Shear Modulus of Fibers

Equation 15.35 only gives accurate shear modulus values of circular fibers, i.e., those with circular cross-sections. When a non-circular fiber is studied, the shape factor should be used. Table 15.8 shows typical shear moduh and shape factors of some fibers. 

For fibers, the relationship between shear modulus and tensile modulus is interesting. Earlier in this Chapter, it has been mentioned the shear modulus of isotropic materials has a simple relationship with tensile modulus and Passion s ratio (v). The ratio of tensile modulus to shear modulus is  

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The shear modulus of fibers in torsional deformation has been measured for cotton, jute, ramie, and sisal, etc., as shown in Table 8.22. The data for ramie appear to be contradictory the reason for the contradiction may lie in the differences in the theoretical models (hence assumptions) and the experimental techniques used by the two sets of investigators. Therefore, caution is advised in the use of such data. 

Salavatian, M., Smith, L. An improved analytical model for shear modulus of fiber reinforced laminates with damage. Compos. Sci. Technol. 105, 9-14 (2014)... 

As discussed in section 15. 4.1, the shear modulus of fibers is important, and it determines the fiber torsional rigidity. The shear modulus of a fiber can be measmed by using the torsion pendulum technique. Figure 15.32 shows the schematic of a... 

Fig. 2. Effect of temperature on the shear modulus of dry nylon-6,6 (—) and nylon-6,6 plus 30% glass fiber (-). To covert MPa to psi, multiply by 145.

The variation of the ia-plane shear modulus normalised with respect to the matrix modulus as a function of the fiber volume fraction is shown ia Figure 11. As noted eadier, it is generally difficult to measure the shear modulus of the fibers, which may themselves be anisotropic. The equation should be used with caution. 

The in-plane shear modulus of a lamina, G12. is determined in the mechanics of materials approach by presuming that the shearing stresses on the fiber and on the matrix are the same (clearly, the shear deformations cannot be the samel). The loading Is shown in the representative volume element of Figure 3-15. By virtue of the basic presumption,... 

Fig. 6.12. Toughness maps depicting contours of predicted fracture toughness (solid lines in kJ/m ) for (a) glass-epoxy composites as a function of fiber strength, Uf, and frictional shear stress, tf and (b) Kevlar-cpoxy composites as a function of at and clastic modulus of fiber, Ef. The dashed line and arrows in (a) indicate a change in dominant failure mechanisms from post-debonding friction, Rif, to interfacial debonding, Sj, and the effect of moisture on the changes of Of and Tf, respectively. Bundle debond length...

The increase in fiber-matrix interfacial shear strength can be predicted from a purely mechanistic viewpoint. Rosen [20], Cox [21], and Whitney and Drzal [22] have shown that the square root of the shear modulus of the matrix appears explicitly in any model of the interfacial shear strength. It has been demonstrated experimentally [23, 24] that the fiber-matrix interfacial shear strength has a dependence on both the product of the strain-to-failure of the matrix times the square root of the shear modulus and on the difference between the test temperature and Tg when the interfacial chemistry is held constant. 

Figure 9.3 Elastic constants of an anisotropic fiber the longitudinal Young s modulus of fiber, or, the transverse Young s modulus or j., and the principal shear modulus, G 2 or Not shown are the two Poisson s ratios i/jj or the longitudinal Poisson s ratio of the fiber and or the transverse or inplane Poisson s ratio of the fiber cross-section.

Fibers are frequently twisted in a variety of operations. Thus, it is important to know the torsional or shear modulus of a fiber. In Fig. 9.7 a cylindrical fiber being twisted under the action of a torque shows the shear modulus corresponding to a torsion angle of 0. The shear strain, y, is given by... 

One can determine the shear modulus of a fiber from a torque per unit area versus twist curve. In practice, a simple apparatus called a torsion penduliun is used more commonly. An experimental setup to measure the shear modulus of small fibers is shown in Fig. 9.8 (Mehta, 1996). The torsional pendulum, placed in a vacuum oven, allows the measurement of shear modulus as a func-... 

F ure 9.8 An experimental setup to measure shear modulus of small fibers (after Mehta, 1996). 

The results of the Eshelby inclusion model of Chow (1978) are summarized in Figs. 4.4 and 4.5 for a prominent application in which the heterogeneities are much stiffer than the matrix and the results are evaluated for the special system of glass fiber or disks in an epoxy-resin polymer matrix where the shear modulus and bulk modulus of the glass are 30.6 GPa and 44.4 GPa, respectively, and those of the epoxy-resin matrix are 1.30 GPa and 3.90 GPa, respectively. For this system the dependence on volume fraction (p of filler of the normalized shear modulus of the heterogeneous composite is given in Fig. 4.4, with p being either the transverse shear modulus p 2 axial-radial shear modulus of the com-... 

The stiffness constants and Young s moduli of 30 wt% glass fiber-reinforced Vectra A are given in Tables 14.9 and 14.10, respectively. The elastic moduli of 30 wt% glass fiber-reinforced polyphenylene sulfide (PPS), taken from reference 41, are also shown for comparison. PPS is isotropic and has a Young s modulus and shear modulus of 4.0... 

Figure 20.26 Variation of longitudinal compressive strength of carbon fiber epoxy resin laminate with temperature showing transition from shear mode to buckling mode failure. Failure depends on the shear modulus of the matrix and shear strength of the fibers and a similar effect is observed with the uptake of water. Source Reprinted with permission from Ewins PD, Potter RT, Phil Trans R Soc London, A294, 507-517, 1980. Copyright 1980, The Royal Society of Chemistry.

Torsional rigidity n. Of a fiber, wire, bar, tube, or profile shape subjected to twisting of one end relative to the other, the torque required to produce a twist of 1 rad. This rigidity is proportional to the shear modulus of the material and is strongly dependent on all dimensions, especially section thickness. 

Area of concrete cross-section, net of steel reinforcement Area of FRP reinforcement Area of FRP shear reinforcement Area of one stirrup leg Area of steel reinforcement Young s modulus of elasticity of concrete Young s modulus of elasticity of FRP reinforcement Young s modulus of elasticity of the fiber itself Young s modulus of elasticity of steel reinforcement Design value of the maximum anchorage force transferred by the FRP reinforcement bonded on a masomy structure in the presence of a force perpendicular to the bonded surface area Shear modulus of adhesive Shear modulus of concrete Moment of inertia of transformed section... 

FIG. 7-7. Measurement of dynamic shear modulus of a fiber by forcedtorsional vibrations. (Wakelin... 

The adherends do not deform in shear, implying that the shear modulus of the adherends is much greater than that of the adhesive. This assumption becomes especially suspect with anisotropic materials such as wood- or fiber-reinforced composites. 

In these expressions, a and E- are the axial and transverse moduli of the fiber, G A is the axial shear modulus of the fiber, Va and vx are the axial and transverse Poisson s ratios of the fiber, a and ax are the axial and transverse thermal expansion coefficients of the fiber, and Em-, Gm, v>m and a , are the tensile modulus, shear modulus, Poisson s ratio, and thermal expansion coefficient of the matrix. 

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