On edge dislocation in a multilayered laminate

We investigate the elastic field induced by an edge dislocation in a multilayered laminated composite composed of (N ? 2) thin bonded elastic layers sandwiched between two semi-infinite elastic media. A simple closed-form solution is obtained when all the phases have equal shear modulus but different Poisson's ratios, and when the dislocation is located in the upper semi-infinite phase. The image force acting on the dislocation due to its interaction with the multilayered structure is also derived. Several specific examples are discussed in detail to illustrate the mobility of the edge dislocation. Some interesting behaviors of the dislocation are observed. Our results indicate that it is possible to find at most (N ? 2) equilibrium positions for the edge dislocation in an N-phase composite structure.     

具有叉指式电极的1-3型压电复合材料层合板力电耦合行为的三维精确分析

基于三维弹性理论和压电理论,导出了含有1-3型压电复合材料层的有限长矩形层合简支板的静力平衡方程和边界条件,给出了该层合板在叉指式电极和外力共同作用下力电耦合特性的三维精确解.数值算例的计算结果与有限元解进行了对比,取得了很好的一致性.研究了压电矩阵各向异性和刚度矩阵各向异性以及电势等因素对其挠曲面扭率最大值的影响.数值结果表明层合板扭率最大值的绝对值随压电矩阵各向异性系数Rd的增大而增大并随刚度矩阵各向异性系数Rc的减小而增加.     

Plastic size effect analysis of lamellar composites using a discrete dislocation plasticity approach

Plastic size effect analysis of lamellar composites consisting of elastic and elastic-plastic layers is performed using a discrete dislocation plasticity approach, which is based on applying periodic homogenization to the superposition method for discrete dislocation plasticity. In this approach, the decomposition of displacements into macro and perturbed components circumvents the calculation of superposing displacement fields induced by dislocations in an infinitely homogeneous medium, resulting in two periodic boundary value problems specialized for the analysis of representative volume elements. The present approach is verified by analyzing a model lamellar composite that includes edge dislocations fixed at interfaces. The plastic size effects due to dislocation pile-ups at interfaces are also analyzed. The analysis shows that, strain hardening in elastic-plastic layers arises depending on two factors, namely the thickness and stiffness of elastic layers; and the gap between slip planes in adjacent elastic-plastic layers. In the case where the thickness of elastic layers is several dozen nm, strain hardening in elastic-plastic layers is restrained as the gap of the slip planes decreases. This particular effect is attributed to the long range stress due to pile-ups in adjacent elastic-plastic layers.     

Computing image stress in an elastic cylinder

We develop numerical methods that efficiently compute image stress fields of defects in an elastic cylinder. These methods facilitate dislocation dynamics simulations of the plastic deformation of micro-pillars. Analytic expressions of the image stress have been found in the Fourier space, taking advantage of the translational and rotational symmetries of the cylinder. To facilitate numerical calculation, the solution is then transformed into the stress field in an infinite elastic body produced by a distribution of body forces or image dislocations. The use of the fast Fourier transform (FFT) method makes the algorithms numerically efficient.     

State space solution to 3D multilayered elastic soils based on order reduction method

Starting with the governing equations in terms of displacements of 3D elastic media, the solutions to displacement components and their first derivatives are obtained by the application of a double Fourier transform and an order reduction method based on the Cayley-Hamilton theorem. Combining the solutions and the constitutive equations which connect the displacements and stresses, the transfer matrix of a single soil layer is acquired. Then, the state space solution to multilayered elastic soils is further obtained by introducing the boundary conditions and continuity conditions between adjacent soil layers. The numerical analysis based on the present theory is carried out, and the vertical displacements of multilayered foundation with a weak and a hard underlying stratums are compared and discussed.     

Analytic Solution for a Line Edge Dislocation in a Bimaterial System Incorporating Interface Elasticity

We examine the plane strain deformations of a bimaterial system consisting of a line edge dislocation interacting with a flat interface between two dissimilar isotropic half-planes in which the additional effects of interface elasticity are incorporated into the model of deformation. The entire system is assumed to be free of any external loading. Despite the fact that it is generally accepted that the separate interface modulus describing interface elasticity is permitted to take negative values, we show that simple closed-form solutions for the dislocation-induced stress field and the image force acting on the dislocation are available only when the interface modulus is assumed to be positive; the corresponding system admits no valid solutions when the interface modulus is negative. We present several numerical examples to illustrate our solutions. Additionally, we show that the influence of interface elasticity on the dislocation-induced interfacial stress field decays with increasing hardness of the adjoining half-plane (free of the dislocation). Moreover, we find that for a given (positive) in-plane interface modulus, the corresponding interface effects on the image force (acting on the dislocation) can reach maximum or minimum values when the Burgers vector of the dislocation is either parallel or perpendicular to the interface.     

