Multiple 3D inhomogeneous inclusions in a half space under contact loading

A semi-analytic solution is given for multiple three-dimensional inhomogeneous inclusions of arbitrary shape in an isotropic half space under contact loading. The solution takes into account interactions between all the inhomogeneous inclusions as well as the interaction between the inhomogeneous inclusions and the loading indenter. In formulating the governing equations for the inhomogeneous inclusion problem, the inhomogeneous inclusions are treated as homogenous inclusions with initial eigenstrains plus unknown equivalent eigenstrains, according to Eshelby’s equivalent inclusion method. Such a treatment converts the original contact problem concerning an inhomogeneous half space into a homogeneous half-space contact problem, for which governing equations with unknown contact load distribution can be conveniently formulated. All the governing equations are solved iteratively using the Conjugate Gradient Method. The iterative process is performed until the convergence of the half-space surface displacements, which are the sum of the displacements due to the contact load and the inhomogeneous inclusions, is achieved. Finally, the obtained solution is applied to two example cases: a single inhomogeneity in a half space subjected to indentation and a stringer of inhomogeneities in an indented half-space. The validation of the solution is done by modeling a layer of film as an inhomogeneity and comparing the present solution with the analytic solution for elastic indentation of thin films. This general solution is expected to have wide applications in addressing engineering problems concerning inelastic deformation and material dissimilarity as well as contact loading.     

Two circular inclusions with inhomogeneous interfaces interacting with a circular Eshelby inclusion in anti-plane shear

Summary An analytical solution in infinite series form for two circular cylindrical elastic inclusions embedded in an infinite matrix with two circumferentially inhomogeneous imperfect interfaces interacting with a circular Eshelby inclusion in anti-plane shear is derived by employing complex variable techniques. All of those coefficients in the series can be uniquely determined in a simple and transparent way. Numerical examples are given to illustrate the effect of imperfection and circumferential inhomogeneity of the two interfaces as well as the size, location and elastic properties of the two circular inclusions on the stress fields induced within the two circular inclusions and the Eshelby inclusion.     

Numerical modeling of elastohydrodynamic lubrication in point or line contact for heterogeneous elasto-plastic materials

A semi-analytic solution is developed for heterogeneous elasto-plastic materials with inhomogeneous inclusions under elastohydrodynamic lubrication in point contact or line contact. The inhomogeneous inclusions within a material are homogenized as homogeneous inclusions with properly determined eigenstrains based on the equivalent inclusion method, and the surface displacements induced by these eigenstrains are then introduced into the gap between the contact bodies to update surface geometry. The accumulative plastic deformation is iteratively obtained by a procedure involving a plasticity loop and an incremental loading process. The model takes into account the interactions among the contact bodies, the embedded inclusions, and the plastic zones, thus leading to a solution of the surface pressure distributions, film thickness profiles, plastic zones, and subsurface stress fields. This solution is of great importance for the analysis of elasto-plasto damage of heterogeneous materials subject to lubricated contact.     

The expanding spherical inhomogeneity with transformation strain

Within the context of linear elastodynamics, the radiated fields (including inertia) for a spherical inhomogeneous (of different elastic constants) inclusion with dilatational transformation strain (or eigenstrain), expanding in general motion under applied loading, can been obtained on the basis of Eshelby’s equivalent inclusion method by using the strain field of the expanding homogeneous spherical inclusion (as a function of the eigenstrain) to determine the equivalent eigenstrain. With the equivalent dynamic eigenstrain (which is dependent on the boundary motion), the radiated fields for the inhomogeneous spherical expanding inclusion can be obtained. Based on them, the “driving force” (self-force) on the moving boundary can be computed, and this is the rate of mechanical work (with inertia) required to create an incremental region of inhomogeneity with eigenstrain, i.e with the elastic constants changing as the region of the eigenstrain expands. The self-force depends on the history of the motion, and, in the presence of external loading the driving force yields a Peach-Koehler type force, which exhibits coupling of the applied loading to the history of the motion of the boundary velocity.     

