An analytical solution for the stress field and stress intensity factor in an infinite plane containing an elliptical hole with two unequal aligned cracks

doi:10.1007/s10483-018-2356-6

Abstract

Abstract:

The existing analytical solutions are extended to obtain the stress fields and the stress intensity factors (SIFs) of two unequal aligned cracks emanating from an elliptical hole in an infinite isotropic plane. A conformal mapping is proposed and combined with the complex variable method. Due to some difficulties in the calculation of the stress function, the mapping function is approximated and simplified via the applications of the series expansion. To validate the obtained solution, several examples are analyzed with the proposed method, the finite element method, etc. In addition, the effects of the lengths of the cracks and the ratio of the semi-axes of the elliptical hole (a/b) on the SIFs are studied. The results show that the present analytical solution is applicable to the SIFs for small cracks.

Key words: finite element, 4-node quadrilateral element, 8-node quadrilateral element, unequal crack, complex variable, stress intensity factor (SIF), elliptical hole, conformal mapping

2010 MSC Number:

74A50

M. HAJIMOHAMADI, R. GHAJAR. An analytical solution for the stress field and stress intensity factor in an infinite plane containing an elliptical hole with two unequal aligned cracks. Applied Mathematics and Mechanics (English Edition), 2018, 39(8): 1103-1118.

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References

[1] MUSKHELISHVILI, N. I. Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhooff Ltd., The Netherlands (1962)
[2] SAVIN, G. N. Stress Concentration Around Holes, Pergamon Press, Oxford (1961)
[3] THEOCARIS, P. S. and PETROU, L. Stress distributions and intensities at corners of equilateral triangular holes. International Journal of Fracture, 31, 271-289(1986)
[4] LEI, G. H., NG, C. W., and RIGBY, D. B. Stress and displacement around an elastic artificial rectangular hole. Journal of Engineering Mechanics, 127, 880-890(2001)
[5] REZAEEPAZHAND, J. and JAFARI, M. Stress concentration in metallic plates with special shaped cutout. International Journal of Mechanical Sciences, 52, 96-102(2010)
[6] LOUHGHALAM, A., IGUSA, T., PARK, C., CHOI, S., and KIM, K. Analysis of stress concentrations in plates with rectangular openings by a combined conformal mapping-finite element approach. International Journal of Solids and Structures, 48, 1991-2004(2011)
[7] Zhou, C. P., Hu, C., Ma, F., and Liu, D. K. Dynamic stress concentrations in thick plates with two holes based on refined theory. Applied Mathematics and Mechanics (English Edition), 35(12), 1591-1606(2014) https://doi.org/10.1007/s10483-014-1883-6
[8] BOWIE, O. L. Analysis of an infinite plate containing radial cracks originating at the boundary of an internal circular bole. Journal of Mathematical Physics, 35, 60-71(1956)
[9] TWEED, J. and ROOKE, D. P. The distribution of stress near the tip of a radial crack at the edge of a circular hole. International Journal of Engineering Science, 11, 1185-1195(1973)
[10] TWEED, J. and ROOKE, D. P. The stress intensity factor for a crack at the edge of a loaded hole. International Journal of Solids and Structures, 15, 899-906(1979)
[11] KIM, J. and HILL, M. R. Weight functions for a finite width plate with single or double radial cracks at a circular hole. Engineering Fracture Mechanics, 168, 112-130(2016)
[12] HASEBE, N. and CHEN, Y. Z. Stress intensity solutions for the interaction between a hole edge crack and a line crack. International Journal of Fracture, 77, 351-366(1996)
[13] MIAO, C., WEI, Y., and YAN, X. A numerical analysis of a center circular-hole crack in a rectangular tensile sheet. Applied Mathematics and Computation, 250, 356-367(2015)
[14] JUNBIAO, L., XING, Z., and SCHIJVE, J. An investigation of a hole-edge crack problem by a combined complex variable and least square method. Engineering Fracture Mechanics, 39, 713-737(1991)
[15] ABDELMOULA, R., SEMANI, K., and LI, J. Analysis of cracks originating at the boundary of a circular hole in an infinite plate by using a new conformal mapping approach. Applied Mathematics and Computation, 188, 1891-1896(2007)
[16] SAIMOTO, A., FUJIKAWA, M., MAKABE, C., YAMANAKA, T., and RANA, M. S. Stress intensity factors for cracks initiated from a center-holed plate with unsymmetrical lengths under tension. Engineering Failure Analysis, 17, 838-847(2010)
[17] WANG, Y. J. and GAO, C. F. The mode Ⅲ cracks originating from the edge of a circular hole in a piezoelectric solid. International Journal of Solids and Structures, 45, 4590-4599(2008)
[18] CHEN, Y., LIN, X., and WANG, Z. A semi-analytic solution for multiple curved cracks emanating from circular hole using singular integral equation. Applied Mathematics and Computation, 213, 389-404(2009)
[19] ZHAO, J., XIE, L., LIU, J., and ZHAO, Q. A method for stress intensity factor calculation of infinite plate containing multiple hole-edge cracks. International Journal of Fatigue, 35, 2-9(2012)
[20] HASEBE, N. and HORIUCHI, Y. Stress analysis for a strip with semi-elliptical notches or cracks on both sides by means of rational mapping function. Ingenieur-Archiv, 47, 169-179(1978)
[21] KUJAWSKI, D. Estimations of stress intensity factors for small cracks at notches. Fatigue & Fracture of Engineering Materials & Structures, 14, 953-965(1991)
[22] PHILIPPS, A. G., KARUPPANAN, S., CHURCHMAN, C. M., and HILLS, D. A. Crack tip stress intensity factors for a crack emanating from a sharp notch. Engineering Fracture Mechanics, 75, 5134-5139(2008)
[23] NEWMAN, J. C. An improved method of collocation for the stress analysis of cracked plates with various shaped boundaries. NASA TN D-6376, 1-45(1971)
[24] WANG, Y. H. Stress intensity factors of a plate with two cracks emanating from an arbitrary hole. Applied Mathematics and Mechanics (English Edition), 11(8), 723-732(1990)
[25] WANG, Y. H. The solution of a crack emanating from an arbitrary hole by boundary collocation method. Applied Mathematics and Mechanics (English Edition), 11(7), 669-678(1990)
[26] GUO, J. H. and LIU, G. T. Analytic solutions to problem of elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals. Applied Mathematics and Mechanics (English Edition), 29(4), 485-493(2008) https://doi.org/10.1007/s10483-008-0406-x
[27] HASEBE, N. and UEDA, M. Crack originating from a corner of a square hole. Engineering Fracture Mechanics, 13, 913-923(1980)
[28] LIU, S. and DUAN, S. Analytical solutions of cracks emanating from an elliptical hole under shear. Chinese Journal of Aeronautics, 27, 829-834(2014)
[29] LIU, S. and DUAN, S. Analytical solutions of cracks emanating from an elliptic hole in an infinite plate under tension. Chinese Journal of Mechanical Engineering, 27, 1057-1063(2014)
[30] YANG, J., ZHOU, Y. T., MA, H. L., DING, S. H., and LI, X. The fracture behavior of two asymmetrical limited permeable cracks emanating from an elliptical hole in one-dimensional hexagonal quasicrystals with piezoelectric effect. International Journal of Solids and Structures, 108, 175-185(2017)
[31] WANG, X. and PAN, E. On a finite crack partially penetrating two circular inhomogeneities and some related problems. Applied Mathematical Modelling, 36, 1766-1775(2012)