Effect of interface stresses on the elastic deformation of an elastic half-plane containing an elastic inclusion

The effect of the interface stresses is studied upon the size-dependent elastic deformation of an elastic half-plane having a cylindrical inclusion with distinct elastic properties. The elastic half-plane is subjected to either a uniaxial loading at infinity or a uniform non-shear eigenstrain in the inclusion. The straight edge of the half-plane is either traction-free, or rigid-slip, or motionless, which represents three practical situations of mechanical structures. Using two-dimensional Papkovich–Neuber potentials and the theory of surface/interface elasticity, the elastic field in the elastic half-plane is obtained. Comparable with classical result, the new formulation renders the significant effect of the interface stresses on the stress distribution in the half-plane when the radius of the inclusion is reduced to the nanometer scale. Numerical results show that the intensity of the influence depends on the surface/interface moduli, the stiffness ratio of the inclusion to the surrounding material, the boundary condition on the edge of the half-plane and the proximity of the inclusion to the edge.     

Dynamic stress intensity factors for two parallel interface cracks between a nonhomogeneous bonding layer and two dissimilar elastic half-planes subject to an impact load

Some composite materials are constructed of two dissimilar half-planes bonded by a nonhomogeneous elastic layer. In the present study, a crack is situated at the interface between the upper half-plane and the bonding layer of such a material, and another crack is located at the interface between the lower half-plane and the bonding layer. The material properties of the bonding layer vary continuously from those of the lower half-plane to those of the upper half-plane. Incoming shock stress waves impinge upon the two interface cracks normal to their surfaces. Fourier transformations were used to reduce the boundary conditions for the cracks to two pairs of dual integral equations in the Laplace domain. To solve these equations, the differences in the crack surface displacements were expanded in a series of functions that are zero-valued outside the cracks. The unknown coefficients in the series were solved using the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors were defined in the Laplace domain and were inverted numerically to physical space. Dynamic stress intensity factors were calculated numerically for selected crack configurations.     

Two-dimensional transient dynamic response of orthotropic layered media

In this study, two-dimensional transient dynamic response of orthotropic plane layered media is investigated. The plane multilayered media consist of N different generally orthotropic, homogeneous and linearly elastic layers with different ply angles. In the generally orthotropic layer, representing a ply reinforced by unidirectional fibers with an arbitrary orientation angle, the principal material directions do not coincide with body coordinate axes. The solution is obtained by employing a numerical technique which combines the use of Fourier transform with the method of characteristics. The numerical results are displayed in curves denoting the variations of stress and displacement components with time at different locations. These curves clearly reveal, in wave profiles, the scattering effects caused by the reflections and refractions of waves at the boundaries and at the interfaces of the layers, and also the effects of anisotropy caused by fiber orientation angle. The curves properly predict the sharp variations in the response at the neighborhood of the wave fronts, which shows the power of the numerical technique employed in the study. By suitably adjusting the elastic constants, the results for multilayered media with transversely isotropic layers, or layers with cubic symmetry, or isotropic layers can easily be obtained from the general formulation. Furthermore, solutions for some special cases, including Lamb’s problem for an elastic half-space, are obtained and compared with the available solutions in the literature and very good agreement is found. Preliminary version presented at the Second International Congress on Mechatronics (MECH2K3), Graz, Austria, July 14-17, 2003.     

A wave propagation problem for non-uniform screw dislocation motion in a viscoelastic half-plane

The wave propagation problem for a largely arbitrary anti-plane displacement discontinuity imposed along a line perpendicular to the surface of a stress-free linearly viscoelastic half-plane is considered. The general Laplace transform solution is obtained and then inverted for the case of a screw dislocation moving at an arbitrary speed in a Maxwell material. It is shown that the material viscoelasticity alters the coefficient of the dislocation edge stress singularity and damps the surface displacements from the elastic values. The surface damping increases with time, distance from the dislocation path and dislocation speed, whether sub- or supersonic.     

