Issue 29

R. Dimitri et alii, Frattura ed Integrità Strutturale, 29 (2014) 266-283; DOI: 10.3221/IGF-ESIS.29.23 281 relative normal displacement g N for mode I, where the specimen is first loaded while following the exponential law up to a value p N =0.9N/mm 2 , then linearly unloaded and then reloaded along the secant stiffness to the origin for displacements values ranging between 0.0006mm and 0.0022mm , then unloaded again along the secant stiffenss, continuing in the compressive regime up to g N =-0.0002mm along the virgin stiffness. The specimen is finally reloaded up to the complete failure conditions. Local pressures in mode II and varying mixed-modes (i.e. varying u/v ratios) are evaluated in a similar fashion for which a complete cycle of loading, unloading up to compression, and reloading up to failure is globally considered (see Fig. 16b,c,d). As also shown in the traction-separation curves of Fig. 16a,b, the maximum normal and tangential tractions p Nmax and p Tmax are correspondingly attained at the characteristic separations g Nmax and g Tmax , whereas a clear reduction in pure-mode interface strengths is visible in Fig. 16c,d due to the non-zero separation history value in the other debonding mode. (a) (b) (c) (d) Figure 16: Computational results in mode I (a) , mode II (b) and mixed mode (c, d) . - Matrix-fiber mixed debonding The second example considers a matrix-fiber debonding process caused by transverse loads under the assumption of plane stress. The problem geometry is reduced in the plane perpendicular to the fiber axis, to a circular fiber with radius a surrounded by a matrix perfectly bonded along its interface, see Fig. 17a. Both matrix and fiber are assumed to be linear elastic materials, with material properties E f =5GPa ,  f =0 for the fiber, and E m =500MPa ,  m =0.25 for the matrix. The fiber volume fraction f is associated with the unit cell size b and the radius of the fiber a . A regular packing is considered with a high volumetric fiber content equal to 7 8.5% leading to b/a=2 , see the geometry modeled in the FEM simulation in Fig. 17a. The fiber-matrix interface debonding is modeled by the mixed exponential CZM describing the nonlinear behaviour of the interface, in terms of normal and tangential cohesive tractions ( p N , p T ), where the adjectives normal and tangential refer to the local axes of the curvilinear reference system along the interface. Correspondingly, a normal and tangential relative separation g N and g T are admitted. The cohesive parameters are the same as in the previous example. Horizontal displacements are imposed on the vertical boundaries of the unit square cell as shown in Fig. 17, and the homogenized stress-strain response is numerically computed. The average stress  is determined as the sum of the horizontal reactions divided by the length of the vertical cell side b , while the average strain  is evaluated as the imposed relative displacement 2u divided by the original length of the horizontal side of the cell b . Fig. 17b shows the deformed mesh at the last time step, along with the contours of the shear stress, as well as the local and the global reactions, whereas the stress-strain history curve is reported in Fig. 18. In the initial loading phase, the fiber and matrix are perfectly bonded throughout the