# Three Dimensional Distribution of Sensitive Field and Stress Field Inversion of Force Sensitive Materials under Constant Current Excitation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Conductivity Mechanism

#### 2.2. Current Field Model

#### 2.3. Sensitive Field Analysis

#### 2.4. Stress Field Inversion

- (1)
- The sensitive field is divided into a plurality of elements by finite element subdivision, and the sensitive field is normalized;
- (2)
- Construct precision objective function. Image reconstruction according to the voltage value of the measurement electrode is equivalent to solving the normalized gray value $G$. This problem can be evolved into an optimization problem, assuming that any given normalized gray value $G$, according to $U=S\times G$, a theoretical calculated voltage value, then solve the gray value can be attributed to the optimization problem of the objective function;
- (3)
- Set the iteration coefficient $k$; Update and calculating ${G}_{k}$ accord to an iteration formula;
- (4)
- Judge whether $f\left(G\right)$ reach a specified precision threshold value or not, and if so, taking ${G}_{k}$ as a final gray value; If the specified precision threshold is not reached, ${G}_{k+1}={G}_{k}$ until the precision requirements are met.

## 3. Results and Discussion

#### 3.1. The Pressure Distribution in the Conductive Silicone Pad

#### 3.2. Inversion Results of Three Dimensional Stress Field

#### 3.3. Demonstration of Force Direction Detection Device

## 4. Materials and Methods

#### 4.1. Preparation of Conductive Silicon Rubber

#### 4.2. Measuring Device

#### 4.2.1. Measurement of Physical Parameters

^{3}specification. And the mechanical parameters of conductive silicone rubber were determined in the dynamic thermal analyzer (DMA 242, NETZSCH, Bavaria, Germany), which is a periodic oscillation force applied to the conductive silica gel sample under certain temperature control program. The corresponding deformation amplitude and hysteresis of conductive silica gel are measured to calculate and obtain the relevant characteristic parameters such as elastic modulus, loss factor and so on. Figure 16 and Figure 17 are respectively the measurement principle of DMA and a real live-action of the conductive silica gel mechanics test.

#### 4.2.2. Voltage Measurement between Conductive Silica Electrodes

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Figure A1.**The arrangement of the single-row electrodes uniformly arranged counterclockwise along the circumferential direction of the conductive silicone rubber.

**Figure A2.**The double-layer electrode arrangement. Two layers of electrodes, and 8 electrodes in per layer and 16 electrodes in total, are uniformly arranged counterclockwise along the circumferential direction of the side surface of the conductive silicone rubber.

## Appendix B

Measured Sequence Number | Excitation Electrode Pair | Measuring Electrode Pair |

