Fiber-to-matrix modulus ratio

Values of EvE are given in Table 3-1 for three values of the fiber-to-matrix modulus ratio. 

Predicted results for E2 are plotted in Figure 3-10 for three values of the fiber-to-matrix-modulus ratio. Note that if Vj = 1, the modulus predicted is that of the fibers. However, recognize that a perfect bond between fibers is then implied if a tensile <32 is applied. No such bond is implied if a compressive 02 is applied. Observe also that more than 50% by volume of fibers is required to raise the transverse modulus E2 to twice the matrix modulus even if E, = 10 x E ,l That is, the fibers do not contribute much to the transverse modulus unless the percentage of fibers is impractically high. Thus, the composite material property E2 is matrix-dominated. 

Fig. 12. A unidirectional lamina under transverse tension. The points of stress concentration are at the dots. The micromechanical analysis shows that the stress concentration factor increases with volume fraction of fiber and fiber to matrix modulus ratio. From Adams et al.70)...

A7.1.3 Effect of Fiber-to-Matrix Modulus Ratio on Effective Fiber Length/Diameter Ratio... 

Eiber lengths between 5 and 10 mm are conveniently selected for the microindentation test (111). Eor a carbon fiber/epoxy system, as the fiber volume fraction increases from 10 to 50 vol%, the indentation displacement distance decreases from 44 to 36 pm but the interfacial shear strength increases from 33 to 46 MPa. When the interphase-to-matrix modulus ratio increases from 1.0 to 7.5, the interfacial shear stress increases by only 10%. Likewise, the interphase thickness and fiber diameter have marginal effects on the interfacial shear stress. Three types of thermoplastic polymers (polyester, polyamide, and polypropylene) were tested for their interfacial shear strength to the glass fiber by Desaeger and... 

Figure 12.1. Modulus ratio as a function of filler concentration for filler-to-matrix moduli ratios of 10 and 100 according to Kerner s (19566) equation. Bottom curves or loops are for cases where the filler (dispersed phase) is the more rigid phase, while the upper curves are for the inverted case, where the more rigid material is the continuous phase. The top curve is the elastic Young s modulus of a material filled with very long fibers aligned in the direction of the applied tensile load. (Nielsen, 1967u.)...

Equation (2.13) has also been used to predict the modulus of flake (platelet) composites containing planar oriented reinforcement for uniform arrays of flakes, Eq. (2.14), and for random overlap, Eq. (2.15) [10, 12, 14, 19]. Equations for the parameter u are somewhat different from those used for fibers, but thqr still contain the important parameters affecting the modulus of the composite, that is, aspect ratio, volume fraction, and flake/matrix modulus ratio. Equation (2.18) has also been used to predict the modulus of platelet-reinforced plastics [17, 21]. 

For a fiber immersed in water, the ratio of the slopes of the stress—strain curve in these three regions is about 100 1 10. Whereas the apparent modulus of the fiber in the preyield region is both time- and water-dependent, the equiUbrium modulus (1.4 GPa) is independent of water content and corresponds to the modulus of the crystalline phase (32). The time-, temperature-, and water-dependence can be attributed to the viscoelastic properties of the matrix phase. 

Dow and Rosen s results are plotted in another form, composite material strain at buckling versus fiber-volume fraction, in Figure 3-62. These results are Equation (3.137) for two values of the ratio of fiber Young s moduius to matrix shear modulus (Ef/Gm) at a matrix Poisson s ratio of. 25. As in the previous form of Dow and Rosen s results, the shear mode governs the composite material behavior for a wide range of fiber-volume fractions. Moreover, note that a factor of 2 change in the ratio Ef/G causes a factor of 2 change in the maximum composite material compressive strain. Thus, the importance of the matrix shear modulus reduction due to inelastic deformation is quite evident. 

