Modified equations of finite-size layered plates made of orthotropic material. Comparison of the results of numerical calculations with analytical solutions

This paper describes the modified bending equations of layered orthotropic plates in the first approximation. The approximation of the solution of the equation of the three-dimensional theory of elasticity by the Legendre polynomial segments is used to obtain differential equations of the elastic layer. For the approximation of equilibrium equations and boundary conditions of three-dimensional theory of elasticity, several approximations of each desired function (stresses and displacements) are used. The stresses at the internal points of the plate are determined from the defining equations for the orthotropic material, averaged with respect to the plate thickness. The construction of the bending equations of layered plates for each layer is carried out with the help of the elastic layer equations and the conjugation conditions on the boundaries between layers, which are conditions for the continuity of normal stresses and displacements. The numerical solution of the problem of bending of the rectangular layered plate obtained with the help of modified equations is compared with an analytical solution. It is determined that the maximum error in determining the stresses does not exceed 3 %.     

Canonical moduli and general solution of equations of a two-dimensional static problem of anisotropic elasticity

Equations of a two-dimensional static problem of anisotropic elasticity are brought to a simple form with the use of orthogonal and affine transformations of coordinates and corresponding transformations of mechanical quantities. It is proved that an arbitrary matrix of elasticity moduli containing six independent components can be always converted by a congruent transformation to a matrix with two independent components, which are called the canonical moduli. Depending on the relations between the canonical moduli, the determinant of the matrix of operators of equations in displacements is presented as a product of various quadratic terms. A general presentation of the solution of equations in displacements in the form of a linear combination of the first derivatives of two quasi-harmonic functions satisfying two independent equations is given. A symmetry operator (i.e., a formula of production of new solutions) is found to correspond to each presentation. In a three-dimensional case, the matrix of elasticity moduli with 21 independent components is congruent to a matrix with 12 independent canonical moduli.     

Interaction between a compliant disk-shaped inclusion and a crack upon incidence of an elastic wave

The propagation of harmonic elastic wave in an infinite three-dimensional matrix containing an interacting low-rigidity disk-shaped inclusion and a crack. The problem is reduced to a system of boundary integral equations for functions that characterize jumps of displacements on the inclusion and crack. The unknown functions are determined by numerical solution of the system of boundary integral equations. For the symmetric problem, graphs are given of the dynamic stress intensity factors in the vicinity of the circular inclusion and the crack on the wavenumber for different distances between them and different compliance parameters of the inclusion.     

Well-posedness of the fundamental boundary value problems for constrained anisotropic elastic materials

We consider the equations of linear homogeneous anisotropic elasticity admitting the possibility that the material is internally constrained, and formulate a simple necessary and sufficient condition for the fundamental boundary value problems to be well-posed. For materials fulfilling the condition, we establish continuous dependence of the displacement and stress on the elastic moduli and ellipticity of the elasticity system. As an application we determine the orthotropic materials for which the fundamental problems are well-posed in terms of their Young's moduli, shear moduli, and Poisson ratios. Finally, we derive a reformulation of the elasticity system that is valid for both constrained and unconstrained materials and involves only one scalar unknown in addition to the displacements. For a two-dimensional constrained material a further reduction to a single scalar equation is outlined.This paper is dedicated to Professor Joachim Nitsche on the occasion of his sixtieth birthday     

Classical and sequential limit analysis revisited

Classical limit analysis applies to ideal plastic materials, and within a linearized geometrical framework implying small displacements and strains. Sequential limit analysis was proposed as a heuristic extension to materials exhibiting strain hardening, and within a fully general geometrical framework involving large displacements and strains. The purpose of this paper is to study and clearly state the precise conditions permitting such an extension. This is done by comparing the evolution equations of the full elastic–plastic problem, the equations of classical limit analysis, and those of sequential limit analysis. The main conclusion is that, whereas classical limit analysis applies to materials exhibiting elasticity – in the absence of hardening and within a linearized geometrical framework –, sequential limit analysis, to be applicable, strictly prohibits the presence of elasticity – although it tolerates strain hardening and large displacements and strains. For a given mechanical situation, the relevance of sequential limit analysis therefore essentially depends upon the importance of the elastic–plastic coupling in the specific case considered.     