1 | 1,2 | 3,4 |

2 | 1,2 | 4,5 |

$\cdots $ | $\cdots $ | $\cdots $ |

5 | 1,2 | 7,8 |

$\cdots $ | $\cdots $ | $\cdots $ |

13 | 1,2 | 15,16 |

14 | 2,3 | 4,5 |

$\cdots $ | $\cdots $ | $\cdots $ |

25 | 2,3 | 15,16 |

26 | 2,3 | 16,1 |

27 | 3,4 | 1,2 |

28 | 3,4 | 5,6 |

$\cdots $ | $\cdots $ | $\cdots $ |

39 | 3,4 | 16,1 |

40 | 4,5 | 1,2 |

41 | 4,5 | 2,3 |

42 | 4,5 | 6,7 |

$\cdots $ | $\cdots $ | $\cdots $ |

52 | 4,5 | 16,1 |

53 | 5,6 | 1,2 |

54 | 5,6 | 2,3 |

55 | 5,6 | 3,4 |

56 | 5,6 | 7,8 |

$\cdots $ | $\cdots $ | $\cdots $ |

65 | 5,6 | 16,1 |

66 | 6,7 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

69 | 6,7 | 4,5 |

70 | 6,7 | 8,9 |

$\cdots $ | $\cdots $ | $\cdots $ |

78 | 6,7 | 16,1 |

79 | 7,8 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

83 | 7,8 | 5,6 |

84 | 7,8 | 9,10 |

$\cdots $ | $\cdots $ | $\cdots $ |

91 | 7,8 | 16,1 |

92 | 8,9 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

97 | 8,9 | 6,7 |

98 | 8,9 | 10,11 |

$\cdots $ | $\cdots $ | $\cdots $ |

100 | 8,9 | 12,13 |

$\cdots $ | $\cdots $ | $\cdots $ |

104 | 8,9 | 16,1 |

105 | 9,10 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

111 | 9,10 | 7,8 |

112 | 9,10 | 11,12 |

$\cdots $ | $\cdots $ | $\cdots $ |

117 | 9,10 | 16,1 |

118 | 10,11 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

125 | 10,11 | 8,9 |

126 | 10,11 | 12,13 |

$\cdots $ | $\cdots $ | $\cdots $ |

130 | 10,11 | 16,1 |

131 | 11,12 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

139 | 11,12 | 9,10 |

140 | 11,12 | 13,14 |

$\cdots $ | $\cdots $ | $\cdots $ |

143 | 11,12 | 16,1 |

144 | 12,13 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

153 | 12,13 | 10,11 |

154 | 12,13 | 14,15 |

$\cdots $ | $\cdots $ | $\cdots $ |

156 | 12,13 | 16,1 |

157 | 13,14 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

167 | 13,14 | 11,12 |

168 | 13,14 | 15,16 |

169 | 13,14 | 16,1 |

170 | 14,15 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

181 | 14,15 | 12,13 |

182 | 14,15 | 16,1 |

183 | 15,16 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

195 | 15,16 | 13,14 |

196 | 16,1 | 2,3 |

$\cdots $ | $\cdots $ | $\cdots $ |

208 | 16,1 | 14,15 |

## Appendix C

Measured Sequence Number | Excitation Electrode Pair | Measuring Electrode Pair |

