2.2. Special Case II

We can even have a situation where λ = λ = λ, i.e., all the principal stress components are equal.

In this case, all three principal stress components are the same, indicating a special condition in stress states. When λ1, λ2, and λ3 are equal, it simplifies the stress analysis significantly. The stress conditions become isotropic, meaning that the stress is the same in all directions, leading to uniform behavior in materials. When we visualize this scenario using Mohr's Circle, it becomes evident that the entire region collapses to a single point.

Imagine a perfectly balanced balloon filled with water, where the water pressure is uniform in all directions. Just like the pressure inside the balloon, where there is no preferred direction for the pressure to exert force, the state of stress here is uniform and equal, showing the same effect on all planes. This is akin to how all sides of the balloon are equally distended.

In this case, the whole region shrinks to a point as shown in Figure 7. So, for any arbitrary plane, we can only have σ = λ and τ = 0.

When all principal stresses are equal, the interactions between forces on any slice of material behave similarly. This means that the normal stress (σ) on any arbitrary plane equals the equal principal stress, and there is no shear stress (τ) acting on those planes. Since τ represents the component of stress that causes deformation, its being zero means that no shear forces are induced on the material, representing a state of pure normal stress.

Consider a wet sponge that has been squeezed uniformly from all sides. Despite the pressure (σ) keeping it compressed, there are no strong forces trying to slide the layers of the sponge over each other (no τ) - thus, it does not shear or distort in any particular direction, much like our isotropic stress state.

In one of the past lectures, we had discussed the state of stress in a fluid in static equilibrium. In such fluids, all planes have only pressure acting on them and it acts in the normal direction. This case of λ = λ = λ is also called hydrostatic state of stress.

The hydrostatic state of stress describes a situation where the fluid is at rest and uniformly distributes pressure in all directions with no shear stress. Because fluids don't support shear stress, they only exert normal stress against the walls of their containers. This uniform pressure condition illustrates a fundamental principle in fluid mechanics and material science, showing how materials respond to pressure in an isotropic manner.

Think about a deep swimming pool. The deeper you go, the greater the water pressure that acts uniformly against your body from all sides. This situation mimics the hydrostatic stress condition, where the pressure is equal in all directions, demonstrating how fluids behave under static conditions, exerting pressure only in the normal direction (no shear stress exists).

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