Biomedical Engineering Reference
In-Depth Information
The deviatoric strain-displacement matrix [
B
D
] is given by:
ε
=
B
D
{
d
}
in which
{
d
}
is the vector of nodal displacements. In element formulation, material con-
stitutive law has to be used to create the relation between stress increment and strain
increment. The constitutive only reflects the stress increment due to straining. In this case,
however, the Cauchy stress cannot be used because it is affected by the rigid body rotation
and is not objective (not frame invariant). Therefore, an objective stress is needed in order
that it can be applied in constitutive law. The Jaumann - Zaremba rate of the Cauchy stress,
expressed by McMeeking and Rice [17], is one of them and is defined as follows [18]:
σ
ij
=
σ
ij
−
σ
ik
ω
jk
−
σ
jk
ω
ik
(6.23)
is
the spin tensor and
σ
ij
is the time rate of the Cauchy stress. Alternatively, using the
constitutive law, the stress change due to straining can be expressed as:
σ
ij
2
∂v
i
∂v
j
∂x
i
1
σ
ij
where
is the Jaumann - Zaremba rate of the Cauchy stress,
ω
=
−
∂x
j
=
C
ijkl
d
kl
(6.24)
where
C
ijkl
is the material constitutive tensor and
d
kl
is the rate of deformation tensor,
given by:
∂v
i
∂x
j
+
∂v
j
∂x
i
1
2
d
ij
=
=
d
ji
(6.25)
in which
v
i
is the velocity. By substituting Equation 6.24 into Equation 6.23 , the Cauchy
stress tensor can be obtained as:
σ
ij
=
C
ijkl
d
kl
+
σ
ik
ω
jk
+
σ
jk
ω
ik
(6.26)
In the present study, in order to model the portion of tissue that is being held by
an MIS grasper, a cube containing a stiffer object is considered for the geometry under
study. The bulk soft tissue, as well as the tumor, was modeled using a 10-node tetrahedral
Solid187 element which has a quadratic displacement behavior and is well suited to
modeling irregular meshes. Each node has three DOF; translations in the nodal
x
,
y
,and
z
directions. This element has mixed formulation capability for simulating deformation
of nearly incompressible elastoplastic materials, and fully incompressible hyperelastic
materials. Although soft tissue is assumed to be hyperelastic, tumor is considered to be
isotropic. The strain energy function (
ψ
) for soft material was selected to be the three-
term Mooney - Rivlin model [19] due to its satisfactory performance in compression states
of stress. The three coefficients of the model for incompressible isotropic hyperelastic
materials were obtained from the implemented experimental stress - strain data using the
least squares optimization method.
6.4 The Parametric Study
As mentioned in Chapter 5, a number of parameters have a significant effect on the
output of sensors positioned on the contact surface. In general, some combinations of such
parameters produce similar stress distributions on the contact surface. Therefore, some
