Analytical Solutions of Dynamic Crack Models of Bridging Fiber Pull-Out in Unidirectional Composite Materials
Affiliation(s)
College of Mechanical Engineering, Liaoning Technical University, Fuxin, China. Department of Civil Engineering, Northeastern University, Shenyang, China. School of Material Science and Engineering, Shenyang Ligong University, Shenyang, China. Department of Astronautics and Mechanics, Harbin Institute of Technology, Harbin, China.
College of Mechanical Engineering, Liaoning Technical University, Fuxin, China. Department of Civil Engineering, Northeastern University, Shenyang, China. School of Material Science and Engineering, Shenyang Ligong University, Shenyang, China. Department of Astronautics and Mechanics, Harbin Institute of Technology, Harbin, China.
ABSTRACT
An elastic analysis of an internal central crack with bridging fibers parallel to the free surface in an infinite orthotropic anisotropic elastic plane was analyzed, and the crack extension should occur in the format of self-similarity. When the fiber strength is over its maximum tensile stress, the fiber breaks. By means of complex variable functions, the problem considered can be easily translated into Reimann-Hilbert mixed boundary value problem. Utilizing the built dynamic model of bridging fiber pull-out in unidirectional composite materials, analytical solutions of the displacements, stresses and stress intensity factors under the action of increasing loads Pt5/x5, Px5/t4 are obtained, respectively. After those analytical solutions were used by superposition theorem, the solutions to arbitrary complex problems were acquired.
Cite this paper
Wang, Y. , Cheng, Y. , Lü, N. and Cheng, J. (2013) Analytical Solutions of Dynamic Crack Models of Bridging Fiber Pull-Out in Unidirectional Composite Materials. Modern Mechanical Engineering, 3, 191-201. doi: 10.4236/mme.2013.34026.
Wang, Y. , Cheng, Y. , Lü, N. and Cheng, J. (2013) Analytical Solutions of Dynamic Crack Models of Bridging Fiber Pull-Out in Unidirectional Composite Materials. Modern Mechanical Engineering, 3, 191-201. doi: 10.4236/mme.2013.34026.
References
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http://dx.doi.org/10.1016/0020-7225(65)90021-2 [21] N.-C. Lü, X.-G. Li, Y.-H. Cheng and J. Cheng, “Fracture Dynamics Problem on Mode I Semi-Infinite Crack,” Archive of Applied Mechanics, Vol. 81, No. 9, 2011, pp. 1181-1193. http://dx.doi.org/10.1007/s00419-010-0476-x [22] N. C. Lü, Y. H. Cheng. X. G. Li and J. Cheng, “Dynamic Propagation Problem of Mode I Semi-Infinite Crack Subjected to Superimpose Loads,” Fatigue & Fracture of Engineering Materials & Structures. Vol. 33, No. 3, 2010, pp. 141-148. [23] N. C. Lü, Y. H. Cheng. X. G. Li and J. Cheng, “An Asymmetrical Dynamic Model for Bridging Fiber PullOut of Unidirectional Composite Materials,” Meccanica, Vol. 47, No. 5, 2012, pp. 1247-1260.
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http://dx.doi.org/10.1023/A:1007621500585 [34] X.-G. Li, Y.-H. Cheng, N.-C. Lv, G.-D. Hao and J. Cheng, “A Dynamic Asymmetrical Crack Model of Bridging Fiber Pull-Out in Unidirectional Composite Materials,” Journal of Mechanical Science and Technology, Vol. 25, No. 9, 2011, pp. 2297-2309.
http://dx.doi.org/10.1007/s12206-011-0526-5 [35] N. C. Lv, Y. H. Cheng. H. L. Si and J. Cheng, “Dynamics of Asymmetrical Crack Propagation in Composite Materials,” Theoretical and Applied Fracture Mechanics, Vol. 47, No. 3, 2007, pp. 260-273.
http://dx.doi.org/10.1016/j.tafmec.2007.01.004 [36] N. C. Lü, Y. H. Cheng and J. Cheng, “Mode I Crack Tips Propagating at Different Speeds under Differential Surface Tractions,” Theoretical and Applied Fracture Mechanics, Vol. 46, No. 3, 2006, pp. 262-275.
http://dx.doi.org/10.1016/j.tafmec.2006.09.004 [37] A. S. Kobayashi, “Dynamic Fracture Analysis by Dynamic Finite Element Method. Generation and Prediction Analyses,” In: Nonlinear and Dynarnic Fracture Mechanics, American Society of Mechanical Engineers, New York, 1979, pp. 19-36. [38] K. Ravi-Chandar and W. G. Knauss, “An Experimental Investigation into Dynamic Fracture: Part 1, Crack Initiation and Arrest,” International Journal of Fracture, Vol. 25, No. 41, 1984, pp. 247-262.
http://dx.doi.org/10.1007/BF00963460 [39] K. Ravi-Chandar and W. G. Knauss, “An Experimental Investigation into Dynamic Fracture: Part 2, Microstructural Aspects,” International Journal of Fracture, Vol. 26, No. 11, 1984, pp. 65-80.
http://dx.doi.org/10.1007/BF01152313 [40] K. Ravi-Chandar and W. G. Knauss, “An Experimental Investigation into Dynamic Fracture: Part 3, on SteadyState Crack Propagation and Crack Branching,” International Journal of Fracture, Vol. 26, No. 2, 1984, pp. 141152. http://dx.doi.org/10.1007/BF01157550 [41] L. A. Galin, “Contact Problems in Elasticity Theory,” GITTL, Moscow, 1953.
