Infinite Isotropic Medium Research Articles

Propagation of SH waves in an regular non homogeneous monoclinic crustal layer lying over a non-homogeneous semi-infinite medium

The present paper discusses the dispersion equation for SH waves in a non-homogeneous monoclinic layer over a semi infinite isotropic medium. The wave velocity equation has been obtained. In the isotropic case, when non-homogeneity is absent, the dispersion equation reduces to the standard SH wave equation. The dispersion curves are depicted by means of graphs for different values of non-homogeneity parameters for the layer and semi-infinite medium.

Thermoelastic interactions in an isotropic unbounded medium due to moving heat source using GNIII model

In this work, we study a problem of thermoelastic interaction due to moving heat source in an isotropic infinite medium under Green and Naghdi model of type III (GNIII). The form of vector-matrix differential equation in the Laplace transform domain, the basic equations have been written, which is then solved by an eigenvalue technique. The analytical solution in the Laplace transforms domain with eigenvalue approach has been obtained. Numerical results for the displacement, temperature and the stress distributions are represented graphically. Some comparisons have been shown in figures to estimate the effect of heat source velocity and time in the physical quintets.

An integral equation formulation of two- and three-dimensional nanoscale inhomogeneities

An integral equation formulation of two- and three-dimensional infinite isotropic medium with nanoscale inhomogeneities is presented in this paper. The Gurtin---Murdoch interface constitutive relation is used to model the continuity conditions along the internal interfaces between the matrix and inhomogeneities. The Poisson's ratios of both the matrix and inhomogeneities are assumed to be the same. The proposed integral formulation only contains the unknown interface displacements and their derivatives. In order to solve the nanoscale inhomogeneities, the displacement integral equation is used when the source points are acting on the interfaces between the matrix and inhomogeneities. Thus, the resulting system of equations can be formulated so that the interface displacements can be obtained. Furthermore, the stresses at points being in the matrix and nanoscale inhomogeneities can be calculated using the stress integral equation formulation. Numerical results from the present method are in good agreement with those from the conventional sub-domain boundary element method and the analytical method.

Localized cracking around pigment particles in tablet film coatings: a theoretical approach.

The problem of localized cracking around pigment particles in tablet film coatings has been treated as being analogous to stress cracking in two phase ceramics. For this purpose a simple model of a spherical cavity in an infinite isotropic medium has been used. As well as allowing the estimation of the stresses generated by the differences in thermal expansion of the dispersed pigment and the polymer matrix, the model predicts the effect of some formulation variables. The predictions are in agreement with the observed cracking on film-coated tablets.

Electromagnetic scattering by cylindrical orthotropic waveguide irises

Abstract The paper is devoted to the mathematical analysis of scattered time-harmonic electromagnetic waves by an infinitely long cylindrical orthotropic waveguide iris. This is modeled by an orthotropic Maxwell system in a cylindrical waveguide iris for plane waves propagating in the x 3-direction, imbedded in an isotropic infinite medium. The problem is equivalently reduced to a 2-dimensional boundary-contact problem with the operator div M grad+k 2 inside the domain and the (Helmholtz) operator Δ+k 2 = div grad+k 2 outside the domain. Here M is a 2 × 2 positive definite, symmetric matrix with constant, real valued entries. The unique solvability of the appropriate boundary value problems is proved and the regularity of solutions is established in Bessel potential spaces.

Incompatibility of strains and its application to mesoscopic studies of plasticity

Structural transitions are invariably affected by lattice distortions. If the body is to remain crack-free, the strain field cannot be arbitrary but has to satisfy the Saint-Venant compatibility constraint. Equivalently, an incompatibility constraint consistent with the actual dislocation network has to be satisfied in media with dislocations. This constraint can be incorporated into strain-based free energy functionals to study the influence of dislocations on phase stability. We provide a systematic analysis of this constraint in three dimensions and show how three incompatibility equations accommodate an arbitrary dislocation density. This approach allows the internal stress field to be calculated for an anisotropic material with spatially inhomogeneous microstructure and distribution of dislocations by minimizing the free energy. This is illustrated by calculating the stress field of an edge dislocation and comparing it with that of an edge dislocation in an infinite isotropic medium. We outline how this procedure can be utilized to study the interaction of plasticity with polarization and magnetization.

