4. Macroscopic bands (the Danilov-Zuev relaxation wave type)

Now, in the dislocation density balance Eq. (2), we take into account correlation forces [7] originating from redistribution of energy between the interaction dislocations more exactly:

$$f\_{cor} = A\_1 \frac{\partial \rho}{\partial \mathbf{x}} + A\_2 \frac{\partial^3 \rho}{\partial \mathbf{x}^3} \tag{8}$$

where <sup>A</sup><sup>1</sup> <sup>¼</sup> Gb<sup>2</sup> 4πρ<sup>0</sup> , <sup>A</sup><sup>2</sup> <sup>¼</sup> <sup>b</sup>σextL 8πρ<sup>2</sup> 0 , and L is an average relaxation length of a dislocation ensemble [7]. The right side of Eq. (2) can be represented as <sup>J</sup>ð Þ¼ <sup>ρ</sup> <sup>k</sup><sup>1</sup> � <sup>k</sup>2ρ<sup>2</sup> (see [6]) where k<sup>1</sup> characterizes a dislocation source and k<sup>2</sup> is responsible for interaction of the dislocations which can annihilate in particular.

Supposing ρ<sup>1</sup> ¼ ρ0u xð Þ ; t , one obtains the dimensionless equation for the relative density u xð Þ ; t of dislocations from Eq. (2):

$$\frac{\partial u}{\partial t} + \theta \frac{\partial u}{\partial \mathbf{x}} + u \frac{\partial u}{\partial \mathbf{x}} + \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^4 u}{\partial \mathbf{x}^4} = -\chi\_1 u - \chi\_2 u^2,\tag{9}$$

where θ is the numerical coefficient and χ1, χ<sup>2</sup> are responsible for the velocity of dislocation formation and their annihilation, respectively. In the work [8], it is shown that χ1, χ<sup>2</sup> are extremely small for the parameter numerical values of a real standard metal. Due to this, the right side of Eq. (9) is considered to be equal to zero. Then Eq. (9) goes to Kuramoto-Sivashinsky's equation type:

$$\frac{\partial u}{\partial t} + \theta \frac{\partial u}{\partial \mathbf{x}} + u \frac{\partial u}{\partial \mathbf{x}} + \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^4 u}{\partial \mathbf{x}^4} = \mathbf{0},\tag{10}$$

solution [9] of which describes spatial quasi-periodical structures. For Eq. (10), the Cauchy problem was solved at the different initial conditions for the dimensionless function u xð Þ ; t . The same level values of u xð Þ ; 0 were set at the initial time moment under a random distribution in x on the segment [0, 30]. In Figure 3, the plot I corresponds to the level u xð Þ¼ ; 0 0:2; the plots II and III do u xð Þ¼ ; 0 0:5 and

#### Figure 3.

The plots of the Eq. (10) solution constructed for the time interval Δt ¼ 7:3. The plot I corresponds to u xð Þ¼ ; 0 0:2; the plots II and III do u xð Þ¼ ; 0 0:5 и u xð Þ¼ ; 0 1, respectively.

u xð Þ¼ ; 0 1, respectively. These plots show that the spatially inhomogeneous quasi-periodical dislocation structures are formed in a sample in a certain time interval Δt (here Δt ¼ 7:3). The level of the u xð Þ ; 0 initial density corresponds qualitatively to the certain irradiation dose level of the material. From Figure 3, one can see that the localized dislocation structures are formed faster and become more striking in increasing the irradiation dose. The similar wavy deformation distribution is experimentally got in a material sample [10] (Figure 4 [11] shows the pattern of the deformation distribution along a sample during the initial straining stages).

The appreciable effect of the right side of Eq. (2) on a solution form u xð Þ ; t begins from the χ1, χ<sup>2</sup> values of 10�<sup>3</sup> . Figure 5 shows relaxation of the random dislocation distribution u xð Þ ; 0 during time on the x-interval equal to 45 of relative units.

The right side of Eq. (10) not equal to zero determines so-called "point" kinetics of dislocation interactions involving microlevels of plastic deformation. It may say the structures are formed when the point kinetics is absent.

Generalizing and following a concept of structure levels, it is possible to expect that switching-on micro- and meso-levels affects positively plastic deformation of irradiated and nonirradiated materials. So in materials deformed in the conditions of superplasticity, the effect of "running cell" is observed. When even extension exhausts its supply (it is small in these materials), the first neck stage must evolve, that is, deformation of a sample as a whole occurs (macro level N), the mechanism of grain border sliding (macro level N-1) is switched on and does not allow for long

Figure 5.

