**4.2 Formulation of the model**

We consider the model in which the temperature dependence can be interpreted as a result the of phase transition between two plastic deformation structure levels that is characterized by specific values of the athermal stress component.

Changing yield strength in dependence on temperature *T* is characterized by derivative *d/dT*. As the phase transition is considered between two plastic deformation structure levels then this function must take the form that can be approximated by a parabolic dependence on . Such dependence has to be equal zero when yield strength coincides with theoretical quantities of the first (high temperature) 1 *th* and the second (low temperature) 2 *th* athermal plateau. As a result, it can be written a phenomenological equation of the structure phase transition in question as follows

$$\frac{d\sigma}{dT} = \frac{(\sigma\_{1\mu}^{\text{th}} - \sigma)(\sigma\_{2\mu}^{\text{th}} - \sigma)}{(\sigma\_{1\mu}^{\text{th}} - \sigma\_{2\mu}^{\text{th}})\Delta T\_0},\tag{4.1}$$

where *T*0 – the specific temperature interval in that yield strength increases occur on the magnitude of the thermo activated component of flow stress.

Solution of Eq. (4.1) takes the form:

158 Polycrystalline Materials – Theoretical and Practical Aspects

by point kinetics processes of dislocations. The area (2) on Fig. 4.1. is characterized by availability of athermal component which is mainly determined by long – range elastic internal stresses forming due to interaction of dislocations moving in parallel or crossing sliding planes. At temperatures of *Т* ≥ 0,45*Т*m (the area (3) of Fig. 4.1.), edge dislocation creeping conditioned by diffusion processes and forming crew dislocation jogs are determined by the thermo activated flow stress component. In the area (4) of Fig. 4.1., intensification of grain boundary processes of plastic deformation takes place and forms a

Connection of \* and changes with temperature dependence of radiation embrittlement in a wide enough experience temperature interval including certain areas shown on Fig. 4.1. can be studied by a method of modeling neutron irradiation action by relativistic electron beams with energies exceeding nuclear reaction threshold (so called ((*e*, ) – beams). Such irradiation as well as reactor one leads to forming different radiation defects ( for instance,

Main preference of such beams is a possibility to create for short time (for some of hours) radiation damages equivalent to ones obtained for some of years of irradiation in reactors. Besides modeling experiments can be fulfilled under severely controlled conditions that has paramount importance to clear up mechanisms of phenomena in nuclear and thermonuclear

The beams of electrons and - quanta having a large track length in materials make it possible to create homogeneous radiation damages in samples assigned for investigating mechanical properties. Investigations of mechanical prosperities of materials irradiated by (*е*,) – beams showed availability of their employment for modeling reactor damages and

When high energy electrons get through substance an electromagnetic avalanche develops. In increasing electron penetration depth in to a material sample a number of avalanche particles increases, energy of electron decreases and the X-ray bremsstrahlung increases.

Irradiation of materials by high energy electrons leads to accumulation of large amount of helium due to secondary (,) – reactions, which is accountable for high temperature radiation embrittlement. The unique feature of (*е*,) – beams is a possible to receive samples with distinct ratio of helium accumulation rate to rate of forming displacements at the same

Changing elastic and inelastic properties of polycrystalline materials are caused by lattice damages under irradiation and their next interaction with dislocations. Diffusion of point defects plays important role in process of pinning dislocations in connection with that it can be obtained significant information about radiation defects investigating influence of

We consider the model in which the temperature dependence can be interpreted as a result the of phase transition between two plastic deformation structure levels that is characterized

defects of diclocation loop type) besides nuclear reaction products) [3, 16].

second athermal plateau *G.* 

reactor materials under exploitation.

selection of construction material [17].

**4.2 Formulation of the model** 

experience temperature on a quantity of radiation damage.

by specific values of the athermal stress component.

experiment.

$$\sigma(T) = \frac{\sigma\_{1\mu}^{\text{fl}} + \sigma\_{2\mu}^{\text{fl}}}{2} - \frac{\sigma\_{1\mu}^{\text{fl}} - \sigma\_{2\mu}^{\text{fl}}}{2} \tanh\left(\frac{T - T\_c}{2\Delta T\_0}\right) \tag{4.2}$$

where *Тс* is temperature corresponding to the average value of athermal stresses of the high temperature and low temperature plateau.

To describe the yield strength experiment dependences of the irradiated materials on temperature it is convenient to rewrite Eq. (4.2) as follows

$$\sigma(T) = \sigma\_c - \sigma\_m \tanh\left(\frac{T - T\_c}{2\Delta T\_0}\right) \tag{4.3}$$

where 1 2 ( )/2 *th th <sup>c</sup>* , 1 2 ( )/2 *th th <sup>m</sup>* .

Empirical parameters *с*, *m*, *T*0, *Tс* of the model have next phenomenological meaning. Temperature *T*0 is connected with activation energy *Qe* of the plastic deformation transition on a higher structure level after that the material goes to a stage of radiation embrittlement: *Qe* = *T*0*kB*/2 where *kB* is Boltzmann constant. Parameters *с* and *Тс* are stress and temperature of the transition, respectively, between the structure levels of plastic deformation of irradiated materials, characterized by the known experimental values of athremal stress.

