Associated computational plasticity schemes for nonassociated frictional materials

Abstract

A new methodology for computational plasticity of nonassociated frictional materials is presented. The new approach is inspired by the micromechanical origins of friction and results in a set of governing equations similar to those of standard associated plasticity. As such, procedures previously developed for associated plasticity are applicable with minor modification. This is illustrated by adaptation of the standard implicit scheme. Moreover, the governing equations can be cast in terms of a variational principle, which after discretization is solved by means of a newly developed second-order cone programming algorithm. The effects of nonassociativity are discussed with reference to localization of deformations and illustrated by means of a comprehensive set of examples.K. KRABBENHOFT ET AL. difficult to obtain than in the case where the flow rule is associated [7][8][9][10]. These complications have a tendency to be more pronounced for high (but realistic) values of the friction angle and the degree of nonassociativity. Similarly, for fixed material parameters, one usually observes a degradation of the performance as the number of finite elements in the model is increased.These facts motivate a closer look at the physical origins of nonassociated flow rules and the numerical methods used to solve problems of frictional plasticity. In the following, inspired by the micromechanical origins of friction and its modeling in terms of plasticity theory, a new approach to computational plasticity for frictional (and generally nonassociated) materials is presented. The resulting scheme essentially approximates the original nonassociated problem as one of associated plasticity. Consequently, all the well-established numerical procedures for standard associated plasticity are applicable with little modification.The paper is organized as follows. The governing equations are briefly summarized in Section 2 before the new approach of approximating general nonassociated plasticity models in terms of equivalent associated ones is presented in Section 3. In Section 4, two different solution algorithms are presented. The first one is a slight modification of the common fully implicit scheme by Simo and his coworkers [11,12]. Secondly, following recent work of the authors [13-16], we formulate the governing equations in terms of a mathematical program. For certain yield criteria, notably Drucker-Prager, the resulting discrete programs may be solved very efficiently using a second-order cone programming solver, SONIC, recently developed by the authors. Next, in Section 5, the consequences of nonassociativity in terms of the ultimate load bearing capacity are discussed before the new numerical schemes are tested on some common boundary value problems in Section 6. These problems also highlight the consequences of nonassociated flow rules in terms of localization of deformations. Finally, conclusions are drawn in Section 7.Matrix notation is used throughout with bold uppercase and lowercase letters representing matrices and ve...

Slope stability analysis by means of finite element limit analysis and finite element strength reduction techniques. Part I: Numerical studies considering non-associated plasticity

“…When using structured finite element meshes with a non-associated flow rule, it has been observed that the failure surface tends to propagate along element boundaries [4]. This effect can also be seen in Figs.…”

“…Nordal [11]). This issue has also been studied extensively by Krabbenhoft et al [4,8]. The main problem with a non-associated flow rule is that the solution of the governing equations is not unique.…”

A projected Newton algorithm for the dual convex program of elastoplasticity

“…Krabbenhoft et al . also used this SOCP approach to approximate a non‐associated flow model by a convex one. In all of these primal approaches, a concept similar to the Sherman–Morrison–Woodbury formula is used to reduce the linear system to be solved .…”

“…Although we consider only quasi-static problems here, the concepts developed have implications in dynamic analyses [3,11,23,26,55,56]. They are also relevant in the finite deformation case and in instances where convexity does not apply, because these can often be approached using a succession of convex subproblems [20,[57][58][59][60][61].…”

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“…Incremental formulations using SOCP for continuum and discrete modelling approaches which take into account non-associative behaviour have also been recently developed (Krabbenhoft et al, 2012). Even so, considering the different field of application, also in these cases crushing is not taken into account.…”