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.

1. Introduction

In the recent years new classes of ultra-dispersive and nanocrystalline materials [13], which require a different modern approach than that of a classical continuous media, have been proposed. For example, the nanocrystalline material is represented by a regular or quasiregular lattice with small size bodies (domains, granules, fullerenes, nanotubes, or clusters nanoparticles) possessing internal degrees of freedom occupying the lattice sites [4]. The situation mentioned is also a characteristic for various problems of nanomechanics [5, 6], because the transition of a material to the nanostructural state is accompanied by dimensional effects in its mechanical properties. Models established on a classical continuous media cannot govern high frequency vibrations, material behavior in the vicinity of cracks and on the fronts of destruction waves, and during phase transitions [7]. It is not possible to reach and overcome bifurcation points, that is, thresholds of lattice stability under catastrophic deformations [8]. Wave dispersion in granular materials [911] represents an important example of microstructure effects too. Microstructural effects are essential for correct description of softening phenomena [12] in damage mechanics [1316] and in the theory of plasticity [17, 18]. As it is mentioned in [19], “From the behaviour of calcium waves in living cells to the discontinuous propagation of action potentials in the heart or chains of neurons and from chains of chemical reactors or arrays of Josephson junctions to optical waveguides, dislocations and the DNA double strand, the relevant models of physical reality are inherently discrete.”

The effects mentioned may be analyzed within the frame of discrete models, using molecular dynamics [20], quasicontinuum analysis [21, 22], or other numerical approaches. However, a sought result is difficult to obtain using high tech computers in an economical way. For example, modern practical problems are still intractable for molecular dynamics-based analysis, even if the highest computing facility is at disposal.

Situation can be described by Dirac’s words [23]: “The physical laws necessary for the mathematical theory of a large part of physics and whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be solvable. Therefore, it becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computations.”

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 [24], but it does not take into account spatial dispersion.

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 [2531]. Phenomenological approach is very useful and accurate enough in the applied sciences when it is necessary to solve problems of practical significance [3234]. But progress of natural sciences makes us look for ways of substantiated derivation of constitutive relations from “first principles.” In this connection one can recall the 6th Hilbert’s problem (mathematical treatment of the axioms of physics) [35]: “Boltzmann’s work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua. Conversely, one might try to derive the laws of the rigid bodies’ motion by a limiting process from a system of axioms depending upon the idea of continuously varying conditions of a material filling all space continuously, these conditions being defined by parameters. The question on the equivalence of different systems of axioms is always of great theoretical interest.”

Statistical approach: starting from an inhomogeneous classical continuum, average values of the state variables are computed to produce enhanced field equations [36]. Unfortunately, great mathematical problems can not give possibility of wide usage of this approach.

Homogenization approach is based on Γ- and G-limit technique [3743] and gave many important pure mathematical results, but not new insights for physics.

Continualization procedures are based on various approximations of local (discrete) operator by the nonlocal one. For this aim Taylor series [44, 45], one- and two-point Padé approximants [4662], composite equations [63, 64], and other approaches are used. Independently of the strategy employed, the constitutive relation of the homogenized material becomes nonlocal. Mathematically this implies integral relations between stresses and strains, in which the stress in a point is not only related to the strain in the same point but also instantly to strains in the neighbouring points. We will analyse these approaches further.

The paper is organized as follows. Section 2 is devoted to some general remarks concerning a chain of elastically linear coupled masses. In Sections 3 and 4 the classical continuous approximation and so-called “splash effect” for 1D case are described. Section 5 is devoted to anticontinuum limit. Improved continuous approximations are studied is Sections 6 and 7 for natural and forced oscillations. Waves in 1D discrete and continuous media are compared in Section 8. 2D problems are analysed in Section 9. Nonlinear phenomenons are studied in Section 10. In Section 11 we analyse some possible generalization of described approaches. Section 12 is devoted to Navier-Stokes equations. Section 13 presents brief concluding remarks. In appendixes to the paper we describe one- and two-point Padé approximations, continuum limit of Toda lattice, and correspondence between functions of discrete arguments and approximating analytical functions.

2. A Chain of Elastically Coupled Masses

In this section we follow paper [65]. We study a chain of 𝑛+1 material points with the same masses𝑚, located in equilibrium states in the points of the axis 𝑥 with coordinates 𝑗(𝑗=0,1,,𝑛,𝑛+1) and suspended by elastic couplings of stiffness 𝑐 (Figure 1).

Owing to the Hooke’s law the elastic force acting on the jth mass is as follows: 𝜎𝑗𝑦(𝑡)=𝑐𝑗+1(𝑡)𝑦𝑗𝑦(𝑡)𝑐𝑗(𝑡)𝑦𝑗1𝑦(𝑡)=𝑐𝑗1(𝑡)2𝑦𝑗(𝑡)+𝑦𝑗+1,(𝑡)𝑗=1,2,,𝑛,(2.1) where 𝑦𝑗(𝑡) is the displacement of the 𝑗th material point from its static equilibrium position.

Applying the 2nd Newton’s law one gets the following system of ODEs governing chain dynamics: 𝑚𝑦𝑗𝑡𝑡𝑦(𝑡)=𝑐𝑗1(𝑡)2𝑦𝑗(𝑡)+𝑦𝑗+1(𝑡),𝑗=1,2,,𝑛.(2.2)

System (2.2) can be cast into the following form: 𝑚𝜎𝑗𝑡𝑡𝜎(𝑡)=𝑐𝑗+12𝜎𝑗+𝜎𝑗1,𝑗=1,,𝑛.(2.3)

Let the chain ends be fixed𝑦𝑜(𝑡)=𝑦𝑛+1(𝑡)=0.(2.4)

In general, the initial conditions have the following form: 𝑦𝑗(𝑡)=𝜑𝑗(0);𝑦𝑗𝑡(𝑡)=𝜑𝑗(1)for𝑡=0.(2.5)

As it has been shown in [65], for any solution of the BVP (2.2), (2.4), (2.5) the total energy is constant. Besides, the solutions mentioned so far are nonasymptotic and stable due to the Lyapunov stability definition.

A solution to the BVP (2.2), (2.4), (2.5) can be expressed by elementary functions applying the discrete variant of the method of variables separation. For this purpose normal vibrations are constructed of the form 𝑦𝑗(𝑡)=𝐶𝑗𝑇(𝑡),𝑗=1,,𝑛,(2.6) where constants 𝐶𝑗are defined via solution of the following eigenvalue problem: 𝜆𝐶𝑗=𝐶𝑗+12𝐶𝑗+𝐶𝑗1,𝑗=1,,𝑛,𝐶0=𝐶𝑛+1=0.(2.7)

Function 𝑇(𝑡) satisfies the following equation:𝑚𝑇𝑡𝑡+𝑐𝜆𝑇=0.(2.8)

Eigenvalues of the problem (2.7) follow [65]𝜆𝑘=4sin2𝑘𝜋2(𝑛+1),𝑘=1,2,,𝑛.(2.9)

A solution to (2.8) has the form 𝑇=𝐴exp(𝑖𝜔𝑡). Hence, (2.8) and (2.9) yield the following Lagrange formula for frequencies 𝜔𝑘 of discrete system: 𝜔𝑘=2𝑐𝑚sin𝑘𝜋2(𝑛+1),𝑘=1,2,,𝑛.(2.10)