反平面弹性中边缘内分叉裂纹的奇异积分方程解法

利用复变函数和奇异积分方程方法,求解反平面弹性中半平面边缘内分叉裂纹问题。提出了满足半平面边界自由的由分布位错密度表示的半平面中单裂纹的基本解,此基本解由主要部分和辅助部分组成。将半平面边缘内分叉裂纹问题看作是许多单裂纹问题的叠加,建立了以分布位错密度为未知函数的Cauchy型奇异积分方程组。然后,利用半开型积分法则求解奇异积分方程,得到了裂纹端处的应力强度因子。文中给出两个数值算例的计算结果。     

Determination of stress intensity factors in half-plane containing several moving cracks

The dynamic stress intensity factors in a half-plane weakened by several finite moving cracks are investigated by employing the Fourier complex transformation. Stress analysis is performed in a half-plane containing a single dislocation and without dislocation. An exact solution in a closed form to the stress fields and displacement is ob- tained. The Galilean transformation is used to transform between coordinates connected to the cracks. The stress components are of the Cauchy singular kind at the location of dislocation and the point of application of the the influence of crack length and crack running force. Numerical examples demonstrate velocity on the stress intensity factor.     

地磁场作用半平面铁磁结构广义Flamant问题的弹性变形扰动静磁场解

采用Fourier变换方法对广义Flamant载荷作用半平面铁磁结构弹性变形诱导的静磁场分布进行求解,得到的表达式经退化与集中力作用情形的扰动静磁场分布表达式相同。数值结果表明, 载荷分布形式通过影响结构的弹性变形进而影响扰动磁场分布的形貌和大小,扰动静磁场分布的形貌与载荷极大值的分布区域有直接关联,扰动静磁场分布的最值与作用力的大小正相关;扰动静磁场强度的法向分量的最值随着与结构表面的距离的减小而增大,随载荷分布尺寸的增大而增大。     

Inclusions in a two-phase elastic space—plane circular and rectangular inclusions

This paper is concerned with the elastic field generated in two bonded isotropic half-planes containing either a circular or a rectangular inclusion, each having the same elastic properties as those of the surrounding half-plane. The circular inclusion undergoes a transformation which in the absence of the surrounding material would be an arbitrary uniform stress-free strain, while that imposed on the rectangular inclusion corresponds to a pure dilatation.     

求解多层弹性半空间轴对称问题的精确刚度矩阵法

本文首先从弹性力学的基本方程出发，利用Hankel积分变换等数学手段，推导出了单层弹性半空问轴对称问题的刚度矩阵，然后按传统的有限元方法组成总体刚度矩阵。通过求解由总体刚度矩阵所构成的代数方程和Hankel积分逆变换就可解出静荷载作用下多层弹性半空间轴对称问题的精确解。由于刚度矩阵的元素中不含有正指数项，计算时不会出现溢出的现象，从而克服了传递矩阵法的缺点。由于在推导过程中摒弃了应力函数的选择，使得问题的求解更加理论化和合理化，同时也为进一步研究这类问题如温度场，动力学等方向奠定了理论基础。最后，文中还给出了计算实例来证明推导结果的准确性。     

A layered composite containing a crack in its lower layer loaded by a rigid stamp

The elastostatic plane problem of a layered composite containing an internal or edge crack perpendicular to its boundaries in its lower layer is considered. The layered composite consists of two elastic layers having different elastic constants and heights and rests on two simple supports. Solution of the problem is obtained by superposition of solutions for the following two problems: The layered composite subjected to a concentrated load through a rigid rectangular stamp without a crack and the layered composite having a crack whose surface is subjected to the opposite of the stress distribution obtained from the solution of the first problem. Using theory of Elasticity and Fourier transform technique, the problem is formulated in terms of two singular integral equations. Solving these integral equations numerically by making use of Gauss–Chebyshev integration, numerical results related to the normal stress σ_x(0,y), the stress-intensity factors, and the crack opening displacements are presented and shown graphically for various dimensionless quantities.     

Thermal stresses around two parallel cracks in two bonded dissimilar elastic half-planes

Summary Thermal stresses around two parallel cracks in two bonded dissimilar elastic half-planes are determined. One of the cracks lies in the upper half-plane, while the other is in the lower half-plane. Uniform heat flow is assumed to be at right angles to the interface. Application of the Fourier transform technique reduces the problem to that of solving dual integral equations. To solve the equations, the difference of the crack surface temperature and those of the crack surface displacements are expanded in a series of functions which are automatically zero outside the cracks. The unknown coefficients in the series are solved by the Schmidt method. The stress intensity factors are calculated numerially for composite materials featuring a ceramic upper half-plane and a steel lower half-plane.

