Acta Geotechnica manuscript No. (will be inserted by the editor) Energy distribution during the quasi-static confined comminution of granular materials Pei Wang · Chloe´ Arson Received: date / Accepted: date Abstract A method is proposed to calculate the distribu- 1 Introduction tion of energy during the quasi-static confined comminution of particulate assemblies. The work input, calculated by in- Particle breakage upon quasi-static comminution is a topic tegrating the load-displacement curve, is written as the sum of interest in civil engineering, powder technology and the of the elastic deformation energy, the breakage energy and mineral industry. Many aspects of soil behavior and prop- the redistribution energy. Experimental results obtained on erties, such as dilation, yielding, shear strength and perme- samples subjected to compression stresses ranging between ability are related to particle breakage [16, 22, 40]. Under 0.4 MPa and 92 MPa are used to calibrate the model. The high compressive stress, particles break and decrease in size. elastic energy stored in the samples is obtained by simulat- During this process, the total input energy (δW ) is stored ing the compression test on the final Particle Size Distribu- in the form of elastic energy in the grains, and dissipated tions (PSD) with the Discrete Element Method (DEM), and by breakage, friction and redistribution, i.e., the production by extracting the contact forces. A PSD evolution law is pro- of kinetic energy triggered by crushing [37]. A good un- posed to account for particle breakage. The PSD is related to derstanding of how the energy dissipates during compres- the total particle surface in the sample, which allows calcu- sion is necessary to formulate sound constitutive models for lating the breakage energy. The redistribution energy, which granular materials. For instance, McDowell and Bolton were comprises the kinetic energy of particles being rearranged able to explain compressibility changes by relating parti- and the friction energy dissipated at contacts, is obtained by cle crushing to the variations of void ratio, and by intro- subtracting the elastic energy and breakage energy from the ducing the breakage energy in the Cam clay model [23]. work input. Results show that: (1) At least 60% of the work Based on thermodynamic principles, Collin introduced non- input is dissipated by particle redistribution; (2) The fraction associative flow rules to model friction-induced plasticity of elastic deformation energy increases and the fraction of in granular materials [8]. Recently, the breakage mechan- redistribution energy decreases as the compression stress in- ics theory was formulated to predict the thermodynamic re- creases; (3) The breakage energy accounts for less than 5% sponse of crushable granular materials within continuum me- of the total input energy, and this value is independent of chanics [12, 13]. In the theory of breakage mechanics, the the compressive stress; (4) The energy dissipated by redis- evolution of the particle size distribution is used to predict tribution is between 14 to 30 times larger than the breakage the amount of energy dissipated by breakage, which is fully energy. coupled to the redistribution energy. Within this framework, further developments were proposed to analyze creep, per- Keywords Particulate mechanics · Comminution · Grain meability, cementation and the brittle-ductile transition of breakage · Energy dissipation · Thermodynamics granular materials [11, 29, 49, 52]. How to calculate the relative fraction of the energy compo- nents dissipated during commiunition has been studied for P. Wang · C. Arson decades both at the grain and at the sample scales, with School of Civil and Environmental Engineering, analytical, numerical and experimental methods. Bolton et Georgia Institute of Technology, Atlanta, GA, USA al. used a Discrete Element Method (DEM) to analyze the E-mail: [email protected], [email protected] energy distribution during a crushing test performed on a 2 Pei Wang, Chloe´ Arson single particle, modeled as an assembly of bonded spheres 2 Energy decomposition [5]. The breakage energy was calculated as the total elas- tic energy stored at the contacts before bonds failed. The Based on the theory of critical state soil mechanics, Roscoe energy dissipated by breakage was estimated to be about et al. [35] established the following energy balance equation 10% of the total input energy. The remainder energy was for a soil sample subjected to an increment of mean stress δp converted into kinetic energy of fragments and then lost by and an increment of deviatoric stress δq: friction and damping. By contrast, in another DEM simula- κδp qδ" + pδν = + Mpδ"; (1) tion of particle crushing, the breakage energy was found to 1 + e be 30% of the total input energy [2]. This difference may come from the properties of the contact bond model, which where δ" and δν are the shear and volumetric strain, κ is the is difficult to calibrate, and yet largely