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    Granularmaterials_Bri_省略_idsandturbulent.pdf

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    Granularmaterials_Bri_省略_idsandturbulent.pdf

    THEORETICAL accepted 4 February 2013; published online 10 March 2013Abstract Granular materials exhibit abundant dissipations due to uctuations in both granularmotions and con gurations i.e., granular skeleton evolutions. Twin granular temperatures Tkand Te are introduced accounting for two types of uctuations, and the so-called twin granulartemperatures theory is established as an extension of granular solid hydrodynamics. By usingsimulations, the nona ne deformations in a 2D assembly are simulated by using discrete elementmethods. By analogizing with microdamages in deformed solids, double scalar damage variables, DPand Dq, are proposed to describe the deformed granular solid under triaxial compressions. Granularows are found intrinsically turbulent due to the presence of Tk and the Naiver{Stokes equation isobtained for granular ows. c⃝ 2013 The Chinese Society of Theoretical and Applied Mechanics.[doi10.1063/2.1302108]Keywords non-equilibrium thermodynamics, granular temperature, damage mechanics, turbulentowGranular materials possess di erent phases thatmay roughly be referred to as gas, liquid and solid.The underlying physics of such granular phases are sub-stantially di erent from ordinary phases, because theelement in granular materials is macroscopic particlesrather than microscopic atoms or molecules. The ba-sic feature is dissipative inter-particle interaction andcorrelated granular con gurations which indeed causedi culties in developing thermodynamics and statis-tical mechanics for granular materials. The majorityof theoretic considerations are focused on highly ex-cited gaseous state, and a number of kinetic theorieshave been established.1 The thermodynamics relationfor elastic solid deformations was rstly proposed byLandau and Lifshitz,2 by introducing the state variableelastic stress πij and the conjugate elastic strain tensoruij. In contrast, for quasi-static deformation of granularsolids and slow granular ows, the innumerable contin-uum mechanical models that are more a rendition ofcomplex empirical data, less a re ection of the under-lying physics. As a uni ed theory, granular solid hy-drodynamics GSH qualitatively account for a broadrange of granular phenomena.3 The total energy den-sity w is a function of entropy s, density ρ and elas-tic strain uij. The conjugate variables are temperatureT ≡ ∂w/∂s, chemical potential ≡ ∂w/∂ρ, and elasticstress πij ≡ ∂w/∂uij. More importantly, GSH intro-duces a granular temperature Tg to quantify the extentof agitation and the conjugate variable is granular en-tropy sg ≡ ∂w/∂Tg.Sun et al.4 extended Tg to a pair of variables Tk andTe, which denote the uctuating kinetic energy as suchin kinetic theories and uctuating potential energy, re-spectively. The velocity of the assembly consists of anaverage value vi ∑j vi;j/N and a random uctuationv′i, where vi;j is the velocity of the jth particle. Therandom motion of particles could roughly correspondaCorresponding author. Email qcsuntsinghua.edu.cn.Granular solids Granular fluidsElasticsolidsDamage andfailureQuasi-staticflowsSlowflowsRapidflowsTedominantTkdominantTe and TkdominantFig. 1. Granular temperatures in various phases of granu-lar materials. For fully elastic granular solids and granulargases e.g., rapid ows, granular elasticity theory and gran-ular kinetic theory are established. For granular damage andfailure and other ow regimes, theories are far from devel-oped. Note that the nona ne deformation of con gurationsis analogized with microdamages in damaged solids.to eddies in a turbulent ows. The uctuating kineticdensity can be denoted by TkTk ∝ ⟨v′iv′i⟩. 1For unjammed systems, Tk C_γ2l2/1 − e2, whereC 1 for two dimensions and C 3 for three dimen-sions, l is the mean free path, _γ is the shear rate and e isthe restitution coe cient. In jammed ows, con gura-tions frequently fail and reestablish which leads elasticenergy uctuations around the averaged value of πijuij.The new granular temperature Te would measure thise ect. Obviously, Te can be excited by πij, vij andTk. If πij 0,vij 0 and Tk 0, the con gurationswill last inde nitely, i.e., Te 0. Rigorous treatment ofcon guration evolutions and determination of Te requiretools not yet well developed due to the strong dissipa-tive nature. Some successes have been made over theseyears. Edwards et al.5 were the rst to propose that astatistical mechanics approach might be feasible to de-scribe dense granular media where the main observablevariable is the volume i.e., V -ensemble. The gran-ular temperature is de ned as compactivity χ, and thegranular entropy S ∂V/∂χ is related to the number of021008-2 Q. Sun, S. Song, and J. Liu, et al. Theor. Appl. Mech. Lett. 3, 021008 2013mechanically stable states for a