Effective high-order approximations of layered coatings and linings of anisotropic elastic,viscoelastic and nematic materials

Beginning at low frequencies, asymptotically exact models of anisotropic coatings and linings with a small ratio of the half-thickness to the longitudinal deformation scale are constructed. The requirement on the conditions of contact with the substrate, where at least one of the boundary conditions must contain a displacement component of the strain in explicit form is “non-classical” here. The action of the coating/lining on a thicker body is approximated by impedance boundary conditions at the interface. The error of the model is reduced to the third order for layered packets and to the sixth order for a single layer. The physical limit of the applicability is the frequency of the first quasi-resonance in the corresponding deformed system. A comparison with the propagator matrix and numerical testing for partial waves shows satisfactory accuracy, comparable with the accuracy of the theory of classical plates of similar order. The results can be used in contact problems and for rapid algorithms for calculating the spectrum of the eigenwaves in half-spaces and thick layered plates with any number of coatings and linings. An extension to the case of viscoelastic materials and nematic elastomers is given.     

Calculation of rigidly clamped circular plates in the statement of the problem on contact interaction in a two-layer package

The use of a mechanicomathematical model of bending of thick transversely isotropic plates is illustrated, where the plate is divided in an arbitrary number of equally thick conditional layers. This model allows one to approximately reduce the problem of determination of stresses and displacements in the thick plate to a corresponding contact problem for a bent pack age of layers. The axisymmetric bending of a rigidly clamped package consisting of two plates rigidly fastened together is considered. The results of numerical calculations are presented, which are compared with those obtained within the framework of a refined bending model of plates (with account of transverse compression and shear) and of the Timoshenko model, as well as in the statement of the three-dimensional theory of elasticity. The accuracy of satisfying the boundary conditions in each model is analyzed. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 93–108, January–February, 2007.     

Behavior of surface waves at transition through a junction line on the boundary of an elastic homogenous isotropic body

A homogeneous elastic body with stress-free boundary is considered. The boundary of the body, which consists of a smooth cylindrical surface and a half-plane, has a continuous tangent plane, but the curvature of the normal section of the boundary has a discontinuity of the first kind at each point of the junction line. The behavior of two kinds of “whispering gallery” transversal surface waves at transition through the junction line is studied. For waves of the first kind (corresponding to Dirichlet boundary conditions), the displacement vector is normal to the boundary, whereas for waves of the second kind (corresponding to Neumann boundary conditions) the displacement vector is tangent to the boundary and normal to the ray, similarly to the case of Love waves. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 86–102. Translated by N. Ya. Kirpichnikova.     

en

The case of a linearly elastic plate with free boundary conditions on the lateral side is investigated as the half-thickness ɛ tends to zero. As in hard clamped plates, the generic leading term of the asymptotic expansion of the scaled displacement is a Kirchhoff-Love field with in-plane generating functions satisfying classical bending and membrane problems of Neumann type (compare with [1]). The first boundary layer profile is of bending type, so that in the case of a membrane load the convergence of the three-dimensional solution to the two-dimensional limit one is of improved accuracy. Conditions under which the asymptotic expansion ‘starts later’ are given and the structure of the first non-vanishing term is studied.     

Instability of thermoelastic interaction between a half-space and a rigid base through a thin liquid layer

We investigate the instability of thermoelastic interaction between elastic and rigid half-spaces through a liquid interlayer under the conditions of heat transfer across the interfaces. Due to the small thickness of the liquid layer, its influence on the temperature field is taken into account by the thermal resistance of the contact between the bodies, which depends on the normal displacement of the boundary of the elastic body. The pressure inside the liquid is equal to the external pressure applied to the bodies. We determined the critical value of the external heat flow for which the instability becomes possible in such a system and studied the dependence of this value on the parameters of the elastic half-space, the thickness of the liquid layer, and its thermal conduction. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 76–82, April–June, 1998.     

The contact of a solid with an elastic half-space through a thin coating

More accurate equations of the deformation of thin plates, which are more convenient for solving contact problems for bodies with coatings and containing, as a special case, the equations of all known applied theories, are derived by an asymptotic analysis of the first fundamental problem of the theory of elasticity. The equations of the deformation of thin-walled elastic bodies are classified, their qualitative correspondence to the equations of the theory of elasticity is clarified, and the forms of the features that arise along the shift lines of the boundary conditions in the corresponding contact problems are established. A criterion for selecting approximate models to describe the properties of the coatings depending on the geometrical and mechanical characteristics of the coating and the substrate and also on their degree of adhesion is given.     