Two‐scale computational modelling of water flow in unsaturated soils containing irregular‐shaped inclusions

The focus of this paper is two‐dimensional computational modelling of water flow in unsaturated soils consisting of weakly conductive disconnected inclusions embedded in a highly conductive connected matrix. When the inclusions are small, a two‐scale Richards’ equation‐based model has been proposed in the literature taking the form of an equation with effective parameters governing the macroscopic flow coupled with a microscopic equation, defined at each point in the macroscopic domain, governing the flow in the inclusions. This paper is devoted to a number of advances in the numerical implementation of this model. Namely, by treating the micro‐scale as a two‐dimensional problem, our solution approach based on a control volume finite element method can be applied to irregular inclusion geometries, and, if necessary, modified to account for additional phenomena (e.g. imposing the macroscopic gradient on the micro‐scale via a linear approximation of the macroscopic variable along the microscopic boundary). This is achieved with the help of an exponential integrator for advancing the solution in time. This time integration method completely avoids generation of the Jacobian matrix of the system and hence eases the computation when solving the two‐scale model in a completely coupled manner. Numerical simulations are presented for a two‐dimensional infiltration problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Equivalent inclusion approach and effective medium approximations for the effective conductivity of isotropic multicomponent materials

Most effective medium approximations for isotropic inhomogeneous materials are based on dilute solutions of some typical inclusions in an infinite matrix medium, while the simplest approximations are those for the composites with spherical and circular inclusions. Practical particulate composites often involve inhomogeneities of more complicated geometry than that of the spherical (or circular) one. In our approach, those inhomogeneities are supposed to be substituted by simple equivalent spherical (circular) inclusions from a comparison of their respective dilute solution results. Then the available simple approximations for the equivalent spherical (circular) inclusion material can be used to estimate the effective conductivity of the original composite. Numerical illustrations of the approach are performed on some 2D and 3D geometries involving elliptical and ellipsoidal inclusions.     

Elastizitätstheoretische Lösung des kreisförmigen Einschlusses in unendlicher Scheibe mittels der Beziehungen zwischen Loch‐ und Einschlussproblem

The method applied for the derivation of an elastic solution of an infinite plate with a circular inclusion of mismatching stiffness under uniform remote tension stress is based on the special deformation characteristic of this inclusion geometry. Outgoing from the analysis of the plate with a circular hole is the exact solution of the general inclusion problem obtained by superposition of a specific inner stress field.     

The elastic strain energy of a coherent inclusion with deviatoric misfit strains

The elastic strain energy of a misfitted coherent inclusion is discussed using the Eshelby method for ellipsoidal inclusions. Deviatoric and tetragonal misfit strains in an elastically inhomogeneous spheroidal inclusion are particularly considered. The variation of the elastic strain energy is evaluated as a function of the shape and orientation of the inclusion. Results obtained are compared with those for a coherent inclusion with purely dilatational misfit strains. Under certain deviatoric misfit strains and elastic moduli of the inclusion, the least elastic strain energy is achieved by a spheroid with an intermediate shape between a plate and a sphere or between a sphere and a needle.     

On the transient elastic field for an expanding spherical inclusion in 3-D solid

Summary. The three-dimensional transient elastic field of an infinite isotropic elastic medium is investigated when a phase transformation is nucleated from a point and proceeds through the crystal dynamically. The phase transformation keeps the spherical shape and expands at a speed of arbitrary time profile. This process is modeled by an expanding spherical inclusion with a spatially uniform eigenstrain. The objective of this paper is to present a general method to determine the transient displacement field for points either covered or not covered by the transformation area. This method can be applied to investigate the nucleation and expanding mechanism of phase transformation. Using a Green's function approach, an explicit procedure is presented to evaluate the 3-D displacement field when the expanding history of the spherical inclusion is given. As numerical examples, the explicit formulations are given for the transient elastic fields, when the spherical inclusion expands at a constant or an exponent damping speed with a pure dilatational eigenstrain or pure shear eigenstrain. It is found that the elastic field inside the expanding inclusion remains constant with respect to time, which is consistent with the well-known Eshelby solution for a static inclusion case.     

