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In this study, 3-D Lattice Solid Model (LSMearth or LSM) was extended by introducing particle-scale rotation. In the new model, for each 3-D particle, we introduce six degrees of freedom: Three for translational motion, and three for orientation. Six kinds of relative motions are permitted between two neighboring particles, and six interactions are transferred, i.e., radial, two shearing forces, twisting and two bending torques. By using quaternion algebra, relative rotation between two particles is decomposed into two sequence-independent rotations such that all interactions due to the relative motions between interactive rigid bodies can be uniquely decided. After incorporating this mechanism and introducing bond breaking under torsion and bending into the LSM, several tests on 2-D and 3-D rock failure under uniaxial compression are carried out. Compared with the simulations without the single particle rotational mechanism, the new simulation results match more closely experimental results of rock fracture and hence, are encouraging. Since more parameters are introduced, an approach for choosing the new parameters is presented.

In this paper we propose a new algorithm to simulate the dynamics of 3-D interacting rigid bodies. Six degrees of freedom are introduced to describe a single 3-D body or particle, and six relative motions and interactions are permitted between bonded bodies. We develop a new decomposition technique for 3-D rotation and pay particular attention to the fact that an arbitrary relative rotation between two coordinate systems or two rigid bodies can not be decomposed into three mutually independent rotations around three orthogonal axes. However, it can be decomposed into two rotations, one pure axial rotation around the line between the centers of two bodies, and another rotation on a specified plane controlled by another parameter. These two rotations, corresponding to the relative axial twisting and bending in our model, are sequenceindependent. Therefore all interactions due to the relative translational and rotational motions between linked bodies can be uniquely determined using such a two-step decomposition technique. A complete algorithm for one such simulation is presented. Compared with existing methods, this algorithm is physically more reliable and has greater numerical accuracy.an auxiliary body-fixed frame of particle 2, obtained by directly rotating X 2 Y 2 Z 2 at T = 0 such that its Z 2 0 -axis is pointing to particle 1. There is no relative rotation betweenVectors f total force acting on the particle, measured in the space-fixed system XYZ s b total torque acting on the particle expressed in body-fixed frame f r normal force between two particles f s1 , f s2 shear forces between two particles s t torque cause by twisting or torsion between two particles s b1 ; s b2 torques cause by relative bending between two particles Da t relative angular displacement caused by twisting motion Da b1 ; Da b2 relative angular displacements caused by bending motion Du r relative displacement in normal direction Du s1 , Du s2 relative displacements in tangent directions r position vector of a particle, measured in XYZ x b angular velocities measured in the bodyfixed frame r 10 , r 20 initial position of particle 1 and particle 2, measured in XYZ r 1 , r 2 current positions of particle 1 and particle 2, measured in XYZ r 0 initial position of particle 1 relative to particle 2, measured in X 2 Y 2 Z 2 (or XYZ) r c current position of particle 1 relative to particle 2, measured in X 2 Y 2 Z 2 r f current position of particle 1 relative to particle 2, measured in XYZ Dr translational displacement of particle 1 relative to particle 2, measured in X 2 Y 2 Z 2 f s t shear force caused by translational motion of particle 1 relative to particle 2, measured in X 2 Y 2 Z 2 s t s torque generated by f s t , measured in X 2 Y 2 Z 2 f s r shear force caused by the rotation of particle 1 relative to particle 2, measured in X 2 Y 2 Z 2 s r s torque generated by f s r , measured in X 2 Y 2 Z 2 s r b bending torque cause by the rotation of particle 1 relative to particle 2, measured in X 2 Y 2 Z 2 s r t twisting torque cause by the rota...

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