Acta Mechanica Sinica

, Volume 33, Issue 1, pp 120–131 | Cite as

In situ strengths of matrix in a composite

Research Paper


A major obstacle to achieving reasonable strength prediction of a composite only from its constituent information is in the determination of in situ strengths of the matrix. One can measure only the original strengths of the pure matrix, on the basis of which the predicted transverse strengths of a unidirectional (UD) composite are far from reality. It is impossible to reliably measure matrix in situ strengths. This paper focuses on the correlation between in situ and original strengths. Stress concentrations in a matrix owing to the introduction of fibers are attributed to the strength variation. Once stress concentration factors (SCFs) are obtained, the matrix in situ strengths are assigned as the original counterparts divided by them. Such an SCF cannot be defined following a classical approach. All of the relevant issues associated with determining it are systematically addressed in this paper. Analytical expressions for SCFs under transverse tension, transverse compression, and transverse shear are derived. Closed-form and compact formulas for all of the uniaxial strengths of a UD composite are first presented in this paper. Their application to strength predictions of a number of typical UD composites demonstrates the correctness of these formulas.


Composites Strength formulas Micromechanics Stress concentration factor Matrix in situ strength 



The project was supported by the National Natural Science Foundation of China (Grants 11272238, 11472192) and the Doctoral Fund of the Ministry of Education of China (Grant 20120072110036). The authors greatly appreciate discussions of this work with our colleague Prof. Y.-D. Xue at Tongji University.


