Research Papers

A Constitutive Model for Soft Materials Incorporating Viscoelasticity and Mullins Effect

[+] Author and Article Information
Tongqing Lu

State Key Lab for Strength and
Vibration of Mechanical Structures,
Shaanxi Engineering Laboratory for Vibration
Control of Aerospace Structures,
Department of Engineering Mechanics,
Xi'an Jiaotong University,
Xi'an 710049, China

Jikun Wang, Ruisen Yang

State Key Lab for Strength and
Vibration of Mechanical Structures,
Department of Engineering Mechanics,
Xi'an Jiaotong University,
Xi'an 710049, China

T. J. Wang

State Key Lab for Strength and
Vibration of Mechanical Structures,
Department of Engineering Mechanics,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: wangtj@mail.xjtu.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 8, 2016; final manuscript received November 4, 2016; published online November 22, 2016. Editor: Yonggang Huang.

J. Appl. Mech 84(2), 021010 (Nov 22, 2016) (9 pages) Paper No: JAM-16-1493; doi: 10.1115/1.4035180 History: Received October 08, 2016; Revised November 04, 2016

Soft materials including elastomers and gels are widely used in applications of energy absorption, soft robotics, bioengineering, and medical instruments. For many soft materials subject to loading and unloading cycles, the stress required on reloading is often less than that on the initial loading, known as Mullins effect. Meanwhile, soft materials usually exhibit rate-dependent viscous behavior. Both effects were recently reported on a new kind of synthesized tough gel, with capability of large deformation, high strength, and extremely high toughness. In this work, we develop a coupled viscoelastic and Mullins-effect model to characterize the deformation behavior of the tough gel. We modify one of the elastic components in Zener model to be a damageable spring to incorporate the Mullins effect and model the viscous effect to behave as a Newtonian fluid. We synthesized the tough gel described in the literature (Sun et al., Nature 2012) and conducted uniaxial tensile tests and stress relaxation tests. We also investigated the two effects on three other soft materials, polyacrylate elastomer, Nitrile-Butadiene Rubber, and polyurethane. We find that our presented model is so robust that it can characterize all the four materials, with modulus ranging from a few tens of kilopascal to megapascal. The theory and experiment for all tested materials agree very well.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Mullins, L. , and Tobin, N. , 1957, “ Theoretical Model for the Elastic Behavior of Filler-Reinforced Vulcanized Rubbers,” Rubber Chem. Technol., 30(2), pp. 555–571. [CrossRef]
Mullins, L. , and Tobin, N. , 1965, “ Stress Softening in Rubber Vulcanizates—Part I: Use of a Strain Amplification Factor to Describe the Elastic Behavior of Filler–Reinforced Vulcanized Rubber,” J. Appl. Polym. Sci., 9(9), pp. 2993–3009. [CrossRef]
Bhuiyan, A. , Okui, Y. , Mitamura, H. , and Imai, T. , 2009, “ A Rheology Model of High Damping Rubber Bearings for Seismic Analysis: Identification of Nonlinear Viscosity,” Int. J. Solids Struct., 46(7), pp. 1778–1792. [CrossRef]
Heinrich, G. , Klüppel, M. , and Vilgis, T. A. , 2002, “ Reinforcement of Elastomers,” Curr. Opin. Solid State Mater. Sci., 6(3), pp. 195–203. [CrossRef]
Tantayanon, S. , and Juikham, S. , 2004, “ Enhanced Toughening of Poly (Propylene) With Reclaimed‐Tire Rubber,” J. Appl. Polym. Sci., 91(1), pp. 510–515. [CrossRef]
Diani, J. , Fayolle, B. , and Gilormini, P. , 2009, “ A Review on the Mullins Effect,” Eur. Polym. J., 45(3), pp. 601–612. [CrossRef]
Johnson, M. , and Beatty, M. , 1993, “ The Mullins Effect in Uniaxial Extension and Its Influence on the Transverse Vibration of a Rubber String,” Continuum Mech. Thermodyn., 5(2), pp. 83–115. [CrossRef]
Johnson, M. , and Beatty, M. , 1993, “ A Constitutive Equation for the Mullins Effect in Stress Controlled Uniaxial Extension Experiments,” Continuum Mech. Thermodyn., 5(4), pp. 301–318. [CrossRef]
Beatty, M. F. , and Krishnaswamy, S. , 2000, “ A Theory of Stress-Softening in Incompressible Isotropic Materials,” J. Mech. Phys. Solids, 48(9), pp. 1931–1965. [CrossRef]
