A fractional nonlocal time-space viscoelasticity theory and its applications in structural dynamics

Li, Li; Lin, Rongping; Ng, Teng Yong

doi:10.1016/j.apm.2020.03.048

Cited by 48 publications

(9 citation statements)

References 70 publications

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“…Here, DF n is the applied force normal to the differential area DS, which in turn means that the length of interest should be ''small enough at macro-scale'' but ''large enough at micro-scale'' to obey the continuum axiom [78]. In the case of nanoscale, materials or structures are naturally discrete, and therefore the classical mechanics is questionable.…”

Section: Nonlocal Elastic Theorymentioning

confidence: 99%

Nonlocal vibration of functionally graded nanoplates using a layerwise theory

Belarbi

Houari

et al. 2022

Mathematics and Mechanics of Solids

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This work studies the size-dependent free vibration response of functionally graded (FG) nanoplates using a layerwise theory. The proposed model supposes not only a higher-order displacement field for the core but also the first-order displacement field for the two face sheets, thereby maintaining an interlaminar displacement continuity among layers. Unlike the conventional layerwise models, the number of variables is kept fixed and does not increase for an increased number of layers. This is a very important feature compared to conventional layerwise models and facilitates significantly the engineering analyses. The material properties of the FG nanoplate are graded continuously through the thickness direction in accordance with a power-law function. The Eringen’s nonlocal elasticity theory is here adopted to relax the continuum axiom required in classical continuum mechanics and hence hopeful to capture the small size effects of naturally discrete nanoplates. The equations of motion of the problem are obtained via a classical Hamilton’s principle. The present layerwise model is implemented with a computationally efficient C0- continuous isoparametric serendipity elements and applied to solve a large-scale discrete numerical problem. The robustness and reliability of the developed finite element model are demonstrated by a comparative evaluation of results against predictions from literature. The comparative studies show that the proposed finite element model is: (a) free of shear locking, (b) accurate with a fast rate convergence for both thin and thick FG nanoplates, and (c) excellent in terms of numerical stability. Moreover, a detailed parametric analysis checks for the sensitivity of the vibration response of FG nanoplates to the aspect ratio, length-to-thickness ratio, nonlocal parameter, boundary conditions, power-law index, and modes shapes. Referential results are also reported, for the first time, for natural frequencies of FGM nanoplates which will serve as benchmarks for further computational investigations.

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Section: Nonlocal Elastic Theorymentioning

confidence: 99%

Nonlocal vibration of functionally graded nanoplates using a layerwise theory

Belarbi

Houari

et al. 2022

Mathematics and Mechanics of Solids

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“…From the above, we can clearly observe the G and B are full rank matrices, then the numerical schemes Equations (19) and (20) are uniquely solvable.…”

Section: Solvability Of the Implicit Difference Schemementioning

confidence: 86%

“…In order to prove the unique solvability of the numerical equation, we rewrote Equations (19) and (20) in the matrix form as follows:…”

Section: Solvability Of the Implicit Difference Schemementioning

confidence: 99%

“…For the models discussed above, the space fractional derivative or the time fractional derivative the fractional parameters were usually used respectively. Recently, the time-space fractional models have been applied as a more effective tool than single-time or space fractional models [18][19][20]. Chen et al [21] investigated the unsteady boundary layer flow with a time-space fractional derivative on a moving surface.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Solutions of Unsteady Boundary Layer Flow with a Time-Space Fractional Constitutive Relationship

et al. 2020

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In this paper, we develop a new time-space fractional constitution relation to study the unsteady boundary layer flow over a stretching sheet. For the convenience of calculation, the boundary layer flow is simulated as a symmetrical rectangular area. The implicit difference method combined with an L1-algorithm and shift Grünwald scheme is used to obtain the numerical solutions of the fractional governing equation. The validity and solvability of the present numerical method are analyzed systematically. The numerical results show that the thickness of the velocity boundary layer increases with an increase in the space fractional parameter γ. For a different stress fractional parameter α, the viscoelastic fluid will exhibit viscous or elastic behavior, respectively. Furthermore, the numerical method in this study is validated and can be extended to other time-space fractional boundary layer models.

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“…x ε in one dimension) could also be considered as suggested by the last author [2] and partially implemented by others [23,24], but this is outside the scope of the present article. This is also the case for other possible models including the fractional Laplacians of both ε and ε, as well as σ and σ.…”

Section: A Fractional Extensionmentioning

confidence: 99%