Deformation non-affine

Johnson, M. W. and Segalman, D., 1977. A model for viscoelastic fluid behaviour which allows non-affine deformation. J. Non-Newtonian Fluid Mech. 2, 255-270. 

Drawn isotactic polypropylene (iPP) fibres have been studied by using deuterated n-decane as a 2H NMR probe of the chain deformation ratio in the amorphous regions. It is observed that the slope P=A/(X2-X 1) (defined in the limit of low deformations) indeed depends on the annealing temperature [82]. Thus, annealing above the melting temperature Tm of the crystallites allows chains to relax to some extent. Then, the local deformation ratio in the amorphous phase Xt becomes lower than the macroscopic one X and depends on the annealing temperature, i.e., on the amount of chain relaxation. Therefore, such systems have strongly non-affine deformation at the chain scale. 

At least, even if the Gordon-Schowalter derivative is obviously an improper tool for the description of non-affine deformation, its basic meaning retains some consistency with experimental observations, especially concerning the loss of junctions in materials with very different molecular structures. Indeed, going back to the significance of a, its low value in the case of LD is in agreement with the assumption of some kind of resistance to slip of junctions in branched materials, whereas the opposite trend is observed in the case of linear polymers (LLD and HD), for which higher and rather similar values are found (Table 4). 

The simplest explanation is that there is a rubber-like network present and that this has a maximum extensibility due to the degree of entanglement, which is constant for a given grade of polymer and depends on its molar mass and method of polymerisation. This limiting extensibility is not to be confused with the limit of applicability of the affine rubber model for predicting orientation distributions discussed in section 11.2.1 because the limiting extension can involve non-affine deformation. 

This results in a non-affine deformation process with the long chains taking up most of the elongation. This work by Mark et al clearly indicates the importance of the distribution of the molecular weight between crosslinks in determining the characteristics of a network. 

In the limit of the very non-affine deformation which would be exhibited by a phantom network, A is given by... 

Experimental results indicate that the response to deformation of a network generally falls between the affine and phantom limits [31-34]. At low deformations, chain-junction entangling suppresses the fluctuations of the junctions and the deformation is relatively close to the affine limit. This is illustrated in Fig. 1.8, which shows schematically some of the results of the constrained-junction theory based on this qualitative idea [32-34]. In the case of the two limits, the affine deformation and the non-affine deformation in the phantom-network limit, the reduced stress should be independent of a. Because of junction fluctuations, the value for the... 

Affine deformation Bend Dielectrostriction Direct piezoelectricity Electric polarization Electromechanical materials Electrostriction Flexoelectricity Gibbs free energy Inverse piezoelectrieity Non-affine deformation ... 

Fig. 4 Non-affine deformation of an elastically heterogeneous dielectric with k atrix k ipoie leads to a dipole-moment change (positive signal so-called primary piezoelectricity)...

N.W Johnson, J. Segalmann A model for viscoelastic fluid behaviour which allows Non- Affine deformation Journal of Non-Newtonian Fluid Mechanics, 1977, Vol. 2, p. 255-270. 

This book emphasises the empirical approach and it is not intended to include a review on the theoretical approach here. The theoretical approach at this stage contains many crude assumptions for example, gum rubber is considered to be an inelastic, viscous fluid and consequently the steady-state viscosity is used. The fill factor is ignored the presence and importance of the vacant space in the chamber is not recognised. Generally, slip at the rubber-metal interface is not considered. Non-affine deformation at the microscopic level for the filled system is not treated. 

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