S. Potapenko

On the Solution of Mixed Problems in Anti-Plane Micropolar Elasticity

In the present paper we apply the real boundary integral equation method to solve interior and exterior mixed boundary value problems arising in a linear theory of anti-plane elasticity which includes the effects of material microstructure.

Generalized Fourier series solution of torsion of an elliptic beam with microstructure

In this paper, we use a generalized Fourier series to approximate the solution of a boundary value problem describing the torsion of an elliptic micropolar beam. We provide an example demonstrating the effect of material microstructure.

Steady Thermoelastic Oscillations in a Linear Theory of Plane Elasticity with Microstructure

In the present paper we consider the problem of steady thermoelastic oscillations in a linear theory of plane elasticity with microstructure where disturbance is represented by a train of harmonic waves. The corresponding boundary value problems of Dirichlet and Neumann type are formulated and solutions are obtained in the form of integral potentials using the real boundary integral equation method. Appropriate Sommerfeld radiation conditions are prescribed in the case of the exterior domain and the uniqueness result is established.

Propagation of torsional waves in a linear unbounded Cosserat continuum

In the present work we formulate uniqueness theorems for the problem of propagation of longitudinal monochromatic waves in a linear theory of elasticity with microstructure where disturbance is represented by a train of harmonic waves. The corresponding boundary value problems of Dirichlet and Neumann type are formulated together with appropriate Sommerfeld radiation conditions in the case of the exterior domain and the uniqueness results are established.

An integral representation for the solution of the inclusion problem in the theory of antiplane micropolar elasticity

In the present paper, we apply the real boundary integral equation method to obtain the solution of the inclusion boundary-value problem with an imperfect interface arising in the theory of antiplane elasticity with significant microstructure. We find the solution in the form of the integral potential and employ the boundary element method to derive an approximate representation for the corresponding integral density. Finally, we consider an example of an elliptic inclusion with a homogeneously imperfect interface and find the distribution of shear stresses along the inclusion interface to demonstrate the effectiveness of the method. The method is very general in nature so it can be applied for the treatment of problems relating to inclusions of arbitrary shape, different types of interface and general forms of applied loading.

Variational Formulation of Crack Problems in Three-dimensional Classical Elasticity

In this paper we consider a crack of arbitrary shape in a homogeneous elastic media in the absence of body forces, formulate variational Dirichlet and Neumann crack problems in a linear three-dimensional elasticity in Sobolev spaces and prove the existence and uniqueness of the corresponding (weak) solutions.

Integral Representation of the Solution of Torsion of an Elliptic Beam with Microstructure

The theory of micropolar elasticity [1] was developed to account for discrepancies between the classical theory and experiments when the effects of material microstructure were known to significantly affect a body’s overall deformation. The problem of torsion of micropolar elastic beams has been considered in [2] and [3]. However, the results in [2] are confined to the simple case of a beam with circular cross section while the analysis in [3] overlooks certain differentiability requirements that are essential to establish the rigorous solution of the problem (see, for example, [4]). In neither case is there any attempt to quantify the influence of material microstructure on the beam’s deformation. The treatment of the torsion problem in micropolar elasticity requires the rigorous analysis of a Neumann-type boundary value problem in which the governing equations are a set of three second-order coupled partial differential equations for three unknown antiplane displacement and microrotation fields. This is in contrast to the relatively simple torsion problem arising in classical linear elasticity, in which a single antiplane displacement is found from the solution of a Neumann problem for Laplace’s equation [5]. This means that in the case of a micropolar beam with noncircular cross section it is extremely difficult (if not impossible) to find a closed-form analytic solution to the torsion problem. In this paper, we use a simple, yet effective, numerical scheme based on an extension of Kupradze’s method of generalized Fourier series [6] to approximate the solution of the problem of torsion of an elliptic micropolar beam. Our numerical results demonstrate that the material microstructure does indeed have a significant effect on the torsional function and the subsequent warping of a typical cross section.