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Within the framework of a discrete model of the nuclei of linear and planar defects, the variational principles of sliding in translational and rotational plasticity, fracture by separation (cleavage) and shear (shearing) in crystalline materials are considered. The analysis of mass transfer fluxes near structural kinetic transitions of slip bands into cells, cells into fragments of deformation origin, destruction by separation and shear for fractal spaces using fractional Riemann-Liouville derivatives, local and global criteria of destruction is carried out. One of the possible schemes of the crack initiation and growth mechanism in metals is disclosed. It is shown that the discrete model of plasticity and fracture does not contradict the known dislocation models of fracture and makes it possible to abandon the kinetic concept of thermofluctuation rupture of interatomic bonds at low temperatures.

The analysis of works [

At large plastic deformations leading to the formation of stable fragmented structures up to critical ones, the average electron densities n e p and n p h e and the corresponding plasma frequencies Ω e p and Ω p h e are quantities of the same order, which leads to a fundamentally new distribution of the dielectric constant tensor ε α β ( ω , k ) both in space and in time. In this case, a new branch of the spectrum associated with the presence of beams is added to the main branch of the spectrum of longitudinal oscillations of the intrinsic plasma, where Ω e p reflects the collective natural oscillations, and Ω p h e -oscillations and rotations in the additional potential relief of the nuclei of linear and plane defects. Here it is necessary to note the fundamental difference in the nature of the motion of electrons in these subsystems: in their own plasma, the directions of thermal motion of electrons are equally distributed in the total solid angle, the values of their velocities in metals are not lower than v F ; and in dielectrics and semiconductors in the volumes of shock waves at electric fields near the breakdown tend to the rates of local metallization v m s , v m d . On the contrary, in the plasma of beams, the alternating (intermittent) field creates dynamic anisotropy, while the directions of the velocities V p h e lie in the slip planes of single crystals, and in polycrystals the appearance of a subsystem of beams is possible at threshold values of the projections of these velocities on the slip plane in individual crystallites.

Optical and electronic micro-fractography of the surface of fatigue brittle and viscous-plastic fractures of specimens from a wide range of metals and their alloys [

It is also known that the macroscopic curves of tension [

Currently, there are several ways to describe the structural kinetic transitions “cell-fragment”, “fragment-microcrack”: 1) within the framework of a synergetic approach using scale invariance [

The aim of this work is to build a physical and mathematical discrete model of structural kinetic transitions taking into account the fractality of deformation processes.

Let us consider variations in the potential relief of a crystal V c ( r → , t ) as a functional of external currents J o s c e , J t u r n e , J o s c c a t , J t u r n c a t [

δ V c ( J μ ν ) = 0 , ν = e , c a t ; μ = o s c , t u r n (1)

On the contrary, the processes of translational, rotational plasticity and destruction are transient processes caused by the non-equilibrium of the system from the influence of external and internal electromagnetic fields when the threshold values of extraneous currents of photoelectrons (e) and cations (cat) are successively reached: J e o s c t h r ( t r p l ) and J c a t o s c t h r ( t r p l ) -currents of oscillations and rotations with translational plasticity; J e t u r n t h r ( r o t p l ) and J c a t t u r n t h r ( r o t p l ) -currents of oscillations and rotations with rotational plasticity ( r o t p l ) ; J ν μ t h r ( d s t ) -currents of oscillations and rotations during destruction. Here equality (1) turns into an inequality, and the variations δ J μ ν ( η ) are connected in pairs

δ J o s c e ( t r p l ) ⇄ δ J o s c c a t ( t r p l ) (2.1)

δ J t u r n e ( t r p l ) ⇄ δ J t u r n c a t ( t r p l ) (2.2)

δ J o s c e ( r o t p l ) ⇄ δ J o s c c a t ( r o t p l ) (2.3)

δ J t u r n e ( r o t p l ) ⇄ δ J t u r n c a t ( r o t p l ) (2.4)

δ J o s c e ( d s t ) ⇄ δ J o s c c a t ( d s t ) (2.5)

