Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 281

for all states of the mixture. The implementation in the inequality ( 10.75 ) of the
constraint of incompressibility is accomplished by introducing the Lagrange
multipliers p (for this use of Lagrange multipliers, see Example 6.4.1 concerning
pressure as a Lagrange multiplier in incompressible fluids; for another method of
imposing the incompressibility constraint in poroelasticity, see Chap. 9 ). The
Lagrange multiplier p for incompressibility is introduced by multiplying the fol-
lowing form of ( 10.76 ),

!

X

Ns
b¼ 1 f ðbÞ r

r v s þ u ðbÞ þ u ðbÞ rf ðbÞ ¼ 0 ; (10.81)

by p and adding the result to ( 10.75 ), thus ( 10.75 ) becomes

X

N
a¼ 1 er½r ðbÞ u ðbÞ þ

D s D s

D t r y

D t

1
y

T eff r q ry þ : D ðsÞ

X

Ns
b¼ 1 f½

þ T ðbÞ þ f ðbÞ p 1 þ r ðbÞ C ðbÞ 1 : ½r u ðbÞ g

X

Ns
b¼ 1 f

þ u ðbÞ ½r T ðbÞ þ p rf ðbÞ rr ðbÞ C ðbÞ g 0 ; (10.82)

where the effective stress has been introduced:

T eff ¼ T þ p 1 : (10.83)

The development of four, relatively simple, constitutive relations was described
in Chap. 6 . The process of developing constitutive relations was described in
Chap. 3 . The steps in this process consisted of the constitutive idea and the
restrictions associated with the notions of localization, invariance under rigid object
motions, determinism, coordinate invariance, and material symmetry. In that devel-
opment, restrictions on the coefficients representing material properties were devel-
oped without recourse to the second law of thermodynamics; ad hoc arguments
equivalent to those obtainable from the second law were employed. The present
development proceeds by making constitutive assumptions that are consistent with
the restrictions of localization, invariance under rigid object motions, determinism
and coordinate invariance by assuming a general form for their functional depen-
dence on localized tensorial variables that are invariant under rigid object motions,
and that have no dependence upon time except for the present time. Then the entropy
inequality is employed to restrict the constitutive assumptions in the manner of
Coleman and Noll ( 1963 ) based on the philosophy described in the opening quote for
this Chapter due to Noll ( 2009 ). This is a straightforward process that often appears
complex due to the notation for the many factors that must be accounted for in a
mixture. To ease the reader into this method a simpler example is first presented.