Apparent Transverse Youngs Modulus

The total transverse elongation of the element W due to Oj is the combined transverse stretching of fibers and matrix  

FIGURE 8.9 Representative element subjected to uniaxial stress in the transverse direction. 

In order to relate the fiber and matrix strains to the constituent material properties, note that transverse normal stresses across the fiber-matrix interface must be continuous, that is, 

Assuming that fibers and matrix are also subjected to transverse stress only, then Hooke s Law for these materials gives 

Substituting the results of Equation 8.20 and Equation 8.17 into Equation 8.18 gives the desired result 

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Obviously, the assumptions involved in the foregoing derivation are not entirely consistent. A transverse strain mismatch exists at the boundary between the fiber and the matrix by virtue of Equation (3.8). Moreover, the transverse stresses in the fiber and in the matrix are not likely to be the same because v, is not equal to Instead, a complete match of displacements across the boundary between the fiber and the matrix would constitute a rigorous solution for the apparent transverse Young s modulus. Such a solution can be found only by use of the theory of elasticity. The seriousness of such inconsistencies can be determined only by comparison with experimental results. 

In order to find the transverse Young s Modulus, the representative element is subjected to a transverse uniaxial stress Ot with Ol = 0, and Xlt = 0, as shown in Figure 8.9. Since Oy is the only macroscopic stress acting on the element, then the apparent transverse Young s modulus is... 

Thus, it is apparent that the composite longitudinal Young s modulus and major Poisson s ratio are strongly influenced by the fiber elastic response whereas the composite transverse Young s modulus and shear modulus behavior is dominated by the matrix elastic response, except at large fiber volume fractions. 

In addition to chemical analysis a number of physical and mechanical properties are employed to determine cemented carbide quaUty. Standard test methods employed by the iadustry for abrasive wear resistance, apparent grain size, apparent porosity, coercive force, compressive strength, density, fracture toughness, hardness, linear thermal expansion, magnetic permeabiUty, microstmcture, Poisson s ratio, transverse mpture strength, and Young s modulus are set forth by ASTM/ANSI and the ISO. 

The apparent Young s modulus, E2, of the composite material in the direction transverse to the fibers is considered next. In the mechanics of materials approach, the same transverse stress, 02, is assumed to be applied to both the fiber and the matrix as in Figure 3-9. That is, equilibrium of adjacent elements in the composite material (fibers and matrix) must occur (certainly plausible). However, we cannot make any plausible approximation or assumption about the strains in the fiber and in the matrix in the 2-direction. 

Experimental load deflection curves (Fig. 3.) illustrate the large difference in crack propagation observed in each case. A difference in stiffness between both bonded specimens is observed and results from either a difference in the bond line quality or from interfacial conditions. For both specimens, adherends were made from the same sample of wood. Both wood substrates contained no apparent defects and had the same longitudinal Young s modulus (14500 MPa). Both also had the same growth characteristics (oven dry specific density, annual growth rings), and as a consequence very close values of transverse and shear modulus adjacent to the bond line. Thus, any difference in stiffness is likely to be due to... 

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