Biomedical Engineering Reference
In-Depth Information
(
x
,
y
,
z
). We will start with the compliance form of Hooke's law:
ν
1
E
x
yx
E
y
−
ν
zx
E
z
ε
x
ε
−
0
0
0
σ
x
σ
ν
xy
E
x
ν
zy
E
z
1
E
y
−
−
0
0
0
y
y
ν
yz
E
y
−
ν
xz
E
x
1
E
z
ε
−
0
0
0
σ
z
z
=
·
(6.41)
1
G
xy
2
ε
0
0
0
0
0
σ
xy
xy
1
G
yz
ε
0
0
0
0
0
σ
2
yz
yz
1
G
xz
0
0
0
0
0
2
ε
σ
xz
xz
Here,
E
x
,
E
y
,and
E
z
are Young's moduli for the principal axes, and
ν
ij
are Poisson's
ratios for these axes. The Poisson's ratio
ν
xy
, for example, represents the ratio
between strains
ε
x
and
ε
y
when the material is subjected to uniaxial stress in the
y-direction, that is
σ
y
=
σ
=−
ε
x
ν
(6.42)
xy
ε
y
The coefficients
G
xy
,
G
yz
,and
G
xz
represent the shear moduli for the
x
-
y
,
y
-
z
,and
x
-
z
planes, respectively. The compliance matrix is symmetric, and therefore the
following relations must be satisfied (observe the fact that Poisson's ratios are not
symmetric):
ν
ν
,
ν
ν
,
ν
ν
xy
E
x
yx
E
y
yz
E
y
zy
E
z
xz
E
x
zx
E
z
=
=
=
(6.43)
From the previous equation, it follows that the compliance matrix comprises nine
independent material constants and that the following equation holds
7)
:
ν
ν
ν
=
ν
ν
ν
(6.44)
xy
yz
zx
xz
yx
zy
The stiffness form of Hooke's law is obtained by inverting the compliance matrix as
1
−
ν
yz
ν
zy
E
y
E
z
D
ν
yx
+
ν
zx
ν
yz
E
y
E
z
D
ν
zx
+
ν
yx
ν
zy
E
y
E
z
D
000
σ
ε
x
x
ν
yx
+
ν
zx
ν
yz
E
y
E
z
D
ν
zy
+
ν
xy
ν
zx
E
x
E
z
D
000
1
−
ν
xz
ν
zx
E
x
E
z
D
σ
ε
y
y
ν
zx
+
ν
yx
ν
zy
E
y
E
z
D
ν
zy
+
ν
xy
ν
zx
E
x
E
z
D
1
−
ν
xy
ν
yx
E
x
E
y
D
000
σ
ε
z
z
=
·
(6.45)
σ
2
ε
0
0
0
G
xy
00
xy
xy
σ
2
ε
0
0
0
0
G
yz
0
yz
σ
xz
yz
2
ε
xz
0
0
0
0
0
G
xz
where
1
−
ν
ν
−
ν
ν
−
ν
ν
−
2
ν
ν
ν
xy
yx
xz
zx
yz
zy
yx
xz
zy
=
D
(6.46)
E
x
E
y
E
z
is the determinant of the compliance matrix in Eq. (6.41) multiplied by
G
xy
G
yz
G
xz
.
7)
Used to simplify the expression for the de-
terminant of Eq. (6.41).
