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A direct traction boundary integral equation method for three-dimension crack problems in infinite and finite domains

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Abstract

This paper presents a direct traction boundary integral equation method (TBIEM) for three-dimensional crack problems. The TBIEM is based on the traction boundary integral equation (TBIE). The TBIE is collocated on both the external boundary and one of the crack surfaces. The displacements and tractions are used as unknowns on the external boundary and the relative crack opening displacements (CODs) are introduced as unknowns on the crack surface. In our implementation, all the surfaces of the considered structure are discretized into discontinuous elements to satisfy the continuity requirement for the existence of finite-part integrals, and special crack-front elements are constructed to capture the crack-tip behavior. To calculate the finite-part integrals, an adaptive singular integral technique is proposed. The stress intensity factors (SIFs) are computed through a modified COD extrapolation method. Numerical examples of SIFs computation are presented to demonstrate the accuracy and efficiency of our method.

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Acknowledgments

This work was supported in part by National Science Foundation of China under Grant Numbers 11172098, in part by National 973 Project of China under grant number 2010CB328005 and in part by Hunan Provincial Natural Science Foundation for Creative Research Groups of China (Grant No. 12JJ7001).

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, China

    Guizhong Xie, Jianming Zhang, Cheng Huang, Chenjun Lu & Guangyao Li

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  1. Guizhong Xie
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  2. Jianming Zhang
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  3. Cheng Huang
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  4. Chenjun Lu
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  5. Guangyao Li
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Corresponding author

Correspondence to Jianming Zhang.

Appendices

Appendix 1

$$\begin{aligned} M^{0}&= 1.254611404\xi +1.672815209\xi ^{2} -3.763834213\xi \sqrt{1+\eta } \\&-5.018445625\xi ^{2}\sqrt{1+\eta }-5.018445625\xi (1+\eta ) \\&+3.345630416\xi ^{2}(1+\eta ) \\ M^{1}&= -1.254611404\xi +1.672815209\xi ^{2}+3.763834213\xi \sqrt{1+\eta } \\&-5.018445625\xi ^{2}\sqrt{1+\eta }-2.509222810\xi (1+\eta ) \\&+3.345630416\xi ^{2}(1+\eta ) \\ M^{2}&= -2.143500293\xi +2.858000394\xi ^{2}+3.763834213\xi \sqrt{1+\eta } \\&-5.018445623\xi ^{2}\sqrt{1+\eta }-1.620333921\xi (1+\eta ) \\&+2.160445229\xi ^{2}(1+\eta ) \\ M^{3}&= 2.143500292\xi +2.858000396\xi ^{2}-3.763834212\xi \sqrt{1+\eta } \\&-5.018445628\xi ^{2}\sqrt{1+\eta }+1.620333920\xi (1+\eta ) \\&+2.160445232\xi ^{2}(1+\eta ) \\ M^{4}&= 1.881917111+0.5e-8\xi +-3.345630412\xi ^{2} \\&-5.645751329\sqrt{1+\eta }-0.18e-7\xi \sqrt{1+\eta } \\&+10.03689123\xi ^{2}\sqrt{1+\eta }+3.763834218(1+\eta ) \\&+0.13e-7\xi (1+\eta )-6.691260818\xi ^{2}(1+\eta ) \\ M^{5}&= 2.731445032\xi -3.641926709\xi ^{2}-7.527668433\xi \sqrt{1+\eta } \\&+10.03689124\xi ^{2}\sqrt{1+\eta }+4.129556734\xi (1+\eta ) \\&-5.506075642\xi ^{2}(1+\eta ) \\ M^{6}&= 3.215250440+0.3e-8\xi -5.716000784\xi ^{2} \\&-5.645751320\sqrt{1+\eta }-0.10e-7\xi \sqrt{1+\eta } \\&+10.03689124\xi ^{2}\sqrt{1+\eta }+2.430500880(1+\eta ) \\&+0.6e-8\xi (1+\eta )-4.320890456\xi ^{2}(1+\eta ) \\ M^{7}&= -2.731445031\xi -3.641926715\xi ^{2}+7.527668426\xi \sqrt{1+\eta } \\&+10.03689125\xi ^{2}\sqrt{1+\eta }+-4.129556728\xi (1+\eta ) \\&+ -5.506075646\xi ^{2}(1+\eta ) \\ M^{8}&= -4.097167551+7.283853429\xi ^{2}+11.29150265\sqrt{1+\eta } \\&-20.07378250\xi ^{2}\sqrt{1+\eta }-6.194335099(1+\eta ) \\&-0.2e-8\xi (1+\eta )+11.01215129\xi ^{2}(1+\eta ) \\ \end{aligned}$$

