An implicit integration algorithm with a projection method for viscoplastic constitutive equations
01 Oct 1989-International Journal for Numerical Methods in Engineering (Wiley)-Vol. 28, Iss: 10, pp 2397-2421
TL;DR: In this article, the solution of the non-linear system of algebraic equations arising from time discretization of the constitutive equations is determined using a projection method in combination with Newton's method.
Abstract: Robinson's viscoplastic model, a representative of the so-called overstress models, is integrated by use of the generalized midpoint rule. The solution of the non-linear system of algebraic equations arising from time discretization of the constitutive equations is determined using a projection method in combination with Newton's method. Consistent tangent moduli are calculated and the quadratic convergence of the global Newton equilibrium iteration is shown. The time increment size is controlled by the convergence behaviour of the equilibrium iteration and the accuracy of the numerical integration. Various numerical examples are considered to demonstrate the efficiency of the methods.
Citations
More filters
TL;DR: In this article, the equivalence of stress algorithms, based on a Backward-Euler-step applied on viscoplastic models of Chaboche-type, and their elastoplastic counterpart, is discussed.
Abstract: This paper deals with two main topics. The first one concerns the equivalence of stress algorithms, based on a Backward-Euler-step applied on viscoplastic models of Chaboche-type, and their elastoplastic counterpart. Generally, the stress algorithm yields a system of non-linear algebraic equations and the corresponding consistent tangent operator, occurring in the principle of virtual displacements, leads to a system of linear equations. This procedure can be obtained utilizing only numerical methods. The second topic concerns a special constitutive relation based on a kinematic hardening model using a sum of Armstrong/Frederick terms, which is equivalent to a multi-surface plasticity model. Applying this model a so-called problem-adapted stress algorithm is derived, where only one non-linear equation must be solved. This result is independent of the number of terms in the hardening model. Furthermore, only the viscoplastic algorithm must be implemented, since it includes the elastoplastic constitutive model as a special case. © 1997 by John Wiley & Sons, Ltd.
95 citations
TL;DR: In this article, a geometric linear viscoplastic model is generalized to finite strain and its numerical application is presented. But the numerical integration of the constitutive model involves the solution of only one nonlinear equation for one scalar unknown.
63 citations
TL;DR: In this paper, a new approach for cyclic viscoplastic problems is described, which is an iterative procedure that accounts for the whole loading process in a single time increment.
62 citations
TL;DR: The mathematical structure underlying the rate equations of a recently developed constitutive model for the coupled viscoplastic-damage response of anisotropic composites is critically examined in this paper, where a number of tensor projection operators have been identified and their properties were exploited to enable the development of a general computational framework for their numerical implementation using the Euler fully-implicit integration method.
Abstract: The mathematical structure underlying the rate equations of a recently-developed constitutive model for the coupled viscoplastic-damage response of anisotropic composites is critically examined. In this regard, a number of tensor projection operators have been identified, and their properties were exploited to enable the development of a general computational framework for their numerical implementation using the Euler fully-implicit integration method. In particular, this facilitated (i) the derivation of explicit expressions of the (consistent) material tangent stiffnesses that are valid for both three-dimensional as well as subspace (e.g. plane stress) formulations, (ii) the implications of the symmetry or unsymmetry properties of these tangent operators from a thermodynamic standpoint, and (iii) the development of an effective time-step control strategy to ensure accuracy and convergence of the solution. In addition, the special limiting case of inviscid elastoplasticity is treated. The results of several numerical simulations are given to demonstrate the effectiveness of the schemes developed.
52 citations
TL;DR: In this article, a time-integration algorithm for solving a non-linear viscoelastic-viscoplastic (VE-VP) constitutive equation of isotropic polymers is presented.
Abstract: The present study introduces a time-integration algorithm for solving a non-linear viscoelastic–viscoplastic (VE–VP) constitutive equation of isotropic polymers. The material parameters in the constitutive models are stress dependent. The algorithm is derived based on an implicit time-integration method (Computational Inelasticity. Springer: New York, 1998) within a general displacement-based finite element (FE) analysis and suitable for small deformation gradient problems. Schapery's integral model is used for the VE responses, while the VP component follows the Perzyna model having an overstress function. A recursive-iterative method (Int. J. Numer. Meth. Engng 2004; 59:25–45) is employed and modified to solve the VE–VP constitutive equation. An iterative procedure with predictor–corrector steps is added to the recursive integration method. A residual vector is defined for the incremental total strain and the magnitude of the incremental VP strain. A consistent tangent stiffness matrix, as previously discussed in Ju (J. Eng. Mech. 1990; 116:1764–1779) and Simo and Hughes (Computational Inelasticity. Springer: New York, 1998), is also formulated to improve convergence and avoid divergence. Available experimental data on time-dependent and inelastic responses of high-density polyethylene are used to verify the current numerical algorithm. The time-integration scheme is examined in terms of its computational efficiency and accuracy. Numerical FE analyses of microstructural responses of polyethylene reinforced with elastic particle are also presented. Copyright © 2009 John Wiley & Sons, Ltd.
42 citations
References
More filters
TL;DR: In this paper, it is shown that consistency between the tangent operator and the integration algorithm employed in the solution of the incremental problem plays crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon Newton's method.
1,702 citations
TL;DR: In this article, an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations was carried out, where attention was confined to infinitesimal deformations.
Abstract: An accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors, is carried out in this paper. For simplicity, attention is confined to infinitesimal deformations. The integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations. General rate-independent and rate-dependent behaviour, with plastic hardening or softening, associated or non-associated flow rules and nonlinear elastic response can be efficiently treated within the present framework. Isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding.
800 citations
TL;DR: In this paper, an unconditionally stable algorithm for plane stress elastoplasticity is developed, based upon the notion of elastic predictor-return mapping (plastic corrector). Enforcement of the consistency condition is shown to reduce to the solution of a simple nonlinear equation.
Abstract: An unconditionally stable algorithm for plane stress elastoplasticity is developed, based upon the notion of elastic predictor-return mapping (plastic corrector). Enforcement of the consistency condition is shown to reduce to the solution of a simple nonlinear equation. Consistent elastoplastic tangent moduli are obtained by exact linearization of the algorithm. Use of these moduli is essential in order to preserve the asymptotic rate of quadratic convergence of Newton methods. The accuracy of the algorithm is assessed by means of iso-error maps. The excellent performance of the algorithm for large time steps is illustrated in numerical experiments.
662 citations