Biomedical Engineering Reference
In-Depth Information
A
F
|
l
|
F
-
A
⊥
Tensile
stress
=
l
o
A
l
o
Δ
l
l
o
||
Tensile
strain
=
A
||
=
Shear stress
l
Δ
l
l
=
Shear strain
F
⊥
Fig. 3.1.2-5 Shear stress and shear strain.
Fig. 3.1.2-4 Tensile stress and tensile strain.
Elastic constants
Tension and compression
By using these definitions of stress and strain, Hooke's
law can be expressed in quantitative terms:
In tension and compression the area supporting the load
is perpendicular to the loading direction (tensile stress),
and the change in length is parallel to the original length
(tensile strain).
If weights are used to provide the applied load, the
stress is calculated by adding up the total number of
pounds-force (lb) or newtons (N) used and dividing by the
perpendicular cross-sectional area. For regular specimen
geometries such as cyclindrical rods or rectangular bars,
a measuring instrument, such as a micrometer, is used to
determine the dimensions. The units of stress are pounds
per inch squared (psi) or newtons per meter squared
(N/m
2
). The N/m
2
unit is also known as the pascal (Pa).
The measurement of strain is achieved, in the simplest
case, by applying reference marks to the specimen and
measuring the distance between with calipers. This is the
original length,
l
o
. A load is then applied, and the distance
between marks is measured again to determine the final
length,
l
f
. The strain, 3, is then calculated by:
s
¼ E
3
;
tension or compression
;
(3.1.2-2a)
s
¼ G
g
;
shear
:
(3.1.2-2b)
E
and
G
are proportionality constants that may be
likened to spring constants. The tensile constant,
E
, is the
tensile (or (Young's) modulus and
G
is the shear modu-
lus. These moduli are also the slopes of the elastic portion
of the stress versus strain curve (
Fig. 3.1.2-6
). Since all
geometric influences have been removed,
E
and
G
rep-
resent inherent properties of the material. These two
moduli are direct macroscopic manifestations of the
strengths of the interatomic bonds. Elastic strain is
achieved by actually increasing the interatomic distances
in the crystal (i.e., stretching the bonds). For materials
with strong bonds (e.g., diamond, Al
2
O
3
, tungsten), the
moduli are high and a given stress produces only a small
strain. For materials with weaker bonds (e.g., polymers
and gold), the moduli are lower (
Hummel, 1997
). The
tensile elastic moduli for some important biomaterials
are presented in
Table 3.1.2-2
.
3
¼
l
f
l
o
l
o
¼
Dl
l
o
:
(3.1.2.1)
This is essentially the technique used for flexible
materials like rubbers, polymers, and soft tissues. For
stiff materials like metals, ceramics, and bone, the de-
flections are so small that a more sensitive measuring
method is needed (i.e., the electrical resistance strain
gage).
Isotropy
The two constants,
E
and G, are all that are needed to
fully characterize the stiffness of an isotropic material
(i.e., a material whose properties are the same in all
directions).
Single crystals are anisotropic (not isotropic) because the
stiffness varies as the orientation of applied force changes
relative to the interatomic bond directions in the crystal. In
polycrystalline materials (e.g., most metallic and ceramic
specimens), a great multitude of grains (crystallites) are ag-
gregated with multiply distributed orientations. On the
Shear
For cases of shear, the applied load is parallel to the area
supporting it (shear stress, s), and the dimensional
change is perpendicular to the reference dimension
(shear strain, g
)
(
Fig. 3.1.2-5
).
