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Deviatoric Stress Analysis in a Pressurized Cylindrical Shell

Consider a cylindrical shell with an internal radius ri and an external radius ro$,$ subjected to internal

pressure pi and external pressure po$.$ The cylinder is made of a homogeneous, isotropic material

with a known Young's modulus E and Poisson's ratio $.$ The shell is thick$-$walled, so plane stress or

plane strain assumptions are not sufficient, and a more advanced approach is required.

$1.$ Find the Deviatoric Stress Components:

Compute the deviatoric stress components at any point within the cylindrical shell.

Deviatoric stress is defined as the difference between the total stress and the hydrostatic

$($mean$)$ stress component. Express the deviatoric stresses in terms of the radial and

circumferential stresses and illustrate how these stresses vary across the thickness of the

shell.

$2.$ Analyze the Deviatoric Stress:

Plot the distribution of the deviatoric stress across the thickness of the cylindrical shell.

Identify the locations within the shell where the deviatoric stress reaches its maximum and

minimum values.

Your solution shall be detailed and contains

$$ Calculation and visualization of the deviatoric stress.

$$ Interpretation of the physical meaning and engineering importance of deviatoric stress.

$($NOTE: This solution is for mechanical engineet master degree level so take your time to solve it$)$