNm"2] (A-B) (D-E) x -7 (l 0**2m] 2 m/sec] MN [ MN ' (10 .

9 DPS10-1 6 10 " no.38" o .583T "O.58I5 2.91 10.14 full sii ppi ng

8 DPS 10-2 6 10 10.38 0.7062 0.6234 3.53 10.13 full sticki ng

Table 4.1V FE results for direct extrusion, plane strain, R 10.

Geometry as mesh IV (Fig 4.1) with Al = 0.01, A2 0.5, A3 = A4 = A5 = A6 = 0.02 [lo"2m] . -2 Ram velocity 10 [m/sec). -2. Flow stress = 173.2[ MNm J

Note: Forces expressed per unit of width length

/3 (1.45 In _1 ) = 2.891 Hill (176)

2k 2 Y2

0.63 + 0.95 1 n __1 = 2.817 slip.

2k Y2 ) Johnson (177)

_p_= 0.63 + 1.28 ln J_1 = 3.557 stick. 2k Y„

80 4 * 4 4 4 *

- " " I > 1 I

^ V N \ \ \ | " y \ \ \ \ I

i 11 11 tmnt x x X X X x -f xx-h-XF X X X X X X X XX XX 4f

XXX X X X X XX XXXf XXX X X X X XX XX -H-

X XX X X X X XX XX XX XXX x x X X X X X f XX

X X X X X X X * * X 4 XX- f 11 if h 1M! !5i

fig A.6-Smooth Container Walls Velocity Field .Principal Stresses .Flow Lines and Strain Rate Isolines for Plane Strain

X Direct Extrusion R=10 .perfectly plastic material V -icrtn/sec 0 0 9 9

- - - V \ \ \ • •

- - - - - ^ \ \ \ \ • 1 * A - I T l P 1 iiiii mil 11/1 / ill/ /

CP rJ £ ^ •/• "X x y- + f -f X X X + •f -f ++-H-* X X x x •f + + +-H-* X XX XX X X + ++-V* X XX XX XX f + X XX XX XX X + +JR JK X XX XX XX /IH* X X j * * I M» rtt i m m m M 1 i if:

fig 4.7 -Rough Container Walls Velocity Field .Principal Stresses .Flow Lines and Strain Rate Isolines for Plane Strain Direct Extrusion R=10 .perfectly plastic material V =10~^m/sec. H '

X the container is allowed, the shape and size of that region is com-

pletely different (Fig 4.7). The strain rate isolines show also a

different behavi our than for R — 2. When friction is present, the

material coming from the surface close to the container has accumulated

a very large strain, due to its travel along a high shearing area, as

the strain rate isolines of Fig 4.7 indicates. Such a situation does

not occur in the case of frictionless walls. As the plots in Fig 4.6

show, the flow does not move along but rather crosses a shear line.

Besides, the strain rate level achieved at M^ (Fig 4.6) is quite low,

and the material only finds an appreciable strain rate when reaching

the exit. A general agreement with the location of those lines in Slip

Line Field construct ions is found; more elements are necessary to

improve the resolution of the strain rate contours. The main lines of

Fig 4.7 resemble the solution presented by Alexander (Fig 264 (178));

also both Mj and M2 lines in Fig 4.6 are located as Fig 260 (178).

The principal stresses picture indicates that the assumed boundary

conditions are satisfied, their axes lying either at 0° or at 45° of

the interface, for frictionless or sticking respectively. Still, the

problem mentioned in section 4.2.2.2 in connection with sticking

borders remains. The effects have been emphasized in Fig 4.7 allowing

the tensile stresses to be represented by dotted lines; the container

border appears having oscillatory stresses, alternating tensile and

compressive states.

4.3 Indirect Plane Strain Extrusion Through Square Dies

4.3.1 Computational Conditions

Examples of extrusion ratio R ~ 2 and R = 10 were solved, using

meshes whose configurations are detailed in Fig 4.1. The material was -2

considered to be perfectly plastic, having a flow stress of 173-2 MNm

A cut off value of 10 was selected to reset the strain rates when that level was lower. 0 0* + 0 0 + 0

( fx =0 . Uy = 0 )

> It ( b )

(f =0.u =0) x y

Oo ( U :0,u:0) U :V,U = 0 ) X y

o II

~ \ o o 11 \1 II f \ ( C ) (d) > V AIi=0V0)

-'-I

( fx = 0, Uy- 0 ) fx =0. uy=0 )

fig .A .8-Indirect Extrusion . Boundary Conditions imposed directly on nodes. The boundary conditions explored are shown in Fig 4.8. Depending

on the experimental arrangement, two different sets are possible. In

Fig 4.8c the container moves jointly with the ram, squeezing the billet

against the fixed pad holding the die; in Fig 4.8d all parts remain

fixed and the moving tool is the frontal pad pushed backwards.

Frictional effects were introduced as before, operating on the

tangential component of either the force or velocity at the billet-tool

interfaces, in order to simulate the extreme cases of either slipping

or sticking.

4.3.2 Indirect Extrusion R = 2 and R = 10

* Table 4.V summarizes the various conditions employed and the

results obtained. One interesting feature is the nature of the boundary

conditions at the back wall for the situation of full lubrication. It

would seem natural to adopt there the restriction of no tangential forces

^ (Fig 4.8a). However, the p/2k value obtained for that hypothesis ( IPS-l)

is high compared with the slip line result 1.29 given by Hill (171)? 9

closer examination at the principal stresses (Fig 4.9)>shows that the

condition imposed on that interface is not recovered in the solution:

main stress directions lay far apart from the 0° degree characteristic

^ of no surface deformation, and high shearing is detected near the axis.

This result proves that even for full lubrication, the billet tends to

stick at the extrusion ram. It shows how the imposition of the boun-

dary forces condition does not guarantee that such a result is

^ actually achieved in the numerical solution (see section 2.2.6). V/hen

the tangential velocity was cancelled at the ram wall as in Fig 4.8b,

the results IPS-2 indicated the pressure improved and the stresses

behaved coherently with that restriction imposed. Reduction of the

layer A4 for run IPS2-3 improved both pressure and exit velocity.

The mechanics of Fig 4.8c and 4.8d only differ by a constant com-

ponent of u velocity. In spite of the difference of both flow patterns

(Fig 4.10), in any derivative-connected field (eg strain rate, stress,

^ load), that constant does not produce any perturbation as can be seen

in the resulting loads for IPS-4 and I PSV .

35 4

fig 4.9 - Indirect Plane Strain Extrusion R-2 Effect of Boundary Conditions at Rom for fig 4.8(a)

fig 4.10 - Plane Strain Indirect Extrusion R = 2. Flow Patterns for the two ways of operation

PC Code Mesli BC Extrus ion Number Penalty Billet 10 2 Ram Die Wall p 1 " H Average u^ fig rat io of 1ength force Force ram [MNm"2j iter. (A-B) (D-E) 2k at x=F R = Il [l0'2m] Al A2 A3 A<4 A 5 A 6 M (MNJ 2 [10~ m/sec] Y2 8 IPS2-1 <:.7a 2 6 ,o 11.9 0.10 2.<45 0.10 2.00 0.10 - 0.2956 0.29<47 1.

8 IPS2-2 1 1 <4. 7b 2 6 ,o 11.9 0.10 2.<45 0.10 2.00 0.10 - 0.28<47 0.2851 1 .<425 1.06<4 , SL

6 IPS2-3 \i <4.7b 2 6 in 11.9 0.10 1.25 0.10 0.10 0.10 0.10 0.2702 0.2702 1.351 1 .00<4 SL

8 IPS2-A 1 1 <4. 7d 2 6 ,o 11.9 0.10 1 .75 0.10 0.10 0.10 - 0.2597 0.3215 1.299 1.028 ST

8 1PS2-V 1 1 *4.7C 2 6 ,o 11.9 0.10 n. in 0.10 0.10 - 2.028 1.75 0.2597 0.3215 1.299 i ST j

6 IPS2-5 V <4.7d 2 6 10 11.9 0.10 1.25 0.10 0.10 0.10 0.10 0.270<4 0.3055 1.352 1.007 j ST

8 1 1 PS 10-1 IV <4. 7d 10 6 ,o 10.<4 0.01 0.50 0.02 0.02 0.02 0.02 0.62<42 0.6888 3.12 9.120 1 ST

Table <4. V FE results for Indirect Extrusion, Plane Strain

flote: all load expressed per unit of width length Both ram and die loadsare very close for the frictionless container

case. However, when the relative movement between billet and tool is

fully restricted (Fig 4.8c or 4.8d), the ram force almost retains the

former value but the die reaction increases. The first effect allows

the consideration of the ram load as a frictionless estimation, in

agreement with the practical rule of elimination of friction to which

the indirect process is associated. The amount of frictional forces,

given by the difference between ram and die wall values, is lower than

for the direct extrusion case.

A better understanding of the mechanics involved can be gained

from the plots of Fig 4.11 to 4.13- For R = 2, full lubrication still

produces a dead metal zone (Fig 4.11). Two isolines, M^ and M^, of

maximum strain rate spread reaching both container wall and axial line iO * / * 2 • 2. at=45 . Along these lines, the deformation - (£ +Y ) is mainly Z. x e tangential; the normal component x is very small due to the

low 3u /3x changes. Therefore M.. and M_ are maximum shear lines and X * £M

are in agreement with the slip lines proposed by Hill (171) . However,

when the relative movement of the billet-container is eliminated, the

dead metal zone almost disappears (Fig 4.12); this effect is much

more pronounced for R = 10 (Fig 4.13). The dotted lines superimposed

on the strain rates in Fig 4.12 shows the excellent agreement with the

slip line field proposed by Lee (Fig 42 in (13), and (17°)), and

Johnson (Fig 17 in (35))- It is apparent that both solutions are

possible, depending on the type of friction assumed. For R =10 the

picture is not clear, and a critical analysis will require more data

points for contouring, and hence more finite elements. However,

certain information is available; two main shearing directions depart

from the die corner (IHJ and Mq ) , and a back line (l^ ) is also observed; all these results agree with Lee's(Fig 42 (13)) proposed solution.

In both R= 2 and R =10 solutions for sticking walls, a valley appears

in the region just above the exit orifice. In spite of it exhibiting almost zero strain rate, its role is completely different than the similar dead metal zone previously found for the direct case. Here the material actually moves across such region, as the flow patterns show. Indirect extrusion has been regarded as a process developing a deformation more homogeneous and localized than the direct one. Apart 2yr - 2./3. t

t It I t t t f

t CD ////in* UD / /1 M 111— i i n nin, i 1111111«»

fig 4.11-Smooth Container Walls. Velocity Field,Principal Stresses ,Mesh Deformation and Strain Rate Isolines for Plane Strain Indirect Extrusion CD R = 2 perfectly plostlc material ,V=10~2 m/sec

x 4 * %

f ig 4.12-Rough Container Walls. Velocity Field .Principal Stresses ,Mes^ Deformation and Strain Rate isollnes 4c Plane Strain indirect Extr,jsIon perfectly plastic material . V-10 m/sec.

r * 4 0 4

N X \\ \ \

XXX + \ \ $ X + r X X + ++H** X X X X X X-f-R-Vk* XXX X X X X-H-P** XXX X X X XX+-VP*. XXX X X X X XX +-V X XX X X X X XX +V X X X X X x xxy f +

1 If II H II !! !J:

f igA.13-Rough Container Walls Velocity Field .Principal Stresses .Flow L^es and Strain Rate isolines for Plane Strain Indirect Extrusion R=10 .perfectly plastic, material V=10~2m/sec x from the phenomena just described, a comparison among the direct strain

rate isolines of Fig A.12 and A.5 indicates that in the indirect case,

the material reaches the exit without having accumulated a large total

strain; almost all deformation actually happens at the main maximum

0 line M^ departing from the corner die. A further analysis of these

topics will be carried out for the case of rod extrusion treated later.

The distortion of a vertical line parallel to the ram, as it moves

forward under steady state conditions indicates that a flatter profile

is obtained for sticking (Fig A.12) than for slipping walls (Fig A.11),

with dx/Dg values of 0.9032 and 1.3870 respectively. Therefore, the

amount of inhomogeneity found among distorted squares of the grid at a

cross-section of extrude is reduced when friction increases. It is

% worth noting the coincidence of dx/D^ for direct slipping with indirect

s t i ck i ng wa11s.

A.A Alternative Introduction of Friction

A.A.I Computational Conditions

A different approach was implemented in order to develop a general

procedure which could handle both maxima and minima as well as inter-

mediate frictional situations. Following Zienkiewicz et al (138, 1A1)

a narrow layer of elements was placed at the wall interface; the outer

border of such elements are in contact with the container and both

degrees of freedom are fully restricted as in Fig A.2b; the flow

stress is described by:

mote if Ima e ' < a (A.3) v 1 v y

o i f Lot e I > a (A.A) y v y

where* a = Yield stress of the billet material y ote^ - Hydrostatic pressure (a = penalty, e^ = volumetric strain

rate)

0 ^ m ^ 1 fm = 0 ful 1 s 1 i ppi ng

[m =1 ful1 s t i cki ng

92 Initially, the boundary was modelled with the same considerations

as the bulk, namely using nine noded isoparametric elements and

selective integration. The solution obtained for m = 0 did not give

an approximation to the cases previously dealt with, by nodal force

specification as in 4.2.2.1. The load results were too high and the

principal stresses at the border failed to satisfy the non-shearing

condition imposed through m = 0. A drastic improvement was obtained when the penalty matrix term of the border elements was fully inte-

grated at the 3X3 Gauss points. The load results obtained with this

method (DEPS2) are compared with the previous solution DPS2-5 given by

the nodal specification approach in Table 4.VI, showing the close

agreement of both.

The problem mentioned above ,in connection with oscillating stresses

at the interface for the condition of rough walls, still persisted.

A close examination at the corner ram-container-billet shows a strong

transition in the x-velocity from its value at ram to zero at the con-

tainer wall. The thin element is not able to cope with such singularity

numerically and the perturbation propagates, giving rise to spurious

strain and hence stresses along the interface. By freeing the last

node in the line between ram and container, the problem was removed.

In spite of the theoretical feasibility of eq(4.3) in dealing with intermediate friction, difficulties were found. The loads obtained with such a scheme for m = 0.2 did not show differences with the extreme case m = 1. Further examination of the solution revealed that at the critical corner ram-container, the incompressibi1ity easily fails, and then the hydrostatic pressure becomes very high. This fact makes any m factor slightly different from zero to degenerate in the full sticking way of eq(4.4). Thus, the corner elements holds the slipping allowed by the low m.

A modification was then made, following the more pragmatic view adopted in upper bound techniques (49.) of operating directly on

the flow stress:

a = ma (4.5) y

93 CODE Mesh Penaltv BL m Uniaxial Ram u at P P* x e (4.7) P/2k comparisons with' (Fig [MM] yield speed R = —1 2 exi t 2k 2k previous work 1 4.1) K «i stress _2 Y m [10 m/sec] 2 (l/sec] [MNm~2] rd

DPS2-5 11 10.3 0 2.026 1.36 1.37 (a) = 1.29 (e) = 1.36

8 2 DEPS1 VI ID 11.9 1 173-200 1.0 2.017 1.53 1.30 (b) = 1.29

0EPS2 VI 11.9 0 2.016 1.3*4 1.3*4 (c) = 1.52

DEPS7 1 3.607 0.828 3.28 2.09 (b) = 2.05 h- J20 7 O DEPS8 VI 1 !0 9-5 0.5 107.256 0.8 3-605 0.807 3.36 2.57 (c) = 2.55 LU a. DEPS9 0 3.601 0.837 2.30 2.28 (e) =1.88 O i 7 /80 DEPS10 VI 1 10 6.2 1 127.269 0.8 7.306 1.563 3-75 2.9*4 (b) = 2.71 (c) = 3.43 (e) = 2.75 ;

8 10 DPSIO-2 IV 10 10. 4 1 173.200 t.O 10.13 3-53 3- 12 (b) = 2.82 (c) = 3-58 (e) = 2.89

IPS2-5 6 1 1.007 1.35 1.53 (d) = 1.29 2 11 10 11.9 173.200 1.0 1 IPS2-3 1.004 1.35 1.35 0

I0PS1 1 3.610 0.91 2.21 2.46 (d)T = 2.16 l- 7 O ifzo IDPS2 VI 1 10 9-5 0.5 107.256 0.8 3.607 0.855 2.21 2.37 • al IDPS3 0 3.60*4 0.867 2.17 2.17 O i

7 /80 IDPS4 VI 1 10 9.5 1 127.269 0.8 7.306 1.707 2.96 3-30 (d)t = 3.20 I

i 8 10 1 DPS 10- IV ,0 10. 4 1 173.20 1.0 9 -120 3-12 3.4*4 (d)T = 3-25

Table A.VI FE results for Direct and Indirect Plane Strain Extrusion

(a) HM! (27)

(b) Johnson l^yyj [case (i): p/2k = 0.63 + 0.95 InR

(d) Hill (171)

situation of Fig 4.2b is recovered. On the other hand, for m = 0 the

plastic ratio H(2.37) becomes zero; so does the border shear Z. and

its associated tangential force, which reproduces the slipping inter-

face conditions of section 2.2.6.

A useful comparison parameter used throughout all this thesis is provided by the ram equivalent to the die wall pressure defined as:

p* _ Reaction at the die wall (4.6) Ram area

4.4.2 Mean Strain Rate

These estimations were intended to be exploratory, and as a byproduct of the FE solution. Taking advantage of the availability of the strain rate during calculations at element level (before assembling all of them), an average along the central line of the 3X3 Gauss points was introduced. This value was then weighted by the cross-sectional area of the element. When all elements were screened, an overall average was produced as:

alt 1 elements where A^ = ram area, as in eq(4.2)

AA = 2W (Y - Y . ) for plane strain max min r AA * 2 TT (Y + Y . ) . (Y - Y . ) for axisymmetric max min max min ' 2

Y and Y . stand for the extreme radial ordinates max min of each element.

N = number of elements with significant strain rate

in the flow direction.

An assessment of N can be obtained from the width of the peak (in

Fig 4.14 and 4.21) and dimensions in mesh VII of Fig 4.1. Such examination of the strain rate profiles gave N = 3 for all the examples in

95 Tables 4,VI and 4.VIII. The definition in eq(4.7) implies a volume

average of the strain rate at the exit. The method is similar to

the one used by Paterson (8), with the difference that here no weighted

average was introduced in the flow direction. This limitation is set

by the—impossibi 1 ity when operatincr-air-element 1 evet—aF"determln?ng

the path length, if the starting point is outside the element. An

alternative procedure much similar to the one in (8), can be easily

designed for operation on the assembled set of data during the post-

processing and histoline recording step. In Chapter 5 differences

among mean values and actual heights of the strain rate peaks are

examined.

4.4.3 Plane Strain Extrusion

Results obtained with this procedure are displayed in Table 4.VI, and compared with both the nodal approach of Fig 4.2 and slip line

solutions. In all of them, FE values retain their character of upper bounds to the latter, with quite a good agreement. Further improvements are still possible, in the solutions obtained through meshes VI and VII,

reducing the critical dimensions A1 to A5 and collapsing coordinates of nodes C and D. Nevertheless, that operation was not carried out, because at the time of solving the examples in Table 4.VI the aim was

to analyze the strain rate field through contouring of isolines, and as discussed in section 4.1, the presence of closer Gauss points becomes disadvantageous. This direction of work was abandoned later, after the successful outcome of the technique of recording the history along flow paths.

An analysis of the internal mechanics for plane strain extrusions,

R = /20 is presented in Fig 4.14. In the direct mode, the material flowing along both axial L^ and middle L^ lines, is affected by slightly different strain rates, independent of the wall friction.

These main peaks are associated with the directions Mj of Fig 4.4 and

4.5, and their coordinate shift constitutes another way of visualizing the spreading backwards of the M^ slip line. For the line L^ running close to the border of the dead metal zone the strain rate rises earlier, namely as soon as the direction is reached, fig 4.14 - Dependence of strain rate,strain and hydrostatic pressure with mode of deformation and frlctlonal conditions .Plane strain extrusion ,R=4.47, perfectly plastic material 2 oy= 107.2( MN/m ) Starting coordinates of flow lines (10~ Vi) LINE Ll{—) L2 (-—) L3 ( ) Y: 0.35 1.86 3.45 DIRECT INDIRECT m = 0. 0 m= 1 .0 m= 1 .0

Length of flow line (10 m) 45'

£ + X

•V .A' .v

fig. 4.15- Detail of Principal Stresses at exit. Plane Strain Indirect Extrusion 1 mm = 8.47 MN/m2 R =/20 , 0=107. 256 MN/m2 1 mm = 2.17. 10"5 m c ompress i ve tensile and the rise is faster for rough walls? the high peak corresponds to

the strong transition around the die corner. The accumulated strain

also illustrates the different behaviour of the outer line L,.

—Foi^ -the i nd i rect-mode, i n-both--l wies-U, -afld-Lj-f lowi ng -f.com-above_

the outlet, a plateau followed by a sharp increase of strain rate is

detected. These results are due to the flow crossing the strain rate

valley observed in Fig 4.12 and 4.13- Worthwhile noting is the simila-

rity in the total deformation accumulated for the three lines, which

contrasts with the heterogeneity of the direct process when m = 1.

