Nonlocal Continuum Field Theories

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A. Cemal Eringen

Nonlocal Continuum Field Theories

With 72 Figures

A. Cemal Eringen Emeritus Professor, Princeton University 15 Red Tail Drive Littleton, CO 80126-5001 USA

Library of Congress Cataloging-in-Publication Data Eringen, A. Cemal.

Nonlocal continuum field theories/A. Cemal. Eringen.

p. cm. Includes bibliographical references and index. ISBN 0-387-95275-6 (alk. paper)

1. Field theory (Physics). 2. Continuum mechanics. I. Title. QC173.7.E76 2001 530–dc21 200102044

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Preface

This book is devoted to the development of the nonlocal continuum field theories of material bodies where nonlocal intermolecular attractions are important. There exists a large class of problems in classical physics and continuum mechanics (classical field theories) that fall outside their domain of applications. Fracture of solids, stress fields at the dislocation core and at the tips of cracks, singularities present at the point of application of concentrated loads (forces, couples, heat, etc.), sharp corners and discontinuities in bodies, the failure in the prediction of short wavelength behavior of elastic waves, and several decades of the viscosity rise in fluids flowing in microscopic channels are but few major anomalies that defy classical treatment. In the field of electromagnetic theory, polaritons, gyrotropic effects, and superconductivity cannot be treated by the classical field theories. For example, several branches of high-frequency waves are not predicted and short wavelength regions deviate grossly from experimental observations. To cast light on the materials behavior in these areas, physicists often seek help from atomic lattice dynamics.

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. To this end, we have relied on the natural extensions of the two fundamental laws of physics to nonlocality: (i) the energy balance law is postulated to remain in global form; and (ii) 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 (memory-dependence).

vi Preface

Solutions of the field equations are given for many critical problems in these fields, displaying unexpected favorable agreements with the atomic theories and/or the experimental observations.

Presently, there exists no published treatise in this field, except some papers and limited reviews that are scattered in the literature. We believe this book offers rich material for research ideas in nonlocal solid and fluid mechanics, electromagnetic theory, and the nonlocal theories of media with microstructure and nonlocal theory of liquid crystals. It provides mathematical methods for the exploration of many failures of classical field theories and for the discoveries of new physical phenomena and/or explanations of the old ones. It can be used for graduate study in the departments of applied physics, applied mathematics, engineering sciences, and for research.

We express our thanks to Jennifer Page who typed the nonmathematical parts of the manuscript. My daughter Meva who freed much time for me by undertaking many chores of life and life itself. To her goes my everlasting love.

Littleton, Colorado December 2000

A. Cemal Eringen

Introduction

Nonlocal continuum field theories are concerned with the physics of material bodies whose behavior at a material point is influenced by the state of all points of the body. Following the classical notions, material points of a body are considered to be continuous and are assigned some physically independent objects (variables) (e.g., mass, charge, electric field, magnetic field).The state of the body, at a material point, is described by the relations of the response objects that constitute another class (e.g., stress, internal energy, heat) as functions of the independent objects. These relations are called constitutive equations. The nonlocal theory generalizes the classical field theory in two respects: (i) the energy balance law is considered valid globally (for the entire body); and (ii) the state of the body at a material point is described by the response functionals . This means that we need complete knowledge of the independent variables at all points of the body to describe the state of the body at each point. In this book the terminology of nonlocal field theories is to be understood in this sense. In fact, we have also included, in this description, nonlocality in time which is known a s memory-dependence .

As a subclass, the present definition includes gradient theories which possess limited nonlocality . Here various order gradients of the independent objects, at a materialpointandtheirtimerates,enterintoresponsefunctionsatthesamematerial points, Eringen [1966].A third category of nonlocality involves constitutive ersatzs between the spatial statistical moments of dependent and corresponding sets of independent variables. Such a group of theories are known as director theories, micromorphic, microstretch, and micropolar theories. These theories are explored elsewhere, Eringen [1999], [2001].

viii Introduction

The present book is devoted to the development of the nonlocal field theories described by the fundamental notions (i) and (ii) above.

