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A Further Improved Maximum Tangential Stress Criterion for Assessing Mode I Fracture of Rocks Considering Non-singular Stress Terms of the Williams Expansion

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Abstract

Previous studies showed that the modified maximum tangential stress (MMTS) criterion considering two stress terms (containing r−1/2 and r1/2) of the well-known Williams series expansion can better assess the mode I fracture toughness (KIc) of rocks than the traditional maximum tangential stress (MTS) criterion. However, this study indicates that in some cases, only using the two stress terms cannot fully describe the tangential stress at the critical distance for rock specimens, and the higher order, non-singular stress terms can also play important roles in the tangential stress. The MMTS fracture criterion might still induce non-negligible errors for assessing mode I rock fracture. Thus, we propose a further improved MTS (FIMTS) criterion. The FIMTS criterion emphasizes that the number of non-singular stress terms used in a MTS-based criterion should be carefully chosen according to the following principle: the tangential stress at the critical distance can be accurately described using the selected stress terms. Mode I fracture tests on two kinds of rocks are conducted using a newly proposed V-notched short rod bend (VNSRB) specimen. The specimen- or loading-configuration-dependence of KIc indicated by the experimental data is theoretically assessed using the traditional MTS, the MMTS and the FIMTS criteria, and the last criterion is shown to be the best. Therefore, considering non-singular stress terms of the Williams expansion can better assess mode I fracture of rocks. Our study calls for more attention to the effects of non-singular stress terms on mode I fracture of rocks, especially for small-size specimens and for rocks with relatively large critical distances.

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Abbreviations

A 1, A 2, A 3, …, A n, B n :

Coefficients of the crack tip asymptotic field

A :

Crack length

a 0 :

Initial length of V-shaped crack/notch

a 1 :

Final length of V-shaped crack/notch

a c :

Critical crack length

B :

Thickness of specimen

CB:

Chevron bend

CCNBD:

Cracked chevron notched Brazilian disc

CCNSCB:

Cracked chevron notched semi-circular bend

CSTBD:

Cracked straight-through Brazilian disc

D :

Diameter of specimen

ECT:

Edge-cracked triangular

FBD:

Flattened Brazilian disc

FEM:

Finite element modeling

FIMTS:

Further improved maximum tangential stress

H :

Height of specimen

ISRM:

International Society for Rock Mechanics

K I :

Mode I stress intensity factor

K Ic :

Mode I fracture toughness

MMTS:

Modified maximum tangential stress

MR:

Modified ring

MTS:

Maximum tangential stress

n :

The ordinal number of a term

P :

Load on specimen

P max :

Maximum load

R :

Radius of specimen

R 0 :

The reference specimen radius (= 1 m)

r :

Distance from the crack tip

r c :

Critical distance

S :

Support span

SCB:

Semi-circular bend

SIF:

Stress intensity factor

SNDB:

Straight-notched disc bend

SR:

Short rod

Y :

Normalized stress intensity factor

Y c :

Critical normalized stress intensity factor

α :

Normalized crack length

α 0 :

Normalized initial length of V-shaped crack/notch

α 1 :

Normalized final length of V-shaped crack/notch

α c :

Critical normalized crack length

β :

Angle between the notch edge and the end face of the specimen

σ t :

Tensile strength

σ x, σ y, τxy :

Stresses

τ θθ :

Tangential stress

τ θθc :

Critical tangential stress

θ :

Direction of fracture in polar coordinates

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Acknowledgements

The authors are grateful for the financial support from the National Program on Key Basic Research Project (no. 2015CB057903), the National Natural Science Foundation of China (no. 51779164) and the Graduate Student’s Research and Innovation Fund of Sichuan University (no. 2018YJSY012).

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Authors and Affiliations

  1. State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resource and Hydropower, Sichuan University, Chengdu, 610065, Sichuan, China

    Ming-Dong Wei, Feng Dai, Jia-Wen Zhou, Yi Liu & Jing Luo

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  1. Ming-Dong Wei
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  2. Feng Dai
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Correspondence to Feng Dai.

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Wei, MD., Dai, F., Zhou, JW. et al. A Further Improved Maximum Tangential Stress Criterion for Assessing Mode I Fracture of Rocks Considering Non-singular Stress Terms of the Williams Expansion. Rock Mech Rock Eng 51, 3471–3488 (2018). https://doi.org/10.1007/s00603-018-1524-z

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