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Microcontinuum field theories extend classical field theories to
microscopic spaces and short time scales. This volume is concerned
with the kinematics of microcontinua. It begins with a discussion
of strain, stress tensors, balance laws, and constitutive
equations, and then discusses applications of the fundamental ideas
to the theory of elasticity. The ideas developed here are important
in modeling the fluid or elastic properties of porous media,
polymers, liquid crystals, slurries, and composite materials.
Continuum physics is concemed with the predictions of deformations,
stress, temperature, and electromagnetic fields in deformable and
fluent bodies. To that extent, mathematical formulation requires
the establishment of basic balance laws and constitutive equations.
Balance laws are the union of those of continuum thermomechanics
and MaxweIl's equations, as coIlected in Chapter 1. To dose the
theory it is necessary to formulate equations for the material
response to extemal stimuli. These equations bring into play the
material properties of the media under consideration. In their
simplest forms these are the constitutive laws, such as Hooke's law
of dassical elasticity, Stokes' law of viscosity of viscous fluids,
Fourier's law of heat conduction, Ohm's law of electric conduction,
etc. For large deformations and fields in material media, the
constitutive laws become very complicated, in vol ving all physical
effects and material symmetry. The present work is concemed with
the material symmetry regulations arising from the crystaIlographic
symmetry of magnetic crystals. While there exist some works on the
thirty-two conventional crystal dasses, exduding the linear case,
there exists no study on the nonlinear constitutive equations for
the ninty magnetic crystal dasses. Yet the interaction of strong
electromagnetic fields with deformable solids cannot be explained
without the material sym metry regulations relevant to magnetic
crystals. In this monograph, we present a thorough discussion of
magnetic symmetry by means of group theory. We consider onlyone
scalar function which depends on one symmetric second-order tensor
(e. g."
The electrodynamics of continua is a branch ofthe physical sciences
concerned with the interaction of electromagnetic fields with
deformable bodies. De formable bodies are considered to be continua
endowed with continuous distributions of mass and charge. The
theory of electromagnetic continua is concerned with the
determination of deformations, motions, stress, and elec
tromagnetic fields developed in bodies upon the applications of
external loads. External loads may be of mechanical origin (e.g.,
forces, couples, constraints placed on the surface of the body, and
initial and boundary conditions arising from thermal and other
changes) and/or electromagnetic origin (e.g., electric, magnetic,
and current fields). Because bodies of different constitutions
respond to external stimuli in a different way, it is imperative to
characterize properly the response functions relevant to a given
class of continua. This is done by means of the constitutive
theory. For example, an elastic dielectric responds to
electromagnetic fields in a totally different way than a magnetic
fluid. The present book is intended to present a unified approach
to the subject matter, based on the principles of contemporary
continuum physics."
This is the second volume of a two-volume set presenting a unified
approach to the electrodynamics of continua, based on the
principles of contemporary continuum of physics. The first volume
was devoted mainly to the development of the theory and
applications to deformable solid media. This volume extends the
developments of the first volume to richer and newer grounds. It
contains discussions on fluid media, magnetohydrodynamics,
eletrohydrodynamics and media with more complicated structures.
With the discussion, in the last two chapters, of memory-dependent
materials and non-local E-M theory, the authors account for the
nonlocal effects arising from motions and fields of material points
at past times and at spatially distant points. This discussion is
included here to stimulate further research in these important
fields, which are presently in development stages. The second
volume is self-contained and can be studied without the help of
volume I. A section summarizing the constitutive equations and the
underlying physical ideas, which were presented in more detail in
the first volume, is included. This volume may be used as a basis
for several graduate courses in engineering schools, applied
mathematics and physics departments. It also contains fresh ideas
and will stimulate further research in the directions the authors
outline.