1 | 1,2 | 3,4 |

2 | 1,2 | 4,5 |

$\cdots $ | $\cdots $ | $\cdots $ |

5 | 1,2 | 7,8 |

6 | 1,2 | 9,10 |

$\cdots $ | $\cdots $ | $\cdots $ |

12 | 1,2 | 15,16 |

13 | 1,2 | 16,9 |

14 | 2,3 | 4,5 |

$\cdots $ | $\cdots $ | $\cdots $ |

17 | 2,3 | 7,8 |

18 | 2,3 | 8,1 |

19 | 2,3 | 9,10 |

$\cdots $ | $\cdots $ | $\cdots $ |

26 | 2,3 | 16,9 |

27 | 3,4 | 1,2 |

28 | 3,4 | 5,6 |

$\cdots $ | $\cdots $ | $\cdots $ |

30 | 3,4 | 7,8 |

31 | 3,4 | 8,1 |

32 | 3,4 | 9,10 |

$\cdots $ | $\cdots $ | $\cdots $ |

39 | 3,4 | 16,9 |

40 | 4,5 | 1,2 |

41 | 4,5 | 2,3 |

42 | 4,5 | 6,7 |

43 | 4,5 | 7,8 |

44 | 4,5 | 8,1 |

45 | 4,5 | 9,10 |

$\cdots $ | $\cdots $ | $\cdots $ |

52 | 4,5 | 16,9 |

53 | 5,6 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

55 | 5,6 | 3,4 |

56 | 5,6 | 7,8 |

$\cdots $ | $\cdots $ | $\cdots $ |

65 | 5,6 | 16,9 |

66 | 6,7 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

69 | 6,7 | 4,5 |

70 | 6,7 | 8,1 |

71 | 6,7 | 9,10 |

$\cdots $ | $\cdots $ | $\cdots $ |

78 | 6,7 | 16,9 |

79 | 7,8 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

83 | 7,8 | 5,6 |

84 | 7,8 | 9,10 |

$\cdots $ | $\cdots $ | $\cdots $ |

91 | 7,8 | 16,9 |

92 | 8,1 | 2,3 |

$\cdots $ | $\cdots $ | $\cdots $ |

96 | 8,1 | 6,7 |

97 | 8,1 | 9,10 |

$\cdots $ | $\cdots $ | $\cdots $ |

100 | 8,1 | 12,13 |

$\cdots $ | $\cdots $ | $\cdots $ |

104 | 8,1 | 16,9 |

105 | 9,10 | 1,2 |

$\cdots $ | $\cdots $ | |

112 | 9,10 | 8,1 |

113 | 9,10 | 11,12 |

$\cdots $ | $\cdots $ | $\cdots $ |

117 | 9,10 | 15,16 |

118 | 10,11 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

125 | 10,11 | 8,1 |

126 | 10,11 | 12,13 |

$\cdots $ | $\cdots $ | $\cdots $ |

130 | 10,11 | 16,9 |

131 | 11,12 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

139 | 11,12 | 9,10 |

140 | 11,12 | 13,14 |

$\cdots $ | $\cdots $ | $\cdots $ |

143 | 11,12 | 16,9 |

144 | 12,13 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

153 | 12,13 | 10,11 |

154 | 12,13 | 14,15 |

$\cdots $ | $\cdots $ | $\cdots $ |

156 | 12,13 | 16,9 |

157 | 13,14 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

167 | 13,14 | 11,12 |

168 | 13,14 | 15,16 |

169 | 13,14 | 16,9 |

170 | 14,15 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

181 | 14,15 | 13,14 |

182 | 14,15 | 16,9 |

183 | 15,16 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

195 | 15,16 | 14,15 |

196 | 16,9 | 1,2 |

$\cdots $ | $\cdots $ | $\cdots $ |

203 | 16,9 | 8,1 |

204 | 16,9 | 10,11 |

$\cdots $ | $\cdots $ | $\cdots $ |

208 | 16,9 | 14,15 |

## References

- Al-Oqla, F.M.; Sapuan, S.M.; Anwer, T.; Jawaid, M.; Hoque, M.E. Natural fiber reinforced conductive polymer composites as functional materials: A review. Synth. Met.
**2015**, 206, 42–54. [Google Scholar] [CrossRef] - Nguyen, T.D.; Han, H.S.; Shin, H.Y.; Nguyen, C.T.; Phung, H.; Van Hoang, H.; Choi, H.R. Highly sensitive flexible proximity tactile array sensor by using carbon micro coils. Sens. Actuators A Phys.
**2017**, 266, 166–177. [Google Scholar] [CrossRef] - Büscher, G.H.; Kõiva, R.; Schuermann, C.; Ritter, H.J. Flexible and stretchable fabric-based tactile sensor. Robot. Auton. Syst.
**2015**, 63, 244–252. [Google Scholar] [CrossRef] - Khanbareh, H.; Boom, K.D.; Schelen, B.; Scharff, R.B.N.; Wang, C.C.L.; van der Zwaag, S.; Groen, P. Large Area and Flexible Micro-Porous Piezoelectric Materials for Soft Robotic Skin. Sens. Actuators A Phys.
**2017**, 263, 554–562. [Google Scholar] [CrossRef] - Petković, D.; Issa, M.; Pavlović, N.D.; Pavlović, N.T.; Zentner, L. Adaptive neuro-fuzzy estimation of conductive silicone rubber mechanical properties. Expert Syst. Appl.
**2012**, 39, 9477–9482. [Google Scholar] [CrossRef] - Ziraki, S.; Zebarjad, S.M.; Hadianfard, M.J. A study on the tensile properties of silicone rubber/polypropylene fibers/silica hybrid nanocomposites. J. Mech. Behav. Biomed. Mater.
