Biomedical Engineering Reference
In-Depth Information
In the following an interpretation of Hooke's law will be given. For this purpose
we focus on the matrix formulations
σ
=
σ
(
ε
) and
ε
=
ε
(
σ
). The symmetrical
matrices
σ
and
ε
are composed according to:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
σ
xx
σ
xy
σ
xz
ε
xx
ε
xy
ε
xz
σ
=
σ
yx
σ
yy
σ
yz
,
ε
=
ε
yx
ε
yy
ε
yz
.
(12.17)
σ
zx
σ
zy
σ
zz
ε
zx
ε
zy
ε
zz
Using
ε
=
ε
(
σ
) it follows from Eq. (
12.16
) for the diagonal components (the
strains in the
x
-,
y
- and
z
-directions):
1
9
K
−
(
σ
xx
+
σ
yy
+
σ
zz
)
+
1
6
G
1
2
G
σ
xx
ε
xx
=
1
9
K
−
(
σ
xx
+
σ
yy
+
σ
zz
)
+
1
6
G
1
2
G
σ
yy
ε
yy
=
1
9
K
−
(
σ
xx
+
σ
yy
+
σ
zz
)
+
1
6
G
1
2
G
σ
zz
.
ε
zz
=
(12.18)
Eq. (
12.18
) shows that the strains in the
x
-,
y
- and
z
-directions are determined
solely by the normal stresses in the
x
-,
y
- and
z
-directions, which is a consequence
of assuming isotropy. It is common practice to use an alternative set of material
parameters, namely the Young's modulus
E
and the Poisson's ratio
ν
. The Young's
modulus follows from
1
9
K
−
3
K
+
G
9
KG
1
E
=
1
6
G
1
2
G
=
+
,
(12.19)
and therefore
9
KG
3
K
+
G
.
E
=
(12.20)
The Poisson's ratio is defined by
1
9
K
−
E
=−
1
6
G
3
K
2
G
18
KG
−
=
,
(12.21)
so
3
K
2
G
6
K
+
2
G
.
−
ν
=
(12.22)
From
K
>
0 and
G
>
0 it can easily be derived that
E
>
0 and
−
1
<ν<
0.5.
Using
E
and
ν
leads to the commonly used formulation for Hooke's equations:
