The problem of thermal conduction for two ellipsoidal inhomogeneities in an anisotropic medium and its relevance to composite materials

Summary We consider the problem of thermal conduction for an unbounded medium containing two ellipsoidal inhomogeneities subjected to a remote homogeneous boundary condition of temperature. The constituents are anisotropic and the ellipsoids could be at arbitrary orientations. In the formulation we first introduce some appropriate transformations into the heterogeneous medium and transform the problem into an isotropic matrix consisting of two analogous ellipsoidal inhomogeneities. Next, we replace the effect of inhomogeneities by some polynomial types of equivalent eigen-intensities by the concept of equivalent inclusion. These procedures allow us to write the local fields in terms of harmonic potentials and their derivatives. Numerical results show that linear approximations of eigen-fields yield accurate results in comparison with existing solutions by Honein et al. [2] for moderately separated inhomogeneities. Solutions of this type are used to estimate the overall thermal conductivity of composites with periodic microstructure. Finally, we present results for composites consisting of spherical inclusions with body-centered cubic, face-centered cubic, body-centered orthorhombic, and face-centered orthorhombic arrays.     

Stresses in an infinite medium with two similar circular cylindrical inclusions

Summary Stresses around two similar circular cylindrical inclusions in an infinite medium under the generalised plane strain conditions subjected to uniform far-field stresses are investigated in this paper. The analysis is based on the complex stress potentials of Muskhelishvili [1]. Stresses can be found with any clearance between the two inclusions. Special treatment has been made for the case in which the two inclusions are in contact with each other, leading to a closed form solution to the local stresses at the contact point. Stress singularities are established in two extreme cases of either rigid or void inclusions, complementing the results for the anti-plane problem [2]. It has been shown that for inclusions of finite modulus no stress singularity arises but different degrees of stress concentration around the contact point can be found instead depending on the Young's modulus ratio between the inclusion and the medium and the loading condition. Other effects, such as the clearance between the inclusions and the Young's modulus ratio between the inclusion and the medium on the distribution of the interfacial stresses, are also examined when the two inclusions are in contact or separate. Numerical results are shown and discussed and they tend to imply a wider applicability of the conclusions obtained in this paper than the idealised case as analysed.     

On the interaction between an edge dislocation and two circular inclusions in an infinite medium

The interaction between an edge dislocation near two circular inclusions, placed on the axis of symmetry, in an infinite medium is investigated within the frame-work of the linear theory of elasticity. The Airy stress function with the help of the bipolar coordinates will be used for convenience. The solution is constructed by considering the relation between the real edge dislocation and the imaginary one placed at the inverse point. Results obtained by numerical calculation indicate that, for the region near interface, the interactions are greatly affected by the geometrical relations and the combinations of elastic constants. The dislocation has a stable equilibrium position or an unstable equilibrium position at some intermediate distance from the inclusions under some combinations of elastic constants.     

The onset of Brinkman ferroconvection in an anisotropic porous medium

Onset of ferroconvection in an anisotropic porous layer heated from below is investigated theoretically using modified Brinkman extended-Darcy equation with fluid viscosity different from effective viscosity. The isothermal bounding surfaces of a porous layer are considered to be either free or rigid-paramagnetic/ferromagnetic. The eigenvalue problem is solved exactly for free boundaries, while for realistic rigid-paramagnetic or rigid-ferromagnetic boundaries the critical stability parameters are obtained numerically using the Galerkin method. It is seen that the stability of the system depends on the nature of boundaries and rigid-paramagnetic boundaries are found to be preferred to the ferromagnetic ones as well as free boundaries in controlling ferroconvection in an anisotropic porous layer. It is observed that increase in the value of thermal anisotropy parameter and viscosity ratio is to delay the onset of ferroconvection, while increase in the value of mechanical anisotropy parameter and magnetic number is to hasten the onset of ferroconvection. Moreover, increasing the value of thermal anisotropy parameter and decreasing the value of mechanical anisotropy parameter is to narrow the convection cells.     

Hypersingular integral equations for a thermoelastic problem of multiple planar cracks in an anisotropic medium

The problem of calculating the thermoelastic stress around an arbitrary number of arbitrarily located planar cracks in an infinite anisotropic medium is considered. The cracks open up under the action of suitably prescribed heat flux and traction. With the aid of suitable integral solutions, we reduce the problem to solving a system of Hadamard finite-part (hypersingular) integral equations. The hypersingular integral equations are solved for specific cases of the problem.     

Thermal stresses in an anisotropic plate with angular inclusions

We reduce the plane problem of anisotropic thermoelasticity for bodies with angular inclusions to a Fredholm system of integral equations. We construct potentials with power singularities at the angular points in the explicit form and present an example of application of the obtained results to the investigation of the dependence of local stresses in a plate with elastic stringer on its physicomechanical characteristics. Lutsk State Technical University, Lutsk. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 3, pp. 59–68, May–June, 1999.     

Collinear periodic cracks in an anisotropic medium

The problem of collinear periodic cracks in an anisotropic medium is examined in this paper. By means of Stroh formalism and the conformal mapping method, we obtain general periodic solutions for collinear cracks. The corresponding stress intensity factors, crack opening displacements and strain energy release rate are found.     

