Near-surface buckling in stability of a system consisting of a moderately rigid substrate, a viscoelastic bond layer, and an elastic covering layer

Within the frame work of the three-dimensional linearized theory of stability of deformable bodies (TLTSDB), the near-surface buckling instability of a system consisting of a half-plane (substrate), a viscoelastic bond layer, and an elastic covering layer is suggested. The equations of the TLTSDB are obtained from the three-dimensional geometrically non linear equations of viscoelasticity theory by using the boundary-form perturbation technique. By employing the Laplace transform, a method for solving the problem is developed. It is supposed that the covering layer has an insignificant initial imperfection. The stability of the system is considered lost if the imperfection starts to increase and grows indefinitely. Numerical results for the critical compressive force and the critical time are presented. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 4, pp. 517–530, July–August, 2006.     

Near-surface failure of layered viscoelastic materials

Within the framework of a piecewise homogeneous body model, with the use of exact equations of the geometrically nonlinear theory of viscoelastic bodies, the distribution of near-surface self-balanced normal stresses in a body consisting of a viscoelastic half-plane, an elastic locally curved bond layer, and a viscoelastic covering layer is investigated. A method for solving the problem considered by employing the Laplace and Fourier transformations is developed. Numerical results for the self-balanced normal stresses caused by a local curving (imperfection) of an elastic bond layer upon tension and compression of the body mentioned along the free face plane are presented and analyzed. The viscoelasticity of the materials is described by the Rabotnov fractional-exponential operators. A macroscopic failure criterion is proposed, and the validity of this criterion is examined.     

3D Analyses of the symmetric local stability loss of the circular hollow cylinder made from viscoelastic composite material

The 3D approach was employed for investigations of the symmetric local stability loss of the circular hollow cylinder made from the viscoelastic composite materials. This approach is based on investigations of the development of the initial rotationally symmetric infinitesimal local imperfections of the circular hollow cylinder within the scope of 3D geometrically nonlinear field equations of the theory of viscoelasticity for anisotropic bodies. The numerical results of the critical force and critical time are presented and discussed. For comparison and estimation of the accuracy of the results given by the 3D approach, the same problem is also solved by using various approximate shell theories. The viscoelasticity properties of the plate material are described by the fractional–exponential operator. The numerical results and their discussion are presented for the case where the cylinder is made of a uni-directional fibrous viscoelastic composite material. In particular, it is established that the difference between the critical times obtained by employing 3D and third order refined shell theories becomes more non-negligible if the values of the external compressive force are close to the critical compressive force which is obtained at t = ∞ (t denotes a time).     

Local buckling of compressed elements of layered viscoelastic material

The problem of local surface buckling of compressed elements of layered material is solved using the equations of the theory of multilayer plates [1, 2, 5]. The local buckling deformations are described by means of the solutions of differential-difference equations damping into the interior of the medium. The effect of compressive forces, buckling wavelength, and the elastic and viscoelastic constants of the material on the behavior of the deformations and on the thickness of the layer to which significant buckling extends is investigated. Numerical estimates are presented for the buckling parameters of compressed laminated-plastic elements.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 816–821, September–October, 1968.     

FEM Analysis of the Three-Dimensional Buckling Problem for a Clamped Thick Rectangular Plate Made of a Viscoelastic Composite

The buckling instability of a thick rectangular plate made of a viscoelastic composite material is studied. The investigation is carried out within the framework of the three-dimensional linearized theory of stability. The plate edges are clamped and the plate is compressed through the clamps. Moreover, it is assumed that the plate has an initial infinitesimal imperfection, and, as a buckling criterion, the state is taken where this imperfection starts to increase indefinitely at fixed finite values of external compressive forces. From this criterion, the critical time is determined. The corresponding boundary-value problems are solved by employing the three-dimensional FEM and the Laplace transform. The material of the plate is assumed orthotropic, viscoelastic, and homogeneous. Numerical results related to the critical time are presented.     

Fiber Buckling in a Viscoelastic Matrix

Within the framework of the three-dimensional linearized theory of stability, an approach for investigating fiber buckling in the structure of unidirectional fibrous viscoelastic composites is developed. For simplicity, a small fiber concentration is considered, and the buckling problem for a single elastic fiber in an infinite viscoelastic matrix is investigated. In this case, it is assumed that the fiber has an insignificant initial periodical imperfection, and the growth of this imperfection with time is studied. The state where this imperfection starts to grow indefinitely is taken as a fiber-buckling criterion, and the critical time is determined from this criterion.     

