Breakage mechanics—Part I: Theory

Abstract

Different measures have been suggested for quantifying the amount of fragmentation in randomly compacted crushable aggregates. A most effective and popular measure is to adopt variants of Hardin's [1985. Crushing of soil particles. J. Geotech. Eng. ASCE 111(10), 1177–1192] definition of relative breakage ‘$B_r$’. In this paper we further develop the concept of breakage to formulate a new continuum mechanics theory for crushable granular materials based on statistical and thermomechanical principles. Analogous to the damage internal variable ‘$D$’ which is used in continuum damage mechanics (CDM), here the breakage internal variable ‘$B$’ is adopted. This internal variable represents a particular form of the relative breakage ‘$B_r$’ and measures the relative distance of the current grain size distribution from the initial and ultimate distributions. Similar to ‘$D$’, ‘$B$’ varies from zero to one and describes processes of micro-fractures and the growth of surface area. However, unlike damage that is most suitable to tensioned solid-like materials, the breakage is aimed towards compressed granular matter. While damage effectively represents the opening of micro-cavities and cracks, breakage represents comminution of particles. We term the new theory continuum breakage mechanics (CBM), reflecting the analogy with CDM. A focus is given to developing fundamental concepts and postulates, and identifying the physical meaning of the various variables. In this part of the paper we limit the study to describe an ideal dissipative process that includes breakage without plasticity. Plastic strains are essential, however, in representing aspects that relate to frictional dissipation, and this is covered in Part II of this paper together with model examples.

Introduction

The mechanics of particle crushing, or particle breakage, is one of the most intractable problems in geosciences. The topic is of interest to many research disciplines including powder technology, minerals and mining engineering, geology, geophysics and geomechanics. Prior studies have normally been aimed towards characterising the evolution of the particle size distribution with increase in energy. Another motivation, which is of particular interest in the field of geomechanics, is to link between the crushing of particles and the mechanical response of the soil through behavioural constitutive models. Existing constitutive models are based on simple curve-fitting parameters, which are taken in isolation from unique stress–strain tests, following paths appropriate to the case in hand. The result is that geotechnical engineers often distrust the use of modern constitutive models, which presents one of the most pressing problems of soil mechanics today (Bolton, 2000). For example, none of the current soil mechanics constitutive models takes into account the effect of the grain size distribution, although this distribution is routinely measured at almost every engineering site. The ability of models to represent the evolution of this distribution during the crushing of particles under any mechanical boundary conditions, may potentially form the physical foundations for clarifying many geo-phenomena, and improve the confidence of geotechnical engineers in using constitutive models for sands.

A practical way of introducing the vulnerability of a collection of particles to crush is by constructing an enriched continuum model that mesoscopically averages the micro-crushing events. An exciting example was recently proposed by McDowell et al. (1996). Their theory aimed to explain the reasons behind isotropic hardening in critical state soil mechanics. By adopting a new work equation that includes dissipation from the fracture and creation of new surface area, their proposition was to use the term ‘clastic hardening’, rather than ‘isotropic hardening’, to highlight that it is the fragmentation process that causes the hardening phenomenon. In the particular case of one-dimensional compression, the fracture term may be degenerated and be substituted by a useful plastic dissipation term. When more general loading conditions apply, their approach converts to a typical plasticity formulation, essentially omitting the fact that plastic and fracture dissipations are separate. Consequently, their theory was limited to providing physical justification to the choice of a value for the hardening parameter $\lambda$ in critical state soil models, rather than developing new models. Another interesting approach was undertaken by Ueng and Chen (2000), associating the energy consumption from crushing linearly with the increase of surface area. Unlike the clastic hardening approach that was aimed for one-dimensional compression, Ueng and Chen's approach was concerned only with shear deformations. Indraratna and Salim (2002) have later extended Ueng and Chen's approach by linking the energy consumption with Marsal's (1973) breakage index rather than the increase of surface area. This enabled Indraratna and Salim to extend the analysis for triaxial loading conditions. However, this required adding phenomenological parameters from curve-fitting triaxial stress–strain tests.

Another approach to understand the collective fragmentation process of an aggregate could be given by tracking the changes in the particle size distribution. Hardin (1985) underpinned the necessity for an adequate measure of the crushing in establishing continuum stress–strain models:

In order to understand the physics of the strength and stress–strain behaviour of soils and to devise mathematical models that adequately represent such behaviour, it is important to define the degree to which the particles of an element of soil are crushed or broken during loading.

By advancing the earlier works of Lee and Farhoomand (1967) and Marsal (1973), Hardin developed the concept of relative breakage. Although the relative breakage concept has been widely applied to ‘define the degree to which the particles of an element of soil are crushed or broken during loading’ (e.g., Coop and Lee, 1993, Lee and Coop, 1995, Nakata et al., 1999), the issue of creating ‘mathematical models that adequately represent such behaviour’ by rigorously incorporating the relative breakage concept has yet to be satisfactorily addressed.

The purpose of this paper is to establish a soundly based continuum theory that incorporates the concept of breakage. The pressing problems of the lack of physical meaning of the various parameters is addressed by consistently incorporating the evolution of grain size distribution in constitutive models by developing the concept of breakage.