Determination of Apparent Longitudinal Youngs Modulus

Consider first the case where the representative volume defined earlier is snbjected to uniaxial loading in the longitudinal (fiber) direction only as shown in Fignre 8.5. Then, the macroscopic (average) stresses acting on the element are 

FIGURE 8.5 Mechanics of materials approach in viewing deformation of representative heterogeneous volume elements, (a) Deformation assumption and (b) force balance. 

Now let the macroscopic strain of the element in the longitudinal direction be l. Then, by definition the apparent (macroscopic) Young s modulus of the element, Ei, is given by the equation 

the mechanics of materials deformation assumption states that the fibers and matrix elongate by the same amount in the longitudinal direction. Therefore, they experience the same longitudinal strain. That is. 

Moreover, since the macroscopic transverse and shearing stresses, and Xlt, are zero, the fibers and matrix must also be in a state of uniaxial longitudinal stress. Hence, Young s moduli for the fibers and matrix are related to their respective stresses and strains as follows  

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