Biomedical Engineering Reference
In-Depth Information
Taking into account that
ε
(
τ
=
0)
=
0
it appears that the 'internal elastic energy
per unit volume'
int
0
at time
t
is fully determined by the components of the
strain tensor at that specific time
t
; this means that the indication
t
in this case is
redundant and without any problem it can be written:
1
1
2
K
(
2
K
tr
2
(
d
d
)
v
)
2
d
d
) .
int
0
(
ε
)
=
ε
)
+
G
tr(
ε
·
ε
=
ε
+
G
tr(
ε
·
ε
(12.34)
It can be established that in this energy density the hydrostatic (volumetric)
part and the deviatoric part are separately identifiable; just like in Hooke's law
there is a decoupling. Both parts deliver an always positive (better: non-negative)
contribution to the energy density, for every arbitrary
ε
.
Using Hooke's law, the energy density
int
0
(
ε
), according to the above given
equation, can be transformed to
int
0
(
σ
) resulting in
1
2
1
1
4
G
tr(
K
p
2
d
d
) .
int
0
(
σ
)
=
+
σ
·
σ
(12.35)
Finally, it should be remarked that from the previous it can be concluded that a
cyclic process in the deformation or in the stress will always be energetically neu-
tral. This means that the 'postulated' constitutive equation (Hooke's law) indeed
gives a correct description of elastic material behaviour. Every cyclic process is
reversible. No energy is dissipated or released. If we compare the second term
on the right-hand side of Eq. (
12.35
), representing the 'distortion energy density'
with Eq. (
8.75
), defining the von Mises stress, it is clear that both are related. In
other words, if we would like to define some threshold based on the maximum
amount of distortional energy that can be stored in a material before it becomes
damaged, the von Mises stress can be used for this purpose.
12.4
Elastic behaviour at large deformations and/or large rotations
In this section the attention is focussed on constitutive equations for elastic, com-
pressible, isotropic material behaviour at large deformations. The formulation in
Sections
12.2
and
12.3
is no longer valid, because under those circumstances the
linear strain tensor
ε
cannot be used.
The Cauchy stress tensor
σ
is objective (
σ
transforms in a very specific way
when an extra rigid body rotation is enforced, see Section
9.7
). This implies that
σ
can certainly not be coupled to an invariant measure for the strain, such as the
right Cauchy Green deformation tensor
C
or the related Green Lagrange strain
tensor
E
). But it is allowed to relate
to the objective left Cauchy Green tensor
B
, or the associated strain tensors
A
(Almansi Euler) and
σ
ε
F
(Finger).