Qi, H. , and Boyce, M. , 2004, “ Constitutive Model for Stretch-Induced Softening of the Stress–Stretch Behavior of Elastomeric Materials,” J. Mech. Phys. Solids, 52(10), pp. 2187–2205. [CrossRef]
Blanchard, A. , and Parkinson, D. , 1952, “ Breakage of Carbon-Rubber Networks by Applied Stress,” Ind. Eng. Chem., 44(4), pp. 799–812. [CrossRef]
Bueche, F. , 1960, “ Molecular Basis for the Mullins Effect,” J. Appl. Polym. Sci., 4(10), pp. 107–114. [CrossRef]
Simo, J. , 1987, “ On a Fully Three-Dimensional Finite-Strain Viscoelastic Damage Model: Formulation and Computational Aspects,” Comput. Methods Appl. Mech. Eng., 60(2), pp. 153–173. [CrossRef]
Gong, J. P. , Katsuyama, Y. , Kurokawa, T. , and Osada, Y. , 2003, “ Double‐Network Hydrogels With Extremely High Mechanical Strength,” Adv. Mater., 15(14), pp. 1155–1158. [CrossRef]
Webber, R. E. , Creton, C. , Brown, H. R. , and Gong, J. P. , 2007, “ Large Strain Hysteresis and Mullins Effect of Tough Double-Network Hydrogels,” Macromolecules, 40(8), pp. 2919–2927. [CrossRef]
Zhang, T. , Lin, S. , Yuk, H. , and Zhao, X. , 2015, “ Predicting Fracture Energies and Crack-Tip Fields of Soft Tough Materials,” Extreme Mech. Lett., 4, pp. 1–8. [CrossRef]
Zhao, X. , 2014, “ Multi-Scale Multi-Mechanism Design of Tough Hydrogels: Building Dissipation Into Stretchy Networks,” Soft Matter, 10(5), pp. 672–687. [CrossRef] [PubMed]
Lee, K. Y. , and Mooney, D. J. , 2001, “ Hydrogels for Tissue Engineering,” Chem. Rev., 101(7), pp. 1869–1880. [CrossRef] [PubMed]
Hong, S. , Sycks, D. , Chan, H. F. , Lin, S. , Lopez, G. P. , Guilak, F. , Leong, K. W. , and Zhao, X. , 2015, “ 3D Printing of Highly Stretchable and Tough Hydrogels Into Complex, Cellularized Structures,” Adv. Mater., 27(27), pp. 4035–4040. [CrossRef] [PubMed]
Sun, J.-Y. , Zhao, X. , Illeperuma, W. R. , Chaudhuri, O. , Oh, K. H. , Mooney, D. J. , Vlassak, J. J. , and Suo, Z. , 2012, “ Highly Stretchable and Tough Hydrogels,” Nature, 489(7414), pp. 133–136. [CrossRef] [PubMed]
Gong, J. P. , 2010, “ Why are Double Network Hydrogels so Tough?,” Soft Matter, 6(12), pp. 2583–2590. [CrossRef]
Nakajima, T. , Kurokawa, T. , Ahmed, S. , Wu, W.-L. , and Gong, J. P. , 2013, “ Characterization of Internal Fracture Process of Double Network Hydrogels Under Uniaxial Elongation,” Soft Matter, 9(6), pp. 1955–1966. [CrossRef]
Tang, J. , Xu, G. , Sun, Y. , Pei, Y. , and Fang, D. , 2014, “ Dissipative Properties and Chain Evolution of Highly Strained Nanocomposite Hydrogel,” J. Appl. Phys., 116(24), p. 244901. [CrossRef]
Tang, J. , Chen, X. , Pei, Y. , and Fang, D. , 2016, “ Pseudoelasticity and Nonideal Mullins Effect of Nanocomposite Hydrogels,” ASME J. Appl. Mech., 83(11), p. 111010. [CrossRef]
Wang, X. , and Hong, W. , 2011, “ Pseudo-Elasticity of a Double Network Gel,” Soft Matter, 7(18), pp. 8576–8581. [CrossRef]
Foo, C. C. , Cai, S. , Koh, S. J. A. , Bauer, S. , and Suo, Z. , 2012, “ Model of Dissipative Dielectric Elastomers,” J. Appl. Phys., 111(3), p. 034102. [CrossRef]
Karimi, A. , Navidbakhsh, M. , and Beigzadeh, B. , 2014, “ A Visco-Hyperelastic Constitutive Approach for Modeling Polyvinyl Alcohol Sponge,” Tissue Cell, 46(1), pp. 97–102. [CrossRef] [PubMed]
Seitz, M. E. , Martina, D. , Baumberger, T. , Krishnan, V. R. , Hui, C.-Y. , and Shull, K. R. , 2009, “ Fracture and Large Strain Behavior of Self-Assembled Triblock Copolymer Gels,” Soft Matter, 5(2), pp. 447–456. [CrossRef]
Long, R. , Mayumi, K. , Creton, C. , Narita, T. , and Hui, C.-Y. , 2014, “ Time Dependent Behavior of a Dual Cross-Link Self-Healing Gel: Theory and Experiments,” Macromolecules, 47(20), pp. 7243–7250. [CrossRef]
Sasso, M. , Chiappini, G. , Rossi, M. , Cortese, L. , and Mancini, E. , 2014, “ Visco-Hyper-Pseudo-Elastic Characterization of a Fluoro-Silicone Rubber,” Exp. Mech., 54(3), pp. 315–328. [CrossRef]
Qi, H. , and Boyce, M. , 2005, “ Stress–Strain Behavior of Thermoplastic Polyurethanes,” Mech. Mater., 37(8), pp. 817–839. [CrossRef]
Tauheed, F. , and Sarangi, S. , 2015, “ Rheological Model for Stress-Softened Visco-Hyperelastic Materials,” Int. J. Damage Mech., 24(5), pp. 728–749. [CrossRef]
Ogden, R. , and Roxburgh, D. , 1999, “ A Pseudo–Elastic Model for the Mullins Effect in Filled Rubber,” Proc. R. Soc. London. A, pp. 2861–2877.


