# Analytical Description of Plastic Deformation Distribution in the Neck of a Flat Tensile Specimen

Yevgeny Ye. Deryugin, Natalya Antipina

► To cite this version:

Yevgeny Ye. Deryugin, Natalya Antipina. Analytical Description of Plastic Deformation Distribution in the Neck of a Flat Tensile Specimen. Mechanics, Materials Science & Engineering Journal, 2015, 10.13140/RG.2.1.2372.4568 . hal-01305943

HAL Id: hal-01305943

https://hal.science/hal-01305943

Submitted on 22 Apr 2016

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Analytical Description of Plastic Deformation Distribution in the Neck of a Flat Tensile Specimen

Yevgeny Ye. Deryugin¹, Natalya Antipina²

1 – Institute of Strength Physics and Materials Science of the SB RAS, Tomsk, Russia

2 – National Research Tomsk Polytechnic University, Tomsk, Russia

a – [dee@ispms.tsc.ru](mailto:dee@ispms.tsc.ru)

**Keywords:** neck of a flat specimen, plastic deformation, analytical description

**ABSTRACT.** This work presents an analytical description of the non-uniform field distribution of plastic deformation in a flat specimen, which determines distortion of the specimen in the necking zone. The proposed method enables to be simulated the real non-uniform distributions of plastic deformation and neck distortion according to experimental measurements data. Analytical expressions are suitable for calculation of gradients and concentration of stress in the neck of a flat specimen made of real material, using well-known analytical and numerical methods: finite element methods, boundary element methods, relaxation element methods etc.

**1. Introduction.** Tensile test of the material is one of the basic test types that can outline the most important mechanical properties of the materials in engineering applications. The peculiarity of many structural metals and alloys is the descending segment on the conventional stress-strain diagrams associated with plastic deformation localization in the neck emerging before the material fracture. The specimen cross-sectional area at point of necking reduces suddenly followed by load decrease required for further specimen deformation before fracture. To analyze accurately the physical mechanisms of plastic deformation and material strengthening at the pre fracture stage, we need to calculate the dependence σ – ε corresponding to the material response in the neck local zone, where the plastic deformation is maximum and develops at highest rate. The true load diagram in the given local volume is not reflected if to take into account only a cross section reduction in the neck, since the plastic deformation distribution in the neck is extremely non-uniform. As experience shows [1, 2], the maximum degree of plastic deformation and the critical state of the material is reached at the center of the minimum cross section of a flat specimen. There are certain difficulties in experimental measurement of the geometric shape and plastic deformation distribution in the necking zone [3]. Experimental techniques for measuring local deformation of solid bodies under different boundary loading conditions are now being developed [1–8]. Therefore, description of the specimen plastic distortion related to non-uniform plastic deformation distribution in the emerging neck is one of topical problems in mechanics of the deformed solid body.

In this paper, we propose a universal method for the analytical setting of a smooth field with plastic deformation gradients in the local zone of a flat tensile specimen that determines the geometric shape of the neck and plastic deformation distribution in it. The proposed form of analytical description allows one to obtain distributions, which agree with experimental measurements data qualitatively and quantitatively, by variation of geometrical parameters in equations. An analytical setting of plastic deformation distribution in the neck zone allows calculating distribution and concentration of stresses in a solid body at the pre fraction stage using the methods of continuum theory of defects [9-11] and numerical methods for the deformed solid mechanics [12-14]. Calculation of stresses in the necking zone is an independent task and is not presented in this work.

**2. Description of plastic deformation distribution in the neck of a flat tensile specimen.**
Experimental studies of plastic deformation distribution emerging in the neck of a flat tensile specimen show that generally, this distribution is extremely non-uniform and at the macro-scale level satisfy the following conditions [2 - 4, 15]:

a) Maximum degree of plastic deformation is observed in the center of a symmetric neck;

b) Plastic deformation degree decreases with distance from this center;

c) In the minimum cross-section of a specimen, the material is subjected to the more severe plastic deformation;
d) Plastic deformation zone boundary in the neck is generally expanding as it approaches to the lateral edges of a flat specimen, i.e. plastic deformation localization (LPD) zone in the neck is characterized by the X-type shape. Besides, the LPD zones in the shape of a cross band are observed in the experiments.

