Modeling Plastic Deformation in Irradiated Materials

Nicholas Kamyshanchenko, Vladimir Krasil'nikov, Alexander Parkhomenko and Victor Robuk

## Abstract

The classification of physical models of plastic deformation localization phenomena in the temperature range including the low-temperature radiation embrittlement effect is done. The new approach for the dislocation channeling mechanism description considering the collective behavior of dislocations and their interaction with radiation defects is proposed. The dislocation collective behavior model in materials irradiated, for example, by reactor radiation including neutron and accompanying gamma radiation is proposed on the basis of the evolution equation for dislocation density taking into account Burgers type nonlinearity. It is shown that the localized structures such as Danilov-Zuev's relaxation waves can be described. The possibility of localization effects (embrittlement) decreasing by the plastic deformation microlevel switching-on is demonstrated in relation with the thermal activated processes. The model describing dose dependence of uniform elongation of irradiated materials is constructed. This model is in good agreement with the experimental data of low-activated alloy based on chromium under (е, γ)—beam radiation.

Keywords: radiation, gamma-quants, embrittlement, deformation wave

#### 1. Introduction

Since Paul Ulrich Villard's discovery of gamma rays, there are manifold phenomena occurring under the influence of high intensive flows of particles, including gamma radiation, in solid state physics. Gamma radiation is shortwave electromagnetic radiation with wavelength of λ < 1 nm. If an atomic nucleus is in an excited state, then its transition from the excited state into the normal state causes gammaquant emission. Also gamma radiation can appear, for example, as a result of decay and annihilation of elementary particles and bremsstrahlung. Gamma radiation can be considered as a gamma-quant (photon) beam that demonstrates characteristic corpuscular properties due to its high energy. For example, gamma-quants with energy of from 10 keV to 10 MeV are produced in nuclear reactors.

Note, intensity and spectrum of gamma radiation uniquely characterize radioisotopes. Since decay of many nuclides, for example, <sup>137</sup>Cs, <sup>60</sup>Co, <sup>22</sup>Na, <sup>152</sup>Eu, is accompanied by gamma radiation, then knowledge of gamma radiation spectrum allows to identify composition and a number of radioactive isotopes in materials and also to define the degree of burnout of nuclear fuel in the reactors.

A structure and properties of the substances exposed to radiation experience significant changes called radiation damages. The radiation damages created in solid crystalline bodies destroy the proper crystal structure due to a displacement of atom from the host crystal lattice and formation of radiation defect complexes (point defects, clusters, dislocation loops, pore lattices, and so on). Besides, nuclear reactions can originate under radiation; therefore transmutants (new elements) can arise.

i. Stationary dissipative structures (dislocation channeling)

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

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

2. Stationary dissipative structures (dislocation channeling)

obstacles in dislocation channel regime is obtained:

(concentration of the defects arising from radiation).

<sup>∂</sup>ρð Þ <sup>x</sup>; <sup>t</sup> ∂t

band type)

203

<sup>q</sup> <sup>¼</sup> exp �j j <sup>v</sup> <sup>m</sup>þ<sup>1</sup>

synergetic point of view [4].

with any crystal lattice type. We consider briefly these phenomena with the

ii. Moving fronts of deformation localization (the Chernov-Luders band type)

In the works [1–3], it was shown these effects were observed in the materials

Electron microscope investigations of deformed irradiated materials showed that its structure has some features as long channels in order of several parts of micron width, without any radiation defects (see, for instance, [2]). In the work [5], it was proposed a new approach to describing mechanisms of dislocation channel formation on the basis of considering collective processes of interaction between dislocation ensembles and radiation defects. Upon that, it is supposed by the experimental facts that in the irradiated deformed material, ensemble dislocations move with velocities closed to 0.1 of the sound velocity, that is, in dynamic regime. On the basis of the general kinetic approach to evolving the ensemble of dislocations interacting with obstacles, the expression for part of dislocations q overcoming

where v is an initial velocity of the dislocations running across obstacles, a is dislocation acceleration, m < � 1, and constant A is proportional to a radiation hardening power A � σirr=σinit (σirr is stress after radiation; σinit is stress before radiation). At j j v ! ∞ (or increasing a), this fraction goes to unity, that is, at the high velocities (energies), the dislocations bypass obstacles without stopping. Thus, in irradiated deformed materials, the effect of sharp increasing the part of dislocations overcoming obstacles in a dynamic regime can be observed. Upon that this effect can be gotten in lower deformation velocities in increasing a hardening power