The hydrostatic pressure is compressive in all the deformation

area. Only the line L^ emerging from the border reaches tensile

values, and a further detail is presented in Fig 4.15. The horizontal

plateau of a^ for small coordinates, which determines the extrusion

pressure,also shows the requirement of higher load for frictional

direct extrusions.

4.4.4 Axisymmetric Rod Extrusion

4.4.4.1 Direct extrusion, low ratios

The FE results for ram pressure p presented in Table 4.VI I were obtained for the hypothesis of rough container walls. The results in

section 4.2 for plane strain conditions suggested that the frictionless

ram load could be estimated by the die wall reactions. Thus, the ram

equivalent to the die wall pressure p* (eq(4.6)), should give an

estimate of the required ram pressure under fully lubricated conditions.

This prediction is found here to approximate the experimental results

for lubricated extrusion of lead and super pure aluminium given by

Johnson (180). The values provided by his fitting relationship

p/a = 0.8 + 1.5 Ln (R^R^) are slightly higher than the FE friction-

less estimate. The difference can be attributed to lubrication in the experimental setup still allowing for some frictional forces,while

the FE model for m = 0 does not. However, outside the perfectly plast.ic idealization, another explanation is possible in terms of changes in flow patterns due to work hardening (section 5.8). Table 4.VI I Comparison of results for direct rod extrusion, low

reductions

Kobayash i Hal 1i ng Adie Johnson FE FE and and (182) (180) P Thornsen Mitchell P* a a (18D (49)1

0.8 3.05 3.19 3.19 3.21 3.00 5.05

0.9 3.79 4.32 4.23 4.25 3.58 5.99

"[: recomputed in ref (182)

p* = pressure at die wall converted to rem area (4.6)

Pg = ram pressure

2 cry = 107.26 MNm"

100 Comparison with reported Upper Bound solutions are also given in

Table 4.VI I. Smooth container walls and dead metal zone in the upper corner, were assumed by Thomsen and Kobayashi (181) and Adie (182).

Hailing and Mitchell (49),adopting the same boundary condition for a

90° die concluded that no dead metal _zQne_occu_rred for_.r:edjJCJtion.

(Y1/Y2) >» 3.6363, behaviour which has been found in the present work

for plane strain situations (transition from Fig 4.4 to 4.6). The

FE frictionless estimation (p*/a) gives lower Values than the mentioned

Upper Bound results.

4.4.4.2 Direct and indirect extrusion, high ratios

Numerical results for a wide range of extrusion ratios, up to

R =150 are presented in Table 4.VIII. Quasi-steady-state analysis was produced for R = 12.4 ancl R = 20, considering any elastic and acceleration effects to be negligible.

In the case of direct extrusion, the incidence of frictional forces on the resulting ram pressure p along its displacement is noticeable (Fig 4.16a). On the other hand, the estimated frictionless pressure p*, eq(4.6) remains almost constant, only slightly rising when the billet length inside the container becomes short enough to interfere with the steady state.

A similar analysis was done for direct and indirect mode R = 20, under the same overall conditions (Fig 4.16b). As indicated by practical experience, ram pressure results are lower for indirect than for direct extrusions. The frictional effects are also lower for indirect mode, as can be seen by comparing the slopes of curves I. and III in

Fig 4.16b. In general, Fig 4.16 indicates that in a perfectly plastic isothermal analysis, the frictional forces exhibit a linear variation with the billet length.

In Fig 4.17, the pressure dependence with the extrusion ratio R is examined for the direct extrusion case. The frictionless pressure p*

(4.6) obtained by FE, is lower than Johnson's empirical relationship

(180), and the separation increases with the extrusion ratio. It is here recalled what has been said in section 4.4.4.1 regarding the full

101 Extr. Code Frfct Billet Ram Uniaxial u at Mean I Ram force Die WalI F^ :/ P /a p- crv Ratio 1ength speed yield X equi val. (A-B) FR force (D-E) y •2 exi t,x»F s t ra i n ra t _2 _2 stress (MNr ! 3 3 (4.6) [l0 m ] [l0 m/s] J[l0"2m] (4-7) fseCM (9.8 10~ MN] (9.8 IO~ MN) 5 DEAX1 1 9-5 0.8 107.256 <4.053 0.849 242.28 114.98 5.05 2.<40

10 DEAX2 9^5 0.8_ m/L256~ _-8_1,33 _1.156 787 1 <4 154 32 5.99 3.22.

DEAX3 1 27.50 3-538 0.205 1253-34 600.81 1M 3.28

12. <4 DEAX<4 1 20.32 0.27<4 81.065 3-538 0.207 1079.36 601.20 6.<42 3.28

DEAX5 1 1<4. 18 3.538 0.213 936.35 603.<48 5.57 3.30

0EAX6 J 8.00 3.538 0.250 796.52 619.80 <4. 7

DEAX7 1 9.5 16.25 1.565 329.24 197.146 6.87 <4.12

DEAX8 0.5 9-5 16.23 1.497 288.36 200.72 6.01 20 0.8 107.256

DEAX10 1 3.8 16.25 1.995 255.84 204.19 5 - 34 <4.26

60 0EAX11 l 9.5 0.8 107.256 <49.16 2.<489 393.44 258.16 8.21 5.39

00 DEAX12 1 6.2 0.8 127.269 62.72 2.939 <4 <47.20 3141.17 7.86 6.00

150 DEAX13 1 9-5 0.8 107.256 12<(. 30 3.598 '158.13 319.95 9.56 6.66

5 IDAX1 1 9.5 0.8 107.256 <4.067 0.943 127.31 153.43 2.65 3.20

10 I0AX2 1 9-5 0.8 107.256 8.1<45 1.263 173-56 199.84 3.62

IDAX3 ,1 9.5 16.25 1.668 219 -14 238.54 14.57 <4.97

IDAX<4 0.5 9.5 16.23 1.551 212.07 4 .42 <4.72 20 0.8 107.256 226.45 IDAX5 9.5 16.21 1.579 208.76 208.51 1|.35 4-35

IDAX6 1 3.8 16.25 2.166 199.19 235.'4l 14.15 <4.91

60 I0AX7 1 9.5 0.8 107.256 <49.16 2.581 281.51 300.64 5.87 6.27

80 IDAX8 1 6.2 0.8 127.269 66.69 3.122 356.72 386.63 6.27 6.32

150 IDAX9 9.5 0.8 107.256 12

Table A.VI I I FE Solutions for Axisymmetric Extrusion, Set B, Mesh VII (Fig <4.1),

6 Iterations, Penalty = 1(2

Al = 0.025, A2 = 0.035, A3 = A'l = A5 = A6 = 0.02 10_2m

= (o. Ibmfor R = 12.<4, 0.075m all other examples 'com. 1 lubrication at walls achieved in the numerical model. FE results are

also compared with Minimized Upper Bound solutions. For smooth con-

tainer walls (Fig 4.17a), the bounds given by single and double

triangular arrangements of MUB (54, 55) are higher than the FE estimate

p*. Lower dependence with In (ft)~Is^fuund-in FE~ttian~trrMUB;—this outcome is particularly important when it is recalled that no a-priori assumptions in the shape and position of discontinuity lines are made

in the case of FE.

For direct extrusion with the assumption of rough container walls,

the ram pressure obtained by FE is lower than the MUB one (Fig 4.17b), separation which increases with the extrusion ratio. The points on the vertical line at R = 12.4 in Fig 4.17b show the FE bound dependence with the billet length. On the other hand, for a constant billet

length, the parallelism between the FE pressure lines for frictionless and rough walls, suggests that no alteration of the flow pattern at the container border as R increases takes place.

For indirect mode, the results in Table 4.VIII show that the ram

load Fg is normally lower than the one acting on the die wall F^.

Again, as in the plane strain situation, a close agreement between both values is found when m = 0. When friction increases, a reasonable closeness is still kept among ram values Fg(nflt0) with the ones had been found in the frictionless situation Fg(m=0). Even if the dis- persion is somewhat higher here than that for plane strain, the general behaviour discussed in section 4.3.2 is still valid, allowing the extension of the association with the "elimination of friction" to that

in which the indirect extrusion is practically linked. Thus, for the

indirect mode the frictionless pressure is given directly by the ram value p . FE results produce lower load values than MUB (Fig 4.18), and this separation increases at high R. Another interesting feature is that the amount of frictional forces connected with the difference between die wall and ram pressures is reduced with the increase of R.

This suggests that at higher ratios the M^ shear line (Fig 4.12) moves more into the volume, reducing the small amount of border shearing.

103 ? 4 6 B 10 12 M 16 IB 20 22 24 Displacement (.MO"2 m) R= 1 2 . 4 ,

I DIRECT, p o II DIRECT .p* \ 7 H INDIRECT p a. jy — INDIRECT p*

-I

( b ) i

? .ft 4.7 4 .9 S .6 6.3 *> .0 Displacemenl (10^ m) 7 R 20 , 0COnt - 5 mm, Initial l. B Tile t 95 mm 1 m-i ,Vrar^ 8 mm/sec ,oy-• 07.76 MN/m-

fig 4.16 _ Dependence of pressures on ram displacement Axisymmetric direct and indiret rod extrusion

104 ^mcotr; Contemner Walls Rough Conta°rior Wcib

...3r 13 : Fl ,es'.'ma:"cn 'rem ore press (4.6) Single trlonq. ,MUBt mr I ) tutcner (55 > prison C I&0) O FE .Frlctlonless estimation from a'e wan V r ID-Sl- bl^ge tr-onq.Mug, Tutcce' (55) cl l? :— FE.

; ° en

( b)

?.bl :.B 7.5 ifr .-9 ?•? ?.6 3.0 3.4 •<•6 5.0 5.4 Ln (R)

fig-. 4.17 - Comparison of FE and Upper Bound Results Axisymmetric Direct Rod Extrusion [ Paterson ( 8 > o r Stng. rrfong. .DMZ.Tunnrcltfle t54> I s:ng. Tr-rang. .NO OMZ. Tunn/cO'fe (54) • f"E.p = rarn Dressu^e ; 'E.p* (4.6) •X FE results for 2 BL = 95 mm , oy = l 07 . 26 MN/m [ m=1.0 X

2.2 2.6 3.0 3.4 3.8 4 .2 4.6 S.C 5-4 Ln ( R )

fi g.4.1 8-Comparison of FE ana Upper Bound Results

Axisymmetric Indirect Rod- Extrusion The connection between the mean equivalent strain rate, evaluated as described in section 4.4.2 and In R is shown in Fig 4.19. Indirect mode produces higher values than the direct one. A general agreement with the linear relationship proposed by Feltham (20) for direct extrusion is-found, whiclr improves-at-larger extra sJon-rat4os^—However, both mean parameters are averages of heterogeneous strain rate distributions, which will be examined below.

DIRECT INDIRECT

m=0.0

m=0. 5

m=l .0

fig 4.20-Effect of frictional con ditions on the flow patterns.Axisymmetric rod extrusion R= 20

Restriction on the relative movement of the container-billet operates differently for the direct mode than for the indirect one, as can be seen in the flow patterns of Fig 4.20. While in the former an extension of the dead metal zone with the increase of m is observed, almost no influence appears in the latter one.

An analysis of the internal mechanics has been produced from the

record of variations along flow lines L^, l_2 and L^. The strain rate plots of Fig 4.21 reveal the marked increase in the height of the peaks with respect to the same geometry, material and mechanical

107 o Feltham (20) dJ l/l . FEW.7) + Indirect a> v = 8.10 [m/sec ram L 23 c a

c 0/ *>d D 2 cr o c o «/ x

•loo I- O 1 E+ r A

i L. 1-5 1.9 2-3 2.7 3 -1 3-5 3-9 t. 3 4.7 5-1 5. Ln( R > fig. 4.19 - Strain Rate Dependence with Extrusion Ratio

Ax I symmetric Direct and Indirect Rod Extruslcr parameters than for the plane strain situation of Fig 4.14. The profiles for the lines L^ and l_2 are very heterogeneous in the direct case, a situation which is noticeably improved for the indirect one. The position of the rod extrusion slip line equivalent to the M^ of plane strain (Figs 4.4, 4.5, 4.6, 4.7), can be traced by the relative shift of the z coordinates associated to the peaks of Fig 4.21. The path along l_2 (Fig 4.20) is longer than for L^; besides, the first also shortens when m increases from zero. The profiles in Fig 4.21 indicate that the coordinates z^ and z2 of those peaks approach themselves when friction increases, which indicates M^ shifts forward from the frictionless location. This phenomenon seems a particular feature of axisymmetric direct extrusion. For the equivalent plane strain situation, the line M^ was found to shift slightly backwards (Fig 4.4 and 4.5) with the rise of friction. For the line L^ running along the container wall and eventually dead metal zone, the history for direct and indirect mode is different. In the direct case, the strain rate increases early, and this is more marked for rough walls; a better indication of that is featured by the total strain picture of Fig 4.22. For indirect, the line L^ exhibits a plateau in the strain rate when crossing the upper part nearby the container corner; this result can be associated with the similarly located undeformed valley found in the case of plane strain (Fig 4.12 and 4.13).

In spite of the quality of the recording of the hydrostatic pressure oj^ (Fig 4.23), it is evident a dependence with the nature of the billet-container wall interface in the case of direct extrusion. On the other hand, for indirect mode the hydrostatic level for frictionless and sticking is almost the same.

A comparison between the hydrostatic (from Fig 4.23) and total pressures (from Table 4.VIll) at the ram, for direct extrusion, gives:

-2 m ~ 0 Q. = 480 MNm -2 h p = 470.31 MNm

a -2 -2 m = 1 L - 600 MNm p = 736.85 MNm

109 f ig 4.21-Dependence of stroin rote with mode of deformation and frictional conditions. AxTsymmetrTc extrusion _ 2 R=20 .perfectly plastic material oy= 107.2 MN/m Vram=8.(103.m/sec) Starting coordinates of flow lines (fig 4.20) LINE L1 (—) L2(—) L3 (—) z 0.00 0.00 0.00)/ iQ-2m \ . r 0.35 1.86 3.45 DIRECT INDIRECT

Length of flow line (10 -m)

110 fig 4.22-Oependence of strain with mode of deformation and frictional conditions.Axisymmetric extrusion

2 R=20 , perfectly plastic material ,oy = 107.2 MN/m Starting coordinates of flow lines (fig 4.20) LINE L1(- '-) L2( ) L3(-q Z 0.00 0.00 2 r 0.35 1 .66 ;^)(10- m)

DIRECT INDIRECT 1.16

O II E

in o II E

n E

ength of flow line (10 . m)

111 f ig 4.23-Dependence of hydrostatic pressure with mode of deformation and frictional conditions Axisymmetric extrusion ,R=20, perfectly 2 plastic material-Ttry = 167.2 MN/m ^ - — Starting coordinates of flow lines LINE ll( ) L2(—) L3 ( J

Z .00 .00 .00l/1n-2 » 0.35 1.86 3.W10 m> DIRECT INDIRECT

112 which shows that for smooth walls the contribution at the ram interface of both deviatoric stress components and frictional forces are very

lowr As friction increases, the distribution of o^ along the ram wall becomes heterogeneous, line L^ having a more compressive (negative) va 1 ue. —The pract i ca 1 devrce^ trf~a 11 owing i^fowijackwards to reduce the required power to extrude, as used in combined can-rod extrusion (182) can be interpreted here in terms of releasing that increase in the hydrostatic component. The difference in the values of cr^ and p for m = 1 shown above, is due to the contribution of the container wall to the frictional forces in p .

For direct extrusion with rough walls, the axial line L^ exhibits a less compressive state, hence a tensile value with respect to the outer lines. Therefore, the elastic expansion of the material at the axial line is less favoured than the off-axis one. Thus, if these hydrostatic pressure gradients were allowed to operate nearby the ram wall, the known practical effect of "piping" should appear. However, further research is necessary in order to determine if this effect is produced either by a less restrictive boundary condition at the ram (Fig A.2) or by incidence of hydrostatic pressure gradients in the distributed body forces eq(2.50). Both border and inner flow would allow the relief of the a^ gradients appearing in Fig A.23. The second manner seems more probable: cases of piping are reported for lead (13, 35), where the ratio shear bulk modulus (209) is: G/k = 56/458 = 0.122. On the other hand, for aluminium (209) G/K = 262/752^= 0.35, and the piping effect is almost nil (8). The bulk modulus is connected with the hydrostatic pressure, and the shear with the deviatoric (plastic) one. Thus, piping appears to be controlled by hydrostatic pressure effects which are more relevant for softer alloys or in general for low G/k ratios.

A.5 Conclus ions

1) For plane strain situations, FE pressure results are a slight overestimation of slip line predictions, provided the exit velocity is over the one given by the mass balance value. Better agreement with them is obtained for higher values of penalty parameter.

113 For axisymmetric geometry, FE pressures are lower than the best upper bound, and the billet length dependence appears naturally.

In FE the dead metal zone and the main shear directions are not assumed a-prior|_as in s 1 i p 1 ine or Upper Bound methods, and they come out as a result of the imposed boundary conditions. Agree- ment is found with the location of the Slip-Lines assumed in those techniques.

For both direct and indirect mode, it is found:

(4.8) P* a m=0 m=0 where p* is given by eq(4.6). This agreement improves at lower extrusion ratios.

Frictionless load or pressure estimations can be obtained from rough container solutions, using the observed behaviour:

Di rect: = P* (4.9) a a m^O m=0

Ind i rect: (4.10)

i^O m=0

The above properties can be used to assess the effect of the frictional component in the load.

For the direct mode, the ram load and therefore the friction~at the walls is very sensitive to the billet length inside the container. Overall, frictional effects are much lower in the indirect mode, and so is the required power to extrude.

114 Bilinear interpolation performed from FE results along flow lines proved to be a fast numerical tool for post-analyzing the internal mechanics involved.

Strain rate prof i 1 es-show- a strong- transit ion-assaci.ated to a main slip line just before the exit. The distribution across that area is very heterogeneous for the direct mode, notably improving for indirect. Absolute values are higher for the indirect case. Mean equivalent strain rates agree in very general terms with FeHham predictions.

115 CHAPTER FIVE: ANALYSIS OF EXTRUSION II THERMOMECHANICAL WORKING 5.1 Introduction

In this chapter, the FE techniques developed for the perfectly plastic case are expanded to the analysis of non-isothermal deformation of real materials. Considerations of large extrusion ratios, squared container bore and die geometries, and allowance for friction, make the mesh updating requirement for a complete dynamic analysis prohibi- tive. Thus, a fixed mesh description, for only steady state purposes was initially aimed at. However, relevant information to realistic practical situations became available later. The pure steady state analysis of non-linear material is not exempt of difficulties. Its assumptions, applicability and achievements are presented below.

5.2 Mechanics

5.2.1 Material behaviour

The perfectly plastic idealization of Chapter 4, implicitly assumes no sensitivity of the flow stress to any variable of the internal mechanics of deformation. Strictly speaking, such assumption no longer holds when dealing with real materials. In particular-^ for fhot working, it is now widely accepted that the alloy follows a constitutive law of the form (8, 55, 188):

where z = e (5.2) is the Zener-Hol lomon parameter (185) or Temperature Compensated Equivalent (eqAI.25) Strain Rate. For the material parameters, the physical interpretation given by Jonas et al (186) is:

AH = activation energy of the rate controlling mechanism

a = reciprocal value of stress at which Z changes from power to exponential stress dependence

117 A = structure factor at high stress levels, and is a measure of the activateable sites in the material and the probability that a deformation event will occur at that site (188).

n = measure of the stress senslfTvity~of"the~l

G = 8.314 Joule/(gr mol °K)

T = Absolute temperature ( K)

Of the two alloys selected in order to analyze the performance of the extrusion model, A15052 is the one closer to a perfectly plastic behaviour; on the other hand, A15456 is harder and more sensitive to strain rate (Fig 5.1). The hot working parameters adopted for both alloys are shown in Table 5.1.

Alloy a(m2/MN] A AH fJoule n Igr mol, (218) A15052 0.017 0.325 x 1011 1.54567 X 105 5.11

(55) Air, rz 0.014 8 1.06428 X 105 A15456 0.230 x 10 2.61

Table 5.1 Hot working parameters

They have been produced by the analysis of Sheppard and Wright (188, 192) for processing data obtained from torsion testing of rod specimens, with a coupled finite difference scheme for temperature raise computations (192, 218). In order for the linear variation of strain in the radial direction of the specimen not to introduce a spurious effect into the associated shear z, which must reflect the response to only one strain level, the basic analysis of Nadai (184) is modified. Thus, the surface shear stress is obtained as:

- _ M (3 + n* + m*) (5-3) " ~~ 3 2 ira^

118 fig 5.1-Hot Working behavior of two Al alloys where M is the torque, a the specimen radius and

n* - 9Ln M (5.A) ~ 8Ln 9 8,T

m- 9Ln_M (5.5) 3Ln 9 9,T the parameters n* and m* are obtained by a set of separate experiments at constant twist rates Band temperature. Thus, the dynamic loading response of the material

= Z (e, e,T) (5.6) is constructed from (5*3), with the outcome of eq(5*1).