At the macro scale level, non-uniform field of plastic deformation in the neck of a flat specimen can be presented as a smooth field with plastic deformation gradients in continuous medium, which finally should meet the above-specified conditions. The specified plastic deformation distributions hold provide equal displacement to fall the points of upper and lower ends of a flat specimen.

First we define the geometrical parameters of a neck zone, which should qualitatively describe the X-shape of the LPD boundary. Let the distance $d(y)$ in the specimen in length $a$ and width $b$ (Fig. 1) from the origin of coordinates in the center of symmetric neck to the boundary of LPD zone in direction, which is parallel to the tension axis $x$, depends on the coordinate $y$ according to the equation

$$l(y) = a_2 + (a_1 - a_2) \left( \frac{2|y|}{b} \right)^{\gamma+1} \quad (1)$$

Here $a_1$ is the maximum value of this function at the specimen's edge, when $y = \pm b/2$, $a_2$ is the minimum value of this function at $y = 0$. In this representation, the values of $a_1$, $a_2$, $b$ and $\gamma$, obviously, will affect the value and geometrical shape of the LPD zone.

Figure 1 presents an example of the LPD zone, when $a_1$, $a_2$, $b$ are equal to 120, 80 and 120 conventional units, respectively, $\gamma = 1$. The specimen length is described by the value of $a$ equal to 480 conventional units. In case, when $\gamma$ is 1, the LPD zone boundary is described by a parabolic shape. It is seen that dependence (1) qualitatively determines the X-shaped LPD zones.

<figure><img src="image_2.png" /><figcaption>Figure 1. LPD zone in the flat specimen neck. $\gamma = 1$</figcaption></figure>

Figure 2 illustrates the effect of $\gamma$ in the equation (1) on the LPD zone distortion. It is seen that the higher $\gamma$ is, the faster the X-shaped LPD zone transforms into the LPD band across the specimen. At $\gamma \to \infty$, we have a case of LPD cross band in width of $2a_2$.

Figure 2 b shows the effect of the difference $a_1 - a_2$ on the shape of the LPD zone. At $a_2 \to a_1$, the X-shaped LPD zone gradually transforms into the LPD cross band in width of $2a_1$. Obviously, this LPD zone distortion will be also observed at $a_1 \to a_2$. However, in the latter case, the cross band width will be equal to $2a_2$.

Thus, it is possible to change shape and dimensions of the LPD zone in a wide range by varying the parameters $a$, $b$, $a_1$, $a_2$, and $\gamma$ in Eq. (1).

To build a continuous (with no jumps) plastic deformation distribution in the neck, the LPD zone was divided into two areas I and II (Fig.3). The boundary of area I was positioned at a distance of $\alpha l(y)$ from the boundary of the LPD zone, where $0 < \alpha < 1$. In this area, the normal component distribution $\epsilon_x$ of plastic deformation along the tension axis $x$ was specified according to the equation
$$ \epsilon_{1x}(x, y) = \epsilon_0(y) \left[ 1 - \frac{1}{(1-\alpha)^\beta} \left( \frac{|x|}{l(y)} \right)^{\beta+1} \right], \quad (2) $$

and in the area II—according to the equation

$$ \epsilon_{2x}(x, y) = \frac{\epsilon_0(y)}{\alpha^\beta} \left( 1 - \frac{|x|}{l(y)} \right)^{\beta+1}. \quad (3) $$

<figure><img src="image_3.png" /><figcaption>Figure 2. The LPD zone distortion with increase in γ (a) and reduction in the difference of a₁ – a₂ (b) in Eq. (1): β = 1</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3. Half of the LPD zone in the flat specimen neck</figcaption></figure>

In Eqs. (2) and (3)
$$ \epsilon_0(y) = \frac{\Delta L(\beta + 2)}{2(\beta + 1 - \alpha\beta)l(y)}. \qquad (4) $$

Here $\beta$ is the parameter determining the value and change in plastic deformation gradients in the LPD zone, $\Delta L$ is the specimen elongation due to the plastic deformation in the neck.

Equations (2)–(4) provide a smooth field of plastic deformation with no jumps at the boundaries and inside the LPD zone. The maximum gradients of plastic deformation are observed at the boundary between the areas I and II.

