Fractional-Order Shell Theory: Formulation and Application to the Analysis of Nonlocal Cylindrical Panels

Abstract

We present a theoretical and computational framework based on fractional calculus for the analysis of the nonlocal static response of cylindrical shell panels. The differ-integral nature of fractional derivatives allows an efficient and accurate methodology to account for the effect of long-range (nonlocal) interactions in curved structures. More specifically, the use of frame-invariant fractional-order kinematic relations enables a physically, mathematically, and thermodynamically consistent formulation to model the nonlocal elastic interactions. In order to evaluate the response of these nonlocal shells under practical scenarios involving generalized loads and boundary conditions, the fractional-Finite Element Method (f-FEM) is extended to incorporate shell elements based on the first-order shear-deformable displacement theory. Finally, numerical studies are performed exploring both the linear and the geometrically nonlinear static response of nonlocal cylindrical shell panels. This study is intended to provide a general foundation to investigate the nonlocal behavior of curved structures by means of fractional order models.

BVP with a Load in the Form of a Fractional Integral

“…Lemma . Boundary value problems (15) and (16) are equivalently reduced to a Volterra integral equation of the second kind.…”

“…Lemma 5. For BVP (15), (16), there is continuity in the order β in the loaded term of equation (15).…”

“…Boundary value problems with spatial fractional integrals are also interesting for the current study and are planned for further research. From the applied side, for example, works by authors in [14,15] are relevant.…”