Transversely isotropic

In this paper, 2D simulations of the ultrasonic testing of different types of (transversely isotropic) austenitic V-butt welds are presented they have been obtained with an EFIT-code developed by Marklein [5]. 

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... 

If at every point of a material there is one plane in which the mechanical properties are equal in all directions, then the material is called transversely isotropic. If, for example, the 1-2 plane is the plane of isotropy, then the 1 and 2 subscripts on the stiffnesses are interchangeable. The stress-strain relations have only five independent constants ... 

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. 

A variation on the exact soiutions is the so-caiied seif-consistent modei that is explained in simpiest engineering terms by Whitney and Riiey [3-12]. Their modei has a singie hollow fiber embedded in a concentric cylinder of matrix material as in Figure 3-26. That is, only one inclusion is considered. The volume fraction of the inclusion in the composite cylinder is the same as that of the entire body of fibers in the composite material. Such an assumption is not entirely valid because the matrix material might tend to coat the fibers imperfectiy and hence ieave voids. Note that there is no association of this model with any particular array of fibers. Also recognize the similarity between this model and the concentric-cylinder model of Hashin and Rosen [3-8]. Other more complex self-consistent models include those by Hill [3-13] and Hermans [3-14] which are discussed by Chamis and Sendeckyj [3-5]. Whitney extended his model to transversely isotropic fibers [3-15] and to twisted fibers [3-16]. 

It is important to note that comparable information to that obtained from infra-red spectroscopy can in principle be obtained from refractive index measurements. It has been shown that for a transversely isotropic film, the relationship equivalent to 11(c) is... 

For a transversely isotropic aggregate these Equations reduce to... 

For this situation of a transversely isotropic aggregate of transversely isotropic units, the Legendre addition theorem gives... 

It is also comparatively straightforward to-calculate P200, P220, P420 and P o for a biaxially oriented aggregate of transversely isotropic units in terms of the principal extension ratios Xx, X2 and (with X,X2 3 = 1). 

Fig. 10. Comparison of the measured refractive indices for PET film with values calculated from the orientation functions determined from n.m.r. assuming transversely isotropic structural units. I, Experimental points predicted values. Reproduced from Polymer by permission of the publishers, Butterworth Co (Publishers) Ltd. (C)...

Total Sn0 for transversely isotropic structural units in trans configuration + 0.00424 —0.00227 + 0.00276... 

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. 

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 ... 

A bed of aligned fibers is approximately transversely isotropic and flow parallel to the fibers is greatly favored over perpendicular flow. One could thus argue that Equation 11.7 should be generalized to incorporate a permeability tensor. Due to its lack of scientific... 

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

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. 

For aligned transversely isotropic elements the self consistent method gives (Walpole58 ) the relations... 

The statistical independence of the eigenvalues at a 0.05 confidence level (a) is also tested. In this way, it is possible to distinguish among orthotropic structure (in which all the three eigenvalues differ each other), transverse isotropic structure (with two eigenvalues statistically equivalent), and isotropic structure (where all the three eigenvalues are equivalent). 

Abstract A poromechanics formulation for transversely isotropic chemically active poroelastic media under non-isothermal conditions is presented. The formation pore fluid is modeled as a two-species constituent comprising of the solute and the solvent. The model is applied to study the thermo-chemical effects on the stress and pore pressure distributions in the vicinity of an inclined borehole drilled in a chemically active transversely isotropic formation under non-isothermal conditions. 

The borehole is assumed to be infinitely long and inclined with respect to the in-situ three-dimensional state of stress. The axis of the borehole is assumed to be perpendicular to the plane of isotropy of the transversely isotropic formation. Details of the problem geometry, boundary conditions and solutions for the stresses, pore pressure and temperature are available in [7], The solution is applied to assess the thermo-chemical effects on stresses and pore pressures. Both the formation pore fluid and the wellbore fluid are assumed to comprise of two chemical species, i.e., a solute fraction and solvent fraction. The formation material properties are those of a Gulf of Mexico shale [7] given as E = 1853.0 MPa u = 0.22 B = 0.92 k = 10-4 md /r = 10-9 MPa.s Ch = 8.64 x 10-5 m2/day % = 0.9 = 0.14 cn = 0.13824 m2/day asm = 6.0 x 10-6 1°C otsf = 3.0 x 10-4 /°C. A simplified example is considered wherein the in-situ stress gradients are assumed to be trivial and pore pressure gradients of the formation fluid and wellbore fluid are assumed to be = 9.8 kPa/m. The difference between the formation temperature and the wellbore fluid temperature is assumed to be 50°C. The solute concentration in the pore fluid is assumed to be more than that in the wellbore fluid such that mw — mf> = —1-8 x 10-2. 

Abousleiman, Y. and Cui, L. (1998) Poroelastic solutions in transversely isotropic media for wellbore and cylinder. Int. J. Solids Structures 35, 4905-4929... 

Abousleiman, Y. and Ekbote, S. (2005) Porothermoelastic solution for an inclined borehole in transversely isotropic media. J. Appl. Mech. 72, 102-114... 

General Form of the Raman Tensor for Transversely Isotropic Systems... 

This expression is identical in form to equation (5.20). In the case of Raman scattering, however, it is necessary to compute the average Raman tensor, (. For a transversely isotropic system, the segment is free to spin about the r. axis, and the vector ni is averaged over the unit circle normal to r . In addition to (n() = 0 and equation (5.22), we require the result,... 

Figure 1 Effect of dimensionality on the shape of the Fermi surface. The surfaces shown are for (a) tb, tc = 0 (one-dimensional), (b) tb ta, tc = 0 (quasi-one-dimensional), (c) transverse isotropic quasi-one-dimensional, (d) isotropic two-dimensional, (e) tc ta, tb (quasi-two-dimensional), and (f) isotropic three-dimensional. (Courtesy of J. Lefebvre.)...

Fig. 11.1 Examples of transversely isotropic whisker-reinforced ceramic composites.

Insertion of Eqn. (19) into Eqn. (4) along with Eqn. (20) through Eqn. (23) yields a noninteractive reliability model for a three-dimensional state of stress in a transversely isotropic whisker-reinforced ceramic composite. The noninteractive representation of reliability for orthotropic ceramic composites would follow a similar development. The analytical details for this can be found in Duffy and Manderscheid.20... 

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