207

Figure 4.

work [11]).

The spatially temporary evolution of the dislocation density, according to Eq. (10) with the right side being

The pattern of the deformation distribution along a sample during the initial straining stages (Figure 1 of the

Modeling Plastic Deformation in Irradiated Materials DOI: http://dx.doi.org/10.5772/intechopen.82635

equal to 10<sup>3</sup> in relative units. The account time is 0.3 (a), 1 (b), and 5 s (c).

Figure 4.

u xð Þ¼ ; 0 1, respectively. These plots show that the spatially inhomogeneous quasi-periodical dislocation structures are formed in a sample in a certain time interval Δt (here Δt ¼ 7:3). The level of the u xð Þ ; 0 initial density corresponds qualitatively to the certain irradiation dose level of the material. From Figure 3, one can see that the localized dislocation structures are formed faster and become more striking in increasing the irradiation dose. The similar wavy deformation distribution is experimentally got in a material sample [10] (Figure 4 [11] shows the pattern of the deformation distribution along a sample during the initial straining stages). The appreciable effect of the right side of Eq. (2) on a solution form u xð Þ ; t

The plots of the Eq. (10) solution constructed for the time interval Δt ¼ 7:3. The plot I corresponds to

u xð Þ¼ ; 0 0:2; the plots II and III do u xð Þ¼ ; 0 0:5 и u xð Þ¼ ; 0 1, respectively.

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dislocation distribution u xð Þ ; 0 during time on the x-interval equal to 45 of

the structures are formed when the point kinetics is absent.

The right side of Eq. (10) not equal to zero determines so-called "point" kinetics of dislocation interactions involving microlevels of plastic deformation. It may say

Generalizing and following a concept of structure levels, it is possible to expect that switching-on micro- and meso-levels affects positively plastic deformation of irradiated and nonirradiated materials. So in materials deformed in the conditions of superplasticity, the effect of "running cell" is observed. When even extension exhausts its supply (it is small in these materials), the first neck stage must evolve, that is, deformation of a sample as a whole occurs (macro level N), the mechanism of grain border sliding (macro level N-1) is switched on and does not allow for long

. Figure 5 shows relaxation of the random

begins from the χ1, χ<sup>2</sup> values of 10�<sup>3</sup>

relative units.

206

Figure 3.

The pattern of the deformation distribution along a sample during the initial straining stages (Figure 1 of the work [11]).

Figure 5.

The spatially temporary evolution of the dislocation density, according to Eq. (10) with the right side being equal to 10<sup>3</sup> in relative units. The account time is 0.3 (a), 1 (b), and 5 s (c).

where Eq. (11) is a known criterion of plastic stability loss of samples stretched by uniaxial stress with constant speed when a plastic flow localization occurs and a neck is formed. This equation defines a limit value of uniform elongation εp. <sup>d</sup><sup>σ</sup>

coefficient of deformation hardening, σ plastic flow stress, and ε relative deformation. Eq. (12) determines the plastic flow stress dependence on dislocation density ρ as a function of deformation ε and the dislocation loop density Nð Þ Φt on dose (d—a size of a dislocation loop). Coefficient α is of the order of unity. Eq. (13) is an empiric formula of dislocation density dependence on relative deformation ε (see, e.g., [16]). Here ρ<sup>0</sup> is the value of dislocation density of undeformed material, and ρ<sup>∞</sup> is the saturation value of dislocation density at large values of ε. The values of ρ<sup>0</sup>

Further, there are the experimental dependences of deformation hardening coefficient dσ=dε (hardening speed) on deformation ε at different dose values of

Computer processing of the curves depicted in Figure 7 leads to Eq. (14) for the dependence of deformation hardening coefficient on relative plastic deformation ε: the values of parameters A, B, γ of Eq. (14) for corresponding doses are given in Table 1. Eq. (15) is the dose dependence of radiation defect density N. In this model, N is considered as volume density of dislocation loops. Consider the options of the dose

Nð Þ¼ Φt N<sup>0</sup> 1 � e

The dependence of deformation hardening coefficient on relative plastic deformation ε: The curve 1 corresponds to unirradiated nickel; the curves 2, 3, 4, 5, 6, and 7 correspond to nickel irradiated by electrons with energy of

