This protocol is extracted from research article:
Tracking time with ricequakes in partially soaked brittle porous media
Sci Adv, Oct 12, 2018; DOI: 10.1126/sciadv.aat6961

Assuming again small solid deformations, it is also possible to resolve analytically the overall displacement over time. At any given time t, the last carriage n experiencing crushing could be identified with the help of Eq. 12Embedded Image(14)while during that time t, the total deformation D(t) (with t = 0 denoting the point of soaking the medium) can be calculated by summing up the contribution from all the n crushed carriages and the Nn uncrushed onesEmbedded Image(15)

Since the deformation in each crushed carriages di is independent of time, the overall contribution to the deformation from all the n crushed carriages is given asEmbedded Image(16)

On the other hand, the length at time t of an uncrushed carriage i > n can be calculated from Eq. 4 as l(n) = lm(1 − p*/K(n)), where K(n) is given by Eq. 5 in terms of the integration of its corresponding effective saturation. The deformation from the time of soaking in that uncrushed carriage is then Embedded Image. Under the assumption of small deformation, which implies that each of the carriages carries a constant effective saturation that is only a function of the initial position through Eq. 11, we may therefore approximate the contribution from all the (Nn) uncrushed carriages to the overall deformationEmbedded Image(17)such that by combining Eqs. 9, 15, and 17, and since the initial height of the pack is simply H = Nlm (with Hlm), the total deformation at time t after the soaking event could be calculated asEmbedded Image(18)while n grows discontinuously according to Eq. 14, with n = 0 at time t = 0.

The analytic expression of the deformation over time in Eq. 18 is discontinuous due to its dependence on the typical size of the micropore units lm and the integer n marking the transition level between crushed and uncrushed carriages. The overall deformation curve is given by two-phase response modes, including a discrete crushing phase with n and D jumping distinctly, and a smooth creep phase. It is possible to smoothen the discrete part of the deformation by replacing the sum operation with an integral equation. This could be done by taking lm = dz as the infinitesimal integration width only within that discontinuous summation part of the equation. Out of this summation, we keep lm to be finite and take H = lmN as the sample height. In addition, with the help of Eq. 12, the transition integer n and transition level zn+1 can be replaced by the smooth continuous time Embedded Image as Embedded Image and Embedded Image (such that at t = 0, we have both n = 0 and z1 = 0, with Embedded Image as a nondimensional continuous time). Using this step, the approximated smooth analytic expression for the overall deformation D(t) takes the formEmbedded Image(19)

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