**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]:

$$
\Delta \sigma = \sum\_{i=1}^{N} \Delta \sigma\_i \tag{2.1}
$$

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 21016 n/cm2), 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

( )*<sup>n</sup> a t* , (2.2)

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

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

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

At the large irradiation doses, a yield strength increment can be represented by the sum of

*N*

1

*i* 

where index *i* is a barrier type, *N* is a number of barrier types affecting the yield strength,

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 21016 n/cm2), the hardening occurs due to forming the slowly increasing interstitial clusters. The vacancy clusters begin

In these stages, the yield strength increment is described with sufficient approximation of

( )*<sup>n</sup>* 

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

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

*i*

, (2.1)

*a t* , (2.2)

**2. Mutual recombination and clusterization effect of the vacancy and** 

**interstitial barriers on radiation hardening materials** 

athermic surmounting obstacles by dislocations [3].

*i* is the yield strength increment of *i'* barrier type.

to contribute to the hardening in increasing dose.

type and the irradiation condition from 0.25 to 0.75 [3].

the dependence of the form

barrier contributions of different types [3]:

**2.1 Formulation of the model** 

theory.

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:

$$
\Delta \sigma = \Delta \sigma\_1 + \Delta \sigma\_2 \,. \tag{2.3}
$$

For all of the obstacle types, the metal yield strength increment conditioned by dislocation deceleration is described as [2]:

$$
\Delta \sigma\_i = \alpha\_i \mu b (C\_i d\_i)^{1/2} \text{ , i = 1, 2, \dots} \tag{2.4}
$$

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, *Сi* the volume density of *i*' barrier type, *di* 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:

$$\begin{cases} \frac{d\mathbf{C}\_1}{d\tau} = K\_1 - \frac{\mathbf{C}\_1}{\tau\_1} - \gamma\_{12}\mathbf{C}\_1\mathbf{C}\_2 - \gamma\_1\mathbf{C}\_1^2, \\\frac{d\mathbf{C}\_2}{d\tau} = K\_2 - \frac{\mathbf{C}\_2}{\tau\_2} - \gamma\_{12}\mathbf{C}\_1\mathbf{C}\_2 - \gamma\_2\mathbf{C}\_2^2. \end{cases} \tag{2.5}$$

where = Ф*t*, Ф is density of particle flux, *t* irradiation time, *Ki*, *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 <sup>1</sup> *i* can be represented by the form: <sup>1</sup> *i* = *KiVi*, where *Vi* 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,

Influence of Irradiation on Mechanical Properties of Materials 145

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

= 0.9, (*b*) –

 = 1, 0 2 

> 12 0 0 *С*

= 1.5, and different values of

are the

= 0.1 where *С*<sup>0</sup> , 0

Fig. 2.1. Dependences of relative barrier densities of vacancy type (the curve 1) and interstitial type (the curve 2) on irradiate dose at fixed values of the dimensionless

> = 1.1, 20 0 ( ) *K C*

measure scales of barriers and dose (fluence) respectively the concrete selection of that is

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 <sup>12</sup> 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

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

11 0 22 0

 

0 1/2 1/2

0 1/2 0

where *d* is an average size of the barrier clusters of all types (here it can be determined as

{( / ) ( / ) } *d C dC d C dC* , (2.8)

*bC d* ( ) (2.9)

 12 0 0 *С* 


parameters 10 0 ( ) *K C*

half-sum of *d*1 and *d*2).

the recombination coefficient: (*a*) –

conditioned by the concrete problem.

the monotonic quantity (see Fig.2.1, a) and b)).

figures, it is convenient to represent Eq (2.3) as follows

 

 = 2.5, 0 1 

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:

$$\mathbf{C}\_{i}(\mathbf{0}) = \mathbf{C}\_{i}^{(0)}, \ i = \mathbf{1}, \ \mathbf{2}. \tag{2.6}$$

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).

#### **2.2 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 <sup>2</sup> *i i C* in every equation of the system (2.5).