Dose dpa <sup>0</sup> <sup>10</sup>�<sup>3</sup> <sup>10</sup>�<sup>2</sup> <sup>10</sup>�<sup>1</sup> <sup>2</sup>�10�<sup>1</sup> <sup>5</sup>�10�<sup>1</sup>

γ 0.24 0.09 0.09 0.11 0.13 0.27

, 5�10�<sup>1</sup>

) 816.59 222.84 59.33 257.88 171.96 430.47

) 1454.48 1731.54 2653.8 2268.2 2399.68 2684.6

, 2�10�<sup>1</sup>

, 10�<sup>2</sup>

, 10�<sup>1</sup>

�ξΦ<sup>t</sup> 2 <sup>1</sup>

2

, (16)

, and 5�100 (dpa), respectively).

and ρ<sup>∞</sup> are taken from the experimental data [16, 17].

Modeling Plastic Deformation in Irradiated Materials DOI: http://dx.doi.org/10.5772/intechopen.82635

material radiation in the proposed model (see Figure 7).

dependence Eq. (15) of dislocation loop density N(Фt):

1. The monotonic dependence obtained in [18]

Figure 7.

А (kg/mm<sup>2</sup>

В (kg/mm<sup>2</sup>

Table 1.

209

225 MeV (dose of radiation is 10�<sup>3</sup>

Parameters of Eq. (4) for different doses.

<sup>d</sup><sup>ε</sup> is the

#### Figure 6.

The temperature dependence of relative extension of a low alloyed chromium irradiated with a fluence of approximately 0.1 displacements per atom (dpa) (1), the initial material (2), and the material irradiated and deformed by bends (3).

wave modes to be established. The neck only starting to arise relaxes due to the mechanism of grain border sliding.

The next example of the effect of switching-on (N-1) level is based on the investigation results of low alloyed chromium [12]. Figure 6 shows this material when irradiated is absolutely brittle in the rather wide temperature range up to 500°C (plot 1). A part of amount of samples after irradiation was deformed by bend in 30–40°. As a result, new dislocations were created there, and plasticity became not equal to zero after radiation tests (plot 3). So in this case too, switching on the lower structure level (here microlevel) of plastic deformation leads to essential lowering material embrittlement. Thus, the synergetic law conception and applying the structure level conceptual design to study processes of material radiation embrittlement give us not only the investigation direction but the pointing of the ways to solve the specific problems for the material radiation embrittlement to be reduced.

#### 5. Modeling dose dependence of uniform elongation of materials

Experiment shows it is possible for both the monotonous decrease [13] and non-monotonic behavior [14] of uniform elongation dependence ε<sup>р</sup> to take place on radiation dose Фt as result of radiation. It does not take into consideration possible deformation hardening in traditional approaches to description of this phenomenon.

Here a new approach is represented allowing to explain peculiar properties of the εр(Фt) dependence of irradiated materials considering deformation hardening [15].

The approach is based on the following equation system:

$$\frac{d\sigma}{d\varepsilon} = \sigma$$

$$
\sigma = aGb\sqrt{\rho(\varepsilon) + N(\Phi t)d} \tag{12}
$$

$$
\rho(\varepsilon) = \rho\_{\infty} - (\rho\_{\infty} - \rho\_0)e^{-\beta x} \tag{13}
$$

$$\frac{d\sigma}{d\varepsilon} = A + B e^{-\gamma \varepsilon} \tag{14}$$

$$N = N(\Phi t) \tag{15}$$

Modeling Plastic Deformation in Irradiated Materials DOI: http://dx.doi.org/10.5772/intechopen.82635

where Eq. (11) is a known criterion of plastic stability loss of samples stretched by uniaxial stress with constant speed when a plastic flow localization occurs and a neck is formed. This equation defines a limit value of uniform elongation εp. <sup>d</sup><sup>σ</sup> <sup>d</sup><sup>ε</sup> is the coefficient of deformation hardening, σ plastic flow stress, and ε relative deformation. Eq. (12) determines the plastic flow stress dependence on dislocation density ρ as a function of deformation ε and the dislocation loop density Nð Þ Φt on dose (d—a size of a dislocation loop). Coefficient α is of the order of unity. Eq. (13) is an empiric formula of dislocation density dependence on relative deformation ε (see, e.g., [16]). Here ρ<sup>0</sup> is the value of dislocation density of undeformed material, and ρ<sup>∞</sup> is the saturation value of dislocation density at large values of ε. The values of ρ<sup>0</sup> and ρ<sup>∞</sup> are taken from the experimental data [16, 17].

