Strain dependence polymers

A8. The Helmholtz elastic free energy relation of the composite network contains a separate term for each of the two networks as in eq. 5. However, the precise mathematical form of the strain dependence is not critical at small deformations. Although all the assumptions seem to be reasonably fulfilled, a simpler method, which would require fewer assumptions, would obviously be desirable. A simpler method can be used if we just want to compare the equilibrium contribution from chain engangling in the cross-linked polymer to the stress-relaxation modulus of the uncross-linked polymer. The new method is described in Part 3. 

It became clear in the early development of the tube model that it provided a means of calculating the response of entangled polymers to large deformations as well as small ones [2]. Some predictions, especially in steady shear flow, lead to strange anomaUes as we shall see, but others met with surprising success. In particular the same step-strain experiment used to determine G(t) directly in shear is straightforward to extend to large shear strains y. In many cases of such experiments on polymer melts both Hnear and branched, monodisperse and polydisperse,the experimental strain-dependent relaxation function G(t,Y) may be written... 

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

The above theory reveals that the piezoelectricity of polymer film can be classified into four cases at its origin (A) The intrinsic piezoelectricity due to internal strain in the crystal, (B) the intrinsic piezoelectricity due to strain-dependence of spontaneous polarization, (Q the piezoelectricity originated from the polarization charge qp arising from strain-independent persistent polarization P0, and (D) the piezoelectricity from the true charge gt embedded in the film. It must be emphasized that in cases (Q and (D) heterogeneous strain Au = (u — Sl/T) must exist in the film. 

In the case of other polar polymers which have been polarized under a static field, the origin of the piezoelectricity may be either an embedded polarization charge plus heterogeneous strain, or strain dependence of spontaneous polarization. The problem is still open to questioa... 

The electrostriction effect, beside its effect on the piezoelectricity, gives a new insight onto relaxations in polymers when the electrostriction constant is obtained over a wide frequency range. It provides us with a knowledge of the strain dependence of relaxation time. 

Dependencies of longitudinal viscosity upon time at different extension velocities coincide up to the start of deviation from linear behavior. The higher the extension velocity, the earlier deviation is observed. An interesting fact is that in all cases the deviation from linearity is observed at one and the same critical strain independent of the molecular weight of the polymer and temperature. The value of critical strain depends only upon the nature of the polymer and lies within the limits of 0.4 — 1. 

The mathematical relationship between the stress and the strain depends on material properties, temperature, and the rate of deformation. Many materials such as metals, ceramics, crystalline polymers, and wood behave elastically at small stresses. For tensile elastic deformation, the linear relation between the stress, a, and strain, e, is described by Hooke s law as... 

The above results, which were formulated in the early thirties (TVeloar 1958), explain the main features of the elastic behaviour of polymer networks. Nevertheless, there are notable discrepancies between empirical data and the cited results. Further investigations demonstrated that the values of shear modulus and stress-strain dependence are determined substantially by topological constraints due to the proximity of the chains. The theory was improved by taking into account the discussed issue (Edwards 1967a, 1967b, 1969 Flory 1977 Erman and Flory 1978 Priss 1957, 1980, 1981). More recent developments are summarised in the work of Panyukov and Rabin (1996), where many additional relevant references could be found. 

In view of an illustration of the viscoelastic characteristics of the developed model, simulations of uniaxial stress-strain cycles in the small strain regime have been performed for various pre-strains, as depicted in Fig. 47b. Thereby, the material parameters obtained from the adaptation in Fig. 47a (Table 4, sample type C60) have been used. The dashed lines represent the polymer contributions, which include the pre-strain dependent hydrodynamic amplification of the polymer matrix. It becomes clear that in the small and medium strain regime a pronounced filler-induced hysteresis is predicted, due to the cyclic breakdown and re-aggregation of filler clusters. It can considered to be the main mechanism of energy dissipation of filler reinforced rubbers that appears even in the quasi-static limit. In addition, stress softening is present, also at small strains. It leads to the characteristic decline of the polymer contributions with rising pre-strain (dashed lines in... 

