3. Relating stress and strain in cylindrical coordinate system for isotropic materials

Once we have stress and strain matrices in cylindrical coordinate system, let us relate them for an isotropic material.

In this part of the lecture, we start examining how stress (the internal forces per unit area within materials) and strain (the deformation caused by stress) relate to each other when dealing with isotropic materials—materials where properties are the same in all directions. Specifically, we are focusing on the cylindrical coordinate system, which is useful for problems involving objects with circular symmetry, like pipes or cylinders.

Imagine a rubber band being stretched. Whether you pull it sideways or from one end, if it’s the same rubber band, it will respond in the same way regardless of how you pull it. This concept relates to isotropic materials, as their response (stress and strain) remains consistent regardless of how they are oriented.

We know how to relate stress and strain in Cartesian coordinate system. We also know that for an isotropic material, all material properties are independent of the direction.

The properties of isotropic materials do not change with orientation. This means that whether we measure stress and strain in the x, y, or z directions (as in a Cartesian coordinate system) or in cylindrical coordinates, the fundamental relationships between stress and strain remain the same. The direction of measurement doesn’t affect how the material behaves under load.

Consider a basketball. No matter how you spin or hold it, it behaves identically when pressure is applied—if you press on it, it will squish in the same way regardless of how you are holding it. This illustrates the isotropic nature of the material.

The relationship between stress and strain components must also be independent of the coordinate system. This means that we could choose any set of three perpendicular directions and resolve our stress and strain tensors in those directions, but the mathematical form of their relationship would remain unchanged.

When working with isotropic materials, we can choose any convenient axis system since the mathematical relationships will not vary based on our choice. For instance, whether we use the equations derived for cylindrical coordinates or rectangular coordinates, the fundamental equations governing stress and strain will hold true. This allows engineers and scientists to simplify calculations based on the symmetry and shape of the structure they are analyzing.

Think of a chair. You can describe how strong it is (stress and strain properties) from different angles or perspectives (say, top-down, side-on, etc.). No matter how you analyze it, the strength and behavior of the chair remain the same, highlighting the freedom in direction selection with isotropic materials.

We can obtain all other relations in a similar way leading to the emphasized conclusion that the above relationship would have a different mathematical form if the material were not isotropic.

Once we understand that the relationships between stress and strain remain consistent across different coordinate systems for isotropic materials, we can derive various mathematical expressions that link these two concepts. However, it's key to note that if the material were anisotropic (having directional dependencies), the relationships would become more complex and change based on the specific directions of loading and properties of the material.

Imagine if a balloon only stretched more in certain directions when blown up, depending on how it was inflated. This would signify an anisotropic material. In contrast, if it stretched evenly in all directions regardless of how it was blown up, as in the case of our previous discussions, it is isotropic, simplifying our relationships significantly.

Having obtained stress and strain components and their relation in cylindrical coordinate system, we will learn in the next few lectures how using them for deformation of cylindrical bodies leads to simplified forms of equations.

With a firm understanding of the stress-strain relationship in cylindrical coordinates for isotropic materials, students will soon explore practical applications that involve cylindrical objects. This knowledge will pave the way for solving real-world engineering problems involving deformation in structures like pipes and cylinders, allowing simplifications which make calculations easier and more manageable.

For example, consider a pipe carrying water under pressure. By knowing how the material deforms under the pressure (stress-strain relations), engineers can design pipes that are efficient and safe, ensuring they don’t burst under pressure. This understanding will be built upon in the upcoming lectures.

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