Mullins effect deformations

After a test specimen of an elastomer has been subjected to repeated deformation, the modulus at some lesser deformation is lower than the initial value. The modulus after such stress softening has been termed the degraded modulus. See Mullins Effect. 

It was shown that the stress-induced orientational order is larger in a filled network than in an unfilled one [78]. Two effects explain this observation first, adsorption of network chains on filler particles leads to an increase of the effective crosslink density, and secondly, the microscopic deformation ratio differs from the macroscopic one, since part of the volume is occupied by solid filler particles. An important question for understanding the elastic properties of filled elastomeric systems, is to know to what extent the adsorption layer is affected by an external stress. Tong-time elastic relaxation and/or non-linearity in the elastic behaviour (Mullins effect, Payne effect) may be related to this question [79]. Just above the melting temperature Tm, it has been shown that local chain mobility in the adsorption layer decreases under stress, which may allow some elastic energy to be dissipated, (i.e., to relax). This may provide a mechanism for the reinforcement of filled PDMS networks [78]. 

To our knowledge, very few results have been published concerning the morphology of star SB copolymers. In 1975, Pedemonte et al,153) studied two copolymers (SB)4Si called Europrene T 161 and Europrene T 162 containing, respectively, 35% and 49% polystyrene. For the annealed samples, a disordered cylindrical structure was found in T 161 and a lamellar structure in T 162. In 1976, Pedemonte et al.ls4) published the results of a more detailed study performed on compression-moulded films of T 162. These films deformed increasing step by step the maximum strain value exhibit the Mullins effect. If in the original compression moulded film rods of polystyrene were arranged almost perpendicular to the compression plane, the de-... 

The data in Table 1 indicates that the deformation energy is lower for the second deformation compared with the first one. This effect is called the Mullins effect or stress-softening. Two explanations of this effect are given for the case of complete elastic recovery of a sample before its second deformation [39-43] ... 

Equation (1.5) has also been used to estimate the force at which a rubber molecule will become detached from a particle of a reinforcing filler (e.g., carbon black) when a filled rubber is deformed (Bueche, 1960, 1961). In this way, a general semiquantitative treatment has been achieved for stress-induced softening (Mullins effect) of filled mbbers (shown in Figure 1.5). 

Fig. 19 Mullins effect at small and moderate deformations observed on a 50 phr carbon-black filled SBR submitted to cyclic uniaxial tension...

Another softening phenomenon which manifests the dependence of the stress upon the entire history of deformation is the so-called Payne effect. Like the Mullins effect, this is a softening phenomena but it concerns the behavior of carbon blackfilled rubber subjected to oscillatory displacement. Strain dependence of the storage and loss moduli (Payne effect) at 70 °C and 10 Hz for a rubber compotmd with different concentration of carbon black filler [7] (Fig. 26). Indeed, the dynamic part of the stress response presents a rather strong nonlinear amplitude dependence, which is actually the Payne effect [8, 16, 43]. 

The Mullins effect, which can be considered as a hysteretic mechanism related to energy dissipated by the material during deformation, corresponds to a decrease in the number of elastically effective network chains. It results from chains that reach their limit of extensibility by strain amplification effects caused by the inclusion of undeformable filler particles [24,25]. Stress-softening in filled rubbers has been associated with the rupture properties and a quantitative relationship between total hysteresis (area between the first extension and the first release curves in the first extension cycle) and the enei-gy required for rupture has been derived [26,27]. 

Mullins L (1969) Softening of rubber by deformation. Rubber Chem Tec/i 42 339-362. Bueche F (1960) Molecular basis for the Mullins effect, J Appl Polym Sci 4 107-114. Clement F, Bokobza L and Monnerie L (2001) On the Mullins effect in silicarfilled polydiniethylsiloxane networks. Rubber Chem Tech 74 847-870. 

Non-linear constitutive equations are developed for highly filled polymeric materials. These materials typically exhibit an irreversible stress softening called the "Mullins Effect." The development stems from attempting to mathematically model the failing microstructure of these composite materials in terms of a linear cumulative damage model. It is demonstrated that p order Lebesgue norms of the deformation history can be used to describe the state of damage in these materials and can also be used in the constitutive equations to characterize their time dependent response to strain distrubances. This method of analysis produces time dependent constitutive equations, yet they need not contain any internal viscosity contributions. This theory is applied to experimental data and shown to yield accurate stress predictions for a variety of strain inputs. Included in the development are analysis methods for proportional stress boundary valued problems for special cases of the non-linear constitutive equation. 

Experience indicates that highly filled polymers do not fall into the category of fading memory materials even at small strains below detectable dewetting [1,2] or volume dilatation. These materials suffer from the "Mullins Effect" [2-8], which is a stresssoftening that occurs with deformation, and causes a permanent hysteresis on repeat loading. 

There is considerable evidence that all the hysteresis effects observed in these materials and most of the viscoelastic behavior can be caused by the time dependent failure of the polymer on a molecular basis and are not due to internal viscosity [1,2]. At near equilibrium rates and small strains filled polymers exhibit the same type of hysteresis that many lowly filled, highly cross-linked rubbers demonstrate at large strains [1-8]. This phenomenon is called the "Mullins Effect" and has been attributed to micro-structural failure. Mullins postulated that a breakdown of particle-particle association and possibly also particle-polymer breakdown could account for the effect [3-5]. Later Bueche [7,8] proposed a molecular model for the Mullins Effect based on the assumption that the centers of the filler particles are displaced in an affine manner during deformation of the composite. Such deformations would cause a highly non-uniform strain and stress gradient in the polymer... 

There have been various models and mechanisms proposed for the Mullins Effect . Bueche proposed a model based on chains failing due to physically non-homogeneous local deformations [7,8]. His... 

Figure 5,54. Results showing the relative percent of shear deformation dne to end effects in the three-point bending experiment in relation to sample length to thickness ratio a homogeneous sample (curve A) has a low end effect, while a unidirectional fiber composite (curve B) has a much more serious end effect. 6s/6p is the relative percent of shear deformation compored to flexural deformation. (Mullin and Knoell, 1970, reprinted with permission, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428.)...

The strain amplification proposed here is very different from that given by Mullins and Tobin. Uncrosslinked rubbers are used and therefore they are not in equilibrium deformation. The material behaviour is time dependent, i.e., viscoelastic. Therefore, the modulus ( e) is a function of strain and strain rate, e. If the stress is uniform throughout the specimen. Equation (7.5) may be adopted for the dynamic situation. If the matrix is a glass, a stress concentration may occur in the vicinity of fillers. When the matrix is a rubber, the stress concentration dissipates quickly. If the rate of dissipation is much faster than the deformation rate, the stress may be regarded as uniform, and this is the approximation used. At large deformations close to failure, stress concentration may occur and the approximation may not be valid. In such a case the amplification defined by Equation (7.5) includes the effect of the non-uniform stress. The equation is rewritten for the dynamic behaviour as... 

Within identical validity limits, Mullins and Tobin have shown that the stress-strain behavior of black-loaded rubber vulcanizates corresponds to the stress-strain response of pure gum vulcanizates multiplied by a suitable strain amplification factor X, which expresses the fact that the average strain supported by the rubber phase, is increased by the presence of filler. In other terms, the effective strain of the elastomer matrix X is given by X =X.xX, where X is the overall measured deformation of the filled material. 

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