VTT PUBLICATIONS 312 F19800095
Thermomechanics of solid materials with application to the Gurson-Tvergaard material model
Kari Santaoja
VlT Manufacturing Technology .
TECHNICAL RESEARCH CENTRE OF FINLAND ESP00 1997 312
Kari Santaoja
Thermomechanics of solid materials with application to the Gurson-Tvergaard material model
TECHNICAL RESEARCH CENTRE OF FINLAND ESP00 1997 ISBN 951-38-5060-9 (soft back ed.) ISSN 123.54621 (soft back ed.) ISBN 95 1-38-5061-7 (URL: http://www.inf.vtt.fi/pdf/) ISSN 14554849 (URL: http://www.inf.vtt.fi/pdf/) Copyright 0 Valtion teknillinen tutkimuskeskus (VTT) 1997
JULKAISIJA - UTGIVARE - PUBLISHER
Valtion teknillinen tutkimuskeskus (VTT), Vuorimiehentie 5, PL 2000, 02044 VTT puh. vaihde (09) 4561, faksi (09) 456 4374 Statens tekniska forskningscentral (VTT), Bergsmansvagen 5, PB 2000,02044 VTT tel. vaxel(O9) 4561, fax (09) 456 4374 Technical Research Centre of Finland (VTT), Vuorimiehentie 5, P.O.Box 2000, FIN42044 VTT, Finland phone internat. + 358 9 4561, fax + 358 9 456 4374
VIT Valmistustekniikka, Ydinvoimalaitosten materiaalitekniikka, Kemistintie 3, PL 1704,02044 VTT puh. vaihde (09) 4561, faksi (09) 456 7002 VTT Tillverkningsteknik, Material och strukturell integritet, Kemistvagen 3, PB 1’704,02044 VTT tel. vaxel(O9) 4561, fax (09) 456 7002 VTT Manufacturing Technology, Maaterials and Structural Integrity, Kemistintie 3, P.O.Box 1704, FIN42044 VTT, Finland phone internat. + 358 9 4561, fax + 358 9 456 7002
Technical editing Kerttu Tirronen
VTT OFFSETPAINO. ESP00 1997 Santaoja, Kari. Thermornechanics of solid materials with application to the Gurson-Tvergaard material model. Espoo 1997. Technical Research Centre of Finland, UTPublications 312. 162 p. + app. 14 p. UDC 536.7 Keywords thermomechanical analysis, thermodynamics, plasticity, porous medium, Gurson- Tvergaard model
ABSTRACT
The elastic-plastic material model for porous material proposed by Gurson and Tvergaard is evaluated. First a general description is given of constitutive equations for solid materials by thermomechanics with internal variables. The role and definition of internal variables are briefly discussed and the following definition is given: The independent variables present (possibly hidden) in the basic laws for thermomechanics are called controllable variables, The other independent variables are called internal variables. An internal variable is shown always to be a state variable. This work shows that if the specific dissipation function is a homogeneous function of degree one in the fluxes, a description for a time-independent process is obtained.
When damage to materials is evaluated, usually a scalar-valued or tensorial variable called damage is introduced in the set of internal variables, A problem arises when determining the relationship between physically observable weakening of the material and the value for damage. Here a more feasible approach is used. Instead of damage, the void volume fraction is inserted into the set of internal variables, This allows use of an analytical equation for description of the mechanical weakening of the material.
An extension to the material model proposed by Gurson and modified by Tvergaard is derived. The derivation is based on results obtained by thermomechanics and damage mechanics. The main difference between the original Gurson-Tvergaard material model and the extended one lies in the definition of the internal variable 'equivalent tensile flow stress in the matrix
3 material' denoted by oM. Using classical plasticity theory, Tvergaard elegantly derived an evolution equation foroM.This is not necessary in the present model, since damage mechanics gives an analytical equation between the stress tensor u and oM. Investigation of the Clausius-Duhem inequality shows that in compression, states occur which are not allowed.
4 PREFACE
The present publication was prepared for the RAKE project (Structural Analyses for Nuclear Power Plant Components). The main objective of the RAKE project is to create, evaluate and apply effective and reliable structural analysis methods for the safety and availability assessment of nuclear power plant applications. In particular, they are applied on pressure vessels and piping. There are three target research areas: to develop fracture assessment tools, to assess component behaviour under realistic loading cases, and to verify the methods using large scale experiments. This work forms part of the first item on the above list.
The RAKE project belongs to the Finnish research programme on the 'Structural Integrity of Nuclear Power Plants' (RATU2). The RATUZ programme is set for the period 1995- 1998.
The author would like to thank professor Martti Mikkola - Helsinki University of
Technology (Espoo, Finland) - for his review and valuable comments and the extensive work he put into this report. The comments given by professor V. N. Kukudianov - the Institute for Problems in Mechanics, Russian Academy of
Sciences, Moscow - are appreciated. I am grateful to professor Nguyen Quoc Son - Ecole Polytechnique (91 128 Palaiseau Cedex, France) - for our fruitful discussions during his visit to Helsinki (Finland), on September 30 - October 4. 1996.
I am grateful to Mrs Hilkka Hanninen and to Mr Tuomo Hokkanen for their drawings.
The RATU2 research programme has been funded mainly by the Ministry of Trade and Industry (KTM), the Finnish Centre for Radiation and Nuclear Safety (STUK), lmatran Voima (IVO), Teollisuuden Voirna Oy (TVO) and the Technical Research Centre of Finland (VTT). This task was funded by KTM and VTT. Their financial support is greatly appreciated.
6 CONTENTS
ABSTRACT ...... 3 PREFACE ...... 5 LIST OF SYMBOLS 10
1 INTRODUCTION 13
2 THERMOMECHANICAL PRELIMINARIES . . 17
3 CONTINUUM MECHANICS ..... 27 3.1 Law of conservation of mass ...... 27 3.2 Law of balance of momentum ...... 27 3.3 Law of balance of moment of momentum ... 29
4 THERMODYNAMICS .. ... 33 4.1 General remarks ...... 33 4.2 Thermostatics ...... 37 4.3 Thermodynamics ...... 40
5 THERMOMECHANICS ...... 43 5.1 Major dialects of thermodynamics ...... 41 5.2 Thermodynamics with internal variables ...... 46 5.3 Variables describing the modelled process ...... 51 5.4 Law of caloric equation of state and the axiom of local accompanying state ...... 55 5.5 First law of thermodynamics ...... 58 5.5.1 Heat equation ...... 63 5.6 Second law of thermodynamics ...... 65
7 5.7 Clausius-Duhem inequality ...... 68 5.8 Principle of maximal rate of entropy production ...... 72 5.8.1 Normality rule in a general case ...... 72 5.8.2 Normality rule for the specific
complementary dissipation function (pc ...... 79 5.8.3 Normality rule for thermoplastic material behaviour ...... 82
6 FOURIER'S LAW OF HEAT CONDUCTION ...... 91
7 GENERAL FORMULATION FOR THE THEORY OF THERMOPLASTICITY ...... 93 7.1 Kelvin-Voigt type of material models ...... 96 7.2 Maxwell type of material models ...... 98 7.3 State equations and normality rule for Maxwell type of material models ...... 101 7.4 Maxwell type of material models with elastic strain tensor ...... 104
8 MATERIAL MODELS FOR ISOTROPIC AND KINEMATIC HARDENING 109 8.1 Rates of the internal forces 'p' and 'p2 ...... 109 8.2 Consistency condition and the multiplier ...... 111 8.3 Clausius-Duhem inequality ...... 113 8.4 Particular material models ...... 113 8.4.1 Duhamel-Neumann form of Hooke's law ...... 113 8.4.2 Special models for plastic flow ...... 115
9 GURSON-TVERGAARD MATERIAL MODEL ...... 121
10 EXTENSION FOR THE GURSON-TVERGAARD MATERIAL MODEL . 133 10.1 Specific complementary Helmholtz free energy itc ...... 134 10.2 Effective stress tensor 0 ...... 139
8 10.3 Yield function F and evolution equations ... . . , 141 10.4 Clausius-Duhem inequality ...... 144 10.5 Plasticity multiplier ...... 146
11 DISCUSSION AND CONCLUSIONS ... 141
REFERENCES ...... 157
APPENDICES
A Double-dot product of a skew-symmetric third--order tensor and a symmetric second-order tensor B Legendre transformation C Legendre partial transformation D Divergence of the dot product of a second-order tensor and a vector E Stress power per unit volume F Partial derivatives of the second deviatoric invariant JJa- p’)
9 LIST OF SYMBOLS
surface area of a subsystem in the initial configuration deviatoric part of the internal force B deviatoric part of the plastic strain tensor 8 yield function void volume fraction scalar-valued internal force function showing the inhomogeneity of the material fourth-order identity tensor For every second-order tensor A holds, I A = A : I = A kinetic energy function showing the inhomogeneity of the material outward unit normal power input heat input rate material parameters in the Gurson-Tvergaard material model heat flux vector heat source per unit mass entropy fourth-order compliance tensor for Hookean material compliance tensor for Hookean material with spherical voids
S deviatoric stress tensor
S specific entropy specific entropy rate specific entropy production rate specific entropy production rate (thermal part) specific entropy production rate (mechanical part) absolute temperature time surface traction vector
10 U internal energy
U specific internal energy - U displacement of a material point volume of a subsystem in the initial configuration representative volume - V velocity of a material point 1 second-order identity tensor -- For every vector v holds 1 . v = v . 1 = internal state variable (a second-order tensor) tensorial internal state variable (for kinematic hardening) scalar-valued internal state variable (for isotropic hardening) internal force (a second-order tensor) tensorial internal force (for kinematic hardening) scalar-valued internal force (for isotropic hardening) fouflh-order damage effect tensor
E strain tensor
Ee elastic strain tensor
EP" void-plastic strain tensor
EP plastic strain tensor
EV viscous strain tensor h Lagrange multiplier i plasticity multiplier P density in the initial configuration stress tensor dissipative stress tensor equivalent tensile flow stress in the matrix material quasiconservative stress tensor microscopic equivalent tensile stress effective stress tensor
cp specific dissipation function (potential)
(Pc specific complementary dissipation function (potential)
11 'PCO" specific dissipation function for plastic yield (thermal)
'PlocP specific dissipation function for plastic yield (mechanical) w specific Helmholz free energy of the form y~(E, ...) i$ specific Helmholz free energy of the form i$(~- E~,...) = ...) -VT/T thermal force
Special notations
first invariant of the second-order tensor ( ) second invariant of the second-order deviatoric tensor ( ) material derivative operator vector operator del material derivative operator material derivative operator vector second- and fourth-order tensors are denoted by boldface letters Kronecker delta = 1 if i = j and = 0 otherwise (x) = x if x 2 0 and (x) = 0 if x < 0
(J : & = 6..E.. 11 11 s=K:a e s..T.r.=K.. 6 IJ I J ijkl kl 7.5I j boundary of the volume V av i a&= av i aEI, i,I, integral integral over a closed domain
12 1 INTRODUCTION
The aim of the present work is the investigation of the Gurson-Tvergaard material model presented by Gurson (1977) and Tvergaard (1981). This widely used constitutive equation describes the mechanical response of porous materials under stressistrain loading. Temperature effects are not included in the material model. Gurson derived his yield function by describing the mechanical response of the matrix material with a rigid-perfectly plastic constitutive equation. Tvergaard made a minor change to the yield function proposed by Gurson in order to describe the hardening of the matrix material. He then added linear elastic strain to the total deformation of the porous material. This means that the present model is prepared for conditions where creep effects are negligible. Void nucleation and growth mechanisms with elastic-plastic material behaviour are important phenomena during ductile fracture of materials.
The detailed target of the evaluation of the Gurson-Tvergaard material model is the investigation of the foundation of this constitutive model and the formulation of the Gurson-Tvergaard material model within the thermodynamical (thermomechanical) framework. The latter will be done to verify whether the model satisfies the Clausius-Duhem inequality and for further enhancement (not in this study) of the model to a form which does not show localisation. In order to do this the general theory of thermomechanics is first presented.
This publication therefore studies the thermomechanics of solid materials and derives the necessary results for evaluation of the Gurson-Tvergaard material model. Particular attention is paid to the theory of thermoplasticity, as this extends the theory of plasticity by taking into account thermal effects. Thermomechanics is a science that combines thermodynamics and mechanics. This work looks at continuum mechanics, or more exactly the mechanics of deformable bodies. The general formulation of thermoplasticity with applications to some constitutive equations is studied from a thermomechanical point of view.
13 The material models under consideration are the material models for isotropic and kinematic hardening and the Gurson-Tvergaard material model. This study is therefore restricted to the thermomechanical response of the material, i.e. other mechanisms such as electrical, chemical, etc. are assumed to be negligible.
Derivation of the normality rule is the key problem in the formulation of thermoplasticity within the framework of thermomechanics. The solution is obtained by first deriving a general thermomechanical theory for solid materials and then by studying thermoplasticity as a special case.
In the theory of plasticity the normality rule has traditionally been written as [see e.g. Kachanov (1974, p. 84 and Chapt. 13)]:
aa, deP = dh -a. where @(a) is the plastic potential. Expression (1) can be cast into the rate form, viz.
where the 'circle' above h indicates that is a rate but not a material derivative of h.
Plasticity theory also introduces the yield function F. The case in which the yield function F is itself a plastic potential Q, is called an associated flow rule [see e.g. Lubliner (1990, p. 119)]. In the French literature materials obeying an associated flow rule are usually called standard materials. Thus, based on Normality Rule (2) the associated normality rule takes the following form:
14 The above discussion is for the classical ttr8oty of plasticity. Thermomechanics gives a normality rule having the form {see e.g. Santaoja [ 1994, Eq. (1 87)]}:
where Ev is the viscous (viscoelastic + viscoplastic) strain rate tensor and (pc is the specific complementary dissipation function. which is a function of forces in similar manner to the yield function F.
Although Expressions (3) and (4) have a similar form, casting plasticity theory and thermomechanics together is not a simple task. The key problem is that the standard formulation of thermomechanics is for description of time-dependent processes whereas plastic yield is a time-independent deformation. The latter is of course a model of a natural event. Santaoja (1994), among others has formulated a viscous material model using a thermomechanical approach. The difference between viscous and plastic deformation is remarkable from the normality rule point of view. At a certain state (e.g. when stress has a constant value) viscous strain takes place. This is described by Normality Rule (4). However, in general at constant stress no plastic yield occurs (except for ideally plastic material behaviour). Changes in the value of stress can cause plastic flow. This phenomenon is described by Normality Rule (3).
Several writers have investigated the formulation of the theory of plasticity in terms of thermomechanics, see e.g. Kondaurov and Kukudjanov (1979), Lehmann (1981), Simo (1988), Lemaitre and Chaboche (1990, Chap. S), Lubliner (1990) and Maugin (1992). However, the author is not fully satisfied with the expressed formulations and therefore this work takes a slightly different approach. The derived theory will be verified in Chapter 8 by introducing certain traditional constitutive equations for isotropic and kinematic hardening.
15 The second purpose of this publication is to investigate the Gurson-Tvergaard material model. Chapter 9 describes this constitutive equation in more detail. Chapter 10 introduces the extended Gurson-Tvergaard material model proposed by the author. This new constitutive equation is based on an investigation which utilises thermomechanics and damage mechanics.
16 2 THERMOMECHANICAL PRELIMINARIES
This chapter is a review of the basic laws, axioms and definitions for thermomechanics. Some preliminary definitions and equations which appear in subsequent chapters are also covered. Here deformations and rotations are assumed to be small, and the (sub)systems studied are closed. The material can be inhomogeneous. Material behaviour related to an elastic response does not evolve with time, whereas during plastic flow strain hardening occurs which leads to varying plastic material behaviour.
In the present work the term ’basic law’ is used for concepts that are generally accepted and can be experimentally verified to hold for a wide range of natural events. It should be kept in mind that the following basic laws are not ’facts’ but form a picture of nature which has been observed to be a good approximation of ’reality’. The law of conservation of mass, for example, can be replaced by the expression of mass and energy E = mc2 derived by Einstein. However, the processes which will be modelled here obey the law of conservation of mass and therefore this basic law is acceptable in the present context. ’Axiom’, on the other hand, refers to an idea which is not widely acceptediverified to hold but which with the basic laws forms the foundation of a theory.
The list of the basic laws, axioms and definitions for thermomechanics is as follows. The notations A, D and BL refer to an axiom, a definition, and a basic law, respectively.
Continuum mechanics (mechanics of deformable bodies) is based on the following basic laws:
BL. 1 The law of conservation of mass, BL.2 The law of balance of momentum, BL.3 The law of balance of moment of momentum
17 Basic Laws BL.2 and BL.3 are based on Newton's second law of motion which can be expressed as:
-D- DT = -(mv) = -(mu) F Dt Dt
where the notation D/Dt and the dot ' stand for the material (time) derivative. In - Equation (5) the quantities F, m, 7 and u are the force vector, the mass, the velocity of a material point and the displacement of a material point, respectively. Instead of the above vectors continuum mechanics uses the following quantities; the stress tensor ci, the strain tensor E and the rotation tensor o.In case of small deformations and rotations the latter two second-order tensors are defined by {see e.g. Malvern [1969, Eq. (4.2.9), (4.7.1) and p. 2391):
D. 1 and I -.. - - D.2 0 := -(uV -Vu) (7) 2
where the vector operators del, denoted by and 0, are defined by:
and
where the hat (-) on V and &ax, indicate that the vector operator acts on the preceding quantity.
Figure I(a) shows a fixed coordinate system (x,, x2, x3), the basis of which is (T,, -- I~,I~), and a body Vb or a system in thermodynamic language. An arbitrary part of the system is called a subsystem and is denoted by V. The volumes and boundaries of the above media are denoted by Vb, dVb, V dV as Figure l(a) shows. According to Figure I(b) the outward unit normal to the volume V is denoted by and the surface tractions are denoted by7 Surface traction vector
18 - t represents the force F in Equation (5). The stress tensor IY is defined by: - - D.3 t := n.a (9)
System (body)
Subsystem
av
Figure 1. (a) Body Vb, called a system, and an arbitrary part of the body denoted by V and called a subsystem. (b) Outward unit normal to the volume V denoted by n and surface tractionsifor the definition of the stress tensor cr.
Thermostatics is based on the following three basic laws:
BL.4 The law of caloric equation of state, BL.5 The first law of thermodynamics, BL.6 The second law of thermodynamics.
Thermodynamics, which is an extension to thermostatics, requires more fundamental information. The following concept may not be the final one but today seems to be the most popular. The two axioms for thermodynamics beyond thermostatics read:
A. 1 The axiom of local accompanying state,
19 A.2 The principle of maximal rate of entropy production.
There are several variations of the entropy production theme, but here the author wishes to refer to the work by Ziegler (1963, p. 14).
The difference between thermostatics and thermodynamics is discussed in more detail in Chapter 4.
The basic laws, axioms and definitions expressed above are not adequate to describe the thermomechanical aspects of nature. Several primitive undefined concepts, such as length, mass, time, temperature, entropy and state must first be introduced. Of these, length, mass, time, temperature and entropy are primitive quantities, and state is a primitive concept. Other quantities such as stress and strain are derived from primitive quantities.
However something is still missing. The following discussion is by Astarita (1989, pp. 13-14), with comments by the author between the braces {}.
Every branch of physical science is based on two sets of fundamental equations. The first set is that of basic laws {and axioms as well as many definitions} of physics, which are postulated to hold valid for all bodies under all conceivable circumstances; the principles {laws} of conservation of mass, of linear momentum, and of angular momentum {i.e. moment of momentum} are typical examples. In thermodynamics, the basic laws are the first and second laws. {The present work also expresses the law of caloric equation of state as a basic law of thermodynamics.} The large majority of basic laws of physics are principles of conservation of some quantity (mass, linear momentum etc.); the first law of thermodynamics falls into this category, but the second law is an exception, since it is not a principle of conservation.
The second set of fundamental equations are the constitutive equations: these
20 are relationships which are not supposed to hold for all bodies, but only to describe the behaviour of some restricted class of bodies, or possibly of a larger class of bodies for a more restricted class of phenomena. A good example is that of the mechanics of rigid bodies; it is of course obvious that there are many bodies in nature which are not rigid (and perhaps one could argue that there are in actual fact no bodies which are truly rigid); however, the theory of rigid bodies is a useful abstraction which describes satisfactorily some phenomena as observed in nature. Constitutive equations are assumptions which may, or may not, adequately describe the behaviour of real bodies.
To the above comments by Astarita the author adds the following: The form of the constitutive equation is dependent on (a) the material, (b) the loadingienvironment and (c) the required accuracy of the model.
(a) The material and (b) the loading/environment aspects have e.g. the following effects on material models: The response of structural steel, for example, can be simulated by Hooke’s law, if the stress is below the yield strength of the material and temperature is close to or below the room temperature. For higher stress, however, the plastic yield must be described and at elevated temperatures creep may be the dominant deformation mechanism. On the other hand, at a stress level where plastic flow is remarkable in structural steels, ceramics obey Hooke’s law.
(c) The required accuracy of the constitutive equation is dependent on the application of the model. When simulating hysteresis loops caused by cyclic loading, the material model needs to be far more accurate than when evaluating the elastic-plastic bending of a beam.
