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

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 ΔT, determine the fiber stress, σf , and the matrix stress, σm, 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 αf and αm, the cross-sectional areas Af and Am, the Young’s moduli Ef and Em, and the temperature change, ΔT, where the subscripts f and m refer to fiber and matrix, respectively