Biomedical Engineering Reference
In-Depth Information
β
R
β
β
β
α
h
h c
h
h c
a
a
(A)
(B)
FIGURE 17.3
Contact geometry: a - pointed indenter, b - spherical indenter.
measured value of some property is the same for any depth in a homogeneous specimen. This is use-
ful for larger depths of penetration and for the study of properties distribution in nonhomogeneous
specimens. The stresses and strains under a spherical indenter ( Figure 17.3B ) increase gradually. This
enables the study of elastic as well as elastic-plastic response, determination of yield strength, and
construction of stress-strain curves.
2
. c for Berkovich or
Vickers pyramid, and A 2 πRh c for a spherical tip and small depth of penetration. However, the tip
shape of small indenters is often not ideal, and this can cause significant errors in the measurement.
Calibration is therefore necessary, giving the mean contact radius as a function of the distance from
the tip, a a ( h c ). In such calibration, the indenter is pressed into a suitable reference material (usually
fused silica), and the load and displacement are measured for the chosen depth range, P P ( h ). Also
the contact stiffness is determined as a function of depth, S ( h ), either using CSM mode or by fitting
the unloading curves for a series of nominal loads and making the derivatives ( S d P /d h ). Then, con-
tact depths are calculated via Eq. (17.2) as a function of indenter penetration, h c ( h ). Finally, the con-
tact radius for these depths is obtained (using the known elastic modulus of the reference material) as:
In ideal case, the contact area is related to the contact depth as A
24 5
h
a h
(
)
S h
(
) /(
2
E
)
(17.10)
c
c
r,ref
which follows from Eq. (17.4). The calibration curve is usually approximated by a simple analytical
expression, such as [14] a kh
c m or a polynomial. Also the calibration curve for contact area, A ( h c ),
can be created in a similar way [4] .
17.3 CHARACTERIZATION OF INELASTIC PROPERTIES
17.3.1 Stress-Strain Diagram
In common solids, including teeth, low stresses cause elastic reversible deformations, with strains pro-
portional to the stress. If the stress exceeds the yield strength σ Y , the material deforms faster, as irrevers-
ible deformations appear in addition to the elastic ones. Yield strength, which is related to hardness, is
thus an important material parameter. Further information is provided by the stress-strain diagram σ ( ),
 
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