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arxiv: 1906.09626 · v1 · pith:5POWKXNKnew · submitted 2019-06-23 · ⚛️ physics.app-ph

Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams

Raffaele Barretta , Marko Canadija , Raimondo Luciano , Francesco Marotti de Sciarra This is my paper

Pith reviewed 2026-05-25 17:38 UTC · model grok-4.3

classification ⚛️ physics.app-ph
keywords nonlocal thermoelasticitynanobeamsstress-driven modelHelmholtz kernelintegral formulationdifferential equivalenceconstitutive boundary conditions
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The pith

A stress-driven nonlocal integral model for thermoelastic nanobeams equals differential equations with higher-order boundary conditions when the Helmholtz kernel is used.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a nonlocal integral model for analyzing the thermoelastic response of nano- and microbeams that relies on stress-driven elasticity instead of the usual strain-driven formulations. It establishes that this integral model becomes equivalent to a system of differential equations together with higher-order constitutive boundary conditions precisely when the Helmholtz averaging kernel appears in the integral convolution. Exact solutions follow directly from the differential form, and these solutions serve as new reference cases for checking numerical schemes in nonlocal thermoelastic problems. A reader would care because the approach removes documented inconsistencies of strain-driven nonlocal models while still capturing size-dependent thermal effects in small-scale beams.

Core claim

The new thermoelastic nonlocal integral model is proven to be equivalent to an adequate set of differential equations, accompanied by higher-order constitutive boundary conditions, when the special Helmholtz averaging kernel is adopted in the convolution.

What carries the argument

The stress-driven nonlocal integral model with the Helmholtz averaging kernel, which converts the integral formulation into an equivalent differential problem supplied with higher-order constitutive boundary conditions.

If this is right

  • Exact closed-form nonlocal solutions become available for a range of thermoelastic nanobeam problems.
  • New benchmark cases are generated that can be used to verify numerical implementations of nonlocal thermoelasticity.
  • The formulation applies directly to nonisothermal structural analysis of elastic nano- and microbeams.
  • Higher-order constitutive boundary conditions must be enforced alongside the usual equilibrium and kinematic conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The differential form may simplify the construction of finite-element discretizations for larger thermoelastic nanostructures.
  • The same kernel-based equivalence could be examined for other structural members such as plates or frames.
  • If the boundary conditions prove stable in practice, the model could be inserted into existing beam-analysis software with minimal additional coding.

Load-bearing premise

The stress-driven elasticity theory effectively circumvents issues associated with strain-driven formulations.

What would settle it

Solve the integral model and the derived differential model for the same cantilever nanobeam under a uniform temperature change and check whether the two displacement and stress fields coincide exactly, including satisfaction of the higher-order boundary conditions.

Figures

Figures reproduced from arXiv: 1906.09626 by Francesco Marotti de Sciarra, Marko Canadija, Raffaele Barretta, Raimondo Luciano.

Figure 1
Figure 1. Figure 1: Coordinate system of a Bernoulli-Euler nanobeam [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Distribution of u0 along the beam loaded with the axial force and uniform temperature for various values of λ the nonlocal parameter effectively increases the beam stiffness. The nonlocal strain is obtained by differentiation as: ε = ∂xu0 = α∆θ + P AE − P 2AE  e x−L Lλ + e −x Lλ  . (48) Straightforward application of the Remark 2 enables simple calculation of stress from the equilibrium equation as σ = P… view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of u0 along the doubly clamped beam loaded with uniform temper￾ature for various values of λ [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A three-dimensional detail of distribution of [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dependence of the reaction in supports P on the nonloal parameter λ. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 λ u 0 (third term) P=-1 P=f(λ) [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Dependence of the third term in Eq. (53) on λ for a constant axial force and the one where P is dependent on the nonlocal parameter [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Distribution of the normal strain ε along the doubly clamped beam loaded with uniform temperature, λ = 0.1, lateral view. Inset in the top right corner - normal stress distribution σ = −1.33 used to obtain the normal strain. ature field as: ΦN = 1 A R Ω α∆θdA = αa1(e a2h−1) a2h , ΦM = 1 Iy R Ω α∆θzdA = 6αa1(a2h+e a2h(a2h−2)+2) a2 2h3 . (56) Since the temperature field does not depend on the longitudinal co… view at source ↗
Figure 8
Figure 8. Figure 8: Distribution of the axial displacement u0 along the doubly clamped beam loaded with the non-uniform temperature for λ ∈  10−4 , 0.05, 0.25, 5, 10 nm 0.0 0.2 0.4 0.6 0.8 1.0 -0.008 -0.006 -0.004 -0.002 0.000 L w 0.0001 0.05 0.25 5. 10 [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of the transverse displacement [PITH_FULL_IMAGE:figures/full_fig_p035_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of the bending angle ϕ along the doubly clamped beam loaded with the non-uniform temperature for λ ∈  10−4 , 0.05, 0.25, 5, 10 nm [PITH_FULL_IMAGE:figures/full_fig_p035_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Distribution of the curvature χ along the doubly clamped beam loaded with the non-uniform temperature for λ ∈  10−4 , 0.05, 0.25, 5, 10 nm [PITH_FULL_IMAGE:figures/full_fig_p036_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Distribution of the axial displacement u(x, z) along the doubly clamped beam loaded with the non-uniform temperature, lateral view. λ = 0.25 nm 0.0 0.2 0.4 0.6 0.8 1.0 -0.4 -0.2 0.0 0.2 0.4 -0.117 -0.039 0.039 0.117 0.195 0.273 0.351 0.429 [PITH_FULL_IMAGE:figures/full_fig_p036_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Distribution of the normal strain ε = εth + εel along the doubly clamped beam loaded with the non-uniform temperature, lateral view. λ = 0.25 nm [PITH_FULL_IMAGE:figures/full_fig_p036_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Distribution of the normal stress σ along the doubly clamped beam loaded with the non-uniform temperature, lateral view. λ = 0.25 nm 6. Conclusions A new thermoelastic integral model for nanobeams has been developed by adopting the nonlocal elasticity theory by G. Romano and R. Barretta [25]. Effectiveness of the presented methodology has been tested by examining selected case-studies. Some closing remark… view at source ↗
read the original abstract

