Study of mixed mode I/II cohesive zone models of different rank coals
Introduction
Understanding the fracture behaviors of coals is of paramount importance to mining engineering [1], [2] and coalbed methane (CBM) extraction [3], [4]. As distinct from brittle materials like hard rocks, coals demonstrate apparent quasi-brittle fracture behaviors. Hard rocks typically involve the complete loss of tensile stress between fracture surfaces and the sudden failure incurred by rapid crack growth immediately after peak load is reached. Quasi-brittle fractures of coals are distinguishable from brittle fractures by a progressive failure process characterized by a softening branch of the load–displacement relationship in the post-peak stage [5], [6]. Despite the enormous success in modeling brittle fractures, Linear Elasticity Fracture Mechanics (LEFM) [7], [8], [9], [10], [11], [12], [13], [14], [15] fails to give accurate predictions for quasi-brittle materials, such as stress distribution, peak load, and softening behavior. In quasi-brittle materials, energy dissipation processes take place in a nonlinear damage zone around crack tips [16], name as fracture process zone (FPZ) as shown in Fig. 1, which is in discrepancy with LEFM which postulates that energy is dissipated at one point while the rest of the body remains elastic. Cohesive zone model (CZM), pioneered by Barenblatt [17], Dugdale [18] and Hillerborg [19], has proven a useful tool for analyzing the fracture process of quasi-brittle materials. CZM idealizes FPZ as a fictitious crack surface, whose separation is resisted by the interface cohesive stress that is related to the relative displacement across cohesive crack surface through CZM constitutive equations (cohesion-separation law). CZM constitutive equations represent phenomenologically the physical process occurring in FPZ, for example, the initiation, growth, and coalescence of micro-cracks. Hence, compared to LEFM, CZM is able to capture the softening behavior of quasi-brittle fractures, meanwhile eliminating the physically unrealistic stress singularity at crack tips and producing smooth crack faces closure. Moreover, CZM removes the nonlinear degeneracy problems originating from the fluid pressure singularity at the crack tip in hydraulic fracturing modeling.
In the authors’ previous study [20], the mode I CZMs of five different rank coals were studied using disk-shaped compact tension (DC(T)) experiments on the basis of Karihaloo’s polynomial cohesion-separation law. The applicability of the established CZMs was validated by a comparison between numerical results and experimental data. The mixed mode I/II (tension/shear) crack growth, though, is more likely to occur in realistic scenarios [21]. However, to the authors’ best knowledge, previous investigations [22], [23] still rely on LEFM to study mixed mode I/II crack propagation in coals, which cannot reflect their quasi-brittle fracture characteristics as aforementioned [24], [25]. Hence, the present paper seeks to provide a systematic investigation of the mixed mode I/II CZMs of coals.
