Tensor Properties of Anisotropic Materials

As you might guess, when describing the properties of anisotropic materials you now get into tensors and all sorts of horrible looking equations. Don t worry, we re not going there 1 We ll just finish this little review by considering a subtle feature of shear forces. 

With elastically anisotropic materials the elastic behavior varies with the crystallographic axes. The elastic properties of these materials are completely characterized only by the specification of several elastic constants. For example, it can be seen from Table 10.3 that for a cubic monocrystal, the highest symmetry class, there are three independent elastic-stiffness constants, namely, Cn, C12, and C44. By contrast, polycrystalline aggregates, with random or perfectly disordered crystallite orientation and amorphous solids, are elastically isotropic, as a whole, and only two independent elastic-stiffness coefficients, C44 and C12, need be specified to fully describe their elastic response. In other words, the fourth-order elastic modulus tensor for an isotropic body has only two independent constants. These are often referred to as the Lame constants, /r and A, named after French mathematician Gabriel Lame (1795-1870) ... 

Nye, J.F. (1985) Physical Properties of Crystals, Clarendon Press, Oxford. This is the standard reference for tensor representation. Chapter XI covers transport properties including electrical conductivity. The representation of o by tensors is not necessary to understand the electrical behavior of materials. Its significance becomes clear when we want to specify certain properties of anisotropic single crystals. 

The Cauchy stress tensor cr and Green Lagrange strain tensor Cgl are of second order and may be connected for a general anisotropic linear elastic material via a fourth-order tensor. The originally 81 constants of such an elasticity tensor reduce to 36 due to the symmetry of the stress and strain tensor, and may be represented by a square matrix of dimension six. Because of the potential property of elastic materials, such a matrix is symmetric and thus the number of independent components is further reduced to 21. For small displacements, the mechanical constitutive relation with the stiffness matrix C or with the compliance matrix S reads... 

The essential feature of Ohm s law is that J is directly proportional to the applied field E and (J, being a property of the material, is independent of the field. Note that J and E are vector quantities while cr is a scalar (tensor of rank zero) for an isotropic media however, it will be a tensor of rank 2 for an anisotropic material such as a single crystal. 

Many materials properties are anisotropic they vary with direction in the material. When anisotropic materials properties are characterized, the values used to represent the properties must be specified with respect to particular coordinate axes. If the material remains fixed and the properties are specified with respect to some new set of coordinate axes, the properties themselves must remain invariant. The way in which the properties are described will change, but the properties themselves (i.e., the material behavior) will not. The components of tensor quantities transform in specified ways with changes in coordinate axes such transformation laws distinguish tensors from matrices [6]. 

A delta function, 5(f),is a distribution that equals zero everywhere except where its argument is zero, where it has an infinite singularity. It has the property f f r)5(r- ro)dr = /(ff0) so it also follows that /S(r — ro)dr = 1. The singularity of 5(f — fo) is located at F o-BThis technique can be used to measure the diffusivity in anisotropic materials, as described in Section 4.5. Measurements of the concentration profile in the principal directions can be used to determine the entire diffusion tensor. 

Piezoelectric materials are materials that exhibit a linear relationship between electric and mechanical variables. Electric polarization is proportional to mechanical stress. The direct piezoelectric effect can be described as the ability of materials to convert mechanical stress into an electric field, and the reverse, to convert an electric field into a mechanical stress. The use of the piezoelectric effect in sensors is based on the latter property. For materials to exhibit the piezoelectric effect, the materials must be anisotropic and electrically poled ie, there must be a spontaneous electric field maintained in a particular direction throughout the material. A key feature of a piezoelectric material involves this spontaneous electric field and its disappearance above the Curie point. Only solids without a center of symmetry show this piezoelectric effect, a third-rank tensor property (14,15). 

Most engineering materials are isotropic in nature, and thus they have the same properties in all directions. For such materials we do not need to be concerned about the variation of properties with direction. But in anisotropic materials such as (he fibrous or composite materials, (he properties may change with direction. For example, some of the properties of wood along the grain are different than those in (he direction normal to the grain. In such cases the thermal conductivity may need to be expressed as a tensor quantity to account for the variation with direction. I he treatment of such advanced topics is beyond the scope of tlus text, and we will assume the thermal conductivity of a material to be independent of direction. 

Since second-order nonlinear optical materials are anisotropic, their optical properties are described by tensors as discussed previously in Sect. 2.1.2. For a nonlinear optical process, the -th order nonlinear polarization is due to n interacting electric held vectors and is described by an (n -I-1) rank tensor composed of 3"+ tensor elements. In nonlinear optics, several fields with different frequencies l can be present simultaneously so that the electric field and the polarization are represented by... 

Liquid crystals are anisotropic materials, hence their linear optical properties are determined by a symmetrical dielectric tensor, e, rather than a scalar refractive index. In certain cases it is possible to prepare uniformly oriented liquid crystal films. In these films e is a constant and light propagation can be described by the well-known laws of crystal optics. ... 

Theory and Physics of Piezoelectricity. The discussion that follows constitutes a very brief introduction to the theoretical formulation of the physical properties of crystals. If a solid is piezoelectric (and therefore also anisotropic), acoustic displacement and strain will result in electrical polarization of the solid material along certain of its dimensions. The nature and extent of the changes are related to the relationships between the electric field (E) and electric polarization (P). which are treated as vectors, and such elastic factors as stress Tand strain (S), which are treated as tensors. In piezoelectric crystals an applied stress produces an electric polarization. Assuming Ihe dependence is linear, the direct piezoelectric effect can be described by the equation ... 

For such materials, which may be smectic or nematic liquid crystals, the dielectric properties of the LC phase are anisotropic. Fora uniaxial LC phase, the dielectric tensor is diagonal such that... 

Liquid crystals as anisotropic fluids exhibit a wide range of complex physical phenomena that can only be understood if the appropriate macroscopic tensor properties are fully characterized. This involves a determination of the number of independent components of the property tensor, and their measurement. Thus a knowledge of refractive indices, electric permittivity, electrical conductivity, magnetic susceptibilities, elastic and viscosity tensors are necessary to describe the switching of liquid crystal films by electric and magnetic fields. Development of new and improved materials relies on the design of liquid crystals having particular macroscopic tensor properties, and the optimum performance of liquid crystal devices is often only possible for materials with carefully specified optical and electrical properties. 

In this book, vector quantities such as x and y above are normally column vectors. When necessary, row vectors are indicated by use of the transpose (e.g., r). If the components of x and y refer to coordinate axes [e.g., orthogonal coordinate axes ( i, 2, 3) aligned with a particular choice of right, forward, and up in a laboratory], the square matrix M is a rank-two tensor.9 In this book we denote tensors of rank two and higher using boldface symbols (i.e., M). If x is an applied force and y is the material response to the force (such as a flux), M is a rank-two material-property tensor. For example, the full anisotropic form of Ohm s law gives a charge flux Jq in terms of an applied electric field E as... 

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