Biomedical Engineering Reference
In-Depth Information
Examples of transversely isotropic materials include hexagonal close-packed crys-
tals, some piezoelectric materials (e.g. PZT-4, barium titanate), and fiber-reinforced
composites, where all fibers are in parallel.
By convention, the five elastic constants in transverse isotropic constitutive
equations are Young's modulus and Poisson's ratio in the
x
-
y
symmetry plane
(index 'p'),
E
p
and
ν
p
, Young's modulus and Poisson's ratio in the
z
-direction,
E
pz
and
ν
pz
, and the shear modulus in the
z
-direction,
G
zp
.
The compliance form of Hooke's law takes the form
ν
ν
p
E
p
zp
E
z
1
E
p
−
−
0
0
0
ε
σ
x
ε
x
σ
ν
p
E
p
ν
zp
E
z
1
E
p
−
−
0
0
0
y
y
ν
pz
E
p
ν
pz
E
p
1
E
z
−
−
0
0
0
ε
σ
z
2
ε
xy
2
ε
yz
2
z
σ
xy
σ
yz
σ
=
·
(6.52)
+
ν
p
E
p
1
0
0
0
0
0
1
G
pz
0
0
0
0
0
1
G
pz
ε
0
0
0
0
0
xz
xz
where Poisson's ratios are not symmetric, but satisfy
ν
ν
pz
E
p
=
zp
E
z
(6.53)
The stiffness form of Hooke's law is obtained by inverting the compliance matrix
as
1
−
ν
ν
ν
+
ν
ν
ν
+
ν
ν
zp
E
p
E
z
D
pz
p
pz
E
p
E
z
D
zp
zp
zp
E
p
E
z
D
000
p
σ
ε
x
x
ε
ν
p
+
ν
zp
ν
pz
E
p
E
z
D
1
−
ν
pz
ν
zp
E
p
E
z
D
ν
zp
+
ν
p
ν
zp
E
p
E
z
D
000
σ
y
y
p
E
p
D
1
−
ν
σ
ν
zp
+
ν
p
ν
zp
E
p
E
z
D
ν
zp
+
ν
p
ν
zp
E
p
E
z
D
ε
000
z
z
=
·
(6.54)
σ
ε
2
E
p
xy
xy
0
0
0
+
ν
p
00
1
σ
2
ε
yz
yz
0
0
0
0
G
pz
0
σ
2
ε
xz
xz
0
0
0
0
0
G
pz
where
(1
−
ν
p
)(1
−
ν
−
2
ν
ν
zp
)
p
pz
=
D
(6.55)
E
p
E
z
Table 6.7 summarizes the different formulations of Hooke's law and the assigned
number of independent variables.
6.3.5
Plastic Behavior, Failure, and Limit Surface
The three essential ingredients of plastic analysis are the yield criterion, the
flow rule, and the hardening rule, cf. [19]. The yield criterion relates the state
of stress to the onset of yielding. The flow rule relates the state of stress
σ
ij
to
p
ij
when an increment of plastic
the corresponding increments of plastic strain d
ε