According to Coburn (1 955, p. 74) the generalised Gauss’s theorem is: If g is a continuous field (scalar, vector etc.) and if g possesses continuous partial derivatives in the closed subsystem V, then:
21 v*g dV = f h*gdA jV av where the 'star product' * represents either the dot product . or the cross product x.
Malvern (1969, p. 21 I) gives the Reynolds transport theorem, i.e. the material derivative of a volume integral:
ospg dV= spsdV Dt Dt V V Malvern [1969, Eq.- (2.3.24)j gives the following expression for the cross product of vectors a and b: - - - a x b = em,, a, b, i where the permutation symbol emnris defined as having the value 0, +1, or -1 as follows {see Malvern [1969, Eq. (2.3.22)]}:
0 when any two indices are equal
+I when m, n, r are 1,2,3 or an even < (13) emnr := permutation of 1,2, 3
-1 when m,n,r are an odd permutation of 1,2,3
The partial derivative of the axis xi with respect to the axis xi reads:
axi _-- ljii a xi where the notation ljii stands for the Kronecker delta and is defined to be zero when i and j differ and to be unity when i = j, viz.
22 0 when i f j 6.. : = 'I 1 when i = j
Utilising Property (15) the Kronecker delta 6ilcan be shown to operate as follows:
a. = ai, (16) 6.I1 I" where the quantity a can be a tensor of any order.
In this work the partial derivatives of certain scalar-valued functions with respect to vectors or second-order tensors are considered. For example, the partial derivative of the specific Helmholtz free energy w with respect to the elastic strain tensor (a second-order tensor) is a second-order tensor components of which are the partial derivatives of the specific Helmholtz free energy y~ with respect to the elastic strain tensor components Mathematically this can be written in the form:
Correspondingly it is obtained:
The Heaviside function (unit-step function) is defined by:
0 when x < 0 H(x) := 1 when x 2 0 The component l,, of the second-orderI identity tensor 1 is defined by:
23 1..II := 6..II (20)
The component lijkl5 of the fourth-order symmetric identity tensor Is is defined by:
The second invariant of the deviatoric stress tensor Jz(a) is defined by {see e.g. Lin (1968, p. 15)):
1 Jz(a) := -S:S 2 where s is the deviatoric stress tensor.
The deviatoric stress tensor s and the deviatoric internal force b1are defined by:
s := K:a and b' := K:P' where the fourth-order tensor K is defined by:
K := I--11I 3
In Definition (24) the second-order identity tensor 1 is defined by Definition (16) and the component lijk, of the foufih-order identity tensor I is defined by:
:= 6. 6. 1..ilk1 ik JI
Let A be an arbitrary second-order tensor. The following equalities hold:
aA _--I I:I = I and aA
The inverse of the second-order tensor A is a second-order tensor denoted by A-I. It is defined by:
The first invariant of the stress tensor Il(a) is defined by {see e.g. Lin [1968, Eq.
24 ( 1.4.2)]}:
Il(U) := 1:o = Oii = 011io2* +cJ33 (28)
The McAuley brackets have the following property:
0 when x < 0 (x) := x when x 2 0
25 NEXT PAQC(S) I kftBLANK I 3 CONTINUUM MECHANICS
The purpose of the present chapter is to introduce the basic laws of continuum mechanics and derive local forms and consequences from these basic laws.
3.1 LAW OF CONSERVATION OF MASS
The density of material is assumed to be a piecewise continuous quantity. The mass of an arbitrary subsystem V sketched in Figure 1 can be written:
m := J’ p dV V
Since the Lagrangian description is used, the integration in Equation (30) is carried out in the reference configuration. The law of conservation of mass reads:
D D1 --Dm -DIpdV=-SpldV=jp-dV-O Dt Dt Dt v V where Reynolds transport theorem, see Theorem (1 l), is exploited. Thus the law of conservation of mass is satisfied. When Lagrangian description is used the variables present in the integrands of the basic laws of thermomechanics are dependent on their values in the current configuration but they are referred to the reference configuration. Based on the assumption that deformations and rotations are small, the difference between the values of the quantities in initia! and current configuration is negligible.
3.2 LAW OF BALANCE OF MOMENTUM
Figure 2 shows the differential forces TdA and p b dV acting on an arbitrary subsystem V. The notation dA refers to the differential area, dV is the differential
27 volume element, and the external body force vector is denoted by 6.
Figure 2. Forces acting on an arbitrary subsystem V.
Based on Newton’s second law of motion, Expression (9,the law of balance of momentum reads {see e.g. Malvern [1969, Eq. (5.3.1)]}: - - fTdA +JpbdV = EJpvdVDt av V V
Applying the definition for the stress tensor u, Definition (9), and the generalised Gauss’s theorem, Theorem (lo), the first term of Basic Law (32) can be manipulated as follows:
Based on Reynolds transport theorem, Theorem (I I), it is arrived at:
-jpidVD = jp-!!idV Dt Dt (34) V V
Substituting Expressions (33) and (34) into Basic Law (32) gives the following form:
where the notation 6stands for a vector the components of which vanish.
28 Because the volume of subsystem V is an arbitrary part of the system, Integral (35) must vanish for every V within the system Vb. Thus, the term in parentheses must vanish at every point of the material. This gives:
Equation (36) is Cauchy’s equation of motion.
3.3 LAW OF BALANCE OF MOMENT OF MOMENTUM
According to Malvern [1969, Eq. (5.3.6a)l the law of balance of moment of momentum takes the following appearance: - f(?xT)dA+J(Fxpb)dV = (?xpv) dV Dt (37) r)V V “IV where 7 is a vector from a an arbitrary point Q to the material point P. This - means that in Basic Law (37) the moment of forcestdA and p b dV about point Q is calculated.
Vector 7 can be expressed as the sum of two vectors, namely, vectors and x, where is the vector from point Q to the origin of the coordinate system (x,. x,, 5 -- x3) and x is the position vector of material point P (see Figure 2). This is r = c + x which provides the following form for Basic Law (37): x [ f i dA + p b dV - !. jp v dV av V Dt V -1 + f (YxT) dA + ~(XXP~,dV = oJ’(xxpv) dV Dt av V v
Expression (38) exploits the fact that vector is constant, which means it can be moved outside the integral sign.
29 Due to the law of balance of momentum, Basic Law (32), the first row of Expression (38), vanishes. Thus, using the expression for the cross product x, Expression (1 2), Basic Law (37) can be written as: - f ( eiik Xi tk) dA ii + J’ (eijk X, p bk) dV Ti JV V (39)
By utilising the definition for the stress tensor u, Definition (9), in the dyadic form tk = ns ask, and the generalised Gauss’s theorem, Theorem (lo), the first integral of Expression (39) can be changed to a volume integral to give the following:
Applying Results (14) and ( 16) gives:
-3x1 (3,k = 6js ‘Jsk = (3jk 2% and Equation (40) reduces to:
Reynolds transport theorem, Theorem ( 1 l), provides the following manipulation for the right-hand side of Expression (39):
30 where also the fact Dx, / Dt = v, is exploited.
Appendix A Equation (A 1) shows that the double-dot product of a third-order tensor c which is skew-symmetric in the last two indices (Le. c,,~= - clkl) and a symmetric second-order tensor h (Le. h,, = h,,) vanishes. Since the permutation symbol elJk is skew-symmetric in the indices jk [see Definition (13)] and the tensor v, vk is symmetric, the following holds:
and Term (43) collapses to:
Substitution of Expressions (42) and (45) into Expression (39) yields:
According to Cauchy’s equation of motion. Equation (36b). the term in parentheses in the second integral of Expression (46) vanishes and the following is arrived at: - - / eiik ojk ii dV = 0 (47) V where the order of the terms has been changed.
Since subsystem V is an arbitrary part of the system Vb, the integrand in - Expression (47) has to vanish everywhere within the system Vb. The notation 0
31 stands for a vector the components of which vanish. Thus Expression (47) gives:
i = 1,2,3
Definition of the permutation symbol eijk, Definition (13), allows Condition (48) to be written as:
i=l
' i=2 (49)
i=3 which can also be written as:
T Oil = (5- u= u I' or
Result (50) means that the stress tensor u is symmetric.
32 4 THERMODYNAMICS
4.1 GENERAL REMARKS
This section gives some remarks on and definitions of thermostatics and thermodynamics and outlines the difference between thermostatics and thermodynamics.
A thermodynamic system, or system for short, is any collection of matter imagined isolated from the rest with the aid of a clearly defined boundary. It is not necessarily assumed that the boundary is rigid. on the contrary. in most cases the boundary will deform when the system is subjected to some process which we have to understand and analyze. (Kestin 1978. I, pp. 22 and 23.)
In the present work the term system refers to the body Vb. The surface of the system is denoted by dVb. An arbitrary part of the body is called subsystem and denoted by V the boundary of which is dV. Since subsystem V is an arbitrary part of the system Vb, it can cover the whole system Vb. Therefore the subsequent definitions, which are expressed for the subsystems. hold also for the system Vb.
A subsystem V is called closed when its boundary is not crossed by matter. In the contrary case it is called open. The present study investigates closed subsystems. The present work studies thermomechanical systems (Le. thermomechanical response of materials) which means that the influence of e.g. electrical and chemical effects are neglected. A thermomechanically closed subsystem V exchanges energy with its surrounding by exchange of heat and by work done by the volume forces and by the surface forces acting on the boundary dV.
33 Consider a given system. When all the information required for a complete characterization of the system for a purpose at hand is available, it will be said that the sate of the system is known. For example, for a certain homogeneous elastic body at rest, a complete description of its thermodynamic state 8requires a specification of its material content, Le., the quantity of each chemical substance contained; its geometry in the natural or unstrained state &; its deviation from the natural state or strain field; its stress field; and, if some physical properties depend on whether the body is hot or cold, one extra independent quantity which fixes the degree of hotness or coldness. These quantities are called state variables. If a certain state variable can be expressed as a single-valued function of a set of other state variables, then the functional relationship is said to be an equation of state, and the variable so described is called a state function. The selection of a particular set of independent state variables is important in each problem, but the choice is to a certain extent arbitrary (Fung 1965, p. 341).
If the state 8of a material point P within a subsystem V is independent of the location of the point P i.e. the state 8has the same value everywhere within the subsystem V and if the material of the subsystem has the same properties everywhere, the subsystem V is called a homogeneous subsystem. A subsystem V is said to be in thermodynamic equilibrium if this subsystem V is isolated and it does not evolve with time (Maugin & Muschik, 1994, p. 219). A thermodynamic subsystem V in which there is no energy exchange with its surrounding is said to be isolated. If the state variables vary with time the system is said to undergo a thermodynamic process or briefly, a process (Fung, 1965, p. 341). Maugin (1992, p. 263) gives the following definition for reversible and irreversible process. A thermodynamic process is said to be reversible if the inverse evolution of the system in time - i.e. the succession of thermodynamic states 8that the system has gone through - implies the reversal in time of the action of external stimuli. Otherwise the thermodynamic process is said to be irreversible.
34 Thermostatics is the science that compares subsystems in thermodynamic equilibrium. For example, it describes the transition from a state 5 of equilibrium to another state iT2 of equilibrium. Thermodynamics, in its main sense, is the study of phenomena outside a state of equilibrium, but actually not far outside this equilibrium. Everybody of course agrees that in the years 1890 to 1920 thermostatics was developed in an elegant mathematical form by Clausius, Gibbs, Duhem and Caratheodory, in harmony with the experiments. Unfortunately, we cannot say the same about thermodynamics outside equilibrium; schools strongly disagree with each other on this subject. (Maugin, 1992, pp. 263 and 264.)
The above discussion gives rise to the question about the form of transition from a state g,of equilibrium to another state of equilibrium. According to Lavenda (1993, p. 4) the transition is a quasi-static process which is a process that is carried out infinitely slowly so that the system can be considered as passing through a continuous series of equilibrium states. The processes described by Lavenda are not (yet) finished, since they are infinitely slow and since the age of the universe is finite. This may cast suspicion of the elegance of the formulation of thermostatics (cf. above discussion by Maugin).
This work presents thermostatics as a theory for homogeneous subsystems, whereas the field theory lies within the framework of thermodynamics. Ziegler (1983. p. 56) argues that the lack of a thermodynamic field theory even at the end of the first half of the present century seems to be more surprising as ... . Maugin (1992, p. 263) states that already in 1920 thermostatics was developed in an elegant mathematical form. The comments by Ziegler and by Maugin support the above concept that thermostatics is not a field theory. However, it has to be kept in mind that the assumption that thermostatics is a field theory would lead to the same description for thermodynamics as is presented here. Only the path of the derivation would be different. Unfortunately thermodynamics is usually the term which covers thermostatics and thermodynamics. The basic laws of thermodynamics are good examples of this terminology; the reader must know whether the discussion is about thermostatics or thermodynamics. The problem is real, since - as discussed above by Maugin - thermostatics is a well-defined theory, whereas there are still some open questions in the theory of thermodynamics which has led to the introduction of several different schools of thermodynamics.
The introduction of entropy and internal energy are the main notions of thermodynamics beyond mechanics. The following discussion by Narasimhan
( 1993, pp. 345 and 346) clarifies the idea of entropy.
The concept of entropy occupies the core of thermodynamics. Before developing an analytical formulation of entropy, we present a motivation for the entropy concept. The first law of thermodynamics, namely, the law of conservation of energy, essentially states that the energy of a material system cannot be created or destroyed, but can only be transformed from one form into another. The first law does not, however, specify the manner in which this transformation may occur. For instance, there is no information furnished by the first law as to whether the energy transfer is reversible or irreversible. The latter question of reversibility of energy transfer becomes important in material systems in order to keep track of the amount of energy available for use. The information as regards the manner in which the transformation of energy occurs is furnished by the law of entropy, which is also referred to as the second law of thermodynamics.
Consider the example of gasification of coal, in which the latter is heated to produce an energy source as coke oven gas. But a part of this available energy in the coal is transformed into hydrogen sulphide, ammonia, and other gases which escape into external atmosphere. Although the total energy is not lost, the burned coal cannot be reheated to obtain a further amount of work. This means that the energy transfer occurs only in one direction, from an available into an
36 unavailable form. Entropy may to be interpreted as a measure of the loss in the amount of energy that is transformed irreversibly from a usable to an unusable form, in which it cannot be converted into work again. Similarly, a physical system with some initial order prevailing in its internal constituents tends to lose that order in an irreversible way upon heating, corresponding to the transformation of energy from available to unavailable form. A transformation from an ordered to a disordered state Bis interpreted as an increase in entropy. Hence, for physical systems, the addition of energy which is drawn by them from their environment contributes to an increase in entropy. Production of entropy in a physical system, therefore, implies that the system has undergone irreversible changes and conversely, irreversible changes in the system imply entropy production. A constant entropy implies only reversible changes in the system. The law of entropy is essentially the embodiment of the statement of increase of entropy in physical systems.
4.2 THE R MOSTATICS
This section sketches the foundation of thermostatics. This is done by investigation of the basic laws of thermostatics one by one.
BL.4 The law of caloric equation of state
Thermostatics is based on the assumption that the state 8of a subsystem V is described by a finite set of mutually independent variables uy y = 1, ... n (which are mechanical, electrical, chemical etc. depending on the modelled processes) and one thermal variable which in the present work is the entropy S. Some scientists use thermodynamic temperature T instead of entropy S in the description of the state 8(see e.g. Malvern, 1969, p. 233). State 8is expressed by the internal energy U through the caloric equation of state, viz.
37 u = U(UrS) where in a general case the quantities uY and S are dependent on time t.
Figure 3 sketches a simple example of some gas inside a cylinder. The volume and the pressure of the gas are denoted by V and p. The investigated system V in this example is defined as the volume bounded by the walls of the cylinder and the piston head.
For this particular example the internal energy U takes the form:
u = U(V,S)
Differentiation of Expression (52) gives:
Figure 3. Gas inside a cylinder.
BL.5 First law of thermodynamics
According to Maugin (1992, pp. 266 and 267) the differential form for the first law of thermodynamics for a mechanical, (i.e. electrical, chemical etc. processes are neglected) closed and homogeneous system reads:
38 dU = Q+O (54)
where R is the elementary work received and @ is the elementary heat received. Since Basic Law (54) is formulated for thermostatics describing quasi-static processes no kinetic energy term is present [cf. Basic Law (76) which is written for thermodynamics].
For the above example the elementary work received 62 takes the following appearance:
0 = pdV (55)
BL.6 The second law of thermodynamics
Maugin (1992, p. 267) gives the second law of thermodynamics for closed systems and reversible processes. It has the following appearance:
@ = TdS where T is the thermodynamic temperature or absolute temperature. This work refers to T as temperature.
Substitution of Expression (55) and Basic Law (56) into Basic Law (54) yields:
dU = pdV + TdS (57)
This is Gibbs equation, which can be written in the following rate form:
U = pi/ +TS
Comparison of Forms (53) and (57) for the differential internal energy dU gives the following results:
and T=aucv, s) as (59) which are the state equations.
39 4.3 THERMODYNAMICS
This section gives the extension for thermodynamics beyond thermostatics. Chapter 2 introduced two axioms, which for this extension are briefly studied.
A. 1 The axiom of local accompanying state.
By applying the axiom of local accompanying state, points of non-equilibrium state space are associated with points of equilibrium state space by means of a projection (Maugin & Muschik, 1994, p. 226). Figure 4 sketches this projection.
nonequilibrium
accompanying process state space
Figure 4. Projection P maps the (non-equilibrium) process Z(.) point by point onto the equilibrium subspace represented as a hypersurface in the state space. Consequently the accompanying reversible process PZ(.) is parametrised by 1. (Modified from Muschik, 1992, Figure 1.16)
According to Maugin and Muschik (1994, p. 226) the axiom of local accompanying state is the most commonly accepted viewpoint which consists in replacing the axiom of the local equilibrium state of classical irreversible thermodynamics with a somewhat straightforward generalisation known under the name of the axiom of the local accompanying state.
The key concept of the axiom of local accompanying state is that it allows the introduction of the state equations [c.f. State Equations (59) of thermostatics] also
40 outside the equilibrium. Thus, state equations are derived for an accompanying process in the equilibrium state space by using thermostatics, and the obtained state equations are assumed to hold for the real nonequilibrium process.
The second extension of thermodynamics beyond thermostatics is:
A.2 The principle of maximal entropy production
With the introduction of the specific dissipation function cp the principle of maximal entropy production provides the evolution equation for the mechanical state variables and for the heat flux vector i.
41 5 THERMOMECHANICS
The purpose of this chapter is to introduce the theory of thermomechanics and the formulation of this theory for preparation of the constitutive equation. A fundamental set of variables is introduced for many thermomechanical material models of solid materials. The obtained results are generally valid for almost any thermomechanical process. The number of internal variables can be decreased or increased yet the results of this chapter remain valid, with only some terms disappearing or new terms having to be added. The new terms have the same form as those already existing.
5.1 MAJOR DIALECTS OF THERMODYNAMICS
This section discusses different theories describing the processes outside the thermodynamic equilibrium.
The main problem of thermodynamics is the definition of temperature T and entropy S outside equilibrium. There are several variations of the theme to solve this problem and to formulate an elegant thermodynamical theory to model processes (far) outside the equilibrium. The following discussion by Lebon et al. (1992, pp. 41 and 42) and Lebon (1992, p. 7) sketches the concepts of the main theories. Comments in braces { 1 are those of the author.
Since the Second World War, two lines of thought have been developed in the field of non-equilibrium thermodynamics. The first is known as the classical thermodynamic theory of irreversible processes, in short classical irreversible thermodynamics (CIT), the second one is referred to by its founders Truesdell, Coleman and Noll (Truesdell 1969, 1988) as rational thermodynamics (RT) . The foundations of CIT were laid down by Lars Onsager (1931) in two celebrated papers published in the Physical Review but the theory owes much of its
43 success to the Brussels school directed by llya Prigogine (1947, 1961). It is worth recalling that both Onsager and Prigogine were awarded the Nobel Prize in Chemistry in 1968 and 1977 respectively. Rational and classical thermodynamics aim at the same objective: to derive constitutive equations of material systems driven out of equilibrium {thermodynamic equilibrium}.
CIT borrows several results from classical thermodynamics { thermostatics} and is concerned with the class of materials and processes described by the "local equilibrium hypothesis". The latter states that the local and instantaneous relations between the thermal and mechanical properties of a material system are the same as for a uniform system at equilibrium. CIT is mainly applicable to situations "not too far" from equilibrium. More explicitly stated, it is supposed that the constitutive equations are expressed by means of linear relations between cause (also named force) and effect (the flux, according to the CIT terminology). An important pillar of the theory is provided by the Onsager symmetry relations between transport coefficients of coupling processes. Onsager's reciprocal relations were derived from the statistical theory of fluctuations and the hypothesis of microscopic reversibility. They receive a microscopic confirmation from the theory of gases and the theory of linear response; they were also confirmed by several experimental observations mainly in thermoelectricity and thermodiffusion.
