Load transmission in layered materials

Load transmission in reinforced plastic heterogeneous media is investigated on the basis of the equations of the theory proposed in [1–3]. The stress distribution problem is solved for a layered half-space with loads applied to one of the "hard" layers in the plane of that layer. Simplified stress formulas are presented. The corresponding error is estimated by working a numerical example. The results are compared with the corresponding problem of the theory of elasticity for a homogeneous orthotropic body.Mekhanika Polimerov, Vol. 4, No. 2, pp. 322–327, 1968     

Forced Vibration of a Prestretched Two-Layer Slab on a Rigid Foundation

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 forced vibration of a prestretched two-layer slab resting on a rigid foundation is studied. To the upper plane of the slab, a harmonic point force is applied. It is assumed that the layer materials are incompressible, and their elastic properties are characterized by Treloar’s potential. Numerical results are presented for the case where the material stiffness of the lower layer is greater than that of the upper one. The influence of prestretching the layers on the frequency dependences of the normal stresses operating on the interface between the layers and between the slab and the rigid foundation are analyzed.__________Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 3, pp. 335–350, May–June, 2005.     

Torsion of laminated cylindrical shells with adhesive interlayers

Based on refined equations of the Timoshenko-type shell theory, the contact stresses in torsion of a two-layer cylindrical shell with an adhesive interlayer are numerically studied. The effect of the geometric and physical-mechanical parameters of the load-carrying layers and adhesive interlayer of the shell on the distribution of the interlaminar tangential stress is analyzed. The results are presented graphically.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Pidstryhach Institute of Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 4, pp. 501–506, July–August, 1998.     

Finite element construction for simulating delamination of sandwich composite plates and masses

Displacements and transverse normal stresses in sandwich plates and masses have been approximated by the Ambartsumyan iterative approach to constructing mathematical models of the stress-strain state of sandwich structures. A linear distribution of the displacements in the sandwich structure is set up as the first step of the iterative process, while in the subsequent steps the displacement approximations with higher-order polynomials are obtained. The approximation of the compression stresses is based on Hooke's law using the expression of the tangential displacements in the second step and the normal displacements in the third step of the iterative process. Two shear functions are introduced. The finite element is rectangular and has four nodes. The number of degrees of freedom of finite elements is independent of the quantity of the layers that may be orthotropic. The finite element allows us to simulate delamination by a thin low-modulus interlayer. In doing so, the quantity of the layers increases, while the order of the resolving set of equations does not grow. A number of numerical experiments were carried out. It has been shown that the delamination can greatly increase the level of the stresses in the structure. This effect is especially significant for thin structures. The stresses are somewhat lower when taking into account the interlaminar friction.Submitted to the 10th International Conference on Mechanics of Composite Materials (Riga, April 20–23, 1998).Ukrainian Transport University, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 251–263, March–April, 1998.     

The theory of orthotropic layered cylindrical shells

The author examines orthotropic layered cylindrical shells for which the moduli of elasticity of the load-carrying layers substantially exceed the shear modulus between layers. This class of structure includes, in particular, shells made of orthotropic glass-reinforced plastic. In this case the classical theory based on the Kirchhoff-Love hypotheses requires refinement; the corresponding equations obtained as a result of approximating the distribution of shear stresses or displacements over the thickness of the shell by a certain known function are presented in [4, 7, 8]. In this paper equations that make it possible to construct the stress distribution over the shell thickness are obtained within the framework of the engineering theory on the basis of the hypothesis of the incompressibility of a normal element.Mekhanika Polimerov, Vol. 4, No. 1, pp. 136–144, 1968     

Allowance for the effect of hydrostatic pressure in calculating the deformation of polymer structural elements

Simple equations of state of hereditary type are introduced and the results of analyzing certain experiments are presented. A solution of the problem of determining the stresses and strains in the case where the average stress does not depend on the coordinates is given for the equations of state in question.Central Scientific-Research Institute of Chemistry and Mechanics, Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 50–57, January–February, 1973.     

Prediction of the bearing capacity of composite flanges for aircraft-engine casing parts

Computed results of the stress-strain state of four different composite flanges for aircraft engines are presented for assigned external-load parameters, and their most heavily loaded regions are defined. The static strength of the structures is found from the stresses in the layers using the maximum-stress criterion and Hill's modified criterion. It is established that transverse tensile stresses are most critical. The service life of the flanges is estimated on the basis of the fatigue-failure criterion of the weakest link.Perm' State Technical University. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 3, pp. 360–369, May–June, 1997.     

