On nonlocal integral models for elastic nano-beams

Romano, Giovanni; Barretta, Raffaele; Diaco, Marina

doi:10.1016/j.ijmecsci.2017.07.013

Cited by 155 publications

(46 citation statements)

References 24 publications

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“…However, this also follows from (7) owing to the fact that the operator L 2 (Ω; R n ) u → Ω K(x, x )u(x ) dx ∈ L 2 (Ω; R n ) is compact, hence also a completely continuous operator (as any Hilbert-Schmidt integral operator) [14]. Proposition 2 implies that V A cannot be continuously embedded into L 2 (Ω; R n ).…”

Section: Discussion Of Eringen's Modelmentioning

confidence: 99%

“…3 In view of the previous discussion, in this case we cannot guarantee that the linear functional in the right hand side of (4) is bounded and therefore also existence of solutions to (4). More generally, equation (7) shows the equivalence between (4) and a Fredholm integral equation of the first kind with kernel K(x, x ) = −∆Ã(|x − x |). Fredholm equation of the first kind is a canonical ill-posed problem [16].…”

Section: Discussion Of Eringen's Modelmentioning

confidence: 99%

“…An even easier way of seeing this is by noting that −∆ A(| · |) = 2δ(·) (or 2δ(·) − δ(· − 1) − δ(· + 1) if A(| · |) is extended by zero outside of Ω − Ω), in the sense of distributions. This can then be used in(7) immediately resulting in (u, v) A = (u, −∆ A(| · |) * v) L 2 (R) = 2(u, v) L 2 (R) = 2(u, v) L 2 (Ω) (or, respectively, in (u, v) A = (u, −∆ A(| · |) * v) L 2 (R) = 2(u, v) L 2 (R) − (u, v(· − 1)) L 2 (R) − (u, v(· + 1)) L 2 (R) = 2(u, v) L 2 (Ω) ).…”

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confidence: 99%

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From non-local Eringen’s model to fractional elasticity

Evgrafov

Bellido

2018

Mathematics and Mechanics of Solids

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Eringen's model is one of the most popular theories in nonlocal elasticity. It has been applied to many practical situations with the objective of removing the anomalous stress concentrations around geometric shape singularities, which appear when the local modelling is used. Despite the great popularity of Eringen's model in mechanical engineering community, even the most basic questions such as the existence and uniqueness of solutions have been rarely considered in the research literature for this model. In this work we focus on precisely these questions, proving that the model is in general ill-posed in the case of smooth kernels, the case which appears rather often in numerical studies. We also consider the case of singular, non-smooth kernels, and for the paradigmatic case of the Riesz potential we establish the well-posedness of the model in fractional Sobolev spaces. For such a kernel, in dimension one the model reduces to the well-known fractional Laplacian. Finally, we discuss possible extensions of Eringen's model to spatially heterogeneous material distributions.

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Section: Discussion Of Eringen's Modelmentioning

confidence: 99%

Section: Discussion Of Eringen's Modelmentioning

confidence: 99%

mentioning

confidence: 99%

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From non-local Eringen’s model to fractional elasticity

Evgrafov

Bellido

2018

Mathematics and Mechanics of Solids

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“…According to the stress-driven approach, the nonlocal elastic strain field is the convolution between the stress field and a suitable averaging kernel. Properties and merits of the stress-driven strategy in comparison with strain-driven formulations can be found in [26,27]. Transverse free vibrations of Bernoulli-Euler nanobeams are investigated in [1] by stress-driven integral approach.…”

Section: Introductionmentioning

confidence: 99%

Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams

Barretta

Čanađija

Sciarra

2018

International Journal of Engineering Science

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129

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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,

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“…Romano and Barretta, introduced the stress‐driven nonlocal integral model to study torsion of nanobeams, vibration of nanobeam and nanorods . In addition, the stress‐driven model was used in analysis of Euler‐Bernoulli nanobeams without taking the effect of chirality into account.…”

Section: Introductionmentioning

confidence: 99%

Doublet mechanical analysis of bending of Euler‐Bernoulli and Timoshenko nanobeams

2018

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By taking the effect of the scale parameter explicitly into account, bending behavior of Euler‐Bernoulli and Timoshenko nanobeams is studied using doublet mechanics. In addition, the effect of chirality on the softening or hardening behavior of an Euler‐Bernoulli nanobeam in bending is studied by taking the effect of the chiral angle explicitly into account. For the bending of the Timoshenko nanobeam the effect of the scale parameter and chirality on the axial and shear stresses is studied and the results are compared with the gradient and Eringen‐like models. It is shown that as the chiral angle increases, a simply supported Timoshenko nanobeam under a uniform loading changes from a softening to a hardening nanobeam indicating that chirality has a pronounced effect on the bending response of the nanobeams.

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On nonlocal integral models for elastic nano-beams

Cited by 155 publications

References 24 publications

From non-local Eringen’s model to fractional elasticity

From non-local Eringen’s model to fractional elasticity

Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams

Doublet mechanical analysis of bending of Euler‐Bernoulli and Timoshenko nanobeams

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