Network polymer stress-strain relation

From X and X, together with stress-strain measurements in extension from the state of ease, or simply from the equilibrium stress at X, the concentration of trapped entanglement strands can be calculated and compared with the entanglement strand density estimated from transient measurements on the uncrosslinked polymer. To obtain consistent results, especially for stress-strain relations in large extensions of the dual network from its state of ease, it is necessary to attribute deviations from neo-Hookean elasticity to the trapped entanglement network, as described by the... 

In the field of rubber elasticity both experimentalists and theoreticians have mainly concentrated on the equilibrium stress-strain relation of these materials, i e on the stress as a function of strain at infinite time after the imposition of the strain > This approach is obviously impossible for polymer melts Another complication which has thwarted the comparison of stress-strain relations for networks and melts is that cross-linked networks can be stretched uniaxially more easily, because of their high elasticity, than polymer melts On the other hand, polymer melts can be subjected to large shear strains and networks cannot because of slippage at the shearing surface at relatively low strains These seem to be the main reasons why up to some time ago no experimental results were available to compare the nonlinear viscoelastic behaviour of these two types of material Yet, in the last decade, apparatuses have been built to measure the simple extension properties of polymer melts >. It has thus become possible to compare the stress-strain relation at large uniaxial extension of cross-linked rubbers and polymer melts ... 

Assuming that for highly crosslinked polymer networks an elastic stress strain relation can be obtained in the following form ... 

Here, the stress-strain relations, or the equations of state, of networks is described by considering the homogeneous deformation of a cube of sides Lq in the reference state and L, Ly, and in the deformed state (6). More general states of deformation are discussed in several textbooks (2,5,13). The reference state is defined as the state in which the network has been formed. The deformation ratios are defined as the ratio of the final to the reference state, such as A. = LJLq, etc. The state of the network during formation is of particular importance in obtaining the correct expression for the stress. If the network is prepared in the presence of a diluent, the reference volume Vo = equals the sum of the dry volume Vd of bulk polymer, and the volume of solvent Vg-... 

From the viewpoint of the mechanics of continua, the stress-strain relationship of a perfectly elastic material is fully described in terms of the strain energy density function W. In fact, this relationship is expressed as a linear combination erf the partial derivatives of W with respect to the three invariants of deformation tensor, /j, /2, and /3. It is the fundamental task for a phenomenologic study of elastic material to determine W as a function of these three independent variables either from molecular theory or by experiment. The present paper has reviewed approaches to this task from biaxial extension experiment and the related data. The results obtained so far demonstrate that the kinetic theory of polymer network does not describe actual behavior of rubber vulcanizates. In particular, contrary to the kinetic theory, the observed derivative bW/bI2 does not vanish. 

The equilibrium small-strain elastic behavior of an "incompressible" rubbery network polymer can be specified by a single number—either the shear modulus or the Young s modulus (which for an incompressible elastomer is equal to 3. This modulus being known, the stress-strain behavior in uniaxial tension, biaxial tension, shear, or compression can be calculated in a simple manner. (If compressibility is taken into account, two moduli are required and the bulk modulus. ) The relation between elastic properties and molecular architecture becomes a simple relation between two numbers the shear modulus and the cross-link density (or the... 

In attempting to predict the direction that future research in carbon black technology will follow, a review of the literature suggests that carbon black-elastomer interactions will provide the most potential to enhance compound performance. Le Bras demonstrated that carboxyl, phenolic, quinone, and other functional groups on the carbon black surface react with the polymer and provided evidence that chemical crosslinks exist between these materials in vul-canizates (LeBras and Papirer, 1979). Ayala et al. (1990, 1990) determined a rubber-filler interaction parameter directly from vulcanizatemeasurements. The authors identified the ratio a jn, where a = slope of the stress-strain curve that relates to the black-polymer interaction, and n = the ratio of dynamic modulus E at 1 and 25% strain amplitude and is a measure of filler-filler interaction. This interaction parameter emphasizes the contribution of carbon black-polymer interactions and reduces the influence of physical phenomena associated with networking. Use of this defined parameter enabled a number of conclusions to be made ... 

The Vc and Me values for crosslinked polymer networks can also be evaluated from stress-strain diagrams on the basis of theories for the rubber elasticity of polymeric networks. In the relaxed state the polymer chains of an elastomer form random coils. On extension, the chains are stretched out, and their conformational entropy is reduced. When the stress is released, this reduced entropy makes the long polymer chains snap back into their original positions entropy elasticity). Classical statistical models of entropy elasticity affine or phantom network model [39]) derive the following simple relation for the experimentally measured stress cr ... 

This chapter is devoted to the molecular rheology of transient networks made up of associating polymers in which the network junctions break and recombine. After an introduction to theoretical description of the model networks, the linear response of the network to oscillatory deformations is studied in detail. The analysis is then developed to the nonlinear regime. Stationary nonhnear viscosity, and first and second normal stresses, are calculated and compared with the experiments. The criterion for thickening and thinning of the flows is presented in terms of the molecular parameters. Transient flows such as nonhnear relaxation, start-up flow, etc., are studied within the same theoretical framework. Macroscopic properties such as strain hardening and stress overshoot are related to the tension-elongation curve of the constituent network polymers. 

In reality a polymer mass will only be an effective rubber if the individual chains are joined into a network structure. This virtually eliminates the unlimited slippage of one chain past another causing viscous flow or creep. Subject to a number of assumptions it is possible to derive expressions relating stress-strain relationships in such a network structure. 

The theoretical equations presented above can be used to interpret stress-strain measurements in uniaxial extension and thus to fully characterize elastomeric networks. In this regard, equations (124) and (125) are of particular interest since they relate the parameter k, quantifying the entanglement constraints in the Flory and Erman model, to the polymer microstructure and conformational properties and to the network topology. An illustrative analysis of stress-strain data due to Queslel, Thirion and Monnerie is reported below. 

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. 

In Eq. (1), a is the equilibrium stress (Nm 2) supported by the swollen specimen a is the stretched specimen length divided by the unstretched length (extension ratio) v2 is the volume fraction of dry protein and p is the density of dry protein. In the common case of tetrafunctional crosslinks, the concentration of network chains n (mol network chains/g polymer) is exactly one-half the concentration of crosslinks, so that n = 2c. The hypothesis that a specimen behaves as if it were an ideal rubber can be confirmed by observing a linear relation with zero intercept between a and the strain function (a — 1/a2) and by establishing a direct proportionality between a and the absolute temperature at constant value of the extension ratio, as stipulated by Eq. (1). 

To determine the tensor of recoverable strains, we have to equate the stress tensor for a deformed polymer network (given in the simplest case by equation (1.43)) with the elastic part of the stress tensor for a polymer liquid, given in the general case by equation (9.19) or, in the simplified case by equation (9.48). The latter case leads to the relation... 

A polymer is more likely to fail by brittle fracture under uniaxial tension than under uniaxial compression. Lesser and Kody [164] showed that the yielding of epoxy-amine networks subjected to multiaxial stress states can be described with the modified van Mises criterion. It was found to be possible to measure a compressive yield stress (Gcy) for all of their networks, while the networks with the smallest Mc values failed by brittle fracture and did not provide measured values for the tensile yield stress (Gty) [23,164-166]. Crawford and Lesser [165] showed that Gcy and Gty at a given temperature and strain rate were related by Equation 11.43. 

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