Influence of Irradiation on Mechanical Properties of Materials

The one of actual problems of radiation material science is to reveal plastic deformation laws, hardening and fracture ones of materials under intense external action, particularly irradiation. Herein we imply different kinds of irradiation, for instance, such a (e, ) beam irradiation, ion or neutron irradiation and so on. Evolution of a construction material microstructure at a high temperature trial operation is substantially conditioned by free migrating defects [1]. The processes of interaction of point defects with each other, with dislocations and interface underlie all of metal radiation hardening mechanisms [2]. In the section 1, the nonlinear model of dose dependence saturation of the yield strength is proposed on the base of the vacancy and interstitial barrier interaction. Processes of mutual recombination of vacancy and interstitial barriers and formation of vacancy and interstitial clusters are taken into consideration.


Introduction
The one of actual problems of radiation material science is to reveal plastic deformation laws, hardening and fracture ones of materials under intense external action, particularly irradiation.Herein we imply different kinds of irradiation, for instance, such a (e, ) beam irradiation, ion or neutron irradiation and so on.Evolution of a construction material microstructure at a high temperature trial operation is substantially conditioned by free migrating defects [1].The processes of interaction of point defects with each other, with dislocations and interface underlie all of metal radiation hardening mechanisms [2].In the section 1, the nonlinear model of dose dependence saturation of the yield strength is proposed on the base of the vacancy and interstitial barrier interaction.Processes of mutual recombination of vacancy and interstitial barriers and formation of vacancy and interstitial clusters are taken into consideration.
A series of different radiation defects (retardation barriers of dislocations), and their sizes, and a form of their volume distribution contribute into a yield strength increment for all of sorts of irradiation.The contribution of a barrier type is determined by conditions of irradiation and tests.At the low temperature irradiation (at the test temperatures up to 0.3 T m , T m is melting temperature), interstitial atoms, and vacancies, and their clusters contribute mainly into the hardening.In the section 2, evolution of radiation barrier (vacancies and interstitials) clusters is analyzed under low temperature radiation in the presence of the most important secondary effectes: recombination and formation of divacancy complexes.It is proposed a barrier hardening model in that mechanisms of mutual annihilation of the vacancy and interstitial barriers and their clusterization play a main role.
In the section 3 unlike two preceding sections where the dose dependences are considered, the phenomenological model is formulated to describe a yield strength temperature dependence of polycrystalline materials that have undergone irradiation and mechanical experiences in a wide temperature interval including structure levels of plastic deformation.In this section, a new phenomenological model is proposed to give a suitable description of yield strength temperature dependence of some of irradiated materials in a temperature interval including plastic deformation structure levels.

