Generalised homogenisation procedures for granular materials

Journal of Engineering Mathematics

Volumn 52, Issue 1, 2005, Pages 199-229

  Pasternak, Elena ^a   Muhlhaus H B ^a,b  

^a UNIVERSITY OF WESTERN AUSTRALIA   (Australia)

^b UNIVERSITY OF QUEENSLAND   (Australia)

Author keywords

Cosserat continuum; Homogenisation; Non local continuum; Rotational degrees of freedom; Wave propagation

Indexed keywords

CONTINUUM MECHANICS; DEFORMATION; DEGREES OF FREEDOM (MECHANICS); DISCRETE FOURIER TRANSFORMS; FINITE DIFFERENCE METHOD; MATHEMATICAL MODELS; ROTATION;

COSSERAT CONTINUUM THEORY; HOMOGENIZATION; INTEGRAL HOMOGENIZATION THEORY; TAYLOR EXPANSIONS;

GRANULAR MATERIALS;

GRANULAR MEDIUM;

EID: 22844441338     PISSN: 00220833     EISSN: 15732703     Source Type: Journal    
DOI: 10.1007/s10665-004-3950-z     Document Type: Article

References (74)

1. E. Cosserat and F. Cosserat 1909 Théorie des Corps Déformables A Herrmann et Fils Paris 226pp
2. W. Nowacki 1974 The linear theory of micropolar elasticity In: W. Nowacki and W. Olszak (eds.) Micropolar Elasticity Springer-Verlag Wien, New York 1-43
3. R.D. Mindlin 1964 Micro-structure in linear elasticity Arch. Ration. Mech. Anal 16 51-78
4. A.V. Dyskin R.L. Salganik and K.B. Ustinov 1992 Multi-scale geomechanical modelling In: T. Szwedziki G.R. Baird and T.N. Little (eds.) Proceedings of Western Australian Conference of Mining Geomechanics Curtin University WASM Kalgoorlie, Western Australia 235-246
5. L.N. Germanivich and A.V. Dyskin 1994 Virial expansions in problems of effective characteristics. Part I. General concepts. J. Mech. Compos. Mater 30 2 222-237
6. Mühlhaus H.-B., Dyskin A., Pasternak E. and Adhikary D. (1999). Non-standard continuum theories in geomechanics: Theory, experiments and analysis. In: Picu R.C., Kremple E. (ed). Proceedings of the Fourth International Conference on Constitutive Laws for Engineering Materials. Troy New York, Rensselaer Polytechnic Institute, pp. 321-324
7. R.D. Mindlin and H.F. Tiersten 1962 Effects of couple-stresses in linear elasticity Arch. Ration. Mech. Anal 11 415-448
8. A.C. Eringen 1966 Linear theory of micropolar elasticity J. Math. Mech 15 909-923
9. A.C. Eringen and C.B. Kafadar 1976 Polar field theories In: A.C. Eringen (ed.) Continuum Physics, Volume IV, Part I Academic Press New York 4-73
10. P. Germain 1973 La méthode des puissances virtuelles en mécanique des milieux continues. Première partie. Théorie. du second gradient J. Mécanique 12 235-274
11. P. Germain 1973 The method of virtual power in continuum mechanics. Part 2: Microstructure SIAM J. Appl. Math 25 556-575
12. G.A. Maugin 1980 The method of virtual power in continuum mechanics: application to coupled fields Acta Mech 35 1-70
13. I.A. Kunin 1982 Elastic Media with Microstructure 1. One-dimensional Models Springer-Verlag Berlin, Heidelberg, New York 291
14. E. Kröner 1970 The problem of non-locality in the mechanics of solids: Review of the present status In: J.A. Simmons R. Wit de and R. Bullough (eds.) Fundamental Aspects of Dislocation Theory. National Bureau of Standards Special Publication 317. Vol. II National Bureau of Standards Washington 729-736
15. E. Kröner B.K. Datta 1970 Non-local theory of elasticity for a finite inhomogeneous medium - A derivation from lattice theory In: J.A. Simmons R. Wit de and R. Bullough (eds.) Fundamental Aspects of Dislocation Theory. National Bureau of Standards Special Publication 317. Vol. II National Bureau of Standards Washington 737-746
