Modelling and simulation of the localized plastic deformation by relaxation element method

DOI 10.1007/s00466-006-0141-y ORIGINAL PAPER Modelling and simulation of the localized plastic deformation by relaxation element method Ye. Ye. Deryugin G. V. Lasko S. Schmauder Received: 20 February 2006 / Accepted: 30 October 2006 Springer-Verlag 2006 Abstract In the present work a new and a little known method of modelling and simulation of plastic strain localization in a loaded solid - relaxation element method (REM) is presented. The fundamental property of solid: Plastic deformation of solid under loading is accompanied by stress relaxation in the local volume of it lies in the basis of the method. The theoretical grounding of the method from the point of view of constitutive equations of theory of elasticity and continuum theory of defects is conducted. For the plane-stress state the technique of construction of local sites of plastic deformation by the method on the mesoscopic scale level is demonstrated. The technique of calculation of stress state of the plane with the sites of different shape and gradients of plastic deformation is described. The new type of relaxation elements, different from ones used in the works [1 6] has been applied. This allowed to extend the method on the new class of objects. The examples and the results of simulation of the processes of plastic strain localization, accompanied by the effect of Lüder s band propagation and Portevin Le Chatelier Effect are considered. The examples show, that REM is Ye. Ye. Deryugin G. V. Lasko (B) Institute of Strength Physics and Material Science, Siberian Branch of Russian Academy of Science, Pr. Academicheskii 2/1, Tomsk 634021, Russia e-mail: galina_lasko@hotmail.com Ye. Ye. Deryugin e-mail: dee@ispms.tsc.ru G. V. Lasko S. Schmauder Institut für Materialprüfung, Werkstoffkunde und Festigkeitslehre (IMWF), Pfaffenwaldring 32, Stuttgart, 70569, Germany a comfortable tool when simulating the effect of plastic strain localization on meso- and macro- level. 1 Introduction The development of plastic deformation in a solid is known to proceed inhomogeneously in space and irregular in time. This property results in the process of plastic strain localization, developing during the loading of material. The evolution of the inhomogeneous distribution of plastic deformation develops under the action of the various stress concentrators, caused by the inhomogenuity of the initial structure of the material and by change in the stress state of the loaded system under the given boundary conditions. In the course of plastic deformation of the local volumes, continuous changing of the field of internal stresses in the volume of solid occurs. The technological and strength properties of the material depend strongly on the character of plastic strain localization and material degradation. In this connection there is an actual problem of the description of the stress strain state of a medium with the sites of plastic deformation and the simulation of the process of localization of deformation in a material under loading. The uncertainty of the situation lies in the fact that it is not possible to formulate the universal physical law of the connection between the plastic deformation and the stresses in the solid because of the relaxation nature of the former. In the present work the original method of modelling and simulation of localized plastic deformation in a loaded solid is represented. The fundamental property of a solid: plastic deformation of solid under loading is

accompanied by stress relaxation in the local volumes of solid lies in the basis of the method. In the papers [1 6] the separated solutions, obtained within the framework of REM have been considered. In the paper the explanation of the new method is represented in details. Besides that, a number of new results of simulations with application of new analytical expressions are presented. The examples and the results of modeling of the process of plastic deformation, accompanied by effects of Luder s band propagation and Portevin-Le-Chatelier effect are considered. 2 Some consequencies from the theory of elasticity and continuum theory of defects 2.1 Experimental validation of the relaxation nature of plastic deformation Classical experiments on stress relaxation in a solid are realized in the mode of uniaxial loading, ensuring the prescribed increase in the length l of the specimen. The experiments show, that the mode of the constant length of the specimen after loading above the yield stress is realized under decreasing external applied stress (Fig. 1). The physical reason of stress relaxation is the plastic deformation of the specimen. Really, the total deformation of the specimen at the initial instant of time t = 0, is caused only by elastic deformation and is equal to ε = σ 0 /E = const, where σ 0 is an external load, and E is the Young s modulus. In the course of time the same value is defined by the contribution of elastic and plastic deformation: ε = ε e (t) + ε p (t) = σ 0 /E. (1) Fig. 2 An interaction between elastic and plastic deformation in the experiments on stress relaxation The contribution of elastic deformation at an arbitrary instant of time is defined by Hooke s law: ε e = σ(t)/e. (2) The linear dependence of plastic deformation on the external stress follows from Eqs. (1) and (2): ε p (t) = (σ 0 σ(t))/e = σ (t)/e, (3) where σ (t) is the value of stress drop. Figure 2 illustrates the interaction between elastic and plastic deformation in the experiments on relaxation. The point A at the straight line defines the values of elastic ε e (t) and plastic ε p (t) deformations at the moment t, matching the stress relaxation in the value σ (t). Plastic deformation increases not under increasing, but under decreasing external stress, i.e. it is proportional not to actual, but to disappeared stress, the stress which existed but disappeared. Correspondingly the dissipation of elastic energy of the specimen took place in the value: U = 0.5 σ ε p ls/e = 0.5 (ε p ) 2 V, where l is the length, S is the cross-section, and V is the volume of the specimen. In the mode of constant external load σ = const, i.e. in the experiments on creep a continuous increase in the length of the specimen takes place. In the present case the dissipation of elastic energy is compensated by the work of external stress. A = σε p ls = σε p V. Fig. 1 Temporal dependence of the external stress in the experiments on relaxation In the mode of continuous loading due to stress relaxation in the local volume of material under plastic

deformation the deviation of the curve from the straight line of elastic loading takes place. This examples proves the fact of connection between plastic deformation and stress relaxation. However, a one-dimensional approximation doesnot give the idea about the stress strain state and about the distribution of plastic deformation in the local volume of solid, since the development of plastic deformation always occurs inhomogeneously in space and irregular in time. In this connection there is an actual problem of description of the stress strain state of the medium with the site of plastic deformation. Under the site we will understand the stationary field of plastic deformation, localized in a separate volume of the solid. The correct statement of the problem requires the accounting of the connection of plastic deformation, localized in the bounded region with stress relaxation in the given volume. Apparently, in the common case this connection should be expressed in tensor form. 2.2 Constitutive equations and postulates of continuum theory of defects The dependence of plastic deformation on the stresses in the local volume of solid is defined by the statements and postulates of continuum theory of defects [7 11]. The first postulate reads, that the tensor of total deformation ε ij is equal to the sum of the tensors of elastic εij e and ε p ij plastic deformation: ε ij = ε e ij + εp ij. For the total deformation the compatibility (continuity) condition for the given boundary conditions of loading is always obeyed. For the two-dimensional approximation it is written in the form of the following equation: 2 ε yy x 2 + 2 ε xx y 2 = 2 2 ε xy x y. The presence of plastic deformation breaks the compatibility condition for the elastic deformation ε e ij. The condition (2) means, that the incompatibility of the tensor of elastic deformation is fully compensated by the incompatibility of plastic deformation ε p ij. The stresses are connected with the elastic deformation by the generalized Hooke s law: σ ij = C ijkl ε e kl, where C ijkl are the coefficients of elasticity of the material. The system of equations σ ij,j + f i = 0 (4) defines the condition of force equilibrium in the volume of solid. Here f i are the volume forces. The second postulate reads, that the presence of the field of plastic deformation within the local volume is adequate to the operation of the fictitious volume forces, which can be represented in the following manner f i = C ijkl ε p mn,j. Thus, Eq. (4) are represented in the form σ ij,j + f i = C ijkl ε e kl,j C ijklε p mn,j = 0, or C ijkl ε e kl,j = C ijmnε p mn,j. (5) There is a special technique [8] to solve Eq. (5). The above equations are enough to unambiguously define the connection between incompatible plastic deformation and stresses within a volume of solid. The Relaxation Element Method [1 6] simplifies the solution of the problem, connected with the theoretical calculation of the stress fields in a continuous medium with the sites of plastic deformation. This will be demonstrated by the examples given below. 3 Stress strain state of a medium with sites of plastic deformation 3.1 Preliminary notes To make clear the technique of REM, let us consider the general case of the plane under the operation of an external tensile stress σ = const. Let us suppose that in a local volume, surrounded by the close contour S 0, as a result of plastic deformation ε p ij the σ ij stress is relaxed by σ ij value (Fig. 3). The prescription of the relaxation tensor σ ij unambiguously defines the boundary conditions in displacement on the contour S 0, i.e. the new shape of contour S (Fig. 3a). Let us delete the material inside the contour S, keeping its shape unchangeable (Fig. 3b). The material beyond the S contour is deformed elastically. Therefore the theoretical calculation of stresses beyond the contour doesnot cause principle difficulties, since under the uniaxial loading conditions the displacements of the points on the contour are known. The stress field