Figure 4a illustrates the plastic deformation distribution in the specimen neck, which is obtained under the following values of parameters in equations (1)–(4): $\gamma = \beta = 1$, $\alpha = 0.4$, $a/b = 4.5$, $a_1/b = 1.5$, $a_2/b = 1$, $\Delta L/b = 0.125$. As seen from the presented results, the proposed description of the plastic deformation field in the neck qualitatively satisfies the distributions observed in the experiments [2–4, 15].

<figure><img src="image_5.png" /><figcaption>Figure 4. Examples of plastic deformation distributions in the flat specimen neck. The calculated parameters are specified in the text</figcaption></figure>

When varying the geometrical parameters of the LPD zone, one can obtain different variants of distributions. We will give a few examples varying only some of the calculated parameters, specified above. Figure 4b presents the distribution for $\beta=4$ under the same conditions. Comparison with Fig. 4a shows that increase in $\beta$ leads to the increase in plastic deformation gradients in front of LPD.

The condition $a_1=a_2$, as it was specified above, provides the variant of the LPD cross band. Figure 5a presents the case, when $a_1/b = a_2/b = 1$ and $\beta = 1$. In this band, the plastic deformation profile $\epsilon_x$ does not change along the y axis. Figure 5b illustrates the LPD cross band at $\beta = 4$. Comparison of Figs. 4 and 5 shows that $\beta$ determines the maximum degree value of plastic deformation $\epsilon_x^{\max}$ irrespective of $a_1-a_2$. At the same time, increase in $\beta$ results in some decrease in $\epsilon_x^{\max}$.

According to the continuum theory of defects [9–11], certain gradients and concentrations of stresses correspond to the specified non-uniform field of plastic deformation in the continuous medium. When setting the concrete plastic deformation distribution for plane stress state with Esq. (1)–(4), one can calculate numerically or analytically the non-uniform field of stresses in the volume of a solid body using the well-known methods of finite elements, boundary elements, relaxation elements etc. [12–14].
<figure><img src="image_6.png" /><figcaption>Figure 5. Examples of LPD cross bands: β= 1 (a), 4 (b).
The calculated parameters are specified in the text</figcaption></figure>

It should be emphasized that Eqs. (2)–(4) determine only non-uniform field of plastic deformation. Generally, a uniform field of plastic deformation should be added to these equations over the entire specimen area, characterized by the plastic deformation value outside the neck zone. It is known that uniform field of plastic deformation (without gradients) does not affect the field of internal stresses.

**3. The neck distortion in a flat tensile specimen.** Macro non-uniform distribution of plastic deformation with gradients determines a significant distortion of a flat specimen. As a result, configuration occurs in local zone of a specimen, which is determined in scientific literature as “neck”. Obviously, shape and dimensions of the neck depend on qualitative and quantitative distribution characteristics in the LPD zone.

To describe the neck shape, it is sufficient to calculate the displacement of points at the edges of a flat specimen. This requires knowing all the plastic deformation tensor components in the specimen, namely $\varepsilon_x$, $\varepsilon_y$ and $\varepsilon_{xy}$. It will be assumed that for a symmetric neck, $\varepsilon_{xy}$ is 0. The values for the plastic deformation normal component $\varepsilon_x$, directed along the tension axis, are set by Eqs. (2)–(6). To calculate $\varepsilon_y$ component, the additional assumptions are required. It is usually assumed that there is no change in the material volume under plastic deformation. For simplicity, let us assume that under tensile deformation $\varepsilon_x = \varepsilon$, the compressive deformation takes place along the x axis, that is proportional to the value of $\varepsilon$, identical for width and thickness of the specimen, i.e. $\varepsilon_y = -v\varepsilon$ and $\varepsilon_z = -v\varepsilon$, where v is the proportionality coefficient, or the Poisson's ratio, as it is determined for elastic deformation. For the elementary volume $\Delta v$ is $\Delta x\Delta y\Delta z$, so the equality can be written as

$$\Delta x \Delta y \Delta z = \Delta x(1+\varepsilon)\Delta y(1-v\varepsilon)\Delta z(1-v\varepsilon), \text{ or } 1 = (1+\varepsilon)(1-v\varepsilon)^2.$$

