# Finite deformation plasticity in principal axes: from a ...w3.lmt.ens- ELSEVIER Comput. Methods Appl

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Computer methods in applied

moGhaniGs and engineering

ELSEVIER Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369

Finite deformation plasticity in principal axes: from a manifold to the euclidean setting

A d n a n I b r a h i m b e g o v i c * , Fadi G h a r z e d d i n e CompiOgne University of Technology, Department GSM, Division MNM, Lab. G2MS, UPRES A 6606 CNRS, BP-20.529,

60205 CompiOgne, France

Received 19 February 1998; revised 1 May 1998

Abstract

This contribution presents a link between early developments in finite deformation plasticity, where the notion of a covariant formulation is introduced and employed, and more recent developments, where rediscovery of a fundamental work of Hill on the method of principal axes led to a very efficient implementation scheme. More precisely, we demonstrate how to develop a covariant theory of finite deformation plasticity in an invariant form, by making use of the elastic principal stretches. We also show how to implement principal axis formulation in the framework of manifold, to carry out all the necessary manipulations by exploiting the Lie derivative formalism and eventually to simplify the final result to the Euclidean setting.

Much of our work on numerical implementation reflects the fruitful cross-fertilization of ideas with those from theoretical formulation. In particular, we show how the operator split method, which is typically used to s~mplify the plastic flow computation, can also be used to reduce the computational cost related to the special finite element interpolation schemes based on incompatible modes. The latter proves to be an indispensable ingredient lbr accommodating the near-incompressibility constraint arising in the finite deformation deviatoric plasticity. An important advantage of the proposed formulation as opposed to alternative remedies (e.g. B-bar method) is that the basic structure of the governing equations need not be modified. © 1999 Elsevier Science S.A. All rights reserved.

I. Introduction

The main objective of this work is to fill the gap between initial works on numerical implementation of finite deformation plasticity on a manifold (see e.g. [1-3] and more recent works using the principal axis representation, which is restricted to the Euclidean setting (see e.g, [4-8], among others).

Initial works on numerical implementation of finite deformation plasticity (see [1-3]) have indicated that the most appropriate setting for exploiting the full generality of multiplicative decomposition of the deformation gradient (see e.g. [9]) and developing a consistent numerical implementation procedure, is a differential manifold ~. The latter offers a clear geometric interpretation of the role of the operator split method in separating the computation of a new deformed configuration (i.e. computation of the total displacements and total deformations) from the computation of a new intermediate configuration (i.e. computation of the current values of the internal variables which control the plastic flow). A differential manifold also provides a general framework in which one can construct a consistent formulation of the finite deformation plasticity which is independent of a particular choice of the reference configuration, yet referred as covariant formulation (see e.g. [10]). Finally, it circumvents (see e.g. [11]) many conflicting issues in finite deformation plasticity (see e.g.

* Corresponding author.

A differential manifold embodies the idea of allowing general coordinate systems and sufficient continuity of the mappings, (see e.g. ll0]).

0045-7825/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 9 8 ) 0 0 2 1 5 - 1

342 A. lbrahimbegovi~, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369

[12]), including the most notable one pertinent to the question of the appropriate stress rates, by simply appealing to the notion of the Lie derivative.

In developing an invariant representation of the finite defi~rmation plasticity on a manifold for the case of interest of isotropic elastic response 2, the initial works (see e.g. [1-3]) have invariably resorted to using the invariants of the chosen strain measures. The potential developments in terms of the principal values of the chosen strain measures, or elastic principal stretches were not attempted, judging (see e.g. [10, p. 220]) that they might be too complicated.

Having shown in our previous work (see [11]) that one can indeed provide a very sound formulation of a covariant theory for finite deformation plasticity in the space of principal axes, we hope to show in this work that the same methodology leads to a very efficient numerical implementation. The latter has also been recognized in several recent works on numerical implementation of the finite deformation plasticity (see e.g. [4-8] among others), where it has been observed that a judicious choice of the logarithmic strain measure simplifies drastically the internal variable computation, reducing the latter to the procedure which is fully equivalent to the one used in the case of infinitesimal kinematics. All these recent works follow the fundamental work of Hill 114] and rely crucially on the Euclidean setting.

In this work we show that the same desirable simplification of the internal variable computation can be obtained when departing from a more general finite plasticity tbrmulation set on a manifold. When working with manifolds, one needs to include the metric tensor in the list of state variables, and thus use a general linear eigenvalue problem to define the principle axes (see [I 1]). Whenever possible (e.g. 3d, plane strain and axisymmetric problems), such an eigenvalue problem is simplified to the standard one, by choosing a particular coordinate representation of the metric tensor given as the unit-matrix 3 and recovering some of the recent results on finite defornlation plasticity obtained in the Euclidean setting (see e.g. [4-8]).

However, recovering some of the results recently obtained in the Euclidean setting in an alternative manner, departing from a more general setting of manifolds, is not our only goal. More importantly, we show that the proposed general framework of manifolds offers the only appropriate geometric interpretation which facilitates the development of the method of incompatible modes (see e.g. [16,40]) in finite strain regime. The latter is used as a very effective remedy to eliminate the locking problems, associated with the incompressibility constraint on the plastic deformation field. More precisely, the variational basis of the method of incompatible modes for finite deformation plasticity is developed in a manifold setting, its geometric interpretation is discussed and the simplifications ~tppropriate for the Euclidean setting are then carried out. The feature of a special interest for the proposed implementation of the incompatible mode method relates to exploiting the operator split procedure to simplify the implementation and significantly reduces the secondary storage requirements. The particular implementation of the method of incompatible modes has much in common with the operator split procedure proposed for finite plasticity implementation. This should not come as a surprise, since both problems are of similar natures; Namely, they are both the evolution problems in a constrained configuration space: the constraint to be imposed in plasticity pertains to the plastic admissibility of state variables (in the sense of yield criterion), whereas the constraint to be imposed in the incompatible mode method is pertinent to the satisfaction of the momentum balance equation associated with the variation of the incompatible mode parameters. Similarity of two problems extends further since in both cases imposing the constraint is reduced to a local computation, over a numerical integration point in plasticity, or over a single element in incompatible mode method.

The outline oil" the paper is as follows. In the next section, we briefly recall the fundamental ingredients of the finite deformation plasticity model in principal axes. For a more detailed discussion we refer to Ibrahimbegovic [11]. In Section 3, we discuss the numerical implementation details of the internal variable computations. The method of incompatible modes for finite deformation plasticity is described in Section 4. In Section 5, we present and discuss several illustrative numerical simulations. Some closing remarks are given in Section 6.

2 As noted by Drucker [13], the isotropy of the elastic response is quite a realistic assumption for most polycrystalline metals and alloys. 3 In our previous work (see [15]), we have presented the numerical implementation of finite deformation plasticity for the cases where no

simplification to the Euclidean setting can be obtained (e.g. space-curved membranes or shells).

A. lbrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369 343

2. Formulat ion of finite deformation plasticity

2.1. Phenomenological model in terms o f spatial strain measures

Following the fundamental works of Lee [9] and Mandel [18], in the case when both elastic and plastic deformations are of unrestricted size, one can assume that the multiplicative decomposition of the deformation gradient, F.

F = F ~ F p , (1)

gives rise to a stress-free intermediate configuration 4, with F e and F p referred to, respectively, as the elastic and the plastic deformation gradient. See Fig. 1.

Among