On solutions of the problem in stresses with the use of Maxwell stress functions

The spatial problems of elasticity are mainly solved in displacements [1, 2], i.e., the Lamé equations are taken as the initial equations. This is related to the lack of general solutions for the system of basic equations of elasticity expressed in stresses. In this connection, a new variational statement of the problem in stresses was developed in [3, 4]; this statement consists in solving six generalized equations of compatibility for six independent components of the stress tensor, while the three equilibrium equations are transferred to the set of boundary conditions. This method is more convenient for the numerical solution of problems in stresses and has been tested when solving various boundary value problems. In the present paper, analyzing the completeness of the Saint-Venant identities and using the Maxwell stress functions, we obtain a new resolving system of three differential equations of strain compatibility for the three desired stress functions φ, ξ, and ψ. This system is an alternative to the three Lamé equilibrium equations for three desired displacement components u, v, w and is simpler in structure. Moreover, both of these systems of resolving equations can be solved by the new recursive-operator method [5, 6]. In contrast to well-known methods for constructing general solutions of linear differential equations and their systems, the solutions obtained by the recursive-operator method are constructed as operator-power series acting on arbitrary analytic functions of real variables (not necessarily harmonic), and the series coefficients are determined from recursive relations (matrix in the case of systems of equations). The arbitrary functions contained in the general solution can be determined directly either from the boundary conditions (the obtained system of inhomogeneous equations with a right-hand side can also be solved by the recursive-operator method [6]) or by choosing them from various classes of analytic functions (elementary, special); a complete set of particular solutions can be obtained in the same function classes, and the coefficients of linear combinations of particular solutions can be determined by the Trefftz method, the least-squares method, and the collocation method.     

Mixed boundary value problem of elasticity for a quarter space

We consider the static elasticity problem for a quarter space with zero displacements on one of its surfaces and with given stresses on the other. The method for solving this problem is based on the use of newunknown functions in the formof a linear combination of the desired displacements, which reduces the system of three Lamé equations to two equations to be solved simultaneously and one equation to be solved separately. The exact solution of this problem was obtained earlier by the same method [1]. But it was shown in [2] that such a solution is exact only under certain restrictions on the given functions. In the present paper, the solution of this problem is constructed without restrictions on the given functions, which necessitates solving a one-dimensional integro-differential equation; this can be done approximately by the orthogonal polynomial method. We present numerical results obtained on the basis of our solution.     

Displacement function and simplifying of plane elasticity problems of two-dimensional quasicrystals with noncrystal rotational symmetry

The paper systematically investigates the plane elasticity problems of two-dimensional quasicrystals with noncrystal rotational symmetry. First, applying their independent elastic constants, the equilibrium differential equations of the problems in terms of displacements are derived and it is found that the plane elasticity of pentagonal quasicrystals is the same as that of decagonal. Then by introducing displacement functions, huge numbers of complicated partial differential equations of the problems are simplified to a single or a pair of partial differential equations of higher order, which is called governing equations, such that the problems can be easily solved. Finally, some solving methods of these governing equations obtained are introduced briefly.     

A high-order theory for functionally graded axially symmetric cylindrical shells

A high-order theory for functionally graded axially symmetric cylindrical shell based on expansion of the axially symmetric equations of elasticity for functionally graded materials into Legendre polynomials series has been developed. The axially symmetric equations of elasticity have been expanded into Legendre polynomials series in terms of a thickness coordinate. In the same way, functions that describe functionally graded relations has been also expanded. Thereby, all equations of elasticity including Hook’s law have been transformed to corresponding equations for coefficients of Legendre polynomials expansion. Then system of differential equations in terms of displacements and boundary conditions for the coefficients of Legendre polynomials expansion coefficients has been obtained. Cases of the first and second approximations have been considered in more details. For obtained boundary-value problems’ solution, a finite element has been used and numerical calculations have been done with COMSOL MULTIPHYSICS and MATLAB.     

Classification of variational principles in elasticity

In this paper, variational principels in elasticity are classified according to the differences in the constraints used in these principles. It is shown in a previous paper [4] that the stress-strain relations are the constraint conditions in all these variational principles, and cannot be removed by the method of linear Lagrange multiplier. The other possible constraints are four of them: (1) equations of equilibrium, (2) strain-displacement relations, (3) boundary conditions of given external forces and (4) boundary conditions of given boundary displacements. In variational principles of elasticity, some of them have only one kind of such constraints, some have two kinds or three kinds of constraints and at the most four kinds of constraints. Thus, we have altogether 15 kinds of possible variational principles. However, for every possible variational principle, either the strain energy density or the complementary energy density may be used. Hence, there are altogether 30 classes of functional of variational principles in elasticity. In this paper, all these functionals are tabulated in detail.     

