 （12）
以h1代替h，其余数值不变，重新利用Olive-Pharr方法和Ma方法计算硬度和弹性模量，并和原来的结果进行比较，就可定量分析出仪器柔度标定误差对两种压入测试方法的精度影响。图3和图4分别显示两种材料压入载荷为100N时仪器柔度标定无误差和+10%误差下的载荷—压入深度曲线。
图3 σy=35MPa材料的压入载荷-深度曲线
Fig.3 Load-depth curves for thematerial of σy=35MPa
图4 σy=560MPa材料的压入载荷-深度曲线
Fig.4 Load-depth curves for the material of σy =560MPa
表1-表8分别列出了两种材料在不同压入载荷和不同压入深度下的计算结果。表中hm1和E1分别为计入仪器柔度标定误差后的最大压入深度和材料杨氏模量。表中比较了仪器柔度标定误差为-10%时杨氏模量的变化情况。为便于分析，将两种杨氏模量和标准值70Gpa也进行了比较。
表1 σy=35MPa不同压入载荷Olive-Pharr方法计算结果
Table 1 The values obtained by Oliver-Pharr method for the material of σy=35MPa
|
Pm
/N
|
hm
/μm
|
hm1
/μm
|
H
/GPa
|
E
/GPa
|
E1
/GPa
|
(E-70)
/70
|
(E1-70)
/70
|
(E1-E)
/E
|
|
0.5
|
8.675
|
8.670
|
0.276
|
92.45
|
97.38
|
32.1%
|
39.1%
|
5.3%
|
|
5
|
27.43
|
27.38
|
0.276
|
92.45
|
110.06
|
32.1%
|
57.2%
|
19.0%
|
|
50
|
86.75
|
86.25
|
0.276
|
92.45
|
186.80
|
32.1%
|
166.9%
|
102.1%
|
|
100
|
122.68
|
121.68
|
0.276
|
92.45
|
322.64
|
32.1%
|
360.9%
|
249.0%
|
表2 σy=560MPa不同压入载荷Olive-Pharr方法计算结果
Table 2 The values obtained by Oliver-Pharr method for the material of σy=560MPa
|
Pm
/N
|
hm
/μm
|
hm1
/μm
|
H
/GPa
|
E
/GPa
|
E1
/GPa
|
(E-70)
/70
|
(E1-70)
/70
|
(E1-E)
/E
|
|
0.5
|
3.164
|
3.159
|
2.42
|
84.27
|
85.64
|
20.4%
|
22.3%
|
1.6%
|
|
5
|
10.01
|
9.96
|
2.42
|
84.27
|
88.76
|
20.4%
|
26.8%
|
5.3%
|
|
50
|
31.64
|
31.14
|
2.42
|
84.27
|
100.25
|
20.4%
|
43.2%
|
19.0%
|
|
100
|
44.75
|
43.75
|
2.42
|
84.27
|
108.74
|
20.4%
|
55.3%
|
29.0%
|
表3 σy=35MPa不同压入载荷Ma方法计算结果
Table 3 The values obtained by Ma method for the material of σy=35MPa
|
Pm
/N
|
hm
/μm
|
hm1
/μm
|
Hn
/GPa
|
E
/GPa
|
E1
/GPa
|
(E-70)
/70
|
(E1-70)
/70
|
(E1-E)
/E
|
|
0.5
|
8.675
|
8.670
|
0.271
|
80.95
|
84.92
|
15.6%
|
21.3%
|
4.9%
|
|
5
|
27.43
|
27.38
|
0.271
|
81.25
|
95.02
|
16.1%
|
35.7%
|
17.0%
|
|
50
|
86.75
|
86.25
|
0.271
|
80.90
|
152.70
|
15.6%
|
118.1%
|
88.8%
|
|
100
|
122.68
|
121.68
|
0.271
|
80.98
|
244.44
|
15.7%
|
249.2%
|
201.9%
|
表4 σy=560MPa不同压入载荷Ma方法计算结果
Table 4 The values obtained by Ma method for the material of σy=560MPa
|
Pm
/N
|
hm
/μm
|
hm1
/μm
|
Hn
/GPa
|
E
/GPa
|
E1
/GPa
|
(E-70)
/70
|
(E1-70)
/70
|
(E1-E)
/E
|
|
0.5
|
3.164
|
3.159
|
2.04
|
78.88
|
79.97
|
12.7%
|
14.2%
|
1.4%
|
|
5
|
10.01
|
9.96
|
2.04
|
78.88
|
82.46
|
12.7%
|
17.8%
|
4.5%
|
|
50
|
31.64
|
31.14
|
2.04
|
78.88
|
91.49
|
12.7%
|
31.1%
|
16.0%
|
|
100
|
44.75
|
43.75
|
2.04
|
78.88
|
98.05
|
12.7%
|
40.1%
|
24.3%
|
表5 σy=35MPa不同压入深度Olive-Pharr方法计算结果
