Numerical method and models for anti-plane strain of a system with a thin elastic wedge

Abstract

The paper presents a general method to find asymptotics for a (multi-)wedge system containing a thin wedge. It employs separation of the symmetric and anti-symmetric parts of the boundary displacements and tractions of the wedge. The method is applicable when the angle of the thin wedge turns to zero. A physical interpretation of the derived equations is obtained by using power expansions of non-polynomial functions, which appear after the Mellin transform. We establish that the first term in the expansion of the symmetric part corresponds to shear, while the first term of the anti-symmetric part describes deflection of the wedge axis. Numerical experiments, performed by using a code developed on the basis of the theory, show that using only the first terms of the expansions insignificantly influence accuracy: the approximate results coincide with the exact values of roots to the third significant digit even for the wedge angle of 30 • .

Interface conditions simulating influence of a thin elastic wedge with smooth contacts

“…3b) of m linearly elastic isotropic wedges with arbitrary angles, Poisson's ratios and shear modules. This figure, as well as the next one, differs from the analogous figure of the paper on anti-plane strain [13] by adding the Poisson's ratio in the list of wedge properties.…”

“…Then, the problem becomes partly similar to that for the Laplace's equation what suggests using and possibly improving the approach of [13]. The investigation below contains the study of this problem.…”

“…It has been done by distinguishing the physically significant symmetric and anti-symmetric parts of the solution, an appropriate re-arrangement of terms and expanding trigonometric functions into truncated power series. Meanwhile, our attempt to immediately extend the method of [13], which looked quite general, to plane strain (or plane stress) problems failed. It has appeared that there is need in a deeper analysis of the equations and the characteristic determinant obtained.…”

“…Although the study [15] concerned with a particular case of anti-plane strain problem for a semi-infinite crack with a thin wedge ahead of its tip, the starting concept looks promising for micromechanical analysis of interface conditions. This impelled us to extend the results of [15] to an arbitrary anti-plane problem [13] for a (multi-) wedge system. It has been done by distinguishing the physically significant symmetric and anti-symmetric parts of the solution, an appropriate re-arrangement of terms and expanding trigonometric functions into truncated power series.…”

“…The elastic constants of the wedge are μ 0 and ν 0 . We use the approach developed in [13] for anti-plane strain to avoid degeneration when 0 → 0. Consider separately the thin wedge (Fig.…”