Stress analysis of multiple anisotropic elliptical inclusions in composites

A volume integral equation method is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing interacting multiple anisotropic elliptical inclusions subject to uniform remote tension or remote in-plane shear. This method is applied to two-dimensional problems involving long parallel elliptical cylindrical inclusions. A detailed analysis of the stress field at the interface between the matrix and the central elliptical inclusion is carried out for square and hexagonal packing of anisotropic inclusions. The effects of the number of anisotropic inclusions and various inclusion volume fractions on the stress field at the interface between the isotropic matrix and the central elliptical cylindrical inclusion are investigated in detail. The stress field at the interface between the isotropic matrix and the central elliptical inclusion is also compared with that between the isotropic matrix and the central circular inclusion.     

Elastic field of cylindrical inclusion

The elastic field of a cylindrical inclusion with plane or antiplane transformation strain is found under assumptions of linear isotropic homogeneous theory of elasticity. Different integral expressions are obtained from expressions known for the three-dimensional case and interpreted in terms of the theory of dislocations. The surface integrals correspond to a volume distribution of infinitesimal dislocation dipoles inside the inclusion, the line integrals to a surface distribution of dislocations in the boundary between the inclusion and matrix. The case of a shear transformation in a rectangular region is discussed in more detail in connection with twinning.     

A solution of a non-homogeneous orthotropic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems

A solution of a non-homogeneous orthotropic elastic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions as well as the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to the homogeneous ones. Then by virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally presented, which can degenerate in a rather straightforward way to the solution for a homogeneous orthotropic cylindrical shell and isotropic solid cylinder as well as that for a non-homogeneous isotropic cylindrical shell. Using the present method, integral transform can be avoided. It is fit for a cylindrical shell with arbitrary thickness subjected to arbitrary thermal loads. It is also very convenient to deal with dynamic thermoelastic problems for different boundary conditions. Besides, the numerical calculation involved is very easy to be performed. Several examples are presented.     

Effective dynamic properties of 3D composite materials containing rigid penny-shaped inclusions

The propagation of time-harmonic plane elastic waves in infinite elastic composite materials consisting of linear elastic matrix and rigid penny-shaped inclusions is investigated in this paper. The inclusions are allowed to translate and rotate in the matrix. First, the three-dimensional (3D) wave scattering problem by a single inclusion is reduced to a system of boundary integral equations for the stress jumps across the inclusion surfaces. A boundary element method (BEM) is developed for solving the boundary integral equations numerically. Far-field scattering amplitudes and complex wavenumbers are computed by using the stress jumps. Then the solution of the single scattering problem is applied to estimate the effective dynamic parameters of the composite materials containing randomly distributed inclusions of dilute concentration. Numerical results for the attenuation coefficient and the effective velocity of longitudinal and transverse waves in infinite elastic composites containing parallel and randomly oriented rigid penny-shaped inclusions of equal size and equal mass are presented and discussed. The effects of the wave frequency, the inclusion mass, the inclusion density, and the inclusion orientation or the direction of the wave incidence on the attenuation coefficient and the effective wave velocities are analysed. The results presented in this paper are compared with the available analytical results in the low-frequency range.     

Shielding effect of a nano-circular inclusion acting on semi-infinite wedge cracks

The model of a screw dislocation near a semi-infinite wedge crack tip inside a nano-circular inclusion is proposed to investigate the shielding effect of nano inclusions acting on cracks. Utilizing the complex function method, the closed-form solutions of the stress fields in the matrix and the inclusion region are derived. The stress intensity factor, the image force, as well as the critical loads for dislocation emission are discussed in detail. The results show that the nano inclusion not only enhances the shielding effect exerted by the dislocation, but also provides a shielding effect itself. Moreover, dislocations may be trapped in the nano inclusion even if the matrix is softer than the inclusion. This helps the dislocation shield crack, and reduces the dislocation density within the matrix.     

