Crack Singularities for General Elliptic Systems

We consider general homogeneous Agmon‐Douglis‐Nirenberg elliptic systems with constant coefficients complemented by the same set of boundary conditions on both sides of a crack in a two‐dimensional domain. We prove that the singular functions expressed in polar coordinates (r, θ) near the crack tip all have the form r^{k + 1/2}φ(θ) with k ≥ 0 integer, with the possible exception of a finite number of singularities of the form r^k log r φ(θ). We also prove results about singularities in the case when the boundary conditions on the two sides of the crack are not the same, and in particular in mixed Dirichlet‐Neumann boundary value problems for strongly coercive systems: in the latter case, we prove that the exponents of singularity have the form with real η and integer k. This is valid for general anisotropic elasticity too.