The effect of interphase on residual thermal stresses. 2. Unidirectional fiber composite materials

In real composite materials an additional phase may exist between the fiber and the matrix. This phase, commonly known as the interphase, is a local region that results from the matrix bonds with the fiber surface or the fiber sizing. The differing thermal expansions or contractions of the fiber and matrix cause thermally induced stresses in composite materials. In the present study, a four-cylinder model is proposed for the determination of residual thermal stresses in unidirectional composite materials. The elastic modulus of the interphase is a function of the interphase radius and thickness. The governing equations in terms of displacements are solved in the form of expansion into a series [1]. The effective elastic characteristics are obtained using the finite element approach. The effect of the interphase thickness and different distributions of the interphase Young's modulus on the thermal residual stress field in unidirectional composite materials is investigated.For Pt. 1, see [1].Published in Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 200–214, March–April, 1997.     

Resistance of three-layered structures to static and cyclic bending

Conclusions The above studies of two types of three-layer structural elements showed that the types have different resistances to static deformation in bending. Regardless of the materials, the use of structures which are symmetrical in regard to stiffness makes it possible to obtain a stiffness and strength for the structure which are 10–15% lower than the stiffness and strength of the external plates if the thickness of the latter does not account for more than 25% of the thickness of the structure. This finding, in turn, permits a substantial reduction in the weight of the structure by the use of a lower-density material for the internal layer. Resistance to static bending is determined mainly by the resistance of the structure to shear stresses. The mechanism of fatigue fracture differs appreciably from the fracture mechanism in static deformation. Regardless of the thickness of the structural elements, fatigue fracture for both types of structure occurs as a result of the acting normal compressive stresses. The endurance limit of the hybrid structure is determined by the fatigue resistance of the external layers, and its value is nearly equal to the resistance of the pure materials.Presented at the Sixth All-Union Conference on the Mechanics of Polymer and Composite Materials (Riga, November, 1986).Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 878–882, September–October, 1986.     

Reduction of elasticity theory boundary problems for inhomogeneous media to sets of integral equations

The boundary problem of elasticity theory in stresses or displacements for materials which are continuously inhomogeneous along one coordinate is reduced by means of Laplace and Helmholtz equations to a set of four integro-differential equations, two of which are singular. Each of the equations contains integrals for the contour of the transverse section of a body which is assumed to be piecewise-smooth, and integrals for a region coincident with the section of the body.Sumy. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 20–23, 1990.     

Second-approximation shear theory for shallow layered shells and plates

Analysis of a second-approximation refined shear model for shallow layered composite shells and plates with a substantially inhomogeneous structure over the thickness is presented. The tangential displacements and corresponding normal stresses are expressed in the form of a polynomial of the fith degree in the transverse coordinate and contain squared rigidity characteristics. In this way, the accuracy of results and practical coincidence with the 3D solutions is ensured. Based on the refined model, a theory of shallow layered shells is developed. A system of resolving equations of sixteenth power together with appropriate boundary conditions was obtained and solved analytically. It is shown that the area of application of the formed model is extended as compared with the model of the first approximation. The model proposed allows us to examine the stress-strain state of layered composite structures of substantially different thickness and physical-mechanical characteristics of the layers, including the possibility of simulating relatively large shear deformations of rigid layers separated by a low-modulus thin interlayer pliable to transverse shear.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Ukrainian Transport University, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 363–370, May–June, 1998.     

Solutions of one-dimensional problems of elasticity and thermoelasticity for cylindrical piecewise-homogeneous bodies

We propose a method of direct integration of the equilibrium and continuity equations in stresses for one-dimensional problems of elasticity and thermoelasticity for piecewise-homogeneous cylinders and disks with an arbitrary number of layers. The solutions are reduced to finding the constants of systems of algebraic equations with nearly triangular matrices of coefficients, making it possible to find the unknown constants in a closed form that is functionally dependent on the bulk forces and temperature field.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 139–148.     

Model of a layered plate considering nonideal contact between layers

We propose a mathematical model for doing calculations for layered plates, allowing for both rigid and sliding contact in the presence of frictional forces between the sliding layers. The model takes into account the distribution of tangential and normal displacements across the thickness of the sliding layered stack, and also the distribution of transverse normal stresses. The strain tensor is obtained using the Cauchy relations; the stress tensor is obtained based on Hooke's law. Tne Lagrange variational principle allows us to obtain the resolvent system of differential equations and the corresponding boundary conditions. The spatial model for deformation of a layered plate has a number of special features compared with familiar models. The system of differential equations has operators no higher than second order. It is described relative to displacements on the faces of the stack. This is convenient in solving problems involving sliding of layers with and without friction.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 671–676, September–October 1995.     