Effective Anisotropic Dielectric Properties of Crystal Composites

Transformation fieM method （TFM） is developed to estimate the anisotropic dielectric properties of crystal composites having arbitrary shapes and dielectric properties of crystal inclusions, whose principal dielectric axis are different from those of anisotropic crystal matrix. The complicated boundary-value problem caused by inclusion shapes is circumvented by introducing a transformation electric field into the crystal composites regions, and the effective anisotropic dielectric responses are formulated in terms of the transformation field. Furthermore, the numerical results show that the effective anisotropie dielectric responses of crystal composites periodically vary as a function of the rotating angle between the principal dielectric axes of inclusion and matrix crystal materials. It is found that at larger inclusion volume fraction the inclusion shapes induce profound effect on the effective anisotropic dielectric responses.     

Thermal stresses around a circular inclusion with functionally graded interphase in a finite matrix

Based on the theory of the complex variable functions, the analysis of non-axisymmetric thermal stresses in a finite matrix containing a circular inclusion with functionally graded interphase is presented by means of the least square boundary collocation technique. The distribution of thermal stress for the functionally graded interphase layer with arbitrary radial material parameters is derived by using the method of piece-wise homogeneous layers when the finite matrix is subjected to uniform heat flow. The effects of matrix size, interphase thickness and compositional gradient on the interfacial thermal stress are discussed in detail. Numerical results show that the magnitude and distribution of interfacial thermal stress in the inclusion and matrix can be designed properly by controlling these parameters.     

Micromechanical modelling of an arbitrary ellipsoidal multi-coated inclusion

The present work aims to provide a general framework to deal with an elementary heterogeneous problem, where the inhomogeneity consists of an n-layered inclusion composed of n concentric ellipsoids made of anisotropic elastic materials. The methodology is based on a combination of Green's function techniques with interface operators, illustrating the stress and strain jump conditions at the interfaces between two adjacent coatings, which are considered perfectly bonded. The model is validated in the case of double-coated spherical inclusions made of isotropic materials, where the obtained analytical results cover the exact solution of Hervé and Zaoui. The model can be applied, after adequate choice of scale-transition methods, to describe the overall behaviour of real composite materials with complex microstructures that are significantly influenced by the presence of interphase layers between constituents (fillers and matrix). Such composites are widely employed in automotive and aerospace industries. As a typical example one can consider a composite with an epoxy matrix reinforced by glass beads coated using a thin soft polymeric phase or syntactic foams particulate composites obtained by filling a polymeric matrix with hollow solid inclusions.     

Anisotropic ellipsoidal inclusion with an anisotropic shell in an isotropic medium subjected to a uniform electric field

An electrostatic problem has been solved for a dielectric inclusion that consists of an anisotropic core and an anisotropic shell. The inclusion is immersed in a uniform isotropic medium (matrix) subjected to a uniform electric field. It is assumed that the outer boundaries of the core and shell are ellipsoidal and become confocal after a linear nonorthogonal transformation that removes the anisotropy of the dielectric properties of the shell. Analytical expressions have been derived for the potential and strength of the electric field in the matrix and also in the shell and core of the inclusion, and an expression for the polarizability tensor of the inclusion has been deduced. It has been shown that the results agree with the well-known solutions in partial (limiting) cases.     

Acoustic scattering by a modified Werner method

A modified integral Werner method is used to calculate pressure scattered by an axisymmetric body immersed in a perfect and compressible fluid subject to a harmonic acoustic field. This integral representation is built as the sum of a potential of a simple layer and a potential of volume. It is equivalent to the exterior Helmholtz problem with Neumann boundary condition for all real wave numbers of the incident acoustic field. For elastic structure scattering problems, the modified Werner method is coupled with an elastodynamic integral formulation in order to account for the elastic contribution of the displacement field at the fluid/structure interface. The resulting system of integral equations is solved by the collocation method with a quadratic interpolation. The introduction of a weighting factor in the modified Werner method decreases the number of volume elements necessary for a good convergence of results. This approach becomes very competitive when it is compared with other integral methods that are valid for all wave numbers. A numerical comparison with an experiment on a tungsten carbide end-capped cylinder allows a glimpse of the interesting possibilities for using the coupling of the modified Werner method and the integral elastodynamic equation used in this research.     

