Exploration of Double Elasticity and Dynamics in Rock Using the Bonded Particle Model
Authors: Arnold L. A. 
Published: September 2023

ABSTRACT
The ability to model the dynamic behavior of rock is dependent on our ability to model its deformation under dynamic loading conditions. Unlike static loading conditions, dynamic loads may be cyclic in nature (e.g., during seismic shaking) and involve the unloading and reloading of material. As a cemented granular material, rock is inherently double elastic, meaning that it has different stiffnesses in tension and compression – the tensile stiffness is lower than compressive. Loading cases that involve both tension and compression will therefore result in greater deformation than predicted based purely on compressive stiffness (which is most commonly measured in the laboratory). For example, a cantilevered rock column, shaken at its base, will exhibit a lower frequency dynamic response than predicted based on a uniform stiffness matching the column's behavior in compression. In the bonded particle model (BPM), the relative stiffness between particle-to-particle contacts and bonds simulates the imperfect cementation of granular minerals within rock, which results in double elasticity. In this paper, a simple mathematical model of a double elastic cantilever rock column is evaluated. By solving equations of equilibrium, kinematics, and constitutive relations for this system, a relationship between micro-level stiffnesses and emergent macro-level stiffnesses is developed. The relationship is validated by its ability to predict the dynamic macro-scale flexural stiffness of a 2D BPM rock column from the micro-scale bond and particle parameters. Both 2D BPM and simple mathematical models are shown to be in agreement with experimental data of the dynamic flexural behavior of a cemented granular synthetic rock.

INTRODUCTION
Studying and modeling the dynamic behavior of rock is critical to predicting rock behavior during dynamic loading, such as from blasting and seismic loading. Dynamic loading from seismic sources is particularly hazardous due to its unpredictability, and uncontrolled nature. Seismically-induced rock slope failures are one of the most dangerous and deadly of all co-seismic hazards, capable of inducing extensive damage and killing tens of thousands in a single event (Keefer & Larsen, 2007; Sun et al., 2012; Xu et al., 2009; Massey et al., 2016). This paper focuses on one aspect of dynamic loading from earthquakes in rock: the potential for reversal of stress magnitude from compression to tension (and back again). This reversal can occur near the ground surface, particularly when topographic features (e.g., slopes) and geologic features (e.g., joints) allow for geometries, such as in toppling slopes, lending themselves to flexural behavior (Adhikary et al., 1997). To model this dynamic behavior, we must capture the dynamic deformation in both tension and compression. The dynamic deformation not only controls the maximum strains experienced in the rock but also the natural frequency of the body of interest, which in turn, controls its internal stress response to dynamic loading (Makdisi & Seed, 1978). The dynamic deformation is a parameter influencing stability that cannot be ignored.