Considered is a sample of cohesionless granular material, in which the individual granules are regarded rigid, and which is subjected to overall macroscopic average stresses. On the basis of the principle of virtual work, and by an examination of the manner by which adjacent granules transmit forces through their contacts, a general representation is established for the macroscopic stresses in terms of the volume average of the (tensorial) product of the contact forces and the vectors which connect the centroids of adjacent contacting granules. Then the corresponding kinematics is examined and the overall macroscopic deformation rate and spin tensors are developed in terms of the volume average of relevant microscopic kinematical variables. As an illustration of the application of the general expressions developed, two explicit macroscopic results are deduced: (1) a dilatancy equation which both qualitatively and quantitatively seems to be in accord with experimental observation, and (2) a noncoaxiality equation which seems to support the vertex plasticity model. Since the development is based on a microstructural consideration, all material coefficients entering the results have well-defined physical interpretations.

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SUMMARYThe notion of overall macroscopic stress in granular masses is examined from a fundamental point of view by a statistical consideration of the contact forces that are transmitted by the contacting granules at the microscale. This examination leads in a natural way to relations between the macroscopic stress and the resulting granular fabric. The overall stresses are expressed in terms of the contact forces in two different but complementary ways: (1) by a statistical averaging over the sample volume of contact forces and 'branches' which are vectors connecting the centroids of two contacting granules; and (2) by defining the overall tractions transmitted across an interior imagined plane as the sum of the contact forces which represent the mechanical effect of granules on one side of a unit area of this plane, upon those on the other side. Conditions under which the two representations of overall stresses are equivalent, are examined in detail. In addition, explicit results are given, which define stresses in terms of the fabric and other microstructural characteristics of the granular mass.

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