Numerical analysis of pre- and post-critical response of elastic continua at finite strains

Abstract Based on the method of finite-elements, and the exact formulation of finite and incremental elasticity, the paper presents techniques for the numerical analysis of the pre- and post-critical behavior of elastic continua which are subjected to conservative loads and which undergo large deformations with finite strains. A combined incremental and iterative scheme is used in such a manner that convergence to the equilibrium state is attained at each state of loading, independently of the inaccuracy which may be involved in the characterization of the preceding state. The initial undeformed configuration of the elastic continuum is divided into a finite number of tetrahedra, and a linear variation of the displacement field within each tetrahedron is assumed. The first and second variations of the total potential energy at each loading stage are then developed in the form of a finite number of equations which characterize the response (the total response from the undeformed state, as well as the incremental one) and stability of the continuum. At each loading stage a pseudo-eigenvalue problem is formulated, by means of which an estimate can be made of the increment of load required to reach the critical state. The results are illustrated by means of two examples for compressible rubber-like materials for which a computer program has been developed. The first example demonstrates a snap-through instability, while the second example demonstrates a bifurcation-type instability. A formution for incompressible materials in plane strain is also presented.