Method of homogeneous solutions in the mixed problem for a finite circular cylinder : PMM vol. 43, no. 6, 1979, pp. 1073–1081

The mixed axisymmetric problem of elasticity theory on the torsion of a finite circular cylinder by a stamp is considered. The stamp is fixed rigidly to one plane face of the cylinder, the other plane face is fixed, and conditions for no displacements or stresses are given on the cylinder surface. The problem is investigated by the method of homogeneous solutions [1], which permits obtaining its approximate solution for practically any values of the parameters. Such efficiency of the method is determined by the fact that the solution of the problem reduces to investigating an infinite algebraic system of the Poincaré — Koch normal systems type. When the ratio of the cylinder height to the radius of the stamp is sufficiently large, the system coefficients, the contact stresses, and the other characteristics of the problem are evaluated to any degree of accuracy, and effective asymptotic expressions are obtained for small values of this ratio. Results of numerical computations are presented.

A solution of the problem for the case of a large value of the ratio (R − a) /h and small values of the ratio λ = h / a is obtained in [2].     

A theory of elasticity with spatial distribution of matter. Three-dimensional complex structure

References [1 and 2] consider a theory of elasticity with spatial distribution of matter for a medium having simple structure and for a one-dimensional medium having complex structure. In the present article the general case of a three-dimensional medium with complex structure is examined. The general scheme of the one-dimensional case [2] is retained; chief attention is directed toward the specific character of the three-dimensional problem. The original micro-model is a complex crystal lattice [3]. In Section 1 this model is generalized to the case of a continuous distribution of matter. The displacements of the mass centers of the unit cells and the micro-strains of the cells are introduced as the kinematic variables. The force variables are the micro-moments. The transition to an exact continuous representation is carried out, and the equations of an elastic medium of complex structure with spatial distribution of matter are derived. The operators corresponding to the continuous theory are expressed in terms of the original microparameters. It is shown that the well known conditions of symmetry of the tensor of elastic constants, which are usually interpreted as the condition of absence of initial stresses [3 and 4], are consequences of the invariance of the elastic energy under translation and rotation. In Section 2 some special models are examined, and the equations of a medium are obtained for the approximation of weak dispersion of matter. These equations contain as a special case the equations of linear nonsymmetric elasticity (couple-stress theory) [5 to 7]. However, in the latter it turns out that the orders of approximation are inconsistent in the various equations from the point of view of the theory of spatial distribution.

In Section 3 the equations of a medium having complex structure are transformed in the acoustic range into equations, one of which contains only a single kinematic variable (the displacement of the mass centers) and the others of which are explicitly solvable for the remaining kinematic variables. The first equation of this set coincides in form with the equation for a medium with simple structure, but differs from it by the presence of a timewise dispersion which is unrelated to energy dissipation. Expressions are written for the energy density, and it is shown that it is possible to introduce a symmetric stress tensor, as in the case of a simple structure.     

Effect of random irregularities of the creep of reinforced layered plastics

The authors investigate the creep of inhomogeneous materials consisting of a large number of stiff orthotropic elastic layers alternating with layers of linear isotropic viscoelastic material. The elastic layers are assumed to be almost plane; the functions describing the irregularities (curvature) form a random field. The averaged characteristics of the medium are found together with the variation of the averaged displacements and strains in time. An analogous problem was previously considered in [1, 6] on the assumption that the binder layers are elastic. The present paper is based on the equations of [1] and the elastic-viscoelastic correspondence principle [4]. When the correlation scales of the irregularities are small as compared with the dimensions of the body and the characteristic distances over which the averaged parameters of the stress-strain state vary appreciably is considered in detail. A relation is established between the creep functions for simple cases of the state of stress and the parameters characterizing the properties of the components, the properties of the random field of initial irregularities, etc. The development of perturbations with different wave numbers is investigated. The theory is used to describe the creep of reinforced layered plastics.Mekhanika Polimerov, Vol. 2, No. 5, pp. 755–762, 1966     

Plane strain deformation of an initially unstressed elastic medium

Selim and Ahmed [1] used the eigenvalue approach by assuming distinct eigenvalues to calculate the elastic deformation due to an inclined load at any point as a result of an inclined line load of initially stressed orthotropic elastic medium. They studied the plane strain problem and obtained the corresponding results for an unstressed orthotropic medium as a particular case. In the present paper, it is shown that all the eigenvalues do not remain distinct, but become repeated when the elastic medium is free from the initial compressive stresses. Further, the displacements and stresses for an unstressed elastic medium have been independently obtained. The variation of the displacements and stresses due to normal and tangential line load are also shown graphically.     