Solutions to Eshelby’s problems of non-elliptical thermal inclusions and cylindrical elastic inclusions of non-elliptical cross section

Eshelby’s inclusion problem is solved for non-elliptical inclusions in the context of two-dimensional thermal conduction and for cylindrical inclusions of non-elliptical cross section within the framework of generalized plane elasticity. First, we consider a two-dimensional infinite isotropic or anisotropic homogeneous medium with a non-elliptical inclusion subjected to a prescribed uniform heat flux-free temperature gradient. Eshelby’s conduction tensor field and its area average are first expressed compactly in terms of two boundary integrals avoiding the usual singularity and then specified analytically for arbitrary polygonal inclusions and for inclusions characterized by the finite Laurent series. Next, we are interested in a three-dimensional infinite isotropic or transversely isotropic homogeneous medium with a cylindrical inclusion of a non-elliptical cross section that undergoes uniform generalized plane eigenstrains. The solution to this problem is obtained by decomposing a generalized plane eigenstrain tensor into a plane strain part and an anti-plane strain part, exploiting the mathematical similarity between two-dimensional thermal conduction and anti-plane elasticity, and combining the relevant results of Zou et al. (Zou et al. 2010 J. Mech. Phys. Solids 58 , 346–372. ( doi:10.1016/j.jmps.2009.11.008 )) with those derived in the present work for Eshelby’s conduction tensor field and its area average.

Mechanoelectromagnetic interaction in isotropic dielectrics with regard for the local displacement of mass

On the basis of the earlier proposed model of electromagnetothermomechanics of polarizable bodies, which takes into account the process of local displacement of mass, key equations for the corresponding scalar and vector potentials are written. The generalization of the Lorentz gauge, at which the equations for calculating the scalar and vector potentials of an electromagnetic field become uncoupled, is proposed. We write a resolving system of equations for potentials in the dimensionless form and obtain a parameter of interrelation of the processes of local displacement of mass and deformation. With the use of this system, the propagation of a plane harmonic wave in an infinite isotropic medium is investigated. It is shown that the model describes the dispersion of a modified elastic wave in the high-frequency region. The obtained results agree with known data presented in the literature and obtained from relations of the gradient theory of piezoelectrics.

Electroelastic fields in interacting piezoelectric inhomogeneities by the electromechanicalequivalent inclusion method

Consider two piezoelectric ellipsoidal inhomogeneities of arbitrary size, orientation andmaterial constants, which in turn are surrounded by an infinite isotropic medium. Thesystem under consideration is subjected to far-field non-uniform electromechanical loadings.Based on the extension of the electromechanical equivalent inclusion method (EMEIM), thepresent paper develops a unified solution for determination of the associated electroelasticfields in the vicinity of interacting inhomogeneities. Accordingly, each of the piezoelectricinhomogeneities is broken down into two equivalent inclusions with proper polynomialeigenstrains and eigenelectric fields. The robustness and efficacy of the present solution aredemonstrated through consideration of several boundary value problems. As a specialcase encompassed by the presented formulation, the interaction of a piezoelectricinhomogeneity and a lamellar inhomogeneity for two- and three-dimensional problems isaddressed. For a particular case involving interaction of a slit-like crack and apiezoelectric circular fiber, comparison with the other available results in the literatureattests to the validity of the proposed method. Subsequently, the effect of someparameters such as geometry and stiffness of each phase on the quantitative value ofstress intensity factors (SIFs) are examined for far-field non-uniform loadings.

On Green's function for cylindrically symmetric fields of polarized radiation

Analytic expressions for Green's function describing the process of transfer of polarized radiation in homogeneous isotropic infinite medium in case of cylindrical symmetry and nonconservative scattering are obtained. The solution is based on the set of systems of Abel integral equations of the first kind obtained using the principle of superposition, and the known expression of Green's function for radiation fields with plane-parallel symmetry. Eigenvalue decompositions for the corresponding matrices of generalized spherical functions are found. Using this result the systems of Abel integral equations are diagonalized, and the final solution is obtained.

Non-uniform Eshelby's tensor inside a spherical inclusion in a functionally graded space in transport phenomena

Within the framework of thermal conduction, we consider a functionally graded isotropic infinite medium containing a spherical inclusion which undergoes prescribed uniform heat flux-free temperature gradient. In this research the thermal conductivity is assumed to be exponentially varied in space. Analytical expressions in series form for the temperature and the second-order Eshelby's conduction tensor inside and outside the spherical inclusion are obtained. Our analytical results indicate that the second-order Eshelby's conduction tensor is non-uniform within the spherical inclusion and that it is in general not symmetric. Furthermore our numerical results quantitatively demonstrate how the Eshelby's tensor within the spherical inclusion is non-uniformly distributed due to the spatially varying thermal conductivity.