Determination of contact stresses in problems on bending of a package of transversely isotropic bars

A mechanomathematical model for bending of packages of transversely isotropic bars of rectangular cross section is proposed. Adhesion, slippage, and separation zones between the bars are considered. The resolving equations for deflections and tangential displacements are supplemented with a system of linear differential equations for determining the normal and tangential contact stresses, and boundary conditions are formulated. A scheme for analytical solution of two contact problems—a package under the action of a distributed load and a round stamp—is considered. For these packages, a transition is performed from the initial system of differential equations for determining the contact stresses, where the unknown functions are interrelated by recurrent relationships, to one linear differential equation of fourth order and then to a system of linear algebraic equations. This transition allows us to integrate the initial system and get expressions for the contact stresses.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 761–778, November–December, 2004.     

A linearized theory of thermoviscoelasticity

A generalized linearized theory of thermoviscoelasticity, including the effect of heat formation, is presented. The linearized equations of motion, of state, and for the energy are given together with the linearized boundary conditions for large initial deformations. Attention is drawn to the fact that the equations which have been derived can be used for the solution of problems concerning the stability of viscoelastic bodies, the propagation of waves in viscoelastic materials which are subjected to deformation, and problems concerning the stress-deformed state of viscoelastic elements. The problem of the propagation of plane waves in viscoelastic materials which are subjected to deformation is considered as an example.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 2, pp. 214–221, March–April, 1972.     

Theory of a reinforced layered medium with random initial irregularities

The author examines an elastic medium reinforced with slightly distorted elastic layers. The basic equations are obtained by the method proposed in [1,2]. It is assumed that the functions describing the initial distortions of the reinforcing layers form a random field. With the help of the method of canonical expansions [3], expressions are derived for the statistical characteristics of the stresses, strains and displacements in the reinforced medium. The theory is used to account for the known experimental fact of the reduction in the moduli of elasticity of layered glass-reinforced plastics as compared with the values calculated for an ideal reinforced medium. In particular, it is shown that this reduction may be considerable even when the initial irregularities are relatively small.Mekhanika Polimerov, Vol. 2, No. 1, pp. 11–19, 1966     

Results of an experimental investigation of the residual stresses in wound glass-reinforced plastics

The results of an experimental investigation of the residual stresses in wound glass-reinforced plastic rings are presented. The residual stresses were determined by the Davidenkov method. The dependence of the maximum tensile and compressive circumferential stresses on ring thickness and polymerization temperature is investigated. The experimental data are compared with the results of calculations based on the theory proposed in [1, 2].Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1116–1119, November–December, 1970.     

Concentration of electroelastic fields in a composite piezoceramic plate with a hole intersecting the interface of materials

A plane problem of electroelasticity is considered for an infinite compound plate with a hole located in both constituents of the plate. The corresponding boundary value problems is reduced to a system of singular integral equations of second kind, which is solved in numerical quadratures. Calculation results are presented that describe the concentration of electroelastic fields near the hole upon action at infinity of the vectors of mechanical stresses and electric field strength.Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 3, pp. 359–366, May–June, 1999.     

Refined stability theory for sandwich shells with transversally stiff core. 1. Nonlinear equilibrium equations

A refined version of geometrically nonlinear relationships is proposed for the static thermoelastic response of sandwich shells with face sheets made of composite or homogeneous materials and a transversally stiff core. This theory has primary importance for studying mixed forms of buckling of the bearing sheets, which are mainly realized in the zones of a momentary stress-deformed state of the shell on the whole. An iteration procedure was developed for construction of the model. In the first step, assuming that the core is transversally soft, expressions are derived for the components of the displacement vector after integration of the three-dimensional equilibrium equations. In the second step, the tangential stresses are determined assuming a transversally stiff core to obtain the in-plane stresses and highly accurate transverse normal stresses. The proposed model admits a formal changeover to the model of a shell with a transversely soft core.Center for the Study of Dynamics and Stability. A. N. Tupolev Kazan State Technical University, Kazan, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 4, pp. 513–524, July–August, 1996.     

General structural equilibrium solutions of boundary-value problems in the mechanics of composites

Based on the theory of R-functions, we have constructed general structural equilibrium solutions for stresses identically satisfying both the differential equilibrium equations and the contact conditions at the three-identically boundaries of inhomogeneities of arbitrary shape. These solutions can be used in developing the next generation of numerical and analytical methods in the mechanics of inhomogeneous bodies and the micromechanics of composite materials.Kharkov State Technical University of Construction and Architecture, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 474–481, July–August, 1997.