Abstract
A linear, periodic, three-dimensional shear-lag model of unidirectionally-reinforced composites that allows for fibre breakage, and matrix failure is proposed. Matrix failure can take the form of matrix splitting or interfacial debonding. A computationally efficient scheme for its solution is developed. This scheme exploits the translation invariance of the elastostatic fields due to failed elements in the periodic cell, and is asymptotically faster than the classical eigensolution-based approach. The new computational scheme is used to illustrate the influence of matrix failure on the elastostatic fields induced by small clusters of fibre breaks in several test problems. Monte Carlo simulations of fracture in model three-dimensional composite specimen with Weibull-distributed fibre segment strengths are also performed. Matrix failure is found to considerably alter fracture development, to weaken the median specimen, and to reduce the variability in composite strength.
Similar content being viewed by others
References
Anderson DG (1965) Iterative procedures for nonlinear integral equations. J ACM 12(4):547–560
Bai ZZ (2006) Structured preconditioners for nonsingular matrices of block two-by-two structures. Math Comput 75(254):791–815
Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numerica 14:1–137
Beyerlein IJ, Landis CM (1999) Shear-lag model for failure simulations of unidirectional fiber composites including matrix stiffness. Mech Mater 31(5):331–350
Beyerlein IJ, Phoenix SL (1996) Stress concentrations around multiple fiber breaks in an elastic matrix with local yielding or debonding using quadratic influence superposition. J Mech Phys Solids 44(12):1997–2039
Beyerlein IJ, Phoenix SL (1997) Statistics of fracture for an elastic notched composite lamina containing Weibull fibers—part I. Features from monte-carlo simulation. Eng Fract Mech 57(2–3):241–265
Beyerlein IJ, Phoenix SL, Sastry AM (1996) Comparison of shear-lag theory and continuum fracture mechanics for modeling fiber and matrix stresses in an elastic cracked composite lamina. Int J Solids Struct 33(18):2543–2574
Briggs WL, Henson VE (1995) The DFT: an owners’ manual for the discrete Fourier transform. SIAM, New Delhi
Cormen TH, Leiserson CE, Rivest RL, Stein C (2009) Introduction to algorithms. MIT Press, Cambridge
Cox H (1952) The elasticity and strength of paper and other fibrous materials. Br J Appl Phys 3(3):72
Curtin W (2000) Dimensionality and size effects on the strength of fiber-reinforced composites. Compos Sci Technol 60(4):543–551
Elman HC, Golub GH (1994) Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J Numer Anal 31(6):1645–1661
Evans A, Zok F (1994) The physics and mechanics of fibre-reinforced brittle matrix composites. J Mater Sci 29(15):3857–3896
Fukunaga H, Chou TW, Fukuda H (1984) Strength of intermingled hybrid composites. J Reinf Plast Compos 3(2):145–160
Golub GH, Van Loan CF (2012) Matrix computations, 3rd edn. JHU Press, Baltimore
Greenbaum A (1997) Iterative methods for solving linear systems, vol 17. SIAM, New Delhi
Gupta A, Mahesh S, Keralavarma SM (2017) A fast algorithm for the elastic fields due to a single fiber break in a periodic fiber-reinforced composite. Int J Fract 204(1):121–127
Gupta A, Mahesh S, Keralavarma SM (2018) A fast algorithm for the elastic fields due to interacting fibre breaks in a periodic fibre composite. Int J Fract 211(1–2):295–303
Habeeb CI, Mahesh S (2015) Strength distribution of planar local load-sharing bundles. Phys Rev E 92(2):022125
Harlow DG, Phoenix S (1981) Probability distributions for the strength of composite materials i: two-level bounds. Int J Fract 17(4):347–372
Hedgepeth JM (1961) Stress concentrations in filamentary structures. Tech. Rep. RN D-882, NASA
Hedgepeth JM, Van Dyke P (1967) Local stress concentrations in imperfect filamentary composite materials. J Compos Mater 1(3):294–309
Ho N, Olson SD, Walker HF (2017) Accelerating the Uzawa algorithm. SIAM J Sci Comput 39(5):S461–S476
Hull D, Clyne TW (1996) An introduction to composite materials. Cambridge University Press, Cambridge
