Analytical Solutions of a Symmetrical Dynamic Crack Model of Bridging Fibers in Unidirectional Composites

Affiliation(s)

School of Material Science and Engineering, Shenyang Ligong University, Shenyang, China.

Department of Civil Engineering, Northeastern University, Shenyang, China.

College of Mechanical Engineering, Liaoning Technical University, Fuxin, China.

Department of Astronautics and Mechanics, Harbin Institute of Technology, Harbin, China.

School of Material Science and Engineering, Shenyang Ligong University, Shenyang, China.

Department of Civil Engineering, Northeastern University, Shenyang, China.

College of Mechanical Engineering, Liaoning Technical University, Fuxin, China.

Department of Astronautics and Mechanics, Harbin Institute of Technology, Harbin, China.

ABSTRACT

When composite materials occur crack, their fibrous locations will
produce bridging fibers. A symmetrical dynamic crack model of bridging fibers in
unidirectional composite materials are not probed as deeply by virtue of the complexity,
cockamamie and difficulty in mathematical operations. In the light of the
theory of complex variable functions, the problems discussed can be facilely translated
into Remann-Hilbert problems. Analytical solutions of the displacements,
stresses and stress intensity factors under the action of variable loads *Pt*^{6}*/x*^{6}, *Px*^{6}*/t*^{5} are attained,
respectively. After those analytical solutions were used by superposition
theorem, the solutions of arbitrary complex problems were acquired.

Cite this paper

N. Lü, Y. Cheng, Y. Wang and J. Cheng, "Analytical Solutions of a Symmetrical Dynamic Crack Model of Bridging Fibers in Unidirectional Composites,"*World Journal of Mechanics*, Vol. 3 No. 5, 2013, pp. 22-32. doi: 10.4236/wjm.2013.35A004.

N. Lü, Y. Cheng, Y. Wang and J. Cheng, "Analytical Solutions of a Symmetrical Dynamic Crack Model of Bridging Fibers in Unidirectional Composites,"

References

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[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. doi: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.

[4] D. B. Marshall and B. N. Cox, “Tensile Fracture of Brittle Matrix Composites: Influence of Fiber Strength,” Acta Metallurgica, Vol. 35, No. 11, 1987, pp. 2607-2619. doi:10.1016/0001-6160(87)90260-4

[5] Z.-M. Wang, “Mechanics and Structural Mechanics of Composite Materials,” Publisher of Machinery Industry, Beijing, 1991, (in Chinese).

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[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. doi:10.1016/0955-7997(93)90038-M

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[15] K. B. Broberg, “The Propagation of a Brittle Crack,” Arkiv for Fysik, Vol. 18, No. 2, 1960, pp. 159-192. doi:10.1016/0022-5096(60)90006-5

[16] Y. W. Craggs, “The Growth of a Disk-Shaped Crack,” International Journal of Engineering Science, Vol. 4, No. 2, 1966, pp. 113-124. doi:10.1016/0013-7944(80)90086-7

[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, 2001, pp. 55-578. doi:10.1016/0020-7225(66)90019-X

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[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 doi:10.1016/j.amc.2011.04.028

[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 Pull-Out of Unidirectional Composite Materials,” Meccanica, Vol. 47, No. 5, 2012, pp. 1247-1260. doi:10.1016/j.tafmec.2012.05.007

[24] N. I. Muskhlishvili, “Singular Integral Equations,” Nauka, Moscow City, 1968.

[25] N. I. Muskhlishvili, “Some Fundamental Problems in the Mathematical Theory of Elasticity,” Nauka, Moscow City, 1966.

[26] F. D. Gakhov, “Boundary-Value Problems,” Fitzmatigiz, Moscow City, 1963.

[27] R. F. Hoskins, “Generalized Functions,” Horwood, Ellis, 1979.

[28] X. S Wang, “Singular Functions and Their Applications in Mechanics,” Scientific Press, Beijing, 1993 (in Chinese).

[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.

[31] Editorial Group of Mathematics Handbook, “Mathematical Handbook,” Advanced Education Press, Beijing, 2002 (in Chinese).

[32] Teaching Office of Mathematics of Tongji University, “Advanced Mathematics,” Advanced Education Press, Beijing, 1994 (in Chinese).

[33] K. C. Wu, “Dynamic Crack Growth in Anisotropic Material,” International Journal of Fracture, Vol. 106, No. 1, 2000, pp. 1-12. doi:10.1016/S0022-5096(00)00012-0

[34] X.-G. Li, Y.-H. Cheng, N.-C. Lü, G.-D. Hao and J. Cheng, “A Dynamic Asymmetrical Crack Model of Bridging Fiber Pull-Out in Uniderectional Composite Materials,” Journal of Mechanical Science and Technoogy, Vol. 25, No. 9, 2011, pp. 2297-2309. doi:10.1016/j.ymssp.2011.07.013

[35] N. C, Lü, 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. doi: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.

