Solving the three-dimensional equations of the linear theory of elasticity

The three-dimensional Lamé equations are solved using Cartesian and curvilinear orthogonal coordinates. It is proved that the solution includes only three independent harmonic functions. The general solution of equations of elasticity for stresses is found. The stress tensor is expressed in both coordinate systems in terms of three harmonic functions. The general solution of the problem of elasticity in cylindrical coordinates is presented as an example. The three-dimensional stress–strain state of an elastic cylinder subjected, on the lateral surface, to arbitrary forces represented by a series of eigenfunctions is determined. An axisymmetric problem for a finite cylinder is solved numerically     

Stress distribution in multilayered composites near loaded edges

The edge effect in layered composite material is studied using the piecewise-homogeneous body model and the exact equations of the theory of elasticity. It is assumed that continuously distributed normal forces act at the edges of the reinforcing layers. A plain strain state is considered and the stresses are expressed in terms of the solutions of a system of dual singular integral equations. The singularity of the stresses is determined by the solution procedure. The concentration of the reinforcing layers is assumed low and the interaction between them is not taken into account. A numerical algorithm is developed and numerical results on the stress distribution are presented Published in Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 134–144, April 2008.     

Anisotropic bodies with inhomogeneities: The case of a set of cracks

We use the Betti theorem to obtain the integral equations of the dynamic theory of elasticity for a multilayer convex body with an arbitrary elastic anisotropy of layers containing plane infinitely thin cracks. The systems of integral equations relate the displacement jumps to the stresses on the crack lips and are stated numerically in terms of Fourier transforms. For the case of plane-parallel layers with a set of plane cracks on the interfaces between the layers, we propose a simple numerical-analytic method for constructing the Fourier symbol, i.e., the matrix of the kernel of the system of integral equations. The method is stable for an arbitrary combination of continuous and discontinuous conditions on the layer boundaries. Numerical examples are given for a packet of four heterogeneous anisotropic layers.     

厚板的高阶剪切变形理论研究

已有多种厚板理论和高阶剪切变形模型,但仍需要进一步研究以更加完善.首先根据平均转角及上下表面剪应力自由这两个条件,提出了具有统一高阶剪切变形模型的中面位移模式,并将之表示为正交分解形式.根据正交特性,定义了板的广义应力;运用板问题应变能密度表示的等价性,提出了与广义应力功共轭的广义应变表示形式,建立了板的本构关系.证明了不同转角定义时虚功原理板理论表示的客观性,以及与三维弹性理论表示的等价性.运用虚功原理,建立了变分自洽的高阶厚板理论和变分渐近的低阶厚板理论,推导了相应的平衡方程及边界条件,分析了与已有板理论的异同.以广义应力形式建立了厚板理论的平衡方程,厘清了不同转角表示时板理论间的关系、低阶厚板理论与高阶厚板理论间的关系以及剪切系数计算等若干基本问题.对圣维南扭转问题的求解证明了该理论的正确性.      

Relation between the stress and couple-stress tensors in the microcontinuum theory of elasticity

The stress tensor is expressed in terms of an arbitrary symmetric tensor field of second rank and the couple-stress tensor. The stress and couple-stress tensors are represented by arbitrary tensor fields satisfying the homogeneous equilibrium equations. These tensors are also given in the form of the expressions satisfying the inhomogeneous equilibrium equations used in the microcontinuum theory of elasticity. The stress tensor functions are considered.     

Compatibility conditions and equations of motion in the linear micropolar theory of elasticity

An analog of Cesàro’s formula and several compatibility conditions are given for the three-dimensional and two-dimensional linear micropolar theory of elasticity in the form different from that used in the literature. A number of formulas are obtained to determine the antisymmetric part of the strain (stress) tensor in terms of the symmetric part of the strain tensor and the symmetric part of the bending-torsion (stress and couple-stress) tensor and to determine the antisymmetric part of the bending-torsion (couple-stress) tensor in terms of the symmetric part of the bending-torsion (couplestress) tensor. Some integro-differential equations of motion expressed in terms of the symmetric parts of the stress and couple-stress tensors are proposed for the micropolar theory of elasticity.     

Nonlinear equations of elastic deformation of plates

A method for constructing nonlinear equations of elastic deformation of plates with boundary conditions for stresses and displacements at the face surfaces in an arbitrary coordinate system is proposed. The initial three–dimensional problem of the nonlinear theory of elasticity is reduced to a one–parameter sequence of two–dimensional problems by approximating the unknown functions by truncated series in Legendre polynomials. The same unknowns are approximated by different truncated series. In each approximation, a linearized system of equations whose differential order does not depend on the boundary conditions at the face surfaces which can be formulated in terms of stresses or displacements is obtained.     

Version of the linear elasticity theory with a structural parameter

It is shown that, in the planar case, the system of constitutive equations of the linear elasticity theory should contain five independent equations. In the classical theory, only three equations are formulated, while the other two are contained in implicit form in the postulate of diffeomorphism, which is the assumption of smoothness of the displacement field. A closed elasticity model is constructed without the assumption of diffeomorphism, and it contains a structural parameter having a dimension of length. It is shown that, in a static version, macrodeformations depend on stresses and second derivatives of stresses with respect to the coordinates, while there is dispersion of longitudinal and transverse waves in the dynamics.     

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 %.     

