Singularity Analysis for Notches in Orthotropic Composite Plates With the Interpolating Matrix Method

GE Ren-yu; CHENG Chang-zheng; YANG Zhi-yong; NIU Zhong-rong

doi:10.3879/j.issn.1000-0887.2014.04.011

Volume 35 Issue 4

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GE Ren-yu, CHENG Chang-zheng, YANG Zhi-yong, NIU Zhong-rong. Singularity Analysis for Notches in Orthotropic Composite Plates With the Interpolating Matrix Method[J]. Applied Mathematics and Mechanics, 2014, 35(4): 459-470. doi: 10.3879/j.issn.1000-0887.2014.04.011

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Singularity Analysis for Notches in Orthotropic Composite Plates With the Interpolating Matrix Method

doi: 10.3879/j.issn.1000-0887.2014.04.011

¹College of Civil Engineering and Architecture, Anhui Polytechnic University, Wuhu, Anhui 241000, P.R.China;²School of Civil Engineering, Hefei University of Technology, Hefei 230009, P.R.China

Funds: The National Natural Science Foundation of China(11272111；11372049)

Received Date: 2013-08-08
Rev Recd Date: 2013-09-24
Publish Date: 2014-04-15

Abstract

Abstract

Based on asymptotic expansion of generalized displacement field at the V-notch tip, a new method for analyzing the stress singularity exponents of the notches in orthotropic composite plates was proposed. Through introduction of the typical terms in asymptotic expansion of the generalized displacement functions into the basic elastic equations of the plate, the eigenvalue problem of a set of nonlinear ordinary differential equations(ODEs) about the stress singularity exponents of the notch was obtained, then the nonlinear eigenvalue problem was transformed into a linear one by means of variable substitution, and the interpolating matrix method was employed to solve the problem to determine the stress singularity exponents and associated characteristic functions at the notch tip in the orthotropic bi-material plate. With the present method, both the stress singularity exponents and the associated characteristic angle functions can be acquired simultaneously, and the stress singularity exponents can be easily distinguished between plane and anti-plane singularities according to the corresponding characteristic angle functions. Validity of the present method is confirmed in comparison with the existing results through numerical calculation.
- orthotropic plate,
- V-notch,
- asymptotic expansion,
- stress singularity,
- interpolating matrix method

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References(21)

References

[1]	Li Y T, Song M. Method to calculate stress intensity factor of V-notch in bi-materials[J].Acta Mechanica Solida Sinica,2008,21(4): 337-346.
[2]	Williams M L. Surface stress singularities resulting from various boundary conditions in angular corners of plates under bending[C]// Proceedings of the First US National Congress of Applied Mechanics. Chicago, 1951: 325-329.
[3]	Sih G C, Paris P C, Erdogan F. Crack-tip, stress-intensity factors for plane extension and plate bending problems[J].Journal of Applied Mechanics,1962,29(2): 306-312.
[4]	Knowles J K, Wang N M. On the bending of an elastic plate containing a crack[J].Journal of Mathematics and Physics,1960,39(5): 223-236.
[5]	柳春图. 承受弯曲的板在裂纹顶端附近的应力和变形[J]. 固体力学学报, 1983,9(3): 441-448.(LIU Chun-tu. Stresses and deformations near the crack tip for bending plate[J].Acta Mechanica Solida Sinica,1983,9(3): 441-448.(in Chinese))
[6]	Delate F, Erdogan F. The effect of transverse shear in a cracked plate under skew-symmetric loading[J].Journal of Applied Mechanics,1979,46(3): 618-624.
[7]	Murthy M V V, Raju K N, Viswanath S. On the bending stress distribution at the tip of a stationary crack from Reissner’s theory[J].International Journal of Fracture,1981,17(6): 537-552.
[8]	Boduroglu H, Erdogan F. Internal and edge cracks in a plate of finite width under bending[J].Journal of Applied Mechanics,1983,50(3): 621-629.
[9]	Sosa H A, Eischen J W. Computation of stress intensity factors for plate bending via a path-independent integral[J].Engineering Fracture Mechanics,1986,25(4): 451-462.
[10]	Hui C Y, Zehnder A T. A theory for the fracture of thin plates subjected to bending and twisting moments[J].International Journal of Fracture,1993,61(3): 211-229.
[11]	Young M J, Sun C T. Cracked plates subjected to out-of-plane tearing loads[J].International Journal of Fracture,1993,60(1): 1-18.
[12]	Su R K L, Leung A Y T. Mixed mode crack in Reissner plates[J].International Journal of Fracture,2001,107(3): 235-257.
[13]	Sih G C, Rice J R. The bending of plates of dissimilar materials with cracks[J].Journal of Applied Mechanics,1964,31(3): 477-482.
[14]	Sih G C. Flexural problems of cracks in mixed media[C]// Proceedings of the First International Conference on Fracture. Sendai, 1965,1: 391-409.
[15]	Ang D D, Williams M L. Combined stresses in an orthotropic plate having a finite crack[J].Journal of Applied Mechanics,1961,28(3): 372-378.
[16]	Sih G C, Chen E P.Cracks in Composite Materials [M].Mechanics of Fracture .6. Hague, Boston, London: Martinus Nijhoff Publishers,1981: 76-87.
[17]	Yuan F G, Yang S. Asymptotic crack-tip fields in an anisotropic plate subjected to bending, twisting moments and transverse shear loads[J].Composites Science and Technology,2000,60(12/13): 2489-2502.
[18]	Li J. Singularity analysis of near-tip fields for notches formed from several anisotropic plates under bending[J].International Journal of Solids and Structures,2002,39(23): 5767-5785.
[19]	牛忠荣. 轴对称变厚度扁球壳的非线性弯曲问题[J]. 应用数学和力学, 1993,14(11): 971-978.(NIU Zhong-rong. Nonlinear bending of the shallow spherical shells with variable thickness under axisymmetrical loads[J].Applied Mathematics and Mechanics,1993,14(11): 971-978.(in Chinese))
[20]	NIU Zhong-rong, GE Da-li, CHENG Chang-zheng, YE Jian-qiao, Recho N. Evaluation of the stress singularities of plane V-notches in bonded dissimilar materials[J].Applied Mathematical Modelling,2009,33(3): 1776-1792.
[21]	CHENG Chang-zheng, NIU Zong-rong, Recho N. Effect of non-singular stress on the brittle fracture of V-notched structure[J].International Journal of Fracture,2012,174(2): 127-138.