The Contact Problems of the Mathematical Theory of Elasticity for Plates with an Elastic Inclusion

Abstract

The contacts problem of the theory of elasticity and bending theory of plates for finite or infinite plates with an elastic inclusion of variable rigidity are considered. The problems are reduced to integral differential equation or to the system of integral differential equations with variable coefficient of singular operator. If such coefficient varies with power law we can manage to investigate the obtained equations, to get exact or approximate solutions and to establish behavior of unknown contact stresses at the ends of elastic inclusion.

The adhesive contact problem for a piecewise-homogeneous orthotropic plate with an elastic patch

“…Stiffness of the patch and glue varies linearly, i.e., E(x) = hx, k 0 (x) = k 0 x, 0\x\1 (Figure 2). Equation (10) and the corresponding boundary conditions take the form…”

“…The solutions of static contact problems for different domains, reinforced with elastic thin inclusions and patches of variable stiffness and the behavior of the contact stresses at the ends of the contact line, have been investigated as a function of the law of variation of the geometrical and physical parameters of these thin-walled elements [1–13]. The first fundamental problem for a piecewise-homogeneous plane, when a crack of finite length arrives at the interface of two bodies at the right angle, was solved in Khrapkov [14]; a similar problem for a piecewise-homogeneous plane when acted upon by symmetrical normal stresses at the crack sides was solved in Bantsuri [15] and Ungiadze [16], as well as the contact problems for a piecewise-homogeneous plane with a semi-infinite and finite inclusion were solved in Bantsuri and Shavlakadze [17], Shavlakadze et al [18], and Shavlakadze et al [19].…”

Some effective solutions for Prandtl's type integro‐differential equation

“…Exact and approximate solutions of static contact problems for different domains, reinforced with 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 law of variation of the geometrical and physical parameters of these elements [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. As is known, stringers and inclusions, such as rigid punch and cuts, represent the stress concentrates.…”

Adhesive Interaction Of A Piecewise-Homogeneous Orthotropic Plate With An Elastic Beam

“…The solutions of static contact problems for different domains, reinforced with elasticthin inclusions andpatches of variable stiffness were obtained, and the behavior of the contact stresses at the ends of the contact line has been investigated, depending on the geometrical and physical parameters of these thin-walled elements [1][2][3][4][5][6][7][8][9][10]. The first fundamental problem for a piecewise-homogeneous plane was solved, when a crack of finite length arrives at the interface of two bodies at the right angle [11], and also a similar problem for a piecewise-homogeneous plane when acted upon by symmetrical normal stresses at the crack sides [12,13], as well as the contact problems for piecewisehomogeneous planes with a semi-infinite and finite inclusion [14].…”