Rubber stress-strain relations

Figure 18.1 is the typical stress-strain curves of the filled rubber (SBR filled with fine carbon black, HAF),

However, in the model Figure 18.3, we cannot give any reasonable answers to questions 2 and 3. It is well established that the matrix cross-hnked rubber in the filled system is almost the same as the unfilled cross-linked rubber. It means that the stress-strain relation of the matrix rubber in the filled... 

Figure 18.17 shows that the characteristics of the stress-strain curve depend mainly on the value of n the smaller the n value, the more rapid the upturn. Anyway, this non-Gaussian treatment indicates that if the rubber has the idealized molecular network strucmre in the system, the stress-strain relation will show the inverse S shape. However, the real mbber vulcanizate (SBR) that does not crystallize under extension at room temperature and other mbbers (NR, IR, and BR at high temperature) do not show the stress upturn at all, and as a result, their tensile strength and strain at break are all 2-3 MPa and 400%-500%. It means that the stress-strain relation of the real (noncrystallizing) rubber vulcanizate obeys the Gaussian rather than the non-Gaussian theory. 

Moreover, we must pay attention to the points that in the cross-linked rubber, the cross-link stops the sliding of molecules and has its own excluded volume. Generally, in the calculation of the stress-strain relation, the four-chain model is used for the cross-link junction and recently the eight-chain model is considered to be more realistic and available. Thus, it is quite reasonable to consider that the bulky excluded volume that a cross-link junction possesses must be a real obstacle for the orientation of molecules, just like the case observed in Figure 18.16B. 

Stress upturn in the stress-strain relation of carbon black-fiUed rubbers can be reasonably revealed in terms of the non-Gaussian treatment, by regarding the distance between adjacent carbon particles as the distance between cross-links in the theory. 

This consists of experimental measurements of stress-strain relations and analysis of the data in terms of the mathematical theory of elastic continua. Rivlin7-10 was the first to pply the finite (or large) deformation theory to the phenomenologic analysis of rubber elasticity. He correctly pointed out the above-mentioned restrictions on W, and proposed an empirical form... 

Van der Hoff,B.M.E. The stress-strain relation of swollen rubbers. Polymer (London) 6,397-399(1965). 

Eq. (IV-9) as well as the stress-strain relation, which follows easily from Eq. (IV-10), lead to an upward curvature at large strains. Such an upward curvature, due to finite extensibility, is indeed found experimentally, as demonstrated in Figs. 23 and 24, taken from work of Mullins (128). In swollen rubbers the effect comes into play at an earlier stage because of the pre-stretching due to the swelling. 

M.H.Wagner, J.Schaeffer, Rubbers and polymer melts Universal aspects of nonlinear stress-strain relations, J. Rheol. 31 (1993), 643-661. 

In the pretransitional region above the transition it has been shown that a stress-strain relation similar to that of common rubbers occurs [3]. From the behavior of stress-strain curves just below the phase transition, but above the transition to a monodomain state it has been demonstrated that one can linearly extrapolate to stress zero and extract the length /q of a corresponding monodomain sample [3]. When... 

Because the strain energy function for rubber is valid at large strains, and yields stress-strain relations which are nonlinear in character, the stresses depend on the square and higher powers of strain, rather than the simple proportionality expected at small strains. A striking example of this feature of large elastic deformations is afforded by the normal stresses tn,(22,133 that are necessary... 

Stress-strain measurements at uniaxial extension are the most frequently performed experiments on stress-strain behaviour, and the typical deviations from the phantom network behaviour, which can be observed in many experiments, provided the most important motivation for the development of theories of real networks. However, it has turned out that the stress-strain relations in uniaxial deformation are unable to distinguish between different models. This can be demonstrated by comparing Eqs. (49) and (54) with precise experimental data of Kawabata et al. on uniaxially stretched natural rubber crosslinked with sulphur. The corresponding stress-strain curves and the experimental points are shown in Fig. 4. The predictions of both... 

Figure 5 demonstrates the different behaviour resulting from Eqs. (49) and (54) in the case of uniaxial compression. We also tested the elastic potential (Eq. (44)) in the two cases v = 1/2 and v = —1/4 by comparing the corresponding stress-strain relations with biaxial extension experiments which cover relatively small as well as large deformation regions for an isoprene rubber vulcanizate. In the rectan-... 

The stress-strain relations of rubber-like substances and that of isotropic cellulose. 