Since all values 𝜆𝑘 are distinct, then all eigenvalues are different. Therefore, each of the eigenvalues is associated with one eigenvector 𝐂𝑘(𝐶1(𝑘),𝐶2(𝑘),,𝐶𝑛(𝑘)) of the form𝐂𝑘=cos𝑒𝑐𝑘𝜋𝑛+1sin𝑘𝜋𝑛+1,sin2𝑘𝜋𝑛+1,,sin𝑛𝑘𝜋𝑛+1,𝑘=1,2,,𝑛.(2.11)

Eigenvectors are mutually orthogonal; whereas ||𝐂𝑘||2=𝑛+12cos𝑒𝑐2𝑘𝜋𝑛+1,𝑘=1,2,,𝑛.(2.12)

Each of the eigenfrequencies (2.10) is associated with a normal vibration 𝑦𝑗(𝑘)(𝑡)=𝐶𝑗(𝑘)𝐴𝑘𝜔cos𝑘𝑡+𝐵𝑘𝜔sin𝑘𝑡,𝑘=1,2,,𝑛.(2.13)

A general solution of the BVP (2.3)–(2.5) is obtained as a result of superposition of normal vibrations 𝑦𝑗(𝑡)=𝑛𝑘=1𝐶𝑗(𝑘)𝐴𝑘𝜔cos𝑘𝑡+𝐵𝑘𝜔sin𝑘𝑡,𝑗=1,,𝑛.(2.14)

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 (2.3) with the following boundary and initial conditions: 𝜎𝑜(𝑡)=1,𝜎𝑛+1𝜎(𝑡)=0,𝑗(𝑡)=𝜎𝑗𝑡(𝑡)=0for𝑡=0.(2.15)

In what follows the initial BVP with nonhomogeneous boundary conditions (2.3), (2.15) will be reduced to that of homogeneous boundary condition for (2.3) with nonhomogeneous initial conditions, and then the method of superposition of normal vibrations can be applied. The formulas of normal forms obtained so far can be applied in a similar way with exchange of 𝑦𝑗(𝑡) for𝜎𝑗(𝑡). In effect, the following exact solution to the BVP (2.3), (2.15) is obtained [65]:𝜎𝑗=1𝑛+1𝑛𝑘=1sin𝜋𝑘𝑗𝑛+1𝑐𝑡𝑔𝜋𝑘𝜔2(𝑛+1)1cos𝑘𝑡,𝑗=1,2,,𝑛.(2.16)

3. Classical Continuous Approximations

For large values of n usually continuous approximation of discrete problem is applied. In our case, described by (2.3), (2.15), it takes the form of𝑚𝜎𝑡𝑡(𝑥,𝑡)=𝑐2𝜎𝑥𝑥(𝑥,𝑡),(3.1)𝜎(0,𝑡)=1,𝜎(𝑙,𝑡)=0,(3.2)𝜎(𝑥,0)=𝜎𝑡(𝑥,0)=0,(3.3) where 𝑙=(𝑛+1).

BVP (3.1)–(3.3) can be used, for example, for modeling of stresses in van couplings of the rolling stocks [66].

Having in hand a solution to continuous BVP (3.1)–(3.3), one obtains a solution of a discrete problem due to the following formulas:𝜎𝑗(𝑡)=𝜎(𝑗,𝑡),𝑗=0,1,,𝑛,𝑛+1.(3.4)

Formally, the approximation described so far can be obtained in the following way. Let us denote the difference operator occurring in (2.3) as 𝐷, that is, 𝑚𝜎𝑗𝑡𝑡(𝑡)=𝑐𝐷𝜎(𝑡).(3.5)

Applying the translation operatorexp(𝜕/𝜕𝑥), one gets [67]𝜕𝐷=exp𝜕𝜕𝑥+exp𝜕𝑥2=4sin2𝑖2𝜕𝜕𝑥.(3.6)

Let us explain (3.6) in more details. The McLaurin formula for infinitely many times differentiable function 𝐹(𝑥) has the following form:𝜕𝐹(𝑥+1)=1++1𝜕𝑥𝜕2!2𝜕𝑥2𝜕+𝐹(𝑥)=exp𝜕𝑥𝐹(𝑥).(3.7)

Observe that exp(𝜕/𝜕𝑥) belongs to the so-called pseudodifferential operators. Using relations (3.5)–(3.7), we cast (2.3) into pseudo-differential equation of the following form: 𝑚𝜕2𝜎𝜕𝑡2+4𝑐sin2𝑖2𝜕𝜕𝑥𝜎=0.(3.8)

On the other hand, splitting the pseudo-differential operator into the McLaurin series is as follows: sin2𝑖2𝜕1𝜕𝑥=2𝑘=12𝑘𝜕(2𝑘)!2𝑘𝜕𝑥2𝑘=24𝜕2𝜕𝑥21+2𝜕122𝜕𝑥2+4𝜕3604𝜕𝑥4+6𝜕100806𝜕𝑥6+.(3.9)

Keeping in right hand of (3.9) only the first term, one obtains a continuous approximation (3.1). Note that an application of the McLaurin series implies that displacements of the neighborhood masses differ slightly from each other. From a physical point of view, it means that we study vibrations of the chain with a few masses located on the spatial period (Figure 2); that is, the long wave approximation takes place. Note that the vertical axis in Figure 2 represents the displacement in the 𝑥 direction, since the problem is 1D.

Continuous system (3.1) possesses the following discrete infinite spectrum: 𝛼𝑘=𝜋𝑐𝑚𝑘𝑛+1,𝑘=1,2,.(3.10)

Relations (3.10) relatively good approximate low frequencies of discrete system (2.10), whereas the 𝑛th frequency 𝛼𝑘 of a continuous system differs from the corresponding 𝑛th frequency 𝜔𝑘 of a discrete system of order of 50%. Accuracy of continuous approximations can be improved, what will be discussed further.

4. Splashes

One can obtain an exact solution to the BVP (3.1)–(3.3), using the d’Alembert method matched with operational calculus [65]: ||||𝜋𝜎(𝑥,𝑡)=𝐻𝑛arcsinsin2𝑛𝑐𝑚𝑡||||𝑥,(4.1) where 𝐻() is the Heaviside function.

From (4.1) one obtains the following estimation: ||||𝜎(𝑥,𝑡)1.(4.2)

It was believed that estimation (4.2) with a help of relation (3.4) can be applied also to a discrete system [68]. However, analytical as well as numerical investigations [66, 6973] indicated a need to distinguish between global and local characteristics of a discrete system. In other words during investigation of lower frequency part a transition into continuous model is allowed. However, in the case of forced oscillations solutions to a discrete system may not be continuously transited into solution of a wave equation for 0[67]. Numerical investigations show that for given masses in a discrete chain quantity the 𝑃𝑗=|𝜎𝑗(𝑡)| may exceed the values of 1 in certain time instants (reported in [69, Table  5.1]).

Observe that splash amplitude does not depend on the parameter m/c. In addition, the amplitude of the chain vibrations increases with an increase of 𝑛; whereas its total energy does not depend on 𝑛. However, this is not a paradox. Namely, amplitude of vibrations has an order of sum of quantities𝜎𝑗(𝑡); whereas its potential energy order is represented by a sum of squares of the quantities mentioned [65].