It must be realised that CIT is unable to describe materials with memory and is not adequate for studying processes taking place far from equilibrium, in particular high frequency and short wave-length phenomena. These restrictions are inherent to the local equilibrium hypothesis. According to CIT the disturbances will propagate at an infinite velocity, in contradiction with the principle of causality, which demands that the cause preceds the effects: in CIT, cause and effect happen simultaneously.
The objective of RT is more ambitious than that of CIT as it seeks to describe a
44 wider class of materials driven far from equilibrium. But this goal is achieved at the price of a greater mathematical complexity. In RT, absolute temperature and entropy are introduced as primitive concepts without a sound physical interpretation. The notion of state 8 expressing that any property evaluated at time t can be written in terms of the state parameters given at the same time t is given ~ipand replaced by the notion of history or memory. Accordingly, the properties of any system are not only affected by the values of the variables at the present time but the values of the variables at the present time may also depend on their values in the past. As a consequence, the constitutive equations take the form of time-functionals. These functionals cannot, however, take any arbitrary possible form: there are restrictions placed by the second law of thermodynamics such as the positiveness of the heat conductivity and the viscosity and the criterion of material frame-indifference, demanding that the constitutive equations are independent of the motion of any observer. The theory has met an impressive success among mathematicians and theoretical mechanicians because of its generality and its mathematical rigour.
Classical Rational irreversible irreversible thermodynamics thermodynamics thermodynamics with internal variables Figure 5. Major theories for description of processes outside the thermodynamic equilibrium. The third approach is called extended irreversible thermodynamics (EIT) and it provides a mesoscopic and causal description of non-equilibrium processes: it was born out of the double necessity to go beyond the hypothesis of local 45 equilibrium and to avoid the paradox of propagation of disturbances at infinite velocity. At the present time, the theory may be considered as fully developed and it is formulated in the book by Jou et al. (1993). These three major theories and the thermodynamics of internal variables are outlined in Figure 5. 5.2 THERMODYNAMICS WITH INTERNAL VARIABLES The aim of this section is to sketch the foundation for thermodynamics with internal variables. The present work utilises a variation of CIT which can be called thermodynamics with internal variables, Maugin and Muschik (1994, pp. 222 and 233) gives the following description for this theory. A fourth thermodynamics is the one that introduces internal variables as state variables. It stands somewhere in between the CIT and RT but in fact it is the simplest generalisation of CIT. The origin of the internal-variable thermodynamics may first be traced back in the kinetic description of physicochemical processes of evolution, but its spectacular development is related to rheological models and the elasto-viscoplasticity of deformable materials of the metallic type (alloys, polycrystals) [see e.g. Lemaitre & Chaboche (1990)l. As already mentioned, this approach is adopting a somewhat intermediary line between the two thermodynamics already sketched out. Essentially, it provides a new characterisation of continuous media which, in order to define the thermodynamic state 8 of a subsystem V, introduces, in addition to the usual observable state variables (x; e.g. temperature and elastic strain), a certain number of internal variables collectively denoted by a, which are supposed to describe the internal structure [hidden to the eye of the (untrained) external observer who can only see a black box - hence also the alternate name of hidden variables; but this naming will be avoided as it sometimes creates 46 confusion with variables so christened in certain re-interpretations of quantum mechanics]. It follows that the value, at moment t, of the dependent variables (e.g. the stress) becomes simultaneously a function both of the values of the independent observable variables and of the internal variables. This constitutive equation, say o(x, a)where x represents, as before, the controllable variables of state, must be complemented by an evolution equation which describes the temporal evolution of the variable a. For instance, we can write the following: and (61) Cw = f( x, a) + g(x, a)ir. evolution equation In fact, we may suppose that we have been able to select the a's in such a way that g(x, a) might be identically zero and that an instantaneous variation of K does not cause any instantaneous variation in the a's [if x is a strain, the hypothesis g(x, a) = 0 corresponds to the fact that instantaneous strains are elastic or zero]. The author's comments on the above discussion by Maugin and Muschik are the following: First, the author cannot agree with the above comment on hidden variables. When mentioning internal variables which can be hidden to the eye of the (untrained) external observer, the writers may have been thinking of e.g. dislocation movement (plasticity or creep) in metals. Since metallic materials are not transparent all internal deformation mechanisms are "hidden". On the other hand, damage associated with microcracking is often modelled by an internal variable called damage [see e.g. Lemaitre & Chaboche (1990, pp. 58 and 346)j. However, microcracks in ice are visible to everyone, as Figure 6 clarifies. This implies that the division of variables into observable variables and internal variables is not acceptable. Section 5.3 looks more detailed at the terminology of 47 independent variables in thermodynamics with internal variables. Figure 6. Visible microcracks (Le. internal variable) in ice. A fine-grained compression test specimen (a) before and (b), (c) after the test. (Currier, Schulson & St. Lawrence, 1983. Figs. 10, 31 and 32). Second, the above style for introduction of Equations (60) and (61) may lead to 48 confusion about the idea of modelling materials. When the theory of internal variables is used for preparation of constitutive equations one does not assume any equations to be similar to Equations (60) and (61) but introduces explicit forms for two functions, namely the specific Helmholtz free energy and the specific dissipation function. Together with results similar to Expressions (59a) and (59bj the explicit form for the specific Helmholtz free energy gives the explicit relation for the stress (3. The introduction of the explicit form for the specific dissipation function applied to the normality rule, which is a consequence of the principle of maximal rate of entropy production, yields evolution equations for internal variables, such as Equation (61). As already mentioned, thermodynamics with internal variables is the simplest generalisation of the classical thermodynamic theory of irreversible processes CIT. According to Maugin and Muschik (1994, p. 226) it is the most commonly accepted viewpoint which consists in replacing the axiom of the local equilibrium state of classical thermodynamic theory of irreversible processes with a somewhat straightfotward generalisation known under the name of axiom of the local accompanying state (for short L.A.S.). The author stresses that the above axioms are concepts for introduction of the formalism and results of thermostatics also for processes outside the thermodynamic equilibrium. This viewpoint is followed also here. The introduction of internal variables has several advantages of great practical importance. First, the history dependence of deformation is described by the internal variables, and the obtained constitutive equation is a set of differential equations, as given by Expression (5.2). Rational thermodynamics, however, uses for description of the history dependent deformation models having the functional form, viz. 49 t E” = ... dt 0 The classical theory of viscoelasticity [see e.g. Flugge (1975)Jis a good example of functional constitutive equations. For a varying state of stress this theory gives formulations which are difficult to solve, since they use computing time excessively and require a considerable amount of computer memory, as noticed by Santaoja (1987) and Santaoja (1990, Sects. 4.4 ... 4.9 and Apps. 3 and 4). Strictly speaking, in general performing a structural analysis with a history dependent material model by using a finite difference or a finite element method is not possible if the constitutive equation has a functional appearance. Second, the classical thermodynamic theory of irreversible processes CIT cannot describe processes with memory. Plastic yield is a path-dependent process and therefore displays memory effects. Internal variables are introduced for modelling of history dependence. This is the major power of thermodynamics with internal variables over CIT. Finally, the foundation of the classical thermodynamic theory of irreversible processes is the assumption that the (generalised) forces and (generalised) fluxes have a linear mutual relationship. The celebrated work of Onsager (1931a) and (193 1b) showed that the coefficients of the above-mentioned relation satisfy the Onsager reciprocal relations. This assumption has not been adopted into thermodynamics with internal variables but is replaced by introduction of the dissipation potential and the principle of maximal rate of entropy production. This approach allows the formulation of several constitutive equations which do not fit CIT. Little criticism has been raised against the thermodynamics of internal variables. Lebon et al. (1992, p. 50) pointed out the following: In the thermodynamics of internal variables the internal variables do not appear in the balance equations 50 of momentum and energy. Moreover, since the selection of internal variables is not regulated by strict rules, the same class of materials can be described by several formalisms. 5.3 VARIABLES DESCRIBING THE MODELLED PROCESSES The aim of this section is to introduce the variables for description of the modelled processes. At the risk of sounding repetitive, the advantages of using of thermodynamics with internal variables and the requirements for these internal variables are discussed briefly now. 1 An internal variable is a variable describing the internal response of a material. The value of an internal variable can be measured but its evolution cannot be directly controlled. In other words, an internal variable is measurable but not controllable. Although internal variables are sometimes called hidden variables they can be visible. as Figure 6 illustrated. The present work describes the following internal mechanisms: dislocation movement (plasticity), as well as void nucleation, and growth that causes damage. 2 By introduction of a set of internal variables the constitutive equations can be expressed in rate form, in which no time functionals appear. This is a major advantage over the traditional theory of viscoelasticity [see e.g. Flugge (197S)l or the rational thermodynamics of Truesdell, Coleman and Noll [Coleman ( 1964), Truesdell and Noll (1965) and Truesdell (1969, 1988)], because numerical analysis with time functionals may require a great amount 51 of computer capacity [see e.g. Santaoja (1987, Table 3) or Santaoja (1990, Table 4)l. 3 The author cannot agree with the comment by Maugin that an internal variable may or may not be a state variable [Muschik (1992a, p. 48)j. In his studies Muschik may have used an incorrect set of internal variables which led him to this erroneous conclusion. Section 7.4 and Chapter headed "Discussion and Conclusions" evaluates this problem. All the internal variables present in this work are of course also state variables. 4 Internal variables need a model or an (microscopic or molecular) interpretation [Muschik (1992a, p. 53)1. The key problem in the mathematical modelling of material behaviour is the choice of a set of independent variables. The "correct" set of independent variables is dependent on the deformation mechanisms modelled and the sought accuracy of the constitutive equation. Thus any generally valid set of independent variables cannot be given. The expertise of a physicist is displayed by his capability to select a successful set of independent variables. As assumed by Astarita (1989, Chapt. 2), the set of state variables may also contain rate terms [see also Malvern (1969, p. 257)].Furthermore, according to Maugin (1990), divergence of the state variables may be necessary to include in the set of state variables. The above item 4 requiring that an internal variable needs a microscopic or molecular interpretation is difficult to fulfil within the gradient theory which uses gradient of damage as a variable to prevent localisation. 52 The author suggests the terminology shown in Figure 7 for the independent variables in the thermodynamics with internal variables, The following pages will show how the specific entropy s is replaced by the absolute temperature T. State variables \ Thermal variables Mechanical variables s (and T) J Controllable IInternal variables variables E a Figure 7. Different types of independent variables in thermodynamics with internal variables, It is important to note that controllable variables are only controllable to some extent. Thus, e.g. the value for the strain tensor E is controllable only at the boundaries of the system. A uniaxial tensionicompression is investigated as an example. If the researcher had an extremely strong and stiff loading machine, heishe could control the strain of the specimen on the surfaces where the specimen is attached to the loading device. The same holds for the absolute temperature. The above means that the adjective 'controllable' cannot agree fully with the quantities called controllable variables. The phenomenon 'controllable' is not the key concept for separation of the variables into two categories. The following separation by Kukudianov ( 1996) gives an extremely clear definition for the two categories of variables. A variable present in the conservation laws is a controllable variable. The other variables are internal variables. In Chapter -7 Basic Law BL.l is the law of conservation of mass and Basic Law BL.5 is often called the law of conservation of energy. Furthermore, according to Lebon et al. (1992, p. SO) the internal variables (they call them hidden variables) do not appear in the balance equations of momentum and energy. Those expressions are denoted by BL.2, BL.3 and BL.5 in Chapter 2. 53 Investigation of the basic laws reveals that the strain tensor E does not explicitly appear in those basic laws. For example, Cauchy’s equation of motion, Expression (36). which is a consequence of Basic Law BL.2, has the velocity as an independent variable. The velocity v is a material derivative of the displacement u. Definition D. I, Expression (6), gives the relationship between the displacement and the strain tensor E. Thus the strain tensor E is present in Basic Law BL.2, although hidden. As a conclusion the following definition is expressed: D.4 The independent variables present (possibly hidden) in the basic laws of thermomechanics are called controllable variables. The other independent variables are called internal variables. In light of the above the number of state variables may be considerably high, which makes thermodynamics with internal variables more complicated and - hopefully - more capable. On the other hand, the choice of an expressive set of state variables is a constitutive assumption and shows - in the case where it is done correctly - the professionalism of the physicist. Use of an unnecessary large set of variables may lead to an overly complicated theory with many (physically unclear) variables. A theory with too few variables is not capable of describing the event. The controllable independent variables for thermomechanical processes are: E The strain tensor. It is a second-order tensor describing both mechanical and thermal deformation. S Specific entropy. 54 Besides the above controllable variables there is a set of internal variables denoted by ai, i = l...n. The internal variable ai can have a scalar, vector or tensor form. The number of internal variables and their form are determined by the model under consideration. Thus the set of internal variables is denoted here by a second-order tensor a.The results which will be derived for a are generally valid and therefore the reader can easily extend the results of this work for his/her set of independent variables. This extension will also be done here when evaluating the particular material models for thermoplasticity. In some special cases, e.g. when the gradient theory is evaluated, a slightly different approach may be necessary and therefore this investigation may not be adequate to describe all thermomechanical processes. 5.4 LAW OF CALORIC EQUATION OF STATE AND THE AXIOM OF LOCAL ACCOMPANYING STATE The aim of this section is to formulate the law of caloric equation of state for thermodynamics and to introduce the state equations outside thermodynamic equilibrium by utilising the axiom of local accompanying state. In contrast to thermostatics, thermodynamics is a field theory. Therefore [c.f. Equation (SI)] the field variables, the specific entropy s (i.e. entropy per unit mass) and the specific internal energy u (i.e. internal energy per unit mass) will be introduced in the caloric equation of state. Instead of writing Equation (51) the following form for the caloric equation of state is expressed: u = U(E,CC,S, h(x)) where the notation h(x) indicates that the system V may be thermodynamically inhomogeneous. If the internal energy u does not depend on h(x), the system is said to be thermodynamically homogeneous, i e. the mechanical as well as thermal properties are the same everywhere within the system V. It is important to note the similarity of the two caloric equations of state; namely Expressions (52) and (63). The internal variable a is the extension due to the introduction of thermodynamics with internal variables. Of course the theory of thermostatics could be extended by introduction of internal state variables. Since the traditional form of thermostatics does not contain internal variables, they are not adopted here. Entropy S and internal energy U are defined by: U(t) := J PU dV and S(t) := ps dV ( 64) V V Applying Reynolds transport theorem, Theorem (1 l), it is arrived at: U(t) = J' p u dV and S(t) = J' ps dV (65) V V By applying the axiom of local accompanying state the corresponding equations for Equations (59a) and (59b) are written: and furthermore In Expression (66a) the notation oq refers to the quasiconservative stress tensor [due to Ziegler (1983, p. 61)). State Equations (66) ...( 67) define a new set of variables; (aq, p, T) which are called state functions. The minus sign in the right-hand side of Equation (66b) is due to the definition of positive value for internal force p. A different definition is of course possible. The idea behind the above sign definition is that it leads to a more aesthetic form 56 for the heat equation and for the Clausius-Duhem inequality, as Sections 5.4.1 and 5.6 show. The origin of the density p in State Equations (66) can be shown by a simple dimensional analysis. State Equation (59a) for thermostatics is recalled and the following dimensional analysis is carried out: State Equation (66a) can be evaluated as follows: Energy per unit mass Pure number kg Nmikg =[z] A new state function, namely the specific HelmholtzI=[$] free energy tq is introduced. It is defined by: It is worth noting that the specific Helmholtz free energy w does not contain any new information, because it is represented by the two already introduced state functions (u and T) and a state variable (s). The idea of introducing the specific Helmholtz free energy v, is based on the fact that it has almost the same appearance as the strain-energy density w. Therefore the existing library of strain energy densities w for different material behaviour can be utilised. Other kinds of state functions, e.g. Gibbs free energy, can be introduced. According to Appendices B and C the specific Helmholtz free energy v, is a Legendre partial transformation of the specific internal energy u. Comparison of 57 Definition (70) and Transformation (C.3)in Appendix C gives b = -1 and a = I, as well as the fact that the specific entropy s is the active variable of the transformation whereas the quantities E, a and h(x) are passive ones. Based on Results (C.6) in Appendix C the following is achieved: and and furthermore Substitution of Expressions (72) and (73) into Expressions (66) and (67) yields: and 5.5 FIRST LAW OF THERMODYNAMICS In this section the first law of thermodynamics is considered and its consequence the energy equation is derived. According to Narasimhan (1992, p. 321) the principle of conservation of energy, also referred to as the first law of thermodynamics, can now be stated: The time rate of change of the sum total of the kinetic energy K and the internal energy U 58 in the body is equal to the sum of the rates of work done by the surface and body loads in producing the deformation (or flow) together with heat energy that may leave or enter the body at a certain rate. Thus, the following is obtained: D -(K + U) = Pext+ Q which yields + U = Pext+ Q (76) Dt K where the dot denotes the material derivative. In Basic Law (76b) 'K is the rate of the kinetic energy, 'U is the rate of the internal energy, Pextis the power input of the external forces and Q is the heat input rate. It is worth noting that Forms (76) are for thermomechanical materials where the other influences, such as electrical, chemical etc. are negligible in comparison with thermomechanical effects. Forms (76) also assume a closed subsystem V [see Kestin (1578. Vol. I, Chapt. 6)]. The total energy of the subsystem will be considered as the sum of two parts, the kinetic energy K and the internal energy U. By kinetic energy, it is meant the macroscopic kinetic energy associated with the usual macroscopically observable velocity of the continuum. The kinetic energy of the random thermal motions of molecules, associated with temperature measurements instead of velocity measurements, is considered a part of the internal energy. The internal energy also includes stored elastic energy and possibly other forms of energy not specified explicitly. (Malvern 1969, p. 230) The kinetic energy K is defined by {see Narasimhan [1952, Eq. (5.3.8)]): which when using Reynolds transport theorem, Theorem (7). gives: 59 -KD = K = p'G.VdV Dt IV The power input Ped has the form {compare with Malvern [1969, Eq. (S.S.2)1}: Pext = J' i.v dA + J' p b.: dV (79) JV V Definition of the stress tensor 0, Definition (3,reads: - - t := n.0 By replacing the star product 4: by the dot product . and denoting g = (J . 7 the generalised Gauss's theorem, Theorem (lo), gives the equality: The dot product . satisfies the following axiom [see for example Malvern (1969. p. 36)]: - n.(a.V) = (ii.u).V Taking into account Equations (80) ...