  1. 1.
    Hinton, M.J., Soden, P.D.: Predicting failure in composite laminates: background to the exercise. Compos. Sci. Technol. 58, 1001–1010 (1998)CrossRefGoogle Scholar
  2. 2.
    Maimi, P., Camanho, P.P., Mayugo, J.A., et al.: A continuum damage model for composite laminates: part I—constitutive model. Mech. Mater. 39, 897–908 (2007)CrossRefGoogle Scholar
  3. 3.
    Kaddour, A.S., Hinton, M.J.: Maturity of 3D failure criteria for fiber reinforced composites: comparison between theories and experiments: Part B of WWFE-II. J. Compos. Mater. 47, 925–966 (2013)CrossRefGoogle Scholar
  4. 4.
    MIL-HDBK-17-3F.: Department of defense handbook Composite materials handbook. Volume 3: polymer matrix composites materials usage, design, and analysis. Department of Defense, USA (2002)Google Scholar
  5. 5.
    Soden, P.D., Hinton, M.J., Kaddour, A.S.: Lamina properties, lay-up configurations and loading conditions for a range of fiber-reinforced composite laminates. Compos. Sci. Technol. 58, 1011–1022 (1998)CrossRefGoogle Scholar
  6. 6.
    Kaddour, A.S., Hinton, M.J.: Input data for test cases used in benchmarking triaxial failure theories of composites. J. Compos. Mater. 46, 2295–2312 (2012)CrossRefGoogle Scholar
  7. 7.
    Kaddour, A.S., Hinton, M.J., Smith, P.A., et al.: Mechanical properties and details of composite laminates for test cases used in the third world-wide failure exercise. J. Compos. Mater. 47, 2427–2442 (2013)CrossRefGoogle Scholar
  8. 8.
    Chamis, C.C.: Fracture and fatigue. In: Brautman, J. (ed.) Micromechanics Strength Theories, 94–148. Academic, London (1974)Google Scholar
  9. 9.
    Gotsis, P.K., Chamis, C.C., Minnetyan, L.: Prediction of composite laminate fracture: micromechanics and progressive fracture. Compos. Sci. Technol. 58, 1137–1150 (1998)CrossRefGoogle Scholar
  10. 10.
    Mayes, S.J., Hansen, A.C.: Composite laminate failure analysis using multicontinuum theory. Compos. Sci. Technol. 64, 379–394 (2004)CrossRefGoogle Scholar
  11. 11.
    Pindera, M.J., Bansal, Y.: On the micromechanics-based simulation of metal matrix composite response. ASME J. Eng. Mater. Technol. 129, 468–482 (2007)CrossRefGoogle Scholar
  12. 12.
    Asp, L.E., Berglung, L.A.: Effects of a composite-like stress state on the fracture of epoxies. Compos. Sci. Technol. 53, 27–37 (1995)CrossRefGoogle Scholar
  13. 13.
    Fiedler, B., Hojo, M., Ochiai, S., et al.: Failure behavior of an epoxy matrix under different kinds of static loading. Compos. Sci. Technol. 61, 1615–1624 (2001)CrossRefGoogle Scholar
  14. 14.
    Huang, Z.-M., Liu, L.: Assessment of composite failure and ultimate strength without experiment on composite. Acta Mech. Sin. 30, 569–588 (2014)Google Scholar
  15. 15.
    Liu, L., Huang, Z.-M.: Stress concentration factor in matrix of a composite reinforced with transversely isotropic fibers. J. Compos. Mater. 48, 81–98 (2014)Google Scholar
  16. 16.
    Pinho, S., Iannucci, L., Robinson, P.: Physically based failure models and criteria for laminated fiber-reinforced composites with emphasis on fiber kinking. Part II: FE implementation. Compos. Part A 37, 63–73 (2006)CrossRefGoogle Scholar
  17. 17.
    Aragonés, D.: Fracture micromechanisms in c/epoxy composites under transverse compression. [Ph.D. thesis], Universidad Politécnica de Madrid, Spain (2007)Google Scholar
  18. 18.
    Lowe, A.: Transverse compressive testing of T300/914. J. Mater. Sci. 31, 1005–1011 (1996)CrossRefGoogle Scholar
  19. 19.
    Totry, E., Gonzalez, C., Llorca, J.: Prediction of the failure locus of C/PEEK composites under transverse compression and longitudinal shear through computational micromechanics. Compos. Sci. Technol. 68, 3128–3136 (2008)CrossRefGoogle Scholar
  20. 20.
    Vaughan, T.J., McCarthy, C.T.: Micromechanical modeling of the transverse damage behavior in fiber reinforced composites. Compos. Sci. Technol. 71, 388–396 (2011)CrossRefGoogle Scholar
  21. 21.
    Huang, Z.-M., Liu, L.: Predicting strength of fibrous laminates under triaxial loads only upon independently measured constituent properties. Int. J. Mech. Sci. 79, 105–129 (2014)CrossRefGoogle Scholar
  22. 22.
    Markov, K., Preziosi, L.: Heterogeneous Media: Micromechanics Modeling Methods and Simulations. Birkhauser, Boston (2000)CrossRefMATHGoogle Scholar
  23. 23.
    Pindera, M.J., Khatam, H., Drago, A.S., et al.: Micromechanics of spatially uniform heterogeneous media: a critical review and emerging approaches. Compos. Part B 40, 349–378 (2009)CrossRefGoogle Scholar
  24. 24.
    Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problem. Proc. R. Soc. A 241, 376–396 (1957)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Hashin, Z., Rosen, B.W.: The elastic moduli of fiber-reinforced materials. ASME J. Appl. Mech. 86, 223–232 (1964)CrossRefGoogle Scholar
  26. 26.
    Hashin, Z.: On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry. J. Mech. Phys. Solids 13, 119–134 (1965)CrossRefGoogle Scholar
  27. 27.
    Mori, T., Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21, 571–574 (1973)CrossRefGoogle Scholar
  28. 28.
    Benveniste, Y.: A new approach to the application of Mori-Tanaka’s theory in composite materials. Mech. Mater. 6, 147–157 (1987)CrossRefGoogle Scholar
  29. 29.
    Huang, Z.-M., Zhou, Y.X.: Strength of Fibrous Composites. Zhejiang University Press, Zhejiang (2011)Google Scholar
  30. 30.
    Wang, Y.C., Huang, Z.-M.: A new approach to a bridging tensor. Polym. Compos. 36, 1417–1431 (2015)CrossRefGoogle Scholar
  31. 31.
    Hibbeler, R.C.: Mechanics of Materials, 5th edn. Pearson Education Inc., Prentice Hall, Upper Saddle River (2003)MATHGoogle Scholar
  32. 32.
    Ryan, S., Wicklein, M., Mouritz, A., et al.: Theoretical prediction of dynamic composite material properties for hypervelocity impact simulations. Int. J. Impact Eng. 36, 899–912 (2009)CrossRefGoogle Scholar
  33. 33.
    Shaw, A., Sriramula, S., Gosling, P.D., et al.: A critical reliability evaluation of fibre reinforced composite materials based on probabilistic micro and macro-mechanical analysis. Compos. Part B 41, 446–453 (2010)CrossRefGoogle Scholar
  34. 34.
    Younes, R., Hallal, A., Fardoun, F., et al.: Comparative review study on elastic properties modeling for unidirectional composite materials. In: Composites and Their Properties, InTech, 391–408 (2012). doi: 10.5772/50362
  35. 35.
    Hinton, M.J., Kaddour, A.S., Soden, P.D.: A further assessment of the predictive capabilities of current failure theories for composite laminates: comparison with experimental evidence. Compos. Sci. Technol. 64, 549–588 (2004)CrossRefGoogle Scholar
  36. 36.
    Soden, P.D., Hinton, M.J., Kaddour, A.S.: Biaxial test results for strength and deformation of a range of E-glass and carbon fiber reinforced composite laminates: failure exercise benchmark data. Compos. Sci. Technol. 62, 1489–1514 (2012)CrossRefGoogle Scholar
  37. 37.
    Wang, Y.C., Huang, Z.-M.: Bridging tensor with an imperfect interface. Eur. J. Mech. A/Solids 56, 73–91 (2016)Google Scholar
  38. 38.
    Hashin, Z.: The interphase/imperfect interface in elasticity with application to coated fiber composites. J. Mech. Phys. Solids 50, 2509–2537 (2002)Google Scholar
  39. 39.
    MIL-HDBK-17-1F.: Department of defense handbook composite materials handbook. Volume 1: polymer matrix composites guidelines for characterization of structural materials. Department of Defense, USA (2002)Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Key Laboratory of the Ministry of Education for Advanced Civil Engineering MaterialsSchool of Aerospace Engineering and Applied Mechanics, Tongji UniversityShanghaiChina

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