Understanding Modulus of Elasticity

The modulus of elasticity, often denoted by $E$, is a fundamental material property that measures its stiffness. It describes the relationship between stress and strain in the elastic region of deformation.

Stress-Strain Relationship

The relationship is defined by Hooke's Law for uniaxial stress:

$ \sigma = E \epsilon $

where:

• $ \sigma $ is the applied stress
• $ E $ is the modulus of elasticity
• $ \epsilon $ is the resulting strain

Implication of Infinite Modulus of Elasticity

The question states that a material has an infinitely large modulus of elasticity ($E \to \infty$). We can rearrange Hooke's Law to solve for strain:

$ \epsilon = \frac{\sigma}{E} $

If $ E $ approaches infinity ($ E \to \infty $) while the stress $ \sigma $ remains finite, the strain $ \epsilon $ must approach zero:

$ \epsilon \to \frac{\sigma}{\infty} \to 0 $

A strain of zero means there is no deformation or change in shape, regardless of the applied stress.

Material Classification

Based on the implication of zero strain:

• A material that does not deform under any amount of stress is defined as perfectly rigid.
• Perfectly elastic materials have a finite $E$ and return to their original shape.
• Perfectly plastic materials deform permanently after yielding.
• Perfectly flexible materials would deform easily, implying a very low or zero modulus of elasticity.

Conclusion

Therefore, a material with an infinitely large modulus of elasticity ($E \to \infty$) is considered perfectly rigid because it experiences zero strain under any applied stress.