Further, there are the experimental dependences of deformation hardening coefficient dσ=dε (hardening speed) on deformation ε at different dose values of material radiation in the proposed model (see Figure 7).

Computer processing of the curves depicted in Figure 7 leads to Eq. (14) for the dependence of deformation hardening coefficient on relative plastic deformation ε: the values of parameters A, B, γ of Eq. (14) for corresponding doses are given in Table 1.

Eq. (15) is the dose dependence of radiation defect density N. In this model, N is considered as volume density of dislocation loops. Consider the options of the dose dependence Eq. (15) of dislocation loop density N(Фt):

1. The monotonic dependence obtained in [18]

$$N(\Phi t) = N\_0 \left(\mathbf{1} - e^{-\xi \frac{\Phi t}{2}}\right)^{\frac{1}{2}},\tag{16}$$

Figure 7.

wave modes to be established. The neck only starting to arise relaxes due to the

The temperature dependence of relative extension of a low alloyed chromium irradiated with a fluence of approximately 0.1 displacements per atom (dpa) (1), the initial material (2), and the material irradiated

The next example of the effect of switching-on (N-1) level is based on the investigation results of low alloyed chromium [12]. Figure 6 shows this material when irradiated is absolutely brittle in the rather wide temperature range up to 500°C (plot 1). A part of amount of samples after irradiation was deformed by bend in 30–40°. As a result, new dislocations were created there, and plasticity became not equal to zero after radiation tests (plot 3). So in this case too, switching on the lower structure level (here microlevel) of plastic deformation leads to essential lowering material embrittlement. Thus, the synergetic law conception and applying the structure level conceptual design to study processes of material radiation embrittlement give us not only the investigation direction but the pointing of the ways to solve the

specific problems for the material radiation embrittlement to be reduced.

5. Modeling dose dependence of uniform elongation of materials

The approach is based on the following equation system:

Experiment shows it is possible for both the monotonous decrease [13] and non-monotonic behavior [14] of uniform elongation dependence ε<sup>р</sup> to take place on radiation dose Фt as result of radiation. It does not take into consideration possible deformation hardening in traditional approaches to description of this phenomenon. Here a new approach is represented allowing to explain peculiar properties of the εр(Фt) dependence of irradiated materials considering deformation hardening [15].

dσ

<sup>σ</sup> <sup>¼</sup> <sup>α</sup>Gb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ρ εð Þ¼ ρ<sup>∞</sup> � ρ<sup>∞</sup> � ρ<sup>0</sup> ð Þe

dσ

<sup>d</sup><sup>ε</sup> <sup>¼</sup> <sup>σ</sup> (11)

ρ εð Þþ <sup>N</sup>ð Þ <sup>Φ</sup><sup>t</sup> <sup>d</sup> <sup>p</sup> (12)

<sup>d</sup><sup>ε</sup> <sup>¼</sup> <sup>A</sup> <sup>þ</sup> Be�γε (14)

N ¼ Nð Þ Φt (15)

�βε (13)

mechanism of grain border sliding.

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Figure 6.

208

and deformed by bends (3).

The dependence of deformation hardening coefficient on relative plastic deformation ε: The curve 1 corresponds to unirradiated nickel; the curves 2, 3, 4, 5, 6, and 7 correspond to nickel irradiated by electrons with energy of 225 MeV (dose of radiation is 10�<sup>3</sup> , 10�<sup>2</sup> , 10�<sup>1</sup> , 2�10�<sup>1</sup> , 5�10�<sup>1</sup> , and 5�100 (dpa), respectively).


#### Table 1.

Parameters of Eq. (4) for different doses.

Figure 8. The nonmonotonic dependence of dislocation loop density on radiation dose.

where ξ is a numerical coefficient and N<sup>0</sup> is the saturation value of loop density

2. The nonmonotonic dependence of a type presented in Figure 8 [19].

The dependence of dislocation loop density on radiation dose can be described by the following analytic formula:

$$N(\Phi t) = N\_0 e^{-(\Phi t - \Phi t\_0)^2} \tag{17}$$

The experimental nonmonotonic dependence εр(Фt) of low-activated alloy based on chromium VCh-2 K irradiated by (е, γ)—beams in a dose interval of

The experimental dose dependence of elongation of low-activated alloy based on chromium VCh-2 K under

1. Evolution of the ensemble of dislocations interacting with obstacles in irradiated materials is analyzes, and the expression for part of dislocations overcoming obstacles is obtained on the basis of the general kinetic approach.

