Abstract
This paper presents a stochastic fracture analysis method for random non-homogenous cracked structures using the scaled boundary finite element method (SBFEM). The analyzed cracked domain is divided into polygon-based sub-domains such that the material property of each sub-domain can be modeled differently in the SBFEM. The spatial variability of the material property is represented as random field which is then incorporated directly into the coefficient matrices of the SBFEM. The stochastic global stiffness matrix and the stochastic equilibrium equation of the SBFEM through which the stochastic structural responses can be successfully calculated are further formulated. The stochastic stress intensity factors (SIFs) considering the spatial variability of the material property are then directly extracted from the stress solution of the cracked polygon. Numerical examples are given to demonstrate the validity of the proposed method and investigate the effect of the spatial distribution of the material property on the SIFs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chiong I, Ooi ET, Song CM, Tinc-Loi F (2014) Scaled boundary polygons with application to fracture analysis of functionally graded materials. Int J Numer Methods Eng 98(8):562–589. doi:10.1002/nme.4645
Chowdhury MS, Song CM, Gao W (2011) Probabilistic fracture mechanics by using Monte Carlo simulation and the scaled boundary finite element method. Eng Fract Mech 78(12):2369–2389. doi:10.1016/j.engfracmech.2011.05.008
Deeks A, Wolf J (2002) A virtual work derivation of the scaled boundary finite-element method for elastostatics. Comput Mech 28(6):489–504. doi:10.1007/s00466-002-0314-2
Erdogan F, Sih GC (1963) On the crack extension in plates under plane loading and transverse shear. J Basic Eng 85(4):519–525. doi:10.1115/1.3656897
Ghanem RG (1999) The nonlinear Gaussian spectrum of log-normal stochastic processes and variables. J Appl Mech 66(4):964–973. doi:10.1115/1.2791806
Ghanem RG, Spanos PD (1991a) Spectral stochastic finite-element formulation for reliability analysis. J Eng Mech. doi:10.1061/(ASCE)0733-9399(1991)117:10(2351)
Ghanem RG, Spanos PD (1991b) Stochastic finite elements: a spectral approach. Springer, Berlin
Huang SP, Quek ST, Phoon KK (2001) Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes. Int J Numer Methods Eng 52(9):1029–1043. doi:10.1002/nme.255
Irwin GR (1948) Fracture dynamics. Fract Met 147:166
Jiang C, Long XY, Han X, Tao Y, Liu J (2013) Probability-interval hybrid reliability analysis for cracked structures existing epistemic uncertainty. Eng Fract Mech 112:148–164. doi:10.1016/j.engfracmech.2013.10.009
Koutsourelakis PS, Deodatis G (2006) Simulation of multidimensional binary random fields with application to modeling of two-phase random media. J Eng Mech 132(6):619–631. doi:10.1061/(ASCE)0733-9399(2006)132:6(619)
Lin S-C, Abel JF (1988) Variational approach for a new direct-integration form of the virtual crack extension method. Int J Fract 38(3):217–235. doi:10.1007/BF00034286
Long XY, Jiang C, Han X, Gao W (2015) Stochastic response analysis of the scaled boundary finite element method and application to probabilistic fracture mechanics. Comput Struct 153:185–200. doi:10.1016/j.compstruc.2015.03.004
Long XY, Jiang C, Han X, Gao W, Bi R (2014) Sensitivity analysis of the scaled boundary finite element method for elastostatics. Comput Methods Appl Mech Eng 276:212–232. doi:10.1016/j.cma.2014.03.005
Ooi ET, Yang Z (2011) Modelling dynamic crack propagation using the scaled boundary finite element method. Int J Numer Methods Eng 88(4):329–349. doi:10.1002/nme.3177
Ooi ET, Song CM, Tinc-Loi F, Yang Z (2012) Polygon scaled boundary finite elements for crack propagation modelling. Int J Numer Methods Eng 91(3):319–342. doi:10.1002/nme.4284
Ooi ET, Song CM, Tin-Loi F (2014) A scaled boundary polygon formulation for elasto-plastic analyses. Comput Methods Appl Mech Eng 268:905–937. doi:10.1016/j.cma.2013.10.021
Phoon K, Huang S, Quek S (2002) Implementation of Karhunen–Loeve expansion for simulation using a wavelet-Galerkin scheme. Probab Eng Mech 17(3):293–303. doi:10.1016/S0266-8920(02)00013-9
