The elastic torsion problem for a nonhomogeneous and transversely isotropic half-space

Summary The statical Reissner-Sagoci problem for nonhomogeneous transversely isotropic elastic half-space is investigated. The modulus of rigidity is assumed asC _ii (r,z)= _ii r f(z),i=4, 6 and 0, where _ii are constants. The expressions for stresses, displacement and torque are given. The analysis is based on the assumption that the tangential displacement is prescribed within the areaz=0,r1 and shearing stress is zero in the outside arear>1,z=0. The integral equation which is obtained is solved in the case whenf(z)=cosh² kz and 0, wherek is a constant.With 7 Figures     

Torsion of a nonhomogeneous transversely isotropic half space

The statical Reissner Sagoci problem for a transversely isotropic, nonhomogeneous elastic solid is investigated. The modulus of rigidity of the medium is assumed to be variable as a power of the radial coordinate in the form $\text{r}{}^{\mathrm{\beta }}\text{(β ? 0)}$. The expressions for stresses, displacement and torque are given.     

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.     

A fundamental solution for transversely isotropic and nonhomogeneous media

The purpose of this paper is to consider the concept of a ring of sources or forces using the integral transform techniques to derive the axisymmetric fundamental solution for nonhomogeneous transversely isotropic elastic media. Firstly, the formulation of the problem in homogeneous media to derive the fundamental solutions is shown. In the case of a nonhomogeneous medium, the shear modulus of the material varies with the z-coordinate exponentially.     

Interaction between internal loading and surface indentation for a transversely isotropic elastic half-space

Summary An analytical solution is given for the axisymmetric indentation problem of a half-space loaded by internal forces. Closed form results are presented for an interior point force and a flat-ended indenter. The interaction between internal and external loadings is discussed using numerical results and graphs.With 2 Figures     

Annular crack surrounding an elastic fibre in transversely isotropic elasticity

The axisymmetric problem of an infinitely long transversely isotropic elastic fibre perfectly bonded to a dissimilar transversely isotropic elastic matrix containing an annular crack is considered. The annular crack, surrounding the fibre, is subjected to prescribed longitudinal tension. A potential function approach is used to find the solution of the basic equations. The mixed boundary value problem is reduced to the solution of a singular integral equation, which is further reduced, by using Chebyshev polynomials, to a system of algebraic equations.     

Stresses in a constrained transversely isotropic elastic solid caused by a moving dislocation

Summary An unbounded, transversely isotropic elastic solid whose elastic constants obey the constraint (c ₁₁–c ₄₄) (c ₃₃–c ₄₄)=(c ₁₃+c ₄₄)² is excited by a suddenly applied moving dislocation. Explicit expressions are obtained for the plane stress field induced by the (constant speed) dislocation moving parallel to the material symmetry axis. The speed of the dislocation may be sub-, tran-, or super-sonic with respect to the material wave speeds. An unexpected result is the existence of a special dislocation speed, in the transonic range, which causes the Mach head wave to be annihilated.With 2 Figures     

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.     

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.     

Tangential contact problems for several transversely isotropic elastic layers bonded to an elastic foundation

We consider a system consisting of n elastic layers made of different transversely isotropic materials bonded to each other and the last layer bonded to an elastic half-space made of a different transversely isotropic material. An arbitrary tangential displacement is prescribed over a domain S of the first layer, while the rest of the layer’s surface is stress free. The tangential contact problem consists of finding the complete stress and displacement fields in this system. The Generalized Images method developed by the author is used to get an elementary solution to the problem. We first consider the case of two layers and then generalize it for the case of n layers. The same problem is solved by the integral transform method, and it is shown that an integral transform can be interpreted as a sum of generalized images. The results are valid for the case of isotropy as well.     

Penny-shaped crack in a transversely isotropic solid

An analytical solution is given for the displacement and stress distribution produced in the interior of a transversely isotropie solid containing a penny-shaped crack situated in an elastic symmetry plane and axially-loaded. Curves of numerical results are presented for the stress intensity factor and the normal displacement. They show the influence of this type of anisotropy.     

