Biomedical Engineering Reference
In-Depth Information
N
(
x
)
dN
dx
1
N
(
x
+
D
x
)
N
x
x
x
+
D
x
Figure 6.2
First-order derivative of a function
N
(
x
).
dN
dx
N
(
x
+
x
)
≈
N
(
x
)
+
x
.
(6.4)
The result of this transformation is that the force equilibrium relation Eq. (
6.2
)
may be written as
dN
dx
+
q
=
0.
(6.5)
If the load per unit length
q
equals zero, then the equilibrium equation reduces to
dN
dx
=
0,
(6.6)
which means that the force
N
is constant throughout the bar. It actually has to be
equal to the force
F
applied to the right end of the bar see Fig.
6.1
(a). Conse-
quently, if the slice of Fig.
6.1
(b) is considered the force
N
(
x
) equals
N
(
x
x
).
In other words, the force in the bar can only be non-constant if
q
can be neglected.
+
6.3
Stress and strain
The equilibrium equation (
6.5
) derived in the previous section does not give infor-
mation about the deformation of the bar. For this purpose a relation between force
and strain or strain rate must be defined, similar to the force-strain relation for an
elastic spring, discussed in Chapter
4
. For continuous media it is more appropriate
to formulate a relation between force per unit area (stress) and a deformation mea-
sure, such as strain or strain rate. The concepts of stress and strain in continuous
media are introduced in this section.
In the one-dimensional case discussed in this chapter, the force
N
acting on
a cross section of the bar is assumed to be homogeneously distributed over the