Hence we find the value of the Poisson's ratio $v$:

$$v = [1 - 1/(1+\varepsilon)^{1/2}]/\varepsilon. \qquad (5)$$

As seen from the equation, $v$ depends on the deformation degree. At small deformation degrees, when $\varepsilon \to 0$, $v1 \to 0.5$. One can verify it, expanding the expression $1/(1+\varepsilon)^{1/2}$ in a series by the value of $\varepsilon$ [16], being limited by the first two members $1/(1+\varepsilon)^{1/2} = 1 - \varepsilon/2$. As $\varepsilon$ increases, $v$ decreases. In examples that we have seen in Figs. 4 and 5, the maximum degree of plastic deformation is equal to $\varepsilon_{max} = 0.213$. When substituting this value into Eq. (5), we shall obtain $v = 0.43$. Further, for simplicity of calculations, we will consider $v$ as a constant and equal to 0.4.
The change of the specimen width occurs due to the deformation component $\varepsilon_y$. The elementary volume contribution to the displacement of point $(x, b/2)$ on the side face of the specimen is equal to

$$du_y^I(x) = -\nu\varepsilon_1(x, y)dy, \qquad (6)$$

if it is found in area I (see Fig. 3). If the elementary volume is found in the area II, it provides the displacement of point $(x, b/2)$ by an elementary value

$$du_y^{II}(x) = -\nu\varepsilon_2(x, y)dy. \qquad (7)$$

Total displacement of points $(x, b/2)$ is determined by the integration of elementary displacements (6) and (7) along the $y$ variable in areas I and II, respectively. We shall write this as two integrals, summing displacements separately in areas I and II

$$u_y(x) = du_y^I(x) + du_y^{II}(x) = \nu \int_{y_1^I}^{y_2^I} \varepsilon_1(x, y)dy + \nu \int_{y_1^{II}}^{y_2^{II}} \varepsilon_2(x, y)dy$$

When substituting the expressions (1) – (4) into Eq. (6) if $\nu$ = const, for the first integral we shall obtain

$$u_y^I(x) = \frac{-\nu\Delta L(\beta + 2)}{2[\beta + 1 - \alpha\beta]} \left[ \int_{y_1^I}^{y_2^I} \frac{1}{l(y)} \left[ 1 - \frac{1}{(1-\alpha)^\beta} \left( \frac{|x|}{l(y)} \right)^{\beta+1} \right] dy \right]. \qquad (8)$$

Similar substitution of the expressions (1)–(4) into the second integral results in the equation

$$u_y^{II}(x) = \frac{-\nu\Delta L(\beta + 2)}{2[\beta + 1 - \alpha\beta]\alpha^\beta} \left[ \int_{y_1^{II}}^{y_2^{II}} \frac{1}{l(y)} \left[ 1 - \frac{|x|}{l(y)} \right]^{\beta+1} dy \right]. \qquad (9)$$

Let us consider a concrete example, when $a_1(1-\alpha) < a_2$, presented in Fig. 6a, where four regions with different limits of integration are distinguished. In the region 1, only Eq.(8) is integrated, with integration limits of $y_1^I = 0$, $y_2^I = b/2$. The boundary between the areas I and II determines the lower limit of integration $y_1^I$ for Eq. (8) and upper limit of integration $y_2^I$ for Eq. (9) for the region 2. From $|x| = l(y)(1-\alpha)$, we find $y_1^I = y_2^I = \frac{b}{2} \cdot \gamma^{1+1} \sqrt{\frac{|x| - a_2(1-\alpha)}{(a_1 - a_2)(1-\alpha)}}$. In this case, $y_2^I = b/2$ and $y_1^I = 0$. In the region 3, only Eq. (9) is integrated, with integration limits of $y_2^I = b/2$ and $y_1^I = 0$. Finally, in the region 4, the LPD zone boundary determines the lower limit of integration for Eq.(9), equal to
$y_1^{II} = \frac{b}{2} \cdot \gamma + 1 \sqrt{\frac{|x| - a_2}{a_1 - a_2}}$, according to the condition $|x| = l(y)$. The upper limit of integration for Eq. (9) thereby, is equal to $y_2^{II} = b/2$.

