Biomedical Engineering Reference
In-Depth Information
1
E
(
ε
xx
=
σ
xx
−
νσ
yy
−
νσ
zz
)
1
E
(
−
νσ
xx
+
ε
yy
=
σ
yy
−
νσ
zz
)
1
E
(
−
νσ
xx
−
νσ
yy
+
ε
zz
=
σ
zz
) .
(12.23)
The strain in a certain direction is directly coupled to the stress in that direction
via Young's modulus. The stresses in the other directions cause a transverse strain.
The reverse equations can also be derived:
(1
+
ν
)(1
−
2
ν
)
(1
νε
zz
E
σ
xx
=
−
ν
)
ε
xx
+
νε
yy
+
)
νε
zz
E
σ
yy
=
νε
xx
+
(1
−
ν
)
ε
yy
+
(1
+
ν
)(1
−
2
ν
(1
+
ν
)(1
−
2
ν
)
ε
zz
.
E
σ
zz
=
νε
xx
+
νε
yy
+
(1
−
ν
)
(12.24)
For the shear strains, the off-diagonal components of the matrix
ε
, it follows from
Eq. (
12.16
):
1
2
G
σ
xy
=
1
2
G
σ
yx
ε
xy
=
ε
yx
=
1
2
G
σ
xz
=
1
2
G
σ
zx
ε
xz
=
ε
zx
=
1
2
G
σ
yz
=
1
2
G
σ
zy
.
ε
zy
=
ε
yz
=
(12.25)
It is clear that shear strains are coupled directly to shear stresses (due to the
assumption of isotropy). The inverse relations are trivial. If required, the shear
modulus can be written as a function of the Young's modulus
E
and the Poisson's
ratio
by means of the following equation (follows from Eqs. (
12.20
) and (
12.21
)
by eliminating
K
):
ν
E
2( 1
+
ν
)
.
G
=
(12.26)
12.3
The stored internal energy
It is interesting to study the stored internal energy during deformation of a material
that is described by means of Hooke's law for linearly elastic behaviour. In Section
11.4
the balance of power was derived for an infinitesimally small element with
reference volume
dV
0
and current volume
dV
JdV
0
. By integrating the inter-
nally stored power
dP
int
over the relevant time domain 0
=
≤
τ
≤
t
, the process
