Biomedical Engineering Reference
In-Depth Information
configuration is reduced to a time and mass invariant equilibrium that can be
expressed by the divergence of the Cauchy stress tensor.
di
v
˃
=
0
(1)
This expression is under-constrained and additional definitions are required. First of
all, the continuity of the field quantities is postulated, meaning that all deformations
are physically objective and infinitesimal small material particles are not allowed to
penetrate each other or fluctuate. This uniqueness is obtained by the definition of the
deformation tensor.
dx
dX
F
=
(2)
In terms of the physical objectivity, the material behavior of elastic bodies (also
denoted as Cauchy elasticity) now demand tensor compatibility of stress and defor-
mation.
˃
=
f
(
F
)
(3)
Such compatibility is given by the Rivlin-Ericksen theorem for isotropic behavior
[
11
].
a
1
F
T
F
F
T
F
2
˃
=
a
0
+
+
a
2
(
)
(4)
Thereby, the coefficients
a
0
,
a
1
and
a
2
are arbitrary scalar functions of the invariants
of the deformation tensor. Usually, this dependence is formulated in relation to the
strain energy density
. Thus, the Green or hyper-elastic material behavior can be
defined by a pure scalar function
ˈ
ˈ
=
f
(
I
1
,
I
2
,
J
)
, with the constitutive law:
J
−
1
d
d
F
F
T
˃
=
(5)
However, for this assumption isotropy is presumed, meaning that not every material
can be modeled by this constitutive law. In order to consider anisotropic effects as
well, the constitutive law is expressed by using the geometric linearized form of the
right Cauchy-Green tensor.
1
2
(
F
T
F
ʵ
=
−
1
)
(6)
This modified tensor can be used to derive a simplified relation between stress and
deformation. In analogy to (
4
) an equalization of
f
(
F
)
∼
f
(ʵ)
leads to Hooke's law
in continuum mechanics
1
2
(ʻ
˃
=
tr
ʵ
)
+
2
G
ʵ
(7)
