13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- respective direction was calculated as the ratio between the total reaction force and the boundary area, i.e. σi = ΣFi / 400L 2. For the cases where nodal displacements were determined from analysis, the macroscopic strain in the respective direction was calculated as the ratio between the average displacement and the model length, i.e. εi = ΣUi / (21 2×20L). The evolution of damage was simulated by failure of bonds, controlled by an in-house code, and repetitive solution for equilibrium performed by Abaqus with constant applied displacements. The values of di were selected so that the strain energy density in the system prior to damage was one for the four cases for the purpose of comparison. At each step the in-house code obtains the forces and moments in all bonds and calculates the propensity for failure, Π, for each bond. The bond with maximum Π is then removed and the updated lattice is solved for equilibrium. This leads to redistribution of forces for the continuous damage evolution. The magnitudes of Π at which consecutive failures occurred can be used to cut-back the applied strain and resulting stress and obtain a macroscopic stress-strain response. The focus of this work is not on determining the stress-strain response, but on the relation between damage and crack population. To this end we define four damage parameters, measuring the relative changes of the hydrostatic stress and the three components of the stress deviator by: ( ) ( ) , 1,2,3 , 0 1 ; 0 1 = − = = − = − i S where S S D D i H i i i i H H H σ σ σ σ . (4) Note that for isotropic deformation and damage these parameters must be equal and equivalent to the damage parameter defined via relative reduction of Young’s modulus or shear modulus. 2.4. Crack population analysis A bond failure is thought of as a micro-crack nucleation, specifically as a separation between the adjacent cells in the cellular structure along their common face. Initially, the micro-cracks may be dispersed in the model reflecting the random distribution of pore sizes and the low level of interaction due to force redistribution. Interaction and coalescence may follow as the population of micro-cracks increases. The structure of the failed surface can be represented with a mathematical graph, where graph nodes represent failed faces and graph edges exist between failed faces with common triple line in the cellular structure, i.e. where two micro-cracks formed a continuous larger crack. Generally, the graph of a failed surface is a disconnected set of sub-graphs or components, some of which could be single nodes as at the start of damage evolution, while others could be connected sets representing larger micro-cracks as the coalescence develops. For the analysis, nodes are equipped with weights equal to the failed face areas. Edges are equipped with weights equal to the shortest path along connected faces between their centres. The components of a failed surface graph are sorted into sets according to their areas A1 < A2 … < Ak, so that each set contains Ni disconnected components of area Ai. The linear size of a component is approximated with the square root of its area so that the moment of the crack population is formed using (compare to Eq. (1)) ∑ = = k i i i T A N A M 1 3 2 1 , (5) where AT is the total area of the faces in the cellular structure. This can in principle be replaced with a linear measure to conform to Eq. (1). A realistic choice is to use the component diameter which is the maximal shortest path between component’s nodes calculated with the weighted edges. The process, however, is computationally expensive and does not lead to noticeable changes in the
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