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 ith element pressure [p(i)] and its elastic strain [ε(i), positive in compression] is defined asEmbedded Image(4)where K(i) is the ith constrained elastic modulus, with K0 being its initial value before any interaction with fluid; l(i) is the compressed length of the ith carriage; and Embedded Image is its corresponding equilibrium length. In the following, the equilibrium length will have two possible values for two distinct behavioral phases of the material: prior and upon and after crushing (Fig. 4C).

Prior to crushing. The equilibrium length is given by the uncrushed characteristic micropore length lm, while the stiffness degrades with fluid activity Embedded ImageEmbedded Image(5)where the fluid activity is defined in terms of the time (t) integrated effect of the chemical reaction of the porous medium with the fluid, i.e., in terms of the effective degree of saturation Embedded Image of node (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) Embedded Image [where A, Ea, R, and T(i) are the activation rate constant, activation energy, ideal gas constant, and absolute temperature of the ith carriage, respectively]. Under the constant room temperature (≈20°C) maintained in our experiments, α is therefore constant in both space and time and thusEmbedded Image(6)

Accordingly, for fixed degrees of saturation in time, one should expect a linear relationship between aS and Se, which is also consistent with experimental observations with cereals (33).

Upon and after crushing. The equilibrium length changes to a new smaller constant value lcr. The crushing of the ith carriage occurs when its fluid activity reaches a critical value Embedded Image. At this point, the constrained modulus attains a constant fully degraded valueEmbedded Image(7)where Embedded Image denotes the critical time for the crushing of the ith carriage. Under constant temperature, its value can be found from the following implicit relationEmbedded Image(8)

Accordingly, the onset of crushing of the ith carriage depends on how its effective saturation Embedded Image evolves with time. Experimentally, however, it is noticed that in between ricequakes, the capillary wetting front remains quite static, whereas immediately after a quake, its motion establishes a new equilibrium position fairly quickly (at the order of a second or less). Therefore, in the interest of providing the simplest explanation for the recurring ricequakes, it could be safely assumed that the saturation is fixed in space relative to the initial position of the wetting front, while the solid points consolidate, and thus, the effective saturation in the ith 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 lcr can be related to the other constants by combining Eqs. 4 and 8 [for p(i) = p*]Embedded Image(9)

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