δ J t u r n e ( d s t ) ⇄ δ J t u r n c a t ( d s t ) (2.6)

where η = t r p l ; r o t p l ; d s t and in the region of structural kinetic transitions asymptotically tend to step functions. A natural question arises: What is the physical and mathematical model of such transitions, taking into account the fractality of deformation processes? Here we assume that the deformed volume of the material is considered as a fractal space, where the equations of mass transfer with the help of electron and ion plasma waves ( [

( D a + α f ) ( x ) = 1 Γ ( 1 − α ) d d x ∫ a x f ( t ) d t ( x − t ) α (3)

and right-handed operators

( D b − α f ) ( x ) = − 1 Γ ( 1 − α ) d d x ∫ x b f ( t ) d t ( t − x ) α (4)

where f ≡ f i n j , f p h e , f c a t are the distribution functions of injected electrons, photoelectrons and cations, respectively; Γ ( 1 − α ) —Gamma function. The convolution integral is written on the right-hand sides of (3) and (4); therefore, it is more convenient to consider fractional operators in ω , k —space. If we introduce the linear operator d / d x under the sign of this integral, then its Fourier transform as a → − ∞ and b → + ∞ leads to the product of the Fourier components of the function f μ ν and the power hyperbolic function F g p with fractional exponent. This transformation is applicable only in undeformed dielectrics and pure undoped semiconductors, where, due to the low density of free carriers, relaxation processes proceed extremely slowly. At the same time, such processes in metals are fast, spatio-temporal intervals [ a , b ] ≈ τ r e , ( 5 ÷ 10 ) a 0 , which is caused by the equations of selective selection of frequencies ω p w and wave vectors k p w of plasma waves [

ω p w − k p w V e = 0 (5)

ε l ( k → p w , ω p w ) = 0 (6)

when generating linear defects. The dielectric constant near plane defects has a tensor representation ε α β ( ω , k ) . Here, for the low-angle boundaries of inclination and rotation in the principal (normal and tangential) axes of the matrix ε α β , two functions ε n and ε t can be distinguished and, accordingly, two equations of selective selection. For high-angle “interfragment” boundaries of deformation origin, in particular, multi-wall Chalmers boundaries ( [

In ω , k —space near the regions of structural kinetic transitions, by analogy with [

α i = A n i ( n l i n g l i ) ⋅ [ 1 − ( δ n ν i n ν g l i ) β ] + A V i ( V p h e i V g l i ) ⋅ [ 1 − ( δ V ν i V ν g l i ) β ] , ν = e , c a t (7)

where the distribution functions of photoelectrons f p h e and cations f c a t , from which the photoelectrons were knocked out, are averaged over the local test (volumes near slip bands, boundaries of blocks, grains, fragments) volume V l and global (fragment, grain, crystal) volume V g l : δ n ν i = 〈 f ν i 〉 V l − 〈 f ν i 〉 V g l ≡ n ν l i − n ν g l i ; variations in the velocities of propagation of charged particles δ V ν i = 〈 V ν i 〉 V l − 〈 V ν i 〉 V g l ≡ V ν l i − V ν g l i for the i-th structural kinetic transition. Here, the numerical coefficients are 0 < A n i , A V i < 1 , and the power exponent β depends primarily on the density of conduction electrons n e c . On the other hand, β plays the role of the Hausdorff-Besicovich fractal dimension ( [

At the first stage of the theory, let us return to the well-known island model of the hereditary boundaries of a polycrystal in metals [

〈 f e c + f p h e 〉 V l b ≡ ( n l e c + n l p h e ) > n g l ≡ 〈 f e c 〉 V c (8)

where V l b and V c are the volumes adjacent to the interfragment boundary with the average fragment diameter L f r ≈ 100 ÷ 200 nm [

Hence, the mechanism of motion of the boundary separating fragment 1 and fragment 2 is clear, if the average density of electrons in the first n g l 1 < n g l 2 is the same density in the second, parallel to itself towards the first fragment. Adjacent fragments form edges and junctions, at which the selective frequencies and wave vectors of plasma waves must be matched with similar values at intersecting boundaries. It should be noted that the interaction of the deformation plasma of the beams and the intrinsic plasma of a solid at large plastic deformations leads to a significant increase in the large-scale correlation energy of the relative rotation of injected electrons and photoelectrons with respect to the distribution of conduction electrons. Here, in the region of the volume of fragment boundaries, the sizes of vacancy volumes and trajectories of rotation of injected electrons and photoelectrons are of the same order [

