Granular Mechanics of Geomaterials (presentation in Cassino, Italy, 2006)

Transcription

1 Granular Mechanics of Geomaterials (presentation in Cassino, Italy, 2006) Takashi Matsushima University of Tsukuba 1

2 Where am I from? 2

3 What is granular mechanics? Particle characteristics (size, shape, crushability etc.) Particle interaction Direct simulation (by DEM etc.) Constitutive model as solid or fluid No element test!? Simulation as continua (by FEM, SPH etc.) 3

4 Lunar exploration Mechanical properties of Lunar soil is needed. But only TWO TC test results is available. Lunar regolith: very angular well graded sand Lunar sourcebook,

5 Why granular mechanics? *TRUE prediction from particle information eg. Mechanics of crushable soil Mechanics of unsaturated soil Mechanics of cemented soil and more *UNIQUE constitutive model more convincing, more rational 5

6 Contents: 1. Theory of granular mechanics Chang and Misra 1990, Matsushima and Chang Micro X-ray CT experiment Matsushima et al Image-based DEM simulation Matsushima incl. recent application to Lunar exploration Matsushima

7 Basic theories Molecular dynamics Statistical mechanics for molecules (sparse, non-frictional ) Mechanics of solid frictional particles Kinetic theory (sparse behaves as a fluid) Mechanics of geomaterials (dense behaves as a solid) + 7

8 Uniform strain model (Chang and Misra 1990) stress strain contact contact force, F displ. *Grain controids stick to continuum deformation field *Grain rotation coincides with continuum rotation contact displ. 8

9 9 Contact displ. Contact force Stress Global local coordinates δ R δ T L L s n s n L k k f f δ f 0 0 Force-displ. relation at contact L f R f Global local coordinates δ ref l l c t t t R t T V ) ( 1 f l σ Stress Sum of contacts in VR work at contact = work as continuum

10 Elastic solution Assuming equal-sized isotropic sphere packing E 2 2 kn 4 k N V r N 15V n ks k (2k s n 3k s ) 5kn 4k k n 3n n :coordination number 3 4 r (1 e) s e-n relation is necessary Overall Young s modulus and Poisson ratio can be described by contact stiffness 10

11 11 Elastic solution (continued..) Applying Hertz-Mindlin contact law (Johnson,Contact mechanics) n n n n n k f G r f 1/3 2/3 1 3 s s s n s n s k f f k f 1/3 tan 1 2 ) 2(1 ) 3 (5 2 2/3 1/3 0 2 ) (1 ) (9 ) 3 3(5 ) 4 (5 2 V GN r E 1/3 0 2/3 ) ( 1 ) ( ) 2(1 e e n A E G 3 2/ 3 1/3 2 ) (1 4 3 (9) ) 15(2 ) 4 (5 2 r G r A G is a function of confining pressure, 0 and void ratio, e e e n ) ( Smith et al. (1929) by the combination of closest and loosest packing

12 Elastic solution (continued..) G max (kpa) =100(kPa) Chang (A=280) H-R (B=840) experiment (Kokusho et al.) experiment (Kuribayasi et al.) void ratio Cf. Hardin & Richart G B (2.17 e) 1 e 2 ( ) 0 1/ 2 12

13 Elastic solution (continued..) e=0.68 G max (kpa) Chang (A=280) H-R (B=840) experiment (Iwasaki et al.) confining pressure (kpa) Cf. Hardin & Richart G B (2.17 e) 1 e 2 ( ) 0 1/ 2 13

14 Elastic solution (continued..) Note: A=280 corresponds to G=73(GPa), =0.25 (sand particle as a solid) Estimation of is not good. (Further research is necessary.) 14

15 Nonlinear model Loss of contact is considered by tension-free model f n Contact slip (plasticity) f s f f n n ( elastic) ( sliding) f s f n sign( f s ) n Analytical solution is not available. A set of branch vectors are assumed and the solution is obtained numerically. y x 15

