Low-amplitude strain

Standard geotechnical test reports address typical static properties of soil such as shear strength and bearing capacity but may not provide dynamic properties unless they are specifically requested. In these situations, it is necessary to use the static properties. Dynamic soil properties which are reported may be based on low strain amplitude tests which may or may not be applicable to the situation of interest. Soils reports will generally provide vertical and lateral stiffness values for the foundation type recommended. These can be used along with ultimate bearing capacities to perform a dynamic response calculation of the foundation for the applied blast load. 

FIGURE 15. Panel a shows the strain amplitude sweep experiment on the 2D film of Ag nanopartides. The storage modulus, C p), is higher than the loss modulus, C" (O) at low strain amplitudes. Panel b shows the frequency dependence of interfacial storage, C ( ), and loss, C" (o), moduli of the film. Reproduced from ref 33. Copyright 2007 American Chemical Society. 

Many materials, particularly polymers, exhibit both the capacity to store energy (typical of an elastic material) and the capacity to dissipate energy (typical of a viscous material). When a sudden stress is applied, the response of these materials is an instantaneous elastic deformation followed by a delayed deformation. The delayed deformation is due to various molecular relaxation processes (particularly structural relaxation), which take a finite time to come to equilibrium. Very general stress-strain relations for viscoelastic response were proposed by Boltzmann, who assumed that at low strain amplitudes the effects of prior strains can be superposed linearly. Therefore, the stress at time t at a given point in the material depends both on the strain at time t, and on the previous strain history at that point. The stress-strain relations proposed by Boltzmann are [4,39] ... 

The parameter t]0 is the limiting viscosity at low co. The reciprocal (t-1) of relaxation time % marks the midpoint co for the transition from a power-law exponent of 0 at low co to - 1 at high co. Interpretation of these low strain amplitude parameters in nonlinear fabrication shear flows is enabled by the Cox-Merz rule [43]. 

The modulus of strongly flocculated gels tends to be highly strain-dependent, with linear behavior confined to very low strain amplitudes (Buscall et al. 1988). Figure 7-16 shows the low-frequency modulus in creep versus normalized strain amplitude for strongly flocculated polystyrene particles with volume fractions (j> between 0.1 and 0.25. In... 

In the ordered state, lamellar block copolymers frequently show departures from linear viscoelastic behavior at low strain amplitudes of around 1% (Rosedale and Bates 1990 Winey et al. 1993a). Homogeneous polymers, on the other hand, typically show departures... 

The response of unvulcanized black-filled polymers (in the rubbery zone) to oscillating shear strains (151) is characterized by a strong dependence of the dynamic storage modulus, G, on the strain amplitude or the strain work (product of stress and strain amplitudes). The same behavior is observed in cross-linked rubbers and will be discussed in more detail in connection with the dynamic response of filled networks. It is clearly established that the manyfold drop of G, which occurs between double strain amplitudes of ca. 0.001 and 0.5, is due to the breakdown of secondary (Van der Waals) filler aggregation. In fact, as Payne (102) has shown, in the limit of low strain amplitudes a storage modulus of the order of 10 dynes/cm2 is obtained with concentrated (30 parts by volume and higher) carbon black dispersions made up from low molecular liquids or polymers alike. Carbon black pastes from low molecular liquids also show a very similar functional relationship between G and the strain amplitude. At lower black concentrations the contribution due to secondary aggregation becomes much smaller and, in polymers, it is always sensitive to the state of filler dispersion. 

In the course of this work we realized that the available instruments are not suitable to carry out simultaneously mechanical and rheological measurements. For example, instruments which are suitable for determining polymer transitions operate at low-strain amplitudes and thus cannot be used to carry out fatigue experiments to rupture polymer specimens of normal and medium strength. In addition, most of the rheological instruments cannot measure the viscoelastic properties if the... 

The more versatile instruments are equipped to carry out simultaneously the fatigue and nonlinear viscoelastic experiments. Nevertheless, the use of these expensive instruments is not recommended for longterm fatigue experiments in which a specimen may have to be cycled for several days or weeks under same conditions. We decided, therefore, to construct an instrument which combines the essential characteristic of the low strain amplitude Rheovibron with those of sophisticated fatigue apparatus. The apparatus described in this chapter has the following capabilities compared with the commercially available rheological and fatigue instruments ... 

Field tests which are most commonly employed in geotechnical earthquake engineering can be grouped into those that measure low-strain properties and those that measure properties at intermediate to high strains. Low-strain field tests t5q)ically induce seismic waves in the soil and seek to measure the velocities at which these waves propagate. Due to low strain amplitudes the measured shear wave velocity (V ) along with soil density (p) is used to compute low-strain shear modulus. 

The modulus at minimum and low strain amplitudes is due to the so-called filler network and it is accepted that the filler surface area, as well as the surface activity, play a major role in establishing a filler network, determining the effective contact area between filler particles and between filler particles and the elastomer matrix. The stress assisted disruption of the filler network causes the reduction of the modulus as the strain amplitude increases, giving rise to the non-linearity of the dynamic-mechanical behaviour of the rubber composite. This phenomenon is known as the Payne effect and it is (to a certain extent) reversible. The disruption and re-formation of the filler network is... 

Examples of dislocation structures after fatigue deformation appear in a series of microstructures (Figs. 7.55-7.59) obtained by TEM. Subramanian [27] claims that these dislocation substructures are similar to that of unidirectionaUy-stressed MgO. The micrographs presented below are of single-crystal magnesia which underwent a large number of cycles (in the millions) of low strain amplitude. The maximum strain was about 0.1 % per cycle. The characteristics of the dislocation structure in MgO, having a rock-salt structure, is as follows ... 

Dispersion of Carbon Black in Unvulcanized Natural Rubber The effects of increases in straim amplitude (from the lowest value of 1.4 /o) are to give decreases in both G and G" Figs. 17 and 18). This has, of course, been previously observed [3,24-36]. It was found that as strain amplitude increases, the value of G rather quickly decreases from one plateau value to another (but G" increases to maximum) at very low strain amplitudes (Fig. 21, adapted from Medalia). In the present work, we do not find the first... 

Providing tests are performed at low strain amplitude, small enough for the complex modulus to exhibit no strain dependency, then dynamic testing yields in principle linear viscoelastic functions. This implies that, with an unknown material, a preliminary strain sweep test is performed in order to experimentally detect the maximum strain amplitude for a linear response to be observed [i.e. G lo, f(Y)]-As illustrated in Fig. 6 with data from Dick and Pawlowsky [20], such a requirement is practically never met within the available experimental window with filled rubber materials, whose linear region tends to move back to a lower and lower strain range as the filler content increases. 

Finally, the crack initiation mechanisms at E=Eq are very similar to those observed in air. No pitting occurs, but cracks initiate in the a-phase at high strain amplitudes and in the y-phase at low strain amplitudes (sometimes also at a-y boundaries). 

Stress relaxation is highly dependent on the previous parameters as shown in Fig. 6.35(a) At 550°C, for instance, almost no relaxation is observed at low strain amplitude whatever the hold time duration, whereas the relaxed stress increases with increasing hold time. 

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