**3.1 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:

$$
\Delta \sigma = \Delta \sigma\_1 + \Delta \sigma\_2 + \Delta \sigma\_3 \,. \tag{3.1}
$$

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

Influence of Irradiation on Mechanical Properties of Materials 153

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

breakdown of vacancy clusters and development of divacancy clusters. Effect of these contributions are appreciable if intensity of secondary processes of developing divacancy

To find the volume densities of the nonequilibrium barriers it is necessary to set up their

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).

The model values of parameters (are given in captures of Figures) and the initial conditions (0) *Ci* <sup>5</sup><sup>1013</sup> <sup>3</sup> *cm* .are used 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. 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

Here is shown the specified form of dose dependences of relative densities for vacancy barriers *C C* 1 0 / , and interstitial barriers*C C* 2 0 / , and more large vacancy complexes *C C* 3 0 / where *С*0 is a measure scale of barrier density taken to be equal 1015 <sup>3</sup> *cm* in this case on

in units measured by the scale τ0 (it is convenient to select a minimal fluence of

 = 1, 0 2 

 = 0.5, 0 3 

= 1.25,

 = 1, 0 1 

 = 1.5, 20 0 ( ) *K C* 

3 

and 1 characterize intensities of

(0) (0) *C C <sup>i</sup> <sup>i</sup>* , *i* = 1, 2, 3 (3.4)

vacancy barrier density. The kinetic coefficients <sup>1</sup>

complexes predominates over their breakdown.

**3.2 Numerical analysis of the model results** 

initial values:

parameters 10 0 ( ) *K C*

 12 0 0 *С* 

dose 0 

 = 0.9, <sup>2</sup> *C*0 1 <sup>=</sup> <sup>2</sup> *C*0 2 =1.

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:

$$\begin{cases} \frac{\partial \mathbf{C}\_{1}}{\partial \tau} = D\_{1} \Delta \mathbf{C}\_{1} + \mathbf{K}\_{1} - \mathbf{C}\_{1} \left(\tau\_{1} - \boldsymbol{\gamma}\_{12} \mathbf{C}\_{1} \mathbf{C}\_{2} - \boldsymbol{\gamma}\_{1} \mathbf{C}\_{1}^{2}\right) \\ \frac{\partial \mathbf{C}\_{2}}{\partial \tau} = D\_{2} \Delta \mathbf{C}\_{2} + \mathbf{K}\_{2} - \mathbf{C}\_{2} \left(\tau\_{2} - \boldsymbol{\gamma}\_{12} \mathbf{C}\_{1} \mathbf{C}\_{2} - \boldsymbol{\gamma}\_{2} \mathbf{C}\_{2}^{2}\right) \\ \frac{\partial \mathbf{C}\_{3}}{\partial \tau} = D\_{3} \Delta \mathbf{C}\_{3} + \boldsymbol{\gamma}\_{1} \mathbf{C}\_{1}^{2} - \mathbf{C}\_{3} \left/\tau\_{3}\right. \end{cases} \tag{3.2}$$

Here = Ф*t*, Ф is the 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), the coefficients <sup>1</sup> *i* can be represented by the form: 1 *i* = *KiVi*, where *Vi* 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:

$$\gamma\_{12} = \frac{4\pi r (D\_1 + D\_2)}{\Omega} e^{-\frac{E\_r}{k\_B T}}$$

where atom volume, *Er* activation energy of recombination of vacancy and interstitial barriers, *r* recombination radius, *T* test temperature, *kB* Boltzmann constant, *Di* diffusion coefficients of the non-equilibrium barriers of the given types: 0 exp( / ) *D D E kT i i iB* , *i* = 1, 2, where *Ei* energy of activation and migration of respective barriers, *Di*0 = *a2*, *а* and are length and barrier jumping frequency for migration, respectively.

