**4. Phenomenological model of yield strength dependence on the temperature of irradiated materials**

#### **4.1 Temperature intervals of radiation embrittlement with taking into account two components of material flow stress**

The results of experimental studying radiation embrittlement effects and the temperature dependences of such durable material characteristics as specific elongation and yield strength have been given in a series of the works [10-15]. In Refs [12, 15], it is shown that a deformation process connected with dislocation collective behavior in irradiated deformed materials is characterized by availability of the different structure deformation levels.

As known under irradiation material plastic properties undergo strong changes. In particular a radiation embrittlement phenomenon takes place [11]. Upon that plastic properties of irradiated materials depend essentially on temperature. It is interesting to analyze phenomena of radiation embrittlement and radiation hardening of reactor materials with taking into account their durable characteristics on temperature.

To analyze radiation embrittlement it is necessary to take into account availability of two components of material flow stress ( ): the thermal (thermo activated) component \* created by short – rang forces and the athermal one determined by long – range forces of slowing – down dislocations and no experiencing influence of temperature. These components are shown on the plot of a temperature dependence of material flow stress (Fig.4.1.).

Fig. 4.1. Generalized scheme of temperature dependence of flow stress in polycrystalline materials. Area 1 corresponds to low temperature range T<0,15 Tm; area 2 is characterized by athermal component up to ~0,45 Tm; area 3 corresponds to thermo activated component of flow stress; area 4 corresponds to second athermal plateau ~G.

The first activated area (1) on Fig. 4.1. covers a low temperature interval *Т* ≤ 0,15*Т*m in which a activative volume quantity of plastic deformation is as *b*3 where *b* is the Burgers vector modulus. This corresponds to a microscale level of dislocation interactions that is realized

Influence of Irradiation on Mechanical Properties of Materials 159

*/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

1 2

*th th th th*

( )

where *T*0 – the specific temperature interval in that yield strength increases occur on the

 

*dT T* 

12 12

 

 

where *Тс* is temperature corresponding to the average value of athermal stresses of the high

To describe the yield strength experiment dependences of the irradiated materials on

( ) tanh <sup>2</sup>

is fulfilled if *T* > *Tc* where 1

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

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

 . 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

 

( ) tanh

120

   

( )( )

22 2 *th th th th T Tc <sup>T</sup>*

> *<sup>c</sup> c m T T <sup>T</sup>*

in dependence on temperature *T* is characterized by derivative

*th* 

, (4.1)

0

, (4.3)

*th* 

 and 2 *th* 

*th* 

> *m th* / 2 .

*с* is valid if *T* < *Tc*

are the theoretical

*T*

0

*T*

and the second (low

, (4.2)

. Such dependence has to be equal zero when yield strength coincides

athermal plateau. As a result, it can be written a phenomenological

 

Changing yield strength

*th* 

Solution of Eq. (4.1) takes the form:

where 1 2 ( )/2 *th th* 

athremal stress.

 *<sup>c</sup>* 

and the equality *m* = *<sup>с</sup>* <sup>2</sup>

 

 

dependence on

temperature) 2

*d* with theoretical quantities of the first (high temperature) 1

*d*

equation of the structure phase transition in question as follows

magnitude of the thermo activated component of flow stress.

temperature it is convenient to rewrite Eq. (4.2) as follows

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

From Eq (4.3) follows at (*TTc*)/2*T*0 >> 1 that an equality *m* = <sup>1</sup>

thermo activated component of irradiated materials stress by \*

*th* 

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

temperature and low temperature plateau.

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 second athermal plateau *G.* 

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

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 reactor materials under exploitation.

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 selection of construction material [17].

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 experiment.

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 experience temperature on a quantity of radiation damage.