On the other hand, a vibration amplitude of a mass with a fixed number is bounded for 𝑛, but amplitude of vibrations of a mass with a certain number increasing with increase of 𝑛 tends to infinity for 𝑛 following ln𝑛 [65]. “In the language of mechanics what we just said means that when analyzing the so-called “local properties” of a one-dimensional continuous medium, one cannot treat the medium as the limiting case of a linear chain of point masses, obtained when the number of points increases without limit” [66].

It should be emphasized that a rigorous proof of the mentioned properties has been obtained for a case, when 𝑛 + 1 is a simple digit or it is a power of 2. However, this assumption is not necessary, as it is mentioned in [65].

Earlier the same effect of continualization was predicted by Ulam, who wrote [74, pages 89, 90]: “The simplest problems involving an actual infinity of particles in distributions of matter appear already in classical mechanics.” A discussion on these will permit us to introduce more general schemes which may possibly be useful in future physical theories.

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 finite systems enabling one to use the algorithms of the calculus. The usual introduction of the continuum leaves much to be discussed and examined critically. The derivation of the equations of motion for fluids, for example, runs somewhat as follows. One images a very large number N of particles, say with equal masses constituting a net approximating the continuum, which is to be studied. The forces between these particles are assumed to be given, and one writes Lagrange equations for the motion of N particles. The finite system of ordinary differential equations becomes in the limit 𝑁= one or several partial differential equations. The Newtonian laws of conservation of energy and momentum are seemingly correctly formulated for the limiting case of the continuum. There appears at once, however, at least possible objection to the unrestricted validity of this formulation. For the very fact that the limiting equations imply tacitly the continuity and differentiability of the functions describing the motion of the continuum seems to impose various constraints on the possible motions of the approximating finite systems. Indeed, at any stage of the limiting process, it is quite conceivable for two neighbouring particles to be moving in opposite directions with a relative velocity which does not need to tend to zero as N becomes infinite; whereas the continuity imposed on the solution of the limiting continuum excludes such a situation. There are, therefore, constraints on the class of possible motions which are not explicitly recognized. This means that a viscosity or other type of constraints must be introduced initially, singling out “smooth” motions from the totality of all possible ones. In some cases, therefore, the usual differential equations of hydrodynamics may constitute a misleading description of the physical process.

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 [75] the continuous limit was derived based on the hypothesis that the microscopic displacements are equal to the macroscopic ones. In [75] authors supposed that the displacements of the particles which are connected by elastic springs are small in the following sense:|𝑦𝑖𝑦𝑗|𝑐𝜀. This inequality can be justified only for lower part of spectrum for natural oscillations.

5. Anticontinuum Limit

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 3).

In this case one has𝜎𝑘=(1)𝑘Ω, and equation for Ω reads 𝑚Ω𝑡𝑡+4𝑐Ω=0.(5.1)

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” [7679] (Figure 4). Namely, first we use staggered transformation 𝜎𝑘=(1)𝑘Ω𝑘,(5.2) and then (2.3) and (2.15) are reduced to the following BVP: 𝑚Ω𝑘𝑡𝑡+𝑐4Ω𝑘+Ω𝑘12Ω𝑘+Ω𝑘+1Ω=0,(5.3)0=1,Ω𝑛+1Ω=0,(5.4)𝑘=Ω𝑘𝑡=0for𝑡=0,𝑘=1,2,,𝑛.(5.5)

Then, the following relations are applied:Ω𝑘12Ω𝑘+Ω𝑘+1=4sin2𝑖2𝜕𝜕𝑥Ω=2𝜕2𝜕𝑥2+4𝜕124𝜕𝑥4+6𝜕3606𝜕𝑥6+Ω,𝑘=0,1,2,,𝑛,𝑛+1.(5.6)

Using (5.6), (5.3), and considering 2 as a small parameter one gets (we take zeroth and first-order approximations only) 𝑚Ω𝑡𝑡+4𝑐Ω+c2Ω𝑥𝑥=0.(5.7)

Appropriate boundary and initial conditions for (5.7) follow Ω=1for𝑥=0,Ω=0for𝑥=𝑙,Ω=Ω𝑡=0for𝑡=0.(5.8)

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.

6. Improved Continuous Approximations

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 [80]. In what follows we apply the gradient formulation approach.

If, in the series (3.9), we keep three first terms, the following model is obtained: 𝑚𝜕2𝜎𝜕𝑡2=𝑐2𝜕2𝜕𝑥2+2𝜕124𝜕𝑥4+4𝜕3606𝜕𝑥6𝜎.(6.1)

However, a nontrivial problem regarding boundary conditions for (6.1) appears [81, 82]. The conditions mentioned can be defined only assuming the chain dynamics behavior for 𝑘=1,2,3; 𝑘=𝑁+2,𝑁+3,𝑁+4. In other words, a boundary point is replaced by a boundary domain [44, 45]. In particular, in the case of periodic spatial extension “simple support” one gets 𝜎=𝜎𝑥𝑥=𝜎𝑥𝑥𝑥𝑥=0for𝑥=0,𝑙.(6.2)

Assuming 𝜎𝑘(𝑡)=0 for 𝑘=1,2,3; 𝑘=𝑁+2,𝑁+3,𝑁+4, instead of the boundary conditions (6.2) we have “clamping” 𝜎=𝜎𝑥=𝜎𝑥𝑥𝑥=0for𝑥=0,𝑙.(6.3)

Comparison of 𝑛th frequency of a continuous system (6.1), (6.2) with that of a discrete system exhibits essential accuracy improvement (applying coefficient 2.1 instead of 2 in an exact solution yields an error of ~5%). Observe that an estimation of the accuracy of continuous approximation on the basis of comparison of discrete and continuous systems frequencies rather simple but yielding reliable results.

In a general case, keeping in series (3.9) 𝑁 terms, one gets equations of the so-called intermediate continuous models [70] 𝑚𝜕2𝜎𝜕𝑡2=2𝑐𝑁𝑘=12𝑘𝜕(2𝑘)!2𝑘𝜎𝜕𝑥2𝑘.(6.4)

Boundary conditions for (6.4) have the following form: 𝜕2𝑘𝜎𝜕𝑥2𝑘=0for𝑥=0,𝑙,𝑘=0,1,,𝑁1(6.5) or 𝜕𝜎=0,2𝑘1𝜎𝜕𝑥2𝑘1=0for𝑥=0,𝑙,𝑘=1,,𝑁1.(6.6)

The corresponding BVPs are correct (and also stable during numerical solution) for odd 𝑁. In this case (6.4) is of hyperbolic type [70]. Application of intermediate continuous models allows catching the mentioned splashes effect.

For𝑁=2𝑘, 𝑘=0,1, intermediate continuous models (6.4) are unstable.

One also can mention the momentous elasticity theories [14, 2527, 8387]. For example, taking into account dependence of energy of deformation from the higher gradients of displacements leads to the concept of momentous stresses [88]. Le Roux was the first who showed the importance of the higher gradients of displacements [28]. In momentous theories of elasticity the method of macrocells is also widely used [8991]. The dynamics of the continuous media is described by the equations of displacement of the centre of mass of a macro cell and by the equations of moments of various orders. The spectrum of this media tends to the complete spectrum of a linear chain with increasing number of moments. Critical reexamination of this approach can be found in [3].

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 [52, 92] (for description of PA see Appendix A). Collins [53] proposed to construct continuous models using PA. Then this approximation was widely used by Rosenau [5560]. More exactly, Collins and Rosenau did not use PA straightforward; this was done later [61, 62, 93]. Sometimes these continuous models are called quasicontinuum approximations.

