Neutral spherical inhomogeneities in a twisted circular cylindrical bar

We consider the torsion problem of a circular cylindrical bar which is filled up with composite spherical inclusions. The composite inclusions consist of a core and coating both of which are spherically orthotropic with the volume fractions of the core being the same in every composite inclusion. The center points of the spherical inhomogeneities are on the axis of revolution of the circular cylinder. The neutral inhomogeneity in the considered problem of elastic equilibrium is defined as a foreign body (inclusion) which can be introduced in a host body without disturbing the elastic field (displacements, stresses) in it. The conditions of the neutral inhomogeneity for the twisted circular cylindrical bar are derived, and some special cases of inhomogeneity are analyzed. The present paper gives a new example for neutral inhomogeneity in the field of elasticity.     

The interaction between a penny-shaped crack and a spherical inhomogeneity in an infinite solid under uniaxial tension

Summary This paper presents an approximate three-dimensional analytical solution to the elastic stress field of a penny-shaped crack and a spherical inhomogeneity embedded in an infinite and isotropic matrix. The body is subjected to an uniaxial tension applied at infinity. The inhomogeneity is also isotropic but has different elastic moduli from the matrix. The interaction between the crack and the inhomogeneity is treated by the superposition principle of elasticity theory and Eshelby's equivalent inclusion method. The stress intensity factor at the boundary of the penny-shaped crack and the stress field inside the inhomogeneity are evaluated in the form of a series which involves the ratio of the radii of the spherical inhomogeneity and the crack to the distance between the centers of inhomogeneity and crack. Numerical calculations are carried out and show the variation of the stress intensity factor with the configuration and the elastic properties of the matrix and the inhomogeneity.     

The stress field and intensity factor due to a craze formed at the equator of a spherical inhomogeneity

The problem of a hoop-like craze formed at the equator of a spherical inhomogeneity has been investigated. The inhomogeneity is embedded in an infinitely extended elastic body which is under uniaxial tension. Both the inhomogeneity and the matrix are isotropic but have different elastic moduli. The craze is treated as a crack with parallel fibrils connecting the top and bottom surfaces. The analysis is based on the superposition principle of the elasticity theory, Hankel transform and Eshelby's equivalent inclusion method. The stress field inside the inhomogeneity and the stress intensity factor on the boundary of the craze are evaluated in the form of integral equations which are solved numerically. The result obtained is in good agreement with experimental results given in the literature. By setting the elastic moduli of the inhomogeneity the same as those of the matrix, the stress intensity factor for a thin hoop-like crack embedded in an isotropic matrix can be obtained as a deduction.     

Solution of the Eshelby-type anti-plane strain polygonal inclusion problem based on a simplified strain gradient elasticity theory

The Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion of arbitrary-shape polygonal cross-section is analytically solved using a simplified strain gradient elasticity theory that incorporates one material length scale parameter. The Eshelby tensor (with four nonzero components) is obtained in a general form in terms of two scalar-valued potential functions. These potential functions, as area integrals over the polygonal cross-section, are first converted to two line (contour) integrals using Green’s theorem, which are then evaluated analytically by direct integration. The newly derived Eshelby tensor is separated into a classical part and a gradient part. The former does not contain any elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle (inclusion) size effect. For homogenization applications, the area average of the new position-dependent Eshelby tensor over the polygonal cross-section is also provided in a general form. To illustrate the newly obtained Eshelby tensor, five types of regular polygonal inclusions (i.e., triangular, quadrate, hexagonal, octagonal, and tetrakaidecagonal) are quantitatively studied by directly using the general formulas derived. The components of the induced strain and the averaged Eshelby tensor inside the inclusion are evaluated. Numerical results reveal that the induced strain varies with both the position and the inclusion size. The values of the induced strain components in a polygonal inclusion approach from below those in a corresponding circular inclusion when the inclusion size or the number of sides of the polygonal inclusion increases. The results for the averaged Eshelby tensor components show that the size effect is significant when the inclusion size is small but may be neglected for large inclusions.     

A new finite element for treating plane thermomechanical heterogeneous solids

This study is concerned with the development and implementation of a new finite element which is capable of treating the problem of interacting circular inhomogeneities in heterogeneous solids under mechanical and thermal loadings. The general form of the element, which is constructed from a cell containing a single circular inhomogeneity in a surrounding matrix, is derived explicitly using the complex potentials of Muskhelishvili and the Laurent series expansion method. The newly proposed eight‐noded plane element can be used to treat quite readily the two‐dimensional steady‐state heat conduction and thermoelastic problems of an elastic circular inclusion embedded in an elastic matrix with different thermomechanical properties. Moreover, the devised element may be applied to deal with arbitrarily and periodically located multiple inhomogeneities under general mechanical and thermal loading conditions using a very limited number of elements. The current element also enables the determination of the local and effective thermoelastic properties of composite materials with relative ease. Three numerical examples are given to demonstrate its versatility, accuracy and efficiency. Copyright © 1999 John Wiley & Sons. Ltd.     