Analysis of thermoelastic response in a functionally graded spherically isotropic hollow sphere based on Green–Lindsay theory

This paper is concerned with the investigation of thermoelastic displacements and stresses in a functionally graded spherically isotropic hollow sphere due to prescribed temperature in the context of the linear theory of generalized thermoelasticity with two relaxation time parameters (Green and Lindsay theory). Both the surfaces of the body are free from radial stresses, and the inner surface is subjected to a time-dependent thermal shock whereas the outer one is maintained at constant temperature. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain which is then solved by an eigenvalue approach. The numerical inversion of the transforms is carried out using a method of Bellman et al. The displacements and stresses are computed and presented graphically. It is found that the variation of the thermophysical properties of a material as well as the thickness of the body strongly influence the response to loading. A comparative study with the corresponding homogeneous material has also been made. The solution of the problem of a spherically isotropic infinite medium containing a spherical cavity has been derived theoretically by tending the outer radius to infinity, as a particular case.

Study of two dimensional visco-elastic problems in generalized thermoelastic medium with heat source

In this paper, a thermo-viscoelastic problem in an infinite isotropic medium in two dimensions in the presence of a point heat source is considered. The fundamental equations of the problems of generalized thermoelasticity including heat sources in a thermo-viscoelastic media have been derived in the form of a vector matrix differential equation in the Laplace-Fourier transform domain for a two dimensional problem. These equations have been solved by the eigenvalue approach. The results have been compared to those available in the existing literature. The graphs have been drawn for different cases.

The arithmetic mean property for rotational symmetrical inclusions with rotational symmetrical eigenstrains

AbstractEshelby proved in 1957 that the strain field within a homogeneous ellipsoidal inclusion embedded in an infinite isotropic media is uniform, when the eigenstrain prescribed in the inclusion is uniform. This uniform strain field property was conjectured to hold exclusively for the ellipsoidally shaped inclusions. In recent studies the authors have discovered the arithmetic mean properties of the strains for N‐fold (N is greater than 2 but unequal to 4) rotational symmetrical inclusions with uniform eigenstrains, called the arithmetic mean theorem of the Eshelby tensors. In this paper, we investigate the two‐dimensional rotational symmetrical inclusions with rotational symmetrical eigenstrains. It is shown that for N‐fold (N is greater than 4) rotational symmetrical inclusions with the same fold rotational symmetrical eigenstrains which are constant on the boundary of the inclusion, the arithmetic mean of the strains at N rotational symmetrical points presents special properties: For the points outside the inclusion, the arithmetic mean vanishes; for the points inside the inclusion, the arithmetic mean is in direct proportion to the eigenstrain on the boundary of the inclusion, with the coefficient tensor equal to the Eshelby tensor for circular inclusions. It is followed that the average strain over the inclusion domain and the line average of strains along a concentric circle inside the inclusion also equal the eigenstrains on the boundary of the inclusion multiplied by the Eshelby tensor for circular inclusions. These properties obtained in this paper can be safely reduced to the results for uniform eigenstrains shown in our former works.

Ellipsoidal Domains: Piecewise Nonuniform and Impotent Eigenstrain Fields

In association with multi-inhomogeneity problems, a special class of eigenstrains is discovered to give rise to disturbance stresses of interesting nature. Some previously unnoticed properties of Eshelby’s tensors prove useful in this accomplishment. Consider the set of nested similar ellipsoidal domains {Ω1, Ω2,⋯,ΩN+1}, which are embedded in an infinite isotropic medium. Suppose that $$\Omega_{t}=\{\; {\textbf{x}} \mid {\textbf{x}} \in \mathbb{R}^{3} , \quad \sum_{p=1}^3\frac{x_{p}^2}{a_{p}^{2}}\leqslant\xi_{t}^2 \;\},$$ in which \(0\leqslant\xi_{1} < \xi_{2} < \cdots < \xi_{N+1}\) and ξtap, p=1,2,3 are the principal half axes of Ωt. Suppose, the distribution of eigenstrain, eij*(x) over the regions Γt=Ωt+1−Ωt , t=1,2,⋯,N can be expressed as $$\epsilon_{ij}^{*} \left( {\textbf{x}} \right)=\begin{cases} f_{ijkl \cdots m }^{(t)}\biggl( \sum\limits_{p=1}^3\dfrac{ x_{p}^2 }{a_{p}^{2}} \biggr) \;x_{k}x_{l} \cdots x_{m}, \qquad {\textbf{x}} \in \Gamma_{t},\\ \quad 0, \qquad\qquad\qquad\qquad\qquad\ \qquad {\textbf{x}} \in \Omega_1\bigcup \left(\mathbb{R}^3-\Omega_{N+1}\right), \end{cases}$$ (‡) where xkxl ⋯xm is of order n, and fijkl ⋯m(t) represents 3N(n+2)(n+1) different piecewise continuous functions whose arguments are ∑p=13xp2 /ap2. The nature of the disturbance stresses due to various classes of the piecewise nonuniform distribution of eigenstrains, obtained via superpositions of Eq. (‡) is predicted and an infinite number of impotent eigenstrains are introduced. The present theory not only provides a general framework for handling a broad range of nonuniform distribution of eigenstrains exactly, but also has great implications in employing the equivalent inclusion method (EIM) to study the behavior of composites with functionally graded reinforcements.

