FINITE DEFORMATION OF AN ELASTO-PLASTIC SOIL

SUMMARY Presented in this paper is a formulation and a numerical solution method for problems which involve finite deformations of an elasto-plastic material. The governing equations are cast in rate form and the constitutive laws are formulated in a frame indifferent manner. Particular reference is made to the finite deformation of soil. Plastic failure is described by a general yield condition and plastic deformation by an arbitrary flow rule. Several examples are treated numerically. INTRODUCI'ION In the formulation of theories in applied mechanics and in particular soil mechanics, it has been a common practice to assume that strains, both elastic and plastic are infinitesimal and, that the initial geometry of a deforming body is not appreciably altered during the deformation process. These assumptions are less justified for soil than for such materials as steel and concrete. Theories of finite strain that relax some of these restrictive assumptions have been developed and there exists a considerable body of literature on what might be called the classical elastic large strain theory'-5 (e.g. the large deformation of materials like rubber has been In contrast to the methods of these early investigators, many more recent studies have preferred an incremental appr~ach"~ to facilitate the analysis of the more general class of inelastic materials whose constitutive laws are expressible in terms of incremental or rate quantities. For such formulations the solution of a given problem is found by following a specified loading path. In most cases the governing equations cannot be solved analytically and it is necessary to adopt an approximate numerical technique." Much recent work has been devoted to formulating analyses for plate and shell problems involving large displacement but small strains (a survey is given by Marcal"). As has been noted'* these formulations are inappropriate for applications to bulky geometries such as occur in many problems in soil mechanics. Several attempts have been made to formulate a finite deformation theory suitable for use with soils. An example is that of Thoms and Arman13 who directly applied a technique of ArgyrisI4 to the problem of an embankment constructed on soft clay. The analysis was restricted to an elastic material and theoretical results were compared with results from photo-elastic model tests. Davidson and Chen" have given some solutions for the problem of footings on clay while Fernandez and Christian16 examined flexible footing and retaining wall problems. In this paper a formulation is given for the solution of problems of finite elasto-plastic flow without restricting deformation magnitude. Plastic failure is described by a general yield condition and plastic deformation by an arbitrary flow rule. The theory is developed for a general constitutive law which relates an objective stress rate to the strain rate. This theory has applications to such problems as: the penetration of embankments into very soft soil; the behaviour of layers of normally consolidated clay in which both elastic modulus and undrained