Acta Mechanica

, Volume 230, Issue 1, pp 281–301 | Cite as

Mixed-mode I/II fracture criterion for crack initiation assessment of composite materials

  • Mahdi FakoorEmail author
  • Hannaneh Manafi Farid
Original Paper


This paper presents a mixed-mode I/II fracture criterion to investigate the crack initiation in orthotropic materials in which the crack is directed along fibers. The minimum strain energy density criterion is extended to investigate the cracked orthotropic materials. The reinforced isotropic solid model based on collinear crack propagation along fibers is proposed as an advantageous model to study the fracture behavior of composites. This model introduces fibers as reinforcements of the isotropic matrix in orthotropic materials in which their effects are qualified by defining reinforcement factors at tension and shear modes. The proposed criterion can predict the crack initiation phenomenon. Therefore, in this study a new concept of linear fracture toughness for orthotropic materials is proposed. Experimental data are used to validate the output of the proposed criterion. The coincidence of fracture limit curves and experimental data indicates the ability of the new criterion to predict crack initiation in orthotropic materials.

List of symbols

\(F_\mathrm{Y}, F_\mathrm{U}\)

Yield, ultimate force at F–D diagram

\(K_{\mathrm{Ic}}^{\mathrm{L}} ,K_{\mathrm{IIc}}^{\mathrm{L}}\)

Mode I and mode II linear fracture toughness

\(K_{\mathrm{Ic}}^{\mathrm{NL}} ,K_{\mathrm{IIc}}^{\mathrm{NL}}\)

Mode I and mode II nonlinear fracture toughness

\(K_{\mathrm{Ic}} ,K_{\mathrm{IIc}}\)

Mode I and mode II fracture toughness

\(K_\mathrm{I} ,K_{\mathrm{II}}\)

Mode I and mode II stress intensity factors

\(\rho _{\mathrm{NL}} ,\rho _{\mathrm{L}}\)

Nonlinear and linear damage factor

\(w_{\mathrm{m}} ,w_{\mathrm{f}} ,w\)

The width of matrix, fiber, RVE

\(l, \delta l\)

RVE’s Length and longitudinal displacement


The thickness of the RVE

\(\varepsilon _{\mathrm{m}} ,\varepsilon _{\mathrm{f}} ,\varepsilon \)

Strain of the matrix, fiber, RVE

\(\sigma _{\mathrm{m}} ,\sigma _{\mathrm{f}} ,\sigma \)

Normal stress of the matrix, fiber, RVE

\(E_{\mathrm{m}} ,E_{\mathrm{f}} ,E_{xx} ,E_{yy}\)

Elastic module of matrix, fiber, RVE along fiber, RVE across fiber

\(G_{\mathrm{m}} ,G_{\mathrm{f}} ,G_{xy} ,G_{yx}\)

Shear module of matrix, fiber, RVE in the xy plane

\(F_{\mathrm{m}} ,F_{\mathrm{f}} ,F\,\mathrm{or} F_\mathrm{c}\)

The force applied to matrix, fiber, RVE

\(A_{\mathrm{m}} ,A_{\mathrm{f}} ,A\)

The area in which the force is applied to matrix, fiber, RVE

\(\gamma _{\mathrm{m}} ,\gamma _{\mathrm{f}} ,\gamma \)

The shear angle of the matrix, fiber, RVE

\(\tau _{\mathrm{m}} ,\tau _{\mathrm{f}} ,\tau \)

Shear stress of matrix, fiber, RVE

\({\Delta }_{\mathrm{m}} ,{\Delta }_{\mathrm{f}} ,{\Delta }\)

The longitudinal displacements of matrix, fiber and RVE in pure shear loading

\(V_{\mathrm{m}} ,V_{\mathrm{f}}\)

Matrix and fiber fraction in a composite

\(E_{ij}, G_{ij} ,\nu _{ij} \)

Elastic, shear modulus and Poisson’s ratio of a composite in different directions

\({\xi }_1 ,{\xi }_2 ,{\xi }_3 \)

Reinforcement, ReSt, factors

\(\sigma _{ij}, \,\varepsilon _{ij} \)

Stress and strain functions

\(\sigma _{ij}^{\mathrm{iso}} ,\sigma _{ij}^{\mathrm{ortho}} \)