**2016**, 57, 289–296. [Google Scholar] [CrossRef] [PubMed] - Wang, Y.; Yang, R.; Shi, Z.; Zhang, L.; Shi, D.; Wang, E.; Zhang, G. Super-Elastic Graphene Ripples for Flexible Strain Sensors. ACS Nano
**2011**, 5, 3645–3650. [Google Scholar] [CrossRef] [PubMed] - Tuukkanen, S.; Hoikkanen, M.; Poikelispää, M.; Honkanen, M.; Vuoriner, T.; kakkonen, M.; Vuorinen, J.; Lupo, D. Stretching of solution processed carbon nanotube and graphene nanocomposite films on rubber substrates. Synth. Met.
**2014**, 191, 28–35. [Google Scholar] [CrossRef] - Sencadas, V.; Mutlu, R.; Alici, G. Large area and ultra-thin compliant strain sensors for prosthetic devices. Sens. Actuators A Phys.
**2017**, 266, 56–64. [Google Scholar] [CrossRef] - Vatani, M.; Engeberg, E.D.; Choi, J.W. Force and slip detection with direct-write compliant tactile sensors using multi-walled carbon nanotube/polymer composites. Sens. Actuators A Phys.
**2013**, 195, 90–97. [Google Scholar] [CrossRef] - Matos, C.F.; Galembeck, F.; Zarbin, A.J.G. Multifunctional and environmentally friendly nanocomposites between natural rubber and graphene or graphene oxide. Carbon
**2014**, 78, 469–479. [Google Scholar] [CrossRef] - Tai, Y.; Mulle, M.; Aguilar, V.I.; Lubineau, G. A highly sensitive, low-cost, wearable pressure sensor based on conductive hydrogel spheres. Nanoscale
**2015**, 7, 14766–14773. [Google Scholar] [CrossRef] [PubMed] - Seyedmehdi, S.A.; Zhang, H.; Zhu, J. Influence of production method, silicone type and thickness on silicon rubber superhydrophobic coatings. Prog. Org. Coat.
**2016**, 90, 291–295. [Google Scholar] [CrossRef] - Ohmukai, M.; Kami, Y.; Matsuura, R. Electrode for Force Sensor of Conductive Rubber. J. Sens. Technol.
**2012**, 2, 127–131. [Google Scholar] [CrossRef] - Valentini, L.; Bon, S.B.; Pugno, N.M. Severe graphene nanoplatelets aggregation as building block for the preparation of negative temperature coefficient and healable silicone rubber composites. Compos. Sci. Technol.
**2016**, 134, 125–131. [Google Scholar] [CrossRef] - Jamal, E.M.A.; Joy, P.A.; Kurian, P.; Anantharman, M.R. Synthesis of nickel–rubber nanocomposites and evaluation of their dielectric properties. Mater. Sci. Eng. B.
**2009**, 156, 24–31. [Google Scholar] [CrossRef] - Lee, S.E.; Sang, S.J.; Park, H.; Park, S.-H.; Han, I.; Mizusaki, S. Large reduction in electrical contact resistance of flexible carbon nanotube/silicone rubber composites by trifluoroacetic acid treatment. Compos. Sci. Technol.
**2017**, 143, 98–105. [Google Scholar] [CrossRef] - Spitalsky, Z.; Tasis, D.; Papagelis, K.; Galiotis, C. Carbon nanotube–polymer composites: Chemistry, processing, mechanical and electrical properties. Prog. Polym. Sci.
**2010**, 35, 357–401. [Google Scholar] [CrossRef] - Kemaloglu, S.; Ozkoc, G.; Aytac, A. Properties of thermally conductive micro and nano size boron nitride reinforced silicon rubber composites. Thermochim. Acta
**2010**, 499, 40–47. [Google Scholar] [CrossRef] - Ismail, A.M.; Mahmoud, K.R.; Salam, A.E. Electrical conductivity and positron annihilation characteristics ofternary silicone rubber/carbon black/TiB 2, nanocomposites. Polym. Test.
**2015**, 48, 37–43. [Google Scholar] [CrossRef] - Xiao, W.; Lei, Y.; Xia, Z.; Chen, X.; Han, Y.; Nie, J. Effect of silver plating time on the properties of conductive silicone rubber filled with silver-coated carbonyl nickel powder. J. Alloys Compd.
**2017**, 724, 24–28. [Google Scholar] [CrossRef] - Saleem, A.; Frormann, L.; Soever, A. Fabrication of Extrinsically Conductive Silicone Rubbers with High Elasticity and Analysis of Their Mechanical and Electrical Characteristics. Polymers
**2010**, 2, 200–210. [Google Scholar] [CrossRef] - Brigandi, P.J.; Cogen, J.M.; Pearson, R.A. Electrically conductive multiphase polymer blend carbon-based composites. Polym. Eng. Sci.
**2014**, 54, 1–16. [Google Scholar] [CrossRef] - Zakaria, M.Y.; Sulong, A.B.; Sahari, J.; Suherman, H. Effect of the addition of milled carbon fiber as a secondary filler on the electrical conductivity of graphite/epoxy composites for electrical conductive material. Compos. Part B Eng.
