# Characterization of Shear Strain on PDMS: Numerical and Experimental Approaches

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Constitutive Models

^{T}is the transposed deformation gradient and p is a multiple of Lagrange obtained according to the state of tension T.

_{3}, is 1 and Equation (5) is defined as:

_{1}).

_{1}, C

_{2}and C

_{3}are constants of the material that are determined from the experimental tests.

_{1}and I

_{2}are the constant strain and C

_{10}and C

_{01}are the material constants.

_{i}and α

_{i}are material constants that can be positive, negative, integer or not.

_{1}and I

_{2}, of the left Cauchy-Green deformation tensor, with Cij denoting material constants. The strain energy is given by [34]:

_{1}− 3.

## 3. Materials and Methods

#### 3.1. Experimental Tests

^{®}by enterprise Dow Corning from Wiesbaden, Germany, To obtain the PDMS resin, it was necessary to mix the curing agent with the prepolymer; for each 10 g of prepolymer, 1 g of the curing agent was added (10: 1).

^{3}) was determined which is very similar to other references [40].

#### 3.1.1. Tensile Testing

#### 3.1.2. Shear Testing

^{3}dimensions were used. The bars’ surfaces were sanded and degreased with acetone to improve the adhesion of the PDMS to the steel. After the cleaning, the PDMS bonding was done using a cyanoacrylate structural adhesive Locttice Super Glue 3 (Germany), presenting the following configuration Figure 3.

^{®}DIC system made it Dutchman Blvd. Irmo, SC, USA was used to measure the strain field in-plane and out-plane of the PDMS surface.

#### 3.2. Numerical Simulation

^{®}R18.1 academic research mechanical license.

## 4. Results and Discussion

#### 4.1. Experimental

#### 4.2. Numerical

^{−3}.

