As shown earlier: 

* (AIM- ab initio mundu (lat.) – from the beginning of the world).

Thus, the AIM theory is an open FK model with an increasing number of particles and with “running” ones, i.e. J-dependent; periods, masses and potential amplitudes.

In systems with periodic potentials, inhomogeneous dynamic solutions inevitably arise that are not destroyed. Consequently, in a system with potential (1) it is impossible to obtain a homogeneous solution as the final result. 

In this regard, it is necessary to introduce additional terms into Lagrangian (1), ensuring the destruction of nonlinear excitations. We believe that the initial terms of the “destruction mechanism” should be the first and second harmonics V(J) with the main and doubled periods alternating on (off) depending on the parity of J. 

We expect that in a system with the first and second harmonics alternately turning on (off) the previous one-dimensional solitons stop, but over time two-dimensional non-decaying dynamic excitations are formed.

After some time J_K ~ J₀ in (1), the following harmonics V_K(J) are turned on (off). At points J = J_K on the temporary dislocation chain of the AIM system, phase transitions occur with a change in the spatial dimension of dynamic excited states.

To summarize, for the AIM model we write: (1), 

where is the “destruction mechanism”, with each moment of time divided into K-instants, with the corresponding harmonics of the external potential. 

From general considerations it follows that V_K(J) ~ V₀(J), V₀(J₀) = 0. The phase transition points J_K are determined by inequalities (1).

From a cosmological point of view, the number of moments of the time is equal to the optimally round number, i.e. .

Let’s compare the generally accepted concepts with the concepts of the AIM model:

1. “Matter” – energy excitations on the time chain;

2. “Dark energy” - one-dimensional phase of matter (stationary); 

3. “Dark matter” - two-dimensional phase of matter (stationary);

4. “Visible matter” - three-dimensional phase of matter (dynamic);

5. The next phase is four-dimensional, etc.

Let us estimate the phase composition of matter at different stages of the development of the Universe.

Let X be the dynamic weight part of the Universe, then for a state of K phases we write:

  X+KX+K²X+K³X+…+K^K-1X=1; those. X= (K-1)/(K^K-1).

When K=3, X+3X+9X=1; X₃≈8%. Based on estimates of modern cosmology, for time intervals T_k we have:

T₃=30 billion years; T₂=90 billion years; T₁=180 billion years; T₄=7.5 billion years; T₅=1.5 billion years...; T = ∑T_k ≈ 310 billion years.

The AIM+ model assumes the return of the emitted atoms of the DFK –chain to the point of their departure, with the formation of two interacting subsystems in the AIM model.

A sublattice model of metal alloys and the hypothesis that alloys are described by commensurate phases of the DFK model are put forward. The chemical composition of its grains is predicted for the A_xB_1-x alloy.

A metal alloy is a collection of crystalline grains with an average size R, the space between which is filled with impurities. 

Classical theories of metal alloys are based on the idea of a “random phase”: - atoms in a crystal lattice according to chemistry composition can be arranged randomly.

The DFK model puts forward the idea of a “commensurate phase”, which is realized by the strong interaction of crystal sublattices: for example - an alloy AB of equiatomic composition is considered as a commensurate crystal with AB molecules (N = L). If the main periods of the sublattices are equal, respectively for crystal A - a, for crystal B - b, then the period of the AB alloy crystal is equal to . When heated (T>V₀), the sublattices become independent and return to the original periods (a, b).

Having asked the question about the atomic composition of the grain of the A_xB_1-x alloy, we proceed from the main hypothesis - metal alloys are described by commensurate phases of the DFK structure. Let us project an alloy of cubic symmetry A_xB_1-x (x≥0.5) onto a one-dimensional DFK model. Assuming that the alloy is a commensurate phase of the DFK model with CH(A_x) and CH(B_1-x) sublattices and strong interchain interaction, V₀~1 - we will show that x can only take discretely defined values x=x₀.

To the alloy grain A_xB_1-x (x ≥ 0.5) we associate elastically periodic chains CH(A_x) and CH(B_1-x) of N and L atoms of the same size, respectively, then if the period of the chain CH(A_x) = 1, then the period of the chain CH(B_1-x) is equal to .

Thus, x can only take discrete values x₀:

                                                                                                                              (1)

The chemical composition of the A_xB_1-x alloy grain has the form A_x0B_1-x0. We choose in (1) the first fractional-rational values ≥ 1, with the smallest denominators, because commensurate phases [1-4] with strong interaction can only be realized with them. From (1) we have: x₀ = 0.50; 0.70; 0.89, i.e. very limited number of options depending on V₀.

The dropped atoms, with density Δx=x-x₀, are located between the grains, determining their size R:

                                                              R~1/ Δx.                                                                        (2)

It is interesting to note that the chemical composition of many metal alloys A_xB_1-xC, with high-temperature superconductivity (HTS), lies near the values of x₀ = 0.50; 0.77; 0.89; 0.96.