Mutual recombination and clusterization effect of the vacancy and interstitial barriers on radiation hardening materials 2.1 Formulation of the model
In initial stages of low temperature irradiation (up to 0.3T m where T m is melting temperature) at small doses, inhibiting processes of initial dislocations and their sources dominate.The point defects are connect with a dislocation and form dislocation jogs and steps but interacting with each other form interstitial -vacancy -impurity clusters.As a result, energy and geometry dislocation characteristics change substantially.A phenomenological description of these mechanisms is within the usual Granato -Lukke theory.
At large irradiation doses of metals, the processes of an elastic and contact interaction of sliding dislocations with different potential barriers begin to play a main role.Besides the separated point defects in irradiated material, the dislocations are to surmount interstitial and vacancy clusters and dislocation loops, the interstitial -vacancy -impurity clusters, precipitates, voids.The dislocation can cut a barrier, can be bent by the barrier and can go round an obstacle by a dislocation climb subject to barrier intensity and a distance between barriers.These mechanisms of barrier hardening are described by the Orowan model of athermic surmounting obstacles by dislocations [3].
At the large irradiation doses, a yield strength increment can be represented by the sum of barrier contributions of different types [3]: where index i is a barrier type, N is a number of barrier types affecting the yield strength,  i is the yield strength increment of i' barrier type.
Up to now, there is many of experimental data and numerical theoretical models are developed to describe a dependence of barrier concentration of different types on radiation hardening power and behavior pattern of point defect clusters [4].It is known that on earlier stage at small irradiation doses (by neutron radiation up to 210 16 n/cm 2 ), the hardening occurs due to forming the slowly increasing interstitial clusters.The vacancy clusters begin to contribute to the hardening in increasing dose.
In these stages, the yield strength increment is described with sufficient approximation of the dependence of the form where а is a parameter depending on irradiation conditions and the research material type, Ф is density of particle flux, t is irradiation time, exponent n changes against the material type and the irradiation condition from 0.25 to 0.75 [3].
A saturation nature of the hardening is still not clear finally.Probable causes of the yield strength increment saturation can be such as overlapping stresses field created by radiation defects of certain their concentration, creation round the volume defects of defect-free zones, the beginning of the dislocation channeling and surmounting obstacles processes and so on.
In the Ref [5], the model is proposed to describe the dose dependence of the copper yield strength increment where the saturation is explained by a decreasing velocity of the forming clusters with increasing irradiate dose due to interaction between the available clusters and newly forming ones.
Here the model is proposed to describe the dose dependence of the yield strength increment taking into account of vacancy and interstitial barrier interaction.
We consider that vacancies and interstitial atoms make a main contribution to the yield strength increment of a certain material at some of irradiation conditions.They are barriers to play the main role in the hardening at the low temperature irradiation.Therefore, in the proposed model N=2; the index values of i=1 and 2 correspond to the vacancies and interstitial atoms (and their clusters).Then in this model Eq (2.1) takes the form: For all of the obstacle types, the metal yield strength increment conditioned by dislocation deceleration is described as [2]: where  i is the parameters characterizing i' barrier intensity (a fixed quantity for some of barrier types, material and irradiation condition),  is the shear modulus, b the Burgers vector length, С i the volume density of i' barrier type, d i their average size.For instance, the vacancy and interstitial have the average size ~ 10 nm, and the parameters characterizing barrier intensity has the value about 0.2 [2].
The present model is based on the system equations for the volume densities of the radiation-induced nonequilibrium vacancies and interstitial barriers C 1 , C 2 : where  = Фt, Ф is density of particle flux, t irradiation time, K i , i = 1, 2, the intensities of forming the radiation -induced vacancy and interstitial barriers,  i are the coefficients of barrier recombination and characterize forming the clusters of acceptable barrier type (it can be named as clusterization coefficients),  12 the coefficient of mutual recombination of the annihilating vacancy and interstitial barriers, the coefficients 1 i   can be represented by the form: , where V i are the effective volumes of interaction of the certain barriers with each other.
The first terms of the equation system (2.5) describe the intensity of increasing the volume barrier densities of the acceptable type, the second ones correspond to decreasing the volume barrier densities due to absorbing the barriers on natural sinks: voids, dislocations, www.intechopen.comPolycrystalline Materials -Theoretical and Practical Aspects 144 dislocation network, grain boundaries and so on.In the proposed model, the mechanisms of the mutual annihilation of vacancy and interstitial barriers and their clusterization are assigned.The third terms of the system (2.5) describe decreasing the barrier densities due to of the mutual annihilation of two different type barriers, and the fourth ones do due to forming the clusters of two barriers of the same type.
To find the volume densities of the radiation -induced nonequilibrium vacancies and interstitial atoms it is necessary to set up their the initial values: Thus, the mathematical formulation of the present model is the Cauchy problem for the system of nonlinear differential equations (2.5).The volume barrier densities found as a result of solution of the Cauchy problem (2.5), (2.6) determine the yield strength increment according to Eqs (2.3), (2.4).

