An approach based on distributed dislocations and disclinations for crack problems in couple-stress elasticity

The technique of distributed dislocations proved to be in the past an effective approach in studying crack problems within classical elasticity. The present work is intended to extend this technique in studying crack problems within couple-stress elasticity, i.e. within a theory accounting for effects of microstructure. This extension is not an obvious one since rotations and couple-stresses are involved in the theory employed to analyze the crack problems. Here, the technique is introduced to study the case of a mode I crack. Due to the nature of the boundary conditions that arise in couple-stress elasticity, the crack is modeled by a continuous distribution of climb dislocations and constrained wedge disclinations (the concept of ‘constrained wedge disclination’ is first introduced in the present work). These distributions create both standard stresses and couple stresses in the body. In particular, it is shown that the mode-I case is governed by a system of coupled singular integral equations with both Cauchy-type and logarithmic kernels. The numerical solution of this system shows that a cracked solid governed by couple-stress elasticity behaves in a more rigid way (having increased stiffness) as compared to a solid governed by classical elasticity. Also, the stress level at the crack-tip region is appreciably higher than the one predicted by classical elasticity.     

Uniqueness for plane crack problems in dipolar gradient elasticity and in couple-stress elasticity

The present work deals with the uniqueness theorem for plane crack problems in solids characterized by dipolar gradient elasticity. The theory of gradient elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the second gradient of the displacement (additional term). Specific cases of the general theory employed here are the well-known theory of couple-stress elasticity and the recently popularized theory of strain-gradient elasticity. These cases are also treated in the present study. We consider an anisotropic material response of the cracked plane body, within the linear version of gradient elasticity, and conditions of plane-strain or anti-plane strain. It is emphasized that, for crack problems in general, a uniqueness theorem more extended than the standard Kirchhoff theorem is needed because of the singular behavior of the solutions at the crack tips. Such a theorem will necessarily impose certain restrictions on the behavior of the fields in the vicinity of crack tips. In standard elasticity, a theorem was indeed established by Knowles and Pucik [Knowles, J.K., Pucik, T.A., 1973. Uniqueness for plane crack problems in linear elastostatics. J. Elast. 3, 155–160], who showed that the necessary conditions for solution uniqueness are a bounded displacement field and a bounded body-force field. In our study, we show that the additional (to the two previous conditions) requirement of a bounded displacement-gradient field in the vicinity of the crack tips guarantees uniqueness within the general form of the theory of dipolar gradient elasticity. In the specific cases of couple-stress elasticity and pure strain-gradient elasticity, the additional requirement is less stringent. This only involves a bounded rotation field for the first case and a bounded strain field for the second case.     

Higher order fields for damaged nonlinear antiplane shear notch, crack and inclusion problems

Damaged nonlinear antiplane shear problems with a variety of singularities are studied analytically. A deformation plasticity theory coupled with damage is employed in analysis. The effect of microscopic damage is considered in terms of continuum damage mechanics approach. An exact solution for the general damaged nonlinear singular antiplane shear problem is derived in the stress plane by means of a hodograph transformation, then corresponding higher order asymptotic solutions are obtained by reversing the stress plane solution to the physical plane. As example, traction free sharp notch and crack, rigid sharp wedge and flat inclusion, and mixed boundary sharp notch problems are investigated, respectively. Consequently, higher order fields are obtained, in which analytical expressions of the dominant and second order singularity exponents and angular distribution functions of the near tip fields are derived. Effects of the damage and hardening exponents of materials and the geometric angle of notch/wedge on the near tip quantities are discussed in detail. It is found that damage leads to a weaker dominant singularity of stress, but to little stronger singularities of the dominant and second order terms of strain compared to that for undamaged material. It is also seen that damage has important effect on the angular distribution functions of the near tip stress and strain fields. As special cases, higher order analytical solutions of the crack and rigid flat inclusion tip fields are obtained, respectively, by reducing the notch/wedge tip solutions. Effects of damage and hardening exponents on the dominant and second order terms in the solutions of the crack and inclusion tip fields are discussed.     

Variational analysis of gradient elastic flexural plates under static loading

Gradient elastic flexural Kirchhoff plates under static loading are considered. Their governing equation of equilibrium in terms of their lateral deflection is a sixth order partial differential equation instead of the fourth order one for the classical case. A variational formulation of the problem is established with the aid of the principle of virtual work and used to determine all possible boundary conditions, classical and non-classical ones. Two circular gradient elastic plates, clamped or simply supported at their boundaries, are analyzed analytically and the gradient effect on their static response is assessed in detail. A rectangular gradient elastic plate, simply supported at its boundaries, is also analyzed analytically and its rationally obtained boundary conditions are compared with the heuristically obtained ones in a previous publication of the authors. Finally, a plate with two opposite sides clamped experiencing cylindrical bending is also analyzed and its response compared against that for the cases of micropolar and couple-stress elasticity theories.     

