An RVE from a particle-reinforced composite is shown in Figure 3.12. The particle has a cross-sectional area \(A_{\mathrm{p}}(x)\) that varies with the distance \(x\), and the stresses and strains in particle and matrix materials also vary with \(x\). Find the expression for the effective Young's modulus of the composite, \(E_{x}\) along the \(x\) direction. The answer should be left in terms of an integral involving the length, \(L\); the particle modulus, \(E_{\mathrm{p}}\); the matrix modulus, \(E_{\mathrm{m}} ;\) and the particle area fraction \(a_{\mathrm{p}}(x)=A_{\mathrm{p}}(x) / A_{\mathrm{c}}\), where \(A_{\mathrm{c}}\) is the total composite crosssectional area. Assume that both the particle and the matrix are isotropic.

Transcribed Image Text:

FIGURE 3.12 0x Particle x Matrix L Matrix -Ap(x)