The stress strain behavior of a material is shown in figure. Its

SOLUTION

Concept:

Modulus of resilience:

Resilience per unit volume is known as the modulus of resilience which is the area under the stress-strain curve up to the elastic limit.

Modulus of toughness:

Toughness per unit volume is known as the modulus of toughness which is the total area under the stress-strain curve.

Calculation:

Given:

\({U_{res}} = \frac{1}{2}{\sigma _e}\epsilon = \frac{1}{2} \times 70 \times {10^6} \times 0.004 = 14 \times {10^4}\;Nm/{m^3}\)

 

\({U_{tough}} = \) The total area under the stress-strain diagram.

\( = \left[ {\frac{1}{2} \times 70 \times 0.004 + \frac{1}{2} \times \left( {120 - 70} \right)\left( {0.012 - 0.004} \right) + 70\left( {0.012 - 0.004} \right)} \right] \times {10^6}\)

\(= \left[ {\frac{1}{2} \times 70 \times 0.004 + \frac{1}{2} \times 50 \times 0.008 + 70 \times 0.008} \right] \times {10^6} = 90 \times {10^4}\;Nm/{m^3}\)

 

Proof resilience

• It is defined as the maximum strain energy stored in a body.
• So, it is the quantity of strain energy stored in a body when strained up to the elastic limit (ability to store or absorb energy without permanent deformation).

Modulus of resilience

• It is defined as proof resilience per unit volume.
• It is the area under the stress-strain curve up to the elastic limit.

Modulus of toughness

• It is the ability to absorb energy up to fracture.
• From the stress-strain diagram, the area under the complete curve gives the measure of modules of toughness.