Stability analysis of gradient elastic circular cylindrical thin shells

The governing equation of equilibrium of gradient elastic circular cylindrical thin shells under axial compressive forces is explicitly derived. This is accomplished by appropriately combining the equations of equilibrium and the strain-displacement relations according to Donnell's theory and the stress-stain equations of a simple strain gradient elastic theory with just one constant (the internal length squared) in addition to the classical elastic moduli. The resulting partial differential equation in terms of the radial displacement of the shell is of the tenth order instead of the eighth, for the classical elastic case. The boundary value problem of buckling of a circular cylindrical shell with simply supported ends is solved analytically and the critical loading is obtained analytically-numerically. An assessment of the effect of the gradient coefficient (internal length) on the buckling load is made by comparing that load against the classical one.