Figure 6b presents another variant, when $a_1(1-\alpha) > a_2$. In this case, four regions with different limits of integration are also distinguished. As seen from the comparison of Figs. 6a and 6b, in the regions 1, 2 and 4, the integration limits did not change. The difference of integration limits is observed only

in the region 3. In the last example, Eq (9) is integrated, with integration limits of $y_1^{II} = \frac{b}{2} \cdot \gamma + 1 \sqrt{\frac{|x| - a_2}{a_1 - a_2}}$ and $y_2^{II} = \frac{b}{2} \cdot \gamma + 1 \sqrt{\frac{|x| - a_2(1-\alpha)}{(a_1 - a_2)(1-\alpha)}}$, and also Eq. (8) with integration limits of $y_1^I = \frac{b}{2} \cdot \gamma + 1 \sqrt{\frac{|x| - a_2}{a_1 - a_2}}$ and $y_2^I = \frac{b}{2}$.

In the region 3. In the last example, Eq (9) is integrated, with integration limits of $y_1^{II} = \frac{b}{2} \cdot \gamma + 1 \sqrt{\frac{|x| - a_2}{a_1 - a_2}}$ and $y_2^{II} = \frac{b}{2} \cdot \gamma + 1 \sqrt{\frac{|x| - a_2(1-\alpha)}{(a_1 - a_2)(1-\alpha)}}$, and also Eq. (8) with integration limits of $y_1^I = \frac{b}{2} \cdot \gamma + 1 \sqrt{\frac{|x| - a_2}{a_1 - a_2}}$ and $y_2^I = \frac{b}{2}$.

<figure><img src="image_7.png" /><figcaption>Figure 6. The scheme of integration regions in the neck zone: a is the first variant, b is the second variant.</figcaption></figure>

Now let us define the displacements of points of the specimen's side face along the tension axis. For this purpose, one should integrate the elementary displacement contributions of points at the edges of the specimen along the tension axis.

After integration we will come to the following result

$$u_x = \varepsilon * \begin{cases} (1-\alpha)|x| \left[ 1 - \frac{1}{\beta+2} \left( \frac{|x|}{a_1(1-\alpha)} \right)^{\beta+1} \right] + \alpha|x|, & \text{if } 0 \le x \le a_1(1-\alpha); \\ \frac{\alpha^2 a_1}{\beta+2} \left[ 1 - \left( \frac{a_1-|x|}{\alpha a_1} \right)^{\beta+2} \right] + \alpha(1-\alpha)a_1 + a_1(1-\alpha)^2 \frac{\beta+1}{\beta+2}, & \text{if } a_1(1-\alpha) \le x \le a_1; \\ \frac{a_1(\beta+1-\alpha\beta)}{\beta+2}, & \text{if } x \ge a_1. \end{cases} \quad (10)$$

$$\text{Here } \varepsilon* = \varepsilon_0(b/2) = \frac{\Delta L(\beta + 2)}{2 \cdot a_1 \cdot (\beta + 1 - \alpha\beta)}.$$
It is easily to verify that in the interval $x > a_1$, all the points on the side face of the specimen are displaced by the same value of $u_x = \Delta L/2$.

The displacement components $u_x$ and $u_y$, set by Eqs. (9)–(10), completely determine the distortion of the flat specimen. Figure 7 presents the calculation data for the configuration change of the neck as the specimen length $\Delta L$ increases due to the non-uniform plastic deformation in the LPD zone. It is seen that plastic deformation accumulation in the LPD zone leads to the minimum cross-section reduction and increase in the specimen length.

<figure><img src="image_8.png" /><figcaption>Figure 7. The neck distortion with increase in specimen length: $\Delta L/b = 0.0125, 0.25, 0.5, 1.0, 1.5$; $\beta = \gamma = 1$</figcaption></figure>

Figure 8 shows the effect of $\beta$ on the specimen shape. Increase in $\beta$ results in more rapid decrease in the specimen width. In this case, the minimum cross-section changes insignificantly.

<figure><img src="image_9.png" /><figcaption>Fig. 8. The neck distortion with increase in $\beta$ in Eqs. (8)-(9): $\beta=1, 4, 30, 150$; $\Delta L/b = 1$</figcaption></figure>

**Summary.** This work presents an analytical description of the non-uniform field distribution of plastic deformation in a flat specimen, which determines distortion of the specimen in the necking zone. The proposed method enables to be simulated the real non-uniform distributions of plastic deformation and neck distortion according to experimental measurements data. Analytical expressions are suitable for calculation of gradients and concentration of stress in the neck of a flat specimen made of real material, using well-known analytical and numerical methods: finite element methods, boundary element methods, relaxation element methods etc.