From Eq (4.3) follows at (*TTc*)/2*T*0 >> 1 that an equality *m* = <sup>1</sup> *th с* is valid if *T* < *Tc* and the equality *m* = *<sup>с</sup>* <sup>2</sup> *th* is fulfilled if *T* > *Tc* where 1 *th* and 2 *th* are the theoretical magnitudes of the first (low temperature) and the second (high temperature) athermal plateau, respectively (see Fig.4.1.). This implies that parameter *m* is connected with the thermo activated component of irradiated materials stress by \* *m th* / 2 .

#### **4.3 Discussion of model results and experimental data**

The values of empirical parameters *с*, *m*, *T*0, *Tс* are fit by the best coinciding the values of the function (4.3) for corresponding experience temperatures with experimental values of

Influence of Irradiation on Mechanical Properties of Materials 161

Fig. 4.3. Yield strength temperature dependences of irradiated bcc-materials: 1 – vanadium;

There are the approximating function (4.3) values of empirical parameters for the different materials, relative error of approximation and confidence quantity *R*2 of approximation (determination coefficient) for all of the dependences in the table. The determination coefficient is close to unit. It means good agreement the proposed theoretical dependences with experimental data for all of the considered materials in the wide experience temperature interval. It should be noted that the relative errors for experimental data to be approximated by the function (4.3) in the case of main fcc-materials (X18H10T steel, copper)

Material *с*, MPa *m*, MPa *T*0, *С Тс*, *С* , % *R*<sup>2</sup> 0Х18Н10Т steel 197.71 171.99 61.7284 634.87 1.457 0.9997 copper 87.76 35.48 58.1395 376.70 1.799 0.9965 nickel 132.35 113.78 116.2791 357.74 4.100 0.9972 15Х2МФА steel 148.78 135.35 12.9534 196.45 1.083 0.9996 vanadium 241 0.9295 243.9024 505 9.158 0.9638 chromium 277 0.5812 91.4077 251 2.848 0.9810

Also the yield strength temperature dependences of no irradiated materials can be approximated by Eq. (4.3) reasonably enough. For instance, the empirical parameters of no irradiated X18H10T steel are *с* = 59.45MPa, *m* = 26.1 MPa, *Tс* = 771.65 *С*, =4.405 %, *R*2 = 0.9814. Further, it is given the results of comparison of increments for thermo activated \*, and athermal high temperature 1 and low temperature 2 of stress components

are lower than in the case of main bcc-materials (vanadium, chromium).

Table 1. Empirical parameters of the dependence (4.3).

2 – chromium; 3 - 15Х2MФА steel.

the material yield strength. Criterion of fitting the empirical parameter values has been minimization of a quadratic deviation sum of the yield strength experiment values from ones calculated by Eq (4.3) at corresponding experience temperatures for all of the specific materials.

0Х18Н10Т steel samples have been irradiated by (*е*,) – beams with energy of 225 MeV up to dose of 1025 el/cm2 at temperatures of 170-190С. For mechanical experiences, the planar samples of test portion sizes of 1020.3 mm have been experienced in vacuum at temperatures of 20-1200С with deformation velocity of 0.003 c−1[15]. The copper samples of vacuum – induced remelting (purity of 99.98) have been irradiated by -beams with energy of 225 MeV up to fluence of 0.1 dpa [15]. The nickel samples have been irradiated by electrons with energy of 225 MeV up to dose of 1019 el/cm2 [11]. The 15Х2MФА steel samples have been irradiated by neutrons up to fluence of 31020 neutron/cm2 [11, 12]. The vanadium samples (purity of 99.9) have been irradiated by high energy (of 225 MeV) (*е*,) – beams up to fluence of 0.01 dpa [12, 15]. Chromium single crystals have been irradiated by (*е*,) – beams up to fluence of 1025 el/cm2 [15]

The summary experimental results for the yield strength temperature dependence of the irradiated materials are shown with Fig.4.2. (fcc lattice materials) and Fig.4.3. (bcc lattice materials) where experimental points are marked by corresponding labels.

Fig. 4.2. Yield strength temperature dependences of irradiated fcc-materials (point label – experiment, solid lines – theoretical plots calculated by Eq (4.3)). 1 – X18H10T steel; 2 – nickel; 3 – copper.

the material yield strength. Criterion of fitting the empirical parameter values has been minimization of a quadratic deviation sum of the yield strength experiment values from ones calculated by Eq (4.3) at corresponding experience temperatures for all of the specific

