Constitutive Equations of Nonlinear Electromagnetic-Elastic Crystals

Although magnetic crystals have been the subject of research and conjecture for some time, it is only relatively recently that this field has become an active one for experimental and theoretical work. The basic and broad approach to the study of magnetic materials is the investigation of their constitutive behavior. Constitutive equations are the relations between dependent (response) and independent (state) tensors. For a given physical situation, the problem is to determine the restrictions to be imposed on the form of these constitutive relations as dictated by the symmetry of the material.

In this chapter we present a summary of the basic equations of the electromagnetic theory of deformable and fluent bodies. These consist of balance laws and constitutive equations. The purpose of this summary is to close the theory, without a thorough discussion of the derivations which would require a long, separate account. Interested readers may consult Eringen [1980, Chap. 10] and Eringen and Maugin [1989] for deformable bodies; De Groot and Suttorp [1972] for the statistical foundations of electromagnetic theory; and Jackson [1975] and Landau and Lifshitz [1960] for the classical theories.

A three-dimensional point group is a group of symmetry operators which acts at a fixed point O, and leaves invariant all distances and angles in a three-dimensional Euclidean space. The symmetry operators having these properties are rotations about the axes through O, or products of such rotations and the inversion. Such products, of course, include reflections in planes through O.

In addition to the geometrical symmetries present in the lattice structure of the crystals, the atoms of the lattice in magnetic materials are endowed with atomic magnetic moments (spins). The usual spatial symmetry operations, rotations and rotation-inversions, while preserving the geometrical properties of the lattice, may reverse the orientation of the spins.

Most physical properties of crystals are defined by the relationship between two or more tensors and are therefore themselves represented by tensors. If a crystal is subjected to an influence presented by a tensor \( {I_{{j_1}}} \ldots {j_n} \), which produces a physical effect \( {E_{{i_1}}} \ldots {i_m} \), then a linear relationship between the influence and the effect is given by where the tensors E and I are called physical (field) tensors, and m is the material tensor. Note that, in higher-order effects, we may have more than one influence, i.e., as, say, in the piezomagnetoelectrism e _ij = A _ijkr P _k M _r , where e is the strain (or stress) tensor and P and M are the electric and magnetization vectors polarization, respectively.

Let the matrices {M¹, M²,…, M ^g } be the symmetry group {M} of the material under consideration. Since {M} is obtained from {G} by {M} = {H} + τ{G − H}, M ^α is either an ordinary symmetry element S ^α , or it is a complementary element in the form τS ^α (α = 1,…,g).

In general, a linear relationship between the influence I and the effect E is given by (4.1), i.e.,

In this chapter we tabulate the elements of the integrity basis which are invariant under the magnetic symmetry group of the electromagnetic crystalline solids. For the sake of simplicity, the electromagnetic solids considered here are assumed to have the constitutive equations for the internal energy. We have similar expressions for the entropy, the electric field, the magnetic induction, the heat flux, and the stress tensor. The arguments E _KL , P _K , M _K , and θ appearing in (7.1) are, respectively, the material description of the strain tensor (a true, symmetric, and i-tensor), the polarization (a polar and time-symmetric vector), the magnetization (an axial, time-asymmetric vector), and the temperature (a true time-symmetric scalar). The dependent quantities such as the entropy, the electric field, the magnetic induction, and the stress tensor are derived from the internal energy, W, since these materials are conservative (cf. eq. (1.40)).

Here we give a number of examples on the application of the results and tables obtained in the previous chapters.