2.3. Torsional Energy

In case of torsion, the state of stress in cylindrical coordinates system is

$\tau_{r\theta} = \frac{T}{J} \cdot r$

where $\tau_{r\theta}$ represents the shear stress, $T$ is the internal torque or twisting moment, and $J$ is the polar moment of inertia.

This chunk introduces the concept of torsional stress, which is the stress experienced by a material when it is twisted. The formula given signifies that the shear stress at any point within a cylindrical object is proportional to the torque applied and depends on the distance from the center of the object. Here, $r$ is the radial distance from the centroid to the point where the shear stress is being considered, and $J$ is a geometric property that depends on the shape of the cross-section.

Imagine twisting a wet towel. The harder you twist, the more stress you put on the towel fibers. Similarly, when a cylindrical object, like a metal rod, is twisted, the material undergoes shear stresses which can eventually lead to failure if the torque exceeds its capacity.

We had also derived earlier that

$\phi = \frac{1}{2} \tau_{r\theta} \cdot \gamma_{r\theta}$

Here $\phi$ is the energy density related to torsion, and $\gamma_{r\theta}$ is the shear strain.

In this chunk, the energy density due to torsional deformation is presented. It states that the energy stored in a material when twisted is proportional to both the shear stress and the corresponding shear strain. This means that the greater the twisting, or the higher the stress, the more energy is stored in the material because of its deformation.

Think about winding up a rubber band. As you twist it more, it gains energy. This energy can be released when you let go, sending the band flying. Similarly, materials store energy when subjected to torsion, and this can be released as mechanical energy.

Thus, the cross-sectional energy will be

$E_{T} = \int_{0}^{L} \phi \, dV$

where $V$ is the volume of the cross-section under consideration.

The formulation for the total torsional energy represents the total energy stored in an object subjected to twisting. The integration takes into account the energy density across the entire volume of the material. By integrating from the start to the end of the length (L) of the object, we can assess the total energy stored due to torsional deformation.

Consider a piece of twisted wire. If we want to find out how much energy is stored in that wire due to its twisted shape, we would look at each small section of the wire, calculate the energy for that section, and then add it all together, just like summing numbers in a calculation.

The final integration over the cross-section yields:

$E_{T} = \int_{0}^{A} \phi \, dA$

where $A$ is the area of the cross-section.

This expression summarizes the process of finding the total torsional energy stored within a cross-section. By integrating the energy density over the area, you ensure that all parts of the cross-section are accounted for in terms of how much energy they contribute due to torsion.

Imagine covering a surface with paint. To know how much paint is required, you need to know the total area. In the same manner, when calculating the torsional energy, we assess the entire cross-section area to determine the total energy storage capacity due to twisting.

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