(a) Write an expression for the modulus of elasticity for a hybrid composite in which all fibers of both types are oriented in the same direction. (b) Using this expression, compute the longitudinal modulus of elasticity of a hybrid composite consisting of aramid and glass fibers in volume fractions of 0.25 and 0.35 respectively, within a polyester resin matrix \(\left[E_{m}=4.0 \mathrm{GPa}\left(6 \times 10^{5} \mathrm{psi}\right)\right]\)

The Rule of Mixtures for anisotropic materials states that the modulus of elasticity of the composite (E_c) can be calculated by summing the product of each constituent's modulus of elasticity (E_i) and its volume fraction (V_i): \(E_c = \sum_i E_i V_i\) In this case, we have two types of fibers embedded in a matrix. We can consider the matrix as a third component in the composite. Therefore, the expression for the modulus of elasticity of a hybrid composite becomes: \(E_c = E_{f1} V_{f1} + E_{f2} V_{f2} + E_{m} V_{m}\) Where: \(E_c\) = Modulus of elasticity of the hybrid composite; \(E_{f1}\) = Modulus of elasticity of fiber type 1; \(E_{f2}\) = Modulus of elasticity of fiber type 2; \(E_m\) = Modulus of elasticity of the matrix; \(V_{f1}\) = Volume fraction of fiber type 1; \(V_{f2}\) = Volume fraction of fiber type 2; \(V_m\) = Volume fraction of the matrix. (Note: \(V_{f1} + V_{f2} + V_{m} = 1\)) #b) Calculate the longitudinal modulus of elasticity#

We are given the following information about the composite material: 1) Volume fraction of aramid fibers: \(V_{f1} = 0.25\); 2) Volume fraction of glass fibers: \(V_{f2} = 0.35\); 3) Volume fraction of the polyester resin matrix: \(V_m = 1 - (V_{f1} + V_{f2})\); 4) Modulus of elasticity of the matrix (in GPa): \(E_m = 4.0 \mathrm{GPa} (6 \times 10^{5} \mathrm{psi})\). In addition, we can look up the modulus of elasticity for the aramid and glass fibers in the literature. Here, we will use the following values: 1) Modulus of elasticity of aramid fibers: \(E_{f1} = 70 \mathrm{GPa}\); 2) Modulus of elasticity of glass fibers: \(E_{f2} = 73 \mathrm{GPa}\).

Now, we can substitute all these values into our derived expression from part (a) to find the longitudinal modulus of elasticity for the hybrid composite: \(E_c = E_{f1} V_{f1} + E_{f2} V_{f2} + E_{m} V_{m}\) \(E_c = 70 \times 0.25 + 73 \times 0.35 + 4.0 \times (1 - (0.25 + 0.35))\)

Now, we can simply calculate the value for \(E_c\): \(E_c = 17.5 + 25.55 + 1.6 = 44.65 \mathrm{GPa}\) Hence, the longitudinal modulus of elasticity of the given hybrid composite consisting of aramid and glass fibers within a polyester resin matrix is 44.65 GPa.