Analytical Solution for a Pressurized Thick-Walled Spherical Shell Based on a Simplified Strain Gradient Elasticity Theory

The problem of a pressurized thick-walled spherical shell is analytically solved using a simplified strain gradient elasticity theory. The closed-form solution derived contains a material length scale parameter and can account for microstructural effects, which qualitatively differs from Lame's solution in classical elasticity. When the strain gradient effect (a measure of the underlying material microstructure) is not considered, the newly derived strain gradient elasticity solution reduces to Lame's classical elasticity solution. To illustrate the new solution, a sample problem with specified geometrical parameters, pressure values and material properties is solved. The numerical results reveal that the magnitudes of both the radial and tangential stress components in the shell wall given by the current strain gradient solution are smaller than those given by Lame's solution. Also, it is quantitatively shown that microstructural effects can be large and Lame's solution may not be accurate for materials exhibiting significant microstructure dependence.