16 Basic response (biaxial compression) stress ratio 1 / elastic elastic, no tension slip plasticity(=0.5) no tension shear strain, = 1-3 shear strain, = stress-strain curve volumetric strain v = elastic slip plasticity(=0.5) no tension elastic, no tension dilation curve Material does NOT yield! 16

17 Basic response (biaxial compression) Why? Contact of loading direction does NOT slip. slip no tension, =0.5 contact normal force (N) Contact normal force distribution 17

18 DEM study stress ratio 1 / No grain rotation grain rotation free periodic boundaries at both directions volumetric strain v maximum shear strain No grain rotation grain rotation free maximum shear strain

19 DEM study dense, rotation free contact normal force (N) Force chain in granular assembly Contact normal force distribution 19

20 Buckling of granular column (Matsushima and Chang, 2006) Assign minimum aspect ratio for contact force distribution Determine average contact force such that the stored energy in contact keeps constant insufficient confining pressure buckling 20

21 Response of Buckling model (Matsushima and Chang, 2006) Buckling resistance controls the yield strength related to particle properties 21

22 Possibility of granular mechanics C.E. Rational incorporation of particle properties (size, shape, stiffness, crushability, contact cement, etc.) Validation not only with macro response but also with particle-level information Detailed comparison with Particle visualization experiment More realistic DEM is needed. 22

23 Particle visualization by Micro X-ray CT At University Joseph Fourier, Grenoble,

24 Objective to obtain 3-D micro properties (grain properties, microstructure and its change due to external loading) of some standard sands with micro X-ray CT in SPring-8 Typical standard sands: Toyoura sand: D50 =0.167(mm) Ottawa sand: D50 =0.174(mm) Hostun sand: D50 =0.408(mm) S.L.B. sand: D50 =0.681(mm) (high resolution system is necessary) 24

25 Micro X-ray CT at Spring-8 Storage Ring Synchrotron (Electron is accelerated up to 8 GeV) The world's largest third-generation synchrotron radiation facility 25

26 Storage ring facility Experimental room 47 beamlines (BL) X-ray CT is available at BL20B2 and BL47XU. 26

27 Outline of Spring-8 CCD camera X-ray comes from here Stage Detector 27

28 X-ray CT system at SPring-8 CCD Monochromator X-rays specimen rotation stage visible light fluorescent screen mirror Resolution: BL20B2: 13 m BL47XU: 1.5 m 28

29 Toyoura sand dense 29

30 Example of CT image (BL20B2) Toyoura sand dense medium dense loose 30

31 Example of CT image (BL20B2) Glass beads Hostun sand Ottawa sand 31

32 Example of CT image (BL47XU) Ottawa sand Toyoura sand Wakasa sand Roundish very angular Lunar soil simulant(fjs-1) 32

33 Example of CT image (BL47XU) Masado (crushable sand) unsaturated condition 33

34 Micro Triaxial test (BL20B2)(1) Load cell specimen pressure cell 4.3mm pressure tank loading motor membrane specimen diameter 4.3mm height 10mm 34

35 Micro Triaxial test (BL20B2)(2) CT scan of initial condition loading stop loading and CT scan (1.5 hours) axial stress (kpa) Toyoura sand confining pressure=100 (kpa) dense (e ini =0.719) loose (e ini =0.984) axial strain (%) stress-strain curve 35

36 Micro Triaxial test (BL20B2)(3) Toyoura (medium dense) void increase within a shear band 36

37 Micro Triaxial test (BL20B2)(4) Masado (cruchable sand) mm (medium dense) particle crushing is NOT predominant 37

38 Micro 1-D compression test axial strain (%) 0 1-D compression Masado 10 ( mm) axial stress (MPa) particle crushing after yield stress 38

39 Processing of CT image Original image After binarization After pore-filling After 1st erosion After 2nd erosion After 3rd erosion Cluster labeling Attribution 39

