Theory Of Elasticity For Bodies Research Articles

Asymptotic Approach in the Dynamic Problems of the Theory of Elasticity for Bodies with Thin Elastic Inclusions

Asymptotic Approach in the Dynamic Problems of the Theory of Elasticity for Bodies with Thin Elastic Inclusions

Asymptotic approach in dynamic problems of the elasticity theory for bodies with thin elastic inclusions

Asymptotic approach in dynamic problems of the elasticity theory for bodies with thin elastic inclusions

The problem of infinite plate loaded with normal force following a complex trajectory*

Introduction. A method for solving the problem of an infinite plate on an elastic foundation is proposed. The plate is affected by a periodic load in the form of a force following an arbitrary closed path. The work objective is to develop a numerical method for solving problems of the elasticity theory for bodies under a moving load. Materials andMethods. Given the periodicity of the load under consideration, it is decomposed in a Fourier series in a time interval whose length is equal to the load period. The solution to the original problem is constructed by superposition of the solutions to the problems corresponding to the load specified by the terms of the Fourier series described above. The final solution to the problem is presented as a segment of a series. In this case, each term corresponds to the solution of the problem of the impact on an infinite plate of a load distributed along a closed curve (the trajectory of the force motion). To find these solutions, the fundamental solution to the equation of vibration of an infinite plate lying on an elastic base is used.Research Results. A new method is proposed for solving problems on the elasticity theory for bodies with a load following a closed path of arbitrary shape. The problem of an infinite plane along which a concentrated force moves at a constant speed is solved. It is determined that the trajectory of motion is a smooth closed curve consisting of circular arcs. The behavior of displacements and stresses near a moving force is considered. The energy propagation of the elastic waves is studied. For this purpose, the coordinates of the Umov – Poynting vector are calculated. The effect of the force motion speed on the length of the Umov – Poynting vector is investigated.Discussion and Conclusions. The method is applicable when considering more complex objects (plates of complex shape, layered plates, viscoelastic plates). Its advantage is profitability since the known problem solutions are used to build the solution. The final decision is expressed in a convenient form – as the sum of curvilinear integrals. The results obtained can be used in the road design process. Studying the energy propagation of elastic waves from moving vehicles will enable to evaluate the impact of these waves on buildings near the road. The wear of the pavement is estimated considering data on the behavior of displacements and stresses

О сведении уравнений Максвелла в волноводах к системесвязанных уравнений Гельмгольца

The investigation of the electromagnetic field in a regular homogeneous waveguide reducesto the investigation of two independent boundary value problems for the Helmholtz equation,corresponding to TE- and TM-modes. In the case of an inhomogeneous waveguide TE- andTM-modes are connected to each other, which in numerical experiments can not always be fullytaken into account. In this paper we show how to rewrite the Helmholtz equations in vectorform to express this relationship explicitly.In the article the cylindrical waveguide with perfectly conducting walls is considered, but wedon’t make any assumptions about filling of waveguide. The introduced approach is based ontwo-dimensional analogue of the theorem known in the theory of elastic bodies as the Helmholtzdecomposition. On its basis, we introduce four potentials, instead of two potentials, usuallyused in the theory of hollow waveguides. It is proved that any solution of Maxwell’s equationsin a waveguide that satisfies the boundary conditions of ideal conductivity on the boundariesof a waveguide can be represented with the help of these potentials. The system of Maxwell’sequations is written with respect to these potentials and it is shown that this system has theform of two independent Helmholtz equations in the case of a hollow waveguide.

The solution of a certain class of dual integral equations with the right-hand side in the form of a Fourier series and its application to the solution of contact problems for inhomogeneous media

Abstract Using the bilateral asymptotic method, a semi-analytical solution of a dual integral equation with its right-hand side in the form of a Fourier series is constructed. This equation arises in the solution of a number of contact problems of elasticity theory for bodies with inhomogeneous coatings. The efficiency of the method is illustrated in the example of the solution of the plane contact problem on bending of a beam lying on a functionally graded strip with arbitrary variation of the elastic moduli with depth. It is assumed that the strip is perfectly bonded to an elastic half-plane. Numerical results are presented for a strip whose Young's modulus varies harmonically with depth. In this case, Young's modulus of the substrate is 100 times greater than at the lower boundary of the coating.

A Result regarding the Seismic Dislocations in Microstretch Thermoelastic Bodies

The aim of our study is to derive a relation of De Hoop-Knopoff type for displacement fields within context of thermoelastic microstretch bodies. Then, as a consequence, an explicit expression of the body loadings equivalent to a seismic dislocation is obtained. The results are extensions of those from the classical theory of elastic bodies.