#### 4.3. Comparison

## 5. Conclusions

- From experimental tests, the hyperelastic behavior of PDMS was verified using a simple shear test. The DIC tests showed that the most significant shear strain occurred in the centre of the PDMS plate. These experimental tests also verified that the PDMS rupture occurred at the bonding interface between the steel and PDMS plates. The rupture happened for different PDMS thicknesses. For 4 mm of PDMS, the specimen broke up at a 154 N of shear force. The specimen with 2 mm of PDMS suffered rupture at a shear force of 80 N. For an equal value of shear force, a higher value of displacement happened for the higher PDMS thickness.
- The numerical simulations were done using four hyperelastic constitutive models (Mooney–Rivlin, Yeoh, Gent and polynomial). All the hyperelastic constitutive models presented similar results despite some critical differences. The main reason to obtain similar results in shear stress was due to the appliance of low displacement values and the constitutive models, for this displacement level, have identical behavior. The values that show a greater difference in shear strain occurred for the Mooney–Rivlin constitutive model when compared with the other constitutive models (see Table 2). The numerical model with 2 mm of PDMS results in higher values of shear strain than with 4 mm. So, for low displacement levels it is possible to use any of the hyperelastic constitutive to simulate the shear test.
- Qualitatively, in the central region of the specimen, the numerical and experimental results have similar behavior, and the values of shear strain are close. Nevertheless, the maximum relative error between the numerical and experimental results is very different for the two thickness values. It is possible to conclude that, for higher values of displacement and thicknesses, the numerical simulation results move further away from experimental values.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Martin, S.; Bhushan, B. Transparent, wear-resistant, superhydrophobic and superoleophobic poly(dimethylsiloxane) (PDMS) surfaces. J. Coll. Int. Sci.
**2017**, 488, 118–126. [Google Scholar] [CrossRef] [PubMed] - Hassler, C.; Boretius, T.; Stieglitz, T. Polymers for neural implants. J. Polym. Sci. Part B Polym. Phys.
**2011**, 49, 18–33. [Google Scholar] [CrossRef] - Roh, C.; Lee, J.; Kang, C. Physical Properties of PDMS (Polydimethylsiloxane) Microfluidic Devices on Fluid Behaviors: Various Diameters and Shapes of Periodically-Embedded Microstructures. Materials
**2016**, 9, 836. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kim, J.H.; Lau, K.T.; Shepherd, R.; Wu, Y.; Wallace, G.; Diamond, D. Performance characteristics of a polypyrrole modified polydimethylsiloxane (PDMS) membrane based microfluidic pump. Sens. Actuators A Phys.
**2008**, 148, 239–244. [Google Scholar] [CrossRef] - Lin, Y.H.; Kang, S.W.; Wu, T.Y. Fabrication of polydimethylsiloxane (PDMS) pulsating heat pipe. Appl. Therm. Eng.
**2009**, 29, 573–580. [Google Scholar] [CrossRef] - Casanova-Moreno, J.; To, J.; Tony Yang, C.W.; Turner, R.F.; Bizzotto, D.; Cheung, K.C. Fabricating devices with improved adhesion between PDMS and gold-patterned glass. Sens. Actuators B Chem.
**2017**, 246, 904–909. [Google Scholar] [CrossRef] - Jewkes, R.; Burton, H.; Espino, D. Towards Additive Manufacture of Functional, Spline-Based Morphometric Models of Healthy and Diseased Coronary Arteries: In Vitro Proof-of-Concept Using a Porcine Template. J. Funct. Biomater.
**2018**, 9, 15. [Google Scholar] [CrossRef] [Green Version] - Teixeira, A.; Hernández-Rodríguez, J.; Wu, L.; Oliveira, K.; Kant, K.; Piairo, P.; Diéguez, L.; Abalde-Cela, S. Microfluidics-Driven Fabrication of a Low Cost and Ultrasensitive SERS-Based Paper Biosensor. Appl. Sci.
**2019**, 9, 1387. [Google Scholar] [CrossRef] [Green Version] - Farfán-Cabrera, L.I.; Pascual-Francisco, J.B.; Gallardo-Hernández, E.A.; Susarrey-Huerta, O. Application of digital image correlation technique to evaluate creep degradation of sealing elastomers due to exposure to fluids. Polym. Test.
**2018**, 65, 134–141. [Google Scholar] [CrossRef] - Bashirzadeh, Y.; Qian, S.; Maruthamuthu, V. Non-intrusive measurement of wall shear stress in flow channels. Sens. Actuators A. Phys.
**2018**, 271, 118–123. [Google Scholar] [CrossRef] - Rodrigues, R.O.; Pinho, D.; Bento, D.; Lima, R.; Ribeiro, J. Wall expansion assessment of an intracranial aneurysm model by a 3D Digital Image Correlation System. Measurement
**2016**, 88, 262–270. [Google Scholar] [CrossRef] [Green Version] - Huang, Y.M.; Tsai, N.C.; Lai, J.Y. Development of tactile sensors for simultaneous, detection of normal and shear stresses. Sens. Actuators A Phys.
**2010**, 159, 189–195. [Google Scholar] [CrossRef] - Unver, O.; Uneri, A.; Aydemir, A.; Sitti, M. Geckobot: A Gecko Inspired Climbing Robot Using Elastomer Adhesives. In Proceedings of the 2006 IEEE International Conference on Robotics and Automation (2006 ICRA), Orlando, FL, USA, 15–19 May 2006. [Google Scholar]
- Xu, F.; Li, X.; Shi, Y.; Li, L.; Wang, W.; He, L.; Liu, R. Recent Developments for Flexible Pressure Sensors: A Review. Micromachines
