Data processing: measuring – calibrating – or testing – Measurement system in a specific environment – Mechanical measurement system
Reexamination Certificate
2002-07-15
2004-08-17
Bui, Bryan (Department: 2863)
Data processing: measuring, calibrating, or testing
Measurement system in a specific environment
Mechanical measurement system
C702S035000, C702S043000, C073S081000
Reexamination Certificate
active
06778916
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a ball indentation tester and testing technique being used to measure the material properties when tensile test cannot be applied; welding parts with continuous property variation, brittle materials with unstable crack growth during preparation and test of specimen, and the parts in present structural use. More particularly, indentation test is non-destructive and easily applicable to obtain material properties. A new numerical indentation technique is invented by examining the finite element solutions based on the incremental plasticity theory with large geometry change. The load-depth curve from indentation test successfully converts to a stress-strain curve.
2. Background of the Related Art
While indentation test is non-destructive and easily applicable to obtain material properties, the test result is difficult to analyze because of complicated triaxial stress state under ball indenter. For this reason, the indentation test is inappropriate to measure various material properties. Thus, it is used to obtain merely hardness. Recently, however, this kind of difficulty is greatly overcome both by finite element analyses of subindenter stress and deformation fields, and by continuous measurement of load and depth. As a result, stress-strain relation can be obtained from analysis of load-depth curve.
An automated indentation test gives a stress-strain curve from measured load-depth data.
FIG. 1
shows a schematic profile of indentation. Here h
t
and d
t
are ideal indentation depth and projected diameter at loaded state, and h
p
and d
p
are plastic indentation depth and projected diameter at unloaded state. With an indenter of diameter D, the following relation is delivered from spherical geometric configuration.
d
t
=2{square root over (
h
t
D−h
t
2
)} (1)
Assuming that “projected” indentation diameter at loaded and unloaded states remains the same as shown in
FIG. 2
, Hertz expressed d (=d
t
=d
p
under this assumption) as follows.
d
=
2.22
{
P
2
r
1
r
2
r
2
-
r
1
(
1
E
1
-
1
E
2
)
}
1
/
3
(
2
)
where r
1
and r
2
are indentation radius of indenter and specimen at unloaded state, and E
1
and E
2
are Young's modulus of indenter and specimen, respectively. If the indenter is rigid, r
1
=D/2 and r
2
is a function of d and h
p
.
Substituting these into Eq. (3) gives:
d
=
[
0.5
C
D
{
h
p
2
+
(
d
/
2
)
2
}
h
p
2
+
(
d
/
2
)
2
-
h
p
D
]
1
/
3
(
3
)
where C is 5.47P (E
1
−1
+E
2
−1
).
Tabor brought the experimental conclusion that equivalent (plastic) strain “at the (Brinell and Micro Vickers) indenter contact edge” is given by:
ϵ
p
=
0.2
(
d
D
)
(
4
)
where d is calculated from Eq. (2). But, Haggag et al. ignored pile-up and sink-in of material. They simply calculated the indentation diameter d with Eq. (1) at loaded state and plastic diameter d
p
with Eq. (3) at unloaded state, and plastic strain with Eq. (4) by substituting d
p
for d.
Mean contact pressure p
m
is defined by p
m
=4P(&pgr;/d
2
), where P is the compressive indentation load. Then constraint factor &psgr;, which is a function of equivalent plastic strain, is defined as the ratio between mean contact pressure and equivalent stress.
&psgr;(&egr;
p
)≡
p
m
/&sgr; (5)
Hence, the equivalent stress is expressed in the form:
σ
=
4
P
π
d
2
ψ
(
6
)
Note that, in a strict sense, both equivalent plastic strain and equivalent stress are functions of location within the subindenter deformed region as well as deformation intensity itself. Thus constraint factor &psgr; is also a function of location. Francis classified the indentation states into three regions and presented the empirical formula for &psgr; with indentation test results for the various materials taken into consideration.
(1) Elastic region with recoverable deformation
(2) Transient region with elastic-plastic deformation
(3) Fully plastic region with dominant plastic deformation
Haggag et al. calculated stress-using d
p
instead of d in Eq. (6), and they modified Francis' constraint factor considering that constraint factor is a function of strain rate and strain hardening.
ψ
=
{
1.12
φ
≤
1
1.12
+
τlnφ
1
≤
φ
≤
27
ψ
max
φ
≥
27
(7a)
&psgr;
max
=2.87&agr;
m
(7b)
&tgr;≡(&psgr;
max
−1.12)/ln 27 (7c)
where &agr;
m
is constraint factor index. It is proportional to strain rate, and has the value of 1 for the material with low strain rate. By investigating the experimental results, Francis suggested a normalized variable &phgr; in the form:
φ
=
ϵ
p
E
2
0.43
σ
(
8
)
Since equivalent strain in Eq. (4) is the value at the indenter contact edge, all the values in Eqs. (5)-(8) implicitly mean values also at the indenter contact edge.
For spherical indenter, the following relation called Meyer's law holds between applied load P and indentation projected diameter d.
P=kd
m
(9)
where k and m are material constants when indenter diameter D is fixed, and m is Meyer's index generally in the range of 2 to 2.5.
Meyer's experiment reveled that index m is independent of diameter D, and k decreases with increasing D.
A=k
1
D
1
m
−2
=k
2
D
2
m−2
k
3
D
3
m−2
= . . . (10)
where A is a constant. Substituting this into Eq. (9) gives:
P
d
2
=
A
(
d
D
)
m
-
2
(
11
)
Equation (6) converts to Eq. (12) by Eq. (11).
σ
=
4
A
πψ
(
d
D
)
m
-
2
(
12
)
After replacing d with d
t
in Eq. (11), Haggag et al. calculated yield strength &sgr;
0
from the following relation of yield strength and slope A that George et al. obtained from experiment.
&sgr;
0
=&bgr;
m
A
(13)
where &bgr;
m
is a material constant. The value of &bgr;
m
in steel is about 0.229, which comes from analysis of tensile yield strength and A.
Rice and Rosengren proposed a stress-strain relation in piecewise power law form.
ϵ
t
ϵ
o
=
{
σ
σ
o
fo
r
σ
≤
σ
o
(
σ
σ
o
)
n
fo
r
σ
>
σ
o
;
1
<
n
≤
∞
(
14
)
where &sgr;
0
is yield strength, &egr;
0
=&sgr;
0
/E yield strain and n strain hardening exponent. Total strain &egr;
t
is decomposed into elastic and plastic strains (&egr;
t
=&egr;
e
+&egr;
p
).
FIG. 3
shows the calculation process of the material properties by Haggag's indentation method. In the approach of Haggag et al., each repetition of loading and unloading provides one point of stress-strain data points. Thus a single indentation test usually picks up total only 6-7 data point. The approach also requires prior material constants from extra tensile tests.
The Haggag's model for the SSM system adopts the indentation theories of Francis and Tabor established on the experimental observations and some analyses. Haggag's approach requires prior material constants from extra tensile tests, which is one of the shortcomings.
The SSM system gives stress-strain curves through regression of load-depth data obtained from 6-7 times repetitive loading and unloading. This insufficient number of data often leads to inaccurate regression. Above all, the most critical issue in Haggag's approach is that subindenter stress field from deformation theory is far from the real one.
SUMMARY OF THE INVENTION
In order to overcome the disadvantages as described above, the present invention provides an automated indentation system for performing a compression test by loading a compressive indentation load (P). Then, an elastic modulus (E), a yield strength (&sgr;
0
), and a hardening exponent (n) are calculated based on measured indentation depth (h
Bui Bryan
G W
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