4.2. Application of boundary conditions

Whenever we solve a differential equation, we get unknown integrating constants. To obtain those constants, one has to apply boundary condition.

When we work with differential equations, especially in mechanics and physics, there are often constants that arise that we cannot determine without additional information. These constants are known as 'integrating constants'. To find these constants, we need to apply what are known as 'boundary conditions'. These conditions reflect the constraints or limits of the system we are studying and help us solve for the unknowns.

Think of boundary conditions like the rules of a game. Just as players must follow specific rules to play the game properly, we must adhere to boundary conditions to ensure our mathematical model accurately represents the physical situation.

Similarly, we need to identify the boundary conditions for our deformation problem. Figure 1 shows our cylinder which is subjected to pressure on its inner surface. The pressure acts as an externally applied known traction and thus can be used as one of the boundary conditions.

In a hollow cylinder subjected to pressure, the inner surface experiences a force due to the internal pressure. This pressure is a known quantity, acting as a boundary condition that helps us establish the relationship between stress and displacement within the cylinder. In addition to this, the outer surface of the cylinder typically experiences no external traction, providing us with another boundary condition.

Consider blowing up a balloon. The air inside applies pressure on the inner walls, causing them to stretch. The pressure represents an external condition that must be considered when understanding how the balloon will behave as it inflates.

We had seen earlier that the internal traction due to stress at the surface point equals the externally applied traction (t_app) through the following relation: σ_n = t_app.

The relation between internal stress (σ_n) and externally applied traction (t_app) is essential in understanding the balance of forces acting on the surface of the hollow cylinder. At any point on the surface of the cylinder, the stress from the internal pressure must match the traction applied from the outside to maintain equilibrium. This relationship helps in solving for unknowns in our equations.

Imagine a water-filled balloon again. The pressure from the water inside pushes outward, while the balloon material pushes back to keep its shape. The balance in this scenario is like ensuring that internal stress matches any external force applied.

The outwards surface normal of the inner curved surface (at r = r_1) points in -r direction, i.e., (22). Similarly, the external traction acts radially outward there, i.e., in the +r direction.

Understanding the direction of these forces is critical for setting up our equations correctly. The inward normal on the inner surface points towards the center, while the external traction on the outer surface points away. This directionality is important as it helps us determine the sign convention in our stress equations and ensures the correct physical interpretation of the model.

Visualize a tightrope walker pulling down on the ropes. The force (traction) he applies on one side (outward) must be countered by the tension on the other side (inward) for him to maintain his balance. Here, inward and outward forces balance each other, just like the forces acting on the cylinder.

Upon writing equation (21) in cylindrical coordinate system and further substituing the above two results, we get: or, σ(r_1) = -P, τ(r_1) = 0, τ(r_1) = 0.

By switching to a cylindrical coordinate system that aligns with the geometry of the hollow cylinder, we can express the boundary conditions more naturally. Here, we quantify the internal radial stress (σ) at the inner surface as equal to the negative of the internal pressure, while shear stresses (τ) at this boundary are zero, indicating no sliding at the surface.

Think about pouring syrup into a cylindrical glass. The syrups pressing the walls of the glass with a certain force (the negative of the pressure) while you also have no friction if the syrup is not touching the glass at the top, suggesting no other forces are acting at that point.

Similar analysis for the outer surface where no external traction is present yields σ(r_2) = 0, τ(r_2), τ(r_2) = 0.

For the outer surface of the cylinder, which is not under any external force, the stress must equal zero. This means that there is no additional pressure acting on the outer surface, leading to three equations that define the conditions necessary to solve for the integrating constants within our equations. These conditions ensure that we are correctly representing the physical constraints on the cylinder.

Imagine the exterior of an unpressurized soda can. There’s no force working on the outside like in a pressurized container, resulting in the can's surface not being stressed by an external force — hence the stress on the outer surface can be defined as zero.

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