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Robust Integration Schemes for Generalized Viscoplasticity with Internal-State Variables

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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 accu- racy and eciency of the overall ®nite element sol- ution. 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
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