Biomedical Engineering Reference
In-Depth Information
6.4
Elastic stress-strain relation
Recall, that the force-strain relation for an elastic spring at small, infinitesimal
displacements is given by
c
a
·
(
u
B
−
u
A
)
l
0
F
=
a
.
(6.15)
fibre strain
Here,
c
represents the stiffness of the spring, while the unit vector
a
denotes the
orientation of the spring in space. In analogy with this, the (one-dimensional)
stress-strain relation for linearly elastic materials is defined as
σ
=
E
ε
,
(6.16)
where
E
is the so-called
Young's modulus
. Using the definition of the strain in
terms of the derivative of the displacement field, this may also be written as
σ
=
E
du
dx
.
(6.17)
Example 6.1
For a given displacement field
u
(
x
), the stress field can be computed. Suppose, for
instance, that the Young's modulus is constant and that
u
is given by a polynomial
expression, say:
u
=
a
1
x
+
2
a
2
x
2
+
5
a
3
x
3
,
with
a
1
,
a
2
and
a
3
known coefficients. Then the stress will be
E
du
15
a
3
x
2
).
σ
=
dx
=
E
(
a
1
+
4
a
2
x
+
6.5
Deformation of an inhomogeneous bar
In case of a one-dimensional bar, the stress at each cross section is uniquely
defined according to
N
A
.
σ
=
(6.18)
Substitution of
N
=
A
σ
into the equilibrium equation Eq. (
6.5
) yields
d
(
A
σ
)
dx
+
q
=
0.
(6.19)
Subsequently, the stress-strain relation Eq. (
6.17
) is substituted such that the fol-
lowing second-order differential equation in terms of the displacement field
u
(
x
)
is obtained:
