13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- subjected to an external load ( F or ) and a thin wedge shape PZ domain with variable width 0 of original material that undergoes cold drawing under a distributed closing tractions dr as shown in Figure 1. Figure 1. Decomposition of Crack Layer Model. Accordingly, CL growth is decomposed into two closely coupled processes: 1) the PZ growth into the surrounding original material; 2) the crack penetration into the PZ [13]. The thermodynamic forces for such elementary processes are conventionally presented as the derivatives from Gibbs potential of the system with respect to the corresponding CL geometrical parameters such as the crack and the PZ lengths. Performing the calculations one can show that the thermodynamic force driving PZ growth PZ X is expressed as [13]: 2 PZ PZ tr tot PZ K V X E , (1) where tot K is the total SIF presented as the sum of SIF due to remote load and SIF due to traction dr along the PZ boundary; E plane strain Young’s modulus; tr the specific energy of transformation, i.e., the work required to transform a unit mass of original material into an equal mass of oriented unstressed state plus the difference of strain energy densities in the original and drawn states; PZ V PZ volume; and PZ process zone size. Similarly, the crack driving thermodynamic force cr X has the following form [13], 1 2 CR cr X J , (2) where 1 cr J is the energy release rate due to crack extension into the PZ, when PZ is stationary, and 2 is surface (fracture) energy per unit length. A stationary CL configuration takes place, when the thermodynamic forces are not positive, i.e., 0 PZ X and 0 CR X . The equilibrium is achieved, when the both forces equal zero. At a small deviation from equilibrium, a thermodynamic system has a tendency to return to the equilibrium state. However, fracture is an essentially irreversible process: there is no “healing”, when the PZ fibers are broken, or PZ advances into the original material via cavitation followed by cold drawing of the material between the cavities and formation of membranes and fibers. Thus, when CL departs from one stationary state, it moves into next stationary configuration. Such process of crack layer propagation continues by crack and process zone assisting mutual growth. The described CL propagation is formalized in the following system of coupled ordinary differential equations with respect to the crack length ( cr ) and the PZ size ( pz ):
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