[37] A. S. Kobayashi, “Dynamic Fracture Analysis by Dynamic Finite Element Method. Generation and Prediction Analyses,” In: Nonlinear and Dynamic Fracture Mechanics, New York Publisher, New York, 1979, pp. 19-36.

[38] J. F. Kalthof, J. Beinert and S. Winkler, “Measurements of Dynamic Stress Intensity Factors for Fast Running and Arresting Cracks in Double-Cantilever-Beam Specimens,” In: Fast Fracture and Crack Arrest, PA Publisher, Philadelphia, 1977, pp. 161-176.

[39] 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,

[40] K. Ravi-Chandar and W. G. Knauss, “An Experimental Investigation into Dynamic Fracture: Part 2, Microstructural Aspects,” International Journal of Fracture, Vol. 18, No. 7, 1984, pp. 735-738. doi:10.1016/0036-9748(84)90330-2

[41] K. Ravi-Chandar and W. G. Knauss, “An Experimental Investigation into Dynamic Fracture: Part 3, on Steady-State Crack Propagation and Crack Branching,” International Journal of Fracture, Vol. 26, No. 2, 1984, pp. 141-152.

[42] L. B. Freund, “Crack Propagation in an Elastic Solid Subjected to General Loading—I. Constant Rate of Extension,” Journal of the Mechanics and Physics of Solids, Vol. 20, No. 3, 1972, pp. 129-140. doi:10.1016/0022-5096(72)90006-3

[43] N. I. Sneddon, “Fourier Transform,” McGraw-Hill, New York, 1951.

[44] N. I. Muskhelishvili, “Some Basic Problems of the Mathematical Theory of Elasticity,” Noordoff, Groningen, 1953.

[45] L. A. Galin, “Contact Problems in Elasticity Theory,” GITTL, Moscow City, 1953.

[46] N.-C. Lü, Y.-H. Cheng, X. G. Li and J. Cheng, “Fracture Dynamics Problem on Mode I Semi-Infinite Crack for Anisotropic Orthotropic Body,” Nonlinear Dynamics, Vol. 67, No. 4, 2012, pp. 2381-2396.

[47] N.-C. Lü, Y.-H. Cheng, Y.-T. Wang and J. Cheng, “Dynamic Extension Problems Concerning Asymmetrical Mode III Crack,” Applied Mathematical Modelling, Vol. 35, No. 5, 2011, pp. 2499-2507. doi:10.1016/j.apm.2010.11.058

[48] N.-C. Lü, Y.-H. Cheng, Y.-T. Wang and J. Cheng, “Fracture Dynamics Problems of Orthotropic Solids under Anti-Plane Shear Loading,” Nonlinear Dynamics, Vol. 63, No. 4, 2011, pp. 793-806.

[49] N.-C. Lü, Y.-H. Cheng, X. G. Li and J. Cheng, “Asymmetrical Dynamic Propagation Problems Concerning Mode Ⅲ Interface Crack,” Composite Interfaces, Vol. 17, No. 1, 2010, pp. 37-48.

[50] N.-C. Lü, J. Cheng and Y.-H. Cheng, “Self-Similar Solutions of Fracture Dynamics Problems on Axially Symmetry,” Applied Mathematical and Mechanics, Vol. 22, No. 12, 2001, pp. 1429-1435.

[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. doi: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. doi: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.

[4] D. B. Marshall and B. N. Cox, “Tensile Fracture of Brittle Matrix Composites: Influence of Fiber Strength,” Acta Metallurgica, Vol. 35, No. 11, 1987, pp. 2607-2619. doi:10.1016/0001-6160(87)90260-4

[5] Z.-M. Wang, “Mechanics and Structural Mechanics of Composite Materials,” Publisher of Machinery Industry, Beijing, 1991, (in Chinese).

[6] G.-L. Shen, “Mechanics of Composite Materials,” Tsinghua University Press, Beijing, 1996, (in Chinese).

[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. doi:10.1016/0955-7997(93)90038-M

[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. doi: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. doi: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. doi:10.1016/0013-7944(94)90092-2

[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. doi:10.1016/S1359-835X(02)00143-4

[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. doi: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. doi: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,” Arkiv for Fysik, Vol. 18, No. 2, 1960, pp. 159-192. doi:10.1016/0022-5096(60)90006-5

[16] Y. W. Craggs, “The Growth of a Disk-Shaped Crack,” International Journal of Engineering Science, Vol. 4, No. 2, 1966, pp. 113-124. doi:10.1016/0013-7944(80)90086-7

[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, 2001, pp. 55-578. doi:10.1016/0020-7225(66)90019-X

[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, 665-690. doi:10.1016/0020-7225(74)90043-3

[19] G. P. Charepanov, “Mechanics of Brittle Fracture,” Nauka, Moscow City, 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. doi:10.1016/0020-7225(65)90018-2

[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 doi:10.1016/j.amc.2011.04.028

[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 Pull-Out of Unidirectional Composite Materials,” Meccanica, Vol. 47, No. 5, 2012, pp. 1247-1260. doi:10.1016/j.tafmec.2012.05.007

[24] N. I. Muskhlishvili, “Singular Integral Equations,” Nauka, Moscow City, 1968.

