3. Moving fronts of deformation localization (the Chernov-Luders band type)

Recently, the synergetic approach is employed more and more to describe the evolution of plastic deformation in materials. As known, the balance equation for a local dislocation density ρð Þ x; t underlies the synergetic models. In the work [6], one of these is considered. There, the balance equation was written as

$$\frac{\partial \rho(\mathbf{x},t)}{\partial t} + div(\mathbf{v}\rho(\mathbf{x},t) - D\nabla \rho(\mathbf{x},t)) = J(\rho(\mathbf{x},t))\tag{2}$$

where v is velocity vector of dislocation sliding, D is dislocation diffusion coefficient, and Jð Þ ρð Þ x; t is the dislocation density functional determined by interaction of dislocation with each other. The velocity of sliding dislocations v can be represented from three parts: v = vext + m( fint + fcor), where vext is velocity from

external stress, m is dislocation mobility, and fint origins from internal stress fint = bσint supposing internal stress σ ¼ αbG ffiffiffi ρ p , where b is Burgers vector quantity, α is a numerical coefficient, and G is shear modulus. fcor is correlation force arising from mutual disposition of dislocations. We used the expression for it from [7]

$$f\_{cor} = A\_1 \frac{\partial \rho}{\partial \mathbf{x}}\tag{3}$$

where <sup>A</sup><sup>1</sup> <sup>¼</sup> Gb<sup>2</sup> 4πρ<sup>0</sup> and ρ<sup>0</sup> is an average stationary dislocation density. Inserting expressions for forces in Eq. (2) and neglecting its right side (J(ρ) ffi 0 (argumentation in [6])), we obtain the basic equation of our model:

$$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial \mathbf{x}} \left( V\_{ext} \rho + m b^2 \mathbf{G} \rho \sqrt{\rho} + \left( m \frac{\mathbf{G} b^2}{4 \pi \rho\_0} - D \right) \frac{\partial \rho}{\partial \mathbf{x}} \right) = \mathbf{0} \tag{4}$$

The solution of Eq. (4) is looked for as

$$
\rho = \rho\_0 + \rho\_1(\mathbf{x}, t) \tag{5}
$$

4. Macroscopic bands (the Danilov-Zuev relaxation wave type)

Three-dimensional plot of step propagation of plastic deformation in sliding plane.

forces [7] originating from redistribution of energy between the interaction

f cor ¼ A<sup>1</sup>

dislocations more exactly:

Figure 2.

205

where <sup>A</sup><sup>1</sup> <sup>¼</sup> Gb<sup>2</sup>

4πρ<sup>0</sup>

, <sup>A</sup><sup>2</sup> <sup>¼</sup> <sup>b</sup>σextL 8πρ<sup>2</sup> 0

Modeling Plastic Deformation in Irradiated Materials DOI: http://dx.doi.org/10.5772/intechopen.82635

density u xð Þ ; t of dislocations from Eq. (2):

∂u ∂t þ θ ∂u ∂x þ u ∂u ∂x þ ∂2 u ∂x<sup>2</sup> þ

of the dislocations which can annihilate in particular.

∂u ∂t þ θ ∂u ∂x þ u ∂u ∂x þ ∂2 u ∂x<sup>2</sup> þ

Now, in the dislocation density balance Eq. (2), we take into account correlation

∂ρ ∂x þ A<sup>2</sup>

ensemble [7]. The right side of Eq. (2) can be represented as <sup>J</sup>ð Þ¼ <sup>ρ</sup> <sup>k</sup><sup>1</sup> � <sup>k</sup>2ρ<sup>2</sup> (see [6]) where k<sup>1</sup> characterizes a dislocation source and k<sup>2</sup> is responsible for interaction

Supposing ρ<sup>1</sup> ¼ ρ0u xð Þ ; t , one obtains the dimensionless equation for the relative

where θ is the numerical coefficient and χ1, χ<sup>2</sup> are responsible for the velocity of

solution [9] of which describes spatial quasi-periodical structures. For Eq. (10), the Cauchy problem was solved at the different initial conditions for the dimensionless function u xð Þ ; t . The same level values of u xð Þ ; 0 were set at the initial time moment under a random distribution in x on the segment [0, 30]. In Figure 3, the plot I corresponds to the level u xð Þ¼ ; 0 0:2; the plots II and III do u xð Þ¼ ; 0 0:5 and

dislocation formation and their annihilation, respectively. In the work [8], it is shown that χ1, χ<sup>2</sup> are extremely small for the parameter numerical values of a real standard metal. Due to this, the right side of Eq. (9) is considered to be equal to

zero. Then Eq. (9) goes to Kuramoto-Sivashinsky's equation type:

∂<sup>4</sup>u

∂<sup>3</sup>ρ

, and L is an average relaxation length of a dislocation

<sup>∂</sup>x<sup>4</sup> ¼ �χ1<sup>u</sup> � <sup>χ</sup>2u<sup>2</sup>

∂<sup>4</sup>u

<sup>∂</sup>x<sup>3</sup> (8)

<sup>∂</sup>x<sup>4</sup> <sup>¼</sup> <sup>0</sup>, (10)