General Solution for the Two-Dimensional System of Static Lame’s Equations with an Asymmetric Elasticity Matrix

We study a two-dimensional system of equations of linear elasticity theory in the case when the symmetric stress and strain tensors are related by an asymmetric matrix of elasticity moduli or elastic compliances. The linear relation between stresses and strains is written in an invariant form which contains three positive eigenmodules in the two-dimensional case. Using a special eigenbasis in the strain space, it is possible to write the constitutive equations with a symmetric matrix, i.e., in the same way as in the case of hyperelasticity. We obtain a representation of the general solution of two-dimensional equations in displacements as a linear combination of the first derivatives of two functions which satisfy two independent harmonic equations. The obtained representation directly implies a generalization of the Kolosov–Muskhelishvili representation of displacements and stresses in terms of two analytic functions of complex variable. We consider all admissible values of elastic parameters, including the case when the system of differential equations may become singular. We provide an example of solving the problem for a plane with a circular hole loaded by constant stresses.     

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     

Influence of geometrical nonlinearity on the waves propagating through a thin, free plate : PMM vol. 43, no. 6, 1979, pp. 1089–1094

Exact wave solutions of the equations of motion of a thin plate are obtained for the one-dimensional case of a first approximation model. The dependence of the velocity of propagation of flexural-longitudinal waves on their frequency is computed for different values of the amplitude of the bending components. The nonlinear character of the relation connecting the deformations and displacements in the theory of thin shells may give rise to effects which cannot be described in terms of the linear approximation even in those cases when the Hooke's law still holds [1]. The estimation of the magnitude of the displacements at which the effects caused by the geometrical nonlinearity become apparent, is of interest.     

Structural macroscopic theory of stiff and soft composites. Invariant description

A structural macroscopic theory of stiff and soft composites, which generalizes the theory in [1] constructed with application of a model of one-dimensional stressed state of reinforcing fibers in the current configuration of a composite is presented. The theory combines the micro- and macromechanics of composite materials. The two trends in the mechanics of composites are based on the idea of a field of macroscopic displacements and the concept of macroscopic stresses of the composite material when changes in the metrics of the matrix and reinforcing fibers in the current state of a composite medium are taken into consideration. The fibers of the reinforcing systems and matrix are analyzed on the basis of a general 3D model of deformation. No limits on the stiffness of the materials of the structural components are imposed. The analysis of the composite medium, on the macromechanical level, includes a definition of macrodisplacement and macrodeformation fields, as well as parametric structural fields in the current configuration. On the micromechanical level, the fields of macroscopic stresses in the medium, together with the fields of microscopic strains and stresses in the structural components, are defined on the basis of information obtained from the analysis of the field of the macroscopic displacements. With the corresponding interpretation of the field of macroscopic displacements, the structural macroscopic theory is applied to composite media with fibrous, laminated, and matrix structures.     

Continual mechanics of monodisperse suspensions. rheological equations of state for suspensions of moderate concentration : PMM vol. 37, n6, 1973, pp. 1059–1077

Rheological relationships linking mean and moment stresses and, also, the force and moment of interphase reaction in a macroscopic flow of small solid sphere suspension with the kinematic characteristics of the flow are derived. This makes it possible to close the system of equations of suspension hydrodynamics. Coefficients of viscosity and of moment viscosity of a suspension are obtained and calculated.The equations of conservation of mass, momentum and moment of momentum of suspension and of its phases, considered (from the macroscopic point of view) to be coexistent continuous media, were formulated in a general form in [1]. These equations contain unknown vectors and tensors which define the interaction between the considered continuous media and, also, stresses and moment stresses appearing when these are in motion. To close the equations of conservation it is necessary to express all these quantities in terms of unknown variables of these equations (mean concentration of suspension, pressure in the fluid phases, and phase velocities). This problem is the second of the fundamental problems of hydromechanics of suspensions indicated in [1].Here this problem is solved with the use of a kind of self-consistent field theory, which is essentially an extension and generalization of methods developed in [2 – 7]. Expressions for all of the quantities mentioned above are derived. They can be considered to be rheological equations of state for suspensions. Expressions for the various coefficients of these equations and their dependence on parameters of phases and on the flow frequency spectrum are also considered.     

