A method for the determination of the elastic equilibrium of isotropic bodies with curvilinear inclusions. Part 3. Antiplane deformation

Within the framework of the approach proposed by L. P. Mazurak, L. T. Berezhnyts'kyi, and P. S. Kachur [“Method for determination of elastic equilibrium of isotropic bodies with curvilinear inclusions. Part 1. Mathematical foundations,”Fiz.-Khim. Mekh. Mater.,33, No. 6, 21–31 (1997)], we construct a new method for the determination of elastic equilibrium of cylindrical bodies with noncanonical curvilinear foreign elastic inclusions under conditions of longitudinal shear. Unlike the method of perturbation of the form of a boundary, this method imposes no restrictions on the form of inclusions. The method is based on a procedure of determination of contour integrals of the Cauchy type by using the Faber polynomials. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 2, pp. 21–26, March–April, 1999.     

Numerical modeling of development of fracture in anisotropic composite materials at low-velocity loading

The problem of normal interaction of the steel isotropic compact cylindrical projectile with the orthotropic plate on ballistic limit in range of velocities of impact from 50 to 400 m/s is considered. Target material is as organoplastic with initial orientation of mechanical properties, and the material, which properties received by turn on 90° relative to the axis OY of an initial material. Fracture of targets is investigated; the comparative analysis of efficiency of their protective properties depending on orientation of elastic and strength properties of an anisotropic material is carried out. The task is solved numerically, using the method of finite elements in three-dimensional statement. The behavior of a material of the projectile is described by elastic–plastic model; the behavior of anisotropic material of the target is described by elastic–brittle model with various ultimate strength limits on pressure and tension.     

A method for the determination of the elastic equilibrium of isotropic bodies with curvilinear inclusions. Part 2. Plane problem

Within the framework of the approach proposed in the first part of the work, the two-dimensional boundary-value problem for an isotropic body with noncanonical elastic inclusion is reduced to a finite system of linear algebraic equations. It is shown that the solution of this problem for elastic inclusions with small radius of curvature at the tip and/or cusps describes the intensity and concentration of stresses in the composition. For some special examples, we reveal the influence of elastic properties of the components of the composition and configuration of the inclusions on its local stress-strain state. It is also established that, unlike the method of perturbation of the shape of the boundary, this method is applicable to the determination of the concentration and intensity of stresses in the vicinity of the tips of elastic inclusions with small radii of curvature, including the inclusions whose elastic properties are close to the elastic properties of the matrix. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 1, pp. 16–26, January–February, 1999.     

Optimisation of 3D RVE for anisotropy index reduction in modelling thermoelastic properties of two-phase composites using a periodic homogenisation method

In the present paper, a periodic homogenisation method is used to predict the thermomechanical properties of heterogeneous ceramics containing randomly dispersed inclusions (or pores) by modelling this complex microstructure by a Representative Volume Element (RVE) composed of a small number of inclusions (or pores). The combination of suitable RVE geometries is used to find 3D arrangements as less anisotropic as possible, in the case of ceramics with glass matrix and either alumina inclusions or pores. It is shown that anisotropy indices, in terms of elastic stiffness and thermal expansion of face-centred cubic (F.C.C.) and hexagonal close-packed (H.C.P.) arrangements are very close to the isotropic case. Therefore, these arrangements have, then, been assumed to be isotropic for the calculation of the isotropic thermoelastic properties of the model materials. A good agreement with the experimental and analytical results is observed, validating the proposed methodology.     

Vibration control of an elastic strip by a singular force

Vibration characteristics of an elastic plate in the shape of an infinite strip are changed by applying a lateral concentrated force to the plate. The homogeneous, isotropic, elastic plate is infinite in the x-direction and the sides are simply supported. The size of the force is changed in proportion to the displacement measured at a certain point of the plate. The proportionality constant serves as the control parameter. The mathematical formulation of this distributed control problem and its analytical solution in terms of the vibration frequencies of the plate are given. The vibration frequencies are plotted as a function of the control parameter.     

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.     

The problem of two plastic and heterogeneous inclusions in an anisotropic medium

We demonstrate an integral equation for the total local strain ε^T in an anisotropic heterogeneous medium with incompatible strain ε^p and which is at the same time submitted to an exterior field. The integral equation is solved in the case of an heterogeneous and plastic pair of inclusions, for which we calculate the average fields in each inclusion as well as the different parts of the elastic energy stocked in the medium.The solution is applied to the case of two isotropic and spherical inclusions in an isotropic matrix loaded in shear. The results are compared with those deduced from a more approximate method based on Horn's approximation of the integral equation. In appendix we give a numerical method for calculating the interaction tensors between anisotropic inclusions in an anisotropic medium as well as the analytic solution in the case of two spherical inclusions located in an isotropic medium.     