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:

$$\begin{cases} \frac{\partial \mathbf{C}\_1}{\partial \tau} = \mathbf{K}\_1 - \mathbf{C}\_1 / \tau\_1 - \gamma\_{12} \mathbf{C}\_1 \mathbf{C}\_2 - \gamma\_1 \mathbf{C}\_1^2, \\\frac{\partial \mathbf{C}\_2}{\partial \tau} = \mathbf{K}\_2 - \mathbf{C}\_2 / \tau\_2 - \gamma\_{12} \mathbf{C}\_1 \mathbf{C}\_2 - \gamma\_2 \mathbf{C}\_2^2, \\\frac{\partial \mathbf{C}\_3}{\partial \tau} = \mathbf{\gamma}\_1 \mathbf{C}\_1^2 - \mathbf{C}\_3 / \tau\_3. \end{cases} \tag{3.3}$$

In this equation system, the first two equations coincide with the system (2.5) completely (see the preceding section).

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

> 1 2 1 1 1 1 1 12 1 2 1 1

*<sup>C</sup> D C K C CC C*

/ ,

 

> 

(3.2)

can be represented by the form:

(3.3)

/ ,

2 2 2 2 2 2 2 12 1 2 2 2

*<sup>C</sup> D C K C CC C*

Here = Ф*t*, Ф is the 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

 = *KiVi*, where *Vi* 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

1 2

*rD D k T <sup>e</sup>*

4( ) *<sup>r</sup>*

,

where atom volume, *Er* activation energy of recombination of vacancy and interstitial barriers, *r* recombination radius, *T* test temperature, *kB* Boltzmann constant, *Di* diffusion coefficients of the non-equilibrium barriers of the given types: 0 exp( / ) *D D E kT i i iB* , *i* = 1, 2, where *Ei* energy of activation and migration of respective barriers, *Di*0 = *a2*, *а* and

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

> 1 2 1 1 1 12 1 2 1 1

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

/ ,

 

> 

/ ,

2 2 2 2 2 12 1 2 2 2

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

/ .

 

In this equation system, the first two equations coincide with the system (2.5) completely

distributions are spatially homogeneous. In this case, the system (3.2) takes the form:

11 3 3

/ .

*i* 

> *B E*

 

3 3 11 3 3

by bimolecular mechanism of the vacancy barriers:

barriers. It can be valued as follows:

1 *i* 

named as clusterization coefficients), the coefficients <sup>1</sup>

3 2

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

12

are length and barrier jumping frequency for migration, respectively.

3 2

(see the preceding section).

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

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 <sup>1</sup> 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:

$$\mathbf{C}\_{i}(\mathbf{0}) = \mathbf{C}\_{i}^{(0)}, \text{ i = 1, 2, 3} \tag{3.4}$$

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).

#### **3.2 Numerical analysis of the model results**

The model values of parameters (are given in captures of Figures) and the initial conditions (0) *Ci* <sup>5</sup><sup>1013</sup> <sup>3</sup> *cm* .are used 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. 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 ( ) *K C* = 1.5, 20 0 ( ) *K C* = 1, 0 1 = 1, 0 2 = 0.5, 0 3 = 1.25, 12 0 0 *С* = 0.9, <sup>2</sup> *C*0 1 <sup>=</sup> <sup>2</sup> *C*0 2 =1.

Here is shown the specified form of dose dependences of relative densities for vacancy barriers *C C* 1 0 / , and interstitial barriers*C C* 2 0 / , and more large vacancy complexes *C C* 3 0 / where *С*0 is a measure scale of barrier density taken to be equal 1015 <sup>3</sup> *cm* in this case on dose 0 in units measured by the scale τ0 (it is convenient to select a minimal fluence of

Influence of Irradiation on Mechanical Properties of Materials 155

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)

> 1 2 1 1 1 11

 

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

/ ,

(3.6)

. The obtained

/ .