自然弯扭梁广义翘曲坐标的求解

提出了自然弯扭梁受复杂载荷作用时静力分析的一种理论方法,重点在于对控制方程的求解,其中考虑了与扭转有关的翘曲变形和横向剪切变形的影响．在特殊的情况下,可以比较容易地得到这些方程的解答,包括各种内力、应力、应变和位移的计算．算例给出了平面曲梁受水平和垂直分布载荷作用时广义翘曲坐标的求解方法．计算结果表明,求得的应力和位移的理论值和三维有限元结果非常接近．此外,该理论不限于具有双对称横截面的自然弯扭梁,同样可推广至具有一般横截面形状的情况．     

Estimates for deviations from exact solutions to plane problems in the cosserat theory of elasticity

A functional a posteriori estimate is obtained for control of the accuracy of approximate solutions to plane problems arising in the Cosserat theory of elasticity. We deal with the case where displacements and independent rotation are given on the boundary of the domain, a continuous medium is isotropic, and there is a linear dependence between stresses and strains. The proposed method is based on the duality theory of Calculus of Variations and can be also applied to the case of anisotropic media.     

Generalization of the method of functionally invariant solutions for dynamic problems of the plane theory of elasticity of anisotropic media

A new approach for constructing functionally invariant solutions for dynamic problems of the plane theory of elasticity of anisotropic media is proposed. Solutions of the equations of motion in displacements and potentials, which express plane waves and waves from a point source, and also complex solutions of a general type are obtained and investigated. The problem of the reflection of plane waves from the boundary of a half-space is solved for comparison with earlier results [1]. The solutions obtained agree with the physical meaning of the problems and with the solutions for isotropic media.     

Inverse problems of optimal control in creep theory

The statement of the inverse quasistatic problems of creep theory is given in the form of a variational principle and optimal control with the restrictions on displacements and stresses. Also the necessary conditions of optimality are presented. While solving some certain examples, we obtain a continuous function of optimal loading which depends on two parameters. Some method of determining the parameters from the given conditions of the problem is constructed and numerically realized.     

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.     

Refined stability theory for sandwich shells with transversally stiff core. 3. Simplest one-dimensional problems

The applicability and accuracy of stability equations of the refined theory for sandwich shells with a transversally stiff core proposed in [1] are investigated. The model problem of calculating the critical loads and stress fields in the core at mixed forms of the loss of stability is solved for an infinitely wide sandwich plate with an orthotropic core and composite load-carrying layers subjected to in-plane edge loads. The case of pure bending of the plate is considered in detail. The results obtained by variation of the physical-mechanical parameters are compared with the solutions of the three-dimensional theory for the core [2]. It is shown that the version of the refined theory [1] is more accurate than the other two-dimensional theories.For Pt. 2 see [1].Center for Study of Dynamics and Stability, Tupolev Kazan State Technical University, Kazan, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 1, pp. 57–65, January–Feburary, 1998.     

A theory of thin shells with finite displacements and deformations based on a modified Kirchhoff–Love model

A refined classical Kirchhoff–Love theory of thin shells with finite displacements and deformations is given that takes account of deformation in a transverse direction by introducing an additional unknown function to describe it. It is shown that the last of the three equilibrium equations for the moments obtained from the variational equation of the principle of virtual displacements serves to determine it. Constitutive relations are constructed for the internal forces and moments introduced into the treatment based on the introduction of the true Novoshilov stresses and strains into the discussion. The solution of problem of the static stability of a cylindrical shell made of a rubber-like incompressible material inflated by an internal pressure is given using the equations constructed. Chernykh's constitutive relations are used in its formulation.     

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     

Creep strength of anisotropic plates taking account of transverse shear strains

Cylindrical creep buckling of an orthotropic glass-reinforced plastic plate is investigated with consideration of shear creep in the plane xz. In this case the rheology of buckling is not described by the Kirchhoff-Love model, therefore the investigation is carried out using the modified theory of plates [3]. The long-time and instantaneous critical forces are obtained.Mekhanika Polimerov, Vol. 1, No. 5, pp. 114–117, 1965     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

Time-dependent magnetothermoelastic creep modeling of FGM spheres using method of successive elastic solution

Magnetothermoelastic creep behavior of thick-walled spheres made of functionally graded materials (FGM) placed in uniform magnetic and distributed temperature fields and subjected to an internal pressure is investigated using method of successive elastic solution. The material creep, magnetic and mechanical properties through the radial graded direction are assumed to obey the simple power law variation. Using equations of equilibrium, stress-strain and strain-displacement a differential equation, containing creep strains, for displacement is obtained. A semi-analytical method in conjunction with the Mendelson’s method of successive elastic solution has been developed to obtain history of stresses and strains. History of stresses, strains and effective creep strain rate from their initial elastic distribution at zero time up to 55 years are presented in this paper. Stresses, strains and effective creep strain rate are changing in time with a decreasing rate so that after almost 50 years the time-dependent solution approaches the steady state condition when there is no distinction between stresses and strains at 50 and 55 years.