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The paper focuses on continuous models derived from a discrete microstructure. Various continualization procedures that take into account the nonlocal interaction between variables of the discrete media are analysed.

In the recent years new classes of ultra-dispersive and nanocrystalline materials [

The effects mentioned may be analyzed within the frame of discrete models, using molecular dynamics [

Situation can be described by Dirac’s words [

Hence, refinement of the existing theory of continuous media for the purpose of more realistic predictions seems to be the only viable alternative. In connection with this, one of the most challenging problems in multiscale analysis is that of finding continuous models for discrete, atomistic models. In statistical physics, these questions were already addressed 100 years ago, but many problems remain open even today. Most prominent is the question how to obtain irreversible thermodynamics as a macroscopic limit from microscopic models that are reversible. In this paper we consider another part of this field that is far from thermodynamic fluctuation. We are interested in reversible, macroscopic limits of atomic models. Debye approach is the simplest model of this type [

Therefore, continuous modeling of micro- and nanoeffects plays a crucial role in mechanics. It seems that the simplest approach to realize this idea relies on a modification of the classical modeling keeping both hypothesis of continuity and main characteristic properties of a discrete structure. Here four strategies exist.

Phenomenological approach: additional terms are added to the energy functional or to the constitutive relation. The structure and character of these terms are postulated [

Statistical approach: starting from an inhomogeneous classical continuum, average values of the state variables are computed to produce enhanced field equations [

Homogenization approach is based on

Continualization procedures are based on various approximations of local (discrete) operator by the nonlocal one. For this aim Taylor series [

The paper is organized as follows. Section

In this section we follow paper [

A chain of elastically coupled masses.

Owing to the Hooke’s law the elastic force acting on the

Applying the 2nd Newton’s law one gets the following system of ODEs governing chain dynamics:

System (

Let the chain ends be fixed

In general, the initial conditions have the following form:

As it has been shown in [

A solution to the BVP (

Function

Eigenvalues of the problem (

A solution to (

Since all values

Eigenvectors are mutually orthogonal; whereas

Each of the eigenfrequencies (

A general solution of the BVP (

Let us study now the problem of chain masses movement under action of a unit constant force on the point number zero. Motion of such system is governed by (

In what follows the initial BVP with nonhomogeneous boundary conditions (

For large values of

BVP (

Having in hand a solution to continuous BVP (

Formally, the approximation described so far can be obtained in the following way. Let us denote the difference operator occurring in (

Applying the translation operator

Let us explain (

Observe that

On the other hand, splitting the pseudo-differential operator into the McLaurin series is as follows:

Keeping in right hand of (

Solution form

Continuous system (

Relations (

One can obtain an exact solution to the BVP (

From (

It was believed that estimation (

Observe that splash amplitude does not depend on the parameter

On the other hand, a vibration amplitude of a mass with a fixed number is bounded for

It should be emphasized that a rigorous proof of the mentioned properties has been obtained for a case, when

Earlier the same effect of continualization was predicted by Ulam, who wrote [

Strictly speaking, one has to consider a true infinity in the distribution of matter in all problems of the physics of continua. In the classical treatment, as usually given in textbooks of hydrodynamics and field theory, this is, however, not really essential, and in most theories serves merely as a convenient limiting model of

Splash effect was observed numerically in 2D linear and 1D nonlinear case (A. M. Filimonov, private communication). By the way, due to this effect many papers “justifying” usual continualization can be treated as naive. For example, in [

A classical continuous approximation allows for relatively good description of a low part of the vibration spectrum of a finite chain of masses. In what follows we study now another limiting case, so-called anticontinuum limit, that is, completely uncoupled limit for lattice (Figure

Saw-tooth vibrations of a mass chain.

In this case one has

This is so-called anticontinuum limit.

In the case of vibrations close to the saw-tooth one, the short-wave approximation is applied “envelope continualization” [

Envelope continualization.

Then, the following relations are applied:

Using (

Appropriate boundary and initial conditions for (

Observe that practically the whole frequency interval of discrete model is well approximated for two limiting cases, that is, for the case of the chain and for the case of the envelope.

