Constitutive model for stretch-induced softening of the stress–stretch behavior of elastomeric materials

Abstract

Elastomeric materials experience stretch-induced softening as evidenced by a pre-stretched material exhibiting a significantly more compliant response than that of the virgin material. In this paper, we propose a fully three-dimensional constitutive model for the observed softening of the stress–strain behavior. The model adopts the Mullins and Tobin concept of an evolution in the underlying hard and soft domain microstructure whereby the effective volume fraction of the soft domain increases with stretch. The concept of amplified strain is then utilized in a mapping of the macroscopic deformation to the deformation experienced by the soft domain. The strain energy density function of the material is then determined from the strain energy of the soft domain and thus evolves as the volume fraction of soft domain evolves with deformation. Comparisons of model results for cyclic simple extension with the experimental data of Mullins and Tobin show the efficacy of the model and suggest that an evolution in the underlying soft/hard domain microstructure of the elastomer captures the fundamental features of stretch-induced softening. Model simulations of the cyclic stress–strain behavior and corresponding evolution in structure with strain for uniaxial tension, biaxial tension and plane strain tension are also presented and demonstrate three-dimensional features of the constitutive model.

Introduction

The equilibrium stress–strain behavior of elastomeric materials is observed to undergo a softening with strain history. This phenomenon is referred to as stress-softening, stretch-induced softening, and/or cyclic softening. It is observed in conventional rubbery materials (Mullins and Tobin 1957, Mullins and Tobin 1965; Harwood et al., 1965; Harwood and Payne 1966a, Harwood and Payne 1966b; Mullins, 1969; Bueche, 1961) as well as in thermoplastic elastomers such as thermoplastic polyurethanes (Yokoyama, 1978; Bonart and Muller-Riederer, 1981; Qi and Boyce, 2004) and thermoplastic vulcanizates (Boyce 2001a, Boyce 2001b). In rubbery materials, the softening of the equilibrium stress–strain curve is referred to as the “Mullins’ effect”, so named due to the comprehensive study of this behavior by Mullins on unfilled and filled rubbers during the 1950s and the 1960s (Mullins and Tobin 1957, Mullins and Tobin 1965; Harwood et al., 1965; Harwood and Payne 1966a, Harwood and Payne 1966b; Mullins, 1969). Fig. 1 illustrates the characteristic features of the softening of the equilibrium nominal stress-nominal strain behavior of elastomeric materials as observed in a uniaxial tension test. Three stress–strain curves, each on the same material to a final nominal strain of 3.0 are depicted: Curve 1 gives the monotonic stress–strain behavior of the material having had no prior strain history; Curve 2 gives the reloading stress–strain behavior of the material after having been initially subjected to a prior strain of 1.2; Curve 3 gives the reloading stress–strain behavior after having been subjected to a prior strain of 2.0. The reloading curves exhibit several key features:

• After having been subjected to a prior strain, the material exhibits a more compliant response at strains smaller than the maximum strain incurred in its prior strain history.

• During reloading, as the reloading strain approaches the maximum strain seen in its prior strain history, the stress–strain behavior begins to stiffen and rejoin the reference virgin curve; upon reaching the reference virgin curve, the stress–strain behavior follows that of the virgin stress–strain behavior.

• A larger prior strain gives a larger increase in compliance (greater softening of the response) upon reloading.

Although Mullins identified softening to occur in both unfilled and filled elastomers, its effect is far more pronounced in filled elastomers and therefore is frequently identified to be a filled elastomer phenomenon.

At present, most softening theories are based on two concepts. The first theory originates from Blanchard and Parkinson (1952) and Bueche 1960, Bueche 1961, who considered the increase in stiffness produced by stiff filler particles to be a result of rubber-filler attachments providing additional restrictions on the crosslinked rubber network. They attributed softening to result from the breakdown or loosening of some of these attachments. Bueche 1960, Bueche 1961, Dannenberg (1974), and Rigbi (1980) generalized the softening to be a result of strain-induced relative motion of carbon and rubber, and in some cases local separation of carbon black particles and rubber. Simo (1987), Govindjee and Simo 1991, Govindjee and Simo 1992, and Miehe and Keck (2000), and Lion 1996, Lion 1997 extended the Bueche concept and developed damage-based constitutive models to simulate the material behavior within the framework of large strain continuum mechanics.

