Construction and Analysis of an Analytical Solution for the Surface Rayleigh Wave within the Framework of the Cosserat Continuum

Abstract

The problem of propagation of an acoustic surface Rayleigh wave in an infinite half-space is considered within the framework of the asymmetric theory of elasticity (Cosserat medium). It is assumed that material deformation is described not only by the displacement vector but also by an independent rotation vector. A global analytical solution of the problem in displacements is obtained. A comparative analysis of the solution obtained and the corresponding solution for the classical elastic medium is performed. Macroparameters characterizing the difference of the stress-strain state from that predicted by the classical theory of elasticity are introduced.Introduction. The model of the medium whose deformation is described not only by the displacement vector u but also by a kinematically independent rotation vector ω, which are functions of coordinates and time, has riveted attention of researchers for a long time. This theory was called the moment or asymmetric theory of elasticity. The deformation behavior of elastic bodies in this theory has some specific features, namely, the elastic body, beginning from a certain characteristic scale and (or) at high gradients of stresses or strains, can acquire a stress-strain state significantly different from that predicted by the classical (symmetric) theory of elasticity.In the theory of the Cosserat medium [1, 2], the vector ω characterizes small turns of particles, and the tensors of stressesσ and moment stressesμ are asymmetric. The dynamic behavior of an elastic isotropic medium with no allowance given to temperature effects is characterized by eight constants: two Lamé constants, four elastic constants characterizing the microstructure, density, and a parameter responsible for the measure of inertia of the medium during its rotation (density of the moment of inertia).In many publications, the analysis of the moment behavior of the material is considered within the framework of a simplified medium called the medium with constrained rotation or the Cosserat pseudomedium [3]. Simplification is reached by using the dependence ω = (1/2) rot u, which, in particular, allows one to reduce the number of physical parameters from eight to five. Because of the drawbacks considered in detail in [1], however, the Cosserat pseudomedium model will not be further used in the present work.A large amount of exact analytical solutions have been obtained within the framework of the asymmetric theory of elasticity (especially in the case of constrained rotation). In many papers, these solutions are analyzed and compared with the corresponding solutions of the classical theory of elasticity. In such a comparison, new physical constants determining the contribution of the moment components are normally chosen from the range of their energetically admissible values. This is caused by the lack of information on material constants with a microstructure, which is one of the main factors preventing the development of models of asymmetric media.There are several papers where the physical constant...

Constructing an analytical solution for lamb waves using the cosserat continuum approach

“…Applying the method described in detail in [11] to Eqs. (2.2) and (2.3), we obtain the general dimensionless solution of the equations of motion (1.4).…”

“…(2.2) and (2.3), we obtain the general dimensionless solution of the equations of motion (1.4). It should be noted that unlike in the case of surface waves, a solution for which is considered in [11], in the given case, it is necessary to retain all particular solutions and not only those that damp with depth. Thus, the general solution in displacements is written as …”

“…Substitution of solution (2.4) into boundary conditions (3.1) yields two homogeneous systems of algebraic equations, and from the resolvability condition for these equations, we obtain the following dispersion equations for the two types of waves: 1) for Rayleigh waves with components u x , u z , and ω y [11],…”

Analysis of the wave solution of the elastokinetic equations of a Cosserat continuum for the case of bulk plane waves

“…Dispersion of Rayleigh elastic surface waves was found in [9,10] (in the classical theory of elasticity, Rayleigh waves do not exhibit dispersion). A detailed analysis of the components of the displacement and rotation vectors of Rayleigh waves was performed in [11]. An effect due to the propagation of a surface transverse wave with horizontal polarization was found.…”

“…The basic equations of a Cosserat continuum and the general equation of plane waves are also given in the paper. A particular solution for bulk waves is constructed and compared with the solutions obtained in [11][12][13].…”