Strain regimes

The model describes the characteristic stress softening via the prestrain-dependent amplification factor X in Equation 22.22. It also considers the hysteresis behavior of reinforced mbbers, since the sum in Equation 22.23 has taken over the stretching directions with ds/dt > 0, only, implying that up and down cycles are described differently. An example showing a fit of various hysteresis cycles of silica-filled ethylene-propylene-diene monomer (EPDM) mbber in the medium-strain regime up to 50% is depicted in Figure 22.12. It must be noted that the topological constraint modulus Gg has... 

A comparison of the experimental data for the first stretching cycle of the samples to the simulation curve in Fig. 45c shows no good agreement. Significant deviations are observed especially in the low strain regime, shown in the insert, where an extrapolation of the function X(E) is used. The reason for the deviations may partly lie in the application of the power law approximation Eq. (53) for X(E), instead of the micro-mechanically motivated Eq. (39). This may also lead to the unphysical negative value of X at extrapolated infinite strain (X00=-1.21). 

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... 

Elastoplastic materials Elastoplastic materials deform elastically for small strains, but start to deform plastically (permanently) for larger ones. In the small-strain regime, this behavior may be captured by writing the total strain as the sum of elastic and plastic parts (i.e., e = e -I- gP, where e and gP are the elastic and plastic strains, respectively). The stress in the material is generally assumed to depend on the elastic strain only (not on the plastic strain or the strain rate), and hence, no unique functional relationship exists between stress and strain. This fact also implies that energy is dissipated during plastic deformation. The point at which the material starts to deform plastically (the yield locus) is usually specified via a yield condition, which for one-dimensional plasticity may be stated as (38)... 

As yet, our reflections on the elastic properties of solids have ventured only so far as the small-strain regime. On the other hand, one of the powerful inheritances of our use of microscopic methods for computing the total energy is the ease with which we may compute the energetics of states of arbitrarily large homogeneous deformations. Indeed, this was already hinted at in fig. 4.1. 

Dynamic torsional shear experiments were conducted on a Rheometric Scientific ARES rheometer. The samples were cut 6.35 cm in length strips, 0.30 cm thick. The single frequency temperature ramp test was taken at 1 Hz from -100°C to 150°C at 2°C/min in the linear strain regime (0.01 to 2.00%). 

The theory behind linear viscoelasticity is simple and appealing. It is important to realize, however, that the applicability of the model for fluoropolymers is restricted to strains below the yield strain. One example comparing predictions based on linear viscoelasticity and experimental data for PTFE with 15 vol% glass fiber in the very small strain regime is shown in Fig. 11.4. 

TheKelvin model represents a parallel combination of two linear elements, i.e., of elasticity and viscosity (Fig. IX-10). In this case both strains are the same, while the shear stresses are additive, i.e. x= tg + t. An interesting strain regime in this model is the one in which constant shear stress is applied, i.e. t = t0 = const. 

I dP <0. In instances such as the phase diagram of water, where the melting curve does not display an initial maximum, it is predicted that this occurs in the metastable tensile-strained regime where the equilibrium state is the vapour. 

The above mathematical models (and later derivatives) define constitutive relationships for the plastic strain regime and they all assume a linear elastic behavior terminated by a yield point that is rate dependent. Hence the yield surface of the material is rate dependent. Since the purpose of these models are to develop methods to calculate deformations which are rate dependent beyond the yield point of a material they are often referred to by the term viscoplasticity, (see Perzyna, (1980), Christescu, (1982)). This practice is analogous to referring to methods to calculate deformation beyond the yield point of an ideal rate independent elastic-plastic material as classical plasticity. However, more general theories of viscoplasticity have been developed in some of which no yield stress is necessary. See Bodner, (1975) and Lubliner, (1990) for examples. 

Since these material constants are defined for a perfectly isotropic and homogeneous material within an ideally elastic, small strain regime, application of such mechanical concepts to single molecules should only be done with reservations. 

At the molecular level, polymer chains are highly anisotropic and inhomogeneous, and their deformations under externally applied forces are not confined fo fhe small strain regime. For example, a forced extension of an otherwise coiled chain under an externally applied tensile stress can make it reach more than ten times its original dimensions and the force-extension (analogous to stress-strain curve) is decidedly non-linear. 

The data from DMA was close to flexural creep data for the first four decades of time. This is probably due to linear viscoelastic (low stresses and infinitesimal strains) regime of testing in DMA [9] and the actual flexural tests which makes the two conparative. 

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