The asymmetric displacement of a rigid spheroidal inclusion embedded in a transversely isotropic medium

Summary An explicit analytical solution is presented for the problem of a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium, where the inclusion is given a constant displacement in a direction perpendicular to the axis of symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. The loadfeflection relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of both the aspect ratio of the spheroid and the anisotropy on the translational stiffness.With 5 Figures     

Asymmetric displacements of a rigid disc inclusion embedded in a transversely isotropic elastic medium of infinite extent

Asymmetric problems related to a penny-shaped rigid inclusion embedded in bonded contact with a transversely isotropic elastic medium are investigated. The asymmetric displacements of the rigid circular inclusion correspond to a rotation about a diametral axis and an in-plane lateral translation. These problems are formulated in terms of Hankel integral transforms and reduced to systems of dual integral equations. The rotational and translational stiffnesses for the embedded rigid circular disc inclusion are obtained in exact closed forms.     

Stress concentration around a strongly oblate spheroidal cavity in a transversely isotropic medium

Expressions for the Eshelby tensor of a strongly oblate spheroidal region in a transversely isotropic medium are given explicitly. Based on the equivalent inclusion method, three dimensional stress concentration around the spherodal cavity subjected to remote uniform loading is analyzed and the associated stress concentration factor is determined. Analogously to two-dimensional blunted cracks, the so called stress rounding factor is introduced so that the connection between the crack tip stress and the stress intensity factor in linear fracture mechanics is established. Numerical values of the stress rounding factor for several representative cases of transversely isotropic symmetry are given.     

3D point force solution for a permeable penny-shaped crack embedded in an infinite transversely isotropic piezoelectric medium

This paper considers the non-axisymmetric three-dimensional problem of a penny-shaped crack with permeable electric conditions imposed on the crack surfaces, subjected to a pair of point normal forces applied symmetrically with respect to the crack plane. The crack is embedded in an infinite transversely isotropic piezoelectric body with the crack face perpendicular to the axis of material symmetry. Applying the symmetry of the problem under consideration then leads to a mixed–mixed boundary value problem of a half-space, for which potential theory method is employed for the purpose of analysis. The cases of equal eigenvalues are also discussed. Although the treatment differs from that for an impermeable crack reported in literature, the resulting governing equation still has a familiar structure. For the case of a point force, exact expressions for the full-space electro-elastic field are derived in terms of elementary functions with explicit stress and electric displacement intensity factors presented. The exact solution for a uniform loading is also given.     

轴对称横观各向同性土体中桩的扭转振动响应研究

基于单相弹性土体的运动方程以及横观各向同性材料的本构关系,研究了轴对称条件下端承桩在横观各向同性土体中的耦合扭转振动响应问题.在柱坐标下推导得到横观各向同性土体受谐和扭转荷载作用的动力控制方程,并采用分离变量法求得了土体扭转振动位移形式解.依据桩土接触面上的衔接条件,求解了桩身的动力平衡方程,并在频域内得到了桩身转角、扭矩和桩顶复刚度的解析解.最后分析了横观各向同性土体的力学参数对动力响应的影响.结果表明,横观各向同性土体的力学参数对桩顶复刚度、桩身转角和扭矩沿深度方向的变化均有着显著影响.     

On the stressed state of a transversely isotropic piezoceramic material with arbitrarily oriented spheroidal inhomogeneity

We consider a stressed state problem of a piezoelectric medium containing an arbitrarily oriented spheroidal inclusion under uniform mechanical and electrical loads. The problem has been solved by using a generalized Eshelby method of equivalent inclusion for the case of piezoceramic material. Testing of the approach for the case of a spheroidal cavity (when the cavity rotation axis coincides with the material polarization axis), for which an exact solution of the problem exists, confirms its high efficiency. Numerical investigations have been carried out, and the stress distribution along the surface of an arbitrarily oriented spheroidal cavity has been studied. __________ Translated from Problemy Prochnosti, No. 2, pp. 112–120, March–April, 2008.     

Transient generalized thermoelastic waves in transversely isotropic medium with a cylindrical hole

The distribution of temperature, displacement, and stress in an infinite homogeneous transversely isotropic elastic solid having a cylindrical hole has been investigated by taking (i) unit step in stress and zero temperature change, and (ii) unit step in temperature and zero stress, at the boundary of the cylindrical hole. The Laplace transform on time has been used to obtain the solutions. Because of the short duration of the second sound effects, the small time approximations have been considered. The temperature and stress are found to be discontinuous at the wave fronts in case (i) whereas these quantities are continuous in case (ii). However, as expected, the displacement is found to be continuous in both the cases.     

Torsion of a transversely isotropic elastic layer bonded to a transversely isotropic half-space

Torsional stresses and displacement of a transversely isotropic elastic layer of finite thickness for which torsional shearing forces are prescribed on its boundary surface are considered. The solutions are given for a few particular cases.     