If only two terms are left in the series (3.9), then the PA can be cast into the following form:𝜕2𝜕𝑥2+2𝜕124𝜕𝑥4𝜕2/𝜕𝑥212𝜕/122/𝜕𝑥2.(6.7)

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

The corresponding quasicontinuum model reads 𝑚12𝜕122𝜕𝑥2𝜎𝑡𝑡𝑐2𝜎𝑥𝑥=0.(6.8)

The boundary conditions for (6.8) have the form 𝜎=0for𝑥=0,𝑙.(6.9)

The error regarding estimation of n-th frequency in comparison to that of a discrete chain is of ~16.5%. However, (6.8) is of lower order in comparison to (6.1).

Equation (6.8) is usually called Love equation [94] (but as Love mentioned [95] this equation was obtained earlier by Rayleigh [96, 97]). Term 𝜎𝑥𝑥𝑡𝑡 can be treated as the lateral inertia. Equation (6.8) has hyperbolic type.

Kaplunov et al. [98] refer to these type theories as theories with modified inertia.

Passage to (6.8) can be treating as regularization procedure. The model governed by (6.8) is unconditionally stable and propagating waves cannot transfer energy quicker than the velocity c. However, the model governed by (6.8) predicts that short waves transfer elastic energy with almost zero speed [99].

It is worth noting that (5.7) also can be regularized using PA as follows [100102]:𝑚124𝜕2𝜕𝑥2Ω𝑡𝑡+𝑐Ω=0.(6.10)

Finally, having in hand both long and short waves asymptotics, one may also apply two-point PA (description of two-point PA see Appendix A). In what follows we construct two-point PA using the first term of the series (3.9) as a one of limiting cases. We supposesin2𝑖2𝜕1𝜕𝑥=2𝑘=12𝑘𝜕(2𝑘)!2𝑘𝜕𝑥2𝑘=24𝜕2𝜕𝑥21+2𝜕122𝜕𝑥2+4𝜕3604𝜕𝑥4+2𝜕/42/𝜕𝑥21𝑎2𝜕2/𝜕𝑥2.(6.11)

The improved continuous approximation is governed by the following equation: 𝑚1𝑎22𝜕2𝜕𝑥2𝜎𝑡𝑡𝑐2𝜎𝑥𝑥=0.(6.12)

Now we require the n-th frequency of a continuous and discrete system to coincide𝜔𝑛=2𝑐𝑚sin𝑛𝜋2(𝑛+1).(6.13)

For large value of 𝑛one may approximately assume that 𝛼𝑛2𝑐/𝑚(6.14) with the boundary conditions (6.9).

Oscillations frequencies defined by the BVP (6.12), (6.9) are𝛼𝑘=𝜋𝑐𝑚𝑘(𝑛+1)2+𝑎2𝑘2,𝑘=1,2,.(6.15)

Now, using (6.14) one gets 𝑎2=0.25𝜋2.

The highest error in estimation of the eigenfrequencies appears for 𝑘=[0.5(𝑛+1)] and consists of 3%.

Observe that equations similar to the (6.14) are already known. Eringen [103106] using a correspondence between the dispersion curves of the continuous approximation and Born-von Kármán model [82, 107] obtained 𝑎2=0.1521. This value is very close to that proposed in [108, 109] on the basis of a certain physical interpretation, where also model (6.14) is referred to as the “dynamically consistent model.”

Interesting, that Mindlin and Herrmann used very similar to two-point PA idea for construction their well-known equation for longitudinal waves in rod [110, 111].

Dispersion relation (6.15) does not satisfy condition 𝑑𝛼𝑘𝑑𝑘=0(6.16) at the end of first Brillouin zone [112]. That is why in [105, 113] so-called bi-Helmholtz type equation was proposed. In [113] this equation is𝑚1𝑎22𝜕2𝜕𝑥2+𝑎214𝜕4𝜕𝑥4𝜎𝑡𝑡𝑐2𝜎𝑥𝑥=0,(6.17) where 𝑎1=1/𝜋.

Boundary conditions associated with this equation are 𝜎=𝜎𝑥𝑥=0for𝑥=0,𝑙(6.18) or 𝜎=𝜎𝑥=0for𝑥=0,𝑙.(6.19)

7. Forced Oscillations

In order to study forced vibrations we begin with a classical continuous approximation. A solution to (3.1) is sought in the following form: 𝑥𝜎=1𝑙+𝑢(𝑥,𝑡),(7.1) and the function 𝑢(𝑥,𝑡) is defined by the following BVP: 𝑚𝜕2𝑢𝜕𝑡2=𝑐2𝜕2𝑢𝜕𝑥2,𝑥𝑢(0,𝑡)=𝑢(𝑙,𝑡)=0,𝑢(𝑥,0)=1+𝑙,𝑢𝑡(𝑥,0)=0.(7.2)

Solution to the BVP (7.2) is obtained via Fourier method 𝑥𝜎=1𝑙2𝜋𝑘=11𝑘sin𝑘𝜋𝑥𝑙𝛼cos𝑘𝑡,(7.3) where 𝛼𝑘 defined by (3.10).

For the approximation the motion of the chain, one must keep in an infinite sum only n first harmonics 𝑥𝜎=1𝑙2𝜋𝑛𝑘=11𝑘sin𝑘𝜋𝑥𝑙𝛼cos𝑘𝑡.(7.4)

Observe that solutions (2.16) and (7.4) differ regarding frequencies 𝛼𝑘 and 𝜔𝑘 (relations (2.8) and (3.10), resp.). In addition, the coefficients in series (2.16) and (7.4) differ from each other, that is, projections onto normal vibrations of discrete and continuous systems, (1/(𝑛+1))𝑐𝑡𝑔(𝜋𝑘/2(𝑛+1)) and 2/𝜋𝑘, respectively, are strongly different for 𝑘1. This is because during projection into normal vibrations for the discrete system one must use summation due k from 0 to n, whereas for the continuous system—integration due 𝑥 from 0 to l. The problem occurring so far can be overcome using the Euler-McLaurin formulas [114]𝑛+1𝑘=0𝑓(𝑘)=0𝑛+11𝑓(𝑥)𝑑𝑥+2[]+𝑓(0)+𝑓(𝑛+1)𝑗=1(1)𝑗+1𝐵𝑗+1𝑗𝑑𝑗𝑓(𝑛+1)𝑑𝑥𝑗𝑑𝑗𝑓(0)𝑑𝑥𝑗,(7.5) where 𝐵𝑖 are the Bernoulli numbers, having the following values: 𝐵0=1,𝐵1=1/2,𝐵2=1/6,𝐵3=0,.

