An arbitrarily oriented,rigid, elliptic inclusion in a finite anisotropic medium

This analysis presents an effective method of addressing rigid inclusion problems in an anisotropic medium with finite boundaries. For an arbitrarily oriented, rigid, elliptic inclusion, a solution is obtained by using the modified mapping collocation method. Within the two-dimensional theory of elasticity, this method provides nearly exact solutions for the stress and displacement fields and for the rigid-body translation and rotation concerning rigid, elliptic inclusions in an anisotropic medium. The effectiveness of the method is demonstrated through several examples and their comparison with other solution methods. The validity and accuracy of the solutions are established by comparing the results from this analysis with those obtained analytically and by other numerical techniques.     

任意分布多个椭圆形刚性夹杂的反平面问题

对于硬夹杂与软基体的复合材料,考虑夹杂间的相互影响,采用坐标变换和复变函数的依次保角映射方法,构造任意分布且相互影响的多个椭圆形刚性夹杂模型的复应力函数,同时满足各个夹杂的边界条件,利用围线积分将求解方程化为线性代数方程,推导出了在无穷远双向均匀剪切,椭圆形刚性夹杂任意分布的界面应力解析表达式,算例分析给出了单夹杂模型与多夹杂模型的夹杂形状对界面应力最大值的影响规律,并进行了对比,描绘出了曲线。     

Calculation of electro and thermo static fields in matrix composite materials of regular or random microstructures

In the work, a numerical method for calculation of electro and thermo static fields in matrix composite materials is considered. Such materials consist of a regular or random set of isolated inclusions embedded in a homogeneous background medium (matrix). The proposed method is based on fast calculation of fields in a homogeneous medium containing a finite number of isolated inclusions. By the solution of this problem, the volume integral equations for the fields in heterogeneous media are used. Discretization of these equations is carried out by Gaussian approximating functions that allow calculating the elements of the matrix of the discretized problem in explicit analytical forms. If the grid of approximating nodes is regular, the matrix of the discretized problem proves to have the Toeplitz structure, and the matrix-vector product with such matrices can be calculated by the Fast Fourier Transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. In the case of an infinite medium containing a homogeneous in space random set of inclusions, our approach combines a self-consistent effective field method with the numerical solution of the conductivity problem for a typical cell. The method allows constructing detailed static (electric or temperature) fields in the composites with inclusions of arbitrary shapes and calculating effective conductivity coefficients of the composites. Results are given for 2D and 3D-composites and compared with the existing exact and numerical solutions.     

Effective properties for single size, rigid spherical inclusions in an elastic matrix

The main problem is that of determining the effective moduli for a compressible isotropic elastic medium containing single size, rigid, spherical inclusions at non-dilute concentrations. A solution is synthesized from available rigorous elasticity results that have been found under asymptotic conditions. Preliminary to obtaining this result for the compressible medium case, results are first found for the incompressible case over a range of rigid particle size distributions. All results extend from the dilute condition up through the full packing limit, which depends upon the size distribution.     

Thermal analysis of carbon-nanotube composites using a rigid-line inclusion model by the boundary integral equation method

The boundary integral equation (BIE) method is applied for the thermal analysis of fiber-reinforced composites, particularly the carbon-nanotube (CNT) composites, based on a rigid-line inclusion model. The steady state heat conduction equation is solved using the BIE in a two-dimensional infinite domain containing line inclusions which are assumed to have a much higher thermal conductivity (like CNTs) than that of the host medium. Thus the temperature along the length of a line inclusion can be assumed constant. In this way, each inclusion can be regarded as a rigid line (the opposite of a crack) in the medium. It is shown that, like the crack case, the hypersingular (derivative) BIE can be applied to model these rigid lines. The boundary element method (BEM), accelerated with the fast multipole method, is used to solve the established hypersingular BIE. Numerical examples with up to 10,000 rigid lines (with 1,000,000 equations), are successfully solved by the BEM code on a laptop computer. Effective thermal conductivity of fiber-reinforced composites are evaluated using the computed temperature and heat flux fields. These numerical results are compared with the analytical solution for a single inclusion case and with the experimental one reported in the literature for carbon-nanotube composites for multiple inclusion cases. Good agreements are observed in both situations, which clearly demonstrates the potential of the developed approach in large-scale modeling of fiber-reinforced composites, particularly that of the emerging carbon-nanotube composites.     