A simplified performance index for stiffness is readily obtained from the essentials of micromechanics theory (see, for example. Chapter 3). The fundamental engineering constants for a unidirectionally reinforced lamina, ., 2, v.,2, and G.,2, are easily analyzed with simple back-of-the-envelope calculations that reveal which engineering constants are dominated by the fiber properties, which by the matrix properties, and which are not dominated by either fiber or matrix properties. Recall that the fiber-direction modulus, is fiber-dominated. Moreover, both the modulus transverse to the fibers, 2, and the shear modulus, G12. are matrix-dominated. Finally, the Poisson s ratio, v.,2, is neither fiber-dominated nor matrix-dominated. Accordingly, if for design purposes the matrix has been selected but the value of 1 is insufficient, then another more-capable fiber system is necessary. Flowever, if 2 and/or G12 are insufficient, then selection of a different fiber system will do no practical good. The actual problem is the matrix systemi The same arguments apply to variations in the relative percentages of fiber and matrix for a fixed material system. 

Fig. 7.11. Normalized interface shear stress distributions along the fiber length for composites with and without PVAL coating coating thickness t = 5 pm and Young s modulus ratio of coating to matrix...

The microductile/compliant layer concept stems from the early work on composite models containing spherical particles and oriented fibers (Broutman and Agarwal, 1974) in that the stress around the inclusions are functions of the shear modulus and Poisson ratio of the interlayer. A photoelastic study (Marom and Arridge, 1976) has proven that the stress concentration in the radial and transverse directions when subjected to transverse loading was substantially reduced when there was a soft interlayer introduced at the fiber-matrix interface. The soft/ductile interlayer allowed the fiber to distribute the local stresses acting on the fibers more evenly, which, in turn, enhanced the energy absorption capability of the composite (Shelton and Marks, 1988). 

The Halpin-Tsai equations represent a semiempirical approach to the problem of the significant separation between the upper and lower bounds of elastic properties observed when the fiber and matrix elastic constants differ significantly. The equations employ the rule-of-mixture approximations for axial elastic modulus and Poisson s ratio [Equations. (5.119) and (5.120), respectively]. The expressions for the transverse elastic modulus, Et, and the axial and transverse shear moduli, Ga and Gf, are assumed to be of the general form... 

Consider the same unidirectional lamina with the stresses now applied perpendicular to the fiber axis as shown in Fig. 12. The local stress at the fiber matrix interface can be calculated and compared to the nominally applied stress on the whole lamina to give K, the stress concentration factor. The plot of the results of this analysis shows that the interfacial stresses at the point of maximum principal stress can range up to 2.6 times the applied stress depending on the moduli of the constituents and the volume fraction of the reinforcement. For a typical graphite-epoxy composite, with a modulus ratio of 70 and a volume fraction of 70 % the stress concentration factor at the interface is about 2.4. That is, the local stresses at the interface are a factor of 2.4 times greater than the applied stress. 

Imposing a shear stress parallel to the fiber axis of a unidirectional composite creates an interfacial shear stress. Because of the disparity in material properties between fiber and matrix, a stress concentration factor can develop at the fiber-matrix interface. Linder longitudinal shear stress as shown by the diagram in Fig. 13, the stress concentration factor is interfacial. The analysis shows that the stress concentration factor can be increased with the constituent shear modulus ratio and volume fraction of fibers in the composite. Under shear loading conditions at the interface, the stress concentration factor can range up to 11. This is a value that is much greater than any of the other loadings have produced at the fiber-matrix interface. 

Here we have conducted experiments to develop an understanding of how the commercial size interacts with the matrix in the glass fiber-matrix interphase. Careful characterization of the mechanical response of the fiber-matrix interphase (interfacial shear strength and failure mode) with measurements of the relevant materials properties (tensile modulus, tensile strength, Poisson s ratio, and toughness) of size/matrix compositions typical of expected interphases has been used to develop a materials perspective of the fiber-sizing-matrix interphase which can be used to explain composite mechanical behavior and which can aid in the formulation of new sizing systems. 

---