Dynamic (Harmonic) Interfacial Stress Field in a Half-Plane Covered with a Prestretched Soft Layer

A half-plane covered with a prestretched layer is considered under the action of a periodic dynamic (harmonic) lineal load applied to the free surface of the layer. Within the framework of a piecewise homegeneous body model, with the use of equations of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the problem of stress state is formulated. It is assumed that the materials of the layer and half-plane are linearly elastic, homogeneous, and isotropic, and a plane strain state is considered. The corresponding boundary-value problems are solved analyticaly by employing the exponential Fourier transformations. Numerical results are obtained in the case where the elastic modulus of the half-plane material is greater than that of the layer material. It is established that, because of softening of the layer material, the stresses on the interplane increase mainly in the vicinity of the acting force and this increase has a local character. Moreover, it is established that the prestretching of the cover layer decreases the absolute values of these stresses.     

Mixed problem of plane orthotropic elasticity in a half-plane

We consider a mixed problem of plane isotropic elasticity in a half-plane in which the displacement vector and the normal component of the stress tensor are alternately specified on successive intervals of the real axis. We derive a closed-form expression for the solution of this problem, which is similar to the well-known Keldysh–Sedov formula for the half-plane.     

Singular integral equation method for contact problem for rigidly connected punches on elastic half-plane

In the contact problem of a rigid flat-ended punch on an elastic half-plane, the contact stress under punch is studied. The angle distribution for the stress components in the elastic medium under punch is achieved in an explicit form. From obtained singular stress distribution, the punch singular stress factor (abbreviated as PSSF) is defined. A fundamental solution for the multiple flat punch problems on the elastic half-plane is investigated where the punches are disconnected and the forces applied on the punches are arbitrary. The singular integral equation method is suggested to obtain the fundamental solution. Further, the contact problem for rigidly connected punches on an elastic half-plane is considered. The solution for this problem can be considered as a superposition of many particular fundamental solutions. The resultant forces on punches are the undetermined unknowns in the problem, which can be evaluated by the condition of relative descent between punches. Finally, the resultant forces on punches can be determined, and the PSSFs at the corner points can be evaluated. Numerical examples are given.     

Diffraction by an acoustically transmissive or an electromagnetically dielectric half-plane

This work gives a mathematical model for an acoustically penetrable or electromagnetically dielectric half-plane. An approximate boundary condition is used that depends on the thickness of, and the material constants for, the half-plane. A solution is obtained, by using the approximate boundary condition, for the problem of a line source field diffracted by a penetrable/dielectric half-plane. The asymmetry of the approximate boundary condition results in a matrix Wiener–Hopf problem, which is solved explicitly.     

Stress intensity factors for three cracks at the interfaces of a graded layer bonding two different materials

We consider two dissimilar elastic half-planes bonded by a nonhomogeneous elastic layer in which there is one crack at the lower interface between the elastic layer and the lower half-plane and two cracks at the upper interface between the elastic layer and the upper half-plane. The stress intensity factors for these three cracks are solved for when tension is applied perpendicular to the interface cracks. The material properties of the bonding layer vary continuously between those of the lower half-plane and those of the upper half-plane. The differences in the crack surface displacements are expanded in a series of functions that are zero outside the cracks. The unknown coefficients in the series are solved by the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors are calculated numerically for selected crack configurations.     

The diffusion of a discontinuity of the shear stress at the boundary of a viscoplastic half-plane

For the problem of the diffusion of a discontinuity of the shear stress at the boundary of a half-plane, which is a special case of the general problem of the diffusion of a vortex layer, the classes of media and types of assignment of boundary conditions for which self-similar solutions exist are discussed. For a viscoplastic medium in a half-plane the problem reduces to the problem in a layer of time-variable thickness, the solution of which does not possess the property of analyticity. The long-term asymptotic of this problem are investigated. In the case where, at an accessible boundary, it is possible simultaneously to measure both the shear stress and the horizontal velocity, an algorithm is proposed for finding a quantity that is difficult to measure, A namely, the thickness of the zone of viscoplastic flow.     

Thermoelastic instability of functionally graded coating with arbitrarily varying properties considering contact resistance and frictional heat

Using the homogeneous multi-layered model, this paper studies the thermoelastic instability (TEI) of the functionally graded material (FGM) coating with arbitrary varying properties considering the frictional heat and thermal contact resistance. A homogeneous half-plane slides on an FGM coated half-plane at the out-of-plane direction under a uniform pressure. The perturbation method and transfer matrix method are used to deduce the characteristic equation of the TEI problem, which is then solved to obtain the relationship between the critical sliding speed and critical heat flux. The effects of the gradient index and varying form of material properties of the FGM coating on the stability boundaries are examined. The results show that FGM coating can adjust the thermoelastic contact stability of sliding systems.     

