Behavior at Small Strains

Studies on the mechanical properties of glassy polymer-solvent or, more generally, polymer-diluent mixtures have been primarily concerned with the deformation behavior at small strains which is governed by the viscoelastic properties of the material. From these studies it is well known that diluents significantly affect relaxation processes in glassy polymers, as clearly evidenced by phenomena such as plasticization and antiplasticization... 

A fundamental difficulty in the study of the linear viscoelastic behavior of filled rubbers is the secondary aggregation of filler particles, which greatly influences the behavior at small strains, where the response is linear. The effect of this aggregation is overcome at large strains, but now non-linearity and a number of other complications become problems. 

Since these double-base proplnts consist essentially of a single phase which bears the total load in any application of force, their mechanical property behavior is significantly different from composite proplnts. In the latter formulations, the hydrocarbon binder comprises only about 14% of the composite structure, the remainder being solid particles. Under stress, the binder of these proplnts bears a proportionately higher load than that in the single phase double-base proplnts. At small strain levels, these proplnts behave in a linear viscoelastic manner where the solids reinforce the binder. As strain increases, the bond between the oxidizer and binder breaks down... 

The lengths of the molecular chains dominate large strain behavior and crack propagation in contrast to their minimal influence at small strain levels. Consequently, thermosets are characterized by small deformation zones (Fig. 8.2) and brittle fracture. 

The work of Becker25 quoted above marked a very important step in this field, for it probably provided the first detailed information about dW/dlj evaluated from general biaxial extension experiments. From this work the behavior of these derivatives of W at small strains was found to be very complex. Our data displayed in the next section conform in many points to his important findings. 

Butter and milk fat exhibit viscoelastic behavior at small stresses (Chwiej, 1969 Pijanowski et al., 1969 Shama and Sherman, 1970 Sherman 1976 Shukla and Rizvi, 1995). To probe this behavior, a very small stress or deformation is applied to a sample and the relationships between stress, strain and time are monitored. Viscoelastic testing is performed in the linear viscoelastic region (LVR) where a linear relationship between stress and strain exists and where the sample remains intact. Depending on the material, this region lies at a strain of less than 1.0% (Mulder and Walstra, 1974) or even less than 0.1% (Rohm and Weidinger, 1993). Figure 7.10 shows the small deformation test results for milk fat at 5°C. 

So far, we have considered the elasticity of filler networks in elastomers and its reinforcing action at small strain amplitudes, where no fracture of filler-filler bonds appears. With increasing strain, a successive breakdown of the filler network takes place and the elastic modulus decreases rapidly if a critical strain amplitude is exceeded (Fig. 42). For a theoretical description of this behavior, the ultimate properties and fracture mechanics of CCA-filler clusters in elastomers have to be evaluated. This will be a basic tool for a quantitative understanding of stress softening phenomena and the role of fillers in internal friction of reinforced rubbers. 

Brown JD (1997) Nonlinear dynamic behavior of filled elastomers at small strain amplitudes. PhD Thesis, Rensselaer Polytechnic Institute, Troy, New York Chazeau L, Brown JD,Yanyo LC, Sternstein SS (2000) Polym Compos 21 202... 

Secondary aggregation of carbon black in rubber has important effects on the rheological behavior of vulcanizates at small strains and is discussed at a later point in this review. 

The effects of secondary aggregation of small particle carbon blacks on the elastic modulus at small strains are large. They have been studied primarily in dynamic oscillatory loading experiments and are discussed in Section VII, dealing with viscoelastic behavior. The effects of prior deformation on stress-strain relationships (stress softening) are also time-dependent phenomena, consideration of which is postponed to a later point in this review. 

A rubber-like solid is unique in that its physical properties resemble those of solids, liquids, and gases in various respects. It is solidlike in that it maintains dimensional stability, and its elastic response at small strains (<5%) is essentially Hookean. It behaves like a liquid because its coefficient of thermal expansion and isothermal compressibility are of the same order of magnitude as those of liquids. The implication of this is that the intermolecular forces in an elastomer are similar to those in liquids. It resembles gases in the sense that the stress in a deformed elastomer increases with increasing temperature, much as the pressure in a compressed gas increases with increasing temperature. This gas-like behavior was, in fact, what first provided the hint that rubbery stresses are entropic in origin. 

The applied stress results in the shear strain of the cube, i.e. the top face becomes shifted with respect to the bottom one by distance y. This displacement is numerically equal to the tangent of a tilt angle of the side face, i.e. it is equal to the relative shear strain, y, and at small strains tany y. The relationship between shear stress, x, and shear strain, y, and the rates of change in these quantities with time, dx/dt=x, dy/dt=y, represent mechanical behavior, which is the main subject in rheology. One usually begins the description of mechanical behavior with three elementary models, namely elastic, viscous, and plastic behavior. 

At small strains, polymers (both amorphous and crystalline) show essentially linear elastic behavior. The strain observed in this phase arises from bond angle deformation and bond stretching it is recoverable on removing the applied stress. The slope of this initial portion of the stress-strain curve is the elastic modulus. With further increase in strain, strain-induced softening occurs, resulting in a reduction of the instantaneous modulus (i.e., slope decreases). Strain-softening phenomenon is attributed to uncoiling... 

A sample which shows such time anomaly only is said to be viscoelastic and can be analyzed as described in this section. At small strains most polymers show such viscoelastic behavior. 

Considerable success has also been achieved in fitting the observed elastic behavior of rubbers by strain energy functions that are formulated directly in terms of the extension ratios Xi, X2, X2, instead of in terms of the strain invariants /i, I2 [22]. Although experimental results can be described economically and accurately in this way, the functions employed are empirical and the numerical parameters used as fitting constants do not appear to have any direct physical significance in terms of the molecular structure of the material. On the other hand, the molecular elasticity theory, supplemented by a simple non-Gaussian term whose molecular origin is in principle within reach, seems able to account for the observed behavior at small and moderate strains with comparable success. 

For concentrated emulsions and foams, Princen (1983, 1985) proposed a stress-strain theory based on a two-dimensional cell model. Consider steady-state shearing of such a system. Initially, at small strains, the stress increases linearly as in an elastic body. As the strain increases, the stress reaches the yield value, then at stiU higher deformation, it catastrophically drops to negative values. The reason for the latter behavior is the creation of unstable cell structure that generates a recoil mechanism. The predicted dependencies for modulus and the yield stress were expressed as... 

The discussion that follows, of sound propagation in a lossy polymer, is limited to the case where the stress-strain relation in the polymer is linear. The effect of loss mechanisms on the mechanical response of polsrmers is included by modifying the stress-strain relations (eq. 9). At small strains, at which the behavior of the polymer is linear, the stress-strain relations are modified according to the... 

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