The question arises, why nonlocality? The domain of the applicability of classical field theories is intimately connected to the length and time scales. If L denotes the external characteristic length (e.g., crack length, wavelength, a length over which applied loads are smooth) and l the internal characteristic length (e.g., granular distance, lattice parameter), then in the region L/l ≫ 1, classical field theories predict sufficiently accurate results. On the other hand, when L/l ∼ 1, local theories fail and we must resort to either atomic or nonlocal theories that can account for the long-range interatomic attractions. For the dynamical case, there will be a similar scale T/τ where T is the external characteristic time (e.g., the time scale of the applied loads) and τ is the internal characteristic length (e.g., the time scale of transmission of signal from one molecule to the next). Again, classical theories fail when T/τ ∼ 1. Thus we see that the physical phenomenon in space–time requires nonlocality and memory effects scaled by L/l and T/τ .

The concept of nonlocality is inherent in solid state physics where the nonlocal attractions of atoms are prevalent. Here the material is considered to consist of discrete atoms connected by distant forces from other neighboring atoms. Hence, we expect relations between the nonlocal field theories and the lattice dynamics (see Section 6.9).

Real materials do not resemble perfect crystal lattices or lattices with few imperfections (e.g., dislocations, foreign atoms, etc.) which enable a physicist to describe the state of the body. Unfortunately, amorphous and engineering materials do not lend themselves to such ideal descriptions, while nonlocal continuum theories make it possible to describe material properties very well. In fact, this description is faithful in microscopic scales all the way to the size of the lattice parameter (see Sections 6.10 to 6.17).

Thus, nonlocal continuum theory should be considered a viable discipline in physics and applied science.

Prior to the 1960s some elements of continuum theory had entered into the discussion of various physical phenomena that required mathematical descriptions in the forms of the integro-partial differential equations. For example, Chandrasekhar’s [1950] radiative transfer equation is of this type. Other historical references in this category are given in several articles in Continuum Physics , Vol. IV, edited by Eringen [1976]. The formulation of the nonlocal elastic constitutive equations by means of lattice dynamics, was given by Krumhansl [1965], [1968], Kröner and Datta [1966], and Kunin [1966]. The continuum approach to memory-dependent nonlocal elasticity was made by Eringen [1966] and to nonlocal micropolar elasticity by Eringen [1965]. These early theories did not include the aspect (i), namely, the global nature of the balance laws. This aspect along with the nonlocal constitutive equations was formulated by Eringen and Edelen [1972] and Eringen [1972a,b].A review and development of the theory, and historic reference up to 1976, are contained in Continuum Physics ,Vol. IV, edited by Eringen [1976].

Introduction ix

The theory of nonlocal elasticity has attracted the attention of many writers, perhaps because of its early success in resolving an old problem in fracture mechanics. It is well known that classical elasticity predicts infinite stress at the tip of a sharp crack. This led researchers to invent several fracture criteria (e.g., Griffith, fracture toughness, energy release rate, J -integral, etc.) independent of the state of stress at the tip of a crack. The nonlocal elasticity solution of Eringen and Kim [1974a,b] and Eringen et al. [1977] showed that the stress at the tip of a crack is finite, it rises to a maximum and then diminishes with the distance from the crack tip (cf. Section 6.14). Similarly, the nonlocal solution of the discreet dislocation problem (Eringen [1977a,b]) leads to vanishing stress at the eye of the dislocation, rising to a maximum in the interior of the body and approaches the classical solution with the distance. These results led to the introduction of a natural fracture criterion, based on the maximum stress hypothesis and the accurate prediction of the cohesive stress (cf. Sections 6.12 to 6.17). Moreover, nonlocal solutions also lead to dispersion relations for harmonic waves that are faithful to the lattice dynamical results throughout the entire Brillouin zone.