Fig. 3 To the analysis of the stress strain state of the plane with the site of plastic deformation Fig. 4 The representation of the total solution (a) ofthe task on the stress-strain state ofaplanewiththesiteof plastic deformation in the form of superposition of two simpler solutions (b) and(c) we are looking for is defined by the solution of a mixed problem of theory of elasticity, where the boundary conditions in displacements are prescribed on the contour S 0, and at the external surface these displacements are defined by applied stress σ = const. Let us consider now the stress strain state of the deleted part of a specimen with accounting for the displacements, caused by plastic deformation. Let us prescribe at the external S 0 boundary of the selected volume the displacements of the points till the exact coincidence with the contour S (Fig. 3c). The tensor of total deformation ε ij, ensuring the prescribed displacements is easy to define, suppose that at that time the material is deformed only elastically. Plastic deformation ε p ij, according to Eq. (5), unambiguously defines the residual field of elastic deformation εij e = εij e εp ij, and thus results in a stress field σij = C ijkl εkl e C ijklε p kl = σ ij σ ij inside the contour. Hence the tensor of stress S relaxation σ ij matches a definite tensor of plastic deformation ε p ij and vice versa. In such a manner, there is a definite connection between the localized plastic deformation and stresses in the volume of solid (via relaxation tensor σ ij ). Let s insert the selected volume in it s previous place. The discontinuity of material will be restored, since the boundary of the cut part exactly coincides with the contour of the cut-out. We obtain the full pattern of stress strain state of material with the site of plastic deformation. Thus, under the known field of plastic deformation (or at the known value of relaxation tensor σ ij ) the stress strain state of a solid with sites of plastic deformation is unambiguously defined by the methods of linear theory of elasticity and continual theory of defects. The Relaxation Element Method deals only with localized plastic deformations connected with stress relaxation. 3.2 The site of plastic deformation of round shape Let us consider a simple example. Let under the external tensile stress σ = const in an elastic plane within the local region in the form of a circle the stress relaxation of value σ (Fig. 4) takes place. Given conditions are sufficient enough for the definition of the corresponding stress state of the whole system. Inside the round region the stress state is known by definition, it is equal to σ σ. Having substracted the trivial homogeneous stress field from the total solution, we obtain the stress field in the plane, on the round contour of which there is no normal and tangential stresses. A similar problem is known as a Kirsch problem [12] for the plane with a hole. In the system of coordinates at the center of the circle and the 0 y-axis along the tensile stresses, the components of stress field beyond the round contour

Fig. 5 Spatial distribution of σ y stress in the plane with the site of plastic deformation are equal: σ y = σ a 4r 2 σ x = σ a 4r 2 ( 1 + 3a2 + 10y 2 r 2 + 24a2 y 4 ) r 6 + σ ; ( 1 3a2 + 18y 2 24a2 y 4 r 6 r 2 ( 8y2 3a 2 + 2y 2) r 4 ( + 8y2 3a 2 + 2y 2) r 4 ) ; (6) ( σ ayx σ xy = 4r 4 3 2(3a2 + 4y 2 ) r 2 + 12a2 y 2 ) r 4. However, in the given case the analogous stress state of the plane is caused by plastic deformation of the material in the circle, and the region without stresses should not be considered as a region where there is no material. In such a case it should be noted that under the unloading the material will deform elastically as in the whole volume of material (because of the definition plastic deformation is irreversible). Unloading means the relief of the homogeneous stress field σ. It is not difficult to prove that at that time beyond the round region the non-homogeneous stress field will exist (7) without σ stress. Within the site of plastic deformation the material will be in a compression state σ. Shown in Fig. 5 is the spatial distribution of the σ y component of the stress field in the plane with the site of plastic deformation of the considered type [see Eq. (6)]. It is seen, that an inhomogeneous stress field exists beyond the site. Stress relaxation σ in the value σ creates around itself a zone of elevated stress concentration. The maximum value of stress σ y (3 σ ) exceeds the value of the external applied stress in 2 σ value. We