13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- ; , , , ; , 1 cr cr AZ AZ tot dr AZ eff x x LW F V B . (13) Since the WZ does not change if ℓ is fixed, the derivative of the third term in Eq. (6) with respect to ℓ is zero. Finally, putting together Eq. (6), Eq. (8), Eq. (10) and Eq. (13) we obtain the expression for : 2 ( ) ' 1 cr tr AZ tot tot x K X x E . (14) The AZ reaches equilibrium when the thermodynamic force is vanishing. Therefore, the equilibrium AZ length AZ eq is determined by the following equation: 2 ( ) 0 ' 1 cr tr tot tot x K x E . (15) For a given external load, and crack length, the equilibrium AZ size depends on and drawing stress dr . The cold drawing phenomenon in PE is long studied using macroscopical specimens. It is well established that the drawing stress dr increases with the loading rate [7]. Unfortunately, it is difficult to evaluate dr on microfibers. Therefore, we use the linear relations between dr and log strain rate obtained in macroscopical tests. Since the loading rate of the AZ material changes with CL growth, the dr acting along AZ boundary is also varies with CL growth and can be estimated on the basis of either a combination of COD rate and macroscopical ~ln dr relations or indirectly on duration of AZ stationary state. Regarding , the transformation of the original continuum media into highly oriented fibrillated structure within PZ is a complex process. It starts with cavitation ahead of the crack front as a precursor of drawing. The cavitation makes it possible to draw thin membranes between the cavities. Further drawing leads to splitting some of the membranes into thin fibers. For such micron scale of processes, a surface energy of microfibers and membrane becomes comparable to the bulk material energy. It complicates a theoretical evaluation of . A comparison of the value of AZ size AZ eq resulting from solution of the Eq. (16) with actual AZ eq observed in the experiment allows evaluating of an effective (phenomenological) value of . This estimation has been performed and reported in a separate paper accompanying this one. 3.2. Kinetic Equations of Crack Layer Growth in PE A stationary CL configuration takes place, when the CL thermodynamic forces are not positive, i.e., 0 AZ X and 0 CR X . The equilibrium is achieved, when the forces equal 0 (for example Eq. (15) for AZ equilibrium). 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 return, when the crack is moving into AZ by breaking fibers, or AZ 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 equilibrium stationary state, it evolves to the next stationary configuration.
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