The Arithmetic Mean Theorem of Eshelby Tensor for Exterior Points Outside the Rotational Symmetrical Inclusion

In 1957, Eshelby proved that the strain field within a homogeneous ellipsoidal inclusion embedded in an infinite isotropic media is uniform, when the eigenstrain prescribed in the inclusion is uniform. This property is usually referred to as the Eshelby property. Although the Eshelby property does not hold for the non-ellipsoidal inclusions, in recent studies we have successfully proved that the arithmetic mean of Eshelby tensors at N rotational symmetrical points inside an N-fold rotational symmetrical inclusion is constant and equals the Eshelby tensor for a circular inclusion, when N⩾3 and N≠4. The property is named the quasi-Eshelby property or the arithmetic mean theorem of Eshelby tensors for interior points. In this paper, we investigate the elastic field outside the inclusion. By the Green formula and the knowledge of complex variable functions, we prove that the arithmetic mean of Eshelby tensors at N rotational symmetrical points outside an N-fold rotational symmetrical inclusion is equal to zero, when N⩾3 and N≠4. The property is referred to as the arithmetic mean theorem of Eshelby tensors for exterior points. Due to the quality of the Green function for plane strain problems, the fourfold rotational symmetrical inclusions are excluded from possessing the arithmetic mean theorem. At the same time, by the method proposed in this paper, we verify the quasi-Eshelby property which has been obtained in our previous work. As corollaries, two more special properties of Eshelby tensor for N-fold rotational symmetrical inclusions are presented which may be beneficial to the evaluation of effective material properties of composites. Finally, the circular inclusion is used to test the validity of the arithmetic mean theorem for exterior points by using the known solutions.

An electro-magneto-thermo-visco-elastic problem in an infinite medium with a cylindrical hole

A magneto-thermo-visco-elastic problem of an infinite isotropic medium is considered in this problem. The analysis encompasses Lord–Shulman and Green–Lindsay theories of generalized thermo-elasticity to account for the finite velocity of heat equation. The problem is solved by taking Laplace transform techniques and the inversion of the Laplace transform is carried out using a numerical approach.

Numerical analysis of doubly periodic array of cracks/rigid-line inclusions in an infinite isotropic medium using the boundary integral equation method

In this paper, the boundary integral equation approaches are used to study the doubly periodic array of cracks/rigid-line inclusions in an infinite isotropic plane medium. For the doubly periodic rigid-line inclusion problems, the special integral equation containing the axial and shear forces within the rigid-line inclusion is used. The doubly periodic crack problems are dealt with using the displacement discontinuous integral equation approach. Stress intensity factors, effective elastic properties for doubly periodic array of cracks/rigid-line inclusions are calculated and compared with the available numerical solutions.

A new integral equation formulation of two-dimensional inclusion–crack problems

A new integral equation formulation of two-dimensional infinite isotropic medium (matrix) with various inclusions and cracks is presented in this paper. The proposed integral formulation only contains the unknown displacements on the inclusion–matrix interfaces and the discontinuous displacements over the cracks. In order to solve the inclusion–crack problems, the displacement integral equation is used when the source points are acting on the inclusion–matrix interfaces, whilst the stress integral equation is adopted when the source points are being on the crack surfaces. Thus, the resulting system of equations can be formulated so that the displacements on the inclusion–matrix interfaces and the discontinuous displacements over the cracks can be obtained. Based on one point formulation, the stress intensity factors at the crack tips can be achieved. Numerical results from the present method are in excellent agreement with those from the conventional boundary element method.

An Approach for Calculating Strain Distributions inArbitrarily Shaped Quantum Dots

We study strain distribution inside and outside an arbitrarily-shapedquantum dot (QD) buried in an infinite isotropic medium. By defining avery simple vector, we derive a compact formula for stress fieldsexpressed by an integral over the interface between the QD and itssurrounding material. Using this method, the analytical solution for acuboidal QD is obtained, which is different from the previous result.It is shown that our solution satisfies the tractioncontinuity condition on the interface. Based on this solution,it is found that thestrain field in the cuboidal QD in the semiconductor heterostructure issensitive to its height. In addition, the strain distribution around ahemispherical QD is also calculated and demonstrated.