[25] N. I. Muskhlishvili, “Some Fundamental Problems in the Mathematical Theory of Elasticity,” Nauka, Moscow City, 1966.

[26] F. D. Gakhov, “Boundary-Value Problems,” Fitzmatigiz, Moscow City, 1963.

[27] R. F. Hoskins, “Generalized Functions,” Horwood, Ellis, 1979.

[28] X. S Wang, “Singular Functions and Their Applications in Mechanics,” Scientific Press, Beijing, 1993 (in Chinese).

[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.

[31] Editorial Group of Mathematics Handbook, “Mathematical Handbook,” Advanced Education Press, Beijing, 2002 (in Chinese).

[32] Teaching Office of Mathematics of Tongji University, “Advanced Mathematics,” Advanced Education Press, Beijing, 1994 (in Chinese).

[33] K. C. Wu, “Dynamic Crack Growth in Anisotropic Material,” International Journal of Fracture, Vol. 106, No. 1, 2000, pp. 1-12. doi:10.1016/S0022-5096(00)00012-0

[34] X.-G. Li, Y.-H. Cheng, N.-C. Lü, G.-D. Hao and J. Cheng, “A Dynamic Asymmetrical Crack Model of Bridging Fiber Pull-Out in Uniderectional Composite Materials,” Journal of Mechanical Science and Technoogy, Vol. 25, No. 9, 2011, pp. 2297-2309. doi:10.1016/j.ymssp.2011.07.013

[35] N. C, Lü, 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. doi: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.

[37] A. S. Kobayashi, “Dynamic Fracture Analysis by Dynamic Finite Element Method. Generation and Prediction Analyses,” In: Nonlinear and Dynamic Fracture Mechanics, New York Publisher, New York, 1979, pp. 19-36.

[38] J. F. Kalthof, J. Beinert and S. Winkler, “Measurements of Dynamic Stress Intensity Factors for Fast Running and Arresting Cracks in Double-Cantilever-Beam Specimens,” In: Fast Fracture and Crack Arrest, PA Publisher, Philadelphia, 1977, pp. 161-176.

[39] 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,

[40] K. Ravi-Chandar and W. G. Knauss, “An Experimental Investigation into Dynamic Fracture: Part 2, Microstructural Aspects,” International Journal of Fracture, Vol. 18, No. 7, 1984, pp. 735-738. doi:10.1016/0036-9748(84)90330-2

[41] K. Ravi-Chandar and W. G. Knauss, “An Experimental Investigation into Dynamic Fracture: Part 3, on Steady-State Crack Propagation and Crack Branching,” International Journal of Fracture, Vol. 26, No. 2, 1984, pp. 141-152.

[42] L. B. Freund, “Crack Propagation in an Elastic Solid Subjected to General Loading—I. Constant Rate of Extension,” Journal of the Mechanics and Physics of Solids, Vol. 20, No. 3, 1972, pp. 129-140. doi:10.1016/0022-5096(72)90006-3

[43] N. I. Sneddon, “Fourier Transform,” McGraw-Hill, New York, 1951.

[44] N. I. Muskhelishvili, “Some Basic Problems of the Mathematical Theory of Elasticity,” Noordoff, Groningen, 1953.

[45] L. A. Galin, “Contact Problems in Elasticity Theory,” GITTL, Moscow City, 1953.

[46] N.-C. Lü, Y.-H. Cheng, X. G. Li and J. Cheng, “Fracture Dynamics Problem on Mode I Semi-Infinite Crack for Anisotropic Orthotropic Body,” Nonlinear Dynamics, Vol. 67, No. 4, 2012, pp. 2381-2396.

[47] N.-C. Lü, Y.-H. Cheng, Y.-T. Wang and J. Cheng, “Dynamic Extension Problems Concerning Asymmetrical Mode III Crack,” Applied Mathematical Modelling, Vol. 35, No. 5, 2011, pp. 2499-2507. doi:10.1016/j.apm.2010.11.058

[48] N.-C. Lü, Y.-H. Cheng, Y.-T. Wang and J. Cheng, “Fracture Dynamics Problems of Orthotropic Solids under Anti-Plane Shear Loading,” Nonlinear Dynamics, Vol. 63, No. 4, 2011, pp. 793-806.

[49] N.-C. Lü, Y.-H. Cheng, X. G. Li and J. Cheng, “Asymmetrical Dynamic Propagation Problems Concerning Mode Ⅲ Interface Crack,” Composite Interfaces, Vol. 17, No. 1, 2010, pp. 37-48.

[50] N.-C. Lü, J. Cheng and Y.-H. Cheng, “Self-Similar Solutions of Fracture Dynamics Problems on Axially Symmetry,” Applied Mathematical and Mechanics, Vol. 22, No. 12, 2001, pp. 1429-1435.