PSI - Issue 7

Robert Goldstein et al. / Procedia Structural Integrity 7 (2017) 222–228 R.Goldstein, M. Perelmuter / Structural Integrity Procedia 00 (2017) 000–000

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on the crack propagation rate, and kinetic e ff ects are essential only in the crack cohesive zone, where bonds are significantly stressed. We had extended this approach for interfacial bridged cracks with spring-like bonds deformation law and nonuniform bonds stresses distribution (Goldstein and Perelmuter, 2012). It is also assumed that the bridged zone is not small compared to the whole crack length. The alliance of the Zhurkov thermofluctuational approach and the crack bridged zone model (the condition of zero on the total stress intensity factor is not imposed, (Cox and Marshall, 1994)) is used for bonds destruction and cracks nucleation modelling. An analysis of the bonds destruction in the crack bridged zone and kinetic characteristics of the interfacial junction are based on the following assumptions: • at the initial time instant there is a region of weakened bonds on the interface between materials; • bonds density in that region varies in time according to the thermal fluctuation mechanism; • bonds rigidity in that region is proportional to their density at each point of the crack bridged zone; • the defect nucleation occurs near the center of the weakened bond region; • the defect nucleation condition is related to decreasing the average bond density to the critical value on a part of the weakened bond region. The paper is organized as follows: first, we shortly describe the modelling approach and the solving procedure and secondly, some new results of numerical modelling are presented. Thermofluctuational model of fracture was developed and studied by Zhurkov with co-authors (Zhurkov, 1965; Regel et al., 1974). Introducing this approach into fracture mechanics was initiated by Barenblatt et al. (1966, 1967). Some recent applications based on these ideas can be found in (Goldstein et al., 1997; Bazant and Pang, 2007; Romero de la Osa et al., 2009). The durability of materials under the action of the tensile stress σ satisfies the following experimentally established formula (Zhurkov, 1965; Regel et al., 1974) τ = µτ 0 e U ( σ ) kT (1) where τ is the specimen durability, k is the Boltzmann constant, T is the absolute temperature, τ 0 ≈ h / ( kT ) is a constant of the order of the atomic thermal vibration period, h is the Planck constant, µ is a dimensionless coe ffi cient depending on the type of the material (polymer, metal, or ceramics), and U ( σ ) is the fracture activation energy. For a su ffi ciently wide interval of external loads the function U ( σ ) is linear, U ( σ ) = U 0 − A ( σ ) , A ( σ ) = γσ (2) where U 0 is the interatomic bond breakage activation energy and γ is structure-sensitive parameter characterizing the material strength properties. It follows from expression in Eq. (2) that term A ( σ ) decreases the bond breakage energy barrier and it can be treated as the work done by external stresses for the body fracture. Formula similar to that in (1) also holds for the average life time (durability) of a loaded interatomic bond (Regel et al., 1974). The external stresses work in this case depends on the stress value in the bond. Nonuniform stresses distribution within bonds in an actual body results in local damage accumulation and defects formation. The most intensive damage accumulation occurs in the weakened interatomic (intermolecular) bond regions in the material. The weakened bond region in the material (or on the material interface) will be treated as a crack filled with bonds (bridges) whose properties vary in time according to the thermal fluctuation mechanism. It is assumed that the expression in Eq. (1) also holds for bonds in the crack bridged zone but the work A ( σ ) contained in Eq. (2) is the bond deformation work, which is determined with accounting for the inhomogeneous stresses distribution within the bonds in the bridged zone. The bond deformation work and the bond durability in the bridged zone of a rectilinear crack on the material interface (see Fig. 1) depends on the bonds tension in the bridged zone σ b (for details, see (Goldstein and Perelmuter, 2012)) and the bond coordinate location along this zone, x such that A = A ( σ b , x ). Let’s assume that the time variation of the bond density n ( x , t ) in the crack bridged zone is described by the following equation of chemical destruction n ( x , t ) = n 0 exp − t τ b ( σ b , x ) (3) 2. Alliance of bonds kinetics and the crack bridged zone models