Materials & Structure Cheat Sheet
Chapter 2: Atomic Structure & Bonding
\(E = \int F \, dr\)
\(F = \frac{dE}{dr}\)
\(E_A = -\frac{A}{r}\)
\(E_R = \frac{B}{r^n}\)
\(F_A = \frac{(|Z_1|e)(|Z_2|e)}{4\pi\varepsilon_0 r^2}\)
\(\%IC = \{1 - \exp[-(0.25)(X_A - X_B)^2]\} \times 100\)
Chapter 3: Crystal Structure
\(a = 2R\sqrt{2}\)
\(\text{APF} = \frac{V_S}{V_C} = \frac{\text{volume of atoms in a unit cell}}{\text{total unit cell volume}}\)
\(a = \frac{4R}{\sqrt{3}}\)
\(\rho = \frac{nA}{V_C N_A}\)
\(q = \frac{a}{P_x}\)
\(u = n\left(\frac{x_1 - x_2}{a}\right)\)
\(u = \frac{1}{3}(2U - V)\)
\(U = n\left(\frac{a_1''-a_1'}{a}\right)\)
\(h = \frac{n a}{A}\)
\(\text{LD} = \frac{\text{number of atoms centered on direction vector}}{\text{length of direction vector}}\)
\(\text{PD} = \frac{\text{number of atoms centered on a plane}}{\text{area of plane}}\)
\(n\lambda = 2d_{hkl}\sin\theta\)
\(d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}\)
Chapter 4: Imperfections in Solids
\(N_v = N \exp\left(-\frac{Q_v}{kT}\right)\)
\(N = \frac{N_A \rho}{A}\)
\(C_1 = \frac{m_1}{m_1 + m_2} \times 100\)
\(C_1' = \frac{nm_1}{nm_1 + nm_2} \times 100\)
\(C_1' = \frac{C_1 A_2}{C_1 A_2 + C_2 A_1} \times 100\)
\(C_1 = \frac{C_1' A_1}{C_1' A_1 + C_2' A_2} \times 100\)
\(C_1'' = \frac{C_1}{\left(\frac{C_1}{\rho_1} + \frac{C_2}{\rho_2}\right)} \times 10^3\)
\(\rho_{\text{ave}} = \frac{100}{\frac{C_1}{\rho_1} + \frac{C_2}{\rho_2}}\)
\(A_{\text{ave}} = \frac{100}{\frac{C_1}{A_1} + \frac{C_2}{A_2}}\)
\(l = \frac{L_T}{P_M}\)
\(n = 2^{G-1}\)
\(n_M = \left(\frac{100}{M}\right)^2 (2^{G-1})\)
Chapter 5: Diffusion
\(J = \frac{M}{At}\)
\(J = -D\frac{dC}{dx}\)
\(\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2}\)
\(\frac{C_x - C_0}{C_s - C_0} = 1 - \text{erf}\left(\frac{x}{2\sqrt{Dt}}\right)\)
\(D = D_0 \exp\left(-\frac{Q_d}{RT}\right)\)
Chapter 6: Mechanical Properties
\(\sigma = \frac{F}{A_0}\)
\(\epsilon = \frac{l_i - l_0}{l_0} = \frac{\Delta l}{l_0}\)
\(\sigma = E\epsilon\)
\(\nu = -\frac{\epsilon_x}{\epsilon_z} = -\frac{\epsilon_y}{\epsilon_z}\)
\(\%\text{EL} = \frac{l_f - l_0}{l_0} \times 100\)
\(\%\text{RA} = \frac{A_0 - A_f}{A_0} \times 100\)
\(\sigma_T = \frac{F}{A_i}\)
\(\epsilon_T = \ln\frac{l_i}{l_0}\)
\(\sigma_T = K\epsilon_T^n\)
\(\text{TS(MPa)} = 3.45 \times \text{HB}\)
\(\text{TS(psi)} = 500 \times \text{HB}\)
\(\sigma_w = \frac{\sigma_y}{N}\)
Chapter 7: Dislocations and Strengthening Mechanisms
\(\tau_R = \sigma \cos\phi \cos\lambda\)
\(\tau_{\text{crss}} = \sigma_y(\cos\phi \cos\lambda)_{\text{max}}\)
\(\sigma_y = \sigma_0 + k_y d^{-1/2}\)
\(\%\text{CW} = \frac{A_0 - A_d}{A_0} \times 100\)
\(d^n - d_0^n = Kt\)
Chapter 9: Phase Diagrams
\(W_L = \frac{C_\alpha - C_0}{C_\alpha - C_L}\)
\(W_\alpha = \frac{C_0 - C_L}{C_\alpha - C_L}\)
\(V_\alpha = \frac{v_\alpha}{v_\alpha + v_\beta}\)
\(V_\alpha = \frac{\frac{W_\alpha}{\rho_\alpha}}{\frac{W_\alpha}{\rho_\alpha} + \frac{W_\beta}{\rho_\beta}}\)
\(W_\alpha = \frac{V_\alpha \rho_\alpha}{V_\alpha \rho_\alpha + V_\beta \rho_\beta}\)
\(W_e = \frac{P}{P + Q}\)
\(W_\alpha' = \frac{Q}{P + Q}\)
\(W_\alpha = \frac{Q + R}{P + Q + R}\)
\(W_\beta = \frac{P}{P + Q + R}\)
\(P + F = C + N\)
\(W_p = \frac{C_0' - 0.022}{0.74}\)
\(W_\alpha' = \frac{0.76 - C_0'}{0.74}\)
\(W_p = \frac{6.70 - C_1'}{5.94}\)
\(W_{\text{Fe}_3\text{C}'} = \frac{C_1' - 0.76}{5.94}\)
Chapter 10: Phase Transformations
\(r^* = -\frac{2\gamma}{\Delta G_v}\)
\(\Delta G^* = \frac{16\pi\gamma^3}{3(\Delta G_v)^2}\)
\(r^* = \frac{2\gamma T_m}{\Delta H_f}\left(\frac{1}{T_m - T}\right)\)
\(\gamma_{IL} = \gamma_{SI} + \gamma_{SL}\cos\theta\)
\(r^* = -\frac{2\gamma_{SL}}{\Delta G_v}\)
\(\Delta G^* = \frac{16\pi\gamma_{SL}^3}{3\Delta G_v^2}S(\theta)\)
\(y = 1 - \exp(-kt^n)\)
\(\text{rate} = \frac{1}{t_{0.5}}\)
Chapter 12: Structures and Properties of Ceramics
\(\rho = \frac{n'(\Sigma A_C + \Sigma A_A)}{V_C N_A}\)
\(\sigma_{fs} = \frac{3F_f L}{2bd^2}\)
\(\sigma_{fs} = \frac{2F_f L}{\pi R^3}\)
\(E = E_0(1 - 1.9P + 0.9P^2)\)
\(\sigma_{fs} = \sigma_0 \exp(-nP)\)