Power-law breakdown

In specimens deformed to several percent strain (or more) at low to intermediate temperatures and stresses, where neither work-hardening nor recovery processes predominate, dislocations tend to tangle into localized walls (Kirby and McCormick 1979 McCormick 1977 McLaren et al. 1970 Morrison-Smith et al. 1976). These walls behave as optical phase objects and give rise to the deformation lamellae that are commonly observed in deformed crystals by optical microscopy (see Section 1.3 and McLaren et al. 1970). Similar walls of tangled dislocations develop in metals in the power-law-breakdown creep regime where both recovery-controlled and glide-controlled deformation mechanisms are operative (see, e.g., Drury and Humphreys 1986). 

A drawback of the power law equation is the so-called power law breakdown effect. As the applied stress increases in magnitude, the stress exponent, n, no longer remains constant and increases with stress level. Therefore, it becomes necessary to use multiple power law equations to describe the creep behavior over an extended range of applied stresses. 

Q. Zhang, et al. performed shear creep tests on 95.5Sn-3.9Ag-0.6Cu solder joints (Ref 73). The gap thickness was 0.180 mm (0.007 in.). The tests were performed at three temperatures 25, 75, and 125 °C (77, 167, and 257 T). The shear stresses ranged from 4 to 30 MPa (0.58 to 4.35 ksi). Initially, the authors observed the power law breakdown effect when the steady-state creep rate, shear stress and temperature were assessed with the power law function. Subsequently, the authors obtained a more consistent fit when all of the steady-state creep data were analyzed with the sinh law equation ... 

The kinetic observations reported by Young [721] for the same reaction show points of difference, though the mechanistic implications of these are not developed. The initial limited ( 2%) deceleratory process, which fitted the first-order equation with E = 121 kJ mole-1, is (again) attributed to the breakdown of superficial impurities and this precedes, indeed defers, the onset of the main reaction. The subsequent acceleratory process is well described by the cubic law [eqn. (2), n = 3], with E = 233 kJ mole-1, attributed to the initial formation of a constant number of lead nuclei (i.e. instantaneous nucleation) followed by three-dimensional growth (P = 0, X = 3). Deviations from strict obedience to the power law (n = 3) are attributed to an increase in the effective number of nuclei with reaction temperature, so that the magnitude of E for the interface process was 209 kJ mole-1. 

In practice, the trap-filled limit is difficult to observe as it is often preceded by electrical breakdown of the sample. The transition from the linear to square law (Child s Law) dependence of current on voltage is usually not sharply defined. Thus samples may display an intermediate power law over a considerable voltage range. This, and the uncertainty of the trapping factor, render the measurement of current-voltage characteristics unsuitable for tire determination of carrier mobility. 

In the case of intrinsically rigid polyelectrolytes, such as DNA, experimental results [67] show that electrostatic persistence length calculated from the data shows no unique power law dependence on cs. Compared to the OSF theory [60,61], a much better agreement with these data was achieved later by the calculation of Le via numerical solution of the Poisson-Boltzmann equation for a toroidal polyion geometry [59,62], These calculations showed that the exponent ft in the scaling Rg cs p varies from -1 to -1/4 upon increase of cs. A breakdown of the OSF theory for flexible chains (unless Le /.p) was indicated by taking into account fluctuations in the chain configuration [63]. 

Steady shear measurements were used to determine flow properties and to estimate the degree of structure breakdown with shear (Elliott and Ganz, 1977). The power law equation (Eq. 3) has been used to describe the shear stress-shear rate behavior of salad dressings (Figoni and Shoemaker, 1983 Paredes et al, 1988, 1989). The flow behavior index of five commercial salad dressings at different temperatures and storage times of up to 29 days were all less than one, indicating that they were pseudoplastic fluids. The consistency index (/f) decreased with the increase in product temperature. 

When the applied voltage reaches a threshold value known as the breakdown voltage a large current flows, following a power law ... 

Recently, Tanner et al. (2008) introduced the use of a simple Lodge-type model (Lodge, 1964), including a power law memoiy function, and a damage function (assumed to be a function of strain) to represent the breakdown of the molecular structure within the dough, for a description of the rheological behavior of wheat dough. The model produced a... 

---