Page 112 - Engineered Interfaces in Fiber Reinforced Composites

Chapter 4. Micromechanics of stress trnnsfer 95

at the interface between the fiber and matrix, and the Poisson contraction in the
lateral direction is the same in the fiber and matrix, to be detailed in Section 4.2.2.
Later, Dow (1963) modified Cox’s model, assuming that the matrix axial
displacement is not constant as opposed to the original assumption of Cox, and
there is no matrix present at the end of the fiber. Rosen (1964, 1965) further refined
the models by Cox and Dow by considering that the matrix encapsulating the fiber is
in turn surrounded by a material having the average properties of the composite. It
was assumed, however, that the fiber and the average composite material carried
only a tensile stress while the matrix carried only shear stresses. Rosen (1964)
quantitatively defined the ‘ineffective fiber length’ by specifying a fraction (say,
900/,) of the undisturbed stress field value below which the fiber will be considered
‘ineffective’. Although the shear-lag analysis of this type is not accurate nor
completely adequate to predict the gross mechanical performance of composites, it
has provided a firm basis to help understand the fundamental micromechanics of
load transfer at the interface region and to assist further development of more
rigorous, reliable micromechanics models in various specimen geometry.
Besides the foregoing early shear-lag models, there have been a number of
micromechanics analyses developed to quantify the stress transfer between the
composite constituents, aiming specifically at describing the mechanical properties
of the composites on the one hand, and estimating the interface bond quality using
the fiber fragmentation experiments on the other. These studies on the single fiber
(or simulated multiple fiber) composites include: Muki and Sternberg (1969, 1971),
Sternberg and Muki (1970), Russel (1973), Berthelot et al. (1978, 1993), Piggott
(1980), Eshelby (1982), Aboudi (1983), Whitney and Drzal (1987), Lhotellier and
Brinson (1988), Hsueh (1989), Chiang (1991), Lacroix et al. (1992), Lee and Daniel
(1992), Nairn (1992), Kim et al. (1993b), Zhou et al. (1995a). Feillard et al. (1994)
has recently presented an in-depth review of the theoretical aspects of the fiber
fragmentation test, with particular emphasis on practical applications of the models
to predict interface bond quality.
Notably, Russel (1973) developed a slender body theory where the idealized
composite consists of an elastic matrix containing elastic fibers aligned unidirec-
tionally at concentrations dilute enough to neglect the interactions between the
neighboring fibers. Russel derived a solution for the critical transfer length as a
function of the inverse of Young’s modulus ratio, Ef/Em, and found it sensitive to
the Poisson ratio of the matrix, vm. Using the composite model similar to that
employed by Rosen (1964), Whitney and Drzal (1987) also proposed a two-
dimensional thermo-mechanics model of stress transfer based on the superposition
of solutions for two axi-symmetric problems of the exact far field solution and the
approximate local transient solution. They obtained a solution for the critical
transfer length as a function of the elastic properties of the composite constituents.
The critical transfer length was defined as the fiber length required for the fiber axial
stress (FAS) to reach 95% of that for infinitely long fiber (of its far field value).
In an approach similar to that adopted in the work of Greszczuk (1969) on the
fiber pull-out test, Piggott (1980) has obtained solutions for the stress fields in the
fiber for several different cases of fiber-matrix interface, including the perfectly