Nonlocal continuum field theories are concerned with material
bodies whose behavior at any interior point depends on the state of
all other points in the body -- rather than only on an effective
field resulting from these points -- in addition to its own state
and the state of some calculable external field. Nonlocal field
theory extends classical field theory by describing the responses
of points within the medium by functionals rather than functions
(the "constitutive relations" of classical field theory). Such
considerations are already well known in solid-state physics, where
the nonlocal interactions between the atoms are prevalent in
determining the properties of the material. The tools developed for
crystalline materials, however, do not lend themselves to analyzing
amorphous materials, or materials in which imperfections are a
major part of the structure. Nonlocal continuum theories, by
contrast, can describe these materials faithfully at scales down to
the lattice parameter. This book presents a unified approach to
field theories for elastic solids, viscous fluids, and
heat-conducting electromagnetic solids and fluids that include
nonlocal effects in both space and time (memory effects). The
solutions to the field equations agree remarkably well with atomic
theories and experimental observations.
Nonlocal continuum field theories are concerned with material bodies whose behavior at any interior point depends on the state of all other points in the body -- rather than only on an effective field resulting from these points -- in addition to its own state and the state of some calculable external field. Nonlocal field theory extends classical field theory by describing the responses of points within the medium by functionals rather than functions (the "constitutive relations" of classical field theory). Such considerations are already well known in solid-state physics, where the nonlocal interactions between the atoms are prevalent in determining the properties of the material. The tools developed for crystalline materials, however, do not lend themselves to analyzing amorphous materials, or materials in which imperfections are a major part of the structure. Nonlocal continuum theories, by contrast, can describe these materials faithfully at scales down to the lattice parameter. This book presents a unified approach to field theories for elastic solids, viscous fluids, and heat-conducting electromagnetic solids and fluids that include nonlocal effects in both space and time (memory effects). The solutions to the field equations agree remarkably well with atomic theories and experimental observations.
Microcontinuum field theories constitute an extension of classical field theories -- of elastic bodies, deformations, electromagnetism, and the like -- to microscopic spaces and short time scales. Material bodies are here viewed as collections of large numbers of deformable particles, much as each volume element of a fluid in statistical mechanics is viewed as consisting of a large number of small particles for which statistical laws are valid. Classical continuum theories are valid when the characteristic length associated with external forces or stimuli is much larger than any internal scale of the body under consideration. When the characteristic lengths are comparable, however, the response of the individual constituents becomes important, for example, in considering the fluid or elastic properties of blood, porous media, polymers, liquid crystals, slurries, and composite materials. This volume is concerned with the kinematics of microcontinua. It begins with a discussion of strain, stress tensors, balance laws, and constitutive equations, and then discusses applications of the fundamental ideas to the theory of elasticity.
Microcontinuum Field Theories constitutes an extension of classical
field theories - of elastic solids, viscous fluids,
electromagnetism, and the like - to microscopic length and time
scales. Material bodies are viewed as collections of a large number
of deformable particles (sub-continua), suitable for modeling
blood, porous media, polymers, liquid crystals, slurries, and
composite materials.This volume extends and applies the ideas
developed in the first volume, Microcontinuum Field Theories:
Foundations and Solids, to liquid crystals, biological fluids, and
other microstretch and micomorphic fluids. The theory makes it
possible to discuss properties of such materials that are beyond
the scope of classical field theories and may provide a basis for
the resolution of some outstanding problems, such as turbulence.
Microcontinuum Field Theories constitutes an extension of classical field theories - of elastic solids, viscous fluids, electromagnetism, and the like - to microscopic length and time scales. Material bodies are viewed as collections of a large number of deformable particles (sub-continua), suitable for modeling blood, porous media, polymers, liquid crystals, slurries, and composite materials. This volume extends and applies the ideas developed in the first volume, Microcontinuum Field Theories: Foundations and Solids, to liquid crystals, biological fluids, and other microstretch and micomorphic fluids. The theory makes it possible to discuss properties of such materials that are beyond the scope of classical field theories and may provide a basis for the resolution of some outstanding problems, such as turbulence.
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