In addition, the following formulas hold [115]:𝐵𝑛1=𝑛+1𝑛𝑘=1𝐶𝑘+1𝑛+1𝐵𝑛𝑘,(7.6)𝑛+1𝑘=0sin2𝑘𝜋𝑗=𝑛+1𝑛+12,(7.7)𝑛+1𝑘=0𝑗1𝑛+1sin𝑘𝜋𝑗=1𝑛+1𝑛+1𝑐𝑡𝑔𝑗𝜋.2(𝑛+1)(7.8)

The corresponding integrals read 0𝑛+1sin2𝜋𝑗𝑥𝑛+1𝑑𝑥=𝑛+12,(7.9)0𝑛+1𝑥1𝑙sin𝜋𝑗𝑥2𝑛+1𝑑𝑥=.𝜋𝑗(7.10)

Observe that the values of sum (7.7) and integral (7.9) coincide. Using the Euler-McLaurin formula one obtains 𝑛+1𝑘=0𝑗1𝑛+1sin𝑘𝜋𝑗=𝑛+10𝑛+1𝑥1𝑙sin𝜋𝑗𝑥1𝑛+1𝑑𝑥+2[]sin0+sin(𝑗𝜋)𝑗𝜋6=2(𝑛+1)cos0+𝜋𝜋𝑗12𝑗212(𝑛+1)2.(7.11)

Owing to (7.11), one can construct a simple expression relatively good approximating sum (7.8) for arbitrary j values from j = 1 to j = n. For this purpose we change second term of the right-hand side of (7.11) in the following way: 𝑛+1𝑘=0𝑗1𝑛+1sin𝑘𝜋𝑗2𝑛+1𝑗𝜋𝑗12(𝑛+1)2.(7.12)

8. Wave Processes in Discrete and Continuous 1D Media

Let us study wave propagation in the infinite 1D medium (Figure 5). Assuming that when time instant t = 0, a force 𝑃 acts on a mass with number 0 in direction of the axis 𝑥. Governing equations can be written in the following way [116]:𝑚𝑑2𝑦𝑗𝑑𝑡2𝑦=𝑐𝑗+12𝑦𝑗+𝑦𝑗1𝑚𝑑,𝑗0,2𝑦0𝑑𝑡2𝑦=𝑐12𝑦0+𝑦1+𝑃.(8.1)

We reduce (8.1) to the dimensionless form 𝜕2𝑌𝑗𝜕𝜏2=𝑌𝑗+12𝑌𝑗+𝑌𝑗1,𝜕2𝑌0𝜕𝜏2=𝑌12𝑌0+𝑌1+𝑃1.(8.2)

Here 𝜏=𝑡𝑐/𝑚;𝑌𝑗=𝑦𝑗/;𝑃1=𝑃/(𝑐).

Due to the linearity of the problem one can suppose 𝑃1=1.

Let us apply Fourier transform to (8.2) [116]. Namely, multiplying left-and right-hand sides of (8.2) by exp(𝑖𝑞𝑗),𝑗=0,±1,±2,and adding them one obtains𝑑2𝑈𝑑𝜏2+4sin2𝑞2𝑈=1,(8.3) where 𝑈=𝑗=𝑌𝑗exp(𝑖𝑞𝑗).

Observe that the term 4sin2(𝑞/2) can be developed in the vicinity of either 𝑞=0(classical continuous approximation)4sin2𝑞2=𝑞2+(8.4) or 𝑞=𝜋 (envelope continualization)4sin2𝑞2=4+.(8.5)

Solving (8.3) and applying inverse Fourier transform one obtains wave velocity propagation in discrete media: 𝑑𝑌𝑗(𝜏)=𝐴𝑑𝜏1(𝑗,𝜏)𝜋=1𝜋0𝜋/2sin(2𝜏sin𝑠)sin𝑠cos(2𝑗𝑠)𝑑𝑠.(8.6)

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 (8.2), the following wave equation with the Dirac delta-function in right hand-side is obtained:𝜕2𝑌𝜕𝜏2𝜕2𝑌𝜕𝑧2=𝛿(𝑧).(8.7)

Here 𝑧=𝑥/.

The following relation between discrete and continuous systems holds:𝑌𝑗(𝜏)=𝑌(𝑗,𝜏),𝑗=0,±1,±2,.(8.8)

Now, applying the Fourier transform in 𝑥 and the Laplace transform in 𝜏, one obtains𝜕𝑌𝜕𝜏=0.5𝐻(𝜏|𝑧|).(8.9)

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:𝜕2𝑌𝜕𝜏2𝜕2𝜕𝑧2+1𝜕124𝜕𝑧4+1𝜕3606𝜕𝑧6𝑌=sin(𝜋𝑧)𝜋𝑧.(8.10)

Let us explain the occurrence of term sin(𝜋𝑧)/(𝜋𝑧)in the right hand of (8.10) instead of Dirac delta-function (for detail description see Appendix C).

Applying both Fourier transform in 𝑧 and Laplace transform in 𝜏, the following equation governing wave velocity propagation is obtained: 𝜕𝑌(𝑧,𝜏)=𝐴𝜕𝜏2(𝑧,𝜏)𝜋=1𝜋0𝜋/2sin2𝜏𝑠1(1/3)𝑠2+(2/45)𝑠4𝑠1(1/3)𝑠2+(2/45)𝑠4cos(2𝑠𝑧)𝑑𝑠.(8.11)

The quasicontinuum model is as follows:11𝜕122𝜕𝑧2𝜕2𝑌𝜕𝜏2𝜕2𝑌𝜕𝑧2=11𝜕122𝜕𝑧2sin(𝜋𝑧)𝜋𝑧.(8.12)

The associated wave velocity propagation follows: 𝜕𝑌(𝑧,𝜏)=𝐴𝜕𝜏3(𝑧,𝜏)𝜋=1𝜋0𝜋/21+(1/3)𝑠2sin2𝜏𝑠/1+(1/3)𝑠2𝑠cos(2𝑠𝑧)𝑑𝑠.(8.13)

Model (6.14) for our case has the following form:1𝑎2𝜕2𝜕𝑧2𝜕2𝑌𝜕𝜏2𝜕2𝑌𝜕𝑧2=1𝑎2𝜕2𝜕𝑧2sin(𝜋𝑧)𝜋𝑧.(8.14)

In this case the associated wave velocity propagation reads𝜕𝑌(𝑧,𝜏)=𝐴𝜕𝜏4(𝑧,𝜏)𝜋=1𝜋0𝜋/21+4𝑎2𝑠2sin2𝜏𝑠/1+4𝑎2𝑠2𝑠cos(2𝑠𝑧)𝑑𝑧.(8.15)

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 𝐴1(𝑗,𝜏) will serve us as the standard solution used for estimation of other solutions. Results of computations are shown on Figures 6(a)6(c) and Figures 7(a)7(c). One can conclude that above described improved continuous models give possibility qualitatively taking into account effect of media discreteness: (i) occurrence of oscillations, being gradually damped at x = const; (ii) infinite velocity of perturbations propagation (owing to assumption of instantaneous masses interaction during their draw near); (iii) occurrence of the quasi-front domain, where the stresses increase relatively fast, but without jump, as on the wave front set; velocities and deformations exponentially decay while the quasi-front 𝜏|𝑧| increases finally becoming negligibly small.

Equation (8.10) has sixth order in spatial variable, (8.12) and (8.14) have second order in spatial variable, so, they are simpler for practice applications. Equations (8.12), and (8.14) give for case of wave motion qualitatively the similar results.

