Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 2, pp. 589-607, Warsaw 2012 50th Anniversary of JTAM SIMULATION OF DUCTILE FRACTURE OF S235JR STEEL USING COMPUTATIONAL CELLS WITH MICROSTRUCTURALLY-BASED LENGTH SCALES Paweł Grzegorz Kossakowski Faculty of Civil and Environmental Engineering, Kielce University of Technology, Kielce, Poland e-mail: kossak@tu.kielce.pl The paper presents results of a numerical modelling of ductile fracture and failure of elementsmade of S235JR steel subjected to static tension. The analysis was performed using the Gurson-Tvergaard-Needleman (GTN) model, which takes into consideration the material structure. The GTNmaterial parameters of S235JR steel were assumed basing on the results of itsmicrostructural and strengthproperties. The aimof the research was to specify and test the procedure that enables one to per- form engineering calculations of tensioned structural elements using the GTNmaterialmodel. During numerical simulations, themesh size-effect was analysed, basing onmicrostructurally-based length scales methods. The results show high sensivity of strength curves in the range from the maximum force to the failure depending on the method and mesh size applied. Key words: Gurson-Tvergaard-Needlemanmodel, voids, numerical cal- culations, S235JR steel 1. Introduction The analysis of plastic ranges is a fundamental issue undertaken in both rese- arch as well as in ordinary engineering practice. From practical point of view, it is particularly important to estimate the limit loads beyond the elastic range for structural elements, especially when operating in the pre-failure condition. The failure of amaterial is dependent onmore than onemajor stress, and the safety of a steel structure can be assessed using the so called Huber-Mises- Hencky (HMH) strength hypotheses.Anumber of studies show that theHMH model is not always suitable to analyse the plastic state in and beyond the 590 P.G. Kossakowski range of deformations corresponding to the necking of an element subjected to tension.Assuming the continuumof thematerial, one cannot apply thismodel to establish the influence of microstructural defects on thematerial strength. The processes of crack initiation and propagation in polycrystalline struc- tures are closely related to thematerialmicrostructure.There are several basic types of fracture mechanisms, i.e. plastic, cleavage and brittle intergranular, shear or void-sheeting and ductile (Teirlinck et al., 1988). In shear and ductile fracture, the cracking is attributable to the nucleation, growth and coalescence of voids (Fig.1). Fig. 1. Scheme of void evolution (basing on Sedlacek et al. (2008)) Damage in form of voids occurs on the existing inclusions and second- phase particles. The growth and coalescence of these voids (Fig.2) result in the development of localised plastic deformations. Fig. 2. Types of void coalescence (basing on Sedlacek et al. (2008)) The phenomenon described above can be analysed using damage models taking into account the influence of microstructural defects on the material strength by defining a relationship between the particular failure stages and the strength of the material. The Gurson (1977) model for a porous solid is one of such models. It is a modified Huber-Mises-Hencky criterion, defining the influence of an increase in the void volume fraction on the strength of the material. This model was further modified by Tvergaard (1981) and then Simulation of ductile fracture of S235JR steel... 591 by Tvergaard and Needleman (1984), and Needleman and Tvergaard (1984), who developed a method for the calculation of failure loads resulting from ductile fractureby considering selectedmicrostructural parameters andplastic properties of a material. The Gurson-Tvergaard-Needleman model is referred to as the GTNmodel. It should be emphasised that the GTN model is a basic damage model recommended to use in the analysis of emergency condition for building struc- tures according to the current standards, for instancePN-EN1993-1-10 (2007) and its commentary by Sedlacek et al. (2008). A number of studies carried out by authors (also by Kossakowski, 2009, 2010;Kossakowski andTrąmpczyński, 2011) show that, for alloys and structu- ral steels, the GTNmodel ensures a better consistency of the strength curves obtained by numerical calculation (modelling) with experimental results than the HMH criterion. Most results are obtained by the authors by conducting simple strength tests, mainly tensile strength tests. Numerical analyses are performed using the method of best fit to material constants (the GTN mo- del) for elements with predetermined geometries and strength properties. The GTN model parameters are frequently determined basing on the fitting cu- rve technique. Another drawback is the lack of standardised microstructural parameters to develop a GTN model for steels used most commonly in civil engineering. Numerical strength analysis of structural elements in the nonlinear range is another problem.When simulating ductile fracture using the finite element method, in particular, using the GTN model, size effects are encountered. It reveals the softening of the final part of the strength curve. Several methods have been tested (eg. Tvergaard andNeedleman, 1995; Ramaswamy andAva- ras, 1998; Borino et al., 2003) to minimize the softening effect, but many of them cannot be applied due to limitations in available engineering software. All these problems require development of a procedure for numerical mo- delling of the load limit for structural elements, taking into account the effect of microfailure. From practical point of view, it is particularly important to specify the procedures to carry out engineering calculations using the GTN material model. Therefore, the aim of the research was to specify and test the procedure that enables one to perform engineering calculations of tensioned structural elements using the GTN material model. This paper includes results of a numerical modelling of elements made of S235JR steel subjected to static tension, basing on the Gurson-Tvergaard-Needleman (GTN) model, taking into consideration the real material structure. S235JR steel was selected for 592 P.G. Kossakowski the tests because it is a common structural material in Poland as well as in other European countries. ForS235JRsteel, the elastic-plasticmaterialmodelwas elaborated through testing of various approximation stress-strain curves. The GTN material pa- rameters were assumed basing on the results of microstructural and strength properties of S235JR steel and the results of analysis performed before by Kossakowski (2009, 2010), and Kossakowski and Trąmpczyński (2011). During numerical simulations, themesh size-effect was analysed, basing on themicrostructurally-based length scales methods. It should be noticed that the presented datamay be used for any analyses and expertise connectedwith the assessment of the load-carrying capacity and safety of structural elements, especially made of S235JR steel. 