Elastic fields in two imperfectly bonded half-planes with a thermal inclusion of arbitrary shape

A general method is presented for the rigorous solution of Eshelby’s problem concerned with an arbitrary shaped inclusion embedded within one of two dissimilar elastic half-planes in plane elasticity. The bonding between the half-planes is considered to be imperfect with the assumption that the interface imperfections are uniform. Using analytic continuation, the basic boundary value problem is reduced to a set of two coupled nonhomogeneous first-order differential equations for two analytic functions defined in the lower half-plane which is free of the thermal inclusion. Using diagonalization, the two coupled differential equations are decoupled into two independent nonhomogeneous first-order differential equations for two newly defined analytic functions. The resulting closed-form solutions are given in terms of the constant imperfect interface parameters and the auxiliary function constructed from the conformal mapping which maps the exterior of the inclusion onto the exterior of the unit circle. The method is illustrated using several examples of an imperfect interface. In particular, when the same degree of imperfection is realized in both the normal and tangential directions between the two half-planes, a thermal inclusion of arbitrary shape in the upper half-plane does not cause any mean stress to develop in the lower half-plane. Alternatively, when the imperfect interface parameters are not equal, then a nonzero mean stress will be induced in the lower half-plane by the thermal inclusion of arbitrary shape in the upper half-plane. Detailed results are presented for the mean stress and the interfacial normal and shear stresses caused by a circular and elliptical thermal inclusion, respectively. Results from these calculations reveal that the imperfect bonding condition has a significant effect on the internal stress field induced within the inclusion as well as on the interfacial normal and shear stresses existing between the two half-planes especially when the inclusion is near the imperfect interface.     

The state of stress of an anisotropic half-plane with an elliptic inclusion

One solves the problem of the determination of the state of stress in an anisotropic half-plane in the presence of an elliptic inclusion of the same material, subjected to a preliminary strain. This problem is equivalent to the problem of the determination of the state of stress in a half-plane, due to a thermal patch, coinciding with the inclusion domain. The solution of the problem is given in complex potentials. The case of an orthotropic half-plane with a circular inclusion is considered in detail. The results are illustrated by graphs.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 79–86, 1987.     

The stress state of planar surface of a nanometer-sized elastic body under periodic loading

The paper is concerned with the model of an elastic body in the form of a half-plane whose boundary is subjected to periodic loading. It is assumed that there exists an additional surface stress, which is characteristic of nanometer-sized bodies and which obeys the laws of surface elasticity theory. With the use of the boundary properties of analytical functions and the Goursat-Kolosov complex potentials, the boundary value problem in its general setting with an arbitrary load is reduced to a hypersingular integral equation with respect to the derivative of the surface stress. For a periodic load, the solution of this equation is obtained in the form of a Fourier series. The effect of the surface stress upon the stress state of the boundary of the half-plane is examined with independent action of periodically distributed tangential and normal loads. In particular, the size effect was discovered, which is manifested in the dependence of stresses versus the period of loading within several dozens of nanometers. Normal loads are shown to be responsible for tangential stresses on the boundary, which are zero in the classical solution.     

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.     

Interaction of elastic waves with a curvilinear crack in a half-plane

We consider the problem of the interaction of monochromatic displacement waves with a curvilinear crack-cut in a half-plane. We find integral representations of the solution. The boundary-value problem is reduced to a system of singular integral equations. A parametric investigation is carried out for the effect of the form of the load, the fastening conditions on the boundary of the half-plane, and the curvature of the crack on the dynamic coefficients of the stress intensity.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 77–82, 1988.     

The critical speed of a moving time-harmonic load acting on a system consisting a pre-stressed orthotropic covering layer and a pre-stressed half-plane

The dynamic response of a system consisting of an initially stressed covering layer and an initially stressed half-plane to a moving time-harmonic load is investigated within the scope of the piecewise-homogeneous body model utilizing three-dimensional linearized wave propagation theory in the initially stressed body. It is assumed that the material of the layer and half-plane is orthotropic. It is also assumed that the velocity of the line-located time harmonic moving load which acts on the covering layer is constant. The investigations were carried out were for the plane-strain state under subsonic velocity of the moving load for two types of contact conditions, namely: complete and incomplete. An algorithm is developed for the determination of the values of the moving load’s critical velocity. For various values of the problem parameters the numerical results were presented and discussed.     

Solutions of the generalized kinetic model of annihilation for a mixture of particles of two types

The evolution of the concentrations of particles of two types that annihilate at collision is considered. The kinetic model describing the dynamics of the mixture is represented by a system of two first-order nonlinear partial differential equations. It is shown that the solutions of this model are related to the solutions of the inhomogeneous transport equations by the Bäcklund transform. Analytic solutions of the problem about penetration of particles of the first type from the left half-plane into the right half-plane occupied by the particles of the second type (the two-dimensional penetration problem or molecular beam problem) and of the problem of outflow of the particles of the first type from a circular source into a domain occupied by the particles of the second type are obtained. Possible generalizations of the model are discussed.     