**2015**, 83, 75–80. [Google Scholar] [CrossRef] - Gao, L.; Yang, X.; Hu, J.; He, J. ZnO microvaristors doped polymer composites with electrical field dependent nonlinear conductive and dielectric characteristics. Mater. Lett.
**2016**, 171, 1–4. [Google Scholar] [CrossRef] - Ma, P.C.; Tang, B.Z.; Kim, J.K. Effect of CNT decoration with silver nanoparticles on electrical conductivity of CNT-polymer composites. Carbon
**2008**, 46, 1497–1505. [Google Scholar] [CrossRef] - Boudenne, A.; Mamunya, Y.; Levchenko, V.; Garnier, B.; Lebedev, E. Improvement of thermal and electrical properties of Silicone–Ni composites using magnetic field. Eur. Polym. J.
**2015**, 63, 11–19. [Google Scholar] [CrossRef] - Yoshimura, K.; Nakano, K.; Hishikawa, Y. Flexible tactile sensor materials based on carbon microcoil/silicone-rubber porous composites. Compos. Sci. Technol.
**2016**, 123, 241–249. [Google Scholar] [CrossRef] - Kumar, V.; Lee, D.J.; Lee, J.Y. Studies of RTV silicone rubber nanocomposites based on graphitic nanofillers. Polym. Test.
**2016**, 56, 369–378. [Google Scholar] [CrossRef] - Witt, N.; Tang, Y.; Ye, L.; Fang, L. Silicone rubber nanocomposites containing a small amount of hybrid fillers with enhanced electrical sensitivity. Mater. Des.
**2013**, 45, 548–554. [Google Scholar] [CrossRef] - Giffney, T.; Bejanin, E.; Kurian, A.S.; Travas-Sejdic, J.; Aw, K. Highly stretchable printed strain sensors using multi-walled carbon nanotube/silicone rubber composites. Sens. Actuators A Phys.
**2017**, 259, 44–49. [Google Scholar] [CrossRef] - Petković, D.; Issa, M.; Pavlović, N.D.; Zentner, L. Intelligent rotational direction control of passive robotic joint with embedded sensors. Expert Syst. Appl. Int. J.
**2013**, 40, 1265–1273. [Google Scholar] - Cho, C.; Ryuh, Y. Fabrication of flexible tactile force sensor using conductive ink and silicon elastomer. Sens. Actuators A Phys.
**2016**, 237, 72–80. [Google Scholar] [CrossRef] - Kim, H.; Eom, T.S.; Cho, W.; Woo, K.; Shon, Y.; Wie, J.J.; Shim, B.S. Soft Electronics on Asymmetrical Porous Conducting Membranes by Molecular Layer-by-Layer Assembly. Sens. Actuators B Chem.
**2017**, 254, 916–925. [Google Scholar] [CrossRef] - Petković, D.; Issa, M.; Pavlović, N.D.; Zentner, L. Potential of adaptive neuro-fuzzy inference system for contact positions detection of sensing structure. Measurement
**2015**, 61, 234–242. [Google Scholar] [CrossRef] - Wang, L.; Ding, T.; Peng, W. Influence of carbon black concentration on piezoresistivity for carbon-black-filled silicone rubber composite. Carbon
**2009**, 47, 3151–3157. [Google Scholar] - Bauhofer, W.; Kovacs, J.Z. A review and analysis of electrical percolation in carbon nanotube polymer composite. Compos. Sci. Technol.
**2009**, 69, 1486–1498. [Google Scholar] [CrossRef] - Ding, S.; Han, B.; Dong, X.; Yu, X.; Ni, Y.; Zheng, Q.; Qu, J. Pressure-sensitive behaviors, mechanisms and model of field assisted quantum tunneling composites. Polymer
**2017**, 113, 105–118. [Google Scholar] [CrossRef] - Mclachlan, D.S.; Hwang, J.H.; Mason, T.O. Evaluating Dielectric Impedance Spectra using Effective Media Theories. J. Electroceram.
**2000**, 5, 37–51. [Google Scholar] [CrossRef] - Zhao, G.F. Developing a four-dimensional lattice spring model for mechanical responses of solids. Comput. Methods Appl. Mech. Eng.
**2017**, 35, 881–895. [Google Scholar] [CrossRef] - Zhao, G.F.; Fang, J.; Zhao, J. A 3D distinct lattice spring model for elasticity and dynamic failure. Int. J. Numer. Anal. Methods Geomech.
**2011**, 35, 859–885. [Google Scholar] [CrossRef] - Zhao, S.; Wang, W.; Guo, W.; Zhang, C. A Human Body Pressure Distribution Imaging System Based on Wavelet Analysis and Resistance Tomography. Sensors
**2017**, 17, 2634. [Google Scholar] [CrossRef] [PubMed] - Yogeswaran, N.; Dan, W.; Navaraj, W.T.; Shakthivel, D.; Khan, S.; Polat, E.O.; Gupta, S.; Heidari, H.; Kaboli, M.; Lorenzelli, L.; et al. New materials and advances in making electronic skin for interactive robots. Adv. Robot.
**2015**, 29, 1359–1373. [Google Scholar] [CrossRef]