At high dislocation densities ρ = 10 10 ÷ 10 11 cm − 2 [

The most general global criteria for the destruction of crystalline materials by detachment and shearing are

〈 f i n j + f p h e 〉 V l ≡ n i n j + n p h e > n e c ≡ 〈 f e c 〉 V g l (9)

n i n j + n p h e > n e f f (10)

V p h e , V i n j > v F , v m s , v m d , x j ∈ S (11)

for a part of the surface of the interfragment boundary, the cleavage plane. Here n e f f is the limiting density of the electronic subsystem including photoelectrons, injected electrons, conduction electrons, reflecting the small-scale correlation energy due to the Pauli principle; at interelectronic distances r e f f ≤ ( 0.1 ÷ 0.2 ) nm [

In dielectrics, when fractured by cleavage along the cleavage planes, criteria (10) and (11) are satisfied. Similarly, in metals, the same criteria take place during the initiation of microcracks at interfragment boundaries ( [

Δ l c r v c a t ( t ) ≥ Δ L e e V e ( t ) , e ≡ p h e , i n j (12)

where v c a t ( t ) = 1 M c a t ∫ 0 Δ t o p ∂ U c a t c a t ( r , t ) ∂ r c a t c a t d t ; V e ( t ) = 1 m e ∫ 0 Δ t o p ∂ U e e ( r , t ) ∂ r e e d t ; Δ t o p is

time interval of crack opening. Analysis (12) shows that at Δ l c r ≈ 0 , 7 ÷ 12 nm ( [

The fractal shape of the emerging crack surfaces is the most acceptable for its stabilization ( [

At first glance, the huge variety of structures and associated deformation and relaxation processes, which take place in a wide range of loads, deformations, up to destruction, seems absolutely amazing.

To date, there is no unified theory in the literature for their description with a seemingly rather simple combination of electronic and cationic subsystems for a metal bond, semiconductor atoms for a covalent bond, cations and anions of dielectrics for an ionic bond. Nevertheless, the nature of plasticity and fracture in crystalline materials still remains completely undisclosed.

In this work, only a qualitative description of deformation processes is presented. For a quantitative consideration, it will be necessary to solve systems of equations by numerical methods.

A fundamental question arises: Is a fragment of deformation origin a quantum dot [

On the contrary, the distribution of injected, photo and conductivity electrons inside a fragment with L f r = 100 ÷ 300 nm has a continuous spectrum, while the interfragment boundary, in contrast to the free (impenetrable) surface, is partially permeable to these electrons. The ratio of the reflection coefficients ϰ r e f and transmission ϰ t r m of electrons in the beams, and separately, depending on the misorientation angle for low-angle and high-angle boundaries of deformation origin, has yet to be found.

The scheme of the crack initiation and growth mechanism contains a number of fundamental differences:

• Based on a discrete model of charged particles;

• Reflects the interatomic potential and the type of bond between particles inherent in a given material;

• Does not contradict the well-known dislocation models of brittle fracture by Zener, Straw, Cottrell [

• Allows abandoning the kinetic concept of thermal fluctuation rupture of interatomic bonds [

The model of plasticity and fracture presented in this work shows that the pumping energy under shock loads is redistributed as follows: the generalized space of rectangular pulses along the lines is replaced by a similar space along planes and curved surfaces, and the superposition of step functions in the form of terraces and steps is replaced by a superposition of undifferentiated peaks and ridges in the form of grooved and pit relief of fractures , at the same time, in real conditions, most often there are mixed structures from both types of relief; with the growth of cracks, exactly self-similarity of deformation processes appears. As a result, according to the destruction criteria (9) - (11), material objects are divided into separate fragments, the properties inside which are preserved. This is precisely the fractality of these objects under deformation.

The author declares no conflicts of interest regarding the publication of this paper.

Busov, V.L. (2021) Discrete Model of Plasticity and Failure of Crystalline Materials. Applied Mathematics, 12, 147-156. https://doi.org/10.4236/am.2021.123010