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A micro–macro method for evaluating progressive and direct tensile fractures in brittle rocks

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Abstract

A micro–macro method for evaluating the behaviors of direct tensile fractures during progressive loadings in brittle rocks is proposed in this study. The method consists of the suggested equation of the stress intensity factor of the mode-I crack that considers crack initiation, growth, and coalescence subjected to triaxial tensile loadings and the expression of axial strain relating to the extended length of the wing crack. The direct tensile correlation of stress and strain for depicting the complete initial elasticity, strain hardening, strain softening, and fracture stages is also studied. The reasonability of the presented method is proved by contrasting published results of experiment. Furthermore, the sensitivities of the density, inclination angle and size of the initial crack on the axial stress–strain curve, axial stress–crack length curve, tensile strength, crack initiation stress, and elastic modulus are determined. The tensile peak stress, tensile stress at crack initiation, and tensile elastic modulus descend with the increment of the inclination angle or size of the initial crack. The tensile peak stress initially descends and then remains constant, finally reaching a critical value with the increment of the density of the initial crack. The tensile elastic modulus descends with the increment of the density of the initial crack. The calculated results have a great significance for the safety evaluation of surrounding rocks in deep-buried underground engineering.

Article highlights

  • A micro–macro method evaluating the direct tensile fracture of brittle rocks is proposed.

  • The complete stress–strain constitutive relationship of brittle rocks under tensile loadings is studied.

  • The effect of microcrack geometry on the properties of direct tensile fracture is analyzed.

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Abbreviations

a :

Size of the initial crack, i.e., the three-dimensional penny-shaped crack radius

D o :

Damage of the initial microcrack

E :

Initial elastic modulus

F V :

Wedge force

K I :

Mode-I crack stress intensity factor

K IC :

Mode-I fracture toughness

l :

Length of wing crack extension

l coa :

Length of crack coalescence between the adjacent wing cracks

m :

Material constant

N V :

Initial microcrack number per unit volume (i.e., crack density)

r :

Averagely occupied radius of the initial crack per unit volume

S :

Averagely occupied area of the initial crack plane per unit volume

α :

Sine value of initial crack inclination angle φ

β :

Material constant

φ :

Initial crack inclination angle

σ 1 :

Maximal principal tensile stress

σ 3 :

Minimal principal tensile stress

σ 1ci :

Crack initiation stress

σ i 1 :

Internal stress between the tips of the adjacent wing cracks

σ n :

Normal stress at the plane of initial crack

τ :

Shear stress at the plane of initial crack

ε 1 :

Axial strain

ε 1ci :

Axial strain at crack initiation

ε 1f :

Axial failure strain

ε o :

Material constant

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51708016 and 12172036), the Scientific Research Program of Beijing Municipal Education Commission (KM202110016014), the Pyramid Talent Training Project of Beijing University of Civil Engineering and Architecture (Grant No. JDYC20200307), and the Fundamental Research Funds for Beijing Universities (Grant No. X20129).

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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  1. School of Civil and Transportation Engineering, Beijing University of Civil Engineering and Architecture, Beijing, 100044, China

    Xiaozhao Li, Xing Che, Huaiwei Yan & Chengzhi Qi

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  1. Xiaozhao Li
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Correspondence to Xiaozhao Li.

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Li, X., Che, X., Yan, H. et al. A micro–macro method for evaluating progressive and direct tensile fractures in brittle rocks. Geomech. Geophys. Geo-energ. Geo-resour. 8, 133 (2022). https://doi.org/10.1007/s40948-022-00450-x

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