Biomedical Engineering Reference
In-Depth Information
fully incompressible deformation, such as for fully incompressible hyperelastic materi-
als. In the mixed u-p formulation, in which the hydrostatic pressure P is interpolated on
the element level, and solved on the global level independently in the same way as dis-
placement, these difficulties are eliminated. In this method, the contribution of hydrostatic
pressure, referred to as 'volumetric response,' is separated from the 'deviatoric response.'
Therefore, the stress, for instance has to be updated by:
σ ij
P
= σ ij δ ij
(6.15)
where the prime indicates the deviatoric component of the Cauchy stress tensor. Alterna-
tively, the deviatoric component of the deformation tensor e ij can be expressed as:
1
3 δ ij e v
ε ij
= ε ij
(6.16)
where e v = δ ij e ij = e ii and:
∂u i
∂x j +
∂u j
∂x i
1
2
e ij
=
(6.17)
in which u i and x i are the displacement and coordinate in the current configuration,
respectively. In addition, for a fully incompressible hyperelastic material, the volume
constraint is the incompressible condition [13 - 15]:
1 J
= 0
(6.18)
where, J
is the determinant of deformation gradient
tensor and V 0 is the original volume. The constraint Equation 6.18 is introduced to the
internal virtual work by the Lagrangian multiplier P . Finally, differentiating the aug-
mented internal virtual work gives the stiffness matrix in the form:
K uu K up
K pu K pp
=| F ij |= dv/dV 0 ,inwhich
| F ij |
u
P
R
0
=
(6.19)
where u and P are displacement and hydrostatic pressure increments, respectively.
The stiffness submatrices can be obtained from the following equations [16]:
B D T C B D dV
K uu =
(6.20)
V
B V T N p dV
K up = K up =−
(6.21)
V
N p T 1
k N p dV
K pp =−
(6.22)
V
where [ C ] is the stress-strain matrix for the deviatoric stress and strain components. The
matrix [ N p ] can be obtained from the additional interpolation introduced for hydrostatic
pressure:
p = N p P .
 
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