2. Formation of the front Chernov-Luders band is due to the presence of a nonlinear term called the Burgers nonlinearity in the evolution equation for

3. The description of the process of formation of space–time self-organizing dislocation structures is offered. The qualitative agreement of a dislocation density distribution is shown along sample length with the experimentally detected deformation distribution called the Danilov-Zuev relaxation waves in

4.The description of the dependences of radiation embrittlement of reactor materials on radiation dose is suggested. It is found out that monotonic and nonmonotonic dependences of uniform elongation of irradiated materials are determined by the form of the dislocation loop density dependence on

Comparison graphs of εр(Фt) in Figures 9 and 10 show alignment of modeling

10<sup>3</sup> ÷ 10<sup>1</sup> (dpa)—is represented in Figure 10.

Modeling Plastic Deformation in Irradiated Materials DOI: http://dx.doi.org/10.5772/intechopen.82635

and experimental results.

dislocation density.

irradiated materials.

radiation dose.

211

6. Conclusions

Figure 10.

(е, γ)—Beam radiation.

in the area of maximum. The nonmonotonic dependence of dislocation loop density on radiation dose is associated with the processes of growing a loop size. At the beginning of its evolution, the loops increase. Their density increases too. Upon reaching a certain radiation dose as a function of temperature and the speed of a displacement, the loops begin to interact with each other. As a result, the process of decreasing its density and loss of its defectiveness begins what is observed by [19].

Eq. (11) considering Eqs. (12)–(14) takes the form

$$\left(\frac{A + B\varepsilon^{-\gamma\varepsilon}}{aGb}\right)^2 = \rho\_0 + N(\Phi t) - (\rho\_\infty - \rho\_0)\varepsilon^{-\beta\varepsilon} \tag{18}$$

Eq. (18) admits a numerical solution only. The numerical solution of Eq. (18) leads to the dependence of uniform elongation ε<sup>p</sup> on dose for Eqs. (16) and (17) to be used. The respective curves are represented by Figure 9.

Figure 9.

The dependences of uniform elongation on radiation dose. The curve 1 corresponds to the monotonic dependence of the loop density (16) on dose. The curve 2 corresponds to the nonmonotonic dependence (17).

Modeling Plastic Deformation in Irradiated Materials DOI: http://dx.doi.org/10.5772/intechopen.82635

#### Figure 10.

where ξ is a numerical coefficient and N<sup>0</sup> is the saturation value of loop density

The dependence of dislocation loop density on radiation dose can be described

in the area of maximum. The nonmonotonic dependence of dislocation loop density on radiation dose is associated with the processes of growing a loop size. At the beginning of its evolution, the loops increase. Their density increases too. Upon reaching a certain radiation dose as a function of temperature and the speed of a displacement, the loops begin to interact with each other. As a result, the process of decreasing its density and loss of its defectiveness begins what is observed by [19].

Eq. (18) admits a numerical solution only. The numerical solution of Eq. (18) leads to the dependence of uniform elongation ε<sup>p</sup> on dose for Eqs. (16) and (17) to

The dependences of uniform elongation on radiation dose. The curve 1 corresponds to the monotonic dependence

of the loop density (16) on dose. The curve 2 corresponds to the nonmonotonic dependence (17).

�ð Þ <sup>Φ</sup>t�Φt<sup>0</sup> <sup>2</sup>

¼ ρ<sup>0</sup> þ Nð Þ� Φt ρ<sup>∞</sup> � ρ<sup>0</sup> ð Þe

(17)

�βε (18)

2. The nonmonotonic dependence of a type presented in Figure 8 [19].

Nð Þ¼ Φt N0e

Eq. (11) considering Eqs. (12)–(14) takes the form

The nonmonotonic dependence of dislocation loop density on radiation dose.

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be used. The respective curves are represented by Figure 9.

<sup>A</sup> <sup>þ</sup> Be�γε αGb <sup>2</sup>

by the following analytic formula:

Figure 8.

Figure 9.

210

The experimental dose dependence of elongation of low-activated alloy based on chromium VCh-2 K under (е, γ)—Beam radiation.

The experimental nonmonotonic dependence εр(Фt) of low-activated alloy based on chromium VCh-2 K irradiated by (е, γ)—beams in a dose interval of 10<sup>3</sup> ÷ 10<sup>1</sup> (dpa)—is represented in Figure 10.

