Stress state of freely supported multilayered elliptical plates of anisotropic materials

The problem of a stressed state in elliptic plates has been considered in general for a rigid contour fixation. It is much more difficult to obtain a solution for the freely supported plates, even for isotropic materials. In this paper we suggest an approach for defining the stressed state of thin elliptic plates with layered structure under the condition of a freely supported contour. The solution is obtained in a rectangular cartesian coordinate system. The displacements, which are the fundamental unknowns, are given in the form of polynomials with unknown coefficients defined by a system of algebraic equations. The resolving equations and three out of the four boundary conditions are satisfied precisely. One boundary condition, is satisfied by means of collocation method of separate points of the contour. Estimation of the accuracy of the suggested approach is carried out by comparing the obtained results with the known ones. The problem of deformation of a twolayered plate has been discussed, in which the principal direction of elasticity does not coincide with the coordinate directions.S. P. Timoshenko Institute of Mechanics, National Academy of Science of the Ukraine, Kiev. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 496–504, July–August, 1997. Original article submitted March 19     

Stress state near loaded holes in anisotropic plates

Special representations of the solution are constructed and solving integral equations of the problem of the elastic equilibrium of a finite anisotropic plate weakened by an elliptical hole or rectilinear crack are derived. The absence of the unknown function for the boundary of the internal hole (crack) makes it possible to propose an effective algorithm for the problem's numeric solution. The results of calculations, which illustrate the effect of the external boundary and material anisotropy on the stress distribution near loaded holes of different sizes, are presented. Direct comparison with the finite-element method indicates that the proposed algorithm significantly lowers the amount of input data, the computer time, and the required volume of memory with comparable accuracy.Novosibirsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 45–51, 1990.     

Action of a shear stress pulse on the surface of a viscoelastic anisotropic half-space

The boundary value problem of the behavior of a viscoelastic half-space subjected to a surface shear stress pulse is solved in the linear formulation on the basis of dynamical correspondence principle. The medium occupying the half-space possesses the property of transverse isotropy. The exact solution of the problem is obtained using integral transforms. The solution is analyzed with reference to a Maxwell-Voigt model.Moscow Region. Translated from Mekhanika Polimerov, No. 5, pp. 933–937, September–October, 1969.     

State of stress of flexible viscoelastic plates with an opening

The problem of the state of stress of flexible plates with an opening is investigated with allowance for the viscoelastic properties of the material. The basic equations and the corresponding boundary conditions are formulated. The boundary value problem is solved by numerical integration. The effect of the creep properties of the material on the stress redistribution in the plate is investigated with reference to polymethyl methacrylate. It is shown that the angle of rotation, the ring moment, and the ring forces at the edge of the opening vary with time and as a function of the load.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 4, pp. 708–713, July–August, 1971.     

Equilibrium conditions and vector variational inequalities: a complex relation

Since a look to the present literature shows that the notion of vector equilibrium is not univocally defined, the relationship between vector equilibrium problems and vector variational inequalities deserves a specific analysis, deeper than in the scalar case. We analyse and compare various definitions and clarify how they depend on the modelling of the problem under consideration. Furthermore, we show that at least one vector variational inequality formulation of the problem is possible, which is equivalent to a Wardrop-type equilibrium condition. The analysis is illustrated through the concrete example of a multiclass and multiobjective supply demand network.      

Determination of the viscoelastic state of an anisotropic plate with rigid inclusions

A method for the determination of the viscoelastic state of multiconnected anisotropic plates with absolutely rigid inclusions is proposed. As an example, we consider the solution of the problem of viscoelasticity for a plate with an elliptic inclusion which, in a special case, turns into a linear one. For the case of tension of the plate, results of numerical investigations of the stress state depending on geometric parameters of the inclusion and on the time of application of load are described.     