AN ACCURATE TWO-DIMENSIONAL THEORY OF VIBRATIONS OF ISOTROPIC,ELASTIC PLATES

An infinite system of two-dimensional equations of motion of isotropic elastic plates with edge and corner conditions are deduced from the three-dimensional equations of elasticity by expansion of displacements in a series of trigonometrical functions and a linear function of the thickness coordinate of the plate. The linear term in the expansion is to accommodate the in-plane displacements induced by the rotation of the plate normal in low-frequency flexural motions. A system of first-order equations of flexural motions and accompanying boundary conditions are extracted from the infinite system. It is shown that the present system of equations is equivalent to the Mindlin’s first-order equations, and the dispersion relation of straight-crested waves of the present theory is identical to that of the Mindlin’s without introducing any corrections. Reduction of present equations and boundary conditions to those of classical plate theories of flexural motions is also presented.     

基于Hamilton体系的弹性力学问题的比例边界有限元方法

比例边界有限元方法是求解偏微分方程的一种半解析半数值解法。对于弹性力学问题,可采用基于力学相似性、基于比例坐标相似变换的加权余量法和虚功原理得到以位移为未知量的系统控制方程,属于Lagrange体系。但在求解时,又引入了表面力为未知量,控制方程属于Hamilton体系。因而,本文提出在比例边界有限元离散方法的基础上,利...     

Formulation of Problems in the General Kirchhoff—Love Theory of Inhomogeneous Anisotropic Plates

In this paper we study the procedure of reducing the three-dimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate to a two-dimensional problem in the median plane. The plate is in equilibrium under the action of volume and surface forces of general form. À notion of internal force factors is introduced. The equations for force factors (the equilibrium equations in the median plane) are obtained from the thickness-averaged three-dimensional equations of elasticity theory. In order to establish the relation between the internal force factors and the characteristics of the deformed middle surface, we use some prior assumptions on the distribution of displacements along the thickness of the plate. To arrange these assumptions in order, the displacements of plate points are expanded into Taylor series in the transverse coordinate with consideration of the physical hypotheses on the deformation of a material fiber being originally perpendicular to the median plane. The well-known Kirchhoff—Love hypothesis is considered in detail. À closed system of equations for the theory of inhomogeneous anisotropic plates is obtained on the basis of the Kirchhoff—Love hypothesis. The boundary conditions are formulated from the Lagrange variational principle.     

A Representation Theorem for Material Tensors of?Weakly-Textured Polycrystals and?Its Applications in?Elasticity

Material tensors pertaining to polycrystalline aggregates should manifest also the influence of crystallographic texture on the material properties in question. In this paper we make use of tensors which form bases of irreducible representations of the rotation group and prove a representation theorem by which a given material tensor of a weakly-textured polycrystal is expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a set of undetermined material parameters. Once the irreducible basis tensors that appear in the formula are determined, the representation formula, which is valid for all texture and crystal symmetries, will delineate quantitatively the effect of crystallographic texture on the material tensor in question. We present an integral formula and an orthonormalization process which serve as the basis for a procedure to determine explicitly the irreducible basis tensors required in the representation formula. For applications we determine a set of irreducible basis tensors for the elasticity tensor and a set for fourth-order tensors that define constitutive equations in incompressible elasticity and Hill’s quadratic yield functions in plasticity. We show that orientation averaging of a tensor can be done easily if we have in hand a set of irreducible basis tensors for the decomposition of the tensor in question. As illustration we derive a formula, which is valid for all texture and crystal symmetries, for the elasticity tensor under the Voigt model.     

有限尺寸多钉连接件钉传载荷计算的解析方法

提供一种确定多钉连接件中钉传载荷的解析方法，这个方法将被连接件看作弹性体，以经典结构力学以及弹性理论平面问题复变函数解法为基础，建立了求解钉传载荷的线性代数方程组并给出了若干算例。这个方法不仅具有合理的力学模型，而且具有计算的简捷性与适用的广泛性。     

Equations of motion and boundary conditions of physical meaning of micropolar theory of thin bodies with two small cuts

