7.1. Special case: composite cylinder

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.

A composite cylinder consists of two or more different materials. In this case, the inner section, which we refer to as the inner cylinder, is made of Aluminium, while the outer section is made of Steel. This setup is interesting because each material has distinct mechanical properties, which can significantly impact how the cylinder behaves under applied loads.

Think of a sandwich, where the bread is the strong outer layer (Steel), and the filling is the softer inner layer (Aluminium). Just like a sandwich has different layers that respond differently to pressure, a composite cylinder behaves according to the properties of both materials when forces are applied.

If we now twist the cylinder, the cross section is again going to rotate. The shear strain γ will be the same as earlier since it is completely prescribed by applied deformation and thus varies continuously.

When the composite cylinder is twisted, all parts of the cylinder rotate as a whole. The shear strain (γ) that occurs due to this twisting is uniform across the entire cylinder, meaning it changes continuously from the center to the edge of the cylinder. This is indicative of how composite structures can effectively respond to similar types of forces.

Imagine twisting a towel. As you twist from one end, every part of the towel rotates together. No matter how you twist, the strain is consistent throughout the length, similar to how the composite cylinder behaves when twisted.

However, when we calculate τ by equation (46), we will have different shear moduli for aluminium and steel. Thus, there will be a discontinuity in the shear stress at r = r1.

The difference in materials means they respond uniquely to stress. The equation for shear stress (τ) depends on the material’s shear modulus. Since Aluminium and Steel have different shear moduli, this leads to a sudden change or discontinuity in the shear stress at the interface of the two materials, specifically at the radius where they meet (r1).

Think of two people holding hands, where one is much stronger than the other. If they both pull in a direction, the stronger person may exert much more force than the weaker person can handle, leading to a situation where the weaker person's hand would begin to slip. This slipping represents a discontinuity – just like in our composite cylinder, where the different materials respond differently to the same force.

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.

In the graphical representation of shear strain (γ) and shear stress (τ), we see how γ varies smoothly throughout the cylinder, indicated by a continuous line. In contrast, τ has a piecewise linear representation indicating that it abruptly changes due to the differing materials at the interface. The slopes of these lines represent the shear modulus of Aluminium and Steel, highlighting how each material carries different amounts of stress.

Think of driving on a bumpy road and smooth pavement. On the bumpy road (discontinuity), you notice a sudden change in how your car handles, while on smooth pavement, it feels consistent. Similarly, the composite cylinder experiences a continuous shear strain across both materials but shows a sudden shift in shear stress where they meet.

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