16. I.A. Kunin and A.M. Waisman 1970 On problems of the non-local theory of elasticity In: J.A. Simmons R. Wit de and R. Bullough (eds.) Fundamental Aspects of Dislocation Theory. National Bureau of Standards Special Publication 317 Vol. II National Bureau of Standards Washington 747-759
17. I.A. Kunin 1983 Elastic Media with Microstructure I1. Three-dimensional Models Springer-Verlag Berlin, Heidelberg, New York 272
18. A.C. Eringen 1987 Non-local continuum description of lattice dynamics and application In: J. Chandra and R.P. Srivastav (eds.) Constitutive Models of Deformation SIAM Philadelphia 59-80
19. A.C. Eringen 1976 Non-local polar field theories A.C. Eringen Continuum Physics. Volume IV Part III Academic Press New York 205-264
20. A.C. Eringen 1978 Non-local continuum mechanics and some application In: A.O. Barut (ed.) Non-linear Equations in Physics and Mathematics D. Reidel Publishing Company Dordrecht 271-318
21. R. Borst de A. Benallal and R.H.J. Peerlings 1997 On gradient-enhanced damage theories In: N.A. Fleck and A.C.F. Cocks (eds.) IUTAM Symposium on Mechanics of Granular and Porous Materials Kluwer Academic Publishers Dordrecht 215-226
22. G. Pijaudier-Cabot and Z.P. Bazant 1987 Non-local damage theory J. Engng. Mech 113 1512-1533
23. Z.P. Bazant and G. Pijaudier-Cabot 1988 Non-local continuum damage, localization instability and convergence J. Appl. Mech 55 287-293
24. G. Pijaudier-Cabot 1995 Non-local damage In: H.-B. Mühlhaus (ed.) Continuum Models for Materials with Microstructure. (Chapter 4) John Wiley & Sons Chichester, New York, Brisbane, Toronto, Singapore 105-143
25. H.-B. Mühlhaus 1993 Continuum models for layered and blocky rock In: J.A. Hudson (ed.) Comprehensive Rock Engineering: Principles, Practice & Projects. Invited Chapter for Vol. II: Analysis and Design Methods Pergamon Press Oxford, New York 209-230
26. N.V. Zvolinskii and K.N. Shkhinek 1984 Continual model of laminar elastic medium Mech. Solids 19 1 1-9
27. D.P Adhikary and A.V. Dyskin 1997 A Cosserat continuum model for layered materials Comp. Geotechn 20 15-45
28. H.-B. Mülhaus 1995 A relative gradient model for laminated materials In: H.-B. Mühlhaus (ed.) Continuum Models for Materials with Microstructure. (Chapter 13) John Wiley & Sons Chichester, New York, Brisbane, Toronto, Singapore 451-482
29. H.-B. Mühlhaus and P. Hornby 1998 A relative gradient theory for layered materials J. Phys. IV France 8 269-276
30. H.-B. Mühlhaus and P. Hornby 1998 A beam theory gradient continua In: R. Borst de E. Giessen van der (eds.) Material Instabilities in Solids. (Chapter 32) John Wiley & Sons Chichester, New York 521-532
31. J. Sulem and H.-B. Mühlhaus 1997 A continuum model for periodic two-dimensional block structures Mech. Cohesive-Frictional Mater 2 31-46
32. H.-B. Mühlhaus and I. Vardoulakis 1987 The thickness of shear bands in granular materials Géotechnique 37 271-283
33. H.-B. Mühlhaus R. Borst de and E.C. Aifantis 1991 Constitutive models and numerical analyses for inelastic materials with microstructure In: G. Beer J.R. Booker and J. Carter (eds.) Computing Methods and Advances in Geomechanics Balkema Rotterdam 377-385
34. C.S Chang and L. Ma 1992 Elastic material constants for isotropic granular solids with particle rotation Int. J. Solids Struct 29 1001-1018