given volume. Recently,a force-moment ∑ensemble was proposed by using theconcept of angoricity. A scalar eld distribution of theAiry stress function is related to the local stress tensor,which can be expanded to a mean stress value and theuctuations around that value. Note that a value forthe granular entropy S ∑ has been determined.6Figure 1 schematically illustrates the e ects of Tkand Te in jammed and unjammed states. In unjammedrapid ows, particles undertake binary and instanta-neous collisions, and Tk is responsible for bulk viscos-ity, thermal di usivity and collisional energy sinks. Forwell-jammed systems undertaking quasistatic deforma-tions, granular con gurations and the evolutions play agreat role in determining bulk properties, i.e., Te is dom-inant. Most conventional tests in soil mechanics belongto this regime. For intermediate shear rates and packingfractions, i.e., in quasistatic ows and slow ows, gran-ular motion and con guration evolution are of similarimportance, and both compromise to control the bulkbehaviors.For a granular assembly moving at velocity vi, thecomplete set of state variables are proposed in the totalenergy density equationdw Tdsdρ vidpi −πijduij Tedse Tkdsk, 2where granular entropy sk ≡ ∂w/∂Tk, se ≡ ∂w/∂Te.pi ρvi is the momentum density. The systempressure can be derived as p Ts ρ pivi −πijuij Tksk Tese − w. Equilibrium conditions are{∇kT,vij,πij,Tk,Te} 0. As a stability condition,∂πij/∂uij 1 for both microcrackingdamages and microvoid damages. The granular solidsmay contain multiple damage mechanisms, dependingupon the features of con gurations e.g. topology andcharacteristic sizes, boundary conditions and loadinghistory. The ratio of DP/Dq is plotted for states beyondthe densest fraction at ε 2. It is found that as ϱbecomes less DP/Dq would become larger. The ttingfunction isDP/Dq ≃−19.07 40.01ϱ−20.29ϱ2−1 . 20Note that as ϱ 1, DP/Dq ≃ 1.60 and granular solidswould be well jammed which may be treated as ordinarysolids. Extensive comparisons of microdamages withnona ne deformations will be conducted soon.The unsteady granular motion generates an addi-tional stress similar to the Reynolds stress in turbu-lent ows due to emerging eddies which is proposionalto ⟨v′iv′i⟩. Thus, Tk would introduce a similar turbu-lent stress in granular ows. In Tk unjammed ows,021008-5 Granular materials Bridging damaged solids and turbulent fluids Theor. Appl. Mech. Lett. 3, 021008 2013p0 ρ vipi − w Ts Tksk ≃ 0, we obtain σij ≃σmij σkij ηmijklvij ηkijklvij, where ηmijkl ηmδilδjk ςm −ηm/3δijδlk. ηkijkl ηkδilδjk ςk −ηk/3δijδlk.Substituting σij into the momentum balance equationρdtvi ∇kσij ρgi gives the Naiver{Stokes equationρdtvi −∇kp0δij ∇kηmv∗ij −ςmvkkδij ∇kηkv∗ij −ςkvkkδij ρgi, 21where v∗ij vij − vkkδij/3 is the traceless part of vij.The meaning of each term is given by 1 the pressuregradient ∇kp0δij and p0 ρ2∂w/ρ/∂ρ, 2 viscos-ity ∇kηmv∗ij −ςmvkkδij, and 3 external forces e.g.gravity ρgi. ∇kηkv∗ij−ςkvkkδij denotes the turbulencee ects. If Tk is weak and the uid viscosity is ignor-able, Eq. 21 can be simpli ed into the Euler equationρdtvi ∇kp0δijρgi. Note that in realistic granularows where enduring con gurations are formed, all theterms in Eq. 17 should be considered and a much morecomplicated equation than Eq. 21 can be obtained.Mechanical behaviours of granular materials arecollective representations of abundant irreversible pro-cesses which are resulted from unsteady granular mo-tion and con guration evolutions. As belonging to aclass of principle theories, the twin granular temper-atures TGT theory established uctuations, dissipa-tions and the relations to hydrodynamic variables, espe-cially Cauchy stress and irreversible ows. The stress{strain relations in Te dominant granular solids can bestudied with TGT, and the degradations in stresses areanalyzed with double damage variables by comparingwith the treatment in damaged solids. For Tk dominantrapid granular ows, the Naiver{Stokes equation is de-rived as well. Hence, TGT theory touches the frontiersof damage mechanics and turbulent ow mechanics, andprovides a universal theory for granular behaviours. Itis true that the physical meanings of each quantities,in particular, Tk,Te and conjugate sk,se, are not veryclear. They are exactly the key concepts in the statis-tical mechanics of granular matter which is at infancystate similar with the thermodynamics for granular ma-terials.1. C. K. K. Lun, J. Fluid Mech. 233, 539 19912. L. D. Landau, and E. M. Lifshitz, Theory of ElasticityButterworth-Heinemann, Oxford, 19863. Y. Jiang and M Liu, Granular Mat. 11, 139 20094. Q. Sun, S. Song, and F. Jin, et al., Theoret. Appl. Mech. Lett.2, 021002 20125. S. F. Edwards, and R. B .S. Oakeshott, Physica A 157, 108019896. S. Henkes, and B. Chakraborty, Phys. Rev. E 79, 061301 20097. A. Tordesillas, M. Muthuswamy, and S. D. C. Walsh, J. Eng.Mech. 134, 1095 2008

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