Composite Materials Research Progress

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 41
performed, introducing various numerical stress states (compatible with the constitutive
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hypotheses of the present work) for σ c . The tests showed that the microscopic strength
coefficients are, as expected, independent from the choice of the initial macroscopic stress
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state σ c : one set of coefficients only is found as the unique solution of system (60). This
demonstrates that the inverse model presented here is reliable from a numerical point of view.
The obtained results for the ultimate uniaxial stresses of 3501-6 and N5208 epoxies are
close together (Table 12), whereas the macroscopic strength present significant discrepancies
(Table 8). As an example, the relative deviation between the macroscopic longitudinal tensile
ultimate stress of the two composites reaches around 25% when the relative deviation
between the longitudinal tensile ultimate stress of the two epoxies is limited to 6%. Moreover,
the representation of the microscopic failure envelopes are rather similar for the two
considered resins, (Figure 5), whereas the macroscopic failure envelopes differ from one
composite to the other (Figure 5, also). This could be interpreted as follows: for the
considered composites, the observed deviation in the macroscopic failure envelopes comes
from the choice of the reinforcing fibers and not from the choice of the resin. This is
remarkable, since the considered epoxies exhibit a very different elastic mechanical behaviour
(see Tables 1 and 5).
Moreover, the predicted microscopic ultimate uniaxial stresses are coherent with
experimental results measured on plain resins. For instance, reference (Fiedler et al., 2001)
reports a strength value of 117 MPa in compression, and elastic limits reaching respectively
29 MPa in tension and 31 MPa in torsion for small specimen of plain unreinforced Bisphenol-
A type resin (i.e. “small” denotes a significantly reduced sized in normal and transverse
directions compared to “bulk” specimen). These measured strength are of the same order of
magnitude than the strength, calculated in the present work, for 3501-6 and N5208 epoxies.
At the opposite, the strengths determined on bulk specimens of 5208 and 3501-6 plain
epoxies are approximately two times higher than the values obtained in the present work, for
the strength of the corresponding epoxies embedded in thin composite plies. This last result is
also compatible with both the experimental comparison achieved in reference (Fiedler at al.,
2001) on various sized pure epoxies and the practical comparisons of the failure mechanisms
exhibited by composites structures and their constitutive epoxy resin (see Garett and Bailey,
1977; Christensen and Rinde, 1979). The present work allows to represent the scale effects
observed in practice on the composite constituents strengths, because the composite ply
strengths involved in the calculations do actually depend on both the constituents properties
and microstructure.
6. Conclusions
The present work dealt with the question of scale transition modelling of polymer matrix
composites and its application to several fields of investigation. Therefore, Mori-Tanaka and
Eshelby-Kröner self-consistent models, taking advantage of arithmetic averages, were both
considered for achieving the determinaiton of the homogenized properties of composite ply as
a function of the properties of its constituents (on the one hand, the matrix , and on the second
hand, the reinforcements).