13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- To link the cell energies to kinematic quantities we calculate particle translations and rotations using three unit vectors, n1, n2 and n3 normal to three orthogonal faces of a particle, see Fig. 4. After deformation these vectors remain orthogonal as the particles are rigid, given with t1, t2 and t3 in Fig. 4. The coordinates of these, arranged in columns, form the transformation matrix, T, for the particle. The particle motion can be represented by a single rotation, θ given by Eq. (16), around a normalized axis, α given by Eq. (17). The components of the rotation vector for a particle are calculated by Eq. (18). The relative rotations between central and any other particle are expressed in the coordinate system defined by the particular pair using the corresponding transformation. ߠ ൌܿ ିݏ ଵ ቀ௧ଶሺ܂ሻି ଵቁ (16) ߙ ൌଶ ୱ୧ଵ୬ሺఏሻ ܶ ଷଶ െܶ ଶଷܶ ଵଷ െܶ ଷଵܶ ଶଵ െܶ ଵଶ൩ (17) ߱ ൌ ߠ ൈ ߙ , ൌ1,2,3 (18) Figure 4. Orthogonal vectors, (n1, n2, n3) and (t1, t2, t3) describing the orientation of P0 before and after deformation respectively. 3. Results and Discussion For the particle arrangement used (a central particle and all 14 particles of the site-bond model) we have used four different particle sizes relative to the cell size in order to investigate the effect of cell to particle size ratio, . The ratios are 3 (large particles), 4, 6, and 12 (small particles). In the case of hydrostatic loading no rotations of particles were observed and the relative displacements between P0 and any principal or octahedral particle were only axial, conforming to Eq. (10). In the site-bond model these relative displacements should be resisted by axial springs with stiffness coefficients Kn p and Kn o, respectively. The relative axial displacements were found independent of , scaling with the applied displacements. The cell energies were found to be dependent on , as shown in Figure . For small particles, = 12, the cell energy approached 0; the presence of particles has negligible effect. The cell energy and displacements from this case can be used to calibrate a linear combination of Kn p and Kn o. Note, that in general this case is not sufficient for calibrating stiffness coefficients separately. However, the particle size effect appears to be very
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