Dose saturation features of the yield strength subject to annihilation effects of the vacancy and interstitial barriers
At the beginning, we consider the yield strength behavior against the material irradiation dose taking into account of only the mutual recombination of the vacancy and interstitial barriers that is their annihilation.Upon that, the effects of forming the barrier clusters of the same type are not considered.The barrier annihilation effects but no their clusterization play the main role in the radiation hardening.Therefore  12 >>  i , i = 1, 2, and it is possible to neglect the last terms of  in every equation of the system (2.5).
In this approximation the system (2.5) takes the form: .
As Eqs (2.7) has not an analytic solution, we illustrate a dependence pattern of the barrier density by a numerical analysis example of the Cauchy problem solution in which the parameter simulating values and the initial defect densities (0) i C  510 13 cm -3 (they are pointed out in the figure captures following then) are used.The numerical solution results of the Cauchy problem (2.6), (2.7) reduced to the dimensionless form attached to the parameter simulating values are shown in Fig. 2.1.
Here it is shown the dependence typical behavior of the barrier relative densities of the vacancy 10 / CC and interstitial 20 / CC types on the relative irradiate dose 0   where С 0 is a measure scale of the defect densities (here it is equal 10 15 cm -3 ).The measure units are accepted relative to the selected scale τ 0 (fluence) the numerical value of which is determined by a specific problem (it is convenient to select a minimal fluence of the specific problem as the dose measure scale; for instance, in the ion irradiation the value of τ 0 can be equal 10 14 ion/cm 2 or 10 22 n/m 2 as the contemporary neutron fluence).This dependence pattern is universal and independent of the selected measure scales of the specific physics parameters.In increasing dose it is happened the saturation of the material by the radiation -induced barriers (their densities do not increased far more).The numerical analysis shows that at K 1 > K 2 and small doses, the barrier density of the vacancy type exceeds the one of the interstitial type.At opposite inequality, a situation becomes reverse that is natural as it physically means that increasing the irradiation intensity leads to enlarging a number both the vacancy and the interstitial barriers.Upon that the higher the velocity of forming the radiation barriers of any type is the larger their volume density.It is necessary to mark that the vacancy barrier density usually is higher than the interstitial barrier density in the real materials near a sample surface irradiated.
The numerical analysis also shows that increasing the mutual recombination coefficient  12 leads to decreasing the saturation values of the barrier densities.It leads as well to changing the dose dependence pattern of the barrier density of the corresponding type that becomes the monotonic quantity (see Fig. 2.1, a) and b)).Substitution of the found numerical solution of the system (2.7) to Eq (2.
3) allows getting the dose dependences of the material yield strength increment.For plotting these dependence figures, it is convenient to represent Eq (2. 3) as follows where d is an average size of the barrier clusters of all types (here it can be determined as half-sum of d 1 and d 2 ).

146
The results of numerical modeling the behavior of the yield strength are represented in Fig. strength increment value at  12 = 0 that is under the condition of total neglecting interaction between the vacancy and interstitial barriers).It is well seen that the relative yield strength increment decreases with the growth of the mutual recombination coefficient, this dependence being nonlinear and monotonic.
A stationary point of the systems (2.7) determines the saturation values of the barrier density: where The saturation values of the barrier density (2.10) and (2.11) are achieved quickly enough (see Fig. 2.1.).
In the case when the recombination of the vacancy and interstitial barriers is negligibly small the equations of the system (2.7) become independent at 12  = 0.If to suppose in addition that the radiation barriers are virtually absent at the initial time (0) 0 i C  , (i = 1, 2) then as a result the known expression of the volume barrier density is obtained [3]: where Substitution of Eq (2.12) to Eq (3) gives the expression of the yield strength increment at  12 = 0: ) where In this case, at large doses it is followed from Eq (2.13) that.
It should be noted that at small doses it is followed the well known law of Eq (2.2) at n = ½: where In the absence of the mutual recombination of the vacancy and interstitial barriers but at the arbitrary initial barrier densities and the initial conditions (2.6) and at  12 = 0, the solution of the equation system (2.7) leads to the expression ) is used to plot the yield strength dependence on the intensity quantity of the vacancy and interstitial barrier recombination at fixed dose in Fig. 2.3.

Dose saturation features of the yield strength subject to the effects of clustering barriers
Let us consider now the main contribution to the radiation hardening is given by the effects of forming the clusters of two barriers of the same type and the mutual recombination of the vacancy and interstitial barriers is negligibly small.Therefore,  12 <<  i , i = 1, 2, and in every equation of the system (2.5) it can be neglected by the last terms of 12 1 2 CC  .After this, the system equations become independent and therefore in what follows the indexes 1 and 2 of notations can be omitted.
Further we consider the contribution to the radiation hardening only from the barriers of the same type (either vacancy or interstitial) and then instead Eqs (2.3) and (2.4) the next expression is used where  is the parameter characterizing the barrier intensity (a fixed quantity for some of barrier types), С the volume barrier density of the same type, d their average size.
In this case it is convenient to go to single equation of the system (2.5): where  i of Eqs (2.5) is changed by the new notation according to the relationship  i = K and all of the indexes are omitted.
The solution of Eq (2.18) with the initial condition C (0) = 0 at  > 0 takes the form  It follows from (2.19) that at low irradiate doses that is when  <<   (accordingly (2.19) it corresponds to low density of radiation defects) the dependence of the volume density of the radiation-induced barriers on irradiate dose is linear: () Cq V q K t   .Substituting this expression into (2.17)we obtain the dependence of At high irradiate doses that is when  >>   the volume density of the radiation-induced barriers is saturated and tends to the constant () / 2 Cq V    .This value of the volume density is the stationary point of Eq (2.18).Substituting this expression into Eqs (2.17) we obtain the saturation value of the yield strength:  is determined by Eq (2.9).
If the clusterization effects can be neglected (as well as the mutual annihilation of the barrier of the different types has been already neglected too) then it can be obtained from Eq (2.19) the known expression of the volume barrier density as Eq (2.12).This dependence corresponds to the curve 1 in Fig. 2.4.
After substituting such volume density into Eq (2.17) it is obtained the dose dependence of the yield strength in the case when the barrier clusterization makes negligibly small contribution to the velocity of forming the radiation -induced barriers [3]: On the base of the obtained dependence of the yield strength saturation by Eq (2.21) on the barrier recombination intensity, it can be drawn the next conclusion.The saturation quantity of the material yield strength at fixed dose decreases monotonically with increasing the intensity of radiation -induced barrier clusterization.This behavior is shown in Fig. 2.6.where the plots of Eq (2.21) are presented.