In early works, a and n appeared to be temperature dependent. In the method of Ref (188), as an infinite number of experimental points from the deformation field are available for the optimization, values of a and n can be separated satisfactorily. The hot working constants are expected to vary with material purity and alloying additions; the general trend, has been found to be a reduction of A and n as alloying additions increase (186, 188, 210). . . •

Several important points must be brought in here in connection with the transfer of material properties from torsion testing to the forming operation:

I In torsion, the concentric nature of the flow produces a steady state in which any piece of rotating material is subjected in time to almost the same conditions (193). Thus, in spite of spatial variations of strain rate and temperature in the axial direction, both distributions in the circumferential one are quite steady almost up to the failure of the specimen. On the other hand, in a process like extrusion the material faces near the exit a sharp strain rate gradient due to the presence of the M^ slip line (eg Fig A.21) and a step in temperature (eg Fig 5.2c).

120 In tensile tests, a sudden change from a constant strain rate to a different level bas been found to produce two main effects (157, 158). The first is a time delay for the flow stress to approach the nominal value for e^. Secondly, in the abrupt change situation the flow stress is slightly different than in the continuous one, and the (ln a-e) plot is shifted along the strain axis. In order to describe that, Knocks and Jonas (157) introduced a parameter N evaluated at "constant state", given by:

N = 91 na + I gfi^ (5.7) 91 ne c 91 ne

The first term in the RHS(m°)tis the instantaneous sensitivity (eg m* eq(5«5)). The second is proportional to the rate sensitivity of the strain hardening, a quantity which is positive or zero. In the latter case, the (ln a-c ) curves for different rates are shifted parallel to the stress axis and there is no delay in response. This is the basic assumption underlying the torsion analysis eq(5»3). They (157) concluded that m° is of importance for the initial behaviour, but the long-term trends are influenced by the strain hardening behaviour. Thus, the accuracy of the analysis in eq(5-3) becomes worse at low temperatures.

So far, the total strain influence on the flow stress (n*) has not been incorporated into eq(5.l). Two basic forms can be produced from the dynamical loading curve of eq(5«6); the one implicit in eq(5.6) lays on the use of the total accumulated strain as current parameter. Another alternative is given by the use of incremental or rate forms for the flow stress (eq 2.38 and Chapter 6) which compels the explicit use of the strain sensitivity of the flow stress dcr/d e.

The values of n* appear to be more noticeable at very low strain rate regimes and low temperatures (188, 212). Stuwe's (212) experimental hot torsion stress-strain results for commercial pure aluminium, deformed at a constant strain rate of 0.02 sec indicates an increase in torque by a factor of 6 when the preheating homologous temperature is reduced from 0.72 to 0.57, with pronounced n* developments.

121 The gradient n* is usually positive at the low strains and negative at the higher ones, (188, 212); at high temperatures n* tends to become negative over a complete deformation range; this has been suggested to be a direct consequence of the softening due to temperature rise. Thus, when fitting constitutive relationships, it is customary to use either the peak value from the stress-strain record or the stress at some constant strain. However, if the strain is to be incorporated into the constitutive relationship, the handling of n* can no longer be avoided.

The transfer of the strain influence on the flow stress from torsion testing to any forming operation is not direct. For extrusion in particular, the large compressive hydrostatic pressures present within the container (Fig 4.23) prevent the material separation even if moderate strain levels are reached (Fig 4.22). This situation does not happen during a failure of the torsion specimen , taking place under a tensile hydrostatic pressure. It appears that more reliable information for describing the behaviour of the loaded material inside the container should be produced by compression tests; besides, the order of strain levels for these regions (ie maxima along M^, Figs 4.7, 4.13, 4.21) do not forbid the use of such an experimental set-up. On the contrary, at the location of the main slip line M^ (ie Fig 4.21) abrupt changes in strain happen, but they take place simultaneously with the hydrostatic unloading. Thus, for this zone the use of torsion data may be satisfactory. A further examination of the role of hydrostatic pressure is carried out in section 6.4.

The flow stress (5.6) must reflect only material behaviour. Spurious contributions due to geometrical non-linearities introduced by rigid body motions must be eliminated. In spite of them being in general small for the test situation, some work has been produced accounting for them (213, 214).

122 IV So far, the practical control of temperature variations in the torsion specimen gives only an approximate homogeneous preheating. The greater peripheral heating produced by the induction coil device is compensated by the heat losses to the air. As mentioned in (I) the thermal history for the flow in extrusion is affected by a temperature gradient not present in conventional torsion tests. Some work lately produced indicates that a sudden temperature decrement during a tensile test, produces similar delaying effects (158) to that discussed for the strain rate gradients in (I). Thus, the transition from isothermal curves in Fig 5.1 can be considered only as an instantaneous response, with similar 1imi tat ions to m*.

The use of numerical methods for simulation of forming problems allows more realistic material behaviour than perfectly plastic to be considered, which has posed new requirements for experimental refi nements.

5.2.2 Acceleration Terms

The incidence of acceleration effects cannot always be neglected. The additional inertial forces eq(2.15) entering into the dynamic description for axisymmetric rod extrusion, take the form:

c C 9 b = b + b - p( iMz + iMr ) + £ (U Y + u Y ) (5.8) - - - 3t 3t f -z zr -r rz

where the approximation stands for considering the tangential deformation 7* being much bigger then the compressive strains e , , which is a characteristic of shear flow.

The bt term in (5.8) is only relevant at the onset of the deformation, outweighting the geometrical acceleration forces given by and the stress gradients in (2.1A), (which at this stage are mainly elastic). As the velocity field increases and rotates from the undeformed state, b*" gradually diminishes until vanishing. Their effects in the equilibrium eq(2.lA) are gradually taken over by the b^

123 forces and stress gradients (now due to plastic regime) which later remain holding the steady state. The proper time tracing of b* when a fixed grid is used impels a resort to step marching schemes, and is examined in Chapter 6. As for the b9 forces, their importance increases at higher ram velocities and extrusion ratios, if z is the axial direction, only the b9 component is worth being considered. For the most critical situation of the high shearing area near the outlet, assuming the velocity in that location has achieved its nominal maximum value:

b^.e.fzrRVR (5.9)

where R is the extrusion ratio and VD the ram speed. K

A rough estimation of the contribution of (5*9) to the balance of the internal rate of work, gives for aluminium:

T 9 9 3 2 2 2 u . b = uz b = 0.135 • 10~ R VR Yrz (MN/jii sec)] (5.10)

-1 In a very unfavourable case, with low strain rate level - 1 sec and high temperature T = 550°C, the softer alloy A15052 exhibits a flow stress of the order of = 36 MNm (Fig 5.1). An estimate of the plastic work for such a condition gives:

T 2 (e .g) - 2 Yrz /3 = 125 Yp2 [MN/(m sec)) (5.11)

-3 As an example, for R = 80 the level is reached at about = 2 . 10 m/sec eq(1.1). Thus, the ratio of the work due to the acceleration forces (5.10) to the plastic one (5.11) gives:

RGP = (v -9) = 2.77 X 10~7 (5.12) (eT.a )

124 Larger velocity gradients, produced by increasing either the extrusion ratio or the ram speed, increases the weight of acceleration effects. -3 Thus, for the previous example, when VR = 8.2 . 10 m/sec * 5 -2 1/sec , flow stress ^ 45 MNm :

g 6 rgp= (J. b ) ^ 3.74 X 10" T (£ . o )

For alloys which are more sensitive to strain rate as A15456, the rise

of Rrn with ram speed is lower, due to the faster increase of the flow ur stress 1evel.

At very high strain rates, b^ might become important in the steady state. In the previous example of A15052, a strain rate 20 1/sec , -3 attainable for R = 80 with an impractical V ^ 32.8 . 10 m/sec , -2 -5 gives a flow stress of 60 MNm , and * 4.54 . 10 which is still negligible. Nevertheless, for such conditions the small strain analysis of the present work is inlikely to hold as a good approximation and both considerations of the rotational strain rate components (6.7) and time integration (section 6.2.5) become compulsory.

5.2.3 Applicability of the Steady State Approach 1: Assumptions and General Considerations

By the steady state extension of the plastic law eq(2.34) to complex behaviour like (5.1), emerging differences in neighbouring local stresses da are implicitly accounted for through the internal forces term in the R.H.S. of the Newton-Raphson scheme (2.58). The same occurs with the term introduced by the non-geometric part of the accelerations b*" in (5.8). Thus, convergences of the numeric algorithm, material non-linearities and dynamics evolve simultaneously to the final solution. Time does not enter explicitly into either the solution algorithm or the mechanical or thermal boundary conditions. The ram

is considered to move with the constant VD achieved after the peak pressure, and the container is assumed to be a reservoir at constant temperature; with the above simplifications, the problem becomes determined by the boundary conditions. The lack of control in the time step of this approach, may produce strong differences in local stresses and accelerations that are spatially neighbouring, which could lead

125 to a spurious dynamic response. Nevertheless, for laws of the form (a-e) as (5-1) such behaviour is only likely to happen at large deformations, For such regimes, compensation due to rigid body movements (6^11) compels the application of time integration (section 6.2). This is normally designed as an incremental-time step algorithm, by far more- stable than equilibrium schemes like (2.89). However, for steady state problems, Dawson (128) even avoided the former algorithm, by making the stress rate corrections (6.11), using the interpolated values of stresses and spin rates from the flow pattern set in a previous iteration.

If the total strain is included, the simplistic steady state approach may not be too successful. The system may evolve by internal force compensation to a steady solution, but there is no guarantee that the stresses achieved at one stage of the iteration (2.58) are a picture of a common time. Neither is there a guarantee of a coincidence of the resulting final state from (2.58), with the one that would be achieved as an asymptotic trend of the transient. The total strain distribution is more spread in volume (Fig 4.22) than the strain rate (fig 4.21) or temperature ones. Thus, any significant flow stress susceptibility to strain introduces additional forces along the flow path, which are not so localized as the ones due to strain rate.

Pacheco (137) dealt with conical die steady extrusion of a work hardening material by intercalation at each equilibrium iteration (2.58) an evaluation of total strain along flow lines. This is subjected to the same criticism as Dawson's work (128); such strain distribution is a byproduct of the steady state solution driven by the boundary conditions, while the consideration of the complete dynamics could lead to a different response compatible with the same border assumptions. In this case a proper assessment at explicit time steps becomes necessary.

The former uncertainty does not appear when the stress level is controlled by only current values, which happens in two-cases. The first is when operating in the upper range of the hot working region, where the strain dependence of the flow stress (n*, eq 5.4) is negligible. The second occurs up to moderate reductions where

126 rotational corrections (6.11) can be neglected. For both of them, the assumption that the solution evolves only driven by the boundary conditions is a reasonable one. The effects which a consideration of the strain sensitivity of the flow stress would produce are examined in the light of experimental res ul ts_. i_n_ sect ion 5.8.

5.3. One Dimensional Thermal Model

Within the steady-state framework, no capability of dealing directly with the 8T/8t derivative in (2.74) exists. However, if equilibrium schemes like (2.58) are used for (2-74) , the nodal temperature perturbations (as in (2.57)) should lead to a setting of a steady state profile, which evolves simultaneously with the numerical convergence. Such an approach has been found to produce reasonable results for low extrusion ratio and ram speed (127, 128). However, early work in this theses revealed that at higher ratios of axisymmetric extrusion that procedure leads to spurious response.

On the other hand, Jain (122), Onate (141) and Zienckiewicz et al (138-140) used a direct scheme for both velocities and temperatures, and solved plane strain problems. A similar approach, as given by (2.89) was selected for the present work. Here, as the perturbation of nodal parameters in (2.57) is preset to zero, the temperature derivative 8T/9t becomes zero throughout all iterations.

The physical implications involved in the choice of the solution algorithm do not appear to be fully explored. With this aim, a one dimensional model is introduced below and examined from both perspectives.

A solid body moves with a constant velocity V (both in direction and magnitude). At a location x^, over all the infinite plane transverse to the movement, an uniformly distributed source of strength Q, exists. The arrangement resembles the large amount of heat produced at the sharp transition of strain rate near the exit (Fig 4.21). The billet is initially at a constant temperature Tg. For that particular set up, the geometry produces transverse derivatives much smaller than the ones in the moving direction; therefore they can be neglected in (2.74), and the problem becomes one-dimensional: 127 pc 9x 9t A' (5.13)

This is a particular case of the Fokker-Plank-equation-(190). -By the transformation:

A,x - Pat T (x, t) = T" (x, t). (5.14)

(5-13) becomes:

ail* 9j =-D 6(X-XA) K 9x2 " 91 (5.15)

When:

Ai JL 2K

A| = K Af (5.16a)

K being the specific conductivity:

k (5.16b) K=PT

The general solution of (5.13) can be obtained by Green functions techniques (191), from convolution of its RHS I(x, t) with the response G (x - x' , t - t* ) of the same LHS to an instantaneous and localized source 6 (x - x' , t - t* ) : ff T (x, t) = TB + / / G (x - x' , t - t' ). I (x* , t' )dx* dt' •i 'o (5.17)

The Green function satisfies the equation:

9 G 9G 9G = -Q 6 (x-x1 , t-t') (5.18) 9x 9t

128 The availability of localized source - initial value problem solutions for eq (5.15), allows the construction of G by the use of (5.1*0:

A1(x - x') - A!(t - t') A G(x - x\ t - t') = e . G (x - x', t - t')

(5.19) where:

(x-x')

w AKU-t') 1 1 (5.20) G"(X-X , t-t ) = Q. [4K(t-t')]^

accounts for the heat advance by conduction only, in a solid at rest (147); when V = 0 the other exponential in (5.19) becomes = 1. The temperature distribution (5.20) evolves in time widening the hot area and decreasing its peak-from the-t = t1 value. However, when the source is not "on" only at t = t* but continuously as in (5.13) both height and width of G reach the stationary distribution allowed by the boundary conditions. Such initial transient, and specially the spread of heat from the main shear line have influence on the applied stress driving the mechanics. The coupled nature of both problems makes it impossible for them to deal with the present simplistic model. Instead it is assumed here that after tQ seconds, a steady temperature rise of the form:

( i \ 2 G* (x-x1, t-t') = AT e"B (X~X 3 (5.21)

has been achieved, and the more important effect of the first exponential in (5.19) will be examined.

129 Introducing (5.21) into (5.19), and substituting this into (5-17):

T(x, t) =TD+ F I AT e-BW^A^x-x'ljy^, e"Af(t-f )dx< dt,

g ives:

B(x XA) T(x,t) = Tb + AT e" " ! g(x) . f(t) (5.22)

where the tQ shape (5.21) is modulated in space and time by:

A1(x-Xa) (5.23) g(x) = e and

f(t) - 1 - e~A* (t"to) (5.24)

A2 2

In spite of (5.21) simplification for the evolution of the conduction part ,(5.22) accounts for the transient introduced by the convection due to the movement of the solid. As inspection of the behaviour of f(t) Fig 5.2.a) indicates its steady value is achieved very fast. At larger extrude velocities V, the step becomes sharper and its height decreases; however, in real materials this is counteracted by the increase in AT brought about by the larger strain rate. The function g(x) shows a gradient increase with V (Fig 5.2.b). As a consequence, the steady temperature distribution of (5.22), achieved once f(t) reaches the plateau is noticeably dependent on the speed. At the lower V, the solution is close to the static conduction profile (5.20), becoming projected forward into the extrude as V rises (Fig 5-2.c). Therefore, ahead of the heat generation, the advance of the hot front due to the velocity-controlled convection is much faster than the one that is taking place by conduction. So far, the adoption of (5-21) neglects the conduction transient; in fact (5.20) and f(t) evolve simultaneously; the drop in temperature behind the source produced by g(x) operation

130 (Fig 5-2.c) is compensated by a conduction still compatible with the boundary conditions. Therefore, at the back of the source, the departure from the pure conduction problem introduced by the movement of the solid is not as serious as at the front of it.

AT(x t0).g(x)

V > V 2 1

fig. 5.2-Solution for the one-dimensional thermal model

On the other hand, the implications of a premature setting of 3T/3t = 0 in (5-13) as in eq (2.89) are examined below. Such conditions forces T" eq (5.14) to satisfy:

3T 2 *

(5.25) and now (5-15) becomes:

K ' A2 T" = 6(X'XA' (5.26)

131 which generates (190) the Green function:

G* (x-x-)=,Te-B^')2.efe(X"X,)l (5-27)

instead of (5.21). Introducing (5.27) into (5.19) and this one into (5.17):

x .t 2 j-B(x-x<)-^(x'x,)+A' (x"x,)] T(x,t)=TBH- / AT.eT x /i<

2 • , 4 .«(x'-x J.e-^tt- V dt

B(x X )2 T(x.t) =TB+AT.e- - A .f(t) (5.28)

Two main properties become apparent: first, as A2 eq (5.16a) depends on the normally large extrude velocity, practically only the t > to contribution is kept in (5.28). Secondly, the temperature rise at the source position (x = x^) is the same in eq (5*28) and in the more accurate treatment of eq (5.22). Thus the instantaneous temperature rise AT at the source location given by neglecting 3T/9t is a correct prediction. The implicit g(x) = 1 constant factor in (5.28),makes the temperature distribution fails to reproduce two effects: the drop at the back of the source and the projection forward into the extrude as Fig 5.2.c . The first effect is not so drastic as the latter one. These consequences will be discussed in connection with realistic extrusion situations in sections 5.5 and 5.7.

132 5.4. Extrusion Thermal Modelling

5.4.1. General Considerations:

In all industrial forming operations the situation becomes more complex than the one-dimensional model of section 5*3. The velocity field exhibits strong variations along both longitudinal and transverse directions; the strength of the heat source has a non-linear dependence on the strain rate and temperature. Changes in operational parameters which tend to raise the peak temperature, such as increase of ram speed or reduction of the billet preheating, are counteracted by the drop in flow stress induced by that higher temperature. If the shear lines (where heat generation is most concentrated) move during the process, an additional convective term due to the moving source should be considered in eqs(2.74) and (5-13). Even for the steady state situation, the problem is complicated enough to make almost impossible its analytical resolution.

The alternative of numerical models which combine high speed computing and iterative methods to handle the non-linear situations,becomes a useful tool of simulation of the real process. The advantages are evident for internal parameter1 assesments which cannot be measured experimentally. Correlations between the recorded history of deformation obtained by means of numerical modelling and product structure may help the understanding of metallurgical changes occurring during the process. Further, as the former is determined by the hot working tool parameters, the establishment of such dependences can offer considerable gains in control of the final structure. However, some problems appear in the simulation itself, with assumptions, simplifications, computing limitations ~and consequences which must be properly weighted. Besides, from the point of view of the behaviour of the material, it is quite likely that difficulties or dependences underestimated for the controlled test situation used to determine the constitutive relationship, can be more relevant in the real process. In such cases the discrepancy between theoretical prediction and experimental result can pose new needs into test analysis for feedback to the process model in a trial and error sequence. 5.4.2 Applicability of the Thermal Model

The discussion in section 5-3 indicated that the advance of the heat profile by conduction is very slow compared to the convective 4| one-due-to-the movement of the material. As K becomes smaller, the ^shape of Fig 5.2.c. approaches a step function: before the source its

value is T0 and is not influenced by backward conduction. The profile past the source retains the height achieved at the temperature peak, % with no decay by transverse dissipation. In this limit, an approximation can be achieved neglecting conduction effects; such has been the approach used by Johnson and Tanner (56) drawn from an upper bound solution of the mechanics; each time a new discontinuity line is crossed the associated

temperature rise is added up and projected forward as the new T0. No temperature variations were allowed after exiting the deformation area. Conceptually, the construction assumes a steady temperature distribution, but with an implicit 3T/3t = 0 setting after the convection front has operated.

The behaviour of Fig 5.2.c has been obtained as a result in FE modelling of transient rolling by Argyris et al (99). In an early paper, also the extension of the hot zone in front of the source has been reported for a very low ratio extrusion (98). In both cases, mesh updating and time integration were carried out. This last device was not available for the present thesis work, it being considered as a further development of the code to be carried out jointly with the time-dependent mechanical description.

The first attempts to solve (2.83), were done using a Newton- Raphson equilibrium iteration scheme as for the mechanical equations (2.57) and (2.58). The temperature distribution was updated with the rise from its initial uniform value Tg. Strong numerical instabilities were present, and only at low reductions (extrusion ratio and ram speeds), numerical convergence was achieved. Nevertheless an examination of the solution revealed an excess dependence of conduction effects, both in a backwards advance of the hot front and in cooling at walls. As no control in time is possible with this algorithm the time is reaching the final temperature easily goes beyond the actual squeezing out time of the billet.

134 As a second design, a more pragmatic approach was adopted, as a compromise situation able to provide some meaningful results and leaving the code in an advantageous position to interface a time marching scheme in the future. Equations (2.83) were solved by a direct method as in (2.89), which proved to be much"more"5tabde from— the numerical convergence point of view. Such setting is equivalent to keep almost only the instantaneous generation profile, as discussed when examining eq (5.28). Therefore, as pointed out in section 5.3 the temperature rise evaluation at the main slip line M^ is correct. The fact that the resulting local stresses there, are the ones defining the frictionless tool load, allows to relay on the estimate for the latter one. The temperature defficiencies at the neighbourhood (as in Fig 5.2.c) play a secondary role at the rear of that line, with increasing influences at the front as the extrude velocity rises.

