An Implicit integration scheme for generalized viscoplasticity with dynamic recovery

Abstract

This paper is concerned with the development of a general implicit time-stepping integrator for the¯ow and evolution equations in a recent representative class of generalized viscoplastic models, involving both hardening and dynamic recovery mechanisms. To this end, the computational framework is developed on the basis of the unconditionally stable, backward Euler difference scheme. Its mathematical structure is of suf®cient generality to allow a systematic treatment of several internal variables of the tensorial and scalar types. The matrix forms developed are directly applicable in general (three-dimensional) situations as well as subspace applications (i.e., plane stress/ strain, axisymmetric, generalized plane stress in shells). The closed-form expressions for residual vectors and the algorithmic, (consistent) material tangent stiffness array are given explicitly, with the maximum matrix sizes``optimized'' to depend only on the number of independent stress components, but not the number of internal state variables involved. Several numerical simulations are given to assess the performance of the developed schemes. IntroductionThe analysis of various structural components operating under complex thermomechanical and multiaxial loading conditions require large-scale, nonlinear numerical simulations, usually involving the ®nite element method. From the computational standpoint, the global solutions for the ®nite element method typically use an incremental/iterative (either full or semi-) Newton-Raphson procedure. The key ingredient in a nonlinear material analysis is the local (integration point) time integration of the viscoplastic (rate-dependent) constitutive models which are sets of ®rst-order, differential equations that are highly nonlinear, coupled and mathematically stiff. The accuracy, stability properties and the implementation of the local integrator directly in¯uences both the accuracy and ef®ciency of the global ®nite element solution.Thus, the development of stress integration algorithms has received considerable attention in the recent literature on computational plasticity and viscoplasticity. Implicit methods of time integration have been particularly emphasized because of improved stability as compared to explicit integration methods (Wang and Atluri 1994). For example, a number of well-established (fully-and semi-) implicit schemes are presently available for inviscid plasticity and viscoplasticity; see Caddemi and Martin () that, among several other implementation details, the following two are of utmost importance for the effective use of implicit integration for viscoplasticity, i.e., (i) consistent linearization of the nonlinear stress response function resulting from the integration algorithm, and (ii) convergence of the local iterative (Newton-type) solution. Considering item (i), and within the standard strain-driven format of implementation (i.e., displacement model in ®nite elements), we note that the integrated stress ®eld is de®ned as a function of the speci®ed strain history over a...

Thermomechanical analysis of direct chill casting using finite element method

“…The total strain E can be additively decomposed into three components as (10) where E e is the elastic strain tensor, E t is the thermal strain tensor, and E vp is the viscoplastic strain tensor. The volumetric nature of the thermal strain tensor can be defined as,…”

“…To ac-celerate the convergence in the global level, a linesearch algo-rithm is used within the Newton-Raphson iterations. Further, in the integration point level, the constitutive equations are in-tegrated [10] iteratively again using the line-search algorithm within the Newton-Rahphson iterations. The flow chart of the algorithm used to solve the DC casting problem is shown in Fig.…”