40 Adequate erosion cycles number of resultng clusters Toyoura sand, dense a glass beads, dense d Toyoura sand, dense a number of erosion after 3rd erosion maximum equivalent diameter (m) Toyoura sand, dense a glass beads, dense d Toyoura sand, dense a number of erosion after 5th erosion frequency frequency equivalent diameter of cluster (m) equivalent diameter of cluster (m) 40

41 Grain identification 3-D image after attribution process 41

42 Grain identification Identified Toyoura sand grains 42

43 Grain size distribution percent passing by weight Glass beads loose Toyoura sand dense Glass beads dense Equivalent diameter (m) 43

44 z z z z z Contact area and their normals 160 Contact between No.1 and No Contact between No.4 and No x y x y 160 Contact between No.4 and No Contact between No.4 and No Contact between No.4 and No x y x coordination number, fabric tensor, etc. y x y 44

45 Coordination number (1) frequency (%) an030525a-3 Toyoura sand dense(n=0.409) frequency (%) coordination number, N C coordination number, N C Smith et al. (1929) lead balls loose(n=0.447) at030525e-3 glass beads loose (n=0.451) like a normal distribution 45

46 Coordination number (2) 12 coodination number N C Smith et al.(1929) Glass beads dense Toyoura sand dense Glass beads loose porosity n e n e 1 46

47 Next step *Detection of each grain motion (translation and rotation) (Chang, Matsushima, Lee, J. Eng. Mech. ASCE, 2002.) *Detection of grain crushing 47

48 Image-based DEM 48

49 Background * Rapid increase of computer abilities DEM simulations with large number of grains * For quantitative discussion Precise modeling of grains (contact model, grain shape, crushability, etc.) is necessary. This study deals with GRAIN SHAPE MODELING 49

50 Image-based modeling (Matsushima and Saomoto 2002) Irregular grain shape is described by a rigid connection of elements (circles or spheres) To find an optimum positions and radii of the elements time marching computation (Dynamic optimization) Each element moves and expands (or shrink) to get a better fitting to the target grain shape 50

51 Dynamic Optimization algorithm Each surface point of a target grain gives an attraction to the closest element which directs from the centroid of the element to the surface point and its magnitude is proportional to the distance. 51

52 2D example (1) percent passing by weight Toyoura Ottawa Diameter (mm) Grain density 2.64(g/cm 2 ) Spring constant (normal) 1.0e9 (g/s 2 ) (tangential) 2.5e8 (g/s 2 ) Damping coefficient (normal) 2.0e2 (g/s) (tangential) 1.0e2 (g/s) Friction coefficient between grains 27 (deg.) Time increment 2.5e-8 (sec.) Toyoura sand (sub-angular) Ottawa sand (sub-rounded) 52

53 2D example (2) 200 grains (Each grain is modeled with 10 elements) Periodic boundary at both sides Constant confining pressure(10kn/m), Lateral displacement is imposed at the bottom grains 53

54 2D example (3) Dense specimen (eini=0.20) mobilized friction angle mob (deg.) Toyoura Ottawa Circles simple shear test with 200 grains (e ini =0.20) shear strain Toyoura>Ottawa>Circles for peak strength 54

55 2D example (4) Peak strengths with different initial void ratio peak friction angle peak (deg.) Toyoura 15 Ottawa Circles initial void ratio e ini The modeling is verified qualitatively. 55

56 2D example (5) Granular column structure during shear Two grains are in contact with plural points Moment transmission Higher strength 56

57 3D example: Toyoura sand (1) X-ray CT result 3D modeling Toyoura sand model 57

58 3D example: Toyoura sand (2) percent passing by weight DEM model CT data for 400 grains grain diameter (mm) Grain size distribution: DEM model = original CT data Simple shear simulation 58