In this approximation the system (2.5) takes the form:

$$\begin{cases} \frac{d\mathbf{C}\_1}{d\tau} = K\_1(1 - V\_1\mathbf{C}\_1) - \gamma\_{12}\mathbf{C}\_1\mathbf{C}\_{2'}\\ \frac{d\mathbf{C}\_2}{d\tau} = K\_2(1 - V\_2\mathbf{C}\_2) - \gamma\_{12}\mathbf{C}\_1\mathbf{C}\_{2'} \end{cases} \tag{2.7}$$

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) *Ci* 51013cm-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 *C C* 1 0 / and interstitial *C C* 2 0 / types on the relative irradiate dose <sup>0</sup> where *С*0 is a measure scale of the defect densities (here it is equal 1015 cm-3). The measure units are accepted relative to the selected scale τ0 (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 τ0 can be equal 1014 ion/cm2 or 1022 n/m2 as the contemporary neutron fluence). This dependence

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

To find the volume densities of the radiation - induced nonequilibrium vacancies and

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

**2.2 Dose saturation features of the yield strength subject to annihilation effects of the** 

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

*C* in every equation of the system (2.5).

*dC K VC CC*

*dC K VC CC*

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) *Ci* 51013cm-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

Here it is shown the dependence typical behavior of the barrier relative densities of the

measure scale of the defect densities (here it is equal 1015 cm-3). The measure units are accepted relative to the selected scale τ0 (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 τ0 can be equal 1014 ion/cm2 or 1022 n/m2 as the contemporary neutron fluence). This dependence

vacancy *C C* 1 0 / and interstitial *C C* 2 0 / types on the relative irradiate dose <sup>0</sup>

1 1 1 12 1 2

(1 ) ,

(2.7)

 

where *С*0 is a

2 2 2 12 1 2

(1 ) .

(0) (0) *C C <sup>i</sup> <sup>i</sup>* , *i* = 1, 2. (2.6)

forming the clusters of two barriers of the same type.

according to Eqs (2.3), (2.4).

neglect the last terms of <sup>2</sup>

*i i*

parameter simulating values are shown in Fig.2.1.

In this approximation the system (2.5) takes the form:

1

*d*

*d*

2

**vacancy and interstitial barriers** 

interstitial atoms it is necessary to set up their the initial values:

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).

Fig. 2.1. Dependences of relative barrier densities of vacancy type (the curve 1) and interstitial type (the curve 2) on irradiate dose at fixed values of the dimensionless parameters 10 0 ( ) *K C* = 2.5, 0 1 = 1.1, 20 0 ( ) *K C* = 1, 0 2 = 1.5, and different values of the recombination coefficient: (*a*) – 12 0 0 *С* = 0.9, (*b*) – 12 0 0 *С* = 0.1 where *С*<sup>0</sup> , 0 are the measure scales of barriers and dose (fluence) respectively the concrete selection of that is conditioned by the concrete problem.

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 <sup>12</sup> 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

$$
\Delta \sigma = \Delta \sigma\_{\alpha}^{0} \left| \left( d\_{1} \mathbf{C}\_{1} \;/\ d \mathbf{C}\_{0} \right)^{1/2} + \left( d\_{2} \mathbf{C}\_{2} \;/\ d \mathbf{C}\_{0} \right)^{1/2} \right|, \tag{2.8}$$

$$
\Delta \sigma\_{\alpha}^{0} = a \mu b (\mathbf{C}\_{0} d)^{1/2} \tag{2.9}
$$

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).

Influence of Irradiation on Mechanical Properties of Materials 147

*KV C* 

, 1 1

The saturation values of the barrier density (2.10) and (2.11) are achieved quickly enough

In the case when the recombination of the vacancy and interstitial barriers is negligibly

addition that the radiation barriers are virtually absent at the initial time (0) <sup>0</sup> *Ci* , (*i* = 1, 2)

 

where *i* = (*KiVi*)-1. Substitution of Eq (2.12) to Eq (3) gives the expression of the yield

It should be noted that at small doses it is followed the well known law of Eq (2.2) at *n* = ½:

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

Substituting Eq (2.15) into (2.3) it can be obtained the expression for the yield strength

1 1 2 2 ( ) (0) 1/2 ( ) (0) 1/2 0 1 1 1 2 2 <sup>2</sup> {1 ( 1) } {1 ( 1) } *s s K V K V CV e CV e* 

(2.16)

<sup>0</sup> 

<sup>1</sup> (0) ( ) {1 ( 1) } *K Vi i*

*i i i i C CV e V*

1 2 ( ) / / 1/2 ( ) 1/2 <sup>0</sup> 1 2 (1 ) (1 ) *s s e e*

1/2

then as a result the known expression of the volume barrier density is obtained [3]:

<sup>1</sup> / ( ) (1 )*<sup>i</sup>*

*i C e V*

> 

 

*s*

*a*

*b K V <sup>C</sup> <sup>V</sup>* .