Discussion of experiment data
As known, since 1988 the EU, USA, Japan and Russia joint works have been fulfilled within the intergovernmental agreement approved by IAEA (International Atomic Energy Agency).According to technological design of the thermonuclear reactor ITER, one of main constructional material is austenitic steel 316(N)-IG [6].
In the Ref [9], it is developed the model of the radiation hardening of the concrete irradiation material, 316(N)-IG steel.This model is based on the equation to be equivalent to Eq (2.18) for the potential barrier density C the role of which plays the stacking fault tetrahedral observed by electron microscopy as black dots.Experiment reveals that the concentration of these barriers grows with increasing irradiation dose.
Authors of the work [9] make a comparison with experimental data for 316(N)-IG steel on the base of the equation analogous to above Eq (2.22) to be the particular case of Eq (2.18) that is not taking into account of the barrier annihilation.They receive relatively good fit with the experimental data in temperature ranges 20 -150 and 230 -300C.For the higher temperatures (330 -400C) this equation does not obey an adequate description of the yield strength and ultimate stress increment.To describe the experimental data the authors of Ref [7] fit an exponent in the dependence of the form (2.17) pointing out this exponent to be varied approximately in the interval 1.4 -2.7 for the best agreement with the experimental data.It is possible that accounting the barrier recombination of the different types leads to invariability of the exponent ½ in Eq (2.17).
In the work [8], the equation analogous to Eq (2.22) it is used for fitting the experimental data for strength and ductility of corrosion -resisting austenitic 06Х18 10Т steel irradiated by the WWER -440 reactor up to damaging dose ~ 21 d.p.a. at different testing temperatures.Authors of Ref [8] find out that the radiation hardening saturation of 06Х18 10Т steel irradiated in WWER-440 reactors takes place at the damage dose of ~ 10  15 d.p.a.(11,510 26 n/cm 2 ).