The fundamental issue central to mixed mode I/II CZM is determining cohesive interactions between fracture surfaces, that is, mixed mode I/II constitutive equations between the cohesive stress and the relative displacement of fracture surfaces. Lee et al. [26] established a linear mixed mode I/II CZM for iron composite materials through co-cured single leg bending tests. Campilho et al. [27] utilized the trapezoidal traction–separation relationship in the mixed-mode I/II CZM to simulate the crack growth in carbon-fiber-reinforced plastics. Song et al. [28] employed bilinear CZM to investigate mixed mode I/II fracture behaviors of asphalt concrete. Moroni and Pirondi [29] proposed an exponential cohesive relationship to simulate the mixed-mode I/II crack propagation in adhesively bonded joints. The above mixed-mode I/II CZMs fall in to the category of non-potential-based models [30] which are relatively simple to develop without needing to guarantee symmetric systems. Nevertheless, its main limitations are that one model does not account for all possible separation of the quasi-brittle materials, and the computational burden of solving the governing equation is heavy due to the unsymmetrical tangential stiffness. On the other hand, the potential-based models [30], i.e., the mixed-mode I/II fracture energy distribution in conjunction with the relative separation displacement of fractured surfaces, use the first derivative of the fracture potential energy function to provide the cohesive stress over fractured surfaces and its second derivative yields the constitutive relationship. Compared with the non-potential-based models, one single potential function characterizes the mixed mode I/II fracture behaviors in various quasi-brittle materials and can be extended and generalized by advanced numerical algorithms [31], [32]. Needleman [33] established a polynomial function-based potential for describing void nucleation by inclusion debonding. Beltz and Rice [34] developed exponential function-based potential to analyze mixed mode I/II crack growth. Xu and Needleman [35] advanced the exponential potential by considering normal and tangential cohesive interactions. However, the above potentials have several limitations [36], especially when mode I and mode II fracture energy are not equal, including unsymmetrical boundary conditions, the vague fracture parameters that are difficult to determine, and the initial slope which cannot be controlled. As a remedy, Park et al. [37] developed a new potential-based constitutive model, called the Park–Paulino–Roesler (PPR) model, for mixed-mode I/II cohesive fractured. In the PPR model, the mode I and mode II fracture energy are obviously distinguishable, the different initial sloped and cohesive strengths are considered, and the softening parameters are proposed in order to represent a wide range of failure responses in a variety of quasi-brittle materials.
The present study aims to establish mixed mode I/II CZMs for coals based on PPR models. To this end, we performed disk-shaped compact tension (DC(T)) tests and punch-through shear (PTS) tests to obtain the fracture parameters of PPR models for mixed mode I/II cohesive cracks in various rank coals, such as weakly caking coals, fat coals, and anthracite. Then, to validate the established PPR-based CZM, we implemented the model within the framework of the Extend Finite Element Method (XFEM) to simulate the mixed mode I/II crack propagation in single-edge notched beams (SENB) made of weakly caking coals, fat coals, and anthracite, respectively. Finally, the size effect of the mixed mode I/II fractures in coals were discussed based on the PPR CZM.
Section snippets
Park–Paulino–Roesler (PPR) model
For completeness, we briefly recall the PPR model in this section. In PPR model, both tangential and normal cohesive stress, Tt, and Tn are functions of the tangential and normal separation (Δt, Δn). The PPR model enforces the following boundary conditions to the cohesive crack (Fig. 2):
- (1)
When Δn reaches the maximum width of normal crack opening (δn) or Δt reaches tangential conjugate final crack opening displacement (), Tn vanishes on crack surface, standing for complete normal failure of
Experimental methods and process
Our study adopted DC(T) test methods [39] to establish the constitutive relationship of mode I CZM. The DC(T) test method possesses certain advantages over other methods [40], [41] in that the specimen preparation process is convenient, the large potential area for crack propagation guarantees steady crack propagation in front of the preset crack tip, and the flexural bending effect is avoided. The DC(T) specimen is shown in Fig. 3, and the experimental setup is illustrated in Fig. 4. And the
Results and discussion
In this section, the parameters of the PPR CZM are determined based on the experimental data. Although some parameters can be straightforwardly measured in the laboratory experiments, but many need to be calculated according the theory introduced in Section 2. After that, we implemented the model in XFEM to simulate mix-mode fracture of coals, and compare the numerical results with experimental data. We also discuss the underlying mechanism of size effects, which demonstrates the capability of
Conclusion
In this paper, the key parameters of the mixed mode I/II PPR cohesive zone models (CZMs) of three coal ranks were determined using DC(T) tests and PTS tests. For the verification of the developed models, numerical simulation results obtained from extended finite element methods (XFEM) were compared with the experimental results of mixed mode I/II single-edge notched beam (SENB) tests. Also, the size effect of crack propagation in coals was discussed. The following conclusions are drawn:
- (1)
In
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors thank the financial supports from the National Natural Science Foundation of China (NSFC) under Grant No. 52004203 and 51904202.
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