STATISTICS OF FRACTURE FOR AN ELASTIC NOTCHED COMPOSITE LAMINA CONTAINING WEIBULL FIBERS-PART I. FEATURES FROM MONTE-CARLO SIMULATION

Abstract Monte-Carlo simulation is used to study the effects of the statistics of fiber strength on the fracture process, the fracture resistance, and the overall strength distribution for an elastic composite lamina with an internal transverse notch of N contiguous, broken fibers (0 ≤ N ≤ 51). To isolate the effects of variability in fiber strength, we assign individual fiber strengths drawn from a Weibull distribution with shape parameter γ ≥ 3 typical of commercial fibers, and we consider a simple case where fiber strength does not vary along the fiber length. The latter forces fibers to fail in the notch plane, eliminating the need to consider staggered breaks, debonding and fiber pullout. So under an increasing tensile load, failure develops through a progression of random fiber fractures governed by an interplay of stress concentrations and variations in fiber strength along the notch plane. Calculation of the fiber stresses for every configuration of surviving and broken fibers that occurs as the load is increased up to catastrophic failure is performed by an efficient, shear-lag based, break influence superposition (BIS) technique. Results show that the mean strength relative to the deterministic value (y = ∞) and mean number of new fiber fractures up to crack instability all increase with N regardless of γ, whereas variability in strength decreases. For smaller γ, we identify mechanisms responsible for flaw intolerance in the short notch regime and for toughness in the long notch regime, and show that variability in fiber strength can manifest as a nonlinear mechanism in an otherwise elastically deforming composite. Indeed as N increases we observe R-curve behavior, which is most pronounced for the smallest γ values where fracture resistance increases markedly and where mean fracture strength scales inversely with the initial notch size slower than the usual power of 1 2 . Compared to simulation results, a weakest-link or first failure model and unique fiber strength model severely underestimate fracture strength, failing to capture the statistical aspects of composite fracture.

[1]  B. Lawn Fracture of Brittle Solids by Brian Lawn , 1993 .

[2]  Duxbury,et al.  Fracture of heterogeneous materials with continuous distributions of local breaking strengths. , 1994, Physical review. B, Condensed matter.

[3]  P. Becher,et al.  Microstructural Contributions to the Fracture Resistance of Silicon Nitride Ceramics , 1994 .

[4]  J. R. Willis IUTAM Symposium on Nonlinear Analysis of Fracture , 1997 .

[5]  G. R. Leverant,et al.  BORON FIBER METAL MATRIX COMPOSITES BY PLASMA SPRAYING. , 1966 .

[6]  van der Erik Giessen,et al.  SIMULATION OF MATERIALS PROCESSING: THEORY, METHODS AND APPLICATIONS , 1998 .

[7]  K. Goda,et al.  Reliability Approach to the Tensile Strength of Unidirectional CFRP Composites by Monte-Carlo Simulation , 1993, Reliability and Optimization of Structural Systems.

[8]  H. T. Corten,et al.  FRACTURE MECHANICS OF COMPOSITES , 1972 .

[9]  William A. Curtin,et al.  Failure of fiber composites: A lattice Green function model , 1995 .

[10]  R. Pipes,et al.  Strength size effect of laminated composites , 1995 .

[11]  Z. Suo,et al.  Notch Ductile-to-Brittle Transition Due to Localized Inelastic Band , 1993 .

[12]  R. S. Gross,et al.  Stresses in a three-dimensional unidirectional composite containing broken fibers , 1980 .

[13]  P. Beaumont,et al.  Crack-tip energy absorption processes in fibre composites , 1985 .

[14]  S. Leigh Phoenix,et al.  Stress concentrations around multiple fiber breaks in an elastic matrix with local yielding or debonding using quadratic influence superposition , 1996 .

[15]  H. Herrmann,et al.  Statistical models for the fracture of disordered media. North‐Holland, 1990, 353 p., ISBN 0444 88551x (hardbound) US $ 92.25, 0444 885501 (paperback) US $ 41.00 , 1990 .

[16]  S. Batdorf,et al.  Use of shear lag for composite microstress analysis Rectangular array , 1985 .

[17]  M. Wisnom,et al.  Three-dimensional finite element analysis of the stress concentration at a single fibre break , 1994 .

[18]  Zhigang Suo,et al.  Remarks on Crack-Bridging Concepts , 1992 .

[19]  T. Yamamuro,et al.  Compositional Dependence of Bioactivity of Glasses in the System CaO-SiO2-P2O5 , 1991 .

[20]  R. L. Smith,et al.  A comparison of probabilistic techniques for the strength of fibrous materials under local load-sharing among fibers☆ , 1983 .

[21]  C. W. Beadle,et al.  The Stochastic Finite Element Simulation of Parallel Fiber Composites , 1976 .