Sounding of finite solid bodies by way of topological derivative

This paper is concerned with an application of the concept of topological derivative to elastic‐wave imaging of finite solid bodies containing cavities. Building on the approach originally proposed in the (elastostatic) theory of shape optimization, the topological derivative, which quantifies the sensitivity of a featured cost functional due to the creation of an infinitesimal hole in the cavity‐free (reference) body, is used as a void indicator through an assembly of sampling points where it attains negative values. The computation of topological derivative is shown to involve an elastodynamic solution to a set of supplementary boundary‐value problems for the reference body, which are here formulated as boundary integral equations. For a comprehensive treatment of the subject, formulas for topological sensitivity are obtained using three alternative methodologies, namely (i) direct differentiation approach, (ii) adjoint field method, and (iii) limiting form of the shape sensitivity analysis. The competing techniques are further shown to lead to distinct computational procedures. Methodologies (i) and (ii) are implemented within a BEM‐based platform and validated against an analytical solution. A set of numerical results is included to illustrate the utility of topological derivative for 3D elastic‐wave sounding of solid bodies; an approach that may perform best when used as a pre‐conditioning tool for more accurate, gradient‐based imaging algorithms. Despite the fact that the formulation and results presented in this investigation are established on the basis of a boundary integral solution, the proposed methodology is readily applicable to other computational platforms such as the finite element and finite difference techniques. Copyright © 2004 John Wiley & Sons, Ltd.     

负泊松比微结构拓扑优化设计

在构建负泊松比结构拓扑优化模型时,直接用负泊松比的数学表达式构造目标函数,将使得目标函数高度非线性,迭代过程敏度分析困难。采用线性拟合法,构建了具有线性特征的负泊松比微结构拓扑优化目标函数,基于能量法和均匀化方法,结合拓扑优化理论,构建了一种可以快速准确求解负泊松比的拓扑优化设计模型,求解该模型得到了一种优化的拓扑构型及相应的负泊松比值。根据优化求解得到的结构模型,参考国家标准GB/T 22315-2008《金属材料弹性模量和泊松比试验方法》,利用有限元软件对其泊松比进行仿真计算,然后采用激光加工方式制造试样,并测试其泊松比,经过与优化模型求解得到的泊松比值对比分析,验证了所构建优化模型的正确性。本文方法既避免了以负泊松比表达式为优化函数时会出现的高度非线性问题,也降低了求解的复杂程度,为负泊松比微结构的设计提供了一种参考方法。     

APPLICATION OF TOPOLOGICAL OPTIMIZATION TECHNIQUES TO STRUCTURAL CRASHWORTHINESS

The topological optimization of components to maximize crash energy absorption for a given volume is considered. The crash analysis is performed using a DYNA3D finite element analysis. The original solid elements are replaced by ones with holes, the hole size being characterized by a so-called density (measure of the reduced volume). A homogenization method is used to find elastic moduli as a function of this density. Simpler approximations were developed to find plastic moduli and yield stress as functions of density. Optimality criteria were derived from an optimization statement using densities as the design variables. A resizing algorithm was constructed so that the optimality criteria are approximately satisfied. A novel feature is the introduction of an objective function based on strain energies weighted at specified times. Each different choice of weighting factors leads to a different structure, allowing a range of design possibilities to be explored. The method was applied to an automotive body rear rail. The original design and a new design of equal volume with holes were compared for energy absorption.     

Non-uniform eigenstrain induced anti-plane stress field in an elliptic inhomogeneity embedded in anisotropic media with a single plane of symmetry

A closed-form solution is derived for an anti-plane stress field emanating from non-uniform eigenstrains in an elliptic anisotropic inhomogeneity embedded in anisotropic media with one elastic plane of symmetry. The prescribed eigenstrains are characterized by linear functions of the inhomogeneity in Cartesian coordinates. By means of the polynomial conservation theorem, use of complex function method and conformal transformation, explicit expressions for stresses at the interior boundary of the matrix and the strain energy for the elastic inhomogeneity/matrix system are obtained in terms of coefficients in the linear functions. The coefficients are evaluated analytically using the principle of minimum potential energy of the elastic system, leading to the anti-plane stress field. The resulting solution is verified by means of the continuity condition for the shear stress at the interface between the elliptic inhomogeneity and matrix. The present solution is shown to reduce to known results for uniform eigenstrains with illustration by numerical examples.     

