The Reissner-Sagoci problem for a half-space with a surface constraint

In this paper, the Reissner-Sagoci problem for a half-space with a surface constraint is considered. The problem is reduced to the triple integral equation by use of integral transforms. The triple integral equations are solved for small values of parameters characterizing the geometry of the problem. An expression for the torque required to rotate the disc through a fixed angle is obtained.     

Torsional wave propagation in a pre-stressed circular cylinder embedded in a pre-stressed elastic medium

Within the framework of the piecewise homogeneous body model with utilization of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the mathematical modeling of the torsional wave propagation in the initially stressed infinite body containing an initially stressed circular solid cylinder (case 1) and circular hollow cylinder (case 2) are proposed. In these cases, it has been assumed that in the constituents of the considered systems there exist only the normal homogeneous tensional or compressional initial stress acting along the cylinder, i.e. in the direction of wave propagation. In the case where the mentioned initial stresses are not present, the proposed mathematical modeling coincides with that proposed and investigated by other authors within the classical linear theory of elastic waves. The mechanical properties of the cylinder and surrounding infinite medium have been described by the Murnaghan potential. The numerical results related to the torsional wave dispersion and the influence of the mentioned initial stresses on this dispersion are presented and discussed.     

The breaking of a non-homogeneous fiber embedded in an infinite non-homogeneous medium

The torsion of an infinite non-homogeneous elastic cylindrical fiber, containing a penny-shaped crack embedded in an infinite non-homogeneous elastic material is considered. The cylinder and elastic medium have different shear moduli. Using integral transformation techniques the solution of the problem is reduced to the solution of dual integral equations. Later on the solution of the dual integral equations is transformed into the solution of a Fredholm integral equation of the second kind, which is solved numerically. Closed form expressions are obtained for the stress intensity factor and numerical values for the stress intensity factors are graphed to demonstrate the effect of non-homogeneity of the fiber and infinite medium. In the end the stress singularity is obtained when the crack touches the infinite non-homogeneous medium (matrix).     

The problem of a viscoelastic cylinder rolling on a rigid half-space

The problem of a viscoelastic cylinder rolling on a rigid base, propelled by a line force acting at its centre, is solved in the noninertial approximation. The method used is based on a decomposition of hereditary integrals developed by the authors in previous work, and on the viscoelastic Kolosov-Muskhelishvili equations which are used to generate a Hilbert problem. In this formulation, the problem reduces to a nonsingular integral equation in space and time, which simplifies under steady-state conditions and for exponential decay materials, to algebraic form. There are also two subsidiary conditions.In the case of a standard linear model, explicit analytic results and numerical examples are given for the pressure function, for surface displacements, and also for hysteretic friction.     

Recovery of residual stress in a vertically heterogeneous elastic medium

We study the problem of identifying residual stress within athin subsurface layer in an elastic medium occupying a region = {(x₁, x₂, x₃) , R³: 0 < x₃ < L, where L } in space,where all parameters depend only on the depth x₃. Under thetheoretical framework of linear elasticity with initial stress,the incremental elasticity tensor of each material point iswritten as a sum of two terms, namely the elasticity tensorand the acoustoelastic tensor, both of which are taken hereas isotropic functions of their arguments. By imposing impulsiveloads and measuring the displacements at the boundary x₃ = 0,we recover the residual stress and its gradient there. If theresidual stress has a diagonal form, we can recover the residualstress inside the subsurface layer.     

Dispersion of elastic waves in the contact-impact problem of a long cylinder

Numerical dispersion of two-dimensional finite elements was studied. The outcome of the dispersion study was verified by the numerical and analytical solutions to the longitudinal impact of two long cylindrical bars. In accordance with the results of the dispersion analysis it was demonstrated that the quadratic elements showed better accuracy than the linear ones.     

The conjunction problem for thin elastic and rigid inclusions in an elastic body

Under consideration is the conjunction problem for a thin elastic and a thin rigid inclusions that are in contact at one point and placed in an elastic body. Depending on what kind of conjunction conditions are set at the contact point of inclusions, we consider the two cases: the case of no fracture, where, as the conjunction conditions, we take the matching of displacements at the contact point and preservation of the angle between the inclusions, and the case with a fracture in which only the matching of displacements is assumed. At the point of conjunction, we obtain the boundary conditions for the differential formulation of the problem. On the positive face of the rigid inclusion, there is delamination. On the crack faces, some nonlinear boundary conditions of the type of inequalities are set, that prevent mutual penetration of the faces. The existence and uniqueness theorems for the solution of the equilibrium problem are proved in both cases.     

The inverse bending problem for a thin plate with an elastic imperfection

We consider the inverse problem consisting of determining the unknown shape of an elastic imperfection contained in a thin plate from the condition of equal strength in the stressed state along the phase interface surface. It is shown that such a state is attained in the case of an elliptic imperfection whose shape depends on the values of the applied moments and the mechanical properties of the component phases. It is established that for the geometry found for the imperfection the sum of the moments is constant and the second invariant of the deviator of the stress tensor is superharmonic over the entire plate. Numerical computations are carried out. In special cases the results obtained coincide with known data. One figure. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 34–40, 1991.     