An elasticity problem for an isotropic medium containing thin tunnel inclusions with complex shapes

Summary The state of planar deformation has been examined in an isotropic medium containing a thin tunnel inclusion whose rigidity does not exceed that of the matrix, while the cross section is bounded by a smooth curve of oval type. The treatment involves singular integrodifferential equations for the displacement steps at the inclusion surfaces. Analytic solutions are obtained for these equations and the stresses in the inclusion are determined, as well as the stress concentrations in the matrix around the vertices. The local state of stress around the vertices for an inclusion with a compound configuration can be determined by solving for the corresponding equivalent elliptic tunnel inclusion.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 5, pp. 23–28, September–October, 1989.     

Scattering of radiation in an anisotropic turbulent medium

Scattering of laser radiation on density fluctuations in propagation of radiation through an anisotropic turbulent medium is analyzed. It is shown that the deviation angles in turbulent gas flows at atmospheric pressure equal ∼10⁻⁵–10⁻⁴ rad and can be detected by means of speckle photography. A statistical analysis of two-dimensional fields of deviation angles makes it possible to evaluate three-dimensional density correlation functions in a turbulent flow. It is shown that taking account of the turbulence anisotropy leads to distributions of the laser-radiation intensity over deviation angles that deviate substantially from the Gaussian distribution. Academic Scientific Complex “A. V. Luikov Institute of Heat and Mass Transfer of the National Academy of Sciences of Belarus,” Minsk, Belarus. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 72, No. 1, pp. 96–101, January–February, 1999.     

Longitudinal shear of an anisotropic body with angular inclusions

We study local stresses near the tips of angular inclusions in an anisotropic body subjected to antiplane deformation and deduce explicit expressions for the indicated stresses in the vicinities of the tips. We also present characteristic equations for the order of singularities of stresses and perform their numerical analysis. Lutsk Industrial Institute, Lutsk. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 4, pp. 86–90, July–August, 1996.     

Two-dimensional static deformation of an anisotropic medium

The problem of two-dimensional static deformation of a monoclinic elastic medium has been studied using the eigenvalue method, following a Fourier transform. We have obtained expressions for displacements and stresses for the medium in the transformed domain. As an application of the above theory, the particular case of a normal line-load acting inside an orthotropic elastic half-space has been considered in detail and closed form expressions for the displacements and stresses are obtained. Further, the results for the displacements for a transversely isotropic as well as for an isotropic medium have also been derived in the closed form. The use of matrix notation is straightforward and avoids unwieldy mathematical expressions. To examine the effect of anisotropy, variations of dimensionless displacements for an orthotropic, transversely isotropic and isotropic elastic medium have been compared numerically and it is found that anisotropy affects the deformation significantly.     

Electro-optic modulation in an anisotropic artificial Kerr medium

We describe the performance of intensity and phase modulators that use an aqueous suspension of polytetrafluoroethylene (PTFE) microparticles. In this medium, the electro-optic effect is caused by the reorientation of anisotropic microparticles in an applied electric field. The intensity modulator was constructed in the Kerr geometry by the use of a sample path length of 20 μm. The response time of the modulator is less than 25 ms, and the depth of modulation was measured to be 28 dB for a switching voltage of 134 V(rms). The switching voltage necessary to achieve a π-phase shift with the phase modulator is less than 30 (Vrms).     

On the interaction between an edge dislocation and two circular inclusions in an infinite medium (continued report)

To continue the work presented in the previous paper [1] the interaction between two circular inclusions and an edge dislocation, which are arranged in the order of inclusion-edge dislocation-inclusion and are placed on the axis of symmetry, is investigated within the frame-work of the linear theory of elasticity. Results obtained by numerical calculation indicate that, for the region near interface, the interactions are greatly affected by the geometrical relations and the combinations of elastic constants.It is also interesting to note that the dislocation has a stable equilibrium position or an unstable equilibrium position at some intermediate distance by the combinations of elastic constants.     

Riemann problem for the one-velocity model of a heterogeneous medium

The internal interfractional-interaction forces have been allowed for in the one-velocity model of a heterogeneous medium, and a complete solution of the Riemann problem has been obtained. The shock adiabat of the mixture, consistent with the model’s equations, has been used in constructing the solution with shock waves. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 81, No. 6, pp. 1133–1141, November–December, 2008. Original article submitted February 27, 2006; revision submitted March 30, 2008.     

Mechanical response of an anisotropic liquid-saturated porous medium

Summary.  An eigenvalue approach following Laplace transformation has been employed to study the mechanical response of an anisotropic liquid-saturated porous solid. Analytical solutions are obtained in the transformed domain. A numerical inversion technique is used for inverting the Laplace transforms and to get the results in the physical domain, numerically. The results are taken for two types of surface loadings: (i) Impulsive loading, (ii) continuous loading, for a particular model and are discussed graphically. Received July 30, 2001; revised December 9, 2002 Published online: May 20, 2003     

Homogenization of a degenerate parabolic problem in a highly heterogeneous periodic medium

We consider the homogenization of a time-dependent heat transfer problem in a highly heterogeneous periodic medium made of two connected components having heat capacities c_α(x) and heat conductivities a_α(x), α=1,2 of order one, separated by a third material with thickness of the same order ε than the basic periodicity cell having heat capacity c₃(x) and conductivity λa₃(x) where a₃ is of order one and λ tends to zero with the size ε of the elementary cell. We assume only that c_i(x)?0, i=1,2,3 almost everywhere, such that the problem can be degenerate (parabolic-elliptic). We show that the critical value of the problem is and identify the homogenized problem depending on δ is zero, strictly positive finite or infinite.