On the Antiplane Dynamical Problem of Elasticity Theory in a Domain with Crack

Abstract

We consider an antiplane dynamical problem in elasticity theory. Let infinite isotropic elastic medium have a cut {(x, y, z): r > 0, α < ∣φ∣ < π, z ∈ R }, where (r, φ) are the polar coordinates on the (x, y) — plane with origin at the corner point. The displacement W satisfies $$ \begin{gathered} {\partial^2}W/\partial {t^2} - \Delta W = 0,\quad (x,y) \in K,\quad t > 0, \hfill \\ \partial W/\partial n = 0,\quad (x,y) \in \partial K,\quad t > 0, \hfill \\ W(x,y,0) = f(x,y),\quad {\partial_t}W(x,y,0) = g(x,y) \hfill \\ \end{gathered} $$ where K = {(r, φ): r > 0, 0 < ∣φ∣ < α } is an angle of opening 2α. We consider plane waves travelling from infinity to the corner point (or to the tip of the crack, if α = π). Our main purpose is to study the displacement W after the collision of travelling wave with the tip of the crack. We obtain the asymptotics of W near the corner point. Then we calculate the coefficients of the asymptotics of the displacement W and study their properties after the collision. Besides we find out the range of use of the asymptotics. To this end we compare an explicit solution of some model problem with the asymptotics. Here we essentially use the results of [1], [2], [3]. These articles are devoted to the boundary value problem for the wave equation in domains with edges and conical points under Dirichlet or Neumann boundary condition. The articles contain the theory of solvability of this problem in scales of weighted spaces and asymptotic formulas of the solutions near edges and conical points.