ASYMPTOTIC ANALYSIS OF AN END-LOADED, TRANSVERSELY ISOTROPIC, ELASTIC, SEMI-INFINITE STRIP WEAK IN SHEAR

We use linear elasticity to study a transversely isotropic (or specially orthotropic), semi-infinite slab in plane strain, free of traction its faces and at infinity and subject to edge loads or displacements that produce stresses and displacements that decay in the axial direction. The governing equations (which are identical to those for a strip in plane stress, free of traction on its long sides and at infinity, and subject to tractions or displacements on its short side) are reduced, in the standard way, to a fourth-order partial differential equation with boundary conditions for a dimensionless Airy stress function, f. We study the asymptotic solutions to this equation for four sets of end conditions-traction, mixed (two), displacement-as epsilon, the ratio of the shear modulus to the geometric mean of the axial and transverse extensional moduli, approaches zero. In all cases, the solutions for f consist of a "wide" boundary layer that decays slowly in the axial direction (over a distance that is long compared to the width of the strip) plus a "narrow" boundary layer that decays rapidly in the axial direction (over a distance that is short compared to the width of the strip). Moreover, we find that the narrow boundary layer has a "sinuous" part that varies rapidly in the transverse direction, but which, to lowest order, does not enter the boundary conditions nor affect the transverse normal stress or the displacements. Because the exact biorthogonality condition for the eigenfunctions associated with f can be replaced by simpler orthogonality conditions in the limit as epsilon --> 0, we are able to obtain, to lowest order, explicit formulae for the coefficients in the eigenfunction expansions of f for the four different end conditions.