Grahic Jump Location
Fig. 1

(a) Nominal stress–stretch curves of the tough gel under uniaxial cyclic loading at different loading rates and (b) nominal stress–stretch curves of the tough gel under multiple uniaxial cyclic loading

Grahic Jump Location
Fig. 2

Rheological model with Mullins effect: (a) a damageable spring in series with a Kevin unit and (b) a damageable spring in parallel with a Maxwell unit

Grahic Jump Location
Fig. 3

(a) Samples of four different materials and (b) Shimadzu tensile tester

Grahic Jump Location
Fig. 4

Nominal stress–stretch curves for uniaxial tension tests at a nominal strain rate 3.33×10-3/s : (a) tough gel, (b) VHB, (c) NBR, and (d) PU. The dots are experimental data and the solid lines are theoretical predictions.

Grahic Jump Location
Fig. 5

Representative profile of the history of applied stretch in stress relaxation tests

Grahic Jump Location
Fig. 6

Experimental results and theoretical predictions for stress relaxation tests of the tough gel under different loading rates (a), (c) stress–stretch curves and (b), (d) stress-time history

Grahic Jump Location
Fig. 7

Experimental results and theoretical predictions for stress relaxation tests of VHB under different loading rates

Grahic Jump Location
Fig. 8

Experimental results and theoretical predictions for stress relaxation tests of NBR under different loading rates

Grahic Jump Location
Fig. 9

Experimental results and theoretical predictions for stress relaxation tests of PU under different loading rates

Grahic Jump Location
Fig. 10

Comparisons of stress–stretch curves for the tough gel under different loading rates. The curve under the loading rate 3.33×10-3/s is from the uniaxial tensile test. The curves under the other two loading rates are from the stress relaxation tests.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In