Further improvements in the temperature field can still be produced after exiting the FE solution at each iteration; by global operations in the assembled network of Gauss points (as done in histoline recording) the temperature rise can be extended forward to the extrude as in Ref (56). Such modification obviously will give a more realistic temperature and flow stress ahead of the source. Nevertheless, for many cases such area is virtually undeformed and the present approximation is enough. In fact, the histoline recording was produced after all the FE results of the present work were ready, and was designed as a post-processor. It now emerges clearly as an improvement in steady state problems: interfacing such device at each (2.89) iteration. The solution of (2.89) is not exempt of numerical convergence difficulties; axisymmetric geometries bring about sharper changes in velocity distributions than the plane strain cases in which upwinding (143, 144) has been reported to succeed (138-141).

The procedure developed in the present thesis (section 2.5) was applied to direct and indirect extrusion at ratios much higher than previous work. An assessment of numerical limitations and comparisons with experimental results are carried out for different ratios, alloys and process conditions.

135 5.5 Direct Rod Extrusion, R = 12.4, Al 5456

At this low R, once the condition 3u/3t = 0 is achieved, acceleration effects are negligible in all range of practical ram speed. Assuming first that the lowest strain rate level "where" 3u/3t = 0 is a very unfavourable, one: t = 1 (sec the associated ram speed * "3 from (1.1) is VR = 6 x 10 (m/sec) and the flow stress from Fig 5-1

2 for T = 550 (°C) is cr (MN/m ). An estimate of the ratio RPD (5-12) -11 gives: R_p - 3.12 x 10 . On the other extreme, for a very large ram -3 . -1 speed V = 40 x 10 (m/sec), the strain rate level is - 2.2 (sec ), o 2 the flow stress for T = 550 ( C) is a - 50 (MN/m ), and therefore -6 Rgp - O.96 x 10 . Thus, for both extreme situations the geometric acceleration terms in (5.8) can be neglected.

In analogy with Slip-Line techniques, it is considered here that the steady state is only determined by the boundary conditions. In such quasi-steady-state analysis, it is assumed that the initial transient would drive asymptotically the flow to a stationary state which is analogous to the one that is produced by neglecting in (2.45) the acceleration terms (not only geometric, but the very important ones during upsetting given by the first term in the RHS of eq (5.8)). However, as the FE problem (2.58) is posed as a Boundary Value one rather than an initial Value problem, allowance for velocity changes is given. Thus, the difficulty in separating with such scheme dynamic results from the numerical convergence, as discussed in 5-2.3, does not prevent reaching an asymptotic state. An examination of the boundary conditions of Fig 4.2.b reveals the only external condition which requires updating is the billet length,which was implemented as detailed in Fig 5.3- The friction was dealt with by border elements as described in 4.4.1.

For the thermal problem, the boundary conditions are detailed in Fig 5.4. According to the mechanical description, each displacement step is allowed to reach the nominal value of the velocity field which determines both heat generation and convection, with no reference to the temperature distribution of the previous position.

136 < c ) BL = 1 42 mm

fig.5.3 Mesh Geometry for FE Analysis of Direct Rod Extrusion R=12 . 4 ,^ = 160mm a < d ) Grid " 455 nodes .102 9-noded BL = 80 mm isoparametric elements

137 Experimental results were obtained in a 2500 tons industrial press. The surface temperature was controlled by a NANMAC thermocouple placed in the land die area.

The performance of the numerical model, explored for the geometry of Fig 5-3.8 and different ram speeds is summarized in Table 5.11. An index of the oscillating behaviour of the solution at faster velocities is given by the deviation of two consecutive iterations.

Two main numerical assumptions are worth mentioning here. First, the nonsymmetric shape functions of the upwinding were produced in connection with an homogeneous equation, with generation term zero (142, 143, 144). For that particular problem, the exact solution is known and the performance was excellent, but the situation is far from the massive heat generation terms which appear at high ram speeds. Secondly, in the Peclet (2-86) number used to control the upwind with optimal parameters, the average velocity (2^88) may become a bad approximation wherever the local velocity field exhibits large gradients, as may happen in axisymmetric reductions. The only available parameter to attack the difficulty is the mesh size, but the discretization can not be conducted too far, being limited by the computing cost involved in using more elements.

A comparison with experimental load results in table 5.11 indicates an error increase with ram speed. The divergence seems to be caused by the deficiency of the present model to cope with the softening of material just in front of the strain rate peak due to the advance of the hot front; as Fig 5.2.c indicates, such effect is more pronounced at higher velocities.

The good numerical performance and its good agreement with experimental results at low velocities was used to explore the effect of friction and heat transfer parameters on the load along the ram displacement positions of Fig 5-3. Increase of either the m factor or heat transfer to container coefficient a^ , operates changing the flow stress of the layer of billet in contact with the container.

138 Code Ram D i stance Experimental FE Results Ram fjorce- Speed Ram-D i e Load error't % ] Ram Load FR Die Wal1 Reaction R^ 3 Wal 1 _3 i [ 10~ m/sec] [ 9•8x10 MN] 6th Deviation 6th Deviation [10~3m ] F (6) F (5) (6) (5) 1teration 1 terat ion R -R R B D RD 3 3 [9.8x10" MN] 19.8x10~ MNl [ 9.8x10~ MN] 19.8x10 MN] i i j. i D12479 8.50 80.4 877 1078" -291.31 924" -295.51* +22.9 (*) i

D12473 5.09 80.4 723 851 +1.58 695 + 1.63 +17. 7

D12480 2.75 80.4 718 736 +0.98 622 +1.09 +2. 5

Table 5.I I Comparison of experimental and FE results Direct Rod Extrusion, R = 12.4, A15456. m = 1, ots = 0.l4x10x(kcal)/(sec.m2 °C)

Tb= 454°C, Tc= 404°C, ha = 0.0, c = 0.25 (kcal)/(kg °C), k = 0.5787 10_lx (kcal)/(sec m °C) p = 2.71xl03 (kg/m3) (*) FE value taken as an average of the last two iterations. The former constrains the flow stress to behave similar to the bulk of the material, hence intensifying the absence of relative motion imposed as a boundary condition as m rises; the latter one making that layer colder, hence harder. In both situations only frictional forces increase, as can-be seen in Fig TTT5 "compafTrTg"with the basIc line of frictionless estimation. The full sticking condition m = 1 appears giving a better fit to the experimental results, when some transference is allowed. For the smallest ram displacement (longer billet lengths), numerical oscillations were detected; this problem may be connected with the incidence on the Peclet (2.86) of the increase of mesh size, and the above mentioned departure from the average (2.88) of the local velocity field. The plotted value in Fig 5.5 for that position is an average of the last two iterations. On the other hand, the numerical simulation gives reasonable results for larger displacements, situation which will be exploited in next sections to explore higher extrusion ratios.

The initial billet temperature has a strong influence on the frictionless contribution p (4.6) to the total pressure p (Fig 5.6.b).

On the other hand, the effect of TD on the frictional forces is lower, B * as can be seen in Fig 5*6.c from the difference between p and p values. This is also apparent from Fig 5.6.a: the changes in slope (-index of friction-) and in absolute value of the pressure with the preheating

temperature TR are such:

(5.29)

I being the ram displacement. The increase of the gap between pressures in Fig 5.6.c at lower temperatures of operation seems to be related to a larger temperature difference between the container border and the area heated up by mechanical work located at the exit. Such separation is likely to be lower in a real extrusion, due to the total strain influence on both the flow stress and temperature, and will be examined in section 5»7« fig.5.5-Direct Rod Extrusion R=12.4,AL-5456 .Effect of Heat Transfer to Container and Friction on Load-Displacement 3 Curve. Tb=454' C',Tc=400 C • , VROM = 2 . 75(10 m/sec) - -600,

riujt <\J ® Prom .••P^'lm.ntol A Prom.EE o,=0. M E - \ 4 Prom.EE o,*0.00 z X P*(4.6J .re 350

O i/l l/l QJ U Q_

m=1 .0

25 50 75 100 125 150 175 200 225 250 275 Ram Displacement ( mm )

450 O Prom* .»»P»/"

u Z3 1/7 l/l QJ Qe. —* I X

m=0 .5 ;

Ram Displacement (I0"3m ) Billet: diameter =0.16m initial length = 0.285m

141 zii

2 p* f r i r. t ionless pressure (M N/ m ) ram presure (MN/rr/l The internal mechanics, recorded along the flow lines L^ to L^ (Fig 5-7) is shown in Fig 5.8 for the larger ram displacement of Fig 5.3. Strain rates are very low other than at the exit; the height of the peak decreases and becomes wider towar'ds the axial line. As discussed in section 5.3, the temperature profiles retain only the instantaneous generation value; in Fig 5.8, the temperature rise along the axial line Li is lower than at the outer L3. This fact is connected with the major amount of mechanical work near the edge of the die; the temperature along the lines running closer to the container (Lj| and L5 are lower. Thus, the correct extension of the peaks into the extrude as discussed in sections 5.3 and 5*4 predicts a peripheral heating of the outcoming rod. In the real situation, the fall of the exiting temperature along the radius after the maximum (T[_3 in Fig 5.8), is likely to be less pronounced, specially at low TR. Both heat due to the strain hardening of the material running along the DMZ border and decrease of outward flux eq (2.76) due to heating of the container, are factors promoting an additional rise on the outer lines. In the present work the container was considered to keep a constant temperature Tc. Modelling of a row of static element which could simulate the container behaviour is a simple matter, and as only one degree of freedom (temperature) need to be solved in this new.area the increase in computing time is small; however proper time step control is necessary to account for the behaviour during the actual time lapse the billet is being squeezed. In spite of some improvements still being possible for the the temperature field, as examined in 5*4.2, both previous problems favour the need for time integration of (2.74). Thus, the forming equivalent to the simple (5-22) solution should be achieved,making it possible to obtain a proper assessment of the factors controlling the surface temperature of the outcoming product. In any case, it is meaningless to consider convective heat transfer to the air as in Ref (138, 139, 140) until the more important one accounting for the moving solid is allowed to evolve.

For the same experimental set of Fig 5-5, the NANMAC thermocouple record of the surface temperature was Ts = 496°C. The FE predictions of Fig 5.8, extended forward as discussed in sections 5.3 and 5.4 give: 9 0 0*00 + 4

Direct Rod Extrusion ,R=12.4 ,mesh (4.3.d )

AL 5456 ,Tb=454 C-,Tc=404 C'.a^O.M, 3 VrQm = 2.75(10~ m/sec) - 0 + 00** 0 0 i

375 480

3 60-r 465

345 450

c_n 52 0 UStartto, c»or«»atti ho m ) (c) t'H r-O.) Lt !•() #••,.» Lj I'M «"V4 1-. .-4.7 I'M .«T.»

T CC) fig b Tc CC ) c

~~(c) 500 454 0.14 2.75~~

~~490 fig 5.9-Dependence of the Temperature1 25 30 35 Rise on both Initial Billet and length of flow line ( 10"^. m) Container Temperatures.. TL3 = 464°C~~

~~TL4 = 463°C~~

approximately 6.5% lower than the experimental j^esult. At this high __ Tg, such difference can be attributed to a wall temperature gradient reduction due to the heating up of both die and container, which is not allowed in the present numerical model.

For a given ram speed, hence strain rate, a lower preheating Tg involves harder material (Fig 5.1); therefore more mechanical work is dissipated as heat eq (2.72) at the deforming region, and the temperature rises. Thus, the material softens and the internal friction producing heat drops; this feeds back a cause for a temperature rise. The net result set after equilibrium is an increase in temperature. This explains the marked dependence of the peak temperature with the

initial one TD (Fig 5*8 and 5.9). Their heights, and the difference D among the maxima diminishes when the initial temperature of operation Tg is higher, which explains the behaviour of the load results in Fig 5.6.c. If no transference to walls is allowed (as = 0) only the nearest line to that surface seems to be affected. The variation of the temperature rise with the preheating Tg agrees with the findings of Sheppard and Wright (188) and Wada et al (193) for torsion testing.

The profiles of the temperature compensated strain rate (Z) have the same limitations as the temperature itself, the height of the peak being a correct prediction. Direct extrusion involves hetereogeneous strain rate along the main shear line Mf. The large differences in the strain rates of the outer flow lines are lowered in the Ln Z profiles due to the associated colder temperatures; but in general, the logarithmic dependence smooths out the strain rate distribution for material exiting from different flow lines. Such a result is .important in connection with the metallurgical substructure dependence of the product on Ln Z (186).

An indication of the incidence of the preheating temperature Tg at constant ram speed, on the Ln Z values obtained from histoline recording from the FE solution, is given In Table 5.NI. Reference to the geometry of Fig 5.7 shows that 66% of the rod area has to cross a similar barrier.

~~146 For the remaining annular ring the height is greater by only h% somewhat larger for the line creeping along the die corner.~~

~~-Ln Z Flow Line~~

~~Tb =350°, Tc =300° Tg =454°, Tc =404° Tg =500°, Tc =454°~~

~~L 17.85 2 15.15 14.15~~

~~L 18.55 15.65 4 14.75~~

~~Table 5»I 1 I: FE resultsjLn Z dependence with temperature across the _ o rod section for VD = 2.75 x 10 (m/sec)~~

The resulting velocity field and stresses are shown in Fig 5.10. The elongation of principal stresses at the rear of the container wall clearly indicates the presence of a shearing border. The strain accumulated along the border flowline, characteristic of direct extrusion, is very sensitive to ram speed changes (Fig 5.11). Both the internal forces due to the strain sensitivity do/de and the heat generation are then accentuated by the VR increase. These effects are more important at low Tg due to the harder and more strain dependent material. The two phenomena are competitive, and the softening induced by the temperature rise poses the local stresses in a hot working region where the total strain susceptibility is much lower. Such an effect can even occur at the expense of modifications in the flow pattern, as discussed in section 5.8. As regards the heat generation due to the strain sensitivity, the need is clearly felt for an algorithm to allow its convection. Such source is much more spread in the billet and not localized like the one due to strain rate at the main slip line at the exit. Therefore, simplifications as the forward projection of the achieved temperature as in Ref (56) are more difficult to produce due to such spread. A rigorous examination can no longer avoid the time integration. Only then can the achieved value of temperature be reliable, allowing a

~~147 proper inspection of the conditions under which the experimentally defected recrystal1ization of the outer shell of the extrude (194) takes place.~~

In a cylindrical cross section, the hydrostatic pressure a^ becomes more compressive towards the container wall (Fig 5.7) showing in general the same behaviour as for the perfectly plastic law (Fig 4.23). Such a compressive state is released longitudinally and some tensile values appear past the corner (Fig 5-12). In spite of the similarity between the effect these stresses would induce and known practical problems of finishing (as surface-cracking (8, 195), and hot shortness (8) in which the peripheral temperature rise also plays an important role), care must be taken not to extend the calculations out of the limits of the model. Further, this is an area in which the adjustment of correlations between experiment and numerical predictions can lead to more realistic spanned constitutive laws, which should improve the know-how transference to an industrial scale.

So far, only strain rates have been considered in the constitutive law (5.1). But as fracture is strongly governed by strain, its consideration might alter the picture of (Fig 5.12). Even .if the result of (5.12) gives a reasonable qualitative description, no quantitative results can strictly be obtained.

Microstructural changes at the high shearing Slip Line areas are likely to modify the flow stress dependence provided by the mechanical test. This seems not to be critical for the strain rate dependence due to the hill decaying to negligible values in very short distances simultaneously with the unloading. But the total strain reaches the main exit slip line with a large and hetereogeneous accumulated strain (Fig 5.11) which might be partially released before or during the crossing. It is the flow stress corresponding to the new substructure and strain level which facing the tensile field at the edge can reach critical elongation values. Some experimental information is available from hot ductibility assessments by the number of revolutions to failure (8). The general trend is an increase of ductibility with the increase

~~148 0 0~~

~~N S \ \ * \ \ \W~~

/ / / X X X XYX-tfiKX / / X X X X •i 11 \m / x X X X X •i 11 \m X x X X X X i \\>m< X x X X X X 111 * l li tnv.c -1—LL-LLUV X X X X X 1 1 I "IW X X X X X X X X X X * i 9- * wMt X X + i i f t nwm X X i 1 i t i t t + + + X + 4 4 « »»•»• i f ig 5.10-Direct Rod Extrusion R=12.4 Velocity Field and Principal Stresses

I I I * # • * • * * *

~~2.75 lio"3m/sec) 81 (io"3m/sec) 1.15 1.15 StoM'ng coora;r~~

~~c o Co c O> '5 cr IxJ~~

~~35 25 30 Length of Flow Line~~

~~T =454 C-,Ta-454 C*,a =0.0 Tb = 454 C' ,Tr = 404 C*=.as=0.14 r s~~

~~f ig.5.11 - Strain Dependence with Ram Speed Direct Rod Extrusion R= 1 2 .4 . - : compressive~~

~~.tensile~~

~~fig.5.12-DIrect Rod Extrusion R=i2.4 Details of Principal Stresses In the deformation region and die exit corner.-~~

~~/ in temperature (196); that behaviour was found to reverse for some Al-Mg (54) and Al-Zn-Mg-Cu (55) alloys at high strain rates.~~

From the continuum mechanics point of view, another important po i ntemerges. -The-hyd rostat tc-pres^ttre- grad i ents at the outl et (Figs 4.23, 5*8) introduce additional body forces distributed in volume, (2.50) which cannot be neglected as the gravitational ones, the discussion in section 4.4.4 suggests that their incidence is more pronounced wherever the ratio shear/bulk moduli is low. It is evident that such situation can be achieved in hot operations, due to softening of the outcoming product produced by the temperature rise. However contrary to the piping effect examined in section 4.4.4.2, the hydrostatic pressure corresponds here to unloading. Thus, the incidence of its gradients may be reduced.

~~5.6 Direct Rod Extrusion, R = 20, A15052~~

~~Mesh VII of Fig 4.1, in conjunction with the mechanical boundary conditions of Fig 4.2.b.and the thermal ones of Fig 5.4 were used in the present simulation. The results are presented in Table 5.1V.~~

The numerical convergence performance for the shorter billet length 38x10 (m) is very good,upto ram speeds of 8x10 ' (m/sec). The deviation values in table 5-IV produced by the difference.between two consecutive iterations, indicate no oscillation. An asymptotic convergence is also revealed from the negative values of those deviations. This is not the case for longer billet lengths; in some instances there is a positive deviation with an associated large absolute value, which indicates the presence of instabilities.

The results of set I for three friction factors m reproduced the situation already detected in perfectly plastic materials, regarding the proximity of die wall load for any m with the frictionless one at ram FD (m=0). Such property worsens when oscillations are present (set II and IV), but otherwise it kept even for the larger speeds.

152 Set Main hot Bi Met Fri ction FE Run FE Results Experimental Deviation work i ng Length m Ram Load Dev RF j Die Wal 1 Dev. Load FE - DF0 • Reactions parameters 3 DWR _3 Experimental [10" m] -1 1 R I 9.8X10 MN] -3 J Load[ %) 19.8x10 "'MM] [9.8x10 MM) -3 |[ 9.8x10 MN]; 9.8x10 JMNJ

~~0 ETH2001 248 -4.6 247 -4.6~~

~~_3 VR=l4.5x10 sfc 73 0.5 ETH2C02 334 -7.3 242 -0.9~~

~~1 ETH2003 357 -6.1 240 -1.8 306 16.7~~

~~1 T0=4OO°C 0 ETH2004 260 -3.5 258 -2.9 TC=350°C 38 0.5 ETH2005 299 -0.2 259 -0.5 O"s=0 .44 1 ETH2006 309 -0.6 257 -0.8 270 14.4~~

~~_3 +104.2 VR=7.5x10 sfc 0 ETH2007 415 413 +104.6 73 370 +86.3 394 35 Tb=350°C 1 ETH2008 532 +87.4 " TC=300°C 0 ETH2009 338 -3.5 337 -2.8 38 °^=0.44 1 ETH2010 400 -0.2 327 -0.9 365 9.5~~

~~3 0 ETH2011 234 -94 234 -94 Vr=8X10" S^C 73~~

~~Tb=A92°C 1 ETH2012 308 -56 233 -55 44.6 1 1 1 213~~

~~TC=455°C 0 ETH2013 177 -0.9 175 -0.6 38 o< s=0 1 ETH2014 210 -0.6 176 -0.7 209 0.5~~

~~3 Vr=8X10" ^ 95 1 A2800 341 -50.5 240 -51.3~~

~~Tb=A92°C IV 73 1 C2800 306 +60 233 +61.5 213 43.6~~

~~TC^55°C 38 1 B2300 209 +0.7 175 +0.7 209 0.0 <>s=0.14~~

~~Table 5.1V. Direct Axisymmetric Rod Extrusion R = 20, Al5052~~

~~k = 0.5787x1(kcal)/(sec m °C)~~

~~c = 0.25 (kca1)/(kg °C)~~

~~P = 2.71x10(kg/m3)~~

~~Ois in (lOxkcal )/(sec m2 °C)~~

Deviation = FE values (6th Iter. - 5th Iter.). For the ram load, a comparison with experimental results in Fig 5.13 for set I, suggests a better fit for m slightly lower than 1. In fact the heat transfer to the container was somewhat exaggerated; for the same conditions which led to the agreement in Fig 5-5, viz as = 0.14, an m factor closer to one..should. hold.

