Stress, stress asymmetry and couple stress: from discrete particles to continuous fields

Irving J.H., Kirkwood J.G.: The statistical mechanical theory of transport properties IV. The equations of hydrodynamics. J. Chem. Phys. 18, 817–829 (1950)

Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Oxford (1954, reissue 2000)

Reichl, L.E.: A Modern Course in Statistical Physics. Wiley and numerous references therein (1998)

Weber J.: Recherches concernantes les contraintes intergranulaires dans les milieux pulvérantes. Bull. de Liais P. C. 20, 1–31 (1966)

Cambou B., Dubujet P., Emeriault F., Sidoroff F.: Homogenization for granular media. Eur. J. Mech. A Solids 14, 255–276 (1995)

Chang C.S., Gao J.: Kinematic and static hypotheses for constitutive modeling of granulates considering particle rotation. Acta Mech. 115, 213–229 (1996)

Bagi K.: Stress and strain in granular assemblies. Mech. Mat. 22(3), 165–177 (1996)

Babic M.: Average balance equations for granular materials. Int. J. Eng. Sci. 35, 523–548 (1997)

Goddard J.D.: Continuum modeling of granular assemblies: quasi-static dilatancy and yield. In: Herrmann, H.J., Hovi, J.-P., Luding, S. (eds) Physics of Dry Granular Media, pp. 1–24. Kluwer, Dordrecht (1998)

Nemat-Nasser S.: A micromechanically-based constitutive model for frictional deformation of granular materials. J. Mech. Phys. Solids 48, 1541–1563 (2000)

Lätzel M., Luding S., Herrmann H.J.: From discontinuous models towards a continuum description. In: Vermeer, P.A., Diebels, S., Ehlers, W., Herrmann, H.J., Luding, S., Ramm, E. (eds) Continuous and Discontinuous Models of Cohesive-Frictional Materials, Lecture Notes in Physics 568, pp. 215–230. Springer, Berlin (2001)

Bardet J.P., Vardoulakis I.: The asymmetry of stress in granular media. Int. J. Solids Struct. 38, 353–367 (2001)

Zhu H.P., Yu A.B.: Averaging method for granular materials. Phys. Rev. E 66, 021302 (2002)

Ball R.C., Blumenfeld R.: Stress field in granular systems, loop forces and potential formulation. Phys. Rev. Lett. 88, 115505 (2002)

Ehlers W., Ramm E., Diebels S., D’Addetta G.A.: From particle ensembles to Cosserat continua: homogenization of contact forces towards stresses and couple stresses. Int. J. Solids Struct. 40, 6681–6702 (2003)

Kruyt N.P., Rothenburg L.: Kinematic and static assumptions for homogenization in micromechanics of granular media. Mech. Mater. 36(22), 1157–1173 (2004)

Froiio F., Tomassetti G., Vardoulakis I.: Mechanics of granular materials: the discrete and continuum descriptions juxtaposed. Int. J. Solids Struct. 43, 7684–7720 (2006)

Goddard J.D.: From granular matter to generalized continuum. In: Capriz, G., Giovine, P., Mariano, P.M. (eds) Mathematical Models of Granular Matter, Lecture Notes in Applied Mathematics, vol. 1937, Ch. 1, pp. 1–20. Springer, Berlin (2008)

Goddard J.D.: Microstructural origins of continuum stress fields—a brief history and some unresolved issues. In: DeKee, D., Kaloni, P.N. (eds) Recent Developments in Structured Continua, Pitman Research Notes in Mathematics, No. 143, pp. 179–208. Longman/ Wiley, New York (1986)

Goldhirsch I.: Rapid granular flows. Ann. Rev. Fluid Mech. 35, 267–293 (2003)

Glasser B.J., Goldhirsch I.: Scale dependence, correlations and fluctuations of stresses in rapid granular flows. Phys. Fluids 13, 407–420 (2001)

Love A.E.H.: Treatise of Mathematical Theory of Elasticity. Cambridge, UP, London (1927)

Voigt W.: Theoretische Studien über die Elasticitätverhhältnisse der Krystalle. Abhandt. Ges. Wiss. Gött. 34, 3–52 (1887)

Goldenberg C., Goldhirsch I.: Continuum mechanics for small systems and fine resolutions. In: Rieth, M., Schommers, W. (eds) Handbook of Theoretical and Computational Nanotechnology, pp. 330–386. American Scientific, New York (2006)

Murdoch A.I.: On the microscopic interpretation of stress and couple stress. J. Elast. 71(1–3), 105–131 (2003)

Murdoch A.I.: A critique of atomistic definitions of the stress tensor. J. Elast. 88(2), 113–140 (2007)

Goldhirsch I., Goldenberg C.: On the microscopic foundations of elasticity. Eur. Phys. J. E 9(3), 245–251 (2002)

Goldenberg C., Goldhirsch I.: Small and large scale granular statics. Granul. Matter 6(2–3), 87–96 (2004)

Goldhirsch I., Goldenberg C.: Stress in dense granular materials. In: Wolf, D.E., Hinrichsen, H. (eds) The Physics of Granular Media, pp. 3–22. Wiley, Weinheim (2004)

Schofield P., Henderson J.R.: Statistical mechanics of inhomogeneous fluids. Prof. R. Soc. Lond. Ser. A 379(1776), 231–246 (1982)

Wajnryb E., Altenberger A.R., Dahler J.S.: Uniqueness of the microscopic stress tensor. J. Chem. Phys. 103, 9782–9787 (1995)

Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge, UP, Cambridge (1967)

Goldenberg C., Atman A.P.F., Claudin P., Combe G., Goldhirsch I.: Scale separation in granular packings: stress plateaus and fluctuations. Phys. Rev. Lett. 96(16), 168001 (2006)

Goldenberg C., Goldhirsch I.: Effects of friction and disorder on the quasistatic response of granular solids to a localized force. Phys. Rev. E 77(4), 041303 (2008)

Tan M.-L., Goldhirsch I.: Rapid granular flows as mesoscopic systems. Phys. Rev. Lett. 81(14), 3022–3025 (1998)

Arslan H., Sture S.: Evaluation of a physical length scale for granular materials. Comput. Mater. Sci. 42, 525–530 (2008)

Eringen A.C.: Theory of micropolar elasticity. In: Leibowitz, H. (eds) Fracture: An Advanced Treatise, vol. I, Mathematical Fundamentals, pp. 621–729. Academic Press, New York (1968)

Cercignani C.: The Boltzmann Equation and Its Applications, Ser. “Applied Mathematical Sciences”, No. 67. Springer, Berlin (1988)

Goldhirsch I.: Scales and kinetics of granular flows. CHAOS 9(3), 659–672 (1999)

Goldenberg C., Goldhirsch I.: Friction enhances elasticity in granular solids. Nature 435(7039), 188–191 (2005)

Zhang, J., Behringer, R.P., Goldhirsch, I.: Coarse graining of a physical granular system. Prog. Theor. Phys. Suppl. (2010, in press)