Asymptotic solution near the apex of an elastic wedge with curved boundaries

When the boundaries of an elastic wedge are straight lines, the asymptotic solution near the apex $r = 0$ of the wedge is simply a series of eigenfunctions of the form ${r^\lambda }f(\theta ,\lambda )$ in which $(r,\theta )$ is the polar coordinate with origin at the wedge apex and $\lambda$ is the eigenvalue. When the wedge boundaries are curved, the eigenvalues remain the same but the curvatures of the boundaries change the form of the eigenfunctions. The eigenfunction associated with a $\lambda$ contains not only the term ${r^\lambda }$, but also ${r^{\lambda + 1}},{r^{\lambda + 2}},...$ In some cases it also contains the term ${r^{\lambda + 1}}(ln r)$. Therefore, the second and higher order terms of asymptotic solution are not simply the second and next eigenfunctions. Examples are given for the first few terms of asymptotic solution for wedges with wedge angle $\pi$ and $2\pi$. The latter corresponds to a crack with curved free boundaries and we show that there exists a term ${r^{1/2}}(In r)$ besides the familiar terms ${r^{ - 1/2}}$.