Critical strain amplitude

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. 

Similar results were obtained on all three samples over a range of frequencies and temperatures. The upturn in the viscosity (and the downturn in tan5) can be seen to occur at a critical strain amplitude over a wide range of temperatures and frequencies (for a given silicate loading). Three important features are observed in the rheological response of all three samples - (a) the process is reversible (b) there... 

On a global scale, the linear viscoelastic behavior of the polymer chains in the nanocomposites, as detected by conventional rheometry, is dramatically altered when the chains are tethered to the surface of the silicate or are in close proximity to the silicate layers as in intercalated nanocomposites. Some of these systems show close analogies to other intrinsically anisotropic materials such as block copolymers and smectic liquid crystalline polymers and provide model systems to understand the dynamics of polymer brushes. Finally, the polymer melt-brushes exhibit intriguing non-linear viscoelastic behavior, which shows strainhardening with a characteric critical strain amplitude that is only a function of the interlayer distance. These results provide complementary information to that obtained for solution brushes using the SFA, and are attributed to chain stretching associated with the space-filling requirements of a melt brush. 

The linear visco-elastic range ends when the elastic modulus G starts to fall off with the further increase of the strain amplitude. This value is called the critical amplitude yi This is the maximum amplitude that can be used for non-destructive dynamic oscillation measurements... 

The above equations are generally valid for any isotropic material, including critical gels, as long as the strain amplitude y0 is sufficiently small. The material is completely characterized by the relaxation function G(t) and, in case of a solid, an additional equilibrium modulus Ge. 

Using model concentrated suspensions of polyvinyl chloride and titanium dioxide particles in a Newtonian polybutene fluid, small amplitude oscillatory shear and creep experiments were described [2]. It was shown that the gel-like behaviour at very small strain, and strain hardening at a critical strain, are caused by particle interactions and the state of particle dispersion. 

The frequency is fixed say at 1 H z (or 6.28 rad s ) and G, G and G" are measured as a function of strain amplitude this is illustrated in Figure 20.12, where G, G and G" are seen to remain constant up to a critical strain. This is the linear viscoelastic region where the moduli are independent of the applied strain. Above however, G and G start to decrease whereas G" starts to increase with further increase in y this is the nonlinear region. 

Fig. 3. Critical current as a function of the number of fatigue cycles between zero stress and the maximum stress indicated on each curve, et refers to the total strain of the maximum stress, Ac refers to the peak-to-peak strain amplitude as defined in Fig. 4. Critical current was measured by stopping the cyclic loading at maximum.

Next, the ac strain amplitude was plotted as a function of dc bias fields for various driving ac fields, as seen in figure 8. The first-harmonic piezoelectric strain increased with both the ac and dc fields until a maximum of approximately 0.8 x lO m.m-i occurred at 1.1 MV.m-i dc and 1.09 MV.m-i ac peak-to-peak. These results once again confirm studies of the dielectric behaviour of PLZT (9.0/65/35) (Bobnar et al. 1999), in which above a critical field, called Ec, a phase transition from relaxor to ferroelectric occurs in the PLZT structure. It is perhaps the presence of both phases simultaneously that give rise to the piezoelectric strain further increasing of the fields would just render the samples more and more ferroelectric, therefore decreasing the strain. 

Rheological measurements are used to investigate the bulk properties of suspension concentrates (see Chapter 7 for details). Three types of measurements can be applied (1) Steady-state shear stress-shear rate measurements that allow one to obtain the viscosity of the suspensions and its yield value. (2) Constant stress or creep measurements, which allow one to determine the residual or zero shear viscosity (which can predict sedimentation) and the critical stress above which the structure starts to break-down (the true yield stress). (3) Dynamic or oscillatory measurements that allow one to obtain the complex modulus, the storage modulus (the elastic component) and the loss modulus (the viscous component) as a function of applied strain amplitude and frequency. From a knowledge of the storage modulus and the critical strain above which the structure starts to break-down , one can obtain the cohesive energy density of the structure. 

In dynamic (oscillatory) measurements, one applies a sinusoidal strain or stress (with amplitudes yo or < o and frequency co in rad s ) and the stress or strain is measured simultaneously. For a viscoelastic system, the stress oscillates with the same frequency as the strain, but out of phase. From the time shift of stress and strain, one can calculate the phase angle shift <5. This allows one to obtain the various viscoelastic parameters G (the complex modulus), G (the storage modulus, i.e. the elastic component of the complex modulus) and G" (the loss modulus or the viscous component of the complex modulus). These viscoelastic parameters are measured as a function of strain amplitude (at constant frequency) to obtain the linear viscoelastic region, whereby G, G and G" are independent of the applied strain until a critical strain above which G and G begin to decrease with further increase of strain, whereas G" shows an increase. Below y the structure of the system is not broken down, whereas above y the structure begins to break. From G and one can obtain the cohesive energy density of the structure... 

As indicated earlier the ER particle may form a network structure instead of the nbrillaled chain structure once the particle volume fraction exceeds the critical volume fraction 162,83], due to the percolation transition. Under a large amplitude shear field the formation and destruction of network junctions may happen one after the other, and thus the type III LAOS behavior may best describe ER suspensions. Figure 43 shows the simulated storage and loss moduli vs. strain amplitude at different frequencies, using an idealized electrostatic polarization model of ER fluids that was implemented in the particle-level dynamics simulation. Tlie storage modulus G and loss modulus G" remain constant up to a certain strain amplitude (/() 0.4), which defines the linear response region. With further increase... 

In oscillatory measurements one carries out two sets of experiments (i) Strain sweep measurements. In this case, the oscillation is fixed (say at 1 Hz) and the viscoelastic parameters are measured as a function of strain amplitude. This allows one to obtain the linear viscoelastic region. In this region all moduli are independent of the appUed strain amplitude and become only a function of time or frequency. This is illustrated in Fig. 3.50, which shows a schematic representation of the variation of G, G and G" with strain amplitude (at a fixed frequency). It can be seen from Fig. 3.49 that G, G and G" remain virtually constant up to a critical strain value, y . This region is the linear viscoelastic region. Above y, G and G start to fall, whereas G" starts to increase. This is the nonlinear region. The value of y may be identified with the minimum strain above which the "structure of the suspension starts to break down (for example breakdown of floes into smaller units and/or breakdown of a structuring agent). 

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