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Open AccessArticle

A New Approach to Non-Singular Plane Cracks Theory in Gradient Elasticity

1
Institute of Applied Mechanics of RAS, Leningradkiy Prospect 7, Moscow 125040, Russia
2
FRC CSC RAS, Vavilova 40, Moscow 119333, Russia
3
Institute for Problems in Mechanics of RAS, Vernadskogo Prospect 101, Moscow 119526, Russia
*
Author to whom correspondence should be addressed.
Math. Comput. Appl. 2019, 24(4), 93; https://doi.org/10.3390/mca24040093
Received: 30 August 2019 / Revised: 5 October 2019 / Accepted: 24 October 2019 / Published: 26 October 2019
(This article belongs to the Special Issue Related Problems of Continuum Mechanics)
A non-local solution is obtained here in the theory of cracks, which depends on the scale parameter in the non-local theory of elasticity. The gradient solution is constructed as a regular solution of the inhomogeneous Helmholtz equation, where the function on the right side of the Helmholtz equation is a singular classical solution. An assertion is proved that allows us to propose a new solution for displacements and stresses at the crack tip through the vector harmonic potential, which determines by the Papkovich-Neuber representation. One of the goals of this work is a definition of a new representation of the solution of the plane problem of the theory of elasticity through the complex-valued harmonic potentials included in the Papkovich-Neuber relations represented in a symmetric form, which is convenient for applications. It is shown here that this new representation of the solution for the mechanics of cracks can be written through one harmonic complex-valued potential. The explicit potential value is found by comparing the new solution with the classical representation of the singular solution at the crack tip constructed using the complex potentials of Kolosov-Muskhelishvili. A generalized solution of the singular problem of fracture mechanics is reduced to a non-singular stress concentration problem, which allows one to implement a new concept of non-singular fracture mechanics, where the scale parameter along with ultimate stresses determines the fracture criterion and is determined by experiments. View Full-Text
Keywords: singular problems; gradient fracture mechanics; Papkovich-Neuber representation; complex potentials; new form of solutions; regular gradient solutions singular problems; gradient fracture mechanics; Papkovich-Neuber representation; complex potentials; new form of solutions; regular gradient solutions
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Lurie, S.A.; Volkov-Bogorodsky, D.B.; Vasiliev, V.V. A New Approach to Non-Singular Plane Cracks Theory in Gradient Elasticity. Math. Comput. Appl. 2019, 24, 93.

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