Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 262

are not negligible. It is also assumed that the stress tensor associated with each
constituent is symmetric and that there are no action-at-a-distance couples, as there
would be, for example, if the material contained electric dipoles and was subjected
to an electrical field. The restrictions associated with each of these assumptions may
be removed.

10.3 The Second law of Thermodynamics

The development of constitutive equations for theories of mixtures cannot proceed
without such a formal algebraic statement of the irreversibility principle, the second
law of thermodynamics and that is why that topic is introduced in this chapter. Thus
far in the development of the subjects of this topic it has not been necessary to
formulate a specific equation restricting the direction of development or evolution
of material processes. In Chap. 6 , where the linear continuum theories of heat
conduction, elastic solids, viscous fluids, and viscoelastic materials were devel-
oped, direct physical arguments about irreversible processes could be made, with-
out invoking the second law of thermodynamics. These arguments, which were in
fact special applications of the second law, influenced only the signs of material
coefficients and stemmed from intuitively acceptable statements like “heat only
flows from hot to cold.” In this chapter a statement of the entropy inequality (the
second law) is introduced and its use is developed as a method of restricting
constitutive functions to physically acceptable processes using the arguments
introduced by Coleman and Noll ( 1963 ). The basis of that argument is summarized
in the quote from Walter Noll in 2009 repeated at the top of this Chapter.

10.4 Kinematics of Mixtures

In formulation of mixture theory-based poroelasticity presented here, the Eulerian
point used as a model of the continuum point (Fig. 8.9b) for a mixture whose
constituents are all fluids is replaced by a larger RVE introduced by Biot ( 1962 )as
the model of the poroelastic continuum point (Fig. 8.9a). Further, Biot's concept of
the RVE level representation of the fluid velocity as a function of the pore fluid
velocities in the sub RVE pores is employed. Biot related the components of the
relative microvelocity field w micro in the sub RVE pores to the RVE level fluid
velocity vector v by a linear transformation or second order tensor denoted here as J ,

w micro _ ¼ J v :

Biot noted that J depended on the coordinates in the pores and the pore geometry.
The formula (9.35) relates J to the fabric tensor of the RVE. Thus, the mixture