Formulation of the secondary process contribution model
The base of the model is a barrier mechanism of the radiation hardening [2] according to that the yield strength increment can be represented by the sum of barrier contributions of different types (see Eq (2.1) of the preceding section).
We consider that the barriers of vacancy and interstitial types make a main contribution to the yield strength increment of a certain material.These barriers play the main role in the hardening at the low temperature irradiation.Therefore, in the proposed model N=3; the index values of i=1 correspond to the vacancy barriers, i = 2 do to interstitial ones and i = 3 do to more large vacancy complexes.Then in this model Eq (2.1) takes the form: For all of the obstacle types, the metal yield strength increment conditioned by dislocation deceleration is described by the Eq (2.4) (see the preceding section).
Under irradiation, development of radiation defect clusters (barriers) of different types occurs in a region of a primary knocked-on atom.It is proposed that the interstitial barriers have considerably smaller sizes and leave the damage region of a sample sooner than the vacancy barriers do.In connection with this, we formulate the phenomenological model that is based on the equation system for volume densities of the radiation-induced nonequilibrium vacancy C 1 and interstitial barriers C 2 and more large complexes C 3 developed by bimolecular mechanism of the vacancy barriers: Here  = Фt, Ф is the particle flux, t irradiation time, K i , i = 1, 2, the intensities of forming the radiation -induced vacancy and interstitial barriers,  i are the coefficients of barrier recombination and characterize forming the clusters of acceptable barrier type (it can be named as clusterization coefficients), the coefficients 1 i   can be represented by the form: , where V i are the effective volumes of interaction of the certain barriers with each other,  12 the coefficient of mutual recombination of the annihilating vacancy and interstitial barriers.It can be valued as follows: As material structure changes go under irradiation for times large in comparison with relaxation time of point defects then only diffusion barrier processes are considered to be very slow and therefore we neglect diffusion terms in the equations of the system (3.2).In addition, we study evolution of barrier volume densities in time considering their distributions are spatially homogeneous.In this case, the system (3.2) takes the form: In this equation system, the first two equations coincide with the system (2.5) completely (see the preceding section).
The third equation of the system (3.3)describes redistribution of divacancy barrier complexes.Contribution to the hardening due to divacancy barriers is determined by only vacancy barrier density.The kinetic coefficients 1 3   and  1 characterize intensities of breakdown of vacancy clusters and development of divacancy clusters.Effect of these contributions are appreciable if intensity of secondary processes of developing divacancy complexes predominates over their breakdown.
To find the volume densities of the nonequilibrium barriers it is necessary to set up their initial values: Thus, the mathematical formulation of the model proposed in this section is the Cauchy problem for the system of nonlinear differential equations (3.3) with initial conditions (3.4).
The volume barrier densities found as a result of solution of the Cauchy problem (3.3), (3.4) are to be inserted into Eq (2.4) that determines the total yield strength increment (3.1).

Numerical analysis of the model results
The model values of parameters (are given in captures of Figures) and the initial conditions Here is shown the specified form of dose dependences of relative densities for vacancy barriers 10 / CC , and interstitial barriers 20 / CC , and more large vacancy complexes 30 / CC where С 0 is a measure scale of barrier density taken to be equal 10 15   3   cm  in this case on dose 0   in units measured by the scale τ 0 (it is convenient to select a minimal fluence of www.intechopen.comPolycrystalline Materials -Theoretical and Practical Aspects 154 the specific problem as the dose measure scale; for instance, in the ion irradiation the value of τ 0 can be equal 10 14 ion/cm 2 or 10 22 n/m 2 in the neutron irradiation and so on, for the specific problem, respectively).These dependence patterns are universal and independent of the selected measure scales of the specific physics parameters.It is shown that the saturation of the material by the radiation -induced barriers takes place with increasing dose.
The numerical solution of the Cauchy problem (3.3), (3.4) permits to obtain the dose dependences of the total increment of material yield strength which is convenient to represent for construction of graph as follows It is should be noted though the vacancy complexes have lower concentration in comparison with vacancies and interstitial atoms they, due to their larger sizes, contribute more considerably to yield strength increment at dose build-up.

Secondary reaction contribution analysis
Let us consider the case when secondary reactions play a main role that is barrier recombination goes less intensively than developing barrier clusters.In this extreme case, we consider  12 << 1 and  12 << 2 .Then the second equation of the system (3.3)becomes independent and coinciding formally with the first equation.In the result, the system (3.3)consists of two equations: The first equation of the system (3.6)doesn't contain С 3 .Therefore, it is independent.Its solution with zero initial condition takes the form: where Substituting Eq (3.7) into the second equation of the system (3.6) its solution can be written as Further, this expression is used to analysis the vacancy complex contribution to yield strength increment: The increasing of saturation quantity of yield strength increment goes with increasing intensity of the clusterization processes that is with increasing parameter  1 (Fig. 3.3a).
This increasing is nonlinear, enlargement of yield strength increment saturation going less and less considerably with increasing intensity of the clusterization processes.
Growth of saturation quantity of yield strength increment goes with increasing specific times  1 and  3 as well (Fig.   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][11][12][13][14][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  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.

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) where Т с 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.Temperature T 0 is connected with activation energy Q e of the plastic deformation transition on a higher structure level after that the material goes to a stage of radiation embrittlement: Q e = T 0 k B /2 where k B is Boltzmann constant.Parameters  с and Т с 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.

