Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

The nonlocal theory solution for two collinear cracks in functionally graded materials subjected to the harmonic elastic anti-plane shear waves

  • Zhou, Zhen-Gong (Center for fomposite Materials, Harbin Institute of Technology) ;
  • Wang, Biao (School of Physics and Engineering, Sun Yat-Sen University)
  • Received : 2005.10.26
  • Accepted : 2006.12.15
  • Published : 2006.05.10
⟨ Previous Next ⟩

Abstract

In this paper, the scattering of harmonic elastic anti-plane shear waves by two collinear cracks in functionally graded materials is investigated by means of nonlocal theory. The traditional concepts of the non-local theory are extended to solve the fracture problem of functionally graded materials. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress field near the crack tips. To make the analysis tractable, it is assumed that the shear modulus and the material density vary exponentially with coordinate vertical to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present at crack tips.

Keywords

References

  1. Amemiya, A. and Taguchi, T. (1969), Numerical Analysis and Fortran, Maruzen, Tokyo
  2. Bao, G. and Cai, H. (1997), ' Delamination cracking in functionally graded coating/metal substrate systems ', ACTA Materialia, 45, 1055-1066 https://doi.org/10.1016/S1359-6454(96)00232-7
  3. Chen, Y.-F. (1990), ' Interface crack in nonhomogeneous bonded materials of finite thickness ', Ph.D. Dissertation, Lehigh University
  4. Delale, F. and Erdogan, F. (1988), ' On the mechanical modeling of the interfacial region in bonded half-planes ', J. Appl. Mech., ASME, 55, 317-324 https://doi.org/10.1115/1.3173677
  5. Edelen, D.G.B. (1976), Nonlocal Field Theory. In: A.C. Eringen (ed.), Continuum Physics. 4. New York, Academic Press, 75-204
  6. Erdelyi, A. (ed), (1954), Tables of Integral Transforms, McGraw-Hill, New York. 1
  7. Erdogan, F. and Wu, B.-H. (1997), ' The surface crack problem for a plate with functionally graded properties ', J. Appl. Mech., 64, 449-456 https://doi.org/10.1115/1.2788914
  8. Erdogan, F. and Wu, H.-B. (1996), ' Crack problems in FGM layer under thermal stress ', J .Thermal Stress, 19, 237-265 https://doi.org/10.1080/01495739608946172
  9. Eringen, A.C. (1972), ' Linear theory of nonlocal elasticity and dispersion of plane waves ' , Int. J .Eng. Sci., 10, 425-435 https://doi.org/10.1016/0020-7225(72)90050-X
  10. Eringen, A.C. (1976), Nonlocal Polar Field Theory. In: A.C. Eringen (ed.), Continuum Physics. 4. Academic Press, New York, 205-267
  11. Eringen, AC. (1977),' Continuum mechanics at the atomic scale ', Crystal Lattice Defects, 7, 109-130
  12. Eringen, A.C. (1978), ' Linear crack subject to shear ', Int. J. Fracture, 14, 367-379 https://doi.org/10.1007/BF00015990
  13. Eringen, A.C. (1979), 'Linear crack subject to anti-plane shear', Eng. Fracture Mech., 12,211-219 https://doi.org/10.1016/0013-7944(79)90114-0
  14. Eringen, A.C. (1983), ' Interaction of a dislocation with a crack ', J Appl. Phys., 54, 6811-6817 https://doi.org/10.1063/1.332001
  15. Eringen, A.C. and Kim, B.S. (1974), ' On the problem of crack in nonlocal elasticity ', In: P. Thoft-Christensen (ed.): Continuum Mechanics Aspects of Geodynamics and Rock Fracture Mechanics. Dordrecht, Holland: ReideI 81-113