The nonlocal continuum theory of fluent media, including the microstructural effects, were discussed by Eringen [1972b], [1973b], applications to turbulence by Speziale and Eringen [1981], the diffusion of gases by Demiray and Eringen [1978], and magnetohydrodynamics by Eringen [1986]. McCay and Narasimhan [1981] discussed theory of nonlocal electromagnetic fluids, and Narasimhan and McCay [1981] applied their theory to dispersive waves in dielectric fluids. Eringen [1973c], [1984a], [1984b], [1984c], [1986], [1990], [1991] introduced the theories of nonlocal electromagnetic elastic solids and gave solutions of several problems: point charges, optical waves, polaritons, Eddy currents, gyrotropic media, superconductivity,Alfven waves, etc.A book byAgranovich and Ginsburg [1984] covers many aspects of the nonlocal effects on crystal optics. The nonlocality here appears under the terminology of “Spatial Disperson."

In spite of its age, the literature on nonlocal field theories is not extensive. There exist several reviews, conference symposia, and reports.Among them we mention:

Continuum Physics , Vol. IV, edited by A.C. Eringen, Academic Press, 1976.

This book contains accounts of the polar and nonlocal polar theories of elastic solids.

Nonlocal Theories of Material Media , The Polish Academy, 1976.

This book contains reviews on nonlocal elasticity.

Nonlocal Theory of Material Media , CISM Courses and Lectures, No. 268, edited by D. Rogula, Springer-Verlag, 1982.

Here, lattice defects, nonlocal elasticity, and some soluble problems are discussed by several authors. Elastic Media with Microstructure , Vols. I and II, by I.A. Kunin, Springer-Verlag, 1982/1983.

In these books the lattice dynamical approach is employed for the discussion of microstructural problems.

x Introduction

Mechanics, Electromagnetic Materials, and Structures , AMD, Vol. 61, MD-Vol. 42, edited by J.S. Lee, G.A. Maugin, andY. Shindo,American Society of Mechanical Engineers, 1993.

This is a collection of papers presented at a scientific meeting. The articles are mostly in classical fields, except one article entitled “Vistas of Nonlocal Electrodynamics" by Eringen.

EUROMECH Colloquium , Vol. 378, Mulhouse, France 20–22, April 1998, organized and published by Universite de Mulhouse CNRS-CSI.

This report contains several articles on micromechanics, nonlocal elasticity, and the solutions of some problems. Report No. 39 by E.S. ¸Suhubi, March 1987.

This report was used for the lectures by Professor ¸Suhubi at the University of Calgary Canada. I am indebted to Professor ¸Suhubi for sending me a copy of this report.

The unified formulation of the theories presented in this book is new and has not appeared before in the literature. The solutions to a plethora of problems are based mostly on the author’s, and his collaborator’s, work. References to these and works of other authors appear in the appropriate sections. It is not feasible to make a comprehensive survey of the fields that include all the published work. Needless to say, there are many other published works that may have escaped my attention. For this failure, I express my apologies to those authors whose work I missed to give credit.

Themainpurposeofthisbookistopresentaunifiedapproachtotheformulations of the nonlocal field theories for memory-dependent elastic solids, viscous fluids, electromagnetic solids, and fluids that conduct heat. Nonlocal balance laws are obtained by using the axiom of the Galilean invariance of the energy balance law (Chapter 2). Nonlocal constitutive equations are then constructed, based on some postulates and the thermodynamic conditions of admissibility. The resulting equations are nonlinear and include memory and nonlocality. Chapter 3 presents theconstitutiveequationsofnonlinear,nonlocal,memory-dependentthermoelastic solids, and thermoviscous fluids. In Chapter 4, we obtain the nonlocal balance laws of the electromagnetic theory and in Chapter 5 the constitutive equations of nonlinear, nonlocal, memory-dependent electromagnetic thermoelastic solids and thermoviscous fluids. In principle, the basic formulations are now complete.

In Chapter 6 on nonlocal elasticity, we begin to develop the linear theories and we present a multitude of applications to the theory of different fields. The solutions of some critical problems show unequivocally the power and potential of the nonlocal theory. For example, the stress field at the core of a discrete dislocation tends to vanish, reaching a maximum outside the core of dislocation. When this maximum is equated to the cohesive stress that holds the atomic bonds together, we obtain the celebrated Griffith criterion. Alternatively, we are able to predict the cohesive stress exactly, by setting the internal characteristic length to the one based on phonon dispersion (cf. Sections 6.10 to 6.14). The solution of the crack problems, rigid stamp on a half-plane leads us to a Griffith-like fracture criterium