PSI - Issue 13

Lapin V.N. et al. / Procedia Structural Integrity 13 (2018) 1171–1176 Lapin V.N. and Cherny S.G. / Structural Integrity Procedia 00 (2018) 000–000

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necessary to calculate the fracture increment and propagation direction. The direction is defined by kinking angle θ and twisting angle ψ as is shown in Fig. 1. Plane mixed mode criteria such as maximum tensile stress (2DMTS) Erdogan and Sih (1963) or strain energy density Sih (1974) predicts kinking angle only. 3D criteria like proposed in Cooke and Pollard (1996), Schollmann et al. (2002), Richard at al. (2005) gives both angles but take into account stress state and stress intensity factor at the front point under consideration. The influence of the neighbor points is ignored by these criteria.

Рис. 1. a) The kinking angle θ and the twisting angle ψ that define the direction of crack front propagation. b) 3PB-specimen with an inclined initial notch used in Lazarus et al. (2008).

To take the influence of the neighbor points and to increase the stability of the simulation process an implicit criterion was proposed in Cherny at al. (2016) as the part of the 3D hydraulic fracture propagation model. The implicit criterion at each step of the propagation considers the rock stress state both before and after the propagation. It is based on the assumption that the fracture tends to propagate in the direction where II and III modes of stress intensity factor (SIF) are zero. In most cases these both conditions cannot be satisfied simultaneously, so the values of the SIF modes II and III are combined in one function with weight coefficient β and are integrated along the whole front. A few possible propagation directions (and possible fracture form geometries obtained from the stress state after the fracture propagates) are considered and the one is chosen to provide the minimum of the integral. Since criterion parameters has not been chosen in Cherny at al. (2016) here the implicit criterion is applied to the fractures propagating under mixed mode I and III to propose appropriate values for them. Three point bend (3PB) test described in Buchholz et al. (2004) and Lazarus et al. (2008) (Fig. 1) has been chosen for the implicit criterion validation. On the one hand this experiment shows the behavior of the fracture under mode III loading and on the other hand it is quite simple to simulate. Moreover two global type criteria are applied to this problem by Lazarus et al. (2008) and obtained results are appropriate for comparison. In the section 2 a few criteria are described and the section 3 contains the comparison of the criteria.

2. Propagation criteria

2.1. Local criteria

The maximum tangential stress criterion (2DMTS) Erdogan and Sih (1963) states that the crack propagates in the direction where the circumferential tensile stress is maximum. This yields the condition K I sin θ + K II (3 cos θ − 1) = 0 , (1) and the explicit formula for the kink angle θ = 2 arctan   1 − √ 1 + 8 λ 2 4 λ   , λ = K II / K I . (2) The 3D modification of MTS criterion developed by Schollmann et al. (2002) that is named here as 3DMTS (following Lazarus et al. (2008)) assumes that the crack growth occurs in the direction of a