In what follows we are going to construct improved continuous approximations. Modeling of such systems (nonlocal theories of elasticity) requires integral or gradient formulation. The integral formulation may be reduced to a gradient form by truncating the series expansion of the nonlocality kernel in the reciprocal space [

If, in the series (

However, a nontrivial problem regarding boundary conditions for (

Assuming

Comparison of

In a general case, keeping in series (

Boundary conditions for (

The corresponding BVPs are correct (and also stable during numerical solution) for odd

For

One also can mention the momentous elasticity theories [

The construction of intermediate continuous models is mainly based on the development of a difference operator into Taylor series. However, very often application of Padé approximants (PAs) is more effective for approximation [

If only two terms are left in the series (

For justification of this procedure Fourier or Laplace transforms can be used.

The corresponding quasicontinuum model reads

The boundary conditions for (

The error regarding estimation of

Equation (

Kaplunov et al. [

Passage to (

It is worth noting that (

Finally, having in hand both long and short waves asymptotics, one may also apply two-point PA (description of two-point PA see Appendix

The improved continuous approximation is governed by the following equation:

Now we require the

For large value of

Oscillations frequencies defined by the BVP (

Now, using (

The highest error in estimation of the eigenfrequencies appears for

Observe that equations similar to the (

Interesting, that Mindlin and Herrmann used very similar to two-point PA idea for construction their well-known equation for longitudinal waves in rod [

Dispersion relation (

Boundary conditions associated with this equation are

In order to study forced vibrations we begin with a classical continuous approximation. A solution to (

Solution to the BVP (

For the approximation the motion of the chain, one must keep in an infinite sum only

Observe that solutions (

In addition, the following formulas hold [

The corresponding integrals read

Observe that the values of sum (

Owing to (

Let us study wave propagation in the infinite 1D medium (Figure

Infinite chain of masses.

We reduce (

Here

Due to the linearity of the problem one can suppose

Let us apply Fourier transform to (

Observe that the term

Solving (

The obtained solution is exact one and will be further used for the error estimation of improved continuous models.

Applying the classical continuous approximation instead of system (

Here

The following relation between discrete and continuous systems holds:

Now, applying the Fourier transform in

So, a wave propagation in a discrete media strongly differs from this phenomenon in the classical continuous media. In what follows we give solutions on a basis of the theories of elasticity described so far. The intermediate continuous model is as follows:

Let us explain the occurrence of term

Applying both Fourier transform in

The quasicontinuum model is as follows:

The associated wave velocity propagation follows:

Model (

In this case the associated wave velocity propagation reads

Since analytical comparison of the obtained wave velocity propagation is difficult, we apply numerical procedures.

As it has been already mentioned, exact solution of the joined discrete problem governed by relation

Wave propagation for

Wave propagation for

Wave propagation for

Wave propagation for

Wave propagation for

Wave propagation for

Equation (

In order to analyse the 2D case we use 9-cell square lattice (Figure

2D lattice.

Here

The standard continualization procedure for (

Naturally, one can construct equations of higher order, but, as it is shown in [

Using staggered transformations

As in 1D case (see (

The existence of the continuous approximations (

Identify the terms in the differential equations, the neglection of which in the straightforward approximation is responsible for the nonuniformity.

Approximate those terms insofar as possible while retaining their essential character in the region of nonuniformity.

In our case the composite equations will be constructed in order to overlap (approximately) with (

Here

For 1D case from (

Fundamental methods of analysis of linear lattices are those based on either discrete or continuous Fourier transformations [

Besides, one may achieve even a solution to a nonlinear problem using the following observation [

Assume that solution to the ODEs is a traveling wave,

Introducing notation

Term

In result one obtains equation of continuous approximation

We show also a simultaneous application of PA matched with a perturbation procedure using an example of the Toda lattice [

In continuum limit one obtains (for details see Appendix

One can obtain from (

Then, (

On the other hand, there are nonlinear lattices with a special type nonlinearities allowing to achieve exact solutions as soliton and soliton-like solutions (Toda, Ablowitz and Ladik,

