## Constrain Zone

FIGURE 6.16 Three-zone system according to de los Rios et al. [4].

The advantages of such representation are: (a) the model considers plastic displacements throughout the crack system and so the near-tip plasticity effects are included; (b) the model can be effectively applied for short and long cracks; (c) the model accounts also for high crack tip plasticity (large-scale yielding), since it is based on elastoplastic principles; and (d) the friction stresses in the crack wake are not idealized as a continuum closure pressure but as point loads (can apply also for minimal degrees of anisotropy).

In terms of the model (Fig. 6.17), the fiber diameter is represented by d, the interfiber spacing by D, and the crack length by 2a. Considering only the positive half of the crack system, the crack tip is at a, the front of the fiber at iD/2 — d/2, and the back of the fiber at iD/2 + d/2 = c. In terms of dimensionless coordinate system x/c = Z, the crack tip is at n1, the fiber front at n2, and the fiber back is at 1. The factor i represents the number of half fiber spacings in the crack system, that is, i = 1, 3, 5,...

In the case where the crack is subjected to an applied stress a in mode I, the stresses in each zone are: (a) a1 in the crack zone (friction stress due to fiber

FIGURE 6.17 Schematic representation of crack system. Crack length is denoted by 2a, the fiber diameter by d and the fiber spacing by D. Considering only the positive coordinates side, crack tip is positioned at a, the plastic zone extends to the next fiber ahead of the crack tip at iD/2 — d/2 and the fiber plastic constrained zone at iD/2 + d/2 = c, i = 1, 3, 5 ... Dimensionless coordinate Z describes position throughout, in particular, Z = nj at the crack tip, Z = n2 at the plastic zone, and Z = 1 at the end of fiber.

FIGURE 6.17 Schematic representation of crack system. Crack length is denoted by 2a, the fiber diameter by d and the fiber spacing by D. Considering only the positive coordinates side, crack tip is positioned at a, the plastic zone extends to the next fiber ahead of the crack tip at iD/2 — d/2 and the fiber plastic constrained zone at iD/2 + d/2 = c, i = 1, 3, 5 ... Dimensionless coordinate Z describes position throughout, in particular, Z = nj at the crack tip, Z = n2 at the plastic zone, and Z = 1 at the end of fiber.

bridging), (b) a2 is the flow response in the plastic zone (resistance to plastic deformation), and (c) a3 in the fiber zone ahead of the plastic zone (constraint effect provided by the fiber to matrix microplasticity). A typical stress distribution along the crack system is shown in Fig. 6.16. The stress ai is considered zero for a nonbridged crack and nonzero for a bridged crack. The plastic zone is assumed to be blocked by the fibers, which impedes any matrix plastic displacement at the fibers. The effect of this constraint is the development of stress a3 in the matrix between the fibers of a row, which on achieving a critical value, resolved along the fiber-matrix interface, will cause debonding.

The solution of the equilibrium equation of all the forces, internal and external, acting in the three-zone system, was obtained by Navarro and de los Rios [85, 93] and gives the expressions for the COD = ^ over the entire crack system, and for the stress a3. These are as follows:

-(Zb + ni) cosh-1 - (Zb + ni) cosh-1 -(Zb + n2) cosh-1

1 i -1 -1 n i ff3 =-:—|(ff2-ori)sin nj-ffîSin n2 + -a cos-1 n2 v 2 J

n1 - Za 1 - n2Zb n2 - Zb 1 - n2Za n2 - Za ni + Za 1 + n2Zb n2 + Zb

 1 + n2'Ça x\
0 0