Elasticity symmetry

Orthotropic construction Having three mutually perpendicular planes of elastic symmetry. 

Fig. 11. Differences of elastic symmetry constants of La 57Ce and La Ce ,B4 from those of LaBj as functions of temperature. The solid line is a theoretical fit (Winter et al. 1986).

Orthotropic Having three mutually perpendicular planes of elastic symmetry at each point. Out-life Period of time prepreg material remains with unchanged properties, in handleable form outside of the specified storage conditions. 

Orthotropic n. Having three mutually perpendicular planes of elastic symmetry, as in composites having fibers miming in two perpendicular directions, or biaxially oriented sheet. If the fibers or orientation are unidirectional, the material is still orthotropic but also isotropic in the two directions perpendicular to the fibers or oriented polymer chains. Complete textile glossary. Celanese Corporation, New York. Brandrup J, Immergut EH (eds) (1989) Polymer handbook, 3rd edn. Wiley-Interscience, New York. 

Other applications of the Takayanagi model to oriented polymers have included linear polyethylene that was cross-linked and then crystallized by slow cooling from the melt under a high tensile strain [22], and sheets of nylon with orthorhombic elastic symmetry [23]. A fuller discussion is given in the more advanced text by Ward [24]. 

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... 

Note that most textbooks claim that in the case of trigonal and tetragonal monocrystals it is necessary to distinguish two cases of elastic symmetry, depending on the crystal class... 

As for crystals, tire elasticity of smectic and columnar phases is analysed in tenns of displacements of tire lattice witli respect to the undistorted state, described by tire field u(r). This represents tire distortion of tire layers in a smectic phase and, tluis, u(r) is a one-dimensional vector (conventionally defined along z), whereas tire columnar phase is two dimensional, so tliat u(r) is also. The symmetry of a smectic A phase leads to an elastic free energy density of tire fonn [86]... 

Remark. The specific choice of bijki as the inverse of the Uijki for the elliptic regularization appears to be natural, since in the case of pure elastic (with K = [I/ (R)] , respectively p a) = 0), the boundary condition (5.16) reduces to (5.9). However, the proof of Theorem 5.1 works with any other choice of bijki as long as requirements of symmetry, boundedness and coercivity are met. 

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... 

Here g [ ] may be called the elastic compliance tensor, andl [-]maybe called the inelastic compliance tensor. Note that g is a fourth-order tensor which shares the symmetries of t. Again, (5.16) may be written as... 

In solids of cubic symmetry or in isotropic, homogeneous polycrystalline solids, the lateral component of stress is related to the longitudinal component of stress through appropriate elastic constants. A representation of these uniaxial strain, hydrostatic (isotropic) and shear stress states is depicted in Fig. 2.4. Such relationships are thought to apply to many solids, but exceptions are certainly possible as in the case of vitreous silica [88C02]. 

The consequences of this approximation are well known. While E s is good enough for calculating bulk moduli it will fail for deformations of the crystal that do not preserve symmetry. So it cannot be used to calculate, for example, shear elastic constants or phonons. The reason is simple. changes little if you rotate one atomic sphere... 

It is further assumed that the mesophase layer consists of a material having progressively variable mechanical properties. In order to match the respective properties of the two main phases bounding the mesophase, a variable elastic modulus for the mesophase may be defined, which, for reasons of symmetry, depends only on the radial distance from the fiber-mesophase surface. In other words, it is assumed that the mesophase layer consists of a series of elementary peels, whose constant mechanical properties differ to each other by a quantity (small enough) defined by the law of variation of Ej(r). 

Since the wavelength is of the order of lattice distances, electrons that are scattered elastically undergo constructive and destructive interference (as with X-rays in XRD). The back-scattered electrons form a pattern of spots on a fluorescent screen from which the symmetry and structure of the surface may be deduced. 

Considerably better agreement with the observed stress-strain relationships has been obtained through the use of empirical equations first proposed by Mooney and subsequently generalized by Rivlin. The latter showed, solely on the basis of required symmetry conditions and independently of any hypothesis as to the nature of the elastic body, that the stored energy associated with a deformation described by ax ay, az at constant volume (i.e., with axayaz l) must be a function of two quantities (q +q +q ) and (l/a +l/ay+l/ag). The simplest acceptable function of these two quantities can be written... 

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