Conclusion
In this chapter, three models are proposed to describe the properties of irradiated deformed polycrystalline materials.
The first model (see section 2) describes the dose dependence of the yield strength of the irradiated material.It is generalization of the model known earlier (see for instance [5]), as the recombination effects of the radiation -induced vacancy and interstitial barriers and their clusterization are taken into account in it.Within the framework of the model formulated, it is found that the saturation quantity of the yield strength decreases with increasing both the intensity of the mutual barrier recombination and the clusterization intensity.It follows to note that in spite of model's assumption for the mean sizes of the radiation-induced barriers d i and the shear modulus  to be independent the shear modulus increases practically owing to the radiation point defects come on the dislocation and reduce 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 strength of the irradiated material to be modified.
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 different type barriers and its total value in dependence on dose.
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 mechanical properties.

Fig. 2 . 6 .С
Fig. 2.6.Dependences of the relative yield strength increment saturation (2.21) at the fixed parameter values 00 () KC  = 1: (a) dependence on nondimensional intensity of the barrier clusterization 2 0 С  at different values of their volumes: (1) С 0 V = 0.4, (2) С 0 V = 1, (3) С 0 V = 2 in relative units.(b) dependence on V at different values of the clusterization intensity: (1) 2 0 С  = 0.1, (2) volume, E r activation energy of recombination of vacancy and interstitial barriers, r recombination radius, T test temperature, k B Boltzmann constant, D i diffusion coefficients of the non-equilibrium barriers of the given types: where E i energy of activation and migration of respective barriers, D i0 = a 2 , а and  are length and barrier jumping frequency for migration, respectively.

cm
 .areused to fulfill numerical analysis of the Cauchy problem (3.3), (3.4).The results of the numerical solution of the Cauchy problem (3.3), (3.4) (reduced to the dimensionless form) are represented on Fig.3.1.at the indicated parameter values.

Fig. 3 . 1 .
Fig. 3.1.Dependences of relative barrier density of vacancy type (1), interstitial type (2), vacancy complexes (3) and their total density (4) on dose at fixed values of dimensionless parameters 10 0 () KC  = 1.5, Fig. 3.2.Dose dependences of barrier contributions of vacancy type (1), interstitial type (2), vacancy complexes (3) and their total density (4) to yield strength increment at fixed values of parameters the same as in Fig.3.1.and d 1 /d = 0.016, d 2 /d = 0.035, d 3 /d = 0.097.The numerical analysis shows that the yield strength increment gets the saturation quickly enough.The typical monotonic form of the dose saturation plots of the yield strength increment does not change virtually in a broad enough interval of the model parameter values satisfying to the existence condition of the Cauchy problem (3.3), (3.4) solution.
expression (3.7) describes the dependence of vacancy barrier volume density on dose  = Фt.When the processes of vacancy barrier clusterization are absent overall ( 1 =0) it results from (known contribution to yield strength increment follows in the case of hardening by the barrier of a single type:

Fig. 3 .Fig. 3 . 3 .model of yield strength dependence on the temperature of irradiated materials 4 . 1
Fig. 3.3.a. Dose dependences of vacancy complex contribution to yield strength increment at fixed values of parameters 10 0 () KC  = 1, d 3 /d = 0.097, 10   = 1, 30   = 0.4, (1) - 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 .
Fig. 4.1.Generalized scheme of temperature dependence of flow stress in polycrystalline materials.Area 1 corresponds to low temperature range T<0,15 T m ; area 2 is characterized by athermal component   up to ~0,45 T m ; 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  с ,  m , T 0 , T с of the model have next phenomenological meaning.

4 . 3
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 Discussion of model results and experimental dataThe values of empirical parameters  с ,  m , T 0 , T с are fit by the best coinciding the values of the function (4.3) for corresponding experience temperatures with experimental values of 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.0Х1810Т steel samples have been irradiated by (е,) -beams with energy of 225 MeV up to dose of 10 25 el/cm 2 at temperatures of 170-190С.For mechanical experiences, the planar samples of test portion sizes of 1020.3mm have been experienced in vacuum at temperatures of 20-1200С with deformation velocity of 0.003 c Б1[15].The copper samples of vacuuminduced 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 10 19 el/cm 2[11].The 15Х2MФ steel samples have been irradiated by neutrons up to fluence of 310 20 neutron/cm 2[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 10 25 el/cm 2[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.
Фt.The typical plot of the dependence (2.19) is shown in Fig.2.4.at the different values of barrier clusterization intensity and the fixed rest parameters in the relative units (as stated above).

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, T с = 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 www.intechopen.comPolycrystalline Materials -Theoretical and Practical Aspects 162 obtained experimentally (A) and by a theoretical calculation (B) under radiation up to dose of 10 25 el/cm 2 for X18H10T steel.