Mooney-Rivlin strain-energy function

We can also derive an expression on the basis of the Mooney-Rivlin strain energy function for swollen elastomers. A dry elastomer sample will undergo two types of deformation one due to swelling and the other due to extension. The strain energy function per unit volume of swollen elastomer is related to that of the dry sample by... 

Repeat Problem 3 using the Mooney-Rivlin strain energy function [equation (6-80)]. 

FIGURE 29.2. Comparison of typical stress-strain data for PDMS rubber [39] in a Mooney-Rivlin plot with Neo-Hookean and Mooney-Rivlin strain energy function descriptions. (See text for discussion). 

Equation (29.7) makes obvious the reasons for the representation of experimental data in the so-called Mooney-Rivlin plot. If the material has a Mooney-Rivlin strain energy density function then a plot of (cn — straight line with the slope and intercept at A = 1 determining 2C2 and (2Ci -I- 2C2), respectively. 

The stored strain energy can also be determined for the general case of multiaxial stresses [1] and lattices of varying crystal structure and anisotropy. The latter could be important at interfaces where mode mixing can occur, or for fracture of rubber, where f/ is a function of the three stretch rations 1], A2 and A3, for example, via the Mooney-Rivlin equation, or suitable finite deformation strain energy functional. 

There are two common phenomenological strain energy functions that have been used to describe the stress-strain response of rubber [58,59,64]. These are referred to as the Neo-Hookean form and the Mooney-Rivlin form and both can be written as Valanis-Landel forms, although they represent truncated forms of more general strain energy density functions. The Neo-Hookean form is a special form of the Mooney-Rivlin form, so we will begin with the latter. For a Mooney-Rivlin material the strain energy density function is written as ... 

Mooney-Rivlin material Mooney and Rivlin developed a hyperelastic constitutive material model. The strain energy function is written in the form of series expansion of... 

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... 

Mooney—Rivlin model would describe the behaviors exhibited (89—91). The reader is referred to reviews of various proposed strain energy functions (89,92,93) for further information. 

Comparison with values of and of the NAST model described earlier supports this hypothesis [36]. However, unlike the NAST model, the Mooney-Rivlin model fails to predict compression data as the linearity shown in Figure 9.21 continues for a > 1, whereas the experimental data show a maximum and a decrease in the function plotted in Figure 9.21. There have been many other constitutive relations for rubbers based on different representation of the strain energy function... 

For the Neo-Hookean material, the strain energy density function is the same as the Mooney-Rivlin material but with C2 = 0 ... 

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. 

Considerably better agreement with the observed stress-strain relationships has been obtained through the use of empirical equations first proposed by Mooney and subsequently generalized by Rivlin. The latter showed, solely on the basis of required symmetry conditions and independently of any hypothesis as to the nature of the elastic body, that the stored energy associated with a deformation described by ax ay, az at constant volume (i.e., with axayaz l) must be a function of two quantities (q +q +q ) and (l/a +l/ay+l/ag). The simplest acceptable function of these two quantities can be written... 

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