Bipotential versus return mapping algorithms: Implementation of non associated flow rules

Numerical implementation of constitutive laws involves specific incremental methods. The “return mapping” (Simo and Hughes, 1998) and the “bipotential” (de Saxcé, 1992) are one of those, associated respectively to two different classes of materials: the General Standard Materials (GSM) for the return mapping and the Implicit Standard Materials (ISM) for the bipotential.The objective of this paper is then to compare the implementation of those both methods in the case of non associated flow rules in plasticity.In the first section, the properties of the different previous material classes will be recalled and the methods of “return mapping” and “bipotential” will be detailed. The comparison of both methods is realised on the non linear kinematic hardening rule of Armstrong–Frederick (Armstrong and Frederick, 1966) in a second section and the details are given in a third part. The numerical implementation is realised in Abaqus/Standard 6.11 by the means of a UMat subroutine and the practical simple case of tension–compression is analysed in a last section.     

A bipotential-based limit analysis and homogenization of ductile porous materials with non-associated Drucker–Prager matrix

In Gurson's footsteps, different authors have proposed macroscopic plastic models for porous solid with pressure-sensitive dilatant matrix obeying the normality law (associated materials). The main objective of the present paper is to extend this class of models to porous materials in the context of non-associated plasticity. This is the case of Drucker–Prager matrix for which the dilatancy angle is different from the friction one, and classical limit analysis theory cannot be applied. For such materials, the second last author has proposed a relevant modeling approach based on the concept of bipotential, a function of both dual variables, the plastic strain rate and stress tensors. On this ground, after recalling the basic elements of the Drucker–Prager model, we present the corresponding variational principles and the extended limit analysis theorems. Then, we formulate a new variational approach for the homogenization of porous materials with a non-associated matrix. This is implemented by considering the hollow sphere model with a non-associated Drucker–Prager matrix. The proposed procedure delivers a closed-form expression of the macroscopic bifunctional from which the criterion and a non-associated flow rule are readily obtained for the porous material. It is shown that these general results recover several available models as particular cases. Finally, the established results are assessed and validated by comparing their predictions to those obtained from finite element computations carried out on a cell representing the considered class of materials.     

A bipotential approach for plastic limit loads of strip footings with non-associated materials

This paper is concerned with a bipotential approach for estimating the plastic collapse loads of a half-space made with a non-associated Mohr–Coulomb material and indented by a rigid punch. In geotechnics, this problem is called the bearing capacity of shallow strip footing for which the analytical solution is derived by Prandtl (1920) [46] and Hill (1950) [35] in the context of associated plasticity. However, when the plastic model is not associated, no analytical methods have yet been developed. Here we explore this issue in a rigorous mathematical framework coupling the bipotential concept and limit analysis. First, the method proposed makes use of the method of characteristics to build a statically and plastically admissible stress field that enables a lower estimate of the plastic limit loads. Next, the extended kinematic theorem of limit analysis to non-standard plasticity is applied to derive an upper quasi-bound of the collapse loads. For this aim, the internal rate of plastic dissipation is obtained thanks to the bipotential functional depending on both a trial stress field and a Prandtl-like collapse mechanism. The analytic estimates are compared to the formulae and numerical results provided in literature.     

Weak solutions for contact models involving a class of generalized materials

We consider a contact model for a class of generalized materials whose behaviour is described by means of a bipotential. Using the theory of the bipotentials we deliver a weak formulation consisting of a system of three variational inequalities. The unknown is a triple having as components the displacement field, the Cauchy stress tensor and a Lagrange multiplier related to the friction force on the frictional contact zone. We investigate the existence of the weak solutions by using a fixed point theorem for set-valued mappings and a minimization technique.     

Weak solutions via bipotentials in mechanics of deformable solids

We consider a displacement-traction boundary values problem for elastic materials, under the small deformations hypothesis, for static processes. The behavior of the material is modeled by a constitutive law involving the subdifferential of a proper, convex, and lower semicontinuous map. The constitutive map and its Fenchel conjugate allow us to construct a bipotential function. Based on this construction, we propose a weak formulation of our mechanical problem. Furthermore, we prove the existence of at least one weak solution and we investigate the uniqueness of the weak solution. We also comment on the relevance of our variational approach, by considering three significant examples.