Numerical study of contacts between a flat-ended punch and a half-space embedded with inhomogeneities

This paper presented a numerical approach to solving the problem of a flat-ended punch in contact with a half-space matrix embedded with multiple three dimensional arbitrary-shaped inhomogeneities.Based on the semi-analytical method(SAM)and the equivalent inclusion method,numerical procedures were developed and the effects of inclusion shape and distribution were analyzed.Fast Fourier transform technique was implemented to accelerate the calculation of surface deformation and subsurface stress.Interactions of inter-inclusions and inclusion-matrix were taken into account.Numerical results showed the presence of inhomogeneities(i.e.,microstructures in solids)indeed had a great effect on local contact pressure and a strong disturbance to the subsurface stress field in the vicinity of inclusions.The effects were dependent on the shape and distribution of inclusions and inter-inclusion interactions.The physical significance of this study is to provide an insight into the relation between the material microstructure and its response to the external load,and the solution approach and procedures may find useful applications,for example,the analysis of fatigue and crack propagation for composite materials,prediction of stress field in solids containing material defects,and study of the mechanism of chemical-mechanical polish(CMP)for inhomogeneous materials,etc.     

Effective permittivity of a discrete random medium with anisotropic inclusions

We use the method of substitution of field variables in the bilocal approximation to find the effective permittivity of a two-phase composite random medium in the form of an ensemble of small, arbitrarily anisotropic spherical inclusions distributed inside an isotropic matrix. To illustrate the results we calculate the damping of the plane waves of the mean field in such a medium. Zh. éksp. Teor. Fiz. 114, 1188–1201 (October 1998)     

A Simulation of Near Field Optics by Three Dimensional Volume Integral Equation of Classical Electromagnetic Theory

A numerical simulation code for three dimensional problems of near-field optics has been developed using the volume integral equation with the moment method. The object is assumed to be continuous and macroscopic dielectric and can be treated by macroscopic Maxwell#x0027;s equations. The code can treat the large-scale moment method matrix that is obtained by the discretization of the volume integral equation. The resultant matrix equation is solved by an iteration method called the generalized minimum residual method with reasonable computational cost for simple problems of near field optics. Simulation of a simplified model of a scanning near-field optical microscope has been performed and basic polarization characteristics of the system have been investigated in detail. The code is also applied to the collection-mode of a photon scanning tunneling microscope, where the incident wave is the evanescent wave, and basic relation between near-field and far field i.e., output image, is recognized.     

Effect of the non-homogenity on the composite infinite cylinder of orthotropic material

In this Letter, the solution of non-homogeneous orthotropic elastic cylinder for plane strain problems is developed. The dynamical problem of an orthotropic cylinder containing: (i) an isotropic core and (ii) a rigid core are considered. The elastic constants and density are taken as a power function of the radial coordinate. Analytical expressions for the component of the displacement and the components of the stresses in different cases are obtained. The numerical calculations are carried out for the component of displacement and the components of the stresses through the radial of the cylinder. The results indicate that the effect of inhomogeneity is very pronounced. Those cases have been illustrated and discussed by figures.     

Solution of the transport integral equation with anisotropic scattering

The paper deals with the solution of the integral equation for particle transport in homogeneous material systems having plane and spherical symmetry. Emphasis is put on the explicit inclusion of anisotropic scattering (higher Legendre components of the scattering kernel). The present approach is based on a generalization of the Integral Transform method. The solution is represented as an expansion with respect to analytical basis functions with coefficients satisfying a certain linear system. The determination of this linear system and its matrix elements in a form convenient for numerical purposes is the central point of the paper.     

An analytic and numerical solution with spectral Green's function method for transport equation in spherical geometry

Since in many cases curvilinear geometry is more appropriate than cartesian geometry for precise modeling of the complex systems for reactor calculation, we have developed the spectral Green's function (SGF) method which is employed to obtain angular and scalar flux distributions in heterogeneous sphere geometry with isotropic scattering. In this study, we showed that the neutron transport problems of homogeneous spheres could be reduced to the solution of plane geometry equation.Finally, some results are discussed and compared with those already obtained by diamond difference scheme to test the accuracy of the results. The agreement is satisfactory. SGF method is very suitable for the numerical solution of the neutron transport equation with isotropic scattering.     

