3.4.3. Dimensional analysis and results
In the dimensional analysis, an relationship between the curvature the material properties were developed based on the dimensional analysis. Dimensional analysis has proven to be
useful to the study of the contact mechanics for instrumented normal indentation Cheng. et al
, 2004. In Indentation, indenting normally into a power law elasto plastic solid, the load P can be written as Dao, 2001.
P = P h, E, v, E
ª
, v
ª
, σ
¬
, n 3.6
Where E
ª
is Young’s modulus of the indenter, and v
ª
is its Poisson’s ratio. By combining elasticity effects of an elastic indenter and an elasto plastic solid as
P = P h, E
∗
, σ
¬
, n 3.7
Where
E
∗
= }
Q-
®
•
+
Q-
u®
•
u
•
Q
3.8
Alternatively, equation 3.7 can be written as P = P h, E
∗
, σ
¯
, n 3.9 Or
P = P h, E
∗
, σ
¬
, σ
¯
3.10
Applying the theorem in dimensional analysis, equation 3.9 becomes
P = σ
°
h
I
∏ 1
•
∗
²
³
, n 3.11
And thus
C =
µ ¶
®
= σ
¯
∏ 1 ·
•
∗
²
³
, n¸ 3.12
Where ∏ 1 is a dimensionless function. similarly, applying the theorem to equation
3.10, loading curvature C may alternatively be expressed as
C =
µ ¶
®
= σ
¬
∏ 1 n
•
∗
²
¹
,
²
³
²
¹
o 3.13
Where ∏ 1 is a dimensionless function, the dimensionless given in equation 3.12. and
the normalization was taken with respect to E
∗
instead of
6
or Figure 3.9.a shows the relationship between normalised Cv Vs properties material, and
Figure 3.9.b. Show the relationship between normalized Cs Vs properties material strain hardening exponent n and Yielding stress. It is clearly shown that all the data. Curve
fitting has been performed by iterating the relationship between loading curvature indentation and properties material
6
, as following equations
C
-
=
6 J º
⁄
374.14 Ž
J. ½¾ _
3.14
Similarly an equation has been determined for the Spherical indentation
C
¿
=
6 J º
⁄
8011.9 Ž
.½Âº _
3.15 Using the relationship determined, for each combination of Cv and Cs, there are a range of
material data as shown in Figure 3.9.c. The intersection point will represent the predicted material properties. In this case, for
6
= 150, n=0.2, Cv = 30647.664 and Cs = 515752. The predicted value are
′
6
= 149, n’ =0.209. Accuracy studies results were done for some
input value, the results were listed in Table 3.3. The average accuracy error ∆
6 6
= 0.065196 and nn = -0.06296, which is much better than the accuracy for the 3D
mapping approach.
3.4.4. Chart based Intersection approach and results
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