Orientation-dependent piezoelectric Eshelby S-tensor for a lamellar structure in a transversely isotropic medium

Summary.  This paper provides explicit expressions for the orientation-dependent piezoelectric Eshelby S-tensor with a lamellar structure in a transversely isotropic medium. Many piezoelectric or ferroelectric materials prefer to form a lamellar structure during a change of nano-scale microstructure and domain switch; such a lamellar morphology is energetically more favorable than other shapes in a diffusionless process. According to the above observation, this paper seeks to derive the Eshelby tensor in a coupled field with the consideration of lamellar structure. In general, Eshelby's tensor can be obtained through a Green's function technique, but it is usually given in an integral form. Due to the geometric simplicity of a lamellar structure, a simple explicit form of piezoelectric S-tensor can be obtained directly from the interfacial continuous conditions. In a transversely isotropic medium, the angle between the global symmetric axis and one of the local parallel to the surface of the lamellar inclusion is the key geometrical factor in the construction of the S-tensor. Two special cases with 90° and 180° orientation that bear significant relevance to domain switch in ferroelectric ceramics are discussed in detail. Received September 2, 2002; revised December 3, 2002 Published online: May 8, 2003 This work was supported by the National Science Foundation, Surface Engineering and Materials Design Program, under grant CMS-0093808.     

On three coplanar cracks in an infinite transversely isotropic medium

This paper concerns the problem of determining the stress distribution in an infinite transversely-isotropic medium containing three coplanar cracks. The analysis is carried out by using a solution of the equilibrium equations expressed in terms of displacements under plane strain assumption. By the use of Fourier transforms, we reduce the problem to solving a set of four integral equations. An exact solution of these equations is obtained by using finite Hilbert transform and expressions for the quantities of physical interest are obtained in closed form.     

Interaction of SH-waves by a Griffith crack in an infinite transversely isotropic medium

As a follow-on to a previous problem involving imposed P-waves, the diffraction of SH-waves by a finite Griffith crack located on a symmetric surface of an infinite transversely-isotropic medium is considered. A Fredholm integral equation is derived for the determination of diffracted field. From the integral equation length symptotic development of the solution is obtained which is valid for wavelength compared to the crack length. For wavelengths comparable with the size of the crack, the integral equation is solved numerically. The stress-intensity factor and displacement field in the vicinity of the crack are compared for a range of values of frequency. The approximate solution is compared with the exact solution.

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Résumé Comme suite à l'étude d'un problème précédent, dans lequel on imposait un régime d'ondes de type P, on a considéré la diffraction d'ondes SH par une fissure finie de Griffith située sur la surface symétrique d'un milieu transversalement isotrope. On a déduit une équation intégrale de Fredholm pour déterminer le champ de diffraction. On en a tiré une solution en développement asymptotique qui est valable lorsque la longueur d'onde est grande par rapport à la longueur de la fissure. Pour des longueurs d'onde comparables à la longueur de la fissure, on résoud l'équation intégrale par voie numérique. On compare le facteur d'intensité d'entaille et le champ de déplacement dans le voisinage de la fissure pour diverses valeurs de fréquence.La solution approchée est comparée à la solution exacte.

     

Rayleigh-type wave propagation on a transversely isotropic viscoelastic layer with yielding and rigid foundations

The present study investigates the propagation of Rayleigh-type wave in a transversely isotropic viscoelastic layer in effect of yielding foundation and rigid foundation in two different cases for two considered models. Numerical computation and graphical demonstration have been carried out for the case when layer is comprised of transversely isotropic viscoelastic Mesaverde clay shale material (Model I) and simply isotropic viscoelastic material (Model II). Closed-form expression of phase velocity and damped velocity for both the cases are deduced analytically. Obtained result is found in well agreement to the established standard results existing in the literature. Significant effect of dilatational viscoelasticity, volume viscoelasticity and yielding parameter on phase and damped velocities for both the considered models has been traced out. The comparative study has been performed to unravel the effect of viscoelasticity over elasticity and anisotropy over isotropy in the context of the present problem. Moreover, comparison of phase and damped velocities for the case of layer with stress-free foundation, layer with rigid foundation and layer with yielding foundation serve as a major highlight of the present work.     

Coupled horizontal and rocking vibrations of a rigid circular plate on a transversely isotropic bi-material interface

This paper investigates the coupled rocking and horizontal vibratory response of a rigid circular plate embedded in viscoelastic, transversely isotropic, three-dimensional unbounded media. The boundary-value problem corresponding to the case of distributed horizontal and rocking ring loads at a bi-material interface is solved to obtain the required influence functions for the solution of the present problem. The case of an embedded rigid plate is formulated in terms of a discretized integral equation, which couples the rigid body displacements of the plate with the tractions acting over its contact surface through a set of displacement influence functions. The system of resulting discretized integral equations is solved numerically. The solution results in the tractions over each disc element. This paper carefully takes into account the coupling of the rocking and horizontal responses of the plates that is typical of non-homogeneous interfaces, i.e., their horizontal displacements due to rocking moments and their rotations due to horizontal loads. The dynamic direct and cross compliances of the embedded plate are shown for different governing parameters such as frequency of excitation and bi-material configuration. The present results are useful to the study of dynamic response of deeply buried foundations and anchors in non-homogeneous soils.     

A transversely isotropic half-space indented by a flat annular rigid stamp in the presence of adhesion

Conclusions The curves of normal displacementu _z (r, 0) (Fig. 2) in the presence of adhesion coincide approximately with the dashed line. The contact pressure zz(r,0) in the presence of adhesion is slightly larger than the one without friction. Thus,u z(r,0) and zz(r,0) in the actual condition can be evaluated approximately by the results without friction.With 3 Figures