Transversally isotropic elasticity

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. 

In a coordinate system where the 3 axis is the axis of symmetry, the compliance matrix (equation (2.31)) looks like this  

In this case, we have five independent elastic parameters since there is a relation between the Vij due to the symmetry of the compliance matrix, similar to that for orthotropic materials V2i = vn and u i/E = vy-ijE-i. 

1 The crystal lattice itself, however, is only symmetric when rotated by multiples of 60.  

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Thus to find the five transverse isotropic elastic constants, at least five independent measurements are required, for example, a dilatational longitudinal wave in the 2 and 1(2) directions, a transverse wave in the 13(23) and 12 planes, etc. The technical moduH must then be calculated from the full set of C,p For improved statistics, redundant measurements should be made. Correspondingly, for orthotropic symmetry, enough independent measurements must be made to obtain aU 9 C again, redundancy in measurements is a suggested approach. 

Consider the case of uniform stress. This can be imagined as a system of N elemental cubes arranged end-to-end forming a series model (Figure 8.23). Assume that each elemental cube is a transversely isotropic elastic solid, the direction of elastic symmetry being defined by the angle 9, which its axis makes with the direction of applied external stress a. The strain in each cube ci is then given by the compliance formula... 

Mechanical Properties. The hexagonal symmetry of a graphite crystal causes the elastic properties to be transversely isotropic ia the layer plane only five independent constants are necessary to define the complete set. The self-consistent set of elastic constants given ia Table 2 has been measured ia air at room temperature for highly ordered pyrolytic graphite (20). With the exception of these values are expected to be representative of... 

The elasticity approaches depend to a great extent on the specific geometry of the composite material as well as on the characteristics of the fibers and the matrix. The fibers can be hollow or solid, but are usually circular in cross section, although rectangular-cross-section fibers are not uncommon. In addition, fibeie rejjsuallyjsotropic, but can have more complex material behavior, e.g., graphite fibers are transversely isotropic. 

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. 

In a homogeneous isotropic elastic medium it is possible to split acoustic waves in independent longitudinal and transverse waves, each travelling at a speed cL and cT, respectively. As 2 is greater than or equal to zero, cT is lower than or equal to cl/ J2. 

Upper and lower bounds on the elastic constants of transversely isotropic unidirectional composites involve only the elastic constants of the two phases and the fiber volume fraction, Vf. The following symbols and conventions are used in expressions for mechanical properties Superscript plus and minus signs denote upper and lower bounds, and subscripts / and m indicate fiber and matrix properties, as previously. Upper and lower bounds on the composite axial tensile modulus, Ea, are given by the following expressions ... 

These results of Walpole61 include as special cases those of Hill47 and of Hashin and Shtrikman48. For anisotropic phases Walpole58 gives bounds on the five elastic moduli of an aligned array of transversely isotropic elements and for randomly oriented fibrous inclusions in an isotropic matrix. For the former case (alignment) the bounds are expressed in terms of phase concentration q and the quantities k, 1, m, n, p defined as follows k - 1/2 (Cjj + C, m — 1/2 (Cn — C22), = C13, n = C33, p = C44 = C55. 

The LE theory is rather complex since it contains both viscous and elastic stresses. It can best be understood by considering viscous and elastic effects separately. If elastic effects are neglected, the LE equations reduce to Ericksens transversely isotropic fluidy while in the absence of flow the elastic stresses are just those of the Frank-Oseen theory (discussed below in Section 10.2.2). ... 

Material Constants, Elastic wave velocities have been obtained for oil shale by ultrasonic methods for various modes of propagation. Elastic constants can be inferred from these data if the oil shale is assumed to be a transversely isotropic solid (9). This is a reasonable approximation considering the bedded nature of the rock. Many of the properties of oil shale depend on the grade (kerogen content), which in turn is correlated with the density ( 10). The high pressure behavior of oil shale under shock loading has been studied in gas-gun impact experiments (11). 

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... 

Chung and Buessem, 1968] Katz and Meunier [1987] presented a description for obtaining two scalar quantities defining the compressive and shear anisotropy for bone with transverse isotropic symmetry. Later, they developed a similar pair of scalar quantities for bone exhibiting orthotropic symmetry [Katz and Meunier, 1990], For both cases, the percentage compressive (Ac ) and shear (As ) elastic anisotropy are given, respectively, by... 