Stress state of isotropic and orthotropic materials

\(f_{ij} \left( \theta \right) ,\,g_{ij} \left( \theta \right) \)

Angular function in stress state

\(S,S_\mathrm{c} \)

Strain energy density factor, critical strain energy density factor


Strain energy density

\(r, \theta \)

Polar distance from the crack tip, polar angle

\(C_{ij}, \,C_{ij}^{\prime } \)

Compliance matrix for plane stress and plane strain condition

\(A_{11} ,A_{22} ,A_{12} \)

The factors in SED criterion

\(r_\mathrm{c} ,\theta _\mathrm{c} \)

Critical distance from crack tip and the path of crack growth in SED criterion


Longitudinal, radial and tangential axis in wood


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The authors would like to acknowledge the financial support of University of Tehran for this research under Grant No. 28686/01/01.


  1. 1.
    Carraro, P.A., Zappalorto, M., Quaresimin, M.: A comprehensive description of inter fiber failure in fiber reinforced composites. Theor. Appl. Fract. Mech. 79, 91–97 (2015)CrossRefGoogle Scholar
  2. 2.
    Li, Y.D., Xiong, T., Cai, Q.G.: Coupled interfacial imperfections and their effects on the fracture behavior of a layered multiferroic cylinder. Acta Mech. 226(4), 1183–1199 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ross, R.J.: Wood handbook: wood as an engineering material. USDA Forest Service, Forest Products Laboratory, General Technical Report FPL-GTR-190, 509 p. 1 v., 190 (2010)Google Scholar
  4. 4.
    Wu, E.M.: Application of fracture mechanics to anisotropic plates. J. Appl. Mech. 34(4), 967–974 (1967)CrossRefGoogle Scholar
  5. 5.
    Lin, W.H., Tsai, Y.M.: Fracture of hybrid laminates containing a pair of collinear cracks in the central layer. Acta Mech. 82(3), 159–173 (1990)CrossRefGoogle Scholar
  6. 6.
    Leicester, R.H.: Application of linear fracture mechanics in design of timber structures. In: Conference of the Australian, pp. 156–164. Fracture Group, Melbourne (1974)Google Scholar
  7. 7.
    Reiterer, A., Sinn, G., Stanzl-Tschegg, S.E.: Fracture characteristics of different wood species under mode I loading perpendicular to the grain. Mater. Sci. Eng. A 332(1–2), 29–36 (2002)CrossRefGoogle Scholar
  8. 8.
    Hunt, D.G., Croager, W.P.: Mode II fracture toughness of wood measured by a mixed-mode test method. J. Mater. Sci. Lett. 1(2), 77–79 (1982)CrossRefGoogle Scholar
  9. 9.
    Mall, S., Murphy, J.F., Shottafer, J.E.: Criterion for mixed mode fracture in wood. J. Eng. Mech. 109(3), 680–690 (1983)CrossRefGoogle Scholar
  10. 10.
    Erdogan, F., Sih, G.C.: On the crack extension in plates under plane loading and transverse shear. J. Basic Eng. 85(4), 519–525 (1963)CrossRefGoogle Scholar
  11. 11.
    Sih, G.C.: Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fract. 10(3), 305–321 (1974)CrossRefGoogle Scholar
  12. 12.
    Hussain, M.A., Pu, S.L., Underwood, J.: Strain energy release rate for a crack under combined mode I and mode II. In: Fracture Analysis: Proceedings of the 1973 National Symposium on Fracture Mechanics, Part II. ASTM International (1974)Google Scholar
  13. 13.
    Saouma, V.E., Ayari, M.L., Leavell, D.A.: Mixed mode crack propagation in homogeneous anisotropic solids. Eng. Fract. Mech. 27(2), 171–184 (1987)CrossRefGoogle Scholar
  14. 14.
    Carloni, C., Nobile, L.: Maximum circumferential stress criterion applied to orthotropic materials. Fatigue Fract. Eng. Mater. Struct. 28(9), 825–833 (2005)CrossRefGoogle Scholar
  15. 15.
