Numerical solution of an equilibrium problem for an elastic body with a thin delaminated rigid inclusion

Under consideration is a 2D-problem of elasticity theory for a body with a thin rigid inclusion. It is assumed that there is a delamination crack between the rigid inclusion and the elastic matrix. At the crack faces, the boundary conditions are set in the form of inequalities providing mutual nonpenetration of the crack faces. Some numerical method is proposed for solving the problem, based on domain decomposition and the Uzawa algorithm for solving variational inequalities.We give an example of numerical calculation by the finite element method.     

A contact problem for an elastic plate with a thin rigid inclusion

Under study is an equilibrium problem for a plate under the influence of external forces. The plate is assumed to have a thin rigid inclusion that reaches the boundary at the zero angle and partially contacts a rigid body. On the inclusion face, there is a delamination. We consider the complete Kirchhoff–Love model, where the unknown functions are the vertical and horizontal displacements of the middle surface points of the plate. We present differential and variational formulations of the problem and prove the existence and uniqueness of a solution.     

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.     

The passage of a bending wave through an inhomogeneity in an absolutely rigid rib backing an elastic plate

The wave properties of a system consisting of an elastic plate and an absolutely rigid infinite rib with a defect on a segment are examined. An elastic inclusion and a gap are two kinds of defects under study. The Green's function method is applied to the diffraction problem and transforms it to singular integro-differential equations on an interval. For the case of short defects, the nonresonance and resonance asymptotics of the scattering pattern are obtained. These results show that the coefficient of penetration for a gap is much larger than that for an elastic inclusion if the frequency is nonresonant. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210. 1994, pp. 22–29. Translated by I. V. Andronov.     

Contact with break-off under bending of an elastic strip with a rigid disk

We consider the problem of the theory of elasticity of the contact interaction of a rigid circular disk and an elastic strip, which rests upon two supports with disturbance of contact in the middle part of the contact region. On the basis of the Wiener–Hopf method, an integral equation of the problem is reduced to an infinite system of algebraic equations. The size of the zone of break-off of the boundary of the strip from the disk and the distribution of contact stresses are determined.     

Elastic state of a plate with a circular plug and a rectilinear thin elastic inclusion

There is considered the problem of the state of stress of an infinite elastic plane with a bonded circular plug and an arbitrarily located thin elastic inclusion under biaxial tension. Conditions of ideal mechanical contact are satisfied on the line separating the materials. By using the complex Kolosov — Muskhelishvili potentials, the problem is reduced to a system of integro-differential equations which is solved numerically by utilization of a mechanical quadrature method. A numerical analysis is given for the solution of the problem of the elastic equilibrium of a plane with a circular hole and an arbitrarily located thin inclusion.     

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.     

The natural oscillations of an elastic body with a heavy rigid spike-shaped inclusion

It is established that oscillations in the low-frequency range are characteristic for a body with a heavy-rigid spike-shaped inclusion, and corresponding modes mainly occur as flexural deformations of the tip of the spike, localized close to its vertex.     

The indentation of two rigid stamps into an elastic sphere with a spheroidal inclusion

On the basis of the expansion formulas of the vector solutions of the Lamé equations in spherical coordinates with respect to the solutions of the Lamé equations in oblate spheroidal coordinates and on the basis of their inverse formulas, one solves the problem of the compression of an elastic ball with an absolutely rigid inclusion in the form of an oblate spheroid. The problem is reduced to an infinite system of linear algebraic equations of the second kind with a completely continuous operator in ₂. Results of the numerical solution of the infinite system are given and the obtained results are analyzed.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 9–13, 1989.     

The diffraction of plane elastic waves by a delaminated rigid inclusion when there is smooth contact in the delamination region

A solution of the problem of the diffraction of harmonic elastic waves by a thin rigid strip-like delaminated inclusion in an unbounded elastic medium, in which the conditions for plane deformation are satisfied, is proposed. We mean by a delaminated inclusion an inclusion, one side of which is completely bonded to the elastic medium, while the second does not interact in any way with it, or this interaction is partial. It is assumed that the conditions for smooth contact are satisfied in the delamination region. The method of solution is based on the use of previously constructed discontinuous solutions of the equations describing the vibrations of an elastic medium under plane deformation conditions. The problem therefore reduces to solving a system of three singular integral equations in the unknown stress and strain jumps at the inclusion. An approximate solution of the latter enabled formulae to be obtained that are convenient for numerical realization when investigating the stressed state in the region of the inclusion and its displacements when acted upon by incident waves.     

On the contact of a rigid sphere and a thin plate

We consider the problem of the contact between a rigid sphere and a thin initially flat plate. After reviewing some plate theory, we establish that a deformation where a finite piece of the plate takes the shape of the sphere is physically unrealisable, and that the contact region must be a ring. However, for both small deflections using classical linear elastic theory and large deflections using von Kármán theory, looking at some typical parameter values we find that the radius of the ring is so small that for practical purposes it should be considered as a point load.     

On the problem of a thin rigid inclusion embedded in a Maxwell material

We consider a plane viscoelastic body, composed of Maxwell material, with a crack and a thin rigid inclusion. The statement of the problem includes boundary conditions in the form of inequalities, together with an integral condition describing the equilibrium conditions of the inclusion. An equivalent variational statement is provided and used to prove the uniqueness of the problem’s solution. The analysis is carried out in respect of perfect and non-perfect bonding of the rigid inclusion. Additional smoothness properties of the solutions, namely the existence of the time derivative, are also established.     

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.     

The scattering of an harmonic elastic wave by a volume inclusion with a thin interlayer

The boundary element method is used to investigate the propagation of harmonic elastic waves in an infinite matrix with a volume inclusion with a thin interlayer between the inclusion and the matrix. A boundary-integral formulation of the problem is based on a consideration of a two-phase medium, consisting of the matrix and the inclusion, on the contact surface of which conditions of proportional dependence between the forces and jumps in the displacements, which model the interlayer, are satisfied. These conditions are taken into account implicitly in the boundary integral equations obtained, which are subsequently regularized and discretized on the grid of boundary elements introduced. The numerical results obtained demonstrate the effect of the interlayer on the dynamic contact stresses for a spherical inclusion in the field of a plane longitudinal wave.     

Cylindrical bending of an elastic rectangular sandwich plate on a deformable foundation

Cylindrical bending of an elastic rectangular sandwich plate having a rigid filler and resting on an elastic foundation is considered. To describe the kinematics of the plate, asymmetric across its thickness, the hypotheses of broken normal are assumed. The reaction of the foundation is described by the Winkler model. A system of equilibrium equations is derived, and its exact solution is obtained in terms of displacements. A numerical analysis of the solution is presented.     

Unilateral contact of a plate with a thin elastic obstacle

We study the problem of contact of an elastic body with a beam. The most attention is paid to describing boundary conditions on the possible contact set. Moreover, we study asymptotic properties of solutions and the energy functional as the rigidity parameters tend to infinity or the length of the beam (or the zone of possible contact) changes.     

Wedging of an elastic wedge by a rigid plate under conditions of contact with detaching

We consider a problem of wedging of an elastic wedge by a rigid plate along an edge crack that is located on the axis of symmetry of the wedge and reaches its vertex. The detachment of the crack faces from the surfaces of the plate is taken into account. Using the Wiener–Hopf method, we obtain an analytic solution of the problem. The size of the detachment zone, the stress intensity factor, the distribution of stresses on the line of continuation of the crack and in the contact domain, and circular displacements of the crack faces are determined.