have proved that the prescription of the value of stress relaxation σ within the zone of plastic deformation allows to define the stress strain state beyond this zone with the method of theory of elasticity. As a result the stress field in the whole volume of solid with the site of plastic deformation becomes known. Let us find the field of plastic deformation, which corresponds to the relaxation tensor σ. This field ensures the displacements of the points of the circle in Fig. 4c. The displacement of an arbitrary point on the circle (x 0, y 0 ), according to the solution of the Kirsch problem, in the system of coordinates with the beginning of the center of the circle and 0 y-axis along the tensile axis are equal: u y = 3y 0 σ/ E, u x = x 0 σ/ E. The same displacements are ensured by a homogeneous field of plastic deformation. ε p y = 3 σ/e, ε p x = σ/e, ε p xy = 0. (7) A linear dependence between the plastic deformation and the relaxation tensor is in agreement with the idea that under full stress relaxation an elastic deformation is fully transferred into plastic. The considered site of plastic deformation is characterized by a definite shape (round) and the corresponding field of internal stresses (6) beyond the circle without σ and stress σ inside the circle. This field depends exclusively on the value of stress relaxation σ. Hence, the local site of plastic deformation can be considered as a defect of mesoscopic scale in the continuum medium. Its presence doesnot change the elastic properties of the medium, i.e. doesnot influence the solution of the boundary value problems of linear theory of elasticity. That is why for the internal stress fields of similar defects the superpositional principle is valid. In this connection a given defect can be used as an element for the construction of the various fields of localized plastic deformation. Since it is connected with stress relaxation, it is a relaxation element by definition. Relaxation element method allows one to represent the various variants of localized plastic deformation and to simulate the development of localization of material in materials with the help of different relaxation elements at the given boundary conditions of loading and inhomogeneity of the structure of the material. With the help of REM it is not difficult to construct the smooth fields of localized plastic deformation with the gradient of any order. Shown in Fig. 6a are examples of the distributions of plastic deformation in the round

Fig. 6 The examples of the plastic strain distribution in the round zone (a) and corresponding field of stresses (b) region, defined by the equations. ε p y = 3 σ E ( 1 h a) [ ( ) ] r β+1 1 + h a h a, r2 (a h) 2 ( ) (8) h a r β+1, (a h) 2 r 2 a 2. a h At the distance h from the circle the maximum gradient of plastic deformation is observed grad ε p = 3 σ (β + 1)(a/h 1) β / E a. The value of the gradient is seen to be defined by the β-parameter. Equation (8) allows to regulate the distribution of plastic deformation by the variation of the width of h and the value of the parameter β. For the given distributions (8) the corresponding stress fields are obtained (Fig. 6b). By relaxation element method it was obtained a lot of examples with the sites of localized plastic deformation. Some of them are given in monograph [13]. By the relaxation element method a number of original solutions for similar problems have been obtained [14]. Let us consider one of the examples. The band of localized plastic deformation can be composed from the relaxation elements (RE). Let us construct it, for example from RE of considered type, characterized by homogeneous plastic deformation (7). Let us define for each RE an elementary value of stress drop dσ = σ dx/2a, where 2a is the diameter of the circle, and place them evenly at the distance of dx from each other along the 0 x-axis (Fig. 7). At such construction at the arbitrary point (x, y) the stress drop will be equal to ndσ, where n is a number of RE, inside the circle of which the given point is fit. The stress drop in dσ, according to (12), is ensured by an elementary field of plastic deformation 3.3 The band of localized plastic deformation One of the most interesting tasks, to be practically and theoretically in great demand in mechanics of a deformed solid, is the problem about the stress strain state of the plane with a band of localized plastic deformation. Fig. 7 The scheme of construction of LPD bands from the RE of the round shape

