Robust Integration Schemes for Generalized Viscoplasticity with Internal-State Variables

Computers and Structures 74 (2000) 601±628 www.elsevier.com/locate/compstruc

Robust integration schemes for generalized viscoplasticity with internal-state variables

A.F. Saleeb*, T.E. Wilt, W. Li

Department of Civil Engineering, The University of Akron, Akron, OH 44325, USA Received 11 July 1998; accepted 31 August 1998

Abstract

This paper is concerned with the development of a general framework for the implicit time-stepping integrators for the ¯ow and evolution equations in complex viscoplastic models. The primary goal is to present a complete theoretical formulation, and to address in detail the algorithmic and numerical analysis aspects involved in its ®nite element implementation, as well as to critically assess the numerical performance of the developed schemes in a comprehensive set of test cases. On the theoretical side, we use the unconditionally stable, backward Euler dierence scheme. Its mathematical structure is of sucient generality to allow a uni®ed treatment of dierent classes of viscoplastic models with internal variables. Two speci®c models of this type, which are representatives of the present state-of-the-art in metal viscoplasticity, are considered in the applications reported here; i.e., generalized viscoplasticity with complete potential structure fully associative (GVIPS) and non-associative (NAV) models. The matrix forms developed for both these models are directly applicable for both initially isotropic and anisotropic materials, in three-dimensions as well as subspace applications (i.e., plane stress/strain, axisymmetric, generalized plane stress in shells). On the computational side, issues related to eciency and robustness are emphasized in developing the (local) iterative algorithm. In particular, closed-form expressions for residual vectors and (consistent) material tangent stiness arrays are given explicitly for the GVIPS model, with the maximum matrix sizes `optimized' to depend only on the number of independent stress components (but independent of the number of viscoplastic internal state parameters). Signi®cant robustness of the local iterative solution is provided by complementing the basic Newton±Raphson scheme with a local line search strategy for convergence. # 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Viscoplasticity; Numerical integration; Implicit; Finite element; Line search; Anisotropic; Isotropic; Algorithm; Multiax- ial; Tangent stiness

1. Introduction performance of inelastic structural components operat- ing under complex thermomechanical and multiaxial At present, it is widely accepted that large-scale nu- loading conditions. This is largely due to the signi®cant merical simulations, based on the ®nite element progress made over the years in the development of method, provide a viable approach for assessing the theories of plasticity and viscoplasticity for the phe- nomenological representations of inelastic constitutive properties. In particular, the present state-of-the-art in * Corresponding author. Tel.: +1-330-972-7286; fax: +1- mathematical modeling of metal viscoplasticity is very 330-972-6020. well developed, i.e., based on the so-called internal