(82), Expression (79) yields: Appendix D shows the equality: - where h is a second-order tensor and where Vz is the open product of vectors and e [see Appendix D]. By replacing the tensor h by the stress tensor u and the vector e by the velocity vector v Equation (84) takes the following form: 60 Appendix E gives another form for the second term on the right-hand side of Equality (85). This is Equation (E.7) i.e. - - 0:vv = u:i Applying Equations (8.5) and (86), Equation (83) takes the following form: Multiplying the equation of motion. Equation (36a). by the velocity v from the right gives: - - (V.a).i+ pb.; = p';.; (88) which allows the power input Pextto be written in the form: Equation (89) shows that the rate of increase of the kinetic energy of a subsystem is equal to the mechanical power input minus the total stress power. Heat may be transmitted in three ways, by conduction, convection or thermal radiation. Convection is possible only in a fluid medium. The term convection is applied to the transport of heat that occurs as volumes of liquid or a gas fluid medium move from regions of one temperature to those of another temperature. Thermal radiation is the process of heat propagation by means of electromagnetic waves, depending only on the temperature and on the optical properties of an emitter, with its specific internal energy u being converted into radiation. Heat conduction is identified as the process of molecular transport of heat in bodies (or between them), due to temperature variation in the medium considered. [Isachenko et al. (1977, p. IS)] 61 Thus, in light of the above discussion [see also Malvern (1969, p. 228) and Astarita (1989, p. 16)] the heat input rate Q in solids can be written in the following form: where { is the heat flux vector and r is the heat source per unit mass. The negative sign in front of the first term on the right-hand side of Expression (90) is due to being defined as the outward unit normal vector. The heat source r describes the influence of the radiation, as well as other heat sources not specified here. Applying the generalised Gauss’s theorem, Theorem (lo), by replacing the star product * by the dot product . and denoting g = i,Expression (90) yields: Next the investigation reverts to the equation describing the first law of thermodynamics. Substitution of Equations (65), (78), (89) and (91) into Expression (76b) gives: J’p’;.;dV +lpudV = J(pT.v+o:E)dV +f(-V.q+pr)dV V V V V which yields: - /(pu -a:E +V.i-pr)dV = 0 (93) V Because the volume of the subsystem V is an arbitrary part of the system the 62 integral (93) has to vanish for every V within the system Vb. Thus, the term in parentheses has to vanish at every point of the material. This gives: which is called the energy equation (in the nonpolar case) or the equation of balance of energy [Fung (1965, p. 347) and Malvern (1969, p. 230)). Equation (94) is the local form of the first law of thermodynamics. 5.5.1 Heat Equation This section takes a close look at the heat equation. As pointed out by Maugin (1990, p. 178) the heat equation is none other than the energy equation in disguise. Again, the definition for the specific Helmholtz free energy \v. Definition (70). is: w := u - TS (95) Definition (95) gives: Substitution of Equation (96) into the energy Equation (94) yields: - pQ +pis+ pTS - a:E - pr + V.6 = 0 (97) Since the specific Helmholtz free energy w = W(E, a, T, h(x)), the rate of the specific Helmholtz free energy has the following appearance: Since the material properties related to the specific Helmholtz free energy w are not assumed to vary, the rate h(x) vanishes and therefore the last term in Expression (98) also vanishes. This assumption means that e.g. Young's moduli 63 can take different values at different positions in a body but the values do not evolve with time. Substitution of Equation (98) into Equation (97) leads to: - (p--a):dav + p-:aav + p($+s)T + p(TS-r) + V.g = 0 (99) a& aa Equations (71), (74) and (75) are: = -p- (Jq = P-av and p av ( 100) a& da and furthermore Applying Equations (100) and (101) Equation (99) takes the form: - V.a = (u-uq):i + p:a + - p(TS-r) (102) The following notation is introduced: (Jd := (J - 09 where ad is the dissipative stress tensor [following Ziegler (1983, p. 60)]. Based on Notation (103) Equation (102) takes the form: - v.q = od:i + p:ti + - p(~s-r) Equation (101) gives: Together with Equation (98) and assuming that h(x) = 0, Equation (105) allows Equation (104) to be written in the following form: 64 which is the heat equation. In the case of a pure reversible process 15,~E 0, E reduces to E~ and v( E, a,T, h(x)) reduces to I$( E~,T, h(x)) where the superscript r refers to a reversible process. The above-assumed process leads to the following form of heat equation: derived by Parkus [1976, Eq. (5.28)]. The introduction of the specific heat at constant deformation (Le. E = a = 0) denoted by C and defined by: allows the heat equation to be written in the form: - pci 5.6 SECOND LAW OF THERMODYNAMICS This section looks at the second law of thermodynamics. 65 The second law of thermodynamics is a formalization of the intuitive concept of the irreversibility of natural phenomena. In this, it is conceptually very different from other fundamental principles of physics, which are in general formalizations of a conservation principle (mass, linear momentum, energy etc.). A conservation principle states that some physical quantity is neither created nor destroyed [see for example Eq. (76)l; hence balance equations such as Equation (94) can be written down. In balance equations, if the sign of all the terms are changed, the equation still holds true; hence such equations are invariant under the reversal of the direction of time. As a concrete example, all terms appearing in Equation (94) are rates, and they therefore all change sign upon time reversal. The description of irreversibility, which is required from the second law, imposes that it must be written as an inequality, since inequalities fail to hold true when the sign of all terms is changed. The second law of thermodynamics can be written by requiring the rate of change of some quantity to never be less (but it can be more) than that which one would calculate should a conservation principle apply. (Astarita 1989, pp. 21 and 22) Astarita (1989, p. 22) continues: The quantity whose rate of change is involved is called the entropy. Just as is the case for energy in the first law, entropy is a primitive concept; this being so is less easily acceptable simply because, in contrast to ’energy’ the word ’entropy’ is not a part of familiar, everyday language. The second law of thermodynamics can be written in the following form: s 2 + spTdVr av T V where ‘S is the entropy rate. Applying the generalised Gauss’s theorem, Theorem (lo), to the first term on the right-hand side of Expression (1 10) by replacing g by 1/T 4 and the star product * by the dot product . the following is obtained: 66 S 2 = - V.(-q)dV + s-V ;- V which yields: I-- 1-- 1 s 2 l[--V.q +-q.VT +-prldV T T2 T V The relation between the entropy rate 'S and the specific entropy rate 's is given by Equation (65b), i.e. S = JpSdV (1 13) V Substitution of equation (1 13) into Equation (1 12) leads to the following form: jpSdV 2 j[--V.qI-- +-q.VTI-- +-prldV1 (1 14) T T' T V V which yields: I-- I-- 1 s[pS +-V.q --q.VT --pr]dV2O (1 IS) T T2 V T Because the volume of the subsystem V is an arbitrary part of the system the integral (1 15) has to be non-negative for every V within the system Vb. Thus the term in square brackets has to be non-negative at the every point of the material. This leads to: - I-- I ps + 'v q - -q.VT - -pr 2 0 T T2 T If the terms of Equation (1 16) are multiplied by the absolute temperature, the following equation is obtained: 67 - - pTS + V.q- - -vT.< - pr 2 0 T which is the local form of the second law of thermodynamics. The concept of the specific entropy production rate 'si is introduced. It is defined by: - - VT - pTS' = pTS + V.q - -.q - pr T Because the density p and the absolute temperature T are non-negative, according to Inequality (1 18) the specific entropy production rate 'si is also non- negative, i.e. SI 2 0 5.7 CLAU S IUS-DU HEM INEQUALITY The aim of this section is to derive the Clausius-Duhem inequality which is a combination of the first and second law of thermodynamics. Again, the local forms of the first and second law of thermodynamics Expressions (94) and (1 18) are written: - pu = a:i + pr - V.q and - - VT - pTS' = pTS + V.q - -.q -Pr T Substitution of 7.i evaluated from Equation (121) into Equation (120) gives: 68 VT - pTS' = - pU + pTS - --.q (2 0) (122) a:& T Truesdell and Noll (1965, p. 295) introduced the local entropy production (rate) .Islot and the entropy production (rate) by conduction of heat 'scan' defined by: (123) p T Si = p T SI,, + p T Sion where ( 124) pTS;,, = a:& - pU + pTS and and proposed a stronger assumption, requiring separately: p T SI,, 2 0 and p T SLon 2 0 (126) The first of these inequalities corresponds to the physical observation that a substance at uniform temperature free from sources of heat may consume mechanical energy but cannot give it out. The second inequality corresponds to the fact that heat does not flow spontaneously from the colder to the hotter parts of the system (body). The postulate (126) is a special case of the Clausius- Duhem inequality or the principle of dissipation. (Truesdell & Noll 1965, p. 295.) The left hand sides of Equations (124) and (125) are called intrinsic dissipation and thermal dissipation, respectively [see e.g. Maugin (19O0, p. 178)l. According to Expression (96) the material derivative of the specific internal energy u is: 69 U=Q-Ts+TS Substitution of Equation (127) into Equation (122) yields: VT - pTS' = o:L -pq-pT~---.q (2 0) (128) T Inequality (128) is: which is the Clausius-Duhem inequality. The material model must obey Clausius-Duhem Inequality (129). Verification can be done either numerically during the computation procedure or analytically, the latter evidently being much more convenient. In both cases Form (129) is not very useful and therefore has to be recast into a new appearance. This procedure follows. Looking again at the expression for the rate of the specific Helmholtz free energy Q, Expression (98). It is (material properties related to the specific Helmholtz free energy v do not evolve with time i.e. h(x) 0): Substitution of Expression (130) into Expression (128) gives: Equations (71), (74) and (75) are: 70 and furthermore (133) Notation (103) reads: ( 134) Substitution of Results (132) and (133) as well as Notation (134) into Inequality (131) gives. (135) Inequality (135) is: - (136) ad:& + p:a - E., 2 0 T Forms (179) and (136) are two different formulations for the Clausius-Duhem inequality. The difference between them is the fact that the latter introduces a certain set of internal forces (ad, p) and the internal state variable a. In the next section Form (135) is used with the principle of maximal rate of entropy product ion. By following Separation (123) one can write: pTSi,, = ad:& + p:a ( 137) and 71 5.8 PRINCIPLE OF MAXIMAL RATE OF ENTROPY PRODUCTION This section investigates the principle of maximal rate of entropy production and derives its consequence: the normality rule. The principle of maximal rate of entropy production was first proposed by Ziegler (1963, p. 134). It should be pointed out that this principle proposed by Ziegler [see e.g. (1983, pp. 271 and 272)l is not (yet'?)a basic law of physics, compared with those discussed in the previous sections. Therefore in Chapter 2 this principle was put into the list of axioms instead of the list of basic laws. In Ziegler's own words, the principle of the maximal rate of entropy is quite general (Ziegler & Wehrli, 1987, p. 186). In Subsection 5.8.1, necessary and sufficient conditions for a local maximum in a general case are studied. This study utilises the specific dissipation function cp. In Subsection 5.8.2 a similar study is repeated for a case where the specific dissipation function cp is replaced by the specific complementary dissipation function (pc. Finally, Subsection 5.8.3 investigates the normality rule for thermoplastic material behaviour. 5.8.1 Normality rule in a general case According to Ziegler (1983, pp. 271 and 272), the principle- of maximal rate of entropy production is: provided the forces [ad,fl, - (VT)/T] are to be prescribed, the fluxes (E, a, 4) maximise the rate of entropy production S', Equation (5.72) i.e. subject to: From the physical point of view this principle is particularly appealing, since it may be considered an extension of the second fundamental law. In fact, if a 72 closed system tends towards its state of maximal entropy, it seems reasonable that the rate of entropy increase (the specific entropy production rate) under prescribed forces takes a maximum value, ;.e. the system should approach its final state along the fastest (shortest) possible path. (Ziegler 1983, p. 272) Investigation of the expressions for the rate of entropy production Si. Equation (139), shows that si is dependent upon both (dissipative) forces and fluxes (process). For example, in the expression ad : E the term ad represents the dissipative force and E describes the flux (process). Furthermore, Ziegler (1963, p. 129) assumes the existence of the specific dissipation function [see also Ziegler (1983, p. 76)l. where the notation k(x) displays the inhomogeneity of the dissipative material behaviour beyond the internal variables which usually model the inhomogeneity. The concept of the specific dissipation function cp is that in case of an actual process (i.e. when the maximum is present) it contains the same information (except for temperature T) about the state and the dissipative process as the entropy production rate SI. but the arguments of cp are only the fluxes (E, a, 4) [and the state (E, a,T, h(x))], whereas the expression for SI contains also the conjugate variables Le. forces [ad,p, - (VT)/T]. This can be seen in Equations (139) and (141). The specific dissipation function cp is defined by {see Ziegler [1963, Eq. (4.3)) and [1983, Eq. (5.1)]}: 1 cp = TS' j - cp - SI = 0 for an actual process (142) T Ziegler did not cast his above concept into an exact mathematical framework. The author therefore proposes the following formulation for the principle of maximal rate of entropy production: 73 The process is investigated at a certain state (E, a,T) under certain loading a and the values for the fluxes (E, a, 4) have to be determined in order to maximise the rate of entropy production 'SI. The state gives the values for the state functions oq and p through State Equations (74) ...( 75). Since the values for the d stress measures a and uq are known, Notation (103), i.e. a := u - oq, gives the value for the dissipative stress tensor ad. Because the - value for the absolute temperature T is known, -so is the value for the force - (VT)/T. Thus the values for the forces [ad, p, - (VT)/T] are known. Based on the above discussion the problem can be expressed in the following way: It is assumed that the state and loading are known, i.e. the set (E, a,T, a) is assumed to be known. This implies that the values for the forces [ad, p, - (?T)/T] are known. The question is, what are the magnitudes of the fluxes (E, a, 4) which maximise the rate of entropy production SI? At the same time also Equation (142b) has to be satisfied. Based on the above discussion, the principle of maximal rate of entropy production can be written in the following mathematical form: maximise with respect to the fluxes (E, a, 4): - 1 d,. VT - S'(h,U,q) = .e + p:a - -.q) (143) -(aPT T subject to: 1 r = - q( 8, a,i,k(x); e, a,T, h(x)) T where T = 0 is a constraint. Both 'SI and T are assumed to have at least continuous second partial derivatives with respect to the arguments (E, a, 4).It should be pointed out that also Inequality (140) must be satisfied. 74 Applying Luenberger (1973, p. 225) the first-order sufficient condition for the point (6, a, 6)to be a local maximum is: a. a. -(SI + hr) = 0 and -(s'+hr) = 0 (145) ai adr and furthermore - --(Sia +AT) = 0 and r =O aq where h is the Lagrange multiplier. As mentioned by Arfken (1985, p. 946), the method based on Lagrange multipliers will fail if in Expressions (145) and (146a) the coefficients of h vanish at the extremum. Therefore, also special points where must be investigated. The above indicates that there are two different cases for evaluation of the local maximum; utilisation of Expressions (14s) ...( 146) referred to as Case A, and the special case described by Expressions (147) referred to as Case B. Starting with Case A: Substitution of Equations (143) and (144) into Equation (14Sa) gives: 1 1 d - ud + h( -- - - u ) = 0 PT ai p~ which yields the following result: od = h P-acp 1-1 ai Similarly Equations (14Sb) ...( 146b) give: 75 and -VT - h P: aq (150) T h-1 aq By substituting the results in Equation (149) and (150) into Equation (146b) and by re-ordering the obtained equation, the following result is obtained: 1 aq . aq . acp - cp( k. a,G, k(x); E, a,T, h(x)) = - (-:& +-:a + -.q) (151) A-I ai adr JT It is worth noting that the value of the Lagrange multiplier h is dependent on the set (E, a,T, a). This is based on the definition of the maximisation problem which assumed that the value for the set (E, a,T, a) is known and that the values for the corresponding fluxes (E. a, {) have to be determined. This implies that for a certain set (E, a,T, a) a unique value for h is obtained. Thus the following holds; h = A(&, a. T, h(x), a). By extending the definition for homogeneous functions given by e.g. Widder (1989, pp. 19 and 20) the following is achieved: A function @(x,y,z,u,v) is homogeneous of degree w in variables x,y,z in a region R if, and only if, for x,y,z in R and for every positive value of k the following holds: @(kx,ky, kz, U,V) = km@(x,Y,z, U,V) Sometimes the definition is assumed to hold for every real k, and if the values of k are restricted to being positive, the function is said to be a positive homogeneous function. Euler’s theorem on homogeneous functions [see original form in e.g. Widder (1989, p. 20)1 for the above extended definition reads: a@ a@ a@ o@(x,y,z,u,v) = -x + -y + -z (153) ax JY aZ In the special case that the Lagrange multiplier h is a constant the extended Euler’s theorem on homogeneous functions, Theorem (153), and Equation (152) 76 indicate that the specific dissipation function cp is a homogeneous function of degree (A-I) / h in the variables (E, a, y). The following notation is introduced: h P p := - which gives h = - (154) h-1 M-1 Equation (154b) and the above discussion show that the multiplier p = p(~,a, T, h(x), u) can be any real number excluding p = 1. This means that in Case A the specific dissipation function cp cannot be a homogeneous function of degree l/p = 1. Continuing with Case B: The candidates for the extremum points defined by Equations (147) are investigated next. Substitution of Equation (144) into Equation (147a) gives: 155) Correspondingly it is obtained: 156) Comparing Equations (1S5b) and (156) with Equations (149) and (150) shows that the special points defined by Case 6 give the same solution as Case A except that cp is a homogeneous function of degree 1 (= I/p). Instead of Expression (152) Case B gives: acp . acp - cp( i,a, q, k(x); E, a,T, h(x)) = 2 : i + - : a + -. q (157) a& aa ai Concluding from Cases A and B the following can be said: The first-order sufficient condition for the point (E, a, 4) to be a local maximum 77 is that Equations (149) and (150) hold and that the specific dissipation function cp is a homogeneous function of degree l/p (+ 1). This result was obtained by the method of Lagrange multipliers and was referred to as Case A. The Lagrange multipliers technique fails if Conditions (147) hold. In the latter case, called Case B. Equations (1SSb) and (156) proved to be correct and the specific dissipation function cp was shown to be a homogeneous function of degree lip = 1. The other possibility is that the specific dissipation function cp is not a homogeneous function. In this case the coefficient p is not a constant but depends on the set, Le. p = p(~,a, T, h(x), a) and its value can be determined from Equations (152) and (154a). At the start of this section the following problem was set: The forces [ad, p, - (?T)/TJ are assumed to be known and the fluxes (e, a,i) have to be determined. Therefore, strictly speaking, Transformations ( 155b) ...( 156) should be replaced by their inverse mappings. This means that instead of Equations (155b)...( 156) expressions such as t = f( forces ) should be written. However, in practice, when constitutive models are prepared, the explicit form for the specific dissipation function cp is assumed and the forces [ad, p, - (fT)/TI are determined by Equations (155b)...( 156). The results derived in this subsection can be expressed briefly as follows: The following normality rule holds: and and furthermore 78 The specific dissipation function cp has to satisfy: If the multiplier p is a constant cp(k, a, {, k(x): E, a,T, h(x)) is a homogeneous function of any degree (= lip) in the variables (& a, 4). If the coefficient p is not a constant but p = p( E, a,T, h(x), a) the specific dissipation function cp is not a homogeneous function and the value for p can be determined by Equations (152) and (154b). Equations (142), (158) and (159) show that the specific dissipation function cp is a scalar potential and it is therefore also called the specific dissipation potential. The second-order both necessary and sufficient conditions for a local maximum lead to matrices so extensive [see Luenberger (1973, pp. 226 and 227)] that investigating them is very complicated and it hardly provides any practical results. 5.8.2 Normality rule for the specific Complementary dissipation function Sometimes dissipation of the material response cannot be described by the specific dissipation function cp(15, a,y, k(x); E, a,T, h(x)) where the fluxes (k, a, {) are independent variables, but successful modelling requires utilisation of the - specific complementary dissipation function cpc( od, p, -VT/T, k(x); E, a,T, h(x)) which uses the forces [ad,p, - (VT)/T] as independent variables. The aim of this subsection is to derive of the normality rule for the specific complementary - dissipation function cpc( ad,F, -VT/T, k(x); E, a,T, h(x)). 