Comparison graphs of εр(Фt) in Figures 9 and 10 show alignment of modeling and experimental results.

#### 6. Conclusions


References

[1] AM Parshin IM, Neklyudov NV, Kamyshanchenko, et al. Physics of Radiation Phenomena and Radiation Material Science. Belgorod (Rus): BSU

Modeling Plastic Deformation in Irradiated Materials DOI: http://dx.doi.org/10.5772/intechopen.82635

> [9] Alekseev AA, Kudryashov NA. Numerical simulation of selforganization process in dissipativedispersion media with instability. Izv. USSR AN, Mekh. Zhidk (Rus). Gas.

[10] Zuev LB, Danilov VI. A self-excited wave model of plastic deformation in solids. Phil. Mag. A. 1999;79(1):43

[11] Barannikova SA, Zuev LB, Danilov VI. Kinetics of periodic processes during plastic flow. Fiz. Tv. Tel. 1999;41(7):

[12] Neklyudov IM. Twinning role in the

radiation damage and plastic deformation of irradiation crystals. Functional Materials. 2000;7(1):77

[13] Parkhomenko AA. Electron Microscopy and Strength of Crystals. Vol. 9. Kyiev (Rus): Institute of Material

Science Problems; 1998. p. 103

Zs. Phys. 1959;155(2):247

2002. p. 70

1991;3:35

[14] Seeger A, Berner R, Wolf H. The deformation hardening mechanisms.

[15] Kamyshanchenko NV, Krasil'nikov VV, Sirota VV, Neklyudov IM, Ozhigov LS, Parkhomenko AA, Voevodin VN. Mechanisms of localization of plastic deformation in irradiated materials. In: Proceedings of the XV International Conference on Physics of Radiation Phenomena and Radiation Material Science. Alushta (Rus): The Crimea;

[16] Dudarev EF, Kornienko LA, Bakach GP. The effect of the energy of the defect packing at the development of dislocation substructure, strain hardening and ductility of FCC solid solutions. Izv. Vuzov. Fizika (Rus).

1990;4:130

1222

[2] Malygin GA. Self-organization of dislocations and localization of sliding in plastically deformable crystals. Fiz. Tv.

[3] Kamyshanchenko NV, Krasil'nikov VV, Nekliudov IM, Parkhomenko AA. Kinetics of dislocation ensembles in deformable irradiated materials. Fiz. Tv.

[4] Likhachev VA, Panin VE, Zasimchuk YE, et al. Cooperative Deformation

Localization. Kiyev: Publishing House

[5] Kamyshanchenko NV, Krasil'nikov VV, Nekliudov IM, Parkhomenko AA.

dislocations kinetics with allowance for the dislocation velocity distribution. Journal of Nuclear Materials. 1999;271,

[6] Kamyshanchenko NV, Krasil'nikov

Parkhomenko AA. On the mechanism of development of plastic instability in irradiated materials. Izv. RAN. Metally

[7] Khannanov SK. Dislocation density fluctuations (DDF) in the plastic flow of crystals. Fiz. M. Met (Rus). 1994;

[8] Kamyshanchenko NV, Krasil'nikov

Parkhomenko AA. On the mechanism of development of plastic instability in irradiated materials. Izv. RAN. Metally

VV, Sirota VV, Neklyudov IM,

Publishing House; 1998

Tela (Rus). 1995;37(1):3

Tela (Rus). 1998;40(9):1631

Processes and Deformation

Influence of irradiation on the

VV, Sirota VV, Neklyudov IM,

(Rus). 2000;4:110

(Rus). 2001;6:53

Naukova Dumka; 1989

272:84

78(2):31

213

## Author details

Nicholas Kamyshanchenko<sup>1</sup> , Vladimir Krasil'nikov<sup>1</sup> \*, Alexander Parkhomenko<sup>2</sup> and Victor Robuk<sup>3</sup>

1 Belgorod State University, Belgorod, Russian Federation

2 National Science Center Kharkov, Institute of Physics and Technology, Kharkov, Ukraine

3 Joint Institute for Nuclear Research, Dubna, Moscow Region, Russian Federation

\*Address all correspondence to: kras@bsu.edu.ru

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Modeling Plastic Deformation in Irradiated Materials DOI: http://dx.doi.org/10.5772/intechopen.82635
