Biomedical Engineering Reference
In-Depth Information
Alternatively, Eq. (6.22) can be written in the form of Eq. (6.19) as
µ
L
−
1
µ
L
−
1
T
T
σ
=
2
ε
+
λ
1
⊗
1
(ε
1
+
e
)
=
2
ε
+
λε
11
1
+
λ
11
e
m
m
3
0
1
µ
L
−
1
=
2
ε
+
λ
(3
ε
m
)
1
(6.25)
Substituting for
(cf. Table 6.3) gives the alternative
formulation of Eq. (6.23) based on the engineering constants as
λ
and
µ
in terms of
E
and
ν
σ
x
σ
y
σ
z
σ
xy
σ
yz
σ
1
−
νν
ν
000
ε
x
ε
y
ε
z
2
ε
xy
2
ε
yz
2
ν
1
−
νν
000
E
ν
ν
1
−
ν
000
=
·
1
−
ν
2
00
2
(1
+
ν
)(1
−
2
ν
)
0
0
0
1
−
2
ν
0
0
0
0
0
2
1
−
2
ν
0
0
0
0
0
ε
xz
xz
2
(6.26)
In a simple tension test, the only nonzero stress component
σ
x
causes axial strain
ε
x
and transverse strains
ε
=
ε
z
. Thus, Eq. (6.26) yields
y
σ
x
E
=−
νσ
x
E
ε
=
and
ε
=−
νε
(6.27)
x
y
x
By using Eq. (6.27), one can calculate the elastic constants, Young's modulus
E
and Poisson's ratio
ν
, from a uniaxial tension or compression test. Introducing the
shear modulus
G
and the bulk modulus
K
according to Table 6.3 yields a further
formulation of Hooke's generalized law as
4
3
GK
−
2
3
GK
−
2
3
G
000
K
+
σ
x
σ
y
σ
z
σ
xy
σ
yz
σ
ε
x
ε
y
ε
z
2
ε
xy
2
ε
yz
2
2
3
GK
+
4
3
GK
−
2
3
G
000
K
−
2
3
GK
−
2
3
GK
+
4
3
G
000
K
−
=
·
(6.28)
0
0
0
G
00
0
0
0
0
G
0
ε
0
0
0
0
0
G
xz
xz
4
3
G
in Eq. (6.28) is also known as the
constraint modulus
[13].
Decomposing the stress and strain vectors into spherical and deviatoric components
decouples the volumetric response from the distortional response, and Hooke's law
can be expressed in terms of the volumetric and deviatoric strains in the following
form:
6)
The quantity
K
+
o
L
−
1
σ
=
σ
+
s
=
σ
1
+
s
=
3
K
ε
·
1
+
2
G
e
(6.29)
m
m
From the point of view of continuum mechanics, a state under pure shear stress
(result: shear modulus
G
) and a state under pure hydrostatic stress (result: bulk
modulus
K
) should aim to determine the elastic constants, since the constants
are independent of each other in this case. However, the experimental determi-
nation of the elastic constants is mostly based on a simple realizable tension or
6)
This form is also called the canonical form
[14].