9. 𝟐D Lattice

In order to analyse the 2D case we use 9-cell square lattice (Figure 8) [87, 117]. The central particle is supposed to interact with eight neighbours in the lattice. The mass centres of four of them are on horizontal and vertical lines, while the mass centres of the other four neighbouring particles lie along diagonals. Interactions between the neighbouring particles are modelled by elastic springs of three types. Horizontal and vertical springs with rigidity 𝑐1 define interaction forces of extension/compression of the material. The diagonal longitudinal springs have rigidity 𝑐0, and shear axial spring - 𝑐2. Governing equations of motion are [87]𝑚̈𝑢𝑚,𝑛=𝑐1𝑢𝑚,𝑛12𝑢𝑚,𝑛+𝑢𝑚,𝑛+1+𝑐2𝑢𝑚1,𝑛2𝑢𝑚,𝑛+𝑢𝑚+1,𝑛+0.5𝑐0×𝑢𝑚1,𝑛1+𝑢𝑚+1,𝑛1+𝑢𝑚1,𝑛+1+𝑢𝑚+1,𝑛+1+𝑣𝑚1,𝑛1𝑣𝑚+1,𝑛1𝑣𝑚1,𝑛+1+𝑣𝑚+1,𝑛+14𝑢𝑚,𝑛,𝑚̈𝑣𝑚,𝑛=𝑐1𝑣𝑚,𝑛12𝑣𝑚,𝑛+𝑣𝑚,𝑛+1+𝑐2𝑣𝑚1,𝑛2𝑣𝑚,𝑛+𝑣𝑚+1,𝑛+0.5𝑐0×𝑣𝑚1,𝑛1+𝑣𝑚+1,𝑛1+𝑣𝑚1,𝑛+1+𝑣𝑚+1,𝑛+1+𝑢𝑚1,𝑛1𝑢𝑚+1,𝑛1𝑢𝑚1,𝑛+1+𝑢𝑚+1,𝑛+14𝑣𝑚,𝑛.(9.1)

Here 𝑢𝑖𝑗,𝑣𝑖𝑗is the displacement vector for a particle situated at point(𝑥𝑖,𝑦𝑗), 𝑥𝑖=𝑖,𝑦𝑗=𝑗.

The standard continualization procedure for (9.1) involves introducing a continuous displacement field 𝑢(𝑥,𝑦),𝑣(𝑥,𝑦) such that𝑢(𝑥𝑚,𝑦𝑛)=𝑢𝑚,𝑛;𝑣(𝑥𝑚,𝑦𝑛)=𝑣𝑚,𝑛, and expanding 𝑢𝑚±1,𝑛±1,𝑣𝑚±1,𝑛±1 into Taylor series around 𝑢𝑚,𝑛,𝑣𝑚,𝑛. Second-order continuous theory in respect to the small parameter h is [87]𝑚𝜕2𝑢𝜕𝑡2=𝑐12𝜕2𝑢𝜕𝑥2+𝑐22𝜕2𝑢𝜕𝑦2+2𝑐02𝜕2𝑣,𝑚𝜕𝜕𝑥𝜕𝑦2𝑣𝜕𝑡2=𝑐12𝜕2𝑣𝜕𝑥2+𝑐22𝜕2𝑣𝜕𝑦2+2𝑐02𝜕2𝑢.𝜕𝑥𝜕𝑦(9.2)

Naturally, one can construct equations of higher order, but, as it is shown in [87], generally it is not possible to create asymptotic theories that do not possess extraneous solutions in the nonscalar context.

Using staggered transformations𝑢𝑘=(1)𝑘𝑢,𝑣𝑘=(1)𝑘𝑣,(9.3) one obtains in anticontinuum limit.𝑚𝜕2𝑢𝜕𝑡2𝑐=41𝜕2𝑢𝜕𝑥2+𝑐2𝜕2𝑢𝜕𝑦2,𝑚𝜕2𝑣𝜕𝑡2𝑐=41𝜕2𝑣𝜕𝑥2+𝑐2𝜕2𝑣𝜕𝑦2.(9.4)

As in 1D case (see (5.1)) equations (9.4) do not contain parameter2. For this type of 2D lattice equations in anticontinuum limit are decoupled.

The existence of the continuous approximations (9.2) and (9.4) gives a possibility to construct the composite equations, which is uniformly suitable in the whole interval of the frequencies and the oscillation forms of the 2D lattice of masses. Let us emphasize, that the composite equations, due to Van Dyke [118], can be obtained as a result of synthesis of the limiting cases. The principal idea of the method of the composite equations can be formulated in the following way [118, page 195].(i)Identify the terms in the differential equations, the neglection of which in the straightforward approximation is responsible for the nonuniformity.(ii)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 (9.2) for long wave solution and with (9.4) for short wave solution. As a result of the described procedure one gets𝑚1𝑎22𝜕2𝜕𝑥2𝑎22𝜕2𝜕𝑦2𝜕2𝑢𝜕𝑡2=𝑐12𝜕2𝑢𝜕𝑥2+𝑐22𝜕2𝑢𝜕𝑦2𝑐1+𝑐24𝛾2𝜕4𝑢𝜕𝑥2𝜕𝑦2+2𝑐0211+4𝑐1+𝑐2𝜕2𝜕𝑡2𝜕2𝑣,𝑚𝜕𝑥𝜕𝑦1𝑎22𝜕2𝜕𝑥2𝑎22𝜕2𝜕𝑦2𝜕2𝑣𝜕𝑡2=𝑐12𝜕2𝑣𝜕𝑥2+𝑐22𝜕2𝑣𝜕𝑦2𝑐1+𝑐24𝛾2𝜕4𝑣𝜕𝑥2𝜕𝑦2+2𝑐0211+4𝑐1+𝑐2𝜕2𝜕𝑡2𝜕2𝑢.𝜕𝑥𝜕𝑦(9.5)

Here 𝑎2=0.25𝜋2;𝛾2=(4𝜋2+8𝜋2𝛼2)/(4𝜋2).

For 1D case from (9.5) one obtains (6.14) for 𝑢 and the same equation for 𝑣. For small variability in spatial and time variables equations (9.5) can be approximated by (9.2), for large variability in spatial and time variables equations (9.5) can be approximated by (9.4).

10. Nonlinear Case

Fundamental methods of analysis of linear lattices are those based on either discrete or continuous Fourier transformations [44, 45, 119] and the integral Fourier operators method proposed by Maslov [67]. In a nonlinear case the similar methods, in general, are not applicable. However, there exists class of piece-wise acting forces-springs extensions relations, opening the doors for analytical tools application [120].

Besides, one may achieve even a solution to a nonlinear problem using the following observation [61, 62, 119, 121]. Let nonlinear springs in 1D problem have potential energy 𝑈(𝑦), where 𝑦 is a difference between the neighborhood masses regarding their equilibrium positions. In this case the stretching force is 𝜕𝑈(𝑦)/𝜕𝑦. The chain dynamics is governed by the following infinite system of ODEs:𝜕2𝑟𝑛𝜕𝑡2=𝑟𝜕𝑈𝑛+1𝜕𝑟𝑛+1𝑟2𝜕𝑈𝑛𝜕𝑟𝑛+𝑟𝜕𝑈𝑛1𝜕𝑟𝑛1,(10.1) where 𝑟𝑛=𝑦𝑛+1𝑦𝑛.

Assume that solution to the ODEs is a traveling wave, 𝑟𝑛=𝑅(𝑧),𝑧=𝑛𝑣𝑡. Then (10.1) is cast into the following form: 𝑣2𝑅(𝑧)=𝑈(𝑅(𝑧1))2𝑈(𝑅(𝑧))+𝑈(𝑅(𝑧+1)),(10.2) where ()=𝑑()/𝑑𝑟.

