The boundary value problem of asymmetric elasticity theory in quasi-classical approximation: PMM vol.36, n≗2, 1972, pp.282–290

Abstract

Considered is the static boundary value problem of the asymmetric theory of elasticity for media in which the moment effects [couple stresses] supply a small contribution to the elastic energy. The elastic coefficients in the equilibrium equations, having the dimensions of a squared length, are taken to be small in comparison with the squares of the characteristic dimensions of the body. One obtains the solution of the equilibrium equations, containing small parameters in the leading derivatives. By an approximation method one constructs the solution for the field of displacements and rotations in the form of the sum of their classical limits and moment terms having the form of boundary layer functions. Boundary conditions of kinematic type are considered and a scheme is developed in order to satisfy them by the method of successive approximations. Most of the media whose viscoelastic behavior is described within the limits of the asymmetric continuum mechanics (liquid) crystals, ferromagnetics, in a series of cases dislocation media and suspensions) are characterized by a small contribution of the moment terms in the general energetic balance of the deformation and flow processes. However, without taking into account the interaction of the moments one cannot give and interpretation to an entire series of delicate singularities of their viscoelastic behavior (the effects of the elastic distortion in the field of directions of the axes of molecular orientation in liquid crystals, the formation of spin waves in ferromagnetics, the effects of hardening in the plastic deformation, peculiarities of blood flow, etc.) In connection with this there aiises the problem of analyzing the simplifications which can be introduced in the asymmetric mechanics by the investigation of media with weak moments. Below we consider elastic Isotropie media which are characterized by additional coefficients of rotational elasticity λ and moment elasticity η, τ, θ [1]. The energetic contribution of the moment terms to the elastic potential is determined by the ratio between these coefficients and tine moduli λ, μ of the classical elasticity. If these ratios are small (in some sense which will be specified later), then we will say that the medium has weak moments [couple stresses]. For such media we investigate boundary value problem of the asymmetric elasticity theory.