Heat transfer by lattice (phonon) conduction is proportional to the lattice thermal conductivity tensor Kp(W/m-K), i.e., qk = −Kp · ∇T (the Fourier law, Table 1.1), and sensible heat storage is determined by the phonon (lattice) specific heat capacity cυ,p (J/kg-K). The specific heat capacity is also given per unit volume(J/m3-K), or per atom(J/K). Phonons participate in many thermal energy conversion phenomena, including laser cooling of solids, discussed in Chapter 7 [ṡi-j(W/m3) in Table 1.1]. In this chapter, we examine how the atomic structure of a solid influences cυ,p, Kp, and ṡi-j involving phonons.

Phonons are lattice-thermal-vibration waves that propagate through a crystalline solid. Most lattice vibrations have higher frequencies than audible sound, ultrasound, and even hypersound. Figure 4.1 shows the various sound- and vibrational-wave regimes. A single, constant speed (dispersionless, i.e., having a linear frequency dependence on the wave number) of 103 m/s is used for the sake of illustration. In solids, the speed of sound is related to the moduli of elasticity and density and is as high as 18, 350 m/s in diamond and as low as 818 m/s in thallium. As will be shown, the vibrational waves have different modes, and the propagation speed can be strongly frequency dependent. The phonon energy ħω is general less than about 0.2 eV, with the high energy phonons being available more at higher temperatures.

In this chapter, we begin with lattice vibration and the relation between frequency and wave number (the dispersion relation) for a simple, harmonic, one-dimensional lattice.

The macroscopic heat transfer rates use thermal-energy related properties, such as the thermal conductivity, and in turn these properties are related to the atomic-level properties and processes. Heat transfer physics addresses these atomic-level processes (e.g., kinetics). We begin with the macroscopic energy equation used in heat transfer analysis to describe the rates of thermal energy storage, transport (by means of conduction k, convection u, and radiation r), and conversion to and from other forms of energy. The volumetric macroscopic energy conservation (rate) equation is listed in Table 1.1. The sensible heat storage is the product of density and specific heat capacity ρcp, and the time rate of change of local temperature ∂T/∂t. The heat flux vector q is the sum of the conductive, convective, and radiative heat flux vectors. The conductive heat flux vector qk is the negative of the product of the thermal conductivity k, and the gradient of temperature ∇T, i.e., the Fourier law of conduction. The convective heat flux vector qu (assuming net local motion) is the product of ∇cp, the local velocity vector u, and temperature. For laminar flow, in contrast to turbulent flow that contains chaotic velocity fluctuations, molecular conduction of the fluid is unaltered, whereas in turbulent flow this is augmented (phenomenologically) by turbulent mixing transport (turbulent eddy conductivity). The radiative heat flux vector qr = qph (ph stands for photon) is the spatial (angular) and spectral integrals of the product of the unit vector s and the electromagnetic (EM), spectral (frequency dependent), directional radiation intensity Iph,ω, where ω is the angular frequency of EM radiation (made up of photons).

Energy is stored within atoms (electrons and relativistic effects) and in the bonds among them when they form molecules, and also between these molecules when they form clusters and bulk materials. In this chapter we review the intraatomic and interatomic forces and energies (potentials). We examine electron orbitals of atoms and molecules and review ab initio computations of interatomic potentials and potential models. Then we discuss the dynamics (and scales) of molecular (or particle) interactions in classical (Newtonian and Hamiltonian) treatments of multibody dynamics and the relationship between the microscopic and macroscopic states (particle ensembles). We also begin examining the quantum energy states of the simple harmonic oscillator, free-electron gas, and electrons in hydrogen atom orbitals, by seeking exact solutions to the Schrödinger equation.

Interatomic Forces and Potential Wells

Atoms and molecules, at sufficiently low temperatures, can aggregate into linked structures that form what we know as the condensed phases of matter. What causes this aggregation behavior are the attractive forces generated by the opposite electrical charges of each particle (be it individual atom, ion, or polar molecule) in the system. These forces are in turn caused by the innate charges of the constituent subatomic particles of each atom, ion, or molecule. All physical and chemical bonding is the result of such molecular forces that are electrical in origin (the Coulomb forces). Intramolecular (or interatomic) forces keep atoms in a molecule (a compound of atoms) together, such as the H−O bond in a H2O molecule, whereas intermolecular forces keep neighboring molecules together, such as H2O molecules in the gas, liquid, or solid phase.

Fundamentals of Piezoelectricity

The term “piezoelectricity” translates roughly to “pressure electricity” and refers to an effect observed in many naturally occurring crystals; that is, the generation of electricity under mechanical pressure. The effect was first predicted and then experimentally measured by the brothers Pierre and Jacques Curie in 1880. The research was prompted by investigations into a closely related effect, the pyroelectric effect, which is the generation of electricity as a result of a change in temperature. The effect observed by the Curie brothers is also known as the direct piezoelectric effect. A strict definition of the direct effect is “electric polarization produced by mechanical strain, being directly proportional to the applied strain.” A converse piezoelectric effect also exists and is the appearance of mechanical strain as a result of an applied electric field.

The origin of the piezoelectric effect can be traced to fundamental geometric properties of certain crystals. Based on their geometry, crystals are normally classified into seven categories: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. A structure is called centrosymmetric if it has symmetry with respect to a single point. Based on their symmetry with respect to a point, the crystals are further classified into 32 classes, of which only 20 classes can exhibit piezoelectricity. The unit cell of these crystals possesses a certain degree of asymmetry, leading to a separation of positive and negative charges that results in a permanent polarization.

