Limiting equilibrium and fracture in an orthotropic body containing a thin rigid inclusion

The limiting equilibrium and type of fracture of an orthotropic body containing a linear rigid inclusion in tension at infinity along the axis of the inclusion is studied under the conditions of plane problem. Localized process zones (of weakened contact) develop along the boundary of the inclusion from its ends to the central part. The analytic solution is obtained with the help of complex potentials by reducing the analyzed problem to the problem of conjugation. The influence of loading and the orthotropy of the matrix on the development of the process zones, the distributions of contact stresses and axial forces in the inclusion, and the character of fracture of the composition are investigated. We determine the ultimate loads of possible separation or rupture of the inclusion and compute these quantities for some special cases. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 2, pp. 69–79, March–April, 2006.     

On the motion of a heavy rigid body about a fixed point

Summary The paper considers the classical problem of the motion of a heavy rigid body about a fixed point. The Euler-Poisson equations are used for the presentation of that motion. Three first integrals of these equations are well known. To solve this problem completely one further first integral has to be found. The main purpose of the paper is to obtain a necessary and sufficient condition for some functionF(₁,₂,₃, ₁, ₂, ₃) to be a new first integral of Euler-Poisson equations.     

Plane deformation of a body with rigid cylindrical inclusion

We study the problem of plane deformation of an infinite elastic body with thin rigid cylindrical inclusion with oval cross section. The body is loaded by biaxial uniform tensile forces at infinity. The solution of the problem is reduced to two singular integral equations with Cauchy kernels for the jumps of normal and tangential stresses on the surface of the inclusion. The solutions of these equations are obtained in the closed analytic form and, used to deduce the formulas for the concentration of stresses near the inclusion, for stresses inside the inclusion, and for the angle of rotation of the inclusion as a rigid body. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, Ukrainian State University of Forestry Engineering, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 6, pp. 87–92, November–December, 1996.     

Stress analysis of a semi-infinite plate with a thin rigid body

Stress analysis of a semi-infinite plate with an oblique thin rigid body is carried out as a mixed boundary value problem. The complex variable method and a rational mapping function of a sum of fractional expressions are used. A closed solution is obtained for the shape, which is represented by a rational mapping function. Stress distribution, stress singularity at the tip of the thin rigid body, the resultant moment over the thin rigid body, and the rotation angle are investigated.     

Structural stability for a rigid body with thermal microstructure

We investigate a model for a rigid heat conductor which allows for variation of thermal properties at a microstructure level. With increasing use of nanofluids, nanomaterials, and the need to store materials at cryogenic temperatures we believe theories for bodies with (anisotropic) thermal microstructure will become increasingly important. In this article we examine how the solution depends on changes in coupling coefficients between the macro and microthermal level. A precise a priori continuous dependence result is established. The way in which the solution to the microstructural problem converges to that for a rigid body without thermal microstructure is also investigated.     

Fractional-derivative viscoelastic model of the shock interaction of a rigid body with a plate

The impact of a rigid body upon an elastic isotropic plate is investigated for the case when the equations of motion take rotary inertia and shear deformation into account. The impactor is considered as a mass point, and the contact between it and the plate is established through a buffer involving a linear-spring–fractional-derivative dashpot combination, i.e., the viscoelastic features of the buffer are described by the fractional-derivative Maxwell model. It is assumed that a transient wave of transverse shear is generated in the plate, and that the reflected wave has insufficient time to return to the location of the spring’s contact with the plate before the impact process is completed. To determine the desired values behind the transverse-shear wave front, one-term ray expansions are used, as well as the equations of motion of the impactor and the contact region. As a result, we are led to a set of two linear differential equations for the displacements of the spring’s upper and lower points. The solution of these equations is found analytically by the Laplace-transform method, and the time-dependence of the contact force is obtained. Numerical analysis shows that the maximum of the contact force increases, tending to the maximal contact force when the fractional parameter is equal to unity.     

Longitudinal shear of a body with mutually immobile rigid collinear inclusions

We study the problem of antiplane deformation of an elastic body with collinear perfectly rigid inclusions connected to form a single skeleton. The problem is reduced to a system of singular integral equations with an additional condition guaranteeing the absence of mutual displacements of the inclusions. The influence of mutual immobility and arrangement of the inclusions on the distributions of stresses and displacements in the body is analyzed.Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 3, pp. 69–73, May–June, 2004.