G. de Saxcé

New inequality and functional for contact with friction : the implicit standard material approach

Abstract This paper is devoted to the analysis of the two- or three-dimensional elastic contact problem with Coulomb friction, quasi-static equilibrium, and small displacements. The classical approach is based on two minimum principles, or variational inequalities: the first for unilateral contact and the second for friction. In practical applications, this leads to an algorithm of alternately solving the two problems until convergence is achieved. A coupled approach using one principle or one inequality only is presented. This new approach, based on a model of material called implicit standard, allows for extension of the notion of a normality law to dissipative behavior with a nonassociated flow rule, such as surface friction. For numerical time integration of the laws, Moreaus implicit method is considered. Nondifferentiable potentials are regularized by means of the augmented Lagrangian technique. A discretized formulation using the finite element method and numerical applications are reported in a s...

A complete stress update algorithm for the non-associated Drucker-Prager model including treatment of the Apex

Abstract Numerical techniques based on convex analysis are applied to the non-associated Drucker–Prager model (without hardening) for which the plastic behavior is completely described by a unique function, called bi-potential. Among advantages of the present approach, motivated by mechanical considerations, a variational stress update algorithm along with coupled extremum principles can be derived. The time-integration algorithm is considered in detail and it is shown how the method can conveniently treat the singular point present in the Drucker–Prager model (apex). The existence of weak extremum principles allows using Mathematical Programming techniques and thereby obtains a robust algorithm even in the presence of large time increment and strong non-associativity. Numerical examples of incremental limit analysis for both the associated and the non-associated cases are presented.

A new approach to shakedown analysis for non-standard elastoplastic material by the bipotential

Abstract On the basis of the implicit standard materials that introduces a function, called bipotential, depending on both the stress and plastic strain rate, this paper is devoted to present a new approach of shakedown analysis for non standard elastoplastic materials. The bipotential theory was successfully applied to geomaterials with non-associated flow rule and Coulombs dry friction law. The present analysis is different to the Melans potential and it is based on a corner stone inequality satisfied by the bipotential and the existence of time-independent residual stress field. The deduction of bound theorems, static and kinematic, is detailed in the present article.

An improved discrete element method based on a variational formulation of the frictional contact law

An improved algorithm based on the contact dynamics approach is proposed. Like previous developed algorithms it involves two stages. In the first one (local stage) for each particle, forces are computed from the relative displacement using an interaction law, which models frictional contact and shock. In the second stage (global stage) Newtons second law is used to determine, for each particle, the resulting acceleration which is then time-integrated to find the new particle positions. This process is repeated for each time step until convergence is achieved. The two distinguishing features of the present algorithm are the local integration of the frictional contact law and the convergence criterion. By adopting a variational statement of the frictional contact law based on the bi-potential concept, the integration procedure is reduced to a single predictor-corrector step and a new convergence criterion is introduced. Both aspects significantly reduce the computing time and enhance the convergence. Numerical applications show the robustness of the algorithm.

Finite element analysis of the plastic limit load and the collapse mechanism of strip foundations with non-associated Drucker-Prager model

This paper is devoted to the numerical study of the influence of the non-associativity of the Drucker–Prager model on the plastic limit load and the failure mechanism of the strip footings. The attention is mainly focused on the determination (estimation) of the mechanical fields and geometric characteristics of the collapse mechanism. Rough and smooth contact between the punch and the substrate are considered. Analyses were performed by incremental finite element simulations by using of the object oriented computer code Cast3m. The code is first validated against analytical solutions for two problems available in the literature. It is worth noting that during this validation, we provide new numerical results concerning the ultimate load of a pressurised pore with non-associate matrix. Then, the limit plastic load of the weightless soil is computed and the post-treatment of the numerical results permits one to select the relevant mechanical fields. The main result is that, for the Prandtl mechanism associated to the rough footing, the angle base of the triangular wedge under the footing is independent of the dilatancy angle. This property appears to be very interesting and useful when constructing analytical bounds of the plastic limit load within the framework of limit analysis.

Static Shakedown Theorem for Solids with Temperature-Dependent Elastic Modulus

In this paper, an extension of the static shakedown theorem (Melan’s theorem) for elastic-plastic materials with temperature-dependent elastic modulus is presented. Cases of the decrease of yield function and the variation of coefficient of thermal expansion with temperature are included. The proposed extension leads to the introduction of a shakedown safety coefficient. To illustrate the statements of our theorem, step-by-step finite element procedure is applied to study a three-bar problem and a plate with the central hole subjected to thermo-mechanical cyclic loadings.

Crack Closure Studies of Bending Plate Using a Mongrel Singular Finite Element

The influence of crack closure on the stress intensity factors of cracked bending plates is investigated by the combination of the hybrid mongrel singular finite elements and the mixed method of contact problems. Embedding an exact singular stress field in the singular element by the concepts of hybrid mongrel displacement model and a singular strain field by the jacobian determinant of the isoparametric element, the stress intensity factors are directly obtained by the stress unknown parameters or indirectly by the nodal displacement values. For the contact problems, local flexibility matrix involving only the contact nodes is modified at each incremental step instead of the global stiffness matrix. The effects of crack closure on the stress intensity factors are analyzed in the two cases. 1) Line contact: using two dimensional model for a single edge cracked plate loaded by tensile stress and bending moment. 2) Area contact: using three dimensional model for a central through cracked plate subjected to bending. Some numerical tests proved that this approach is in very good agreement with known solutions even with very coarse mesh.