27.4 : Elastic Strain Energy for Shearing Stresses

As discussed in previous lessons, strain energy in a material is the energy stored when it is elastically deformed, a concept crucial in materials science and mechanical engineering. This energy results from the internal work done against the cohesive forces within the material. When a material undergoes shearing stress and corresponding shearing strain, the strain energy density, which is the energy stored per unit volume, is calculated. Within the elastic limit, where the stress is proportional to the strain through the modulus of rigidity, this strain energy density is proportional to the square of the shearing strain and the modulus of rigidity.

For practical applications, such as a shaft subjected to twisting from applied torques, calculating the total strain energy becomes essential. The shearing stress in a shaft can be determined by the internal torque and the shaft's polar moment of inertia. When integrated over the shaft's volume, this stress yields the total strain energy. In cylindrical shafts, this integration involves the cross-sectional area and the length of the shaft, reflecting how geometry and material properties like the modulus of rigidity influence the material's ability to resist deformation and store energy. This understanding is vital for designing mechanical structures that are both resilient and capable of efficiently enduring operational stresses.