0045-7949/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0045-7949(99)00020-6 602 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 variable formalism in the thermodynamics of irrevers- Motivated by these practical considerations, the ible processes [1±6]. A large number of specialized development of stress integration algorithms with forms of these modern uni®ed viscoplastic models improved performance has received considerable atten- (e.g., isotropic or anisotropic, fully associative or non- tion in the recent literature on computational plasticity associative, isothermal or nonisothermal, etc.) have and viscoplasticity. In this, implicit (e.g. backward been successfully applied to dierent metals and com- Euler, generalized midpoint and trapezoidal rules) posite material systems [4±12]. In these, account is methods of time integration have been particularly made for a host of complicated physical phenomena emphasized, mainly because of their superior stability, and deformation mechanisms; e.g., load-rate depen- and wider range of applicability, compared to the ear- dency, creep-plasticity interactions, stress relaxation, lier explicit counterparts. More speci®cally, a number nonlinear kinematic and isotropic hardening, of well-established implicit schemes are presently avail- Bauschinger and allied reversed-load eects, ratchetting able for inviscid plasticity (e.g. classical radial-return under cyclic loading, thermal and dynamic recovery and its generalizations in the form of operator-split/ mechanisms, material damage, etc.. Naturally, the bet- return-mapping algorithms) and viscoplasticity of ter accuracy and improved modeling capabilities in metals; see Refs. [13±27] for a sampling of these these representations have been often acquired at the works. We also refer to Ref. [25] for a recent survey expense of greater mathematical complexities, larger and summary of several proposals. In this regard, we number of internal variables and increased compu- note the very ecient implementations of the radial- tational demands. return schemes for metals corresponding to the special From the computational standpoint, the global sol- case of full isotropy (both elastic and inelastic re- utions for the discretized (e.g. standard displacement- sponses), enabled by the complete reductions of the based or stiness) ®nite element model are typically rate tensor equations to a ®nal single/scalar nonlinear achieved by an incremental/iterative (e.g. Newton or equation; e.g. in terms of the eective stress, or equiva- quasi-Newton) procedure. A common and important lently, the plastic multiplier or the `magnitude' of the problem in all these nonlinear material analyses con- inelastic strain vector. In particular, details of the cerns the temporal integration, in the standard strain- stress-updating in these algorithms have been worked driven formats, over a discrete sequence of time steps, out for inviscid plasticity, combined plasticity with of the rate evolutionary equations (i.e., sets of ®rst- time-/or strain-hardening creep, as well as classical vis- order dierential equations that are highly nonlinear, coplasticity with no (static/dynamic) recovery e.g. coupled and mathematically sti) in the underlying [13,14,17,26,27]. Furthermore, the importance of using material constitutive models. For a given integration the consistent tangent stinesses in these cases has scheme, the result is a set of local nonlinear response been also clearly demonstrated [13,14,16,21,25,]. functions which de®ne the stress tensor (and other Finally, we also note that in contrast to their time- state variables) as a function of the strain history up to independent elastoplastic counterparts, applications of the current time step, at each material/integration the implicit integration to viscoplasticity with nonlinear point. This local stress update constitutes the main kinematic hardening and (static/dynamic) recovery subject of the present paper. mechanisms have been rather limited, especially for The mathematical forms, accuracy and stability general classes of anisotropic materials. Models of this properties, as well as, implementation details, for the latter type are the focus in the present study. selected integrator, will directly impact both the accuracy and eciency of the overall ®nite element solution. For instance, concerning accuracy we note that 2. Objectives and outline the global (driving) residual force vectors are directly formed from the updated local stress ®elds. On the In this paper, we are concerned with the develop- other hand, the linearization of these latter stresses will ment of a general framework for the implicit time-step- also provide the material tangent stiness (iteration ping integrators for the ¯ow and evolution equations Jacobian matrix) used to form and subsequently in generalized viscoplastic models. The primary goal is assemble element stinesses that govern the conver- to present a complete theoretical formulation, and to gence rate of the global solution; e.g. the well known address in detail the algorithmic and numerical analy- property of the (asymptotic) quadratic rate of conver- sis aspects involved in its ®nite element implemen- gence for the Newton scheme can be preserved only if tations, as well as to critically assess the numerical a consistent (algorithmic) material stiness is utilized performance of the developed schemes in a comprehen- [13±15]. Several other important factors here include sive set of test cases. time- step controls for conditional stability of explicit On the theoretical side, the general framework is integration, or the design of appropriate local iterative developed in the speci®c context of the fully implicit schemes ensuring convergence in implicit algorithms. (backward Euler) dierence scheme. In addition to its A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 603 desirable feature of unconditional stability, this scheme the evolution of each following the Armstrong± is also known to exhibit excellent convergence/ accu- Frederick rule [4,5], i.e., the continuous `(linear) hard- racy characteristics in large scale computations that ening minus dynamic recovery' format. Dierent, more typically involve large time steps, especially with com- elaborate, functional forms are also often utilized in plex anisotropic inelastic models [17,19,25]. the ¯ow equations, e.g., exponential or combined Furthermore, compared to other members of its class; hyperbolic sine-power type [5,12]. e.g., semi-implicit, generalized midpoint integration The majority of viscoplastic models currently in use rules, the fully implicit scheme leads to very ecient belong to the NAV class [4,5,12]. Several developments and direct implementation, i.e., entirely bypassing com- in the GVIPS class are currently underway [7±9] as plications associated with the so- called `continuation these models are very appealing in view of their theor- of the time step' calculations [15,17,21,24,]. The latter etically sound basis from the thermodynamics stand- are required for the semi-implicit methods to advance point, and the symmetries exhibited in the the solution from the intermediate (generalized mid- corresponding linearization of the integrated forms of point) time station (where all the viscoplastic rate their ¯ow and evolution equations. Both types of equations are evaluated) to the end of the step. For models have been actually characterized fro real example, in order to preserve the stability and conver- metals, and these are used in the applications reported gence properties of the method, this will necessitate the here. Speci®cally, as representative of the NAV class, enforcement of global balance (equilibrium) conditions we utilize the isotropic form given by Freed and (and correspondingly evaluating the ®nite element re- Walker in Ref. [12]. For GVIPS, we use both isotropic sidual vectors) at the generalized mid-time step [15]. and transversely isotropic forms [6,7,25], with This problem is even more critical in the context of decoupled Gibb's function, in the numerical simu- `nonclassical' implementation of the above semi-im- lations. plicit midpoint rules; e.g. in the form of approximately The rather complex nature of the evolution discretized, asymptotic integration method [24] where a equations in the above models provides for an ideal nonlinear completion is employed. setting to test any stress integration algorithm. This On the constitutive modeling side, the generality of will become evident in several comparisons reported the present framework allows a uni®ed treatment of latter in the Section 5. dierent classes of viscoplastic models with internal In summary, for each of the viscoplastic models con- variables. To this end, two speci®c models which are sidered, and the associated implicit backward Euler in- representatives of the present state-of-the-art in metal tegration method, the following constitutes the major viscoplasticity (of the generalized J2 or von Mises tasks required to achieve the above stated primary type) are considered here. For brevity, these will be goals: subsequently referred to as GVIPS models (for 1. detailed study of the underlying mathematical struc- Generalized Viscoplasticity with Potential Structure) ture of the viscoplastic equations, and the corre- and NAV models (for Non-Associative sponding integrated ®elds of stress and internal state Viscoplasticity). Each of these models include a num- variables. ber of distinctive features, and collectively they span a 2. development and implementation of the implicit, very wide spectrum of characteristic and predictive backward-Euler stress- updating algorithm, and the capabilities that are shared by many other viscoplastic associated (Newton type) nonlinear/iterative models. equation solver, and its further enhancements (e.g., For example, the GVIPS class [6±9] is unique in its in the form of line searches and time step subincre- utilization of a nonlinear kinematic hardening (back menting) stress) mechanism resulting from an assumed thermo- 3. testing of the convergence, stability, and accuracy dynamic potential (Gibb's) function for the èquation- properties of the algorithms. of-state' de®ning the associated (conjugate/dual) internal strain variable, and dissipation function O- Our primary objective in item (1) is to identify the per- form), including thermal (static) recovery contribution, tinent matrix operators needed in the algorithmic for its rate of evolution. A characteristic partitioning, development; i.e., concise forms of the constitutive re- with several regions of discontinuities of the state sidual vectors for the integrated ®eld equations. The (stress-internal variable) space is also used to account implementation in item (2) utilizes closed-form ex- for stress-reversal/cyclic loading. For the ¯ow pressions of the consistently-derived material tangent equations, simple power functions are typically stiness; i.e. model/integrator-dependent. Here a cru- employed. cial consideration, for computational eciency, con- On the other hand, the NAV models are typically cerns a systematic elimination procedure to reduce to a characterized by the use of several internal state vari- minimum the number of unknowns governing the local ables, particularly for the tensorial back stress, with iterations. To this end, with the stress tensor as the pri- 604 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 mary unknown, analytical reduction was utilized lead- present terminology (e.g. phrases such as `fully associ- ing to òptimized' sizes of all the arrays involved with ative', or the equivalent `complete potential structure') maximum sizes equalling the number of independent to become more meaningful in the present context of stress components i.e., six for the full three-dimen- model classi®cation, we make the following remarks at sional (3D) space, or less in subspaces (e.g. four in the outset. plane-strain/axisymmetric; three for plane stress; and Following the recent work in Ref. [6±9], in the gen- ®ve for plate/shell problems). This is in contrast to eral form for fully associative GVIPS, there are two straightforward `standard' implementation in which key ingredients; (1) the thermodynamic potential, or the unknown solution vector includes stress plus all Gibb's function (or conjugate free energy, Helmholtz tensorial and scalar internal variables. Furthermore, function) de®ning the equations of state, and (2) the the line search enhancement of the basic Newton iter- viscoplastic dissipation function O-form) for the ensu- ation will signi®cantly add to the robustness of the ing ¯ow and evolution laws of inelastic strain and in- overall integration algorithm. ternal state parameters. However, this is dierent from In conjunction with (3), an extensive number of dis- usage by other researchers [10±12], of such notions as criminating test problems will be utilized. The objective the general structure of the `thermodynamically based' is to enable ®rm conclusions to be made regarding the constitutive equations where only the O-form (¯ow or `relative' merits and/or limitations of various schemes. Finally, an outline of the remainder of the paper is dissipation potential) was utilized. These latter forms given as follows. In Section 3, we discuss the math- are introduced only for convenience in satisfying the ematical structure underlying both GVIPS and NAV thermodynamic admissibility restriction (based on classes of viscoplastic models, pointing to several poss- simple properties of non- negativeness and convexity ible extensions and generalizations, particularly per- of these functions) in the form of the well-known taining to the use of a number of internal state Clausius±Duhem or dissipative inequality O-form) [2]. parameters and the eect of initial material anisotro- However, such nonlinear kinematic hardening forms pies for both elastic and viscoplastic response com- do not presuppose or automatically imply the existence ponents. We also outline the practically- used forms of the total (integrated) forms of the associated ther- and speci®c expressions in the ¯ow/evolution equations modynamic potentials, e.g. the Helmholtz free energy, for each. or Gibb's function. For distinction, i.e., when using The details of the implicit integrator are given in only the O-forms, the term ìncomplete' potential form Section 4. These include the general discrete forms of seems more appropriate. Since the opposite situation the integrated equations and the local Newton method, typically occurs in NAV models (i.e., implicit or expli- together with speci®c forms for the constitutive re- cit assumption for the existence of linear equation of siduals and consistent (algorithmic) material stiness. state for the conjugate internal variables, but no O- The outline of the line search employed will also be form), we prefer the description nonassociative for given here. In Section 5, we give the results of a large both of these latter formulations. number of numerical simulations using both GVIPS The complete potential-based class of inelastic con- and NAV models. These range from simple uniaxial stitutive equations possess a number of distinct and validation tests under varying conditions (creep, relax- important attributes from both a theoretical and a ation, cyclic, etc.) to multiaxial testing under nonpro- computational standpoint. First, they constitute the portional loading, as well as a structural problem cornerstone of numerous regularity properties and involving nonhomogeneous states of stresses and de- bounding (limit) theorems in plasticity and viscoplasti- formations. The emphasis is placed on evaluating con- city. Second, they result in a suciently general vari- vergence properties, robustness, and computation ational structure, whose properties can be exploited to eciency. For comparison, we also show some of the results obtained using other, explicit, integration derive a number of useful material conservation laws. methods. Third, on the numerical side, the discrete form of this same variational structure is of great advantage in the development of ecient algorithms for ®nite element 3. Mathematical formulations of viscoplastic models implementation, for example, symmetry- preserving material tangent stiness operators are easily obtained 3.1. General in implicit solution schemes (such as those given later). Finally, this complete potential-based framework con- In this section, we describe the mathematical struc- veniently lends itself to intelligent application of sym- ture and basic governing equations, corresponding to bolic manipulation systems that facilitate the the two viscoplastic models considered; i.e., GVIPS construction, implementation, and analysis of new de- and NAV. However, for clarity, and in order for the formation and damage models [6]. A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 605