59 3D example: Toyoura sand (3) stress ratio ( / n ) e ini =0.672 =20 (deg.) 0.4 e TSS test ( n =100 (kpa) ini =0.763 (Pradhan et al. 1988) 0.2 e ini =0.659 e ini = shear strain volumetric strain d TSS test ( n =100 (kpa) (Pradhan et al. 1988) e ini =0.659 e ini =0.793 =20 (deg.) e ini =0.672 e ini = shear strain Stress-strain curve dilation curve Quantitative agreement with experiment if inter-particle friction angle=20(deg.) Matsushima

60 3D example: Toyoura sand (4) internal friction angle d Toyoura sand DEM =20 (deg.) DEM (spheres) =27 (deg.) Toyoura sand DEM =27 (deg.) TSS test (Toyoura sand) ( n =100 (kpa) (Pradhan et al. 1988) initial void ratio e ini Peak strength for various void ratio 60

61 Lunar soil simulant (1) h= 1.1mm 600 grains (10-spheres model) are fell down into a vessel Periodic boundary in y-direction w=1.0mm Bottom grains are fixed into the bottom plate d= 0.2mm Remove the right side wall 61

62 62

63 Lunar soil simulant(3): Parametric study angle of repose(deg) ks=kn/4 exp. FJS-1 10-elements model 1-element model exp. glass beads angle of repose (deg) elements model 1-element model k n (*1e4) Effect of grain hardness effect of interparticle friction angle of repose (deg) restitution 10-elements coefficient model element model Effect of damping inter-particle friction angle(deg) 63

64 Lunar soil simulant(4): model accuracy 2-elements model 3-elements model 4-elements model 7-elements model 64

65 Lunar soil simulant(5): model accuracy 50 exp (FJS-1) angle of repose(deg) exp (glass beads) number of spheres 65

66 Particle shape effect: The mechanism Contact force chain Matsushima, P&G2005 Plural contact points between two grains rolling resistance overall shape is important 66

67 Conclusions Application of Granular Mechanics into geomaterials has been tried for decades. Some breakthrough has been found recently (in my opinion). Recent progress of various technologies makes it push forward. 67

68 References Chang, C.-S. and Misra, A. 1990a. Application of uniform strain theory to heterogeneous granular solids, J. Eng. Mech., ASCE, 116, 10, Chang, C.-S. and Misra, A. 1990b. Packing structure and me-chanical properties of granulates, J. Eng. Mech., ASCE, 116, 5, Smith, W.O., Foote, P.D., Busang, P.F Packing of ho-mogeneous spheres, Physical Review, 34, Heiken, G.H., Vaniman, D.T., French, B.M. editors (1991). Lunar Sourcebook, Cambridge University Press, Matsushima, T. and Chang, C-S Proc. IS-Yamaguchi, (submitted). Matsushima, T. and Saomoto, H. (2002). Discrete element modeling for irregularly-shaped sand grains, Proc. NUMGE2002: Numerical Methods in Geotechnical Engineering, Mestat (ed.), Matsushima, T. (2004). 3-D Image-based Discrete Element Modeling for Irregularly-Shaped Grains, Proc. 2nd International PFC Symposium. 68

69 References Matsushima, T., Saomoto, H., Uesugi, K., Tsuchiyama, A. and Nakano, T. (2004). Detection of 3-D irregular grain shape of Toyoura sand at SPring-8, X-ray CT for geomaterials: Proc. International workshop on X-ray CT for geomaterials (Otani and Obara eds.), Balkema, Chang, C.-S., Matsushima, T., Lee, X.: Heterogeneous Strain and Bonded Granular Structure Change in Triaxial Specimen Studied by Computer Tomography, Journal of Engineering Mechanics, ASCE. Vol. 129, Issue 11, pp , Matsushima, T. et al. (2006.3) Image-based modeling of Lunar soil simulant for 3- D DEM simulations, Earth & Space 2006 (CD-ROM). (to be submitted to J. Aerospace Eng., ASCE). Matsushima, T. (2005) Effect of irregular grain shape on quasi-static shear behavior of granular assembly, Proc. Powders & Grains