Solutions of some boundary value problems for a new class of elastic bodies undergoing small strains. Comparison with the predictions of the classical theory of linearized elasticity: Part I. Problems with cylindrical symmetry

There is considerable evidence that shows that for a large class of materials the relationship between the stress and the strain is nonlinear even in the range of strain that is considered small enough for the classical linearized theory of elasticity to be applicable (see Saito et al. in Science 300:464–467, 2003; Li et al. in Phys Rev Lett 98:105503, 2007; Talling et al. in Scr Mater 59:669–672, 2008; Withey et al. in Mater Sci Eng A 493:26–32, 2008; Zhang et al. in Scr Mater 60:733–736, 2009). A proper description of the experiments requires an alternative theory which when linearized would allow the possibility of such a nonlinear relationship between the stress and the strain. Recently, such a theory of elastic bodies has been put into place (see Rajagopal in Appl Math 48:279–319, 2003; Bustamante in Proc R Soc A 465:1377–1392, 2009; Rajagopal in Math Mech Solids 16:536–562, 2011). In this paper, we consider a special class of bodies that belong to the new generalization of response relations for elastic bodies that have a nonlinear relationship between the linearized strain and the stress. We use the special class of bodies that exhibit limited small strain to study two boundary value problems, the first concerning the telescopic shearing and inflation of a tube and the second being the extension, inflation and circumferential shearing of a tube. The results that we obtain for the models under consideration are markedly different from the predictions of the classical linearized elastic model with regard to the same boundary value problems.

The Shape Derivative of the Energy Integral in Elasticity Theory for Bodies with Rigid Inclusions and Cracks

We study the mixed boundary value problem for equations of two-diemensional theory of elasticity in a domain with a cut with unilateral constraints for the solutions on the cut. It is assumed that the solution belongs to the space of rigid displacements in a part of the domain, i.e., has a given structure. Under rather general sufficiently smooth perturbations of the domain, we obtain an expression for the derivative of the energy functional with respect to the perturbation parameter.

Invariant integrals in a planar problem of elasticity theory for bodies with rigid inclusions and cracks

The article addresses a planar problem of elasticity theory for a body containing a rigid inclusion and a crack at the interface between the elastic matrix and the rigid inclusion. We show that the problem admits J- and M-invariant integrals. In particular, we construct an invariant integral of the Cherepanov-Rice type for rectilinear cracks.

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.

Service life and fracture of solid bodies under the conditions of cyclic contact interaction

A numerical-analytic model is formulated for the investigation of fracture processes and evaluation of the residual service life of solid bodies under the conditions of cyclic contact interaction. The model is based on the criteria of fracture of materials in the process of growth of fatigue cracks and the algorithms of step-by-step construction of the paths crack propagation developed by the method of singular integral equations for two-dimensional contact problems of the theory of elasticity for bodies with curvilinear cracks. The model is realized for contact interactions of rolling and fretting fatigue. The causes and specific features of the formation of typical defects in rolling bodies of revolution, such as pitting, spalling, squat defects, and cracking are analyzed depending on the operating parameters of a rolling couple. It is shown that, both in the case of rolling and under the conditions of fretting fatigue, the character of fracture of the contact surface and the sizes of debris particles are strongly affected by the value of the friction coefficient, the distribution of tangential forces between contacting bodies, the characteristics of cyclic crack-growth resistance of the material, and the service or random products in the contact zone (water, lubricants, powder fretting products, etc.). We present examples of evaluation of the residual service life of rail (RSB12) and roller (9KhF and GH) steels.

The determination of the configuration of a crack in an anisotropic medium

Inverse geometric problems of the anisotropic theory of elasticity for bodies with cracks of arbitrary configuration are investigated using a linear approach. Questions of the uniqueness of solution of the inverse problems that arise are studied, and effective schemes are constructed for solving them by a combination of the boundary element method and a regularized iteration procedure. The example of the reconstruction of a rectilinear crack in an orthotropic layer is examined.

Longitudinal Shear of an Infinite Body with a System of Polygonal Cracks

On the basis of singular integral equations, we propose a new method for the solution of antiplane problems of elasticity theory for bodies with a system of polygonal cracks with regard for the singularities of stresses at the corner points. The modified singular integral equations have continuous regular kernels and right-hand sides, i.e., belong to the same type as in the case of smooth boundary contours. A numerical solution of these equations is found by the method of mechanical quadratures. The values of the stress intensity factors at the corner points and tips of both a three-link polygonal crack and a system of two two-link polygonal cracks in an infinite body are presented. The limiting case of two cracks which form a rhombic hole is also analyzed.

