Model

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

In the following, the crushing wave model is developed. The model is structured as a train of elasto-brittle carriages from *i* = 1 to *N* akin to a vertical series of micropores extending from the wetting surface upward, with *i* = 1 being the first carriage initially positioned adjacent to the wetting surface and *i* = *N* being the last carriage, which touches a loading plate applying a pressure *p** that is initially positioned at height *H* (Fig. 4A). Neglecting inertia effects and any friction between the tested material and the container boundaries, force equilibrium imposes the pressure *p*^{(i)} = *p** for any *i*. Irrespectively, the relation between the *i*^{th} element pressure [*p*^{(i)}] and its elastic strain [ε^{(i)}, positive in compression] is defined as*K*^{(i)} is the *i*^{th} constrained elastic modulus, with *K*_{0} being its initial value before any interaction with fluid; *l*^{(i)} is the compressed length of the *i*^{th} carriage; and

*Prior to crushing*. The equilibrium length is given by the uncrushed characteristic micropore length *l*_{m}, while the stiffness degrades with fluid activity *t*) integrated effect of the chemical reaction of the porous medium with the fluid, i.e., in terms of the effective degree of saturation *i*) at time *t*, and the reaction rate α. The assumed linear decay of the stiffness with water activity agrees with previous experiments on crunchy material for nonnegligible degrees of saturation (*31*). The reaction rate can be expressed most generally as an Arrhenius’ relationship (*32*) *A*, *E*_{a}, *R*, and *T*^{(i)} are the activation rate constant, activation energy, ideal gas constant, and absolute temperature of the *i*^{th} carriage, respectively]. Under the constant room temperature (≈20°C) maintained in our experiments, α is therefore constant in both space and time and thus

Accordingly, for fixed degrees of saturation in time, one should expect a linear relationship between *a*_{S} and *S*_{e}, which is also consistent with experimental observations with cereals (*33*).

*Upon and after crushing*. The equilibrium length changes to a new smaller constant value *l*_{cr}. The crushing of the *i*^{th} carriage occurs when its fluid activity reaches a critical value *i*^{th} carriage. Under constant temperature, its value can be found from the following implicit relation

Accordingly, the onset of crushing of the *i*^{th} carriage depends on how its effective saturation *i*^{th} carriage is dependent on that carriage distance from the wetting front, which slightly shortens over time. In addition, note that the crushing length of the carriages *l*_{cr} can be related to the other constants by combining Eqs. 4 and 8 [for *p*^{(i)} = *p**]

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