influences the magni- slope of swelling line and M = q=p. The left hand side of tude of the breakage energy. A variety of contact bond mod- the equation represents the work done by the mean and de- els were proposed to better capture bond breakage [19, 24, viatoric stresses, while the right hand side is the sum of the 46]. A study combining experiments and numerical simu- increments of internal energy and dissipated energy. Miura lations showed a strong relationship between bond break- and Yamamoto [26] conducted a series of high pressure tri- age and strain softening [21]. Some DEM models account axial tests on quartz sand and studied the relationship be- for the statistical distributions of bond strengths and parti- tween the increase of specific surface area ∆S and the plas- cle strengths. However, the distribution of energy at sam- tic work δWp. The ratio ∆S/δWp was used as an index of ple sccale is still not fully understood [7, 31]. Recently, X- particle crushing, and was related to the dilatancy rate. The ray micro-tomography was used to measure the area of new model was validated later [25] against static triaxial com- material surfaces created during a single particle crushing pression tests and repeated triaxial tests on granitic soil. The test [53]. Zhao et al. showed that as breakage proceeds, the plastic work included both friction dissipation and break- extent of material surfaces increases, therefore the number age energy. McDowell [23] developed further the theory of of potential areas of contact increases, and the proportion Roscoe et al. by introducing the breakage energy, expressed of energy dissipated by friction overtakes the energy dissi- as: pated by breakage. At the macro-scale, Rusell et al. used Γ dS ΦS = (2) simplified models to calculate the ratio between the redis- Vs(1 + e) tribution energy and the breakage energy, which turned out In which Γ is the surface free energy, which remains con- to be stress independent, but dimension dependent: the ra- stant during a confined compression test. tio was 5 to 20 in 1D, 2 in 2D, and 1 in 3D [29, 37]. The Recently, another form of energy dissipation, called redistri- 3D DEM simulation of a triaxial shear test on crushable soil bution energy, was introduced in the energy balance equa- showed that particle breakage contributed to a small amount tion, in order to account for the rearrangement of grains to the total energy dissipated, but promoted energy dissipa- around crushed particles. The dissipation of redistribution tion by inter-particle friction dissipation [47]. Energy dis- energy is coupled to that of breakage energy, because micro- tribution was stress dependent, which is in agreement with structure rearrangement is triggered by breakage events. The the results published in [30]. In general, recent results show complete energy balance equation was given by [36]: that energy is mostly dissipated by particle redistribution and inter-granular friction. But there is still no consensus on δW = δΨe + δΦp + δΦS + δΦredist (3) the relative fraction of the different components of energy where δΨ , δΦ , δΦ and δΦ represent the elastic en- dissipated. e p S redist ergy, plastic dissipation energy, breakage energy and redis- In this paper, we present a novel method to calculate the dis- tribution energy, respectively. Note that plasticity can be due tribution of energy in a granular sample subjected to quasi- to various processes, such as dislocation creep, dynamic re- static confined comminution. We used results of uniaxial crystallization, plastic deformation and micro-fracturing at compression tests performed on cylindrical samples of ground grain scale [42, 43]. We focus on short-term isothermal pro- shale and sands under both low compressive stress (0.4MPa cesses, and thus ignore thermal softening, volume change to 2.1MPa, [30]) and high compressive stress (up to 92MPa, and other material property changes induced by temperature [27]) . The first section explains how we expressed the en- variations [39]. We also ignore the transition from brittle to ergy in the form of elastic deformation energy, breakage ductile behavior at the grain scale as the coordination num- energy and redistribution energy. The second section de- ber gets higher [48]. Particles are thus considered elastic, scribes the breakage energy model and its calibration. Next, i.e. the plastic deformation of the Representative Elementary the method to calculate the other energy components is pre- Volume (REV) is solely due to friction between the grains. sented. The last section provides a discussion of the results Based on a calibration against two oedometer tests per- obtained. formed on silica sands, it was found that the ratio between Energy distribution during the quasi-static confined comminution of granular materials 3 2 the redistribution energy and the breakage energy, R = δΦredist=∆ΦS10, was between 13 and 16. In another study, the breakage en- Test L1 ergy amounted to 25% to 30 % of the total input energy [2].