Thermoelastic plane problems with curvilinear boundaries

Summary Plane temperature and thermoelastic problems of discs and holes or inclusions in an infinite elastic matrix are considered, when the boundary of the problems is mapped on the unit circle by a known conformal mapping. By application of complex variables for curvilinear coordinate systems the temperature and thermoelastic problem is expressed in terms of the holomorphic functions. Using the method of continuation of the complex function for curvilinear boundaries, the problem is reduced to a Hilbert problem, whose solution gives the heat flow, temperature, stresses and the displacements. The analysis is applied to two particular cases (hypotrochoidal hole and hypotrochoidal rigid inclusion) and the results obtained are successfully compared with those existing in the literature.     

On composites with periodic structure

The overall moduli of a composite with an isotropic elastic matrix containing periodically distributed (anisotropic) inclusions or voids, can be expressed in terms of several infinite series which only depend on the geometry of the inclusions or voids, and hence can be computed once and for all for given geometries. For solids with periodic structures these infinite series play exactly the same role as does Eshelby's tensor for a single inclusion or void in an unbounded elastic medium.For spherical and circular-cylindrical geometries, the required infinite series are calculated and the results are tabulated. These are then used to estimate the overall elastic moduli when either the overall strains or the overall stresses are prescribed, obtaining the same results. These results are compared with other estimates and with experimental data. It is found that the model of composites with periodic structure yields estimates in excellent agreement with the experimental observations.     

Macroscopic yield criteria for ductile materials containing spheroidal voids: An Eshelby-like velocity fields approach

Making use of limit analysis theory, we derive a new expression of the macroscopic yield function for a rigid ideal-plastic von Mises matrix containing spheroidal cavities (oblate or prolate). Key in the development of the new criterion is the consideration of Eshelby-like velocity fields which are built by taking advantage of the solution of the equivalent inclusion problem in which the eigenstrains rate are unknown for the plasticity problem. These heterogeneous trial velocity fields contain non-axisymmetric components which prove to be original in the context of limit analysis of hollow spheroid. After carefully computing the macroscopic plastic dissipation and implementing a minimization procedure required by the use of the Eshelby-like velocity fields, we derive, for the porous medium, a two-field estimate of the anisotropic yield criterion whose closed-form expression is provided. This estimate is compared to existing criteria based on limit analysis theory. Interestingly, in contrast to these criteria, the new results predict a significant effect of shear loadings in the particular case of ductile materials weakened by penny-shaped cracks.     

Potential fields caused by point sources in regions with apertures and foreign inclusions

A semi-analytic approach is developed for obtaining potential fields generated by point sources in regions embedded with foreign inclusions and containing holes of various shape. This study promotes an extension of the range of effective implementation of the Green's function version of the boundary integral equation method in mechanics of contemporary materials. Equivalents of Green's functions are obtained to boundary-contact value problems posed for Laplace equation on piecewise homogeneous multiply connected regions. Dirichlet, Neumann and Robin boundary conditions can be imposed on the outer boundary of the region and on contours of the apertures, while the ideal contact conditions are assumed on the interfacial contours. Source points could be located either outside or inside the inclusion. A Green's function modification of the method of functional equations is applied. A number of illustrative examples included to show the potential of the approach.     

Null-field approach for piezoelectricity problems with arbitrary circular inclusions

In this paper, we derive the null-field integral equation for piezoelectricity problems with arbitrary piezoelectric circular inclusions under remote anti-plane shears and in-plane electric fields. Separable expressions of fundamental solutions and Fourier series for boundary densities are adopted to solve the piezoelectric problem with circular inclusions. Four gains are obtained: (1) well-posed model, (2) singularity free, (3) boundary-layer effect free and (4) exponential convergence. The solution is formulated in a manner of semi-analytical form since error purely attributes to the truncation of Fourier series. Two piezoelectric problems with two piezoelectric circular inclusions are revisited and compared with the Chao and Chang's solutions to demonstrate the validity of our method. The limiting case that the two inclusions separate far away leads to the Pak's solution of a single inclusion. Stress and electric field concentrations are calculated and are dependent on the distance between the two inclusions, the mismatch in the material constants and the magnitude of mechanical and electromechanical loadings. The results for the shear and electric loadings in two directions are also compared well with the Wang and Shen's results.     

