Question

σ₁, σ₂, σ₃ are the three mutually perpendicular principal stresses with ε₁,ε₂ , and ε₃ being the strains produced in the respective directions of the stress, the strain energy stored per unit volume in a cube is

• A. $$\frac 12(\sigma_1^2\epsilon_1^2+\sigma_2^2\epsilon_2^2+\sigma_3^2\epsilon_3^2)$$
• B. $$(\sigma_1\epsilon_1+\sigma_2\epsilon_2+\sigma_3\epsilon_3)$$
• C. $$\frac 12(\sigma_1\epsilon_1^2+\sigma_2\epsilon_2^2+\sigma_3\epsilon_3^2)$$
• D. $$\frac 12(\sigma_1\epsilon_1+\sigma_2\epsilon_2+\sigma_3\epsilon_3)$$

Answer 1

Answer:

$$\frac 12(\sigma_1\epsilon_1+\sigma_2\epsilon_2+\sigma_3\epsilon_3)$$

Strain energy is the energy stored in a body due to deformation. For a 3-D body with mutually perpendicular principal stresses $σ_1, σ_2, and σ_3$ and strains$\epsilon_1,\epsilon_2, and \epsilon_3$, the strain energy is given by $1/2 ( \sigma_1 \epsilon_1+ \sigma_2 \epsilon_2+ \sigma_3 \epsilon_3)$ .