Rahman S, Rao B (2002) Probabilistic fracture mechanics by Galerkin meshless methods-part II: reliability analysis. Comput Mech 28(5):365–374. doi:10.1007/s00466-002-0300-8
Rao B, Rahman S (2002) Probabilistic fracture mechanics by Galerkin meshless methods-part I: rates of stress intensity factors. Comput Mech 28(5):351–364. doi:10.1007/s00466-002-0299-x
Song CM (2005) Evaluation of power-logarithmic singularities, T-stresses and higher order terms of in-plane singular stress fields at cracks and multi-material corners. Eng Fract Mech 72(10):1498–1530. doi:10.1016/j.engfracmech.2004.11.002
Song CM, Vrcelj Z (2008) Evaluation of dynamic stress intensity factors and T-stress using the scaled boundary finite-element method. Eng Fract Mech 75(8):1960–1980. doi:10.1016/j.engfracmech.2007.11.009
Song CM, Wolf JP (1997) The scaled boundary finite-element method-alias consistent infinitesimal finite-element cell method-for elastodynamics. Comput Methods Appl Mech Eng 147(3):329–355. doi:10.1016/S0045-7825(97)00021-2
Song CM, Wolf JP (1998) The scaled boundary finite-element method: analytical solution in frequency domain. Comput Methods Appl Mech Eng 164(1):249–264. doi:10.1016/S0045-7825(98)00058-9
Song CM, Wolf JP (2002) Semi-analytical representation of stress singularities as occurring in cracks in anisotropic multi-materials with the scaled boundary finite-element method. Comput Struct 80(2):183–197. doi:10.1016/S0045-7949(01)00167-5
Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Methods in Appl Mech Eng 198(9):1031–1051. doi:10.1016/j.cma.2008.11.007
Steven Greene M, Liu Y, Chen W, Liu WK (2011) Computational uncertainty analysis in multiresolution materials via stochastic constitutive theory. Comput Methods in Appl Mech Eng 200(1):309–325. doi:10.1016/j.cma.2010.08.013
Su C, Zheng C (2012) Probabilistic fracture mechanics analysis of linear-elastic cracked structures by spline fictitious boundary element method. Eng Anal Bound Elem 36(12):1828–1837. doi:10.1016/j.enganabound.2012.06.006
Sudret B, Der Kiureghian A (2000) Stochastic finite element methods and reliability: a state-of-the-art report. Department of Civil and Environmental Engineering, University of California, Berkeley
Sukumar N, Tabarraei A (2004) Conforming polygonal finite elements. Int J Numer Methods Eng 61(12):2045–2066. doi:10.1002/nme.1141
Tomar V, Zhou M (2005) Deterministic and stochastic analyses of fracture processes in a brittle microstructure system. Eng Fract Mech 72(12):1920–1941. doi:10.1016/j.engfracmech.2004.06.006
Wolf JP, Song CM (2000) The scaled boundary finite-element method—a primer: derivations. Comput Struct 78(1):191–210. doi:10.1016/S0045-7949(00)00099-7
Xu X, Graham-Brady L (2005) A stochastic computational method for evaluation of global and local behavior of random elastic media. Comput Methods Appl Mech Eng 194(42):4362–4385. doi:10.1016/j.cma.2004.12.001
Yang ZJ (2006) Fully automatic modelling of mixed-mode crack propagation using scaled boundary finite element method. Eng Fract Mech 73(12):1711–1731. doi:10.1016/j.engfracmech.2006.02.004
Yang ZJ, Frank XuX (2008) A heterogeneous cohesive model for quasi-brittle materials considering spatially varying random fracture properties. Comput Methods Appl Mech Eng 197(45):4027–4039. doi:10.1016/j.cma.2008.03.027
Yang ZJ, Su X, Chen J, Liu G (2009) Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials. Int J Solids Struct 46(17):3222–3234. doi:10.1016/j.ijsolstr.2009.04.013
Yin X, Lee S, Chen W, Liu WK, Horstemeyer M (2009) Efficient random field uncertainty propagation in design using multiscale analysis. J Mech Des 131(2):021006. doi:10.1115/1.3042159
Acknowledgments
This work is supported by the State Key Program of National Science Foundation of China (11232004), the National Natural Science Foundation of China (51175160), and the National Excellent Doctoral Dissertation special fund (201235).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Long, X.Y., Jiang, C., Han, X. et al. Stochastic fracture analysis of cracked structures with random field property using the scaled boundary finite element method. Int J Fract 195, 1–14 (2015). https://doi.org/10.1007/s10704-015-0042-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-015-0042-1