Failure progression
The Fiber Bundle Model (FBM)
A key feature of our modeling approach is the implementation of threshold mechanics where mechanical interactions are represented by conceptual mechanical bonds (interconnecting neighboring soil columns and soil-bedrock interface) with well-defined strength thresholds. In this study, each mechanical bond is represented by a Fiber Bundle Model [Raischel et al., 2006; Kun et al., 2006, Cohen et al., 2009] that breaks at the predefined threshold but includes local failures occurring before the entire mechanical bond fails.
Basics of the FBM
An FBM consists of numerous mechanical elements (denoted as fibers) considered (conceptually) as linking two stiff plates in a parallel arrangement. The individual fiber strengths are drawn from a predefined strength distribution, representing the disorder of heterogeneous materials. During loading of the two end plates, weak fibers break and their loads are redistributed to intact fibers that may subsequently also break, thereby triggering a chain reaction of breaking fibers within a bundle. A review of FBM can be found in Kun et al. [2006]. As pointed out by Cohen et al. [2009], various soil elements including interstitial cements, capillarity, frictional contacts, and biological binders resemble FBMs. However, we make no attempt to differentiate and attribute the mechanical properties of the FBM for each of these elements and simply use a bundle representing the collective strength of the soil matrix and an additional bundle to model roots [Cohen et al., 2009].
Describing Mechanical Bonds with the FBM
The strength of an individual fiber σth is randomly chosen from a Weibull distribution defined by the following probability density function:
with the Weibull shape parameter m and scale parameter k that may differ for FBMs representing soils and roots, respectively. Different constitutive laws are applied to these two types of FBM: fibers at the soil-bedrock interface are described by an elastoplastic constitutive law (a broken fiber of intact strength σth can only hold a residual value of αfiber . σth, with residual strength factor αfiber), whereas fibers at the lateral bonds are brittle, implying that the load on broken fibers drops down to zero immediately at the instant of failure. For both types of FBMs, global load sharing rules are applied, i.e., the excess load of broken fibers is redistributed to all intact fibers. The parameters of the FBM (m, k and αfiber) are tuned according to the constraint that (i) the macroscopic strength of the FBM matches soil/roots strength and (ii) the residual strength of the FBM and the residual soil/roots strength are identical.
Link FBM to acoustic emissions (AE) induced by local failure events in soil
The FBM representation of various soil elements including interstitial cements, capillarity, frictional contacts, and biological binders provides numerous advantages and flexibility for the identification and quantification of various local failure events [Cohen et al., 2009].