13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Instability and failure of particulate materials caused by rolling of non-spherical particles Elena Pasternak1*, Arcady V. Dyskin2 1 School of Mechanical and Chemical Engineering, University of Western Australia, 6009, Australia 2 School of, Civil and Resource Engineering University of Western Australia, 6009, Australia * Corresponding author: arcady@civil.uwa.edu.au Abstract In particulate materials under compression at the peak load the accumulated damage allows particle rolling. For non-spherical particles the moment equilibrium dictates that further increase in displacement requires reduced shear stress producing an effect of apparent negative stiffness; its value depends upon the magnitude of the compressive stress. Dilatancy produced by rolling particles reduces the value negative stiffness, while the contraction phase causes immediate instability. Material with rolling particles is macroscopically modelled as a matrix containing inclusions with negative shear modulus. When the concentration of negative stiffness inclusions is low, the effective shear modulus is positive and the material is stable. When the concentration reaches a critical level the effective shear modulus abruptly becomes negative and the material loses stability. Furthermore, there exists a special value of negative shear modulus of inclusions (and hence the magnitude of compressive stress) when the critical concentration becomes zero, such that the first rolling particle induces the global instability. Keywords Rolling particles, Negative shear modulus, Effective shear modulus, Critical concentration, Dilation 1. Introduction The importance of particle rotations (and the associated rotational degrees of freedom) in the mechanisms of instability and failure of particulate materials has long been recognised (e.g. [1-7]). The particle rotations were observed in physical experiments (e.g. [8-10]) and discrete element simulations (e.g. [11-14]). The modelling of the effect of particle rotation was mainly based on the concept of spherical (circular in 2D) particles, which offered the maximum simplicity. The effect of particle shape was thought to be quantitative, for instance resulting in reduced velocities of particle flow and increased stresses (e.g. [15]). However, the non-spherical particles can interlock – a phenomenon that does not exist in spherical particles [16]. Furthermore, rotations of non-spherical particles cause elbowing [17] that is coupling between the rotations and normal stresses. Both these mechanisms could lead to qualitatively new phenomena, such as the apparent negative stiffness [18-23]. It was further pointed out in [24] that in producing the negative stiffness effect the role of non-spherical particles could be played by clusters of connected spherical particles. The role of non-spherical particles and clusters of spherical particles is also discussed in [25]. Dyskin and Pasternak [20-24] modelled the apparent negative stiffness associated with particle rotations without taking into account the effect elbowing has on dilation/contraction. Here we include the latter into consideration. This will be accomplished in Section 2. Section 3 models the volume elements with apparent negative stiffness as inclusions in a matrix with positive definite elastic moduli and takes into account the interaction between the inclusions. This result gives an insight into the effect of rotating particles on global stability.
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