Method of perturbations of the shape of the boundary in fracture mechanics. Part 1: Notches

We analyze exact and approximate methods used for the determination of the stress-strain state of plates bent according to Kirchhoff and isotropic plates weakened by sharp defects subjected to two-dimensional tension-compression. For a computational scheme to describe the actual stress-strain state more adequately, it is necessary to take into account the variation of anisotropic and orthotropic properties of the material of a plate across its thickness. This can be realized by finding approximate solutions of the problem by the method of perturbations of the shape of the boundary. We propose a modified approach based on the use of complex-valued functions adapted to the possibilities of contemporary computers. This method allows one to construct approximate solutions meaningful for fracture mechanics for the case of sharp holes and perfectly rigid inclusions under plane and antiplane loads within the framework of refined sixth-order theories of bent plates. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 33, No. 3, pp. 7–18, May–June, 1997.     

Thermostressed state of a plate with angular inclusions

We reduce the problems of heat conduction and thermoelasticity for an isotropic plate with foreign elastic inclusions bounded by piecewise smooth curves to problems of conjugation of analytic functions on broken lines. Singular solutions of these problems and the corresponding complex potentials are constructed. Luts'k Industrial Institute, Luts'k. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 34, No. 3, pp. 67–72, May–June, 1998.     

Longitudinal shear of an anisotropic body with elastic strips

We solve the problem of a longitudinal shear of an anisotropic body with thin elastic inclusions in the form of strips and show that the anisotropy of elastic properties significantly affects the intensity of local stresses. Lutsk Industrial Institute, Lutsk. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 33, No. 1, pp. 51–56, January–February, 1997.     

Analytical evaluation of J-integral for elliptical and parabolic notches under mode I and mode II loading

In the present work the J-integral (indicated here as J_Vρ because two parallel flanks are not present) was calculated by using, along the free border, the exact analytical stress distribution for the ellipse and the asymptotic one for parabolic notches. The material was assumed as homogeneous isotropic and linear elastic. First, for an ellipse under remote tensile loading, the expression of J_Vρ has been analytically calculated on the basis of Inglis’ equations. The equations have been used to prove that, in terms of J-integral, the crack is the limit case of an equivalent elliptic notch. Furthermore, by distinguishing the symmetric and skew-symmetric terms, the well-known Stress Intensity Factors (SIF) of mode I and II for a crack in a wide plate under tension are obtained by adding a limiting condition. Second, by means of Creager–Paris’ equations, J_Vρ has been analytically calculated for a parabolic notch of assigned tip notch radius ρ. The asymptotic value of J_Vρ and the relationship between the peak stress and the relative SIF are the same as the ellipse. Finally, as an engineering application, we provide an accurate formula for the evaluation of the Notch Stress Intensity Factors of a crack, mainly subjected to tensile stress, from the peak stress of the equivalent ellipse under the same loading.     

Evaluation of the anisotropic Green's function in three dimensional elasticity

A perturbation expansion technique for approximating the three dimensional anisotropic elastic Green's function is presented. The method employs the usual series for the matrix (I–A)^-1 to obtain an expansion in which the zeroth order term is an isotropic fundamental solution. The higher order contributions are expressed as contour integrals of matrix products, and can be directly evaluated with a symbolic manipulation program. A convergence condition is established for cubic crystals, and it is shown that convergence is enhanced by employing Voigt averaged isotropic constants to define the expansion point. Example calculations demonstrate that, for moderately anisotropic materials, employing the first few terms in the series provides an accurate solution and a fast computational algorithm. However, for strongly anisotropic solids, this approach will most likely not be competitive with the Wilson-Cruse interpolation algorithm.This research was sponsored by the Exploratory Studies Program of Oak Ridge National Laboratory and the Division of Materials Science, U. S. Department of Energy, under contract DE-AC05-84OR21400 with Lockheed Martin Energy Systems, Inc.     

Magnetic crystallographic anisotropy of Fe/GaAs(100) epitaxial films

An investigation was made of the anisotropic properties of Fe films obtained by molecular-beam epitaxy on GaAs(100) substrates. It is shown that at thicknesses t=40–50 Å the Fe films begin to exhibit cubic magnetic anisotropy. At thicknesses t>100 Å the first constant of cubic anisotropy K₁ has values similar to those for “bulk” Fe single crystals. Films of intermediate thickness 50<t<100 Å possess both isotropic and anisotropic phases. Pis’ma Zh. Tekh. Fiz. 25, 88–95 (January 26, 1999)     

Direct boundary integral procedure for a Boltzmann viscoelastic plane with circular holes and elastic inclusions

A direct boundary integral method in the time domain is presented to solve the problem of an infinite, isotropic Boltzmann viscoelastic plane containing a large number of randomly distributed, non-overlapping circular holes and perfectly bonded elastic inclusions. The holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can, in general, be different. The method is based on a direct boundary integral approach for the problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described by Crouch and Mogilevskaya [1], and a time marching strategy for viscoelastic analysis described by Mesquita and Coda [2–8]. Benchmark problems and numerical examples are included to demonstrate the accuracy and efficiency of the method.     