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Wärmespannungen um zwei parallele Risse in zwei verbundenen, verschiedenen, elastischen Halbunendlichplatten

Übersicht Es werden die Wärmespannungen um zwei parallele Risse in zwei verbundenen, verschiedenen, elastischen Halbunendlichplatten bestimmt. Einer der beiden Risse liegt in der oberen Halbunendlichplatte, der andere in der unteren. Es wird angenommen, daß ein gleichmäßiger Wärmefluß senkrecht zur Grenzfläche erfolgt. Die Anwendung der Fourier-Transformation reduziert das Problem auf die Lösung dualer Integralgleichungen. Zur Lösung der Gleichungen werden die Temperatur-sowie die Verschiebungsdifferenzen an der Rißoberfläche in eine Reihe von Funktionen entwickelt, die außerhalb der Risse automatisch zu Null werden. Die unbekannten Koeffizienten dieser Reihe werden dann über das Schmidt-Verfahren bestimmt. Anschließend werden für Verbundmaterialien, bei denen die obere Halbunendlichplatte aus Keramik und die untere aus Stahl besteht, die Spannungsintensitätsfaktoren numerisch berechnet.

     

The Somigliana ring dislocation

A basic elasticity solution applicable to an important class of internal stress problems related, for example, to fiber-matrix composites and spalling of cylindrical coatings is obtained. The basic problem that has been solved is that of the singular stress-displacement field resulting from the introduction of a Somigliana ring dislocation in an isotropic linear elastic solid. The Burgers vector of this dislocation has two components, one being normal to the plane of the circular ring dislocation (Volterra type) and the other being in the radial direction of the ring dislocation everywhere (Somigliana type). The analytical solution, in terms of complete elliptic integrals of the first, second and third kinds, is obtained using the Love stress function and Fourier transform. The results are verified numerically and by examining various limiting cases, including the straight edge dislocation as the radius of the dislocation loop tends to infinity, the orthogonal pair of dipoles as the radius tends to zero, and the Lamé solution of a cylindrical bar and a cylindrical hole in an infinite medium as the axial location of the dislocation tends to minus infinity. The resulting analytical solution is considered as a step towards evaluating both the extended stress field around and interactions among various three-dimensional defects such as cylindrical cracks, fiber-tips and fiber-matrix debonding.     

Three-dimensional continuum analysis for a unidirectional composite with a broken fiber

The three-dimensional problem of a periodic unidirectional composite with a penny-shaped crack traversing one of the fibers is analyzed by the continuum equations of elasticity. The solution of the crack problem is represented by a superposition of weighted unit normal displacement jump solutions, everyone of which forms a Green’s function. The Green’s functions for the unbounded periodic composite are obtained by the combined use of the representative cell method and the higher-order theory. The representative cell method, based on the triple discrete Fourier transform, allows the reduction of the problem of an infinite domain to a problem of a finite one in the transform space. This problem is solved by the higher-order theory according to which the transformed displacement vector is expressed by a second order expansion in terms of local coordinates, in conjunction with the equilibrium equations and the relevant boundary conditions. The actual elastic field is obtained by a numerical evaluation of the inverse transform. The accuracy of the suggested approach is verified by a comparison with the exact analytical solution for a penny-shaped crack embedded in a homogeneous medium. Results for a unidirectional composite with a broken fiber are given for various fiber volume fractions and fiber-to-matrix stiffness ratios. It is shown that for certain parameter combinations the use of the average stress in the fiber, as it is employed in the framework of the shear lag approach, for the prediction of composite’s strength, leads to an over estimation. To this end, the concept of “point stress concentration factor” is introduced to characterize the strength of the composite with a broken fiber. Several generalizations of the proposed approach are offered.     

A numerical spectral approach for solving elasto-static field dislocation and g-disclination mechanics

A spectral approach is developed to solve the elasto-static equations of field dislocation and g-disclination mechanics in periodic media. Given the spatial distribution of Nye’s dislocation density and/or g-disclination density tensors in heterogeneous or homogenous linear elastic media, the incompatible and compatible elastic distortions are respectively obtained from the solutions of Poisson and Navier-type equations in the Fourier space. Intrinsic discrete Fourier transforms solved by the Fast Fourier Transform (FFT) method, which are consistent with the pixel grid for the calculation of first and second order spatial derivatives, are preferred and compared to the classical discrete approximation of continuous Fourier transforms when deriving elastic fields of defects. Numerical examples are provided for homogeneous linear elastic isotropic solids. For various defects, a regularized defect density in the core is considered and smooth elastic fields without Gibbs oscillations are obtained, when using intrinsic discrete Fourier transforms. The results include the elastic fields of single screw and edge dislocations, standard wedge disclinations and associated dipoles, as well as “twinning g-disclinations”. In order to validate the present spectral approach, comparisons are made with analytical solutions using the Riemann–Graves integral operator and with numerical simulations using the finite element approximation.