@C 3.2. Generalized viscoplasticity with potential structure gd 6 D d (GVIPS) @D i where eij is the total strain and eij is the inelastic strain The scope of the following discussions is limited to tensor, respectively. cases of small deformations and isothermal conditions, The above relations are known as the equations of in which the initial state is assumed to be stress-free b d state, in which, sij,aT,D , are the `force-like' thermo- throughout. For generally, the initial state of material b d dynamic state variables while eij,gT,gD, are the dual/ isotropy or anisotropy is left unspeci®ed here; i.e., conjugate `displacement-like' variables (anities). except for the explicit appearance of the corresponding Now assume the existence of a dissipation potential material-directionally tensors in the arguments list for of the form, the potential functions employed, all derivations remain generally applicable. O O s ,ab ,Dd 7 Here, and for the remainder of the paper, a ij T Cartesian reference frame is utilized, and both matrix which, when written in its decoupled form [6], including and index notations (with summation implied on `¯ow' as well as `static (thermal) recovery' terms, gives, repeated subscripts) are used interchangeably, with the same letters indicating tensor components of variables, Xnb Xnd b d b b d d and their vector matrix `counterparts' (e.g. using the O O1 sij,aT,D O2 aT O3 D 8 familiar contracted Voigt representations in vector b1 d1 forms for components of the corresponding symmetric, Adopting `standard' thermodynamic arguments [1± second-order, tensors. 3,6], and the de®nition of the inelastic strain rate e_i , Consider the following general form for the Gibb's ij one can easily derive the Clausius±Duhem or dissipa- potential: tion inequality. That is, C C s ,ab ,Dd 1 b d ij T O sij,aT,D

n n where ab denotes the Tth-order tensorial internal state Xb Xd T s e_i ab :g_b Ddg_d r0 9 b ijij T T D variables (where b 1, ...,nb; T 1, ...,nT), aT can be b1 d1 a second-order tensor (back stress) or higher order tenors, b denotes the number of internal state vari- where `:' indicate scalar multiplications (complete d b b ables; D is the scalar internal state variable (where contraction) of the (identical-order) tensors aT and g_T, d 1, ...,nd and sij the Cauchy stress tensor. This is a and the overdot _ signi®es a rate or time derivative. typical general form for the Gibb's potential, in which The rate of change of the anities may then be as many internal variables as desired may be speci®ed expressed in terms of their corresponding internal state in Eq. (1). The potential may be decoupled into elastic variables. This results in the ¯ow or kinetic laws and inelastic parts; @O e_i 10 ij @s e i b d ij C C sij C aT,D 2 @O or, upon further decoupling of inelastic mechanisms for g_b 11 T @ab practically used forms, T

n n Xb Xd @O C Ce s Ci ab Ci Dd 3 g_d 12 ij a T D D @Dd b1 d1 From Eqs. (3) and (6), It then follows from Eq. (3) that: d @C @ 2C g_b Á a_b 13 @C T b b b T i dt @aT @aT@aT eij eij 4 @sij d @C @ 2C g_d D_ d 14 D d d d @C dt @D @D @D gb 5 T b @aT where `Á' in Eq. (13) signi®es a `T-times' contraction 606 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628

e (i.e., summation over all tensor indices in the Tth- sor C ijkl can be derived in the following convenient form; b order variable aT see Refs. [25,28] for details " # " # 1 1 Ce 2G P aR 2 G G Q @ 2C @ 2C @O t l t a_b Ág_b Á 15 T b b T b b b 20 @aT@aT @aT@aT @aT a 2 l G d d m V V 3 3 t l 2 1 2 1 d @ C @ C @O where the symbol indicates tensor product, and D_ g_d 16 @Dd@Dd D @Dd@Dd @Dd m1 3a 4 Gl Gt b 21 Eqs. (11)±(14) which de®ne the evolution of the internal state. Together, the equations of the state, Eqs. The a,b,l,Gl and Gt are the ®ve generalized Lames (4)±(6) and evolutionary laws, Eqs. (10), (15) and (16), constants with the subscripts `l' and `t' identifying the constitute the ®nal general forms of GVIPS. longitudinal and transverse directions, respectively, From Eqs. (15) and (16), the (generally-nonlinear) with respect to the ®ber direction (normal to plane of compliance operators (generalized hardening moduli) isotropy) for the shear moduli, Gl and Gt: The fourth- are de®ned as follows: order symmetric tensors P,Q and R are the same as those de®ned later in Section 3.3.2 (see Eq. (29)). @ 2C @ 2C Other, more general forms of anisotropy can be writ- Lab , LDd 17 T b b d d @aT@aT @D @D ten similarly. which relate the rates of `force-like' state variables b _ d b d a_T,D to the `displacement-like' variables g_T,g_D), and provide a link between the assumed Gibb's and complementary dissipation potential.

3.3.2. The viscoplastic response 3.3. Speci®c form for the GVIPS model For the viscoplastic response, we take T 2,nb 1,nd 0, so that only one second-order in- In this section, a speci®c form of the GVIPS model ternal variable, in this case the back (internal) stress, is is outlined. For completeness a transversely-isotropic speci®ed. We write the plastic dissipation function, O, elasto-viscoplastic material, i.e., a solid reinforced by a in terms of the `anisotropic' invariants of the stress single family of ®bers as in Refs. [10,11,25,26], is tensor, sij, the internal variable, aij, including separate assumed. ¯ow/hardening and static-recovery contributions, O1 and O2, respectively, 3.3.1. Elastic response Á Á Á The hyperelastic response is assumed to be linear. O O sij,aij,Vij O1 sij,sij,Vij O2 aij,Vij 22 That is, a quadratic strain±energy±density function, U e, is assumed to exist such that the Cauchy (true) O O1 F O2 G 23 stress components sij are given by @U e e e @O @O1 sij C e 18 _i @ee ijkl kl eij 24 ij @sij @sij where a superscript `e' denotes the elastic component, and the following expression, @O g_ij 25 @aij e i eij eij eij 19 where the parentheses are used to indicate functional holds true in accordance with the basic hypothesis of dependence on the arguments inside ( ); and the ma- the additive decomposition of total strain tensor, eij, terial directionality tensor Vij is de®ned as Vij vivj, e i into its elastic, eij, and inelastic components, eij: In the where v is the unit vector de®ning the material ®ber e above, C ijkl denotes the fourth-order symmetric elastic direction. moduli tensor. The F and G are the quadratic functions in terms of the invariants of the deviatoric components of the Remark 1. In terms of its ®ve independent elastic eective stress, sij aij , and the back stress, aij, ten- moduli, the present transversely-isotropic elasticity tensors, respectively. A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 607

1 F s a M s a 1 where H and Kt (stress units), and b (dimensionless) 2 ij ij ijkl kl kl 2K t are (positive) material constants. The function G G aij was de®ned previously in Eq. (26). Together 1 with the strain vector decomposition of Eq. (19), the G a M a 26 2 ij ijkl kl above Eqs. (30) and (31) de®ne the equations of state, 2K t e @C Á1 Mijkl is a symmetric fourth-order tensor which is a e ei C e s , ij ij @s ijkl kl function of the ®ber orientation tensor Vij (material- ij directionality) and is de®ned as, 32 @Ci 1 g p 1 ij @a h ij M P xQ zR 27 ij ijkl ijkl ijkl 2 ijkl in which h and pij are de®ned as, In the above, 0RxR1 and 0RzR1 are material- H strength ratios de®ned as, h , p M a 33 G b ij ijkl kl Z2 1 4 o2 1 x , z De®ning the free energy C ee,g through ordinary Z2 4o2 1 Legener-type transformation (e.g. 2), i.e., C C e C i, where C e Ce for linear elasticity here, K Y and C i a Á g Ci a the dissipation function O s ,s Z l , o l 28 ij ij Kt Yt is now introduced as follows, where the constants Kt (and Kl and Yt (and Yl are _ O sije_ij Cr0 34 threshold strengths of the composite material in trans- _ verse (and longitudinal) shear and tension/com- Substituting for C in terms of s_ ij and a_ij from Eqs. pression, respectively. The fourth-order tensors Pijkl, (30)±(32), one obtains, Qijkl and Rijkl are de®ned as, ! @O @O @ 2Ci 1 1 Á e_i , g_ a_ 35 P I d d , I d d d d @s ij @a ij @a @a kl ijkl ijkl 3 ij kl ijkl 2 ik jl il jk ij ij ij kl

The above rate form connects a_ij and g_ij through the 1 Á Q V d V d V d 2V V hardening-moduli tensor which is the Hessian matrix ijkl 2 ik jl il jk jl ik ij kl of Ci: Using Eq. (35), and the second of Eq. (31), yields Á 1 R 3V V V d V d d d 29 h 0 Â Ã ijkl ij kl ij kl kl ij 3 ij kl a_ h Z m a a g_ L 1g_ 36 ij ijkl h 1 2b ij kl kl ijkl kl

