Origin of Toughness in Composites

The most significant property improvement in fibre reinforced composites is that of fracture toughness. Toughness is quantified in terms of the energy absorbed per unit crack extension and thus any process which absorbs energy at the crack tip can give rise to an increase in toughness. In metallic matrices, plastic deformation requires considerable energy and so metals are intrinsically tough. In fibre reinforced materials with both brittle fibres and brittle matrices, toughness is derived from two sources. Firstly, if the crack can be made to run up and down every fibre in its path the there will be a large amount of new surface created for a very small increase in crack area perpendicular to the maximum principal stress - INTERFACIAL ENERGY - and in order to get the fibres to break they have to be loaded to their fracture strength and this often rquires additional local elastic work, and secondly If the fibres do not break and therefore bridge the gap then work must be done to pull the fibres out of the matrix - FIBRE PULLOUT. Using simple geometric models we can estimate the contribution of each of these processes to the overall toughness of the composite.

Fibre Pullout - Discontinuous Fibres

Consider the propagation of a crack through a matrix containing short fibres of length lc such that the fibres cannot break. The fibres will bridge the crack and for the crack to extend it is necessary to pull the fibres out of the matrix. Thus the stored elastic strain energy must do work pulling out the fibres against friction or by shearing the matrix parallel to the fibres as well as driving the crack through the matrix. We can estimate the work done pulling out a single fibre by integrating the product F(x).x (force x distance) over the distance l_c/2, where F(x) is the force - distance equation given by the shear lag model.

Figure 1. Fibre pullout during crack growth.

In the above equation d is the fibre diameter, s_m the matrix yield strength and l, the fibre length. The number of fibres intersecting unit area of crack is simply dependent on the volume fraction,

hence the total work done, G, in extending the crack unit area is

The longest fibre that can be pulled out is the critical fibre length, lc, which in turn depends on the fibre fracture strength, s_f. Thus a combination of strong fibres in a relatively weak fibre/matrix interface give the best toughness.

Continuous Fibres

What happens when the fibres are continuous - will the Strain energy relaease rate saturate at its maximum value shown above ? In a composite designed such that the majority of the load is carried by the fibres, the stress in the fibres will increase uniformly with strain upto the fibre uts. Since the strains in the presence of a crack are non-uniform, being greatest at the crack tip then the stress in the fibre will be greatest at the crack tip and so the fibre will fracture in the plane of the crack - hence no pullout. If however the fibres contain a population of defects - as real fibres do - with an average strength of s_f* and a spacing of l then it is possible for some of the continuous fibres to fracture within a distance l_c* /2 of the crack plane and be pulled out as the crack advances - note that we must reduce l_c to l_c* on account of the lower effective fibre strength, s_f* . ie.

if the defect has no strength then l_c*=l_c and the fibre acts as a short fibre of length l. If the defect as a strength almost equal to that of the ideal fibre then l_c* =0 and no pullout is possible. The fraction of fibres pulling out will be l_c* /l (recall l is the average distance between defects) then the work done in pulling out the fibres is

We can also use this expression to determine the toughness of short fibre composites of length l since these corespond to continuous fibres with defect spacings of l in which the defects have zero strength . ie for short fibres of length l>l_c we have

Example Problem

Estimate the maximum work of fracture in a carbon short-fibre epoxy composite containing 60 vol% carbon fibres and hence determine the fracture toughness K_Ic - assume only 50% of the fibres are parallel to the loading direction and have a length equal to the critical length..

Data:

Strength of Carbon fibre = 1800 MPa (Amoco Chemicals T650)
Strength of Epoxy = 85 MPa (Ciba Geigy Araldite HM94)
Fibre Diameter = 8µm
Modulus Data from previous example.

(i) Maximum Energy Absorbtion

G = fds_f² / 4s_m = (0.6/2) x 8x10^-6 x (1.8x10⁹)² / (4 x 85x10⁶) = 23 kJm^-2.

(ii) Fracture Toughness:- First evaluate the modulus of the composite

E= f*E_f (1-l_c/2l) + (1-f*) E_m = 0.3x290 (1-1/2) + 2.8x0.7 = 45 GPa

K²=GE = 23x10³ x 45x10⁹ = 1.04x10¹⁵ . ie. K = 32 MPa.m^1/2

The value obtained for the fracture toughness of the composite should be compared with the fracture toughness values of the epoxy (1 MPa.m^1/2 ).