Generalized Continuum Mechanics
A continuum-mechanical theory is presented in which the deformation is described by a number of generalized coordinate fields. The applied force system is described by conjugate generalized force fields distributed throughout the volume of the body and on
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A. E. Green and R. S. Rivlin Newcastle-upon-Tyne (England) and Bethlehem, Pa. (U.S.A.) Summary A continuum·mechanical theory is presented in which the deformation isdescribed by a number of generalized coordinate fields. The applied force system is described by conjugate generalized force fields distributed throughout the volume of the body and on the surface. These arise as the coefficients of the rates of change of the generalized forces in an expression for the rate at which work is done by the force system acting on the body. The differential equations of the theories are obtained by the systematic use of the first and second laws of thermodynamics, together with invariant-theoretical considerations. The theory is motivated by a particle model in which each particle consists of a number of mass-points. It is seen that starting with a given model there is a considerable ambiguity in the choice of the generalized coordinates describing the deformation and hence in the form of the resulting theory.
1. Introduction In classical continuum mechanics, the deformation of a body is described by a single vector field - the displacement field. The manner in which a rigid motion changes the components, in a fixed rectangular cartesian coordinate system, of this vector field is, of course, implied by the geometrical meaning of displacement. In the last few years there has been a flurry of interest in continuummechanical theories in which the deformation of a body is described not only by the vector displacement field but also by other vector or tensor fields. This interest was stimulated largely by the papers of ERICKSEN (1960, a, b, c, 1961), MINDLIN and TrERSTEN (1962), MINDLIN (1964), TOUPIN (1962) and others and by some remarks in the article by TRUESDELL and TOUPIN (1960). GREEN and RIVLIN (1964) developed a rather general theory in which the deformation of a deformable body was described by a number of deformation fields l . One of these was given 1
by
See also ERINGEN and SUHUBI (1964a, b), (1965) on these papers.
ERINGEN
(1964), and the comments
GREEN
H. Parkus et al. (eds.), Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids © Springer-Verlag/Wien 1968
Generalized Continuum Mechanics
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the properties of the usual displacement field while the others were tensor fields of various orders. This theory was called multipolar theory. In.a more recent paper GREEN and RIVLIN (1967) have discussed two further theories - director theory and multiple displacement theory. In both of these the deformation is described by a number of vector fields, but these are assumed to behave differently under rigid body motion in the two theories. In this paper (GREEN and RIVLIN 1967) the director and multiple displacement theories are related to each other and it is shown that the earlier multipolar theory may be regarded as a special case of the director theory. In developing the field equations of their theories, GREEN and RIVLIN rely mainly on the first and second laws of th
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