(a) Verify that Equation 16.11, the expression for the ratio of fiber load to matrix load \(\left(F_{f} / F_{m}\right)\), is valid. (b) What is the \(F_{f} / F_{c}\) ratio in terms of \(E_{f}, E_{m}\), and \(V_{f}\) ?

To verify the validity of Equation 16.11, we need to apply the rule of mixtures to find the relationship between fiber load and matrix load. The rule of mixtures states that the composite modulus \((E_c)\) can be given by the following equation: \(E_c = E_fV_f + E_m(1 - V_f)\) Let's denote the stress in the fiber as \(\sigma_f\) and in the matrix as \(\sigma_m\). Since we have the fiber load-matrix load ratio as \(F_f/F_m\), we need to find the relationship between the stresses which can be obtained from the applied strain. The applied strain is assumed to be the same in both the fiber and the matrix, denoted as \(\epsilon\): \(\epsilon = \frac{\sigma_f}{E_f} = \frac{\sigma_m}{E_m}\) Now the fiber and matrix loads can be found with the following relations: \(F_f = A_f \sigma_f = V_f A \sigma_f\) \(F_m = A_m \sigma_m = (1 - V_f) A \sigma_m\) Using the stress relations we found above, we can rewrite these expressions for loads as: \(F_f = V_f A \frac{E_f}{E_m} \sigma_m\) \(F_m = (1 - V_f) A \sigma_m\) Finally, let's find the ratio \(F_f / F_m\): \(\frac{F_f}{F_m} = \frac{V_f A \frac{E_f}{E_m} \sigma_m}{(1 - V_f) A \sigma_m} = \frac{V_f E_f}{E_m - V_f E_m}\) Thus, we have successfully verified Equation 16.11 using the rule of mixtures.

Now, we need to find the fiber load-composite load ratio \((F_f/F_c)\) in terms of \(E_f, E_m,\) and \(V_f\). To do this, let's first find the stress in the composite \((\sigma_c)\). We can use the rule of mixtures: \(E_c = E_fV_f + E_m(1 - V_f)\) Now, considering the overall strain \(\epsilon\): \(\epsilon = \frac{\sigma_f}{E_f} = \frac{\sigma_c}{E_c}\) We already know the relationship between fiber load and matrix load, so let's rewrite fiber load in terms of composite load: \(F_f = \frac{V_f E_f}{E_m - V_f E_m} F_m = \frac{V_f E_f}{E_m - V_f E_m} (F_c - F_f)\) Rearranging the equation to find \(F_f / F_c\): \(\frac{F_f}{F_c} = \frac{V_f E_f}{(E_m - V_f E_m) + V_f E_f}\) The fiber load-composite load ratio \((F_f/F_c)\) in terms of \(E_f, E_m,\) and \(V_f\) has been found.