On the thermo-elastostatics of heterogeneous materials: II. Analysis and generalization of some basic hypotheses and propositions

One considers linearly thermoelastic composite media, which consist of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated inclusions. Effective properties (such as compliance and thermal expansion) as well as the first statistical moments of stresses in the phases are estimated for the general case of nonhomogeneity of the thermoelastic inclusion properties. At first, one shortly reproduces both the basic assumptions and propositions of micromechanics used in most popular methods, namely: effective field hypothesis, quasi-crystallite approximation, and the hypothesis of “ellipsoidal symmetry”. The explicit new representations of the effective thermoelastic properties and stress concentration factor are expressed through some building blocks described by numerical solutions for both the one and two inclusions inside the infinite medium subjected to both the homogeneous and inhomogeneous remote loading. The method uses as a background the new general integral equation proposed in the accompanying paper and makes it possible to abandon the basic concepts of micromechanics mentioned above. The results of this abandonment are quantitatively estimated for some modeled composite reinforced by aligned continuously inhomogeneous fibers. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.     

Stability of surface integral equation for left-handed materials

In recent years, left-handed materials (LHMS) have been a hot and debated research topic. Electromagnetic simulation is usually performed for prediction as well as for design. The stability of the surface integral equation method through an insightful and rigorous procedure is analysed. The conclusion is that the PMCHWT formulation at perfectly matched LHM-RHM interface has stability problem, which comes from the plasmon resonance condition. Adding a small amount of loss can greatly improve the condition number of the matrix. Physical and consistent numerical results are shown followed by a brief conclusion     

Boundary integral equation formulation for Interface cracks in anisotropic materials

We present a boundary integral formulation for anisotropic interface crack problems based on an exact Green's function. The fundamental displacement and traction solutions needed for the boundary integral equations are obtained from the Green's function. The traction-free boundary conditions on the crack faces are satisfied exactly with the Green's function so no discretization of the crack surfaces is necessary. The analytic forms of the interface crack displacement and stress fields are contained in the exact Green's function thereby offering advantage over modeling strategies for the crack. The Green's function contains both the inverse square root and oscillatory singularities associated with the elastic, anisotropic interface crack problem. The integral equations for a boundary element analysis are presented and an example problem given for interface cracking in a copper-nickel bimaterial.     

On the existence of Kassab and Divo''s generalized boundary integral equation formulation for isotropic heterogeneous steady state heat conduction problems

In this note, we prove that the boundary integral equation formulation for isotropic heterogeneous steady state conduction problems recently suggested Kassab and Divo is not general. We also obtain the complete integral representation for the new formulation which is of the boundary domain type instead of the boundary only type, as originally proposed by the above authors. The resulting integral equation representation possesess the same difficulty than previous BEM formulations of the problem, however the evaluation of the domain integral in the new formulations is simpler than those of previous formulations.     

On an integral equation of viscous flow theory

Summary The integral equation encountered by van de Vooren and Veldman [1] in their study of the Knudsen region near the leading edge of a flat plate is solved by the method of Wiener and Hopf. This exact solution yields the values of certain arbitrary constants which were not determined in [1].     

On an integral equation of radiation-conduction heat transfer

We present a method for solving analytically the linearized integral equation of radiation-conduction heat transfer. Under certain conditions, realized in practice in heat conduction studies of translucent materials, a sufficiently precise solution may be obtained in closed form.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18, No. 3, pp. 474–480, March, 1970.     

Axisymmetric boundary integral equation analysis of interface cracks between dissimilar materials

The numerical boundary integral equation (BIE) method with quadratic quarter-point crack-tip singular elements is used to analyse interface cracks between dissimilar material in axisymmetry. Such crack problems present modelling difficulties using conventional procedures for obtaining the stress intensity factors. This is because of the oscillatorily singular nature of the stresses in the vicinity of the bimaterial interface crack-tip. Analytical expressions for the direct evaluation of the fracture characterising parameters from the BIE numerical results of displacements or tractions are derived. Three different crack problems are investigated, two of which have known solutions in the literature. Excellent agreement between the BIE results and these other established solutions are obtained even with relatively coarse mesh discretisations. The present study illustrates the ease with which the BIE method may be used in the fracture analysis of both straight and curved binaterial interface cracks.     