79 Definition of the specific complementary dissipation function cpc is done by way of introduction of the Legendre partial transformation of the specific dissipation function 9.The transformation is defined by: - d VT - q p cp"( u , p, -VT/T, k(x); E, a, T, h(x)) : = ud : E + p : ti - -. q T (161) - P p cp( 6,a, i,W; E. a,T, h(x)) where the superscript c in notation cpc refers to the complementary function and where the quantity q is a multiplier making Transformation (161) more general. Based on Normality Rule (158) and (159) and Definition (161) the coefficients in Expression (C.3) of Appendix C take the values a = p p and b = q p. Thus, Expressions ('2.6) of Appendix C give the following normality rule: 3qc(ad, p, T/T, k(x);, , ,) 6 =qp -v a and 3cpc( ud,p, TIT, k(xj;, , ,) a =qp -v ap and furthermore - 2 cpc( ad,0, -7 T/T, k(x); . . . j =qP a(- ?T/T) Again, Condition (146b) [or Condition (144)] is: 1 - 1 FT - -cp(i,a,q,k(x);e,a,T,h(x)) - -(ud:i+p:a--.q) = 0 (165) T PT T Multiplying Equation (165) by p p T gives: 80 - . VT - ~~c~(L.,dr,q,k(X);~,a,T,h(x))= p(ud:i: +p:a--.q) T Expressions (161) and (166) yield: q p cpc( ad,p, -iT/T, k(x); e, a, T, h(x)) = Substitution of Normality Rule (162) and (164) into the right-hand side of Equation ( 167) gives: qC(ud, p, -?TIT, k(x); E, a,T, h(x)) a qc where, on the first line on the right-hand side, the order of terms in the pairs has been changed. Comparison of Normality Rule (162) and (164) with Condition (168) shows that Condition (168) does not set any restrictions for the multiplier q, and therefore the value for q can be selected arbitrarily. If the specific complementary dissipation function (pc is a homogeneous function of degree I in the variables [ad,f!, -VT/T], according to Condition (168) and the extended Euler’s theorem on homogeneous functions, Theorem (153, the multiplier p has to vanish. This implies that both Normality Rule (158) and (159) as well as Legendre Partial Transformation ( 161) vanish. Consequently Normality Rule (162) and (164) does not hold. 81 5.8.3 Normality rule for thermoplastic material behaviour The aim of the present subsection is to evaluate of the normality rule for thermoplastic material behaviour. It is a special case, since thermoplastic deformation is a time-independent process whereas the previous subsections are for modelling of time-dependent processes. The evaluation applies the results obtained in Subsections 5.8.1 and 5.8.2. It is reasonable to assume that the fluxes (8, a) are independent of the heat flux vector 6 which means (the reason is given a few pages further on) that the specific dissipation function for thermoplasticity (pp is separable as follows: M.2 where the superscript p refers to plasticity and where the notation M.2 refers to the model. With Separation (169) the maximisation problem is separated into two different ones. This separation follows the concept of separating the rate of entropy production Si into two different parts proposed by Truesdell and Toupin, as discussed in Section 5.6. According to Santaoia (1994, Sect. 3.8.2) separation of the specific dissipation function (pp into two parts as done in Expression (169) and formulation of the maximisation problem with two constraints, namely T,,, = 0 and T~~~ = 0 [ cf. Equation (5.79)], leads to the same result as introducing two separate maximisation problems. Therefore, in this work two independent separation problems are written whereby instead of maximising SI the rates SI,,' and S,,,' are maximised separately. The heat problem, i.e. the second term on the right-hand side of Expression (169) is evaluated first. Based on Expressions (143) and (144) the following maximisation problem is written: maximise (with respect to the flux 4): 82 i- 1 VT - s,,&) = --- -.q) PT T subject to: where T,,, = 0 is a constraint. Both S,,,' and z,,, are assumed to have at least continuous second partial derivatives with respect to the argument 6. Because the appearance of the specific dissipation functions cp and (P,,,,~ given by Expressions (141) and (169) is the only difference between the formulations for the maximal rate of entropy production in the present subsection and in Subsection S.8.1, the results of Subsection 5.8.1 can be applied here without difficulty. Based on Normality Rule (158) and (159) as well as Condition (15 1) and (160) the following is arrived at: (172) and (1 73) If the multiplier pconis a constant cpconP(~, k(x); E, a,T, h(x)) is a homogeneous function of any degree (=Upcon)in the variable 6. If the coefficient peon is not a constant but depends on the state and loading, i.e. p,,, = peon( E, a,T, h(x), u, 0. 'T), the specific dissipation function cpconP is not a homogeneous function and the value of p,,, can be determined by Equation (173). 83 The mechanical part of the maximisation problem, i.e. the first term on the right- hand side of Expression (l69), viz. is evaluated next. The principle of maximal rate of entropy production, described at the start of Subsection 5.8. I is written next. The process is investigated at a certain state (E. a,T) and under known loading u. This implies that the values of the forces [aq, - p, - VTiT] are known. The problem was to define the values for the fluxes (E, a, - q) in order to maximise the rate of entropy production Si. A new quantity called the specific dissipation function (p was also introduced. According to Expression (141) it was assumed to have the following appearance (the heat flux vector is omitted): The above formulation is capable of describing of several different material responses. However, it is not for thermoplasticity, since the amount of plastic yield is dependent on state (E, a,T) and on lcrading 84 cpp = cpp( e, a, k(x); e, a,T, h(x), a,0, 'T ) Based on Expressions (143) and (144) the following can be written: maximise with respect to the fluxes (t,a): S;,,(&,a) = -(od:E1 + p:a) PT subject to: 1P T~~~ = - cplo,( e, a, k(N; E, a,T,h(x)) T (178) I --(ad:i:q3:dr) = 0 PT where = 0 is a constraint. Both 'slot' and T,,, are assumed to have at least continuous second partial derivatives with respect to the arguments (E, a). It is convenient to assume that dissipation functions cp for time-independent processes are homogeneous functions of degree I. The reason is shown later in this subsection. Based on Normality Rule (1SSb) and (156) it is achieved: Usually in the theory of thermoplasticity the specific dissipation function cp is replaced by the yield function F the arguments of which are the forces (ad, p) instead of the fluxes (6, a).Furthermore, the traditions of thermoplasticity give the following property for the yield function F: F := 0 during plastic flow ( 180) which gives F =O during plastic flow 85 Condition (181) is called the consistency condition Based on the above the following Legendre partial transformation is made: where the plasticity multiplier iis defined by: % : = i(E, a, T, h(x), u, u, 'T) (1 83) The notation ' above quantity h indicates that iis not a material derivative but a function also having material derivatives of quantities as independent variables. Based on Normality Rule (179) and Transformation (182) the coefficients in Expression (C.3) of Appendix C take the values a = p and b = A. Thus, Expressions (C.6) of Appendix C give the following normality rule: and d , k(N; E, a,T, h(x)) a= aF( a P, (1 85) aa Normality Rule (184) and (185) shows that due to Separation (169) (Model M.2) the fluxes (E, a) are independent of the heat flux vector 4as discussed above in relation to Equation (169). Two further topics are considered below. First, the background to the fact that for time-dependent processes the specific dissipation function 'p is a homogeneous function of degree 1; second, demonstration that the homogeneity (of degree 1) of the specific dissipation function has no implications for the yield function 86 F. is needed: 1 d~~o,(ki,ka,e,a,...) 3(p,,,(kt.,ka,c.aP ,...) --- :kl = k 3(k E) a(k i) where Expression (26c), i.e. I : A = A, and the fact that the mechanical part of the dissipation function for thermoplasticity (plocp is a homogeneous function of degree 1 in the variables (2, a),have been exploited. Again normality Rule (179a) is: which exploits the fact that a derivative of a function with respect to one of its arguments is a function having no more than the same arguments. Based on Normality Rule (1 87) one can achieve: a(ploc(P k E, k a, e, a, . . .) ud(ki,kdr,E,a,...) = p a(k E) According to Manipulation (1 86) the right-hand sides of Normality Rules (187) and (188) coincide. Therefore the left-hand sides are also equal. Since two arbitrarily different sets of fluxes, Le. (E, a)and (k8, ka), give the same value for the dissipative stress tensor ad, the value of the dissipative stress tensor ad cannot depend on the fluxes (E, a).This means that the fluxes (8, a),i.e. a time- dependent process, have no influence on the forces (ad,p) and through state 87 equations do not affect the state (E, a, T). Thus, modelling the specific dissipation function cp by a homogeneous function of degree 1 in fluxes gives a time- independent process. This is a sufficient condition for time-independence. Below it is shown that although the specific dissipation function qIocpis a homogeneous function of degree I the yield function F can be nonhomogeneous. Properties (152) and (153) for the homogeneous functions are recalled and applied to cplocp which is a homogeneous function of degree 1 in the fluxes (e, a). They read: and Multiplying both sides of Transformation (192) by 9 positive-valued scalar k gives: Utilisation of Normality Rule (188) [and the corresponding normality rule for the internal force B] and Property (189) to Expression (191) gives: 88 k F(ad, p, k(x); e, a,T, h(x)) : = Based on Property (190) the right-hand side of Expression (192) vanishes. This is the condition aimed at, since the theory of thermoplasticity requires that the value for the yield function vanishes during plastic flow, as indicated by Definition (180). (193) Once again Expression (190) ensures that F := 0 during plastic flow The above indicates that although the specific dissipation function ql0: is a homogeneous function of degree 1 the yield function F can be nonhomogeneous. 89 NEXT PAOW) I Iett BLANK I 6 FOURIER'S LAW OF HEAT CONDUCTION In this section the equation of heat transport according to Fourier's law of heat conduction is derived. Applying Ziegler and Wehrli [ 1987, Eq. (3. IO)]. the specific dissipation function 'p,,, 'p,,, for the linear isotropic material (in the sense of heat conduction) can be written in the form: where ~T)is a temperature-dependent positive scalar called thermal conductivity, T is the absolute temperature and 4 is the heat flux vector. The right-hand side of Equation (194) shows that the influence of the state variables E and a as well as of loading E and 'T on the specific conductive dissipation function (pcOnp is neglected. Obviously it is obtained: (195) which implies that 'pconPis a homogeneous function of degree lipcon = 2 in the heat flux vector {. Normality Rule (173) reads: Substitution of Expression (194) into Equation (196) gives: - q = -Y(T)VT which is Fourier's law of heat conduction (see e.g. Malvern [1969, Eq. (7.1.6)]}, According to lsachenko et al. (1977, p. 22) it has been shown experimentally that 91 with accuracy sufficient for practical applications, the dependence of thermal conductivity y(T) on temperature can be assumed to be linear for many materials: where yo is the thermal conductivity at the temperature T, and y, is a material constant. Recalling the Clausius-Duhem inequality for heat conduction, Inequality ( 138) Substitution of - VTiT evaluated from Model (197) into Clausius-Duhem Inequality (199) yields: Since y(T) and T are positive scalars, Inequality (200) holds true and the Clausius-Duhem inequality is satisfied. 92 7 GENERAL FORMULATION FOR THE THEORY OF TH ERMO PLASTlC ITY This chapter collects the assumptions and results derived in the previous chapters for the theory of thermoplasticity. Some new expressions are introduced. A careful reader may have noticed a certain difference between the traditional expressions for modelling of plastic flow and the equations derived here. Usually in the theory of plasticity e.g. the normality rule is formulated in differential form whereas this work writes equations in rate form. This is because the rate form fits well with the thermomechanical framework. The time scale in plastic yield is not dependent on the rate of the material response, since deformation is modelled as instantaneous. The time scale is a consequence of the rate of loading. Thus the minimal difference between these two approaches depends only on the presentation of loading and can be neglected by a simple time integration. The list of basic laws, axioms and definitions for thermomechanics is given at the beginning of Chapter 2. The list, which is not recalled here, evidently also holds for the thermoplasticity formulation presented in this work. Besides the list, the assumptions below were also made during derivation. Following the same concept as in Chapter 2, notation A indicates an assumption and M a model. A. 1 According to Chapter 2 deformations and rotations are small. M.l According to Chapter 2 the (sub)systems are closed. This means that there is no mass transport within the body. M .2 The principle of maximal rate of entropy production is separable into two mutually independent maximisation problems, i.e. the rate of 93 entropy production Si is separated into the local entropy production i rate Slot and the entropy production rate by conduction of heat Sco,' as follows: SI - .I - sloc + SCO" and both the terms are maximised separately. This assumption was made in Subsection 5.8.3. According to Definition (70) the specific Helmholtz free energy w was assumed to have the following appearance: However, the internal state variable a is only an example of a set of internal variables and can be replaced by an other set of internal state variables. All the results derived in the present work hold, after minor modifications, for the new set of internal variables. Recalling State Equations (71), (74) and (75): (203) and and finally Recalling the expression for the rate of entropy production SI, Expression (133, and the Clausius-Duhem inequality, Inequality (136): 94 and - (3 d :E + p:a - vT.q 2 0 (207) T The normality rules for the specific complementary dissipation function (pc and for the thermoplastic material behaviour are recalled. They are Expressions (162) ...( 164) as well as (183), (184) and (185). For general thermomechanical material behaviour the following was obtained: aqf(od, -vT/T,k(x);...) i. =qp b, (208) a and d(pC(ad,P, -VT/T,k(x);...) Q =qp (209) ap and furthermore (210) a( - VTIT) The normality rule for thermoplasticity reads (heat equation not considered): and a=i aF( ud,B, k(x); E, a,T, h(x)) where the plasticity multiplier has the following property: 95 (213) = i(E, a, T, h(x), u, u, 'T) The theory of (thermo)plasticity assumes that during plastic flow the value for the yield function Fp vanishes. This is shown by Definition (NO), viz. during plastic flow during plastic flow Expression (2 15) is called the consistency condition. There are two different groups of constitutive equations for viscous and thermoplastic material behaviour, here called Kelvin-Voigt type of material models and Maxwell type of material models. The difference between them lies in the sets of independent internal state variables. In the Kelvin-Voigt type of material models the viscous strain tensor E" (the plastic strain tensor E~)does not belong to the set of independent state variables, whereas in the Maxwell type of material model it does. The following two sections study this difference. Although mechanical models for the Kelvin-Voigt and Maxwell models will be evaluated, it should be borne in mind that the value of mechanical models is very limited in proving of the phenomena of these two types of material models, because materials are not built upon springs and dashpots. 7.I KELVIN-VOIGT TYPE OF MATERIAL MODELS This section evaluates the basics of the Kelvin-Voigt type of material models. Figure 8 sketches the mechanical model for the Kelvin-Voigt material model which describes viscoelastic material behaviour. As already discussed the key concept for the Kelvin-Voigt type of material models is that the viscous strain 96 tensor E" (the plastic strain tensor cP) is not an independent state variable. Figure 8. Kelvin-Voigt material model for viscoelastic deformation. Figure 8 gives the 'total' stress tensor u as the sum of the quasiconservative stress tensor uq and the dissipative stress tensor ud. The dependency follows Definition (103) which gives: d u = uq+o (2161 The following form for the specific Helmholtz free energy of the uniaxial homogeneous Kelvin-Voigt model can be expressed as follows: 1 w(E,T) = - E E(T) E (217) 2P where E(T) is Young's modulus. Substitution of Model (217) into State Equation (203) yields: The Clausius-Duhem inequality, Expression (207), takes the form: VT - ud:i - -.g L 0 T The yield function F for the uniaxial homogeneous Kelvin-Voigt material model takes the form: 97 (220) Normality Rule (208) and Model (220) give: Substitution of Expression (218) and the dissipative stress od obtained from Expression (221) into Equation (216) yields: 1. CJ = E(T) e + -E K(T) Expression (222) defines the traditionally expressed form for the Kelvin-Voigt material model [see e.g. Flugge (1975, pp. 9 and 22)]. 7.2 MAXWELL TYPE OF MATERIAL MODELS This section aims to evaluate the foundation of the Maxwell type of material models. Subsequent chapters study material models belonging to this type. Figure 9(a) shows the viscous Maxwell model the corresponding elastic-plastic model of which is sketched in Figure 9(b). 0 E Figure 9. (a) Viscous Maxwell model and (b) corresponding elastic-plastic material model. For Maxwell type of material models the dissipative stress tensor ad vanishes 98 and therefore the values for the stress tensor u and for the quasiconservative stress tensor uq coincide, viz. uq = u for Maxwell type of material models This latter phenomenon can be seen in Equation (216). The second characteristic of this type of material model is that the viscous strain tensor E' (the plastic strain tensor E') is an independent state variable. Thus for the Maxwell model (which has no other internal state variables) the set of internal variables (E, a,T) used in the previous chapters is replaced by the set (E, E', T). Thus: M.3 a := E' for viscous models (224) and M.3 a := EP for plasticity (225) The open question is the role of the force p. The third phenomenon is the concept that the state is expressed as a function of the difference E - E' (or for thermoplasticity E - E'). Therefore the specific Helmholtz free energy y~ has the form: M.4 v(E,E',T) := q(~-e",T) (226) The following form for the specific Helmholtz free energy v is written for the isotropic homogeneous Maxwell material model: 1 w(E,E",T)= -(E -E'):C(T):(C-E") (227) 2 where C is the fourth-order constitutive tensor for a Hookean material. Model (227) shows the state expressed through the difference E - E'. Substitution of Model (227) into State Equations (203) and (204) yields: 99 and Comparison of Equations (228) and (229) gives: P=a (230) Result (230) is a standard result for the Maxwell type of material models. Since the dissipative stress tensor ad vanishes, Normality Rule (208) also disappears. From Result (230) the following form for the yield function F for isotropic homogeneous Maxwell material is achieved: I F(o;T) = -a:~(T):a (231) 2rl P where K(T) is a fouflh-order constitutive tensor. By substituting Model (231) into Normality Rule (209) and taking Result (230) into account it is arrived at: A uniaxial Kelvin-Voigt model is evaluated for simplicity. If the value for C(T) does not vary, Equation (228) gives: 1 which gives iv+ 1.cs = E (233) cV+-o = c - C(T) C(T) Equation (233b) and the uniaxial counterpart of Expression (232) give 100 1 1 (3+ 6 =-& (234) K(T)C(T) K(T) Expression (234) defines the Maxwell model [cf. Flugge (1975, pp. 6 and 22)J. 7.3 STATE EQUATIONS AND THE NORMALITY RULE FOR MAXWELL TYPE OF MATERIAL MODELS The purpose of this section is to evaluate the forms for the state equations in a case where the state is expressed by the difference E - E" or by the quantity (E - E'). Since the goal of this report is the evaluation of thermoplastic material models, the plastic strain tensor E' is used in the evaluation. Of course, the results obtained are also valid for constitutive equations of viscous deformation. The second topic of this section is evaluation of the normality rule for the Maxwell type of material models. The material is assumed to be homogeneous. The present section assumes that the state is expressed by the set (E, E', a',a:. T). Thus there are two internal state variables beyond the plastic strain E' for describing the internal deformation mechanisms of the material. Next a model for thermoplasticity is prepared. The following model, which repeats the concept of Model (226), is introduced: which means that in the theory of thermoplasticity the state is expressed as a function of the difference E - and therefore the specific Helmholtz free energy F is dependent on the variables E and cP through the difference E - E~. The subsequent investigation is for evaluation of new forms for the state equations that replace State Equations (203) ...( 205) due to the introduction of Model (235). State Equations (203), (204) and Model (235) give: 101 In the latter one of Equations (236), Notation (225) and Result (230) are exploited. The chain rule allows the derivative in State Equation (236a) to be manipulated as follows: as(e - eP, al, a’, T) - a@(E -ep, a’,a’, T) , a(E - eP) aE ae (237) where the following three facts were exploited: First, the derivative of a second- order tensor with respect to itself is a fourth-order identity tensor I. Second, the derivative of a scalar (here it is +) with respect to a second-order tensor (here E - E~)is a second-order tensor. Finally, a double-dot product between an arbitrary second-order tensor A and a fourth-order identity tensor I gives the second-order tensor A. These results are given in Expressions (26a), (17) and (26c). Substitution of Manipulation (237) into State Equation (236a) yields: dip( E - ep, a’,a2, T) ,q ,q = P (238) 3( e - EP) Correspondingly to Manipulation (237) it is obtained: 102 Manipulation (239) and State Equation (236bj give: a=pa@( e - ep, a',a', T) = (Jq (240) O(& - eP) where State Equation (238) is exploited. Result (240) is identical to Model (223). This means that Equation (223) is a consequence of State Equations (203) and (204) as well as of Models M.3 and M.4. Therefore Equation (223) is not a model. Model (235) provides the following form for State Equations (204) and (205): and = -p E -E', a2, p* a@( a', T) (242) a a' and finally With the introduction of Model (225) and Result (230), and by remembering that ad 5 0, Expressions (206) and (207) can be rewritten for the present set of internal variables as follows: 103 and Normality Rule (211), (212) and (213) is recast in the form (for homogeneous material): and and finally &2 = dF( u. PI, p2; e - cP, a', a2,T) 1 (248) ap2 where the plasticity multiplier takes the form: i = i(~-eP,a',a2,T,u,u,'T) (249) The following form for Consistency Condition (21.5) is arrived at: F(o,p',p';e-~~,a',a~,T)= 0 during plastic flow (2so) 7.4 MAXWELL TYPE OF MATERIAL MODELS WITH ELASTIC STRAIN TENSOR E" The aim of this section is to formulate the state equations for cases where the 104 specific Helmholtz free energy u, is expressed by the elastic strain tensor The material is assumed to be homogeneous, i.e. h(x) = 0, k(x) 0 and g(x) = 0. Often when researchers apply the theory of thermoplasticity they introduce a new variable defined by: (251) and refer to it as the elastic strain tensor. This variable, however. does not give any new information, since it is defined by two already introduced quantities, F and E~.Because of Notation (251) State Equations (238), (240), (242) and (243) take the following forms: dij( ee, a',a', T ) a=p (252) aEe and furthermore dy(ee. a',a2, T) p' = -p a\?l(ee,a',a2,T) and p' = -p (253) aa' aa2 and finally It is important to note that State Equations (252) ...(254) do not indicate that the I state is expressed by the variables (E~,a', a2,T) but the correct set is (E, E~,a , a', T) where E and T are controllable state variables and E~,a' as well as a* are internal state variables. The notation q is used here in two different cases, namely in q(~- E~, ...) and in ...). Although the independent variables in these two cases are different (E-E~or E~)the function ii, is the same. Thus, if the function @(E - E~,...) is known the function P(E~,...) is obtained simply by replacing the quantity E - by the variable E~. 105 For the present case the forms of the rate of entropy production S', the Clausius- Duhem inequality and the normality rule take similar appearances as in the previous section, see Expressions (244) ...( 248). According to Expression (244) the plastic strain rate Ep is a variable of the problem. Since the rate of the internal state variable a' is present in Expression (244), the plastic strain is also at least a variable of the problem. The quantity is not on the list of state variables, but evidently it is not a controllable variable. It has to be an internal variable, but seems not to be a state variable. This kind of argumentation may have led Muschik (1992a, p. 48) to draw the conclusion that 'an internal variable may be a state variable or not'. The author cannot agree with his opinion, but feels that Muschik has been misled by an incorrect set of internal variables, i.e. the set (E~,a', a'. T) instead of the set (E, E', a',a', T). Normality Rule (246) ...