7. Variation of γ and τ in the cross section

We know

(46)

Note that both the quantities vary linearly with r. This means that whenever we twist a cylinder/bar, shear strain and shear stress vary linearly in the cross-section as we go outwards from the center (see Figure 4).

This section starts by establishing that shear strain (γ) and shear stress (τ) increase in a linear fashion from the center of a twisted cylinder outward to the edge. The equation referenced, (46), likely defines the relationship for shear stress in terms of other factors like the radial distance (r). When a cylindrical object, such as a bar or hollow cylinder, is twisted, different points within the material experience different levels of shear stress and strain based on their distance from the center. This principle is crucial in understanding how materials behave under torsional loads.

Imagine twisting a rubber band. If you grasp its center and twist, you’ll notice that the outer parts of the rubber band stretch more than the parts closer to your fingers. In this analogy, the center is like the center of a cylinder, where the shear stress and strain are lower, while the outer parts represent the areas where shear stress and strain are highest.

We can also think of a composite cylinder made up of two different materials as shown in Figure 5. Supposing the inner part (up to radius r₁) is made up of Aluminium and the outer part is made up of Steel.

The materials are glued together.

This chunk introduces the concept of a composite cylinder, where two different materials, such as Aluminum and Steel, are joined together. When this composite cylinder is twisted, even though the shear strain (γ) remains continuous as it is determined by the twisting action, the shear stress (τ) behaves differently. The crucial point is that because Aluminum and Steel have different material properties (like shear modulus), there will be a discontinuity in shear stress at the interface where they meet. This means that at the boundary between the two materials, shear stress does not change smoothly but rather jumps from one value to another, forming a 'discontinuous' plot of shear stress across the cross-section.

Consider a sandwich made from two types of bread with different textures. If you press down on the sandwich, the filling (analogous to shear strain) is uniformly squished regardless of the bread type. However, the pressure experienced by each type of bread (analogous to shear stress) may differ significantly due to their material properties. Just like the change in pressure does not transition smoothly from one bread type to another, shear stress doesn't transition smoothly between materials with different shear moduli.

The plot of variation of γ and τ in the cross-section is shown in Figure 6. The variation of γ is shown by the continuous blue line while the variation of τ is shown by the discontinuous red line (which is piecewise linear): the slopes of the two straight lines are different and proportional to the shear modulus of the corresponding material.

This final chunk discusses how the graphical representation illustrates the concepts previously described. The shear strain (γ) is represented by a continuous line, indicating a smooth variation across the composite cylinder, while the shear stress (τ) is depicted with a piecewise linear graph, showing its discontinuity at the interface of the two materials. The differing slopes of the lines reflect the properties of each material, wherein the slope is proportional to the shear modulus: a steeper slope indicates a higher shear modulus, meaning that material can withstand higher shear stress for the same shear strain.

Think of riding a bike over a bridge made of two different materials—concrete and wood. As you ride over the concrete, it feels solid and stable, but when you hit the wooden part, it might flex a little differently. Just like this experience varies depending on the material, the graphs here demonstrate different behaviors of shear stress and strain in the aluminum and steel sections of the composite cylinder.

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