The numerical prediction error to experiments in Table 5.1V, increased with the extrusion ratio with respect to the previous example of R = 12.4. This can be explained in terms of the increasing weight of the local stresses at the region just past the main slip line M^, deficiently heated as it was discussed in 5.4. The heat propagation by convection would soften that area, reducing the bound to experiments.

That error shows little variation with the displacement for the smallest ram speed (set I), but a marked dependence is observed for the faster ones. The last result has not only been obtained for lower Tg (set ll), but also for hotter ones (set III, IV). Such departure of the numerical prediction from experiments, could be attributed to the additional heat generation due to the total strain sensitivity of the flow stress (not considered in this work). Certainly, such heat production for longer billet is greater due to the lengthening of the slip line attached to the container (Fig 4.7) along which heat is generated and convected. But it was also noted that the die load for those cases presented a larger variation with displacement (ie sets II, III, IV) than the perfectly plastic cases (Fig 4.16b). Therefore, the disparity seems more a bulk temperature phenomena than a border one. One possible explanation arises from the behaviour of the boundary condition at the ram for the steady state simplification. The imposed one in Fig 5-4, assumes an isothermal ram boundary; however, transverse conduction to the container is found to produce a colder area in front of the ram (Fig 5-14). If the proper time evolution were considered, the dropping effect of the container wall should be counteracted by the convective projection of the ram higher temperature, as in Fig 5.2.c. The net effect will be a hotter billet than now allowed, which will reduce the current gap with the experimental results for long billet lengths. As discussed in section 5.3, the faster the speed, the more relevant is the effect of the convective term in rising the temperature

~~154 T =400 'C , T =350 'C B C~~

~~0^ = 0.44, V =4.5 (10 m/sec) s R~~

~~O Rom Force.e«peclmentai results. A Rom Force .FE m = I + Rom Force .FE m = 0.S X Die Won Force .FE f m-0 estimation~~

~~s? Ram Displacement DO,- .2 m) fig. 5.13-DTrect Rod Extrusion R = 2C .AL-5052. Effect of Friction Factor on Loads~~

~~CJ>~~

~~455~~

~~Length of flow line (10 m) fig 5.14-Temperature Historyalong Flow Lines under severe deformation,~~

~~Indirect Rod Extrusion R= 80 , Al 7075,TB =460t,Tc= 410'C a =0.14 V = 4.10"^ (m/sec ) ^ K~~

15 of the colder regions ahead (Fig 5«2.c). This supports the better agreement obtained for the lower velocity (set I). Other evidence which sustains this argument is that a reduction of container incidence by a lower coefficient of heat transfer (set IV) tends to disminish the error.

The previous discussion introduces a question as to the cause of the numerical instabilities that arise when dealing with long billet lengths. It has been previously argued in section 5.5 that even the upwinding is unable to settle a solution when the Peclet (2.86) is large due to long mesh sizes or when the average velocity (2.88) differs too much from the local values. However, a closer examination of the grid for the two positions of table 5-IV reveals not much geometrical changes in the deforming area; only folding of the back columns of elements of mesh VII in Fig 4.1 were produced to deal with shorter lengths. Besides, sustantial velocity field changes only happen near the deformation area where the mesh refinement for both lengths was the same. Therefore, it seems more likely that the presence of oscillations at early ram displacements is due to having avoided the time integration, which makes the no-transfer condition at ram wall difficult to hold when facing the well arising in front of it. The fact that such a result does not occur for the shorter lengths suggests that the solution

is stable, at least up to the limits screened here, R = 20, VD = 8 x 10 m/sec. That important finding indicates that the focus until now posed into the stability of the solution when large convective terms were present (143, 144) must be concentrated into the time integration of the thermal equation in a fixed frame of reference;

~~5-7 Higher Extrusion Ratios~~

The grid of Fig 4.1 was used varying the extrude radius, in connection with the mechanical boundary conditions of Fig 4.2b and the thermal ones of Fig 5.4. The dependence of the FE bound to direct extrusion experiments presented in Table 5.V is examined below. The trend detected from R = 12.4 to R = 20, regarding an increase of the ram load percentage error with extrusion ratio is here confirmed; thus, the explanation given in section 5.6 can be expanded.

~~156 0 +~~

Set Main Code FE Results Experimental Devi at ion Thermomechan i ca1 Load FE - Exper. Parameters Mean equival. Temperature Mini mum Ram load - Deviation Die Wall- Deviation 3 [ % ) (9 *3X10- MN] strain rate rise Temper. 3 DF Reactions dr Fr[9.8X10~ MN] R D -L (4.7 ) (°c) (°C) 3 sec R [9.8X10~ MN] l)

~~R = 30~~

~~1 Tb = 545 °C DTH3001 2.420 49.6 541.3 189 -1.4 145 -0.5 180 5 Tg = 505 °c~~

~~VR = 8.3x10"3 sfe~~

~~R = 30 1 1 T_ = 322 °C DTH3002 2.427 141.4 318.2 489 + 1.6 375 +9.9 368 32.8 D Tg = 280 °C~~

~~3 VR = 8.4x10 5fc~~

~~R = 80~~

~~„• Tb = 494 °C DTH8001 3.242 129 472.29 301 -29.1 242 -26.3 249 20.9 Tg = 444 °c~~

3 VR = 8x10*

~~R = 80~~

~~IV TB = 395 °C DTH8002 3.269 112.32 373.04 492 -76.8 328 -68.9 350 40.6~~

~~Tc = 345 °C~~

~~3 V =8.1x10" sfc~~

~~Table 5.V Direct Axisymmetric Rod Extrusion R = 30 and R = 80~~

Al 5052, k = 0.5787 X 10-1 (kcal)/(sec m °C) Ots = 0.14 X 10 X (kca 1) / (sec m2 °C) c = 0.25 (kca1)/kg °C) B1 = 34 x 10_3m, m = 1 P = 2.71 x 103 (kg/m3) Deviation: FE values (6th iteration - 5th iteration) Low Tg's have been found to produce larger temperature rises than higher ones (Fig 5.9). This result is obtained again for R = 30 (sets I and II), but seems to fail for R = 80; nevertheless, the big load deviation DR^ in set IV suggests that the predicted temperature rise for-lower T-g is more an 4nst-ab-i Ti ty problem -and—the-genara-1—t-rend holds. Comparison of experimental and FE ram load results indicates the error increase of the prediction at low T_. Its frictionless D component R^ is overeva1uated, due to the defficient heating at the front of the strain rate peak M^, in this particular case accentuated by the greater temperature height resulting at low Tg. Regarding the frictional component fr = F^-R^, it is difficult to infer any trend from the current results; a further examination is carried out in paragraph 5.7.^ below. The stability of the FE model at short billet lengths was used to determine the incidences of the main thermomechanica1 parameters on the resulting flow for axisymmetric rod extrusion R = 30 and R = 80, and they are examined below.

~~5.7.1 Strain Rate~~

For the alloy A15052 (whose low strain sensitivity makes it closer to the perfectly plastic behaviour), the local strain rate recorded along flow-lines indicates a marked dependence of the peak height acheived

~~near the exit with Vp (Fig 5-15). The differences in these values at the cross-sectional area they define also increases with V^. This, hetereogeneity is a main characteristic of direct extrusion.~~

The mean equivalent strain rate (estimated as described in section k.h.2) for different situations, is examined below from the plots in Fig 5.16. For R = 30 direct extrusion, no changes are detected by varying either the initial billet temperature (S^ to S^, S^ to S^, S^ to S or the alloy; both A15052 and the more strain rate sensitive A15^56 gave almost the same values. Also, an increase of extrusion ratio has a significant influence on the mean value (eg S^ to S^). The indirect mode provides greater strain rates (S^ and S^) than the direct one for the same ram speeds, which agrees with finding of Patterson (8).

158 ****** * *

~~3.0 •— V Z 3.10 J( m/sec) V = 5 .10"3( m/sec) R R~~

~~en LD~~

~~V = 7. JO (m/sec) V' =8.3.10"3 ( m/sec)> R •VOMIAO t6w«i"«n» (I0"lm) ! i~~

~~1.5~~

~~7 9 Length of flow line GO"2 m)~~

~~fig_5 15a.Dependence of Strain Rate with Ram Speed. Direct Rod Extrusion R = 30 ,AL~5052.~~

~~Tb = 545 C',Tc = 505 C',as = 0.M CD o~~

~~V =4.10 (m/sec) 3 VR = 6.10 (m/sec) J R 1 !1 1 1 1 1 }|«r|to, (irAfiH I '0* 1 ««».» ; i »•!.« 1 \ \ 1 J >1 1 1 1 • I ' ' > 1 'l 1 1 ' ' / 'i / ' 1 / 1'~~

~~1 0 L --•"iV-'" 10 Length of flow line (10"2m) fig. 5-15-b - Dependence of Strain Rate with Ram Speed. Direct Rod Extrusion R=80~~

TB = 460 C' ,Tc = 4 1 0 C * ,as = 0.I 4 T———1 ' T" u (D on \ © DIR R =30 , AL -5052 TB = 545 c • He =505 51 A 30, AL -5052 322 c • =280 S X DIR R = TB = .Tc 2 rt + DIR R =30 , AL -5456 1Q =54 5 c • he = 505 S3 •ILU X DIR R =30 , AL -5456 322 c • =280 5 3.1 TR = He o IND R =30 , AL -5052 TQ =54 5 c • he =505 St

~~X IND R =30 , AL -5052 1B =32 2 c- .Tc =280 Sc X DIR R =80 , AL -5052 T B = 494 c • » T c=44 4 S° z DIR . R =80 . AL -5052 Tn =30 0 c • . Tc =250 S 8~~

~~2.7~~

~~2.3~~

~~1.9 -~~

~~1.5~~

~~1.1~~

~~01 -3 V. (10 m/sec )~~

~~fig. 5.16-Mean IquTvalent Strain Rate Dependence With Rnm Speed..~~

~~161 Direct Indirect~~

~~Direct ,R=30 ,AL5052~~

~~o Fsknorn ( ZO ) A FE £(+7) 2.4- + FE I Lv,L2 awra9* X FE I a> o cr c a <7>~~

~~Direct ,R=80 .AL7075 14. D O" UJ O Fekiom { •to ) A FE t < 4.7) + FE £ Li,L* ave. X FE £ L,~~

~~Indirect ,R = 30 ,AL5052~~

~~i FE { 1-4.7) + FE 4 L, L, a/e. X FE t L3'~~

~~o 10 11 3 ram ( 10' m/sec)~~

fig. 5.17 - Comparison of local and mean measures of strain rate variation with ram speed. 162 The dependence of different measures of equivalent strain rate on the ram speed is explored in Fig 5-17. As previously found in Fig A.19 for the mean equivalent strain rate dependence on extrusion ratio, a close agreement is found here between the figures provided by eq (A.7) and Feltham's (20) predictions. However, the better information provided by the local equivalent strain rate maxima computed along flowlines (Fig 5.15) indicates a different situation. For direct extrusion, only at low extrusion ratios and slow ram speeds the mean values fall within the bounds set by the achieved peak along

axial (Lr L^) and border (L^) flow lines (Fig 5-17a, b). This was also detected in the only indirect extrusion ful1y explored (Fig 5- 17c). A measurement of the inhomogeneity of the deformation at the cross sectional area defined by the exit slip line, is given by the range of heights of those local strain rate maxima. The results in Fig 5-17 show how such differences are affected by the ram speed and extrusion rat io.

~~5.7.2 Temperature~~

The temperature rise AT examined here was obtained as the maximum height from records as the ones in Fig 5.9, therefore it mainly reflects the border overheating. The results for both direct and indirect extrusion in Fig 5.18, indicate an increase in AT with increase of either ram speed V^ or extrusion ratio R.

For direct extrusion, a decrease of the preheating billet temperature Tg is found to produce a greater AT; this result, already detected for R = 12.4, is here found to hold for higher extrusion ratios. Increase of either R or VR or decrease of Tg, raises the flow stress (5.1) because of the greater Z parameter. In spite of this enhancing the heat generation (2.72), its associated softening sets a competitive process: the lower internal friction reduces the strength of the heat source and the material hardens; thus, both processes reach a final equilibrium. The results for direct mode in Fig 5.18 indicate that the general trend is a predominance of the temperature rise over the decay due to softening. However, such trend varies for very severe deformations at high Z;

~~163 240 -T-T—r-r —i—i—|—i—i—i—i—|—i—i—i—r~~

~~© DlR, R =30, AL- 5052 Tb = 545 Tc=505 T, A DlR. R =30.AL - 5052 T =322 T =280 T 220 b c + DIR,R~ 30,AL- 5456 Tb=545 Tc=505~~

~~X DlR, R =30, AL- 5456 T0=322 Tc=280 J/~~

IND , R = 30,AL- 5052 Tb=545 l"c=505 f 200 z <1) * IND. R 30.AL- 5052 Tb = 322 T -280 -P i/l c X DlR. R = SO.AL- 5052 TB-494 C Tc=444 T? cc 5052 T =300 C Z DlR, R = 80,AL- b Ic=250 f (1) 180 L. Q' O I- CL> 1 60 a. a S ~ 0U E O) 140

~~120~~

~~100~~

~~1 0 .-3 (10 m/sec )~~

~~fig 5.18 Dependence of Temperature Rise with Ram Speed.~~

164 the graph of the temperature rise T^ vs VR changes into a curve with a less steep gradient. For the same external thermomechanics, harder materials introduce bigger flow stresses, with a net result of larger temperature rise (T^> T^), which is more noticeable at low preheating

~~TB (!<>> T2) .~~

Two main factors, not included in the numerical model must be recalled here regarding the applicability of those temperature rise predictions. Both heating of the container and additional heat due to the total strain dependence along the shearing border favour an increasing importance of the softening process. The incidence of the first factor does not seem as relevant as the second: at high preheating Tg, the predicted AT in Fig (5.18) falls within practically expected rises

~~On the other hand, at low TD the results appear as an overestimation D of the real situation. As the total strain incidence on the flow stress~~

~~at low d is relevant ( by its inclusion in (5 an improve- Trs 188, 212), .1) ment in the temperature result is possible.~~

The results for indirect extrusion show a different behaviour than for the direct one. Extreme Tg's promote temperature rises (T<-, T^) less separated in the curves (Fig 5-18) than for the same conditions of the

~~direct mode (T,, T0). For the hottest TD the indirect rise (Tr) is slightly I L D o over the equivalent direct case (T^), a difference which increases~~

with V . At colder TD, the indirect appears to exhibit a much lower R D rise (T^) than the direct one (T^); however, only one point at high ram speed was explored, where the softening effect now favoured by:the greater strain rates inherent to the indirect mode might be more relevant.

~~5.7.3 Die Wal1 Force~~

~~It was previously found in chapter k that the die wall reaction~~

~~Rq gives an estimate of the frictionless component of the load. Rg~~

exhibits a slight dependence on the ram speed VR (Fig 5.19). The Rg drop emerging at larger (D^ and D^) is connected with the softening near and at the main shear line M^ caused by the greater temperature

~~rise (T0 and T7, Fig 5.18). The width of the band in which RR varies~~

165 with V is narrow for very hot extrusion of A15052 (D.), but widen K 1 when A15456 (D^) is deformed under the same conditions. Such results can be attributed to both harder and strain rate sensitivity properties of the A15456 flow stress (Fig 5.1). The strong dependence of Rg with temperature, can be visualized by comparing the big gap introduced by only a drop in Tg (D^ to D^). The separation is greater for larger extrusion ratios (D^ to Dg), and for Al5^56 (D^ and D^ more separated than D.j and D^,). Also the incidence for the indirect mode (D,- and D^ is more pronounced than for the direct one (D^ and D^); this result can be understood in terms of the effect on the stresses of the bigger strain rates developed in the indirect extrusion case at the main slip 1 i ne.

~~5.7.4 Total Ram Force~~

Due to the limitations set up by the press capability, the ram force Fr is of much more practical importance than Rg. The critical initial behaviour which originates the peak can not be reproduced unless the complete dynamics is considered, which involves additional work in the numerical model (chapter 6). Some qualitative predictions from FE steady state results and experiments are examined in sections 5.8, and 6.4. The very important role of the billet length has been examined in sections 5»5 and 5*6. What is intended below is an assessment of the incidence of the other main thermomechanical parameters on Fd and frictional forces fr for a given displacement (Fig 5.20 and Table 5-Vl).

For the direct mode, the flow is parallel to the maximum strain rate line running near the container border (Fig 4.7). Thus, any factor raising the local stresses along M^ increases friction. In spite of the short billet length used, some trends appear:

1. An increase of VR raises the strain rate level at the M^ location (eg line L^, Fig 5.15). Thus, any strain rate sensitivity of the alloy raises the border shear, with an increase of fr (Table 5.VI: f^ to f^, f^ to f/j) • This counteracts the temperature-promoted

~~drop in Rg with VR (eg D^, Fig 5-19). Thus, the resultant ram load~~

~~F (eg R„, Fig 5-20) exhibits less variation with VR.~~

~~166 I ' ' ' ' I o DIR 30 AL- 5052 T = 545 =505 D »R = B C' TC 1 A DIR ,R= 30 AL- 5052 TB = 322 c- Tc =280 m~~

+ DIR .R= 30 AL- 5456 TB = 545 c- Tc =505 Dt X DIR ,R = 30 AL- 5456 1B = 322 c- Tc =280 DT IND 30 AL- 5052 545 =505 Dc ,R= TB = c- Tc * IND 30 AL- 5052 322 =280 D? ,R = TB = C* Tc =444 Dy X DIR .R = 80 AL- 5052 Tq = 494 c- Tc z DIR ,R =80,AL - 5052 TB = 300 c- Tc =250 Da

~~as = 0 o I 4 -3 BL = 34M0.m) ¥~~

~~430~~

~~A 390 A A A~~

~~350~~

~~310~~

~~X X~~

~~O o + + (D O~~

~~7 8 9 10 -3 Vram (10 m/sec )~~

~~Mg. 5.19-Die Wall Reaction Dependence with Ram Speed-~~

167 830 © DIR .R = 30 ,AL- 5052 T0= 545 c- TC =505 RF A DIR .R = 30 • AL- 5052 TB= 322 c- Tc =280 R2 DIR ,R = 30 • AL- 5456 545 c- =505 R 780 + TB= Tc 3 X DIR ,R = 30 AL~ 5456 322 C- =280 T0 = Tc R? IND 30 AL- 5052 545 c- =505 ,R = 1B= Tc R5 * IND ,R = 30 AL- 5052 322 C* =280 R

~~630 as = 0 „ 1 4 -3~~

~~>80 BL = 34(10 m)~~

~~530~~

~~A A A A 480~~

~~430~~

~~380~~

~~330 X X X 280 X +~~

~~tig 5.20 Ram Force Dependence with Ram Speed.~~

~~168 Set Thermomechan i ca1 Frictional forces Parameters fri = Ri - Di~~

~~y°c) yRm) Mode R Alloy (Tons)~~

~~f D 30 A15052 44 1 545 8.3~~

~~322 8.4 D 30 A15052 114 f2~~

~~545 8.3 D 30 A15456 45 f3~~

~~322 8.4 D 30 A15456 156 f4~~

~~545 8.3 1 30 A15052 -24 f5 322 8.4 1 30 A15052 f6 -75 494 4 D 80 A15052 52 f7 300 4 D 80 A15052 124 f8~~

~~Table 5.VI: Frictional forces estimations for BL = 34 mm.~~

169 2. An increase of VD raises the total strain level at the M. K L location (eg line L^, Fig 5.11). Thus, if the alloy exhibits appreciable strain sensitivity (eg low Tg regime), the raise in flow stress and greater temperature increase at would promote competitive processes. As that is not accounted for

~~at this stage, the large frictional forces found (f2, f^, f^) are a product of the low container border temperatures, resulting from considering only the heat generation due to the strain rate.~~

For the indirect mode, the flow crosses the shear line M^ transversal 1 y (Figs *4.12, *t.13), which makes almost all changes to occur off the container wall zone where they could affect friction. This explains the very low values of fr for indirect conditions (f^, f^) when compared with the ones obtained for the same direct ones (f^, f^). For equivalent external thermomechanics, the ram load for indirect is lower than for direct (Fig 5-20: Rj- < R^, R^ < R^) . An increase in

~~VD has almost the same effect on F_ as in the direct mode (Fig 5.20: K K Rj- vs R^), estimation produced for the less strain rate sensitive alloy at hot temperature.~~

Lowering Tg raises both FR (R,. to R^) and fr (f,. to f^). Here, as the time lapse during which the flow is in a high strain rate area is very small, the incidence of the flow stress sensitivity to total strain is minimal. Thus, neglecting it is of almost no consequence to

~~the frictional component and heat generation, and FR is a reliable prediction.~~

~~5.8 Applicability of Steady State II: Transient Limit and Steady State Flows~~

~~The FE steady state flow patterns for direct extrusion showed a dead metal zone of elliptical shape (Fig 5.21) more extended backwards~~

than the optimal load triangular arrangement resulting from MUB (5*0 (Fig 5.23a), but failed to reproduce the changes with the billet temperature experimentally detected (Plate III a, b). In general, the

~~FE results agree more with the high temperature (Plate III a, TR = *l60°C)~~

170 experiments than with the colder ones (Plate III b, Tg = 300°C), contrary to the MUB (54) of Fig 5-23. Also, the patterns as the one in Fig 5.21 did not detect differences with the perfectly plastic case. In Ref (54), a maximum container temperature of T^ = 300°C was allowed. Vierod (197) also found the patterns of Plate III a, b with the container 50°C below the billet preheating Tg.