For the isotropic case, x z 0 and Mijkl reduces to where L is the present, single, compliance operator, Pijkl, i.e. simply the classical von Mises-type forms in ijkl 0 functions F and G. and h is a scaled derivative, For the case considered here, nb 1, nd 0, the pre- 1 dh sent purely mechanical theory, (Eq. (1)) has the follow- h 0 37 K 2 dG ing decoupled form of a Gibb's complementary free- t energy density function in terms of stress and internal m In addition, Z ijkl above is de®ned as the `generalized state parameters, i.e., with superscripts `e' and `i' indi- inverse' of the deviatoric, positive semi-de®nite, ma- cating elastic and inelastic parts, m 1 terial tensor operator Mijkl, i.e. Z ijkl Mijkl: We refer to Ref. [25] for details of dierent conditions (three- C s ,a Ce s Ci a 30 ij ij ij ij dimensional, plane-strain/axisymmetric, generalized plane stress, etc.). The terms Ce and Ci are de®ned to have the following forms, Remark 2. Relative to the conjugate (dual) variables ee and s of the elastic response, one de®nes the free-energy e 1 e Á1 C sij sij C skl 2 ijkl and (complementary-energy) `natural' norms, through 31 the positive-de®nite material-dependent moduli tensor 2 i K t 1b Ce, and its inverse, respectively. Analogously, for the vis- C aij G 1 bH coplastic counterpart, we can de®ne the semi-norms for 608 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 stress, ksk, and (conjugate) inelastic strains, keik, Remark 3. The above Eq. (44) provides the simplest through the M and its generalized inverse Zm; i.e., form of internal loading±unloading criterion; i.e., with r the evolution for the internal strain variables, g_ij, during 1 p loading according to Eq. (40), but with a dierent dissi- ksk s Á M Á s, keik 2ei Á Z Á ei 38 2 m pation function during unloading giving a `magni®ed' u u 1 rate of evolution, g_ij Lijkl Lklmn g_mn, to account for These generalized `norms' will be used repeatedly in the `stier' response upon unloading, with the moduli the sequel, thus allowing a uni®ed, and very eective, Lu taken in the same form as for loading in Eq. (36) treatment of both GVIPS and NAV models (e.g. see klmn with G0 replacing G. However, alternative criterion are Section 3.4.2) also possible to handle these observed readjustment of internal state [8], and may actually have direct impact on such cyclic-loading phenomena as ratchetting.

3.3.3. The viscoplastic response: ¯ow and evolution laws From Eqs. (31) and (35), the ®nal expressions for the inelastic ¯ow and evolution equations for the 3.4. Generalized nonassociative viscoplasticity (NAV) speci®c selection of power functional forms for O1 F and O G , Eq. (23) can be straightforwardly derived; 2 Considering the nonassociative models, their popu- i.e., larity in the literature is mainly due to the added modeling ¯exibility to account for experimental phenomena e_i f FG , G M s a 39 ij ij ij ijkl kl kl by using arbitrary rate forms in the ¯ow and evolution equations. In particular, the Freed±Walker viscoplastic l model [12] is used, as an example, to discuss the theor- g_ e_i p , p M a 40 ij ij h ij ij ijkl kl etical basis of the NAV class of models. This model was formulated to account for both Â Ã l short and long-term material behavior, by admitting a_ L 1 e_i p 41 two tensorial internal variables (short and long term) ij ijkl kl h kl in the additive decomposition of the back stress, and where f,h, and l are the ¯ow, hardening, and recovery one scalar internal variable (a drag stress). As with all functions, respectively, de®ned as, `uni®ed' viscoplastic models, creep-plasticity interaction is also accounted for, but the model was shown to ana- F n H f F , h , l RG mb 42 lytically reduce to the limiting cases of pure creep the- 2m G b ory at high temperature (steady-state condition) and to plasticity theory at low temperature (rate-independent in which n,m,H,b,R, and m are (positive) material con- condition). The existence of these creep and plasticity stants with mrb: bounds was made use of in simplifying the process of Several regions of discontinuities in the state space model parameter characterization. were introduced to account for stress-reversal/cyclic- To date, characterizations and applications of this loading/dynamic-recovery eects. This simply amounts model have been limited to the case of isotropy. For to using dierent functional forms for f,h, and l in easy reference to the original presentation in Ref. [12], dierent regions; i.e., this will also be assumed here. Unlike the more general f f if F > 0 nonisothermal treatment in Ref. [12], we restrict our- selves, as before, to the isothermal case. Tests reported later may be given at dierent temperatures, but the f 0 otherwise 43 given temperature does not change during a test. For clarity, we ®rst summarize the original equations in the above reference before transforming into similar forms G G if G > G0 and dev sij aij pij > 0 (and notations) as those used for GVIPS.

G G0otherwise 44 3.4.1. Elastic behavior Recalling the above isotropy restriction, the linear The `small' constant (cut-o value) G is an `adjusted' 0 elastic response is decomposed into the deviatoric and parameter selected to ®t the experimental data, and hydrostatic components; i.e. prevents singularity in h when G40 and dev( ) is the deviatoric part of ( ). i i Sij 2m0 eij eij ekk 0 45 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 609 s 3k e 46 i kk 0 kk e_ij f J,DGij, Gij Mijkl skl akl 54 In the above we de®ne, the elastic shear, m , and bulk, 0 where f is a function of invariant J and the drag stress k0, moduli, as well as, D, i.e., 1 1 p Sij sij skkdij, sij eij ekkdij 47 1 J 3 3 f yA psinh 55 2 J D for the deviatoric stress and strain, respectively. In keeping with the previous framework presented in and J, analogous to the previous de®nitions in F, Eq. Section 3.3, the above equations of the elastic behavior (26), is a quadratic function in terms sij aij), i.e., can of course be generalized to other material-symmetry groups, e.g. see Remark 1 above. 2 1 J ksij aijk sij aij Mijkl skl akl 56 2 3.4.2. Viscoplastic behavior where Mijkl is the fourth-order tensor as de®ned in Eq. The original inelastic ¯ow equation was postulated (27). The special case of isotropy assumed in the orig- in the form [12], inal model [12], is now trivially obtained by simply taking x z 0, and Mijkl reduces to Pijkl (see Eq. i i 1 sij aij e_ij ke_ knij, nij 48 (27)). The inelastic strain tensor modulus (magnitude) 2 ksij aijk can be de®ned analogously by using Zm (the general- in which, ized inverse of M as a metric in the de®nition of the second of Eq. (38); see also Remark 2 above. i ke_ kyZ if ksij aijk < K 49 The (deviatoric, tensor-valued) back stress, aij, accounts for kinematic (¯ow-induced, anisotropic) where, y > 0 is a thermal function, Zr0 is the so- hardening eects, and is taken to be a ®nite sum of the called Zener parameter and K > 0 is the yield stress. In s l s individual back stresses, i.e., aij aij aij, where aij addition, for isotropic viscoplasticity, the `convention- l and aij are the short-term and long-term back stresses, al' eective-stress von Mises norm ksij aijk, is de®ned respectively. Their evolutions are described by the lin- as, ear `hardening-minus-dynamic recovery' forms; r 1 as ks a k s a s a 50 s i ij i ij ij ij ij ij ij a_ij 2Hs e_ij ke_ k 2 2Ls Note that the norms pertaining to the deviatoric tensors of this section are de®ned as, ! al l i ij i r a_ij 2Hl e_ij ke_ k 57 p 1 2Ll kIk 2I I and kPk P P 51 ij ij 2 ij ij The above evolution of the back stress accounts auto- Qwith Iij being any deviatoric `strain-like' tensor, and matically, and in a continuous fashion, for the rapid ij the `dual' deviatoric `stress-like' tensor. change in stiness that is observed during the tran- The temperature-dependent function, y, was taken in sition from elastic to plastic behavior upon reversed the following (T is temperature) Arrhenius form, loading. The constants Hs, and Hl denote the short- and long-term hardening moduli, and Ls and Ll are Q y exp when T RT < T the corresponding thermal limiting states, which are KT t m given as m0 is the elastic shear modulus, and H0,f0,C,D0 and are material constants), Q T y exp ln t 1 when 0 < TRT 52 C D D D KT T t H m , L f 0 t s 0 s 0 dC and the Zener parameter, Z,as m C D D D ksij aijk 0 0 Z A sinh 53 Hl , Ll 1 f0 58 D H0 dC where Q,Tt,Tm,A and n are material constants. Now, As was done for the ¯ow law, the above original evol- Eq. (48) may be recast into the following more general ution equations with hardening/dynamic recovery form, mechanisms may be recast in the following form, 610 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 h i s i s a_ij 2 Hse_ij gspij de®ned by under-curved symbols and vectors by underlined symbols. h i l i l a_ij 2 Hle_ij glpij 59 4.1. General form of Newton iterative scheme where gs and gl are de®ned as; p In the typical nonlinear ®nite element analysis, the Hs J solution to the speci®ed load history is performed gs yA sinh 2Ls D incrementally. This involves projecting a known solution at step n (at time tn), where the current stress p H J and internal variables are known, to time tn1 where g l yA sinh 60 the above quantities are unknown. The implementation l 2L D l is given here following the `standard' deformation-dri- The scalar internal variable D, the drag strength, ven format; i.e., for speci®ed total strain history accounts for isotropic hardening eects with its evol- en 4en1: ution expressed as, In general, within the context of the generalized mid- Á point scheme the update expression for the vector of _ i D h ke_ kyg D0RDRDmax 61 state variables, S, may be expressed as 0RjR1): Â Ã The drag strength is bounded in the interval Sn1 Sn 1 fDSn fDSn1 64 D0RDRDmax, where Dmax is de®ned below; see Eq. (63) For example, when using the explicit, forward Euler Because of its scalar nature, the strain hardening method, the update Sn1 depends only on the known and dynamic recovery terms are combined into a quantities at n, i.e., DSn (with f 0). For the fully im- single, nonlinear, hardening function, h, the hardening plicit backward Euler scheme considered here, the modulus, update is based upon the unknown quantities at tn1, ! i.e., DSn1 (with f 1). Thus the update expression m for the fully implicit backward Euler has the form, D D0 = dC h hD Â Ã 62 sinh D D0 = dC Sn1 Sn DSn1 65 and the other material constants are given as, where S is the vector of state variables and DS is the increment in state variables. The dimension of the S D D g A sinh 0 and DS vectors depend on the particular constitutive dC model, for example, for the GVIPS model, 1 s D C D S 66 max 2 0 a

