DOI QR코드

DOI QR Code

Longitudinal cracks in non-linear elastic beams exhibiting material inhomogeneity

  • Rizov, Victor I. (Department of Technical Mechanics, University of Architecture, Civil Engineering and Geodesy)
  • Received : 2018.09.03
  • Accepted : 2019.03.27
  • Published : 2019.07.25

Abstract

Longitudinal fracture behavior of non-linear elastic beam configurations is studied in terms of the strain energy release rate. It is assumed that the beams exhibit continuous material inhomogeneity along the width as well as along the height of the crosssection. The Ramberg-Osgood stress-strain relation is used for describing the non-linear mechanical behavior of the inhomogeneous material. A solution to strain energy release rate is derived that holds for inhomogeneous beams of arbitrary cross-section under combination of axial force and bending moments. Besides, the solution may be applied at any law of continuous distribution of the modulus of elasticity in the beam cross-section. The longitudinal crack may be located arbitrary along the beam height. The solution is used to investigate a longitudinal crack in a beam configuration of rectangular cross-section under four-point bending. The crack is located symmetrically with respect to the beam mid-span. It is assumed that the modulus of elasticity varies continuously according a cosine law in the beam cross-section. The longitudinal fracture behavior of the inhomogeneous beam is studied also by applying the J-integral approach for verification of the non-linear solution to the strain energy release rate derived in the present paper. Effects of material inhomogeneity, crack location along the beam height and non-linear mechanical behavior of the material on the longitudinal fracture behavior are evaluated. Thus, the solution derived in the present paper can be used in engineering design of inhomogeneous non-linear elastic structural members to assess the influence of various material and geometrical parameters on longitudinal fracture.

