Abstract
In this paper we consider the development of rate type viscoelastic models from two different perspectives, using the Helmholtz potential based on the deformation gradient and the rate of entropy production, and the Gibbs potential based on the stress and the rate of entropy production. These two methods are not equivalent and one cannot always be obtained from the other by using a Legendre transformation. Also, it is not even possible for certain classes of materials to define a Helmholtz potential for bodies belonging to such classes, and for certain other classes to define a Gibbs’ potential for bodies belonging to them. The Helmholtz potential formulation has been shown to be capable of modeling generalizations of the Maxwell, Oldroyd-B, Burgers’ models as well as models that can be cast within the context of the conformation tensor. These models can also describe the evolution of the anisotropy of such materials during the deformation process. The Gibbs’ potential formulation is particularly well suited to the development of models wherein the material moduli that characterize the body are dependent on the invariants of the stress. Thus, such models can be used to describe bodies whose material moduli depend on the mean normal stress and could be very useful in characterizing geological materials. Such a framework would also be particularly relevant to describing the problem of compaction of asphalt concrete as the material property of the compacted body would depend on the mean normal stress which effects the compaction.
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Notes
It is not necessary to restrict ourselves to the requirement that the body possess instantaneous elastic response. In fact, there is considerable experimental evidence to show that certain asphalts do not exhibit instantaneous elastic response. The thermodynamic framework that is outlined in this paper has been used to describe such response [16].
It is not clear whether by pressure in these studies one means the mean normal stress in the fluid or the thermodynamic pressure. In fact, in Andrade’s paper, and the work of Bridgman [21], the pressure of the confining fluid is treated as the pressure that appears in the formula for the viscosity and the not the pressure within the actual fluid of interest. The pressure of the confining fluid is treated as though it were the pressure in the fluid of interest whose dependence of viscosity on pressure is being determined. Moreover, the pressure is considered to be the same everywhere in the flow domain, and while this is clearly incorrect in that the pressure field will vary over the flow domain, it might however be a very good approximation to being that within the confining fluid. It is also worth observing that the viscosity is inferred by assuming that the drag that an object suffers is the Stokes drag.
A viscoelastic solid model whose material moduli depend on the Lagrange multiplier that enforces the constraint of incompressibility (not the mean normal stress) has been investigated by Rajagopal and Saccomandi [25]. However, the model has not been derived within a thermodynamic framework.
The notion of “natural configuration” was discussed by Eckart [52] within the context of the inelastic response of solids. He however did not discuss issues concerning the material symmetry of the natural configuration and how it evolves. The notion of natural configuration plays a critical role in the response of entropy producing bodies and a detailed discussion of the notion can be found in the report by Rajagopal [53].
If the deformations are not homogeneous, one cannot always unload to κ p(t)(B) such that κ p(t)(B) is stress-free and geometrically contiguous, and \({\bf F}_{\kappa_{p({\rm t})}}\) is in general not the gradient of a function. One then needs to work within the context of non-Euclidean geometry to define these concepts [see 38, 39, for a more detailed discussion].
In general G is not the gradient of a mapping but is a mapping from appropriate tangent space at \({\bf X} \in \kappa _{R}(B)\) to the tangent space at \({\bf x} \in \kappa_{t}(B). \)
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Rajagopal, K.R. Modeling a class of geological materials. Int J Adv Eng Sci Appl Math 3, 2–13 (2011). https://doi.org/10.1007/s12572-011-0029-8
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DOI: https://doi.org/10.1007/s12572-011-0029-8