*C*

, (2.10)

, (2.11)

= 0. If to suppose in

12 2

, *i* = 1, 2, (2.12)

 

*a t* ( ) , (2.14)

 

, (2.13)

*bd V* , *i* = 1, 2. In this case, at large doses it is followed from Eq (2.13)

, *i* = 1,2. (2.15)

( ) 1 1 ( ) 1 1 12 2

<sup>2</sup> 1 1 *<sup>s</sup> <sup>b</sup> <sup>a</sup>*

*<sup>C</sup> C C*

2 1 12 1 2 1 2 12 2 2

*K V*

*K K K K VV <sup>C</sup>*

*<sup>K</sup> <sup>C</sup>*

*s*

( )

( ) 2 *<sup>a</sup>*

small the equations of the system (2.7) become independent at 12

*i*

 

 {( ) ( ) } .

increment at 12=0 and the arbitrary initial barrier densities:

the equation system (2.7) leads to the expression

where

(see Fig.2.1.).

strength increment at 12 = 0:

where ( ) 1/2 (/) *<sup>s</sup> <sup>i</sup> i ii*

 

> 

where 1/2 1/2 1 11 2 22 *a b Kd Kd*

 

that. () () 0 1 2 *s s*

The results of numerical modeling the behavior of the yield strength are represented in Fig. 2.2. where the typical dose dependences of the yield strength increment are shown on the base of the obtained Eqs (2.8) and (2.9) (in relative units) at the model parameter values. The corresponding numerical analysis shows that the yield strength gets the saturation quickly enough. The typical monotonic form of the dose saturation plots of the yield strength does not change virtually in a broad enough interval of the model parameter values satisfying to the existence condition of the Cauchy problem (2.5), (2.6) solution.

Fig. 2.2. Dependences of the relative yield strength increment on irradiate dose at fixed the nondimensional parameters 10 0 ( ) *K C* = 2.5, 0 1 = 1.1, 20 0 ( ) *K C* = 1, 0 2 = 1.5, and different values of the recombination coefficient: (1) 12 0 0 *С* = 0.9, (2) 12 0 0 *С* = 0.1 in relative units.

Fig. 2.3. Dependences of the relative yield strength increment on the recombination coefficient at fixed irradiate dose 0 = 4 and the rest parameter values as in Fig. 2.2.

Besides, on the base of the numerical analysis, it is obtained that a growth of the mutual recombination coefficient <sup>12</sup> leads to decreasing the saturation value of the yield strength. It is shown in Fig.2.3. where the dependence of the relative yield strength increment / 0 on the mutual recombination coefficient at fixed dose is shown (here <sup>0</sup> is the yield 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:

$$\mathbf{C}\_{1}^{(s)} = \frac{\mathbf{K}\_{1}}{\mathbf{K}\_{1}V\_{1} + \mathbf{y}\_{12}\mathbf{C}\_{2}^{(s)}}\;'\tag{2.10}$$

$$\mathbf{C}\_{2}^{(s)} = \mathbf{C}\_{a} \left( \mathbf{1} + \sqrt{\mathbf{1} + \frac{\mathbf{C}\_{b}}{\mathbf{C}\_{a}}} \right) \tag{2.11}$$

where

146 Polycrystalline Materials – Theoretical and Practical Aspects

The results of numerical modeling the behavior of the yield strength are represented in Fig. 2.2. where the typical dose dependences of the yield strength increment are shown on the base of the obtained Eqs (2.8) and (2.9) (in relative units) at the model parameter values. The corresponding numerical analysis shows that the yield strength gets the saturation quickly enough. The typical monotonic form of the dose saturation plots of the yield strength does not change virtually in a broad enough interval of the model parameter values satisfying to

Fig. 2.2. Dependences of the relative yield strength increment on irradiate dose at fixed the

 = 1.1, 20 0 ( ) *K C* 

1

= 0.9, (2)

/ <sup>0</sup> 

= 4 and the rest parameter values as in Fig. 2.2.