3. Moving fronts of deformation localization (the Chernov-Luders

of these is considered. There, the balance equation was written as

Recently, the synergetic approach is employed more and more to describe the evolution of plastic deformation in materials. As known, the balance equation for a local dislocation density ρð Þ x; t underlies the synergetic models. In the work [6], one

where v is velocity vector of dislocation sliding, D is dislocation diffusion coefficient, and Jð Þ ρð Þ x; t is the dislocation density functional determined by interaction

of dislocation with each other. The velocity of sliding dislocations v can be represented from three parts: v = vext + m( fint + fcor), where vext is velocity from

þ divðvρð Þ� x; t D∇ρð Þ x; t Þ ¼ Jð Þ ρð Þ x; t (2)

=ð Þ 2Aam j j ð Þ þ 1

, (1)

There are basic physical phenomena in the solid bodies under radiation. This is radiation growth and radiation swelling, radiation-stimulated and radiationinduced processes, low-temperature and high-temperature radiation embrittlement, radiation creep, radiation doping and erosion of the surface, and others.

Radiation affects essentially on the operating ability of materials. For instance, the atomic defect excess created by radiation accelerates diffusion processes and produces selection centers of new phases in alloys that are easy getting old. If it accumulates significant amount of decay products, then they can segregate, for example, as gas bubbles.

Many of radiation phenomena appear directly in the conditions of operation of the nuclear reactors. It is considered that energy is directly realized by uranium fission in the reactor in the middle of the order of 195–200 MeV. Most of this energy (about 168 MeV) is realized as kinetic energy of fission fragments. A part of the energy (about 24 MeV) is realized as β-particles, γ-quants, and neutrino. The rest of the energy is carried away by fission neutrons. It is necessary to consider that gamma radiation accompanies a neutron flow at interaction of reactor radiation with radiation protection materials based by metal hydride compounds and affects radiation doze power behind protection.

In wide test temperature region, radiation embrittlement includes conditionally two temperature intervals:

LTRE: low-temperature radiation embrittlement—at the test temperatures up to 0.4 Tm (Tm is melting temperature).

HTRE: high-temperature radiation embrittlement—at the test temperatures higher than 0.5 Tm.

In the LTRE region, embrittlement can be accompanied by radiation hardening, that is, increasing a yield point of materials. In the HTRE region, embrittlement can be accompanied by hardening relaxation. But there is a common feature of both phenomena; exactly, they are accompanied by the localization effects of plastic flow. In the HTRE, localization evolves mainly along grain borders, and it leads to speeded up formation of wedge-shaped cracks. In the LTRE one, localization goes in a grain body. Upon that, the factor that leads directly to destruction of a deformed irradiated material is formation of localized high density dislocation charges that contain, according to a theory, the beginning of a crack [1].

Manifold complexes of defects and nuclear reaction products are formed in the materials irradiated in reactors by neutrons and γ-quants that are the base of LTRE mechanism models to be constructed. The helium LTRE theory is widely known. Helium is generated by (n,α)- or (γ,α)-reactions in the construction materials under radiation. For instance, helium is formed by the two-stage reaction of the thermal neutron: Ni<sup>58</sup> <sup>þ</sup> <sup>n</sup> ! Ni<sup>59</sup> <sup>þ</sup> <sup>γ</sup> and Ni<sup>59</sup> <sup>þ</sup> <sup>n</sup> ! Fe<sup>56</sup> <sup>þ</sup> <sup>α</sup>:

Helium is practically insoluble in metals and at elevated temperatures migrates to grain boundaries and other defects where gas bubbles are formed that essentially influence embrittlement of the materials.

From all that was said before, it follows the mechanisms of plastic flow of irradiated deformed materials are the LTRE mechanisms in fact. It is possible to propose the following classification of plastic flow localization phenomena in irradiated materials:

A structure and properties of the substances exposed to radiation experience significant changes called radiation damages. The radiation damages created in solid crystalline bodies destroy the proper crystal structure due to a displacement of atom from the host crystal lattice and formation of radiation defect complexes (point defects, clusters, dislocation loops, pore lattices, and so on). Besides, nuclear reactions can originate under radiation; therefore transmutants (new elements)

Use of Gamma Radiation Techniques in Peaceful Applications

There are basic physical phenomena in the solid bodies under radiation. This is

Many of radiation phenomena appear directly in the conditions of operation of the nuclear reactors. It is considered that energy is directly realized by uranium fission in the reactor in the middle of the order of 195–200 MeV. Most of this energy (about 168 MeV) is realized as kinetic energy of fission fragments. A part of the energy (about 24 MeV) is realized as β-particles, γ-quants, and neutrino. The rest of the energy is carried away by fission neutrons. It is necessary to consider that gamma radiation accompanies a neutron flow at interaction of reactor radiation with radiation protection materials based by metal hydride compounds and affects

