Wave propagation in an incompressible transversely isotropic fibre-reinforced elastic media

The boundary conditions at free surface of an incompressible, transversely isotropic elastic half-space are satisfied to obtain the reflection coefficients for the case when outer slowness section is re-entrant. Two quasi-shear waves will be reflected for an angular range of direction of incident wave. The numerical illustrations of reflection coefficients are presented graphically for three arbitrary materials.     

Exact solutions for stress analysis of transversely isotropic elastic layers

Summary The general equations for the elastic analysis of transversely isotropic materials are written in a form which allows derivatives in the thickness direction for all orders to be conveniently calculated. The displacements and stresses are expanded in Taylor series of a form suitable for deriving exact solutions for thick elastic layers with stress-free surfaces. This extends the work of Rao and Das for isotropic materials to the transversely isotropic case. A class of exact solutions are used to obtain results for stress concentration factors due to circular holes in layers.

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Exakte Lösungen der Spannungsberechnung von transversal-isotropen elastischen Schichten

Übersicht Die Grundgleichungen der Elastizität von transversal-isotropen Stoffen werden auf eine für die Berechnung beliebig hoher Ableitungen in eine Dickenrichtung zweckmäßige Form gebracht. Die Verschiebungen und Spannungen werden in für exakte Lösungen dicker Schichten mit spannungsfreien Laibungen geeignete Taylor-Reihen entwickelt. Dies ist eine Weiterführung der Arbeiten von Rao und Das für den isotropen Fall auf den transversal-isotropen Fall. Eine Klasse exakter Lösungen wird benutzt, um die Spannungskonzentrationsfaktoren gelochter Schichten zu bestimmen.

     

End effects for plane deformations of an elastic anisotropic semi-infinite strip

In the linear theory of elasticity, Saint-Venant's principle is used to justify the neglect of edge effects when determining stresses in a body. For isotropic materials, the validity of this is well established. However for anisotropic and composite materials, experimental results have shown that edge effects may persist much farther into the material than for isotropic materials and as a result cannot be neglected. This paper further examines the effects of material anisotropy on the exponential decay rate for stresses in a semi-infinite elastic strip. A linearly elastic semi-infinite strip in a state of plane stress/strain subject to a self-equilibrated end load is considered first for a specially orthotropic material and then for the general anisotropic material. The problem is governed by a fourth-order elliptic partial differential equation with constant coefficients. In the former case, just a single dimensionless material parameter appears, while in the latter, only three dimensionless parameters are required. Energy methods are used to establish lower bounds on the actual stress decay rate. Both analytic and numerical estimates are obtained in terms of the elastic constants of the material and results are shown for several contemporary engineering materials. When compared with the exact stress decay rate computed numerically from the eigenvalues of a fourth-order ordinary differential equation, the results in some cases show a high degree of accuracy. In particular, for strongly orthotropic materials, an asymptotic estimate provides extremely accurate estimates for the decay rate. Results of the type obtained here have several important practical applications. For example, they provide physical insight into the mechanical testing of anisotropic and laminated composite structures (including the off-axis tension test), are useful in assessing the influence of fasteners, joints, etc. on the behavior of composite structures and allow for tailoring a material with specific properties to ensure that local stresses attenuate at a desired rate.     

Dynamic Saint-Venant region in a semi-infinite elastic strip

This paper presents a formulation for the solution of the steady state rosponse of a semi-infinite strip with atress-free semi-infinite edges and a time-harmonie shear and normal stress applied to the end. If the end stresses form a self-equilibrated stress state, the presence or absence of a dynainic Saint-Venant region may be examined. The mathematical analysis is based on the linear equations for generalized plane stress and are solved by a biorthogonal eigenfunction expansion. The formulation is in terms of stresses and a displacement related auxiliary variable of the same differential order as the stress. Numerical solutions are presented as an indication of frequency and stress mode shape dependency.     

