3R4. Nonlocal Continuum Field Theories. - AC Eringen (Prof Emeritus, Princeton Univ, 15 Red Tail Dr, Littleton, CO 80126-5001). Springer-Verlag, New York. 2002. 376 pp. ISBN 0-387-95275-6. $159.00.

Reviewed by JL Wegner (Dept of Mech Eng, Univ of Victoria, Eng Office Wing, Room 537, Victoria BC, V8W 3P6, Canada).

The main purpose of this book is to present a unified foundation for the development of the basic field equations of nonlocal continuum field theories. In order to achieve this goal, the author has relied on the natural extensions of the two fundamental laws of physics to nonlocality: 1) the energy balance law is postulated to remain in global form, and 2) a material point of the body is considered to be attracted by all points of the body, at all past times. By means of these two natural generalizations of the corresponding local principles, theories of nonlocal elasticity, fluid dynamics, and electromagnetic field theories are formulated that include nonlocality in both space and time. It is a well-written, rigorous mathematical treatment of nonlocal continuum field theories, and deals with such concepts such as nonlocal electromagnetic thermoelastic solids and nonlocal electromagnetic thermoviscious fluids.

The volume consists of 15 chapters and an appendix on the derivation of the Riemann Christoffel curvature tensor, and the Laplacian of the stress tensor in curvilinear coordinates.

Chapter one covers kinematics, that is the geometry of motion and the deformation of material bodies. The outline follows the script of most texts on classical continuum mechanics. That is, the concepts of material bodies and the material derivative are introduced. The deformation gradient tensor is introduced, and its physical meaning is described by the transformation of material line, surface, and volume elements. Similar to most treatments on classical continuum mechanics, the appropriate strain and stretch tensors are described. The remainder of the first chapter discusses the principle of objectivity, fundamental to the constitutive theory, and the compatibility conditions. As a departure from classical mechanics, the concept of nonlocal materials is introduced by the compatibility equations governing the dispacement field. Here, it is shown that if the compatibility conditions are satisfied, then there will not be a singled valued solution for the displacement field, and in this case the body will contain dislocations, cracks, inclusions, or other discontinuous fields. The first chapter is fundamental to the mathematical developments presented in later chapters.

Chapter 2 is concerned with the fundamental concept of stress. The energy and mass balance, for the entire body, are postulated. By means of the transport and Green Gauss theorem, the global laws of mass and energy are localized. An important theorem shows that the invariance of the energy balance, under each Galilean group, leads to a balance law. Localization involves residuals, which are shown to be incorporated into the concepts of the stress and energy. This leaves the new residuals only in the jump conditions governing the equations across a discontinuous surface moving through the material surface. In this way, new concepts of stress and energy involving long-range interactions are made part of the classical concepts. The remainder of the second chapter is devoted to thermodynamics, in which the concept of the second law of thermodynamics is first introduced. The Clausius-Duhem (C D) inequality, fundamental to the development of constitutive equations, is presented. Section 2.3, discusses the concept of dissipation. A theorem shows how one may obtain the solution of the C D inequality. Onsager reciprocal relations are also discussed. The solution of the C D inequality is extended to the memory functionals relevant to the discussions of the dissipative media.

In Section 3.1, eight axioms that are fundamental to the development of the constitutive equations are postulated and discussed separately. In Section 3.2, using these axioms, constitutive equations are obtained for memory dependent nonlocal thermoelastic solids, and in Section 3.3, for memory dependent, nonlocal thermoviscous fluids. Thus far, these developments are exact and applicable to nonlinear media undergoing finite deformations and motions.

The interaction of electromagnetic (E M) fields with deformable bodies requires new concepts which are developed in chapter four. Here, the Mawell equations are extended to nonlocal media, ie the global balance laws involving E M fields are written in Section 4.1. The localization involves some E M residuals. Here, also, Eringen is able to incorporate the nonlocal residuals into the E M fields. In this way, there remains one surface residual only, relevant to the jump conditions governing the magnetic field at a moving discontinuity surface through the surface. Electromagnetic force, couples, and power are discussed in Section 4.2, where the expressions of the stress tensor, the E M momentum, and Poynting stress vector that enter into the mechanical balance equations, are derived. The jump discontinuities of the E M force, couples, and power are developed in order to account for jump discontinuities. This is elaborated in Section 4.3. The expressions for the mechanical balance laws for electromechanically active media are derived in Section 4.4. The energy balance equation and the C D inequality are obtained for the development of the constitutive equations E M elastic solids.

The formulation of the constitutive equations of electromagnetic (E M) elastic solids must employ a special type of Helmholtz free energy functions and dissipation potential. To this end, the energy equations, the C D inequality, and constitutive equations, are developed in the material frame of reference. Separate constitutive equations for the static and dynamic members of the constitutive dependent variables are developed by employing an additive functional and the C D inequality. The dynamic members of the constitutive equations require the introduction of the difference histories of the dependent variables. The resulting constitutive equations are nonlocal both in space and time. Section 5.2 develops the constitutive equations of the memory dependent nonlocal E M thermoviscous fluids in the spatial domain.

The remainder of the book, Chapters 6–15, is devoted to applications of the theory developed in Chapter 3. In Chapter 6, certain problems of nonlocal linear elasticity are solved. In order to linearize the equations, the linear strain tensor, ekl, is introduced and small temperature changes T from a constant ambient temperature, TO are considered. The Helmholtz free energy function is written as a quadratic in terms of ekl and T, and a dissipative functional in terms of the gradient of the temperature field. The constitutive equations for nonlocal anisotropic elastic solids are then obtained using the equations developed in Chapter 3. Later, the constitutive equations for isotropic linear elastic solids are obtained, along with various different, but equivalent, forms of these equations. In Section 6.2, the same constitutive equations are obtained through the lattice dynamical approach.

The linear constitutive equations of nonlocal thermoviscous fluids are obtained in Chapter 7. In particular, the channel flow problem is solved where it is shown that the nonlinear theory predicts that the shear stress is 50% of the shear stress predicted by the classical theory. It is shown that the shear stress depends on the ratio of the internal characteristic length to the channel depth. Other problems solved in this chapter include lubrication problems in microscopic channels, and on rotating discs.

Chapter 8 covers nonlocal linear electromagnetic theory. Memory dependent nonlocal thermoelastic solids are included in Chapter 9. The isotropic solids and Kelvin Voigt type models are given as special cases. The constitutive equations for memory dependent nonlocal thermoviscous fluids are obtained in Chapter 10, along with the formulation of mixed boundary initial value problems.

Chapters 11 and 12 discuss memory dependent nonlocal electromagnetic elastic solids and thermofluids, respectively. For nonlocal elastic solids, memory effects include the E M absorptions and viscous dissipation. The formulation for memory dependent nonlocal electromagnetic thermofluids obtained in Chapter 12, is important to the discussions of various physical phenomena connected with magnetohydrodynamics, plasma physics, and atmospheric ionization. Chapters 13–15 investigate the effects of nonlocality in the case of continua with microstructure.

Nonlocal Continuum Field Theories is suitable for a reader knowledgeable in the general area of classical continuum mechanics, but not necessarily familiar with the concept of nonlocality. What is refreshing in the treatment presented by Eringen, is that in addition to providing the requisite mathematical background, he presents a discussion of nonlocal material behavior throughout the volume, which provides a reader unfamiliar with this concept, an intuitive understanding of the physical nature of this phenomenon.