Lattice dynamics from a continuum viewpoint

Charlotte, Miguel; Truskinovsky, Lev

doi:10.1016/j.jmps.2012.03.004

Cited by 32 publications

(17 citation statements)

References 55 publications

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“…Such quasi-continuum models take into account higher order terms for the diffusion operator together with the characteristic length of the system to approach, in a continuum way, the underlying discrete nature of the problem, see [26] for a review. It is worth noting that similar models were also proposed recently in the field of phase transitions [27] and in structural mechanics [28,29].…”

Section: Introductionsupporting

confidence: 54%

Discretization-dependent model for weakly connected excitable media

2018

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Pattern formation has been widely observed in extended chemical and biological processes. Although the biochemical systems are highly heterogeneous, homogenized continuum approaches formed by partial differential equations have been employed frequently. Such approaches are usually justified by the difference of scales between the heterogeneities and the characteristic spatial size of the patterns. Under different conditions, for example, under weak coupling, discrete models are more adequate. However, discrete models may be less manageable, for instance, in terms of numerical implementation and mesh generation, than the associated continuum models. Here we study a model to approach discreteness which permits the computer implementation on general unstructured meshes. The model is cast as a partial differential equation but with a parameter that depends not only on heterogeneities sizes, as in the case of quasicontinuum models, but also on the discretization mesh. Therefore, we refer to it as a discretization-dependent model. We validate the approach in a generic excitable media that simulates three different phenomena: the propagation of action membrane potential in cardiac tissue, in myelinated axons of neurons, and concentration waves in chemical microemulsions.

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Section: Introductionsupporting

confidence: 54%

Discretization-dependent model for weakly connected excitable media

2018

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“…It has been already shown by Kunin (1982) and more recently by Charlotte and Truskinovsky (2012) that a discrete chain may behave dynamically as an equivalent nonlocal continuum whose kernel depends on the level of approximation of the reference discrete medium. However, the link between the differential format of Eringen's nonlocal model (Eringen, 1983) is more recent (see .…”

Section: Introductionmentioning

confidence: 99%

Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua

Challamel

Picandet

Collet

et al. 2015

European Journal of Mechanics - A/Solids

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In this paper, we revisit the capability of numerical approaches such as finite difference methods and finite element methods, in approximating exact one-dimensional continuous eigenvalue problems (such as lateral vibrations of a string, the axial or the torsional vibrations of a bar, and the buckling of elastic columns). The numerical methods analysed in this paper are converted into difference equations. Following a continualization procedure or the method of differential approximation, the difference operators are then expanded in differential operators via Taylor expansion or Pad e approximants. Analogies between the finite numerical approaches and some equivalent enriched continuum are shown, using this continualization procedure. The finite difference methods (first-order or higher-order finite difference methods) are shown to behave as integral-based nonlocal media (or stress gradient media), while the finite element method is found to behave as gradient elasticity media (or strain gradient media). The length scale identification of each equivalent enriched continuum strongly depends on the order of the numerical method considered. For the finite difference methods, the length scale identification of the equivalent nonlocal medium depends on the static versus dynamic analysis, whereas this length scale appears to be independent of inertia effects for the finite element method. Some comparisons between the exact discrete eigenvalue problems and the approximated continuous ones show the efficiency of the continualization procedure. An equivalent enriched Rayleigh quotient can be defined for each numerical method: the integral-based nonlocal method gives a lower bound solution to the exact eigenvalue multiplier, whereas the gradient elasticity method furnishes an upper bound solution.

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“…The discrete system is typically used with finite micro-cells, and the best equivalent continuous medium is sought that can be associated with a finite size of the structure. This methodology for relating discrete problems to continuous ones is typically one of continualization, as already investigated in the case of elasticity [14][15][16][17][18][19][20], for axial systems and examples for bending systems [7]. A continualization procedure was also implemented by Chang [21] for axial damage systems, leading to an equivalent gradient elasticity damage system; with this approach, the strain-damage loading function was assumed to remain local.…”

Section: Introductionmentioning

confidence: 99%

On the failure of a discrete axial chain using a continualized nonlocal Continuum Damage Mechanics approach

Picandet

Hérisson

Challamel

et al. 2015

Num Anal Meth Geomechanics

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SUMMARYThe failure of a discrete elastic-damage axial system is investigated using both a discrete and an equivalent continuum approach. The Discrete Damage Mechanics approach is based on a microstructured model composed of a series of periodic elastic-damage springs (axial Discrete Damage Mechanics lattice system). Such a discrete damage system can be associated with the finite difference formulation of a Continuum Damage Mechanics evolution problem. Several analytical and numerical results are presented for the tensile failure of this axial damage chain under its own weight.The nonlocal Continuum Damage Mechanics models examined in this paper are mainly built from a continualization procedure applied to centered or uncentered finite difference schemes. The asymptotic expansion of the first-order upward difference equations leads to a first-order nonlocal model, whereas the asymptotic expansion of the centered finite difference equations leads to a second-order nonlocal Eringen's approach. To complete this study, a phenomenological nonlocal gradient approach is also examined and compared with the first continualization methods.A comparison of the discrete and the continuous problems for the chains shows the effectiveness of the new micromechanics-based nonlocal Continuum Damage modeling, especially for capturing scale effects. For both continualized approaches, the length scale of the nonlocal models depends only on the cell size, while for the so-called phenomenological approach, the length scale may depend on the loading parameter. This apparent load-dependent length scale, already discussed in the literature with numerical arguments, is found to be sensitive to the postulated structure of the nonlocal model calibrated according to a lattice approach.

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Lattice dynamics from a continuum viewpoint

Cited by 32 publications

References 55 publications

Discretization-dependent model for weakly connected excitable media

Discretization-dependent model for weakly connected excitable media

Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua

On the failure of a discrete axial chain using a continualized nonlocal Continuum Damage Mechanics approach

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