Average wall stress in an aneurysm

[tandoorichicken]

Hints, please?

"An arterial aneurysm is a life threatening condition associated with a balloon like expansion of a local segment of an artery. Using a free body diagram determine the average wall stresses in a closed spherical aneurysm with wall thickness h and inner radius $r_i$.
Ignore blood flow for this purpose."

Intuitively, it makes sense that an increased stress would decrease the wall thickness h, making for a more unstable aneurysm. I would imagine an equation relating the normal wall stress and wall thickness as inversely proportional quantities, but once again as usual I'm having trouble squeezing the actual mathematical statements out of the info given.

A head-start maybe?

 

[OlderDan]

Without knowing the context of your problem, I might be off base here, but it seems to me there are two possible interpretations of this problem. One is that the stress is just the force per unit area on the inside of the sphere, but that is just the pressure of the blood contained and has nothing to do with the thickness of the walls. I'm guessing your problem is a more difficult problem related to the stress within the walls tending to rip it apart. That will depend on the wall thickness and on the blood pressure.

My guess is the place to start is to find the area of a cross section of the wall of the sphere along a plane cutting the sphere in half. The cross section would be a ring or "washer" with an inner radius r and an outer radius r+h. Then for a given blood pressure you would need to integrate over the wall of a hemisphere to find the total force acting on that hemisphere. That force is counteracted by the forces holding the two hemispheres together, and that force is distributed over the area of the ring. The ratio of the force to the area would be the stress.

Not sure about this, but absent any additional context and information, it's the way I interpret it.

 

[kyle8921]

I would approach this problem by first reviewing the basic principles of stress and strain in materials. In this case, the arterial wall is the material being stressed. The stress in a material is defined as the force per unit area applied to it, and can be calculated using the formula σ=F/A, where σ is the stress, F is the applied force, and A is the cross-sectional area of the material.

In the case of an aneurysm, the force acting on the arterial wall is the blood pressure inside the aneurysm. The cross-sectional area can be calculated using the formula A=4πr^2, where r is the inner radius of the aneurysm. So, the average wall stress can be calculated using the equation σ=F/4πr^2.

To determine the force acting on the arterial wall, we can use a free body diagram. In this case, the aneurysm can be approximated as a closed spherical shape. The force acting on the wall will be the blood pressure inside the aneurysm multiplied by the surface area of the aneurysm wall. This can be calculated using the formula F=pA, where p is the blood pressure and A is the surface area of the aneurysm wall.

Putting it all together, we can now calculate the average wall stress in the aneurysm using the equation σ=pA/4πr^2. This equation takes into account both the force acting on the wall and the cross-sectional area of the wall, providing a more accurate representation of the average stress in the aneurysm.

It is important to note that this calculation is a simplified model and does not take into account factors such as the shape and thickness of the aneurysm wall, as well as the blood flow within the aneurysm. These factors can have a significant impact on the stress distribution within the aneurysm and should be considered in a more comprehensive analysis.

In conclusion, by using basic principles of stress and strain, along with a free body diagram, we can calculate the average wall stress in an aneurysm. This information can be useful in understanding the mechanical properties of an aneurysm and potentially developing treatments to reduce the risk of rupture.