On integral equation approaches to the mechanics of fibre-crack interaction

The paper examines the problem of a penny-shaped crack which is formed by the development of a crack in both the fibre and the matrix of a composite consisting of an isolated elastic fibre located in an elastic matrix of infinite extent. The composite region is subjected to a uniform strain field in the direction of the fibre. The paper presents two integral-equation based approaches for the analysis of the problem. The first approach considers the formulation of the complete integral equations governing the associated elasticity problem for a two material region. The second approach considers the boundary integral equation formulation of the problem. Both methods entail the numerical solution of the governing integral equations. The solutions to these integral equations are used to evaluate the stress intensity factor at the boundary of the penny-shaped crack.     

On shape differentiation of discretized electric field integral equation

This work presents shape derivatives of the system matrix representing electric field integral equation discretized with Raviart–Thomas basis functions. The arising integrals are easy to compute with similar methods as the entries of the original system matrix. The results are compared to derivatives computed with automatic differentiation technique and finite differences, and are found to be in an excellent agreement. Furthermore, the derived formulas are employed to analyze shape sensitivity of the input impedance of a planar inverted F-antenna, and the results are compared to those obtained using a finite difference approximation.     

On the convergence of iterative solutions of the integral magnetic field equation

The solution of the integral magnetic field equation by numerical iteration is discussed. Using a simple linear example, it is shown rigorously that relaxation techniques are required to obtain convergence. The range of permissible relaxation parameters is examined and that particular value which yields most rapid convergence is determined. An iterative solution to a simple nonlinear problem is shown to converge rapidly if the relaxation parameter is adjusted appropriately at each step in the iteration. For the general case of a saturable media of complex geometric shape, a relaxation matrix method is proposed in order to achieve rapid convergence.     

A weakly singular boundary integral equation in elastodynamics for heterogeneous domains mitigating fictitious eigenfrequencies

A boundary element formulation applied to dynamic soil–structure interaction problems with embedded foundations may give rise to inaccurate results at frequencies that correspond to the eigenfrequencies of the finite domain embedded in an exterior domain of semi-infinite extent. These frequencies are referred to as fictitious eigenfrequencies. This problem is illustrated and mitigated modifying the original approach proposed by Burton and Miller for acoustic problems, which combines the boundary integral equations in terms of the displacement and its normal derivative using a complex coupling parameter . Hypersingular terms in the original boundary integral equation are avoided by replacing the normal derivative by a finite difference approximation over a characteristic distance h, still leading to an exact boundary integral equation. A proof of the uniqueness of this formulation for small h and a smooth boundary is given, together with a parametric study for the case of a rigid massless cylindrical embedded foundation. General conclusions are drawn for the practical choice of the dimensionless coupling parameter and the dimensionless distance      

Microstructural effects on ductile fracture in heterogeneous materials. Part I: Sensitivity analysis with LE-VCFEM

Ductile failure of heterogeneous materials, such as cast aluminum alloys and discretely reinforced aluminums or DRA’s, initiates with cracking, fragmentation or interface separation of inclusions, that is followed by propagation in the matrix by a ductile mechanism of void nucleation and growth. Damage localizes in bands of intense plastic deformation between inclusions and coalesces into a macroscopic crack leading to overall failure. Ductile fracture is very sensitive to the local variations of the microstructure morphology. This is the first of a two part paper on the effect of microstructural morphology and properties on the ductile fracture in heterogeneous ductile materials. In this paper the locally enhanced Voronoi cell finite element method (LE-VCFEM) for rate-dependent porous elastic–viscoplastic materials is used to investigate the sensitivity of strain to failure to loading rates, microstructural morphology and material properties. A model is also proposed for strain to failure, incorporating the effects of important morphological parameters.     