Stability of a matrix finite-difference scheme for solving the problem of elastic vibrations

The stability of a matrix finite-difference scheme is analyzed. The scheme is based on central differences and is designed for the differential equation governing the vibrations of an elastic system.     

An operator model for the oscillation problem of liquids on an elastic bottom

This paper deals with the problem of small oscillations in a liquid layer of finite depth under the assumption that the bottom is an elastic medium. The system of equations corresponding to the problem is written out and explained. The main aim of the paper is to recast these equations in the form

带裂纹的弹性狭长体基本问题

利用复变方法和积分方程理论,讨论带任意裂纹的各向同性弹性狭长体的基本问题。通过适当的函数分解和积分变换,将问题简化为一正则型奇异积分方程。对方程解的情况和求解方法进行了研究,并导出裂纹尖端的应力强度因子。     

Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces

We study a non-homogeneous boundary value problem in a smooth bounded domain in R^N. We prove the existence of at least two non-negative and non-trivial weak solutions. Our approach relies on Orlicz-Sobolev spaces theory combined with adequate variational methods and a variant of Mountain Pass Lemma.     

The modified model of Koiter’s type for nonlinearly elastic shell

In this paper, we proposed a modified model of Koiter’s type for nonlinearly elastic shell. The change of metric tensor and the change of curvature tensor play an important role in constructing the linearly and nonlinearly elastic shell model of Koiter’s type. The approximate expressions of them once were proposed by Ciarlet. In this paper, the exact full expressions of the change of metric tensor and the change of curvature tensor are provided by tensor analysis. The former coincides with Ciarlet’s expression. And the latter is more exact than Ciarlet’s expression. Thus the modified model is better than Ciarlet’s model. At the same time, a numerical experiment of special hemispherical shell is provided to validate the modified model of Koiter’s type.     

On the first boundary problem for a hyperbolic equation in an arbitrary cylinder

A study is made of the solutions of a second-order hyperbolic equation which vanish on the boundary of an arbitrary domain in the space of the variables x₁..., x_n The degree of smoothness in the initial conditions, necessary and sufficient to guarantee the same degree of smoothness in the solution (considered as a function of x₁..., x_n for all t, is ascertained.Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 181–190, August, 1968.     

An axisymmetric problem of an embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium

This paper investigates the problem of an axisymmetric penny shaped crack embedded in an infinite functionally graded magneto electro elastic medium. The loading consists of magnetoelectromechanical loads applied on the crack surfaces assumed to be magneto electrically impermeable. The material’s gradient is parallel to the axisymmetric direction and is perpendicular to the crack plane. An anisotropic constitutive law is adopted to model the material behavior. The governing equations are converted analytically using Hankel transform into coupled singular integral equations, which are solved numerically to yield the crack tip stress, electric displacement and magnetic induction intensity factors. A similar problem but with a different crack morphology, that is a plane crack embedded in an infinite functionally graded magneto electro elastic medium, was considered by the authors in a previous work (Rekik et al., 2012) [25]. While the overall solution schemes look similar, the axisymmetric problem resulted in more mathematical complexities and let to different conclusions with respect to the influence of coupling between elastic, electric and magnetic effects. The main focus of this paper is to study the effect of material non-homogeneity on the fields’ intensity factors to understand further the behavior of graded magnetoelectroelastic materials containing penny shaped cracks and to inspect the effect of varying the crack geometry.     

Performance modeling and analysis of blood flow in elastic arteries

The effect of magnetic field has been examined on rheological models of blood. One. of them is the Power Law model and the other is the generalized Maxwell model. It is noticed that the magnetic field has a significant effect on the flow phenomena. The investigation shows that the model considered here is capable of taking into account the rheological properties affecting the blood flow and hemodynamic features, which may be important for medical doctors to predict diseases for individuals on the basis of the pattern of flow for an elastic artery in the presence of a transverses magnetic field. The effects of a magnetic field have been used to control the flow, which may be useful in certain hypertension cases, etc.     

The Betti reciprocity principle and the normal boundary component control problem for linear elastic systems

The background for this article is the question of modification of the geometric configuration of an elastic structure by means of “volume” type actuation. In this actuation mode stresses are applied to the elastic body by injection/extraction of a fluid into, or from, a large number of vacuoles in the elastic “matrix” material. Previous articles by the author, and others, have examined this process and studied its effectiveness in the context of a “naive” continuous model. The present paper continues along these lines, exploring “normal boundary component controllability” criterion for determining achievable configurations for the controlled system in the two-dimensional case. Connections with conformal mapping lead to affirmative results for approximate controllability in this sense and Fourier series techniques provide exact controllability results for the case wherein the domain of the uncontrolled system is a two-dimensional disk.      

Existence and multiplicity of positive solutions for an elastic beam equation

This paper investigates the boundary value problem for elastic beam equation of the form