Analysis of nonlocal smart beams following fractional-order constitutive relations

Abstract

In this study, we develop a fractional-calculus based constitutive model for capturing nonlocal interactions over the multiphysics response in solids. More specifically, we develop constitutive relations for nonlocal piezoelectricity incorporating fractional-order kinematic relations to capture the long-range interactions over electrical and mechanical field variables. This study breaks new ground by developing fractional-order constitutive models for a two-way multiphysics (electro-mechanical) coupling, specifically the direct and converse piezoelectric effect. It is expected that long-range interactions over each field variable (elastic and electrical) can be leveraged to develop metastructures with enhanced multiphysics coupling. To better illustrate this, we choose the example of a smart beam composed of a nonlocal substrate and a piezoelectric layer. We establish the analytical and numerical framework to analyze nonlocal smart beams based on variational principles. The fractional-Finite Element (f-FE) numerical solver, facilitating multiphysics coupling, undergoes comprehensive validation through multiple case studies. Finally, detailed studies point towards tuning the multiphysics coupling possible via nonlocal interactions across the domain.

Figures (12)

• Figure 1: Schematic diagram of unimorph piezoelectric smart beam subject to a uniformly distributed transverse load $q_0(x_1)$.
• Figure 2: A schematic illustration of the nonlocal length scales within the piezoelectric patch and the substrate for the smart beam. Note that for the same point along the length of the smart beam, say P or R, the length scales in the piezoelectric patch and substrate are different. This renders differential nonlocal interactions across the piezoelectric patch and the substrate.
• Figure 3: f-FEM solution for (a) converse, and (b) direct piezoelectric effect, in a nonlocal smart beam for the different number of elements. (Transverse displacement in ($\mathrm{m})$ and electric potential in ($\mathrm{V}$))
• Figure 4: Transverse displacement (in $\mathrm{m}$) of the cantilever smart beam for $\alpha=1$ is compared with 3D FEA conducted in COMSOL multiphysics.
• Figure 5: Transverse displacement (in $\mathrm{m}$) of the simply supported smart beam for $q_0(x_1)=100~\mathrm{N/m}$ and $\phi=0~\mathrm{V}$.