Delamination buckling of a rectangular orthotropic composite plate containing a band crack

The delamination buckling problem for a rectangular plate made of an orthotropic composite material is studied. The plate contains a band crack whose faces have an initial infinitesimal imperfection. The subsequent development of this imperfection due to an external compressive load acting along the crack is studied through the use of the three-dimensional geometrically nonlinear field equations of elasticity theory for anisotropic bodies. A criterion of initial imperfection is used in determining the critical forces. The corresponding boundary-value problems are solved by employing the boundary-form perturbation technique and the FEM. Numerical results for the critical force are presented.     

Dynamics of elastic bodies connected by a thin soft viscoelastic layer

A dynamic study was performed on a structure consisting of two three-dimensional linearly elastic bodies connected by a thin soft nonlinear Kelvin–Voigt viscoelastic adhesive layer. The adhesive is assumed to be viscoelastic of Kelvin–Voigt generalized type, which makes it possible to deal with a relatively wide range of physical behavior by choosing suitable dissipation potentials. In the static and purely elastic case, convergence results when geometrical and mechanical parameters tend to zero have already been obtained using variational convergence methods. To obtain convergence results in the dynamic case, the main tool, as in the quasistatic case, is a nonlinear version of Trotter?s theory of approximation of semigroups acting on variable Hilbert spaces. The limit problem involves a mechanical constraint imposed along the surface to which the layer shrinks. The meaning of this limit with respect to the relative behavior of the parameters is discussed. The problem applies in particular to wave phenomena in bonded domains.     

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.     

On the theoretical strength limit of the layered elastic and viscoelastic composites in compression

The theoretical strength limit in compression (TSLC) of composite material with two alternating isotropic homogeneous elastic and viscoelastic layers is studied. The investigation is made within the piecewise-homogeneous body model with the use of the Three-Dimensional Linearized Theory of Stability for elastic and viscoelastic materials. It is assumed that there is an initial imperfection in the local curving form of the reinforcing layers along the direction which is inclined to the layers. As the criterion for determination of the TSLC, the case is taken for which the aforementioned imperfections start to increase indefinitely. The numerical results for the TSLC are presented and discussed. According to these numerical results, in particular, the direction (or the surface) along which the fracture in compression takes place, is determined.     

On the stability of inhomogeneously viscoelastic reinforced bars

There is investigated the stability of inhomogeneously ageing reinforced viscoelastic bars. It is assumed that the strains and stresses in the reinforcement are related by Hooke's law. The properties of the matrix material are described by equations of the theory of viscoelasticity of inhomogeneously ageing solids /1,2/. Under different boundary conditions for the ends of the bar and loading methods an expression is set up for the critical force in stability problems in an infinite time interval. The stability definition taken corresponds to the Liapunov stability definition for the motion of dynamical systems. Estimates of the critical time when the magnitude of the deflection of a viscoelastic bar reaches a given value are obtained for stability problems in a finite time interval. The formulation for the stability problem in a finite time interval starts from the definition of stability of motion of dynamical systems by taking its beginning from the Chetaev work. The dependence of the critical time on the inhomogeneity and the reinforcing parameter is investigated numerically. The stability of viscoelastic unreinforced bars was studied in /3,4/, A survey and bibliography of research associated with the stability problem for viscoelastic bars are available in /5–8/.     

Buckling delamination of a sandwich plate-strip with piezoelectric face and elastic core layers

In this study, the buckling delamination problem of a sandwich plate-strip with a piezoelectric face and elastic core layers is studied. It is assumed that the plate-strip is simply supported and grounded along its two parallel ends and is subjected to uniformly-distributed compressive forces on these ends. Moreover, we suppose that the plate-strip has two interface inner cracks between the face and the core layers and it is also supposed that before the plate-strip is loaded (i.e. in the natural state), the surfaces of these cracks have insignificant initial imperfections. Due to compressive forces acting along the cracks we investigate the evolution of the initial imperfections of the cracks’ surfaces. Hence, the values of the critical buckling delamination force of the considered plate-strip are determined from the criteria, according to which, the considered initial imperfections of the cracks’ surfaces grow indefinitely by the compressive forces. Mathematical modeling of the considered problem is formulated within the scope of the exact nonlinear equations of electro-elasticity in the framework of the piecewise homogeneous body model, the solution of which is found numerically by employing the finite elements method. Numerical results showing the influences of the geometrical and material parameters as well as the coupling of the electrical and mechanical fields on the values of the critical force are presented and analyzed.     