(249) is recast in the form: and and finally (257) where the plasticity multiplier takes the form: k = k( ee, a',a', T, u, u, 'T) 106 Consistency Condition (250) yields: F(~,B',PZ;~e,a',cr',T)= 0 during plastic flow 107 8 MATERIAL MODELS FOR ISOTROPIC AND KIN EM AT IC HARDEN IN G The aim of this chapter is to derive a general formulation for thermoplastic behaviour showing isotropic and kinematic hardening. The constitutive equations presented in this chapter are of the Maxwell type of material models with no more than two internal variables beyond the plastic strain tensor cP. These two internal variables are denoted by a' and a' the conjugate forces of which are p' and p'. The material is assumed to be homogeneous, i.e. h(x) 0, k(x) = 0 and g(x) E 0. At the end of the chapter some particular models are evaluated. 8.1 RATES OF INTERNAL FORCES 'p' AND '$ This section derives expressions for the rates of the internal forces 'p' and 'p' needed in the subsequent section. M8.1 It is assumed that the internal state variables a' and rx2 have no influence on the relationship between the stress tensor u and the elastic strain tensor Therefore Notation (202) in Chapter 7 can be formulated further as follows: E = ee(u,T) + ~~(ar',a~,T) where the notation M8.1 refers to the first assumption for the model made in Chapter 8 and holds only in this chapter. Modifying Lubliner(1972, Chap. 4) [see also Lubliner (1990, p. 63)],the following can be argued: If decomposition of the strain tensor E into elastic and plastic parts is assumed to take Form (260) then such a decomposition is compatible with the existence of the specific Helmholtz free energy a',$, T) if, and only if, + can be decomposed as: 109 +(ee,al,a2,T)= qe(ee,T) + +P(al,a2,T) Based on Decomposition (261) and the fact that the order of the partial derivative operators 313~~and aiaa' can be changed, the following manipulation can be made: where 0 is the fouflh-order zero tensor, the components of which vanish. Correspondingly it is obtained: (263) where 0 is the second-order zero, tensor the components of which vanish. Recalling State Equation (253a) gives: Based on State Equation (264) the rate of the internal force 'PI reads: Normality Rule (256) and (257) has the following appearance: a' - dF a2 - dF - h- and - h- (266) aP1 ap* Substitution of Result (262) and Normality Rule (266) into Expression (265) yields: 110 (267) Correspondingly it is obtained: (268) 8.2 CONSISTENCY CONDITION AND THE PLASTICITY MULTIPLIER This section studies the consistency condition and derives the expression for the plasticity multiplier of the normality rule. Recalling Consistency condition (259) and omitting variables not needed in this chapter the following is achieved: F(a,B',P';T) = 0 The chain rule gives: Substitution of Expression (270) into Consistency Condition (269), and subsequently of Results (267), (268) into the obtained condition the following is arrived at: =0 Derivation of Expression (271) exploits the fact that for second-order tensors a and b, a : b = b : a holds. This fact was utilised in the second term within the first square brackets. Expression (271) gives: where on the right-hand side the first row is the numerator and the second as well as third rows form the denominator. The Heaviside function H(F), see Definition (19), is added to guarantee that the plasticity multiplier vanishes in pure elastic region [cf. Lemaitre & Chaboche (1990, p. 194)l. The McAuley brackets ( ), see Definition (29), are adopted to imply that the plasticity multiplier itakes only positive values {see Nguyen & Bui [1974, Eq. (7)1}. 112 8.3 CLAU S I US-DU H EM I NEQUALITY This section evaluates the restrictions for satisfaction of the Clausius-Duhem inequality. The local part of Clausius-Duhem Inequality (137) is recalled. Equation (245) thus gives: Substitution of Normality Rule (246) ...(248) into Inequality (273) gives: where the order of the terms in square brackets has been changed 8.4 PARTICULAR MATERIAL MODELS The aim of this section is to introduce some models for material behaviour showing isotropic and kinematic hardening. 8.4.1 Duhamel-Neumann form of Hooke's law This subsection provides a material model for description of elastic deformation obeying the Duhamel-Neumann form of Hooke's law. The thermoplastic response can be determined independently of this presentation. The model for plastic flow has to follow the concept given in Chapter 7 and the previous sections of this chapter. Modifying Ziegler and Wehrli [1987, Eq. (4.14)], the specific Helmholtz free energy Qe for linear isotropic material can be written in the form: 113 ijje(ee,T) = ijj;-so(T-To) +-E~:C:E~--I:E~(T-T~)1 K 2P P (275) + -1 a': - -c0 (T - To)2 P To where +: , so, a' and To are the specific Helmholtz free energy, the specific entropy, the stress tensor and absolute temperature in the initial configuration, respectively, where K is a constant governing thermal stresses and C, is the specific heat (at constant deformation) of the material in the initial configuration. The specific heat C is defined in Equation (108). The second-order identity tensor 1 is defined by Equation (20). The foufih-order constitutive tensor C (for Hooke's law) is defined by: c := All + 2HlS where h and p are the Lame elastic constants defined by: vE E h := and p := - (277) (1 +v)(1-2v) 2(1 +v) where E and v are Young's modulus and Poisson's ratio, respectively. The component Iilk,' of the fourth-order tensor Is is defined by Definition (21). Running State Equation (252) through Decomposition (261) gives: a+( e ', a', T ) d\i,"(~~,~) a=p d, =P aEe a ee Substitution of Model (275) into Equation (278) gives: u = C:E~- K(T-T~) 1 + u0 (279) where the symmetry of the strain tensor is exploited [see Appendix E]. Constitutive Equation (279) is the Duhamel-Neumann form of Hooke's law. 114 8.4.2 Special models for plastic flow This subsection provides some constitutive equations for plastic flow. By following the concept presented here the reader can introduce other material models for thermoplasticity. The following yield function showing isotropic and kinematic hardening is evaluated. The model is formulated as: where J,() is the second invariant of the second-order deviatoric tensor (), see Definition (22). Expressions (F.12) and (F.13) of Appendix F and Model (280) give: at= a -au = -[J,(u-~')]au = s - b' and JF a -- - -[Jz(u-pl)] = -(s - b') apl apl where s and b' are the deviatoric stress tensor and the deviatoric part of the internal force, respectively. The definitions for s and b' are expressed by Definitions (23a) and (23b). Based on Model (280) it is obtained: Substitution of Expressions (28 l),(282) and (283) into Normality Rule (246) ...(248) provides the following result: 115 and and finally (286) Based on Definitions (23) and (24) the deviatoric plastic strain rate tensor 'ep is defined by: where ) is the first invariant of (the strain rate) tensor 'E~defined by: Because the plastic strain E' is associated with deformation without volumetric strain, the following holds: I,(&P) = 0 (289) and Expression (287) collapses to: By combining Normality Rule (284) and Result (290) with definition for the second invariant, see Definition (22), the following manipulation is arrived at: During plastic flow F = 0. Therefore Model (280) gives for plastic yield: 1 I6 J2(a - p’) = (p2)’ By substituting J2(o- PI) in Equation (292) into Expression (291) and the obtained result into Normality Rule (286) it is found: (293) Recalling State Equations (253) with application to Decomposition (261) gives: p’ = -p 3ijlp(a’,aZ,T) and ad Two yield criteria are evaluated in more detail below. Case I von Mises yield criterion (von Mises, 1913) It is assumed that the kinematic hardening is negligible, which implies that all the components of the internal force P’vanish. The specific Helmholtz free energy qp is modelled as: where the scalar-valued quantity k is a material parameter. Substitution of Model (295) into State Equation (294b) yields: p2 = -- k a2 which gives 6 By substituting Expression (293) into Material Derivative (296b) it is obtained: 117 ‘p2 = k [J2(kp)]”2 (297) If b2 vanishes in the initial configuration, Expression (297) gives: where the latter integral is called cumulative plastic strain or according to Odqvist ( 1933) the Odqvist parameter. In this case Normality Rule (284) collapses to: &P =is (299) Case II Prager yield criterion (Prager, 1955 and 1956) Although the material model presented here is usually attributed Prager (I955 and 1956) it should be remembered that he did not write this model but investigated plastic flow by drawing movements of yield surfaces due to hardening. The present model is a mathematical formulation of his drawings. The author is not aware of who was first to write the mathematical formulation for the following constitutive equation. It is assumed that isotropic hardening can be neglected, Le. p2 can be approximated to vanish. The specific Helmholtz free energy qPtakes the form: where c is a material parameter. Substitution of Model (300) into State Equation (294a) yields: = -ea which gives p*I = -ca' (301) Substitution of Normality Rule (285) into Derivative (301b) provides the following rule for kinematic hardening proposed by Prager: 6' = c EP (302) Plastic strain rate EP can be evaluated from Normality Rule (284). viz. (303) kP = A(s - bl) Based on Definitions (23a) and (23b) the deviatoric tensors ep and b' for the tensors kp and '9'can be obtained from: &P := K.iP and 1;' := K: 6' (304) where the fourth-order tensor K is given by Definition (24). Multiplying Hardening Rule (302) by the fourth-order tensor K from the left gives: K:p' = cK:kP (305) which according to Expressions (304a) and (304b) reads: (306) bl T c&P Based on Identity (290), i.e. eP = kp, Equation (306) can be written as: *I b =ckP (307) Comparison of Normality Rules (302) and (307) shows: It is reasonable to assume that the values for the quantities b1and p' vanish in the initial configuration. This assumption and Identity (308) lead to the following res uI t : 119 b' = (309) Based on Result (309) Normality Rule (303) can be written as: &P = i(s - PI) (310) The evaluation equation for the internal force p' is expressed by Equation (302). 120 9 GURSON-TVERGAARD MATERIAL MODEL The aim of the present chapter is to evaluate the Gurson-Tvergaard material model. As the original works by Gurson (1977) and Tvergaard (1981) included neither thermomechanical investigation nor damage mechanics evaluation, they are not discussed here. Their results are included in the next chapter which introduces proposals by the author. In Gurson's (1977) constitutive equation for modelling pressure dependent plastic deformation of metallic materials, pressure dependence is due to void nucleation and growth. Gurson evaluated a unit cube of porous material the void volume fraction of which was denoted by the variable f. He assumed that the matrix material, i.e. the material between the voids, can be approximated by the rigid- perfectly plastic material model. Gurson utilised the von Mises yield criterion in determining the onset of plastic yield in the matrix material. In his evaluation Gurson used two types of variables - macroscopic and microscopic. He studied a small piece of porous material which he called 'a "unit" cube of porous material of volume V, large enough to be statistically representative of the properties of the aggregate'. The term unit cube is here replaced by the term 'representative volume' denoted by Vrep.The representative volume of the material vrePis large enough to be used in homogenisation of microscale effects in the macroscale material model, yet is small enough to provide values for continuous field variables at a given point in a structure. In a macroscopic material model, the real structure of the material in the representative volume Vrep is averaged over the representative volume Vrep throughout which the macroscopic field variables take the same values. Values of the microscopic quantities, however, vary realistically within Vrep, In the following text the terms variable, quantity etc. apply to the macroscopic phenomena. Only in the cases where the author wishes to stress the difference between macroscopic and microscopic quantities is the term macroscopic used. 121 By using the upper bound approach Gurson derived macroscopic yield functions for materials having different shapes of voids. The result for spherical voids with fully plastic flow is referred to as the Gurson model. The yield function for the Gurson model is [Gurson, 1977 (Eqs. (3.12) and (4.9)J: where J2(o) is the second invariant of the deviatoric stress tensor [see Definition (22)1, oy is the microscopic equivalent tensile yield stress and I,(u) is the first invariant of the stress tensor [see Definition (28)l. Based on Gurson [1977, Eq. (2.la)l the quantity oy is the uniaxial yield strength of the matrix material. The parameter oy has a fixed value. In Yield Function (31 I) the variables u and f are macroscopic, which means that they describe the response of the whole representative volume Vrep, i.e. they are variables of an ideal material where the behaviour of voids and matrix material is homogenised within the representative volume Vrep. In the appendix of his paper Gurson showed that the normality rule holds also on the macroscopic level. Therefore the normality rule derived in the previous pages of this work can be used with Yield Function (31 1). Some years later Tvergaard (1981) investigated the influence of voids on shear band instabilities. He made his investigation by studying the problem numerically using two different approaches. First, he performed a finite element method analysis using a fine microscale mesh for description of an elastic-plastic medium containing a doubly periodic array of circular cylindrical voids. Second, he used a continuum model with a macroscopic material model for evaluation of the same problem. For his continuum model analysis Tvergaard modified Gurson Yield Function (31 1) by introducing three new material parameters and assuming that the matrix material shows strain hardening. The latter means that Tvergaard replaced the constant oy of Yield Function (31 1) with the variable (describing hardening) denoted by o’. According to Tvergaard, oM represents the equivalent 122 tensile flow stress in the matrix material, disregarding local stress variations. Tvergaard [ 1981, Eq. (5.l)] introduced the following yield function: F(u,o’,f) = He further added to the model elastic deformation, which he assumed to obey Hooke’s law. Tvergaard introduced his modification in order to obtain a better fit between the numerical results of his microscopically detailed finite element analysis and the macroscopic continuum model [Tvergaard (1981, p. 404)j. By assuming that the M constants q’ = q2 = q’ I, that the variable o takes a constant value oy,and that the elastic deformation can be neglected as a small quantity, the Gurson- Tvergaard material model reduces to the Gurson model. Elastic deformation is described by Hooke’s law. Constitutive Equation (279) is recalled by neglecting thermal effects and initial stress ao to give: (313) a = C:ee The evolution equations for the quantity oM.for the void volume fraction f and for the plastic strain eP are derived next. Although Yield Function (31 I) proposed by Gurson contained the void volume fraction f as a variable, he did not give an evolution equation for it. The following expressions for irMand f are by Tvergaard (1981, pp. 402 and 403). The expression for trM is evaluated first. The following introduction is by the author. The (classical) plasticity theory [see e.g. Santaoja (1993, pp. 23 and 24)] derives the expression for uM as follows. A uniaxial stress-strain curve is drawn for 123 elastic-plastic material behaviour as shown in Figure 10. In this work the matrix material is assumed to behave accordingly. The uniaxial strain terms of the matrix material are denoted by E~, and The uniaxial yield stress of the matrix material is denoted by omY.The value for omyvaries with hardening. The notations Em and EmPin Figure 10 refer to Young's modulus and the slope of the uniaxial stress-strain curve beyond the elastic region, Le. the tangent modulus, of the matrix material, respectively. A E D fn fn W K k v) STRAIN E"' Figure 10. Uniaxial stress-strain curve for an elastic-plastic material behaviour. Figure 10 gives: which can be cast into rate form, viz. Next is the derivation by Tvergaard (1981, pp. 402 and 403). For purposes of 124 compatibility with the other parts of this work the notations are that of the author. As above, the superscript m refers to the microscopic variables. At plastic loading the increment of the effective plastic strain in the matrix material with current tangent modulus E' is: Assuming that varies according to the equivalent plastic work expression 0:&PV = (1 -f)oM E.Mp (317) the increment of oM is given by: The above derivation raises several questions The first has to do with the terminology. Tvergaard called macroscopic deformation - which is due to (traditional) plastic flow and to void nucleation and growth - 'plastic strain', and gave to it the superscript p {see Tvergaard [1981, Eqs. (5.1) and (5,6)]}. In order to show the difference between pure plastic flow and the plasticity of porous materials, the author has in Equation (31 7) adopted the notation and called it the void-plastic strain tensor. According to Tvergaard (1981, p. 402) the quantity oM represents the equivalent tensile flow stress in the matrix material, disregarding local stress variations. Thus, oM is a 'semi-macroscopic' variable. It is obtained by reducing the multiaxial stress field into a uniaxial one (e.g. by using the von Mises equivalent stress concept) and averaging it over the matrix material in the representative volume Vrep. The (real) macroscopic variables are averaged over the entire volume of Vrep.Figure 11 clarifies the differences between the terms microscopic, 125 semi-macroscopic and macroscopic. Uniaxialised and averaged values Exact values (not in the voids) Averaged values Microscopic quantities Semi-microscopic Macroscopic quantities quantities a) b) c) Figure 11. (a) Values for the microscopic quantities vary within the matrix material. They are not defined in voids. (b) Semi-macroscopic quantities are obtained by averaging the microscopic values and by reducing the nine tensor components into one scalar-valued variable. They are not defined in the voids. (c) Macroscopic quantities are averaged over the entire volume of the representative volume vrep. In order to make some sense of Expression (316) the quantities EMP and E' must also be semi-macroscopic. The derivation of Expression (316) is lacking. As the author believes that Equation (315) inspired Tvergaard to write Expression (316), the latter must therefore be interpreted as a model. A problem is the determination of the value for the current tangent modulus E'. Since it is a semi- macroscopic variable, it should be some kind of an integral of the microscopic Young's moduli over the matrix material in the representative volume Vrep.Since no stress-strain curve for semi-macroscopic material behaviour is available, researchers may use in their computer analysis the microscopic stress-strain relation (sketched in Figure 10) and they may replace the microscopic stress om by oM and the microscopic strain by E~.Thus in practice the axes are semi- macroscopic but the stress-strain curve is a microscopic one. 126 In the (traditional) theory of plasticity, when isotropic hardening is modelled the following two different concepts are often utilised [Malvern (1969, pp. 364-367) is followed here]. In the following consideration the term 'microscopic' is adopted to show that the quantities presented are microscopic ones in the sense of this work. Malvern did not use this terminology. The next simplest (after perfect plasticity) hardening assumption - that the yield surface maintains its shape, while its size increase is controlled by a single parameter depending on the plastic deformation - is called isotropic hardening. It remains only to specify the size-determining parameter and its dependence on deformation. Two different schemes have been used for this, which are different in general but reduce to the same when used appropriately with the von Mises yield condition. Universal plastic stress-strain curve The first one is the idea of a universal plastic stress-strain curve. With this concept it is assumed that there exists a universal plastic stress-strain curve relating two scalar quantities - the (microscopic) effective stress omeff(measuring the size of the yield surface) and the integral of the (microscopic) effective plastic strain increment dEmPeff.Thus the universal plastic stress-strain curve is defined by: (319) When the von Mises yield condition is used Expression (319) takes the form: omMi = H(lde::) (320) where the subscript 'Mi' refers to the effective stress according to von Mises. 127 Enerqy condition for strain hardeninq The second concept is that the size of the yield surface depends only on the (microscopic) total plastic work WmP.This means that the (microscopic) effective stress omeffis a single valued function of (microscopic) total plastic work WmP.It can be written as: (321) .,"I, = F(Wmp) where (322) When the von Mises yield function is used and the following is obtained [Malvern (1969, p. 367)j: Definition (322) and Expression (323) give: mm arn:dernp = (324) OMi deMi or the same in the rate form: (325) In writing Equation (3 17) Tvergaard may have integrated Result (325) "formally" over the representative volume Vrep. The term (1-f) on the right-hand side of Expression (317) originates from the fact that the quantities oM and EM are not defined in the voids where the macroscopic quantities u and Epv have a value. According to Expression (323) when the von Mises yield condition is used the (microscopic) plastic Work WmP equals the area under the universal plastic stress-strain curve defined by the (microscopic) von Mises effective stress oM,rn 128 and the (microscopic) effective von Mises plastic strain increment dEMimp, In the uniaxial tension/compression the (microscopic) stress om coincides with the (microscopic) von Mises effective stress oMim.The same holds for the strain measures. This means that the uniaxial plastic stress-strain curve can be interpreted to be the universal plastic stress-strain curve. The former is then connected to the multiaxial quantities through Equation (324) which is widely used in the traditional theory of plasticity {see e.g. Santaoja I1993, Eq. (49)) or Zienkiewicz [1977, Eqs. (18.43) and (18.49a)l). Other conditions The Odqvist parameter, Expression (298), can be used as a measure of isotropic hardening [see Kachanov (1974, p. 62)j. The expression for f is evaluated next, starting with the derivation by Tvergaard. As the matrix material is taken to be plastically incompressible, the increment of the void volume fraction is given by: f = (1 -f) I,@") (326) The author's comments are as follows The following derivation is based on private discussions with Mikkola (1996). According to Narasimhan [1993, Eq. (4.10.30)] the material derivative of the representative volume Vrep takes the following appearance: Narasimhan [ 1993, Eq. (4.10.26)] gives the relationship between the divergence of the velocity vector a .vand the rate of deformation tensor d as: 129 For small displacement gradients and velocity gradients the rate of deformation tensor d and the strain rate tensor i: coincide {see e.g. Narasimhan [1993, Eq. (4.10.20)1}, viz. d=E (329) Combining Equations (327) ...