A circular inclusion in a finite domain II. The Neumann-Eshelby problem

Summary This is the second paper in a series concerned with the precise characterization of the elastic fields due to inclusions embedded in a finite elastic medium. In this part, an exact and closed form solution is obtained for the elastic fields of a circular inclusion embedded in a finite circular representative volume element (RVE), which is subjected to the traction (Neumann) boundary condition. The disturbance strain field due to the presence of an inclusion is related to the uniform eigenstrain field inside the inclusion by the so-called Neumann-Eshelby tensor. Remarkably, an elementary, closed form expression for the Neumann-Eshelby tensor of a circular RVE is obtained in terms of the volume fraction of the inclusion. The newly derived Neumann-Eshelby tensor is complementary to the Dirichlet-Eshelby tensor obtained in the first part of this work. Applications of the Neumann-Eshelby tensor are discussed briefly.     

Effective elastic moduli of three-phase composites with randomly located and interacting spherical particles of distinct properties

A micromechanical analytical framework is presented to predict effective elastic moduli of three-phase composites containing many randomly dispersed and pairwisely interacting spherical particles. Specifically, the two inhomogeneity phases feature distinct elastic properties. A higher-order structure is proposed based on the probabilistic spatial distribution of spherical particles, the pairwise particle interactions, and the ensemble-volume homogenization method. Two non-equivalent formulations are considered in detail to derive effective elastic moduli with heterogeneous inclusions. As a special case, the effective shear modulus for an incompressible matrix containing randomly dispersed and identical rigid spheres is derived. It is demonstrated that a significant improvement in the singular problem and accuracy is achieved by employing the proposed methodology. Comparisons among our theoretical predictions, available experimental data, and other analytical predictions are rendered. Moreover, numerical examples are implemented to illustrate the potential of the present method.     

双周期刚性线的纵向剪切问题

夹杂对工程材料特别是复合材料的断裂行为有着重要的影响。从固体非均匀相的观点看,当片状嵌入物的弹性模量比基质材料的弹性模量大得多时,可以将其看成刚性线夹杂,它将引起严重的应力集中。本文研究了双周期刚性线的纵向剪切问题,利用椭圆函数的保角变换和RiemannSchwarz对称原理,求出了问题严格的闭合解。从本文解答的特殊情况,可直接得到已有的一些结果。     

Interaction between a spherical inhomogeneity and two symmetrically placed penny-shaped cracks

Stress analysis is carried out for a three-dimensional elastic solid containing an elastic spherical inhomogeneity and two coplanar penny-shaped cracks. Each of the two cracks is located on either side of the elastic spherical inhomogeneity and the geometry is subjected to uniform tensile stress at infinity. The interaction between the inhomogeneity and the cracks is tackled by the superposition principle of elasticity theory and Eshelby's equivalent inclusion method. Analytical solutions for the stress intensity factors on the boundaries of the cracks and the stress field inside the inhomogeneity are evaluated in series form. Numerical calculations are reported for several special cases, and show the variations of the stress intensity factors and stress field inside the inhomogeneity with the configuration and elastic properties of the solid and the inhomogeneity.     

New anisotropic crack-tip enrichment functions for the extended finite element method

In this paper, the extended finite element method (X-FEM) is implemented to analyze fracture mechanics problems in elastic materials that exhibit general anisotropy. In the X-FEM, crack modeling is addressed by adding discontinuous enrichment functions to the standard FE polynomial approximation within the framework of partition of unity. In particular, the crack interior is represented by the Heaviside function, whereas the crack-tip is modeled by the so-called crack-tip enrichment functions. These functions have previously been obtained in the literature for isotropic, orthotropic, piezoelectric and magnetoelectroelastic materials. In the present work, the crack-tip functions are determined by means of the Stroh’s formalism for fully anisotropic materials, thus providing a new set of enrichment functions in a concise and compact form. The proposed formulation is validated by comparing the obtained results with other analytical and numerical solutions. Convergence rates for both topological and geometrical enrichments are presented. Performance of the newly derived enrichment functions is studied, and comparisons are made to the well-known classical crack-tip functions for isotropic materials.     

Relations between compliances of inhomogeneities having the same shape but different elastic constants

The contributions of inhomogeneities having the same shape but different elastic constants, to the overall elastic properties are interrelated. The utility of these relations lies, in particular, in the possibility to extend available results for pores or rigid inclusions to inhomogeneities of arbitrary elastic properties. The relations are exact for ellipsoids and approximate for non-ellipsoidal shapes. The constructed approximation also constitutes approximate connection between the first Eshelby’s problem (the eigenstrain problem) and the second one (the inhomogeneity problem), for non-ellipsoidal shapes. It also yields approximate formulas for the contribution of a non-ellipsoidal inhomogeneity to effective elastic properties.     

