Neo-Hookean solid

Somewhat beyond the scope of this book is to demonstrate that the theory, when simplified to this point, predicts that a = Gy, even at high strains (see problem 22). Note that the solid defined by the high-strain theory in equation (6-60) reduces to the Hookean solid at low strains, and is referred to occasionally as a neo-Hookean solid (see Section C). 

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 solid is a special case in which g2 = 0 and gi = G, a single material constant. 

We can write this equation in different cocndinate systems using only the stress terms in Table 1.7.1. In the uniaxial extension and simple shear examples, which we worked for the neo-Hookean solid, the stress was homogeneous, so all its derivatives are zero and eq. 1.7.17 is satisfied. However, with more complex sluq)es, such as twisting of a cylinder shown in Figure 1.7.3, the stresses do vary across the sample and die stress balance is required to solve for the tractions on die surface. 

Furthermore, in a number of polymer processing operations, such as blow molding, film blowing, and thermoforming, deformations are rapid and the polymer melt behaves more like a crosslinked rubber than a viscous liquid. Figure 1.1 showed typical deformation and recovery of a polymeric liquid. As the time scale of the experiment is shortened, the viscoelastic liquid looks more and more like the neo-Hookean solid. In Chapters 3 and 4 we develop models for the full viscoelastic response, but in fact in many cases of rapid deformations the simplest and often most realistic model for the stress response of these polymeric liquids is the elastic solid. 

For a rectangular rubber block, plane strain conditions were imposed in the width direction and the rubber was assumed to be an incompressible elastic solid obeying the simplest nonhnear constitutive relation (neo-Hookean). Hence, the elastic properties could be described by only one elastic constant, the shear modulus jx. The shear stress t 2 is then linearly related to the amount of shear y [1,2] ... 

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. 

It involves only two elastic constants. A special case, where fc = 0, is the neo-Hookean material, which can be derived from thermodynamics principles for a simple solid. Exact solutions can be obtained for the cylindrical deformation of a thick-walled tube. In the case where there is no residual strain, we have... 

Guo, Z., Peng, X. and Moran, B. (2006). Mechanical response of neo-hookean fiber reinforced incompressible nonlinearly elastic solids, International Journal of Solids and Structures 44, pp. 1949-1969. 

This familiar equation is more usually represented as a consequence of the molecular theories of a rubber network. Here we see that it follows from purely phenomenological considerations as a simple constitutive equation for the finite deformation of an isotropic, incompressible solid. Materials that obey this relationship are sometimes called neo-Hookean. 

Length, area, and volume change can also be expressed in terms of the invariants of B or C (see eqs. 1.4.45-1.4.47). Note that the Cauchy tensor operates on unit vectors that are defined in the past state. In the next section we will see that the Cauchy tensor is not as useful as the Finger tensor for describing the stress response at large strain for an elastic solid. But first we illustrate each tensor in Example 1.4.2. This example is particularly important because we will use the results direcfiy in the next section with our neo-Hookean constitutive equation. 

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 ... 

The deformation of a material is governed not oidy by a constitutive relation between deformation and stress, like the neo-Hookean equation discussed above, it also must obey the principles of conservation of mass and conservation of momentum. We have already used die mass conservation principle (conservation of volume for an incompressible material) in solving the uniaxial extension example, eq. 1.4.1. We have not yet needed the momentum balance because the balance was satisfied automatically for the simple deformations we chose that is, they involved no gravity, no flow, nor any inhomogeneous stress fields. However, these balances are needed to solve more complex deformations. They are presented for a flowing system because we will use these results in the following chapters. Here we see how they simplify for a solid. Detailed derivations of these equations are available in nearly every text on fluid or solid mechanics. 

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. 

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