  16. Eringen, A.C. and Kim, B.S. (1977), ' Relation betwcen nonlocal elasticity and lattice dynamics ', Crystal Lattice Defects, 7, 51-57
  17. Eringen, A.C., Speziale,C.G. and Kim, B.S. (1977), ' Crack tip problem in nonlocal elasticity', J. Mech. Phy. Solids, 25(4), 339-346 https://doi.org/10.1016/0022-5096(77)90002-3
  18. Forest, B. (1998), ' Modelling slip, kind and shear banding in classical and generalized single crystal plasticity ', ACTA Material, 46, 3265-3281 https://doi.org/10.1016/S1359-6454(98)00012-3
  19. Gradshteyn, I.S. and Ryzhik, I.M. (1980), Table of Integrals, Series and Products, Academic Press, New York. 480
  20. Green, A.E. and Rivilin, R.S. (1965), ' Multipolar continuum mechanics: Functional theory. I ', Proc. of The Royal Society of London A, 284, 303-315
  21. Itou, S. (1978), ' Three dimensional waves propagation in a cracked elastic solid ', Trans. J. Appl. Mech., ASME, 45, 807-811 https://doi.org/10.1115/1.3424423
  22. Itou, S. (2001), ' Stress intensity factors around a crack in a non-homogcneous interface layer between two dissimilar elastic half-planes ', Int. J. Fracture, 110, 123-135 https://doi.org/10.1023/A:1010851732746
  23. Jin, Z.-H. and Batra, R.-C. (1996), 'Interface cracking between functional1y graded coating and a substrate under antiplane shear ', Int. J. Eng. Sci., 34, 1705-1716 https://doi.org/10.1016/S0020-7225(96)00055-9
  24. Morse, P.M. and Feshbach, H. (1958), Methods of Theoretical Physics, McGraw-Hill, New York. 926
  25. Nowinski, J.L. (1984a), ' On nonlocal aspects of the propagation of Love waves ', Int. J. Eng. Sci., 22,383-392 https://doi.org/10.1016/0020-7225(84)90073-9
  26. Nowinski, J.L. (1984b), ' On non-local theory of wave propagation in elastic plates ' , J. Appl. Mech., ASME, 51, 608-613 https://doi.org/10.1115/1.3167681
  27. Ozturk, M. and Erdogan, F. (1996),' Axisymmetric crack problem in bonded materials with a graded interfacial region ', Int. J. Solids Struct., 33, 193-219 https://doi.org/10.1016/0020-7683(95)00034-8
  28. Pan, K.L. (1992), ' The image force on a dislocation near an elliptic hole in nonlocal elasticity ', Archive of Applied Mechacics, 62, 557-564
  29. Pan, K.L. (1994), ' The image force theorem for a screw dislocation near a crack in nonlocal elasticity ', J .Appl.Phy., 27, 344-346 https://doi.org/10.1063/1.1722373
  30. Pan, K.L. (1995),' Interaction of a dislocation with a surface crack in nonlocal elasticity ', Int. J. Fracture, 69, 307-318 https://doi.org/10.1007/BF00037381
  31. Pan, K.L. (1996), ' Interaction of a dislocation and an inclusion in nonlocal elasticity ', Int. J .Eng. Sci., 34, 1657-1688
  32. Pan, K.L. and Fang, J. (1993), ' Nonlocal interaction of dislocation with a crack ', Archive of Appl. Mech., 64, 44-51 https://doi.org/10.1007/BF00792343
  33. Pan, K.L. and Fang, J. (1996), ' Interaction energy of dislocation and point defect in bcc iron ', Radiation Effect Defects, 139,147-154 https://doi.org/10.1080/10420159608211541
  34. Pan, K.L. and Takeda, N. (1997), ' Stress distribution on bi-material interface in nonlocal elasticity ', Proc. of the 39th JSASS/JSME Structure Conf.Osaka, Japan 181-184
