Non-linear study of mode II delamination fracture in functionally graded beams

  • Rizov, Victor I. (Department of Technical Mechanics, University of Architecture, Civil Engineering and Geodesy)
  • Received : 2016.08.09
  • Accepted : 2017.01.03
  • Published : 2017.02.28


A theoretical study was carried-out of mode II delamination fracture behavior of the End Loaded Split (ELS) functionally graded beam configuration with considering the material non-linearity. The mechanical response of ELS was modeled analytically by using a power-law stress-strain relation. It was assumed that the material is functionally graded transversally to the beam. The non-linear fracture was investigated by using the J-integral approach. Equations were derived for the crack arm curvature and zero axes coordinate that are needed for the J-integral solution. The analysis developed is valid for a delamination crack located arbitrary along the beam height. The J-integral solution was verified by analyzing the strain energy release rate with considering material non-linearity. The effects of material gradient, non-linear material behavior and crack location on the fracture were evaluated. The solution derived is suitable for parametric analyses of non-linear fracture. The results obtained can be used for optimization of functionally graded beams with respect to their mode II fracture performance. Also, such simplified analytical models contribute for the understanding of delamination fracture in functionally graded beams exhibiting material non-linearity.


  1. Ahouel, M., Houari, M., Bedia, E. and Tounsi, A. (2016), "Size-dependent mechanical behaviour of functionally graded trigonometric shear deformation nanobeams including neutral surface position conceptˮ, Steel Compos. Struct., Int. J., 20(5), 963-981.
  2. Akbas, S. (2015), "Wave propagation of a functionally graded beam in thermal environmentsˮ, Steel Compos. Struct., Int. J., 19(6), 1421-1447.
  3. Anlas, G., Santare, M.H. and Lambros, J. (2000), "Numerical calculation of stress intensity factors in functionally graded materialsˮ, Int. J. Fracture, 104(1), 131-143.
  4. Atmane, H., Tounsi, A., Bernard, F. and Mahmoud, S. (2015), "A computational shear model for vibrational analysis of functionally graded beams with porositiesˮ, Steel Compos. Struct., Int. J., 19(2), 369-385.
  5. Bedia, A. and Bousahla, A. (2016), "Mechanical and hydrothermal behaviour of functionally graded plates using a hyperbolic shear deformation theoryˮ, Steel Compos. Struct., Int. J., 20(4), 889-912.
  6. Benferhat, R., Daouadji, T., Hadji, L. and Said Mansour, M. (2016), "Static analysis of the FGM plate with porositiesˮ, Steel Compos. Struct., Int. J., 21(1), 123-136.
  7. Bennai, R., Atmabe, H. and Tounsi, A. (2015), "A new higherorder shear and normal deformation theory for functionally graded sandwich beamsˮ, Steel Compos. Struct., Int. J., 19(3), 521-546.
  8. Bohidar, S.K., Sharma, R. and Mishra, P.R. (2014), "Functionally graded materials: A critical reviewˮ, Int. J. Res., 1(7), 289-301.
  9. Bounouara, F., Benrahou, K., Belkorissat, I. and Tounsi, A. (2016), "A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundationˮ, Steel Compos. Struct., Int. J., 20(2), 227-249.
  10. Carpinteri, A. and Pugno, N. (2006), "Cracks in re-entrant corners in functionally graded materialsˮ, Eng. Fracture Mech., 73(6), 1279-1291.
  11. Chakrabarty, J. (2006), Theory of Plasticity, Elsevier Butterworth-Heinemann, Oxford, UK.
  12. Darlimaz, K. (2015), "Vibration analysis of functionally graded material (FGM) grid systemˮ, Steel Compos. Struct., Int. J., 18(2), 395-408.
  13. Galeban, M., Mojahedin, A., Taghavi, Y. and Jabbari, M. (2016), "Free vibration of functionally graded thin beams made of saturated porous materialsˮ, Steel Compos. Struct., Int. J., 21(5), 999-1016.
  14. Kaman, M.O. and Cetisli, F. (2012), "Numerical analysis of center cracked orthotropic fgm plate: Crack and material axes differ by ${\theta}^{\circ}$ˮ, Steel Compos. Struct., Int. J., 13(2), 187-206. DOI: 10.12989/scs.2012.13.2.187
  15. Lubliner, J. (2006), Plasticity Theory (Revised Edition), University of California, Berkeley, CA, USA.
  16. Parvanova, S.L., Dineva, P.S. and Manolis, G.D. (2013), "Dynamic behavior of a finite-sized elastic solid with multiple cavities and inclusions using BIEMˮ, Acta Mech., 224, 597-618.
  17. Parvanova, S.L., Dineva, P.S., Manolis, G.D. and Kochev, P.N. (2014), "Dynamic response of a solid with multiple inclusions under anti-plane strain conditions by the BEMˮ, Comput. Struct., 139, 65-83.
  18. Pei, G. and Asaro, R.J. (1997), "Cracks in functionally graded materialsˮ, Int. J. Solids Struct., 34(1), 1-17.
  19. Petrov, V.V. (2014), Non-linear Incremental Structural Mechanics, M.: Infra-Injeneria.
  20. Rajabi, M., Soltani, N. and Eshraghi, I. (2016), "Effects of temperature dependent material properties on mixed mode crack tip parameters of functionally graded materialsˮ, Struct. Eng. Mech., Int. J., 58(2), 217-230. DOI: 10.12989/sem.2016.58.2.217
  21. Szekrenyes, A. (2012), "J-integral for delaminated beam and plate modelsˮ, Periodica polytechnica, Mech. Eng., 56(1), 63-71.
  22. Tilbrook, M.T., Moon, R.J. and Hoffman, M. (2005), "Crack propagation in graded compositesˮ, Compos. Sci. Technol., 65(2), 201-220.
  23. Upadhyay, A.K. and Simha, K.R.Y. (2007), "Equivalent homogeneous variable depth beams for cracked FGM beams; compliance approachˮ, Int. J. Fract., 144(2), 209-213.
  24. Uslu Uysal, M. (2017), "Virtual crack closure technique on delamination fracture toughness of composite materials based on epoxy resin filled with micro-scale hard coalˮ, Acta Physica Polonica A. [In press]
  25. Uslu Uysal, M. and Güven, U. (2016), "A bonded plate having orthotropic inclusion in adhesive layer under in-plane shear loadingˮ, J. Adhesion, 92(3), 214-235. DOI: 10.1080/00218464.2015.1019064
  26. Uslu Uysal, M. and Kremzer, M. (2015), "Buckling behaviour of short cylindrical functionally gradient polymeric materialsˮ, Acta Physica Polonica A, 127(4), 1355-1357. DOI: 10.12693/APhysPolA.127.1355
  27. Uysal, M. (2016), "Buckling behaviours of functionally graded polymeric thin-walled hemispherical shellsˮ, Steel Compos. Struct., Int. J., 21(4), 849-862.
  28. Zhang, H., Li, X.F., Tang, G.J. and Shen, Z.B. (2013), "Stress intensity factors of double cantilever nanobeams via gradient elasticity theoryˮ, Eng. Fract. Mech., 105(1), 58-64.

Cited by

  1. Elastic wave scattering and stress concentration in a finite anisotropic solid with nano-cavities vol.87, pp.12, 2017,