Stress-strain experiments, rubber elasticity

The theory of rubber elasticity (Section 9.7) assumes a monodisperse distribution of chain lengths. Earher, the weakest link theory of elastomer rupture postulated that a typical elastomeric network with a broad distribution of chain lengths would have the shortest chains break first, the cause of failure. This was attributed to the limited extensibility presumably associated with such chains, causing breakage at relatively small deformations. The flaw in the weakest link theory involves the implicit assumption that all parts of the network deform affinely (24), whereas chain deformation is markedly nonaffine see Section 9.10.6. Also, it is commonly observed that stress-strain experiments are nearly reversible right up to the point of rupture. 

From the viewpoint of the mechanics of continua, the stress-strain relationship of a perfectly elastic material is fully described in terms of the strain energy density function W. In fact, this relationship is expressed as a linear combination erf the partial derivatives of W with respect to the three invariants of deformation tensor, /j, /2, and /3. It is the fundamental task for a phenomenologic study of elastic material to determine W as a function of these three independent variables either from molecular theory or by experiment. The present paper has reviewed approaches to this task from biaxial extension experiment and the related data. The results obtained so far demonstrate that the kinetic theory of polymer network does not describe actual behavior of rubber vulcanizates. In particular, contrary to the kinetic theory, the observed derivative bW/bI2 does not vanish. 

An extension of rubber elasticity (i.e. of the description of large, static and incompressible deformations) to nematic elastomers has been given in a large number of papers [52, 61-66]. Abrupt transitions between different orientations of the director under external mechanical stress have been predicted in a model without spatial nonuniformities in the strain field [52,63]. The effect of electric fields on rubber elasticity of nematics has been incorporated [65]. Finally the approach of rubber elasticity was also applied recently to smectic A [67] and to smectic C [68] elastomers. Comparisons with experiments on smectic elastomers do not appear to exist at this time. Recently a rather detailed review of the model of an-... 

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-... 

Effects of instrument compliance can induce large errors on shear measurements of elastic and viscoelastic properties of materials [1,2]. These effects are caused not only by the transducer but also the machine itself (load frame), and the rheometer fixtures. We present examples of rheometer compliance effects on the measurement of the material properties of small molecule glass formers and a commercially available polydimethysiloxane (PDMS) rubber. A TA Instruments ARES Rheometer was used with a strain gage transducer (Honeywell-Sensotec). Stress relaxation, aging experiments, and dynamic frequency sweep experiments were performed. We also propose a procedure to correct for comphance effects in stress relaxation experiments and dynamic frequency sweep experiments. Suggestions are made for both instrument and experimental design to avoid and/or reduce compliance effects. 

Stress relaxation is the time-dependent change in stress after an instantaneous and constant deformation and constant temperature. As the shape of the specimen does not change during stress relaxation, this is a pure relaxation phenomenon in the sense defined at the beginning of this section. It is common use to call the time dependent ratio of tensile stress to strain the relaxation modulus, E, and to present the results of the experiments in the form of E as a function of time. This quantity should be distinguished, however, from the tensile modulus E as determined in elastic deformations, because stress relaxation does not occur upon deformation of an ideal rubber. 

It is known that up to a certain limiting load, a solid will recover its original dimensions on the removal of the applied loads. This ability of deformed bodies to recover their original dimensions is known as elastic behavior. Beyond the limit of elastic behavior (elastic limit), a material will experience a permanent set or deformation even when the load is removed. Such a material is said to have undergone plastic deformation. For most materials, Hooke s law is obeyed within the elastic limit, that is, stress proportional to strain. However, proportionality between stress and strain does not always hold when a material exhibits elastic behavior. A typical example is rubber, which is elastic but does not show Hookean behavior over the entire elastic region. 

Figure 9.2 shows experimental data for a silicone polymer similar to the one used in the squeeze flow experiment shown in Figure 1.9. The material is viscoelastic, since both the storage modulus and the dynamic viscosity are nonzero. At low frequencies the storage modulus goes to zero and the dynamic viscosity goes to a low-frequency asymptotic value. The deformation at low frequencies is sufficiently slow to allow the individual polymer chains to respond to the imposed strain hence, the response is viscous, and it is clear that the low frequency limit of n must be the zero-shear viscosity, t]q. At high frequencies the individual chains are unable to respond and the stress is entirely the consequence of deformation of the entangled network. In this limit the polymer melt is indistinguishable from a cross-linked rubber network, and the deformation is that of an elastic body, with G going to an asymptotic value and rj to zero. The value of G in this rubbery plateau region is known as the shear modulus and is usually denoted G.

Figure 1.1.3 shows the results of a different kind of experiment on a similar rubber sample. Here the sample is sheared between two parallel plates maintained at the same separation X2. We see diat the shear stress is linear with the strain over quite a wide range howevo additional stress components, notmal stresses Ti i and T22, act on the block at large strain. In the introduction to this part of the text, we saw that elastic liquids can also generate normal stresses (Figure 1.3). In rubber, the normal stress difference depends on the shear strain squared... 

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