[Paper Review] Fractional-Order Structural Stability: Formulation and Application to the Critical Load of Slender Structures

Sai Sidhardh, Sansit Patnaik+1 more|arXiv (Cornell University)|Aug 15, 2020

Nonlocal and gradient elasticity in micro/nano structures47 references5 citations

TL;DR

This paper introduces a fractional-order continuum framework for analyzing structural stability in nonlocal solids, using space fractional-order derivatives to model long-range interactions. It derives a thermodynamically consistent energy-based stability criterion and shows that nonlocal effects simultaneously reduce both material and geometric stiffness, leading to more accurate critical buckling load predictions than classical nonlocal models that affect only one stiffness term.

ABSTRACT

This study presents the framework to perform a stability analysis of nonlocal solids whose response is formulated according to the fractional-order continuum theory. In this formulation, space fractional-order operators are used to capture the nonlocal response of the medium by introducing nonlocal kinematic relations. First, we use the geometrically nonlinear fractional-order kinematic relations within an energy-based approach to establish the Lagrange-Dirichlet stability criteria for fractional-order nonlocal structures. This energy-based approach to nonlocal structural stability is possible due to a positive-definite and thermodynamically consistent definition of deformation energy enabled by the fractional-order kinematic formulation. Then, the Rayleigh-Ritz coefficient for the critical load is derived for linear buckling conditions. The fractional-order formulation is finally used to determine critical buckling loads of slender nonlocal beams and plates using a dedicated fractional-order finite element solver. Results establish that, in contrast to existing studies, the effect of nonlocal interactions is observed on both the material and the geometric stiffness, when using the fractional-order kinematics approach. We support these observations quantitatively with the help of case studies focusing on the critical buckling response of fractional-order nonlocal slender structures, and qualitatively via direct comparison of the fractional-order approach with the classical nonlocal approaches.

Motivation & Objective

• To develop a thermodynamically consistent, energy-based stability analysis framework for nonlocal solids using fractional-order continuum theory.
• To address limitations in classical nonlocal elasticity models, such as ill-posedness and thermodynamic inconsistencies, particularly in stability analysis.
• To enable accurate prediction of critical buckling loads in slender structures by capturing nonlocal effects on both material and geometric stiffness.
• To demonstrate the superiority of fractional-order models over classical integral and differential nonlocal models through case studies on beams and plates.

Proposed method

• Formulates geometrically nonlinear fractional-order kinematic relations using space fractional-order derivatives to model nonlocal interactions.
• Applies the Lagrange-Dirichlet theorem to establish energy-based stability criteria for fractional-order nonlocal structures.
• Derives the Rayleigh-Ritz expression for the critical buckling load under linear buckling conditions using the fractional-order formulation.
• Develops a fractional-order finite element solver to numerically solve eigenvalue problems for beams and plates.
• Uses a positive-definite deformation energy density derived from fractional-order kinematics to ensure thermodynamic consistency.
• Compares results with classical nonlocal models (Eringen’s integral and differential formulations) to highlight differences in stiffness modification.

Experimental results

Research questions

• RQ1 How can a thermodynamically consistent energy-based stability analysis be formulated for nonlocal solids using fractional calculus?
• RQ2 In what way do fractional-order models modify both material and geometric stiffness compared to classical nonlocal models?
• RQ3 What is the impact of nonlocality on the critical buckling load of slender beams and plates when both stiffness terms are affected?
• RQ4 How does the fractional-order formulation resolve the ill-posedness and thermodynamic inconsistencies present in classical nonlocal elasticity theories?

Key findings

• The fractional-order formulation enables a positive-definite and thermodynamically consistent deformation energy density, making energy-based stability analysis feasible.
• Nonlocal effects in the fractional-order model reduce both material stiffness [K^b] and geometric stiffness [G^b], unlike classical models that affect only one.
• Parametric studies show that increasing nonlocality reduces both stiffness terms, leading to a competing effect on the critical buckling load.
• The fractional-order model predicts lower critical loads than classical local elasticity, consistent with experimental observations of nonlocal behavior.
• Compared to Eringen’s integral model, the fractional model avoids ill-posedness and ensures well-posed, self-adjoint operators even in the limit of pure nonlocality.
• The differential model of nonlocal elasticity modifies only the geometric stiffness [G^b], while the fractional model modifies both [K^b] and [G^b], offering a more comprehensive representation of nonlocal effects.