Mooney-Rivlin materials

If ar = 2, pr = 2C, ar = —2 and pr = — 2C2, this equation reduces into Eq. (73) for Mooney-Rivlin materials. A further development of this approach is given... 

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 ... 

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. 

A numerical tool capable of predicting the mechanical behaviour of SWCNTs reinforced rubber. The formulation is based in a micromechanical, non-linear, multi-scale finite element approach and utilizes a Mooney-Rivlin material model for the rubber and takes into account the atomistic nanostructure of the nanotubes. The interfacial load transfer characteristics were parametrically approximated via the use of joint elements of variable stiffness. The SWCNTs improve significantly the composite strength and toughness especially for higher volume fractions. 

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... 

Mooney-Rivlin material. (--------) Neo-Hookean Ci = 0.24 MPa (.). Mooney-Rivlin ... 

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. 

Strain based hyperelastic Mooney-Rivlin material models were used. 

Examining Figure 3, we see how a variety of Mooney-Rivlin material models applied to the model predict the response. MR05 is a Mooney-Rivlin material model based on 0-5% strain. MR15 is based on a stain range of 0-15% and MR50 is based on 0-50% strain. These models are based on the first cycle for each material experiment (tension, compression, and shear). 

In TPE, the hard domains can act both as filler and intermolecular tie points thus, the toughness results from the inhibition of catastrophic failure from slow crack growth. Hard domains are effective fillers above a volume fraction of 0.2 and a size <100 nm [200]. The fracture energy of TPE is characteristic of the materials and independent of the test methods as observed for rubbers. It is, however, not a single-valued property and depends on the rate of tearing and test temperature [201]. The stress-strain properties of most TPEs have been described by the empirical Mooney-Rivlin equation... 

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

If material is neo-Hookean, its Mooney-Rivlin plot ought to give a horizontal line and hence yield C2 = 0. Thus one is tempted to consider that nonzero C2 must be associated in one way or another with the deviation of a given material from the idealized network model, and it is understandable why so many rubber scientists have concerned themselves with evaluating the C2 term from the Mooney-Rivlin plot of uniaxial extension data. However, the point is that a linear Mooney-Rivlin plot, if found experimentally, does not always warrant that its intercept and slope may be equated to 2(9879/,) and 2(91V/9/2), respectively. This fact is illustrated below with actual data on natural rubber (NR) and styrene-butadiene copolymer rubber (SBR). 

Shen 391 has considered the thermoelastic behaviour of the materials described by the Mooney-Rivlin equation and has shown that the energetic component is given... 

It is very important to stress that the decrease of the internal energy contribution with increasing extension ratio is due to a decrease of the intermolecular interaction with deformation, since the intramolecular contribution is independent of the deformation in full accord with the statistical theory. At very high strains, the /.-dependent part of fu/f approaches a limiting value of —0.68 for the Mooney-Rivlin and —0.07 for the Valanis-Landel materials. 

An analytical elastic membrane model was developed by Feng and Yang (1973) to model the compression of an inflated, non-linear elastic, spherical membrane between two parallel surfaces where the internal contents of the cell were taken to be a gas. This model was extended by Lardner and Pujara (1980) to represent the interior of the cell as an incompressible liquid. This latter assumption obviously makes the model more representative of biological cells. Importantly, this model also does not assume that the cell wall tensions are isotropic. The model is based on a choice of cell wall material constitutive relationships (e.g., linear-elastic, Mooney-Rivlin) and governing equations, which link the constitutive equations to the geometry of the cell during compression. 

For the analysis of experimental force-deformation data, it is necessary to use a suitable constitutive equation for the material under test. The constitutive equation relates the stresses and strains that are generated in the wall during compression, and therefore relates the tensions and stretch ratios. For example, Liu et al. (1996) used a Mooney-Rivlin constitutive equation to investigate the compression of polyurethane microcapsules and the functions f, /2 and fa are produced in... 

The Mooney-Rivlin equation is readily available from the Constitutive Equation for isotropic elastic materials ... 

Creep, Swelling, and Extraction Studies. Additional indications of component interaction were found in the results of creep experiments with the xl series shown in Tables VI and VII. Creep in the xl series is fully recoverable, i. e. there is no permanent set, consistent with a PUx continuous phase. The increase in recoverable compliance, however, indicates a reduction in apparent crosslink density with increasing delay time before irradiation. This result is reinforced by the data in Table VII Ci and Cg are the material constants in the Mooney-Rivlin equation. The rubbery plateau modulus and the crosslink density of PUx prepared in BHA, which mimics xl formation, is less than that of PU prepared neat. 

Figure 6-6. Simulated Mooney-Rivlin plot, equation (6-82), for C2/C = 2. Solid line is the undistorted response of the material "data" are the results with a 1% (standard deviation) random error incorporated into both the force and length "measurements."...

In uniaxial extension, the Mooney-Rivlin model predicts Eq. (2) to hold, where Q and C2 are two material constants. 

An estimation of the junction contribution to the network modulus can be performed most unambiguously if topological and junction contributions give different stress-strain behaviour. Then, the parameters and or the Mooney-Rivlin parameters C, and Cj can be determined directly from the stress-strain data. Most of the experimental material is still represented and discussed in terms of the Mooney-Rivlin equation. Therefore, the front factor problem of real networks will be discussed within the framework of this equation. 

Various material models describe the deformation behavior of elastomers, for example Neo-Hooke, Mooney-Rivlin, etc. 

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 phenomenological approach to rubber-like elasticity is based on continuum mechanics and symmetry arguments rather than on molecular concepts [2, 17, 26, 27]. It attempts to fit stress-strain data with a minimum number of parameters, which are then used to predict other mechanical properties of the same material. Its best-known result is the Mooney-Rivlin equation, which states that the modulus of an elastomer should vary linearly with reciprocal elongation [2],... 

Finally, it is worth mentioning another approach used to describe nonlinear viscoelastic solids nonlinear differential viscoelasticity [49, 178, 179]. This theory has been successfully applied to model finite amplitude waves propagation [180-182]. It is the generalization to the three-dimensional nonlinear case of the rheological element composed by a dashpot in series with a spring. Thus in the simplest case, the stress depends upon the current values of strain and strain rate rally. In this sense, it can account for the nonlinear short-term response and the creep behavior, but it fails to reproduce the long-term material response (e.g., relaxation tests). The so-called Mooney-Rivlin viscoelastic material [183] and the incompressible version of the model proposed by Landau and Lifshitz [184] belraig to this class. 

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