An irregular-shaped inclusion with imperfect interface in antiplane elasticity

Despite extensive studies of inclusions with simple shape, little effort has been devoted to inclusions of irregular shape. In this study, we consider an inclusion of irregular shape embedded within an infinite isotropic elastic matrix subject to antiplane shear deformations. The inclusion–matrix interface is assumed to be imperfect characterized by a single, non-negative, and constant interface parameter. Using complex variable techniques, the analytic function that is defined within the irregular-shaped inclusion is expanded into a Faber series, and in conjunction with the Fourier series, a set of linear algebraic equations for a finite number of unknown coefficients is determined. With this approach and without imposing any constraints on the stress distribution, a semi-analytical solution is derived for the elastic fields within the irregular-shaped inclusion and the surrounding matrix. The method is illustrated using three examples and verified, when possible, with existing solutions. The results from the calculations reveal that the stress distribution within the inclusion is highly non-uniform and depends on the inclusion shape and the weak mechanical contact at the inclusion/matrix boundary. In fact, the results illustrate that the imperfect interface parameter significantly influences the stress distribution.     

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。     

Eigenstrain formulation of boundary integral equations for modeling particle-reinforced composites

A novel computational model is presented using the eigenstrain formulation of the boundary integral equations for modeling the particle-reinforced composites. The model and the solution procedure are both resulted intimately from the concepts of the equivalent inclusion of Eshelby with eigenstrains to be determined in an iterative way for each inhomogeneity embedded in the matrix. The eigenstrains of inhomogeneity are determined with the aid of the Eshelby tensors, which can be readily obtained beforehand through either analytical or numerical means. The solution scale of the inhomogeneity problem with the present model is greatly reduced since the unknowns appear only on the boundary of the solution domain. The overall elastic properties are solved using the newly developed boundary point method for particle-reinforced inhomogeneous materials over a representative volume element with the present model. The effects of a variety of factors related to inhomogeneities on the overall properties of composites as well as on the convergence behaviors of the algorithm are studied numerically including the properties and shapes and orientations and distributions and the total number of particles, showing the validity and the effectiveness of the proposed computational model.     

Smart element method I. The Zienkiewicz–Zhu feedback

A new error control finite element formulation is developed and implemented based on the variational multiscale method, the inclusion theory in homogenization, and the Zienkiewicz–Zhu error estimator. By synthesizing variational multiscale method in computational mechanics, the equivalent eigenstrain principle in micromechanics, and the Zienkiewicz–Zhu error estimator in the finite element method (FEM), the new finite element formulation can automatically detect and subsequently homogenize its own discretization errors in a self‐adaptive and a self‐adjusting manner. It is the first finite element formulation that combines an optimal feedback mechanism and a precisely defined homogenization procedure to reduce its own discretization errors and hence to control numerical pollutions. The paper focuses on the following two issues: (1) how to combine a multiscale method with the existing finite element error estimate criterion through a feedback mechanism, and (2) convergence study. It has been shown that by combining the proposed variational multiscale homogenization method with the Zienkiewicz–Zhu error estimator a clear improvement can be made on the coarse scale computation. It is also shown that when the finite element mesh is refined, the solution obtained by the variational eigenstrain multiscale method will converge to the exact solution. Copyright © 2004 John Wiley & Sons, Ltd.     

Modeling of failure near spot welds in lap-shear specimens based on a plane stress rigid inclusion analysis

In this paper, the failure mechanism of resistance spot welds in dual-phase steel lap-shear specimens is investigated based on experimental observations, two-dimensional elasticity theories and two-dimensional finite element analyses. Optical micrographs of the cross sections of spot welds in lap-shear specimens of a dual-phase steel before and after failure are first examined to understand the failure mechanism. The experimental results suggest that under lap-shear loading conditions, a necking failure is initiated near the middle of the nugget circumference in the base metal and then the failure propagates along the nugget circumference in the sheet to final fracture. Based on the stress function approach of the elasticity theory, an analytic solution for an infinite plate containing a rigid inclusion subjected to a resultant shear force is developed and used to investigate the stress and strain distributions near the nugget in lap-shear specimens. The results of the elastic analytic solution and those of a two-dimensional elastic finite element analysis indicate that the initial yielding starts on the two side edges of the inclusion in the sheet. However, the results of a two-dimensional elastic-plastic finite element analysis indicate that as the applied displacement increases, the maximum equivalent plastic strain shifts from the two side edges of the inclusion to the middle of the inclusion along the inclusion circumference in the sheet. The computational results suggest that the location of the initial necking failure should occur near the middle of the nugget circumference in the sheet as observed in experiments based on the forming limit diagram (FLD) for ductile sheet metals.     