Effective elastic properties of solids with irregularly shaped inclusions

A unit rectangular cell is usually cut out from a medium for investigating fracture mechanism and elastic properties of the medium containing an array of irregularly shaped inclusions. It is desirable to clarify the geometrical parameters controlling the elastic properties of heterogeneous materials because they are usually embedded with randomly distributed particulate. The stress and strain relationship of the rectangular cell is obtained by an ad hoc hybrid-stress finite element method. By matching the boundary condition requirements, the effective elastic properties of composite materials are then calculated, and the effect of shape and arrangement of inclusions on the effective elastic properties is subsequently considered by the application of the ad hoc hybrid-stress finite element method through examining three types of rectangular cell models assuming rectangular arrays of rectangular or diamond inclusions. It is found that the area fraction (the ratio of the inclusion area over the rectangular cell area) is one dominant parameter controlling the effective elastic properties.     

Coupled 2D electric and mechanical fields in piezoelectric solids containing cracks and thin inhomogeneities

This paper develops integral equations and boundary element method for determination of 2D electro-elastic state of solids containing cracks, thin voids and inclusions. It proves that stress and electric displacement field near the tip of thin inhomogeneity possesses square root singularity. Thus, for determination of electromechanical fields near thin defects new special base functions are introduced. The interpolation quadratures along with the polynomial transformations are adopted for efficient numerical evaluation of singular and hypersingular integrals. Presented numerical examples show high efficiency and accuracy of the proposed approach.     

Local Electroelastic Field and Effective Electroelastic Moduli of Piezoelectric Nanocomposites with Interface Effect

Due to the large ratio of surface area to volume in nanoscale objects, the property of surfaces and interfaces likely becomes a prominent factor in controlling the behavior of nano-heterogeneous materials. In this work, based on the Gurtin-Murdoch surface/interface elastic theory, a distinct expression is derived for embedded nano-inclusion in an infinite piezoelectric matrix coupled with interface effect. For the problem of a spherical inclusion in transversely isotropic piezoelectric medium, we reach a conclusion that the elastic and electric field are uniform when eigen-strain and eigen-electric field imposed on the inclusion are uniform even in the presence of the interface influence. The electroelastic fields in the inclusion are related to both interface electroelastic parameters and the radius of the inclusion. Then overall properties of the composites are estimated by using the dilute distribution model. Numerical results reveal that the effective electroelastic moduli are function of the interface parameters and the size of the nano-inhomogeneities.     

The integral equation formulations of an infinite elastic medium containing inclusions, cracks and rigid lines

The integral equation formulations of an infinite homogeneous isotropic medium containing various inclusions, cracks and rigid lines are presented. The present integral equation formulations contain the displacements (no tractions) over the inclusion-matrix interfaces, the discontinuous displacements over crack surfaces and the axial and the shear forces along rigid-line inclusions. Besides, the sub-domain boundary element method is also used in the present research. Numerical results from the present method and the sub-domain boundary element method are compared and discussed.     

Interactions between N circular cylindrical inclusions in a piezoelectric matrix

Summary This paper studies the interactions between N randomly-distributed cylindrical inclusions in a piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is subjected to an anti-plane shear stress and an in-plane electric field at infinity. Based on the complex variable method, the complex potentials in the matrix and inside the inclusions are first obtained in form of power series, and then approximate solutions for electroelastic fields are derived. Numerical examples are presented to discuss the influences of the inclusion array, inclusion size and inclusion properties on couple fields in the matrix and inclusions. Solutions for the case of an infinite piezoelectric matrix with N circular holes or an infinite elastic matrix containing N circular piezoelectric fibers can also be obtained as special cases of the present work. It is shown that the electroelastic field distribution in a piezoelectric material with multiple inclusions is significantly different from that in the case of a single inclusion.     

On three-dimensional singular stress/residual stress fields at the front of a crack/anticrack in an orthotropic/orthorhombic plate under anti-plane shear loading

A novel eigenfunction expansion technique, based in part on separation of the thickness-variable, is developed to derive three-dimensional asymptotic stress field in the vicinity of the front of a semi-infinite through-thickness crack/anticrack weakening/reinforcing an infinite orthotropic/orthorhombic plate, of finite thickness and subjected to far-field anti-plane shear loading. Crack/anticrack-face boundary conditions and those that are prescribed on the top and bottom (free, fixed and lubricated) surfaces of the orthotropic plate are exactly satisfied. Five different through-thickness crack/anticrack-face boundary conditions are considered: (i) slit crack, (ii) anticrack or perfectly bonded rigid inclusion, (iii) transversely rigid inclusion (longitudinal slip permitted), (iv) rigid inclusion in part perfectly bonded, the remainder with slip, and (v) rigid inclusion located alongside a crack. Explicit expressions for the singular stress fields in the vicinity of the fronts of the through-thickness cracks, anticracks or mixed crack–anticrack type discontinuities, weakening/reinforcing orthotropic/orthorhombic plates, subjected to far-field anti-plane shear (mode III) loadings, are presented. In addition, singular residual stress fields in the vicinity of the fronts of these cracks, anticracks and similar discontinuities are also discussed.     

