Question

An isotropic brittle material is tested in the universal testing machine. The stress-strain diagram for the material shows a bi-linear elastic behavior as shown in the figure given below. The strain energy density is _________ MJ m\(^{-3}\) (rounded off to 2 decimal places).

 

Hint:

For calculating strain energy density in bi-linear materials, use the area under the curve method, which is the sum of the areas of the individual triangles and rectangles.

Answer 1

The strain energy density for a bi-linear elastic material is given by the area under the stress-strain curve. For a bi-linear curve, the energy density can be calculated as: \[ U = \frac{1}{2} \sigma_1 \epsilon_1 + \frac{1}{2} (\sigma_2 - \sigma_1)(\epsilon_2 - \epsilon_1) \] Substituting the given values:
\( \sigma_1 = 100 \, {MPa} \),
\( \epsilon_1 = 0.01 \),
\( \sigma_2 = 120 \, {MPa} \),
\( \epsilon_2 = 0.015 \),
\[ U = \frac{1}{2} \times 100 \times 0.01 + \frac{1}{2} \times (120 - 100) \times (0.015 - 0.01) = 1 \, {MJ/m}^3 \]