If the 3D space is considered, s,a and are 6 Â 1 vec- h h when T RT < T D 1 t tors. Thus, S would have the dimension of 12 Â 1: The

expression for the state variable increment DSn1, h0 h1 based upon a ®rst-order expansion, can easily be hD h0 when 0 < TRTt 63 Tt shown to be,

1 DSn1 KS Rn1 67 where K is the iterative Jacobian matrix and R is 4. Implicit integration scheme S n1 the residual function. Again, the residual function's dimension depends on the constitutive model, and for We proceed now, based on the above mathematical the GVIPS model, derivations and general forms, with the presentation of the implicit integration scheme. In this section, all Rs equations and expressions are shown using a concise R 68 Ra matrix notation for the integrated stress and internal stress ®elds. The contracted representations in vector Once more, considering the 3D space for GVIPS, R forms for the components of the corresponding sym- would have the dimension 12 Â 1 and the Jacobian KS metric, second-order tensor, are utilized. Matrices are would be 12 Â 12: On the other hand, corresponding A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 611 @R @R dimensions for the NAV model are 19 Â 1 and 19 Â 19 Rk1 Rk a j dsk a j dak 76 a a @s k @a k for R and KS, respectively. In the following section, it will be shown how Note that the subscript n 1 is dropped for clarity, through `partitioning' of the S vector and writing the and the index `k' is the iteration counter. constitutive model in a concise matrix format, the sizes The third and ®nal step is to form the necessary of the necessary arrays can be greatly reduced (e.g., iterative corrections in stress and internal variables, ds, kept to sizes of 6 Â 1 and 6 Â 6 for 3D space), thus and da using Eqs. (73)±(76). During the derivations, an improving the overall eciency of the implicit scheme objective is to factor and group expressions in terms of in terms of storage and computation time. k the stress increment dsn1: The following result is, Á k 1 e1 1 dsn1 J C Rs P4P1 Ra 77 4.2. A model problem Ð constitutive residuals and consistent tangent stiness k 1 k dan1 P1 Ra P2Dsn1 78 In this section are presented the equations for the fully implicit backward Euler integration scheme. The Note that in the above equations, the dimensions of derivation of the equations follow a systematic process the vectors and matrices are of the order of the under- outlined below, using GVIPS equations for illustration. lying physical stress-space dimension. That is, for a 3D The ®rst step is to write the local update expressions space, and accounting for symmetry of the second- for the stress and internal variables, i.e., order tensors s and a, the vectors are 6 Â 1 and the matrices are 6 Â 6: Thus the size has been reduced s s Ds 69 n1 n n1 from a possible 12 Â 12 matrix size to a maximum dimension of 6 Â 6: an1 an Dan1 70 Now Eqs. (77) and (78) along with the iterative form of the update equations, i.e., Using the implicit update equation for S, Eq. (65), construct the update expressions for the stress, s, and k1 k k sn1 sn1 dsn1 79 the internal variable a: That is, with given De en1 en1, and making use of Eqs. (36) and (39)±(41), k1 k k sn1 an1 dan1 80 e Á sn1 sn C De DtfGn1 71 constitute the complete, fully implicit integration algorithm. a a Dt n1 n In Eq. (77), J is the iteration Jacobian matrix, h 0 72 Á Â Z a a hfG lpj J P P P1P Ce1 P I P1P 81 m h 1 2b n1 3 4 1 2 4 1 2 with the following additional matrices de®ned, The second step involves forming residual vectors which are a necessary part of the iterative implicit P Z1 P DtlM scheme. Thus, using Eqs. (71) and (72), the residual 1 m 2 vectors may be de®ned as, 0 0 h Â Ã Dtl Dtl 1 p p e h Rs sn1 sn C De DtfGjn1 73 " # h 0 2 b 1 b r p p 0 4 2 h h K t G Ra M p pjn1 an1 an 82 h 74 h 0 2 h 0 Dt hfG lpj r M p p n1 h h Referring to Eqs. (69) and (70) the ultimate objective h 0 Â p p p is to calculate the increments in stress and the internal h n1 n n1 variables. Thus, consider a ®rst-order Taylor series Ã expansion of the residual vectors, i.e., pn1 pn pn1

k1 k @Rs k @Rs k Á Rs Rs jkds jkda 75 0 @s @a P2 hP4, P4 Dt fM f G G 83 612 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 Á e1 0 e1 P3 C Dt fM f G G C P4 84 vector/matrix expressions for implementing NAV models. We refer to Ref. [29] for further details, not- In the above, the primes indicate `scaled' derivatives as ing the unsymmetry of the stiness matrix in this given below, case; i.e. the nonassociativeness resulting from 1 df 1 dh 1 dl dynamic recovery. f 0 , h 0 , h 0 85 2 2 2 K t dF K t dG K t dG and (with superscript T indicating transposition) the scalar r is de®ned as

T 4.3. Line search scheme r pn1 Á an1 an 86 Note that all the arrays and functions, e.g. As mentioned, for highly nonlinear problems,

