Transversely isotropic material

The geometry and structure of a bone consist of a mineralised tissue populated with cells. This bone tissue has two distinct structural forms dense cortical and lattice-like cancellous bone, see Figure 7.2(a). Cortical bone is a nearly transversely isotropic material, made up of osteons, longitudinal cylinders of bone centred around blood vessels. Cancellous bone is an orthotropic material, with a porous architecture formed by individual struts or trabeculae. This high surface area structure represents only 20 per cent of the skeletal mass but has 50 per cent of the metabolic activity. The density of cancellous bone varies significantly, and its mechanical behaviour is influenced by density and architecture. The elastic modulus and strength of both tissue structures are functions of the apparent density. 

F depends on both the shape and on the symmetry of the specimen. For isotropic or transversely isotropic materials (e.g. hexagonal symmetry)... 

The anisotropy of cortical bone tissue has been described in two symmetry arrangements. Lang [1969], Katz and Ukraincik [1971], and Yoon and Katz [1976a,b] assumed bone to be transversely isotropic with the bone axis of symmetry (the 3 direction) as the unique axis of symmetry. Any small difference in elastic properties between the radial (1 direction) and transverse (2 direction) axes, due to the apparent gradient in porosity from the periosteal to the endosteal sides of bone, was deemed to be due essentially to the defect and did not alter the basic symmetry. For a transverse isotropic material, the stiffness matrix [Qj] is given by... 

An orthotropic material is called transversely isotropic when one of its principal planes is a plane of isotropy, i.e. at every point there is a plane on which the mechanical properties are the same in all directions [2]. Unidirectional carbon fibers packed in a hexagonal array with a relatively high volume fraction can be considered transversely isotropic, with the 2-3 plane normal to the fibers as the plane of isotropy (Figure 22.2). For a transversely isotropic material, it should be noted that the subscripts 2 and 3 (for a 2-3 plane of symmetry) in the material constants are interchangeable. Hence... 

Textile materials can often be characterized by preferential orientation and symmetry in fibre arrangement, so that they can be considered not as general anisotropic, but orthotropic or even transversely isotropic materials (Fig. 1.12). This simplifies the models development and experimental verification where there would be a smaller number of parameters to be measured. For example, the linear elastic behaviour of anisotropic material can be described in a matrix form as follows ... 

To illustrate this last point, let us note that for a bar formed from a transversely isotropic material, (i) the constitutive equations of finite elasticity imply that the difference between S and the ambient hydrostatic pressure is a function of the values of 3z /d2, br/3Z, 3z/dR, and d /3R at the point under consideration, and (ii) the condition that deformations be isochoric implies that... 

As a general rule, the thermal conductivity of a composite material is a complex function of the thermal conductivity of the matrix kj and that of the reinforcement (k). In the particular case of an orthotropic composite material, the thermal conductivity of each component (i.e., matrix, reinforcement) is a tensor quantity [k..] with only three components fcjp k and fcjj along major axes, that is, one in the axial direction (fcjj) and two in the transversal directions k and k ). In the particular case of transversely isotropic materials such as fiber reinforced composites, the axial thermal conductivity of the material, expressed in... 

The typical building block of a composite structure is the lamina, with a typical thickness of 0.125 mm. The lamina stress-strain relationships are described for orthotropic, transverse isotropic and isotropic materials. When a lamina is reinforced with unidirectional fibres it can be assumed to be a transversely isotropic material. In this chapter, theoretical determination of lamina elastic properties, assumed to be a transversely isotropic material, using micromechanics approaches is presented and illustrated with experimental data. 

When the cross-section of fibre-reinforced composite is a plane of isotropy, it is called a transversely isotropic material, as described in Fig. 11.8. 

The stiffness matrix corresponds to a transversely isotropic material with five independent constants. The constitutive equation can be written in matrix form as... 

The compliance matrix for a transversely isotropic material in terms of engineering constants is given as... 

The relationships between elastic constants which must be satisfied for an isotropic material impose restrictions on the possible range of values for the Poisson s ratio of -1 < v <. In a similar manner, there are restrictions in orthotropic and transversely isotropic materials. These constraints are based on considerations of the first law of thermodynamics [15]. Moreover, these constraints imply that both the stiffness and compliance matrices must be positive-definite, i.e. each major diagonal term of both matrices must be greater than 0. 