Table 5 The values obtained by Oliver-Pharr method for the material of σy=35MPa
|
Pm
/N
|
hm
/μm
|
hm1
/μm
|
H
/GPa
|
E
/GPa
|
E1
/GPa
|
(E-70)
/70
|
(E1-70)
/70
|
(E1-E)
/E
|
|
0.0066
|
1
|
0.9999
|
0.276
|
92.0
|
92.63
|
31.4%
|
32.3%
|
0.7%
|
|
0.166
|
5
|
4.998
|
0.276
|
92.45
|
95.20
|
32.1%
|
36.0%
|
3.0%
|
|
16.61
|
50
|
49.83
|
0.276
|
91.25
|
130.47
|
30.4%
|
86.4%
|
43.0%
|
|
66.44
|
100
|
99.34
|
0.276
|
92.0
|
218.84
|
31.4%
|
212.6%
|
137.9%
|
表6 σy=560MPa不同压入深度Olive-Pharr方法计算结果
Table6 The values obtained by Oliver-Pharr method for the material of σy=560MPa
|
Pm
/N
|
hm
/μm
|
hm1
/μm
|
H
/GPa
|
E
/GPa
|
E1
/GPa
|
(E-70)
/70
|
(E1-70)
/70
|
(E1-E)
/E
|
|
0.0499
|
1
|
0.9995
|
2.42
|
84.27
|
84.70
|
20.4%
|
21.0%
|
0.5%
|
|
1.249
|
5
|
4.988
|
2.42
|
84.27
|
86.46
|
20.4%
|
23.5%
|
2.6%
|
|
124.86
|
50
|
48.75
|
2.42
|
84.27
|
112.55
|
20.4%
|
60.8%
|
33.6%
|
|
499.43
|
100
|
95.01
|
2.42
|
84.27
|
168.20
|
20.4%
|
140.3%
|
99.6%
|
表7 σy=35MPa不同压入深度Ma方法计算结果
Table 7 The values obtained by Ma method for the material of σy=35MPa
|
Pm
/N
|
hm
/μm
|
hm1
/μm
|
Hn
/GPa
|
E
/GPa
|
E1
/GPa
|
(E-70)
/70
|
(E1-70)
/70
|
(E1-E)
/E
|
|
0.0066
|
1
|
0.9999
|
0.271
|
80.91
|
81.41
|
15.6%
|
16.3%
|
0.6%
|
|
0.166
|
5
|
4.998
|
0.271
|
80.95
|
83.18
|
15.6%
|
18.8%
|
2.8%
|
|
16.61
|
50
|
49.83
|
0.271
|
80.93
|
110.94
|
15.6%
|
58.5%
|
37.1%
|
|
66.44
|
100
|
99.34
|
0.271
|
80.88
|
177.16
|
15.5%
|
153.1%
|
119.0%
|
表8 σy=560MPa不同压入深度Ma方法计算结果
Table 8 The values obtained by Ma method for the material of σy=560MPa
|
Pm
/N
|
hm
/μm
|
hm1
/μm
|
Hn
/GPa
|
E
/GPa
|
E1
/GPa
|
(E-70)
/70
|
(E1-70)
/70
|
(E1-E)
/E
|
|
0.0499
|
1
|
0.9995
|
2.04
|
78.88
|
79.22
|
12.7%
|
13.2%
|
0.4%
|
|
1.249
|
5
|
4.988
|
2.04
|
78.88
|
80.63
|
12.7%
|
15.2%
|
2.2%
|
|
124.86
|
50
|
48.75
|
2.04
|
78.88
|
100.96
|
12.7%
|
44.2%
|
28.0%
|
|
499.43
|
100
|
95.01
|
2.04
|
78.88
|
141.52
|
12.7%
|
102.2%
|
79.4%
|
由以上数值计算结果可以看出，当材料较软（σy=35Mpa，H=0.276Gpa）、压入载荷达到或超过50N时，10%的仪器柔度标定误差可导致测得杨氏模量超过100%的误差。当材料硬度提高时，仪器柔度的影响会显著降低，但依然不可忽略，当材料σy=560Mpa、H=2.42Gpa，压入载荷达到或超过50N时硬度，10%的仪器柔度标定误差可导致杨氏模量超过19%的误差。在同一压入深度下，仪器柔度标定误差对硬材料和软材料影响差别不大，这是由于压入同一深度时硬材料需要的压入载荷更大。同时由以上数值计算结果可以看出，Olive-Pharr方法计算得出的结果比Ma方法误差更大，对仪器柔度的标定误差更敏感，这可能和接触刚度S对数据比较敏感有关。
4 结论
由以上分析，可得出以下结论：
1）仪器柔度的标定精度直接影响压入测试结果的准确度，当材料较软且压入载荷较大时尤为明显。
2）对同一材料，压入载荷越大，由仪器柔度标定误差引入的压入测试结果误差越大。
3）在同一压入深度下，针对不同材料，由仪器柔度标定误差引入的压入测试结果误差差别不大。
4）就测试方法而言，利用Ma方法测得杨氏模量比Olive-Pharr方法精度更高，对仪器柔度敏感性更低。
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