Analysis of the dispersion relation for an incompressible transversely isotropic elastic plate

Summary The dispersion relation associated with harmonic wave propagation in an incompressible, transversely isotropic elastic plate is derived. Such a material is characterized by only three material constants, contrasting with five in the corresponding compressible case. Motivated by a numerical investigation, asymptotic expansions, giving phase speed and frequency as functions of wave number, are derived in both the long and short wave regimes. These approximations, which owing to the constitutive simplifications are readily available, are shown to provide excellent agreement with the corresponding numerical solution. It is envisaged that the detailed investigation carried out in this paper will aid numerical inversion of the transform solutions often used in impact problems. Additionally, the asymptotic investigation provides the necessary basis for future studies to derive asymptotically approximate models to describe long and short wave motion.     

Green's function solutions for semi-infinite transversely isotropic materials

The Green's function problem of a semi-infinite transversely isotropic medium with the plane boundary parallel to the plane of isotropy is solved by using the potential function method. The Green's function solutions are expressed in terms of harmonic and bi-harmonic functions which are obtained by the separation of variables method. Closed form solutions for point forces applied in the interior of the medium are obtained. The present solution reduces to Sveklo's results when the point force is normal to the plane of isotropy and $\text{√(C}{}_{11}\text{C}{}_{33}\text{)-C}{}_{13}\text{-2C}{}_{44}\text{≠ 0}$. The Green's function solutions of Michell, Lekhnitzki and Hu, which deal with point forces applied at the free surface of a half-plane and $\text{√(C}{}_{11}\text{C}{}_{33}\text{)-C}{}_{13}\text{-2C}{}_{44}\text{≠ 0}$, can also be reproduced from the present approach. Furthermore, the present solution can be reduced to the results of Mindlin for semi-infinite isotropic materials by suitable substitutions of elastic constants.     

Some dynamic properties of incompressible,transversely isotropic elastic media

Summary This paper investigates various dynamic properties of incompressible, transversely isotropic elastic media. The propagation condition for such materials allows the wave speeds to be obtained in explicit form. An examination of the slowness surface and direction of energy flux as the extensional modulus along the fibre tends to infinity is then easily carried out. The paper also includes an investigation of the dynamic response of such materials to a particular line impulsive force. This is done using integral transforms. These transforms are invertible in closed form.     

Elastodynamic analysis of a half plane crack in a transversely isotropic solid under 3-D transient loading

Summary A three-dimensional analysis of a half plane crack in a transversely isotropic solid is performed. The crack is subjected to a pair of suddenly-applied normal line loadings on its faces. Transform methods are used to reduce the boundary value problem to a single integral equation that can be solved by the Wiener-Hopf technique. The Cagniard-de Hoop method is employed to invert the transforms. An exact expression is derived for the mode I stress intensity factor as a function of time and position along the crack edge. Numerical results are given.     

Long wave asymptotic integration in incompressible transversely isotropic elastic structures

Summary A two-dimensional theory is developed for the motion of incompressible transversely isotropic layered structures in the vicinity of their cut-off frequencies. The dynamic asymptotic stress-strain-state is determined in terms of the long wave amplitude by direct asymptotic integration. Leading order (and refined) governing equations are obtained for the long wave amplitude. At both orders these are shown to be asymptotically consistent with the full three-dimensional theory. The leading order governing equation is observed to show possible wave-like behavior for certain material classes, this being connected to the possible existence of negative group velocity in the long wave regime.     

On simple constitutive restrictions for transversely isotropic nonlinearly elastic materials and isotropic magneto-sensitive elastomers

The aim of this paper is to extend the analysis of constitutive restrictions proposed for isotropic nonlinearly elastic materials to transversely isotropic elastic solids and isotropic magneto-sensitive elastomers. These two models are considered because their more general constitutive equations, which are given as strain-energy functions depending on certain invariants, show a similar formulation. The restrictions imposed on the constitutive relations are based on different physically admissible behaviors and given in terms of inequalities referred to simply as constitutive inequalities. The general considerations studied are used to illustrate the constitutive structure of some examples.