SH波对覆盖层下浅埋圆孔和圆夹杂的散射

利用复变函数法和波函数展开法给出了具有地表覆盖层的弹性半空间内圆形孔洞和圆柱形夹杂在稳态SH波作用下动应力集中问题的解。根据SH波散射的衰减特性,该问题采用大圆弧假定法求解,利用半径很大的圆来拟合地表覆盖层的直边界,将具有地表覆盖层的半空间直边界问题转化为曲面边界问题。借助Helmholtz定理预先写出问题波函数的一般形式解,再利用边界条件并借助复数Fourier-Hankel级数展开把问题化为求解波函数中未知系数的无穷线性代数方程组,截断该无穷代数方程组可求得该问题的近似解析解。最后,通过算例讨论了地表覆盖层及圆孔对浅埋圆柱形夹杂动应力集中的影响。结果表明,覆盖层刚度和厚度的变化及圆孔的存在可显著改变圆夹杂周边动应力集中的分布。     

Continuum structural topological optimizations with stress constraints based on an active constraint technique

Stress‐related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. Continuum structural topological optimization with stress constraints is of wide engineering application prospect, in which there still are many problems to solve, such as the stress concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization method of continuum structures with the stress constraints and the objective function being the structural volume has been presented in this paper. To solve the stress concentration issue, an approximate stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient constraint is constructed for the optimized structure under the r?th load case. To obtain stable convergent series solutions and enhance the control on the stress level, two p‐norm global stress constraint functions with different indexes are adopted, and some weighting p‐norm global stress constraint functions are introduced for any load case. And an equivalent topological optimization model with reduced stress constraints is constructed,being incorporated with the rational approximation for material properties, an active constraint technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed, based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level optimization problem with artificial variables and many possible non‐active design variables is proposed by adopting an inequality constrained nonlinear programming method with simple trust regions, based on the primal‐dual theory, in which the non‐smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a two‐level optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed to deal with the aggregation constraints that always are of loose constraint (non active constraint) features in the conventional structural optimization method. A novel structural topological optimization method with stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed method is feasible and very effective. © 2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem

The solution for the Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The Green’s function based on the SSGET for an infinite three-dimensional elastic body undergoing anti-plane strain deformations is first obtained by employing Fourier transforms. The Eshelby tensor is then analytically derived in a general form for an anti-plane strain inclusion of arbitrary cross-sectional shape using the Green’s function method. By applying this general form, the Eshelby tensor for a circular cylindrical inclusion is obtained explicitly, which is separated into a classical part and a gradient part. The former does not contain any classical elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle size effect. The components of the new Eshelby tensor vary with both the position and the inclusion size, unlike their counterparts based on classical elasticity. For homogenization applications, the average of this Eshelby tensor over the circular cross-sectional area of the inclusion is obtained in a closed form. Numerical results reveal that when the inclusion radius is small, the contribution of the gradient part is significantly large and should not be ignored. Also, it is found that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. These components approach from below the values of their counterparts based on classical elasticity when the inclusion size becomes sufficiently large.     

Dispersion of elastic waves in periodically inhomogeneous media

Propagation of time-harmonic elastic waves through periodically inhomogeneous media is considered. The material inhomogeneity exists in a single direction along which the elastic waves propagate. Within the period of the linear elastic and isotropic medium, the density and elastic modulus vary either in a continuous or a discontinuous manner. The continuous variations are approximated by staircase functions so that the generic problem at hand is the propagation of elastic waves in a medium whose finite period consists of an arbitrary number of different homogeneous layers. A dynamic elasticity formulation is followed and the exact phase velocity is derived explicitly as a solution in closed form in terms of frequency and layer properties. Numerical examples are then presented for several inhomogeneous structures.     

Transient creep behavior of a metal matrix composite with a dilute concentration of randomly oriented spheroidal inclusions

The transient creep behavior of a metal matrix composite containing a dilute concentration of randomly oriented spheroidal inclusions is derived explicitly from the constitutive equation of the matrix. This theory can account for the influence of inclusion shape, elastic inhomogeneity between both phases, and the volume fraction of inclusions. The micro-macro transition is carried out by considering the mechanics of incremental creep, which discloses the nature of stress relaxation in the ductile matrix and the connection between the micro and macro creep strains. The transient creep curves of the composite are displayed with several inclusion shapes. Consistent with the known elastic behavior, spherical inclusions are found to provide the weakest reinforcing effect, whereas thin, circular discs possess the most effective strengthening shape. According to this theory and in line with the experimental data, the creep resistance of cobalt at 500°C can improve by more than 80% after adding a mere 5% of rutile particles into it.