The second theory posed to explain the softening phenomenon is due to Mullins and coworkers (Mullins and Tobin 1957, Mullins and Tobin 1965; Harwood et al., 1965; Harwood and Payne 1966a, Harwood and Payne 1966b; Mullins, 1969). They consider softening to be an evolution in soft and hard domains within the elastomer whereby stretch produces a quasi-irreversible rearrangement of molecular networks due to localized nonaffine deformation resulting from short chains reaching the limit of their extensibility. This nonaffine deformation produces a displacement of the network junctions from their initial state, which thus produces some form of rearrangement of hard and soft domains in the elastomeric phase with strain, acting to increase the effective volume fraction of soft domain. The concept of a phase transition with strain has been used by Wineman and coworkers (Rajagopal and Wineman, 1992; Wineman and Huntley, 1994) to capture the destruction and rebuilding of the underlying molecular network upon reaching critical strain values. The concept of hard/soft domain reorganization with strain has been used as motivation in the models of Beatty and coworkers (Johnson and Beatty 1993a, Johnson and Beatty 1993b; Beatty and Krishnaswamy, 2000) who consider molecular chains to be pulled out from clusters and transformed into soft regions, and of Marckmann et al. (2002) who propose a network alteration whereby molecular chain density decreases and the average number of monomer segments in a molecular chain increases, and of Ogden and coworkers (Ogden and Roxburgh 1999a, Ogden and Roxburgh 1999b; Dorfmann and Ogden, 2003; Horgan et al., 2004) who combine the concept of hard/soft domain reorganization with the damage approach.

In the Mullins approach, filled rubbers were treated as a composite system and the concept of amplified strain was used to explain the enhanced softening phenomenon observed in filled rubbers. In filled elastomers, the average strain (or alternatively, stretch) in the elastomeric domains is necessarily amplified over that of the macroscopic strain since the stiff filler particles accommodate little of the macroscopic strain. For uniaxial tension loading, the amplified elastomer stretch is taken to be Λ=1+X(λ−1), where X is an amplification factor dependent on particle volume fraction and distribution and λ is the macroscopic axial stretch. These researchers (Harwood et al., 1965) proposed that cyclic softening was a property of the unfilled vulcanizate and was magnified through the amplified strain for filled rubbers, thus producing an apparently greater degree of softening at any given macroscopic strain when compared to the corresponding unfilled elastomer.

Based on the concept of amplified strain, Mullins and Tobin (1957) in their very early work suggested that the softening in rubber vulcanizates was due to the decrease of volume fraction of effective hard domain, v_f, as a result of conversion of hard domain to soft domain. Recently, micro-mechanics studies on filled elastomers and filled polymers conducted by Boyce and coworkers (Bergstrom and Boyce 1999, Bergstrom and Boyce 2000; Boyce 2001a, Boyce 2001b) have provided some additional insights into possible hard/soft transition mechanisms. Micromechanical modeling of rigid particle filled elastomers by Bergstrom and Boyce (1999) reveals the entrapment of rubber domains within aggregates of stiff particles, thus resulting in the effective volume fraction of stiff particles to be larger than the physical fraction, i.e., the “occluded volume” effect postulated by earlier workers (Medalia and Kraus, 1994). Evolution in particle distribution with deformation could release occluded volumes of rubber and thus soften the material. Regions of stiffer vs. more compliant elastomer domains in unfilled elastomers could be thought to evolve in a similar manner. In a study of cyclic softening in thermoplastic vulcanizates (TPVs), where the vulcanizates are the filler particles, Boyce 2001a, Boyce 2001b showed that the softening is due to the gradual evolution in particle/matrix configuration due to straining during previous loading cycles. The plastic deformation of the contiguous thermoplastic phase acted to “release” vulcanizate particles creating a pseudo-continuous vulcanizate phase and thus a softer response during subsequent cycles. Although the material in the TPV study is a system of soft fillers/hard matrix, these micromechanical simulations demonstrate how an evolution in soft/hard microstructures can result in softening of the macroscopic mechanical response.

In this paper, we pursue Mullins’ early concept and propose a constitutive model where the softening of the equilibrium stress–stretch behavior is due to an evolution of the effective volume fraction of the soft domain during the deformation process whereupon occluded soft material domains are released during deformation due to the relative motions and deformation of the hard domains. Model predictions of the evolution in structure and corresponding stress–stretch behaviors under cyclic uniaxial loading are shown first; comparisons between model and experimental data from the literature are then presented. The three-dimensional nature of the model is illustrated in simulations of uniaxial tension, equibiaxial tension and plane strain tension.