**Figure 2.**(

**a**) The schematic diagram of a conductive silicone rubber under the external force; (

**b**) The finite element model of conducting silicon rubber under the external force; (

**c**) The three-dimensional stress isometric surface of the conductive silicon rubber under the external force. the blue area in the (

**c**) represents the stress equivalent plane; (

**d**) The conductivity isometric surface (The red area in the (

**d**) represents equivalent surface of conductivity).

**Figure 5.**(

**a**) The way of measuring the voltage when the 1,2 electrode are electrified; (

**b**) The way of measuring the voltage when the 8,9 electrode are electrified; (

**c**) The way of measuring the voltage when the 16,1 electrode are electrified; (

**d**–

**i**) The result of the sensitive distribution at 8,16,24,32,40,48 measurements (The blue area in (

**d**–

**i**) represents the sensitive area of the disc conductive silicone rubber).

**Figure 6.**(

**a**) is the results of stress distribution of finite element analysis of conductive silica disks; (

**b**) is the results of resistivity inversion imaging of conductive silica pad; (

**c**) is the voltage of the measuring electrode pair in the adjacent excitation mode of the conductive silica gel. The x axis is the serial number of the measurement, and the sequence number is related to the excitation mode. The y-axis is the voltage of the measured electrode. Its unit is volts.

**Figure 7.**(

**a**) The result of the sensitivity distribution of the cylindrical conductive silicone rubber at the 5th measurement; (

**b**–

**d**) The sensitivity profile at 0.5, 1.0 and 1.5 cm from the bottom surface of the cylindrical conductive silicone rubber at the 5th measurement respectively; (

**e**) The result of the sensitivity distribution of the cylindrical conductive silicone rubber at the100th measurement; (

**f**–

**h**) The sensitivity profile at 0.5, 1.0 and 1.5 cm from the bottom surface of the cylindrical conductive silicone rubber at the 100th measurement, respectively.