Asymptotics of solutions of the three-dimensional elasticity equations for compressible and incompressible bodies

We analyze the leading terms in the general asymptotic expansions of solutions of the first boundary value problem of three-dimensional elasticity in displacements. The cases of compressible and incompressible bodies, which have substantially different statements, are considered separately. The minimum-to-maximum ratio of characteristic dimensions of the elastic body is a natural small asymptotic parameter. The third dimension can be of any “intermediate” order, including the endpoints. For example, such a geometry is typical of bodies that simultaneously have characteristic macro-, micro-, and nano-dimensions in three coordinate axes.     

Asymptotic theory of the Boltzmann system,for a steady flow of a slightly rarefied gas with a finite Mach number: General theory

A steady rarefied gas flow with Mach number of the order of unity around a body or bodies is considered. The general behaviour of the gas for small Knudsen numbers is studied by asymptotic analysis of the boundary-value problem of the Boltzmann equation for a general domain. The effect of gas rarefaction (or Knudsen number) is expressed as a power series of the square root of the Knudsen number of the system. A series of fluid-dynamic type equations and their associated boundary conditions that determine the component functions of the expansion of the density, flow velocity, and temperature of the gas is obtained by the analysis. The equations up to the order of the square root of the Knudsen number do not contain non-Navier–Stokes stress and heat flow, which differs from the claim by Darrozes (in Rarefied Gas Dynamics, Academic Press, New York, 1969). The contributions up to this order, except in the Knudsen layer, are included in the system of the Navier–Stokes equations and the slip boundary conditions consisting of tangential velocity slip due to the shear of flow and temperature jump due to the temperature gradient normal to the boundary.     

Stress singularities for three-dimensional corners using the boundary integral equation method

The boundary integral equation method is developed to study three-dimensional asymptotic singular stress fields at vertices of a pyramidal notch or inclusion in an isotropic elastic space. Two-dimensional boundary integral equations are used for the infinite body with pyramidal notches and inclusions when either stresses or displacements are specified on its surface. Applying the Mellin integral transformation reduces the problem to one-dimensional singular integral equations over a closed, piece-wise smooth line. Using quadrature formulas for regular and singular integrals with Hilbert and logarithmic kernels, these integral equations are reduced to a homogeneous system of linear algebraic equations. Setting its determinant to zero provides a characteristic equation for the determination of the stress singularity power. Numerical results are obtained and compared against known eigenvalues from the literature for an infinite region with a conical notch or inclusion, for a Fichera vertex, and for a half-space with a wedge-shaped notch or inclusion.     

Corner stress singularities in an FGM thin plate

Describing the behaviors of stress singularities correctly is essential for obtaining accurate numerical solutions of complicated problems with stress singularities. This analysis derives asymptotic solutions for functionally graded material (FGM) thin plates with geometrically induced stress singularities. The classical thin plate theory is used to establish the equilibrium equations for FGM thin plates. It is assumed that the Young’s modulus varies along the thickness and Poisson’s ratio is constant. The eigenfunction expansion method is employed to the equilibrium equations in terms of displacement components for an asymptotic analysis in the vicinity of a sharp corner. The characteristic equations for determining the stress singularity order at the corner vertex and the corresponding corner functions are explicitly given for different combinations of boundary conditions along the radial edges forming the sharp corner. The non-homogeneous elasticity properties are present only in the characteristic equations corresponding to boundary conditions involving simple support. Finally, the effects of material non-homogeneity following a power law on the stress singularity orders are thoroughly examined by showing the minimum real values of the roots of the characteristic equations varying with the material properties and vertex angle.     

Uniqueness in linear elastostatics for problems involving unbounded bodies

The problem of the uniqueness of elastostatic solutions to various boundary value problems involving unbounded two- and three-dimensional bodies is considered. The general boundary value problem is first considered for anisotropic bodies on which the elasticity tensor is uniformly positive definite. The displacement problem is then considered for the cases where the body is homogeneous and isotropic and the elasticity tensor is strongly-elliptic. Finally, the traction problem on homogeneous, isotropic bodies is considered with fairly little restriction placed on the Lamé moduli. In the first two cases no restriction is placed on the geometry of the body other than the normal assumptions allowing for the use of the divergence theorem. For the traction problem, however, it is assumed that one component of the outward normal to the boudary of the body almost never vanishes. In each case it is shown that the desired uniqueness results hold with fairly mild restrictions placed on the displacement or stress fields (depending on the problem) in a neighborhood of infinity. Where possible, the results are shown to hold even though the solutions may possess the sort of discontinuities and singularities commonly found in problems involving cracks, corners, and bonded interfaces.Related results involving the insolvability of certain non-trivial boundary value problems and the behavior of certain elastic states are, where appropriate, developed.     

