Biomedical Engineering Reference
In-Depth Information
The maximum bending and shear stresses follow as
¼
M
c
max
s
b
max
ð
4
:
55
Þ
I
where
c
, the distance to the beam's neutral axis, is
h
/2 for this beam:
10
3
m
s
b
max
¼
8
:
22 Nm
½
0
:
5
ð
5
Þ
¼
197 MPa
10
10
m
4
1
:
042
2
t
b
max
¼
V
h
max
ð
4
:
56
Þ
8
I
2
10
3
m
137 N
ð
5
Þ
¼
Þ
¼
4
:
11 MPa
10
10
m
4
8
ð
1
:
042
All of these stresses are well below s
yield
¼
700 MPa for stainless steel.
4.4 VISCOELASTIC PROPERTIES
The Hookean elastic solid is a valid description of materials only within a narrow
loading range. For example, an ideal spring that relates force and elongation by a spring
constant
is invalid in nonlinear low-load and high-load regions. Further, if this spring is
coupled to a mass and set into motion, the resulting perfect harmonic oscillator will vibrate
forever, which experience shows does not occur. Missing is a description of the system's
viscous or damping properties. In this case, energy is dissipated as heat in the spring and
air friction on the moving system.
Similarly, biomaterials all display viscoelastic properties. Different models of viscoelasti-
city have been developed to characterize materials with simple constitutive equations. For
example, Figure 4.19 shows three such models that consist of a series ideal spring and
dashpot (Maxwell), a parallel spring and dashpot (Voight), and a series spring and dashpot
with a parallel spring (Kelvin). Each body contains a dashpot, which generates force in
proportion to the derivative of its elongation. Consequently, the resulting models exhibit
stress and strain properties that vary in time.
k
(a)
Maxwell
(b)
Voight
(c)
Kelvin
FIGURE 4.19
Three simple viscoelastic models: (a) the Maxwell model, (b) the Voight model, and (c) the Kelvin
body or standard linear solid model.
