Heaviside unit step function

Figure C.l The Heaviside unit step function H x), defined as the limit as e — 0 of...

When the slab is subject to a single-step shear history y(t) = yoH(t), where H(t) is the Heaviside unit step function, zero for negative t and unit for t zero or positive, the stress response can be used to characterize the rheological properties. When the materials are subjected to a step strain as shown in Fig. 11a, the different stress responses are obtained as shown in Fig. lib. If the material were perfectly elastic, the corresponding stress history would be of the form t(i) = to//(i), constant for t positive (curve a in Fig. lib). If the material were an ideal viscous fluid, the stress would be instantaneously infinite during the step and then zero for all positive t, like a Dirac delta, 5(t) = H t) (curve b in Fig. 11b). For most real materials, like semisolid foods, the stress response shows that neither of these idealizations is quite accurate. The stress usually decreases from its initial value... 

Differentiation of a function at a finite discontinuity produces a deltafunction. Consider, for example, the Heaviside unit step function ... 

Here, kq = k /s where e is the dielectric constant of the bulk solvent and is the Debye length 9 stands for the Heaviside unit step function, which reflects here the distance of closest approach, I, of ions to the electrode [I is calculated from the edge of the metal skeleton). 

For quiescent crystallization under a constant temperature, in the case of instantaneous nucleation with a constant number density No, one obtains Nq t) = NoH t) for the activated quiescent nuclei number density, where H(t) is the Heaviside unit step function, zero for f < 0 and unity for t > 0. Then the rate of the nuclei number density is Ng = NoS t), with (5(f) being the Dirac delta function concentrated at f = 0. Equations 4.1 and 4.3 lead to the familiar Avrami equation ... 

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