PSI - Issue 13

Stepanova Larisa / Procedia Structural Integrity 13 (2018) 255–260 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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the crack tip. A crack can be represented by a region where the damage state has attained to its critical state, i.e., by the CDZ. Then the development of the crack and its preceding damage can be elucidated by analyzing the local states of stress, strain and damage. The CDZ may be interpreted as the zone of critical decrease in the effective area due to damage development. Inside the completely damaged zone the damage involved reaches its critical value and a complete fracture failure occurs. In view of material damage stresses are relaxed to vanishing (Stepanova and Igonin (2014), Zhao and Zhang (1995), Murakami (2012), Murakami et al (2000), Stepanova and Adylina (2014)). Therefore, one can assume that the stress components in the CDZ equal zero. Outside the zone damage alters the stress distribution substantially compared to the corresponding non-damaging material. Well outside the CDZ the continuity parameter is equal to 1. Thus, asymptotic remote boundary conditions have the form   1/( 1) * , , ( , ), / ( ) n ij ij n r t k n k C BI r         , (7) where * C is the path-independent integral, n I is the constant depending on n . Dimensional analysis of the system formulated shows that the damage mechanics equations must have similarity solutions of the form (Riedel (1987))   1/ ˆ ( , , ) ( , ), m ij ij r t At        ˆ ( , , ) ( , ) r t       ,   ( 1) / * / , n m n r At BI C     where  is the similarity variable. It should be noted that the remote boundary conditions can be formulated in a more general form compared with Eq. 7   1 , , ( , ) s ij ij r t c r n       , where the stress singularity exponent s is an unknown eigenvalue and has to be found from the nonlinear eigenvalue problem, 1 c is the amplitude of the stress field at infinity defined by the specimen configuration and loading conditions. For the power-law constitutive relations, the power law damage evolution equation (4) and the more general remote boundary conditions the similarity variable   1 , 1/ ( ) c r At sm      can be introduced. The multi-term asymptotic solution for the Airy stress function and the continuity parameter outside the CDZ is assumed in the form (Stepanova (2009)) 1 0 0 ( , ) ( ), ( , ) 1 ( ). j j j j j j f g                       (8) The asymptotic approach developed here allows us to find the geometry of the CDZ in the vicinity of the crack tip by means of Eqs. (8) for different values of the mixity parameter and material constants. The configurations of the completely damaged zones encompassing the crack tip are shown in Figs. 1-4 where the number k implies the k -term asymptotic expansion in (8). Altenbach, Matsuda and Okumura (2015) present numerical analysis of a crack-tip field in particulate-reinforced composites with debonding damage and containing various sized particles has been carried out by the FEM. FEM analysis was carried out for the three kinds of composites in the case of no debonding damage (perfect composite) and with debonding damage (composite with damage). On the composites with damage, with increasing particle size, the debonding damage becomes easy to occur and damage zone spreads out widely and a result the macroscopic equivalent stress is drastically reduced. The geometry of the active damage accumulation zones given by Altenbach, Matsuda and Okumura (2015) is very similar to the configurations shown in Fig. 3. Thus, on can conclude that the asymptotic analysis presented here allows us to obtain new asymptotic behavior of the stress filed in the vicinity of the crack tip and take into account the damage evolution process. 4. The geometry of the completely damaged zone in vicinity of the mixed mode crack tip

Fig. 1. The configuration of the completely damaged zone in the vicinity of the crack tip for different values of the mixity parameter.