# Elasticity and Inelasticity of Rubber

## Abstract

In this chapter we focus on the isothermal phenomenological theory of the elasticity and inelasticity of rubber. First, we describe experimental results that characterize the elastic behaviour of rubber, in particular of vulcanized natural rubber. This is followed by illustrations of how the behaviour departs from the purely elastic; we examine stress softening associated with the Mullins effect, and the different degrees of stress softening for different rubbers are highlighted. Other inelastic effects such as hysteretic stress-strain cycling following pre-conditioning of the material (to remove the Mullins effect) are also described.

With this background established we then begin the process of mathematical modelling of these behaviours. We describe the theory of elasticity necessary for the modelling of the elastic behaviour, and for simple homogeneous deformations we illustrate the good agreement between theory and experiment. We then move on from elasticity and discuss the modelling of stress softening and the Mullins effect. For this purpose the (quasi-static) theory of pseudo-elasticity is used since this represents a relatively simple extension of the well-established theory of elasticity and is able to capture the Mullins effect both qualitatively and quantitatively. The theory is described and then used to fit some basic data on the Mullins effect.

Finally, we examine briefly the effects of time and rate dependence and associated with viscoelastic behaviour, and some outstanding problems in the modelling of the inelastic behaviour of rubber are discussed with particular reference to viscoelasticity.

## Keywords

Natural Rubber Deformation Gradient Residual Strain Pure Shear Nominal Stress## Preview

Unable to display preview. Download preview PDF.

