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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP

Exact solutions of some singular integro-differential equations related to adhesive contact problems of elasticity theory Nugzar Shavlakadze, Nana Odishelidze and Francisco Criado-Aldeanueva Abstract. The problem of constructing an exact solution of singular integro-differential equations related to problems of adhesive interaction between elastic thin semi-infinite homogeneous patch and elastic plate is investigated. For the patch loaded with horizontal forces the usual model of the uniaxial stress state is valid. Using the methods of the theory of analytic functions and integral transformation, the singular integro-differential equation is reduced to the Riemann boundary value problem of the theory of analytic functions. The exact solution of this problem and asymptotic estimates of tangential contact stresses are obtained. Mathematics Subject Classification. 74B05, 74K20, 74K15. Keywords. Adhesive contact problem, Elastic patch, Integro-differential equation, Integral transformation, Riemann problem.

1. Introduction Exact or approximate solutions of static contact problems for different domains, reinforced by elastic thin inclusions and patches of variable stiffness were obtained, and the behavior of the contact stresses at the ends of the contact line has been investigated as a function of the geometrical and physical parameters of these elements [1–7,14–24]. One model assumes continuous interaction, while the other assumes the adhesive contact of thin-shared elements (stringers or inclusions) with massive deformable bodies. In [11], a finite-length stringer is attached to a thin elastic sheet subjected to plane stress. The two different materials are joined along the entire stringer length by a thin uniform elastic adhesive layer assumed to be in pure shear state. The bending is neglected and the interaction between the sheet and the stringer is idealized as a line loading of the sheet. In [9], an elastic semi-infinite plate is strengthened by an elastic finite stringer. The contact between the plate and the stringer is achieved by a thin glue layer. Asymptotic estimates, exact and approximate solutions of the associated integro-differential equation are obtained. In the present paper, the exact solution of singular integro-differential equations related to the problems of adhesive interaction between an elastic thin semi-infinite homogeneous patch and an elastic plate is obtained. From the physical point of view, consideration of the case of the adhesive interaction between the plate and the stringer is interesting since the boundedness of the unknown tangential contact stress near the ends of the stringer is proved by rigorous mathematical methods. The limit transition (case A) corresponds to the case of rigid contact, in which the obtained solution, i.e. tangential contact stress, has a singularity at the end of the stringer.

2. Formulation of the problem and reduction to the integro-differential equation Let a semi-infinite patch with modulus of elasticity E1 (x),