A representative volume element (RVE) consisting of a cylindrical isotropic fiber embedded and perfectly bonded in a cylinder of isotropic matrix material is shown in Figure 5.23. If the ends of the RVE at \(x=0\) and \(x=L\) and the outer surface of the RVE are stress free and the RVE is subjected to a uniform temperature change \(\Delta T\), determine the fiber stress, \(\sigma_{\mathrm{f}}\), and the matrix stress, \(\sigma_{\mathrm{m}}\), along the fiber direction at the midpoint of the RVE (at \(x=L / 2\) ). Use a mechanics of materials approach and express answers in terms of the coefficients of thermal expansion \(\alpha_{\mathrm{f}}\) and \(\alpha_{\mathrm{m}}\), the cross-sectional areas \(A_{\mathrm{f}}\) and \(A_{\mathrm{m}}\) the Young's moduli \(E_{\mathrm{f}}\) and \(E_{\mathrm{m}}\), and the temperature change, \(\Delta T\), where the subscripts \(\mathrm{f}\) and \(\mathrm{m}\) refer to fiber and matrix, respectively.

Transcribed Image Text:

FIGURE 5.23 x L Fiber: A, E, , and Of Matrix: Am Em am and m