Vibration of a pre-stressed plate on a transversely isotropic multilayered half-plane due to a moving load

This paper presents an analytical research on the dynamic interaction problem between a pre-stressed plate and a transversely isotropic multilayered half-plane subjected to a moving load. The pre-stressed plate is governed by the Kirchhoff plate theory, and the transversely isotropic multilayered half-plane is solved by the analytical layer-element method. Combining the frictionless contact and displacement compatibility conditions between the plate and the soil, the contact stress and the deflection of plate in the Fourier transform domain are derived. With the aid of the inverse Fourier transform, the actual solutions can be further achieved. Numerical examples are given to illustrate the influence of load speed, the rigidity of plate, the axial force applied on the plate and the stratified character of the soil.     

The critical speed of a moving load on a prestressed plate resting on a prestressed half-plane

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the dynamical response of a system consisting of a prestressed covering layer and a prestressed half-plane to a moving load applied to the free face of the covering layer is investigated. Two types (complete and incomplete) of contact conditions on the interface are considered. The subsonic state is considered, and numerical results for the critical speed of the moving load are presented. The influence of problem parameters on the critical speed is analyzed. In particular, it is established that the prestressing of the covering layer and half-plane increases the critical speed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 2, pp. 257–270, March–April, 2007.     

Solutions of half-space and half-plane contact problems based on surface elasticity

Analytical solutions for the problems of an elastic half-space and an elastic half-plane subjected to a distributed normal force are derived in a unified manner using the general form of the linearized surface elasticity theory of Gurtin and Murdoch. The Papkovitch–Neuber potential functions, Fourier transforms and Bessel functions are utilized in the formulation. The newly obtained solutions are general and reduce to the solutions for the half-space and half-plane contact problems based on classical linear elasticity when the surface effects are not considered. Also, existing solutions for the half-space and half-plane contact problems based on simplified versions of Gurtin and Murdoch’s surface elasticity theory are recovered as special cases of the current solutions. By applying the new solutions directly, Boussinesq’s flat-ended punch problem, Hertz’s spherical punch problem and a conical punch problem are solved, which lead to depth-dependent hardness formulas different from those based on classical elasticity. The numerical results reveal that smoother elastic fields and smaller displacements are predicted by the current solutions than those given by the classical elasticity-based solutions. Also, it is shown that the out-of-plane displacement and stress components strongly depend on the residual surface stress. In addition, it is found that the new solutions based on the surface elasticity theory predict larger values of the indentation hardness than the solutions based on classical elasticity.     

On diffraction in a piezoelectric medium by a half-plane: The Sommerfeld problem

This paper is concerned with the diffraction problem in a transversely isotropic piezoelectric medium by a half-plane. The half-plane obstacle considered here is a semi-infinite slit, or a crack; both its surfaces are traction free and electric absorbent screens. In a generalized sense, we are dealing with the Sommerfeld problem in a piezoelectric medium.¶The coupled diffraction fields between acoustic wave and electric wave are excited by both incident acoustic wave as well as incident electric wave; and the sound soft and electric "blackness" conditions on the screens are characterized by a system of simultaneous Wiener-Hopf equations. Closed form solutions are sought by employing special techniques. Some interesting results have been obtained, such as mode conversions between acoustic wave and electric wave, novel diffraction patterns in the scattering fields, and the effect of electroacoustic head wave, as well as of surface wave-Bleustein-Gulyaev wave.¶Unlike the classical Sommerfeld problem, in which the only concern is the scattering field of electric wave, the strength of material, e.g. material toughness, is another concern here. From this perspective, relevant dynamic field intensity factors at the crack tip are derived explicitly.     

Elastodynamic analysis of a cracked orthotropic half-plane

The solution of elastodynamic volterra-type dislocation in an orthotropic half-plane is obtained by means of the Fourier transforms. The distributed dislocation technique is used to construct integral equations for an orthotropic half-plane weakened by cracks where the domain is under time-harmonic anti plane traction. These equations are of Cauchy singular type at the location of dislocation which is solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to determine stress intensity factors for multiple smooth cracks. Several examples are solved and the stress intensity factors for multiple cracks with different configuration are obtained.     

Connecting a population dynamic model with a multi-period location-allocation problem for post-disaster relief operations

In this study, we propose a mathematical model and heuristics for solving a multi-period location-allocation problem in post-disaster operations, which takes into account the impact of distribution over the population. Logistics restrictions such as human and financial resources are considered. In addition, a brief review on resilience system models is provided, as well as their connection with quantitative models for post-disaster relief operations. In particular, we highlight how one can improve resilience by means of OR/MS strategies. Then, a simpler resilience schema is proposed, which better reflects an active system for providing humanitarian aid in post-disaster operations, similar to the model focused in this work. The proposed model is non-linear and solved by a decomposition approach: the master level problem is addressed by a non-linear solver, while the slave subproblem is treated as a black-box coupling heuristics and a Variable Neighborhood Descent local search. Computational experiments have been done using several scenarios, and real data from Belo Horizonte city in Brazil.