Applications of smart structures technology to various physical systems are primarily focused on actively controlling vibration, performance, noise, and stability. Applications range from space systems to fixed-wing and rotary-wing aircraft, automotive, civil structures, marine systems, machine tools, and medical devices. Early applications of smart structures technology were focused toward space systems to actively control vibration of large space structures [1] as well as for precision pointing in space (e.g., telescope, and mirrors [2]). The scope and potential of smart structures applications for aeronautical systems have subsequently expanded. Embedded or surface-bonded smart material actuators on an airplane wing or helicopter blade can induce alteration of twist/camber of airfoil (shape change), which in turn can cause variation of lift distribution and may help to control static and dynamic aeroelastic problems. For fixed-wing aircraft, applications cover active control of flutter [3, 4, 5, 6, 7], static divergence [8, 9], panel flutter [10], performance enhancement [11], and interior structure-borne noise [12]. Compared to fixed-wing aircraft, helicopters appear to show the most potential for a major payoff with the application of smart structures technology. Given the broad scope of smart structures applications, developments in the field of rotorcraft are highlighted in a subsequent section. Although most current applications are focused on the minimization of helicopter vibration, there are other potential applications such as interior/exterior noise reduction, aerodynamic performance enhancement that includes stall alleviation, aeromechanical stability augmentation, rotor tracking, handling qualities improvement, rotor head health monitoring, and rotor primary controls implementation (e.g., swashplateless rotors) [13].

The previous chapters discuss the properties and behavior of active materials that existed in a solid state. These materials exhibited changes in properties and physical dimensions when subjected to an electric, magnetic, or thermal field. A special class of fluids exists that change their rheological properties on the application of an electric or a magnetic field. These controllable fluids can generally be grouped under one of two categories: electrorheological (ER) fluids and magnetorheological (MR) fluids. An electric field causes a change in the viscosity of ER fluids, and a magnetic field causes a similar change in MR fluids. The change in viscosity can be used in a variety of applications, such as controllable dampers, clutches, suspension shock absorbers, valves, brakes, prosthetic devices, traversing mechanisms, torque transfer devices, engine mounts, and robotic arms. Other applications such as electropolishing do not rely directly on the change in viscosity but rather on the ability to change properties of the fluid locally.

Most mechanical dampers consist of fixed damping that is designed as a compromise between a range of operating conditions. As a result, these devices do not provide an optimum level of damping for any specific operating environment. Using ER/MR fluid dampers, variable damping levels can be obtained, and the system performance can be optimized over a wide range of operating conditions. In such dampers, the resistance to flow and, consequently, the energy dissipation, can be modulated through the applied electric or magnetic field.

The quest for superior capability in both civil and military products has been a key impetus for the discovery of high performance new materials. In fact, the standard of living has been impacted by the emergence of high performance materials. There is no doubt that the early history of civilization is intertwined with the evolution of new materials. For example, different eras of civilization are branded with their material capabilities, and these periods are referred to as the Stone Age, the Bronze Age, the Iron Age, and the Synthetic Material Age. The Stone Age represents the earliest known period of human civilization that stretches back to one million years BC, when tools and weapons were made out of stone. The Bronze Age (sometimes called the Copper Age) spans 3500–1000 BC. Weapons and implements were made of bronze (an alloy of copper and tin) during this period. The alloy is stronger than either of its constituents. Bronze was used to build weapons such as swords, axes, and arrowheads; implements such as utensils and sculptures; and other industrial products. The Iron Age followed the Bronze Age around 1000 BC and was characterized by the introduction of iron metallurgy. Iron ores were plentiful (cheap) but required high-temperature (2800°F) furnaces as compared to copper, which required lower-temperature (1900°F) furnaces. The Iron Age was the age of the industrial revolution, and many of the initial design tools, mechanics-based analyses, and material characterizations were formulated during this period.

Magnetostrictives and electrostrictives are active materials that exhibit magneto-mechanical and electromechanical coupling, respectively. These materials undergo a change in dimensions in response to an applied magnetic or electric field. A common property of both materials is that the induced strain depends only on the magnitude of the applied field and is independent of its polarity. In other words, it can be said that the induced strain has a quadratic dependence on the applied field. It is this behavior that differentiates electrostriction from the piezoelectric effect, which is also caused by an electric field. This chapter discusses the basic mechanisms behind magnetostriction and electrostriction, and it describes how these materials are used to construct practical actuators and sensors. The behavior of magnetic shape memory alloys (SMAs) is also described.

Magnetostriction

A ferromagnetic material placed in a magnetic field generally undergoes a change in shape [1]. The internal structure of a ferromagnetic material consists of randomly oriented magnetic domains. When a magnetic field is applied, the domains rotate to align themselves along the field, causing a change in the material dimensions. This phenomenon is known as “magnetostriction.” The effect is small in most materials but is measurable (on the order of microstrain) in ferromagnetic materials. Some materials, such as Terfenol-D, exhibit magnetostrictive strains on the order of 2000 microstrain (2000 × 10×6). Such materials can be used as both solid-state actuators and magnetic-field sensors. Magnetostrictive materials are available in the form of rods, thin films, and powder.