Composites Design and Manufacture (Plymouth University teaching support materials)
Strength.  Failure mechanisms.  Fractography.  Failure criteria.  Fracture mechanics.
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Strength

Strength is the stress at failure and is normally measured in MPa (MegaPascals = GN/m2). Failure may be either a non-reversible change of state (propagation of cracks within the material) or the ultimate stress where the specimen breaks into two or more pieces.   For a unidirectional composite where the fibres have a lower tensile strain to failure than the matrix, the Kelly-Tyson model [1] to predict the ultimate strength assumes that all fibres have identical strength and that both the fibre and the matrix fail at the failure strain of the fibre:

σc = σfVf + σm*(1-Vf)
where σc = the ultimate tensile strength of the composite, σf is the ultimate tensile strength of the fibre, σm* is the tensile stress in the matrix at the failure strain of the fibre, Vf is the volume fraction of the fibres and the matrix contains no voids. In many practical situations, the equation can be reduced to σc = σfVf. At low fibre volume fractions, the composite strength may be given by σc < σm#(1-Vf) where σm# is the maximum tensile strength of the matrix.  For mild misalignment of quasi-UD composites, Potter [2] has suggested that the equation should be corrected for the misalignment angle θ:

σc = σfVfsec2θ.

Hart-Smith [3] has proposed an empirical "Ten-Percent Rule" for the preliminary sizing of quadriaxial fibrous composite structures.  In it's original form, each 45° or 90° ply was considered to have one-tenth of the axial stiffness and strength of a reference 0° ply.  Each 0° or 90° ply was considered to have one-tenth of the in-plane shear stiffness and strength of an equivalent ±45° ply.  The in-plane shear strength of a cross-plied laminate can be taken to be one-half of the greater of the (a) tensile or (b) compressive strengths of the complementary (i.e. rotated by 45° fibre) pattern.

Limiting strains in stressed composites

The elongation at break of E-glass virgin filaments is 3.37% with a strength of 3.4 GPa. [4].

When the matrix has a lower failure strain than the fibre, then above the matrix failure strain, the matrix starts to undergo microcracking and this corresponds with the appearance of a 'knee' in the stress-strain curve [5].

Hull and Clyne [5] give failure strains for glass fibres, epoxy matrix and polyester matrix as 2.8%, 2% and 2% respectively.  However, when a cross-plied or woven composite is considered the 'knee' in the stress-strain curve may result from fibre/matrix debonding and/or transverse cracking in the plies normal to the applied load.  Hull and Clyne (Figure 8.28) indicate a knee-point at ~0.3% strain for a typical cross-ply laminate in uniaxial tension.

In the design codes for reinforced plastic pressure vessels (BS4994:1987) [6], the following values are given:

and, for the specific case of filament wound structures:

Failure mechanisms

The principal mechanisms of failure in fibre-reinforced composites are:
a:  
b:      c:      d:  

Imetrum have taken high-speed (25,000 frames per second) video (3 MB MPEG file) of the failure of a "carbon composite material" in tension.

Hedgepath [10] undertook theoretical two-dimensional, elastic, small deformation analyses of stress concentrations arising from fibre fracture in a sheet of parallel filaments in both static and dynamic loading.  The dynamic stress concentration factor for suddenly induced discontinuities was found to increase from 1.15 for a single broken fibre to 1.27 for an infinite number of broken fibres.

Fractography and microstructural characterisation

The following micrographs of fracture surfaces appear in books:

Lothar Engel et al [11]

Anne Roulin-Moloney [12]

Statistical considerations

The failure characteristics of most materials have statistical variability which is normally better modelled using Weibull statistics [13, 14] instead of normal distributions.  The use of A-Basis and B-Basis allowables to reduce risk in the structural design of composite materials and components has two statistically based tolerance bounds:

Multi-scale analysis on a hierarchical basis may combine micro-mechanics and macro-mechanics with finite element analysis, damage tracking, fracture and material degradation capability to analyse structures in depth [17].

