13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- ߱ ሾ,ሿ ൌ ߢ ൌ ଵ ଶ ൫߱ , െ ߱ ,൯ൌ߳ ߢ (6) Generally, the energy potential of the rotation gradient leads to a non-symmetric force stress tensor, σji, Eq. (7), and the introduction of a couple stress tensor, μji, Eq. (8). When the force stress is taken as work conjugate to the strain tensor, and the couple stress is taken as work conjugate to the gradient of the rotation, the symmetric part of the couple stress tensor becomes indeterminate. ߪ ൌ ߪ ሺሻ ߪ ሾሿ (7) ߤ ൌ ߤ ሺሻ ߤ ሾሿ (8) Hadjesfandiari and Dargush [16] suggested a solution to this problem by demonstrating that the entire rotation gradient does not have energy potential, only the curvature tensor, Eq. (6). Thus the deformation energy consists of the work done by the force stress on the strain and the work done by the couple stress on the pure curvature. The symmetric part of the rotation gradient, Eq. (5), has no forces associated with it, which solves the problem of the indeterminate spherical part of the rotation gradient. For an isotropic material, the elastic potential is given by [16]: ܹ ሺ ߢ,ߝ ሻൌ ଵ ଶ ߣ ሺ ߝ ሻଶ ߝ ߤ ߝ 8 ߢ ߟ ߢ (9) where λ and μ are Lamé parameters and η is a material couple stress constant. According to this theory, a homogeneous displacement field, such as hydrostatic compression Eq. (10), does not introduce rotations and hence curvatures. ݑ ଵ ൌ ݔ ଵ ݑ ଶ ൌ ݔ ଶ ݑ ଷ ൌ ݔ ଷ (10) A displacement field producing pure twist, Eq. (11), introduces rotations, Eq. (12), but no true curvatures[16]: ݑ ଵ ൌ െ ݔߠ ଶ ݔ ଷ ݑ ଶ ൌ ݔߠ ଵ ݔ ଷ ݑ ଷ ൌ 0 (11) ߱ ଵ ൌ െଵ ଶ ݔߠ ଵ ߱ ଶ ൌ െଵ ଶ ݔߠ ଶ ߱ ଷ ൌ ݔߠ ଷ (12) A displacement field corresponding to pure bending of a beam, Eq. (13), introduces non-zero rotations, Eq. (14), that result in a single non-zero curvature, Eq. (15) , [16]: ݑ ଵ ൌ െ ோ ଵ ݔ ଵ ݔ ଷ ݑ ଶ ൌ െோ ఔ ݔ ଶ ݔ ଷ ݑ ଷ ൌ ଶ ఔ ோሺ ݔ ଶ ଶ െ ݔ ଷ ଶሻെଶ ଵ ோ ݔ ଵ ଶ (13) ߱ ଵ ൌ ఔ௫మோ ߱ ଶ ൌ ௫భோ (14) ߢ ଷ ൌ ଵି ଶோ ఔ (15) where is Poisson’s ratio and R is the radius of curvature of the beam central axis. 2.2 Discrete site-bond model The site-bond model [9] uses a discrete lattice, based on a regular tessellation of material space into truncated octahedral cells, Fig. 1(a). The lattice derives from the cellular structure when material particles, attached to cell centres, interact via deformable bonds. The bond properties relate to their ability to transfer shear and axial forces as well as torsion and bending moments to satisfy the six degrees of freedom of each site. A site requires 14 bonds to connect it to its neighbours, Fig. 1(b): six bonds of length 2L (2L is the cell size) in principal directions (through square faces), and eight bonds of length √3L in octahedral directions (through hexagonal faces). Development of this model involves bond representations with six independent elastic springs resisting three relative
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