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

*Theoretical modelling*

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## 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## Glossary of Symbols

*a*radius of fibre bundle

*C*2

*τ*_{f}*/aE*_{f}- ε
^{*} critical failure strain of fibre bundle

- ε
_{b} bending strain in outer fibre of a bundle

- ε
_{c} background strain in composite

- ε
_{f} axial strain in fibre

- ε
_{s} strain in fibre bundle due to fibre stretching = ε

_{f}- ε(∞)
strain in composite far from crack

*E*Young's modulus of fibre bundle

*E*_{c}Young's modulus of composite

*E*_{f}Young's modulus of fibre

*E*_{m}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

*F*_{s}component of fibre stretching force normal to crack plane

*F*_{b}component of bending force normal to crack plane

*G*_{m}shear modulus of matrix

*h*crack face opening relative to crack mid-point

*h*_{m}matrix contraction contribution to

*h**h*_{f}fibre deformation contribution to

*h**h*_{max}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

*k*_{0}force constant characterizing an elastic foundation (see Equation 7)

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

*L*_{f}half-length of a discontinuous fibre

*m, n*parameters characterizing degree of misalignment

*N*number of bundles intersecting a unit area of crack plane

*P*_{b}bending force normal to bundle axis at crack midpoint

*P*_{s}stretching force parallel to bundle axis in crack opening

*Q*(φ)distribution function describing the degree of misalignment

*s*_{f}fibre axial tensile stress

*s*_{f}^{*}fibre tensile failure stress

*S*stress supported by totality of bridging fibre bundles

*S*_{max}maximum value of bridging stress

*v*fibre displacement relative to matrix

*v*′elongation of fibre in crack bridging region

*u*_{coh}non-slip contribution to fibre elongation

*U*fibre elongation due to crack bridging

*v*overall volume fraction of fibres

*v*_{f}volume fraction of bundles

*v*_{m}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
(

*k*_{0}/4*EI*)^{1/4}, a parameter in cantilever beam analysis- v
_{m} Poisson's ratio of matrix

- ξ
λ

*L*(see Equation 9)- τ
shear stress

- τ
_{*} interlaminar shear strength of bundle

- τ
_{d} fibre/matrix interfacial shear strength

- τ
_{f} 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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## References

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