Recently, Kinney etal. [2004] used the technique of resonant ultrasound spectroscopy (RUS) tomeasure the elastic constants (Qj) of human dentin from both wet and dry samples. As (%) and Ac (%) calculated from these data are included in both Table 47.5 and Figure 47.4. Their data showed that the samples exhibited transverse isotropic symmetry. However, the Qj for dry dentin implied even higher symmetry. Indeed, the result of using the average value for Q i and Cu = 36.6 GPa and the value for C44 = 14.7 GPa for dry dentin in the calculations suggests that dry human dentin is very nearly elastically isotropic. This isotropic-lifce behavior of the dry dentin may have clinical significance. There is independent experimental evidence to support this calculation of isotropy based on the ultrasonic data. Small angle x-ray diffraction... 

The equations which predict the elastic constants of a partially oriented polymer involve orientation functions to define the orientation of the aggregate units. For example, the average extensional compliance S33 for a transversely isotropic aggregate of transversely isotropic structural units is given by S 33 = Su sin 0+S33Cos" 0-l-(2Si3-l-544) sin 0cos 9. 

A multilayer laminate is composed of several laminas (or layers, or plies). Classical laminate theory (CLT) describes the linear elastic response of a thin laminated composite subjected to in-plane loads and bending moments see, e.g., Eckold (1994) and Herakovich (1998). Individual layers are assumed to be homogeneous, orthotropic, or transversely isotropic and in a state of plane stress. The constitutive relation for a thin multilayer laminated composite is given as ... 

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... 

For an isotropic, elastic material two elastic constants are sufficient to describe the material response, the elastic modulus E (eq. 7) and the Poisson s Ratio (v), defined as the ratio of the axial to the (negative) transverse strain (- 11/622). 

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. 

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... 

Approximate formulae for four E, E, v i, G12) of the five elastic properties of a transversely isotropic composite can be developed using simple approaches based on the strength of materials concepts. These concepts do not necessarily satisfy in full all the elasticity requirements. The RVE considered consists of a uniform arrangement of straight, continnons fibres. 

The strength of materials approach provides fonr of the five elastic properties of transversely isotropic nnidirectional composites. Two properties Ey, V12) are well predicted by this simple approach, i.e. nsing the law of mixtures. The other two ( , G 2) require more accurate micromechanics models. The main reason for this is that E i and V12 are independent of fibre packing while d E and G i2 depend strongly on fibre arrangement. 

Kriz and Stinchcomb [32] published experimental data for unidirectional graphite/epoxy composites. These results illustrate the case when the fibres are transversely isotropic. The elastic properties of the matrix are = 5.28 GPa and T = 0.354, and for the fibres E = 232 GPa, E = 15 GPa, 0(2 = 24 GPa, v 2 = 0.279 and v 3 = 0.49. In Figs 11.21-11.25 are plotted the predictions against the experimental data for , , G12, G23 and V23, i.e. the longitudinal or axial modulus, the transverse modulus, the in-plane shear modulus, the transverse shear modulus and the transverse Poisson s ratio, respectively. 

For the transverse shear modulus, the approach designated self-consistent was based on the formula obtained by the self-consistent method for the plane-strain bulk modulus (11.61), on the transverse modulus calculated using the Chamis approach (11.49b) and the in-plane Poisson s ratio given by the rule of mixtures. Except when used to predict the axial modulus and the major Poisson s ratio, the rule of mixtures underestimates the remaining composite elastic properties. The Bridging Model proved to be a very effective theory to account for all five elastic properties for unidirectional composites that are transversely isotropic. 

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. 

Kriz R. D. and Stinchcomb W. W. (1979) Elastic moduli of transversely isotropic graphite fibres and their composites - Equations used to calculate the complete set of elastic transversely isotropic properties for unidirectional fibre-reinforced materials having transversely isotropic fibres are experimentally verified by using improved ultrasonic techniques. Experimental Mechanics, 19(2), 41 9. 

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