Axial modulus

Like the modulus, the tensile and compressive strengths depend mainly on the density (Fig. 26.6). The strength parallel to the grain varies linearly with density, for the same reason that the axial modulus does it measures the strength of the cell wall, scaled by the fraction of the section it occupies, giving... 

In (15.36) rm is the matrix fracture energy, t is the interfacial shear strength, and Ex is the axial modulus of the composite. In (15.37) e refers to the effective properties of the composite, which, for unidirectional fiber reinforcement, can be calculated with good approximation by the rule of mixtures. 

Figure 22 Variation of the axial modulus of new and aged filters with temperature. (Courtesy of the Society of Automotive Engineeis.)...

Figure 4 shows the variation of the laminate axial modulus E, transverse modulus E., shear modulus, major Poisson s ratio (normalised by their undamaged value) as a function of the relative delamination area D " for the [Oj /30j /- 302], laminate Matrix crack density was... 

Fiber Axial modulus GPa (Msi) Transverse modulus GPa (Msi) Inplane shear modulus GPa (Msi) Poisson s Ratio Axial tensile strength MPa (Kslj Transverse tensile strength MPa (Ksij Axial compressive strength MPa (Ksi) Transverse compressive strength MPa (Ksi) Inplane shear strength MPa (Ksi)... 

Theoretical axial MPa Side chain reduced the axial modulus of 6,000 (45)... 

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). 

Fibre type Failure strength (GPa) Failure strain Axial modulus (GPa) Density (kg/ m3)... 

Filler Density (g/cm ) Tensile axial modulus, CPa (average value) Tensile axial strength, MPa (average value)... 

This suggests that the longitudinal modulus of a unidirectional composite is practically dictated by the axial modulus of the fibres. If the load is applied in a transverse direction or in the case of shear loading, the parallel model mentioned previously cannot be applied. Rather the matrix and fibre can be considered to act in series. Assuming an iso-stress condition where the matrix and fibre carry the same load, the transverse modulus of the composite can be expressed by the following equations ... 

In addition to experimental fiber modulus data. Figure 4 also illustrates two axial modulus predictions calculated using the rule of mixtures. A previous investigation (14) has indicated that the modulus of neat PP fibers has a... 

The engineering properties of interest are the elastic constants in the principal material coordinates. If we restrict ourselves to transversely isotropic materials, the elastic properties needed are Ei, Ei, v, and G23, i.e. the axial modulus, the transverse modulus, the major Poisson s ratio, the in-plane shear modulus and the transverse shear modulus, respectively. All the elastic properties can be obtained from these five elastic constants. Since experimental evaluation of these parameters is costly and time-consuming, it becomes important to have analytical models to compute these parameters based on the elastic constants of the individual constituents of the composite. The goal of micromechanics here is to find the elastic constants of the composite as functions of the elastic constants of its constituents, as... 

Kriz and Stinchcomb [32] published experimental data for unidirectional graphite/epoxy composites. These results illustrate the case when the fibres are transversely isotropic. The elastic properties of the matrix are = 5.28 GPa and T = 0.354, and for the fibres E = 232 GPa, E = 15 GPa, 0(2 = 24 GPa, v 2 = 0.279 and v 3 = 0.49. In Figs 11.21-11.25 are plotted the predictions against the experimental data for , , G12, G23 and V23, i.e. the longitudinal or axial modulus, the transverse modulus, the in-plane shear modulus, the transverse shear modulus and the transverse Poisson s ratio, respectively. 

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. 

FIG. 3 C33 axial modulus of polyethylene obtained using different force fields and computational methods. All calculations are classical. Open symbols are from optimizations (circles, triangles, and diamond are self-consistent QHA lattice dynamics, all others are plotted at temperature of experimental lattice parameters potential energy minimizations are plotted at OK). Filled symbols are Monte Carlo (circles) or molecular dynamics (diamond, triangles). Force field of Ref. 55 open and filled circles (from Ref. 46) force field of Ref. 30 open (from Ref. 33) and filled (from Ref. 68) triangles pcff force field of Biosym/MSI open and filled diamonds (from Ref. 66) x (from Ref. 42) -I- (from Ref. 43) open inverted triangle (from Ref. 44) open square (from Ref. 29). Curves are drawn as guides to the eye. 

A central issue in the development of effective processes has been the increase in axial modulus with deformation ratio (i.e. draw ratio or extrusion ratio). It has been shown that all the solid phase deformation processes give oriented materials which are equivalent to a very good approximation. In polyethylene, it is of some interest to compare the measured mechanical anisotropy at 20°C and -YJ5 C with that predicted theoretically (Table 1). 

It can be seen that the highly oriented polymer is very anisotropic, and that the measured values are quite similar to the predicted values. The mechanical behaviour (Figure 12) shows a very pronounced dependence on temperature and even at the highest draw ratios the a and y relaxations are clearly observed in both extension and shear. The axial modulus approaches the theoretical value for the chain axis, yet there is strong temperature dependence. These observations have been explained along the following lines. 

The increase in axial modulus can therefore be most simply explained on the basis of a Takayanagi model where there... 

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