Introducing notation Φ(𝑧)=𝑈(𝑅(𝑧))and applying the Fourier transformation to (10.2) one gets 𝑅(𝑞)=𝑅(𝑧)exp(𝑖𝑞𝑧)𝑑𝑧,Φ(𝑞)=Φ(𝑧)exp(𝑖𝑞𝑧)𝑑𝑧.(10.3) In the conjugated space one obtains 𝑣2𝑅(𝑞)=𝑓(𝑞)Φ(𝑞),(10.4) where 𝑓(𝑞)=4(sin2(𝑞/2)/𝑞2).

Term 𝑓(𝑞)can be approximated by PA or two-point PA in the following way:1𝑓(𝑞)=1+𝑞21/12or𝑓(𝑞)=1+𝑎2𝑞2,(10.5) where 𝑎2=0.25𝜋2.

In result one obtains equation of continuous approximation 𝑣211𝑑122𝑑𝑧2𝑅(𝑧)=𝑈(𝑅(𝑧))(10.6) or 𝑣21𝑎2𝑑2𝑑𝑧2𝑅(𝑧)=𝑈(𝑅(𝑧)).(10.7)

We show also a simultaneous application of PA matched with a perturbation procedure using an example of the Toda lattice [122]𝑚𝑑2𝑦𝑛𝑑𝑡2𝑏𝑦=𝑎exp𝑛1𝑦𝑛𝑏𝑦exp𝑛𝑦𝑛+1.(10.8)

In continuum limit one obtains (for details see Appendix B)𝜕𝑦1𝜕𝑡1+𝜕1𝜕𝑥1+𝜕242𝜕𝑥2𝑦1𝑦1𝜕𝑦1𝜕𝑥=0.(10.9) where: 𝑡1=(𝑎𝑏/𝑚)𝑡;𝑦1=(𝑏/2)𝑦.

One can obtain from (10.9) the Korteweg-de Vries (KdV) equation [122]. Here, we illustrate how one may get regularized long-wave equation [123125]. Using PA one obtains 11+𝜕242𝜕𝑥211𝜕242𝜕𝑥21.(10.10)

Then, (10.9) yields (up to the highest terms)11𝜕242𝜕𝑥2𝜕𝑦1𝜕𝑡1+𝜕𝑦1𝜕𝑥𝑦1𝜕𝑦1𝜕𝑥=0.(10.11)

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, Langmuir, Calogero, and so forth, [119, 126130]) in the case of infinite lattices or in the case of occurrence of physically unrealistic boundary conditions. In many cases nonintegrable systems like a discrete variant of sine-Gordon equation possess soliton-like solution.

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 [100102, 131]. Improved continuous models give possibility to construct such type of localized solutions [100102].

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 [130], who analysed Toda lattice with variable coefficients 𝑚𝑘𝑑2𝑦𝑘𝑑𝑡2=𝑎𝑘exp𝑏𝑘𝑦𝑘𝑦𝑘11+𝑎𝑘+1exp𝑏𝑘+1𝑦𝑘+1𝑦𝑘,1𝑘=0,±1,±2,.(10.12)

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.

11. Some Generalization

Let us show that PA for constructing of improved continuous models can be used iteratively. For three terms in expansion (3.9) one has PA𝜕2/𝜕𝑥21+2𝜕/202/𝜕𝑥212𝜕/302/𝜕𝑥2,(11.1) and continuous model𝑚12𝜕302𝜕𝑥2𝜎𝑡𝑡𝑐21+2𝜕202𝜕𝑥2𝜎𝑥𝑥=0(11.2) or 𝑚𝜕2𝜎𝜕𝑡2𝑐2𝑚1+𝜕30𝑐2𝜕𝑡2+2𝜕202𝜕𝑥2𝜎𝑥𝑥=0.(11.3)

Now we use PA as follows:𝑚1+𝜕30𝑐2𝜕𝑡2+2𝜕202𝜕𝑥2=𝜕1+(𝑚/30𝑐)2/𝜕𝑡22𝜕1+(𝑚/30𝑐)2/𝜕𝑡22𝜕/202/𝜕𝑥2(11.4) and obtain new improved continuous model𝑚172𝜕602𝜕𝑥2𝜕2𝜎𝜕𝑡2+𝑚230𝑐12𝜕302𝜕𝑥2𝜕4𝜎𝜕𝑡4𝑐2𝜕2𝜎𝜕𝑥2=0.(11.5)

Now let us consider the continuous models of the chain with two different particles in primitive cell, that is, the chain with alternating masses𝑚𝑦𝑗𝑡𝑡𝑦(𝑡)𝑐𝑗1(𝑡)2𝑦𝑗(𝑡)+𝑦𝑗+1=(𝑡)(𝑚𝑀)𝑦2𝑗𝑡𝑡(𝑡);𝑗=1,2,3,.(11.6)

Continuous approximation for (11.6) is (rod with localized masses)𝑚𝑦𝑡𝑡(𝑥,𝑡)𝑐2𝑦𝑥𝑥(𝑥,𝑡)=(𝑚𝑀)𝑘=𝛿(𝑥2𝑘)𝑦𝑡𝑡(𝑥,𝑡).(11.7)

Improved continuous approximation for (11.6) can be written as follows:𝑚12𝜕122𝜕𝑥2𝑦𝑡𝑡(𝑥,𝑡)𝑐2𝑦𝑥𝑥(𝑥,𝑡)=(𝑚𝑀)12𝜕122𝜕𝑥2𝑘=sin𝜋(𝑥2𝑘)𝑦𝜋(𝑥2𝑘)𝑡𝑡(𝑥,𝑡)(11.8) or 𝑚1𝑎2𝜕2𝜕𝑥2𝑦𝑡𝑡(𝑥,𝑡)𝑐2𝑦𝑥𝑥(𝑥,𝑡)=(𝑚𝑀)1𝑎2𝜕2𝜕𝑥2𝑘=sin𝜋(𝑥2𝑘)𝑦𝜋(𝑥2𝑘)𝑡𝑡(𝑥,𝑡).(11.9)

Homogenization and other asymptotic approaches can be use for this continuous system [132, Section  3]).

12. Modified Navier-Stokes Equations

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 [133]. Many famous mathematicians (Smale et al. [134], Yudovich [135]) also wrote about this problem as one of the most important problems of the 21st century. The 3D problem remains open until today [136], although in the 1950s Ladyzhenskaya obtained the key result of global unique solvability of the initial boundary value problem for the 2D Navier-Stokes Equations Ladyzhenskaya believed that in the 3D case the Navier-Stokes Equations, even with very smooth initial data, do not provide uniqueness of their solutions on an arbitrary time interval. She introduced the modified Navier-Stokes equations [133, 137139] (see also [140]):𝑢𝑡𝜕𝜕𝑥𝑘𝜈0+𝜈1𝑢2𝑥𝑢𝑥𝑘+𝑢𝑘𝑢𝑥𝑘=grad𝑝+𝑓(𝑥,𝑡),div𝑢=0,𝑘=1,2,3(12.1)


Equations (12.1), (12.2) differ from the classical ones only in large-velocity gradients and coincide with them for𝜈1=0. For (12.1) and (12.2) Ladyzhenskaya proved global unique solvability under reasonably wide assumptions.

