Some Recent Results in the Theory of Fading Memory [antikvár]

SOME RECENT RESULTS IN THE THEORY OF FADING MEMORY Bernard D. Coleman Mellon Institute and Departmenfof Mathematics, Carnegie-Mellon University, Pittsburgh, Pa 15213, U.S.A. ABSTRACT An outline is given of the phenomenological theory of fading memory recently explored by V. J. Mizel and the author. The theory provides a general framework in which one can derive the restrictions which the second law of thermodynamics places on the constitutive equations of materials with memory. 1. INTRODUCTION In theories of the dynamical behaviour of continua, there are several ways of describing the dissipative effects which, in addition to heat conduction, accompany deformation. The oldest way is to employ a viscous stress which depends on the rate of strain, as is done in the theory of Navier-Stokes fluids. In another description of dissipation, one postulates the existence of internal state variables which influence the stress and obey differential equations in which the strain appears. A third approach is to assume that the entire past history of the strain influences the stress in a manner compatible with a general postulate of smoothness or 'principle of fading memory'. Experience in high-polymer physics shows that the mechanical behaviour of many materials, including polymer melts and solutions, as well as amorphous, crosslinked solids and semi-crystalline plastics, is more easily described within the theory of materials with fading memory than by theories of the viscous-stress type, which do not account for gradual stress-relaxation, or by theories which rest on a finite number of internal state variables and which, therefore, give rise to discrete relaxation spectra when linearized. Some years ago, Walter Noll and I proposed a systematic procedure for rendering explicit the restrictions which the second law places on constitutive relations'. The procedure was easily applied in theories of materials of the viscous-stress type'' ^ and in theories which employ evolution equations for internal state variables'; these applications did not yield results which a physicist would consider surprising and were presented as attempts at clarification, with the emphasis laid upon logical relations. Implementation of the procedure in the theory of materials with memory was a different matter, however, for it there led to conclusions'^ which, although not anticipated by other arguments, have recently been shown to have important bearing on wave propagation' and dynamical stability®-Here I should 321