Lattice discreteness allows the existence of new types of localized structures that would not exist in the continuum limit. Nonlinearity transfers energy to higher wavenumbers but it can be suppressed and balanced by the bound of the spectrum of discrete systems. Such balance can generate highly localized structures, that is, the discrete breathers [

On the other hand, it is more appropriate changing a system of Fermi-Pasta-Ulam not by KdV equation, but by Toda lattice. In the latter case one gets an asymptotic regarding nonlinear parameter instead of applying direct substitution of a discrete system by a continuous one. Occurrence of exact solutions of a discrete system allows applying the following approach; fast changeable solution part is constructed using a discrete model, whereas a continuous approximation is applied for slow components. The latter observation is very well exhibited by Maslov and Omel’yanov [

They construct soliton solutions in the following way: rapidly changing part of soliton is constructed using Toda lattice with constant coefficients, and for slowly part of solution continuous approximation is used. Then these solutions are matched.

Let us show that PA for constructing of improved continuous models can be used iteratively. For three terms in expansion (

Now we use PA as follows:

Now let us consider the continuous models of the chain with two different particles in primitive cell, that is, the chain with alternating masses

Continuous approximation for (

Improved continuous approximation for (

Homogenization and other asymptotic approaches can be use for this continuous system [

The Navier-Stokes equations have a long and glorious history but remain extremely challenging, for example, the issue of existence of physically reasonable solutions of these equations in 3D case was chosen as one of the seven millennium “million dollar” prize problems of the Clay Mathematical Institute [

or

Equations (

As it is mentioned in [

More investigations in direction of improvement of the Navier-Stokes equations are carried out in references [

It should be emphasized a role of mathematical approaches in proving various physical theories [

Therefore, the modified Navier-Stokes equations are more suitable for fluid dynamics description for large Re from a point of view of theoretical mathematics. For small Re they are more close to the Navier-Stokes equations. In the latter case one gets a unique global solutions to the initial-boundary value problems as well as the solvability of the stationary boundary value problems for arbitrary Re numbers.

Note that the modified Navier-Stokes equations (

Classical molecular dynamics simulations have become prominent as a tool for elucidating complex physical phenomena, such as solid fracture and plasticity. However, the length and time scales probed using molecular dynamics are still fairly limited. To overcome this problem it is possible to use molecular dynamics only in localized regions in which the atomic-scale dynamics is important, and a continuous simulation method (e.g., FEM) everywhere else [

Let us compare an impact of the methods illustrated and discussed so far on the improved continuous models. Continuous models based on the PA and two-point PA can be applied in the case of 1D problems (in spite of some artificially constructed examples [

It will be very interesting to use for study investigation of 2D lattices 2D PA [

Let us finally discuss problems closely connected with brittle fracture of elastic solids [

Splashes.

8 | 16 | 32 | 64 | 128 | 256 | ||
---|---|---|---|---|---|---|---|

1.7561 | 2.0645 | 2.3468 | 2.6271 | 2.9078 | 3.1887 |

Let us consider Padé approximants, which allow us to perform, to some extent, the most natural continuation of the power series. Let us formulate the definition [_{i}_{i}_{mn}_{mn}_{mn}

Now we give the notion of two-point Padé approximants [

The TPPA is represented by the rational function

Here we describe construct of the continuum limit of the Toda lattice

We follow [

Equation (

For wave spreading in right direction one obtains

Formally expanding function

Using variables transform

Let us suppose system

Further we follow [

In continuum limit we must construct a function

For function

For

The authors thank Professors H. Askes, A. M. Filimonov, W. T. van Horssen, L.I. Manevitch for their comments and suggestions related to the obtained results, as well as Dr. G. A. Starushenko for help in numerical calculations. This work was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft), Grant no. WE 736/25-1 (for I.V. Andrianov). The authors are grateful to the anonymous reviewers for valuable comments and suggestions, which helped to improve the paper.