NONLIONEAR DEFORMATION THEORY USING FEM FOR ELASTOSTATIC ANALYSIS

Abstract

Stress-strain analysis forms an integral part of the design of mechanical structures. In the stress-strain graph linear elasticity is followed till the proportionality limit. After that there is an intermediate region till the point where the yield or plasticity begins, this region is referred to as the nonlinear elastic region. The nonlinear strains for this region onwards are considered to be large and can no more be neglected as is the practice in linear elastic region. For the linear region the constitutive equations of stress and strain are well established, but for nonlinear region the development of the constitutive is relatively recent and sparsely known. In the following text an attempt is made to explain the nonlinear deformation theory and its departure from the linear analysis. The focus will be on the geometric nonlinearity rather than the material nonlinearity which is a property of the material only. Only elastostatic analysis is considered here, though elastodynamics may be considered as an application of the basic theory. The theory of virtual displacement or simply virtual work is used as the basis of the development of the theory. Suitable examples and figures are provided to better understand the physical significance. The Finite Element Method (FEM) is employed because eventually the algorithm needs to be computationally implemented. The text is a simplified assortment of the various literatures available on the subject. At the end a simulation based on a computational code given by Messer’s Owen and Hinton is also presented based on the theory presented. The development of the theory is as follows. In the first chapter the whole theory is presented in a concise way to show the complete algorithm. In the second chapter some mathematical and other required formulations are given. In the third chapter the linear elastic analysis is presented in detail. In the fourth chapter finite element discretization for the linear case is developed giving the example of the one dimensional truss. In the fifth chapter the basis for the nonlinear analysis is given, again in detail. In the sixth chapter the finite element discretization for the non-linear case is given, again with an example for the one dimensional truss. In the seventh chapter the simulation of the numerical method employed using computational code is given.