The engineering properties of interest are the elastic constants in the principal material coordinates. If we restrict ourselves to transversely isotropic materials, the elastic properties needed are Ei, Ei, v, and G23, i.e. the axial modulus, the transverse modulus, the major Poisson s ratio, the in-plane shear modulus and the transverse shear modulus, respectively. All the elastic properties can be obtained from these five elastic constants. Since experimental evaluation of these parameters is costly and time-consuming, it becomes important to have analytical models to compute these parameters based on the elastic constants of the individual constituents of the composite. The goal of micromechanics here is to find the elastic constants of the composite as functions of the elastic constants of its constituents, as... 

This chapter began by describing briehy the elasticity of anisotropic materials, providing the fundamental relationships and the allowed simplihcations by the existence of material planes of symmetry. The current unidirectional composites are usually classihed as transversely isotropic materials. In this case, only hve independent elastic constants are necessary to fully characterize unidirectional composites. The micromechanics provides the analytical and numerical approaches to predict the elastic constants based on the elastic properties of the composite constituents. Several classical closed formulas are revisited and compared with experimental data. Finally, stiffness and compliance transformations are given in the context of unidirectional composites. Experimental data are used to assess theoretical predictions and illustrate the off-axis in-plane elastic properties. 

In a transversally isotropic material, there is a plane in which all properties are isotropic. Perpendicular to this plane, the properties differ. One example for such a material is a hexagonal crystal which is transversally isotropic with respect to its mechanical properties.Other technically important materials may also be transversally isotropic, for example directionally solidified metals in which the grains have a preferential orientation (see also section 2.5), or composites (chapter 9) with fibres oriented in one direction, but aligned arbitrarily (or hexagonally) in the perpendicular plane. 

When a cube is subjected to normal strains or stresses in any of the three spatial directions, one of the electrostatic fields, contingent upon the electric boundary conditions, is induced parallel or anti-parallel to the polarization direction. Thus, without additional information, the directions of mechanical stimulus cannot be distinguished by such a sensor. For the different cases of electric boundary conditions, the electrostatic fields developing in a piezoelectric cube subjected to normal loads are shown in Figure 4.5 together with the corresponding deformations of the transversely isotropic material. 

An experimental investigation was undertaken to assess the failure behaviour of a series of PU foams of variable specific density. All the foam materials examined were transversely isotropic materials, whose axis of symmetry was, however, the weak axis of the medium. Simple uniaxial tension and compression tests were executed and the experimental data were introduced into the elliptic... 

In a more recent paper. Ho and Drzal [94] used a three-phase nonlinear finite-element analysis to investigate the stress transfer phenomenon in the singlefiber fragmentation test. The effect of fiber properties, interphase properties and thickness on the stress distribution in the vicinity of the fiber break was evaluated. Also, the stress fields for various fiber-matrix interface debonding conditions and the effect of frictional stress transfer were investigated. In this model, the fiber was assumed to be a linear elastic transversely isotropic material, the interphase as an elastic material and the matrix as a nonlinear material. It was found that... 

It is known that bone is a semi-brittle material and will experience fracture with forces greater than its ultimate strength. Based on the maximum amount of stress, the viscoelastic behavior of tibia showed a tendency to propagate the fracture lines and also higher stresses were reached compared to the isotropic and transversely isotropic property cases of tibia during the impact cycle. This indicates the dependency of a viscoelastic material to the time and also the strain rate, as expected. Also, the observed reduction in the maximum amount of stress after removing the impact load can be seen as a consequence of stress relaxation in tibia. The minimum stress was found to be when a transversely isotropic material property was considered for tibia. 

By expanding the summations in equation (18) and equating all the strain components except one to zero, the physical significance of the stiffness components Cp can be explored. By an equivalent procedure the physical significance of the compliance components Sp can be investigated. It is found that Cpq represents the ratio of stress component Op to a given strain component when all other strain components are zero. Similarly it is found that Sp represents the ratio of strain component Sp to a given stress component when all other stress components are zero. It follows that for transversely isotropic materials... 

For the special case of a transversely isotropic material discussed earlier, it follows from equations (23) applying the correspondence principle that... 

For the transverse isotropic material, Thomsen (1986) defined the following parameters ... 

If for a transversely isotropic material as the simplest type of anisotropy ... 

Katahara (1996) published velocity data for kaolinite, illite, and chlorite. Results show a distinct anisotropy for compressional and shear wave. The clay can be described as a transverse isotropic material. 

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