The big difference between normal isotropic liquids and nematic liquids is the effect of anisotropy on the viscous and elastic properties of the material. Liquid crystals of low molecular weight can be Newtonian anisotropic fluids, whereas liquid crystalline polymers can be rate and strain dependent anisotropic non-Newtonian fluids. The anisotropy gives rise to 5 viscosities and 3 elastic constants. In addition, the effective flow properties are determined by the flow dependent and history dependent texture. This all makes the rheology of LCPs extremely complicated. 

The discussion in the Introduction led to the convincing assumption that the strain-dependent behavior of filled rubbers is due to the break-down of filler networks within the rubber matrix. This conviction will be enhanced in the following sections. However, in contrast to this mechanism, sometimes alternative models have been proposed. Gui et al. theorized that the strain amplitude effect was due to deformation, flow and alignment of the rubber molecules attached to the filler particle [41 ]. Another concept has been developed by Smith [42]. He has indicated that a shell of hard rubber (bound rubber) of definite thickness surrounds the filler and the non-linearity in dynamic mechanical behavior is related to the desorption and reabsorption of the hard absorbed shell around the carbon black. In a similar way, recently Maier and Goritz suggested a Langmuir-type polymer chain adsorption on the filler surface to explain the Payne-effect [43]. 

An alternative model is shown schematically in Fig. 10. Just as for the craze tip, the strain-softened polymer being actively deformed at the bulk polymer interface is idealized as a thin layer of non-Newtonion fluid. The velocity v of plastic advance of the craze interface depends on the gradient in hydrostatic tension Van as... 

The complex viscosity can be related to the steady-shear viscosity rf) via the empirical Cox-Merz rule, which notes the equivalence of steady-shear and dynamic-shear viscosities at given shearing rates ri y) = rj (co). The Cox-Merz rule has been confirmed to apply at low rates by Sundstrom and Burkett (1981) for a diallyl phthalate resin and by Pahl and Hesekamp (1993) for a filled epoxy resin. Malkin and Kulichikin (1991) state that for highly filled polymer systems the validity of the Cox-Merz rule is doubtful due to the strain dependence at very low strains and that the material may partially fracture. However, Doraiswamy et al. (1991) discussed a modified Cox-Merz rule for suspensions and yield-stress fluids that equates the steady viscosity with the complex viscosity at a modified shear rate dependent on the strain, ri(y) = rj yrap3), where y i is the maximum strain. This equation has been utilised by Nguyen (1993) and Peters et al. (1993) for the chemorheology of highly filled epoxy-resin systems. 

The low-speed mechanical properties of polymer blends have been frequently used to discriminate between different formulations or methods of preparation. These tests have been often described in the literature. Examples of the results can be found in the references listed in Table 12.9. Measurements of tensile stress-strain behavior of polymer blends is essential [Borders et al., 1946 Satake, 1970 Holden et al., 1969 Charrier and Ranchouse, 1971]. The mbber-modified polymer absorbs considerably more energy, thus higher extension to break can be achieved. By contrast, an addition of rigid resin to ductile polymer enhances the modulus and the heat deflection temperature. These effects are best determined measuring the stress-strain dependence. 

Rheo-optical techniques (46 8) afford information on the strain dependence not only of stress but also of optical quantities associated directly with the structure or molecular morphology. The techniques were developed extensively for crystalline polymers to investigate the molecular deformation mechanism underlying the tensile elongation. In this part, the chain orientation behavior is characterized by infrared dichroism measured simultaneously with tensile deformation at a constant rate of elongation. 

Linear viscoelasticity is valid only imder conditions where structural changes in the material do not induce strain-dependent modulus. This condition is fulfilled by amorphous polymers. On the other hand, the structural changes associated with the orientation of crystalline polymers and elastomers produce anisotropic mechanical properties. Such polymers, therefore, exhibit nonlinear viscoelastic behavior. 

In order to predict the stress-strain dependence for a polymer filled with a non-adhering filler on the basis of known stress-strain curve for pure polymer we have to find the deformed sample effective load bearing cross-section. 

---