弹性圆板下横观各向同性弹性地基的轴对称问题

将与板相关的各力学量展开为Fourier-Bessel级数,利用解析法对弹性圆板下横观各向同性弹性地基的轴对称问题进行了分析。这些级数中的待定系数由板的边界条件、板的控制方程、地基-板的相容条件加以确定。数值计算结果表明:横观各向同性地基的沉降比各向同性地基的沉降要小,而地基反力则变化不大,说明按各向同性地基进行分析偏于保守。该分析方法将原问题简化为求解代数方程组的问题,且该方法可以推广到板的一般弯曲及层状地基与板的相互作用问题。     

Symmetric low-frequency motion in incompressible transversely isotropic elastic plates

The problem of long-wave low-frequency extensional (symmetric) motion in a layer composed of incompressible, transversely isotropic elastic material is investigated. Motivated by appropriate approximations of the dispersion relation, a hierarchy of asymptotically approximate boundary value problems is set up and solved. A leading order system of equations is obtained for the governing extensions, together with a refined system for their second order counterparts. A one-dimensional model problem, involving impact edge loading, is set up and solved in order to illustrate the derived theory.     

Local tangential contact of elastically similar,transversely isotropic elastic bodies

Meccanica - A three-dimensional contact between two linear elastically similar transversely isotropic solids of arbitrary geometry is considered. A simplified version of the classic...     

Crack analysis in semi-infinite transversely isotropic piezoelectric solid. II. Penny-shaped crack near the surface

Making use of the Somigliana identity, the boundary integral equations are obtained for a planar crack of arbitrary shape in an elastic half space. The material is piezoelectric with transversal isotropy. The solution is given for a penny-shaped crack parallel to the free boundary while the loading is axially symmetric.     

Crack analysis in semi-infinite transversely isotropic piezoelectric solid. I. Green’s functions

Three-dimensional fundamental solutions corresponding to a unit point force and point electric charge are obtained for a semi-infinite transversely isotropic piezoelectric solid. The free boundary is parallel to the plane of isotropy. They can be used as the Green’s function for solving the problem of a flat circular crack near the free surface which will be dealt with in Part II of this work.     

Finite elastic deformations of transversely isotropic circular cylindrical tubes

We consider the finite radially symmetric deformation of a circular cylindrical tube of a homogeneous transversely isotropic elastic material subject to axial stretch, radial deformation and torsion, supported by axial load, internal pressure and end moment. Two different directions of transverse isotropy are considered: the radial direction and an arbitrary direction in planes normal locally to the radial direction, the only directions for which the considered deformation is admissible in general. In the absence of body forces, formulas are obtained for the internal pressure, and the resultant axial load and torsional moment on the ends of the tube in respect of a general strain-energy function. For a specific material model of transversely isotropic elasticity, and material and geometrical parameters, numerical results are used to illustrate the dependence of the pressure, (reduced) axial load and moment on the radial stretch and a measure of the torsional deformation for a fixed value of the axial stretch.     

A plane theory of inextensible transversely isotropic elastic composites

A plane strain or plane stress configuration of an inextensible transversely isotropic linear elastic material, with the axis of symmetry in the plane, leads to a harmonic lateral displacement field in stretched coordinates. Various displacement and traction conditions lead to standard and nonstandard boundary value problems of potential theory. Examples for a rectangular plane, half-plane and infinite plate with elliptic hole, are presented in illustration.     

Absorbing boundary conditions for elastic waves in transversely isotropic media

In the numerical simulation of elastic wave propagation in the solid, it is essential to introduce absorbing boundary conditions to limit the large or unbounded domain of computation. In this paper, the absorbing boundaries for transversely isotropic media are composed of simple first-order partial differential operators, and each of the operators can perfectly absorb a plane wave outgoing at a certain angle. To test the absorbing ability, the reflection coefficient formulas for the quasi-P and quasi-S wave on the absorbing boundary are derived based on the potential functions theory of the elastic wave. Numerical examples show that the absorbing effect is good. The boundary conditions given here have a practical meaning.Supported by National Natural Science Foundation of China.     

Torsional oscillation of rigid disk in infinite transversely isotropic elastic cylinder

In the present analysis torsional oscillation of a rigid disk in an infinite transversely isotropic elastic cylinder is considered. The effects of anisotropy in the stress intensity factor are shown graphically.