Calculation of simply supported circular plates in the statement of the problem of contact interaction in a package of two plates

The possibility of applying the mechanicomathematical model of bending of a package of transversely isotropic plates to approximate calculations of thick plates is investigated. Within the frame work of this model, the problem of axisymmetric bending of a package of two identical plates with simply supported edges is considered. The conditions of rigid contact are given between the plates. Based on the analytical result obtained, the unknown distribution of stresses and displacements across the thickness of the plate is approximated by the distribution of corresponding parameters in the package of two plates. For an isotropic body, the results of numerical calculations are compared with those given by the 3D theory of elasticity. In the case of a transversely isotropic body, a comparison with the results found by the refined bending model of plates (taking into account the transverse compression and shear) and the Timoshenko model is carried out. The accuracy to which the boundary conditions in every model are satisfied is analyzed. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 5, pp. 603–616, September–October, 2007.     

A method of solving three-dimensional problems of elasticity for laminated composite plates

An efficient method of solving 3D elasticity problems for thick and thin laminated composite plates is presented. It is based on a new concept of reference surfaces inside the plate. According to this concept, into each nth layer, In arbitrary reference surfaces parallel to the midsurface are introduced, and the displacement vectors of these surfaces are chosen as unknown functions. Such a choice allows one to represent the governing equations of the high-order theory of plates proposed in a very compact form and to derive strain–displacement relationships correctly describing all rigid-body motions of laminated plates.     

A continuous model for investigation of physically nonlinear deformation of orthotropic sandwich plates

A phenomenological yield condition for quasi-brittle and plastic orthotropic materials with initial stresses is suggested. All components of the yield tensor are determined from experiments on uniaxial loading. The reliability estimates of the criterion suggested is discussed. For a plastic material without initial stresses, the given condition transforms into the Marin—Hu criterion. The defining equations of the deformation theory of plasticity with isotropic and “anisotropic” hardening, associated with the yield condition suggested, are obtained. These equations are used as the basis for a highly accurate nonclassical continuous model for nonlinear deformation of thick sandwich plates. The approximations with respect to the transverse coordinate take into account the flexural and nonflexural deformations in transverse shear and compression. The high-order approximations allow us to model the occurrence of layer delamination cracks by introducing thin nonrigid interlayers without violating the continuity concept of the theory. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. pp. 329–340, May–June, 2000.     

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.     

The classical and generalized love problems in a domain of low-frequency interference waves of signal type. I

Low-frequency interference waves propagating through a thin elastic layer, which is in rigid contact with an elastic half-space, are considered. A new method for computing the fields of low-frequency waves suggested by G. I. Petrashen allows us to compute the fields of these waves and to draw some conclusions concerning their behavior. In particular, we may conclude that the thin upper layer significantly distorts a signal sent deep into a medium. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1996, pp. 5–39. Translated by N. S. Zabavnikova.     

Notes to the proof of a weighted Korn inequality for an elastic body with peak-shaped cusps

We present the proof of the weighted anisotropic Korn inequality in a three-dimensional domain with peak-shaped cusps on the boundary. We verify the asymptotic accuracy of distribution of multipliers at the components of displacement vector and their derivatives in the corresponding weighted norm. We indicate conditions on a peak cusp under which the natural energy class is not embedded into a Sobolev or Lebesgue class. In the last case, the operator of elasticity problems possesses the continuous spectrum provoking wave processes in a finite volume (“black holes” for elastic waves). We also discuss possible generalizations of the result and open questions. Bibliography: 39 titles. Illustrations: 9 figures.     

Thermoelastic contact during pressing of a heated elliptical cross-section die into a transversely isotropic piezoceramic half space

The three-dimensional thermoelastic contact as a nonplanar elliptical cross-section die is pressed into a transversely isotropic half space is examined under conditions such that the die has a previously specified temperature while the boundary of the half space outside the die is kept at zero temperature. Contact between the die and the half space is assumed to be ideal. The stressed state in the elastic medium is characterized by potential functions which are constructed explicity. Expressions are given for determining the translational displacement of points on the base of the stamp as well as the major half axis and eccentricity of the contact area. Some possible forms of boundary conditions for the electrical variables at the contact area are examined. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 37–53, 1999.     

The solution of the Hertz axisymmetric contact problem

The main terms of the asymptotic form of the solution of the contact problem of the compression without friction of an elastic body and a punch initially in point contact are constructed by the method of matched asymptotic expansions using an improved matching procedure. The condition of unilateral contact is formulated taking account of tangential displacements on the contact surface. An asymptotic solution of the problem for the boundary layer is constructed by the complex potential method. An asymptotic model is constructed, extending the Hertz theory to the case where the surfaces of the punch and elastic body in the vicinity of the contact area are approximated by paraboloids of revolution. The problem of determining the convergence of the contacting bodies from the magnitude of the compressive force is reduced to the problem of calculating the so-called coefficient of local compliance, which is an integral characteristic of the geometry of the elastic body and its fixing conditions.     

