A. Cemal Eringen

On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves

Integropartial differential equations of the linear theory of nonlocal elasticity are reduced to singular partial differential equations for a special class of physically admissible kernels. Solutions are obtained for the screw dislocation and surface waves. Experimental observations and atomic lattice dynamics appear to support the theoretical results very nicely.

Nonlocal polar elastic continua

Abstract A continuum theory of nonlocal polar bodies is developed. Both the micromorphic and the non-polar continuum theories are incorporated. The balance laws and jump conditions are given. By use of nonlocal thermodynamics and invariance under rigid motions, constitutive equations are obtained for the nonlinear micromorphic elastic solids. The special case, nonpolar, nonlocal elastic solids, is presented.

Nonlinear theory of simple micro-elastic solids—I☆

Abstract The present work is concerned with the formulation of the basic field equations, boundary conditions and constitutive equations of what we call ‘simple micro-elastic’ solids. Such solids are affected by the ‘micro’ deformations and rotations not encountered in the theory of finite elasticity. The theory, in a natural fashion, gives rise to the concept of stress moments, inertial spin and other types of second order effects and their laws of motion. The mechanism of the surface tension is contained in the theory. In a forthcoming paper (Part II) explicit expressions of constitutive equations of several simple micro-elastic solids will be given and applied to some special problems.

Linear theory of nonlocal elasticity and dispersion of plane waves

Abstract The equations of the nonlocal elasticity given in [1]and[2] are linearized. The dispersion relations are obtained for one dimensional plane waves. The nonlocal material moduli are determined to fit exactly the acoustical branch of elastic waves within one Brillouin zone in periodic one dimensional lattices.

Theory of thermomicrofluids

Abstract The simple microfluid theory of Eringen [1] is extended to include the heat conduction and heat dissipation effects. The exact nonlinear theory is presented and restricted by the axioms of constitution and the second law of thermodynamics. The constitutive equations are linearized for dissipative and thermal effects. The complete field equations and accompanying jump conditions are given. Several internally constrained fluids, e.g., micropolar, inertia rateless fluids, are obtained. Boundary and initial conditions are discussed.

Theory of Micropolar Elasticity

In the four previous chapters we have given the complete theory of 3M continua, with and without E-M interactions. Balance laws, jump conditions, and nonlinear constitutive equations were obtained, so that the theory is complete and closed. Beginning with Chapter 5 we explore applications of these theories. By means of mathematical solutions and experimental observations, we try to exhibit new physical phenomena predicted by microcontinuum theories. The aim here is not to be exhaustive with the discussion of all problems for this is neither possible nor desirable, as there is a very large volume of literature in the field. It is not desirable since a large number of solutions tend to hide the main purpose, namely, the new physical phenomena that are not in the domain of predictions of classical field theories.

NONLINEAR THEORY OF MICRO-ELASTIC SOLIDS—II

Abstract The present paper is a continuation of our previous work on the same topic. After the introduction of material and strain measures of micro-elasticity we give the specific form of constitutive equations for isotropic micro-elastic materials, various approximate theories, the linear and the determinate theory of couple stress. Field equations of the linear theory for the latter case are obtained and applied to the study of the Rayleigh surface waves in micro-elasticity.

A CONTINUUM THEORY OF CHEMICALLY REACTING MEDIA—I

Abstract A unified approach is presented for the derivation of conservation of mass, balance of momenta, conservation of energy, balance of entropy and associated jump conditions across a moving surface of discontinuity in a chemically reacting continuum. The thermal, chemical and mechanical volume and surface effects are taken into account in a systematic fashion. A discussion of simultaneous reactions is presented. The content of the present paper is valid for reacting mixtures of fluids as well as solids. In Part II (forthcoming), the thermodynamics of the reacting fluids are studied in detail. Constitutive equations are derived and some illustrative problems are solved.

A unified theory of thermomechanical materials

Abstract The present paper is an attempt to organize and unify the mechanics, thermodynamics and constitutive equations of nonpolar materials subject to thermal effects and mechanical forces. The approach is that of continuum theory. The philosophy is new in that it unifies and extends all known theories under certain common fundamental axioms, some of which are introduced here for the first time. All theories of simple solids and fluids turn out to be special cases of this theory. Several theorems are given on the inexistence of certain classes of non-simple thermo-solids and the restricted forms of the constitutive equations of other thermomechanical materials. A general class of non-simple thermofluids is studied and the specific form of the constitutive equations of second order Stokesian thermo-fluids are obtained.

Theory of thermo-microstretch elastic solids

Abstract Equations of motions, constitutive equations, and boundary conditions are derived for a class of micromorphic elastic solids whose microelements can undergo expansions and contractions or stretch. These solids can have translatory and rotatory motions with spin intertia and therefore can support surface and body tractions and couples, just like micropolar elastic solids. In addition, material points of these solids can stretch and contract independently of their translations and rotatins. This is a continuum model for Bravais lattice with basis. It can also model composite materials reinforced with chopped fibers, porous solids filled with gas or inviscid fluids. Thermodynamical and non-negative strain energy restrictions are studied. Field equations are given and the uniqueness theorem is proved. The theory is illustrated with the solution of one-dimensional waves and compared with lattice dynamical results.