27.4:
Elastic Strain Energy for Shearing Stresses
When the material is subjected to a shearing stress, the strain energy density can be expressed in integral form as the product of the shearing stress and the corresponding shearing strain.
Within the elastic limit, shearing stress is proportional to the shearing strain, with a constant of proportionality being the modulus of rigidity.
Performing the integration, the strain energy density is expressed as a product of the modulus of rigidity and shearing strain squared.
The corresponding strain energy can be calculated by integrating the strain energy density over a small volume element. This equation is valid only for elastic deformations.
Consider a shaft subjected to one or more twisting couples. The shearing stress of a cross-sectional area, A, located at a distance x from a fixed end, can be expressed in terms of internal torque and the polar moment of inertia J.
The stored strain energy in such a shaft can be expressed in integral form in terms of twisting torque. Rewriting the volume element in terms of cross-sectional area simplifies the expression.
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.
