# Toughness of composites and fibre pull–out

For composites to be useful they not only must be strong, but they must also be tough. The fracture energy, which is related to the fracture toughness by Equation 10, is determined by the available energy absorbing mechanisms.

\[{K_c} = {\sigma _*}\sqrt {\pi c} = \sqrt {E{G_c}} (10)\]

For a composite these are:

- Matrix Deformation
- Fibre Fracture
- Interfacial Debonding and Crack Deflection
- Fibre Pull-Out

### Matrix Deformation:

This process is important in ductile matrices, but relatively negligible in brittle matrices. Load transfer onto the fibres reduces the matrix stress and the matrix is unable to deform freely due to the constraint imposed by the fibres. Generally, matrix deformation is unimportant for composites, even in metallic matrices.

### Fibre Fracture:

Metallic and polymeric fibres can undergo plastic deformation before fracture (ductile) and this contributes up to a few kJ m^{ -2} to the overall fracture energy, whereas brittle fibre fracture only provides a few tens of J m^{ -2} . It is important to realize that fibre fracture need not occur in the crack plane. There is a variation in flaw sizes along the fibre and, as a result, there is a variation in strength along the fibre, which can be described by the Weibull Modulus. Typically, fibre fracture in composites makes little contribution to the overall toughness.

### Interfacial Debonding and Crack Deflection:

Click here for Interfacial debonding derivation.

G_{cd} = f s G_{ic}

A composite made from brittle constituents can have a surprisingly high toughness if a crack is repeatedly deflected at fibre/matrix interfaces. The work of debonding per unit crack area, G_{cd} , itself is relatively small, but it allows fibre pull-out, which can potentially contribute significantly to the toughness, to occur.

Example: s = 50, f = 0.5, G_{ic} = 10 J m^{ -2} → G_{ cd} = 0.25 kJ m^{ -2}

### iv) Fibre Pull-out:

The main energy-absorbing mechanism raising the toughness of fibre composites is the pulling of fibres out of their sockets in the matrix during crack advance. This is allowed to take place after interfacial debonding and fibre fracture away from the crack plane have occurred beforehand. The pull-out work per unit crack area is given by:

Click here for Fibre pullout derivation.

G_{cp} = 4 f s^{2} r τ_{i*}

Example: s = 50, f = 0.5, r = 10 μm, τ_{i*} = 20 Mpa → G_{cp} = 1.0 MJ m^{-2}

It is because of these mechanisms that composites exhibit R-curve behaviour.