with the components dε p y = 3dσ/E, dε p x = dσ/e, dε p xy = 0. (9) Integrating this elementary fields (9), we will obtain the following expressions for the component εy p of plastic deformation in a half-infinite band ε p y = ε p y max { (x + a 2 y 2 )/2a, x 2 + y 2 a 2 ; 1 y 2 /a 2, x 2 + y 2 > a 2. (10) Here ε p y max = 3 σ/e. The spatial distribution of the ε p ycomponent of plastic deformation in the band is given in Fig. 8. The maximum value of plastic deformation ε p y = ε p y max is observed along the band axis. As the boundary of plastic region is attained, the degree of plastic deformation decreases up to zero. In analogous manner, by integrating of the elementary fields according to Eq. (6), where σ is substituted by the equation dσ = σ dx/ 2a, we obtain the stress field for the present half-infinite band of localized deformation. The spatial distribution of the σ y component of the stress field is represented in Fig. 9. The field of stresses is seen to be perturbated significantly only at the end of the band. Ahead of the band there exists the stress concentration. At the same time in the band itself the stress is lower than average level of external stress. This example shows, that the stress concentration and high gradients of stresses at the end Fig. 10 Spatial distribution of the shear stress from the band of deformation, formed at an angle of 45 to the tensile axis of the band play the role of the source of the driving force under the formation of PLC band. In the band the stresses are always lower than the average level of applied stresses. It is well illustrated in Fig. 10, in which an example of the finite band of plastic deformation of pure shear is represented, formed at an angle of 45 to the tensile axis. The stress field is perturbated at both ends of the band. Attention should be paid to the following peculiarity of LPD. The zone of stress concentration at the end of the band is always combined with the zone of anti-concentrator inside the band (Fig. 10). 4 Simulation of the process of plastic strain localization by RE method 4.1 Peculiarities of simulation by the relaxation element method Fig. 8 Half-infinite band of localized plastic deformation Fig. 9 Spatial distribution of stress σ y from the half-infinite band of localized deformation Application of relaxation elements as defects, characterizing the interaction between plastic deformation and stresses allows to simulate the process of plastic strain localization and to obtain the dependency of flow stress on the sequence of separate structural elements involvement into plastic deformation. Developed on the basis of REM models operates on the basis of cellular automata [15]. The calculational field is divided into a great number of cells, playing the role of elements of structure. Each element of the simulated medium posseses the ability to switch the state by a discrete jump of plastic deformation, setting by a definite relaxation element. This procedure means that the element of structure can periodically increase its degree of plastic deformation and as a stress concentrator influences the change σ of the stress field in the whole volume of solid. The involvement of structural elements into plastic deformation is realized by definite rules of transition (for

example, at the instance of time of achieving of critical value of shear stress). Interaction of the stress fields from the various structural elements, which have undergone plastic deformation, proceeds automatically according to the procedure, described above. When interpretating the results of simulation one should take into account, that the stress state of deformed system is controlled only by incompatible plastic deformation, connected with stresses in the volume of solid. Relaxation elements in the present case place the role of defects, being responsible for the field of plastic deformation. However, it is experimentally founded that the absolute majority of deformational defects at the stage of developed plastic deformation disappear as a result of annihilation, attraction by the interfaces and exposure at the free surface of solid. The stresses connected with the given defects disappear also. What is remains is the corresponding field of plastic deformation, not connected with stresses, which satisfies the compatibility condition. In such a manner, one can not neglected the compatible plastic deformation. The problem is in fact that compatible plastic deformations can not be represented by definite analytical functions of the coordinates. The same formchanging of the solid can be realized by a number of variants of mass transfer, not influencing the stress field. That means that by REM one can exactly calculate flow stress changing in time. For the definition of the total plastic deformation additional conditions are necessary. They can be formulated when calculating the specific structures, taking known qualitative data from experiment. 