In wide test temperature region, radiation embrittlement includes conditionally

LTRE: low-temperature radiation embrittlement—at the test temperatures up to

In the LTRE region, embrittlement can be accompanied by radiation hardening, that is, increasing a yield point of materials. In the HTRE region, embrittlement can be accompanied by hardening relaxation. But there is a common feature of both phenomena; exactly, they are accompanied by the localization effects of plastic flow. In the HTRE, localization evolves mainly along grain borders, and it leads to speeded up formation of wedge-shaped cracks. In the LTRE one, localization goes in a grain body. Upon that, the factor that leads directly to destruction of a deformed irradiated material is formation of localized high density dislocation charges that

Manifold complexes of defects and nuclear reaction products are formed in the materials irradiated in reactors by neutrons and γ-quants that are the base of LTRE mechanism models to be constructed. The helium LTRE theory is widely known. Helium is generated by (n,α)- or (γ,α)-reactions in the construction materials under radiation. For instance, helium is formed by the two-stage reaction of the thermal

Helium is practically insoluble in metals and at elevated temperatures migrates to grain boundaries and other defects where gas bubbles are formed that essentially

From all that was said before, it follows the mechanisms of plastic flow of irradiated deformed materials are the LTRE mechanisms in fact. It is possible to propose the following classification of plastic flow localization phenomena in

HTRE: high-temperature radiation embrittlement—at the test temperatures

contain, according to a theory, the beginning of a crack [1].

neutron: Ni<sup>58</sup> <sup>þ</sup> <sup>n</sup> ! Ni<sup>59</sup> <sup>þ</sup> <sup>γ</sup> and Ni<sup>59</sup> <sup>þ</sup> <sup>n</sup> ! Fe<sup>56</sup> <sup>þ</sup> <sup>α</sup>:

influence embrittlement of the materials.

radiation growth and radiation swelling, radiation-stimulated and radiationinduced processes, low-temperature and high-temperature radiation embrittlement, radiation creep, radiation doping and erosion of the surface, and others. Radiation affects essentially on the operating ability of materials. For instance, the atomic defect excess created by radiation accelerates diffusion processes and produces selection centers of new phases in alloys that are easy getting old. If it accumulates significant amount of decay products, then they can segregate, for

can arise.

example, as gas bubbles.

two temperature intervals:

higher than 0.5 Tm.

irradiated materials:

202

radiation doze power behind protection.

0.4 Tm (Tm is melting temperature).

i. Stationary dissipative structures (dislocation channeling)

ii. Moving fronts of deformation localization (the Chernov-Luders band type)

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

In the works [1–3], it was shown these effects were observed in the materials with any crystal lattice type. We consider briefly these phenomena with the synergetic point of view [4].

### 2. Stationary dissipative structures (dislocation channeling)

Electron microscope investigations of deformed irradiated materials showed that its structure has some features as long channels in order of several parts of micron width, without any radiation defects (see, for instance, [2]). In the work [5], it was proposed a new approach to describing mechanisms of dislocation channel formation on the basis of considering collective processes of interaction between dislocation ensembles and radiation defects. Upon that, it is supposed by the experimental facts that in the irradiated deformed material, ensemble dislocations move with velocities closed to 0.1 of the sound velocity, that is, in dynamic regime. On the basis of the general kinetic approach to evolving the ensemble of dislocations interacting with obstacles, the expression for part of dislocations q overcoming obstacles in dislocation channel regime is obtained:

$$q = \exp\left(-|\mathbf{v}|^{m+1}/(2A|a(m+1)|)\right),\tag{1}$$

where v is an initial velocity of the dislocations running across obstacles, a is dislocation acceleration, m < � 1, and constant A is proportional to a radiation hardening power A � σirr=σinit (σirr is stress after radiation; σinit is stress before radiation). At j j v ! ∞ (or increasing a), this fraction goes to unity, that is, at the high velocities (energies), the dislocations bypass obstacles without stopping. Thus, in irradiated deformed materials, the effect of sharp increasing the part of dislocations overcoming obstacles in a dynamic regime can be observed. Upon that this effect can be gotten in lower deformation velocities in increasing a hardening power (concentration of the defects arising from radiation).
