Study of mixed mode I/II cohesive zone models of different rank coals

Highlights

• The Cohesive zone models of the mixed mode I/II fractures of various rank coals are studied.

• The cohesive zone models are verified using SENB tests and numerical simulation.

• The size effect in the mixed mode I/II fractures of coals is investigated.

Abstract

The present work studies the Park–Paulino–Roesler (PPR) model, a unified potential-based cohesive zone model (CZM), for mixed mode I/II fractures in different rank coals, including weakly caking coals, fat coals and anthracite, by using disk-shaped compact tension (DC(T)) tests and punch-through shear (PTS) tests. The PTS experiments show that with the increase in coal rank, the initial shear stiffness and shear peak loads grow and the maximum tangential crack opening displacement (δ_t) decreases gradually. Additionally, the post-peak softening curve tends to have a linear shape, and the shear crack surface is rougher and more tortuous for the lower rank coals. The mode II fracture energy (Φ_t) was evaluated by calculating the area under the shear load–displacement curves. Φ_t is remarkably higher than mode I fracture energy (Φ_n) for the same coal rank, and the average value of Φ_t decreases from 145.54 J/m² to 70.11 J/m² as the coal rank of the specimens increases from weakly caking coals to anthracite. In order to verify the obtained PPR CZMs, mixed mode I/II single-edge notched beam (SENB) tests were performed and the experimental results are in good agreement with the numerical simulation. Moreover, we conducted SENB tests with different-size specimens, and the established PPR CZM model demonstrated its ability of describing the size effect of the mixed mode I/II fracture in 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.