A semi‐analytic collocation technique for steady‐state strongly nonlinear advection‐diffusion‐reaction equations with variable coefficients

The aim of this paper is to present a new semi‐analytic numerical method for strongly nonlinear steady‐state advection‐diffusion‐reaction equation (ADRE) in arbitrary 2‐D domains. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. Each basis function used in the algorithm is a sum of an analytic basis function and a special correcting function which is chosen to satisfy the homogeneous boundary conditions of the problem. The polynomials, trigonometric functions, conical radial basis functions, and the multiquadric radial basis functions are used in approximation of the ADRE. This allows us to seek an approximate solution in the analytic form which satisfies the boundary conditions of the initial problem with any choice of free parameters. As a result, we separate the approximation of the boundary conditions and the approximation of the ADRE inside the solution domain. The numerical examples confirm the high accuracy and efficiency of the proposed method in solving strongly nonlinear equations in an arbitrary domain.     

Modal analysis of elastic-viscoplastic Timoshenko beam vibrations

Summary A semi-analytic inelastic Timoshenko beam theory based on a modal solution is developed. Inelastic strains are equivalent to eigenstrains in an identical but entirely elastic background structure. Proper resultants of these eigenstrains, i.e. inelastic curvatures and averaged inelastic shear angels, are defined. Deformations and cross sectional resultants due to these eigenstrain resultants are obtained by means of proper dynamic Green's functions. Since the deformation of the background structure is elastic, linear dynamic solution methods become applicable in a time incremental procedure. In order to enhance the efficiency of this time domain algorithm, an analytic quasistatic protion is separated from the solution. Rate dependence of plastic deformation is considered, and ductile damage in a model of void growth is taken into account. The intensity and distribution of the a priori unknown eigenstrains and imposed shear angles are determined by the constitutive law and calculated in an iterative procedure.     

Particular solutions for axisymmetric Helmholtz-type operators

In this paper, we consider the solution of the axisymmetric heat equation with axisymmetric data in an axisymmetric domain in R³. To solve this problem, we remove the time-dependence by various transform or time-stepping methods. This converts the problem to one of a sequence of modified inhomogeneous Helmholtz equations. Generalizing previous work, we consider solving these equations by boundary-type methods. In order to do this, one needs to subtract off a particular solution, so that one obtains a sequence of modified homogeneous Helmholtz equations. We do this by modifying the usual Dual Reciprocity Method (DRM) and approximating the right-hand sides by Fourier-polynomials or bivariate polynomials. This inevitably leads to analytical solving a sequence of ordinary differential equations (ODEs.) The analytic formulas and their precision are checked using mathematica. In fact, by using an infinite precision technique, the particular solutions can be obtained with infinite precision themselves. This work will form the basis for numerical algorithms for solving axisymmetric heat equation.     

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.     

Analytical solutions for multilayered composite cylinders with harmonic quadratic eigenstrain in arbitrary layers

The eigenstrain problem of multilayered hollow and solid composite cylinders working in a constant magnetic field is investigated analytically in this paper. Each layer of the composite cylinder can undergo a harmonic and spatially varying eigenstrain. The eigenstrain is assumed to be a quadratic polynomial function of the radial coordinate. The closed-form elastic solutions are obtained by solving the inhomogeneous governing Bessel differential equations. Then, the effects of eigenstrain distribution, angular frequency and the intensity of the magnetic field on the radial displacement, radial stress, hoop stress, and axial stress are presented graphically.     

Stress analysis of arbitrarily distributed elliptical inclusions under longitudinal shear loading

This paper deals with an interaction problem of arbitrarily distributed elliptical inclusions under longitudinal shear loading. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are the densities of body forces distributed in the longitudinal directions of infinite bodies having the same elastic constants as those of the matrix and inclusions. In order to satisfy the boundary conditions along the elliptical inclusions, four kinds of fundamental density functions are introduced in a similar way of previous papers treating plane stress problems. Then the body force densities are approximated by a linear combination of those fundamental density functions and polynomials. In the analysis, elastic constants of matrix and inclusion are varied systematically; then the magnitude and position of the maximum stress are shown in tables and the stress distributions along the boundary are shown in figures. For any fixed shape, size and elastic constant of inclusions, the relationships between number of inclusions and maximum stress are investigated for several arrangements.