13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- In couple stress theory (CST) each point within the continuum has three additional degrees of freedom, point rotations. These are associated with couple stresses as the classical (force) stresses are associated with strains. A general CST was proposed early in the 20th century by the Cosserat brothers [11]. It went largely unnoticed until the 1960s when a desire to understand the mechanisms behind micro-crack growth for more accurate crack assessment rejuvenated interest. One branch of CSTs considers point micro-rotations to be independent of the macro-rotations; the rotations derived from the displacement gradient [12, 13]. These are known as micropolar theories or Cosserat models with free rotations [14]. While such a view appears to be well suited for use with discrete lattice methods, it is difficult to establish a link between a continuum and a discrete representation of a material containing features that are following the deformations of the bulk. For such situations it is more plausible to assume that the micro-rotations are equal to the macro-rotation. This assumption led to a branch of CSTs known as Cosserat models with constrained rotations [14, 15]. Initially, these were based on couple stresses work-conjugate to the macro-rotation gradient. As a consequence, the spherical part of the couple stress tensor remained undetermined. Recently, Hadjesfandiari and Dargush [16] proposed a consistent CST using true kinematic quantities to remove the indeterminacy of the couple stress tensor. The consistent CST [16] naturally introduces a length parameter. This is of key importance for the local material behaviour. But the calibration of the CST requires that the length parameter is physically related to the material microstructure; the sizes and distances between characteristic features that disturb the symmetry of the stresses. We report on work in progress investigating whether a medium with features can be used to calculate bond responses to bending and torsion in the discrete model [9] and if this can be used to calibrate the consistent CST. 2. Theory and model 2.1 Generalised continuum The kinematics of a material point under small deformation is given by the displacement gradient, Eq. (1), where comma denotes differentiation, rounded parenthesis denotes symmetric part and square parenthesis denotes skew symmetric part of the tensor. The symmetric (strain tensor eij) and the skew symmetric (rotation tensor ωij) parts are given by Eq. (2) and Eq. (3), respectively. The right hand side of Eq. Error! Reference source not found. gives the rotation tensor as a vector using the permutation tensor. ݑ , ൌ ݑ ሺ,ሻ ݑ ሾ,ሿ (1) ݑ ሺ,ሻ ൌ ݁ ൌ ଵ ଶ൫ ݑ , ݑ ,൯ (2) ݑ ሾ,ሿ ൌ ߱ ൌ ଵ ଶ ൫ ݑ , െ ݑ ,൯ൌ߳ ߱ (3) In classical continuum mechanics, the elastic potential depends solely on the strain tensor. In generalised continuum, additional potential is carried by the gradient of the rotation vector, Eq. (4). The symmetric part of this gradient, χij in Eq. (5), represent “pure” twists, and the skew symmetric part, κij in Eq. (6), represent “pure” curvatures, which can be given by a vector as shown with the right-hand size. ߱ , ൌ ߱ ሺ,ሻ ߱ ሾ,ሿ (4) ߱ ሺ,ሻ ൌ ߯ ൌ ଵ ଶ ൫߱ , ߱ ,൯ (5)
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