J,P1,P2,P3,P4,G,p,f,h etc. in Eqs. (81) to (86) are evalu- although the implicit scheme described above is uncon- k k ditionally stable, its successful application still requires ated based on the last updated state sn1,an1), unless indicated otherwise. Note also that the state an,an the use of an advanced numerical technique such as a corresponds to the converged global solution at the last line search method to produce a robust solution algor- time step; i.e., they are kept constant during the local ithm. The purpose of a line search is to guide the sol- iteration sequence in Eqs. (79) and (80). ution towards convergence, which is accomplished by In keeping with the underlying fully implicit inte- searching for a scalar multiplier that adjusts the gration scheme, one can straightforwardly proceed to amount of the iterative increment vector to be updated dierentiate the updated stress ®eld, sn1, to obtain within each iteration. the algorithmic (also consistent) viscoplastic-moduli or The line search method has been successfully utilized material tangent stiness matrix, Cev, for use in the on the global ®nite element solution level for problems ®nite element stiness calculations. The derivations such as concrete cracking [30]. There is well-documen- will in fact lead to ted evidence that the use of the line search is also Â Ã essential for robust performance of Newton's method, ev 1 e1 1 1 on the global level, for problems dealing with inviscid C J C P4 hP4P1 P4 87 elastoplasticity [13,14,21]. At the global level, the con- e Clearly, C and P1,P2,P3, and P4 are symmetric, result- cept of the line search algorithm pertains to minimiz- ing in ®nal symmetric Cev: ing the potential energy, that is, the work done by the residual force due to the iterative displacement. In this Remarks 4. same spirit, the line search method is applied at the local (material point ) level to drive the residual consti- 1. Note that, as de®ned here, the generalized inverse op- tutive functions to zero. For the line search method, Eq. (65) needs to be erator Zm of Eq. (38); see also [25], is of full rank (i.e., rank 6 in three dimensions ), resulting in a gener- modi®ed. Thus for a typical iteration k4k 1, the ally positive-de®nite matrix in Eq. (82). iterative procedure with the line search is described by 2. The entire algorithmic setting in Eqs. (77)±(86) car- the following expression, ries over, with no change, to the dierent subspaces Sk1 Sk ZdSk 88 ( plane strain/stress, etc.); only the dimensions of the n1 n1 n1 vector/arrays are appropriately reduced. We refer to where Z is a scalar 0RZR1 which adjusts the step Ref. [25] for further details. size to optimize the iterative solution. As mentioned 3. Eqs. (77) and (78) furnish the basis for simultaneous the objective of the line search method is to minimize iterations for both ds and da, known for their greater k k jRn1 Á dSn1j: Here, `Á' indicates the dot product of eciency with general elastoplastic anisotropies. This two vectors and R is the vector-valued residual func- is in contrast to other `specialized' alternatives; e.g. tions of the state variables given, for example, in Eqs. using nested,`®xed-point', or two-level, `alternating' (73) and (74). iterations [20,23,24], often obtained by further re- Now consider, ductions of the equations; e.g. in terms of the magnitude (scalar) and unit-vector direction of the inelastic s Z R Z Á dSk 89 strain vector, under restricted conditions of material symmetries (e.g. a single, scalar `magnitude' equation in which Sk and dSk are ®xed, and R and s are func- for the extreme case of elastoviscoplastic isotropy ). tions of Z: When Z 0, 4. Although far more laborious in its details, the same k k procedure as above can be used to derive the `lengthy' s0 s Z 0 R Z 0 Á dS R0 Á dS 90 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 613

R is the residual function of the state variables at the NAV model. In the GVIPS model, there are regions of end of the previous iteration. discontinuities in the state sij,sij space to account for According to the optimization theory, the best sol- stress-reversal eects, see Remark 3 and Eqs. (43) and ution is s g0, but numerically, this is not realistic (44). For time steps during which these regions of dis- and attempting to achieve this objective would be inef- continuity are traversed, the line search technique is no ®cient. For practical problems, an approximation is longer as powerful, as compared to its implementation thus utilized, i.e., a `slack' line search is used with the with the NAV model. Line search is strictly valid only aim of making the magnitude of the `new' s Z small in if the same `unique' forms of the residual functions comparison with that of its previous estimate s Z 0, apply during the considered step. Consequently, a sub- i.e., incrementing scheme is still needed to re®ne the global step size suciently to capture the point of discontinu- s Z ity. For all other steps, in which the GVIPS model is j j < bls 91 s Z 0 continuous, the line search method will be shown to play an important role in terms of eciency and con- where b is the line search tolerance. Based on past ls vergence. research experience, a suitable value for b is in the ls For all test cases, the material parameters used in order of 0.8 [30]. After the successful search for Z is the GVIPS model are associated with W/Kanthal (a completed, Eq. (88) continues to update the state vari- unidirectional metal matrix composite model material), ables until convergence. For more details of the line whereas the material parameters used in the isotropic search strategy utilized we refer to Ref. [29]. NAV model are for copper. Material constants for both elasto-viscoplastic models are given in Tables 5 4.4. Convergence criteria and 6 (E is the Young's modulus, n is Poisson's ratio). Consequently, no direct comparison between the The typical convergence criterion used here, consid- GVIPS and the NAV models can be made and it is im- ering the case of stress and GVIPS implementation, portant to emphasize this point from the start. has the following form, However, it was attempted to reach the same level of computational eort for each test. For example, for ksk1 sk k n1 n1 < tol 92 the tensile test for both the GVIPS and NAV models, k1 ksn1k0:01Kt the test was performed until `saturation' in stress was attained, thus making the level computational diculty where, here kÁk is the Euclidean norm of vectors, and comparable. Finally, we note that all of the solution `tol' is the tolerance. Other internal variables also have schemes developed were implemented into an in-house a similar convergence criteria. For NAV, we similarly nonlinear ®nite element code in conjunction with a use the yield stress K, Eq. (49), to replace the constant four-node mixed shell element, as described in Ref. Kt in Eq. (92). [31].

5. Applications 5.1. Validation tests A number of numerical tests, utilizing NAV and GVIPS models, are reported here to demonstrate the Here, various tests are used to validate the implicit capabilities, and various aspects, of the implicit itera- integration algorithm for both the GVIPS and NAV tive algorithm; e.g., validation, accuracy, robustness models. These tests include monotonic tension, cyclic, and computational eciency. On the practical side, it creep and relaxation tests performed with dierent is the accuracy, eciency, and a balance between the load steps. An explicit, forward Euler integrator was two, that always constitute major concerns. Validation used in some cases to obtain a `converged' reference of the algorithm may be proven by convergence and solution for comparison. Note that we have made no accuracy, while eciency may be shown by CPU time attempt here to `optimize' this latter explicit solution. comparisons with another, independent; e.g. explicit Note that for all tests of GVIPS, the load is trans- method, or the same implicit scheme but with a dier- verse to the ®ber direction. In the following tables, ent step size, whose solution has the same accuracy. GIT is the average number of global iterations, SUB is In all applications involving NAV, the line search the average number of subincrements, and LIT is the technique outlined earlier in Section 4 is used to average number of local iterations. In addition, all improve the eciency and also guarantee the conver- CPU times are normalized with respect to the smallest gence; see its details in Ref. [29]. The subincrementing CPU time and the CPU is the total time of a complete, technique is not used in the implementation of the single-element, ®nite element analysis. 614 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628

Fig. 1. Tensile load.

5.1.1. Strain controlled tension test This monotonic hardening test is used to demonstrate the accuracy and eciency of the proposed implicit integration algorithm. For both GVIPS and NAV, the uniaxial, strain controlled tension test is as shown in Fig. 1. The maximum strain was 0.20 and the total strain rate was 4:0 Â 105 /s for GVIPS and 7:6 Â 106 /s for NAV. For reference, the 10,000 steps with the forward Euler integrator (explicit) were used Fig. 2. Tensile results for GVIPS. for the `converged' solution, up to the maximum strain Table 1 illustrates the results for three dierent load and 148.82 MPa for the 73, 20 and 10 load steps, re- step increments, utilizing the GVIPS model. The spectively, and the dierences are less than 1%, thus results of the same three dierent cases are as shown validating and further demonstrating the accuracy of in Fig. 2. Note how the 100 step case matches that of the integration algorithm. the 10000 step `converged' solution thus demonstrating With regards to CPU time, it is found that the ratios the accuracy of the integration algorithm and vali- of the explicit method, in the region up to 5% strain, dation of the implementation of the implicit algorithm. and the 73 step implicit method is 200/2.5 = 80, and With regards to CPU time, it is found that the ratios the ratio of the 73 and 10 step cases is 3.2, which of the explicit and the ten step implicit method is 108, demonstrates the eciency of the scheme (Table 2). and the ratio of the 100 and 10 step case is 5, thus Also note that as the time step increases, the amount demonstrating the eciency of the scheme. of local iterations (LIT) also increase as expected For this test case with the NAV model, the explicit integration method failed to reach the maximum strain level. After 10,000 steps, the explicit solution only reached 5% strain (marked by the arrow in Fig. 3). Further attempts, to reach the maximum strain level, required excessive re®ning of the step size, thus render- ing the CPU time prohibitively large. As an aside, the fact that the explicit solution required 10,000 steps is characteristic of explicit methods in general; e.g., as reported by Arya [32] explicit viscoplastic integrations typically required from 40,000 to 100,000 increments. From Fig. 3, the maximum stresses are 148, 148.57

Table 1 Tension results for GVIPS

Method Number of load steps CPU time GIT LIT

Explicit 10,000 108 3 ± Implicit 100 5 4 2 Implicit 10 1 4 3 Fig. 3. Tensile results for NAV. A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 615

Table 2 Tension results for NAV

Method Number of load steps CPU time GIT LIT

Explicit 10,000 > 200 1 ± Implicit 73 3.2 2 7 Implicit 20 1.1 3 8 Implicit 10 1 3 9

(Tables 1 and 2) since more computational eort is required on the local, material point level.

