On the three-dimensional undulation instability of a rectangular viscoelastic composite plate in biaxial compression

Within the frame work of the second version of small precritical deformation in the three-dimensional linearized theory of stability of deformable bodies (TDLTSDB), the undulation instability problem for a simply supported rectangular plate made of a viscoelastic composite material is investigated in biaxial compression in the plate plane. The corresponding boundary-value problem is solved by employing the Laplace transformation and the principle of correspondence. For comparison and estimation of the accuracy of results given by the TDLTSDB, the same problem is also solved by using various approximate plate theories. The viscoelasticity properties of the plate material are described by the Rabotnov fractional-exponential operator. The numerical results and their discussion are presented for the case where the plate is made of a multilayered viscoelastic composite material. In particular, the variation range of problem parameters is established for which it is necessary to investigate the undulation instability of the viscoelastic composite plate by using the TDLTSDB. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 93–108, January–February, 2009.     

Plane problems of stability of composite materials with a finite-size filler

The plane problem of three-dimensional stability is solved for a transversely compressed composite material reinforced with ribbons taking into account the inhomogeneous initial state. An approximate solution of the problems is based on the net method. The effect of the ribbon form factor, the ratio between the elastic moduli of the matrix and filler, and Poisson ratio of the filler on the critical deformation of the material is investigated. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 77–86, January–February, 2000.     

3Development of methods of solving nonlinear boundary problems of stability of viscoelastic thin-walled elements

A new numerical method of solving nonlinear boundary problems based on collocational-type approximations is developed with examples of one- and two-dimensional stability problems for a viscoelastic shell. The method allows us to simplify significantly the arithmetization procedure of differential expressions with subsequent efficient computer implementation of the algorithm. A high degree of calculation accuracy is provided. The estimation of the efficiency of the new numerical scheme is obtained by comparing the results with data where the system was modeled using the method of finite differences and the Galerkin method.Rostov State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 1, pp. 51–58, January–February, 1999.     

Spatial problems of stability theory of stratified,compressible, composite materials

We consider spatial, non-axially-symmetric problems of stability theory of stratified, compressible, composite materials for biaxial uniform compression with the use of three-dimensional, linearized stability theory of deformed bodies. It is shown in general form for arbitrary models of compressible elastic and elastoplastic bodies that the characteristic equation corresponding to spatial non-axially-symmetric problems reduces to the characteristic equation corresponding to planar problems of single-axis compression. In this context all numerical results obtained earlier for the planar problems mentioned are also equally valid for spatial, non-axially-symmetric problems with the corresponding notations for the wave formation parameters.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 24–32, 1988.     

Scientific fundamentals and problems of manufacturing mechanics of structures made of composite materials

Conclusions A number of fundamental investigations have been carried out and the results obtained have been used to solve a large number of applied problems comprising on the whole a clearly defined direction which, by analogy with engineering mechanics, may be referred to as processing mechanics. Since the applied importance of processing mechanics for engineering is expected to increase rapidly, special attention must be paid to carrying out investigations, introducing the results into production, and training of personnel capable of solving the given problems.Translated from Mekhanika Kompozitnykh Materialov, No. 4, pp. 671–699, July–August, 1987.     

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).     

On the stability of inhomogeneously ageing viscoelastic plates

A method of investigation is proposed and conditions are set up for the stability of viscoelastic inhomogeneously ageing plates of arbitrary shape with a common creep kernel. The form of the stability conditions is found as a function of the surface forces. The stability problem is examined numerically in a finite time interval. The paper touches on the investigations in /1–3/. (See the bibliography of research on the stability of homogeneous viscoelastic systems in /1–5/, for example.)     

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/.     

On the mechanism of stability loss

We consider linear systems of differential equations admitting functions in the form of quadratic forms that do not increase along trajectories in the course of time. We find new relations between the inertia indices of these forms and the instability degrees of the equilibria. These assertions generalize well-known results in the oscillation theory of linear systems with dissipation and clarify the mechanism of stability loss, whereby nonincreasing quadratic forms lose the property of minimum.     

Multi-scale method for the quasi-periodic structures of composite materials

In this paper, we shall present a new second-order and two-scale approximate solution of boundary value problems for the linear elasticity systems with quasi-periodic structures by multi-scale analysis. The computation of strain and stress is presented here. This idea can apply to the damage computation of composite materials. Because the computation of the local stress by this method having more precision, it can provide some inspiration for construct optimal design. Finally numerical results show that the method presented in this paper is effective and reliable.