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Duality methods based on modified Lagrangian functional for solving a model crack problem is considered. Without additional assumptions of regularity of the solution of an initial problem duality ratio is established for initial and dual problem. Introduction The classical approach to the crack problem is characterized by the equality type boundary conditions considered at the crack faces, namely, they are assumed to be traction free. This approach cannot guarantee the natural nonpenetration condition between crack faces. Moreover, there are practical examples showing that interpenetration of crack faces may occur. Recently crack models with nonlinear boundary conditions on crack faces are investigated. Suitable boundary conditions are written as inequalities which provide mutual nonpenetration between crack faces. From the standpoint of mechanics such models are more preferable than the linear classical models. 1. Model crack problem The crack problem for the Poisson equation is considered with inequality type boundary conditions on the crack faces. Let Ω ⊂ R be a bounded domain with Lipschitz boundary Γ and γ ⊂ Ω be a cut (crack) inside of Ω. For simplicity we assume γ = {(x1, x2) ∈ Ω: a < x1 < b, x2 = 0} and suppose that both end points (a, 0) and (b, 0) do not belong to the boundary Γ. Denote Ωγ = Ω\γ. Introduce the feasible displacement set K = {v ∈ H(Ωγ) : [v] ≥ 0 on γ, v = 0 on Γ} Received March 22, 2016. 2010 Mathematics Subject Classification. 65F10, 65K10, 49M15, 74G15, 74G65.

MODIFIED DUALITY SCHEME FOR SOLVING MODEL CRACK PROBLEM IN MECHANICS