Biomedical Engineering Reference
In-Depth Information
showing how strain develops with stress in the elastic as well as plastic range. Such diagrams can be
obtained from indentation tests. As the stress field under indenter is nonhomogeneous, it must be char-
acterized by some representative stress and strain,
σ
rep
and
rep
. The mean contact pressure
p
m
is very
suitable for
σ
rep
, while the expression for representative strain will depend also on the indenter shape.
A pointed indenter,
Figure 17.3A
, is not sufficient for the construction of
σ
(
) diagrams, as the
distribution of stresses beneath it is similar for any depth, and only one value of
p
m
is obtained for a
homogeneous material. The representative strain was defined by Tabor
[15]
as
rep
k
tan
β
, where
k
is a constant and
β
is the angle between the indenter and specimen surface (
Figure 17.3A
). Tabor
has recommended
k
0.2, based on the comparison of indentation and tensile tests on metals. For
Berkovich or Vickers indenters,
β
19.7°, so that the measurements with such indenter correspond
to a rather high characteristic strain,
rep
7%.
Under a spherical indenter (
Figure 17.3B
), the mean contact pressure and strains increase with the
depth of penetration and can be used for the construction of stress-strain (
p
m
-
rep
) curves
[15-19]
.
The representative strain is usually expressed as the ratio of contact radius
a
and indenter radius
R
;
rep
k(a
/
R)
; often,
k
0.2 is assumed. For small depths:
F
Rh
F
Rh
p
(17.11)
m
2
2
π
π
(
2
h
)
c
c
c
2
(17.12)
a
2
Rh
h
2
Rh
c
c
c
For elastic deformations, the contact depth is obtained from indenter displacement as
h
c
h
/2.
If the penetration is larger, or if the deformations are elastic-plastic, a more complex procedure
must be used. First, the dependence of contact depth on the indenter displacement,
h
c
(
h
), is obtained
from Eq. (17.2). The contact stiffness
S
(
h
) for these calculations can be measured either directly using
the CSM mode or calculated from the unloading curves for several loads and depths. Then, the con-
tact radius
a
is assigned to the individual depths using the indenter calibration function
a
(
h
c
). The
mean contact pressure and representative strain are then calculated as:
2
p
P A
/
P
/(
π
a
),
ε
0 2
.
a R
/
(17.13)
m
rep
with all quantities expressed as functions of contact depth
h
c
or penetration
h
. Finally, the stress-
strain diagram is obtained by plotting the mean contact pressure
p
m
as a function of representative
strain
rep
(
Figure 17.4
). However, if the indenter is not perfectly spherical and its radius
R
varies, it is
better to use the same definition as for conical indenters,
rep
0.2 tan
β
(the difference between tan
β
and
a
/
R
sin
β
is negligible for small
β
). For the known calibration function
a
f
(
h
c
), tan
β
can be
obtained
[14]
as the reciprocal of its derivative with respect to
h
c
, tan
β
d
h
c
/d
a
. Examples of stress-
strain curves for enamel and other materials used in dentistry are shown in Refs
[6,16,19]
.
17.3.2
Yield Stress
For relatively low stresses,
p
m
is directly proportional to
rep
. As soon as irreversible deformations
appear, the representative strain grows faster. The onset of plastic flow can thus be identified as the
point where the
p
m
rep
curve starts deviating from linearity
[14-18]
(
Figure 17.4
). The correspond-
ing mean contact pressure,
p
m,Y
, is sometimes denoted as yield stress. However, the actual yield stress
σ
Y
is different. Plastic deforming starts
[5,18]
at
p
m
1.1
σ
Y
, in a very small volume beneath the