Appendix 2

$$\begin{aligned} M^{0}&= 31.87450619+14.16644724\xi +14.16644717\eta \\&-80.06422983\sqrt{1+\xi }\sqrt{1+\eta } \\&-16.06324133\xi \sqrt{1+\xi }\sqrt{1+\eta } -16.06324115\eta \\&\times \sqrt{1+\xi }\sqrt{1+\eta }+48.18972364(1+\xi )(1+\eta ) \\&-3.79358801\xi (1+\xi )(1+\eta )-3.79358810\eta \\&\times (1+\xi )(1+\eta ) \\ M^{1}&= 48.55017632+23.31444932\xi +19.18489251\eta \\&-119.7183808\sqrt{1+\xi }\sqrt{1+\eta } \\&-27.44400593\xi \sqrt{1+\xi }\sqrt{1+\eta }-16.06324078\eta \\&\times \sqrt{1+\xi }\sqrt{1+\eta }+71.16820448(1+\xi )(1+\eta ) \\&-3.79358790\xi (1+\xi )(1+\eta )-10.03689095\eta \\&\times (1+\xi )(1+\eta ) \\ M^{2}&= 63.44806842+26.55511681\xi +26.55511668\eta \\&-159.3725312\sqrt{1+\xi }\sqrt{1+\eta } \\&-27.44400556\xi \sqrt{1+\xi }\sqrt{1+\eta }-27.44400519\eta \\&\times \sqrt{1+\xi }\sqrt{1+\eta }+95.92446278(1+\xi )(1+\eta ) \\&-10.03689065\xi (1+\xi )(1+\eta )-10.03689086\eta \\&\times (1+\xi )(1+\eta ) \\ M^{3}&= 48.55017607+19.18489251\xi +23.31444911\eta \\&-119.7183802\sqrt{1+\xi }\sqrt{1+\eta } \\&-16.06324097\xi \sqrt{1+\xi }\sqrt{1+\eta }-27.44400554\eta \\&\times \sqrt{1+\xi }\sqrt{1+\eta }+71.16820413(1+\xi )(1+\eta ) \\&-10.03689076\xi (1+\xi )(1+\eta )-3.79358802\eta \\&\times (1+\xi )(1+\eta )\\ M^{4}&= -67.6957080-67.6957080\xi -29.58750589\eta \\&+170.0419981\sqrt{1+\xi }\sqrt{1+\eta }+ \\&34.97167356\xi \sqrt{1+\xi }\sqrt{1+\eta }+32.12648266\eta \\&\times \sqrt{1+\xi }\sqrt{1+\eta }-102.3462901(1+\xi )(1+\eta ) \\&+7.58717619\xi (1+\xi )(1+\eta )+10.03689064\eta \\&\times (1+\xi )(1+\eta ) \\ M^{5}&= -99.2692723-47.43906644\xi -38.65678635\eta \\&+249.3503046\sqrt{1+\xi }\sqrt{1+\eta }+ \\&54.88801273\xi \sqrt{1+\xi }\sqrt{1+\eta }+34.97167358\eta \\&\times \sqrt{1+\xi }\sqrt{1+\eta }-150.0810323(1+\xi )(1+\eta ) \\&+10.03689065\xi (1+\xi )(1+\eta )+20.07378216\eta \\&\times (1+\xi )(1+\eta ) \\ M^{6}&= -99.2692714-38.65678600\xi -47.43906602\eta \\&+249.3503027\sqrt{1+\xi }\sqrt{1+\eta } \\&+34.97167346\xi \sqrt{1+\xi }\sqrt{1+\eta }+54.88801221\eta \\&\times \sqrt{1+\xi }\sqrt{1+\eta }-150.0810313(1+\xi )(1+\eta ) \\&+20.07378195\xi (1+\xi )(1+\eta )+10.03689071\eta \\&\times (1+\xi )(1+\eta ) \\ M^{7}&= -67.6957080-29.58750589\xi -30.39767274\eta \\&+170.0419981\sqrt{1+\xi }\sqrt{1+\eta } \\&+32.12648266\xi \sqrt{1+\xi }\sqrt{1+\eta }+34.97167356\eta \\&\times \sqrt{1+\xi }\sqrt{1+\eta }-102.3462901(1+\xi )(1+\eta ) \\&+10.03689064\xi (1+\xi )(1+\eta )+7.58717619\eta \\&\times (1+\xi )(1+\eta )\\ M^{8}&= 142.5070281+62.86012379\xi +62.86012350\eta \\&-359.9110701\sqrt{1+\xi }\sqrt{1+\eta } \\&-69.94334743\xi \sqrt{1+\xi }\sqrt{1+\eta }-69.94334662\eta \\&\times \sqrt{1+\xi }\sqrt{1+\eta }+218.4040420(1+\xi )(1+\eta ) \\&-20.07378115\xi (1+\xi )(1+\eta )-20.07378159\eta \\&\times (1+\xi )(1+\eta ) \end{aligned}$$

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Xie, G., Zhang, J., Huang, C. et al. A direct traction boundary integral equation method for three-dimension crack problems in infinite and finite domains. Comput Mech 53, 575–586 (2014). https://doi.org/10.1007/s00466-013-0918-8

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