Page 66 - Fiber Fracture

MODELS OF FIBRE FRACTURE 51

where L, m, and 00 have the meaning given in J3q. 4 and e is the base of the natural
logarithms.
Neglecting the fibre interaction through the matrix is a crude approximation, and
the fibre bundle strength can only be considered a lower limit. An improved model is
obtained by considering that when one microfibril breaks, the elastic unloading in the
microfibril below and above the fracture leads to the development of a shear stress r at
the fibre-matrix interface. This shear stress transmits the load from the matrix back to
the fibre, and the stress in the fibre reaches its former value u at a distance x below and
above the fracture section, which is given by:
U
x=-r
2t
where r is the microfibril radius. Thus, even a soft matrix with low load-bearing
capacity changes the mechanical response of the fibre bundle because one microfibril
break influences the material only at a distance less than x below and above the fracture
section. Each microfibril is then considered to be formed by a chain of links of length
2x; one section of the fibre is a bundle of such links and the overall fibre is modeled
as a series of such bundles. Under such conditions, the fibre strength is given by Q. 9,
where the fibre length L is substituted by 2x. This leads to a significant increase in the
fibre bundle tensile strength, particularly at small values of the Weibull modulus, and
the corresponding estimations are plotted in Fig. 13 for arbitrary values of 2x/L = 1,
0.1 and 0.01.
More sophisticated models for the strength of a multifilamentary fibre were developed
in the context of uniaxially reinforced composites (Curtin, 1999). They assume that fibre
fracture is initiated randomly, but the stress concentration around the fibre breaks leads
to the localisation of damage in the surrounding fibres. Clusters containing broken fibres
grow progressively with deformation until they become unstable and the composite
fails. The degree of damage localisation (and, thus, the composite strength) depends on
the fibre, matrix and interface properties as well as on the fibre volume fraction and
spatial arrangement. These models were very successful in predicting the composite
strength but they have not been applied to multifilamentary fibres, partially because of
the difficulties in measuring the actual properties of the fibre components (microfibrils,
matrix and interface).

Hierarchical Fibres

While the composite fibres described in the previous section present a very simple mi-
crostructure, many biological fibres exhibit an extremely complex hierarchical structure
which is responsible, at least partially, for their excellent combination of strength and
damage tolerance (Renuart and Viney, 2000; Viney, 2000). A good example is the cotton
fibre (Fig. 14), made up of concentric laycrs grown over a small cylindrical lumen at the
fibre centre. Each layer is composed of parallel cellulose microfibrils laid down in spiral.
The helix angle is around 20 to 30". Cellulose molecules can form highly crystalline
structures and, in fact, cotton crystallinity is over 60%. Other vegetal fibres, such as flax,
hemp and ramie, present features similar to those of the cotton fibre, although the con-
tent of cellulose and the helix angle are lower. The mechanical behaviour of these fibres