, (9)

where ρ1ð Þ x; t is a dislocation density fluctuation near the average stationary dislocation density ρ0. For the dislocation density fluctuation ρ1ð Þ x; t , we get the widely known Burgers equation:

$$\frac{\partial \rho\_1}{\partial t} + \rho\_1 \frac{\partial \rho\_1}{\partial \mathbf{x}} = \frac{1}{2Kb} \left( \frac{D}{m} - \frac{Gb^2}{4\pi} \right) \frac{\partial^2 \rho\_1}{\partial \mathbf{x}^2} \tag{6}$$

where K is determined by material constants. As well knowing the solution of Eq. (6) is a step, and ρð Þ x; t takes the form

$$\rho(\mathbf{x},t) = a\delta \left(\mathbf{1} + \tanh\frac{\mathbf{1}}{2}(a\mathbf{x} - a^2t\delta)\right) \tag{7}$$

where <sup>δ</sup> <sup>¼</sup> <sup>1</sup> 2Kb Gb<sup>2</sup> <sup>4</sup><sup>π</sup> � <sup>D</sup> m � � and <sup>a</sup> is constant determined by the bound condition, <sup>ρ</sup>1ð Þ� <sup>x</sup>; <sup>t</sup> tanh <sup>1</sup> <sup>2</sup> ax � <sup>a</sup><sup>2</sup> ð Þ! <sup>t</sup><sup>δ</sup> 0, at <sup>x</sup> � at<sup>δ</sup> ! 0. It corresponds to the edge of Chernov-Luders band or the area of a sharp stepwise transition from a certain value of dislocation density to another value. Irradiation increases the step height. This is shown qualitatively in Figures 1 and 2. There are three plots corresponding to Eq. (7) for three irradiation dose values: p<sup>1</sup> < p<sup>2</sup> < p3.

Figure 1.

The dependence of step height on irradiation dose: p1 < p2 < p3. It corresponds to the edge of Chernov-Luders band.

Modeling Plastic Deformation in Irradiated Materials DOI: http://dx.doi.org/10.5772/intechopen.82635

external stress, m is dislocation mobility, and fint origins from internal stress—

α is a numerical coefficient, and G is shear modulus. fcor is correlation force arising from mutual disposition of dislocations. We used the expression for it from [7]

> ∂ρ ∂x

and ρ<sup>0</sup> is an average stationary dislocation density. Inserting

Gb<sup>2</sup> 4πρ<sup>0</sup>

!

� D

ρ ¼ ρ<sup>0</sup> þ ρ1ð Þ x; t (5)

∂2 ρ1 ∂ρ ∂x

f cor ¼ A<sup>1</sup>

expressions for forces in Eq. (2) and neglecting its right side (J(ρ) ffi 0 (argumenta-

<sup>ρ</sup> <sup>p</sup> <sup>þ</sup> <sup>m</sup>

where ρ1ð Þ x; t is a dislocation density fluctuation near the average stationary dislocation density ρ0. For the dislocation density fluctuation ρ1ð Þ x; t , we get the

where K is determined by material constants. As well knowing the solution of

<sup>ρ</sup>ð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>a</sup><sup>δ</sup> <sup>1</sup> <sup>þ</sup> tanh <sup>1</sup>

D <sup>m</sup> � Gb<sup>2</sup> 4π

2

<sup>2</sup> ax � <sup>a</sup><sup>2</sup> ð Þ! <sup>t</sup><sup>δ</sup> 0, at <sup>x</sup> � at<sup>δ</sup> ! 0. It corresponds to the edge of

Chernov-Luders band or the area of a sharp stepwise transition from a certain value of dislocation density to another value. Irradiation increases the step height. This is shown qualitatively in Figures 1 and 2. There are three plots corresponding to

The dependence of step height on irradiation dose: p1 < p2 < p3. It corresponds to the edge of Chernov-Luders

ax � <sup>a</sup><sup>2</sup> <sup>t</sup><sup>δ</sup> � � � �

and a is constant determined by the bound condition,

!

!

Gρ ffiffiffi

ρ p , where b is Burgers vector quantity,

(3)

(7)

¼ 0 (4)

<sup>∂</sup>x<sup>2</sup> (6)

fint = bσint supposing internal stress σ ¼ αbG ffiffiffi

Use of Gamma Radiation Techniques in Peaceful Applications

tion in [6])), we obtain the basic equation of our model:

The solution of Eq. (4) is looked for as

∂ρ1 ∂t þ ρ<sup>1</sup> ∂ρ1 <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup> 2Kb

Eq. (7) for three irradiation dose values: p<sup>1</sup> < p<sup>2</sup> < p3.

Eq. (6) is a step, and ρð Þ x; t takes the form

Gb<sup>2</sup> <sup>4</sup><sup>π</sup> � <sup>D</sup> m � �

Vext<sup>ρ</sup> <sup>þ</sup> mb<sup>2</sup>

where <sup>A</sup><sup>1</sup> <sup>¼</sup> Gb<sup>2</sup>

4πρ<sup>0</sup>

∂ρ ∂t þ ∂ ∂x

widely known Burgers equation:

where <sup>δ</sup> <sup>¼</sup> <sup>1</sup>

<sup>ρ</sup>1ð Þ� <sup>x</sup>; <sup>t</sup> tanh <sup>1</sup>

Figure 1.

band.

204

2Kb

Figure 2. Three-dimensional plot of step propagation of plastic deformation in sliding plane.