**2018**, 9, 580. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fallahi, H.; Zhang, J.; Phan, H.-P.; Nguyen, N.-T. Flexible Microfluidics: Fundamentals, Recent Developments, and Applications. Micromachines
**2019**, 10, 830. [Google Scholar] [CrossRef] [Green Version] - Banea, M.; Silva, L. The effect of temperature on the mechanical properties of adhesives for the automotive industry. Proc. Inst. Mech. Eng. Part L J. Mat. Des. Applic.
**2010**, 224, 51–62. [Google Scholar] [CrossRef] - Moreira, D.; Nunes, L. Comparison of simple and pure shear for an incompressible isotropic hyperelastic material under large deformation. Polym. Test.
**2013**, 32, 240–248. [Google Scholar] [CrossRef] [Green Version] - Nunes, L.S. Shear modulus estimation of the polymer polydimethylsiloxane (PDMS) using digital image correlation. Mater. Des.
**2010**, 31, 583–588. [Google Scholar] [CrossRef] - Nunes, L.S. Mechanical characterization of hyperelastic polydimethylsiloxane by simple shear test. Mater. Sci. Eng. A
**2011**, 528, 1799–1804. [Google Scholar] [CrossRef] - Post, D.; Han, B.; Ifju, P. High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials; Springer: Berlin/Heidelberg, Germany, 1997; ISBN 978-1-4612-4334-2. [Google Scholar]
- Wang, Y.; Gao, X.; Xie, X.; Wu, S.; Liu, Y.; Yang, L. Simultaneous dual directional strain measurement using spatial phase-shift digital shearography. Opt. Las. Eng.
**2016**, 87, 197–203. [Google Scholar] [CrossRef] - Ribeiro, J.; Fernandes, C.S.; Lima, R. Numerical Simulation of Hyperelastic Behaviour in Aneurysm Models. In Lecture Notes Computational Vision and Biomechanics; Springer: Cham, Switzerland, 2017; pp. 937–944. [Google Scholar] [CrossRef] [Green Version]
- Xue, L.; Pham, J.T.; Iturri, J.; Del Campo, A. Stick-Slip Friction of PDMS Surfaces for Bioinspired Adhesives. Langmuir
**2016**, 32, 2428–2435. [Google Scholar] [CrossRef] - Besson, J.; Cailetaud, G.; Chaboche, J.; Forest, S.; Blétry, M. Non-Linear Mechanics of Materials; Springer: London, UK, 2010. [Google Scholar] [CrossRef]
- Ribeiro, J.; Lopes, H.; Martins, P. A hybrid method to characterize the mechanical behaviour of biological hyperelastic tissues. Comp. Meth. Biomech. Biomed. Eng. Imag. Vis.
**2017**, 5, 157–164. [Google Scholar] [CrossRef] - Holzapfel, G. Nonlinear Solid Mechanics: A Continuum Approach for Engineering; John Wiley & Sons Ltd.: New York, NY, USA, 2000; ISBN 978-0-471-82319-3. [Google Scholar]
- Wriggers, P. Nonlinear Finite Element Methods; Springer: Berlin/Heidelberg, Germany, 2008; ISBN 978-3-540-71000-4. [Google Scholar]
- Yeoh, O.H. Some forms of the strain energy function for rubber. Rubber Chem. Technol.
**1993**, 66, 754–771. [Google Scholar] [CrossRef] - Bhowmick, A.K. Rubber Products Manufacturing Technology; CRC Press: New York, NY, USA, 1994; ISBN 9780824791124. [Google Scholar]
- Guo, Z.; Sluys, J. Application of a new constitutive model for the description of rubber-like materials under monotonic loading. Int. J. Solid Struct.
**2006**, 43, 2799–2819. [Google Scholar] [CrossRef] [Green Version] - Proulx, T. Mechanics of Biological Systems and Materials. In Proceedings of the Annual Conference on Experimental and Applied Mechanics, Uncasville, CT, USA, 13–16 June 2011; Springer Science & Business Media: New York, NY, USA, 2011. [Google Scholar]
- Laksari, K.; Shafieian, M.; Darvish, K. Constitutive model for brain tissue under finite compression. J. Biomech.
**2012**, 45, 642–646. [Google Scholar] [CrossRef] - Hartmann, S.; Neff, P. Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. Int. J. Solid Struct.
**2003**, 40, 2767–2791. [Google Scholar] [CrossRef] - Ritto, T.G.; Nunes, L.S. Bayesian model selection of hyperelastic models for simple and pure shear at large deformations. Comput. Struct.
**2015**, 156, 101–109. [Google Scholar] [CrossRef] - ASTM D412-0:2011. Standard Test Methods for Vulcanized Rubber and Thermoplastic Elastomers—Tension; ASTM International: West Conshohocken, PA, USA, 2016. [Google Scholar]
- Krahmer, D.M.; Polvorosa, R.; Lacalle, L.N.; Alonso-Pinillos, U.; Abate, G.; Riu, F. Alternatives for Specimen Manufacturing in Tensile Testing of Steel Plates. Exp. Tech.
**2016**, 40, 1555–1565. [Google Scholar] [CrossRef] - Silva, C.M.A.; Rosa, P.A.R.; Martins, P.A.F. Innovative Testing Machines and Methodologies for the Mechanical Characterization of Materials. Exp. Tech.
**2016**, 40, 569–581. [Google Scholar] [CrossRef] - Dixit, U.S.; Joshi, S.N.; Davim, J.P. Incorporation of material behavior in modeling of metal forming and machining processes: A review. Mater. Des.
**2011**, 32, 3655–3670. [Google Scholar] [CrossRef] - Aurrekoetxea, M.; Lacalle, L.N.; Llanos, I. Machining Stresses and Initial Geometry on Bulk Residual Stresses Characterization by On-Machine Layer Removal. Materials
**2020**, 13, 1445. [Google Scholar] [CrossRef] [Green Version] - López, M.; Rubio, M.; Sadek, S.H.; Veja, E.J. A simple emulsification technique for the production of micro-sized flexible powder of polydimethylsiloxane (PDMS). Pow. Tecnh.
**2020**, 306, 610–616. [Google Scholar] [CrossRef] - Zhang, J.; Sweedy, A.; Gitzhofer, F.; Baroud, G. A novel method for repeatedly generating speckle patterns used in digital image correlation. Opt. Las. Eng.
**2018**, 100, 259–266. [Google Scholar] [CrossRef] - Cardoso, C.; Fernandes, C.; Lima, R.; Ribeiro, J. Biomechanical analysis of PDMS channels using different hyperelastic numerical constitutive models. Mech. Res. Commun.
**2018**, 90, 26–33. [Google Scholar] [CrossRef] [Green Version] - Ribeiro, J.; Lopes, H.; Martins, P.; César, M.B. Mechanical analysis of PDMS material using biaxial test. AIMS Mater. Sci.
**2019**, 6, 97–110. [Google Scholar] [CrossRef]