This suggests that HTS should be described by a model with incommensurate phases [1-3] and a chemical density wave. In this case the variance of R is minimal.

The DFK model is formed by replacing the external periodic potential of the FK model with a second elastically periodic chain of atoms. The properties of the FK and DFK models are basically the same, but the role of the boundary atoms of the DFK model in the structural phase transitions of the "commensurate- incommensurate " phase (IC) has increased. The IC transitions have become asymmetrical in parameter of incommensurate.

In 1938, the authors: Yakov Ilyich Frenkel and Tatyana Abramovna Kontorova (LFTI) put forward the Frenkel-Kontorova model (FK model) [1], which served as the basis for the creation of many theories of highly nonlinear processes [2].

It is of interest to develop the FK model in order to expand the scope of its natural science applications [3-4].

In order to develop the FK model, the DFK model (Developed Frenkel-Kontorova model) is put forward: - two one-dimensional sequences of N and L point atoms, masses m and M; with coordinates {x_i} and {y_j}, connected by elastic springs with the laws of elastic dispersion Φ₁(x) and Φ₂(y). Chains CH1 and CH2 interact with each other by potential V_i,j.

The Hamiltonian of the DFK model has the form: 

                                    (1)

From the analysis of the ground state of the DFK model (N = L) [3], the following conclusion: - when one of the Hooke’s chains is stretched by force F, an abrupt transition to the incommensurate phase occurs (F>F_c), in which part of the atoms of the stretched chain CH1 leaves the interaction space with CH2. The number of atoms falling out of the V_i,j interaction space , where V₀ = max V_i,j.

With strong interaction (V₀ ~ 1) and strong stretching (F>F_c ~1), the size of the dislocation is 2, and the number of precipitated atoms is N/2. In this case, the incommensurate phase will be a periodic chain of hole dislocations, i.e., commensurate crystal with doubled period.

FK, DFK models and a new theory of metal alloys are used to explain the Portevin-Le Chatelier effect.

Quote from [1,2]: -

“Many experiments measuring the deformation of solids under static loads have revealed sudden yielding and other deviations from normal behavior, now known as the “Portvin-Le Chatelier effect.” If we follow historical truth, then the honor of the discovery of this phenomenon should be associated with the names of Felix Savard (1837) and Antoine Philibert Masson (1841). Masson described a steep, almost vertical (σ-ε diagram) increase in stress, accompanied by very little deformation, up to a value at which there was a sudden sharp increase in deformation at constant stress. In experiments of this type with dead loads used in testing machines in the 19th century, this phenomenon took on the form that later led to the use of the term "staircase effect."

For small and large deformations, this effect has been studied by many over the past two centuries, but a satisfactory explanation has not yet been achieved.

It makes sense to compare the experimental ladders of the Portvin-Le Chatelier effect [1,2] with the already existing theoretical ladders [3,4].

In [1,2], in experiments on stretching AL with a purity of 99.99% shows several detailed graphs of the σ-ε dependence, for example, [1, p. 74] and [2, p. 288].

If you compare the staircase [1, p. 74] with the staircases [3,4], then their similarities are revealed - they almost coincide. But if you look at [1, p. 74] more carefully, especially at the initial stretching section, then qualitative differences are noticeable. First of all, this is the absence of strictly vertical segments in the experimental graphs. Consequently, the FK model is not enough to explain the EPLC, so it needs to be modified and replaced with the DFK model.

From the point of view of the DFK model, the initial stretching segment is associated with the general stretching of two CHs united by the potential V_lj. Further, at a certain critical force F_c, failure occurs with compression of CH2 and abrupt stretching of CH1 followed by interchain capture. The process is repeated until the sample breaks.

The first prediction of the new model is that when stretched, the sample becomes chemically inhomogeneous in length and composition of m and M atoms.

The most important question for the EPLC within the framework of the DFK model arises - the nature of Hooke's chains.

If stretchable CH1 is logically associated with an AL crystal, then the nature of CH2 may be associated with metal impurities. Let's follow this hypothesis.

In metals with a small number of impurities, for example, in ALR_%, the metal impurity R_% is capable of being ordered into a cubic crystal at high temperatures. Impurity period CH2 – one-dimensional projection of the crystal R_% we evaluate as , where % is the number of impurities in the main matrix. Suppose that in our case % = 10^-6, then the period CH2 R ≈ 100.

Comparing the number of steps [1, p. 74] with R=100, we find an approximate match.

As a result of stretching, the period of the R-sublattice changes from r=100 to r=1.

From the temperature graphs of the EPLC [2] it is clear that the EPLC disappears at T> T_c.

EPLC is a special case of phenomena in metal alloys A_xB_1-x.