Stress profile of pressurized bent shell

[ff12120]

Consider a pressurized cylindrical shell of radius r and pressure p, which at equilibrium has a nonvanishing in-plane stress components pr/2 and pr. This result is generically found by force-balancing.

However, if I bent a pressurized torus into this cylinder, then the pressure will still give stresses pr/2 and pr, but there should be a stress due to bending. How does this stress enter into the force balancing at mechanical equilibrium?

In a general elasticity problem, if we applied pressure to a material without changing its shape, the stress cannot be uniquely determined from the strains. Consequently, the stress-strain law only specifies the "deviatoric stress." Is there a "deviatoric stress" for shells, whose in-plane stresses are not the same (and so unlike hydrostatic stress)? In other words, suppose I have both a strain tensor and a pressurized shell. The force balance involves just the geometry of the final deformed state and the pressure. How does the strain tensor (which involves the knowledge of the undeformed state) enter into the force balance, which should uniquely determine the stresses in the shell?

 

[Chestermiller]

In some problems, such as your pressurized cylinder problem, the state of stress is statically determinate. So, once you determine the state of stress, you can then determine the strains.

In other problems, you can't determine the state of stress without taking into account the stress-strain response (Hooke's law).

In the case of your "straightened-out torus," the metal will have to yield to get it into its new configuration, and, when you let go, it will remain in its new stress-free state. If the released configuration happens to be a perfect cylinder and, if you attach end caps, when you pressurize it, the stresses will be pr/2 and pr.

Chet