Potential. Solution of Poisson's Equation, Equations of Continuity and Elasticity

TL;DR

This paper develops a novel method using generalized functions to solve Poisson's and continuity equations, applying it to elasticity problems involving cracks, revealing that infinite cracks always close while surface cracks can be critical.

Contribution

It introduces a new approach to solving Poisson's equation with generalized functions and analyzes crack behavior in elasticity, highlighting differences between infinite and surface cracks.

Findings

Infinite cracks always close, no critical condition.

Surface cracks can reach a critical condition.

Elastic energies of cracks are calculated.

Abstract

The modern theory of the potential does not give a solution of Poisson's equation. In the present work its solution has been found via generalized functions and a nonpotential solution of the continuity equation has been obtained. The method is demonstrated by the solution of elasticity equations using the example of a crack in the infinite specimen and a surface crack. Their elastic energies have been calculated. In has been shown that there is no critical condition for a crack in the infinite specimen and the crack always closes. Only the surface crack possesses the critical condition.