Transition from Discrete to Continuous Media: The Impact of Symmetry Changes on Asymptotic Behavior of Waves

Andrianov, Igor V.; Koblik, Steve; Starushenko, Galina A.

doi:10.3390/sym13061008

Cited by 9 publications

(6 citation statements)

References 40 publications

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“…The exact solution of Eq. ( 1) was alternatively found elegantly during the twentieth century by Schrödinger [11] for general initial conditions in terms of Bessel functions (see more recently [12,13] or [14]) (Fig. 3).…”

Section: Introductionmentioning

confidence: 84%

“…Furthermore, it was shown by Filimonov et al [10] that the solution of the continuous problem may be asymptotically significantly different from the lattice one for large time evolution (even in the linear range). Schrödinger [11] also discussed the limits of the continuous approximation of the wave equation with respect to the lattice wave behavior (see also [12,13] or [14]). However, this brief historical presentation will not address the strong mathematical difficulties related to the connection of nonlinear discrete lattices and nonlinear wave equations, including the soliton phenomenon [24].…”

Section: Introductionmentioning

confidence: 99%

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Discrete and continuous models of linear elasticity: history and connections

Challamel

Zhang

Wang

et al. 2023

Continuum Mech. Thermodyn.

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This paper tracks the development of lattice models that aim to describe linear elasticity of solids and the field equations of which converge asymptotically toward those of isotropic continua, thus showing the connection between discrete and continuum. In 1759, Lagrange used lattice strings/rod dynamics to show the link between the mixed differential-difference equation of a one-dimensional (1D) lattice and the partial differential equation of the associated continuum. A consistent three-dimensional (3D) generalization of this model was given much later: Poincaré and Voigt reconciled the molecular and the continuum approaches at the end of the nineteenth century, but only in 1912 Born and von Kármán presented the mixed differentialdifference equations of discrete isotropic elasticity. Their model is a 3D generalization of Lagrange's 1D lattice and considers longitudinal, diagonal and shear elastic springs among particles, so the associated continuum is characterized by three elastic constants. Born and von Kármán proved that the lattice equations converge to Navier's partial differential ones asymptotically, thus being a formulation of continuous elasticity in terms of spatial finite differences, as for Lagrange's 1D lattice. Neglecting shear springs in Born-Kármán's lattice equals to Navier's assumption of pure central forces among molecules: in the limit, the lattice behaves as a one-parameter isotropic solid ("rari-constant" theory: equal Lamé parameters, or, equivalently, Poisson's ratio υ = 1/4). Hrennikoff and McHenry revisited the lattice approach with pure central interactions using a plane truss; the equivalent Born-Kármán's lattice in plane stress in the limit tends to a continuum with Poisson's Communicated by Andreas Öchsner.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Discrete and continuous models of linear elasticity: history and connections

Challamel

Zhang

Wang

et al. 2023

Continuum Mech. Thermodyn.

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“…Note that the Whittaker-Shannon interpolation formula or sinc interpolation definitions are also used. successful experience of [50].…”

Section: On the Concept Of Concentrated Forces In Media With Microstructurementioning

confidence: 99%

“…The original discrete object is the Lagrange lattice [47][48][49] (for relevant terminology, see Appendix in [50]). The lattice is that of point masses, M , located in equilibrium states at the points of the  -axis with coordinates   0, 1, 2, 3, ... jh j     and coupled with elastic springs of stiffness c (Figure 1).…”

Section: Governing Relationsmentioning

confidence: 99%

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On Aspects of Gradient Elasticity: Green’s Functions and Concentrated Forces

Andrianov

Koblik²,

Starushenko³

et al. 2022

Symmetry

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In the first part of our review paper, we consider the problem of approximating the Green’s function of the Lagrange chain by continuous analogs. It is shown that the use of continuous equations based on the two-point Padé approximants gives good results. In the second part of the paper, the problem of singularities arising in the classical theory of elasticity with affecting concentrated loadings is considered. To overcome this problem, instead of a transition to the gradient theory of elasticity, it is proposed to change the concept of concentrated effort. Namely, the Dirac delta function is replaced by the Whittaker–Shannon–Kotel’nikov interpolating function. The only additional parameter that characterizes the microheterogeneity of the medium is used. An analog of the Flamant problem is considered as an example. The found solution does not contain singularities and tends to the classical one when the microheterogeneity parameter approaches zero. The derived formulas have a simpler form compared to those obtained by the gradient theory of elasticity.

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Dynamics of matter and energy

Кривцов

2022

Z Angew Math Mech

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We propose and examine a potential analogy between mass transfer (in space) and energy transfer (in solids). We adapt classical equations of matter dynamics to describe the dynamics of energy transfer. Such fundamental quantities as the effective mass, momentum, moment of inertia and other quantities typical for bodies of matter are introduced for “bodies” of energy. Along with this, two new concepts of “carrier” and “phantom” are proposed. A carrier is a medium which enables energy transfer. A phantom is a virtual body of matter having its mass distribution equivalent to the energy distribution in the carrier. Using an inhomogeneous chain of particles as a sample system, we show that the phantom motion satisfies the Newton's second law of dynamics. For certain systems, we derive constitutive equations for the net force, which results in a closed system of dynamics equations. We further show that with the relevant properties of the chain it is possible to obtain the dynamics equation for the phantom motion in a gravitational field. We use similar methods to study energy dispersion. To analyze phantom evolution, we introduce the velocities of phantom transfer and dispersion. We show that, depending on the ratio of these velocities, the phantom can behave either as a wave or as a particle. Furthermore, we discuss potential application of energy dynamics to other branches of physics, such as quantum mechanics, electrodynamics and general relativity. We introduce the idea that a body of matter itself can be a phantom in some other carrier, which is a different entity than matter. Possible associations of the phantom/carrier model with current models of physical space are discussed. Based on the presented concept, we propose plausible qualitative answers to several open questions in modern physics.

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Transition from Discrete to Continuous Media: The Impact of Symmetry Changes on Asymptotic Behavior of Waves

Cited by 9 publications

References 40 publications

Discrete and continuous models of linear elasticity: history and connections

Discrete and continuous models of linear elasticity: history and connections

On Aspects of Gradient Elasticity: Green’s Functions and Concentrated Forces

Dynamics of matter and energy

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