Recent developments in the field-boundary element method for finite/small strain elastoplasticity

Abstract

In this paper, some recent works of the authors, in the area of the field-boundary element method for finite/small strain elastoplasticity [Okada, Rajiyah and Atluri (1988a) J. Appl. Mech.55, 786–794; (1988b) Comput. Struct.30, 275–288; (1989) Comput. Mech.4, 165–175; (1990) Int. J. Numer. Meth. Engng29, 15–35], are summarized. Certain new integral representations for displacement (velocity) gradients, which are derived recently by the authors using the weighted residual method, are presented. Early formulations for boundary element methods for linear elasticity and small strain elastoplasticity [see Banerjee and Butterfield (1981) Boundary Element Methods in Engineering Science. McGraw-Hill; Banerjee and Cathie (1980) Int. J. Mech. Sci.22, 233–245; Banerjee and Raveendra (1986) Int. J. Numer. Meth. Engng23, 985–1002; (1987) J. Engng Mech.113, 252–265; Brebbia (1978) The Boundary Element Method in Engineering. Pentech Press; Chandra and Mukherjee (1983) J. Strain Anal.18, 261–270; Cruse (1969) Int. J. Solids Structures5, 1259–1274; Mukherjee and Chandra (1987) Boundary Element Methods in Mechanics, Elsevier Science Publishers; Mukherjee and Kumar (1978) ASME J. Appl. Mech.45, 785–790; Rizzo (1967) Q. Appl. Math.25, 83–95; Swedlow and Cruse (1971) Int. J. Solids Structures7, 1673–1683] are also shown along with these new integral equation representations. These new integral representations have lower order singularities, as compared to those that are obtained by a direct differentiation of the integral representations for displacement (velocity). They are quite attractive from a numerical analysis point of view, and enable the evaluation of the gradients at the boundary of the body, without any difficulties of hyper-singularities as in the conventional approach. In this new approach, the integrals, which have the highest order of the singularity, are evaluated in the sense of Cauchy principal value. Stresses and strains can be obtained directly at the boundary, using these new integral representations, whereas alternate methods or special regularization techniques are required when the conventional type integral equation for the gradients is used. These new integral equations can unify the methods of obtaining stresses as well as strains at the interior and at the boundary of the body. As shown in this article, this is very advantageous in applications to small and finite strain elastoplasticity. The third topic in this paper is a new field-boundary element method for the analysis of a class of problems of finite strain elastoplasticity, that involve bifurcation phenomena in the solution path, such as the buckling of a beam-column, diffused necking of a tensile bar, etc. The field-boundary element method is especially advantageous in the application to the large strain elastoplasticity, since the formulation can cope with the incompressibility of the material in the regime of fully developed plastic flow. This is not true in the case of the finite element displacement method. A full tangent stiffness method has been proposed by the authors to solve such classes of problems. This formulation accounts for all the non-linearities in the problem, and allows for the calculation of the displacement field directly. The problem of diffused necking of a tensile plate is solved for illustrative purposes. It is clearly seen that the full tangent stiffness field-boundary element method is capable not only of capturing the diffused necking bifurcation but also of analysing the post bifurcation diffused necking solution.