This work presents a theoretical description of the non-uniform field distribution of plastic deformation in a flat specimen, which determines distortion of the specimen in the necking zone. Smooth fields with plastic deformation gradients and change in the geometric shape of a flat tensile specimen can be described by the variation of geometrical parameters in the obtained equations. The proposed method enables to be simulated the real non-uniform distributions of plastic deformation and neck distortion according to experimental measurements data. The problem is relevant due to the
solid mechanics associated with transition from the "load-extension" experimental curves to the "stress-strain" loading diagrams for the material in the minimum cross section zone of the specimen, where the plastic deformation develops extremely non-uniformly and at maximum speed. It is the zone the material is subjected to all stages of deformation hardening followed by fracture.

Analytical expressions are suitable for calculation of gradients and concentration of stress in the neck of a flat specimen made of real material, using well-known analytical and numerical methods: finite element methods, boundary element methods, relaxation element methods etc.

Finally, the results can be used in simulation of plastic deformation and ductile fracture of different materials.

**Acknowledgment**

This work was supported by grant No. 10-08-01182-a from the Russian Foundation for Basic Research.

**References:**

[1] M.N. Gusev, I.S. Osipov, The peculiarities of deformation-plastic behavior of metals and alloys irradiated with neutrons in WWR-K and BN-350 reactors, Vestnik Udmurdtskogo Universiteta. Physics. 4 (2007) 104-112.

[2] L.S. Derevyagina, V.E. Panin, A.I. Gordienko, Selforganization of plastic shears in localized deformation microbands in the neck of high-strength polycrystals and its role in material fracture under uniaxial tension, Physical mezomechanics 4 (2007) 59-71.

[3] V. I. Trefilov, V. F. Moiseev et all, Deformation hardening and failure of polycrystal metals, Kiev, Naukova Dumka (1989) 256.

[4] T.M. Poletika, A.P. Pshenichnikov, Nonlinear strain macrolocalization behavior in H.C.P zirconium alloys, ZTF V. 79, 3 (2009) 54-58.

[5] W.H. Peters, W.F. Ranson, Digital imaging technique in experimental stress analysis, Optical Engineering 21 (1982) 427.

[6] M.A. Sutton, W.J. Wolters et all, Determination of displacements using an improved digital image correlation method, Image Vision Computing V. 1, 3 (1982) 133.

[7] V.E. Panin, V.S. Pleshanov, V.V. Kibitkin, S.V. Sapozhnikov, Analysis of displacement vector fields and fatigue failure diagnostics of an aluminum alloy at a mesoscale, Defektoskopiya 2 (1998) 80-87.

[8] S.V. Panin, V.I.Syryamkin, P.S. Lyubutin, Rigid body deformation estimation by surface images, Avtometriya V. 41, 2 (2005) 44-58.

[9] J. D. Eshelby, The continuum theory of lattice defects, Solid State Physics, V. 3, N.Y.: Acad. Press (1956) 79-144.

[10] de Wit R, Linear theory of static disclinations, In: Fundamental aspects of dislocation, ed. by J.A. Simons, R. de Wit, R. Bullough, Nat. Bur. Stand. (US) Spec. Publ. 317, V. I (1970) 651-673.

[11] V.A. Likhachev, A.E. Volkov, V.E. Shudegov, The continuum Theory of Defects., Leningrad: University (1986) 232.

[12] R. Gallager, The finite element method. Moscow: Mir, 1984.

[13] S.L. Crouch, A.M. Starfield, Boundary element methods in solid mechanics. London: George Allen & Unwin (1983).

[14] Ye.Ye. Deryugin, G. Lasko, S. Schmauder, Relaxation Element Method in Mechanics of Deformed Solid. In: Wilhelm U. Oster. Computational Materials, NY: Nova Science Publishers, Inc. (2009) 479-545.

[15] A.V. Panin, A.A. Son, Yu.F. Ivanov, V.I. Kopylov, Features of localization and stage character of plastic deformation of ultrafine-grained armco-iron with the fragmented band substructure, Fizicheskaya Mezomekhanika V. 7, No 3 (2004) 5-16.

[16] M. Ya. Vygotskii, Handbook of Higher Mathematics. Moscow: AST Astrel (2005) 679.