Non-linear analysis of functionally graded plates in cylindrical bending under thermomechanical loadings based on a layerwise theory

A layerwise theory is used to analyze analytically displacements and stresses in functionally graded composite plates in cylindrical bending subjected to thermomechanical loadings. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The non-linear strain–displacement relations in the von Kármán sense are used to study the effect of geometric non-linearity. The equilibrium equations are solved exactly and also by using a perturbation technique. Numerical results are presented to show the effect of the material distribution on the deflections and stresses.     

Dislocations in second strain gradient elasticity

A second strain gradient elasticity theory is proposed based on first and second gradients of the strain tensor. Such a theory is an extension of first strain gradient elasticity with double stresses. In particular, the strain energy depends on the strain tensor and on the first and second gradient terms of it. Using a simplified but straightforward version of this gradient theory, we can connect it with a static version of Eringen’s nonlocal elasticity. For the first time, it is used to study a screw dislocation and an edge dislocation in second strain gradient elasticity. By means of this second gradient theory it is possible to eliminate both strain and stress singularities. Another important result is that we obtain nonsingular expressions for the force stresses, double stresses and triple stresses produced by a straight screw dislocation and a straight edge dislocation. The components of the force stresses and of the triple stresses have maximum values near the dislocation line and are zero there. On the other hand, the double stresses have maximum values at the dislocation line. The main feature is that it is possible to eliminate all unphysical singularities of physical fields, e.g., dislocation density tensor and elastic bend-twist tensor which are still singular in the first strain gradient elasticity.     

Gradient elasticity in statics and dynamics: An overview of formulations,length scale identification procedures,finite element implementations and new results

In this paper, we discuss various formats of gradient elasticity and their performance in static and dynamic applications. Gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher-order spatial derivatives of strains, stresses and/or accelerations. We focus on the versatile class of gradient elasticity theories whereby the higher-order terms are the Laplacian of the corresponding lower-order terms. One of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum. We start with discussing the general Mindlin theory, that in its most general form has 903 constitutive elastic parameters but which were reduced by Mindlin to three independent material length scales. Further simplifications are often possible. In particular, the Aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures. We also address how this can be extended to dynamics. An overview of length scale identification and quantification procedures is given. Finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions. Details are provided for particular formats of gradient elasticity that can be implemented with simple, linear finite element shape functions. New numerical results show the removal of singularities in statics and dynamics, as well as the size-dependent mechanical response predicted by gradient elasticity.     

On equations of the linear theory of shells with surface stresses taken into account

We construct equations of equilibrium and constitutive relations of linear theory of plates and shells with transverse shear strain taken into account, which are based on reducing the spatial elasticity relations with surface stresses taken into account to two-dimensional equations given on the shell median surface. We analyze the influence of surface elasticity moduli on the effective stiffness of plates and shells.     

Near field behavior of in-plane crack extension in nonlinear incompressible material

The nonlinear theory of finite elasticity is applied to obtain the in-plane displacement and stresses in the immediate vicinity of the crack tip. Incompressibility, homogeneity, elasticity and isotropy are assumed for the material while the resultant shear stress and shear strain are assumed to follow a nonlinear hardening/softening behavior. The system of governing differential equations becomes nonelliptical when the strains are sufficiently large.     

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.     

Defects in gradient micropolar elasticity—I: screw dislocation

A gradient micropolar elasticity is proposed based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium. This theory is an extension of the theory of micropolar elasticity with couple stresses together with gradient elasticity in a way that in addition to hyper stresses, hyper couple stresses also appear. In particular, the strain energy, besides its dependence upon the distortion and bend-twist terms of a micropolar medium (Cosserat continuum), depends also on distortion and bend-twist gradients. Using a simplified but rigorous version of this gradient theory, we can connect it to Eringen's nonlocal micropolar elasticity. In addition, it is used to study a screw dislocation in gradient micropolar elasticity. One important result is that we obtained nonsingular expressions for the force and couple stresses. The components of the force stress have maximum values near the dislocation line and those of the couple stress have maximum values at the dislocation line.     

Velocity dispersion in an elastic plate with microstructure: effects of characteristic length in a couple stress model

Microstructural effects become important, when dimensions of the heterogeneous material are comparable to the length scale of microstructure and the state of stress needs to be defined in a non-local manner. Linear theory of elasticity, which is associated with the concept of homogeneity of material and local stresses, cannot describe the behavior of the materials with microstructures. In this study, Couple stress theory of elasticity has been employed to capture the size effects on the propagation of Lamb waves in an elastic plate with microstructure. Effects on the dispersion curves of Lamb waves are studied, when the characteristic length of the material is comparable to cell size. The governing equations of couple stress theory, involving stresses and couple stresses are solved to study the impact of different characteristic lengths, comparable with cell size. Since bone is a material with microstructure, so for numerical calculations and graphical representation of the results, the plate is considered to have mechanical properties typically used for bones.     

A refined theory of elastic thick plates for extensional deformation

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.     

Exact solutions to a problem in terms of stresses for incompressible elastic conical bodies

Exact solutions to the elasticity theory problem in terms of stresses for an incompressible conical body of arbitrary shape under the action of a given concentrated force applied at its vertex are given and analyzed. A solution in terms of stresses with a singularity whose order is higher by one than that in the classical solution is discussed. The surface load at the boundary of the conical body corresponding to such a solution is obtained.