On energies for nonlinear viscoelastic materials of single-integral type

On The Concepts of State and Free Energy in Linear Viscoelasticity

“…Remark. A similar argument shows that the free energy of single-integral type proposed by GURTIN & HRUSA [20]:…”

A note on Onsager’s relations

“…The stress and the entropy relations (4.10) and (4.11) are essentially instances of the results obtained by Coleman [3] within the framework of the general theory of materials with fading memory. The particular form given above is a consequence of the postulated form of the constitutive equations and appears first in Gurtin and Hrusa [7,8]. The heat flux relations (4.12) are instances of the result of Gurtin and Pipkin [9] obtained within the framework of materials with the dependence on the summed history of the gradient of temperature via unspecified functional.…”

“…As far as the dependence on the mechanical variable, F, is concerned, the material exhibits a viscoelastic type of behavior. While there is a vast literature on models qualitatively similar to the one given above (see, e.g., Coleman and Noll [4], Coleman [3]), the above specific form of the dependence on F(£), F(/ -5) has been proposed recently by Gurtin and Hrusa [7,8], who call constitutive assumptions of the type (1.1) single-integral laws. The dependence on the thermal variables is instantaneous in the temperature, and the dependence on the history of temperature through the summed history G' is typical for models that exhibit finite speeds of propagation of thermal disturbances.…”

“…Note that for s -> 0+, the argument of m in (2.5) converges to (F(r), F(t), 9{t), 0, 0) and the limit (2.5) for j-»0+ is therefore consistent with the normalization (2.2), which is assumed to hold only for 5 > 0, as m is not assumed to be defined for 5 = 0. Definition 2.1 differs only in technical details from the definitions given in [7,8], The domination conditions say first of all that the influence of the remote past is small; technically, they are adjusted so as to ensure that all the integrals occurring in the statements and proofs of the results are absolutely convergent. In particular, the assumption that m itself is dominated ensures that the integral in (1.1) is convergent for every process and every time; the dominance of the partial derivatives of m with respect to F, H, 6 , and G ensures the continuous differentiability of p. with respect to t and that the time derivative of the integral in (1.1) is given by the time derivative of the integrand with respect to t. The dominance of sdHm, sdGm permits one first of all to define the kinetic coefficients K, L by formulas (3.6), (3.7) in Sec.…”

“…For simplicity the density of the material element in the reference configuration is assumed to be equal to 1. I refer to Gurtin and Hrusa [7,8] for the discussion of the assumption that the entropy and the free energy are given by the single-integral laws. The stress and the entropy relations (4.10) and (4.11) are essentially instances of the results obtained by Coleman [3] within the framework of the general theory of materials with fading memory.…”