~~For indirect extrusion, the FE pattern (Fig 5.22) generally agreed with the experimental findings of Plate III c (Tg = 460°C) and (Tg = 300°C) As the experiments, they neither showed Tg dependence.~~

Tunnicliffe (54) concluded that relative billet-container motion was responsible for the observed alterations, also indicating the presence of a sheared structure at the edge of the extrude for direct mode. In fact, the arguments given below support more a phenomenon of inner flow within the billet rather than a wall interface movement. It is clear from FE results that distinctive accumulated strain patterns, with different relative orientation respect to material flow operates for direct and indirect modes. The inclusion of such strain into the material law should provide a source of distinct behaviour, and their consequences are examined below.

The large compressive hydrostatic pressures reached inside the container should prevent the material failure as in the collapse tensile situation of the torsion test. A nearly isothermal situation in extrusion during the upsetting is common, wherever the flow moves into new areas, still at preheating temperature. The equivalent isothermal situation in torsion can only be achieved at very slow twist rates, due to the concentric nature of the plastic flow. During pure isothermal deformation, internal energy accumulation in the microstructure is possible, the material work hardens and therefore da/de is positive. The thermomechanics not only evolves differently for different orientation of source lines vs velocity field (2.74), but is also dependent on the hydrostatic stress state (see discussion in section 6.4 ). As the temperature rise becomes relevant, the material softens and da/de becomes negative. It fs obvious that the observed negative slope d^/de in the torque vs twist curves at normal twist rates reflects the adiabatic situation rather than the

~~171 L,~~

~~fig 5.-21 Flow pattern for Direct Extrusion R=80 ,AL~~

~~Tb = 460 C- ,Tc = 410V .a,=0. 1. 3 Vr =3.(1 0" m/sec) (ig 5.22 Flow Pattern for 'Indirect Rod Extrusion .R=80 ,'AL 7075~~

~~Te = 460 C-.Tc = 410 C- ,QS=0.14~~

~~Vrnm=3 mm/sec.- fig .5.23-Minimized Upper Bound constructions for the partial extrusions of Plate III (after Tunniclif f e ( 5A 3) 174 itiilMwl~~

~~5 • u J~~

~~(a) (b)~~

~~(c) (d)~~

~~• Plate III : Extrusion Flow Patterns.- isothermal one. Thus, the turning po.int in the sign of da/de in torsion should be reached much earlier than in extrusion.~~

Any sensitivity of the flow stress to the plastic strain alters the local stress distribution producing additional internal forces. As examined throughly in this chapter, this is promoted by low Tg, fast VR, alloy content and obviously length of flowline along which strain rates are present. All of these parameters have more incidence in the less localized deformation of the direct extrusion mode.

Having set above that both hot temperature and perfectly plastic FE results agree with experiments, an interpretation of the flow alteration observed at low Tg for direct mode is given below. At the early defonmation, the material isothermally moving from the border near the M^ line (Fig 4.7) finds ahead an increasing flow stress due to the da/de (work-hardening)positive sensitivity. Applying the Least Action Principle (198), such path involves greater forces than adjacent ones and hence it must be avoided. Therefore, the flow should move away from the maximum of the steady state slip-line M^ (Fig 4.7) by crossing it at certain angles instead of running parallel to it. Any other adjacent path attempted also faces the same situation, both strain and stresses rising and tending to deviate the flow by reducing the component of the internal forces parallel to M^ Such process is limited by the neighbourhood of another unfavourable path set by the sticking cond i ti on of the billet at the fronta 1 wa11. The heat developed during the movement softens that area, reducing the hardening effect of the strain on the stress. Thus, an alteration of the equilibrium position set in between M^ and the frontal wall takes place, and the final internal border defining the DMZ is set back into the bulk.

~~Some experimental results support the previous discussion:~~

1. The distorted grid for high preheating (Plate III a), reveals that the border flow has run along one path only since the start of the deformation. At low T_, the grid distortion is different D (Plate I I I b); there is a trace of an attempt to flow within what is later established as dead metal zone. Also, the final steady- state position of the border dead metal zone/deformation zone

176 interface is closer to the frontal wall. Both effects strongly suggest that the deformation is dependent upon the strain history or prior strain path, which is minimal when operating with high preheati ng.

2. Exploring hot shortness at low temperature Tg, Patterson (8) found it appearing in an experiment at Tg = 350°C, aftermeter of the rod had been extruded. This delay suggests that only when the final steady state pattern is reached does the temperature rise become critical. In terms of the interpretation presented above, that can be explained as follows. The initial extruded product comes from energetically favourable positions well off the high shearing line M^ and mainly from the inner flow lines; the exiting accumulated strain and its associated heat is not too high. Such is not the case for the final steady state: now both the flow and the M^ line approach a paralellism which gives a product much more straindand hotter.

3. Experiments indicate that appreciable flow changes only occur for direct mode. FE indirect extrusion patterns of perfectly plastic material showed (Fig *f.20) insensi ti vi ty to a softening of the border, as the extreme one produced in the transition from m = 1 to m =s 0. That suggests that even if local stress gradients appear near the so-called DMZ, as the flow is already set at an energetically favourable position, (slip line M^ not parallel to flow, Fig A.12), the changes theoretically expected are minimal.

Alterations of flow patterns with Tg have been reported by Conrad et al (199) for hot lubricated extrusion through conical dies of Al 202*1. They explain the changes in term of a negative strain sensitivity of the flow stress, increasing with Z. The reported source of such data is a modified Fields and Backoffen (187) analysis from torsion data. The previous discussion indicates that such property can not be considered as only an indication of pure strain sensitivity. The interpretation proposed in this thesis assumes that, under compressive hydrostatic stresses, da/de holds positive values for larger strains than in torsion, that the first movements are isothermal, and only after the adiabatic softening operates da/de becomes negative.

177 For the experimental conical die extrusion results reported in Ref (199), an interpretation of the flow changes is also possible in the terms suggested above. For lubricated conical die extrusions, no shear line as M^ (Fig A.7) are present (137). At high temperatures, the widespread strain is not practically affecting the flow stress (not hardening) and the resulting experimental flow looks like a perfectly plastic (137) one, with only one axial maximum of distortion. On the other hand, at low T^ the early isothermal flow tries to avoid the increasing stress ramp da/de which it is facing, which now is larger due to the strain sensitivity. In doing so, the material is deviated against the conical die wall, which is lubricated. Thus, the eventual incidence of heat developed in softening the inner material and setting the flow back into the bulk is much lower than for Plate III conditions. This, and the less blocking geometry of the conical set up, allows the first deviation succeed in reaching a way out running closer to the border, and a second maximum of distortion is obtained (double maximum). It is worth mentioning here that Pacheco's (137) results for steady state isothermal lubricated extrusion through conical dies at low T0 fail to describe the experimental finding of Ref (199). It has been suggested above that for this particular conical set up, thermal effects are not determinant in the final flow pattern. Thus, the critical source of failure of the numerical prediction in (137) comes from the way the current accumulated strain figures are produced, as examined in section

~~6.3.~~

~~5.9 Conclusions and Recommendations for Further Work (I)~~

~~5.9.1 Post-Peak Load Steady State~~

1. From the discussion in section 5.8 it can be concluded that FE quasistatic analysis of indirect extrusion at any temperature and direct at higher ones reproduces the experimentally observed steady state flow patterns; the lower temperature limit is set by the strain sensitivity becoming relevant. For the former cases, neglecting dynamic effects in eq (2.15) is not so serious, and the load estimates are reliables as upper bounds. The discussion in section A.A.A proved FE to give lower load evaluation than

~~178 triangular arrangements of HUB, with no preconditions on shapes of either deformation or dead metal zones.~~

~~At the current development stage of the code, better agreement with load experimental resul ts_axe-ob±alned_aJL_ .lower._extrusLon.~~

ratios R and ram speeds Vp, high billet temperatures TR and short billet lengths BL. As examined throughly in sections 5.5, 5.6 and 5.7, the rise of the predicted error compared to experiments is attributed to defficiency of the thermal model allowing the proper extension of the hot profile both at the frontal area of the main strain rate peak and also at the ram when using long billet lengths. In the latter case the situation deteriorates because of an overtransfer by conduction to the container walls, and numerical convergence instabilities appear. Both these difficulties should be removed by either time Integration (if it became available for the mechanics) algorithms or by some simplification allowing its evolution along flowlines in a device inter-

faced in between the FE (2.89) iterations. The program developed as postprocessor for all histoline recordings of chapter 4 and 5 serves this purpose and can be assembled into the main code. Thus, a simple algorithm can be designed with two steps. The first retaining any previous maximum temperature found along a flowline, which is equivalent to the g(x) factor operation in the correct solution (5-22) (thus the presetting of 3T/3t B 0 in (5-13) is eliminated); the second should use the existing FE set up of eq (2.89). Such improvements should clear up the extension to any

~~BL, VR and R, the mechanical solution of eq (2.58) being by far more stable numerically, as found in chapter 4.~~

Another limit to severe deformations arising from high R and VR is given by rigid body movements becoming important (section 6.2.4) For the preheating conditions of (l), stress corrections due to large spin rates can be produced by taking advantage of the availability of stress and spin rates and the interpolation capabil of histoline recording, now to be interfaced after each iteration of the mechanical equation (2.58).

179 In the case of direct extrusion at low Tg, the strain sensitivity becomes important, driving the flow to an asymptotic steady state well off the perfectly plastic case (section 5.8). Therefore, the inclusion of the plastic strain into the constitutive relationship becomes compulsory. The use-of—accumu l a-ted—strain figures, produced by integration of the current strain rates along steady state flowlines has been found not to reproduce experimental patterns. Further examination of the failure of that algorithm and alternative tracing in time are discussed in section 6.3. Here again, in order to extend the model to larger deformations, both temperature (as in lf) and stress (as in 2.) corrections must be implemented.

So far, in the thermal transient description (2.74), equilibrium thermodynamics is involved. The implications of non-equilibrium considerations, which introdued a free energy variable, are examined in section (6.4). Besides, the widely used constancy of the 3-0.9 factor(82) accounting for conversion of mechanical work into heat in tensile tests, is likely to be inadequate at low temperatures and compressive hydrostatic stresses. Both free energy and 3 are bound to be dependent on temperature and the amount of accumulated strain.

If no changes of microstructure take place, some analytical forms for free energy are available (201). From them, general parametrized expressions can be constructed and the same is possible for 3 Thus, a hybrid experimental-numerical simulation project can be drawn up as follows. Assessments of free energy accumulated in plastic regime can be obtained by post-unloading experiments; one simple choice for that is the residual area under the (e) curve obtained after the removal of plastic strain (eg compression test). On the other hand, by numerical simulation, knowledge of distribution and actual figures of the internal field variables in term of tool parameters becomes available. Thus, by regression analysis the parameters in the proposed expresions for free energy and 8 can be properly adjusted. In the first stage such experiments should be carried out by operations in which the flow does not go along a shearing line to avoid the accumulated strain reaching critical

180 values which could change the microstructure (eg compression tests). Once the background behaviour has been established, analysis of structural change incidences can be designed by the same hybrid scheme; now the forming process itself might be used as an experimental tool. ._

Thus, hybrid schemes become a powerful tool analysis in which the availability of almost all internal mechanical variables provided by the model are complemented by realistic experimental data. Correlations and adjustment of unknown dependences by regression should improve, with obvious implications at the industrial scale.

5. In both direct and indirect extrusion the strain rate rises towards the exit. The exact nature of the sharp peak at the edge of the die detected in FE results needs further investigation in connection with realistic die geometries, boundary conditions, rigid body rotation effects (as in 2.) and compressibility rel i-.ef at the outlet.

6. The discussion in section 5*8 for total strain incidence for direct extrusion is valid for other sensitivities of the flow stress. Thus, the rise of strain rate level along the border makes any factor

accentuating the coupling cr - C Qf (5.1) there (as alloy content, and high Z) to favour flow pattern changes. However, as this coupling is not too high, and the relevant changes of the strain rate happen only at the exit peak, almost no alteration of flow is expected. This explains why for the two different alloys in section 5.7, the same FE flow pattern of Fig (5.22) was obtained.

~~5.9.2 Initial Transient~~

~~The complex nature of the thermomechanics involved in the early deformation adds new relevant problems to the previous quasi-static analysis. The main departures from this are:~~

~~181 Constituti ve~~

The strain sensitivity at incipient plastic regime is high compared with the strain rate one; therefore it must be accounted for in the flow stress. FE results produced at very low R indicate that the main consequence of neglecting the elastic part is a load-ram displacement curve with non-zero ordinate origin. At high ratios, the exact influence on this curve of the strain sensitivity of the flow stress due to both elastic and plastic components must be properly explored.

At small deformations the spin rates (6.7) due to rigid body rotations are of the order of the strain (stretching) rates, therefore, corrections in the flow stress eq(6.1l) must be produced. So far, only very small ratios and non critical geometries (Fig 1.1) have been dealt with.

~~2. Dynamics~~

The rise of the velocity field from zero to the steady state implies that acceleration forces eq (2.15) are important, mainly around the exit. At this initial stage, inertial forces are as important as plastic ones. Most of the work produced tracing the early flow employ a quasi-static analysis at small time steps, at each of which the material position is updated. In spite of osci1lationsemerging due to this mesh updating, the isothermal results in Fig 1.1 for very low R indicate that the general background load-displacement curve was obtained. However, two main experimentally observed facts are not reproduced:

~~i) A prominent peak in the maximum load reached.~~

~~ii) Departures from a linear dependence of the load- displacement loci for postpeak flow (Fig 5«5).~~

It is suggested in this thesis that both of them are thermally promoted. For i) a discussion is carried out in section 6.4. Improvement to account for ii) are possible as proposed in section 5.9.1. 182 3. Therma1

~~Movements of hot areas during the early deformation (section 6.A).~~

~~Further discussion of these topics is_ car_ried_out in chapter 6.~~

~~183 CHAPTER SIX: DYNAMIC ANALYSIS OF THE~~

~~EXTRUSION PROCESS 6.1 Introduct ion~~

The mainstream of work by FEM in this area of metal-forming has been a by product of the highly developed non-linear analysis codes produced for structural analysis; their origin were elasto-plastic material description, but now more general behaviours can be considered. In spite of their complexity, some of the concepts involved are crucial in the simulation of any forming process where a transient situation appears, as laid out in section 5-9 and need to be examined. With this aim, a review of the basis of el as topiastic work is given in section 6.2; from this framework, a critical inspection of the viscoplastic assumptions is carried out in section 6.3. All these set the outlines for the final discussions and proposed further work from the present state of the code in section 6.4.

~~6.2 Elastoplastic model~~

~~6.2.1 General Considerations~~

~~Four main choices define the whole FE algorithm involved in the dynamic description, all of them being interrelated:~~

~~1. variational principle 2. reference frame 3. form of the constitutive relationship 4. time integration~~

~~Their inherent and relative connections are examined below. Emphasis is given to the updated lagrangian scheme, introduced below and its variations.~~

~~6.2.2 Variational Principle~~

~~Differerent internal quantities have been used to construct the variational principle upon which a FE procedure is designed (97) : Internal work (73):~~

~~akm 6e. dv (6.1) 'V km~~

~~Internal power (79, 97):~~

~~skm si, dv„ <6-2a> ,v km and (97) (also viscoplastic (112, 137, I'll))~~

~~okm Si, dv <6-2b> km~~

~~Rate of internal power (53, 88, 97):~~

km (6 3) S Si dv0 ' km km where (u , £^m) stand for small deformation measures as introduced in chapter 2; (S , E^) are the Piola-Ki rchhoff stress and Green-Lagrange strain respectively, defined further on.

~~6.2.3. Frame of Reference~~

~~The quantities entering the variational principle are related to a reference configuration, normally adopted as the FE mesh; the main choices for them have been:~~

I The configuration at time t=0; this approach gave rise to what is known now as total Lagrangian (TL) formulation. Hibitt, Ma real and Rice (79) developed the (6.2a) form, presenting an incremental form for static and quasistatic large deformation elastoplastic analysis, aimed at metal forming.

~~186 II A configuration fixed in space for all times (this has been the standard choice for steady state, flow approach using the viscoplastic model)(122, 137, 138, 141).~~

~~III The current conf i gurat ion~ (deformed-conf-Xgurat ion aALime_t)^^f_rjoriL which the so-called Updated Lagrangian (UL) formulation was developed (88, 89, 92, 94, 100).~~

~~The xj current (spatial) coordinate of a point is given by its initial (material) coordinate Xj and the total displacement u. of the material point:~~

~~x. X. + u. I=i i while the strain rates:~~

~~• ./Su, 3x. Ox, Ou, \ E. . = h — k — k + —k —k ] J \3X. 3Xj 3X. dXj/ (6.5) are the Green-La grange measure in (l) and~~

~~= if^i + 3u : 'J = i 3x. (6.6)~~

~~_ (6.7)~~

~~' j are respectively the stretching and spin rate measures in a frame like I I I; y. is the component of the velocity in the i-direction.~~

~~187 The connection between the second Piola-Kirchhoff stress measure S.. in the configuration (l) and the Cauchy one a., in frame (III) is (79):~~

~~9x 9X. 3X. 5. . = a • r-1 • ~J (6.8) ~9X rs dX dx 'J r s 9x i where is the determinant of the Jacobian matrix.~~

One of the main advantages of TL, describing the motion in a system like (l) is that the integrals entering in the variational principle are performed over the fixed original region, with no boundary changes as would happen in a frame like (ill).

When a static configuration as (I) is used as reference, the deformation history is kept through the Green-Laigrange strain, implicit in (6.5). On the other hand, in a frame like (ill), all strains are available from (6.6) and (6.7) computed from the current distorted mesh, which now acts as the history-recording device. This requirement compels a controlled updating of the FE grid at small time steps:

~~(6.9) x (t+At) =x(t) + u (x, t). At which is particularly important in the setting of plastic flow, due to the strong u changes, as discussed in section 5.9.2).~~

Once the steady state has been reached, the second term in the RHS of (6.9) is null and the kinematic recording by the mesh loses its purpose; this justifies the use of a frame like (II) for only steady state situations, with the dynamic limitations discussed in section 5.8 and 5.9- However, in FE practice badly distorted elements easily appear before the steady state is achieved. This has been the main limitation found when sharp die geometries wer^ treated in connection with sticking friction conditions.

~~188 6,2.4, Constitutive Relationship~~

In principle, both flow stress-strain measures (S,E) and (a,e) from the experimental mechanical test, are appropriate to describe material properties. But in practice,-and-partiCiil-true-out-s-kie t-he—el as-t-i-e--- range, the correlation (S,E) is normally non-linear due to the material behaviour; E itself (6.5) becomes non-linear at large deformation and its relation to the initial configuration, which is one of the basis of that material correlation, becomes very difficult to assess experimentally. On the other hand, current measures (cj,e) relations are simpler to produce; the previous history loading can be accounted for through the gradient d^/d^ (iework hardening parameter):

~~e = \l e + Bo d£ (6.10)~~

The above discussion concerns only material non-linearity. However, in dealing with large deformations, a radical difference may appear, which is generally small or negligible in the controlled and homogeneous situation of the test. The rigid body rotations eq (6.7) affect the stress rate (53, 91, 149, 215), eg in current stress measures:

~~V n (6.11) a..=G..-G. ft . - a. ft. 'J 'J 'P PJ JP P' where a is the co-rotational or spin invariant flow stress rate, If the rotation effects in the mechanical test (mt) can be neglected:~~

~~(6.12) mt imt~~

If not, corrections as (6.11) should be produced (213, 214). Both (6.12) V and the 0 invariance allow the transfer of pure material behaviour to the process geometrical set up, thus eliminating any spurious rigid body contributions:

~~-o (6.13) process Wii mt~~

~~1<3q . _ d_o de v Viz: amt~ d£ dt ~ "^process~~

V -V The choice of a as alternative to—s—01—S~for"aTiy~ttefcnTTiatrtijrT7~wa?""a matter of controversy for some time (88, 91, 96, 125, 126). Corrections based in the rotation of principal stresses directions rather than the spin of the material have also been suggested (96).

~~When either material eq (6.10) or geometrical eq (6.11) non linearities are present, the applied stress must be updated:~~

~~2 (t+At) = fl(t) + cr-At in a way which depends on the program design, as discussed in 6.3.5.~~

~~From the viewpoint of the variational principle, any set of measures~~

(S,E) or (cr,e) and frame can be used. It has been mentioned above that from the point of view of volume integration, fixed frames (l, II) are convenient.From the material focus, this leads to the transformation (6.8), and similarly with e and ft to E ; this involves much more computing, therefore current frames (ill) become more favourable.

