ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

system, it is convenient to write the conservation of mass employing the total derivative
with respect to the motion of the solid phase for the continua:
S
Dt (nS )+ nSdiv(vS ) = 0 (6)
S
Dt (nF )+ div(nFvF )− vSgrad(nF ) = 0 . (7)
Using the Green’s identity on the last term of the left hand side of (7) and summing up
the two conservations of mass we obtain
div(n Fw F )+ div(v S ) = 0 (8)
that, exploiting the identity J / J = div(vS ), can be written as evolution equation for the
determinant of the deformation gradient
( J) * + J * div(n F w F ) = 0 . (9)
From the balance of momentum of the fluid and neglecting elastic stress, we obtain the
Darcy law
nFwF = − k
grad(p) (10)
γ
whereγ is the viscosity and k is the permeability. The global momentum equation can
be obtained summing up the (5) for the two phases.
In order to simulate our mechanical system in a total Lagrangian formulation, the
equations have to be written with respect to the material coordinates so that the
complete system reads:
−Div(FS − JpF −T ) = 0
J − Div((J − n S0 ) k
γ C−1 ∇p) = 0 (11)
where S is the second Piola-Kirchhoff tensor of the solid phase.
3.1 Constitutive laws
A Neo-Hookean formulation is assumed to interpret the mechanical response of solid
phase. According to this choice the second Piola-Kirchhoff tensor has the form
S = µ tr(C)
(I − 2/3
J 3 C−1 )+ 2D1J(J −1)C −1
where D 1 can be related to the volumetric stiffness, while µ is associated to the tissue
shear stiffness.
The permeability k follows the exponential law of the fluid volume fraction proposed
in [2].