Existence and integral representation of solutions for plane deformations of a micropolar elastic solid with surface elasticity

Abstract

We consider a linear theory of elastic boundary reinforcement of a micropolar elastic solid subjected to plane-strain deformations. The reinforcement consists of a thin micropolar elastic coating bonded to part of the boundary of the solid. The elastic properties of the coating incorporate both classical and micropolar bending, extension and twisting effects. Interior and exterior mixed boundary problems are formulated and analyzed using the boundary integral equation method. The boundary value problems are reduced to systems of singular integro-differential equations to which Noether-type theorems are shown to apply. We consider also the corresponding boundary value problems based on an alternative lower-order shell model of the reinforcement. Finally, existence and uniqueness results are presented for the corresponding interior and exterior boundary value problems in the appropriate classical function spaces. K E Y W O R D Sexistence of solution, micropolar boundary reinforcement, micropolar surface effects, plane micropolar elasticity, singular integro-differential equations

The Neumann problem in plane deformations of a micropolar elastic solid with micropolar surface effects

“…where L * is the operator adjoint to L , which is given in [15], and the scalar function t is defined by…”

“…Written in ( τ , n ) coordinates, from equilibrium considerations on Γ , the following relations between prescribed forces, reaction forces from the bulk, and deformations of the surface are obtained [15]:…”

“…Following the approach in [22] and [15], we reduce the interior and exterior boundary value problems to singular integral equations. Without loss of generality, we consider equations (13) and (17) with t 0 ( x ) = 0 on ∂ S t .…”

“…The kinetic theory of the micropolar elastic continuum gives rise to the concept of moment per unit area, also known as couple stress, in addition to the usual force stress. In a series of recent papers, we have proposed an enhancement to the micropolar theory of plane deformations by developing a model that combines the micropolar elastic bulk with a model of surface elasticity capable of incorporating flexural and micropolar twisting resistance [14, 15]. In these papers [14, 15], the boundary integral equation method was adopted to solve mixed boundary value problems from this enhanced micropolar model in the appropriate function spaces.…”

“…In a series of recent papers, we have proposed an enhancement to the micropolar theory of plane deformations by developing a model that combines the micropolar elastic bulk with a model of surface elasticity capable of incorporating flexural and micropolar twisting resistance [14, 15]. In these papers [14, 15], the boundary integral equation method was adopted to solve mixed boundary value problems from this enhanced micropolar model in the appropriate function spaces. In these investigations, we sought to maintain generality by incorporating surface elasticity on part of the boundary of the solid with prescriptions of displacement and stress on the remaining portions of the boundary.…”