Modulus ratio

Experimentally it has been shown that both frictional bridging and whisker pullout play an important role in toughening industrially manufactured composites. Such investigations confirm that to maximize toughness via both mechanisms requires a high volume fraction of whiskers and a high composite modulus to whisker modulus ratio. For example, consider the effect of 20 vol % SiC whisker E = 500 GPa) reinforcement of various matrices on the toughness as presented in Table 7 (53). 

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. 

Results for a square plate under sinusoidal transverse load with a variable modulus ratio, E1/E2, and a 45° lamination angle are shown in Figure 5-17. There, the effect of bending-extension coupling on deflections is significant for all modulus ratios except those quite close to Ei/E2=1. 

It is current practice to select the rotary shoulder connection that provides the balanced bending fatigue resistance for the pin and the box. The pin and the box are equally strong in bending if the cross-section module of the box in its critical zone is 2.5 times greater than the cross-section module of the pin at its critical zone. These critical zones are shown in Figure 4-127. Section modulus ratios from 2.25 to 2.75 are considered to be very good and satisfactory performance has been experienced with ratios from 2.0 to 3.2 [39]. 

The hardness shear modulus ratio in this case is similar to the one for metallic glasses. This suggests that the structure in the KCl-KBr solid solution is highly disordered i.e., glassy. 

The (property/volume cost) ratios show that the self-reinforced polypropylene has a much higher impact resistance ratio than general-purpose GMTs and glass fibre reinforced thermoplastics but a slightly lower modulus ratio. 

Fig. 3.5. Dependence of fiber critical aspect ratio, 2L] /d, on the Young s modulus ratio of fiber to...

Fig. 4.5. Normalized ineffective fiber length, (2i),/2a, as a function of modulus ratio, E jEms for varying fiber volume fraction, Vf. After Rosen (1964).

Fig, 4.35. The relationship between Young s modulus ratio, and radius ratio, b/a, showing the... 

MacLaughlin, T.F. and Barker, R.M. (1972). Effect of modulus ratio on stress near a discontinuous fiber. 

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...

Fig. 7.12. Maximum interface shear stresses plotted (a) as a function of Young s modulus ratio of coating to matrix, Ej/Em for coating thickness t = 50 /tm, and (b) as a function of coating thickness t for Young s modulus ratio of coating to matrix, Ei/Em = 0.5. After Kim et al. (1994c)...

Fig. 7.14. Normalized radial residual stresses as a function of coating thickness, I/a, for varying coefficients of thermal expansion (CTE) of the coating, Oc = 10,70,130 x 10 /°C (a) Young s modulus ratio Ej/Em = 0.333 (b) Ei/En, = 1.0. After Kim and Mai (1996a, b).

Elastomer behavior is depicted by the bottom curve in Figure 3.3. Here the modulus (ratio of stress to strain, as of strength to elongation measure of polymer stiffness) is low, but elongations to several hundred percent are possible before failure. 

Recently, some models (e.g., Halpin-Tsai, Mori- Tanaka, lattice spring model, and FEM) have been applied to estimate the thermo-mechanical properties [247, 248], Young s modulus[249], and reinforcement efficiency [247] of PNCs and the dependence of the materials modulus on the individual filler parameters (e.g., aspect ratio, shape, orientation, clustering) and on the modulus ratio of filler to polymer matrix. 

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. 

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)...

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. 

Sect. 6 a Aa T Do Fsh T X A M R De final radius of cylindrical gel after swelling displacement relaxation time for swelling collective diffusion constant shear energy trace of the strain tensor u,k swelling rate ratio total change of the radius of the gel longitudinal modulus ratio of the shear modulus to the longitudinal modulus effective collective diffusion constant... 

Fig. 6.2.11. Plot showing expanding cavity relation between hardness-to-modulus ratio and the ratio of plastic zone radius to cavity radius. Data points represent experimental measurements of plastic zone dimensions in deft to right) glass, A1203, ZrOj, KC1, ZnS, cold-rolled steel, and hot-rolled brass. (After Lawn et al., 1980b)...

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