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Linear constitutive relations for the response fields D and H in terms of the macroscopic fields E and B of a harmonic plane wave are derived in multipole form up to electric octopole-magnetic quadrupole order. These yield multipole expressions for the permittivity, inverse permeability, and the two magnetoelectric coefficients for both non-magnetic and magnetic media. These four material constants are required to satisfy origin independence, certain symmetries for a non-dissipative medium, and the Post constraint. Derivations of the last two conditions are given. Up to the multipole order considered, all the material constants satisfy the symmetry requirements, but none is origin independent beyond electric dipole order. The Post constraint is involved beyond this order and is not satisfied. These topics are considered further in Chapters 7 and 8.

This chapter first introduces the constitutive relations which are commonly used in electromagnetic theory for the mathematical modelling of complex electromagnetic media. These constitutive relations are to be understood as operators connecting the electric flux density and the magnetic flux density with the electric and the magnetic fields. When they are introduced into the Maxwell equations, this chapter obtains differential equations (PDEs) that govern the evolution of the electromagnetic fields. This chapter also seeks to formulate and discuss the scope of the various problems related to the Maxwell equations that will be treated in this volume. It introduces and formulates in terms of differential equations various problems of interest related to the Maxwell equations: time-harmonic problems, scattering problems, time-domain evolution problems, random and stochastic problems, controllability problems, homogenisation problems, and others.

A review of the basic elements of electricity and magnetism is presented with an introduction to Maxwell’s equations for steady-state in a vacuum. The modifications to these equations necessary to account for time varying sources are shown to produce to a causal unification of magnetic and electric fields. The application of Maxwell’s equations in the presence of matter leads to the concepts of electric and magnetic polarization of matter. Electromagnetic radiation arises directly from Maxwell’s time-dependent equations and the basic response of materials to this radiation is discussed. Finally, electromagnetic conservation laws are derived, including electromagnetic energy and linear and angular momentum.

The initial discussion concerns the need for damage characterization for materials. A little more specifically, cumulative damage is the state of damage that grows until it terminates with the materials failure. Damage states are often characterized by inserting into the constitutive relations explicit damage variables with unknown coefficients to be determined. In the present investigation only damage forms with no adjustable parameters are employed. The main approach considered here is that of kinetic crack growth. In this approach, power-law forms are taken for the rate of growth of the idealized crack representing damage. The crack grows until instability causes macroscopic failure. Three other models are compared with the kinetic crack approach: Miner’s rule, the Hashin–Rotem form, and the Broutman–Sahu form. Three methods of testing the four models are utilized: (i) residual strength, (ii) life after damage, or simply life prediction, and (iii) residual life. Only the kinetic-crack method satisfies the consistency tests of all three testing methods.

Is gravity nonlocal? Einstein interpreted the principle of equivalence of inertial and gravitational masses to mean that there exists a profound relationship between inertia and gravitation. Based on Einstein’s fundamental insight, it would seem natural to extend history dependence to the gravitational domain. However, it is not clear how to develop a nonlocal extension of Einstein’s local principle of equivalence. To go forward, we therefore choose an indirect approach based on a certain analogy with electromagnetism. In a material medium, the electromagnetic constitutive relations are nonlocal and this fact leads to the nonlocal electrodynamics of media. It turns out that general relativity can be formulated in a form that resembles the electrodynamics of media. Making the corresponding gravitational constitutive relations nonlocal would then lead to nonlocal GR. This indirect approach is adopted in the rest of this book.

The theory of light is described by Maxwell’s Equationsand they provide information about the fundamental properties of light. The wave equation is contained within Maxwell’s equations and proof is provided but is an example of a topic that can be skipped. The electromagnetic wave is a transverse wave of both the electric and magnetic field which are also mutually perpendicular. We discuss some of the differences between classical and quantum theory of light but restrict the use of classical wave theory in this text. The classical electromagnetic wave has a momentum that has led to the development of optical twezzers of great use in biological motors. Because the amplitude of the electromagnetic wave is a vector quantity we introduce the concept of polarization to describe the vector properties. We will need the capability in our discussion of reflection.