# Mathematics and Mechanics of Granular Materials by Alexander Soifer

By Alexander Soifer

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Then the macro-motion can be described by x1 = X1 + f1 (X2 , t) , x2 = X2 + f2 (X2 , t) and x3 = X3 , where the spatial co-ordinates xi (i = 1, 2, 3) represent the current position of a 40 E. Bauer (b) (a) (c) Figure 1. Modeling of plane shearing under constant vertical pressure p0 = −σ22 : (a) section of the inﬁnite granular layer between parallel plates with rough surfaces, (b) kinematics of plane shearing with dilatancy and degrees of freedom u1 , u2 and ω3 , (c) stress components σ11 , σ22 , σ33 , σ12 , σ21 and couple-stress components M31 and M32 with respect to the Cartesian co-ordinate system.

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