Plane wave interpolation in direct collocation boundary element method for radiation and wave scattering: numerical aspects and applications

The classical boundary element formulation for the Helmholtz equation is rehearsed, and its limitations with respect to the number of variables needed to model a wavelength are explained. A new type of interpolation for the potential is then described in which the usual boundary element shape functions are modified by the inclusion of a set of plane waves, propagating in a range of directions. This is termed the plane wave basis boundary element method. The modifications needed to the classical procedures, in terms of integration of the element matrices, and location of collocation points are described. The well-known Singular Value Decomposition solution technique, which is adopted here for the solution of the system matrix equation in its complex form, is briefly outlined. The conditioning of the system matrix is analysed for a simple radiation problem. The corresponding diffraction problem is also analysed and results are compared with analytical and classical boundary element solutions. The CHIEF method is adopted to enhance the quality of the solution, particularly in the vicinity of irregular frequencies. The plane wave basis boundary element method is then applied to two problems: scattering of plane waves by an elliptical cylinder and the multiple circular cylinder plane wave scattering problem. In both cases results are compared with analytical solutions. The results clearly demonstrate that the new method is considerably more efficient than the classical approach. For a given number of degrees of freedom, the frequency for which accurate results can be obtained, using the new technique, can be up to three or four times higher than that of the classical method. This makes the method a powerful new addition to our tools for tackling high-frequency radiation and scattering problems.     

Elastic fields induced by non-elastic eigenstrains in a plane elliptical inhomogeneity existing in orthotropic media under uniform tension at infinity

This paper presents an analytical solution for the elastic fields induced by non-elastic eigenstrains in a plane elliptical inhomogeneity embedded in the orthotropic matrix under tension at infinity and inclined at any angle. The conformal transformation and complex function method for the anisotropic elastic material were used to determine the strain energies in the inhomogeneity and matrix, which were expressed by four undetermined coefficients characterizing the equilibrium boundary of the inhomogeneity due to the acting eigenstrains and external load. The use of the principle of the minimum potential energy led to analytical expressions for these coefficients and thus generated a closed-form solution for the elastic strain/stress fields. The resulting stress field in the inhomogeneity was examined and verified by checking the continuity conditions for the normal and shear stresses on the interior boundary of the matrix. Supported by the Program for New Century Excellent Talents in Universities (NCET) of the Ministry of Education of China (Grant No. NCET-04-0373) and the Program for Shanghai Pujiang Talents (Grant No. 05PJ14092)     

A general method for evaluating the current system and its magnetic field of a plane current sheet, uniform except for a certain area of different uniform conductivity, with results for a square area

Summary We derive a linear Fredholm integral equation of the second kind on an arbitrary closed plane contour which divides an infinite plane current sheet into two regions of different uniform integrated conductivities. This integral equation is satisfield along the above-described contour by a certain combination of the limiting values of the electric potential at both sides of the boundary. This electric potential is due to the currents created in the sheet when a uniform electric field is applied to it. The derived integral equation admits exact solutions in closed form for the cases of circular and elliptical insertions. These solutions are identical with those previously obtained, by other methods, for the same cases. A general method is given for the numerical solution of the integral equation. As an illustration, this method is applied to the case of a square insertion where we used the results of Ashour to obtain numerical estimation for the results of the additional magnetic field on and around the square insertion. To speed up publication, the proofs were not sent to the authors and were supervised by the Scientific Commettee.     

Boundary element method for bandgap computation of photonic crystals

A boundary element method for computing bandgap structures of two-dimensional photonic crystals is developed. For photonic crystals composed of a square or triangular lattice of parallel cylinders with arbitrarily shaped cross-sections, the boundary integral equations are formulated for a unit cell. Constant boundary elements are adopted to discretize the boundaries. Applying the periodic boundary conditions and the interface conditions, we obtain a linear eigenvalue equation with relatively small matrices. The solution of the eigenvalue equation yields the Bloch wave vectors for given frequencies. The convergence of the method for the desired accuracy and efficiency is examined by some typical numerical examples. It is shown that the present method is efficient and accurate and thus provides a flexible treatment of electromagnetic waves in periodic structures with inclusions of arbitrary shape.