 

111

 , 

whence the well known contribution to yield strength

/ <sup>1</sup> 0 1/2

<sup>3</sup> ( )/ 2

0 1/2

 

 

Further, this expression is used to analysis the vacancy complex contribution to yield

The increasing of saturation quantity of yield strength increment goes with increasing

This increasing is nonlinear, enlargement of yield strength increment saturation going less

Growth of saturation quantity of yield strength increment goes with increasing specific

 

. (3.8)

, (3.7)

. (3.9)

( /) *d C dC* . (3.10)

Artanh(1 / )

 

11 3 3

The first equation of the system (3.6) doesn't contain *С*3. Therefore, it is independent. Its

1( ) tanh *<sup>а</sup> <sup>b</sup> c*

*C C С* 

1 4*K*

expression (3.7) describes the dependence of vacancy barrier volume density on dose = Ф*t*. When the processes of vacancy barrier clusterization are absent overall (1=0) it results from

1 1 11 <sup>0</sup> { (1 )/ } *d K e dC*

Substituting Eq (3.7) into the second equation of the system (3.6) its solution can be written

 

31 1 0 ( ) ( ) *<sup>s</sup> C e C ds* 

3 33 0

intensity of the clusterization processes that is with increasing parameter 1 (Fig.3.3*a*).

 

and less considerably with increasing intensity of the clusterization processes.

3 2

increment follows in the case of hardening by the barrier of a single type:

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

**3.3 Secondary reaction contribution analysis** 

solution with zero initial condition takes the form:

where *с* = 21/, *Сa*=/211, *Сb*=*Сa*/, <sup>2</sup>

 

consists of two equations:

(3.6) / <sup>1</sup> 1 11 *CK e* ( ) (1 )

 

strength increment:

times 1 and 3 as well (Fig.3.3. *b* and *c*).

as

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 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

$$
\Delta \sigma = \Delta \sigma\_{\infty}^{0} \sum\_{i=1}^{3} \sqrt{d\_i \mathbb{C}\_i / d \mathbb{C}\_0} \tag{3.5}
$$

where <sup>0</sup> is determined by Eq (2.9) (see the preceding section), *d* average size of barrier cluster over all of types. The results of numerical modeling the behavior of the yield strength increment are represented in Fig. 3.2.

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.

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.

#### **3.3 Secondary reaction contribution analysis**

154 Polycrystalline Materials – Theoretical and Practical Aspects

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 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

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

> 3 0

> > 1

cluster over all of types. The results of numerical modeling the behavior of the yield

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

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.

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

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.

considerably to yield strength increment at dose build-up.

*i*

 

0

, (3.5)

*i i* /

is determined by Eq (2.9) (see the preceding section), *d* average size of barrier

*d C dC*

material by the radiation - induced barriers takes place with increasing dose.

represent for construction of graph as follows

strength increment are represented in Fig. 3.2.

where <sup>0</sup> 

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:

$$\begin{cases} \frac{\partial \mathbf{C}\_1}{\partial \tau} = \mathbf{K}\_1 - \mathbf{C}\_1 / \tau\_1 - \boldsymbol{\gamma}\_1 \mathbf{C}\_1^2 \\ \frac{\partial \mathbf{C}\_3}{\partial \tau} = \boldsymbol{\gamma}\_1 \mathbf{C}\_1^2 - \mathbf{C}\_3 / \tau\_3. \end{cases} \tag{3.6}$$

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:

$$\mathcal{C}\_1(\tau) = \mathcal{C}\_a \tanh\left(\frac{\tau}{\tau\_c} + \varphi\right) - \mathcal{C}\_b \, \, \, \, \tag{3.7}$$

where *с* = 21/, *Сa*=/211, *Сb*=*Сa*/, <sup>2</sup> 111 1 4*K* , Artanh(1 / ) . The obtained expression (3.7) describes the dependence of vacancy barrier volume density on dose = Ф*t*.