Couple stress theory for solids

By relying on the definition of admissible boundary conditions, the principle of virtual work and some kinematical considerations, we establish the skew-symmetric character of the couple-stress tensor in size-dependent continuum representations of matter. This fundamental result, which is independent of the material behavior, resolves all difficulties in developing a consistent couple stress theory. We then develop the corresponding size-dependent theory of small deformations in elastic bodies, including the energy and constitutive relations, displacement formulations, the uniqueness theorem for the corresponding boundary value problem and the reciprocal theorem for linear elasticity theory. Next, we consider the more restrictive case of isotropic materials and present general solutions for two-dimensional problems based on stress functions and for problems of anti-plane deformation. Finally, we examine several boundary value problems within this consistent size-dependent theory of elasticity.     

Asymptotic splitting in the three-dimensional problem of elasticity for non-homogeneous piezoelectric plates

A novel asymptotic approach to the theory of non-homogeneous anisotropic plates is suggested. For the problem of linear static deformations we consider solutions, which are slowly varying in the plane of the plate in comparison to the thickness direction. A small parameter is introduced in the general equations of the theory of elasticity. According to the procedure of asymptotic splitting, the principal terms of the series expansion of the solution are determined from the conditions of solvability for the minor terms. Three-dimensional conditions of compatibility make the analysis more efficient and straightforward. We obtain the system of equations of classical Kirchhoff's plate theory, including the balance equations, compatibility conditions, elastic relations and kinematic relations between the displacements and strain measures. Subsequent analysis of the edge layer near the contour of the plate is required in order to satisfy the remaining boundary conditions of the three-dimensional problem. Matching of the asymptotic expansions of the solution in the edge layer and inside the domain provides four classical plate boundary conditions. Additional effects, like electromechanical coupling for piezoelectric plates, can easily be incorporated into the model due to the modular structure of the analysis. The results of the paper constitute a sound basis to the equations of the theory of classical plates with piezoelectric effects, and provide a trustworthy algorithm for computation of the stressed state in the three-dimensional problem. Numerical and analytical studies of a sample electromechanical problem demonstrate the asymptotic nature of the present theory.     

Homogenization of the equations of the Cosserat theory of elasticity of inhomogeneous bodies

The paper deals with the homogenization of a boundary value problem for an inhomogeneous body with Cosserat properties, which is referred to as the original problem. The homogenization process is understood as a method for representing the solution of the original problem in terms of the solution of precisely the same problem for a body with homogeneous properties. The problem for a body with homogeneous properties is called the accompanying problem, and the body itself, the accompanying homogeneous body. As a rule, a constructive homogenization procedure includes the following three stages: at the first stage, the properties of the inhomogeneous body are used to find the properties of the accompanying homogeneous body (efficient properties); at the second stage, the boundary value problem is solved for the accompanying body; at the third stage, the solution of the accompanying problem is used to find the solution of the original problem. This approach was implemented in mechanics of composite materials constructed of numerous representative elements. A significant contribution to the development of mechanics of composites is due to Rabotnov [1–3] and his students. Recently, the homogenization method has been widely used to solve problems for composites of regular structure by expanding the solution of the original problem in a power series in a small geometric parameter equal to the ratio of the characteristic dimension of the periodicity cell to the characteristic dimension of the entire body. The papers by Bakhvalov [4–6] and Pobedrya [7] were the first in the field. At present, there are numerous monographs partially or completely dealing with the method of a small geometric parameter [8–14]. Isolated problems for inhomogeneous bodies with nonperiodic dependence of their properties on the coordinates were considered by many authors. Most of such papers published before 1973 are collected in two vast bibliographic indices [15, 16]. General methods were considered, and many specific problems of the theory of elasticity of continuously inhomogeneous bodies were solved in Lomakin’s papers and his monograph [17]. The theory of torsion of inhomogeneous anisotropic rods was considered in [18]. In 1991, in his Doctoral dissertation, one of the authors of this paper proposed a version of the homogenization method based on an integral formula representing the solution of the original static problem of inhomogeneous elasticity via the solution of the accompanying problem [19, 20]. An integral formula for the dynamic problem of elasticity was published somewhat later [21]. This integral formula was used to develop a constructive method for the homogenization of the dynamic problem of inhomogeneous elasticity, which can be used in the case of both periodic and nonperiodic inhomogeneity of the properties [22]. The integral formula in the case of the Cosserat theory of elasticity was published in [23]. The present paper briefly presents constructive methods for homogenizing the problems of the Cosserat theory of elasticity based on the integral formula.     

