Neo-Hookean behavior

Sharma (90) has examined the fracture behavior of aluminum-filled elastomers using the biaxial hollow cylinder test mentioned earlier (Figure 26). Biaxial tension and tension-compression tests showed considerable stress-induced anisotropy, and comparison of fracture data with various failure theories showed no generally applicable criterion at the strain rates and stress ratios studied. Sharma and Lim (91) conducted fracture studies of an unfilled binder material for five uniaxial and biaxial stress fields at four values of stress rate. Fracture behavior was characterized by a failure envelope obtained by plotting the octahedral shear stress against octahedral shear strain at fracture. This material exhibited neo-Hookean behavior in uniaxial tension, but it is highly unlikely that such behavior would carry over into filled systems. 

During this same period, the equilibrium stress-strain properties of well characterized cross-linked networks were being studied intensively. More complex responses than the neo-Hookean behavior predicted by kinetic theory were observed. Among other possibilities it was speculated that, in some unspecified way, chain entanglements might be a contributing factor. 

If the crosslink is removed, the system reverts to two independent chains. To preserve the symmetry of the system one needs to consider the average for three tetrahedra operating together, one for each of the three possible pairings of the strands. The result is again neo-Hookean behavior but with a lower modulus contribution. 

This result differs somewhat from the expression obtained using the Doi-Edwards model (Eq.40), and it gives a larger departure from neo-Hookean behavior for uniaxial extension (A q>endix II and Fig. 9). In the limit of small deformations the entire contribution to stress comes from the first term in Eq. 62. The entanglement contribution to the infinitesimal shear modulus is predsely the same as the Doi-Edwards expression for the plateau modulus (Eq. 37)... 

A point worth noting here is that several of the molecular models that will be described in the subsequent sections are Neo-Hookean in form. Normally, dry rubbers do not exhibit Neo-Hookean behavior. As for the Mooney-Rivlin form of strain energy density function, rubbers may follow such behavior in extension, yet they do not behave as Mooney-Rivlin materials in compression. In Fig. 29.2, we depict typical experimental data for a polydimethylsiloxane network [39] and compare the response to Mooney-Rivlin and Neo-Hookean behaviors. The horizontal lines represent the affine and the phantom limits (see Network Models in Section 29.2.2). The straight line in the range A <1 shows the fit of the Mooney-Rivlin equation to the experimental data points. 

Because of the limited usefulness of the Mooney-Rivlin equation, it is probably not worthwhile to seek a molecular interpretation of the coefficients C and C2 the deviations from neo-Hookean behavior should be examined in some other framework. However, if the deviations are expressed in terms of the ratio = C2/(C] + C2), this quantity can be correlated rather successfully with the relative numbers of trapped entanglements and cross-links in the network. It may be inferred from this and other studies that both trapped entanglements and cross-links contribute to C, but that C2 is associated with trapped entanglements only. 

A wide, short, and thin sheet of rubber, Lo is clamped along its wide edges and pulled from its original length Lo to L with a force f. This deformation is called planar extension, plain strain, or sometimes pure shear. Assume neo-Hookean behavior, eq. 1.5.2. 

Thus, the uniaxial data, Figs. 13 and l L, indicate neo-Hookean behavior with C2=0 while the biaxial data obtained at still smaller strains indicate a more complex response. Since the fomer data are not accurate in this strain region and our experiments could not be carried to large strains without causing the specimens to crack, it is not possible to say whether a discrepancy exists or whether the apparent C2 value will decrease at still higher strain levels. 

If a Neo-Hookean constitutive equation is used to describe the tensile behavior of the cross-linked interlayer... 

The simplest model is the statistical theory of rubber-like elasticity, also called the affine model or neo-Hookean in the solids mechanics community. It predicts the nonlinear behavior at high strains of a rubber in uniaxial extension with Fq. (1), where ctn is the nominal stress defined as F/Aq, with F the tensile force and Aq the initial cross-section of the adhesive layer, A is the extension ratio, and G is the shear modulus. 

Fig. 29. (a) Reduced stress plot for the Neo-Hookean and Mooney-Rivlin materials of Figure 28. (b) Reduced stress plot for natural rubber and a polydimethylsiloxane (PDMS) rubber as indicated. Plot illustrates that actual rubber behavior may be Mooney-Rivlin-like in tension (>. < 1), but not in compression. Natural rubber PDMS. Plot from Han et al. (89), natural rubber data from Ref. 90, and PDMS data form reference 91. 

For cross-linked polymers at equilibrium (Chapter 14, Section Cl), an equation of this form provides a good empirical fit to stress-strain relations up to moderate extensions," although it is not a proper constitutive equation and is inconsistent with behavior in other types of deformation. " "" The data of Fig. 13-19 and many other examples "" " " " are quite well described by equation 31 with time-dependent Cl and C2 as illustrated in Fig. 13-20. The sum, 6(C + C2), corresponds to the relaxation modulus E(t) of linear viscoelasticity. The term with Ci (which corresponds to neo-Hookean strain dependence) relaxes first and the term with C2 relaxes about two decades later. 

One would be in an ad-hoc fashion to assume that, because of the tendency toward interpenetration, near the crack tips the crack surfaces would come in smooth contact and form a cusp, and the resulting contact region would consist of a single uninterrupted zone rather than the sum of a series of discrete zones as implied by the oscillatory nature of the elastic solution (see Comninou [ll], Atkinson [l2]). Another way is to assume that near the crack tip the linear theory is not valid and to use a large deformation nonlinear theory. An asymptotic analysis using such a theory was provided by Knowles and Sternberg [l3] for the plane stress interface crack problem in two bonded dissimilar incompressible Neo-Hookean materials which shows no oscillatory behavior for stresses or... 

Figure 2. Normal force vs. angle of twist times torque for four polymers showing that PMMA (squares) and PEMA (circles) are far from neo-Hookean in behavior while PSF (upright triangles) and PC (inverted triangles) are close to neo-Hookean. Tests performed at approximately T=Tp (After reference 19).

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