When the processes of vacancy barrier clusterization are absent overall (1=0) it results from (3.6) / <sup>1</sup> 1 11 *CK e* ( ) (1 ) whence the well known contribution to yield strength increment follows in the case of hardening by the barrier of a single type:

$$
\Delta\sigma\_1 = \Delta\sigma\_\alpha^0 \{d\_1 K\_1 \tau\_1 (1 - e^{-\tau/\tau\_1}) / \, d\mathcal{C}\_0\}^{1/2} \,. \tag{3.8}
$$

Substituting Eq (3.7) into the second equation of the system (3.6) its solution can be written as

$$\mathcal{C}\_{\mathfrak{Z}}(\tau) = \gamma\_1 \int\_0^{\tau} e^{(s-\tau)/\tau\_3} \mathcal{C}\_1^2(\tau) ds \,\,. \tag{3.9}$$

Further, this expression is used to analysis the vacancy complex contribution to yield strength increment:

$$
\Delta \sigma\_3 = \Delta \sigma\_\alpha^0 \left( d\_3 \mathbf{C}\_3 \;/\; d\mathbf{C}\_0 \right)^{1/2} \;. \tag{3.10}
$$

The increasing of saturation quantity of yield strength increment goes with increasing intensity of the clusterization processes that is with increasing parameter 1 (Fig.3.3*a*).

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.3.3. *b* and *c*).

Influence of Irradiation on Mechanical Properties of Materials 157

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

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

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

To analyze radiation embrittlement it is necessary to take into account availability of two

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. 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

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

component of flow stress; area 4 corresponds to second athermal plateau

up to ~0,45 Tm; area 3 corresponds to thermo activated

): the thermal (thermo activated) component \*

  determined by long – range

 ~G.

**4.1 Temperature intervals of radiation embrittlement with taking into account two** 

materials is characterized by availability of the different structure deformation levels.

with taking into account their durable characteristics on temperature.

created by short – rang forces and the athermal one

**of irradiated materials** 

**components of material flow stress** 

components of material flow stress (

(Fig.4.1.).

by athermal component

 

Fig. 3.3. *a.* Dose dependences of vacancy complex contribution to yield strength increment at fixed values of parameters 10 0 ( ) *K C* = 1, *d*3/*d* = 0.097, 1 0 = 1, 3 0 = 0.4, (1) – <sup>2</sup> *C*0 1 =5; (2) – <sup>2</sup> *C*0 1 =1.5; (3) – <sup>2</sup> *C*0 1 =0.5.

Fig. 3.3. *b* and *c*. Dose dependences of vacancy complex contribution to yield strength increment at fixed values of parameters 10 0 ( ) *K C* = 1, *d*3/*d* = 0.097,

*b*) 3 0 = 0.4, <sup>2</sup> *C*0 1 =3, (1) – 1 0 = 2; (2) – 1 0 = 1; (3) – 1 0 = 0.5; *c*) 1 0 = 1, <sup>2</sup> *C*0 1 =3, (1) – 3 0 = 0.8; (2) – 3 0 = 0.6; (3) – 3 0 = 0.4.

Fig. 3.3. *a.* Dose dependences of vacancy complex contribution to yield strength increment at

Fig. 3.3. *b* and *c*. Dose dependences of vacancy complex contribution to yield strength

 = 2; (2) – 1 0 

 = 0.8; (2) – 3 0 

= 1, *d*3/*d* = 0.097,

 = 1; (3) – 1 0 

 = 0.6; (3) – 3 0 

= 0.5;

= 0.4.

= 1, *d*3/*d* = 0.097, 1 0

   = 1, 3 0 

= 0.4, (1) –

fixed values of parameters 10 0 ( ) *K C*

increment at fixed values of parameters 10 0 ( ) *K C*

 =3, (1) – 3 0 

 =3, (1) – 1 0 

=5; (2) – <sup>2</sup> *C*0 1

<sup>2</sup> *C*0 1 

*b*) 3 0 

*c*) 1 0 

 = 0.4, <sup>2</sup> *C*0 1 

 = 1, <sup>2</sup> *C*0 1 

=0.5.

=1.5; (3) – <sup>2</sup> *C*0 1
