Symmetry orthotropic

If a circular cylinder of orthotropic symmetry is used with axis along 3, the apparent shear modulus to insert in Eq. (2.1) above is G = 2/(844 + S55) so that for this geometry a unique value for S44 or S55 is unobtainable. 

Theoretioaiiy a maximum of 21 tensor eiements of eiastio stiffness for the triclinic crystal (the lowest-symmetry crystal) can be determined with one specimen. However, it is diffioult to assimilate properties relating to stress waves and elasticity for such a low-symmetry orystal. In praotioe, RUS oan determine nine tensor elements for orthotropic symmetry as well as for higher symmetry (isotropic, cubic, hexagonal or tetragonal). 

The information provided by RUS and reflected ieaky Lamb waves (LLWs) for the determination of uitrasonic wave veiocities and phase veiocity dispersion curves of an inconei 600 piate — a materiai empioyed to produce steam generator tubes for nuciear power piants —were compared [24] and the iongitudinai and the transverse wave veiocities of the specimen determined by RUS, puise-echo and cut-off frequencies of LLWs. The wave veiocities obtained by RUS under the assumption of isotropic symmetry differed from those provided by other methods. However, the wave veiocities in the direction of thickness obtained by RUS under the assumption of orthotropic symmetry were quite simiiar to those obtained with other methods as they measured wave veiocities in the direction of thickness. 

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. 

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... 

The Voigt and Reuss moduli for both transverse isotropic and orthotropic symmetry are given below ... 

The polymer sample is assumed to have at least orthotropic symmetry, i.e. it contains three mutually perpendicular directions such that if it is rotated through 180° about any one of these directions its macroscopic properties are unchanged. Axes OZ1Z2A3 are chosen parallel to these three symmetry directions of the sample. The orientation of a particular structural unit can then be specified in terms of three Euler angles, 6, and jr, as shown in fig. 10.7. 

Thus far four composites listed in Table I have been studied. NbTi/Cu is discussed briefly here. From its microstructure and manufacture, a rectangular cross-section bar, it was assumed that this composite has orthorhombic (orthotropic) symmetry in its physical properties. Materials with this symmetry have nine independent elastic constants. While deviations from elastic behavior were small, nine independent elastic constants were verified. Four specimens were prepared (Fig. 16) and 18 ultrasonic wave velocities were determined by propagating differently polarized waves in six directions, (100) and (110). An example cooling run is shown in Fig. 17 for E33, Young s modulus along the filament axis. These data typify the composites studies a wavy, irregular modulus/temperature curve. 

The quantities Fijp, defined by eqn. (5), which contain information about the distribution of orientations, are in fact the components of a fourth rank tensor [f yp,]. It is clear from the definition of f that interchange of i and j or p and q does not change its value, so that there are at most thirty-six independent components. It follows, however, from the assumed orthotropic symmetry of the specimen that only twelve of these are nonzero, nine of the form Fujj and three of the form Fijij. All the information available about the distribution of orientations from measurements of fluorescence intensities is thus contained in these twelve quantities. 

The second type of explanation for finding values of R less than 3 involves the assumption that the emission and absorption axes of the fluorescent molecule are not coincident. Kimura et al. have considered a model in which the absorption and emission axes each have, independently, a cylindrically symmetric distribution of orientations around a third unique axis in the molecule, and Nobbs et al. have considered a model which includes both this and the possibility that there is a fixed angle between the emission and absorption axes which are otherwise uniformly distributed around a third unique axis. In the more general model at least three parameters are required to specify the relationships between the directions of the emission and absorption axes and that of the unique axis of a fluorescent molecule and these are not generally known. For orthotropic symmetry, five v, are required to characterise the distribution of orientations of the unique axes and if the constant NqIo is included, there is a total of at least nine unknown quantities. No attempt has so far been made to evaluate these from intensity measurements on an orthotropic sample. For a uniaxial sample only two parameters, cos O and cos O, are required for characterising the distribution of orientations and by making various approximations the total number of unknown quantities can be reduced to six. Their evaluation then becomes a practical possibility. 

The scattered intensity thus depends, for chosen values of the / and If, on quantities of the form /o ay-ap,. These may be considered to be the components of a fourth-rank tensor for the sample as a whole, and it follows from the assumption that ay = a, - and the general symmetry restrictions on any fourth-rank tensor describing a material with orthotropic symmetry that only the nine sums of the form ociiXjj or are distinct and different from zero. For a material with uniaxial symmetry it can be shown that... 

Orthorhombic crystals have three mutually perpendicular principal symmetry axes. Since a 180° rotation about each principal axis results in no change, there can be no linear relations between shear stresses and normal strains or between shear stresses and shear strains with different subscripts. This can be proved immediately by observing that, if this were not so, the stated symmetry would not be present. This establishes that in such materials only nine independent elastic compliances (or constants) remain, namely sn, S22, 33, 12, 13, 23, 44, 55, and See- Many technologically important materials, such as rolled metal plates, unidir-ectionally produced polymer films and paper, composite sheet materials, and even wood, have such symmetry, which is referred to as orthotropic symmetry, when it relates to materials rather than crystals. 

The deformation, starting from an initial compression-molded rectangular plate, equilibrated by annealing at 170 °C in vacuum, was performed in an environment of relative humidity 60% at 20 °C to large plastic strains in a channel die similar to the one described for the work on HDPE in Section 9.3.3. The end result of the plane-strain compression history of the Nylon at a CR of 4.0 (se = 1.39), with a similar complement of intermediate-structure probes of TEM, WAXS, and SAXS, was a texture of similar perfection to that of HDPE, with orthotropic symmetry, but incorporating a dual symmetrical set of intermixed monoclinic components of indeterminable scales and form of special aggregation, as depicted in Fig. 9.14. As with the HDPE, the principal direction of molecular alignment... 

Fig. 9.14 A sketch of the quasi-single-crystalline texture of highly oriented Nylon-6 in plane-strain compression depicting the morphological texture. The end result is a material of orthotropic symmetry with a dual set of symmetrical monoclinic contributions (from Lin and Argon (1992) courtesy of the ACS).

The anisotropic viscoelastic properties in shear of the meniscus have been determined by subjecting discs of meniscal tissue to sinusoidal torsional loading [35](Table B2.8). The specimens were cut in the three directions of orthotropic symmetry, i.e. circumferential, axial and radial. A definite correlation is seen with the orientation of the fibers and both the magnitude of the dynamic modulus IG I and the phase angle 8. 

Similar computational methods may also be employed to check the veracity of experimental work. This is useful when large scatter in the data makes them doubtful. For example, orthotropic symmetry considerations ideally require that... 

For materials with orthorhombic symmetry (or orthotropic symmetry), there are three mutually orthogonal axes, each of two-fold rotational symmetry. Translation of the material in the direction of any of these axes leaves the material behavior unaltered, but the translation distances required to recover the same lattice in crystalline materials are distinct, in general. Nine independent constants (cn, C22, C33, C44, C55, cge, C z, C23, C12) are required to specify the elastic response of a general orthorhombic material. For a material in which one of the three symmetry axes has fourfold rotational symmetry (or tw o of the translational invariance distances are equal), the number of independent constants is reduced to six this is achieved by requiring that C22 = cn, C55 = C44 and C23 = C12, for example. 

The response of the material is linearly elastic with orthotropic symmetry, so that... 

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