Strain Energy in a Solid Circular Shaft Subjected to Torsion

Torsion is a type of mechanical loading that causes twisting of an object around its longitudinal axis. When a circular shaft is subjected to torsion, it experiences shear stress distribution across its cross-section, resulting in the deformation of the material. This deformation results in the storage of strain energy within the shaft.

Strain Energy Proportional to 1/GJ

The strain energy stored in a solid circular shaft subjected to torsion is proportional to 1/GJ, where G is the shear modulus of the material and J is the polar moment of inertia of the shaft's cross-section.

Let's understand why the strain energy is proportional to 1/GJ in more detail:

Shear Strain Energy
When a shaft is subjected to torsion, shear stress is induced in the material. This shear stress causes shear strain, which is the angular deformation of the shaft. The work done by the shear stress in deforming the material results in the storage of strain energy.

Shear Modulus (G)
The shear modulus (G) is a material property that represents the ratio of shear stress to shear strain within the elastic range. It quantifies the material's resistance to deformation under shear stress. A higher shear modulus indicates a stiffer material.

Polar Moment of Inertia (J)
The polar moment of inertia (J) is a geometric property of the shaft's cross-section that determines its resistance to torsional deformation. It is analogous to the moment of inertia in bending. A larger J value indicates a greater resistance to torsion.

Proportional Relationship
The strain energy stored in a circular shaft subjected to torsion is directly proportional to the shear modulus (G) and inversely proportional to the polar moment of inertia (J). Mathematically, it can be expressed as:

Strain Energy ∝ 1/(GJ)

This relationship implies that increasing the shear modulus or decreasing the polar moment of inertia will result in a higher strain energy storage in the shaft.

Conclusion
In summary, the strain energy stored in a solid circular shaft subjected to torsion is proportional to 1/(GJ), where G is the shear modulus and J is the polar moment of inertia. This relationship highlights the importance of material stiffness and shaft geometry in determining the amount of strain energy stored during torsional loading.