On bending an elastic plate with a delaminated thin rigid inclusion

Under study is the problem of bending an elastic plate with a thin rigid inclusion which may delaminate and form a crack. We find a system of boundary conditions valid on the faces of the crack and prove the existence of a solution. The problem of bending a plate with a volume rigid inclusion is also considered. We establish the convergence of solutions of this problem to a solution to the original problem as the size of the volume rigid inclusion tends to zero.     

Elastoplastic equilibrium of an isotropic strip with a circular hole under pure bending

The range of variation of external forces over which the plastic region in a bent strip develops from initiation to complete envelopment of a circular aperture is determined. A perturbation method is used to construct a solution for these interactions. Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhnika, No. 30, pp. 105–111, 1999.     

Contact deformation of an elastic composition of a half-plane and a strip of variable width

Asymptotic dependences of the deformation on the contact stresses are derived for a strip of variable width bound to an elastic half-plane. Similar relations were previously obtained in [1] for a strip of constant width.     

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.     

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.     

On an integral equation of the problem for an elastic strip with a slit

A new singular integral equation is obtained that describes the elastic equilibrium of a strip with both an inner and an edge slit (crack) and has a considerable advantage over existing equations /1–9/, etc.) from the viewpoint of a numerical realization and clarification of the analytical relationship with an analogous equation for a half-plane. Numerical results are given of a computation of the stress intensity coefficients at the tips of the inner and edge cracks that refine data in the literature.     

Oscillations of a rigid body containing an elastic element with distributed parameters

The motions of a hybrid (discrete-continual) system, consisting of a carrier rigid body and an elastic element with distributed parameters fastened to it are investigated. Two types of fastening are considered: (1) both ends are clamped, and (2) one of the ends is clamped while the other is free. A closed system of integro-differential equations is obtained which describes the state of the system under arbitrary initial conditions and forces applied to the rigid body. The perturbed motion of the rigid body in the case of a quasi-linear restoring force is investigated using asymptotic methods. The motions are studied both when there is internal resonance between the oscillations of the rigid body and the natural oscillations of the element, and when there are no such resonances. Qualitative effects are found.     

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.     

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.     

Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity

In this paper, mathematical modeling of the propagation of Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity has been considered. The equations of motion have been formulated separately for different media under suitable boundary conditions at the interface of porous layer, elastic half-space under gravity and rigid layer. Following Biot, the frequency equation has been derived which contain Whittaker’s function and its derivative that have been expanded asymptotically up to second term (for approximate result) for large argument due to small values of Biot’s gravity parameter (varying from 0 to 1). The effect of porosity and gravity of the layers in the propagation of Love waves has been studied. The effect of hydrostatic initial stress generated due to gravity in the half-space has also been shown in the phase velocity of Love waves. The phase velocity of Love waves for first two modes has been presented graphically. Frequency equations have also been derived for some particular cases, which are in perfect agreement with standard results. Subsequently the lower and upper bounds of Love wave speed have also been discussed.     

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.     

Circular inclusion in an infinite elastic strip

This paper is concerned with the problem of a circular inclusion undergoing spontaneous dimensional changes in an infinite elastic strip with the straight edges free from displacements. Consequent elastic fields both in the inclusion and the surrounding strip are determined with the aid of complex variable technique. Closed form expressions for two sets of complex potentials and for various stress components are provided. Boundary stresses have been computed and their behaviour examined for varying situations.

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Zusammenfassung Die Arbeit betrifft das Problem eines kreisrunden Fremdkörpers, der sich innerhalb eines unendlichen elastischen Streifens mit verschiebungslosen Rändern deformiert. Die elastischen Felder im Fremdkörper und im Streifen werden mit der komplexen Methode bestimmt, und es werden für zwei Sätze komplexer Potentiale und für verschiedene Spannungskomponenten geschlossene Lösungen angegeben. Die Randspannungen werden berechnet und für verschiedene Fälle untersucht.

     

Contact problem for elastic half-space with an ellipsoidal cavity

We consider the axisymmetric problem of elasticity theory for a space with an elongated ellipsoidal cavity with mixed boundary conditions of smooth contact on its surface. The method of p-analytical functions is applied to reduce the solution of the problem to an infinite quasi-completely regular system of linear algebraic equations with upper bounded free terms that tend to zero as the index increases. The behavior of the normal stress near the contact line of the different boundary conditions is analyzed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 94–103, 1988.     

Damageable contact between an elastic body and a rigid foundation

In this work, the contact problem between an elastic body and a rigid obstacle is studied, including the development of material damage which results from internal compression or tension. The variational problem is formulated as a first-kind variational inequality for the displacements coupled with a parabolic partial differential equation for the damage field. The existence of a unique local weak solution is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, three two-dimensional numerical simulations are performed to demonstrate the accuracy and the behaviour of the scheme.     

The motion of a rigid spherical inclusion in an elastic medium under the action of plane waves

Two problems on the motion of a rigid spherical inclusion in an elastic medium under the action of a nonstationary, longitudinal, plane, compressional wave and a harmonic shear wave are considered. Using the method of vector eigenfunctions and the integral Laplace transform with respect to time their precise solutions are constructed. Numerical results are given which illustrate the dependence of characteristics of motion of the inclusion on parameters of the falling wave.Translated from Dinamicheskie Sistemy, No. 9, pp. 37–47, 1990.     

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.     

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.     

Two problems with mixed boundary conditions for an elastic orthotropic strip

Two problems are considered for an elastic orthotropic strip: the contact problem and the crack problem. Both problems are reduced to integral equations of the first kind with different kernels, containing a singularity: logarithmic for the first problem and singular for the second problem. Regular and singular asymptotic methods are employed to construct approximate solutions of these integral equations. Numerical results are presented.