0Х18Н10Т steel samples have been irradiated by (*е*,) – beams with energy of 225 MeV up to dose of 1025 el/cm2 at temperatures of 170-190С. For mechanical experiences, the planar samples of test portion sizes of 1020.3 mm have been experienced in vacuum at temperatures of 20-1200С with deformation velocity of 0.003 c−1[15]. The copper samples of vacuum – induced remelting (purity of 99.98) have been irradiated by -beams with energy of 225 MeV up to fluence of 0.1 dpa [15]. The nickel samples have been irradiated by electrons with energy of 225 MeV up to dose of 1019 el/cm2 [11]. The 15Х2MФА steel samples have been irradiated by neutrons up to fluence of 31020 neutron/cm2 [11, 12]. The vanadium samples (purity of 99.9) have been irradiated by high energy (of 225 MeV) (*е*,) – beams up to fluence of 0.01 dpa [12, 15]. Chromium single crystals have been irradiated by (*е*,) – beams up to

The summary experimental results for the yield strength temperature dependence of the irradiated materials are shown with Fig.4.2. (fcc lattice materials) and Fig.4.3. (bcc lattice

Fig. 4.2. Yield strength temperature dependences of irradiated fcc-materials (point label – experiment, solid lines – theoretical plots calculated by Eq (4.3)). 1 – X18H10T steel; 2 –

materials) where experimental points are marked by corresponding labels.

materials.

fluence of 1025 el/cm2 [15]

nickel; 3 – copper.

Fig. 4.3. Yield strength temperature dependences of irradiated bcc-materials: 1 – vanadium; 2 – chromium; 3 - 15Х2MФА steel.

There are the approximating function (4.3) values of empirical parameters for the different materials, relative error of approximation and confidence quantity *R*2 of approximation (determination coefficient) for all of the dependences in the table. The determination coefficient is close to unit. It means good agreement the proposed theoretical dependences with experimental data for all of the considered materials in the wide experience temperature interval. It should be noted that the relative errors for experimental data to be approximated by the function (4.3) in the case of main fcc-materials (X18H10T steel, copper) are lower than in the case of main bcc-materials (vanadium, chromium).


Table 1. Empirical parameters of the dependence (4.3).

Also the yield strength temperature dependences of no irradiated materials can be approximated by Eq. (4.3) reasonably enough. For instance, the empirical parameters of no irradiated X18H10T steel are *с* = 59.45MPa, *m* = 26.1 MPa, *Tс* = 771.65 *С*, =4.405 %, *R*2 = 0.9814. Further, it is given the results of comparison of increments for thermo activated \*, and athermal high temperature 1 and low temperature 2 of stress components

Influence of Irradiation on Mechanical Properties of Materials 163

the mean segment length of the dislocation and enhance a degree of the dislocation anchorage. Because of this it can be expected for the obtained dose dependence of the yield

The second model describes barrier hardening polycrystalline materials. It is constructed with taking into account interaction of vacancy and interstitial barrier types. In the frame of the proposed model, it can be estimated both contributions to yield strength increment from

The third model gives possibility to describe the yield strength dependence of the irradiated materials on experience temperature on a quantitative level. It is based on mechanism of yield strength change as the phase transition between two plastic deformation structure levels characterized by certain values of the athermal stress component. The calculations show radiation promotes the transition of plastic deformation on the higher structure level after that the material undergoes radiation embrittlement. General features found permit to forecast embrittlement temperature intervals of reactor materials in dependence on their

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[2] V. Naundorf, "Diffusion in metals and alloys under irradiation," International Journal of

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[4] N. M. Ghoniem, J. Alhajji and Garner F.A., "Hardening of irradiated alloys due to the

[5] K. Khavanchik, D. Senesh and V. A. Shchegolev, "On saturation of yield strength of

[6] ITER Interim Structural Design Criteria (SDC-IC) (ITER Doc.IDoMS G 74 MA 8 01-05-28

[7] G.M. Kalinin, B.S. Rodchenkov and V.A. Pechenkin, "Specification of stress limits for

[8] V. C. Neustroyev, Z. E. Ostrovsky, E. V. Boyev and S. V. Belozerov, "Influence of helium

[9] V.V. Krasil'nikov and S.E. Savotchenko, Russian Metallurgy (Metally), Vol. 2009,No. 2,

[10] Svetukhin V.V., Sidorenko O.G., Golovanov V.N. and Suslov D.N., Fiz. Khim. Obrab.

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strength of the irradiated material to be modified.

mechanical properties.

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[in Russian].

Russian].

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[in Russian].

pp. 172-178.

**6. References** 

different type barriers and its total value in dependence on dose.

Modern Physics B, 6 No. 18 2925--2986 (1992).

Materials 329–333, 1615--1618 (2004).

Mater., 2005, No. 3, pp. 15-20.

on materials@ (ASTM, Philadelphia, 1982), pp.1054--1087.

obtained experimentally (A) and by a theoretical calculation (B) under radiation up to dose of 1025 el/cm2 for X18H10T steel.


According to the data shown at Fig. 4.4., the essential yield strength increment of X18H10T steel is observed after radiarion.

Fig. 4.4. Yield strength temperature dependence of austenic X18H10T steel: 1 – no irradiated, 2 – irradiated by (*е*,) – beams up to dose of 1025 el/cm2. Point labels are experimental; the lines are plots of theoretical dependence calculated by Eq. (4.3).
