The contact problem of elasticity theory for bodies with cracks

The plane contact problem of a stamp impressed into an elastic half-plane containing arbitrarily arranged rectilinear subsurface cracks is formulated and investigated by asymptotic methods. Partial or total overlapping of the crack edges is assumed. The problem reduces to a system of linear singular integrodifferential equations with side conditions in the form of equalities and inequalities. An analytic solution of the problem is obtained in the form of asymptotic power series /1/ in the relative dimension of the greatest crack. Dependences of the first terms of the asymptotic expansions of the desired functions on the mutual location of the cracks and the contact domains, the pressure and friction stress distributions, and the crack size and orientation are determined. Numerical results are presented.

Analysis of the influence of the stress-free boundary of the half-plane on the state of stress and strain of the elastic material near the cracks is presented in /2, 3/. The problem of a crack in an elastic plane whose edges overlap partially is also examined in /3/ by numerical methods.     

On elastic bodies with thin rigid inclusions and cracks

This paper is concerned with the analysis of equilibrium problems for two‐dimensional elastic bodies with thin rigid inclusions and cracks. Inequality‐type boundary conditions are imposed at the crack faces providing a mutual non‐penetration between the crack faces. A rigid inclusion may have a delamination, thus forming a crack with non‐penetration between the opposite faces. We analyze variational and differential problem formulations. Different geometrical situations are considered, in particular, a crack may be parallel to the inclusion as well as the crack may cross the inclusion, and also a deviation of the crack from the rigid inclusion is considered. We obtain a formula for the derivative of the energy functional with respect to the crack length for considering this derivative as a cost functional. An optimal control problem is analyzed to control the crack growth. Copyright © 2010 John Wiley & Sons, Ltd.     

Interactions in an infinite medium of planar cracks with completely rigid planar inclusions

The problem of determining the interactions in an infinite medium of planar cracks with absolutely rigid inclusions leads to a system of integral equations, the regular kernals of which represent the interaction. The system of integral equations is completely determined under boundary conditions for the equilibrium of the inclusions as a rigid body. An approximate solution for the system of integral equations is used. The dependence of the magnitude of the external load on parameters characterizing the distribution in the medium of disc-shaped cracks and inclusions is graphically presented.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 63–68, 1989.     

Junction problem for elastic and rigid inclusions in elastic bodies

An equilibrium problem for an elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. We analyze a junction problem assuming that the inclusions have a joint point. Different equivalent problem formulations are discussed, and existence of solutions is proved. A set of junction conditions is found. We investigate a convergence to infinity and to zero of a rigidity parameter of the elastic inclusion. A delamination of the elastic inclusion is also investigated. In this case, inequality‐type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces. Copyright © 2015 John Wiley & Sons, Ltd.     

Solution of problems of the theory of elasticity for a half-space with planar boundary cracks

We propose a method of solving three-dimensional problems of the theory of elasticity for a half-space containing planar boundary cracks. The problem is reduced to a system of integro-differential equations for determining the functions that characterize the opening of the crack during deformation of the halfspace. The kernels of the equations, besides having poles, also have a fixed singularity at the points of intersection of the surface of the crack with the boundary of the half-space. The equations obtained are solved numerically for the case of cracks that are part of a circular region. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 58–63.     

Contact problem of a rigid indentor in second order elasticity theory

Summary A solution to the contact problem of a rigid indentor in an elastic half space is derived by employing the theory of second order elasticity. The formulae for the distribution of pressure under the punch, shape of the deformed surface, total load on the punch and the depth of penetration are given in the general terms. The results are illustrated by considering the indentation of the half space by a rigid sphere. Amongst other results it is found that for a compressible material the depth of penetration is larger and the total load is smaller as compared to their values in classical elasticity; for an incompressible material the effects observed are exactly reversed to those of the above.