**Figure 8.**(

**a**) The isosurface of stress three-dimensional distribution of the cylindrical conductive silica obtained by finite element analysis under the action of a 20 N force. Different colors in the (

**a**) represent different equivalent surfaces; (

**b**) the sectional view of the stress distribution in the vertical direction (The red area in the (

**b**) represent a region of high stress in a cylindrical conducting silicone rubber); (

**c**) The voltage value of the measuring electrode under load 20 N force and no load force. The x axis represents the measured sequence number and the y axis represents the voltage value of the measuring electrode, and its unit is volts. The red line represents the measured voltage value after loading, and the blue line represents the measured voltage without load; (

**d**) The result of the three-dimensional distribution inversion of the conductivity under a load of 20 N in accordance with the measured voltage (The red area at the top in the (

**d**) represents a region of high stress in a cylindrical conducting silicone rubber).

**Figure 9.**(

**a**) A loading example placing a loaded head with a circular radius of 10 mm on the conductive silicone upper surface; (

**b**) The result of finite element analysis of a cylindrical conductive silicone rubber under stress in a circular region; (

**c**) The sectional view of the stress distribution in the vertical direction (The red region in the (

**c**) represents the region with the greatest stress); (

**d**) The result of the inversion of the conductivity three-dimensional distribution under the circular region load (The red area at the top in the (

**d**) represents a region of high stress in a cylindrical conducting silicone rubber); (

**e**) The loading example placing a loaded head with a square side length of 15 mm on the conductive silicone upper surface; (

**f**) The result of finite element analysis of a cylindrical conductive silicone rubber under stress in a square region; (

**g**) The sectional view of the stress distribution in the vertical direction (The red region in the (

**g**) represents the region with the greatest stress); (

**h**) The result of the inversion of the conductivity three-dimensional distribution under the square region load (The red area at the top in the (

**h**) represents a region of high stress in a cylindrical conducting silicone rubber); (

**i**) A loading example placing a loaded head with two squares of 8 mm × 8 mm on the conductive silicone upper surface; (

**j**) The result of finite element analysis of a cylindrical conductive silicone rubber under stress in two square region; (

**k**) A sectional view of the stress distribution in the vertical direction (The yellow region in the (

**k**) represents the region with the greatest stress); (

**l**) The result of the inversion of the conductivity three-dimensional distribution under two square region load (The red area at the top in the (

**h**) represents a region of high stress in a cylindrical conducting silicone rubber).

**Figure 10.**(

**a**) The result of the sensitivity distribution of the rectangular conductive silicone rubber with a hemispherical hole at the top at the 5th measurement; (

**b**–

**d**) The sensitivity profile at 0.5, 1.0 and 1.5 cm from the bottom surface of the rectangular conductive silicone rubber at the 5th measurement, respectively; (

**e**) The result of the sensitivity distribution of the rectangular conductive silicone rubber with a hemispherical hole at the top at the 100th measurement; (

**f**–

**h**) The sensitivity profile at 0.5, 1.0 and 1.5 cm from the bottom surface of the rectangular conductive silicone rubber at the 100th measurement, respectively.

**Figure 11.**Measured voltage values of rectangular conductive silicone rubber under different direction forces. The x axis represents the measured sequence number, and the y axis represents the voltage value of the measuring electrode, in volts. The blue line represents the measured voltage with no load. The black line represents the measured voltage when the angle between the load direction and the horizontal direction is 30 degrees; the red line represents the measured voltage when the angle is 90 degrees; the purple line represents the measured voltage when the angle between the load direction and the horizontal direction is 120 degrees.