On the Cauchy principal value of the surface integral in the boundary integral equation of 3D elasticity

A simple demonstration of the existence of the Cauchy principal value (CPV) of the strongly singular surface integral in the Somigliana Identity at a non-smooth boundary point is presented. First a regularization of the strongly singular integral by analytical integration of the singular term in the radial direction in pre-image planes of smooth surface patches is carried out. Then it is shown that the sum of the angular integrals of the characteristic of the tractions of the Kelvin fundamental solution is zero, a formula for the transformation of angles between the tangent plane of a suface patch and the pre-image plane at smooth mapping of the surface patch being derived for this purpose.     

Local integral equation method for potential problems in functionally graded anisotropic materials

Efficient computational techniques are developed for 2D potential problems in anisotropic media with continuously variable material coefficients. The method is based on integral relationships considered on local sub-domains and domain-type approximations of the field variable. Three different kinds of integral equations are combined with either a domain element interpolation or a meshless point interpolation. The physical background of the formulation is discussed briefly. The accuracy and the convergence of the proposed techniques are tested by several examples and compared with benchmark analytical solutions.     

Vectorial modal analysis of dielectric waveguides based on a coupled transverse-mode integral equation. I. Mathematical formulation

We propose a rigorous full-vector integral-equation formulation for analyzing modal characteristics of the complex, two-dimensional, rectangular-like dielectric waveguide that is divisible into vertical slices of one-dimensional layered structures. The entire electromagnetic mode field is completely determined by the y-component electric and magnetic field functions on the interfaces between slices. These interfacial functions are governed by a system of vector-coupled transverse-mode integral equations (VCTMIE) whose kernels are made of orthonormal sets of both TE-to-y and TM-to-y modes from each slice. To solve for the unknown functions, we construct sets of suitable expansion functions and turn VCTMIE into a nonlinear matrix equation via orthogonal projection. The eigenvectors of the matrix provide the mode field solutions of the complex dielectric waveguide.     

The computation of the J integral with the traction boundary integral equation

The solutions of the displacement boundary integral equation (BIE) are not uniquely determined in certain types of boundary conditions. Traction boundary integral equations that have unique solutions in these traction and mixed boundary cases are established. For two‐dimensional linear elasticity problems, the divergence‐free property of the traction boundary integral equation is established. By applying Stokes' theorem, unknown tractions or displacements can be reduced to computation of traction integral potential functions at the boundary points. The same is true of the J integral: it is divergence‐free and the evaluation of the J integral can be inverted into the computation of the J integral potential functions at the boundary points of the cracked body. The J integral can be expressed as the linear combination of the tractions and displacements from the traction BIE on the boundary of the cracked body. Numerical integrals are not needed at all. Selected examples are presented to demonstrate the validity of the traction boundary integral and J integral. Copyright © 2004 John Wiley & Sons, Ltd.     

Harmonic wave propagation in viscoelastic heterogeneous materials Part I: Dispersion and damping relations

An analytico-numerical method is presented to study the propagation of plane harmonic waves in infinite periodic linear viscoelastic media. Part I considers only the dispersion and attenuation of acoustical longitudinal and shear waves. To show the accuracy of the method, examples of plane harmonic wave propagation in an infinite homogeneous medium and in a periodic layered viscoelastic medium are presented. The method is then used to calculate the damping and dispersion relations for a fibre-reinforced viscoelastic composite material. The results show clearly the influence of materials' viscoelastic properties and heterogeneities on the propagation of plane harmonic waves through the media.     

Calculation of eigenvalues of the Helmholtz equation by an integral equation

A method is presented to calculate the eigenvalues of the Helmholtz equation Δ²ø + k²ø = 0 in a two-dimensional area when ø vanishes on the boundary. The method is based on an integral equation, which can be easily solved numerically. Results obtained for circular and rectangular geometries are also given and compared to the exact values.     

Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method

Advanced computational method for transient heat conduction analysis in continuously nonhomogeneous functionally graded materials (FGM) is proposed. The method is based on the local boundary integral equations with moving least square approximation of the temperature and heat flux. The initial-boundary value problem is solved by the Laplace transform technique. Both Papoulis and Stehfest algorithms are applied for the numerical Laplace inversion to obtain the time-dependent solutions. Numerical results are presented for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.