Biomedical Engineering Reference
In-Depth Information
From a mathematical perspective these are well manageable relations. The Green
Lagrange strain tensor
E
(with matrix representation
E
) is invariant for extra rigid
body motions of the current state. The components of the (symmetrical, 3
×
3)
Green Lagrange strain matrix
E
can be interpreted as follows:
•
The terms on the diagonal are the Green Lagrange strains of material line segments of
the reference configuration in the
x
-,
y
-and
z
-directions respectively (the component
on the first row in the first column is the Green Lagrange strain of a line segment that
is oriented in the
x
-direction in the reference configuration).
•
The off-diagonal terms determine the shear of the material (the component on the
first row of the second column is a measure for the change of the angle enclosed by
material line segments that are oriented in the
x
-and
y
-direction in the undeformed
configuration).
For the deformation tensor
F
and the displacement vector
u
, both applying to
the current configuration and related to the reference configuration, the following
relation was derived in Section
9.6
:
∇
0
u
T
F
=
I
+
.
(10.39)
Substitution into Eq. (
10.35
) yields
∇
0
u
∇
0
u
T
.
∇
0
u
T
∇
0
u
1
2
E
=
+
+
·
(10.40)
It can be observed that the first two terms on the right-hand side of this equation
are linear in the displacements, while the third term is non-linear (quadratic).
The linear strain
ε
lin
is defined, according to:
e
0
·
F
T
ε
lin
=
ε
lin
(
e
0
)
=
λ
−
=
·
F
·
e
0
−
1
1.
(10.41)
This expression is not easily manageable for mathematical elaborations.
At small deformations and small rotations, for which:
F
≈
I
(and therefore the
components of the tensor
∇
0
u
T
are much smaller than 1), it can be written:
ε
lin
=
ε
lin
(
e
0
)
1
+
e
0
·
F
T
·
F
−
I
·
e
0
−
=
1
F
T
·
F
−
I
1
2
e
0
·
≈
·
e
0
F
T
2
I
1
2
≈
e
0
·
+
F
−
·
e
0
.
(10.42)
The last found approximation for the linear strain is denoted by the symbol
ε
. This
strain definition is used on a broad scale. Therefore, the assumption
F
≈
I
leads
to the following, mathematically well manageable relation:
