Elastic solutions of the end problem for a nonhomogeneous semi-infinite strip in anti-plane shear

The end problem referring to anti-plane shear deformation of a nonhomogeneous semi-infinite strip is investigated here, by using the analogous methodology proposed by Papkovich and Fadle in plane problem. Two types of nonhomogeneity are considered: (i) the shear modulus varies with the thickness coordinate x exponentially; (ii) it varies with the length coordinate y exponentially. The closed elastic solutions in trigonometric series form are derived by the eigenfunctions expansion, and the completeness of the solutions is also proved. Therefore, the elastic field caused by a self-equilibrating traction on the end could be solved in an arbitrary accuracy by taking a necessary number of terms in the series, approximatively, which is usually neglected by invoking Saint-Venant principle. By considering the biggest negative eigenvalue, the Saint-Venant Decay rates of the problem is also estimated in the last section.