Nonlinear elasticity with limiting small strain for cracks subject to non-penetration

Abstract

A major drawback of the study of cracks within the context of the linearized theory of elasticity is the inconsistency that one obtains with regard to the strain at a crack tip, namely it becoming infinite. In this paper we consider the problem within the context of an elastic body that exhibits limiting small strain wherein we are not faced with such an inconsistency. We introduce the concept of a non-smooth viscosity solution which is described by generalized variational inequalities and coincides with the weak solution in the smooth case. The well-posedness is proved by the construction of an approximation problem using elliptic regularization and penalization techniques.

Contacting crack faces within the context of bodies exhibiting limiting strains

“…As δ tends to zero, we can derive the existence theorem below. The proof of Theorem 3 follows the proof given in [17] based on the properties (10)- (12) for the specific case F(σ) = Ψ 2 (∥σ∥)σ.…”

“…Proof For F defined by (15), the upper bound (10) follows from (17) and (20), and the lower bound (12) from (19) and (22), due to…”

“…The brackets ⟨ · : · ⟩ Ωc in (36) and (37) imply a duality pairing between M 1 (Ω c ; Sym(R d×d )) and C c (Ω c ; Sym(R d×d )). By constructing an elliptic regularization of (30) and (31) and penalizing K as follows [17], for fixed δ > 0, there exist a displacement…”

“…In a previous study [17], the nonlinear elasticity problem with limiting small strain was solved for cracks with contacting faces using penalization and elliptic regularization techniques. In the present paper, we extend this approach further in terms of a generalization of nonlinear functions, which are admissible for describing constitutive relations that exhibit limiting small strain behavior.…”

Optimal control of the radius of a rigid circular inclusion in inhomogeneous two-dimensional bodies with cracks

“…It is well known that imposing of linear boundary conditions on the crack may lead to physical inconsistency of mathematical models since mutual penetration of the crack faces may happen [18,24]. In recent years, a crack theory with non-penetration conditions has been under active study [25,26,27,28,29,30]. This approach to solving crack problems is characterized by inequality type boundary conditions at the crack faces, is indeed what we employ in the present paper.…”