Elasticity and Elastic Limit
Whenever an external load acts on a material, the body experiences deformation. If the applied load is removed and the body fully regains its original size and shape (with all deformation disappearing), the material is called an elastic body. This characteristic — the ability of a material to return to its original configuration after the external force is withdrawn — is known as elasticity.
A body will return to its initial form only when the deformation produced by the force remains within a certain allowable range. There exists a specific maximum force up to which the deformation will completely vanish after unloading. This limiting value of force is termed the elastic limit.
If the applied stress goes beyond this elastic limit, the material will lose part of its elastic behavior. Even after removing the force, the body will not fully return to its original shape — a permanent deformation will remain.
Hooke’s Law and Elastic Moduli
Hooke’s Law states that, within the elastic limit of a material, stress is directly proportional to strain caused by the applied load. This implies that the ratio of stress to strain remains constant as long as the material is loaded within its elastic range.
This constant ratio is known as the Modulus of Elasticity or Elastic Modulus or simply the Elastic Moduli.
Modulus of Elasticity (Young’s Modulus)
The ratio of tensile stress or compressive stress to the corresponding strain is called Young’s Modulus or the Modulus of Elasticity. It is represented by the symbol E.
Mathematically:
E = (Tensile stress / Tensile strain)
or
E = (Compressive stress / Compressive strain)
E = σ / ε ...(1.5)
Modulus of Rigidity or Shear Modulus
The ratio of shear stress to the shear strain (within the elastic limit) is known as the Modulus of Rigidity or Shear Modulus. It is denoted by C, G, or N.
C (or G or N) = τ / φ ...(1.6)
Here, τ = shear stress and φ = shear strain.
Factor of Safety
The Factor of Safety (FOS) is defined as the ratio of the ultimate tensile stress of a material to its working (or permissible) stress. It is used to ensure safety in design.
Factor of Safety = Ultimate Stress / Permissible Stress ...(1.7)