Isotropy, transverse

Transverse isotropy Transverse isotropy Transverse isotropy Transverse isotropy... 

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 ... 

For an oriented polymer, the magnitude of the observed second moment static magnetic field H0, which can be conveniently defined by the polar and azimuthal angles A, second moment will depend only on the angle A, there being no preferred orientation in the plane normal to the 3 direction. The treatment follows that originally presented by McBrierty and Ward 9>. 

Each unit of structure in the oriented polymer will also be considered to possess transverse isotropy. Its orientation can therefore be defined by polar and azimuthal angles (0, tp), but the condition of transverse isotropy for the whole sample means that the observed second moment will depend only on functions of 0 (in fact, P200 and P400) the functions involving (p taking fixed average values. 

Finally, a given internuclear vector rjk takes a direction with respect to the 3 axis of the unit of structure defined by polar and azimuthal angles ( , r[). Because each unit possesses transverse isotropy, the second moment will involve functions of h, only, the functions of r being replaced by their average values. 

We may then write the arrays for the elastic constants for various symmetries, the two most useful being hexagonal (also transverse isotropy as in fibre symmetry) and isotropic. Hexagonal gives ... 

Because of the asumed transverse isotropy it follows that = 1/2 (Cu - Cn). The terms Qj are the elastic stiffnesses expressed in the contracted (Voigt) notation. 

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. 

Note that /3 and /4 are stress components in the plane of isotropy and, therefore, have the same Weibull parameters. The parameters i and /3i would be obtained from uniaxial tensile experiments along the material orientation direction, dt. The parameters a2 and /Efe would be obtained from torsional experiments of thin-walled tubular specimens where the shear stress is applied across the material orientation direction. The final two parameters, a3 and /33, would be obtained from uniaxial tensile experiments transverse to the material orientation direction. 

Biaxial orientation leads to isotropic properties in blown film, that is, properties that are eqnal in the two primary directions the film was stretched (i.e., parallel to the flat bit ). Orientation in the machine direction of the film is controlled primarily by take-up ratio, defined above. To control orientation in the transverse direction, we measure something called the blow-up ratio, while the forming ratio provides an indication of the degree of isotropy. 

Determining mechanical characteristics of fibrous materials is far from simple, mainly because of their small diameter. In particular, in the case of anisotropic fibers such as carbon or aramid, we need to determine five elastic constants, assuming isotropy in the cross-sectional plane. Figure 9.3 shows three of the five elastic constants the longitudinal Young s modulus of fiber, E or E, the transverse Young s modulus E22 or Ej, and the principal shear modulus, or Not shown are the two Poisson ratios the longitudinal Poisson s ratio of... 

The liner is an elastic plastic isotropic material. Besides, the laminate behaviour is different from a layer to another and each layer behaves according to the fibre direction. The fibre is assumed to have a transverse isotropy and equivalent properties in the (2-3) plane which normal axis (1) refers to the fibre longitudinal direction, as shown in Figure 1. 

Assuming (1) axis is the longitudinal direction of the fibre, the compliance tensor 5c takes the same form than Si, whereas, taking into account the transversal isotropy, the compliance constants have the following expressions ... 

If one assumes fibre symmetry of the sample (like in uniaxial deformation), transverse isotropy for the molecular units (for instance ai U2 = 03), and additivity of polarizabilities, it is easily shown [9] that the difference in macroscopic polarizabilities along and perpendicular to the fibre axis of the sample is simply ... 

It is evident that the final equations for two transverse components should be the same due to the isotropy of the system. Therefore, one has finally the (u + 2)-component longitudinal set / k and the two component transverse set Pk of dynamic variables describing a multi-component fluid. 

According to von Karmaan and Howarth [179], a consequence of isotropy is that Rij can be expressed in terms of two scalar functions f t,x) and g t, x) identified as the longitudinal and transverse autocorrelation functions, respectively. There are two distinct longitudinal length scales, Af and A/, that can be defined from /, and there are also two corresponding transverse... 

SVI should be measured in both machine and transverse directions to determine the extent of isotropy in the tube. If the SVI values are very low (< 1 %) in both directions or equal, the sample is isotropic. This means that the sample has been adequately sintered and stresses have been relieved. If the values are both high, sintering has been poor. If the transverse value is high and the machine direction value is low, then orientation is primarily in the machine direction and it is unbalanced. 

For isotropic media, there is a transverse isotropy in D, such that the transverse dispersion for n perpendicular to uD is = n D n. 

Since the Qj are simply related to the technical elastic moduli, such as Young s modulus (T), shear modulus (G), bulk modulus (iC), and others, it is possible to describe the moduli along any given direction. The full equations for the most general anisotropy are too long to present here. However, they can be found in Yoon and Katz [1976a]. Presented below are the simplified equations for the case of transverse isotropy. Young s modulus is... 

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... 

Transverse isotropy The symmetry arrangement of structure in which there is a unique axis perpendicular to a plane in which the other two axes are equivalent. The long bone direction is chosen as the unique axis. In crystals this symmetry is called hexagonal. 

Therefore, moisture absorption has a larger effect on the transverse properties of a typical composite system. Despite this, the strength of a 0° composite is also affected by moisture ingress since the reloading of a broken fibre occurs through shear stress transfer Ifom the interphasal matrix. To achieve isotropy, unidirectional plies are stacked at a set of angles such as 0°, 45° and 90° to form a laminate. In this situation, moisture ingress will modify the residual stress state in the individual laminae. 

If this process is repeated, one finds only three values of Poisson s ratio are needed, not six. For fiber-reinforced materials, the number of elastic constants may be further reduced if other symmetries appear. For example, in some materials short fibers are randomly oriented in a plane and this gives transverse isotropy. That is, there is an elastically isotropic plane but the stiffness and compliance constants will be different normal to this plane (five elastic constants are needed). 

Stated in these very general terms, it is by no means obvious that the problem of adequately describing orientation in a polymer can ever be completely solved. Indeed, there are certainly situations where it can be shown to be insoluble in terms of the available information. However, in many if not most of the oriented polymer situations which have been studied in detail, simplifications are possible. The most sweeping simplification is to assume transverse isotropy, i.e. that there is no preferential... 

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