Compressibility strain energy function

Finite Elasticity Theory The VL Representation. While the above description of the finite deformation behavior of elastic materials is very powerful, the limitation on it is that the material parameters W and W2 need to be determined in each geometry of deformation of interest. Hence, the torsional measurements described above only give values of Wi(/i, I2) and W2(/i, I2) for the condition of shear (torsion is anonhomogeneous shear) and that condition is/i = l2 = 3- -y. More measurements need to be made to obtain the parameters in extension, compression, etc. However, in 1967, Valanis and Landel (98) proposed a strain energy function that, rather than being a function of the invariants, is a function of the stretches Xj. The function was assumed to be separable in the stretches as... 

An argument to resolve the discrepancy between the failure envelopes obtained for different modes of straining is indicated by the work of Blatz, Sharda, and Tschoegl [42]. These authors have proposed a generalized strain energy function as constitutive equation of multiaxial deformation. They incorporated more of the nonlinear behavior in the constitutive relation between the strain energy density and the strain. They were then able to describe simultaneously by four material constants the stress-strain curves of natural rubber and of styrene-butadiene rubber in simple tension, simple compression or equibiaxial tension, pure shear, and simple shear. 

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... 

Fig. 1. (a) PL spectra and bandgap from fit to absorption data in InN (b) Comparison of bandgap energy, peak PL emission energy and residual compressive strain as a function of InN thickness (c) Extrapolation of bandgap to zero strain. 

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. 

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... 

Fig. 19.1 Models of pseudomorphic monolayCTs of Pt on three different substrates inducing compressive strain (Ir(lll) and Pd(lll)) and expansive strain (Au(lll)) and activation energies for O2 dissociation and OH formation on differoit PtML/X surfaces as a function of oxygen binding energy. Figure redrawn by the author with permission based on a diagram provided by R. Adzic, based on diagrams in [17]...

Fig, 31. Crack growth rate as a function of strain energy , experimental results for compression, O theoretical results for compression, experimental results for simple tension. Redrawn from... 

The strain energy density function V for a compressible isotropic neo-Hookean material (see Attard Hunt (2004)) is given by ... 

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