The results in this section are presented in terms of coefficient of variations. For each condition (i.e., combination of magnification, particle density, and Poisson’s ratio), 1,000 simulations are run andEandnvalues are extracted for each case.
The coefficient of variation values represent how much the extractedEandnvalues vary over the 1,000 simulations with different noise levels and thus are related to the probability of determining the actualEandnvalue in a single experiment.
In general, a lower coefficient of variation indicates more desirable test conditions.
Results for the numerical simulations over the range of parameters in Table20.1are summarized in Figs.20.2,20.3, and 20.4. Figure20.2shows the coefficients of variation forEandnwith respect tonand the magnification with a fixed particle density of 300 mm2. Figure 20.2ashows that, initially, as magnification increases, the coefficient of variation decreases until it reaches a minimum value between 10and 20. As magnification continues to increase, the coefficient of variation also increases. The coefficient of variation forEis largest when the magnification is 40andnis between 0.3 and 0.5.
Table 20.1 Parameters for
numerical simulations Fixed parameters Parameters varied
n¼0.1–0.5
magnification¼5–40
a¼500mm particle density¼100 mm2to 500 mm2 P¼10 mN uxnoise¼uynoise¼6.7mm/magnification E¼2 MPa uznoise¼10mm/magnification
Field of view¼8.576 mm/magnification by 6.912 mm/magnification
Magnification b
a Coefficient of Variation of ν
10 20 30 40
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Magnification Coefficient of Variation of E
10 20 30 40
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.02 0.03 0.04 0.05 0.06 0.07
Fig. 20.2 (a) Coefficients of variation ofEand (b)nas a function ofnand magnification when the particle density is 300 mm2
144 M.J. Wald et al.
Similarly, Fig.20.2bshows that, for a givenn, the variation ofninitially decreases as magnification increases, reaching a minimum between 10and 20. The largest variation innoccurs when the magnification is 40and Poisson’s ratio is 0.1.
These results suggest that the pixel resolution is too coarse to measure the displacements when the magnification is below 10. For magnifications above 20, there may be too few particles in the field of view for accurate measurements.
Figure20.3shows the relationship between the coefficients of variation ofEandnas a function ofnand particle density when the magnification is fixed at 20. For a constantn, the coefficients of variation forEandndecrease as the particle density increases. This occurs because increasing the particle density increases the number of points the algorithm uses to extractEandn, thereby reducing the error. For the cases examined, the variation ofEis largest when the particle density is 100 mm2and Poisson’s ratio is about 0.4. Variation ofn(Fig.20.3b) is largest when the particle density is 100 mm2and Poisson’s ratio is about 0.1.
The coefficients of variation ofEandnare shown as a function of magnification and bead density for a Poisson’s ratio of 0.45 in Fig.20.4. Similar to the results shown in Figs.20.2and20.3. Figure20.4shows that the variation ofEandndecreases as particle density increases and the variation is smallest for magnifications between 10and 20. Figure 20.5 shows regions in which the coefficients of variation ofEandnare greater than 0.05, between 0.03 and 0.05, and less than 0.03.
These results show that, for a given material, an appropriate combination of particle density and magnification can be chosen to achieve an acceptable coefficient of variation.
Bead Density (mm-2) Coefficient of Variation of E
100 200 300 400 500
0.1 0.2 0.3 0.4 0.5
0.02 0.03 0.04 0.05 0.06
Particle Density (mm-2) Coefficient of Variation of ν
100 200 300 400 500
0.1 0.2 0.3 0.4 0.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Fig. 20.3 (a) Coefficients of variation ofEand (b)nas a function ofnand particle density when the magnification is 20
Particle Density (mm-2)
Magnification
Coefficient of Variation of E
100 200 300 400 500
5 10 15 20 25 30 35 40
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Particle Density (mm-2)
Magnification
Coefficient of Variation of ν
100 200 300 400 500
5 10 15 20 25 30 35 40
0.05 0.1 0.15 0.2 0.25 0.3
a b
Fig. 20.4 (a) Coefficients of variation ofEand (b)nas a function of magnification and particle density whennis 0.45
20 Improved Instrumented Indentation of Soft Materials through Surface Deformation Measurements 145
Because the exact noise of the proposed experiment will depend on the particular system configuration and is thus unknown, a case study was completed in which the added noise was varied. For this case study,nwas 0.45, particle density was 300 mm2, and magnification was 20; 1,000 simulations were run at each noise level. The noise was varied by proportional scaling so that the noise given in Table20.1is the maximum noise and corresponds to a noise level of 1 and no noise corresponds to a noise level of 0. As shown in Fig.20.6, the coefficients of variation forEandnscale linearly with the noise level. Linear fit lines for the coefficients of variation ofEandnas a function of noise level are also shown in Fig.20.6.
The increase in scatter as the noise level increases suggests that 1,000 simulations may not be sufficient to characterize the variations as noise level increases.
The magnitudes of the displacements scale with the applied load. Since the random noise is assumed to not vary with the applied load, as the displacements increase relative to random noise, the effect of the noise on the measured results will be less significant. Conversely, when the measured displacements are smaller, noise may have a larger influence on the variation of the calculatedEandnvalues.
Particle Density (mm-2)
Magnification
Coefficient of Variation of E and ν
100 200 300 400 500
5 10 15 20 25 30 35 40
> 0.05 0.03 - 0.05
< 0.03 Fig. 20.5 Contours showing
regions in which the coefficients of variation are both greater than 0.05, between 0.03 and 0.05 and less than 0.03
0.2 0.4 0.6 0.8 1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Noise Level
Coefficient of Variation
E ν Fig. 20.6 Coefficients of
variation ofEandnversus fraction of the maximum noise for case wherenwas 0.45, magnification was 20, and bead density was 300 mm2
146 M.J. Wald et al.