Failure criteria

BEWARE: The von Mises (VM) failure criterion is "inherently isotropic, and therefore may yield incorrect results for anisotropic [materials]" [18]. The VM failure criterion is "a good option for ductile materials with equal tensile and compressive strength, but it fails with brittle materials" [19].

See Hinton et al [20] and Christensen [21].  Shokrieh and Moshrefzadeh-Sani [22] have questioned the coefficient which is assumed to be constant in the traditional Halpin-Tsai equation for the elastic modulus of randomly oriented composites.  Their research suggests that the coefficient depends on the matrix and reinforcement properties.  By combining the Mori-Tanaka (MT) and Laminated Analogy (LA) methods, the new Mori-Tanaka and Laminated Analogy (MT-LA) model can estimate the properties of randomly oriented (nano-)composites.

Li [23] noted that most real failures do not satisfy the Hashin [24] assumption "It may be argued that in the event that a failure plane can be identified, [then] the failure is produced by the normal and shear stresses on that plane"!

Fracture mechanics (for homogeneous isotropic materials)

In the following equations [25]:

Y  is a dimensionless parameter or function that depends on both the crack and specimen sizes and geometries, as well as the manner of load application,
σ  is the axial stress,
τ  is the shear stress in the appropriate plane,
a  is the half-crack length,
W  is the component width, and
subscript c  implies the critical (failure) condition [26].

BE CAREFUL to ensure that the stress axes and the material axes are accurately specified for composite materials.  Also consider whether the failure mode is relevant to the analysis (has the crack turned to run along a fibre/matrix of interlaminar interface)?

Stress Intensity Factor ( Pa.m1/2 )

KI = Yσ√(πa)
KII = Yτ√(πa)
KIII = Yτ√(πa)

Fracture toughness (critical stress intensity factor, Pa.m1/2 )

KIc = Y(a/w)σ√(πa)
KIIc = Y(a/w)τ√(πa)
KIIIc = Y(a/w)τ√(πa)

Strain energy release rate ( J/m2)

GI = KI2 / E
Critical strain energy release rate ( J/m2 )
Gk = Kk2 / E

For a more detailed description, see [27, 28].  For an excellent review of the subject in the context of composites see the paper by Williams [28].

