13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- dTsin ϕ−T cos ϕ sin ϕ dx l +dPcos ϕ−T sin ϕ cos ϕ dy l =0. (2) We now assume that the change dP in the magnitude of compression is proportional to the vertical displacement such that dP=km dy l . (3) where km is the stiffness of the surrounding parts of the particulate material. Hereafter we refer to the surrounding parts of the particular material as the matrix. Substituting (3) into (2), expressing T via P through the equation of moment equilibrium (1) and taking into account that dy=-dx cos ϕ/sinϕ we obtain dT = −P 1 sin3 ϕ +km cos2 ϕ sin2 ϕ # $ % & ' ( dx l . (4) It is seen that the coefficient between incremental shear force dT and incremental shear strain dx/l can assume negative values, when P>kmsinϕ cos 2 ϕ. We call this effect the apparent negative stiffness. Following [20-23] we model the collective effect of rotating particles by treating them as negative stiffness inclusions (inclusions with negative shear modulus, µincl) embedded in a matrix with positive definite elastic moduli. We then use the theory of effective characteristics in order to determine the elastic moduli of such a composite at macroscale and determine the conditions of global instability. In order to incorporate this phenomenon into a continuum description of the granulate material consider a representative volume element, that is an element of size H>>l. We introduce normal p and shear τ stresses acting on the faces of the element. Therefore we can treat normal and shear forces from (4) as P~pl2 and T~τl2. For the sake of simplicity we will treat the normalised matrix stiffness as the bulk modulus of the matrix, κm~km/l 2. Then we can express the average shear modulus, µincl, associated with the particle rotation as µincl =− p sin3 ϕ +κm cos2 ϕ sin2 ϕ . (5) Here angle ϕ is interpreted as an average angle, which provides a combined description of particle shapes and initial packing. The negative stiffness is achieved when
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