Deformation

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

Tracking time with ricequakes in partially soaked brittle porous media

Procedure

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. 12*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 *N* − *n* uncrushed ones

Since the deformation in each crushed carriages *d*_{i} is independent of time, the overall contribution to the deformation from all the *n* crushed carriages is given as

On the other hand, the length at time *t* of an uncrushed carriage *i* > *n* can be calculated from Eq. 4 as *l*^{(n)} = *l*_{m}(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 *N* − *n*) uncrushed carriages to the overall deformation*H* = *Nl*_{m} (with *H* ≫ *l*_{m}), the total deformation at time *t* after the soaking event could be calculated as*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 *l*_{m} 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 *l*_{m} = *dz* as the infinitesimal integration width only within that discontinuous summation part of the equation. Out of this summation, we keep *l*_{m} to be finite and take *H* = *l*_{m}*N* as the sample height. In addition, with the help of Eq. 12, the transition integer *n* and transition level *z*_{n+1} can be replaced by the smooth continuous time *t* = 0, we have both *n* = 0 and *z*_{1} = 0, with *D*(*t*) takes the form

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