Biomedical Engineering Reference
In-Depth Information
6.3.3
Linear Elastic Behavior: Generalized Hooke's Law for Orthotropic Materials with
Cubic Structure
If the properties of an orthotropic material are identical in all three directions (x, y,
and z), the material is said to have a cubic structure. A huge number of materials
have cubic symmetry, for example, all the face centered cubic (FCC) and body
centered cubic (BCC) metals. Thus, if we have
=
=
=
E
x
E
y
E
z
E
(6.47)
=
=
=
G
xy
G
yz
G
xz
G
(6.48)
ν
=
ν
=···=
ν
(6.49)
xy
yx
the compliance formof Hooke's law for an orthotropicmaterial with cubic structure
is reduced to
ε
x
ε
000
−
ν
1
−
ν
000
−
ν
−
ν
1000
000
G
00
0000
G
0
00000
G
1
−
ν
−
ν
σ
x
σ
y
y
ε
σ
1
E
·
z
z
=
·
(6.50)
2
ε
σ
xy
xy
2
ε
σ
yz
yz
2
ε
σ
xz
xz
where
E
,
ν
,and
G
are the three
independent
material constants. The stiffness form
of Hooke's law is obtained by inverting the compliance matrix as
E
(ν
−
1
)
E
ν
E
ν
σ
−
−
1
000
ε
x
σ
y
σ
x
ε
y
ε
2
2
2
2
ν
+
ν
−
1
2
ν
+
ν
−
1
2
ν
+
ν
−
E
(ν
−
1
)
E
ν
E
ν
−
−
1
000
2
2
2
2
ν
+
ν
−
1
2
ν
+
ν
−
1
2
ν
+
ν
−
E
ν
E
ν
E
(ν
−
1
)
−
−
1
000
z
z
2
2
2
2
ν
+
ν
−
1
2
ν
+
ν
−
1
2
ν
+
ν
−
=
·
(6.51)
σ
2
ε
0
0
0
G
00
xy
xy
σ
2
ε
0
0
0
0
G
0
yz
yz
σ
2
ε
0
0
0
0 0
G
xz
xz
6.3.4
Linear Elastic Behavior: Generalized Hooke's Law for Transverse Isotropic Materials
Special classes of orthotropic materials are those that have the same properties in
one plane (e.g., the
x
-
y
plane) and different properties in the direction normal to
this plane (e.g., the
z
-axis). This implies that the solid can be rotated with respect
to the loading direction about one axis without measurable effect on the solid's
response. Such materials are called
transverse isotropic
, and they are described
by five independent elastic constants, instead of nine for fully orthotropic ones.