4.2 Simulation of Lüder s band propagation Let us represent the result of simulation of the effect of plastic strain localization under the propagation of the Luder s band. (Fig. 11). In the given case the calculational field is consisted from 50 100 points the centers crystallites. The onset of plastic deformation was initiated by relaxation elements of a round shape (see. Sect. 3.2), which was placed at the edge of the calculated field. It corresponds to the plastic deformation of a separate grain, exposing on the edge of polycrystal. Further, a minimum external stress was calculated step by step, at which in any place of a polycrystal (in any of its calculational cells) a maximum shear stress achieved the critical value τ cr = 50 MPa, according to the Tresca criterion. At each step a new Relaxation element of the considered type was put, imitating plastic deformation of the new crystallite. Simulation showed, that under a definite selection of relaxation tensor σ y the mechanism of Luder s band propagation is observed. Along the front of a Luder s band the spontaneous transfer of the process of plastic deformation from grain to grain takes place. The propagation of the process of deformation localization along the front from one edge of the specimen to another result in the movement of the Luder s band front along the specimen in one grain diameter. The presented model predicts the jump-like dependence of external stress on the number of structural element (polycrystal grains) involvements into plastic deformation. Its value (external stress) defines the onset of plastic flow in the grain, where the critical value of shear stress (according to Tresca criterion) is achieved. Since the plastic deformation of every new grain change the field of internal stresses, than the value of external stress oscillates within the corresponding limits. Therefore, the diagram of flow stress changes in jumps with increasing in number of grains involved into pastic deformation. 4.3 Simulation of the Portevin-Le Chatelier effect (PLC) When selecting the corresponding parameters of the model with accounting for strain-hardening the Fig. 11 The stages of the propagation of the Luder s band front (a, b)andthe dependence of the external stress on the number of grains involved into plastic deformation (c)

mechanism of propagation of localized deformation, similar to the mechanism of Luder s band propagation is observed, being characteristic for the alloy with the effect of Portevin-Le Chatelier. In the literature this mechanism is considered as a formation of the bands of localized deformation of type A[16]. Strain-hardening was simulated by the increase of the critical shear stress in the crystallite, undergoing next jump of plastic deformation. The distinction of the mechanism of the formation of the bands of type A from the mechanism of Luder s band propagation in low-carbon steel is done in the following way: in the latter case after the filling out of the whole working part of the specimen by the LPD-bands, the repeated formation of the band is excluded. In alloys with PLCeffect of type A the repeated appearing of the bands, formed by the continuous propagation of the front of deformation localization is observed. The expanding of the band requires an increase in the externally applied stress (Fig. 12e). At that at a definite instance of time the conditions for the initiation of a new band are created, the formation of which at the beginning takes place under decreasing external stress. The following propagation of the front of the band again requires the increase in the external stress and so on. Shown in Fig. 11 (a d) are the new moments of the appearing of the bands of A-type. The arising of a new band is accompanied by stopping of the process of strain localization in the previous band. When simulating we used also other types of RE of the round shape. Each type depends on the relaxation tensor, defining the relaxation of the corresponding components of the stress tensor. Above, we considered a RE from the stress relaxation of the component σ y in the value σ = y. Using of the RE from the stress relaxation of pure shear σ = σ xy in the direction under the angle of 45 to the tensile axis turned out to be comfortable when simulating the stochastic nucleation and formation of the PLC bands in alloys, causes the PLC effect. Shown in Fig. 13 is the sequence of the patterns of PLC band formation, obtained as a result of the simulation of Portevin-le Chatelier effect on the mechanism of stochastic appearing of the bands of localized deformation (type C). On the stress strain dependency (Fig. 14) the corresponding moments are marked. The nucleation of the band of localized plastic deformation requires higher external stresses in comparison with the stress of its further formation. The Portevin-Le Chatelier effect is a bright example of the evolution of instability of plastic flow and Fig. 12 The sequence of formation of the PLC-bands of type A (a d) and corresponding effects of interrupted flow on the curve of dependence of external stress on the number of the elements involved into plastic deformation (e)