**Figure 8.**Image of 2 mm polydimethylsiloxane (PDMS) thickness: (

**a**) before deformation; (

**b**) beginning of rupture; (

**c**) after rupture.

**Figure 9.**Image of 4 mm PDMS thickness: (

**a**) before deformation; (

**b**) beginning of rupture; (

**c**) after rupture.

**Figure 10.**(

**a**) Shear strain field for a 2 mm thickness PDMS plate; (

**b**) Shear strain field for a 4 mm thickness PDMS plate.

**Figure 11.**Shear strain field obtained by numerical simulation using the Mooney–Rivlin constitutive model. (

**a**) 2 mm and (

**b**) 4 mm of PDMS thickness.

**Figure 12.**Shear strain obtained numerically and experimentally along the y-direction for 2 mm of PDMS.

**Figure 13.**Shear strain obtained numerically and experimentally along the y-direction for 4 mm of PDMS.

Numerical Simulation | Displacement [mm] |
---|---|

1 | 0.5 |

2 | 0.2 |

Constitutive Model | Average | Minimum | Maximum | |
---|---|---|---|---|

Mooney-Rivlin 3 parameter | −0.09684 | −0.04678 | −0.10238 | PDMS Thickness 2 mm |

Yeoh 3rd Order | −0.09680 | −0.04722 | −0.10237 | |

Polynomial 3rd Order | −0.09680 | −0.04362 | −0.10514 | |

Gent | −0.09680 | −0.04709 | −0.10240 | |

Mooney-Rivlin 3 parameter | −0.02056 | −0.00585 | −0.02128 | PDMS Thickness 4 mm |

Yeoh 3rd Order | −0.02060 | −0.00585 | −0.02128 | |

Polynomial 3rd Order | −.02060 | −0.00585 | −0.02128 | |

Gent | −0.02060 | −0.00585 | −0.02128 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Souza, A.; Marques, E.; Balsa, C.; Ribeiro, J.
Characterization of Shear Strain on PDMS: Numerical and Experimental Approaches. *Appl. Sci.* **2020**, *10*, 3322.
https://doi.org/10.3390/app10093322

**AMA Style**

Souza A, Marques E, Balsa C, Ribeiro J.
Characterization of Shear Strain on PDMS: Numerical and Experimental Approaches. *Applied Sciences*. 2020; 10(9):3322.
https://doi.org/10.3390/app10093322

**Chicago/Turabian Style**

Souza, Andrews, Eduardo Marques, Carlos Balsa, and João Ribeiro.
2020. "Characterization of Shear Strain on PDMS: Numerical and Experimental Approaches" *Applied Sciences* 10, no. 9: 3322.
https://doi.org/10.3390/app10093322