On the elastic wedge problem within simplified and incomplete strain gradient elasticity theories

In this paper, we investigate the behavior of the strain gradient elasticity solutions around the sharp edges of the body loaded by the concentrated loads. We consider the general Mindlin-Toupin strain gradient elasticity (SGET) and several simplified models, including the dilatation gradient elasticity theory (DGET) and the couple stress theory (CST). In the framework of a plane strain problem for an elastic wedge with prescribed tip displacement we obtain analytical assessments for the displacement field and the stress state around the wedge apex. Involving the Papkovich-Neuber solution we show that the nonsingular solutions for this problem can be obtained within the general SGET and within simplified gradient models that provide the regularization of the dilatational and the rotational parts of the displacement field both. In the incomplete theories such as DGET or CST the singularity in the displacements and stresses at the wedge apex cannot be avoided. This result is validated also based on the full-field numerical solutions for the wedge-type domains. It is shown that the stress field around the wedge apex is bounded in the complete gradient theories, while the mesh dependent solutions are realized in DGET and CST. Generally, obtained results means that the boundary value problems with the edge-type loading should be abandoned within the incomplete gradient theories like DGET and CST or it can be considered only under some additional assumptions (similarly to classical elasticity).