35. H.-B. Mühlhaus and F. Oka 1996 Dispersion and wave propagation in discrete and continuous models for granular materials Int. J. Solids Struct 33 2841-2858
36. Mühlhaus and P. Hornby 1997 On the reality of antisymmetric stresses in fast granular flows In: N.A. Fleck and A.C.F. Cocks (eds.) IUTAM Symposium on Mechanics of Granular and Porous Materials Kluwer Academic Publishers Dordrecht 299-311
37. G.N. Wells and L.J. Sluys 2001 Partition-of-unity for fracture of brittle materials In: H.-B. Mühlhaus A.V. Dyskin and E. Pasternak (eds.) Bifurcation and Localization in Geomechanics Swets & Zeitlinger Lisse 169-176
38. R. Hill 1963 Elastic properties of reinforced solids: Some theoretical principles J. Mech. Phys. Solids 11 357-372
39. T. Mori and K. Tanaka 1973 Average stress in matrix and average elastic energy of materials with misfitting inclusions Acta Metal 21 571-574
40. R.M. Christensen 1979 Mechanics of Composite Materials John Wiley & Sons New York 348
41. Z. Hashin 1988 The differential scheme and its application to cracked materials J. Mech. Phys. Solids 36 719-734
42. M. Kachanov 1992 Effective elastic properties of cracked solids: critical review of some basic concepts Appl. Mech. Rev 45 8 304-335
43. S. Nemat-Nasser and H. Horii (1993). Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland, Amsterdam, London, New York, Tokyo, 687 pp
44. D. Krajcinovic 1996 Damage Mechanics Elsevier Amsterdam, Lausanne, New York, Oxford, Shannon, Tokyo 761
45. B. Cambou 1998 Micromechanical approach in granular materials In: B. Cambou (ed.) Behaviour of Granular Materials. CISM Courses and Lectures No. 385 Springer Wein, New York 171-216
46. G.N. Savin and L.P. Khoroshun 1972 The problem of elastic constants of stoichastically reinforced materials. Mekhanika sploshnoy sredy i rodstvennye problemy analiza. [Mechanics of continuous media and related problems of analysis] Nauka Press Moscow 437-444 (in Russian)
47. L.P. Khoroshun 1978 Methods of theory of random functions in problems of macroscopic properties of microinhomogeneous media Soviet Appl. Mech 14 113-124
48. J. Duffy and R.D. Mindlin 1957 Stress-strain relation and vibrations of granular medium J. Appl. Mech 24 585-593
49. H. Deresiewicz 1958 Stress-strain relations for a simple model of a granular medium J. Appl. Mech 25 402-406
50. S.A. Meguid and A.L. Kalamkarov 1994 Asymptotic homogenization of elastic materials with a regular structure Int. J. Solids Struct 31 303-316
51. G.A. Vanin 1985 Micromechanics of Composite Materials Naukova Dumka Kiev 302 (in Russian)
52. G.A. Maugin 1999 Non-linear Waves in Elastic Crystals Oxford University Press Oxford 314
53. A.S.J. Suiker R. Borst de and C.S. Chang 2000 Micro-mechanically based higher-order continuum models for granular materials In: D. Kolymbas (ed.) Constitutive Modelling of Granular Materials Springer Berlin 249-274
54. A.S.J. Suiker R. Borst de and C.S. Chang 2001 Micro-mechanical modelling of granular material. Part 1: Derivation of a second-gradient micro-polar constitutive theory Acta Mech 149 161-180
55. A.S.J. Suiker R. Borst de and C.S. Chang 2001 Micro-mechanical modelling of granular material. Part 2: Plane wave propagation in finite media Acta Mech 149 181-200
56. M. Satake 1997 Three-dimensional discrete mechanics of granular materials In: N.A. Fleck and A.C.F. Cocks (eds.) IUTAM Symposium on Mechanics of Granular and Porous Materials Kluwer Academic Publishers Dordrecht 193-202
