Biological Systems: Multiscale Modeling Based on Mixture Theory - Multiscale Modeling in Biomechanics and Mechanobiology - page 270

S N L

ˁ =− ˁ ˁ = .

and 0 (11.74)

Next, the momentum of the constituent ˆ ʱ is defined by

m ʱ = ˁ ʱ v d v. (11.75)

ʱ

B ʱ

By including m ʱ in the total change of linear momentum in B ʱ and denoting the

p ʱ , the standard momentum
equation (Cauchy equation of motion) for each constituent becomes

interaction of the momentum of the constituents ˆ ʱ by ˆ

T ʱ + ˁ ʱ ( p ʱ − ˁ ʱ v ʱ =

∇· −

a ʱ ) + ˆ

,

b 0 (11.76)

ˁ ʱ v ʱ where the expression represents the exchange of linear momentum through

ˁ ʱ .Theterm T ʱ ˁ ʱ b

the density supply denotes the partial Cauchy stress tensor,

p ʱ , where

specifies the volume force. In addition, the terms ˆ ʱ = S , L , N , are required

to satisfy the constraint condition

p S p L p N ˆ + ˆ + ˆ = 0 . (11.77)

In the case of either bone remodeling or wound healing, the velocity field is
nearly in steady state. Thus, the acceleration can be neglected by setting a

ʱ =

0. The

resulting system of equations can then be written

T ʱ + ˁ ʱ b p ʱ = ˁ ʱ v

∇· + ˆ

ʱ .

(11.78)

The second law of thermodynamics (entropy inequality) provides expressions for
the stresses in the solid and fluid phases that are dependent on the displacements
and the seepage velocity, respectively. The seepage velocity is a relative velocity
between the liquid and solid phases, which are often obtained from explicit Darcy
velocity expressions for flow through a porous medium (solid phase). Various types
of material behavior can be described in terms of principal invariants of the structural
tensor M and the right Cauchy-Green Tensor C S , where

F S F S ,

= ↗

C S =

M A A and (11.79)

and A is the preferred direction inside the material and F S is the deformation gradient
for a solid undergoing finite deformations. The expressions for the stress in the solid
are dependent on the deformation gradient and consequently the displacements of the
solid. Summation of the momentum conservation equations provides the equation
for the solid displacements. Mass conservation equations, with incorporation of the
saturation condition, provide the equation for interstitial pressure. In addition, the
mass conservation equations for each species provide the equations for the evolution
of volume fractions.