References

  1. Arshad, S.H., Naeem, M.N. and Sultana, N. (2007), "Frequency analysis of functionally graded material cylindrical shells with various volume fraction lawsˮ, Proceedings of the Institution of Mechanical Engineers, Part C J. Mech. Eng. Sci., 221(12), 1483-1495. https://doi.org/10.1243/09544062JMES738. https://doi.org/10.1243/09544062JMES738
  2. Bohidar, S.K., Sharma, R. and Mishra, P.R. (2014), "Functionally graded materials: A critical reviewˮ, J. Res., 1(7), 289-301.
  3. Broek, D. (1986), Elementary Engineering Fracture Mechanics, Springer, Germany.
  4. Carpinteri, A., Paggi, M. and Pugno, N. (2006), "An analytical approach for fracture and fatigue in functionally graded materialsˮ, J. Fracture, 141(2), 535-547. https://doi.org/10.1007/s10704-006-9012-y. https://doi.org/10.1007/s10704-006-9012-y
  5. Wei, D., Liu, Y. and Xiang, Z. (2012), "An analytical method for free vibration analysis of functionally graded beams with edge cracksˮ, J. Sound Vib., 331(2), 1686-1700. https://doi.org/10.1016/j.jsv.2011.11.020. https://doi.org/10.1016/j.jsv.2011.11.020
  6. Dowling, N. (2007), Mechanical Behavior of Materials, Pearson, USA.
  7. Gasik, M.M. (2010), "Functionally graded materials: Bulk processing techniquesˮ, J. Mater. Product. Technol., 39(1-2), 20-29. https://doi.org/10.1504/IJMPT.2010.034257. https://doi.org/10.1504/IJMPT.2010.034257
  8. Hirai, T. and Chen, L. (1999), "Recent and prospective development of functionally graded materials in Japanˮ, Mater Sci. Forum, 308-311(4), 509-514. https://doi.org/10.4028/www.scientific.net/MSF.308-311.509.
  9. Koizumi, M. (1993), "The concept of FGM Ceramic Transˮ, Functionally Gradient Materials, 34(1), 3-10.
  10. Markworth, A.J., Ramesh, K.S. and Parks, Jr.W.P. (1995), "Review: Modeling studies applied to functionally graded materialsˮ, J. Mater. Sci., 30(3), 2183-2193. https://doi.org/10.1007/BF01184560. https://doi.org/10.1007/BF01184560
  11. Mortensen, A. and Suresh, S. (1995), "Functionally graded metals and metal-ceramic composites: Part 1 Processingˮ, Int. Mater. Rev., 40(6), 239-265. https://doi.org/10.1179/imr.1995.40.6.239. https://doi.org/10.1179/imr.1995.40.6.239
  12. Mousavi, S.M. and Paavola, I. (2013), "Analysis of functionally graded magneto-electro-elastic layer with multiple cracksˮ, Theoerital Appl. Fracture Mech., 66(1), 1-8. https://doi.org/10.1016/j.tafmec.2013.11.007. https://doi.org/10.1016/j.tafmec.2013.11.007
  13. Nemat-Allal, M.M., Ata, M.H., Bayoumi, M.R. and Khair-Eldeen, W. (2011), "Powder metallurgical fabrication and microstructural investigations of Aluminum/Steel functionally graded materialˮ, Mater. Sci. Appl., 2(5), 1708-1718.
  14. Neubrand, A. and Rodel, J. (1997), "Gradient materials: An overview of a novel conceptˮ, Zeit. f. Met., 88(4), 358-371.
  15. Rekik, M., El-Borgi, S. and Ounaies, Z. (2014), "An axisymmetric problem of an embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite mediumˮ, Appl. Math. Modell., 38(4), 1193-1210. https://doi.org/10.1016/j.ijsolstr.2011.12.002. https://doi.org/10.1016/j.apm.2013.08.006
  16. Rizov, V. (2017a), "Delamination analysis of a layered elasticplastic beamˮ, J. Struct. Integrity, 8(4), 516-529. https://doi.org/10.1108/IJSI-11-2016-0035
  17. Rizov, V. (2017b), "An analytical solution to the strain energy release rate of a crack in functionally graded beamsˮ, Europ. J. Mech. A/solids, 65(2), 301-312. https://doi.org/10.1016/j.euromechsol.2017.04.005. https://doi.org/10.1016/j.euromechsol.2017.04.005
  18. Rizov, V. (2017c), "Fracture analysis of functionally graded beams with considering material non-linearityˮ, Struct. Eng. Mech., 64(4), 487-494. http://dx.doi.org/10.12989/sem.2017.64.4.487. https://doi.org/10.12989/sem.2017.64.4.487
  19. Rizov, V. (2017d), "Analytical study of elastic-plastic longitudinal fracture in a functionally graded beamˮ, Strength, Fracture Complexity, 10(1), 11-22. https://doi.org/10.3233/SFC-170197. https://doi.org/10.3233/SFC-170197
  20. Rizov, V.I. (2018a), "Lengthwise fracture analyses of functionally graded beams by the Ramberg-Osgood equation", Eng. Rev., 38(2), 309-320. https://hrcak.srce.hr/201161. https://doi.org/10.30765/er.38.3.8
  21. Rizov, V.I. (2018b), "Delamination in multi-layered functionally graded beams - an analytical study by using the Ramberg-Osgood equation", Struct. Integrity Life, 18(1), 70-76.
  22. Rizov, V.I. (2018c), "Analytical study of delamination in multilayered two-dimensional functionally graded non-linear elastic beams," J. Mech., 34(4), 496-504. https://doi.org/10.1017/jmech.2017.104.
  23. Suresh, S. and Mortensen, A. (1998), Fundamentals of Functionally Graded Materials, IOM Communications Ltd., London, United Kingdom.
  24. Wang, B.L. and Noda, N. (2001), "Thermally induced fracture of a smart functionally graded composite structureˮ, Theoretial Appl. Fracture Mech., 35(1), 93-109. https://doi.org/10.1016/S0167-8442(00)00052-5. https://doi.org/10.1016/S0167-8442(00)00052-5