 12 0 0 *C* 

 12 0 0 *С* 

 = 1, 0 2 

 12 0 0 *С* 

= 1.5, and

 / 0

<sup>0</sup> is the yield

= 0.1 in relative

 = 2.5, 0 1 

2

Fig. 2.3. Dependences of the relative yield strength increment on the recombination

on the mutual recombination coefficient at fixed dose is shown (here

Besides, on the base of the numerical analysis, it is obtained that a growth of the mutual recombination coefficient <sup>12</sup> leads to decreasing the saturation value of the yield strength. It is shown in Fig.2.3. where the dependence of the relative yield strength increment

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

A stationary point of the systems (2.7) determines the saturation values of the barrier density:

 

the existence condition of the Cauchy problem (2.5), (2.6) solution.

different values of the recombination coefficient: (1)

 /0

<sup>0</sup> / 

nondimensional parameters 10 0 ( ) *K C*

coefficient at fixed irradiate dose 0

dependence being nonlinear and monotonic.

units.

$$C\_a = \frac{(K\_2 - K\_1)\mathcal{V}\_{12} - K\_1 K\_2 V\_1 V\_2}{2\gamma\_{12} K\_2 V\_2}, \ C\_b = \sqrt{\frac{K\_1 V\_1}{\gamma\_{12} V\_2}} \ .$$

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) <sup>0</sup> *Ci* , (*i* = 1, 2) then as a result the known expression of the volume barrier density is obtained [3]:

$$C\_i(\tau) = \frac{1}{V\_i} (1 - e^{-\tau/\tau\_i}) \quad , \quad i = 1, 2,\tag{2.12}$$

where *i* = (*KiVi*)-1. Substitution of Eq (2.12) to Eq (3) gives the expression of the yield strength increment at 12 = 0:

$$
\Delta\sigma\_0 = \Delta\sigma\_1^{(s)} \left(1 - e^{-\tau/\tau\_1}\right)^{1/2} + \Delta\sigma\_2^{(s)} \left(1 - e^{-\tau/\tau\_2}\right)^{1/2},\tag{2.13}
$$

where ( ) 1/2 (/) *<sup>s</sup> <sup>i</sup> i ii bd V* , *i* = 1, 2. In this case, at large doses it is followed from Eq (2.13) that. () () 0 1 2 *s s* 

It should be noted that at small doses it is followed the well known law of Eq (2.2) at *n* = ½:

$$
\Delta \sigma\_{\phantom{\alpha}0} = a \text{(}\Phi t\text{)}^{1/2}\text{ },\tag{2.14}
$$

where 1/2 1/2 1 11 2 22 *a b Kd Kd* {( ) ( ) } .

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

$$C\_i(\tau) = \frac{1}{V\_i} \{ 1 + (C\_i^{(0)} V\_i - 1)e^{-K\_i V\_i \tau} \} \ , \ i = 1, 2. \tag{2.15}$$

Substituting Eq (2.15) into (2.3) it can be obtained the expression for the yield strength increment at 12=0 and the arbitrary initial barrier densities:

$$
\Delta\sigma\_0 = \Delta\sigma\_1^{(s)} \{ 1 + (\mathbf{C}\_1^{(0)} V\_1 - 1)e^{-K\_1 V\_1 \tau} \}^{1/2} + \Delta\sigma\_2^{(s)} \{ 1 + (\mathbf{C}\_2^{(0)} V\_2 - 1)e^{-K\_2 V\_2 \tau} \}^{1/2} \tag{2.16}
$$

Influence of Irradiation on Mechanical Properties of Materials 149

*C C*/ <sup>0</sup>

1

2

3

1

2 3

/ <sup>0</sup> 

Fig. 2.4. Dose dependences of the volume barrier density (2.19) at the fixed parameter values

In the present model, it is supposed that the average barrier size is a weakly changing function of irradiate dose. The typical plot of the dependence (2.20) is shown in Fig. 2.5 at

the different values of barrier clusterization intensity and the fixed rest parameters.

Fig. 2.5. Dose dependences of the relative yield strength increment (2.20) at the fixed

expression into (2.17) we obtain the dependence of 1/2

density is the stationary point of Eq (2.18). Substituting this expression into Eqs (2.17) we

0

 

 

<sup>0</sup> 2 *q V С*

corresponds to low density of radiation defects) the dependence of the volume density of the radiation-induced barriers on irradiate dose is linear: *C q V qK t* ( ) . Substituting this

= 1, *С*0*V* = 1, and the different values of clusterization intensity:

/ <sup>0</sup> 

the volume density of the radiation-induced

1/2

. This value of the volume

(2.21)

(accordingly (2.19) it

*a t* ( ) where

= 1, *С0V* = 1 and the different values of clusterization intensity: (1) <sup>2</sup> *С*<sup>0</sup> = 0, (2)

0 0 ( ) *K C* 

<sup>2</sup> *С*<sup>0</sup> = 1, (3) <sup>2</sup> *С*<sup>0</sup> = 5 in relative units.

parameter values 0 0 ( ) *K C*

1/2 *a b q V qKd*

{( ) } .