[22]  Takeshi Kawashima,et al.  Grain Size Dependence of the Fracture Toughness of Silicon Nitride Ceramics , 1991 .

[23]  R. S. Gross,et al.  Analysis of a unidirectional composite containing broken fibers and matrix damage , 1980 .

[24]  Anand Jagota,et al.  Element breaking rules in computational models for brittle fracture , 1995 .

[25]  John A. Nairn,et al.  Fracture Mechanics of Unidirectional Composites Using the Shear-Lag Model I: Theory , 1988 .

[26]  E. Reedy Fiber Stresses in a Cracked Monolayer: Comparison of Shear-Lag and 3-D Finite Element Predictions* , 1984 .

[27]  Brian N. Cox,et al.  Tensile fracture of brittle matrix composites: influence of fiber strength , 1987 .

[28]  Duxbury,et al.  Failure probability and average strength of disordered systems. , 1994, Physical review letters.

[29]  M. Sutcu,et al.  Weibull statistics applied to fiber failure in ceramic composites and work of fracture , 1989 .

[30]  T. Chou,et al.  Explicit crack problem solutions of unidirectional composites - Elastic stress concentrations , 1990 .

[31]  H. W. Herring,et al.  Experimental observations of tensile fracture in unidirectional boron filament reinforced aluminum sheet , 1973 .

[32]  H. Fukuda,et al.  On the strength distribution of unidirectional fibre composites , 1977 .

[33]  Phillip M. Duxbury,et al.  Size effect and statistics of fracture in random materials , 1994 .

[34]  W. Curtin,et al.  Strength and reliability of notched fiber-reinforced composites , 1997 .

[35]  M. Shishesaz,et al.  Stress Concentration in Fiber Composite Sheets Including Matrix Extension , 1987 .

[36]  A. Sastry,et al.  Load redistribution near non-aligned fibre breaks in a two-dimensional unidirectional composite using break-influence superposition , 1993 .

[37]  T. Chou,et al.  An Advanced Shear-Lag Model Applicable to Discontinuous Fiber Composites , 1981 .

[38]  M. Sutcu Statistical fibre failure and single crack behaviour in uniaxially reinforced ceramic composites , 1988 .

[39]  K. Reifsnider,et al.  Micromechanical analysis of fiber fracture in unidirectional composite materials , 1996 .

[40]  A. Sastry,et al.  Comparison of shear-lag theory and continuum fracture mechanics for modeling fiber and matrix stresses in an elastic cracked composite lamina , 1996 .

[41]  K. Osamura,et al.  Influences of interfacial bonding strength and scatter of fibre strength on tensile behaviour of unidirectional metal matrix composites , 1988 .

[42]  A. Evans,et al.  Fracture Mechanics of Ceramics , 1986 .

[43]  Michael J. Hoffmann,et al.  Tailoring of mechanical properties of Si[3]N[4] ceramics , 1994 .

[44]  Roux,et al.  Fracture of disordered, elastic lattices in two dimensions. , 1989, Physical review. B, Condensed matter.

[45]  M. G. Bader,et al.  Monte Carlo simulation of the strength of composite fibre bundles , 1982 .

[46]  B. W. Rosen,et al.  Tensile failure of fibrous composites. , 1964 .

[47]  Peter W. R. Beaumont,et al.  Debonding and pull-out processes in fibrous composites , 1985 .

[48]  R. Steinbrech R-Curve Behavior of Ceramics , 1992 .

[49]  J. Hedgepeth,et al.  Stress Concentrations from Single-Filament Failures in Composite Materials , 1969 .

[50]  D. C. Phillips,et al.  Tensile Strengths of Notched Composites , 1972 .

[51]  M. Lienkamp,et al.  A Monte Carlo simulation of the failure of a seven fiber microcomposite , 1993 .

[52]  Peter Schwartz,et al.  Statistics for the strength and lifetime in creep-rupture of model carbon/epoxy composites , 1988 .

[53]  D. Jeulin,et al.  Fracture statistics of a unidirectional composite , 1995 .

[54]  K. Schulte,et al.  Strain concentration factors for fibers and matrix in unidirectional composites , 1991 .

[55]  J. Hedgepeth Stress Concentrations in Filamentary Structures , 1961 .

[56]  John M. Hedgepeth,et al.  Local Stress Concentrations in Imperfect Filamentary Composite Materials , 1967 .

[57]  K. P. Oh A Monte Carlo Study of the Strength of Unidirectional Fiber-Reinforced Composites , 1979 .

[58]  Norman A. Fleck,et al.  A binary model of textile composites—I. Formulation , 1994 .

[59]  Richard L. Smith,et al.  An examination of statistical theories for fibrous materials in the light of experimental data , 1985 .