A Boundary Element Formulation of Problems in Linear Isotropic Elasticity with Body Forces
Traditional formulations of the Boundary Element Method (BEM) applied to elastostatics [1] [2] are extremely convenient when the loading on the body under analysis is limited to surface loading, since it is necessary to discretize only the boundary of the
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A BOUNDARY ELEMENT FORMULATION OF PROBLEMS.IN LINEAR ISOTROPIC ELASTICITY WITH BODY FORCES D.J. Danson Computational Mechanics Centre, Southampton, U.K.
I.
INTRODUCTION
Traditional formulations of the Boundary Element Method (BEM)applied to elastostatia{l] [2] are extremely convenient when the loading on the body under analysis is limited to surface loading, since it is necessary to discretize only the boundary of the body and not the whole domain as must be done when using a technique such as the Finite Element Method (FEM). When body forces are present these have usually been handled by evaluating a domain integral Unfortunately this requires the domain of the problem to be divided into integration cells since for any practical problem the domain integral must be evaluated numerically. This greatly increases the amount of data preparation required and causes the BEM to lose much of its advantage over domain t~e methods. However, Cruse r4J and Cruse, Snow and Wilson ~1 have shown that for certa1n types of commonly encountered body forces the domain integral may be transformed to a boundary integral or boundary integrals which may be evaluated at the same time as the boundary integrals involving the surface displacements and tractions. Ref. Is] is concerned exclusively with axisymmetric geometry; however, the authors' use of the Galerkin vector to achieve the required transformation from domain to boundary integrals provides the key to the present paper which relates to two and three dimensional geometry. The three dimensional case was derived in Ref. lj] without resort to the Galerkin vector. However, the Galerkin vector approach is used below both to demonstrate the power of the technique and to present the results of [4] in a slightly more general form. The two dimensional formulation is also presented.
r3J.
The following commonly encountered body forces will be considered. (i)
The body force due to a body being placed in a constant gravitational field.
C. A. Brebbia (ed.), Boundary Element Methods © Springer-Verlag Berlin Heidelberg 1981
106
(ii)
The body force due to constant rigi.d body rotation about a fixed axis.
(iii) Thermal stresses may be calculated by applying a body force proportional to the temperature gradient ["7]. The domain integral resulting from this body force may be transformed to a boundary integral provided that the temperature distribution is a "steady state" one. 2.
BOUNDARY INTEGRAL FORMULATION
The Somigliana identity for the displacements inside an elastic body may be derived using Maxwell-Betti's reciprocal theorem to give (Ref, [I])
~(x)
J uki(x,y)ti(y)dSY-
I
Tki(x,y)ui(y)dSY
s
s +
J uk.(x,y)b.(y)dV :L :L y
(I)
v Where
is the displacement at an internal point x in the k direction uki(x,y) is the displacement in the i direction at y due to a unit point load in the k direction at x Tki(x,y) is the traction in the i direction at y due to a unit point load in the k direction at x t.(y) is the traction at y u:(y) is the displacement at y b~(y) is the intensity of th
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