Homogenization of elliptic problems with alternating boundary conditions in a thick two-level junction of type 3:2:2

The asymptotic behavior of solutions to boundary value problems for the Poisson equation is studied in a thick two-level junction of type 3:2:2 with alternating boundary conditions. The thick junction consists of a cylinder with ε-periodically stringed thin disks of variable thickness. The disks are divided into two classes depending on their geometric structure and boundary conditions. We consider problems with alternating Dirichlet and Neumann boundary conditions and also problems with different alternating Fourier (Neumann) conditions. We study the influence of the boundary conditions on the asymptotic behavior of solutions as ε → 0. Convergence theorems, in particular, convergence of energy integrals, are proved. Bibliography: 31 titles. Illustrations: 1 figure.     

The indentation of a punch in the form of an elliptic paraboloid into the plane boundary of an elastic body

The three-dimensional contact problem for an elastic body of arbitrary geometry with a single plane face, into which a punch in the shape of an elliptic paraboloid is indented, is considered. The curvilinear boundary of the body is partially clamped, and the remaining boundary (outside the contact region) is stress-free. It is assumed that the dimensions of the contact area are small compared with the characteristic dimension of the body. Using the method of matched asymptotic expansions a model problem of unilateral contact without friction is derived for the boundary layer, which is solved using the apparatus of Hertz's theory. Asymptotic models of the contact interaction of different degrees of accuracy are constructed, including corrections to the geometry and clamping conditions of the elastic body. The sensitivity of the parameters of the elliptic region of the contact to these factors is investigated.     

On the propagation of Lamb waves in a sandwich plate made of compressible materials with finite initial strains

The propagation of flexural Lamb waves in a prestrained sandwich plate made from compressible highly elastic materials is investigated within the scope of a piecewise homogeneous body model by utilizing TLTEWISB. The mechanical relations of layer materials are described by a harmonic-type potential, and numerical results are obtained for the first and second vibration modes. According to the results, the influence of problem parameters and of the initial stretching strain along the layers on the wave propagation speed is examined. The asymptotic values of the speed are considered in the cases of short and long wavelengths, and the influence of the initial strains on these asymptotic (limit) values are also analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 231–244, March–April, 2008.     

Reflection and transmission of elastic waves in the multilayered orthotropic couple-stressed plates sandwiched between two elastic half-spaces

In this paper, a high efficient computational approach, the extended Legendre orthogonal polynomial method (LOPM) is provided to investigate the reflection and transmission of elastic waves in orthotropic couple-stress layered plates sandwiched between two elastic half-spaces. In this approach, the anisotropic couple-stress theory is introduced into the LOPM to calculate the reflection and transmission coefficients of the orthotropic interlayers. The stress components, couple-stress components, rotation vectors and governing equations are derived in terms of the Legendre orthogonal polynomial. The present method does not require calculations of the displacement solutions of each partial wave in anisotropic multilayered microstructures, but expands the displacement vector of each layer into a Legendre orthogonal polynomial series with the expansion coefficients to be calculated. The incident P wave and SH wave are calculated, respectively. The effects of length scale parameters in three different directions are studied. It is found that the reflection and transmission coefficients of the incident P wave are only related to the length scale parameter in the z direction. For incident SH wave, the influence of the length scale parameter along the thickness direction is much more significant than that of the length scale parameter in the y direction.     

Propagation of Rayleigh waves of sv type in transversely isotropic elastic media

The asymptotics of high-frequency surface waves in elastic media is studied for a special case of anisotropy, namely, for transversely isotropic media (where the parameters of elasticity are invariant with respect to rotations about one of the coordinate axes). In the zeroth asymptotic approximation, the slow Rayleigh waves (of SV type) under study are polarized in the plane of the normal section of the surface. The principal term of the asymptotics (which has the form of a space-time (caustic) expansion) is found, and calculations related to the necessity of introducing two additional faster waves with complex eikonals are carried out. The conditions on the elasticity parameters of the medium that insure the origination of the surface waves in question are obtained. Due to the specific structure of the elasticity tensor under consideration, the boundary of the medium is necesarily plane. For appropriate values of elastic parameters, the resulting formulas coincide with the corresponding expressions in the isotropic case. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 278–293. Translated by Z. A. Yanson