## References

- Barenblatt, G.I., and Joseph, D.D. (1996).
*Collected Papers of R.S. Rivlin*. Springer, New York.Google Scholar - Beatty, M.F., and Krishnaswamy, S. (2000). A theory of stress softening in incompressible isotropic elastic materials.
*J. Mech. Phys. Solids*48: 1931–1965.MathSciNetCrossRefMATHGoogle Scholar - Bergström, J.S., and Boyce, M.C. (2000). Large strain time-dependent behaviour of filled elastomers.
*Mech. Materials*32: 627–644.CrossRefGoogle Scholar - Bernstein, B., Kearsley, A., and Zapas, L.J. (1963). A study of stress relaxation with finite strain.
*Trans. Soc. Rheol.*7: 391–410.CrossRefMATHGoogle Scholar - Besdo, D., Schuster, R.H., and Ihlemann, J. (2001).
*Constitutive Models for Rubber II*. Rotterdam: Balkema.Google Scholar - Bouasse, H., and Carrière, Z. (1903). Courbes de traction du caoutchouc vulcanisé.
*Ann. Fac. Sciences de Toulouse*5: 257–283.CrossRefMATHGoogle Scholar - Boyce, M.C., and Arruda, E.M. (2000). Constitutive models of rubber elasticity: a review.
*Rubber Chem. Technol.*73: 504–523.CrossRefGoogle Scholar - Busfield, J.J.C., and Muhr, A.H. (2003).
*Constitutive Models for Rubber III*. Rotterdam: Balkema.Google Scholar - Dafalias, Y.F. (1991). Constitutive model for large viscoelastic deformations of elastomeric materials.
*Mech. Res. Comm.*18: 61–66.MathSciNetCrossRefMATHGoogle Scholar - DeSimone, A., Marigo, J.J., and Teresi, L. (2001). A damage mechanics approach to stress softening and its application to rubber.
*Eur. J. Mech. A/Solids*20: 873–892.MathSciNetCrossRefMATHGoogle Scholar - Dorfmann, A., and Muhr, A. (1999).
*Constitutive Models for Rubber*. Rotterdam: Balkema.Google Scholar - Dorfmann, A., and Ogden, R.W. (2003). A pseudo-elastic model for loading, partial unloading and reloading of particle-reinforced rubber.
*Int. J. Solids Structures*40: 2699–2714.CrossRefMATHGoogle Scholar - Fu, Y.B., and Ogden, R.W. (2001).
*Nonlinear Elasticity: Theory and Applications.*Cambridge University Press.Google Scholar - Govindjee, S., and Simo, J. C. (1991). A micro-mechanically based continuum damage model for carbon black-filled rubbers incorporating the Mullins’ effect.
*J. Mech. Phys. Solids*39: 87–112.MathSciNetCrossRefMATHGoogle Scholar - Govindjee, S., and Simo, J. C. (1992a). Transition from micro-mechanics to computationally efficient phenomenology: carbon black filled rubbers incorporating Mullins’ effect.
*J. Mech. Phys. Solids*40: 213–233.CrossRefMATHGoogle Scholar - Govindjee, S., and Simo, J. C. (1992b). Mullins’ effect and the strain amplitude dependence of the storage modulus.
*Int. J. Solids Structures*29: 1737–1751.CrossRefMATHGoogle Scholar - Hayes, M.A., and Saccomandi, G. (2001).
*Topics in Finite Elasticity*, CISM Courses and Lectures, vol. 424. Wien: Springer.Google Scholar - Holzapfel, G.A. (2000).
*Nonlinear Solid Mechanics*. Chichester: John Wiley.MATHGoogle Scholar - Holzapfel, G.A., and Simo, J.C. (1996). A new viscoelastic constitutive model for continuous media at finite thermodynamical changes.
*Int. J. Solids Structures*33: 3019–3034.CrossRefMATHGoogle Scholar - Horgan, C.O., Ogden, R.W., and Saccomandi, G. (2003). A theory of stress softening of elastomers based on finite chain extensibility.
*Proc. R. Soc. Lond. A*,in press.Google Scholar - Huber, N., and Tsakmakis, C. (2000). Finite deformation viscoelasticity laws.
*Mech. Materials*32: 1–18.CrossRefGoogle Scholar - Johnson, A.R., Quigley, C.J., and Freese, C.E. (1995). A viscohyperelastic finite element model for rubber.
*Comp. Methods Appl. Mech. Eng.*127: 163–180.CrossRefMATHGoogle Scholar - Johnson, M. A., and Beatty, M. F. (1993a). The Mullins effect in uniaxial extension and its influence on the transverse vibration of a rubber string.
*Continuum Mech. Thermodyn.*5: 83–115.MathSciNetCrossRefGoogle Scholar - Johnson, M. A., and Beatty, M. F. (1993b). A constitutive equation for the Mullins effect in stress controlled uniaxial extension experiments.
*Continuum Mech. Thermodyn.*5: 301–318.MathSciNetCrossRefGoogle Scholar - Jones, D.F., and Treloar, L.R.G. (1975). The properties of rubber in pure homogeneous strain.