Even for current configuration (ill), the variational principle can be expressed in terms of either (S-E) (100, 216) or (a~e) (194, 216). By adopting such configuration instantaneously coinciding with the reference one, Uj becomes zero in (6.4) and therefore from (6.5), (6.6), (6.7) and (6.8):

~~Eij=;ij . { (6.16) ft = r ij o j~~

~~(6.17) ij " °ij~~

~~Nevertheless, as soon as geometrical non-linearities emerge the stress rate need to be included as in (6.11) and in general S.. ^ a. .~~

190 The very basis of the UL formulation lays in both the kinematic recording through mesh updating from the original shape and a dynamic loading developed from integration at the same time step of the stress rate (6.15) (88, 91, 92, 93, 9A, 97, 100). The first operation is

~~carried out by (6.9) i f-the-vel«ci t4-es—ere-^the—nodal unknowos-r-o+—Ts a direct outcome of the time step integration scheme (eg 6.20).~~

Argyris et al developed updated schemes for thermocoupled dynamical analysis (98, 99), using "natural" FE and variational principle based in Almansi strain; such measure choice avoids the explicit formulation and storage of the deformation gradient (97) as happens with the Green- Lagrange strain in (6.5).

~~6.2.5. Integration of the Equations of Motion~~

~~Starting from a virtual work principle set from (6.1), Bathe et al~~

~~(92, 3k) developed procedures for TL and UL; for both of them, linearization about the equilibrium configuration at time t adopts the form:~~

*(t+at) "-F(t) "JfWt) (6-'8> where K =K. + K... are stiffness matrices; K. accounts for the linear = =L t=NL =L term in (6.10) and for material (first term in RHS of (6.10)) and geometric (6.11) non linearities. R = vector of external loads F = F(a) vector due to internal stresses M = mass matrix u, u = nodal displacement, nodal accelerations

~~Time integration of (6.18) then proceeds by an implicit method as Newmark or V/ilson 0 (93, 216). It is convenient to make equilibrium iteration within each time step; by introducing:~~

~~k k-1 . k (6 19)~~

~~expression (6.18) is recast into a modified Newton scheme which serves~~

~~this purpose:~~

~~191 K A„(i) - R - F(i-1) - M u(l) S(t).AH - ?(t+At) - (t+At) 2 ^t+At) (6.20)~~

The (6.20) requi.res matrix inversion in the solving of each i-iteration (9*1) , which is greatly time consuming; acceleration of convergence methods (100) have shown to be effective in costs reduction. In any case, the smaller the time step the faster the equilibrium is approached. Once it has been achieved, Cauchy stresses are updated for next step by integration of -(6.1*0 and using (6.11) into (6.15) as:

~~/ Rotational \ -r- de + At x corrections ,, d£ in (6.11) / <6-2,)~~

~~Based in the rate (6.3) form, proposed by Hill (53), Mc Meeking and Rice (88) and Lee (89, 90) developed an UL (initially referred to as "Eulerian") formulation using velocities u as nodal parameters:~~

~~K u = P (6.22) where K = K + K ; K is the incremental stiffness matrix arising from = =c =s =c the constitutive law expressed as:~~

~~(5.23)~~

(of which (6.14) is a particular case), and K the initial stress =s sti ffness (62). P = vector of externally applied and acceleration forces C = C (G, v, h, Zo) G = shear modulus V = Poi sson rat io Zo = effective stress h =0 elastic, h / 0 work hardening parameter; for plastic flow: Zo =Y(yield stress)

1 92 As opposed to (6.18) , the system of ordinary differential eq (6.22) is of first order in the unknowns; its time integration (5) proceeds (88) by updating at small time steps both mesh (6.9) and Cauchy stresses (6.15) using the constitutive relationship (6.23); the rotational terms are corrected through from previous time -val-ue-of— sbres-ses-and—sp4n—— rates.

Both formulation (6.18) and (6.22) are conceptually equivalent, involving updating of mesh nodal positions-and stresses in connection with a time integration scheme. The research above mentioned pointed to general purpose procedures for all range of strain and displacements.

3esides, even for small deformations the rotations (6.7) can become important. Rice (188) pointed out that when the flow stress exhibits a sensitivity dcr/de (either elastic or work hardening (6.14) for the elastop]astic case), of the order of the flow stress , the rotational terms in (6.11) are of the order of the main constititive rate (6.23). Therefore, for the early transient in hot extrusion of soft alloys or at high temperatures, such situations may become relevant.

The main limitation in the application of the above techniques to large reductions, lays in the mesh updating requirements. Badly deformed elements almost inevitably emerge. For sticking boundary conditions, mesh redefinitions become imperative even at low forming ratios, and such alterations interfere in the accumulated history.

The serious obstacle mentioned above has been removed in the procedure introduced by Derbalian Lee, Mc Meekking and Rice (105), by combined use of an Eulerian frame as (II) and UL formulation. The problem of geometrical distortion inherent to any mesh updating disappears, and the refinement of critical areas (as die corners) are not affected during the solution. The procedure follows the outline of (6.22), but it is formulated in a backdated shadow mesh of which the fixed one (ll) is always the next position for each time step:

~~x (t) = x° (t-At) + u (x, t-At). At (6.24)~~

~~193 In order to cope with the tool movement, a convective boundary condition~~

~~is applied to its interface. The updating (6.24) requires the velocity~~

~~of the "retarded" position whose records are kept throughout the code~~

~~memory. This is not possible for the stress updating:~~

~~a (x, t+At) = a° (x-uAt, t)+Aa (6.25)~~

where Aa is given by the Lagrangian solution (>'e:second and third term in the RHS of (6.21)). The first term in the RHS of (6.25) is determined by interpolating the known fixed mesh point stresses at time t (146). By using the conjugate stress approximation of Oden (146), stresses which are continuous across element boundaries are acheived.

~~In its production, an additional set of equations at each time step need to be solved:~~

~~A . = = ~ " (6.26)~~

and from it, o° = o°(X°) 's computed. A is a fundamental matrix computed only once for the FE grid and the shape functions evaluated at the backdated position x°. Solutions for a plane strain radial flow (idealization of drawing) and comparison with classical UL computing times are given (105).

6.3 Viscoplastic Model w It has been shown in chapter 2 that the application of a variational principle (2.20) based in an internal power (6.2b), and current stress- current stretching rate versions of the constitutive law (2.34), lead to the system (2.45): 4

~~K u = F - M u (6.27)~~

A stream of work in transient and asymptotic steady state analysis has % been produced by such an approach. They differ in the particular form of the constitutive law, the frame of description and the integration scheme. Three main ways can be distinguished:

194 By applying explicit forward marching schemes to (6.27), a solution in the velocities u becomes available. The algorithm is controlled by the time step (similarly to (6.20)), and does not require inversion of the matrix K as (2.58). By using this approach, Jain (112)

examined very simple probTems-of--i4Tipu4sJ-ve--load-Lng--of--str4jatures- (fixed beam, fixed bar). He concluded that only pJane stress situations could be successfully dealt with. For plane strain and axisymmetric problems, difficulties were found, and even a reformulation in lagrangian terms is suggested.

In most of the procedures (135~141) 3u/3t is neglected in (2.15). Thus u = 0, and a quasi-static analysis (still retaining the convective or geometric acceleration term if required) becomes available as (2.58). From its solution in the nodal unknown velocities (which involves K matrix inversion), the mesh is updated by (6.9). Thus, in the limit when At is small enough, such neglection becomes a good approximation. The major computing cost of this algorithm, when compared to incremental marching schemes, is compensated by the bigger time step allowed, both factors being competitive. For such design, convergence of the numerical method and time evolution of the problem take place simultaneously, the only control being the time step in (6.9). Modified algorithms are also possible as (6.19) iterating until convergence is improved and only then making the update (6.9). Thus, the availability of the velocity and eventually of temperature at the updated nodes makes it extremely simple to handle constitutive laws relying only upon current values o = cr (e» I ). Limits are set by the rotational corrections (6.11) becoming important, which enforces the use of stress rates and time integration for them. Another limit not sufficiently explored is the time step used for the very initial upsetting. It is normally accepted that even the use of a coarse time step for that initial deformation will lead to the same solution than an accurate smaller one. This seems to be the case for constitutive laws depending only on the current state, because this is determined by the boundary conditions which are also current values (Fig 4.2). However, as soon as the flow stress becomes dependent on the path (eg accumulated strain, or rotational corrections) the situation becomes more difficult to deal with. In this case the discussion in section 5-8 indicates that the flow pattern is not only defined by the border conditions but also by the operation of internal forces which are path-dependent. One of the standard techniques- us-ud when-deal ing-wi th work— — hardening materials is to compute the incremental strain at the updated Gauss point station from the strain rates. Thus, figures of total strain are produced by adding up all contributions while the station moves. Here mesh redefinitions, even within the steady state are dangerous because they can alter the recorded history. In Rcf (137) the work-hardening procedure was only applied to upsetting of a cylinder between infinite flat plates, and for open extrusion-forging where both sharp corner and large flow deviations were present, perfectly plastic material was assumed instead. Besides, no transient load results are reported for the latter one.

For steady state treatments with a fixed mesh like (II), not followed in time as asymptotic trend of its transient, no recording device is available throughout the mesh (6.9). Therefore any path dependence must be introduced in another way. A commonly used approach (128, 137) has been a procedure which intercalates in between each iteration

(2.58) an additional step for computations of hereditary parameters (eg strain) along flow lines. Such has been the design in the so called "initial strain rate" method developed by Thompson and Dawson (125-128), initially aimed at elastoviscoplasticity, and using both velocity and pressure as nodal parameters. It was developed to a more general behaviour by Dawson (128) as:

~~a!. -—c'Y (6.28) 1 j • J G 1J A~~

both additions to (2.3*0 being spin invariant (gYj due to (6.11) and Ej • by(6.16)) and accounting for total strain dependence: the first addition in the elastic range and the second one in the plastic region. The resulting algebraic system was expressed as:

196 where K encompasses the law form (6.23) and an hydrostatic pressure ^ e s matrix; the vectors F and F are introduced by the new terms in (6.28). The results reported for R = A axisymmetric extrusion considered only the pure perfectly plastic (128) behaviour connected with the first term in (6.28). What is pursued below is an examination of the applicability of the method to the steady state resulting from an asymptotical outcome of the transient situation. The Lagrangian strain E.. is obtained from integration of the deformation gradient along a stream line. Because such integration depends on the gradient itself, it is necessary to define the flow line path back to a point with a known deformation gradient. From there on the deformation gradient is integrated to the forward point where it is required. Thus, the initial station is selected close to the ram (128), a choice also used with less sophisticated schemes by Pacheco (137). From there, the station position is updated by

~~(6.9), using the velocities produced by interpolation from the neighbouring nodal values of the fixed mesh.~~

In the light of the discussion of sections 5.8 and 5.9 here is where the source of mistakes is introduced. By the time the advancing time-integration of the stream line reaches the critical area where strain effects at the beginning of the process are important, the boundary conditions have already set another pattern. Actual movements in the neighbourhood of that area at the early stage need a careful tracing in time. This requires integration stations where normally the deformation gradient (or plainly the accumulated strain) is not known and some global interpolation on the assembled network as in Ref (105) is required. Regarding the Fe term, Dawson (128) evaluates the stress flux using for (6.11):

~~(6.30) o = uT.Vo~~

~~197 Therefore, for steady state, no explicit time control is required in stress updating (.contrary to (6.15)) and the whole solution of (6.29) can be carried out without time marching schemes.~~

Another major defficiency of (6.28) is related to the presence of an elastic term in all the deformation range; no hint of its cancellation in the plastic area is given (125_128). Therefore, for larger deformations the contribution of da/de could raise the flow stress to spurious high levels, when in reality it is supposed to be plastically yielding. In fact, the excessive bubbling just before entering the roller reported in Ref (125) seems more due to such assumption than to any large deformation corrections, these being small at the very low reduction ratio employed. Another disadvantage of (125~128) is the use of u/p formulation, which requires the storage and solving of twice the number of nodal parameters than the penalty approach.

With the penalty set-up of the present thesis, an alternative to the above procedure can be drawn, including all strain sensitivity in the form of the second term in (6.28), with proper material /aj j \ parameters IqJ ; for elastic and plastic region and compressibility release out of plastic area. Thus, the evolution of the current state can be traced mainly in two ways. One is by mesh updating, but it limited by its distortion. The second one is to keep the mesh fixed and produce a time integration as in Ref (105). The inclusion of the latter algorithm brings out the capability of dealing with all the refinements considered in elastoplastic treatments and the differences between the two models (viz, elastoplastic and viscoplastic) fade away.

6.4. Conclusions and Recommendations for Further Work (ll): a. In spite of its reported application to conical dies only,the fixed mesh procedure of Lee and coworkers (105) should make a breakthrough in the treatment of sharp geometries possible. Its comparison with the lower computing cost of TL they gave is not applicable to the last case, where mesh distortions go out of control in few steps.

198 In this thesis, it is suggested that such an approach complemented with a thermal analysis should be pursued, with the alterations which follows. Aiming at a code specifically designed for forming problems exhibiting a steady state,two main factors should be explored to reduce the large computing costs. Both of them are related to the global stress interpolation state (a° in (6.25)), operating by matrix inversion (6.26), which is called at each time step:

Elimination of the (6.26) step; the possibility of its conversion to an incremental form controlled with the time step should be explored. Another option is to avoid completely the stress rate time integration, as done partially by Thompson and Dawson and discussed in connection with (6.30). However, this additional calculation increases the computing time and an assessment against the (6.26) option should be done.

Once the steady state is achieved, several operations become meaningless due to their associated negligible values. The code should be designed to detect such situations and both by-pass some of its parts and enlarge the time step during execution. The switching could be built up by checking the convergence of both velocity and temperature against two ab-initio preset bounds.

The implementation of time integration in a fixed mesh description as (a) opens the capability of incorporating BT/9t in the thermal analysis; both final systems of differential equations are of first order in time and can be dealt with by using the same algorithm.

The choice suggested in (a) requires the introduction of strain sensitivity in the form (6.14) or (6.23) in both elastic and plastic range. A control which allows to change from the elastic modulus to the work hardening strain sensitivity dependence ,can be easily designed by checking the local stresses against the yield stress. A slight variation of eq (6.28) can be constructed by setting the V and G connection with the bulk modulus K thus allowing the use of the penalty formulation instead of the u/p one. History dependence of the flow stress as (6.14) is not-only expected for the strain. As discussed in section 5.2.2 both sharp changes of strain rates (as the one taking place in extrusion near the exit) and temperature, affects the plastic flow. A model able to react to any of such susceptibilities can be produced similarly to (6.10):

* a 3a de . 3a de n do dT + Ba . G =Al 3i dt + A2 di &t + A3 3T dt (6,31) where now the coefficients of the 2nd and 3rd term of the RHS must be obtained experimentally from dynamical loading tests, which is no easy task. The zero-order approximation is given by the last term in the RHS of (6.31), flow stress relying only on current values, which is the form of eq (5.1) law. The remaining contributions introduce the previous loading history into the stress and therefore in the internal forces; this fact is not present in the zeroth order. In analogy with (6.21), the stress updating now should take the general form:

~~ft+At~~

~~a = Q + d£+ + dT (t+At) (t) / fi i ^ §T Vj^tational ) ( } N J J t\ ) \correct ions/ '~~

In spite of its application to small extrusion ratios only, a limitation set by the updating mesh requirement, the thermomechanic analysis of Argyris et al (98, 99, 201, 217) is worth examining. The thermal problem is formulated here with allowance for the non equilibrium form of the second law of thermodynamics:

~~S = SG + 5* (6.33)~~

• e where the rate of entropy S is split into two: one part S accounting for the entropy supply from the surrounding medium, and other part S1 for the entropy production. Two extreme situations arise, both connected with a cancellation of local temperature gradients (217). The first, so called isothermal (98, 217) occurs when the temperature changes are extremely slow. On the other hand, adiabatic treatment (98, 201) implies instantaneous change in the physical state of the

~~200 body. Real processes take place in between both extremes,~~

~~requiring a general non-equilibrium thermodynamic treatment, which~~

~~leads to (98):~~

~~cT + StT*-Sc, + £_ g.r {63li) P J where the dot indicates time derivative and: c = spec i f ic heat T = temperature~~

~~Sy= entropy at constant temperature Vq= divergence of the heat flux per unit of mass Jto.ir = fraction of mechanical work converted into heat.~~

~~The S.j. value has to be computed from the specific free energy function of the material f:~~

e 1 l f = f (e®, T) + f (z ?) (6.35) where the superscripts e stands for elastic, and i for inelastic; the subscript F design a current stress-free configuration, achieved analogously in the numerical model by imaginary local unloading. The specific entropy s is given by:

~~S = - (6.36) and from its time rate:~~

~~c = q + is T (6.37)~~

~~5 + 3T 1~~

~~S-j. can be identified and supplied to (6.34).~~

~~For the very particular and limiting case of small strains, some forms of fe are available (98), which produce:~~

~~201 where K is the bulk modulus, J^ the incompressibi1ity rate, aL the coefficient of thermal expansion and p* the density of the stress-free volume. Near incompressibi1ity: or J^ very~~

~~small, and (6.38) takes the form:~~

~~Cu a. /A X s h L (6.39) T P f _ Q where P - P - P is considered constant and ^ is the hydrostatic pressure encompassing the product bulk modulus times volumetric strain rate.~~

~~The entropy eq (6.34) has one thermocoupling link more than the energy one (2.74), through the term S^.. If instead of the updated~~

mesh (as in Ref (98)), a fixed configuration (as in this thesis and improvements suggested in section 6.4.a) is employed in the formulation, the convective heat due to the movement of the solid must be accounted for as in eq (2.73). The production of Sj by (6.37) requires the knowledge of the free energy f' (eq (6.35)) which remains stored within the material when the plastic stresses are removed. From mechanical testing data, the qualitative behaviour of f1 can be examined as follows. The plastic unloading leaves the

material with a residual yield stress. Thus, a parameter Ay, given by the area up to the yield point in the (a~e) plots gives a measurement of the total internal energy (6.35). Therefore, the plastic change f' can be obtained by the difference between the

~~values of Ay fordeformed and original material. The form (6.38) which they (98) suggested depends only on the total accumulated plastic~~

strain; certainly work hardening increases the residual area Ay The temperature also plays an important role on f', with both continuous and discontinuous incidence. At low temperatures the work hardening increases, therefore larger Ay are expected. As a result, a continuous behaviour of f' increasing with the temperature decrease should appear. On the other hand, metallurgical changes introduce discontinuous dependence: recrystal1ization releases free energy and f' drops. And as in the small deformation range, (6.39), the hydrostatic pressure should also be important; compressive (negative) values of a^ prevents release of energy, contrary to what happens for tensile . A proper setting of a

202 general f' expression can be carried out by hybrid experimental - numerical schemes as outlined in section 5.9.1 . Thus, in spite of requiring more information, formulations like (6.3*0 set the basis for a quantitative thermodynamic treatment of the alloy being deformed.

Two current thermomechanica1 states differing only in the hydrostatic pressure evolve differently in both free energy and temperature. This prevents the transfer of path-dependent properties (as strain sensitivity) from a tensile situation like torsion (q positive) to a very compressive one like extrution (cr^ negative!)1. Such perturbation is meant to be more critical when larger values of internal free energy are reached (low T, large accumulated strain). Thus, the explanation given by Conrad et al (199) for extrusion flow changes, is wrongly based on an excessive negative strain sensitivity (da/de)after a maximum shear ( ie torque), which is only developed as a consequence of a tensile situation. A further examination of this matter can be produced by double integration in time (6.37) and temperature (6.36) of S^., which gives:

~~Af °< - oJi AT (6.AO)~~

~~Besides, a pseudo heat generation 0 can be identif i ed in (6.3A) as:~~

~~Q = JL 2-1 - 3 K :aL Ji J (6.A1) PJ P~~

~~The trends valid for very small increments of AT, for two states under the same absolute value of hydrostatic stress can now be i nspected:~~

~~Jj> 0 + Af<0 0 Ml (6.A2)~~

~~Tensile release of free drop in temperature energy (6.a3) Jj< 0 Af>0 Q" + T t Compressive storage of free rise in temperature energy~~

203 An accurate analysis requires the specific form of the free energy of the material. Nevertheless, the (6.43) compressive situation indicates a trend of more storage is counteracted by its temperature rise. Thus, until such temperature rise allows the release of free energy, Af is positive and so it is the strain sensitivity of the flow stress dcr/d£. This is in contrast with the pronounced negative values developed in a tensile situation (6.42) like the torsion test.