Investigation of stress intensity factors near interacting cracks in linear viscoelastic plates

Stress intensity factors as a function of time near interacting cracks during creep in plates made of linear viscoelastic materials are investigated. The interaction of collinear as well as parallel cracks disposed on different levels perpendicular to the tensile stress was studied. It was found that during interaction of collinear cracks, the stress intensity factor K₁(t) grew in creep and varied mainly due to the change in the distance between the tips of interacting cracks. There was practically no increase in the length of the crack. During interaction of parallel cracks, the value of K₁(t) decreased slightly in time, which was also due to an increase in the distance between the tips of the cracks in creep.Institute of Mechanics, Lomonosov Moscow State University, Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 5, pp. 643–650, September–October, 1998.     

The stress state of thick plates under homogeneous boundary conditions of mixed type on the faces

We propose a method of using homogeneous solutions in the case of mixed boundary conditions on the faces of thick plates. We obtain the exact solution of the problem of the stress state of a lamina weakened by a cylindrical cavity. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 3–8.     

A power series solution for vibration and complex modal stress analyses of variable thickness viscoelastic two-directional FGM circular plates on elastic foundations

In the present paper, a power series solution is developed for free vibration and damping analyses of viscoelastic functionally graded plates with variable thickness on elastic foundations. It is assumed that the material properties of the functionally graded material (FGM) vary in the transverse and radial directions, simultaneously. Therefore, the presented solution can be employed for the transversely-graded and radially-graded viscoelastic circular plates, as special cases. In addition to the edge conditions, the plate may be resting on a two-parameter elastic foundation. The complex modulus approach in combination with the elastic–viscoelastic correspondence principle is employed to obtain the solution for various edge conditions. A sensitivity analysis including effects of various edge conditions, geometric parameters, coefficients of the elastic foundation, parameters of the functionally graded material, and material loss factor is carried out. In the present paper, concept of the complex modal stresses of the viscoelastic plates is discussed in detail.     

The stress state of transtropic plates with ends covered by a diaphragm

We obtain homogeneous solutions of the equations of the three-dimensional theory of elasticity for transtropic plates on whose planar faces the normal stress and tangential displacements are zero. We study the problem of the stress state of a transtropic layer with a cylindrical cavity. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 3–12.     

A nonlinear model of a viscoelastic medium with allowance for the nature of the state of stress

A simple model of a nonlinear viscoelastic medium of hereditary type is considered with allowance for the nature of the state of stress. A system of experiments for determining the material functions and constants is described and sufficient conditions for the existence of simple loadings are formulated.Moscow Lomonosov State University. Translated from Mekhanika Polimerov, No. 6, pp. 994–1001, November–December, 1969.     

The stressed state caused by a residual strain field in plates with stress concentrators

We obtain an analytic solution of the problem of determining the stressed state caused by a given residual strain field (while taking account of three-dimensional effects) in a round plate with a concentric foreign inclusion. We study the influence of the geometric parameters of the given system, the nonlinearity of the distribution of the given residual strains over the thickness of the plate, and a possible jump in distortion at the surface of contact on the stressed state of the system. We discover an internal boundary-layer effect that is significant in the case when the given residual strain field is strongly gradient.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 60–66.     

Flattening of a long multilayered viscoelastic cylindrical shell with a varying wall thickness

The stability of a multilayered linearly viscoelastic cylindrical shell with a varying wall thickness under the action of uniformly distributed lateral pressure is investigated. Assuming that the shell is sufficiently long and neglecting the influence of its fastening, the problem posed is reduced to the examination of stability of a compressed ring. The urgency and importance of such problems are connected with the search for reserves of saving of materials with simultaneously increasing the bearing capacity of the structure. In solving problems of this class, the geometric nonlinearity must also be taken into account. The acquisition of efficient analytical solutions here is a rather difficult and sometimes impossible task. This is connected with the integration of nonlinear boundary-value problems with discontinuous coefficients. Therefore, to avoid the existing mathematical obstacles, the problem is solved by using a variational method of mixed type in combination with the Rayleigh–Ritz method. For a three-layer ring with a symmetric structure relative to its median surface, the effect of various geometrically nonlinear theories and of the varying wall thickness on the critical time of stability is revealed numerically.