Doubly periodic arrays of cracks and thin inhomogeneities in an infinite magnetoelectroelastic medium

A two-dimensional boundary element method for the analysis of a magnetoelectroelastic medium containing doubly periodic sets of cracks or thin inclusions is developed in this paper. The integral equations and closed-form expressions for corresponding kernels are obtained. Based on the quasi-periodicity of extended displacement and stress function, the integral representations for average stress, strain, electric displacement, magnetic induction etc. are developed. The algorithm of effective properties determination is given. The numerical examples prove the efficiency and high accuracy of the proposed approach in determination of stress, electric displacement and magnetic induction intensity factors and effective properties of the material containing doubly periodic arrays of cracks or thin inclusions.     

Combining self-consistent and numerical methods for the calculation of elastic fields and effective properties of 3D-matrix composites with periodic and random microstructures

The work is devoted to the calculation of static elastic fields in 3D-composite materials consisting of a homogeneous host medium (matrix) and an array of isolated heterogeneous inclusions. A self-consistent effective field method allows reducing this problem to the problem for a typical cell of the composite that contains a finite number of the inclusions. The volume integral equations for strain and stress fields in a heterogeneous medium are used. Discretization of these equations is performed by the radial Gaussian functions centered at a system of approximating nodes. Such functions allow calculating the elements of the matrix of the discretized problem in explicit analytical form. For a regular grid of approximating nodes, the matrix of the discretized problem has the Toeplitz properties, and matrix-vector products with such matrices may be calculated by the fast fourier transform technique. The latter accelerates significantly the iterative procedure. First, the method is applied to the calculation of elastic fields in a homogeneous medium with a spherical heterogeneous inclusion and then, to composites with periodic and random sets of spherical inclusions. Simple cubic and FCC lattices of the inclusions which material is stiffer or softer than the material of the matrix are considered. The calculations are performed for cells that contain various numbers of the inclusions, and the predicted effective constants of the composites are compared with the numerical solutions of other authors. Finally, a composite material with a random set of spherical inclusions is considered. It is shown that the consideration of a composite cell that contains a dozen of randomly distributed inclusions allows predicting the composite effective elastic constants with sufficient accuracy.     

Binary inclusion model for the overall elasticity of imperfectly bonded composites

Summary. A micromechanical approach is developed to estimate the overall elastic moduli of composite materials with imperfectly bonded spherical fillers. The randomly dispersed particles are assumed to satisfy linear interfacial conditions where both tangential and normal interface displacement discontinuities are linearly related to the respective surface tractions. Using the generalized version of Eshelbys equivalent inclusion method proposed by Furuhashi et al. [6] the analysis of the heterogeneous medium reduces to the study of a corresponding homogeneous medium containing spherical inclusions with a proper distribution of eigenstrain and Somigliana dislocation fields. Based on the estimated pair-wise average of strain fields in two interacting imperfect fillers embedded in the homogeneous infinite matrix, the ensemble phase volume average of field quantities has been evaluated within a representative volume element containing a finite number of imperfect particles. For the case of a constant radial distribution function, results are in reasonable agreement with those based on the generalized self-consistent method and composite sphere assemblage proposed by Hashin [11].     

A generalized self-consistent method for solids containing randomly oriented spheroidal inclusions

Summary In this paper, a generalized self-consistent method is proposed to predict the effective moduli of a material containing single-phase and randomly oriented spheroidal inclusions, with same aspect ratios. This is achieved by using an energy equivalence framework, associated with a generalization of the classical three phase model to spheroidal inclusions. The localization problem (spheroidal duplex inclusion problem) is formulated with the Papkovitch-Neuber approach; this requires expansion formulae for the spheroidal potentials, which are derived in the Appendix. Finally, the determination of the effective moduli is equivalent to solving a system of nonlinear equations. Effective moduli are presented for various types of inclusions, and comparisons are made with the estimations obtained from the self-consistent and Mori-Tanaka methods. Moreover, the effects of inclusion geometry and spatial distribution of inclusions on the effective moduli are investigated and compared to each other.