( 329) gives: The volume of the matrix material within the representative volume can be expressed as: v;;t v;;t = (1 -fj vrep (331) which leads to: By substituting Equation (330) into Equation (332) and rearranging the obtained terms the following is arrived at: . rep f vrep = (1 -f) I,(&)vrep - Vmat (333) which gives: ' rep "mat f = (1 -f) I,(&)- - (334) VreP The plastic deformation of the matrix material does not cause any volume changes. The elastic deformation of the matrix material can be assumed to be very small and therefore the volume change due to elastic deformation can be neglected as a small quantity. This means that the term Vmatrep/Vrep can be 130 assumed to vanish, which implies that the quantity I,(€)is due to void nucleation and growth. Thus Expression (334) gives the following result: f = (1 -f) II(EV) (335) Since pure plastic flow does not contain any dilatation, the strain rate measures I,(Ep") and I,(EV)coincide. Therefore Result (334) has the same form as Equation (326), which is exactly what was aimed at. The following conclusion can be drawn. The introduction of parameters q', q2 and q3 is an ad hoc assumption for the particular problem analysed by Tvergaard. These parameters lack a physical interpretation. Gurson introduced the constant 0' for his yield function only in order to make some quantities dimensionless {see Gurson [ 1977, Eq. (3.12)]}. Gurson called this a normalisation act. This act has to be done, since without it Yield Function (31 1) has a Cosh-function the argument of which has dimension of stress. The independent variable of a Cosh-function has of course to be dimensionless. Gurson may have adopted into his yield function the value oy, which is the microscopic equivalent tensile yield stress, since it provides a form for the yield function where its terms take values close to unit. Since the introduction of CT' has no physical foundation, replacement of the constant CY' by the variable oM can be criticised. The earlier comments on derivation of the evolution equation for the quantity rsM supports this criticism. Furthermore, as discussed by Aravas (1996), the upper bound approach used by Gurson is not valid for a strain hardening case. This means that for a strain hardening material behaviour the yield function can take a totally different form. On the other hand, the evolution equation for the void volume fraction f introduced by Tvergaard is obtained after rigorous mathematical derivation which 131 is based on definitions in mechanics, and is therefore it is strongly recommended by the author. 132 10 EXTENSION FOR THE GURSON-TVERGAARD MATERIAL MODEL The aim of this chapter is to introduce an extension to the Gurson-Tvergaard material model. This new constitutive equation is formulated using the basic laws, axioms and definitions of thermomechanics and the results provided by damage mechanics. The main difference between the present model and that by Gurson-Tvergaard lies in the description of elastic deformation. The extended constitutive equation models the elastic deformation by taking into consideration the stiffness reduction of the material due to voids, whereas the original Gurson-Tvergaard formulation simply uses Hooke's law. The difference between these two descriptions may appear to be negligible, since the elastic deformation plays a minor role in the total deformation of the body. It is true that elastic deformation has a small value by comparison with plastic deformation and therefore the additive deformation due to damage is negligible. The idea of introducing the 'correct elastic stiffness' is that it enables the methods of damage mechanics to be introduced and thereby gives the necessary information to construct the yield function F. The independent state variables of the extension for the Gurson-Tvergaard material model are the strain tensor E, the void-plastic strain tensor and the void volume fraction f. This means that by following the concept of the originai model the thermal effects are neglected. The extended Gurson -Tvergaard material model is a Maxwell type of material model and therefore assumes that the state is expressed by the difference E - E~'. Thus, as indicated by Notation (251), the elastic strain tensor defined by := E- E~"is introduced and the state is expressed by the specific Helmholtz free energy having the form: \i.I = q(ee,f) (336) Once again it should be stressed that Form (336) does not indicate that the 133 elastic strain tensor E" is a state variable. 10.1 SPECIFIC COMPLEMENTARY HELMHOLTZ FREE ENERGY qc This section introduces the explicit form for the specific complementary Helmholtz free energy ijc, and using both it and the state equations, the formulation of the explicit expressions for the elastic strain tensor and the internal force g. The internal force g is a state function conjugate to the independent state variable, void volume fraction f. Eshelby (1957) has studied the elastic field in a Hookean material containing an ellipsoidal inclusion. As a special case he determined the value for the complementary strain-energy density of a material containing 'a volume fraction f' of inhomogeneous spheres. For the purpose of this work the inhomogeneous spheres are 'replaced' by spherical cavities. This is done by assuming that the values for the elastic constants for the cavities vanish. The spherical cavity problem was approached earlier by Mackenzie (1950), but the form of the result by Eshelby better fits the needs of this work. The complementary strain-energy density wC(u,f) takes the form (Eshelby, 1957, p. 390): 1 (1 +Af)[I,(u)l' + -(I1 +Bf)s:s WC(U,f) = - 9(h+-p)2 2P 3 where I,(u) is'I the first invariant of the stress tensor [see Definition (28)l and where h and p are the Lame elastic constants given by Definitions (277) {see Eshelby [1957, Eqs. (2.2) and (2.3)]}. In Model (337) the stress tensor u is defined as a macroscopic stress in the sense of the work by Gurson (1977). For spherical cavities constants A and B take the forms [see Eshelby (1957, pp. 389-3901]: 134 6p+3h 15 (1 A= and B= -u) (338) 4P 7 -5u According to Malvern (1969, pp. 282-283), for the fully recoverable case of isothermal deformation with reversible heat conduction the following holds: W(&") = p \ir(&") (339) where w(E~)is the strain-energy density and where \ir(~") is the specific Helmholtz free energy. Evidently Expression (339) allows the following extension to be written: w(e",f) = p \ir(C",f) (340) which also holds for the complementary functions: pqf(a,f) = WC(U,f) (341) where the initial values for the complementary Helmholtz free energy Qc and the stress tensor o are assumed to vanish and where the temperature effects are neglected [cf. Equation (275jl. By modifying State Equations (252) and (253b) for the purpose of this chapter we get: and &e,f ) (343) 9 = -P a\ircee' which gives -g = P af f, a State Equations (342) and (343) cannot be used without modification, since instead of the specific Helmholtz free energy f) the specific complementary Helmholtz free energy qc(a, f) is available. The specific complementary Helmholtz free energy qc(a,f) is a Legendre partial transformation of the specific 135 Helmholtz free energy f), The former quantity is defined by: ptjC(o,f) := a:&e- p+(&e,f) (344) Comparison of Transformation (344) and State Equations (342) and (343b) with the corresponding Expressions in Appendix C, Expressions (C.2) and Transformation ((2.3) gives a = p and b = p. It is noteworthy that the variable -g in State Equation (343b) is the corresponding variable for the quantity y' in Expression (C.2) of Chapter C. Thus, Expressions (C.6) of Appendix C yield: a+"( 0, f ) &e = p and g=P a. f ) (345) a0 af Substitution of Model (337) into Equation (341) and of the obtained result further into State Equation (34Sa) gives: 1 1 &e = +Af)l,(a) + -(1 +Bf)s (1 1 (346) 9(h+-P)2 2P 3 The second-order identity tensor 1 present in Equation (346) is given by Definition (20). In the derivation of Expression (346) the following facts were exploited: The first invariant of the stress tensor Il(o) is defined by [see Definition (28)]: where Properties (26a) and (26c) are utilised. According to Definitions (23a) and (24) the deviatoric stress tensor s is obtained from: 136 1 s := K:o and K := I--1 1 (348) 3 Thus the following manipulation can be written: = [K:l]:s + s:[K:l] 1 1 = ((I -- 1 l):l]:s+ s:[(I-- 1 1):1] (349) 3 3 1 1 = [I --111:s + s:[l--111 3 3 =s+O+s+0=2s where also Property (26b) and the fact that the first invariant of the deviatoric tensor vanishes {see e.g. Lin [1968, Eq. (1.4.6)]} are exploited. The first invariant of the deviatoric stress tensor Jl(a) is defined by: Jl(o) := 1:s (350) The following facts: I1(o)1 = 1 I](O) = 1 I:o and s := K:u (3.51) allow Constitutive Model (346) to be recast as: 1 1 &.e = (1 +Af)ll:a + -(l+Bf)K:a 2 (352) 9(h +- 2cL 3 P) which can be written as: ee = S(f): o (353) where S(f) is the fourth-order compliance tensor for the Hookean material with spherical voids. It is defined by: 137 1 1 := S(f) (l+Af)ll + -(l+Bf)K (354) 9(h+1.p) 2P 3 The compliance tensor S(f) can be separated into two different parts, viz. S(f) = s + SV(f) (355) and thereby Constitutive Model (352) can be written as: (356) &e = [s + SV(f)] : (J The compliance tensors S and SV(f) are defined by: 1 I s := 11 + -K (357) 9(h+1. p) 2cl 3 and Af Bf SV(f) := I1 +-K 9(h+-P)2 2P (358) 3 Substitution of Model (337) into Equation (341) and of the obtained result into State Equation (345b) yields: A [l,(a)I2 + -s:sB (359) = 7 9(h+-p)2 2P '1 3 Multiplying Definition (358) by the stress tensor o from the leftI and from the right gives: A B o:SV(f):a =f [ll(0)l2+ -s:s (360) 9(A +-2 cl) 2P 3 where a : 1 = 1 : u = I,(u),s = IK : o and a : s = s : s are exploited.I Comparison 138 of Expressions (359) and (360) gives: g = --a:S"(f):a1 (361) 2f 10.2 EFFECTIVE STRESS TENSOR 6 The purpose of the present section is to introduce the effective stress tensor u. This is done using damage mechanics. It is noteworthy that the effective stress u differs from the effective stress beffintroduced in the previous chapter. The following investigation is more or less a copy of the works by Santaoja (1988, Chaps. 2 and 3), Santaoja (1989, Chap. 3) and Santaoja (1990, Sect. 3.3.2). It should be noted that the terminology and notations in the above publications differ from those which used here. This is because prior to the mid 1980s, when the author started his damage mechanics investigations, the notations did not have the form they have today. According to Chaboche (1978, p. 19) the constitutive equation for the damaged Hookean material can be expressed as: (I = C(f):&e and 6 := c:ee (362) where C(f) is the fourth-order constitutive tensor for damaged Hookean material. The above model can also be found in Lemaitre and Chaboche (1990, p. 397) although this latter source contains a misprint. The author has added the variable f for the constitutive tensor C(f). According to Lemaitre and Chaboche (1990, p. 350) the effective stress tensor 6 is the stress calculated over the section (of the damaged material) which effectively resists the forces. The same interpretation for the effective stress tensor 6 can be found in Rabotnov (1968, p. 344), in Hult (1974, p. 139) and in Santaoja (1989, Sect. 3.1). Santaoja (1990, Sect. 3.4) made a simple uniaxial 139 tubeibar investigation to show the above interpretation for the effective stress tensor u. Thus in the material modelled in this work the effective stress tensor 6 describes the stress in the matrix material between the voids. It is of course a homogenised stress over the representative volume Vrep of the damaged material. The inverses of the constitutive tensors C and C(f) are denoted by C-I or S and C-'(f) or S(f), respectively. According to Definitions (27d) and (26c) for a second- order tensor A the following holds A-' : A = I and I : A = A. Thus, by multiplying Equation (362a) from the left by the tensor S(f) the following expression is obtained: (363) Ee = S(f):u which is identical to Expression (353). Substitution of Equation (363) into Model (362b) gives: 0 = c:S(f) :u (364) By introducing the fourth-order damage effect tensor denoted by T(f) Expression (364) can be written in the form: where (365) Separation of the compliance tensor S(f) into two different terms, as in Equation (355),leads to the following equation: (366) r(f) = c:[s + sv(r)] = I + C:sV(f) to describe damage. The damage effect tensor r, the component of which is riik,, is symmetric in the indices ij and kl [Santaoja, 1990, Appendix (A.2)]. 140 The damage effect tensor r(f) in Equation (365a) is quite similar to that proposed by Murakami and Ohno (1980, p. 428) if they applied a second-order tensor to describe damage. However, as their approach led to an unsymmetric effective stress tensor 6 (they called it the net stress tensor) they had to introduce a new symmetric net stress tensor. By introducing the fourth-order damage tensor denoted by D(f) and defined by the equation: [I - D(f)]-' = T(f) (367) Equation (365a) can be written in the form: 6 = [I - D(f)]-' :u In the uniaxial tension oilf 0 Equation (368) leads to the equation: 01 I tiil = 1 - Dill, The component D, I is identical to the scalar valued damage introduced by Hult (1974, p. 139) and applied by many others, e.g. Chaboche (1978, p. 17). As a final remark beyond the papers by the author mentioned in the beginning of this section it should be mentioned that the damage effect tensor T(f) is identical to the fouflh-order tensorial variable called damage operator and denoted by A (Lemaitre and Chaboche 1990 p. 397). 10.3 YIELD FUNCTION F AND THE EVOLUTION EQUATIONS This section introduces the explicit form of the yield function for the extended Gurson-Tvergaard material model. The final result is the evolution equations for the plastic strain tensor E~,for the void volume fraction f and for the effective 141 stress tensor u which is a corresponding variable for the quantity oM in the Gurson-Tvergaard material model. Section 7.4 gave Normality Rule (255) ...(257), viz. and furthermore 0 aF $ - 0 aF a' =A- and - h- (371) ap2 where based on Consistency Condition (259) the yield function F has the following implicit appearance: F = F( (I, p', p2; ee, a',a2, T) (372) According to the beginning of Chapter 10, the extended Gurson-Tvergaard material model is described by the state variables E, and f. The conjugate variable for the void volume fraction is the internal force g. Thus for the extended Gurson-Tvergaard material model Normality Rule (370) and (371) is replaced by: 0 i3F and f =h- (373) ag The extended Gurson-Tvergaard model, however, neglects thermal effects, whereas the state variables E- and f are the parameters of the yield function F. Thus it is written formally as: where the first two variables are the forces and the latter two parameters are the state variables (note that is not a real state variable but represents the difference E - E~'). The extended Gurson-Tvergaard material model is assumed to have the 142 following yield function: F(u,g;Ee,f) = J2(u) + 2 f q' Cosh J2(C:ee) 2 [3 J,(C: (375) - (1 + q' f2) q1 q2 Sinh[ q2 I,[C(f) : Eel g + 3(1-f) ] [3J,(C: 2 [3 J2(C:~~1''~ Substitution of Yield Function (375) into Normality Rule (377a) yields: where the Definition (362b) and the following facts are exploited: The first invariant of the void-plastic strain rate tensor I,(Epv)see Definition (28), takes the form: where the facts that 1 : s = 0 and 1 : 1 = 3 are exploited. Substituting Yield Function (375) into Normality Rule (373b) gives: f = i(l-f)3 q' q2 Snh[ q2 I'(4 1 (379) [ 3 J2(6) ] "2 2 [3 J2(ii)]''2 where Definitions (362a) and (362b) are exploited. 143 Comparison of Expression (378) with Evolution Equation (379) shows that: f = (1 - f) I,(&P") (380) which is exactly the form that was aimed at [see Evolution Equation (326)J. The evolution equation for the effective stress u is considered next. The value for the effective stress 6 can be calculated directly from the value of the void volume fraction f using Equation (364), viz. (381) 6 = c:S(f) :a where the constitutive tensor C for a Hookean material can be obtained e.g. from Equation (276) and the compliance tensor S(f) from Equation (355). 10.4 CLAUSIUS-DUHEM IN EQUALITY This section studies the satisfaction of the Clausius-Duhem inequality for the extended Gurson-Tvergaard material model. The Ciausius-Duhem inequality, Inequality (274), is recalled: (382) For the extended Gurson-Tvergaard material model it takes the following appearance: r 1 20 (383) Based on Expressions (373a) and (376) the first term between the square brackets of Inequality (383) takes the form: 144 where the fact that s : u = s : s is exploited. The first term on the right-hand side of Expression (384) is non-negative. The second term requires brief consideration. It is reasonable to assume that the material parameters q1and q' take non-negative values. The first invariant of the stress tensor Il(o) can take positive and negative values. However, the quantity Il(o) Sinh[l,(a)] is always non-negative. This implies that the second term on the right-hand side of Expression (384) is non-negative. Based on Expressions (373b) and (379) the second term of Clausius-Duhem Inequality (383) yields: where according to Expression (359) the force g has the following form: A l,(a)I2 + -s:sB * 9(h+-p)2 2P g=Ll3 Based on Expression (386) the value of the internal forceI g is always non- negative. Since the first invariant of the stress tensor Il(u)can take positive and negative values, the Expression (385) may be negative. This happens during hydrostatic compression. 145 10.5 PLASTICITY MULTIPLIER h9 The aim of the present section is to derive the form for the plasticity multiplier i. According to Expression (374) the yield function F has the appearance: F = F(u,g;ee,f) (387) which leads to the following consistency condition: (388) Recalling Normality Rule (373b) gives: 0 JF f =h- (389) ag By Combining Expressions (388) and (389) we get: (390) 146 11 DISCUSSION AND CONCLUSIONS The concept of this work was to use thermomechanics and damage mechanics in the evaluation of the material model for porous materials proposed by Gurson (1977) and modified by Tvergaard (1981). This work was separated into two parts. First, a general theory for thermomechanics and especially for thermoplasticity was evaluated. Second, the Gurson-Tvergaard material model was considered. The work started with the investigation of the 'basic laws', 'axioms' and 'definitions' which form the foundation of thermomechanics. There are several variations on the theme of thermodynamics; today the most effective approach for description of constitutive equations for solid materials is thermodynamics with internal variables. The theory of thermodynamics (thermomechanics) with internal variables was derived. Although the expressed formulation is very general, the gradient theory, for example, needs a different approach. The role and the definition of internal variables were discussed briefly and the following definition introduced: 'The independent variables present (possibly hidden) in the basic laws for thermomechanics are called controllable variables. The other independent variables are called internal variables.' As this work showed an internal variable is always a state variable. Muschik (1992a, p. 48) has argued; 'An internal variable may be a state variable or not'. Unfortunately he did not give any examples, but based on Maugin and Muschik [1994, Eq. (4.21)j he seems to have assumed that the elastic strain tensor is an internal variable whereas the plastic strain tensor is not. However, later Maugin and Muschik [1994, Eq. (4.22)) obtained a form for the Clausius-Duhem inequality which had the plastic strain tensor as a variable. This may have (mis)led Muschik to take the above opinion. The author does not agree with Muschik for the reason apparent in this publication. According to Section 7.1, for the Kelvin-Voigt type of materials the 147 plastic strain tensor is not a state variable and therefore clearly does not exist in the Clausius-Duhem inequality (2 19b). The formulation of Maxwell type of material models support this fact. Clausius-Duhem Inequality (245) has the plastic strain tensor as a variable and is therefore also present in the set of state variables [see Expression (235)).For both cases the elastic strain tensor is not a state variable. The reason for adopting the elastic strain tensor for the set of state variables may originate from the fact that for the Maxwell type of material models the state is expressed by the difference E - which is by definition the elastic strain tensor This allows the mathematical derivation to be carried out in a way which seems to assume that the elastic strain tensor is a state variable [see the more detailed discussion in Section 7.41. An important result for thermoplasticity obtained here is the fact that. if the mechanical part of the specific dissipation function (plocp is a homogeneous function of degree 1 in the fluxes, the principle of maximal rate of entropy production gives the normality rule for time-independent processes. This result evidently holds for all time-independent processes. Usually in the theory of thermoplasticity the specific dissipation function 'p,,,p(E, a, etc.) is replaced by its Legendre partial transformation, namely, by the yield function F(o, p, etc.) which is expressed by the forces. It was shown that although the specific dissipation function (plo: is a homogeneous function of degree 1 the yield function F can be nonhomogeneous. The approach used in this work leads to the associated normality rule. This means that besides the consistency condition, the yield function is also present in the normality rule. In Chapter 7 the derived general theory of thermoplasticity was employed in the formulation of two different types of material models for thermoplasticity: the Kelvin-Voigt type material models and the Maxwell type of material models, latter being the more popular. Therefore Chapter 8 introduced two traditional models I48 for plasticity having the Maxwell type of formulation. An important message of this work is related to the role of the damage D which can be a scalar, a vector or a tensor of any order. According to Rabotnov (1968. p. 344) ’the measure of damage of the material can be taken as one of structural parameters’. A similar view is expressed in the books and papers by the French school of thermodynamics. Lemaitre and Chaboche (1990, pp. 400 and 404) introduced a scalar-valued damage D and the fourth-order damage tensor D into the set of internal state variables. The same assumption is made by Maugin [1992, Eq. (10.4)]. The author cannot agree with the above, but believes that instead of damage D a physically relevant variable has to be used. Here the correct variable is the void volume fraction f. For a material with microcracks the internal state variable which ’replaces’ the damage tensor D is a set of vectors. A microcrack is described by vector offering two properties: direction and value. The former gives the direction of the normal to the surface of the microcrack and the value gives the effective area of the microcrack. This approach is supported by Muschik’s (1992, p. 53) comment: ’Internal variables need a model or an (microscopic or molecular) interpretation’. What is an acceptable model for a fourth-order damage tensor D? How does one derive an evolution equation for the damage tensor D?The value for the void volume fraction f, however, can be measured and therefore the validity of the evolution equation can be considered. Furthermore, material science may give expressions for f. The same holds for microcracking. At high strain rates the dominant failure mechanism of fine- grained ice is microcracking. Since the number of microcracks is related to the acoustic emission, it is possible to measure the number of microcracks during the test {see e.g. Currier, Schulson & St. Lawrence [1983, Figs. (26) and (27)]}. The second part of this publication concentrated on the investigation of the Gurson-Tvergaard material model, carried out in Chapters 9 and 10. The idea was to evaluate the foundations of the works by Gurson and Tvergaard and to derive their constitutive equation by using thermomechanics and damage 149 mechanics. Based on the evaluation some criticism was expressed and an extension to the Gurson-Tvergaard material model was derived. The criticism and the basic concepts of the extended Gurson-Tvergaard