The minimum dimensionless stress intensity factor and its upper bound for CCNBD fracture toughness specimen analyzed with straight through crack assumption
Introduction
Over the past two decades there has a great deal of interest and debate on the unique fracture toughness test method suggested by the International Society for Rock Mechanics (ISRM) [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. We thought it is highly unusual that an “apparently typographical error” in the crucial formula for determining mode I (opening mode) fracture toughness of rock using ISRM suggested cracked chevron notched Brazilian disc (CCNBD) specimen (Fig. 1) had been overlooked by the international rock mechanics community for 11 years until it was discovered by Iqbal and Mohanty [2]. If there were such an “apparently typographical error”, it should be discovered much earlier, or at least confirmed officially by the authorities concerned. So far no erratum has been published by the ISRM Testing Commission [1] or by the researchers who initially contributed to developing the test method [3], [4], [5], [6], evidencing the claim to be unfounded. Of course, a faulted formula is detrimental for the test where the formula is used to calculate the output quantity. Our efforts to rectify the errors and misunderstandings are shown in Refs. [7], [8], [9], [10]. Actually, substantial improvement about the formula in the ISRM method, focusing on the coefficients, has been continued for over a decade, contributing to the sound development of this important suggested method. Actually, we were the first to give out a warning that the formula of ISRM was inadequate and inaccurate [7], and also the first to point out the errors of two groups of authors respectively [10], the comprehensive recalibration with hundreds of finite element computations is given in Ref. [11], where the “corrected formula” presented in [2] was shown to overshoot the upper bounds for the coefficients in the formula with exemplification of a wide range of specimen geometries. The upper bound on the coefficient of the formula is set by comparison of variation of stress intensity factor (SIF) for CCNBD during its loading process with the classic SIF value of the counterpart specimen—cracked straight through Brazilian disc (CSTBD) (Fig. 2), the upper bound is determined by the geometry of specified CCNBD itself. The topic is now further addressed using a straightforward analytical method based on straight through crack assumption (STCA), which was adopted to solve SIF for other chevron-notched fracture toughness specimens (e.g. short rod, three-point bend round bar, etc.) used for testing ceramics and metals [12]. The current emphasis is focused on the minimum (critical) dimensionless SIF, which is the coefficient in the formula of [1] for calculating fracture toughness. Although different method is adopted for the analysis, the present conclusion about the rule which must be obeyed by the dimensionless SIF of CCNBD is just the same as that derived by the slice synthesis method [9] and the finite element method(FEM) [8], [11]. The analytical underpinning inherent in the present STCA method is also beneficial in terms of providing a parallel form of analysis, which serves to validate the computational FEM results. Furthermore, the physical significance of the upper bound is further explained by comparison of load bearing capacity for specimens of different geometries. With the upper bound being firmly established, it is obvious that the disputes triggered by some authors are ungrounded; essentially their “correction” is equivalent to replacing the original inaccurate formula in [1] with an unreasonable formula in [2], the latter being even worse because of the violation of the upper bound. Hopefully our recent efforts would further help to resolve the impasse, contribute to eliminating the misunderstanding, and result in making a better ISRM suggested method for testing rock fracture toughness with CCNBD specimens.
Section snippets
Formulation of stress intensity factor for CCNBD with STCA
The ISRM suggested formula for calculating fracture toughness KIC using CCNBD specimen (Fig. 1) is given in [1]:where B is specimen thickness, D is the diameter, is the minimum (critical) dimensionless stress intensity factor (SIF), this minimum value of Y∗ occurs at the critical crack length αm, and corresponds to the maximum load Pmax. Although other value of Y∗ can also be used to calculate fracture toughness, the corresponding load value is not easy to determine
Physical meaning of the upper bound and lower bound for of CCNBD
Let us compare the three specimens with same thickness B and diameter D but with different crack length and configuration in Fig. 5, where (1) is a CCNBD with α = αm, and (2) and (3) are CSTBD with α = α1 and α = αm respectively. Fracture toughness determined by these three specimens is expressed aswhere n = 1, 2, 3, corresponding to Fig. 5(1), (2), and (3) respectively, Pn is the maximum load, Yn is the dimensionless SIF. KIC, B, D are all constants, so that
Comment on methods for deriving of CCNBD
It is important to distinguish stress intensity factor from fracture toughness, the analysis for stress intensity factor should be strictly linearly elastic, and no nonlinear elastic effect is allowed for the underlying elastic crack stress analysis. Our three methods (the slice synthesis method [9], the finite element method [8], [11], and the present STCA method) to analyze SIF of CCNBD are all strictly linearly elastic. However, some researchers used fracture toughness matching method, they
Conclusion
For every CCNBD specimen geometry there is an upper bound on , which is the minimum (critical) dimensionless stress intensity factor and also the coefficient in the ISRM suggested formula for calculating fracture toughness with the specimen [1], the upper bound for is Y(α1), Y(α1) is the dimensionless stress intensity factor of the corresponding CSTBD specimen with crack length of α1, α1 being the final length of the chevron notch of the CCNBD specimen. This conclusion is consistent
Acknowledgements
We are very grateful to the National Natural Science Foundation (Project No. 51179115) and the State Key Laboratory for Geo-Mechanics and Deep Underground Engineering (Project No. SKLGDUEK1020) for the financial support. We thank Reviewer #2 for the suggestions to improve the paper’s quality.
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