13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- location x, 0≤ ≤ℓ within AZ and time t, since AZ evolves in time: 0 = 0( , ). Fig 8a should be AZ instead of PZ Figure 8. Decomposition of the stress analysis problem of CL The width of AZ in actual configuration (after drawing) is = ∙ 0 ( is the natural draw ratio). Thus, the displacement discontinuity in normal to the crack direction due to cold drawing is ( −1) 0. To determine the amount of material that undergoes cold drawing and builds up AZ, i.e., the width 0, we consider crack opening displacement (COD) as an approximation of the opening of a slit in the specimen with CL cutoff. The approximation is quite accurate due to presence of a small parameter: the ratio of the width over the length of the slit. Then, we make use of conventional Fracture Mechanics formalism and compute COD for the boundary value problem depicted in the Figure 8b. Thus, 0 is found from the continuity conditions of the original problem: the COD (x, t) in the specimen with PZ cutoff and the traction along AZ boundary should be equal to the displacement discontinuity in drawing process within AZ: 1 ( , ) ( 1) ( , ) o x t x t . (2) The Eq. (2) establishes the functional relation between the width 0 and COD in the specimen and loading shown in the Figure 8b. The COD depends on applied load ∞, and the specimen dimensions W, crack length ℓ , AZ length ℓ and CL lengths L (L= ℓ + ℓ ). Thus, in this case, we have only two independent geometrical parameters of CL (ℓ and ℓ ), since 0 is not an independent one. That is the reason for simplification of CL in PE: it is reduced to two-parameter model. The stationary state of a solid with two-parameter CL is determined by the minimum of the total Gibbs potential Gtot for the problem of Fig 8a. Based on the decomposition of Fig. 8, the total Gibbs potential is the sum of the Gibbs potentials of the specimen with CL cutoff G0 and the PZ potential GPZ illustrated in Figure 8: 0( , , ; , , ) ( ; , , ) AZ cr AZ tot dr PZ dr G G T L W G T . (3) The Gibbs potential of PZ in turn consists of the wake zone and active zone potentials (see the definitions on WZ in Figure 1). AZ consists of homogeneous drawn material with properties different from that of the surrounding original material. It is also separated from the original by distinct boundary with drawing stress acting along it. As a result, the AZ Gibbs potential can be presented as: tr AZ AZ G V . (4) The multiplier γtr in Eq. (4) stands for the specific energy of transformation of a unit mass of original material into the oriented drawn state under drawing stress : 0 ( ) ( ) tr tr g T g T , where the Gibb potential density is defined as ( , ) ( , ) g T f T w , where ( , ) f T is the strain energy density and w stands for work density [9]. The potential of WZ is different in two ways: 1) it does not contain the strain energy of the stretched
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