57. U. Tüz̈n and D.M. Heyes 1997 Distinct element simulations and dynamic microstructural imaging of slow shearing granular flows In: N.A. Fleck and A.C.F. Cocks (eds.) IUTAM Symposium on Mechanics of Granular and Porous Materials Kluwer Academic Publishers Dordrecht 263-274
58. C. Thornton 2000 Microscopic approach contributions to constitutive modelling In: D. Kolymbas (ed.) Constitutive Modelling of Granular Materials Springer Berlin 193-208
59. H.-B. Mühlhaus L. Moresi and H. Sakaguchi 2000 Discrete and continuum modelling of granular materials In: D. Kolymbas (ed.) Constitutive Modelling of Granular Materials Springer Berlin 209-224
60. G. Gudehus 1996 A comprehensive constitutive equation for granular materials Soils and Foundations 36 1-12
61. P.J. Digby 1981 The effective elastic moduli of porous granular rocks J. Appl. Mech 16 803-808
62. K. Walton 1987 The effective elastic modulus of a random packing of spheres J. Mech. Phys. Solids 35 213-226
63. R.J. Bathurst and L. Rothenberg 1988 Micromechanical aspects of isotropic granular assemblies with linear contact interactions J. Appl. Mech 55 17-23
64. C.S. Chang 1988 Micromechanical modelling of constitutive relations for granular material In: M. Satake and J.T. Jenkins (eds.) Micromechanics of Granular Materials Elsevier Science Publishers B.V. Amsterdam 271-279
65. J.T. Jenkins 1988 Volume change in small strain axisymmetric deformations of a granular material In: M. Satake and J.T. Jenkins (eds.) Micromechanics of Granular Materials Elsevier Science Publishers B.V. Amsterdam 245-252
66. B. Cambou F. Dedecker M. Chaze 2000 Relevant local variables for the change of scale in granular materials In: D. Kolymbas (ed.) Constitutive Modelling of Granular Materials Springer Berlin 275-290
67. N.A. Fleck and A.C.F. Cocks 1997 (eds.) IUTAM Symposium on Mechanics of Granular and Porous Materials Kluwer Academic Publishers Dordrecht 450
68. E. Pasternak and H.-B. Mühlhaus 2001 Non-classical continua for modelling of granulate materials In: J.P. Denier and E.O. Tuck (eds.) The 2001 ANZIAM Applied Mathematics Conference Abstracts University of Adelaide Barossa Valley, South Australia 64
69. E. Pasternak and H.-B. Mühlhaus 2001 Cosserat continuum modelling of granulate materials S. Valliappan N. Khalili Computational Mechanics - New Frontiers for New Millennium Elsevier Science Amsterdam 1189-1194
70. E. Pasternak and H.-B. Mühlhaus 2002 Large deformation Cosserat continuum modelling of granulate materials In: L. Zhang L. Tong and J. Gal (eds.) Applied Mechanics. Progress and Application. ACAM 2002. The Third Australasian Congress on Applied Mechanics World Scientific New Jersey, London, Singapore 389-396
71. E. Pasternak and H.-B. Mühlhaus 2000 Cosserat and non-local continuum models for problems of wave propagation in fractured materials In: X.L. Zhao and R.H. Grzebieta (eds.) Structural Failure and Plasticity (IMPLAST2000) Pergamon Amsterdam 741-746
72. E. Pasternak and H.-B. Mühlhaus 2002 A non-local Cosserat model of heterogeneous materials: 1D structures In: A.V. Dyskin X. Hu and E. Sahouryeh (eds.) Structural Integrity and Fracture Swets & Zeitlinger Lisse 107-114
73. K.F. Graff 1975 Wave Motion in Elastic Solids Dover Publications New York 649
74. J.-P. Kahane 1985 Some Random Series of Functions, 2nd edition Cambridge University Press Cambridge 305