References
  1. A Kelly and WR Tyson, Tensile properties of fibre-reinforced metals: copper/tungsten and copper/molybdenum, Journal of the Mechanics and Physics of Solids, 1965, 13(6), 329-350.
  2. RT Potter, Strength of composites, In A Kelly (editor): Concise Encyclopedia of Composite Materials, Pergamon, Oxford, 1989. ISBN 0-08-034718-5.  PU CSH Library.
  3. LJ Hart-Smith, The ten-percent rule, Aerospace Materials, August-October 1993, 5(2), 10-16.
  4. DR Lovell, Chapter 2A: Reinforcements in NL Hancox, Fibre Composite Hybrid Materials, Elsevier Applied Science, Barking, 1981, page 25.  PU CSH Library
  5. D Hull and TW Clyne, An Introduction to Composite Materials - 2nd edition, Cambridge UP, 1996.  PU CSH Library
  6. British Standard Specification for Design and Construction of Vessels and Tanks in Reinforced Plastics, BS 4994 : 1987 (British Standards Institution, London).
    This standard "covers part of the field, namely, the use of polyester, epoxy and furane resins in wet lay-up systems".  BS4994 is generally accepted to provide a safe, but often over-engineered solution!
  7. F Teklal, A Djebbar, S Allaoui, G Hivet, Y Joliff and B Kacimi, A review of analytical models to describe pull-out behavior ~ fiber/matrix adhesion, Composite Structures, 1 October 2018, 201, 791-815.
  8. R Gutkin, ST Pinho, P Robinson, PT Curtis, Physical mechanisms associated with initiation and propagation of kink-bands, 13th European Conference on Composite Materials (ECCM13), Stockholm, 2 June 2008.
  9. KB Armstrong and RT Barrett, Care and Repair of Advanced Composites, SAE International, Warrendale PA, 1998.  ISBN 0-7680-0047-5. Second edition, 2005.  PU CSH Library.
  10. JM Hedgepath, Stress concentrations in filamentary structures, NASA Technical Note TN-D-882, 20 March 1961.
  11. Lothar Engel, Hermann Klingele, Gottfried W Ehrenstein and Helmut Schaper (translated by MS Welling), An Atlas of Polymer Damage: Surface Examination by Scanning Electron Microscope, Wolfe Science Books, London, 1981.  ISBN 0-7234-0751-7. UOP Library
  12. Anne C Roulin-Moloney, Fractography and Failure Mechanisms of Polymers and Composites, Chapman & Hall (originally Elsevier Applied Science), London, 1988.  ISBN 1-85166-296-0.  PU CSH Library
  13. S van der Zwaag, The concept of filament strength and the Weibull modulus, Journal of Testing and Evaluation, September 1989, 17(5), 292-298.
  14. P Kittl and G Diaz, Weibull's fracture statistics, or probabilistic strength of materials: state of the art, Res Mechanica, 1988, 24(2), 99-207.
  15. R Rice, R Randall, J Bakuckas and S Thompson, Development of MMPDS Handbook Aircraft Design Allowables, 7th Joint DOD/FAA/NASA Conference on Aging Aircraft, New Orleans LA, 8-11 September 2003.
  16. DOT/FAA/AR-03/19, Final Report: Material Qualification and Equivalency for Polymer Matrix Composite Material System: Updated Procedure, US Department of Transportation Federal Aviation Administration - Office of Aviation Research, Washington DC, September, 2003.
  17. MR Talagani, Z Gurdal, F Abdi and S Verhoef, Obtaining A-Basis and B-Basis allowable values for open-hole specimens using virtual testing, AIAAC-2007-127, 4. International Aerospace Conference, Ankara, 10-12 September 2007
  18. CE Korenczuk, LE Votava, RY Dhume, SB Kizilski, GE Brown, R Narain, VH Barocas, Isotropic failure criteria are not appropriate for anisotropic fibrous biological tissues, Journal of Biomechanical Engineering, July 2017, 139(7), 071008 (10 pages). Paper BIO-16-1518.
  19. A Pérez-González, JL Iserte-Vilar and C González-Lluch, Interpreting finite element results for brittle materials in endodontic restorations, BioMedical Engineering OnLine, 2011, 10, article 44.
  20. MJ Hinton, AS Kaddour and PD Soden, Failure criteria in fibre reinforced polymer composites: the world-wide failure exercise, Elsevier, Amsterdam, 2004. ISBN 0-08-044475-x.
  21. Richard M Christensen, Stress Based Failure Criteria for Materials Science and Engineering, 2008, and specifically III: Failure Criteria for Anisotropic Fiber Composite Materials.
  22. MM Shokrieh and H Moshrefzadeh-Sani, On the constant parameters of Halpin-Tsai equation, Polymer, 5 December 2016, 106, 14–20.
  23. S Li, A feeble cornerstone of modern composite failure criteria: the Mohr's criterion and its undue extension to composites, 3rd China International Congress on Composite Materials, Xiaoshan/Hangzhou - China, 21 October 2017.
  24. Z Hashin, Failure criteria for unidirectional fiber composites, Transactions of ASME: Journal of Applied Mechanics, June 1980, 47(2), 329-334.
  25. WD Callister, Materials Science and Engineering - An Introduction - fifth edition, John Wiley & Sons, New York, 2000.  ISBN 0-471-32013-7. Eighth edition, 2011. PU CSH Library.
  26. Chapter 9: Analysis of Fracture, in RF Gibson, Principles of composite material mechanics, McGraw-Hill, 1994, pages 338-373. ISBN 0-07-023451-5. Third edition, 2012. PU CSH Library.
  27. RJ Sanford, Principles of Fracture Mechanics, Prentice Hall, New Jersey, 2003. ISBN 0-13-092992-1.  PU CSH Library.
  28. JG Williams, Fracture mechanics of composite failure, Proceedings of the Institution of Mechanical Engineers Part C: Journal of Mechanical Engineering Science, 1990, 204(4), 209-218.
Recommended further reading

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Updated by John Summerscales on 13-May-2022 12:48. Terms and conditions. Errors and omissions. Corrections.