~~A further examination of the discussion in section 5-8 concerning the changes in flow patterns of Plate III might suggest that negative values of d~~

The previous review demonstrates the complexity of a complete description of the dynamic problem. Achievements for meaningful extrusion ratio are still way ahead. However, a qualitative description of the evolution of the early movements in terms of steady state results seems possible, and it is attempted below. At the beginning of the upsetting stage, only the axial material facing the outlet is allowed to move. The slip line departing from the die corner (at zero degree for sticking friction, Fig 2.2) should prolongate itself towards the billet axis, and reach it normally; the starting exit velocities are perpendicular to such line and parallel to that axis. As the acceleration forces eq (2.15), drive the off-axis material towards the outlet, the extrude velocities rotate, pointing towards the axis. This process is not instantaneous, and certain time should elapse till the overall rotation is completed. Thus, in order to maintain an energetically favourable situation, the component of velocity parallel to the M^ slip line (Fig 4.5, 4.21) must be reduced, which is achieved by a backwards movement of Mj until reaching the steady state position. The hydrostatic pressure behaviour also explains this movement; the release of the relative tensile stress state of the axial fibre (Fig 4.23) tends to shorten it, pulling the plastic border backwards. During all this movement the tool load has risen, because, the complete elimination of the component of the velocities parallel to M^ is impossible. Thus, a stationary load value is achieved wh-rch-eHher - keeps a constant value for frictionless case (curve I, Fig 1.1) or decays due to the reduction of friction with billet length (curve II, Fig 1.1). The temperature results indicate that the instantaneous maximum is achieved at the source location, viz the M^ line (Fig 5-8) where maximum mechanical work is converted into heat. Thus, even for the assumption of instantaneous heat propagation ("adiabatic", in the sense of (e) above), as the source M^ moves against the stream, it is always in a cold T temperature environment d T^ = Tg. The small departure from the preheating Tg is given by backwards conduction effects (Fig 5-2c). During that shifting of M.j, strain rates achieve their nominal steady state value. When M^ settles itself, the heat generation raises its temperature to T, = T + . This happens not instantaneously, but at a finite O D time (as discussed in (e) above). As a consequence, the strained (Fig 5-11) metallurgical structure of the flowing material,which

was allowed to accumulate an amount of internal energy for the Tg = Tj situation, finds itself holding a non-equilibrium value for the Tg _ Tg + AT case. Then, both the release of that excess of energy (ie recovery or recrysta11ization at the end of the extruded rod) and adiabatic decay of local stresses (Fig 5.1) at the M^ line must take place. As the last one determines the non-frictional component of the tool load (Fig 5.19)» this drops.

For the new situation, the almost constant external power applied finds less internal resistance overall, and as a consequence the ram 4 velocity achieves a larger equilibrium value, as practically detected.

The above interpretation explains the presence of the prominent pressure peak Ap in both direct and indirect (where no dead metal zone is present (Fig 4.12, 4.20) cases . Thus, the explanation given by Sheppard and Castle (219) in terms of associating theAp peak to an excess of accumulated deformation in the dislocation

~~205 network is confirmed here on the basis of the transient thermomechanical behaviour.~~

The analysis can be pursued even further. At the main slip line M.j, the strain rate should reach values-close-to the f-ina 1—nominal— state while M. complete its nearly isothermal shifting, and then • • • • = it could be assumed that £a = eb £By where e^y is a mean strain rate value for steady state. Thus, the main differences in the Z parameter after that movement,come from the temperature changes from T to T, . It can easily be shown from eq (5-1) and eq (5.2) 3 d that the difference in local stresses for both states is:

~~1/n AH AT (6.44) a b G TB(TB+AT)~~

On the other hand, the frictional forces involved in both cases are almost the same, due to the thermal phenomenon occurring at the main slip line, well off the container border. Therefore, the changes in tool pressure associated-with the two states are:

~~(6.45) Ap = Ap = b . Ln R.(a - afa) where p is the frictionless eq (4.6) component of the pressure. Thus:~~

~~1/n (6.46) Ap = b . Ln R . _L . AH AT~~

~~G Tb (Tb+AT) from which the following experimental facts are predicted:~~

i) Ap increases with the extrusion ratio R ii) AP increases when operating at low preheating T^, where also the temperature rise AT is bigger iii) AP increases in alloys exhibiting higher activation energy A H.

783 iv) Ap increases when the strain rate sensitivity of the flow stress is larger. Therefore, for perfectly plastic materials, it is expected that Ap = 0, which is the general outcome examined in chapter 1 (eg Fig 1.1). -

The interpretation given in section (5-8) for direct extrusion is based on the relevance of the strain coupling with the flow stress. Any reduction of such link should drop the load required to extrude. The coarsest method seems to break the strain pattern at M^ by alterations in the container bore. This suggests a theoretical explanation to the industrial practice of fitting anular rings inside the container near the die wall. Another consequence of such alteration is the setting of the flow at .certain angle with respect to the maximum strain rate line; this reduced the straining which controls recrysta11ization in the periphery of the outcoming product. Much work on the container and die design can be done with a better understanding of the transient.

~~207 APPENDIX I: COMPUTATIONAL FORM FOR FINITE~~

~~ELEMENT ANALYSIS~~

* APPENDIX I; COMPUTATIONAL FORMS FOR FINITE ELEMENT ANALYSIS

Both sheet and axisymmetric rod extrusion analysis of the flow can be simplified by-a- descr i pt-iorv-i-e- -a-pl-ane—FKmria-l—t-o-the-symme+f^y-^xi-s-:— Thus, only two velocity components are required in the calculus:

~~PLANE STRAIN AXISYMMETRIC~~

~~/ / / /~~

~~Fig A 1.1; Extrusion geometries~~

~~UT - (u ,u ) (A I . 1) u = (u ,u ) (AI.2) — x y — r' z~~

~~The computationally advantageous compact forms for strain rates (2.9) now take the form:~~

~~(Al.3) I1 = (e,,^,e ,Y ) (Al .A) —£ = (ex y ,Y xy ) z' r e rz~~

~~which are allowed by the symmetry of the strain tensor>and the additional ones:~~

~~£ = Y = Y =0 (Al.5) = = 0 (Al.6) z xz yz Ve2 V~~

~~209 In (A 1.3/4), the tangential components y ape introduced a twice 'J the engineering definition (2.1)~~

~~(Al.7) l*r = Y = e x 2 y *y~~

~~The operational matrices L in (2.9), have the explicit forms:~~

~~9 0 " 9 0 9x 3z~~

~~L = 0 9 (Al.8) L = 0 9 (Al.9) By Br 0 1 9 9 r 3x By~~

~~a. 9 . 9z Tr whi 1 e for m in (2.10):~~

~~T T _ m = [ 1 1 0 ] (Al.10) m 11 1 (A I .11)~~

~~The compact form of the deviatoric stress tensor is given as:~~

~~T a't =( a',a',a',r ) (al .13) & = ( a ' i a% » Z 1 — \ > > >~ y °xy z r rz i (Al.12)~~

~~The constitutive matrix D (2.39) is now expressed as:~~

~~D = vr (Al.14) D = tf (Al.15)~~

~~where now can be introduced as ratio of different invariants forms,~~

~~(eq(2.32) and (2.3k)), as follows:~~

~~(j' \2 ,) for (2.34): 1 •2 (Al.16)~~

~~210 where from (2.5);~~

~~2 /T2 =/i(E2 + £* =/i(i2r + + Eg ) +Y ' (AI.17)~~

~~and analogously from (2.12):~~

~~2 a ,z /jj, = + °- + 0 ) + ^ (A I.18)~~

~~Thus,for a pureshear situation:~~

~~/f2 =Y (A I.19)~~

~~/j^ = T~~

~~b) for (2.32) : M* = 2 Z.eff (13, 122) (A I.20) 3 e eff~~

~~2 2 : (e + e + kh + 2 Y (A I.21) r z = 2./L e-« 7f /3 2~~

~~a Z 2 2 ( _ - <7) + - ^9) + (a0 - a ) + 3 r eff V* z r rz (AI.22)~~

~~Thus, for pure shear:~~

~~: 2 Y eff - ,7f-~~

~~_ z (A I.23) a = z. eff ^ ° which coincides with (AI.19)~~

~~The equivalent strain rate, defined for pure tangential strain as £ = 7/3 (AI.24) becomes for the general strain situation~~

~~e - ^ (ai.25) /3~~

~~211 Another differential operator which is sometimes required in the calculations (eg geometrical part of acceleration forces (2.15), thermal convective terms (2.74)) is the gradient:~~

~~(Al.26)~~

The 2-dimensional nature of the calculations forbids considering, in the case of cylindrical problems, an independent circumferential component in (AI.27). Therefore, it is assumed that 3/30 = (1, which implies no gradients of either velocity or temperature are present in the circumferential direction. However, for the axisymmetric case such a condition is strictly valid.

In FE applications, nodal variables as coordinates x (2.55), velocities a (2.42) and temperatures h (2.81) are introduced. Slightly different forms are used for the two cases dealt with in this thesis, as follows:

~~a) I sothermal~~

~~0) i1> .(ii x - * (Al.?.8) £ = it" (Al.29) :<9> :<9» a £L" where for the (i) node:~~

~~PLANE STRAIN AXISYMMETRIC~~

~~id _ (u (i) (Al-30) a = (Al.31) a - u) (it~~

~~M) in 3 z u> _ (A 1.32) a< u = (Al.33) £ u; (i;~~

~~212 The element's shape function matrix (2.55) and (2.42) takes the form:~~

~~n = (n0' (Al.34)~~

~~where for the (i) node:~~

~~n"' 0 N,n = (A I.35) 0 Nt p~~

~~n'" being the shape functions given by (2.42a).~~

~~The B matrix introduced in (2.43) becomes~~

~~1 = L • N = (A I.36)~~

~~the (i)~~

~~li> 111 3N 0 9N_ Ox 9z~~

~~l «i/ 0 (Al-37) B " » 0 M (al.38) 9y 9r 0 1 9y r 9N 9z 9r b) Thermocoupled~~

~~ci> The nodal temperature vector h is introduced by redefining a in (Al.32/33) as:~~

~~M> (ii 3 X en _ 3 (Al.39) (Al.40) y ii / CI)~~

~~213 And now the shape function submatrix (AI.35) is;~~

~~.(I I n 0 0~~

~~m n'" = 0 n 0 (al.41)~~

0 0 •vT where v/" are the shape functions modified by the upwinding. The redefi- nition (AI.39) implies that the temperature has become another degree of freedom. Due to the diagonal form (2.89) no direct coupling among velocities and temperature exists; the coupling is parametric through the flow stress, mechanical work converted into heat and convective terms. For the latter one, the adoption of the velocity distribution from the previous iteration in (2.84) avoids non-diagonal forms in the L.H.S. of (2.89). Thus, the notation in (AI.41) is purely formal; B, m and ^ operate on the (AI.35) part and the thermal matrix H is constructed from the new row in (AI.35) involving the v/u functions.

~~214 0~~

~~APPENDIX II: DATA ADOUISIT I ON AND~~

~~GENERATION~~

~~0 APPENDIX I I : DATA ADCHJISITI ON AND GENERATION~~

The data defining the problem are obtained from an input file with the structure of the sample in Fig-ATI.1,-as.described below. Depending on whether the case is perfectly plastic isothermal (PP) or thermocoupled (TH), the parameters marked with are specified d i fferently.

~~1. Several problems can be solved simultaneously, each starting with the labeling title:FINE... (label)... and finishing with an END macroinstruction.~~

~~2. General parameters of the grid:~~

~~NP = Number of nodes of the network NE = Number of elements MAT = Number of different material sets ND = 2 (spatial dimensions) D0F'f= Number of unknowns per node:~~

~~D0F*(PP) = 2 D0F*(TH) = 3 NEN = Number of nodes per element (In this work adopted as 9) KT* = Thermal nature of the problem KT(PP) = 0 KT(TH) = 1~~

~~3. Coordinate data input: It follows the reading of the COOR macroinstruction, and a typical 1i ne is:~~

~~N = Node number NG = Nodal generator increment X, Y = Nodal coordinates (cm)~~

If the node numbering in the next line Nnext is different than (N + 1), an automatic node generation is started, with numbering varying in NG and an uniform separation along the straight line joining N with Nnext.

~~216 4. Element data input: This starts after the instruction ELEM has been read, and a typical specification for a line is:~~

~~M = Element number MATS = Material set number IX = Nodal connect ions 1ist; it holds the numbering of all the nodes which define the element NG = Automatic element generation~~

Element specifications missing in the input between two consecutive M are generated by adding 1 to the lower M and NG to each node of its list IX. If the geometry breaks or the material set is changed, a new line has to be defined.

~~The choice of the material set depends on the relative position of the external boundary, where surface integrals have to be computed (Fig All.2)~~

~~6 7~~

~~8 9 external border~~

~~Fig A.I 1.2 Material set types for MATS~~

5. Nodal Restriction and Boundary Conditions: The reading and generation after the BOUN instruction sets a code to specify if the associated force under FORC is either a velocity (or temperature) or a force.

N - Boundary node NG = Generation increment for boundary nodes. The missing ones are generated according to the sequence N, N + NG, N + 2NG, by linear interpolation of the values of the boundary condition.

~~217 IDi , i = 1,3. Code for nodal restriction; i = 1,2 are the (x,y ) directions and i = 3 stands for a scalar; the general format is:~~

~~IDi = 0 Force Fi specified (Di = +1 velocity Fi (temperature for i = 3) specified. If NG t 0 negative values must be used for IDi.~~

~~Thus, for the example in Fig AII.1, nodes 1 to 18 have U =0.75 (cm/sec) U = 0, T = A00°C. y~~

~~ID3*(PP) = 0~~

6. Material properties: MATS = Material set identifyer (as Fig AlI.2) IE = Code for the element algorithm defined by (2.89) Yo*(PP) = Yield stress (MN/cm2) Yo(TH) = Control on the p(2.37) value for the first iteration NS = Number of Gauss Points in each direction for numerical integration of K (2.A7, 2.A9) and H (2.8A) NPT = Number of Gauss Points in each direction for the penalty matrix & (2.A8) AL = Penalty parameter cK ICRL = Library control for constitutive lav; 1 Perfectly plastic 2 Hot working law (5.1) BET* = Fraction of mechanical work converted into heat; BET*(PP) = 0

~~Additional material properties (as defined in chapter 2):~~

~~Case PI P2 P3 PA P5 Po P7 P8 P9 P10~~

~~c on 1y if h (TH) P k as f FRIC is - 3 present (PP) ------bel ow - -~~

~~218 where h stands for solid-air heat transfer and T^ =room temp.~~

~~P7 = 1 for MATS 2,4 and 6 to 9 (Fig AII.2)~~

~~P8 =1 for MATS 3,5 and.6 to 9 (Fig _AII.2) P7, P8 = 0 otherwise~~

~~The main data input is closed with an END instruction. If a STOP is set afterwards, the execution finishes, thus allowing for the inspection of the data generated.~~

7. Presetting of the solution algorithm MACR: order to enter the algorithm solution FRIC: allows to use border elements as described in section 4.4. If not included, nodal force specification (as in sections 4.2 and 4.3), commanded only by the boundary condition as detailed in 5- above are assumed. DM3F: choose a direct algorithm to solve the thermal problem (2.83), instead of an iterative scheme (default) as in (2.58).

~~EXTR: define the problem is extrusion and set additional data: Ll = first element in the last column (Fig 4.1) L2 = number of nodes in contact with ram~~

~~THCP" L3] , . , u k only used fo r TH CHTP" L4J~~

~~L3 = number of elements within the container~~

~~" = TB~~

~~UPW: use upwinding for the shape functions 1/ in (2.81); otherwise standard N as (2.42) are used.~~

~~AXYS: define the problem as axisymmetric (default: plane strain).~~

~~LOOP Nmax: define the Nmax iteration in the solution (2.89).~~

~~UTAN: form the LHS in (2.89).~~

~~FORM: form the RHS in (2.89). SOLV: operation of the solver.~~

~~NEXT: close the loop command range.~~

~~219 L print the nodal (velocities, temperature) and Gauss point STRE•f 'J variables (strain rates, stresses, angles and magnitude of principal directions, etc.). END: close the solution presetting.~~

~~If the problem is TH, the hot working constants are read:~~

~~' 2 Joul e Alloy label , A , AH f cx m gr mol MN~~

~~Next the solution proceeds by repeating Nmax times all instructions in the range between LOOP and NEXT, as fully described in section 2.5~~

220 FINE. 391 75 3 2 3 9 NP NC MAT ND DCF NEN KT COOR t t 1.0000 0.0000 5 1 1.0000 0.3100 7 I 1.9000 0.9150 9 t 1.0000 0.6000 17 t 1.0000 3.7150 19 0 1.0000 3.7250 ;o i 1.2000 0.0000 29 I 1.2000 0.3100 26 1 1.2000 . ILHISft 21 t 1.2000 "0.6000' 39 I 1.2000 3.7150 31 0 1.2000 3.7250 39 1 1.9000 0.0000 93 1 1.9000 0.3100 95 t 1.9000 0.9150 97 1 1.9000 0.6000 55 I 1.9000 3.7t50 57 0 1.9000 3.7250 51 1 1.9000 0.0000 92 1 2.9000 0.3100 99 t 1.9000 0.9150 99 1 1.9000 0.6000 79 1 1.9000 3.7150 79 0 1.9000 3.7250 77 t i.tooo 0.0000 It 1 1.1000 0-3100 13 t 1.1000 0.9150 15 1 1.1000 0.6000 93 t 1.1000 3.7150 95 0 t.tooo 3.7250 99 1 9.0000 0.0000 100 1 9.0000 0.3100 102 1 9.0000 0.9150 109 I 9.0000 0.6000 112 1 9.0000 3.7150 119 0 9.0000 3.7250 115 1 9.2000 0.0000 119 1 9.2000 0.3100 121 1 9.2000 0.9150 123 1 9.2000 0.6000- N NG X Y 131 1 9.2000 3.7150 133 0 9.2000 3.7250 139 1 9.9000 0.0000 131 1 9.9000 0.3100 190 1 9.9000 0.9150 192 1 9.9000 0.6000 150 1 9.900C 3.7150 152 0 9.9000 3.7250 153 1 9.9000 0.0000 157 1 9.9000 0.3100 159 I 9.9000 0.9150 191 1 9.9000 0.6000 199 1 9.9000 3.7150 171 0 9.9000 3.7250 172 1 9.7000 0.0000 179 1 9.7000 0.3100 171 1 9.7000 0.9150 110 1 9.7000 0.6000 til 1 9.7000 • 3.7150 190 0 9.7000 3.7250 191 1 9.1000 0.0000' 195 1 9.1000 0.3100 197 1 9.1000 0.9150 19° 1 9.1000 0.6000 207 1 9.1000 3.7150 209 1 9.1000 3.7250 210 1 9.9000 0.0000 219 1 9.9000 0.3100 219 1 9.9000 0.9150 211 1 9.9000 0.6000 229 I 9.9000 3.7150 221 0 9.9000 3.7250 229 1 10.0000 0.0000 233 1 10.0000 0.3100 235 1 10.0000 0.9150 237 1 10.0000 V.QVVV 295 1 10.0000 3.7150 fig All -1 Input Data 297 0 10.0000 3.7250 291 1 10.0050 0.0000 252 1 10.0050 0.3100 259 1 10.0050 0.9150 259 1 10.0050 0.6000 299 1 10.0050 3.7150 299 0 10.0050 3.7250 297 1 10.0100 0.0000 271 I 10.0100 0.3100 273 1 10.0100 0.9150 275 1 10.0100 0.6000 <•3 1 10.0100 3.7150 215 0 10.0100 3.7250 2 It 1 10.0150 0.0000 290 1 10.0150 0.3100 292 0 10.015C 0.9150 293 1 10.0200 0.0000 297 I 10.0200 0.3100 299 0 10.0200 0.9150 300 ; 10.220C 0.0000 309 1 10.2200 0.3I0C 309 0 10.2200 0.9150 307 1 10.9200 0.0000 311 I 10.9200 0.3100 313 0 10.9200 0.9150 319 1 10.7200 0.0000 311 1 10.7200 0.3100 320 0 10.7200 0.9150 321 I 11.0200 0.0000 325 1 11.0200 0.3100 327 0 11.0200 0.9150 321 1 11.2200 0.0000 332 1 1t.2200 0.3100 339 0 11.2200 0.9150 335 1 11.9200 0.0000 339 1 11.9200 0.3100 39! 0 11.9200 0.9150

~~221 LB III~~

~~5° ~ Q_~~

~~CTJ c s If) X<~~

~~UT' £ UJ < -0Q? K> if) o Q I<— N CD 2 O~~

~~I cvi CM Cvl o c.aa T^MyoMior^n^iiNriiii^N~~

~~o —ooooooooooooo oooooooooooooooo -fc- M ••••••••Moi^rirtriiMriM^i'nrnmfnf')**' t/> r-f-ooooooooooooointnoooooooooooooo oooooooooooooooo — — OOOO-* — —-"-"OOOO nn m~~

~~ÔOÔT-OÔ—oo.oô xOPÔhOÔÔOÔv'O IAO lAO lA O or- or- or- w*m .mS -r m - g . —;SRSSÊRfrSKS ACKNOWLEDGMENTS~~

~~The author would like to express his sincere thanks to the following people for their invaluable assistance in the production of this work:~~

Dr T. Sheppard. Staff of the Imperial College Computer Centre. The past and present members of the John Percy Research Group, notably Dr S. Paterson, Dr R. Vierod and Dr R. Parkinson. ALCOA International. Angela and Thaira for their efficient and fast typing of the manuscri pt. And to Bala, who helped me not only in this work.

2 2 3 4 * 4 *

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