Nestor Zouain

Three-dimensional model for solids undergoing stress-induced phase transformations

Abstract A phenomenological model describing the mechanical behavior of polycrystalline solids undergoing stress-induced phase transformations is presented in the setting of three-dimensional (3-D) media. The model, which is developed within the framework of Generalized Standard Materials, has been built on a few basic assumptions, namely an admissibility condition for thermodynamic forces and a locking constraint for phase transformations. The macroscopic kinematic consequences of stress-induced phase transformations are described by a tensor named transformation strain. Numerical integrations of the resulting set of equations show that many features of shape memory alloys obtained by certain authors for CuAlZnMn alloys under non-proportional loadings can be qualitatively described by the present model.

An iterative algorithm for limit analysis with nonlinear yield functions

Abstract A mathematical programming algorithm is proposed for the general limit analysis problem. Plastic behavior is described by a set of linear or nonlinear yield functions. Abstract formulations of limit analysis are first considered and a Quasi-Newton strategy for solving the optimality conditions is then sketched. The structure of the problem arising from a finite element discretization is taken into account in order that the algorithm should be able to solve large scale problems.

An algorithm for shakedown analysis with nonlinear yield functions

Abstract This paper is devoted to propose an algorithm to solve the discrete form of the shakedown analysis problem with a nonlinear yield function. Firstly, variational formulations for shakedown analysis of structures under variable loads are considered. Secondly, the corresponding discrete forms are briefly recalled. Then, the algorithm is derived by combining a Newton formula based on the discrete equality conditions with a return mapping procedure focusing plastic admissibility. The proposed numerical procedure can be applied to finite element models with large number of degrees of freedom because the algorithm is specially designed to take advantage of the structure of such models. The numerical procedure is applied to a square plate with a circular hole under variable traction in two directions, and the obtained approximations are compared with available results. Finally, a rectangular plate with two small holes is considered in both plane strain and plane stress conditions. All application use the Mises condition without linearization.

Extremum principles for bounds to shakedown loads

Bounds to the safety factor for elastic shakedown of structures under variable loads are considered. These bounds represent safety factors in the separate analysis of alternating plasticity and incremental collapse. Extremum principles for these bounds are formulated and compared. The role of simple and combined mechanisms of incremental collapse is emphasized in the theoretical developments and also in two analytical applications.

Bounds to shakedown loads

The shakedown analysis of structures under variable loads is considered, and the corresponding variational principles are presented. Separate formulations for elastic (or classical) shakedown, plastic shakedown and incremental collapse are compared. Two upper bounds for classical safety factors are presented and interpreted as the result of separate analysis concerning alternating plasticity and prevention of simple mechanisms of incremental collapse. As a first application, exact solutions are obtained for a closed tube under variable pressure and temperature. This example is the Bree problem with logarithmic temperature variation across the wall. A finite element procedure for shakedown analysis of tubes is then presented. This numerical procedure is validated by comparison with the exact solutions for the closed tube. Finally, the numerical method is applied to a restrained tube under variable pressure and logarithmic temperature.

A high-cycle fatigue criterion with internal variables

A multiaxial fatigue criterion is derived from a model to high-cycle fatigue. The model is motivated by the description of fatigue endurance as relying upon elastic shakedown in the material representative volume. By analogy, a simplified formal framework for identifying stress ranges which are feasible with respect to fatigue endurance is proposed at the usual macroscopic level of stress analysis, in the setting of continuum mechanics assumptions. There, the modifications of the characteristic volume, which are responsible for the elastic shakedown of the material, are represented by internal tensorial variables. Thus, the resulting fatigue criterion also indicates the directions of induced anisotropy. The model shows good adequacy to experimental data when applied to nine different materials.

Variational Principles for Shakedown Analysis

The safety factor for elastic shakedown of structures under variable loads is considered. Variational principles for shakedown analysis are reviewed in a unified presentation, suitable to finite element discretizations, and considering nonlinear yield functions. Extremum principles for bounds to shakedown loads are also presented. A tube under thermo-mechanical loading is used to show analytical solutions, and numerical results obtained with a mixed finite element formulation.