Strain Energy Release Rate for a Crack Under Combined Mode I and Mode II

Determination of crack initiation and propagation in two disc shaped specimens using the improved maximum tangential stress criterion

“…There are a large number of theoretical models (Erdogan and Sih, 1963;Hussain et al, 1974;Nuismer, 1975;Palaniswamy and Knauss, 1972;Sih, 1974;Theocaris and Andrianopoulos, 1982) and experimental techniques (Grassl and Rempling, 2007;Isaksson and Ståhle, 2002;Lin et al, 2009;Song et al, 2004;Xeidakis et al, 1997) for investigating the mixed mode of crack growth in rock materials. Theoretical failure criteria such as the Maximum Tangential Stress (MTS) criterion (Erdogan and Sih, 1963), the minimum strain energy density criterion (Sih, 1974), and the maximum energy release rate criterion (Hussain et al, 1974) have been frequently used by researchers working on rock and geo-material fields in order to estimate the direction of the mixed mode in the crack growth process. These failure models are usually developed based on stress, strain, and energy conditions in front of the crack tip, and the stress intensity factor is often used to estimate the crack growth direction.…”

Minimum energy multiple crack propagation. Part-II: Discrete solution with XFEM

“…The commonly used criteria in Linear Elastic Fracture Mechanics (LEFM), such as: maximum hoops stress [11], principle of local symmetry [14], minimum strain energy density [30], are heuristic approximations to Griffith's theory. The maximum energy release rate criterion [19,21,18], which implies energy minimality of the cracked solid, is the only criterion that is consistent with Griffith's theory. Even so, the typical numerical implementations of the aforementioned criteria rely on the crack tip stress intensity factors (SIF), which are determined using auxiliary crack tip fields derived from a single-crack problem in an infinite solid [34].…”

“…The m-iterations are continued (updating the working-set as needed) until the solution converges for time-step k. Upon convergence, the solution is advanced to the subsequent time-step k+1 and the iterative solution process repeated for a new set of candidate crack tips I k+1,m=1 tip ⊆ I k tip . For the present algorithm to converge to the globally minimising solution of the energy function E k = E(∆ k , ∆θ k ) using a single trial solution, the function needs to be convex within the admissible solution space, as defined by the constraint equations (18) and (19). The second order sufficiency condition for E k to be minimised with respect to the set I k,m→∞ tip of n k,m tip = |I k,m→∞ tip | of crack tip extensions can be generally given as:…”

“…In summary, the algorithm described in this section is suitable for stable/unstable competing/non-competing crack growth; however, it is required of E(∆ k , ∆θ k ) to be convex within the solution space defined by (18) and (19). If this condition is met, the solution that minimises E(∆ k , ∆θ k ) is determined in a single trial.…”

Modeling of crack initiation, propagation and coalescence in rocks

“…(Whittaker et al, 1992) A closed-form solution for the stress intensity factors of a branched crack was given by Hussain et al (1974): …”