5.1.2. Strain controlled cyclic test This example serves to assess the accuracy of the stress predictions and to demonstrate the robustness of the implicit algorithm. For both GVIPS and NAV, the uniaxial, cyclic (non-monotonic), strain-controlled test, shown in Fig. 4 was simulated. The maximum strain amplitude of was applied at a rate of 0.002/s. Fig. 5. Cyclic results for GVIPS. For the GVIPS model, Fig. 5 shows the explicit converged solution as compared to the implicit 100, 10 and 5 step cases. The fact that all of the implicit cases The predictions for the NAV model are shown in match the converged solution demonstrates the accu- Fig. 6. The 100, 10 and 5 step cases agree reasonably racy, eciency and robustness of the algorithm. In ad- well with the converged explicit solution. For the 5 dition note that even the 5 step case matches the step case, typical stress changes in some steps, of ap- converged solution at all points of the hysteresis loop. proximately 220 to 200 MPa occurred within a single The previously discussed discontinuities within the step, which demonstrates the robustness of the algor- GVIPS formulation occurs along the path from A to B ithm. Comparing the 100 step case with the explicit and from C to D as shown in Fig. 4. Those steps method (requiring 10,000 global steps for one complete which contain the discontinuity point require subincre- cycle), the CPU time ratio is approximately menting in order to cross the discontinuity. Thus, only 340=6 60, and for the 10 step case versus the 10000 these latter two steps required subincrements which step case the CPU ratio is 340, (Table 4). Considering averaged 22 in this case. As such, comparing the 10 the comparable accuracy of these two cases, with step case with the explicit solution (Table 3), the CPU regards to the predicted maximum/minimum stress ratio is approximately 180 which demonstrates a amplitudes, a CPU ratio of 60 and 340 demonstrates a marked saving in CPU time while maintaining compar- signi®cant improvement in eciency and may be able accuracy. attributed to the use of the line search method.

Fig. 4. Cyclic load. 616 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628

Table 3 Table 4 Cyclic results for GVIPS Cyclic results for NAV

Method Number of load steps CPU time GIT LIT Method Number of load steps CPU time GIT LIT

Explicit 10,000 180 3 ± Explicit 10,000 340. 3 ± Implicit 100 5 2 4 Implicit 100 6 2 4 Implicit 10 1.05 4 10 Implicit 10 1.05 3 10 Implicit 5 1 10 20 Implicit 5 1 3 22

5.1.3. Creep test GVIPS and NAV models, the creep test was able to be The creep test Fig. 7, is used to test both the accu- completed using only two steps which demonstrates racy and the robustness (stability) of the implicit inte- the stability of the implicit integrator. It is well known gration algorithm. In a creep test, the primary creep that during steady-state creep, the issue of stability is region is controlled by accuracy due to the high inelas- of major concern. For the GVIPS model, the ®nal tic strain rates involved. In the secondary (steady state) inelastic (creep) strain for the two step case diers creep region, the stability or robustness of the inte- from the `converged' (1600 step) solution by approxi- gration algorithm is tested. Both the GVIPS and NAV mately 8% (see Fig. 8). This result is believed accepta- models will be used. One creep test of 1000 seconds at ble considering the implicit integrator is ®rst-order a constant creep stress of 70 MPa for the GVIPS accurate. On the other hand, for the NAV model the model and a constant creep stress of 24 MPa for the ®nal creep strain for the two step case diers from the NAV model was applied for 1000 seconds. A large `converged' (1600 step) solution by approximately 30% loading rate was used during the initial load-up to the (see Fig. 9). This may be explained by the fact that for target, constant creep stress, in order to simulate a the NAV model, the primary creep response is highly purely elastic loading. nonlinear, as compared to the primary creep region of For both the GVIPS and NAV models, it was deter- the GVIPS response. For the NAV model, since the mined that a `converged' solution was obtained using primary region was basically traversed in one step the implicit method with a total of 1600 global load (Fig. 9), the accuracy in this region was not sucient, steps (Figs. 8 and 9). In addition, for both models, in turn adversely aecting the accuracy of the steady- cases of 100, 50, 10 and 2 steps were used during the state portion of the response. However, note that the constant stress period of the creep test. For all cases NAV model could still be successfully integrated in and both models, 10 steps were used during the initial two steps, again showing the stability of the implicit load-up portion of the test. integrator. Examining Figs. 8 and 9, note that for both the 5.1.4. Relaxation test The stress relaxation test may also be used to assess the accuracy of the integration method. Subsequent to load up, typically the relaxation test, Fig. 10 is characterized by a marked decrease in the initial stress at a high rate. The amount of stress relaxation and corresponding rate is dependent on the speci®c material model. Thus, an accurate integration method is required so as to properly predict the highly nonlinear stress response. Relaxation tests were performed for both the GVIPS and NAV models. The hold strain amplitude is 0.01 and is held for 1000 seconds. As with the creep test, a high initial load-up rate was used to ensure a purely elastic response. In addition, in all cases and for both models, 10 steps were used in the initial load-up region. Cases of 100, 50, 10 and 2 steps were preformed for the hold (relaxation) period. Figs. 11 and 12 show the predicted stress relaxation response for the GVIPS and NAV models, respectively. Note that the GVIPS model response has signi®cant relax- Fig. 6. Cyclic results for NAV. ation, approximately 200 MPa, while the NAV model A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 617

Fig. 7. Creep loading. response has only 4 MPa of stress relaxation. As a 5.2. Nonproportional multiaxial test result, the more nonlinear response of the GVIPS model is more of a challenge for the implicit integra- We next examine the integration algorithm under tor. Note that the relaxation response for both the nonproportional loading conditions. The material el- GVIPS and NAV models could be completed in two ement is subjected to the nonproportional strain path steps (Figs. 11 and 12), demonstrating the robustness shown in Fig. 13. This path is taken from the exper- (stability) of the implicit integrator. This is more sig- imental work of Lamba and Sidebottom [33]. This ni®cant for the GVIPS model whose relaxation re- applied strain path was chosen because of its wide sponse is strongly nonlinear (Fig. 11). From Fig. 11, range of angles between segments, with equal positive the ®nal stress at the end of the relaxation period for and negative peaks. The eight paths combine shear, the two step case diers by only 5% as compared to axial loading and unloading. The end points of the the converged (1600 step) solution. For the NAV path segments were numbered from 0 to 8 and the path model, the two step response is practically identical to was traversed in a numerically increasing sequence. the converged solution. Note that the experiment given in Ref. [33] was con-

Fig. 8. Creep results for GVIPS. 618 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628

Fig. 9. Creep results for NAV. ducted on a saturated specimen, that is, complete The GVIPS model was originally used to character- stabilization with regard to hardening has occurred. ize W/Kanthal whose material constants are given in The stress response of the model to the applied non- Tables 5 and 6. In order to compare the stress re- proportional strain path appears in Fig. 14. This test sponse of the GVIPS model under the nonproportional demonstrates the robustness of the scheme discussed. strain path with the experiment in [33], the material The predicted response, shown in Fig. 14, matches the constants needed to be changed to approximately ®t experimental results (not shown in Fig. 14) given in the copper material used in the experiment. The new Ref. [33] very well. In addition, note that even one material constants are given in Table 7, wherein the step per path (Fig. 14) can successfully complete this material is taken to be isotropic, i.e., o Z 1: (elas- complicated loading history. Because the 50 step and tic constants are given in [33]). Based on the new ma- 10 step cases give identical results, it is concluded that terial constants, the predicted response of the GVIPS 50 steps has converged to the correct solution. Finally model is shown in Fig. 15, with the 10 step and 100 the CPU time for 1, 10 and 50 steps per path are 1 (35 step cases giving almost identical results. Given the s), 3.8 and 21.4, respectively. very crude characterization of the GVIPS model

Fig. 10. Relaxation loading. A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 619

Fig. 11. Relaxation results for GVIPS.

