material engineering
Modulus of Resilience
$$U_r = \int_{0}^{\epsilon_Y}\sigma\mathop{d\epsilon} \approx \frac{\sigma_{YS}^2}{2E}$$
Modulus of Rigidity/Shear Modulus
$$G = \frac{\tau}{\gamma}$$
Modulus of Toughness
$$U_t = \int_{0}^{\epsilon_f}\sigma\mathop{d\epsilon}$$
Poisson's Ratio
$$\nu = - \frac{\epsilon_x}{\epsilon_z} = -\frac{\epsilon_y}{\epsilon_z}$$
Thermal expansion
$$\Delta L = L_0 \alpha \Delta T$$