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Numerical Calibration of Stress Intensity Factor for Transversely Isotropic Central Cracked Brazilian Disk

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Abstract

To obtain the stress intensity factors (SIFs) of transversely isotropic central cracked Brazilian disk (CCBD), an anisotropy coefficient f is introduced based on the isotropic analytical formula. The coefficient f is related to the anisotropy orientation β, the anisotropy ratio of Young’s modulus ξ, and the apparent shear modulus η. The comprehensive calibration of f is accomplished by the finite element method. Then the formulae for mode I and II SIFs are obtained by fitting f to the analytical solutions of the isotropic CCBD, which realizes the generalization and application of the formulae in transversely isotropic rocks based on the isotropic condition. The new formulae presented in this paper can calculate the SIFs for any given elastic modulus, crack length, anisotropy orientation and certain loading angle range (0°–25°). The results show that the relative errors between the formula and the numerical simulation results are within 5%, and the relative errors with other researches are within 2.8%. Both the SIFs and the correlation coefficients M1(β), M2(β), and N1(β) show a variation of sine or cosine concerning the anisotropy orientation β. And the influence of ξ and η on SIFs can be negligible by adjusting β to some specific angles given in this paper.

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Funding

This work was supported by the National Natural Science Foundation of China (nos. 11872042 and 12132019), and China Postdoctoral Science Foundation (2019M653395).

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Authors and Affiliations

  1. Key Laboratory of Deep Underground Science and Engineering, Ministry of Education, College of Architecture and Environment, Sichuan University, 610065, Chengdu, China

    L. Huang, H. Z. Tang, Z. Q. Gan, J. Z. Huang, X. Pan & S. M. Dong

  2. Failure Mechanics and Engineering Disaster Prevention Key Laboratory of Sichuan Province, Sichuan University, 610065, Chengdu, China

    L. Huang, H. Z. Tang, Z. Q. Gan, J. Z. Huang, X. Pan & S. M. Dong

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  1. L. Huang
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  2. H. Z. Tang
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  3. Z. Q. Gan
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  4. J. Z. Huang
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  5. X. Pan
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Corresponding author

Correspondence to S. M. Dong.

Appendices

Nomenclature

a

Half crack length

B

Disk thickness

E, E '

Young’s moduli within and normal to the isotropy plane

f

Anisotropy coefficient defined as the ratio of anisotropic SIFs to isotropic SIFs

FI, FII

Normalized stress intensity factors

G, G '

Shear moduli within and normal to the plane of isotropy

\(G_{{{\text{SV}}}}^{'}\)

Transverse shear modulus approximated from the Saint-Venant relation

h1(x, a), h2(x, a)

Weight function

k c

The correction factor under different relative crack lengths and anisotropy orientation

KI, KIC

Mode I stress intensity factor and fracture toughness

KII, KIIC

Mode II stress intensity factor and fracture toughness

P

Concentrated force

r, θ

Polar coordinate located at the center of uncracked disk

R

Disk radius

x, y

Oriented along with the horizontal and vertical directions

x ', y '

Parallel to and normal to the transverse isotropic plane

α

Relative crack length

β

The angle between the transversely isotropic plane and the horizontal plane

γ

The influence coefficient, ∆FII/∆ξ

η

Anisotropy ratio of apparent shear modulus, G'/\(G_{{{\text{SV}}}}^{'}\)

ξ

Anisotropy ratio of Young’s modulus, E/E'

τθ, τrθ

Normal and tangential stress distribution for uncracked disk

υ, υ '

Poisson’s ratio within and normal to the plane of isotropy

Abbreviations

BEM

Boundary element method

CCBD

Central cracked Brazilian disk

COD

Crack opening displacement

FEM

Finite element method

ISRM

International Society for Rock Mechanics

QPES

Quarter point element stress

SCB

Semi-circular bend

SIF

Stress intensity factor

XFEM

Extended finite element method

APPENDIX A.

Table A1. Correction factors under different relative crack length α and anisotropy orientation β (θ = 0)
Full size table
Table A2. Correction factors under different relative crack length α and anisotropy orientation β (θ = 5°)
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Table A3. Correction factors under different relative crack length α and anisotropy orientation β (θ = 10°)
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Table A4. Correction factors under different relative crack length α and anisotropy orientation β (θ = 15°)
Full size table
Table A5. Correction factors under different relative crack length α and anisotropy orientation β (θ = 20°)
Full size table
Table A6. Correction factors under different relative crack length α and anisotropy orientation β (θ = 25°)
Full size table

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Huang, L., Tang, H.Z., Gan, Z.Q. et al. Numerical Calibration of Stress Intensity Factor for Transversely Isotropic Central Cracked Brazilian Disk. Mech. Solids 57, 835–855 (2022). https://doi.org/10.3103/S0025654422040082

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