Limiting state of a multilayered nonlinearly elastic long cylindrical shell under the action of nonuniform external pressure

The buckling of a long multilayered nonlinearly elastic shell made of different materials and subject to the action of external pressure is investigated. The load is not hydrostatic and greatly varies in value and direction. Neglecting the effect of end fastening of the shell, the problem is reduced to an analysis of the loss of load-carrying ability of a ring of unit width separated from the shell. The solution is based on a variational method of mixed type formulated for heterogeneous nonlinearly elastic bodies, taking into account the geometrical nonlinearity, in a combination with the Rayleigh–Ritz method. The initial analysis is reduced to solving the Cauchy problem for a nonlinear ordinary differential equation resolved for the derivative. Numerically, using the Runge–Kutta method, the effect of the number of layers and of the parameter of nonuniformity of the external pressure on the critical buckling force is revealed. The urgency and importance of the problem are connected with the research of reserves in the saving of materials with a simultaneous possibility of increasing the load-carrying ability of a structure.     

Calculation of the nonlinear viscoelastic oscillations of an antivibration layer

It is proposed to use a viscoelastic layer to protect equipment against vibration. The principal quadratic theory of hereditary viscoelasticity is used as the physical relation between the forces and displacements. The solutions obtained for the integrodifferential vibration equation make it possible to minimize the displacements and accelerations of the protected equipment.Moscow Institute of Electronic-Machine Building. Translated from Mekhanika Polimerov, No. 2, pp. 321–326, March–April, 1972.     

Formation of a transition layer on the fillers of polymer composites

Based on a plane model of composites, the effect of a transition layer on the elastic modulus E_c of the composites is analyzed in the case where, under the action of a load, the transition layer is formed both on the side of matrix and filler. In evaluating E_c, it is assumed that the elastic modulus in the layer grows linearly from the elastic modulus of matrix to that of filler, but pores in the filler are impermeable to matrix macromolecules. Analytic relation ships are found which allow one to determine the volume fractions of the transition layer on the side of matrix and filler if the experimental elastic modulus of the composite is known. These relationships are used to find the magnitude of the layer in epoxy composites with various fillers and to evaluate its effect on the compressive elastic modulus of the composites. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 693–700, September–October, 2006.     

Non-linear near-resonance oscillations of an elastic incompressible layer

Plane one-dimensional waves of small amplitude, propagating transverse to an incompressible elastic layer and reflected successively from its boundaries, are considered. The oscillations are caused by small periodic (or close to periodic) external action on one of the layer boundaries, when the period of the external action is close to the period of natural oscillations of the layer. One of the boundaries of the elastic layer is fixed, while the other performs small specified two-dimensional motion in its plane. In such a near-resonance situation, non-linear effects occur which may build up over time. A system of equations is obtained which describes the slow change in the functions characterizing the oscillations of the medium in each period of the external action. It is assumed that all the quantities depend both on real time, any change of which in the approach considered is limited to one period, and on “slow” time, for which one period of real time serves as a small quantity. It is assumed that the evolution of the solution occurs when the slow time changes, while the role of real time is similar to the role of a spatial variable. This system of equations is obtained by the method of averaging over a period of the quantities representing nonlinear terms and the effect of the boundary conditions in the equations. It contains derivatives with respect to the real and slow times and also values of the functions characterizing the solution averaged over a period of the real time. If the averaged values are known, the equations have a hyperbolic form and their solutions can be both continuous and contain weak and strong discontinuities.     

Deformation of a composite plate on an elastic foundation by local loads

Bending of an elastic annular composite plate with a light filler lying on an elastic foundation is considered. The plate is subjected to local loads. To describe the kinematics of the package, asymmetric across its thickness, the hypotheses of broken normal is accepted. The reaction of foundation is described based on the Winkler model. A system of equilibrium equations is constructed, and its exact solution in displacements is found. Numerical solutions for a metal-polymer sandwich plate are presented. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 109–120, January–February, 2007.     

Propagation of waves in a micropolar elastic layer with stretch immersed in an infinite liquid

The effect of liquid on the propagation of waves in a micropolar elastic layer with stretch has been investigated. The frequency and wave velocity equations for symmetric and antisymmetric vibrations are derived. Propagation of monochromatic waves in a micropolar elastic layer with stretch is discussed. Results of this analysis reduce to those without stretch.     

Wave fields in a thin layer of an (ideal) liquid covering an elastic half-space (the simplest model of a shelf)

The head interference wave associated with the propagation of the P-wave in an elastic half-space is studied by using as an example the propagation of pressure waves in a liquid layer covering an elastic half-space. The attenuation of such a wave with respect to the distance between a source and a receiver is smaller than that in the classical theory. The wave field is considered both in time and frequency domains. The stationary wave field of the head interference wave is of resonance nature. From the mathematical point of view, the resonance peaks occur when the roots of the dispersion equation pass through a branch point. The minimal attenuation of the stationary wave field is observed in a neighborhood of such resonance peaks. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1996, pp. 40–61. Translated by N. S. Zabavnikova.     

A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.