Summary
Using least squares methods consistent averages for stress and strain of a nonhomogeneous granular assembly are derived. The expression for the average of the strain increment serves as a constraint in a functional approach to solving the displacements and spins of particles of a granular assembly in a statistical manner. The method shows directly what microscopic angular distributions are needed to describe the internal state of an assembly. They are the distribution of contacts, the distribution of both the averages and the variations in the interactive properties of the grains. The method is then applied to an assembly with normal-interacting circular grains in two dimensions. The incremental stiffness and the average displacement of the contacts as a function of the angle are obtained for the case of non-rotating strains. The results show clearly that for this case the sliding mode of motion is the most relevant one for deviatoric loading.
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References
Oda, M.: Initial fabrics and their relations to mechanical properties of a granular material. Solis Found.12, 17–36 (1972).
Oda, M.: Deformation mechanisms of sand in triaxial compression tests. Soils Found.12, 45–64 (1972).
Allersma, H. G. B.: Photoelastic stress analysis and strains in simple shear. Proc. IUTAM Conf. on Deformation and Failure of Granular Materials, Delft (Vermeer, P., Luger, D., eds.), pp. 345–354. Rotterdam: Balkema 1982.
Cundall, P. A., Strack, O. D. L.: A discrete numerical method for granular assemblies. Geotechnique29, 47–65 (1979).
Cundall, P. A., Drescher, A., Strack, O. D. L.: Numerical experiments on granular assemblies. Proc. IUTAM Conf on Deformation and Failure of Granular Materials, Delft (Vermeer, P., Luger, D., eds.), pp. 355–369. Rotterdam: Balkema 1982.
Chang, C. S., Misra, A., Sundaram, S. S.: Micromechanical modelling of cemented sands under low amplitude oscillations. Geotechnique40, 251–263 (1990).
Koenders, M. A.: The incremental stiffness of an assembly of particles. Acta Mech.70, 31–49 (1987).
Koenders, M. A.: A two dimensional non-homogeneous deformation model for sand. PhD Thesis, University of London 1984.
Geniev, G. A.: Problems of the dynamics of a granular medium. Akad. Stroit. Archit. SSSR, Moscow 1958.
de Josselin de Jong, G.: The double sliding free rotating model for granular assemblies. Geotechnique21, 155–163 (1971).
Mandel, J.: Sur les equations d'ecoulement des sols ideaux en deformation plane et le concept de double glissement. J. Mech. Phys. Solids14, 303–308 (1966).
Spencer, A. J. M.: A theory of the kinematics of ideal soils under plane strain conditons. J. Mech. Phys. Solids12, 337–351 (1964).
Vermeer, P. A.: Double sliding within an elastic-plastic framework. LGM Mededelingen, Tribute to Professor de Josselin de Jong, Delft Soil Mechanics Laboratory Publications Part XXI no 2, pp. 199–208 (1980).
Spencer, A. J. M.: Mechanics of solids. The Rodney Hill Anniversary Volume (Hopkins, J., Sewell, G., eds.), p. 607, Oxford: Pergamon Press 1981.
Harris, D.: Plasticity models for soil, granular and jointed rock materials. J. Mech. Phys. Solids40, 273–290 (1992).
de Josselin de Jong, G.: Mathematical elaboration of the double sliding, free rotating model. Arch. Mech.29, 561–591 (1977).
Anand, L.: Plane deformation of ideal granular materials. J. Mech. Phys. Solids31, 105–122 (1983).
Koenders, M. A.: Localized deformation using higher order stress/strain theory. Trans. ASME, J. Energy Resources Techn.112, 51–53 (1990).
Arthur, J. R. F., Dunstan, T. F.: Rupture layers in granular media. Proc. IUTAM Conf. on Deformation and Failure of Granular Materials, Delft (Vermeer, P., Luger, D., eds.), pp. 453–460. Rotterdam: Balkema 1982.
Koenders, M. A., Arthur, J. R. F., Dunstan, T. F.: The behaviour of soil at peak stress. In: Yielding, damage and failure of solids (Boehler, J., ed.), pp. 805–818. London: Mechanical Engineering Publication 1990.
Thornton, C.: Deformation of a simple particulate model. Proc. Const. Equations of Soils, IXICSMFE Tokyo, pp. 255–262 (1977).
Thornton, C.: The condition for failure of a face-centred cubic array of uniform rigid spheres. Geotechnique29, 441–459 (1979).
Wilson, A. H.: Thermodynamics and statistical mechanics. Cambridge: Cambridge University Press 1966.
Dantu, P., Weber, J.: Etudes statistique des forces intergranulaire dans un milieu pulverulent. Geotechnique18, 50–55 (1968).
Drescher, A., de Josselin de Jong, G.: Photoelastic verification of a mechanical model for the flow of a granular material. J. Mech. Phys. Solids20, 337–351 (1972).
Walton, K.: The effective elastic moduli of a random packing of spheres. J. Mech. Phys. Solids35, 213–226 (1987).
Becker, E.: Kontinuumsmechanik. Stuttgart: Teubner 1975.
Seber, G. A. F.: Linear regression analysis. New York: Wiley 1977.
van der Giessen, E.: A model of anisotropically hardening materials based upon the concept of a plastically-induced orientational structure. In: Yielding, damage and failure of solids (Boehler, J., ed.), pp. 187–198 London: Mechanical Engineering Publication 1990.
Koenders, M. A.: Analytical estimates for constitutive relations of assemblies of particles in elasto-frictional contact. Proc. Powders and Grains Conference, Birmingham (Thornton, C., ed.), Rotterdam: Balkema 1993.
DERIVE (4th ed.) Soft Warehouse Europe GmbH, Schloß Hagenberg, A-4232 Hagenberg, Austria (1990).
Konishi, J.: Microscopic model studies on the mechanical behaviour of granular materials. US-Japan Seminar Cont. Mech. Stat. Appr. Gran. Mat., pp. 27–46 (1978).
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Koenders, M.A. Least squares methods for the mechanics of nonhomogeneous granular assemblies. Acta Mechanica 106, 23–40 (1994). https://doi.org/10.1007/BF01300942
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DOI: https://doi.org/10.1007/BF01300942