On the Radial Multipliers Method in the Gradient Elastic Fracture Mechanics

A non-singular solution of the gradient elastic fracture mechanics for the cracks of Modes I and II is constructed in this paper. Previously, a similar problem was solved only for cracks of Mode III. A generalized theory of elasticity is used, in which the governing equation in displacements is represented as a product of the Lam\'{e} operator and the Helmholtz operator, and the classical boundary value problem for the total stress is completely distinguished in the problems of crack mechanics. As a result, the determination of local stress fields reduces to solving the Helmholtz equations with the known right-hand side of the equations. The Papkovich-Neuber representation in a complex form is used to construct a solution of the mechanics of cracks. We used also the method of radial multipliers, which allows us to construct fundamental solutions of the Helmholtz equations corresponding to analytical functions with a fractional exponent and, as a result, to find solutions that compensate for the singularities