General Solutions in Gradient Elasticity and Filtration Theory Based on Papkovich–Neuber Potentials

The structure of general solutions for wide class of elasticity problems and mechanics of solids is discussed. The algorithm of formulation of these solutions in the form of generalized Papkovich–Neuber representations through vector and scalar potentials satisfying in the general case of the inhomogeneous Helmholtz equations is demonstrated. It is shown that the generalized Pakovich–Neuber concepts are applicable to obtain general solutions to a wide class of problems in the mechanics of a deformable solid: solutions of the gradient two-parametric theory of elasticity of vector type and dilatation theory of media with a field of defects—porosity, solutions of plane problems of fracture mechanics and problems of Brinkman hydrodynamics. A method for approximating solutions presented using generalized Papkovich–Neuber representations is proposed, convenient for solving specific boundary value problems. It is based on the use of special expansions for solving the Helmholtz equation using a basis system of functions which are a combination of hyperbolic functions and polynomials. These systems form a complete system of functions, analytically exactly satisfy the Helmholtz equation and, in a particular case, transform into a system of harmonic polynomials. As a result, for the class of problems under consideration, a general algorithm for representing the solution in terms of vector potentials satisfying the Helmholtz equations and a general algorithm for constructing solutions to boundary value problems in the form of expansion in a new basis system of functions was proposed.