Transversal modulus calculation

For the transverse shear modulus, the approach designated self-consistent was based on the formula obtained by the self-consistent method for the plane-strain bulk modulus (11.61), on the transverse modulus calculated using the Chamis approach (11.49b) and the in-plane Poisson s ratio given by the rule of mixtures. Except when used to predict the axial modulus and the major Poisson s ratio, the rule of mixtures underestimates the remaining composite elastic properties. The Bridging Model proved to be a very effective theory to account for all five elastic properties for unidirectional composites that are transversely isotropic. 

Example 3.5 Calculate the transverse modulus of the PEEK/carbon fibre composite referred to in Example 3.2, using both the simplified solid mechanics approach and the empirical approach. For PEEK = 0.36. 

Rg. 11.7 A comparison between experimental moduli for low-density polyethylene filaments and values calculated using the pseudo-affine aggregate model E, extensional modulus Ei, transverse modulus and G, torsional modulus (a) experimental, (b) calculated and (c) experimental and calculated. (Reprinted by permission of Kluwer Academic Publishers.)... 

An interpolation procedure applied by Halpin and Tsai [17,18] has led to general expressions for the moduli of composites, as given by Eqs. (2.18) and (2.19). Note that for = 0, Eq. (2.18) reduces to that for the lower hmit, Eq. (2.8), and for = infinity, it becomes equal to the upper limit for continuous composites, Eq. (2.7). By empirical curve fitting, the value of = 2(l/d) has been shown to predict the tensile modulus of aligned short-fiber composites in the direction of the fibers, and the value of = 0.5 can be used for the transverse modulus. Other mathematical relationships for modulus calculations of composites with discontinuous fillers include the Takaya-nagi and the Mori-Tanaka equations [20]. 

There have been no similar attempts to determine comprehensive sets of elastic constants for oriented fibres or monofilaments. Kawabata [21] devised an apparatus that used a linear differential transformer to measure diametral changes of 0.05 iim in single fibres of diameter 5 fxm subjected to transverse compression. Equation (7.2) above was then used to calculate the transverse modulus El = l/ ii. Results were obtained for poly(p-phenylene terephthalamide) (Kevlar) and high-modulus polyethylene (Tekmilon) fibres. Values of E were in the range 2.31 -2.59 GPa for Kevlar and a value of 1 -2 GPa was found for Tekmilon. 

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. 

In many isotropic materials the shear modulus G is high compared to the elastic modulus E, and the shear distortion of a transversely loaded beam is so small that it can be neglected in calculating deflection. In a structural sandwich the core shear modulus G, is usually so much smaller than Ef of the facings that the shear distortion of the core may be large and therefore contribute significantly to the deflection of a transversely loaded beam. The total deflection of a beam is thus composed of two factors the deflection caused by the bending moment alone, and the deflection caused by shear, that is, S = m + Ss, where S = total deflection, Sm = moment deflection, and Ss = shear deflection. 

Person 2 Calculate the elastic modulus transverse to the fiber direction, Ea-... 

Assume that the conductivity of a undirectional, continuous fiber-reinforced composite is a summation effect just like elastic modulus and tensile strength that is, an equation analogous to Eq. (5.88) can be used to describe the conductivity in the axial direction, and one analogous to (5.92) can be used for the transverse direction, where the modulus is replaced with the corresponding conductivity of the fiber and matrix phase. Perform the following calculations for an aluminum matrix composite reinforced with 40 vol% continuous, unidirectional AI2O3 fibers. Use average conductivity values from Appendix 8. 

Table 8.3 The lattice constant (a), bulk modulus (B) and frequency of the transverse optic phonon at the zone centre (TO-T) for MgO from PW-LDA and LCAO-HF calculations compared to observations extrapolated to the...

X-ray analysis indicates that the fired material is amorphous silica. Translucent specimens prepared in this manner have a density of over 99% of the theoretical density of pure fused amorphous silica as calculated from the dimensions and weight of the sample. The modulus of rupture is 5000 p.s.i., as measured by the conventional transverse bend technique of applying a measured increasing load at the middle of the bar which is supported across a 1-inch span. 

In many isotropic materials, the shear modulus G is high compared to the elastic modulus E, and the shear distortion of a transversely loaded beam is so small that it can be neglected in calculating deflection. In a structural sandwich the core shear modulus G, is usually so much smaller than Efof... 

The calculated results can only be accurate when substance laws applied describe the relevant material behavior with sufficient accuracy. Also, the required material parameters, for example modulus of elasticity E, shear modulus G, transversal contraction number must be available plotted against elongation, time, rate, temperature, as well as the effects of mediums or radiation as applicable. Most commercially available databases are still deficient in this respect. 

Indeed it has been found that the average axial modulus of a single cellulose nanofibril was 29—36 GPa (Tanpichai et al., 2012). From the calculations it follows that in this case the MFA of the nanofiber can be 10—12°, which corresponds to TS 0.8—1 GPa. In the transverse direction the mechanical characteristics of nanofibrils are considerably lower than in the axial direction (loelovich, 2012b). 

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