MIXED BOUNDARY-VALUE-PROBLEMS IN NONHOMOGENEOUS ELASTIC-MATERIALS

We consider here some mixed boundary-value problems for inhomogeneous elastic materials where the shear modulus varies with respect to one space variable. The particular modulus variation considered is one where the Poisson ratio is constant and the shear modulus mu(z) varies as (a + b\z\)n, where n is not necessarily an integer. This modulus variation has an advantage over more commonly used power-law (\z\n) variations as the material retains the usual square-root stress singularity at a crack tip and the modulus need not tend to zero at a finite value of z. Some specific problems addressed are semi-infinite mode-3 cracks or notches for special values of n which allow the governing equations to be written in terms of harmonic functions. The crack problems can also be solved for more general values of n and these cases are also considered. In a variety of cases solutions are checked using the application of an invariant integral. The reciprocal theorem is used together with eigensolutions of the functional equations to generalize the crack-tip results we obtain to any loading. The leading-order behaviours of the stresses are evaluated and compared with equivalent homogeneous cases. In particular, for those problems where the displacements are represented in terms of harmonic functions we attempt to compare and contrast the Mellin- and Fourier-transform methods, which both work well for the analogous homogeneous problems. In particular, the Mellin-transform method leads to an inhomogeneous first-order difference equation in the transform parameter. This difference equation is of a quite general character and is solved using a reduction to a Hilbert problem. The difference-equation approach has not been widely used so techniques of solving difference equations are illustrated in some detail. The plane-strain crack problems are also approached for particular cases when the field variables can be written in terms of harmonic functions.