ANALYSIS OF 3-D NOTCHED/CRACKED STRUCTURES BY USING EXTENDED BOUNDARY ELEMENT METHOD 1)

在线弹性理论中,三维 V 形切口/裂纹结构尖端区域存在多重应力奇异性,常规数值方法不易求解. 本文提出和建立了三维扩展边界元法 (XBEM),用于分析三维线弹性 V 形切口/裂纹结构完整的位移和应力场. 先将三维线弹性 V 形切口/裂纹结构分为尖端小扇形柱和挖去小扇形柱后的外围结构. 尖端小扇形柱内的位移函数采用自尖端径向距离 $r$ 的渐近级数展开式表达,其中尖端区域的应力奇异指数、位移和应力特征角函数通过插值矩阵法获得. 而级数展开式各项的幅值系数作为基本未知量. 挖去扇形域后的外围结构采用常规边界元法分析. 两者方程联立求解可获得三维 V 形切口/裂纹结构完整的位移和应力场,包括切口/裂纹尖端区域精细的应力场. 扩展边界元法具有半解析法特征,适用于一般三维 V 形切口/裂纹结构完整位移场和应力场的分析,其解可精细描述从尖端区域到整体结构区域的完整应力场. 作者研制了三维扩展边界元法程序,文中给出了两个算例,通过计算结果分析,表明了扩展边界元法求解三维 V 形切口/裂纹结构完整应力场的准确性和有效性.     

三维切口/裂纹结构的扩展边界元法分析

在线弹性理论中,三维 V 形切口/裂纹结构尖端区域存在多重应力奇异性,常规数值方法不易求解. 本文提出和建立了三维扩展边界元法 (XBEM),用于分析三维线弹性 V 形切口/裂纹结构完整的位移和应力场. 先将三维线弹性 V 形切口/裂纹结构分为尖端小扇形柱和挖去小扇形柱后的外围结构. 尖端小扇形柱内的位移函数采用自尖端径向距离 $r$ 的渐近级数展开式表达,其中尖端区域的应力奇异指数、位移和应力特征角函数通过插值矩阵法获得. 而级数展开式各项的幅值系数作为基本未知量. 挖去扇形域后的外围结构采用常规边界元法分析. 两者方程联立求解可获得三维 V 形切口/裂纹结构完整的位移和应力场,包括切口/裂纹尖端区域精细的应力场. 扩展边界元法具有半解析法特征,适用于一般三维 V 形切口/裂纹结构完整位移场和应力场的分析,其解可精细描述从尖端区域到整体结构区域的完整应力场. 作者研制了三维扩展边界元法程序,文中给出了两个算例,通过计算结果分析,表明了扩展边界元法求解三维 V 形切口/裂纹结构完整应力场的准确性和有效性.      

DYNAMIC ANALYSIS OF A GRADIENT ELASTIC POLYMERIC FIBER

A dynamic analysis of an elastic gradient-dependent polymeric fiber subjected to a periodic excitation is considered.A nonlinear gradient elasticity constitutive equation with strain-dependent gradient coefficients is first derived and the dispersive wave propagation properties for its linearized counterpart are briefly discussed.For the linearized problem a variational formulation is also developed to obtain related boundary conditions of both classical(standard)and non-classical(gradient)type.Analytical solutions in the form of Fourier series for the fiber’s displacement and strain fields are provided.The solutions depend on a dimensionless scale parameter(the diameter to length radio d = D/L)and,therefore,size effects are captured.     

Eigenoscillations of an elastic body with a rough surface

Explicit presentations for the initial terms of the asymptotic solution of the spectral problem of the elasticity theory in a plane region with a rapidly oscillating boundary are obtained. Based on asymptotic formulas, two methods for problem modeling are proposed: with the use of Wenzel’s boundary conditions and with the use of the principle of a smooth image of a singularly perturbed boundary. Various approaches to justification of asymptotic presentations are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 103–114, November–December, 2007.     

The singular field near a sharp notch with mixed boundary conditions for hardening materials under longitudinal shear

The field behavior near a sharp notch tip with mixed homogeneous stress and displacement boundary conditions is examined for a power law hardening material. Using the hodograph transformation, the singularity and the angular distribution of the fields are determined. Special cases as those for linear elastic and perfect plastic materials are discussed.     

极坐标下弹性力学的一个新解答

在极坐标下将Hamilton体系下的分离变量法应用到弹性力学的非齐次边界情况,得到了一个新解答,利用这个新解可以求解一类弹性力学问题。文中给出了具体例子。     

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.