**Figure 12.**The results of finite element analysis under various angles of hinged rods, the slices of stress, and the inversion results of resistivity distribution, respectively (The red region in the (

**b**–

**d**), (

**f**–

**h**), (

**j**–

**k**) indicate the region with greater stress on the conductive silicone rubber); (

**a**) The loading example of the rectangular conductive silicone rubber with a hemispherical hole at the top under the stress tilted to the right; (

**b**) The finite element analysis result of a rectangular conductive silicone rubber under the stress tilted to the right; (

**c**) The sectional view of the stress distribution; (

**d**) The sectional view of the three-dimensional distribution inversion of the electrical conductivity under the stress tilted to the right; (

**e**) The loading example of the rectangular conductive silicone rubber with a hemispherical hole at the top under vertical stress; (

**f**) The result of finite element analysis of a rectangular conductive silicone rubber under the vertical stress; (

**g**) The sectional view of the stress distribution; (

**h**) The sectional view of the three-dimensional distribution inversion of the electrical conductivity under the vertical stress; (

**i**) A loading example of the rectangular conductive silicone rubber with a hemispherical hole at the top under the stress tilted to the left; (

**j**) The finite element analysis result of a rectangular conductive silicone rubber under the stress tilted to the left; (

**k**) The sectional view of the stress distribution; (

**l**) The sectional view of the inversion of the conductivity three-dimensional distribution under the stress tilted to the left.

**Figure 13.**The result of the finite element analysis of the middle section and the inversion result of conductivity distribution when the rectangular conductive silicone rubber with hemispherical grooves on the top is cut along the vertical direction; (

**a**) A finite-element analysis of the middle section of a rectangular conducting silicon rubber with a hemispherical groove at the top, when the hemispherical concave surface is loaded along the direction of 30 degrees; (

**b**) A slice of the inversion result of the conductivity distribution in the middle section of a rectangular conductive silicone rubber; (

**c**) A finite-element analysis of the middle section of a rectangular conducting silicon rubber with a hemispherical groove at the top, when the hemispherical concave surface is loaded along the direction of 90 degrees.; (

**d**) A slice of the inversion result of the conductivity distribution in the middle section of a rectangular conductive silicone rubber; (

**e**) A finite-element analysis of the middle section of a rectangular conducting silicon rubber with a hemispherical groove at the top, when the hemispherical concave surface is loaded along the direction of 90 degrees; (

**f**) A slice of the inversion result of the conductivity distribution in the middle section of a rectangular conductive silicone rubber. (The red region in the (

**a**–

**f**) indicate the region with greater stress on the conductive silicone rubber)

**Figure 14.**The dispersion process of conductive carbon black. The conductive carbon black is dispersed, stirred and dried to become uniformly distributed and non-agglomerated carbon black.

**Figure 15.**The preparation details of the conductive silicone rubber. The dispersed conductive carbon black is mixed with the silicone rubber. The conductive silicone rubber is vulcanized in a vacuum environment.

**Figure 16.**The measurement principle of DMA. The mechanical properties of the material were obtained by measuring the strain of the material under different pressures.

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## Share and Cite

**MDPI and ACS Style**

Zhao, S.; Liu, M.; Guo, W.; Zhang, C. Three Dimensional Distribution of Sensitive Field and Stress Field Inversion of Force Sensitive Materials under Constant Current Excitation. *Sensors* **2018**, *18*, 722.
https://doi.org/10.3390/s18030722

**AMA Style**

Zhao S, Liu M, Guo W, Zhang C. Three Dimensional Distribution of Sensitive Field and Stress Field Inversion of Force Sensitive Materials under Constant Current Excitation. *Sensors*. 2018; 18(3):722.
https://doi.org/10.3390/s18030722

**Chicago/Turabian Style**

Zhao, Shuanfeng, Min Liu, Wei Guo, and Chuanwei Zhang. 2018. "Three Dimensional Distribution of Sensitive Field and Stress Field Inversion of Force Sensitive Materials under Constant Current Excitation" *Sensors* 18, no. 3: 722.
https://doi.org/10.3390/s18030722