The rotation of a rigid spheroidal inclusion embedded in a transversely isotropic medium

The exact three-dimensional elasticity solutions are given for two problems related to a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium. The first is of axisymmetric nature in which the inclusion is given a constant rotation about its axis of revolution which coincides with the axis of symmetry of the material. The second problem is asymmetric where the spheroidal inclusion is given a constant rotation about a direction that is perpendicular to the axis of elastic symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. In both cases, the moment-rotation relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of the aspect ratio of the spheroid on the rotational stiffness.     

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.     

Dynamic potentials and Green’s functions of a quasi-plane magneto-electro-elastic medium with inclusion

The dynamic potentials of a quasi-plane magneto-electro-elastic medium of transversely isotropic symmetry with an inclusion of arbitrary shape are derived, and the dynamic potentials are finally governed by six scalar equations which can be regarded as inhomogeneous wave equations, Laplace and Helmholtz equations. The explicit expressions of the dynamic Green’s functions of this medium are also obtained both in the space–time domain and in the space–frequency domain. Closed-form expressions for the space–frequency representation of the dynamic potentials are given for the case when the inclusion is circular. The results are employed to obtain the generalized displacement fields of a circular inclusion undergoing uniform eigenstrain, eigenelectric field and eigenmagnetic field. In contrast to the corresponding static Eshelby inclusion problem the magneto-electro-elastic fields (i.e. strain, electric and magnetic field) inside the inclusion are non-uniform in the space–frequency domain.     

Computation of sensitivity of periodically excited dynamical systems

An analytical treatment is presented for bonded contact of a rigid disk inclusion embedded in a penny-shaped crack in a transversely isotropic full-space. Theoretical analysis is carried out using a complete potential function method, and with the aid of Hankel transforms. Boundary conditions propel the problem toward a set of triple integral equations, which are solved analytically and then reduced to a pair of Fredholm integral equations of the second kind. Furthermore, the results are evaluated and presented graphically using numerical schemes, and comparison is made with well-known classical solutions in transversely isotropic and isotropic media. This can be obtained as special cases for the problem to reveal the efficacy of the proposed method. Eventually, it can be seen that not only the presence of the crack around the disk inclusion decreases the axial stiffness, but also the extension of its length also reduces the fracture parameter, stress intensity factor, for different degrees of material anisotropy.     

An elasticity solution for transversely isotropic,functionally graded circular plates

This article presents a new elasticity solution for transversely isotropic, functionally graded circular plates subject to axisymmetric loads. It is assumed that the material properties vary along the thickness of a circular plate according to an exponential form. By extending the displacement function presented by Plevako to the case of transversely isotropic material, we derived the governing equation of the problem studied. The displacement function was assumed as the sum of the Bessel function and polynomial function to obtain the analytical solution of a transversely isotropic, functionally graded circular plate under different boundary conditions. As a numerical example, the influence of the graded variations of the material properties on the displacements and stresses was studied. The results demonstrate that the graded variations have a significant effect on the mechanical behavior of a circular plate.     

Effective Poroelastic Properties of Transversely Isotropic Porous Medium with Aligned Spheroidal Inhomogeneities

A two-component poroelastic composite material is considered. This material consists of a homogeneous transversely isotropic poroelastic matrix and aligned spheroidal inclusions filled with material having different poroelastic properties. The effective poroelastic characteristics are obtained by the effective field method. All the formulas are presented in explicit ready-to-use form.     

Dynamic crack problems in three-dimensional transversely isotropic solids

In this paper, a general boundary element approach for three-dimensional dynamic crack problems in transversely isotropic bodies is presented for the first time. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The procedure is based on the subdomain technique, the displacement integral representation for elastodynamic problems and the expressions of the time-harmonic point load fundamental solution for transversely isotropic media. Numerical results corresponding to cracks under the effects of impinging waves are presented. The accuracy of the present approach for the analysis of dynamic fracture mechanics problems in transversely isotropic solids is shown by comparison of the obtained results with existing solutions.     