At high irradiate doses that is when >>

obtain the saturation value of the yield strength:

(1) <sup>2</sup> *С*<sup>0</sup> = 0, (2) <sup>2</sup> *С*<sup>0</sup> = 1, (3) <sup>2</sup> *С*<sup>0</sup> = 5 in relative units.

0 

 

It follows from (2.19) that at low irradiate doses that is when <<

barriers is saturated and tends to the constant*C qV* ( )/2

Eq (2.16) 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.

#### **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 *C C* . 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

$$
\Delta \sigma = a \mu b (\text{Cd})^{1/2} \text{ }, \tag{2.17}
$$

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):

$$\frac{d\mathbf{C}}{d\tau} = \mathbf{K}(\mathbf{1} - \mathbf{V}\mathbf{C} - \mathbf{\gamma}\mathbf{C}^2) \,, \tag{2.18}$$

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

$$C(\tau) = \frac{1}{2\gamma} \left\{ q \text{th} \left( \frac{\tau}{\tau'} + \phi \right) - V \right\},\tag{2.19}$$

$$\text{where } \mathbf{r}' = \mathbf{2} / \,\mathrm{Kq} \,, \quad q = \sqrt{\mathbf{V}^2 + 4\chi} \,, \quad \mathfrak{q} = \mathrm{Arft} \frac{V}{q} = \frac{1}{2} \ln \frac{q + V}{q - V} \,, \quad \text{The } \begin{array}{c} \text{found} \quad \text{expression} \end{array} \tag{2.19}$$

describes the dependence of the volume radiation-induced barrier density on irradiate dose (fluence) = Ф*t*. 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).

Substituting Eqs (2.19) into Eqs (2.17) we obtain the dose dependence of the yield strength increment:

$$
\Delta\sigma = \Delta\sigma\_{\infty}^{0} \frac{1}{\sqrt{2\gamma C\_{0}}} \left\{ \eta \text{th} \left( \frac{\tau}{\tau'} + \Phi \right) - V \right\}^{1/2} \tag{2.20}
$$

Eq (2.16) is used to plot the yield strength dependence on the intensity quantity of the

**2.3 Dose saturation features of the yield strength subject to the effects of clustering** 

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

system equations become independent and therefore in what follows the indexes 1 and 2 of

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

1/2

where is the parameter characterizing the barrier intensity (a fixed quantity for some of

<sup>2</sup> (1 ) *dC K VC C*

where *i* of Eqs (2.5) is changed by the new notation according to the relationship *i* = *K* and

<sup>1</sup> ( ) th <sup>2</sup> *Cq V* 

 

2 *V q V q qV*

describes the dependence of the volume radiation-induced barrier density on irradiate dose (fluence) = Ф*t*. 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

Substituting Eqs (2.19) into Eqs (2.17) we obtain the dose dependence of the yield strength

<sup>2</sup> *q V <sup>C</sup>* 

0 <sup>1</sup> th

  

barrier types), *С* the volume barrier density of the same type, *d* their average size.

The solution of Eq (2.18) with the initial condition C (0) = 0 at > 0 takes the form

, <sup>1</sup> Arth ln

In this case it is convenient to go to single equation of the system (2.5):

*d*

0

 

*b Cd* ( ) , (2.17)

, (2.18)

, (2.19)

. The found expression (2.19)

1/2

(2.20)

12 1 2 *C C* . After this, the

vacancy and interstitial barrier recombination at fixed dose in Fig. 2.3.

equation of the system (2.5) it can be neglected by the last terms of

**barriers** 

notations can be omitted.

all of the indexes are omitted.

2 /*Kq* , <sup>2</sup> *q V* 4

where

(as stated above).

increment:

expression is used

Fig. 2.4. Dose dependences of the volume barrier density (2.19) at the fixed parameter values 0 0 ( ) *K C* = 1, *С0V* = 1 and the different values of clusterization intensity: (1) <sup>2</sup> *С*<sup>0</sup> = 0, (2) <sup>2</sup> *С*<sup>0</sup> = 1, (3) <sup>2</sup> *С*<sup>0</sup> = 5 in relative units.

In the present model, it is supposed that the average barrier size is a weakly changing function of irradiate dose. The typical plot of the dependence (2.20) is shown in Fig. 2.5 at the different values of barrier clusterization intensity and the fixed rest parameters.

Fig. 2.5. Dose dependences of the relative yield strength increment (2.20) at the fixed parameter values 0 0 ( ) *K C* = 1, *С*0*V* = 1, and the different values of clusterization intensity: (1) <sup>2</sup> *С*<sup>0</sup> = 0, (2) <sup>2</sup> *С*<sup>0</sup> = 1, (3) <sup>2</sup> *С*<sup>0</sup> = 5 in relative units.