    Nobile, L., Piva, A., Viola, E.: On the inclined crack problem in an orthotropic medium under biaxial loading. Eng. Fract. Mech. 71(4–6), 529–546 (2004)CrossRefGoogle Scholar
  16. 16.
    Gdoutos, E.E., Zacharopoulos, D.A., Meletis, E.I.: Mixed-mode crack growth in anisotropic media. Eng. Fract. Mech. 34(2), 337–346 (1989)CrossRefGoogle Scholar
  17. 17.
    Jernkvist, L.O.: Fracture of wood under mixed mode loading: I. Derivation of fracture criteria. Eng. Fract. Mech. 68(5), 549–563 (2001)CrossRefGoogle Scholar
  18. 18.
    Jernkvist, L.O.: Fracture of wood under mixed mode loading: II. Experimental investigation of Picea abies. Eng. Fract. Mech. 68(5), 565–576 (2001)CrossRefGoogle Scholar
  19. 19.
    Buczek, M.B., Herakovich, C.T.: A normal stress criterion for crack extension direction in orthotropic composite materials. J. Compos. Mater. 19(6), 544–553 (1985)CrossRefGoogle Scholar
  20. 20.
    Fakoor, M., Rafiee, R.: Fracture investigation of wood under mixed mode I/II loading based on the maximum shear stress criterion. Strength Mater. 45(3), 378–385 (2013)CrossRefGoogle Scholar
  21. 21.
    Romanowicz, M., Seweryn, A.: Verification of a non-local stress criterion for mixed mode fracture in wood. Eng. Fract. Mech. 75(10), 3141–3160 (2008)CrossRefGoogle Scholar
  22. 22.
    Perelmuter, M.: Nonlocal criterion of bridged cracks growth: analytical analysis. Acta Mech. 226(2), 397–418 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Anaraki, A.G., Fakoor, M.: General mixed mode I/II fracture criterion for wood considering T-stress effects. Mater. Des. 31(9), 4461–4469 (2010)CrossRefGoogle Scholar
  24. 24.
    Budiansky, B., O’connell, R.J.: Elastic moduli of a cracked solid. Int. J. Solids Struct. 12(2), 81–97 (1976)CrossRefGoogle Scholar
  25. 25.
    Anaraki, A.G., Fakoor, M.: Mixed mode fracture criterion for wood based on a reinforcement micro-crack damage model. Mater. Sci. Eng. A 527(27–28), 7184–7191 (2010)CrossRefGoogle Scholar
  26. 26.
    Anaraki, A.G., Fakoor, M.: A new mixed-mode fracture criterion for orthotropic materials, based on strength properties. J. Strain Anal. Eng. Des. 46(1), 33–44 (2011)CrossRefGoogle Scholar
  27. 27.
    Fakoor, M., Khansari, N.M.: Mixed mode I/II fracture criterion for orthotropic materials based on damage zone properties. Eng. Fract. Mech. 153, 407–420 (2016)CrossRefGoogle Scholar
  28. 28.
    Van der Put, T.A.C.M.: A new fracture mechanics theory for orthotropic materials like wood. Eng. Fract. Mech. 74(5), 771–781 (2007)CrossRefGoogle Scholar
  29. 29.
    Sih, G.C., Paris, P.C., Irwin, G.R.: On cracks in rectilinearly anisotropic bodies. Int. J. Fract. Mech. 1(3), 189–203 (1965)CrossRefGoogle Scholar
  30. 30.
    Fakoor, M.: Augmented Strain Energy Release Rate (ASER): a novel approach for investigation of mixed-mode I/II fracture of composite materials. Eng. Fract. Mech. 179, 177–189 (2017)CrossRefGoogle Scholar
  31. 31.
    Fett, T., Rizzi, G., Bahr, H.A., Bahr, U., Pham, V.B., Balke, H.: Analytical solutions for stress intensity factor, T-stress and weight function for the edge-cracked half-space. Int. J. Fract. 146(3), 189–195 (2007)CrossRefGoogle Scholar
  32. 32.
    He, Q.L., Wu, L., Li, M., Yu, H.: Prediction of mode I crack growth resistance based on a comparative investigation of J-integral and energy dissipation rate concept. Acta Mech. 215(1–4), 175–191 (2010)CrossRefGoogle Scholar
  33. 33.

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of New Sciences and TechnologiesUniversity of TehranTehranIran

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