Determination and representation of the stress coefficient terms by path-independent integrals in anisotropic cracked solids

Because the elastic T-stress and other coefficients of the higher-order terms play an important role in fracture mechanics such as the stability of crack kinking, crack path, and two-parameter characterization of elastic-plastic crack tip fields, determination of all the coefficients in the crack tip field expansion in an anisotropic linear elastic solid is presented in this paper. Utilizing conservation laws of elasticity and Betti's reciprocal theorem, together with selected auxiliary fields, T-stress and third-order stress coefficients near the crack tip are evaluated first from path-independent line integrals. To determine the T-stress terms using the J-integral and Betti's reciprocal work theorem, auxiliary fields under a concentrated force and moment acting at the crack tip are used respectively. Through the use of Stroh formalism in anisotropic elasticity, analytical expressions for all the coefficients including the stress intensity factors are derived in a compact form that has surprisingly simple structure in terms of one of the Barnett-Lothe tensors, L. The solution forms for degenerated materials, monoclinic, orthotropic, and isotropic materials are also presented.     

Numerical Estimation of Plastic J-Integral by the Load Separation Method for Inclined Cracks Under Tension

The load separation concept is employed to determine the mixed mode plastic η _pl and h^COD_pl{\eta^{COD}_{pl}} factors for the tension plate with an inclined centre throughthickness crack for power law hardening materials. The separation parameters S _ij are calculated from the FE computed load – plastic displacement (or COD) curves for each case. It is shown that the mixed mode plastic η _pl factor is a function of both the crack angle orientation θ (or the applied elastic parameter M _e ) and the strain hardening exponent.     

Mixed-mode fracture mechanics of anisotropic plates by means of the T-criterion

A direct generalization of the T-criterion for the prediction of mixed-mode fracture characteristics of homogeneous anisotropic plates is given in this paper. This fracture criterion, introduced by one of the authors (P.S.T.), was successfully used in various LEFM applications with isotropic materials as well as for cases where small scale yielding (s.s.y) of metallic materials was encountered. Due to the principles of T-criterion, according to which the different role played by the distortional and dilatational strain-energy density in fracture is fully recognized, its eventual use with anisotropic materials was prohibited, since the decomposition of the anisotropic elastic potential was not as yet conceivable. It is in terms of such an analytical decomposition especially for transversely isotropic media, recently derived by the authors, that the present generalization of the T-criterion was under-taken. Although the example problems treated herein assume homogeneously anisotropic media, preliminary comparison of the theoretical predictions with experimental data from fibrous composites revealed satisfactory agreement.     

Plane problem for a piecewise-homogeneous anisotropic body with elastic inclusion in the form of a strip

A plane problem of the theory of elasticity for a piecewise-homogeneous anisotropic plane with elastic inclusions in the form of strips is solved by the method of jump functions. Inclusions are simulated by the jumps of the vector of stresses and the derivative of the vector of displacements on the median surfaces. By using complex potentials, we obtain the dependences of the components of the stress tensor and the vector of displacements on the load and unknown jump functions. In view of the conditions of interaction of a thin inclusion with an anisotropic medium, the problem is reduced to a system of singular integral equations for the jump functions. In the general case, this system is solved by the collocation method. For the cases of a slot and a perfectly rigid inclusion, we deduce the dependences of the generalized stress intensity factors on the concentrated forces and edge dislocations. Franko L'viv University, L'viv; Pidstryhach Institute of Applied Problems of Mechanics and Mathematics. Ukranian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 6, pp. 7–16. November–December, 1999.     

Thermal expansion of bismuth telluride

The results of X-ray and dilatometric measurements of the thermal expansion of bismuth telluride in the temperature range of 4.2–850 K have been critically analyzed. The joint statistical processing of the experimental data has been performed by the least squares method and the most reliable temperature dependences of the linear thermal expansion coefficients along the principal crystallographic axes α _a and α _c and the average linear coefficient [`(a)] _L\bar \alpha _L , as well as the density of Bi₂Te₃, have been recommended. The results indicate that the linear thermal expansion coefficients along the directions parallel and perpendicular to the cleavage planes decrease with an increase in the temperature in a narrow temperature range. At temperatures below 298 K, the character of the temperature dependences of the linear thermal expansion coefficients of Bi₂Te₃ has been analyzed in terms of the anharmonicity of the chemical bonding forces in the layer structure and anisotropy of the elastic constants. In terms of the deviations of the composition of the Bi₂Te₃ compound from the stoichiometric one and the real (defect) structure of the compound, a model has been proposed to explain the minimum in the (α _a T) dependence of Bi₂Te₃ near the melting temperature of bismuth. A method for calculating the temperature dependence of the linear thermal expansion coefficients of the anisotropic layer crystals using the data on the specific heat has been discussed, which provides good agreement of the calculated linear thermal expansion coefficients with the experimental data within the accuracy of the measurements of the linear thermal expansion coefficients.     

Development of a structural model of deformation with hardening on the basis of distributed dislocations

We consider a problem of elastoplastic deformations with strengthening in the case where a “tooth” of yield is present in the (σ-∈) diagram. On simulation of the process of development of plastic deformations with hardening by using distributed screw dislocations, a conclusion on the existence of a structural parameter is drawn. Kherson Industrial Institute, Kherson. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 34, No. 3, pp. 103–107, May–June, 1998.