[1] D. B. Marshall, B .N. Cox and A .G. Evens, “The Mechanics of Matrix Cracking in Brittle-Matrix Fiber Composites,” Acta Metallurgical, Vol. 33, No. 11, 1985, pp. 2013-2021. http://dx.doi.org/10.1016/0001-6160(85)90124-5 [2] B. Budiansky, J. W. Hutchinson and A. G. Evens, “Matrix Fracture in Fiber-Reinforced Ceramics,” Journal of the Mechanics and Physics of Solids. Vol. 34, No. 2, 1986, pp. 167-189.
http://dx.doi.org/10.1016/0022-5096(86)90035-9 [3] M. Ji and H. Ishikawa, “Analysis of an Internal Central Crack with Bridging Fibers in a Finite Orthotropic Plate,” International Journal of Engineering Science, Vol. 35, No. 4, 1997, pp. 549-560.
http://dx.doi.org/10.1016/S0020-7225(96)00099-7 [4] D. B. Marshall and B. N. Cox. “Tensile Fracture of Brittle Matrix Composites: Influence of Fiber Strength,” Acta Metallurgical, Vol. 35, No. 11, 1987, pp. 2607-2619.
http://dx.doi.org/10.1016/0001-6160(87)90260-4 [5] Z.-M. Wang, “Mechanics and Structural Mechanics of Composite Materials,” Publisher of Machinery Industry, Beijing, 1991. [6] G.-L. Shen. “Mechanics of Composite Materials,” Tsinghua University Press, Beijing, 1996. [7] C. W. Woo and Y. H. Wang, “Analysis of an Internal Crack in a Fine Anisotropic Plate,” International Journal of Fracture, Vol. 62, No. 2, 1993, pp. 203-208. [8] J. C. Lee, “Analysis of Fiber Bridged Crack near a Free Surface in Ceramic Matrix Composites,” Engineering Fracture Mechanics, Vol. 37, No. 2, 1990, pp. 209-219.
http://dx.doi.org/10.1016/0013-7944(90)90344-G [9] W. T. Tsai and I. R. Dharani, “Non Self-Similar Fiber Fracture in Unidirectional Composites,” Engineering Fracture Mechanics, Vol. 44, No. 1, 1993, pp. 43-49.
http://dx.doi.org/10.1016/0013-7944(93)90080-C [10] W. N. Liu, “Stress Ahead of the Tip of a Finite-Width Center-Crack in Fiber-Reinforced Composite Specimens: Subjected to Non-Linearly Distributed Bridging Stresses,” International Journal of Fracture, Vol. 70, No. 1, 1994, pp. 31-35. [11] K. Liao and K. Reifsnider, “A Tensile Strength Model for Unidirectional Fiber-Reinforced Brittle Matrix Composite,” International Journal of Fracture, Vol. 106, No. 1, 2000, pp. 95-115.
http://dx.doi.org/10.1023/A:1007645817753 [12] V. Tamuzs, S. Tarasovs and U. Vilks, “Progressive Delamination and Fibre Bridging Modeling in Double Cantilever Beam Composite Specimens,” Engineering Fracture Mechanics, Vol. 68, No. 5, 2001, pp. 513-525.
http://dx.doi.org/10.1016/S0013-7944(00)00131-4 [13] A. Piva and E. Viola, “Crack Propagation in an Orthotropic Media”, Engineering Fracture Mechanics, Vol. 29, No. 5, 1988, pp. 535-547.
http://dx.doi.org/10.1016/0013-7944(88)90179-8 [14] J. De and B. Patra, “Elastodynimic Crack Problems in An Orthotrpic Medium through Complex Variable Approach,” Engineering Fracture Mechanics, Vol. 41, No. 5, 1998, pp. 895-909. [15] K. B. Broberg, “The Propagation of a Brittle Crack,” Arkve Fysik, Vol. 18, No. 2, 1960, pp. 159-192. [16] Y. W. Craggs, “The Growth of a Disk-Shaped Crack,” International Journal of Engineering Science, Vol. 4, No. 2, 1966, pp. 113-124.
http://dx.doi.org/10.1016/0020-7225(66)90019-X [17] J. G. Goree and R. S. Gross, “Analysis of a Unidirectional Composite Containing Broken Fibers and Matrix Damage,” Engineering Fracture Mechanics, Vol. 33, No. 3, 1979, pp. 555-578. [18] G. P. Cherepanov and E. F. Afanasov, “Some Dynamic Problems of the Theory of Elasticity—A Review,” International Journal of Engineering Science, Vol. 12, No. 8, 1974, pp. 665-690.