Erom a practical viewpoint, Eq. (29.4) can be used to describe the stress-strain relation of a material if vi/(A) is known. m/(A) can be obtained in the laboratory in various ways, such as pure shear experiments as described by Valanis and Landel [60], by torsional measurements as described by Kearsley and Zapas [62] and by a combination of tension and compression experiments as also described by Kearsley and Zapas [62]. Treloar and co-workers [63] have also shown that the VL function description of the mechanical response of rubber is a very good one. The reader is referred to the original literature for these methods. 

M. H. Wagner and J. Schaeffer, Rubbers and Polymer Melts Universal Aspects of Nonlinear Stress-Strain Relations J. Rheol. 37, 643-661 (1993). 

Derive the stress-strain relation for a simple extension Ai = A produced by a force applied in the 1 direction, and hence show that the low strain tensile modulus for this rubber is given by is = 6(Ci -1- C2). 

Let us consider in more detail the elasticity of nematic rubbers, which is at the heart of understanding their specific properties. Consider a weakly crosslinked network with junction points sufficiently well spaced to ensure that the conformational freedom of each chain section is not restricted. We recall that for a conventional isotropic network the stress-strain relation for simple stretching (compression) of a unit cube of material can be derived as [62, 63] ... 

The difference between nematic and isotropic elastomers is simply the molecular shape anisotropy induced by the LC order, as discussed in Sect. 2. The simplest approach to nematic rubber elasticity is an extension of classical molecular mbber elasticity using the so-called neo-classical Gaussian chain model [64] see also Warner and Terentjev [4] for a detailed presentation. Imagine an elastomer formed in the isotropic phase and characterized by a scalar step length Iq. After cooling down to a monodomain nematic state, the chains obtain an anisotropic shape described by the step lengths tensor Ig. For this case the stress-strain relation can be written as ... 

In the field of rubber elasticity both experimentalists and theoreticians have mainly concentrated on the equilibrium stress-strain relation of these materials, i e on the stress as a function of strain at infinite time after the imposition of the strain > This approach is obviously impossible for polymer melts Another complication which has thwarted the comparison of stress-strain relations for networks and melts is that cross-linked networks can be stretched uniaxially more easily, because of their high elasticity, than polymer melts On the other hand, polymer melts can be subjected to large shear strains and networks cannot because of slippage at the shearing surface at relatively low strains These seem to be the main reasons why up to some time ago no experimental results were available to compare the nonlinear viscoelastic behaviour of these two types of material Yet, in the last decade, apparatuses have been built to measure the simple extension properties of polymer melts >. It has thus become possible to compare the stress-strain relation at large uniaxial extension of cross-linked rubbers and polymer melts ... 

The general factorable single integral constitutive equation is an equation that describes well the viscoelastic properties of a large class of crosslinked rubbers > and the elasticoviscous properties of many polymer melt8 under various types of deformation An appropriate way to compare the stress-strain relation for cured elastomers and polymer melts therefore is to calculate the strain-dependent function contained in this constitutive equation from experimental results and to compare the strain measures so obtained. 

Ronca and Allegra, and independently Flory, advanced the hypothesis that real rubber networks show departures from these theoretical equations as a result of a transition between the two extreme cases of behaviour. In subsequent papers Floryl >l and Flory and Ermanl derived a theory based on this concept. At small deformations the fluctuations of the network junctions are constrained by the extensive interpenetration of neighbouring, but topologically remote chains. The severity of these constraints is characterized by the value of the parameter k (k - 0 corresponds to the phantom network, k = to the affine network). With increasing deformation these constraints become less restrictive in the direction of the principal extension. The parameter t describes the departures from affine transformation of the shape of the domains of constraints. The resulting stress-strain relation also takes the form of Eq. (7) with... 

The strain measures for dry (unswollen) vulcanizates of a large number of natural rubbers, butadiene-styrene and butadiene-acrylonitrile copolymers, polydimethylsiloxanes, polymethylmethacrylates, polyethylacrylates and polybutadienes with different degrees of crosslinking and measured at various temperatures re confined within the shaded area in Fig. 1. These measures were determined from the stress as a function of extension at (or near) equilibrium, i.e. by applying Eq. (7). Therefore they only reproduce the equilibrium stress-strain relation for the crossllnked rubbers. In all cases the strain dependence of the tensile force (and hence of the tensile stress) was expressed in terms of the well-known Mooney-Rivlin equation, equating the equilibrium tensile stress to ... 

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