The method of potential functions in problems of the theory of elasticity for bodies with a defect

The method of potential functions using a Fourier transformation in the class of slowly increasing distributions, corresponding to the classical method of complex potentials, is proposed for solving well-known problems of the theory of elasticity for bodies with a defect. It is shown that when a Fourier transformation with respect to all the spatial variables is used, the solution of the dynamic problem of the theory of elasticity can also be represented in terms of a jump in the stresses and displacements at the defect. The correctness of the transformed problem is considered (in terms of an analogue of the Lopatinskii condition). The solution of the system of Helmholtz equations, to which the system of Lamé equations is reduced in the case of the two-dimensional dynamic problem, is expressed in terms of the jump in the stresses and displacements at the defect as a result of solving the corresponding singular integral equations.

Crack Growth in Rolling Bodies under the Conditions of Dry Friction and Wetting

Within the framework of a two-dimensional mathematical model, we study the development of a surface macrocrack in the course of unidirectional rolling under the conditions of dry friction and wetting. The damaged body in a rolling couple is modeled by an elastic half plane with an edge cut (crack) and the action of the counterbody is simulated by the translational motion of Hertzian contact forces along the boundary of the half plane with taking into account the friction forces. We compute the crack-growth paths depending on the coefficient of contact friction between the bodies and the orientation and length of the crack. The numerical calculations are performed by the step-by-step method based on the use of the singular integral equations of the theory of elasticity for bodies with curvilinear cracks and the criterion of generalized opening. It is shown that, for high coefficients of friction between the bodies of the rolling couple (f = 0.20–40; dry friction), a branch propagates from the original fatigue shear macrocrack into the bulk of the material of the driven body. In the case of low friction coefficients (f < 0.20; wetting), a branch mainly propagates along the boundary of the body in the direction of motion of the counterbody. We propose an explanation of the mechanism of development of burrow-type defects often encountered in railroad rails.

Estimates for Errors in Two-Dimensional Models of Elasticity Theory

Explicit (computable) estimates for errors in some simplified two-dimensional models of the theory of elastic body are obtained. Such models are usually used if one of the characteristic sizes of a domain is significantly less than two others. In particular, the model of plane stresses is considered. To obtain estimates for the deviation of reduced solutions from exact solutions, the direct and dual variational problems are considered and the duality methods are used. Bibliography: 9 titles.

Solution of three-dimensional problems of the dynamic theory of elasticity for bodies with cracks using hypersingular integrals

Expressions are derived for the singular kernels of a system of integral equations that describe a three-dimensional dynamic problem for a cracked elastic body under harmonic loading. The order of singularity of the kernels of the integral equations is analyzed. Relations that allow reducing hypersingular surface integrals to ordinary contour ones are obtained. Test calculations are conducted.

Equations for a piezoelectric parallelepiped and applications in a gyroscope

Zero-dimensional equations for motions of a rectangular piezoelectric parallelepiped are derived from the three- dimensional equations of linear piezoelectricity by triple power series expansion in the length, width and thickness directions. The resulting equations are ordinary differential equations with time as the only independent variable. The lowest order equa- tions can describe motions with homogeneous shearing and stretching deformations as well as uniform electric fields. It is a generalization of the zero-dimensional theory of elastic bodies in the sense that electric coupling is included. The equations obtained are employed in the analysis of a ceramic plate thickness-shear piezoelectric gyroscope. Results useful to the under- standing and design of the gyroscope are obtained.

A Nonlinear Beam Equation Arising in the Theory of Elastic Bodies

A Nonlinear Beam Equation Arising in the Theory of Elastic Bodies

Cracks and cracklike defects in the preliminarily stressed body

In the present work, we study cracklike defects of various types within the framework of the linearized theory of elasticity for bodies with initial stresses suggested by Guz'. Our approach is based on the solution of the problem of the plane region of discontinuity of displacements and stresses in preliminarily stressed bodies. By using the method of Fourier integral transformations, we construct singular integral equations for cracks with arbitrarily loaded lips and for rigid and elastic inclusions. The exact solutions of the corresponding integral equations are obtained for various types of loading. As a result, we determine the influence of preliminary deformation along plane defects on the stressed state in their vicinity.