Note on the Notion of Incompressibility in Thermodynamic Theories of Porous and Granular Materials

Abstract

We present a simple two‐component model of a porous material based on the constraint assumption that the so‐called true components are incompressible. In my previous work on this subject [1] I pointed out that many such models are not thermodynamically admissible. Namely the second law of thermodynamics led to the conclusion that an additional field of reaction force on the constraint cannot be introduced, and, consequently, the set of field equations was overdetermined. However I speculated as well that an extension of the set of variables may lead to thermodynamic admissibility. Indeed an example presented in this paper supports this speculation. According to results of this work it seems to be necessary to introduce higher gradients to multicomponent models with constraints in order to satisfy the second law of thermodynamics. The latter is exploited by means of Lagrange multipliers eliminating constraints from the entropy inequality. This procedure is known in the literature as the method of Lagrange‐Liu multipliers (see the original work of I‐Shih Liu [2]).

Towards an analytical model of soft biological tissues

“…Wilson et al (2007) included effects of porosity for a specific elastic potential, and we would like to solve this topic for general elastic potentials by treating porosity as a constitutive variable (Wilmanski, 2001) and by using recent developments in multiscale descriptions of transport phenomena in biological tissues (Grillo et al, 2007).…”

On the formulation of boundary value problems with the incompressible constituents constraint in finite deformation poroelasticity

“…Now, by the standard argument of the theory of the Lagrange multiplier, the arbitrariness of p allows one to proceed in an operational fashion as if D .s/ F=Dt, r˝v f and w can be varied independently. ‡ ‡ Thus each of the bracketed terms in Equation (61) must be nonnegative, whereupon it follows that [4,18,33,34], the ideas embodied in the constraint follow from an assumption that each constituent is individually incompressible and that the mixing of the constituents takes place without the creation of voids. Mathematical characterization of each of these statements gives rise to additional balance equations for additional field variables and hence the possibility of numerous Lagrange multipliers if all of these equations enter into the treatment.…”

“…‡‡ ‡‡ Because the different forms in Equation correspond to a single equation, it follows that it is sufficient to introduce a single Lagrange multiplier into the treatment. As discussed in the Introduction and elaborated upon, for example, in , the ideas embodied in the constraint follow from an assumption that each constituent is individually incompressible and that the mixing of the constituents takes place without the creation of voids. Mathematical characterization of each of these statements gives rise to additional balance equations for additional field variables and hence the possibility of numerous Lagrange multipliers if all of these equations enter into the treatment.…”