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: *C q V qK t* ( ) . Substituting this expression into (2.17) we obtain the dependence of 1/2 *a t* ( ) where 1/2 *a b q V qKd* {( ) } .

At high irradiate doses that is when >> the volume density of the radiation-induced barriers is saturated and tends to the constant*C qV* ( )/2 . 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:

$$
\Delta \sigma\_{\infty} = \Delta \sigma\_{\infty}^{0} \left( \frac{q - V}{2 \gamma \mathcal{C}\_{0}} \right)^{1/2} \tag{2.21}
$$

Influence of Irradiation on Mechanical Properties of Materials 151

Agency). According to technological design of the thermonuclear reactor ITER, one of main

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

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 - 300C. For the higher temperatures (330 - 400C) 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

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 06Х18Н10Т 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 06Х18Н10Т steel irradiated in WWER-440 reactors takes place at the damage dose of ~ 10

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

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

> 

For all of the obstacle types, the metal yield strength increment conditioned by dislocation

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

123 . (3.1)

**3. Effect of secondary processes on material hardening under low** 

do to more large vacancy complexes. Then in this model Eq (2.1) takes the form:

**3.1 Formulation of the secondary process contribution model** 

deceleration is described by the Eq (2.4) (see the preceding section).

different types (see Eq (2.1) of the preceding section).

constructional material is austenitic steel 316(N)-IG [6].

invariability of the exponent ½ in Eq (2.17).

15 d.p.a. (11,51026 n/cm2).

**temperature radiation** 

concentration of these barriers grows with increasing irradiation dose.

where <sup>0</sup> 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]:

$$
\Delta \sigma^0 = \Lambda \sigma\_\alpha^0 \left(\frac{1 - e^{-\tau/\mathcal{F}}}{C\_0 V}\right)^{1/2},
\tag{2.22}
$$

where <sup>1</sup> ( ) *KV* . This dependence corresponds to the curve 1 in Fig.2.5. Hence it follows at small irradiate doses the well known law as 1/2 *a t* ( ) where now 1/2 *a b Kd* ( ) .

Fig. 2.6. Dependences of the relative yield strength increment saturation (2.21) at the fixed parameter values 0 0 ( ) *K C* = 1:

(*a*) dependence on nondimensional intensity of the barrier clusterization <sup>2</sup> *С*<sup>0</sup> 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) <sup>2</sup> *С*<sup>0</sup> = 0.1, (2) <sup>2</sup> *С*<sup>0</sup> = 1, (3) <sup>2</sup> *С*<sup>0</sup> = 5 in relative units.

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.

#### **2.4 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

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

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

1/2 /

1

, (2.22)

( ) .

*a t* ( ) where now 1/2 *a b Kd*

3

2

*VC*<sup>0</sup>

0 1 *e С V* 

( ) *KV* . This dependence corresponds to the curve 1 in Fig.2.5. Hence it follows at

 

0 

 

contribution to the velocity of forming the radiation – induced barriers [3]:

(*a*) (*b*)

1

2

3

values of their volumes: (1) *С*0*V* = 0.4, (2) *С*0*V* = 1, (3) *С*0*V* = 2 in relative units.

= 1:

Fig. 2.6. Dependences of the relative yield strength increment saturation (2.21) at the fixed

2 *C*0 

(*a*) dependence on nondimensional intensity of the barrier clusterization <sup>2</sup> *С*<sup>0</sup> at different

(*b*) dependence on *V* at different values of the clusterization intensity: (1) <sup>2</sup> *С*<sup>0</sup> = 0.1, (2)

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.

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

small irradiate doses the well known law as 1/2

0 0

 

 

where <sup>0</sup> 

where <sup>1</sup> 

parameter values 0 0 ( ) *K C*

<sup>2</sup> *С*<sup>0</sup> = 1, (3) <sup>2</sup> *С*<sup>0</sup> = 5 in relative units.

where the plots of Eq (2.21) are presented.

**2.4 Discussion of experiment data** 

is determined by Eq (2.9).

corresponds to the curve 1 in Fig.2.4.

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 - 300C. For the higher temperatures (330 - 400C) 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 06Х18Н10Т 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 06Х18Н10Т steel irradiated in WWER-440 reactors takes place at the damage dose of ~ 10 15 d.p.a. (11,51026 n/cm2).