As it is mentioned in [59], “From the kinetic point of view these equations represent the long-wavelength limit. However, it often happens that the dynamics predicted by the hydrodynamics equations involves relatively short-wavelength scales. This, formally at least, contradicts the conditions under which the macroscopic system was derived.” Note that the modified Navier-Stokes equations (12.1), (12.2) allow for taking into account higher order harmonics influence via artificially separated terms.

More investigations in direction of improvement of the Navier-Stokes equations are carried out in references [133, 137139].

It should be emphasized a role of mathematical approaches in proving various physical theories [141]. Although nowadays a physical experiment plays a crucial role in verification of many novel theories, but it is difficult sometimes to realize a sure and proper experimental verification and validation of the developed theories. In those cases it seems that the proper support of mathematical rigorous approaches may serve as a tool of “experimental” validation of the introduced theories. In other words the rigorous mathematical approaches validating the physical theories can be treated as experimental ones for the so called “theoretical mathematics.”

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 (12.1), (12.2) allow for taking into account higher order harmonics influence via artificially introduced terms. It is clear that it will be very challenging in getting the modified Navier-Stokes equations starting in modeling from a molecular structure, as it has been demonstrated so far using simple examples of discrete lattice. Here we can mention paper by Rosenau, who applied regularization procedure to the Chapman-Enskog expansion [59]. This approach, however, requires further investigations.

13. Conclusion

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 [142]. Then the problem of coupling molecular dynamics and continuum mechanics simulations appears. In this case, one can apply a concept of bridging domain [142144]. We think that in the latter case an application for describing bridging domain of improved continuous models can result in efficient computational time saving.

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 [51]). Intermediate continuous models and BVP obtained on the basis of the composite equations can be used for problems of any dimensions.

It will be very interesting to use for study investigation of 2D lattices 2D PA [145].

Let us finally discuss problems closely connected with brittle fracture of elastic solids [120, 146150]. Observe that a continuous model, which does not include material structure, is not suitable since any material crack occurs on the material structure level. This is a reason why some of the essential properties of damages of the classical continuous model theory are not exhibited. For instance, in a discrete medium the propagated waves transport part of the energy from the elastic body into the crack edges. This is why application of nonlocal theories during explosion process modeling (e.g., for the composite materials) looks very promising. As it is mentioned in [151], “The continuous development of advanced materials ensures that a “one size fits all” approach will no longer serve the engineering community in terms of predicting and preventing fatigue failures and reducing their associated costs.”


A. Padé Approximants

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 [152]. Let𝐹(𝜀)=𝑖=0𝑐𝑖𝜀𝑖,𝐹𝑚𝑛(𝜀)=𝑚𝑖=0𝑎𝑖𝜀𝑖𝑚𝑖=0𝑏𝑖𝜀𝑖,(A.1) where the coefficients ai, bi are determined from the following condition: the first (m + n) components of the expansion of the rational function Fmn (ε) in a McLaurin series coincide with the first (m + n +1) components of the series F(ε). Then Fmn is called the [m/n] PA. The set of Fmn functions for different 𝑚 and 𝑛 forms the Padé table.

Now we give the notion of two-point Padé approximants [152]. Let𝐹(𝜀)=𝑖=0𝑎𝑖𝜀𝑖when𝜀0,𝑖=0𝑏𝑖𝜀𝑖when𝜀.(A.2)

The TPPA is represented by the rational function𝐹(𝜀)=𝑚𝑘=0𝑎𝑘𝜀𝑘𝑛𝑘=0𝑏𝑘𝜀𝑘,(A.3) where k + 1 (k = 0, 1,, n + m + 1) coefficients of a McLaurin expansion, if 𝜀0, and m + n + 1 − k coefficients of a Laurent series, if 𝜀, coincide with the corresponding coefficients of the series (A.2).

B. Continuum Limit of Toda Lattice

Here we describe construct of the continuum limit of the Toda lattice𝑚𝑑2𝑦𝑛𝑑𝑡2𝑏𝑦=𝑎exp𝑛1𝑦𝑛𝑏𝑦exp𝑛𝑦𝑛+1.(B.1)

We follow [122]. Equation (B.1) can be rewritten as pseudodifferential one: 𝑚𝜕2𝑦𝜕𝑡214𝑎sinh2𝜕𝜕𝑥2exp(𝑏𝑦)=0.(B.2)

Equation (B.1) can be approximately factorized𝑚𝜕𝑎𝑏1𝜕𝑡2sinh2𝜕+𝑏𝜕𝑥2𝑦𝜕𝜕𝑥𝑚𝜕𝑎𝑏1𝜕𝑡+2sinh2𝜕𝑏𝜕𝑥2𝑦𝜕𝜕𝑥𝑦=0.(B.3)

For wave spreading in right direction one obtains 𝑚𝜕𝑎1𝜕𝑡+2sinh2𝜕𝜕𝑥2𝑦𝜕𝜕𝑥𝑦=0.(B.4)

Formally expanding function sinh((1/2)(𝜕/𝜕𝑥)) in a Maclaurin series up to the terms of the third-order one obtains from (B.4)𝑚𝑎𝑏𝜕𝑦+𝜕𝑡𝜕𝑦+1𝜕𝑥𝜕243𝑦𝜕𝑥3𝑏2𝑦𝜕𝑦𝜕𝑥=0.(B.5)

Using variables transform 𝑡1=(𝑎𝑏/𝑚)𝑡;𝑦1=(𝑏/2)𝑦 one obtains𝜕𝑦1𝜕𝑡1+𝜕𝑦1+1𝜕𝑥𝜕243𝑦1𝜕𝑥3𝑦1𝜕𝑦1𝜕𝑥=0.(B.6)

C. Correspondence between Functions of Discrete Arguments and Approximating Analytical Functions

Let us suppose system𝑚𝜎𝑗𝑡𝑡𝜎(𝑡)=𝑐𝑗+12𝜎𝑗+𝜎𝑗1+𝑠𝑗(𝑡)𝑠𝑗1(𝑡),𝑗=0,1,,𝑛1,𝑛(C.1)𝑠𝑗(𝑡) is the external force acting on jth point.

Further we follow [44].

In continuum limit we must construct a function s(x,t) representimg a continuous approximation of the function of discrete argument 𝑠𝑗(𝑡). Observe that it is defined with an accuracy to any arbitrary function, which equals zero in nodal points x = j, j = 0,1,2,,n. For this reason, from a set of interpolating functions one may choose the smoothest function owing to filtration of fast oscillating terms. As it is shown in [44], the interpolating function is determined uniquely, if one requires its Fourier image 𝑠(𝑞)𝑠(𝑞)=𝑠(𝑥)e𝑖𝑞𝑥1𝑑𝑥;𝑠(𝑥)=2𝜋𝑠(𝑞)e𝑖𝑞𝑥𝑑𝑥(C.2) to differ from zero only on the segment 𝜋/𝑠(𝑞)𝜋/. From this condition one obtains𝑠(𝑥,𝑡)=𝑛𝑘=1𝑠𝑘(𝑡)sin(𝜋(𝑥𝑘)/)𝜋(𝑥𝑘).(C.3)

For function sin(𝜋𝑥/)𝜋𝑥,(C.4) one has the following relation (from [115, formula])sin(𝜋𝑥/)𝜋𝑥𝑑𝑥=1.(C.5)

For function (C.4) turns into Dirac’s delta function. So for momentous theories of elasticity one must replace Dirac’s delta function to the function (C.4).


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.