http://dx.doi.org/10.1016/0020-7225(74)90043-3 [19] G. P. Charepanov, “Mechanics of Brittle Fracture,” Nauka, Moscow, 1973. [20] C. Atkinson, “The Propagation of a Brittle Crack in Anistropic Material,” International Journal of Engineering Science, Vol. 3, No. 1, 1965, pp. 77-91.
http://dx.doi.org/10.1016/0020-7225(65)90021-2 [21] N.-C. Lü, X.-G. Li, Y.-H. Cheng and J. Cheng, “Fracture Dynamics Problem on Mode I Semi-Infinite Crack,” Archive of Applied Mechanics, Vol. 81, No. 9, 2011, pp. 1181-1193. http://dx.doi.org/10.1007/s00419-010-0476-x [22] N. C. Lü, Y. H. Cheng. X. G. Li and J. Cheng, “Dynamic Propagation Problem of Mode I Semi-Infinite Crack Subjected to Superimpose Loads,” Fatigue & Fracture of Engineering Materials & Structures. Vol. 33, No. 3, 2010, pp. 141-148. [23] N. C. Lü, Y. H. Cheng. X. G. Li and J. Cheng, “An Asymmetrical Dynamic Model for Bridging Fiber PullOut of Unidirectional Composite Materials,” Meccanica, Vol. 47, No. 5, 2012, pp. 1247-1260.
http://dx.doi.org/10.1007/s11012-011-9509-y [24] N. I. Muskhlishvili, “Singular Integral Equations,” Nauka, Moscow, 1968. [25] N. I. Muskhlishvili, “Some Fundamental Problems in the Mathematical Theory of Elasticity,” Nauka, Moscow, 1966. [26] F. D. Gakhov, “Boundary-Value Problems,” Fitzmatigiz, Moscow, 1963. [27] R. F. Hoskins, “Generalized Functions,” Ellis Horwood, Chichester, 1979. [28] X. S. Wang, “Singular Functions and Their Applications in Mechanics,” Scientific Press, Beijing, 1993. [29] G. C. Sih, “Mechanics of Fracture 4. Elastodynamics Crack Problems,” Noordhoff, Leyden, 1977. [30] R. P. Kanwal and D. L. Sharma, “Singularity Methods for Eastostatics,” Journal of Elasticity, Vol. 6, No. 4, 1976, pp. 405-418. http://dx.doi.org/10.1007/BF00040900 [31] Editorial Group of Mathematics Handbook, “Mathematical Handbook,” Advanced Education Press, Beijing, 2002. [32] Teaching Office of Mathematics of Tongji University. “Advanced Mathematics,” Advanced Education Press, Beijing, 1994. [33] K. C. Wu, “Dynamic Crack Growth in Anisotropic Material,” International Journal of Fracture, Vol. 106, No. 1, 2000, pp. 1-12.
http://dx.doi.org/10.1023/A:1007621500585 [34] X.-G. Li, Y.-H. Cheng, N.-C. Lv, G.-D. Hao and J. Cheng, “A Dynamic Asymmetrical Crack Model of Bridging Fiber Pull-Out in Unidirectional Composite Materials,” Journal of Mechanical Science and Technology, Vol. 25, No. 9, 2011, pp. 2297-2309.
http://dx.doi.org/10.1007/s12206-011-0526-5 [35] N. C. Lv, Y. H. Cheng. H. L. Si and J. Cheng, “Dynamics of Asymmetrical Crack Propagation in Composite Materials,” Theoretical and Applied Fracture Mechanics, Vol. 47, No. 3, 2007, pp. 260-273.
http://dx.doi.org/10.1016/j.tafmec.2007.01.004 [36] N. C. Lü, Y. H. Cheng and J. Cheng, “Mode I Crack Tips Propagating at Different Speeds under Differential Surface Tractions,” Theoretical and Applied Fracture Mechanics, Vol. 46, No. 3, 2006, pp. 262-275.
http://dx.doi.org/10.1016/j.tafmec.2006.09.004 [37] A. S. Kobayashi, “Dynamic Fracture Analysis by Dynamic Finite Element Method. Generation and Prediction Analyses,” In: Nonlinear and Dynarnic Fracture Mechanics, American Society of Mechanical Engineers, New York, 1979, pp. 19-36. [38] K. Ravi-Chandar and W. G. Knauss, “An Experimental Investigation into Dynamic Fracture: Part 1, Crack Initiation and Arrest,” International Journal of Fracture, Vol. 25, No. 41, 1984, pp. 247-262.
http://dx.doi.org/10.1007/BF00963460 [39] K. Ravi-Chandar and W. G. Knauss, “An Experimental Investigation into Dynamic Fracture: Part 2, Microstructural Aspects,” International Journal of Fracture, Vol. 26, No. 11, 1984, pp. 65-80.
http://dx.doi.org/10.1007/BF01152313 [40] K. Ravi-Chandar and W. G. Knauss, “An Experimental Investigation into Dynamic Fracture: Part 3, on SteadyState Crack Propagation and Crack Branching,” International Journal of Fracture, Vol. 26, No. 2, 1984, pp. 141152. http://dx.doi.org/10.1007/BF01157550 [41] L. A. Galin, “Contact Problems in Elasticity Theory,” GITTL, Moscow, 1953.