2.2. Significance of ϵ θθ

This strain is also called hoop strain or circumferential strain. This is the elongation of a line element directed along θ direction (circumferential line element) as shown in Figure 2. The circumferential strain has two contributions. The partial derivative term is intuitive because longitudinal strain along a direction is understood as the derivative of displacement in that direction with respect to the same direction.

The hoop strain, represented as ϵ θθ, refers to the deformation experienced by an object when it is subjected to external forces, particularly in a cylindrical structure like a hollow cylinder. It describes how much longer a specific line element oriented around the circumference (θ direction) becomes when the object deforms. The 'two contributions' mentioned refer to two changes in length: one from direct displacement in the θ direction and another possibly from changes in the radial direction impacting how the material elongates in the θ direction instance.

Imagine a balloon. When you blow it up, the rubber will stretch uniformly around the balloon's circumference. The hoop strain is akin to measuring how much longer a particular line around the circumference of that balloon becomes as it inflates. Just like how the balloon visibly expands, so too does any line running around its middle!

The other term is the unusual term which we now try to understand physically. Think of a displacement which has only radial component, i.e., u ≠ 0, u = 0, u = 0 (13). For such a displacement, if we find ϵ using equation (12), we will get θθ.

In this context, when we consider a displacement that only occurs in the radial direction (u), the hoop strain ϵ θθ still manifests, indicating that the material will experience elongation around its circumference even when there is no movement in the θ direction. This illustrates a unique aspect of material deformation; changes in one direction can influence the strains in another due to the interconnectedness of material properties.

Again, consider an inflatable tire. When it is inflated, the inner parts push outwards (radially), but as it expands, the circumference increases as well. Even if you don't twist or stretch it side to side, the tire still fills out and extends, reflecting the hoop strain—showing how changes in one direction (radial) affect dimensions in another (hoop).

Figure 3 shows a typical cross section of a hollow cylinder. For the displacement given in (13), all points in the cross-section simply displace radially. We have also drawn a circumferential line both before and after deformation. All points on this line initially at radial coordinate r displaces to radial coordinate r+u.

This chunk describes how radial displacement influences the geometry of the object, particularly focusing on the circumferential lines. When the object deforms radially, every point moves outward. This displacement creates a scenario where the original circumferential length increases as the object expands. The subsequent increase in length results in measurable hoop strain even when no outward movement directly in the θ direction occurs.

Think of how an elastic band behaves. When you pull it apart from the sides, it becomes longer not just in the pulling direction, but also stretches circumferentially. It's a classic instance in which moving in one direction (radial) also stretches in another (hoop), perfectly exemplifying how hoop strain operates.

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