Back stress

In order to consider the inelastic stress rate relation (5.111), some assumptions must be made about the properties of the set of internal state variables k. With the back stress discussed in Section 5.3 in mind, it will be assumed that k represents a single second-order tensor which is indifferent, i.e., it transforms under (A.50) like the Cauchy stress or the Almansi strain. Like the stress, k is not indifferent, but the Jaumann rate of k, defined in a manner analogous to (A.69), is. With these assumptions, precisely the same arguments... 

It may first be noted that the referential symmetric Piola-Kirchhoff stress tensor S and the spatial Cauchy stress tensor s are related by (A.39). Again with the back stress in mind, it will be assumed in this section that the set of internal state variables is comprised of a single second-order tensor whose referential and spatial forms are related by a similar equation, i.e., by... 

The back stress (or self-stress) ji acting on a pinned dislocation loop is given by... 

As Meyers el al. [3] point out, polymers remain as solid material even when these parts of their chains are rearranging in order to accompany the stress, and as this occurs, it creates a back stress in the material. When the back stress is of the same magnitude as the applied stress, the material no longer creeps. When the original stress is taken away, the accumulated back stresses will cause the polymer to return to its original form. The material creeps, which gives the prefix visco, and the material fully recovers, which gives the suffix elasticity. 

In this expression, q as been substituted for convenience by q = s0/ (ftyo)m with so and yo being below-Tg parameters in Eq. 3, and ft a nondimensional constant. The deformation in the molten state is generally believed to involve chain slippage and temporary entanglements between the moving chains resulting in the non-Newtonian viscosity [ 14]. The details of the deformation process are lumped into the parameters q (or ft) and m so that no back stress contribution is considered above Tg a = a and r = s/l/la a. 

This "back stress" effect has not been considered theoretically in any detail although 1( Is clearly the reason why steps must be poisoned in order to observe etch pits, as has been pointed out in the work of Gilman, Johnston, and Sears (5). 

Let us now turn to the case where the steps are locally poisoned in the presence of low concentration of adsorbable impurities in such a way that between the adsorbed impurities die step becomes festooned and is consequently able to carry along the adsorbed impurities during the course of its motion. Then a step of macroscopic radius r will have a much smaller local radius of curvature, in fact, it will be of the order of die average distance between adsorbed impurities along the step. Thus it follows that c(r) will remain small and the "back stress ... 

It is quite clear, of course, that the dislocations in the metal and those much more numerous in the oxide crystals composing the oxide layer play a role in the growth of the oxide, but because of the "back stress" effect mentioned earlier, they should produce a very nearly uniform film. The formation of these oxide nuclei remains, therefore, unexplained. 

Dislocations emitted from a source pile up against an obstacle, producing a back stress a, that opposes the applied stress a. If the effective stress (a — Oi) acting on the source becomes less than the critical stress ffc nb/2L to activate the source (see Figure 9.6), the source will stop emitting dislocations. To maintain the imposed strain-rate, the applied stress a increases to drive the blocked dislocations over the obstacle, to reactivate the source, or to activate other sources with smaller L. 

Dislocations climb over obstacles in the slip plane. This process lessens hardening by reducing the drag on moving dislocations and by relieving the back stress associated with a pileup of dislocations, thereby allowing the dislocation source to continue to emit dislocations. 

Argon and Salama have demonstrated that this form of the expression accoimts for the measured craze growth kinetics in homogeneous glassy polymers very well, giving for values in the range of 1.5-2.0, which is quite consistent with the definition of this quantity in Eq. (51) and the known extension ratio of 4 in craze matter and back stresses that correspond to this extension ratio. 

The deformation is assumed to result in chain slippage but some chain entanglement will influence the mechanical response. This feature is assumed to be lumped into the parameters q and m so that no back stress contribution appears above Tg. Therefore, the driving stress reduces to a = o and the equivalent shear stress x in (4) is that of the Cauchy stress o. 

For T < Tg, the back stress ft is a function of the plastic stretch and depends on the entanglement density n. The latter is taken to depend on temperature as well, according to the suggestion in [11] that... 

Figure 8.1 shows stress-strain curves of atactic polystyrene (PS) in compression at 295 K for two structures with different initial states well annealed, i.e., furnace cooled from Tg + 20 K to room temperature, and rapidly quenched into ice water (Hasan and Boyce 1993). In both cases there is a gradual transition to fully developed plasticity that is reached at the peak of a yield phenomenon which is more prominent in the annealed material. Both curves show several unloading histories, starting with one close to the upper yield peak. All unloading paths show prominent Bauschinger effects of plastic strain recovery that is independent of the pre-strain. These indicate the presence of strain-induced back stresses and some recoverable stored elastic strain energy. In both cases the flow stress moves toward a unique flow state attained at a strain of around 0.3. 

Using the eight-chain entangled-network model of rubber elasticity (Arruda and Boyce 1993), the back stress is conveniently expressed as... 

Since the programming is the same for the free recovery specimens and fully constrained specimens, the focus will be on step 4 of the thermomechanical cycles. The stress-temperature behavior under a fully constrained recovery condition is shown in Figure 3.19 for the two programming stresses (47 kPa and 263 kPa). The recovery stress-time behavior of the foam programmed at 47kPa pre-stress is also highlighted by the inset in Figure 3.19. The recovery stress comes from two parts thermal expansion stress and entropically stored stress or back stress. Since this is a 1-D fully constrained recovery, the thermal stress can be calculated as... 

In the case of the amorphous phase one may only consider the influence of the inelastic deformation on molecular chain rotations and the subsequent strain hardening effects. The initial texture and pre-orientation effect can be considered by the segmental rotation resistance and network stretching resistance. It is common to neglect the texture effect in elastic deformation and consider it only during inelastic deformation. Dupaix and Boyce [100] relate the initial values of the back stress tensor, a,y, athermal shear resistance, s, network stretch... 

The computed trial elastic stress is relaxed by the plastic-corrector method, as described in Table 5.1. In Table 5.1, Qg is the hardening tensor, 2 is the inelastic multiplier, which shows the magnitude of inelastic deformation, and fi is the shear modulus. The second-order tensor hy defines the direction of the inelastic flow for example, in the case of the associated formulation it is hy = dif//dffy, where the yield surface is = 0. The deviatoric stress and back stress tensors are, respectively, identified by Sy = [Pg.195]

The stress driving the viscoplastic flow of the backstress network is obtained from the same hyperelastic representation tiiat was used to calculate tire back-stress, and has a similar framework as used in the Bergstrom-Boyce representation of crosslinked polymers at high temperatures (Bergstrom and Boyce 1998,2000) ... 

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