A continuous and aligned fibrous reinforced composite having a cross-sectional area of \(970 \mathrm{~mm}^{2}\) (1.5 in. \(\left.^{2}\right)\) is subjected to an external tensile load. If the stresses sustained by the fiber and matrix phases are 215 MPa (31,300 psi) and \(5.38\) MPa (780 psi), respectively, the force sustained by the fiber phase is \(76,800 \mathrm{~N}\left(17,265 \mathrm{lb}_{\mathrm{f}}\right)\), and the total longitudinal composite strain is \(1.56 \times 10^{-3}\), determine the following: (a) The force sustained by the matrix phase (b) The modulus of elasticity of the composite material in the longitudinal direction (c) The moduli of elasticity for fiber and matrix phases

First, we need to find the volume fractions of the fiber and matrix phases. We are given the total stress and stress for fiber and matrix phases. Let the volume fraction of fibers be denoted by \(V_f\) and the volume fraction of matrix be denoted by \(V_m\). Using the rule of mixtures for stress: $$\sigma_{total} = \sigma_fV_f + \sigma_mV_m$$ Where \(\sigma_{total}\) is the total stress, \(\sigma_f\) is the stress in the fiber phase, and \(\sigma_m\) is the stress in the matrix phase. We are given: \(\sigma_{total} = 215\ \mathrm{MPa}\), \(\sigma_f = 5.38\ \mathrm{MPa}\). We can rewrite the equation as: $$V_f = \frac{\sigma_{total} - \sigma_m}{\sigma_f - \sigma_m}$$ Now, we know that \(V_f + V_m = 1\). Therefore, we can calculate the volume fraction of fibers \(V_f\) and of the matrix \(V_m\).

We are given the force sustained by the fiber phase, denoted as \(F_f = 76,800\ \mathrm{N}\). To calculate the force sustained by the matrix phase, we will first calculate the total force carried by the composite. The stress in the composite is given as \(\sigma_{total} = 215\ \mathrm{MPa}\). We are also given the cross-sectional area of the composite as \(970\ \mathrm{mm}^2\). We can calculate the total force on the composite using: $$F_{total} = \sigma_{total} \times A$$ Where \(F_{total}\) is the total force and \(A\) is the cross-sectional area. Next, we can find the force sustained by the matrix phase (\(F_m\)) using: $$F_m = F_{total} - F_f$$

We are given the total longitudinal composite strain, denoted as \(\epsilon_{total} = 1.56 \times 10^{-3}\). We can calculate the modulus of elasticity of the composite material (\(E_c\)) using the following equation: $$E_c = \frac{\sigma_{total}}{\epsilon_{total}}$$

We can use the rule of mixtures for modulus to find the moduli of elasticity for fiber and matrix phases, denoted as \(E_f\) and \(E_m\), respectively. $$\frac{1}{E_c} = \frac{V_f}{E_f} + \frac{V_m}{E_m}$$ We can now solve the equation for \(E_f\) and \(E_m\) using the calculated values in steps 1 and 3.