Neo-Hookean model

The most basic simplification of this generalized equation is to take the first term which gives the neo-Hookean model ... 

In this equation, (j) denotes the solvent fraction in the polymer network, which is dependent on the degree of crosslinking and solvent quality. To put this in perspective, the shear modulus (G) is proportional to crosslinking density (v ) and the solvent fraction according to the Neo-Hookean model of rubber elasticity theory [62]... 

Neo-Hookean material This is an extension of the linear elastic Hooke s law to include large deformations. The main difference between the classical Hooke s law and the Neo-Hookean law is that in the latter the shear modulus is a function of the deformations. The strain energy density for the compressible Neo-Hookean model is given by... 

In the following the solution of the simple tension problem will be presented for three of the most used nonlinear elastic models, viz. Neo-Hooke, Mooney-Rivlin and Yeoh model. The strain energy function in the Neo- Hookean model is... 

Note that when no deformation is present, B = I (recall Example 1.4.2) and eq. l.S.l give r = GI. Because the pressure is arbitrary for an incompressible material, we can set p = G and thus make the total stress T = 0 in the rest state. An alternative way to write the neo-Hookean model is in terms of the strain tensor E (cf. Bird et al., 1987b, p. 365)... 

We can test out our large strain Hookean or neo-Hookean model with the deformations shown in Figure 1.4.2. For uniaxial extension we can calculate the stress components by substituting the components of B given in eq. 1.4.23 into eq. 1.5.2. 

For simple shear, the results from eq. 1.4.24 give B 2 = B21 = y,B = + and B22 = B i = 1. Applying the neo-Hookean model gives quite different results from extension... 

Thus, in simple shear the neo-Hookean model predicts a first normal stress difference that increases quadratically with strain. This also agrees with experimental results for rubber (note Figure 1.1.3). We will see in Chapter 4 that the same kinds of normal stress appear in shear of elastic liquids (recall the rod climbing in Figure 1.3). Note that there is only one normal stress difference, Ni, for the neo-Hookean solid in shear. 

The neo-Hookean model has been applied to many other large strain deformation problems. Several are given in the examples in Section 1.8 and in the exercises in Section 1.10. 

The neo-Hookean model gives a good but not perfect fit to tensile data on real rubber samples. As shown in Figures 1.1.2 and 1.6.1, tensile stress deviates from the model at high extensions. Is there some logical way to generalize the idea of an elastic solid to better describe experimental data ... 

In the neo-Hookean model, stress is linearly proportional to deformation. We can generalize our elastic model by letting stress be a function of the deformation. Thus... 

The strain-energy function that gives the neo-Hookean model is... 

However, because the deformation is not uniform, the stress will not be homogeneous. Aj lying the neo-Hookean model, eq. 1.S.2, gives... 

The solid lines are calculated using eq. 1.7.20 and G = 163 kPa. The excellent agreement between data and calculations indicates that the neo-Hookean model describes this material very well in shear up to x = 0.4. 

The components of the stress tensor were determined from the neo-Hookean model, eqs. 1.5.3and 1.5.4,andthearbitrarypressure p was eliminated by using the boundary tractions tz = ts = 0. Then the unknown traction ti was determined from t) = TuXi. 

A long, thin-walled rubber tube (Figure 1.8.S) is inflated with a gas pressure pa with its length Wo held constant. Assuming that the material obeys the neo-Hookean model, determine how much the tube will inflate. Show that this is a planar extension, identical to Example 1.8.2. 

We gave several specific constitutive equations that fit stress-deformation data for real rubber reasonably well. These are useful fordesignoftiresandothernibbergoods. In Section 1.8 we showed how to attack some elastic boundary value problems. Additional problems are given in the exercises. However, we introduced the neo-Hookean model primarily for its value in developing constitutive equations for viscolastic liquids, particularly in Chapter 4. 

Exercise 1.10.7 derived this result from the neo-Hookean model. 

This result has exactly the same functional dependence as the neo-Hookean model. Thus measurements of Tn in planar extension could not differentiate between the two. However... 

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