Ibnabdeljalil M, Curtin W (1997) Strength and reliability of fiber-reinforced composites: localized load-sharing and associated size effects. Int J Solids Struct 34(21):2649–2668
Landis CM, McMeeking RM (1999) Stress concentrations in composites with interface sliding, matrix stiffness and uneven fiber spacing using shear lag theory. Int J Solids Struct 36(28):4333–4361
Landis CM, Beyerlein IJ, McMeeking RM (2000) Micromechanical simulation of the failure of fiber reinforced composites. J Mech Phys Solids 48(3):621–648
Mahesh S, Mishra A (2018) Strength distribution of Ti/SiC metal-matrix composites under monotonic loading. Eng Fract Mech 194:86–104
Mahesh S, Phoenix S (2004) Lifetime distributions for unidirectional fibrous composites under creep-rupture loading. Int J Fract 127(4):303–360
Mahesh S, Beyerlein IJ, Phoenix SL (1999) Size and heterogeneity effects on the strength of fibrous composites. Phys D Nonlinear Phenom 133(1–4):371–389
Mahesh S, Phoenix SL, Beyerlein IJ (2002) Strength distributions and size effects for 2D and 3D composites with Weibull fibers in an elastic matrix. Int J Fract 115(1):41–85
Mahesh S, Gupta A, Kachhwah US, Sheikh N (2019) A fast algorithm to simulate the failure of a periodic elastic fibre composite. Int J Fract 217(1):127–135
Mason JC, Handscomb DC (2002) Chebyshev polynomials. Chapman and Hall/CRC, Boca Raton
Maxima (2014) Maxima, a computer algebra system. version 5.34.1. http://maxima.sourceforge.net/. Accessed 22 June 2019
Mishnaevsky L Jr, Dai G (2014) Hybrid carbon/glass fiber composites: micromechanical analysis of structure–damage resistance relationships. Comput Mater Sci 81:630–640
Mishra A, Mahesh S (2017) A deformation-theory based model of a damaged metal matrix composite. Int J Solids Struct 121:228–239
Murphy MF, Golub GH, Wathen AJ (2000) A note on preconditioning for indefinite linear systems. SIAM J Sci Comput 21(6):1969–1972
Okabe T, Takeda N, Kamoshida Y, Shimizu M, Curtin W (2001) A 3D shear-lag model considering micro-damage and statistical strength prediction of unidirectional fiber-reinforced composites. Compos Sci Technol 61(12):1773–1787
Rezghi M, Elden L (2011) Diagonalization of tensors with circulant structure. Linear Algebra Appl 435(3):422–447
Saad Y (2003) Iterative methods for sparse linear systems, vol 82. SIAM, New Delhi
Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7(3):856–869
Sastry A, Phoenix S (1993) Load redistribution near non-aligned fibre breaks in a two-dimensional unidirectional composite using break-influence superposition. J Mater Sci Lett 12(20):1596–1599
Sheikh N, Mahesh S (2018) Failure mechanisms and fracture energy of hybrid materials. Int J Fract 213(1):51–81
Smith RL (1982) The asymptotic distribution of the strength of a series-parallel system with equal load-sharing. Ann Probab 137–171
Swolfs Y, Gorbatikh L, Verpoest I (2013) Stress concentrations in hybrid unidirectional fibre-reinforced composites with random fibre packings. Compos Sci Technol 85:10–16
Trefethen LN (2013) Approximation theory and approximation practice, vol 128. SIAM, New Delhi
Uzawa H (1958) Iterative methods for concave programming. In: Arrow KJ, Hurwicz L, Uzawa H (eds) Studies in linear and nonlinear programming. Stanford University Press, Palo Alto, pp 154–165
Weibull W (1952) A statistical distribution function of wide applicability. J Appl Mech Trans ASME 19(2):233–234
Wolla JM, Goree JG (1987) Experimental evaluation of longitudinal splitting in unidirectional composites. J Compos Mater 21(1):49–67
Xia Z, Okabe T, Curtin W (2002) Shear-lag versus finite element models for stress transfer in fiber-reinforced composites. Compos Sci Technol 62(9):1141–1149
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mahesh, S. A fast algorithm for fracture simulations representing fibre breakage and matrix failure in three-dimensional fibre composites. Int J Fract 222, 75–109 (2020). https://doi.org/10.1007/s10704-020-00432-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-020-00432-8