A family of non-conforming crack front elements of quadrilateral and triangular types for 3D crack problems using the boundary element method

Abstract

This study focuses on establishing non- conforming crack front elements of quadrilateral and triangular types for 3D crack problems when the dual boundary element method is applied. The asymptotic behavior of the physical variables in the area near the crack front is fully considered in the construction of the shape function. In the developed quadrilateral and triangular crack front elements, the asymptotic term, which captures the asymptotic behavior of the physical variable, is multiplied directly by the conventional Lagrange shape function to form a new crack front shape function. Several benchmark numerical examples that consider penny-shaped cracks and straight-edge crack problems are presented to illustrate the validity and efficiency of the developed crack front elements.

Keywords

Taylor expansion / crack front elements / stress intensity factors / dual boundary element method

References

[1] Zhang D H, Li Y Q, Xie G Z, Digital image correlation method for measuring deformations of vinyl chloride-coated metal multilayer sheets. Modern Physics Letters B, 2019, 33(5): 1950050

[2] Zhang D H, Li Y Q, Liu J X, A novel 3D optical method for measuring and evaluating springback in sheet metal forming process. Measurement, 2016, 92: 303–317

[3] Zhang D H, Xie G Z, Li Y Q, Strain and mechanical properties of the VCM multilayer sheet and their composites using digital speckle correlation method. Applied Optics, 2015, 54(25): 7534–7541

[4] Zhang D H, Bai D P, Liu J B, Formability behaviors of 2A12 thin-wall part based on DYNAFORM and stamping experiment. Composites Part B: Engineering, 2013, 55: 591–598

[5] Mi Y, Aliabadi M H. Dual boundary element method for three-dimensional fracture mechanics analysis. Engineering Analysis with Boundary Elements, 1992, 10(2): 161–171

[6] Sladek V, Sladek J, Tanaka M. Nonsingular BEM formulations for thin-walled structures and elastostatic crack problems. Acta Mechanica, 1993, 99(1–4): 173–190

[7] Zhang Y M, Gu Y, Chen J T. Boundary element analysis of the thermal behaviour in thin-coated cutting tools. Engineering Analysis with Boundary Elements, 2010, 34(9): 775–784

[8] Yao Z H, Wang H T. Some benchmark problems and basic ideas on the accuracy of boundary element analysis. Engineering Analysis with Boundary Elements, 2013, 37(12): 1674–1692

[9] Zhang J M, Lin W C, Dong Y, A double-layer interpolation method for implementation of BEM analysis of problems in potential theory. Applied Mathematical Modelling, 2017, 51: 250–269

[10] Feng S Z, Han X, Wang G. An efficient on-line algorithm for the optimal design of multi-material structures under thermal loads. International Journal of Thermal Sciences, 2018, 132: 567–577

[11] Feng S Z, Cheng Y H. An element decomposition method for heat transfer analysis. International Journal of Heat and Mass Transfer, 2018, 123: 437–444

[12] Feng S Z, Bordas S P A, Han X, A gradient weighted extended finite element method (GW-XFEM) for fracture mechanics. Acta Mechanica, 2019 (in press)

[13] Cheng C, Niu Z, Recho N. Analysis of the stress singularity for a bi-material V-notch by the boundary element method. Applied Mathematical Modelling, 2013, 37(22): 9398–9408

[14] Cheng C, Ge S, Yao S, et al. Thermal stress singularity analysis for V-notches by natural boundary element method. Applied Mathematical Modelling, 2016, 40(19–20): 8552–8563

[15] Zhu B J, Qin T Y. Application of hypersingular integral equation method to three-dimensional crack in electromagnetothermoelastic multiphase composites. International Journal of Solids and Structures, 2007, 44(18–19): 5994–6012

[16] Henshell R D, Shaw K G. Crack tip finite elements are unnecessary. International Journal for Numerical Methods in Engineering, 1975, 9(3): 495–507

[17] Lv J H, Jiao Y Y, Wriggers P, Efficient integration of crack singularities in the extended finite element method: Duffy-distance transformation and conformal preconditioning strategy. Computer Methods in Applied Mechanics and Engineering, 2018, 340: 559–576

[18] Sáez A, Gallego R, Dominguez J. Hypersingular quarter-point boundary elements for crack problems. International Journal for Numerical Methods in Engineering, 1995, 38(10): 1681–1701

[19] Hong H K, Chen J T. Derivations of integral equations of elasticity. Journal of Engineering Mechanics, 1988, 114(6): 1028–1044

[20] Mi Y, Aliabadi M H. Discontinuous crack-tip elements: Application to 3D boundary element method. International Journal of Fracture, 1994, 67(3): R67–R71

[21] Pan E, Yuan F G. Boundary element analysis of three-dimensional cracks in anisotropic solids. International Journal for Numerical Methods in Engineering, 2000, 48(2): 211–237

[22] Xie G Z, Zhang J, Huang C, A direct traction boundary integral equation method for three-dimension crack problems in infinite and finite domains. Computational Mechanics, 2014, 53(4): 575–586

[23] Li J, Feng W Z, Gao X W. A dual boundary integral equation method based on direct evaluation of higher order singular integral for crack problems. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(2): 387–398 (in Chinese)

[24] Xie G Z, Zhang D, Meng F, Calculation of stress intensity factor along the 3D crack front by dual BIE with new crack front elements. Acta Mechanica, 2017, 228(9): 3135–3153

[25] Ariza M P, Saez A, Dominguez J. A singular element for three-dimensional fracture mechanics analysis. Engineering Analysis with Boundary Elements, 1997, 20(4): 275–285

[26] Ariza M P, Dominguez J. Boundary element formulation for 3D transversely isotropic cracked bodies. International Journal for Numerical Methods in Engineering, 2004, 60(4): 719–753

[27] Aliabadi M H, Rooke D P. Numerical Fracture Mechanics. Boston: Kluwer Academic Publishers, 1991

[28] Shah R C, Kobayashi A S. Stress intensity factor for an elliptical crack under arbitrary normal loading. Engineering Fracture Mechanics, 1971, 3(1): 71–96

[29] Chen L S, Kuang J H. A displacement extrapolation method for determining the stress intensity factors along flaw border. International Journal of Fracture, 1992, 57(4): R51–R58

[30] Liu Y. On the displacement discontinuity method and the boundary element method for solving 3-D crack problems. Engineering Fracture Mechanics, 2016, 164: 35–45

[31] Tada H, Paris P C, Irwin G R. The Stress Analysis of Cracks Handbook. Vol. 130. New York: ASME, 2000

[32] Murakami Y, Keer L M. Stress intensity factors handbook, Vol. 3. Journal of Applied Mechanics, 1993, 60(4): 1063

[33] Raju I S, Newman J C Jr. Three Dimensional Finite-Element Analysis of Finite-Thickness Fracture Specimens. NASA Technical Note, NASA TN D-8414. 1977

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11602229 and 11602082), Hunan Provincial Natural Science Foundation of China (Grant No. 2017JJ3061), and Key Scientific and Technological Project of Henan Province (Grant No. 192102210227).

RIGHTS & PERMISSIONS

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature