On the Numerical Solution of Nonlinear Problems of Continua

Abstract

<p>A state of the art survey of formulation aspects of geometric and material nonlinearity problem is given. The survey covers the formulation methods, the solution of nonlinear equilibrium equations, the incompressibility constraint, and, finally, the software aspects. A consistent Lagrangian, updated Lagrangian, and Eulerian formulation are derived from the energy balance equation transformed to the proper reference configuration. Difficulties opposed the pre-existence of a consistent Eulerian formulation in the literature are critically discussed. The proposed Eulerian formulation includes specific approximations which simplify the numerical treatment, and may restrict the applicability of the formulation to specific classes of problems, but do not alter the nature of the formulation. Differences between the presented Lagrangian and updated Lagrangian formulations and similar ones in the literature are found to be in specific geometric nonlinear terms in the final incremental equation as well as in the definition of the load increment vector. These differences are assessed within the framework of the basic equations of continuum mechanics. Specific forms of constitutive equations for elastic and elastoplastic materials are presented. For elastoplastic applications, it is shown that the use of proper frame indifferent stress rate leads to a constitutive equation which is a function of the incremental displacements and not the corresponding incremental strains.</p> <p>A concise discussion of software aspects is provided which leads to a suggestion of switching from the program package to the programming system concept. A programming system to account for material nonlinear behaviour is developed and tested.</p> <p>In the application, a plasticity theory for porous metals is proposed. A simple model of porous material is analysed by the finite element method. An assessment is made of an existing yield criterion for porous metals. A modified yield criterion and plastic potential function, and consequently, different plasticity equations are given. Reasonable agreement is obtained between the present numerical results and previous experimental and analytical results in the literature.</p>