Axisymmetric time-harmonic response of a transversely isotropic substrate–coating system

By virtue of a method of displacement potentials, an analytical treatment of the response of a transversely isotropic substrate–coating system subjected to axisymmetric time-harmonic excitations is presented. In determination of the corresponding elastic fields, infinite line integrals with singular complex kernels are encountered. Branch points, cuts, and poles along the path of integration are accounted for exactly, and the physical phenomena pertinent to wave propagation in the medium are also highlighted. For evaluation of the integrals at the singular points, an accurate analytical residual theory is presented. Comparisons with the existing numerical solutions for a two-layered transversely isotropic medium under static surface load, and a transversely isotropic half-space subjected to buried time-harmonic load are made to confirm the accuracy of the present solutions. Selected numerical results for displacements and stresses are presented to portray the dependence of the response of the substrate–coating system on the frequency of excitation and the role of coating layer.     

Bending of piezoelectric plates with a circular hole

Departing from the refined theory of transversely isotropic piezoelectric plates, an analytical solution for the bending of an infinite piezoelectric plate with a circular hole is obtained. Expressions of moment, stress and electric displacement concentration factors are presented in closed form. When the piezoelectric coupling is absent, the results reduce to the corresponding solutions for the transversely isotropic elastic plates. Some numerical results for PZT-6B piezoelectric ceramics are obtained and illustrated by figures. These results show that the effect of piezoelectric coupling on the concentration factors is not negligible.     

饱和横观各向同性分数导数黏弹性土中半封闭衬砌振动响应

在频率域内研究了饱和横观各向同性分数导数黏弹性土体中深埋圆形隧道半封闭衬砌振动响应问题。根据土体在长期沉积过程中存在各向异性的特点,将土骨架视为具有分数导数本构关系的横观各向同性黏弹性体,采用饱和多孔介质理论和弹性理论,利用衬砌内边界应力协调以及土体和衬砌界面处应力和位移连续,得到了简谐荷载作用下饱和横观各向同性黏弹性土和弹性衬砌的位移、应力和孔隙水压力解析表达式。考察了饱和经典弹性土、饱和分数导数性黏弹性土和饱和经典黏弹性土三种条件下饱和黏弹性土和衬砌各参数的影响,表明：横观各向同性面的弹性模量和衬砌厚度对系统动力响应的影响与分数导数阶数和土骨架的黏性有关;渗透系数较小时,系统存在明显的共振现象。另外,在三种条件下半封闭衬砌振动响应存在较大差异。     

Tangential contact problem for a transversely isotropic elastic layer bonded to an elastic foundation

A system consisting of an elastic layer made of a transversely isotropic material bonded to an elastic half-space made of a different transversely isotropic material is considered. An arbitrary tangential displacement is prescribed over a domain S of the layer, while the rest of the layer’s surface is stress-free. The tangential contact problem consists of finding a complete field of stresses and displacements in this system. The generalized-images method developed by the author is used to get an elementary solution to the problem. It is also shown that an integral transform can be interpreted as a sum of generalized images. The case of a circular domain of contact is considered in detail. The results are valid for the case of isotropy as well.     

Axisymmetric elasticity solutions for a uniformly loaded annular plate of transversely isotropic functionally graded materials

Summary The axisymmetric problem of a functionally graded, transversely isotropic, annular plate subject to a uniform transverse load is considered. A direct displacement method is developed that the non-zero displacement components are expressed in terms of suitable combinations of power and logarithmic functions of r, the radial coordinate, with coefficients being undetermined functions of z, the axial coordinate. The governing equations as well as the corresponding boundary conditions for the undetermined functions are deduced from the equilibrium equations and the boundary conditions of the annular plate, respectively. Through a step-by-step integration scheme along with the consideration of boundary conditions at the upper and lower surfaces, the z-dependent functions are determined in explicit form, and certain integral constants are then determined completely from the remaining boundary conditions. Thus, analytical elasticity solutions for the plate with different cylindrical boundary conditions are presented. As a promising feature, the developed method is applicable when the five material constants of a transversely isotropic material vary along the thickness arbitrarily and independently. A numerical example is finally given to show the effect of the material inhomogeneity on the elastic field in the annular plate.     

