6.2. Relating axial force and axial strain

To find ϵ, let us obtain the axial force in the cross-section through the integration of σ_{zz}, i.e.,

(42)

Here, A denotes the cross-sectional area.

In order to determine the axial strain (ϵ), we need to calculate the axial force acting on a cross-section of the cylinder. This is done by integrating the stress component σ_{zz} across the area of the cross-section. The integral will add up the stresses distributed over the entire area A to find the total axial force F.

Think of it like calculating the total weight of water in a swimming pool. If you know the pressure at the surface and the area of the pool, you can figure out how much water is pushing down. Similarly, here we use the stress over an area to determine the force acting along the cylinder.

As we know the value of C from equation (29), we are finally able to relate axial force F with axial strain ϵ. In the special case when P=0, the expression of C becomes simpler which yields

(43) where E is the Young’s modulus of elasticity.

Once we have the total axial force, we can relate this force to the axial strain using a constant denoted as C. For instances where no pressure acts on the cylinder (P=0), this relationship becomes simpler and allows us to connect the axial force F directly with the axial strain ϵ. The Young's modulus (E) characterizes the material's stiffness and is an essential factor in determining how much strain results from a given stress.

Imagine stretching a rubber band. The strength of the rubber band (analogous to Young's Modulus) determines how far it will stretch when you pull it. In our case, when we pull on the hollow cylinder, the relationship shows how much it will stretch (strain) based on its material properties and the force applied.

In case of zero pressure, ϵ can also be obtained in a simpler way. We know that when P=0, we get σ_{rr} = σ_{θθ} = 0. Using three-dimensional Hooke’s law, we can then write

(44) The axial force F for such a situation is then

(45)

If there is no internal pressure acting on the cylinder, the stresses σ_{rr} and σ_{θθ} are both zero. This allows for a straightforward application of Hooke’s law, which helps us express the relationships between stress and strain in a simplified manner. The resulting formula can be used to find the axial force F under these conditions, which leads to a better understanding of how the hollow cylinder behaves without pressure.

Consider a balloon that is fully deflated. It might not have any pressure (just like our cylinder'), and thus, it wouldn’t exert any internal stress. If you were to apply a force to stretch it, you would be able to see how the material behaves without any internal tension. Here, we evaluate strain and force without the influence of an internal pressure that would normally complicate things.

We have finally derived the relationship between Ω and T as well as the relationship between ϵ and F. The constant EA is called stretching stiffness while the constant GJ is called torsional stiffness, i.e., The axial strain and twist are constant in the cross-section of the cylinder.

At this point, we've established how the axial strain (ϵ) is related to the axial force (F) and how the axial twist relates to torque (T). The product of Young’s modulus and the area (EA) describes the stiffness of the cylinder against stretching, while the shear modulus and polar moment of area (GJ) describe its resistance to twisting. Recognizing these relationships helps to understand how external forces influence the cylinder’s deformation.

Think of a strong, thick rubber band (EA representing stretching stiffness) and a skinny, weak rubber band (which may have lower EA). The thicker band can withstand more stretching without snapping. Similarly, when you twist these bands, the way they respond differs based on their material properties, highlighting the importance of stiffness related to different types of forces.

--