An anisotropic model of the Mullins effect. (English) Zbl 1115.74009

Summary: The Mullins effect in rubber-like materials is inherently anisotropic. However, most constitutive models developed in the past are isotropic. These models cannot describe the anisotropic stress-softening effect, often called the Mullins effect. In this paper we develop a phenomenological three-dimensional anisotropic model for the Mullins effect in incompressible rubber-like materials. The terms, damage function and damage point are introduced to facilitate the analysis of anisotropic stress softening in rubber-like materials. A material parametric energy function is postulated which depends on the right stretch tensor and written explicitly in terms of principal stretches and directions. The material parameters in the energy function are symmetric second-order damage and shear-history tensors. A class of energy functions and a specific form of constitutive equation are proposed which appear to simplify both the analysis of the three-dimensional model and the calculation of material constants from experimental data. The behaviour of tensional and compressive ground-state Young’s moduli in uniaxial deformations is discussed. To further justify our model, we show that the proposed model produces a transversely anisotropic non-virgin material in a stress-free state after a simple tension deformation. The proposed anisotropic theory is applied to several types of homogeneous deformations, and the theoretical results obtained are consistent with expected behaviour and compare well with several experimental data.


74B20 Nonlinear elasticity
74E10 Anisotropy in solid mechanics
Full Text: DOI


[1] Mullins L (1947) Effect of stretching on the properties of rubber. J Rubber Res 16:275–289
[2] Mullins L, Tobin NR (1957) Theoretical model for the elastic behaviour of filler-reinforced vulcanized rubbers. Rubber Chem Technol 30:551–571
[3] Govindjee S, Simo JC (1991) A micro-mechanically based continuum damage model for carbon black-filled rubbers incorporating the Mullin’s effect. J Mech Phys Solids 39:87–112 · Zbl 0734.73066
[4] Ogden RW, Roxburgh DG (1999) A pseudo-elastic model for the Mullins effect in filled rubber. Proc R Soc London A 455:2861–2877 · Zbl 0971.74011
[5] Beatty MF, Krishnaswamy S (2000) A theory of stress-softening in incompressible isotropic materials. J Mech Phys Solids 48:1931–1965 · Zbl 0983.74014
[6] Horgan CO, Ogden RW, Saccomandi G (2004) A theory of stress softening of elastomers based on finite chain extensibility. Proc R Soc London A 460:1737–1754 · Zbl 1070.74008
[7] Qi HJ, Boyce MC (2004) Contitutive model for stretched-induced softening of the stress-stretch behavior of elastomeric materials. J Mech Phys Solids 52:2187–2205 · Zbl 1115.74311
[8] James AG, Green A (1975) Strain energy functions of rubber. II. Characterisation of filled vulcanizates. J Appl Polym Sci 19:2033–2058
[9] Gough J (2005) Stress-strain behaviour of rubber. PhD Thesis Queen Mary and Westfield College, University of London · Zbl 1079.81042
[10] Pawelski H (2001) Softening behaviour of elastomeric media after loading in changing directions. In: Besdo D, Schuster RH, Ihlemann J (eds). Constitutive models for rubber II. A.A. Balkema Lisse, The Netherlands, pp 27–36
[11] Muhr AH, Gough J, Gregory IH (1999) Experimental determination of model for liquid silicone rubber. In: Muhr A, Dorfmann A (eds) Constitutive models for rubber. A.A. Balkema, Rotterdam, pp 181–187
[12] Shariff MHBM (2005) Multiaxial anisotropic stress-softening constitutive equation. Kauchuk I Rezina 2:16–19
[13] Shariff MHBM, Noor MA (2004) Anisotropic stress-softening model for damaged material. In: Proceedings of the seventh international conference on computational structures technology. Lisbon
[14] Miehe C (1995) Discontinuous and continuous damage evolution in Ogden-type large strain elastic materials. Euro Jl Mech A 14:697–720 · Zbl 0837.73054
[15] Laiarinandrasana L, Layouni K, Piques R (2001) Mullins effect on rubber materials: Damage model driving parameters. In: Besdo D, Schuster RH, and Ihlemann J (eds) Constitutive models for rubber II. A.A. Balkema Publishers Lisse, The Netherlands, pp 149–160
[16] Spencer AJM (1984) Constitutive theory of strongly anisotropic solids. In: Spencer AJM (ed) Constitutive theory of the mechanics of fiber reinforced composites (chapter IX), CISM Courses and Lectures No. 282. Springer, Wien, pp 1–32
[17] Spencer AJM (1971) Theory of invariants. In: Eringen AC (eds) Continuum physics I (Part III). Academic Press, New York, pp 239–353
[18] Valanis KC, Landel RF (1967) The strain-energy function of hyperelastic material in terms of the extension ratios. J Appl Phys 38:2997–3002
[19] Shariff MHBM (2000) Strain energy function for filled and unfilled rubberlike material. Rubber Chem Technol 73:1–21
[20] Shariff MHBM, Parker DF (2000) An extension of Key’s principle to nonlinear elasticity. J Engng Math 37:171–190 · Zbl 0991.74016
[21] Ogden RW (1984) Non-linear deformations. John Wiley, New York
[22] Mullins L (1969) Softening of rubber by deformation. Rubber Chem Technol 42:339–362
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