(essentially ®tting the limit state in a single pure shear problem with nonhomogeneous states of stress and de- test), it is remarkable how accurate it represents the ex- formations. In this context, a perforated plate as used perimental results. previously in [22,25], was analyzed using the NAV model. A quarter of the plate is discretized for the 5.3. A structural problem using the NAV isotropic model analysis (see Fig. 16). A linearly increasing speci®ed displacement, with a maximum value of 0.5% at a rate The ultimate test of the integration method is of of 6:67 Â 105/s, was applied along the upper bound- course to demonstrate its accuracy, robustness and e- ary of the plate. Material constants are the same as ciency when applied to a realistic structural analysis those given in Tables 5 and 6. The results shown in

Fig. 12. Relaxation results for NAV. 620 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628

Fig. 13. Nonproportional loading path. Fig. 14. Nonproportional loading results for GVIPS.

Figs. 17±19 are the eective stress distributions at the implicit integration algorithm. However, the accuracy ®nal load level when 1, 5 and 20 global steps are used. has further decreased (12% dierence in the maximum Also in order to check the convergence and eciency J2) in this latter one step solution. The CPU time for of the implicit integration scheme, an explicit test is explicit scheme and the 20, 5 and 1 step implicit cases run by using 5000 global steps as the reference sol- are 12570, 251, 91 and 25 min, respectively. The CPU ution. The results shown in Figs. 20 and 21 are the ratio of the explicit versus the implicit scheme (20 eective inelastic strain distribution at the ®nal load steps) is more than 50, thus showing the signi®cant e- level by using the implicit scheme (20 global steps) and ciency of the implicit integration scheme. Comparing explicit scheme (5000 global steps), respectively. Note the implicit scheme itself, the CPU ratio of the 20 step that the dierence between these two solutions is less case to that of the 5 step case is approximately 2.7. that 1 percent, i.e., for all practical purposes the im- This is a signi®cant saving in computational time yet plicit 20 step case has completely converged and may the results are almost identical, i.e. the dierence is less be chosen as the reference solution. than 2%. Thus, for the complex structural problem, Comparing the 5 and 20 step cases (Figs. 18 and 19), note that the eective stress distribution of the 5 step case is qualitatively close to the 20 step case which shows reasonable accuracy for even the 5 step case. Also, convergence was achieved using only one global step which demonstrates the robustness of the

Table 7 GVIPS material constants for nonproportional test

Constant Value Units n 2.5 ± m 2:925 Â 1016 s m 3.0 ± b 2.5 ± R 1:94 Â 107 MPa/s H 2:0 Â 104 MPa

G0 2.5 ± o 1.0 ± Z 1.0 ± KFUNCt 0.6823 MPa Fig. 15. Nonproportional loading results for NAV. A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 621

Fig. 16. Structure mesh.

Table 5 NAV Material constants (copper)

Constant Value Units Table 6 GVIPS material constants (W/Kanthal)

Tt 678 K m0 43,000 MPa Constant Value Units m1 17 MPa/K h0 50 MPa n 1.5 ± 7 h1 15 MPa m 9 Â 10 s Q 200,000 J/mol m 1.5 ± K 8.314 J/mol K b 0.5 ± A 20,000,000 1/s R 1:94 Â 108 MPa/s n 4.5 ± H 12.6E4 MPa C 13 MPa G0 0.05 ± D0 0.13 MPa o 2.7 ± f 0.75 ± Z 1.0 ± d 0.035 ± Kt 5.6 MPa m 0.5 ± El 207 MPa H0 20 ± Et 191 MPa n 0.36 ± nlt 0.24 ± 622 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628

Fig. 17. Eective stress distribution (1 step). A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 623

Fig. 18. Efective stress distribution (5 steps). 624 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628

Fig. 19. Eective stress distribution (20 steps). A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 625

Fig. 20. Eective inelastic strain distribution (20 steps). 626 A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628

Fig. 21. Eective inelastic strain distribution (5000 steps, explicit method). A.F. Saleeb et al. / Computers and Structures 74 (2000) 601±628 627 convergence may be reached in an ecient manner ence that convergence is almost always obtained, with without a signi®cant loss in accuracy. the following observations as the time step becomes `excessively' large: (i) the solution accuracy may degrade; (ii) an excessive number of local iterations are 6. Summary and conclusions required; (iii) the local line search attempts to decrease the time step to an unrealistically small step. With The general computational framework described in regards to (ii) a limit on the local iterations has been this paper, within the speci®c context of the backward set and for (iii) a lower limit on the time step size has Euler, fully-implicit, integration method, has been also been set. Of course if the local iteration limit is determined to be successful for uni®ed algorithmic reached, the implicit scheme stops and it is our experi- treatment of both the Generalized Viscoplasticity with ence that a slightly smaller time step will provide an Potential Structure (GVIPS) and the Non-Associative ecient solution. Viscoplastic models (NAV). In this, a number of im- On many occasions, the implicit integration scheme portant enhancements were included, aiming directly at with simple subincrementing, was found to provide computational-cost reduction, improved convergence, ecient solutions for GVIPS. But for NAV models, and better precision. Only these enhanced Newton subincrementing alone was not ecient. That is, many iterative schemes will provide for uniformly valid, con- subincrements and local iterations were still required. vergent, implicit integrations for such complex GVIPS On the other hand, including the line search in the pre- and NAV classes of viscoplastic models. The algor- sent algorithm enabled the NAV class of models to ithms were written in a concise matrix format so as to converge stably with sucient accuracy and signi®- provide sucient generality, are automatically valid cantly improved eciency. for both isotropic and anisotropic materials, both In comparing GVIPS class versus some forms of the cases in full space or subspaces. Exact, closed-form, presently used NAV models, we note that for global linearizations of the constitutive residuals were iterations, GVIPS provides a symmetric (fully-consist- achieved through a systematic elimination procedure, ent) tangent stiness that was used directly, but for resulting in signi®cant reduction of array sizes for e- NAV, only an approximate `symmetrized' tangent sti- ciency. A line-search strategy has complemented the ness was utilized. As a result, from the present numeri- basic local iterations for robustness. cal tests, it was found that NAV exhibited greater Given the present advanced state-of-the-art in com- diculty in global convergence compared to the putational viscoplasticity, we strongly believe that any GVIPS model, i.e., NAV required more global iter- further development in applications to complex ma- ations in order to reach a solution. However, by terial models with dierent mechanisms and material means of a line search scheme, NAV (because of its anisotropy should be well-documented by means of a continuous formulation), can now be integrated more comprehensive set of numerical simulations. To this easily even without subincrementing. The overall e- end, results of a large variety of test problems were ciency of integrating the GVIPS class of viscoplastic reported to assess the performance of algorithms devel- models is impressive when one recalls that, despite the oped. Against their background, several conclusions occasional need for a combination of subincrementing are made here. All numerical tests show that the use of and line search (because of the discontinuity in GVIPS the implicit integration method, for both `small' and evolution equations), all reported results for GVIPS `large' time steps, provides a balanced combination of were often found to be `economical'. stability and accuracy. Even with nonproportional, multiaxial stress-strain states, sucient accuracy and good convergence characteristics were demonstrated Acknowledgements for the implemented scheme. In addition, it has been generally observed that This work was supported by NASA-Lewis Grants explicit schemes either fail completely or become prohi- No. NAG3-1493 and NAG 3-1747. Special thanks are bitively expensive, especially for the NAV class of due to R. Ellis and S. M. Arnold of NASA-Lewis for models. Speci®cally the time step size must be inher- their many helpful suggestions and technical discus- ently very small in order to guarantee convergence, sions. which is not ecient. This point has been clearly demonstrated by several results, where, on average, a total number of time steps ranging from 10,000 to References 80,000 was required in order to obtain a solution. These results appear to be consistent with observations [1] Coleman BD, Gurtin ME. 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