In this chapter we introduce the tensor description of physical properties along with Neumann’s Principle relating symmetry to physical properties. As pointed out in the introduction, many different types of anisotropic properties are described in this book, but all have one thing in common: a physical property is a relationship between two measured quantities. Four examples are illustrated in Fig. 5.1. Elasticity is one of the standard equilibrium properties treated in crystal physics courses. The elastic compliance coefficients relate mechanical strain, the dependent variable, to mechanical stress, the independent variable. For small stresses and strains, the relationship is linear, but higher order elastic constants are needed to describe the departures from Hooke’s Law. Thermal conductivity is typical of the many transport properties in which a gradient leads to flow. Here the dependent variable is heat flow and the independent variable is a temperature gradient. Again the relationship is linear for small temperature gradients. Hysteretic materials such as ferromagnetic iron exhibit more complex physical properties involving domain wall motion. In this case magnetization is the dependent variable responsive to an applied magnetic field. The resulting magnetic susceptibility depends on the past history of the material. If the sample is initially unmagnetized, the magnetization will often involve only reversible domain wall motion for small magnetic fields. In this case the susceptibility is anhysteretic, but for large fields the wall motion is only partly reversible leading to hysteresis. The fourth class of properties leads to permanent changes involving irreversible processes. Under very high electric fields, dielectric materials undergo an electric breakdown process with catastrophic current flow. Under small fields Ohm’s Law governs the relationship between current density and electric field with a well-defined resistivity, but high fields lead to chemical, thermal, and mechanical changes that permanently alter the sample. Irreversible processes are sometimes anisotropic but they will not be discussed in this book. Measured quantities such as stress and strain can be represented by tensors, and so can physical properties like elastic compliance that relate these measurements. This is why tensors are so useful in describing anisotropy.
Nonlinear Dynamic Bending and Domain Wall Motion in Functionally Graded Piezoelectric Actuators under AC Electric Fields: Simulation and Experiment