On the capability of generalized continuum theories to capture dispersion characteristics at the atomic scale

Fafalis, Dimitrios; Filopoulos, S. P.; Tsamasphyros, G.

doi:10.1016/j.euromechsol.2012.02.004

Cited by 53 publications

(35 citation statements)

References 57 publications

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“…According to [18], bothh andḡ should be positive real numbers for thermodynamic consistency, therefore this solution in unacceptable. Regarding the G3 model, which retains three constants, fulfillment of conditions (6) plus the conditionω|k =π/2 = √ 2 at the middle of the Brillouin zone results in the following two sets of values for the length scale parameters [18]:…”

Section: Strain Gradient Theoriesmentioning

confidence: 99%

“…Assuming a plane wave solution we obtain the following dispersion relations (in nondimensional form), respectively [18]:…”

Section: Strain Gradient Theoriesmentioning

confidence: 99%

“…The fulfillment of short-wavelength limit conditions (6) for the G2 model results in the following values for the lengthscale parameters [18]:ḡ…”

Section: Strain Gradient Theoriesmentioning

confidence: 99%

“…In the field of generalized continuum mechanics, several 1D models have been devised following the Mindlin's grade 2 (G2) and grade 3 (G3) strain gradient theories [35,36], including micro-inertia in the kinetic energy density [18], leading to the following motion equations, respectively:…”

Section: Strain Gradient Theoriesmentioning

confidence: 99%

“…Within this category fall the works by Mindlin and Tiersten [2], Kröner [3], Toupin [4,5], Green and Rivlin [6], Mindlin [7,8] and Mindlin and Eshel [9], Yang et al [10], Park and Gao [11,12], as well as the non-local continuum mechanics initiated by Eringen and coworkers [13][14][15], which has been widely used to analyse a variety of problems, such as wave propagation, dislocation, and crack singularities. In this respect, 1D discrete solids have served to provide physical foundation to different axiomatically formulated theories of generalized continuum [16,17], and to test their ability to mimic dispersive curves and to capture size effects [18].…”

Section: Introductionmentioning

confidence: 99%

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One-dimensional dispersion phenomena in terms of fractional media

2016

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Abstract. It is well know that structured solids present dispersive behaviour which cannot be captured by the classical continuum mechanics theories. A canonical problem in which this can be seen is the wave propagation in the Born-Von Karman lattice. In this paper the dispersive effects in a 1D structured solid is analysed using the Fractional Continuum Mechanics (FCM) approach previously proposed by Sumelka (2013). The formulation uses the Riesz-Caputo (RC) fractional derivative and introduces two phenomenological/material parameters: 1) the size of non-local surrounding l f , which plays the role of the lattice spacing; and 2) the order of fractional continua α, which can be devised as a fitting parameter. The results obtained with this approach have been compared with the reference dispersion curve of Born-Von Karman lattice, and the capability of the fractional model to capture the size effects present in the dynamic behaviour of discrete systems has been proved.

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Section: Strain Gradient Theoriesmentioning

confidence: 99%

“…Assuming a plane wave solution we obtain the following dispersion relations (in nondimensional form), respectively [18]:…”

Section: Strain Gradient Theoriesmentioning

confidence: 99%

“…The fulfillment of short-wavelength limit conditions (6) for the G2 model results in the following values for the lengthscale parameters [18]:ḡ…”

Section: Strain Gradient Theoriesmentioning

confidence: 99%

Section: Strain Gradient Theoriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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One-dimensional dispersion phenomena in terms of fractional media

2016

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Nanostructure‐dependent dispersion of carbon nanostructures: New insights into the modified couple stress theory

Khorshidi

Soltani

2020

Math Methods in App Sciences

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This paper presents a size‐dependent analysis of carbon nanostructures to put forward some new insights into the conventional modified couple stress theory. In fact, a micro‐inertia length scale parameter is added to the equation of motion to compensate the deficiency of the theory in describing the size effect of the longitudinal dispersion. In this study, the longitudinal dispersions of carbon nanotubes, graphene, and graphite layers are investigated to obtain the micro‐inertia length scale parameter, and the flexural dispersions of carbon nanotubes and graphene layer are assessed to expose the size dependency of the couple‐stress length scale parameter. The results indicate that the micro‐inertia length scale parameter can directly be related to the lattice size, while the couple‐stress length scale parameter depends on some more factors like geometry, size, and material properties. Finally, some general relations are proposed for the first time to make a connection between these two distinct parameters and other features.

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Computational continua for linear elastic heterogeneous solids on unstructured finite element meshes

Fish

2018

Numerical Meth Engineering

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The computational continua framework, which is a variant of higher-order computational homogenization theories that is free of scale separation, does not require higher-order finite element continuity, and is free of higher-order boundary conditions, has been generalized to unstructured meshes. The salient features of the proposed generalization are (i) a nonlocal quadrature scheme for distorted elements that accounts for unit cell distortion in the parent element domain and (ii) an approximate variant of the nonlocal quadrature that eliminates the cost of computing positions of the quadrature points in the preprocessing stage. The performance of the computational continua framework on unstructured meshes has been compared to the first-order homogenization theory and the direct numerical simulation. KEYWORDScomputational continua, heterogeneous materials, homogenization, multiscale methods, nonlocal quadrature Int J Numer Methods Eng. 2018;115:501-530. wileyonlinelibrary.com/journal/nme

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On the capability of generalized continuum theories to capture dispersion characteristics at the atomic scale

Cited by 53 publications

References 57 publications

One-dimensional dispersion phenomena in terms of fractional media

One-dimensional dispersion phenomena in terms of fractional media

Nanostructure‐dependent dispersion of carbon nanostructures: New insights into the modified couple stress theory

Computational continua for linear elastic heterogeneous solids on unstructured finite element meshes

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