Acta Mechanica Solida Sinica

, Volume 21, Issue 3, pp 221–231

• Xiaoping Zhou
• Qihu Qian
• Yongxing Zhang
Article

## Abstract

An accurate and efficient numerical method for solving the crack-crack interaction problem is presented. The method is mainly by means of the dislocation model, stress superposition principle and Chebyshev polynomial expansion of the pseudo-traction. This method can be applied to compute the stress intensity factors of multiple kinked cracks and multiple rows of periodic cracks as well as the overall strains of rock masses containing multiple kinked cracks under complex loads. Many complex computational examples are given. The dependence of the crack-crack interaction on the crack configuration, the geometrical and physical parameters, and loads pattern, is investigated. By comparison with numerical results under confining pressure unloading, it is shown that the crack-crack interaction under axial-dimensional unloading is weaker than those under confining pressure unloading. Numerical results for single faults and crossed faults show that the single faults are more unstable than the crossed faults. It is found from numerical results for different crack lengths and different crack spacing that the interaction among kinked cracks decreases with an increase in length of the kinked cracks and the crack spacing under axial-dimensional unloading.

## Key Words

interaction among cracks axial-dimensional unloading crack-weakened rock masses the stress-strain relation the Chebyshev polynomial expansion

## References

1. [1]
Wu, G. and Sun, J., The deformation and strength properties of crack-weakened rock masses under unloading. Chinese Journal of Rock Mechanics and Engineering, 1998, 17(6): 615–621.Google Scholar
2. [2]
Zhou, X.P., Analysis of the localization of deformation and the complete stress-strain relation for mesoscopic heterogeneous brittle rock under dynamic uniaxial tensile loading. International Journal of Solids and Structures, 2004, 41(5/6): 1725–1738.
3. [3]
Zhou, X.P., Ha, Q.L. and Zhang, Y.X., The constitutive relation for crack-weakened rock masses under confining pressure unloading with crack interaction effect. Chinese Quarterly of Mechanics, 2002, 23(2): 227–235.Google Scholar
4. [4]
Horii, H. and Nemat-Nasser, S., Brittle failure in compression: splitting, faulting and brittle-ductile transition. Philosophical Transactions of the Royal Society of London, 1986, A319: 337–374.
5. [5]
Niu, J. and Wu, M., Analysis of asymmetric kinked cracks of arbitrary size, location and orientation — Part I. Remote compression. International Journal of Fracure, 1998, 89(1): 19–57.
6. [6]
Deng, H. and Nemat-Nasser, S., Microcrack interaction and shear fault failure. International Journal of Damage Mechanics, 1994, 3(1): 3–37.
7. [7]
Kachanov, M., Elastic solids with many cracks: a simple method of analysis. International Journal of Solids and Structures, 1987, 23(1): 23–43.
8. [8]
Kachanov, M., On the problems of crack interactions and crack coalescence. International Journal of Fracture, 2003, 120(3): 537–543.
9. [9]
Zhan, S., Wang, T. and Han, X., Analysis of two-dimensional finite solids with microcracks. International Journal of Solids and Structures, 1999, 36(25): 3735–3753.
10. [10]
Wang, J., Fang, J. and Karihaloo, B.L., Asymptotics of multiple crack interactions and prediction of effective modulus. International Journal of Solids and Structures, 2000, 37(43): 6221–6237.
11. [11]
Tang, C.A., Lin, P., Wong, R.H.C. and Chau, K.T., Analysis of crack coalescence in rock-like materials containing three flaws—Part II: numerical approach. International Journal of Rock Mechanics and Mining Sciences, 2001, 38(7): 925–939.
12. [12]
Feng, X.Q., Li, J.Y. and Yu, S.W., A simple method for calculating interaction of numerous microcracks and applications. International Journal of Solids and Structures, 2003, 40: 447–464.
13. [13]
Li, M.T., Feng, X.T. and Zhou, H., Cellular automata simulation of the interaction mechanism of two cracks in rock under uniaxial compression. Internation Journal of Rock Mechanics and Mining Sciences, 2004, 41(S1): 484–489.
14. [14]
Carpinteri, A., Brighenti, R. and Vantadori, S., A numerical analysis on the interaction of twin coplanar flaws. Engineering Fracture Mechanics, 2004, 71(4–6): 485–499.
15. [15]
Qing, H. and Yang, W., Characterization of strongly interacted multiple cracks in an infinite plate. Theoretical and Applied Fracture Mechanics, 2006, 46(3): 209–216.
16. [16]
Chen, Y.Z. and Lin, X.Y., Collinear Zener-Stroh crack problem in plane elasticity. Engineering Fracture Mechanics, 2008, 75(6): 1684–1693.
17. [17]
Wang, A.Q. and Qin, T.Y., Finite-part integral and boundary element method for interaction between two parallel planar cracks in three-dimensional finite bodies, Acta Mechanica Solida Sinica, 1998, 11(2): 95–103.Google Scholar
18. [18]
Zhou, X.P. and Zhang, Y.X., The Stress-strain Relationship of the Crack-weakened Rock Mass Under Unloading. Beijing: Science Press, 2007.Google Scholar
19. [19]
Muskhelishvili, N.L., Some Basic Problems in the Mathematical Theory of Elasticity. Netherlands: Noordhoff Groningen, 1953.
20. [20]
Lo, K.K., Analysis of branched cracks. Journal of Applied Mechanics, 1978, 45: 797–802.
21. [21]
Gerasoulis, A., The use of quadratic polynomials for the solution of singular integral equations of cauchy type. Computer &amp; Mathmatics with Applications, 1982, 8: 15–22.