  35. Pan, K.L. and Takeda, N. (1998), ' Nonlocal stress field of interface dislocations ', Archive of Applied Mechacics, 68, 179-184 https://doi.org/10.1007/s004190050155
  36. Pan, K.L. and Xing, J. (1997), ' On presentation of the boundary condition in nonlocal elasticity ', Mechanics Research Communications, 24, 325-330 https://doi.org/10.1016/S0093-6413(97)00030-X
  37. Rice, J.R. (1968), ' A path independent integral and the approximate analysis of strain concentrations by notches and cracks ', J. Appl. Mech., ASME, 35, 379-386 https://doi.org/10.1115/1.3601206
  38. Shbeeb, N.-I. and Binienda, W-K. (1999), ' Analysis of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness ', Eng. Fracture Mech., 64, 693-720 https://doi.org/10.1016/S0013-7944(99)00099-5
  39. Srivastava, K.N., Palaiya, K.N. and Karaulia, D.S. (1983),' Interaction of shear waves with two coplanar Griffith cracks situated in an infinitely long elastic strip ', Int. J. Fracture, 23, 3-14 https://doi.org/10.1007/BF00020153
  40. Sun, Y.G and Zhou, Z.G (2004), ' Stress field near the crack tip in nonlocal anisotropic elasticity ', European J. Mech. A/ Solids, 23(2), 259-269 https://doi.org/10.1016/j.euromechsol.2004.01.006
  41. Xia, Z.C. and Hutchinson, J.W. (1996), ' Crack tip fields in strain gradient plasticity ', J. Mech. Phy. Solids, 44, 1621-1648 https://doi.org/10.1016/0022-5096(96)00035-X
  42. Van, W.F. (1967), ' Axisymmetric slipless indentation of an infinite elastic cylinder ', SIAM J. Appl. Math., 15, 219-227 https://doi.org/10.1137/0115018
  43. Zhou, Z.G and Shen, Y.P. (1999),' Investigation of the scattering of harmonic shear waves by two collinear cracks using the nonlocal theory ', ACTA Mech., 135, 169-179 https://doi.org/10.1007/BF01305750
  44. Zhou, Z.G and Wang, B. (2002),' Investigation of the scattering of harmonic elastic waves by two collinear symmetric cracks using non-local theory ', J. Eng. Math., 44(1), 41-56 https://doi.org/10.1023/A:1020581102605
  45. Zhou, Z.G and Wang, B. (2003), ' Investigation of anti-plane shear behavior of two collinear impermeable cracks in the piezoelectric materials by using the nonlocal theory ' , Int. J. Solids Struct., 39, 1731-1742
  46. Zhou, Z.G, Bai, Y.Y. and Zhang, X.W. (1999),' Two collinear Griffith cracks subjected to uniform tension in infinitely long strip ', Int. J .Solids Struct., 36, 5597-5609 https://doi.org/10.1016/S0020-7683(98)00250-9
  47. Zhou, Z.G, Han, J.C. and Du, S.Y. (1999), ' Investigation of a Griffith crack subject to anti-plane shear by using the nonlocal theory ', Int. J. Solids Struct., 36, 3891-3901 https://doi.org/10.1016/S0020-7683(98)00179-6
  48. Zhou, Z.G, Wang, B. and Du, S.Y. (2003),' Investigation of anti-plane shear behavior of two collinear permeable cracks in a piezoelectric material by using the nonlocal theory ' J .Appl. Mech., ASME, 69, 388-390

Cited by

  1. Two collinear Mode-I cracks in piezoelectric/piezomagnetic materials vol.29, pp.1, 2008, https://doi.org/10.12989/sem.2008.29.1.055
  2. The nonlocal theory solution of a Mode-I crack in functionally graded materials vol.52, pp.4, 2009, https://doi.org/10.1007/s11431-008-0152-3
  3. Investigation of the behavior of a mode-I crack in functionally graded materials by non-local theory vol.45, pp.2-8, 2007, https://doi.org/10.1016/j.ijengsci.2007.03.018