A consistent stress-driven nonlocal integral model for nonisothermal structural analysis of elastic nano- and microbeams is proposed. Most nonlocal models of literature are strain-driven and it was shown that such approaches can lead toward a number of difficulties. Following recent contributions within the isothermal setting, the developed model abandons the classical strain-driven methodology in favour of the modern stress-driven elasticity theory by G. Romano and R. Barretta. This effectively circumvents issues associated with strain-driven formulations. The new thermoelastic nonlocal integral model is proven to be equivalent to an adequate set of differential equations, accompanied by higher-order constitutive boundary conditions, when the special Helmholtz averaging kernel is adopted in the convolution. The example section provides several applications, thus enabling insight into performance of the formulation. Exact nonlocal solutions are established, detecting also new benchmarks for thermoelastic numerical analyses.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper proposes a stress-driven nonlocal integral model for thermoelastic nanobeams that abandons strain-driven formulations in favor of the Romano-Barretta stress-driven approach. It claims that, with the Helmholtz kernel, the integral model is equivalent to a differential system plus higher-order constitutive boundary conditions, and supplies exact solutions for example problems as new benchmarks.

Significance. If the equivalence holds with the thermal term properly incorporated, the work supplies a consistent nonlocal framework for nonisothermal nanobeam analysis and extends the authors' prior isothermal results with verifiable exact solutions.

major comments (1)
  1. The central equivalence claim (integral model to differential system plus higher-order BCs) must be shown to survive the addition of the local thermal strain in the constitutive relation. The abstract states that the proof follows the isothermal Romano-Barretta construction, but the thermal contribution must be tracked explicitly through the inversion step and BC extraction to confirm that no extraneous nonlocal thermal terms appear.
minor comments (1)
  1. Notation for the thermal strain term and its placement relative to the nonlocal stress integral should be clarified in the constitutive equation statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment on the central equivalence claim. We address the point below.

read point-by-point responses

  1. Referee: The central equivalence claim (integral model to differential system plus higher-order BCs) must be shown to survive the addition of the local thermal strain in the constitutive relation. The abstract states that the proof follows the isothermal Romano-Barretta construction, but the thermal contribution must be tracked explicitly through the inversion step and BC extraction to confirm that no extraneous nonlocal thermal terms appear.

    Authors: The equivalence is derived directly in the manuscript for the thermoelastic constitutive relation that includes the additive local thermal strain. Application of the differential operator associated with the Helmholtz kernel inverts the nonlocal integral relation; because the thermal strain is strictly local, it is unaffected by the inversion and appears unchanged as a local forcing term in the resulting differential equation. The higher-order constitutive boundary conditions arise solely from the kernel boundary properties and remain identical to the isothermal case, with no thermal dependence. The derivation therefore introduces no extraneous nonlocal thermal terms. While the abstract notes that the model follows the stress-driven framework of prior isothermal work, the thermoelastic proof is self-contained and does not rely on an unadapted invocation of the isothermal result. revision: no

Circularity Check

1 steps flagged

Equivalence proof extends self-cited isothermal stress-driven framework

specific steps
  1. self citation load bearing [Abstract]
    "Following recent contributions within the isothermal setting, the developed model abandons the classical strain-driven methodology in favour of the modern stress-driven elasticity theory by G. Romano and R. Barretta. This effectively circumvents issues associated with strain-driven formulations. The new thermoelastic nonlocal integral model is proven to be equivalent to an adequate set of differential equations, accompanied by higher-order constitutive boundary conditions, when the special Helmholtz averaging kernel is adopted in the convolution."

    The equivalence proof for the thermoelastic model is presented as following the isothermal framework by G. Romano and R. Barretta (co-author on the present paper). The load-bearing step of mapping the integral constitutive relation (including thermal strain) to differential equations plus higher-order BCs therefore reduces to an extension of the self-cited prior construction rather than an independent derivation.

full rationale

The paper's central claim is an equivalence between the nonlocal integral model and a differential system plus boundary conditions for the thermoelastic case. This follows directly from the stress-driven theory introduced in prior work by co-author R. Barretta (with G. Romano). The abstract explicitly states the model 'follows recent contributions within the isothermal setting' and adopts the 'modern stress-driven elasticity theory by G. Romano and R. Barretta'. No independent derivation or external benchmark for the thermal extension is visible; the equivalence therefore inherits its validity from the self-cited isothermal construction. This is a moderate self-citation load-bearing issue but does not reduce the entire result to a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are detailed. The primary domain assumption is the superiority of the stress-driven approach over strain-driven models.

axioms (1)
  • domain assumption Stress-driven elasticity theory circumvents difficulties of strain-driven formulations
    Invoked in the abstract as the reason for adopting the Romano-Barretta framework.

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Reference graph

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