Computational aspects of a geometrically exact stress resultant model presented in Part I of this work are considered in detail. In particular, by exploiting the underlying geometric structure of the model, a configuration update procedure for the director (rotation) field is developed which is singularity free and exact regardless the magnitude of the director (rotation) increment. Our mixed finite element interpolation for the membrane, shear and bending fields presented in PartII of this work are extended to the finite deformation case. The exact linearization of the discrete form of the equilibrium equations is derived in closed form. The formulation is then illustrated by a comprehensive set of numerical experiments which include bifurcation and post-buckling response, we well as comparisons with closed form solutions and experimental results.

On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory

Numerical analysis and simulation of plasticity

A Geometrically-exact rod model incorporating shear and torsion-warping deformation

The asymptotic expansion method, with the thickness as the parameter, is applied to the equilibrium and constitutive equations of nonlinear three-dimensional elasticity. Then the leading term of the expansion can be identified with the solution of well-known two dimensional nonlinear plate models, such as the von Karman equations. Recent progresses in the application of this method, such as the extension to more general constitutive equations and boundary conditions, the effect of the assumption of polyconvexity, the application to one-dimensional rod models, etc…, are presented. Various open problems, regarding in particular the existence of corresponding three-dimensional solutions and the nature of admissible three-dimensional boundary conditions, are also discussed.

Recent Progress in the Two-Dimensional Approximation of Three - Dimensional Plate Models in Nonlinear Elasticity

The aim of this study is to propose a methodology for optimizing the shape (middle surface and thickness) of an elastic general thin shell (i) under different criteria: minimization of the weight, of the stresses, of the strain energy; (ii) with or without constraints: if any, these can be bounds on some displacements, on some stresses. This is a common problem in real life, the engineer has to construct structures which have the best mechanical behaviour and the best price; the price is often proportional to the weight of the structure. In this paper only the general continuous formulation of the problems is considered. From this strong basis a corresponding discrete formulation and some industrial applications will subsequently be developed.

Shape optimization of an elastic thin shell under various criteria

The effect of the application of an incremental method is the approximation of the three-dimensional nonlinear equations of finite elasticity by a sequence of linear problems. We give here sufficient conditions which guarantee the convergence of such a method.

Hu Jianwei

Papers

On the convergence of the semi-discrete incremental method in nonlinear, three-dimensional, elasticity