Scattering of a point generated field by a multilayered spheroid

Summary The point source excitation acoustic scattering problem by a multilayer isotropic and homogeneous spheroidal body is presented. The multilayer spheroidal body is reached by an acoustic wave emanated by an external point source. The core spheroidal region is inpenetrable and rigid. The exterior interface and the interfaces separating the interior layers are penetrable. The scattered field is determined given the geometrical and physical characteristics of the spheroidal body, the location of the point source and the form of the incident field. The approach is not limited in a certain region of frequencies.     

A Micromechanics Solution of a 3D Internal Instability Problem for a Fiber Series in an Infinite Matrix

A micromechanics solution of an internal instability problem for a fiber series in an infinite isotropic elastic matrix subjected to compression is presented. The theoretical solution permits application of transversely isotropic compressible material models to the reinforcement. Possible instability modes are discussed. Numerical solution procedure is demonstrated on a simple particular case.     

A mixed variational equation for a twelfth-order theory of bending of nonhomogeneous transversely isotropic plates

Departing from a linear-theory statement of the three-dimensional problem of a surface and edge loaded transversely isotropic layer, with material properties which are even functions of the thickness coordinate, we use a variational statement of this problem, with independent displacement and transverse stress variations, for a direct-methods derivation of a corresponding twelfth-order two-dimensional statement, which is believed to be computationally beneficial relative to corresponding statements involving independent displacement variations only.     

Elastic strain of a transversely isotropic half space caused by a rigid punchElastische Berührung eines Stòssels und eines in der Querrichtung isotropen Halbraumes

An analytical investigation of the displacement, strain and stress field in every point of a transversely isotropic solid under the action of a rigid indentor is made.The results are then applied to the case of contact between a flatended cylindrical punch and a transversely isotropic half-space.     

Torsion of a round shaft of variable diameter

Problems for a round shaft of variable diameter subjected to torsion are studied. By transforming the governing equation into oblate spheroidal coordinates, a general solution is obtained in terms of the associated Legendre functions of order 2. Two illustrative problems, one being a shaft with a hyperbolic notch, the other a cylindrical shaft containing a small oblate spheroidal cavity located on its central axis, are solved. On the basis of the ensuing stress concentration, the important connection between the deep notch and crack is exploited. Several previously known solutions can be recovered. Results are extended to the cases of composites and transversely isotropic materials.     

横观各向同性弹性地基塞露蒂问题的解析解

根据横观各向同性弹性体半空间问题的基本方程,利用Hankel变换和Bessel函数理论,基于位移法导出的横观各向同性地基空间问题的一般解,得到了材料特征根不相等情况下横观各向同性地基塞露蒂(Cerruti)半空间问题的基本解。该解答经过退化与各向同性Cerruti问题的经典解相吻合。通过算例对两种不同地基内部位移和应力分布进行了对比分析。结果表明,对于给定的算例,两种地基内部沿深度方向的位移分布基本相同,但两种地基内部应力场的差异比较显著。     

Ductile penny-shaped crack in a transversely isotropic cylinder

The problem of a penny-shaped crack contained in a transversely isotropic cylinder of elastic perfectly-plastic material is considered for the case when the crack is extended by an axial load. The problem is reduced to solving numerically a Fredholm integral equation of the second kind for the width of the plastic zone. Graphical results are presented showing the effect of transverse isotropy upon the width of the plastic zone and these are compared with the results for isotropic materials.     

Boundary element analysis of three-dimensional crack problems in two joined transversely isotropic solids

The present paper presents a boundary element analysis of penny-shaped crack problems in two joined transversely isotropic solids. The boundary element analysis is carried out by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The conventional multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of the stress intensity factors are obtained by using crack opening displacements. The numerical scheme results are verified with the closed-form solutions available in the literature for a penny-shaped crack parallel to the plane of the isotropy of a homogeneous and transversely isotropic solid of infinite extent. The new problem of a penny-shaped crack perpendicular to the interface of a transversely isotropic bi-material solid is then examined in detail. The crack surfaces are subject to the three normal tractions and the uniform shear traction. The associated stress intensity factor values are obtained and analyzed. The present results can be used for the prediction of the stability of composite structures and the hydraulic fracturing in deep rock strata and reservoir engineering.