material model are discussed next. The model proposed by Tvergaard has three internal variables: the void-plastic strain E'~,the void volume fraction f and the equivalent tensile flow stress in the matrix material oM. By the void-plastic strain E"~it is meant a strain that combines the strain due to plastic yield of the matrix material and void nucleation and growth. Tvergaard called E"' the plastic strain and denoted it by E~.The present work adopted the form for the void-plastic strain E'~given by Tvergaard [1981. Eqs. (5.1) and (5.6)j and in fact already obtained by Gurson [1977, Eqs. (4.9) and (A.2)1. The yield function F proposed by Gurson [1977. Eq. (4.9)J and modified by Tvergaard [1981, Eq. (S.1)1 was adopted almost in its original form into the extended Gurson-Tvergaard material model. Two major things, however, were changed. First, an "extra" term for it was added to fulfil the requirements set by the evolution equation for the void volume fraction f. Second, the quantity oM introduced by Tvergaard was replaced by the quantity [3 J,(U)l'''. This work showed that the evolution equation for the void volume fraction f proposed by Tvergaard [1981, Eq. (S.S)]has an acceptable form and is therefore also proposed by the author. However, some researchers {see e.g. Chu and Needleman (1980, Eq. (18)1} have added new terms to the evolution equation of the void volume fraction f. The validity of these new terms is very limited since they may contradict the principles of thermomechanics. This report neglected [as did Tvergaard] the influence of the elastic response of the matrix material on the evolution equation for f [see discussion above Evolution Equation (33S)J. It is the only term which can be added when uniform fields of quantities within the representative volume Vrepare studied. I so The third internal variable in the model by Tvergaard is the equivalent tensile flow stress in the matrix material oM which replaced the constant oy of the Gurson model. Although Gurson called the oy microscopic equivalent tensile yield stress, the role of cy is to refer the argument of the Sinh-function in the yield function F dimensionless (see Gurson [1977, Eqs. (3.12) and (4.9)]}. Tvergaard assumed that the matrix material shows isotropic hardening and elegantly derived (as was attempted in Chapter 9) an evolution equation for the variable oM.As discussed by Aravas (1996) the upper bound approach used by Gurson cannot be used if the matrix material displays hardening. Tvergaard introduced the three material parameters q’, q2 and q’ (and probably also oM) in order to obtain a better fit between the numerical results of his microscopically detailed finite element analysis and the macroscopic continuum model which was based on his yield function [Tvergaard (1981, p. 405)]. Furthermore, Tvergaard studied a two-dimensional medium with circular cylindrical voids. The above means that although the Gurson model is derived from strict analysis based on continuum mechanics, the modification by Tvergaard is an ad hoc implementation to get a better fit in a few examples between two different kinds of numerical analyses. Another set of examples may have led Tvergaard to modify the yield function given by Gurson in a different way. Usually the constitutive equations derived using traditional continuum mechanics satisfy the Clausius-Duhem inequality, whereas the satisfaction of ad hoc material models has to be verified. This was one of the main purposes of this work. Although the author sees the excellence of the derivation for the evolution equation of oM carried out by Tvergaard, the present work replaced the variable oM with the quantity [3 J2(6)]”’ where J2(U) is the second invariant of the deviatoric effective stress tensor. According to Tvergaard oMis the equivalent (in the sense of von Mises) tensile flow stress in the matrix material, disregarding local stress variations (Tvergaard, 1981, p. 402). Based on damage mechanics the effective stress tensor 0 gives the value of the stress in the matrix material (disregarding local stress variations). This implies that oM = [3 J2(ii)]”2. The 151 novelty of the present formulation lies in the fact that damage mechanics gives an analytical relationship between the stress tensor u and the effective stress tensor u. For the case studied in this work the obtained analytical expression is exact if the material behaviour for elastic deformation is isotropic as well as homogeneous, elastic strain obeys Hooke’s law and there is no interaction between the voids [see Eshelby (1957)]. Damage mechanics gives the value for the effective stress tensor u and no special investigation to obtain evolution equation for 6 is necessary. The investigation of the (extended) Gurson-Tvergaard material model was done by performing a thermomechanical study of the constitutive equation. Thermomechanical evaluation allows proof that the material model fulfils the requirements set by thermomechanics. The key role is the satisfaction of the second law of thermomechanics. The standard procedure of this consideration is to combine the first law of thermodynamics with the second law of thermodynamics and to dress the resulting expression as the Clausius-Duhem inequality. Section 10.4 showed that when the hydrostatic part of the stress is tension, the (extended) Gurson-Tvergaard material model always satisfies the Clausius-Duhem inequality. In compression, however, there are states which are not allowed. If this constitutive model is used e.g. in finite element code the program must also perform verification of the satisfaction of the Clausius-Duhem inequality. Furthermore, thermomechanical representation of the model is important in the enhancement of the Gurson-Tvergaard material model for a modified one which does not suffer from localisation. I52 REFERENCES The following two types of citations are applied in the present work. When the writer’s name is mentioned in a sentence it means that this and only this senten- ce is based on the cited source. The following two examples clarify the frequent- ly-used citation practice used in this work. Eshelby (1957) has studied the elastic field in a Hookean material containing an ellipsoidal inclusion. As a special case he determined the value for the complementary strain-energy density of a material containing ’a volume fraction f’ of inhomogeneous spheres. For the purpose of this work the inhomogeneous spheres are ’replaced’ by spherical cavities. This is done by assuming that the values for the elastic constants for the cavities vanish. The spherical cavity problem was approached earlier by Mackenzie (l950), but the form of the result by Eshelby better fits the needs of this work. The complementary strain-energy density wC(qf) takes the form (Eshelby, 1957, p. 390): But when the citation refers to the whole paragraph the author’s name is located after the last full stop of the cited paragraph, as in the following example: A thermodynamic system, or system for short, is any collection of matter imagined isolated from the rest with the aid of a clearly defined boundary. It is not necessarily assumed that the boundary is rigid, on the contrary, in most cases the boundary will deform when the system is subjected to some process which we have to understand and analyze. (Kestin 1978, 1, pp. 22 and 23.) It has to be pointed out that the author does not have the following publications shown by an asterix ’*’. They are listed here, because the author has referred to publications which have a quotation to the paper indicated by the asterix. 153 Aravas, N. 1996. 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Berkshire, England: McGraw-Hill Book Company. 787 p. 162 APPENDIX A DOUBLE-DOT PRODUCT OF A SKEW-SYMMETRIC THIRD-ORDER TENSOR AND A SYMMETRIC SECOND-ORDER TENSOR The aim of the present appendix is to prove the following: - c:h = 0 (A.1) where c is a third-order tensor which is skew-symmetric- in the last two indices and h is a symmetric second-order tensor. The notation 0 refers to a zero vector which means that the values of all the components of vector 0vanish. The vectors and tensors are represented in the orthogonal rectangular Cartesian coordinate system (x~,x2, x3) the basis of which is (T,, T2, TJ. The dyadic representation of vectors and tensors is convenient for calculation of dot products and is therefore written: --- -- c = c..ilk I.'I ij 'k and h = h,, isit The definitions for the skew-symmetry in the last two indices of the tensor c and for the symmetry of the tensor h read: and h = hT or Definition (A.3) leads to: Gill = Ci2* = ci33 = 0 (A.5) Substitution of Forms (A.2) into the left-hand side of Theorem (A. 1) yields: Based on Properties (A.3), (A.4) and (A.5) of the tensors c and h, the following reduced form for Quantity (A.6) is achieved: Result (A.7) proves Theorem (A.1). LEGENDRE TRANSFORMATION APPENDIX B The derivation of the Legendre transformation carried out here follows the presentation given by Lanczos (1949, pp. 161... 163), except for the introduction of coefficients a and b in Equations (B.2) and (B.3) as well as the assumption that the variables are second-order tensors. The investigation is started with a given scalar-valued function F of m second- order tensorial variables u', ... , um : F = F(u', ..., urn) A new set of second-order tensors y', ... ,ym are introduced by means of the following transformation: aF(u', ..., urn) yl = a i = 1, ... ,m aui where the coefficient a is independent of the variables u' and yl (i=l,..., m). In the present derivation variables ui and yi are assumed to be second-order tensors, but certainly they can be tensors of any order. The so-called 'Hessian' - Le., the determinant formed by the second partial derivatives of F - is assumed to be different from zero, guaranteeing the independence of the m variables yl. In that case, the equations (8.2) are solvable for ui as functions of yi. The Legendre transformation R of the function F is defined as follows: m bQ = ui:yi-aF (8.3) i=l where b is a coefficient independent of the variables ui and yi (i=l,..., m). 812 The variables u' are expressed in terms of the tensors yl [Equation (B.2)1and are substituted into Equation (6.3). The function (2 can then be expressed in terms of the new variables yl alone: (2 = R(yi, ... , ym) (B.4) The infinitesimal variation of bR, produced by arbitrary infinitesimal variations of y', is considered next. Because the coefficients a and b are independent of the variables u' and yl, they are constants with respect to the variation. The combination of Equations (B.3) and (8.4) gives: m JC1 .. JF 6(bR) = b-:6y' = (ui:6yi + 6u':y') - al:6u' (B.S) ill dy' i=l 3Ul According to Malvern [ 1969, Eq. (2.4.19a)I the scalar product of two second-order tensors are commutative, i.e. i = I, .._,m Applying Equation (8.6). Equation (B.5) can be re-written in the form: m m ai2 JF I b-:6y' = [ u':6y' - (y' - a-):6u 1 (8.7) I=I ay' 1-1 dU' Since !2 is a function of the tensors yl alone, the variables u' should be expressed as functions of the variables yl. This expresses the variations of the tensors ui in terms of the variations of the tensors yi. However, examination of Equation (8.7) shows that this elimination is rendered unnecessary by the fact that the coefficient of the variation 6u' is automatically zero, since the variables y' are defined according to Equation (B.2). However Equation (8.7) gives at once: 3R(y ..., y") u' = b ', i-I, ...,m 3Y' REFERENCES Lanczos, C. 1949. The variational principles of mechanics. Toronto, Canada: University of Toronto Press. 307 p. Malvern, L. E. 1969. Introduction to the mechanics of a continuous medium. Englewood Cliffs, New Jersey, USA: Prentice-Hall, Inc. 713 p. LEGENDRE PARTIAL TRANSFORMATION APPENDIX C In this work, the Legendre transformation containing both active and passive variables is referred to as the Legendre partial transformation. Here, only such details which differ from those of Appendix 8: 'Legendre transformation', are presented. Also, the study carried out in this appendix follows the concept of Lanczos (1949, pp. 163 and 164). In this case, the scalar-valued function F is assumed to be a function of two independent sets of tensorial variables, which are u', ... ,urnand wl, ... ,w" i.e. F = F(U',..., urn,WI ,..., w") (C.1) The new independent set of second-order tensorial variables y', ... ,ym is assumed to be defined by: aF(u',..., um,w',..., w") y' = a i = 1, ...,m (C.2) a ut where a is a coefficient independent of ui, d and yi (i = 1,..., m and j = 1 ,..., n). The variables ui are called the active variables and the variables d are called the passive variables of the transformation. A new function R,called the Legendre partial transformation, is introduced. It is defined by: m b R(y',..., ym, w1,..., w") = yi: ui - a F(u',..., urn,w1 ,..., w") (c.3) i-I The variables d and yi are given arbitrary variations 6d and 6y'. Thus, Equation (C.3) gives: CI2 which yields rn b-:6y1ai2 + b-:6wJas2 1-1 3yl ]'I 3w' According to Equation (C.2) the first term on the right-hand side of Equation (C.5) vanishes and therefore the following equations are obtained: dQ(y I ,.._)y m ,W1,...,W") U' = b i = 1. ... ,m dyl and (C.6) l3F a a- = -b -R j = I,...,n a WJ awl REFERENCES Lanczos, C. 1949. The variational principles of mechanics. Toronto, Canada: University of Toronto Press. 307 p. APPENDIX D DIVERGENCE OF THE DOT PRODUCT OF A SECOND- ORDER TENSOR AND A VECTOR The aim of the present appendix is to prove the following equality: - V.(h-Z) = (V-h).Z + h:VZ (0.1) - where V is the vector operator del, h is a second-order tensor and e is a vector. The vectors and tensors are represented in the orthogonal rectangular Cartesian coordinate system- (x,, x2, x3) the basis of which is the (TI, i,,T3). In Equation (D.l) the notation V e indicates an open product of the two vectors. The open product of the vectors b and is a second-order tensor called dyad and it is denoted by - - b c. If a = b c, the component aij is (see e.g. Malvern 1969, pp. 36 and 590 - 591): aii = bi cj The dyadic representation of vectors and tensors is convenient for calculating dot products and is therefore written: --a - - V =imp and e = ekik (D.3) ax, Substitution of Expressions (D.3) into the left-hand side of Equality (D. 1) give: V-(h.z) The right-hand side of Equation (D.4) yields: The terms on the right-hand side of Equation (D.l) can be written as follows: D/2 and -- - - - 3- a ek h:Ve = h..i i : im- ekik = hrnk - (0.7) I' ' ' ax, axm Comparison of Equalities (D.4) and (D.5) with Equalities (D.6) and (D.7) proves Expression (D.1). REFERENCES Malvern, L. E. 1969. Introduction to the mechanics of a continuous medium. Englewood Cliffs. New Jersey, USA: Prentice-Hall, Inc. 713 p. STRESS POWER PER UNIT VOLUME APPENDIX E The expression - - u:v v = CYrnt Vt,m obtained in Equation (86) is investigated here in more detail. According to Malvern (1969, pp. 146 and 147) the following equations are obtained: Vt,m = Dtm + Wtm where Dtmis the component of the rate-of-deformation tensor D and W,, is the component of the spin tensor W. The rate-of-deformation tensor D is symmetric and the spin tensor W is skew-symmetric i.e. Dmt = Dtm and The scalar product of the symmetric stress tensor u (see Section 3.3) and the skew-symmetric spin tensor W vanishes, Le. omt Wmt = 0 as pointed out by Flugge [1972, Eq. (1.52)]. Using the information given in Equations (E.2)...( E.4), Equation (E. 1) can be written in the form: - - U: V v = Omt(Dtm+ Wtm) = CYrnt Dtm - CYtm Wtm omtDmt E.5) When the displacements and displacement gradients are very small the following approximation holds (see Malvern, 1969, p. 150): and Equation (E.5) takes the final form: El2 - o:VV = o:i (E.7) The term on the right-hand side of Equation (E.7) is called the stress power (per unit volume). REFERENCES Flugge, W. 1972. Tensor analysis and continuum mechanics. Berlin/Heidelberg, Federal Republic of Germany: Springer-Verlag. 207 p. Malvern, L. E. 1969. Introduction to the mechanics of a continuous medium. Englewood Cliffs, New Jersey, USA: Prentice-Hall, Inc. 713 p. APPENDIX F PARTIAL DERIVATIVES OF THE SECOND DEVIATORIC INVARIANT J,( a-p') This appendix investigates the partial derivatives of the second deviatoric invariant J,(u-B') with respect to the stress tensor u and the internal force B'. Let A be an arbitrary second-order tensor. The following equalities hold: aA -=I , I:I = I and I:A = A:l = A (F.l) aA where the component lijk,of the fourth-order identity tensor I is defined by: lijkl = 6ik 'jl F.2) where tiiiis the Kronecker delta defined to be zero when i and j differ and to be unity when i = j. For a second-order tensor y, the deviatoric tensor of which is denoted by g, the following result is obtained: where 1 is the second-order identity tensor, a component of which is lii, is defined by: The notation I,( g ) in Equation (F.3) is the first invariant of the tensor g and JI( y ) is the first invariant of the deviatoric tensor g. The equivalence = in Equation (F.3) can be found e.g. in Lin [1968, Eq. (1.4.6)]. The second deviatoric invariant of the tensorial variable u-p' is studied next. According to Lin (1968, p. 15) it is defined by: Fi2 1 J2(a-BI) -[s-b']:[s-b'] (F.5) 2 where the deviatoric tensors s and b1are expressed by Equations (23) Le. where the fourth-order tensor K is: 1 K=l--ll (F.7) 3 Substitution of Equation (F.7) into Expressions (F.6) and of the obtained result into Equation (F.5) leads to the following expression: 1 I 1 J(2u -PI) = -[(l--ll):~- (l--ll):P']: 2 3 3 (F.8) 1 1 [(I --1l):u - (I --ll):f+] 3 3 Based on Expression (F.8) we find: a[J,bJ - 1 I au P1)I = - [(I --11):- 1 : au 2 3 aa I 1 [(I--1l):u3 - (I--ll)$]3 + (F.9) 1 I 1 -[(I--1l):u7 - (l--II)$]3 : 2 I [(I --1 I):-]au 3 au Taking Equations (F. la), (F.6) and (F.7) into consideration Equation (F.9) takes the following form: Fl3 d[J*(o - 8'11 = -[I:I--ll:I]:[s-b']1 I + aa 2 3 (F.10) -[s-b']:I [I:I--1 1 1:I] 2 3 which according to Equations (F.lb) and (F.lc) yields: -1 [ s - b' 1 : [ I - -1 I 1 I (F. 1 1) 2 3 I = [I - -1 11: [s-bl] 3 Based on Equation (F.3) the term I : [ s-b' 1 vanishes, and based on Equation (F.lc) I : [s-bl] = [s-b']. Thus Equation (F.ll) collapses to: (F.12) Analogously to the above derivation the following is thus obtained: (F.13) REFERENCES Lin, T. H. 1968. Theory of inelastic structures. New York, USA: John Wiley & Sons. 454 p. Published by Series title, number and report code of publication Vuorirniehentie 5, P.O.Box2000,FIN42044 VTT, Finland V'IT Publications 3 I2 VTT-PUBS-3 12 Phone internat. + 358 9 4561 Fax+35894564374 Date Project number June 1997 V6SUOO320 ruthor(s) Name of project Komponenttien rakenneanaly ysit Santaoja, Kari Commissioned by Ministry of Trade and Industry, Finland (KTM), Technical Research Centre of Finland (VTT) Me Thermomechanics of solid materials with application to the Gurson-Tvergaard material model ibstract The elastic-plastic material model for porous material proposed by Gurson and Tvergaard is evaluated. First a general description is given of constitutive equations for solid materials by thermomechanics with internal variables. The role and definition of internal variables are briefly discussed and the following definition is given: The independent variables present (possibly hidden) in the basic laws for thermomechanics are called controllable variables. The other independent variables are called internal variables. An internal variable is shown always to be a state variable. This work shows that if the specific dissipation function is a homogeneous function of degree one in the fluxes, a description for a time-independent process is obtained. When damage to materials is evaluated, usually a scalar-valued or tensorial variable called damage is introduced in the set of internal variables. A problem arises when determining the relationship between physically observable weakening of the material and the value for damage. Here a more feasible approach is used. Instead of damage, the void volume fraction is inserted into the set of internal variables. This allows use of an analytical equation for description of the mechanical weakening of the material. An extension to the material model proposed by Gurson and modified by Tvergaard is derived. The derivation is based on results obtained by thermomechanics and damage mechanics. The main difference between the original Gurson-Tvergaard material model and the extended one lies in the definition of the internal variable 'equivalent tensile flow stress in the matrix material' denoted by 0 M. Using classical plasticity theory, Tvergaard elegantly derived an evolution equation for o'. This is not necessary in the present model, since damage mechanics gives an analytical equation between the stress tensor d and oM.Investigation of the Clausius-Duhem inequality shows that in compression, states occur which are not allowed. ktivity unit VTT Manufacturing Technology, Materials and Structural Integrity Kemistintie 3, P.O.Box 1704, FIN42044 V'IT, Finland SSN and series title 1235-0621 (soft back edition) VTT PUBLICATIONS 14554849 (URL:http://www.inf.vtt.fi/pdfo SBN Language 951-38-5060-9 (soft back edition) English 951-38-5061-7 (URL: http://www.inf.vtt.fdpdf/) Keywords thermomechanical analysis, thermodynamics, plasticity, porous medium, Gurson-Tvergaard model ;old by VTT Information Service Pages Price group P.O.Box 2000,FIFco2044 VTT, Firdand 162 p. + app. 14 p. D Phone internat. + 358 9 456 4404 Fax + 358 9 456 4374 Preparation of a reliable material model involves the following three major tasks: Investigation of micromechanisms of deformation, experimental work and thermomechanical verification of the model. The latter is the topic of this publication. Micromechanical investigation of the material should lay the foundation of the constitutive equation. Experimental data are used to deter- mine values for the material parameters and to verify the model. Thefmomechanics is a science that combines thermodynamics and mechanics. Every constitutive model must be shown to follow the basic laws of thermomechanics, e.g. Newton’s second law of motion and the first and the second laws of thermodynamics. If the model does not fulfil the require- ments set by the basic laws it must be dropped, as the response it describes cannot exist in nature. The present report gives an introduction to ‘thermomechanics with internal variables’ - the most promising dialect of thermomechanics for preparation of material models. Expressions are given for description of time-dependent and time-independent constitutive equations. The obtained results are widely valid for solids, and can with minor modifications be used by the reader in the verification of hisher own material model. No cumbersome mathematical derivation is required, all the hard work being done here. Some traditional models for thermoplasticity are derived by using thermo- mechanics with internal variables. As a special case the constitutive equation for porous elastic-plastic material proposed by Gurson and modified by Tvergaard is evaluated from the viewpoint of thermomechanics. The main results show that there are states which do not satisfy the second law of thermodynamics and that some extensions to this model oppose the continuity equation (the law of conservation of mass). Both are serious drawbacks of a material model. The first problem may be solved by adding to the computer code a procedure which numerically verifies the second law of thermo- dynamics. FJUluka~sua nq-y Iknna puhlkation alp av This publication IS a\lulahle from VIT ?‘IETOPAL\ZLIJ \IT INFO~V.4TlONSTJ~ST \‘TT INFOIUwIATlON 5FRWCF PL XXK) PR 2MW) P 0 NIX2(KX) 02044 VIT 02044 \TT FIMl2044 VIT Finldnd Puh (09)456 4404 Tel (0)456 4404 Phonc intemt + 3% 9 456 +(!4 Fdksi (0)456 4374 Fax (tr)) 456 4374 Fax + 3% 9 456 4374 ISBN 951-38-50669 (soft back ed.) UDC 536.7 ISSN 1235421 (soft back ed.) ISBN 951-38-5061-7 (URL: http://www.inf.vtt.fdpdf/) ISSN 14554849 (URL: http://www.inf.vtt.fdpdfl)