Fig. 13 The sequence of the PLC-band formation of type C(1 8) self-organizing dissipative structures from micro- to meso- and macroscale levels. 5 Conclusions In the present work the description and simulation of plastic strain localization is performed by the new relaxation element method. The theoretical foundation of the method is performed from the point of view of the principle equations of theory of elasticity and continuum theory of defects. The techniques of the construction of the sites of plastic deformation and the calculation of stresses in the plane with the given sites are demonstrated with the above method. The examples and results of the simulation of the processes of plastic strain localization, accompanied by the effects of Luder s band propagation and Portevin-Le Chatelier are presented. It is shown that the flow stress during the process of plastic deformation evolution is calculated exactly. It should be noted, that for the definition of the total plastic deformation, additional conditions are necessary, since REM accounts only for the part of plastic deformation connected with the presence of the field of internal stresses in the volume of deformed solid. These conditions can be formulated under the calculation of the specific structures, taking the experimental data. In existing models of plasticity a similar problem is solved in the following manner: the σ ε dependencies are prescribed to structural elements taking the macroscopic diagrams of strain-hardening into account. Apparently, that calculated stress strain - curve is the result of averaging of the such introduced diagrams. Such an approach

Acknowledgments The authors aknowledge the support from German Research Society (DFG) under the grant Schm 746/ 52 2. References Fig. 14 Effects of interrupted flow on the curve of dependence of the external stress on the number of grains involved into plastic deformation excluded the appearance of the serrated character of plastic flow and, therefore, it less suitable for the simulation of the effect of plastic strain localization such as Luder s band propagation and Portevin Le Chatelier effect. In reality, plastic flow in the local volumes of solid starts at the elevated stress in the vicinity of a stress concentrator. Further development and accumulation of plastic deformation occurs under the decreasing local stress of the concentrator. The variety of experimental stress strain - diagrams is caused by the different and changing rate of plastic deformation accumulation in the local zones of the deformed solid. High-strength alloys, as a rule are characterized by weakly pronounced yield stress and relatively smooth curves of loading. However, precision experiments on the deformation measurements testify to the fact, that there are no absolutely smooth loading curves. Depending on the nature of the material tested and the condition of testing, the obtained curves are relatively smooth, stepped or serrated. The represented examples show that REM is a comfortable tool when simulating the effects of plastic deformation localization on meso- and macrolevels. 1. Deryugin YeYe, Lasko GV (1995) Relaxation element method in the problem of mesomechanics and calculations of band structures. In: Panin VE (ed), Physical mesomechanics and computer-aided design of materials, vol 1. Nauka, Siberian book-publishing, Novosibirsk pp 131 161 2. Lasko GV, Deryugin YeYe, Schmauder S (2000) Determination of stresses near multiple pores and rigid inclusion by relaxation elements. Theor Appl Fracture Mech 34:93 100 3. Lasko GV, Deryugin YeYe, Schmauder S, Saraev D (2000) Application of the relaxation element method to some problems on stress strain calculations of solids on mesolevel. Comput Mater Sci 19(1 4):35 44 4. Deryugin YeYe, Lasko GV, Schmauder S (2000) Formation and self-organization of the LPD bands within the range from meso- to macrolevel in polycrystals under tensile loading. Comput Mater Sci 15(1):89 95 5. Deruygin YeYe, Lasko GV (2002) Simulation of strain localization in polycrystals. In: International conference on new challenger in measomechanics. Aalborg University, Denmark, pp 41 47 6. Deryugin YeYe, Lasko GV, Schmauder S (2003) Plastic deformation development in polycrystals based on cellular automata and relaxation element method. Comput Mater Sci 26:20 27 7. Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw Hill, New York 8. Muschelishvili II (1977) Some basic problems of mathematical theory of elasticity. Nauka, Moscow 9. Hahn HG (1985) Elastizitatstheorie: Grundlagen der linearen Theorie und Anwendungen auf eindimensionale, ebene und raumliche Probleme B.G. Teubner, Stuttgart 10. Eshelby JD (1956) Continuum theory of defects. Solid State Phys 3:79 11. de Wit R (1970) Linear theory of static disclinations. In: Simons JA, de Wit R, Bullough R (eds) Fundamental aspects of dislocation. Nat Bur Stand US Spec. Publ. 317, vol I, pp 651 673 12. Kirsch G (1898) Die Theorie der Elastizitaet und die Bedurfnisse der Festigkeitslehre. Zentralblatt Verlin Deutscher Ingenieure 42:797 807 13. Deryugin Ye (1998) Relaxation element method, Monograph. Nauka, Siberian book-publishing, Novosibirsk, pp 252 14. Deryugin YeYe (2000) Relaxation element method in calculations of stress state of elastic plane with the plastic deformation band. Comput Mater Sci 19(1 4):53 68 15. Gould H, Tobochnik Ya (1990) An introduction to computer simulation methods. Pt.2: Application to physical systems. Mir, Moscow 16. Neuhäuser H (1990) Plastic instabilities and the deformation of metals. In: Walgraef D, Ghoniem NM (eds) patterns, defects and Material Instabilities, Kluwer, Dordrecht