Nowadays, microcontinuous mechanics (mechanics of media with microstructure) is being developed very intensively, which is testified by recently published papers [1–14] and by many others, as well as by the symposiumdedicated to the hundredth anniversary of the brothers Cosserat monograph [15], held inParis in 2009. A survey of foreign papers is given in [16], and a special place is occupied by earlier publications of Soviet scientists on micropolar theory of elasticity [17–24]. A brief survey of Cosserat theory of elasticity and an analysis and prospects of such theories in mechanics of rigid deformable bodies is given in [21]. It should be noted that, in a majority of cases, the structure strength calculations are based on the classical theory of elasticity. But there are materials such as animal bones, graphite, several polymers, polyurethane films, porous materials (pumice), various synthetic materials, and materials with inclusions which, under certain conditions, exhibit micropolar properties. There are effects which cannot be prescribed by the classical theory. In statics, nonclassical behavior can be observed in bending of thin films and cantilevers, in torsion of thin and thin-walled rods, and in the case of stress concentration near holes, corner points, cracks, and inclusions. For example, thin specimens are more rigid in bending and torsion as is prescribed by the classical theory [25–27]. The stress concentration near holes decreases, and the concentration factor depends on the radius [28]. The stress concentration near cracks also becomes lower. Conversely, the stress concentration near inclusions is higher than predicted by the classical theory [29–31]. If the material has no center of symmetry of elastic properties, then calculations according to the micropolar theory shows that the specimen is twisted in tension [32]. In dynamical problems, several phenomena also differ from the classical concepts. For example, shear waves propagate with dispersion, microrotation waves arise, and the vibration natural modes differ from the classical ones [2, 7, 11–13, 33]. All these phenomena are used to determine material constants of the micropolar theory of elasticity. There are many methods for determining such constants [2, 34]. Since thin bodies (one-, two-, three-, and multilayer structures) are widely used, it is necessary to create new refined microcontinual theories of thin bodies and advanced methods for their computations. In the present paper, various representations of the system of equations of motion are obtained in the micropolar theory of thin bodies with two small parameters in momenta with respect to a system of Legendre polynomials in the case where an arbitrary line is taken for the base. In this connection, a vector parametric equation of the region of a thin body is given for the parametrization under study, different families of bases (frames) are introduced, and expressions for components of the unit tensor of rank two (UTRT) are obtained. Representations of gradient, tensor divergence, equations of motion, and boundary conditions for the considered parametrization are given. Definitions of (m, n)th-order moment of a variable with respect to an arbitrary system of orthogonal polynomials and a system of Legendre polynomials is given. Expressions for themoments of partial derivatives and several expressions with respect to a system of Legendre polynomials and boundary conditions in moments are obtained.     

Boundary Value Problems of Plane Elasticity Involving Orientations of Displacements and Tractions

This paper presents unconventional formulations of boundary problems of plane elasticity formulated in terms of orientations of tractions and displacements on a closed contour separating internal and external domains as the boundary conditions. These are combined with the conditions of continuity of tractions or displacements across the boundary. Therefore the magnitudes of neither tractions nor displacements are assumed on the contour. Four boundary value problems for both external and internal domains are investigated by analyzing the solvability of the corresponding singular integral equations. It is shown that all considered problems can have non-unique solutions expressed as linear combinations of particular solutions and, hence, contain free arbitrary parameters, the number of which is finite and determined by the contour orientations of tractions and/or displacements     

Existence of solutions of boundary-value problems for thin elastic irregular shells

We study the solvability of a geometrically and physically nonlinear boundary-value problem for thin elastic anisotropic irregular shells with clamped edge. For this problem, we suggest a method based on solving the problem in some function space other than the space of displacements and stresses. This method is based on the integral representations of displacement components and strains in terms of auxiliary functions (conventional strains). This method permits obtaining, as the equilibrium equations, nonlinear singular integral equations over a bounded plane domain for the conventional strains. The solvability of these equations is studied by the contraction mapping principle.     

The radially nonhomogeneous elastic axisymmentric problem

The plane axisymmetric problem with axisymmetric geometry and loading is analyzed for a radially nonhomogeneous circular cylinder, in linear elasticity. Considering the radial dependence of the stress, the displacements fields and of the stiffness matrix, after a series of admissible functional manipulations, the general differential system solving the problem is developed. The isotopic radially inhomogeneous elastic axisymmetric problem is also analyzed. The exact elasticity solution is developed for a radially nonhomogeneous hollow circular cylinder of exponential Young’s modulus and constant Poisson’s ratio and of power law Young’s modulus and constant Poisson’s ratio. For the isotropic elastic axisymmetric problem, a general expression of the stress function is derived. After the satisfaction of the biharmonic equation and making compatible the stress field’s expressions, the stress function and the stress and displacements fields of the axisymmetric problem are also deduced. Applications have been made for a radially nonhomogeneous hollow cylinder where the stress and displacements fields are determined.