Biomedical Engineering Reference
In-Depth Information
The left Cauchy Green tensor
B
is, based on the transformation relation Eq.
(
10.32
), called
objective
. In calculating the stretch ratio
λ
with the earlier derived
Eulerian description, the result in the direction
e
in the current state (with asso-
ciated tensor
B
) has to be identical to the result in the direction
e
∗
e
in
the virtual (extra rotated and translated) current state (using tensor
B
∗
). That this
demand is satisfied can be verified easily.
Because the Cauchy stress tensor
=
P
·
σ
transforms in a similar way as tensor
B
(see
Section
9.7
) the tensor
σ
is also objective. The deformation tensor
F
is neither
invariant, nor objective.
10.4
Strain measures and strain tensors and matrices
In the preceding sections the stretch ratio
is considered to be a measure for
the relative length change of a material line segment in the transition from the
reference configuration to the current configuration. If there is no deformation
λ
=
1. Often it is more convenient to introduce a variable equal to zero when there
is no deformation: the strain. In the present section several different, generally
accepted strain measures are treated.
In the previous section it was found, using the Lagrangian description:
λ
2
2
(
e
0
)
=
e
0
·
F
T
λ
=
λ
·
F
·
e
0
=
e
0
·
C
·
e
0
.
(10.33)
Coupled to this relation, the Green Lagrange strain
ε
GL
is defined by:
F
T
·
F
−
I
2
=
λ
−
1
1
2
e
0
·
ε
GL
=
ε
GL
(
e
0
)
=
·
e
0
2
1
2
=
e
0
·
(
C
−
I
)
·
e
0
.
(10.34)
This result invites us to introduce the symmetrical
Green Lagrange strain tensor
E
according to:
2
F
T
·
F
−
I
1
1
2
(
C
−
I
)
,
E
=
=
(10.35)
which implies
ε
GL
=
ε
GL
(
e
0
)
=
e
0
·
E
·
e
0
.
(10.36)
When using matrix notation, Eq. (
10.35
) can be formulated as
2
F
T
F
−
I
2
C
−
I
,
1
1
E
=
=
(10.37)
implying
T
ε
GL
=
ε
GL
(
∼
0
)
=
∼
0
E
∼
0
.
(10.38)