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Résumé La solution du problème de contact d'un corps rigide arrondi s'appuyant sur un demi-espace élastique est dérivée en employant la théorie de l'élasticité du second ordre. Les formules pour la distribution de la pression sous le poinçon pour la configuration de la surface déformée, pour la charge totale sur le poincon et pour la profondeur de pénétration sont données en termes généraux. Les résultats sont illustrés en considérant la cavité due à la pénétration d'une sphère rigide dans le demi-espace. Parmi d'autres resultats on trouve que pour une matière compressible, la profondeur de penetration est plus grande et que la charge totale est plus petite par rapport aux valeurs obtenues à l'aide de la theorie classique de l'elasticite; par une matière incompressible les effets observés sont exactement inverses.

     

Differentiation of energy functionals in the three-dimensional theory of elasticity for bodies with surface cracks

A three-dimensional elastic body with a surface crack is considered. The boundary nonpenetration conditions in the form of inequalities (the Signorini type conditions) are given at the faces of the crack. The convergence is proved of a sequence of equilibrium problems in perturbed domains to the solution of an equilibrium problem in the unperturbed domain in a suitable Sobolev function space. The derivative is calculated of the energy functional with respect to the perturbation parameter of the surface crack.     

Invariant tracking for planar rigid body dynamics

A quasi‐static state feedback is proposed that achieves stable tracking for planar motions of rigid bodies and is invariant with respect to the choice of the inertial frame.     

Integral characteristics of rigid inclusions and cavities in the two-dimensional theory of elasticity

Representations of the components of the elastic-polarization matrices and the Wiener elastic capacity are obtained in terms of the coefficients of the Kolosov-Muskhelishvili complex potentials and the coefficients of the conformal representation, which define the geometry of an infinite elastic solid. A new integral characteristic of a rigid inclusion—the Roben matrix, whose components are dimensiordess, is proposed for use in applied problems. Examples of calculations, which correct formulae published previously elsewhere, are given.     

Complex potentials for the plane problem of elasticity theory for a multiconnected anisotropic body with cracks

Results are given for a study and modernization of basic relationships obtained previously by the author for generalized complex potentials of crack theory generated by solving contact problems in the case of multiconnected anisotropic plates. In contrast to the author's previous work devoted to this question, general presentations of complex potentials, and boundary and additional conditions for finding them, are simplified. Use of them is shown in the case of a crack and concentrated forces, and when a plate has a finite number of cracks along a single line. Independence of stress intensity factor on anisotropy parameters is demonstrated in the last case.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 24–34, 1990.     

Shape optimization of rigid inclusions for elastic plates with cracks

In the paper, we consider an optimal control problem of finding the most safe rigid inclusion shapes in elastic plates with cracks from the viewpoint of the Griffith rupture criterion. We make use of a general Kirchhoff–Love plate model with both vertical and horizontal displacements, and nonpenetration conditions are fulfilled on the crack faces. The dependence of the first derivative of the energy functional with respect to the crack length on regular shape perturbations of the rigid inclusion is analyzed. It is shown that there exists a solution of the optimal control problem.     

Optimal rigid inclusion shapes in elastic bodies with cracks

The paper concerns the control of rigid inclusion shapes in elastic bodies with cracks. Cracks are located on the boundary of rigid inclusions and in the bulk. Inequality type boundary conditions are imposed at the crack faces to guarantee mutual non-penetration. Inclusion shapes are considered as control functions. First we provide the problem formulation and analyze the shape sensitivity with respect to geometrical perturbations of the inclusion. Then, based on Griffith criterion, we introduce the cost functional, which measures the shape sensitivity of the problem with respect to the geometry of the inclusion, provided by the energy release rate. We prove existence of optimal shapes for the problem considered.     

Application of singular integral equations in two-dimensional problems of the theory of elasticity for piecewise-uniform bodies with cracks

Using the method of singular integral equations we solve a two-dimensional problem of the theory of elasticity for an infinite plate containing an elastic inclusion of arbitrary configuration and a system of curvilinear incisions. The numerical solution is found by the method of mechanical quadratures for the case of an elliptic inclusion and a single polygonal crack.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 93–98.