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Journal of Materials Science

, Volume 29, Issue 4, pp 899–920 | Cite as

Effect of fibre misalignment on fracture behaviour of fibre-reinforced composites

Part II Theoretical modelling
  • W. B. Hillig
Papers

Abstract

When a matrix crack encounters a fibre that is inclined relative to the direction of crack opening, geometry requires that the fibre flex is bridging between the crack faces. Conversely, the degree of flexing is a function of the crack face separation, as well as of (1) the compliance of the supporting matrix, (2) the crossing angle, (3) the bundle size, and (4) the shear coupling of the fibre to the matrix. At some crack face separation the stress level in the fibre bundle will cause it to fail. Other bundles, differing in size and orientation, will fail at other values of the crack separation. Such bridging contributes significantly to the resistance of the composite to crack propagation and to ultimate failure. The stress on the composite needed to produce a given crack face separation is inferred by analysing the forces and displacements involved. The resulting model computes stress versus crack-opening behaviour, ultimate strengths, and works of failure. Although the crack is assumed to be planar and to extend indefinitely, the model should also be applicable to finite cracks.

Keywords

Polymer Stress Level Material Processing Ultimate Strength Fracture Behaviour 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Glossary of Symbols

a

radius of fibre bundle

C

2τf/aEf

ε*

critical failure strain of fibre bundle

εb

bending strain in outer fibre of a bundle

εc

background strain in composite

εf

axial strain in fibre

εs

strain in fibre bundle due to fibre stretching = εf

ε(∞)

strain in composite far from crack

E

Young's modulus of fibre bundle

Ec

Young's modulus of composite

Ef

Young's modulus of fibre

Em

Young's modulus of matrix

f(θ)

number density per unit area of fibres crossing crack plane in interval θ to θ + dθ

F

total force exerted by fibre bundle normal to crack plane

Fs

component of fibre stretching force normal to crack plane

Fb

component of bending force normal to crack plane

Gm

shear modulus of matrix

h

crack face opening relative to crack mid-point

hm

matrix contraction contribution to h

hf

fibre deformation contribution to h

hmax

crack opening at which bridging stress is a maximum

I

moment of inertia of fibre bundle

k

fibre stress decay constant in non-slip region

k0

force constant characterizing an elastic foundation (see Equation 7)

L

exposed length of bridging fibre bundle (see Equation 1a)

Lf

half-length of a discontinuous fibre

m, n

parameters characterizing degree of misalignment

N

number of bundles intersecting a unit area of crack plane

Pb

bending force normal to bundle axis at crack midpoint

Ps

stretching force parallel to bundle axis in crack opening

Q(φ)

distribution function describing the degree of misalignment

sf

fibre axial tensile stress

sf*

fibre tensile failure stress

S

stress supported by totality of bridging fibre bundles

Smax

maximum value of bridging stress

v

fibre displacement relative to matrix

v

elongation of fibre in crack bridging region

ucoh

non-slip contribution to fibre elongation

U

fibre elongation due to crack bridging

v

overall volume fraction of fibres

vf

volume fraction of bundles

vm

volume fraction matrix between bundles

w

transverse deflection of bundle at the crack mid-point

x

distance along fibre axis, origin defined by context

X

distance between the end of discontinuous fibre and the crack face

X*

threshold (minimum) value of X that results in fibre failure instead of complete fibre pullout

y

displacement of fibre normal to its undeflected axis

Z(θ)

area fraction angular weighting function

η

tensile strain in fibre relative to applied background strain

η*

critical value of η to cause fibre/matrix debonding

θ

angle at which a fibre bundle crosses the crack plane

λ

(k0/4EI)1/4, a parameter in cantilever beam analysis

vm

Poisson's ratio of matrix

ξ

λL (see Equation 9)

τ

shear stress

τ*

interlaminar shear strength of bundle

τd

fibre/matrix interfacial shear strength

τf

frictional shear slippage stress at bundle/matrix interface

φ

angular deviation of fibre bundle from mean orientation of all bundles

ψ

angle between symmetry axis and crack plane

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Copyright information

© Chapman & Hall 1994

Authors and Affiliations

  • W. B. Hillig
    • 1
  1. 1.General Electric Company, Corporate Research and DevelopmentSchenectadyUSA

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