Effective elastic moduli of periodic granular composite with transversely isotropic phases

We find a rigorous solution describing the macroscopically uniform stress state of a periodic granular composite with transversely isotropic phases. The structure of the composite is modeled by a cube containing a finite number of arbitrarily arranged and oriented, transversely isotropic spherical inclusions. This provides the model with a flexible means of describing the microstructure. Applying periodic vector solutions and local expansion formulas reduces the initial boundary-value problem to a system of linear algebraic equations. By averaging the solution over the unit cell, we derived exact finite expressions for the components of the effective stiffness tensor. The numerical data presented help to evaluate the efficiency of the method and the limits of applicability of available approximate theories.Translated from Prikladnaya Mekhanika, Vol. 40, No. 9, pp. 123–130, September 2004.     

General irreducible representations for constitutive equations of elastic crystals and transversely isotropic elastic solids

By means of the combined invariance restrictions due to material frame-indifference and material symmetry, the present paper provides general reduced forms for non-polynomial elastic constitutive equations of all 32 classes of crystals and transversely isotropic solids.Project supported by National Natural Science Foundation and National Postdoctoral Science Foundation of China.     

POINT FORCE SOLUTION FOR A TRANSVERSELY ISOTROPIC ELASTIC LAYER

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Elastic moduli of transversely isotropic graphite fibers and their composites

This paper demonstrates that it is possible to calculate the complete set of elastic mechanical properties for graphite-epoxy fiber-reinforced materials at any fiber-volume fraction by modifying equations previously developed to include transversely isotropic graphite-fiber properties. Experimental verification of the modified equations is demonstrated by using these equations to curve fit elastic-property data obtained ultrasonically over a range of fiber-volume fractions. Material systems under consideration are T300/5208, AS-3501 and Modomor II/LY558 graphite epoxy. Using the modified equations it is possible to extrapolate for fiber properties. From Modomor II/LY558 ultrasonic data, it is shown that five out of seven extrapolated graphite-fiber properties are consistent with the assumption that graphite fibers are transversely isotropic. Elastic properties for T300/5208 and AS-3501 are ultrasonically evaluated by propagating stress waves through six individual specimens but at various angles from a block of unidirectional material. Particular attention is devoted to specimen dimensions. To demonstrate the need for accurately calculating or experimentally measuring all lamina elastic properties, a brief discussion is included on the effect that variations in lamina elastic properties have on calculating interlaminar stresses.     

Contact problems for several transversely isotropic elastic layers on a smooth elastic half-space

The idea of generalized images, first used by the author for the case of crack problems, is applied here to solve a contact problem for n transversely isotropic elastic layers, with smooth interfaces, resting on a smooth elastic half-space, made of a different transversely isotropic material. A rigid punch of arbitrary shape is pressed against the top layer’s free surface. The governing integral equation is derived for the case of two layers; it is mathematically equivalent to that of an electrostatic problem of an infinite row of coaxial charged disks in the shape of the domain of contact. This result is then generalized for an arbitrary number of layers. As a comparison, the method of integral transforms is also used to solve the problem. The main difference of our integral transform approach with the existing ones is in separating of our half-space solution from the integral transform terms. It is shown that both methods lead to the same results, thus giving a new interpretation to the integral transform as a sum of an infinite series of generalized images.     

Concavities on the zonal slowness section of a transversely isotropic elastic material

The section of the slowness surface of a transversely isotropic elastic material in a zonal plane consists of an ellipse and a quartic curve with two nested branches, the inner of which is convex. Concavities can therefore occur only on the outer branch S and five possibilities arise: (I) S is convex; (II) S has two axial concavities (centred on the points of intersection of S with the axis of transverse isotropy); (III) S has two basal concavities (centred on the points of intersection of S with the basal plane); (IV) S has two axial and two basal concavities; (V) S has four oblique concavities, neither axial nor basal. The first and last of these are commonly realized in actual materials, the others only rarely. A unified treatment of stationary points and concavities on S is given in the course of which some previous results are simplified and their relationship clarified.     

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.     

Non-linear stress-strain equations for incompressible transversely isotropic elastic materials

Non-linear stress-strain equations for incompressible, transversely isotropic elastic materials are developed. In order to obtain these equations, the expressions for a strain energy function is found. The derivation of the strain energy function follows a geometrical approach and a method suggested by Mooney. These stress-strain relations are expressed in terms of three principal stretches to the sixth order.     

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.     

Torsional impact of transversely isotropic solid with functionally graded shear moduli and a penny-shaped crack

The torsional impact response of a penny-shaped crack in an unbounded transversely isotropic solid is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace transform and Hankel transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress fields are obtained. Investigated are the influence of material nonhomogeneity and orthotropy on the dynamic stress intensity factor. The peak value of the dynamic stress intensity factor can be suppressed by increasing the shear moduli's gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface.     

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.

     

Evaluation of the second order elastic moduli

The expressions of the apparent linear elastic moduli and their first and second derivatives, with respect to hydrostatic pressure, are obtained according to the second order elasticity theory. As a particular case when the material is hyperelastic, formulae of the first derivatives of the linear elastic moduli reduce to those obtained by Seeger and Buck.