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Nonlocal elasticity and boundary condition paradoxes: a review

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Abstract

Nonclassical continuum mechanics theories have seen a rise in implementation over the past several years due to the increased research into micro-/nanoelectromechanical systems (MEMS/NEMS), micro-/nanoresonators, carbon nanotubes (CNTs), etc. Typically, these systems exist in the range of several nanometers to the micro-scale. There are several available theories that can capture phenomena inherent to nanoscale structures. Of the available theories, researchers utilize Eringen’s nonlocal theory most frequently because of its ease of implementation and seemingly accurate results for specific loading conditions and boundary conditions. Eringen’s integral nonlocal theory, which leads to a set of integro-partial differential equations, is difficult to solve; therefore, the integral form was reduced to a set of singular partial differential equations using a Green-type attenuation function. However, a so-called paradox has arisen between the integral and differential formulations of Eringen’s nonlocal elasticity. For certain boundary and loading conditions, instead of the expected softening effect inherent in nonlocal particle interactions, some researchers have found a stiffening effect. Still, others have found no variation from those results found using classical theories. As such, the discrepancies between the integral and differential forms have been the subject of debate for nearly two decades, with several proposed resolutions published in recent years. This paper serves to review and consolidate existing theories in nonlocal elasticity along with selected theories in nonclassical continuum mechanics, the utilization of Eringen’s nonlocal elasticity in beams, shells, and plates, the existing discrepancies and proposed solutions, and recommendations for future work.

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Abbreviations

C-C:

Clamped-clamped

CCCC:

All clamped plate boundary conditions

CCFF:

Two sides clamped, two sides free boundary conditions

C-F:

Clamped-free (cantilever)

C-FP:

Cantilever beam with concentrated load P at the free end

C-H:

Clamped-hinged

CNC:

Carbon nanocone

CNT:

Carbon nanotube

CPT:

Classical plate theory

DQM:

Differential quadrature method

DWCNT:

Double-walled carbon nanotube

EBT:

Euler-Bernoulli beam theory

FEM:

Finite element method

FGM:

Functionally graded material

FOPT:

First-order plate theory

H-H:

Hinged-hinged

HHCC:

Two sides hinged, two sides clamped boundary conditions

HHHH:

All hinged plate boundary conditions

HOPT:

Higher-order plate theory

HSDT:

Higher-order shear deformation theory

KPT:

Kirchhoff’s plate theory

MEMS:

Micro-electromechanical system

MPT:

Mindlin’s plate theory

MWCNT:

Multiwalled carbon nanotube

NEMS:

Nanoelectromechanical system

SDM:

Stress-driven model

SGM:

Strain gradient method

SLGS:

Single-layer graphene sheets

SSSD:

Simply supported beam with uniformly distributed load

SWCNT :

Single-walled carbon nanotube

TBT:

Timoshenko beam theory

WRA:

Weighted residual approach

A :

Cross-sectional area

C :

Compliance tensor

E :

Young’s modulus

e 0 :

Material parameter

f :

Body force

F :

Distributed loading

I :

Second area moment of inertia

L :

Length

l c :

Nonlocal length scale parameter

M :

Bending moment

m 0 :

Mass density

P :

Concentrated point force

Q :

Shear force

q :

Axial force

T :

Kinetic energy

U :

Potential energy

u :

Displacement field

V :

Elastic domain

w :

Transverse displacement

α :

Regularizing parameter: gradient model

δ :

Dirac delta

ε :

Strain field

κ :

Nonlocal parameter

λ :

Elastic modulus

μ :

Shear modulus

ξ :

Volume fraction

ρ :

Material density

σ :

Nonlocal stress tensor

τ :

Local stress tensor

ω :

Frequency

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Funding

The authors, S. Ceballes and A. Abdelkefi, received funding support from the New Mexico Consortium, Los Alamos National Laboratory, and the National Science Foundation Graduate Research Fellowship Program.

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  1. Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM, 88003, USA

    S. Ceballes, K. Larkin, E. Rojas, S. S. Ghaffari & A. Abdelkefi

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  1. S. Ceballes
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  2. K. Larkin
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  3. E. Rojas
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  4. S. S. Ghaffari
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  5. A. Abdelkefi
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Correspondence to S. Ceballes.

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This article is part of the topical collection: Nanotechnology Convergence in Africa

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Ceballes, S., Larkin, K., Rojas, E. et al. Nonlocal elasticity and boundary condition paradoxes: a review. J Nanopart Res 23, 66 (2021). https://doi.org/10.1007/s11051-020-05107-y

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