*J. Phys. D*,*Appl. Phys.*8: 1285–1304.CrossRefGoogle Scholar - Krishnaswamy, S., and Beatty, M.F. (2000). The Mullins effect in compressible solids.
*Int. J. Engng Sci.*38: 1397–1414.MathSciNetCrossRefMATHGoogle Scholar - Lion, A. (1996). A constitutive model for carbon black filled rubber: experimental investigation and mathematical representation.
*Cont. Mech. Thermodyn.*8: 153–169.CrossRefGoogle Scholar - Lubliner, J. (1985). A model of rubber viscoelasticity.
*Mech. Res. Comm.*12: 93–99.CrossRefGoogle Scholar - Marckmann, G., Verron, E., Cornet, L., Chagnon, G., Charrier, P., and Fort, P. (2002). A theory of network alteration for the Mullins effect.
*J. Mech. Phys. Solids*50: 2011–2028.CrossRefMATHGoogle Scholar - Muhr, A.H., Gough, J., and Gregory, I.H. (1999). Experimental determination of model for liquid silicone rubber: Hyperelasticity and Mullins’ effect. In Dorfmann, A., and Muhr, A., eds.,
*Proceedings of the First European Conference on Constitutive Models for Rubber*. Rotterdam: Balkema. 181–187.Google Scholar - Mullins, L. (1947). Effect of stretching on the properties of rubber.
*J. Rubber Research*16: 275–289.Google Scholar - Mullins, L. (1969). Softening of rubber by deformation.
*Rubber Chem. Technol.*42: 339–362.CrossRefGoogle Scholar - Mullins, L., and Tobin, N. R. (1957). Theoretical model for the elastic behaviour of filler-reinforced vulcanized rubbers.
*Rubber Chem. Technol.*30: 551–571.Google Scholar - Ogden, R.W. (1972). Large deformation isotropic elasticity: on the correlation of theory and experiment for incompressible rubberlike solids.
*Proc. R. Soc. Lond*. A 326: 565–584.CrossRefMATHGoogle Scholar - Ogden, R.W. (1982). Elastic deformation of rubberlike solids. In Hopkins, H.G., and Sewell, M.J., eds.,
*Mechanics of Solids*, The Rodney Hill 60th Anniversary Volume. Oxford: Perga-mon Press. 499–537.Google Scholar - Ogden, R.W. (1986). Recent advances in the phenomenological theory of rubber elasticity.
*Rubber Chemistry and Technology*59: 361–383.CrossRefGoogle Scholar - Ogden, R.W. (1997).
*Non-linear Elastic Deformations*. New York: Dover Publications.Google Scholar - Ogden, R.W. (2001). Pseudo-elasticity and stress softening. In Fu, Y.B., and Ogden, R.W., eds.,
*Nonlinear Elasticity: Theory and Applications*Cambridge University Press. 491–522.Google Scholar - Ogden, R.W. (2003). On an anisotropic theory of pseudo-elasticity. Manuscript in preparation.Google Scholar
- Ogden, R.W., and Roxburgh, D.G. (1999a). A pseudo-elastic model for the Mullins effect in filled rubber.
*Proc. R. Soc. Lond*. A 455: 2861–2877.MathSciNetCrossRefMATHGoogle Scholar - Ogden, R.W., and Roxburgh, D.G. (1999b). An energy-based model of the Mullins effect. In Dorfmann, A., and Muhr, A., eds.,
*Proceedings of the First European Conference on Constitutive Models for Rubber*. Rotterdam: Balkema. 23–28.Google Scholar - Reese, S., and Govindjee, S. (1998). A theory of finite visocelasticity and numerical aspects.
*Int. J. Solids Structures*35: 3455–3482.CrossRefMATHGoogle Scholar - Schapery, R.A. (2000). Nonlinear viscoelastic solids.
*Int. J. Solids Structures*37: 359–366.MathSciNetCrossRefMATHGoogle Scholar - Treloar, L.R.G. (1944). Stress-strain data for vulcanized rubber under various types of deformation.
*Trans. Faraday Soc.*40: 59–70.CrossRefGoogle Scholar - Treloar, L.R.G. (1975).
*The Physics of Rubber Elasticity*,3rd edition. Oxford University Press.Google Scholar - Truesdell, C.A., and Noll, W. (1965). In Flügge, S., ed.,
*The Nonlinear Field Theories of Mechanics: Handbuch der Physik Vol*.*111/3*. Berlin: Springer.Google Scholar - Valanis, K.C., and Landel, R.F. (1967). The strain-energy function of a hyperelastic material in terms of the extension ratios.
*J. Appl. Phys.*38: 2997–3002.CrossRefGoogle Scholar - Vangerko, H., and Treloar, L.R.G. (1978). The inflation and extension of rubber tube for biaxial strain studies.
*J. Phys. D, Appl. Phys.*11: 1969–1978.CrossRefGoogle Scholar - Yang, L.M., Shim, V.P.W., and Lim, C.T. (2000) A visco-hyperelastic approach to modelling the constitutive behaviour of rubber.
*Int. J. Impact Eng.*24: 545–560.CrossRefGoogle Scholar