2. Gurson-Tvergaard-Needleman (GTN) damage model The first model to take into consideration microdamage (pores, voids) was the Gurson (1977) model, which assumed that the proportion of voids in the plastic potential function was dependent on the void volume fraction f. The original Gurson condition was modified by Tvergaard (1981) and Tvergaard and Needleman (1984) as Φ= (σe σ0 )2 +2q1f ∗cosh ( −q2 3σm 2σ0 ) − (1+q3f∗2)= 0 (2.1) where: σe – von Mises effective stress according to the HMH hypothesis, σ0 – yield stress of the material, σm – hydrostatic pressure (mean stress), f∗ – actual void volume fraction, qi – Tvergaard coefficients describing the plastic properties of the material. In the GTNmodel, the void volume fraction f∗ is defined as follows f∗ =            f for f ¬ fc fc+ fF −fc fF −fc (f−fc) for fc ε pl critical over r > lc (3.1) The fracture criterion is satisfied over a progressively increasing distance until it is satisfied over the characteristic length, and formation of a crack is predic- ted. The solid contour corresponds to the first loading step when the criterion is satisfied over the characteristic length lc, corresponding to fracture initia- tion. To ensure the opportunity to initiate ductile fracture during numerical si- mulation, it is necessary to apply a propermesh size. It is especially important in the region with high stress-strain gradients, for instance near the crack tip. The SMCS fracture criterion can be satisfied for cell dimensions, identified with themesh size, exceeded over the critical volume of material, represented by the characteristic length lc. Thus, the mesh dimension D should be equal to the characteristic length lc at least. Simulation of ductile fracture of S235JR steel... 595 3.2. Cell model-based method Asmentioned at the beginning,manymetals fail due to growth of the void. Thismechanismdisplays amacroscopically planar fractureprocess zone of one or two void spacings in thickness. It is characterised by intense plastic flow in the ligaments between the voids. Outside the fracture process zone, the voids exhibit little or no growth (Xia and Shih, 1995a,b; Gao et al., 1998; Faleskog et al., 1998; Kossakowski and Trąmpczyński, 2011). In order to model the process described above, a layer of the material containing a pre-existing population of similar sized voids is assumed.A single row of computational cells of the linear dimension D represents this layer. Each computational cell contains a single void of some initial volume f0. The thickness of this layer D is interpreted as themean spacing between the voids nucleated from large inclusions. Microvoids nucleated from small inclusions assist the process of hole link-up with the crack tip and can be taken into account in the coalescence phase. Cells near the newly formed crack surface have undergone significant void growth, while cells further away exhibit little or no void growth. The void growth is described by the Gurson constitutive relation for dilatant plasticity, resulting inmaterial softening and, ultimately, loss of stress carrying capacity. The material outside this strip is undamaged by void growth and described by elastic-plastic relation (Fig.4). Fig. 4. The region near the crack tip modelled according to the Cell model method (basing onGao et al. (1998), Faleskog et al., 1998)) 4. Microstructurally-based length scales examinations for S235JR steel The first step of thematerial analysis was to obtainmicrostructural images of S235JR steelwith a ferritic-perliticmatrixwith a large number of non-metallic 596 P.G. Kossakowski inclusions. S235JR steel was reported to have appropriate metallurgical pu- rity. To fully characterize the non-metallic inclusions in the material, it was necessary to determine the basic stereological parameters. The void volume fraction was f0 =0.0017=0.17% (Kossakowski, 2010). As described before, to form a macroscopic crack, the fracture criterion must be satisfied over theminimum volume of thematerial defined in two di- mensions by the characteristic lengthmeasure lc in the region of high stresses and plastic strain. Several approaches have been discussed, but determina- tion of the length scale is subjective (e.g. Panontin and Sheppard, 1995; Chi et al., 2006; Rousselier, 1987). In the presented study, an approach based on that proposed by Hancock and Mackenzie (1976) was used. According to the assumption of this method, the linking of two or more holes formed from coalescing inclusion colonies leads to fracture. Thus a macrocrack formation only occurs when shear localizes betweenmultiple clusters, while void growth expands the cluster of inclusion colonies. The size of these inclusion colonies represents the length scale and is visible in a fractograph. In the presented study, the characteristic length lc was determined as the dimension of plateau and valley on the fracture surface (Fig.5). The measurement was performed for twenty inclusion colonies using the statistical Chauvenet criterion. All aty- pical values were rejected. The characteristic length lc ranged from 130µmto 360µm, with themean value of lc being ≈ 250µm. Fig. 5. Determining the characteristic length l c defined by dimensions of plateaus and valleys While the characteristic length lc was based on dimensions of plateaus and valleys, the length D was defined as the mean spacing between the voids nucleated from large inclusions and determined during the microstructural Simulation of ductile fracture of S235JR steel... 597 examinations as well. As mentioned above, a large number of non-metallic inclusions, whichweremainly sulphides and brittle oxides, were noticed in the microstructure of S235JR steel. Sulfide inclusions were elongated in shape, and their distribution in the volumewas irregular. Theywere arranged in bands.Other non-metallic inclu- sions found in the tested materials are uniaxial brittle oxide inclusions, which may co-occur with sulfide inclusions. In order to determine the length D, a statistical procedurewas used similarly as for the characteristic length lc. The length of D ranged from 40µm to 180µm, with the mean value of D being ≈ 100µm (Fig.6). Fig. 6. Determining the length D defined bymean spacing between the voids nucleated from large inclusions 5. Determination of GTN model parameters The first step was to perform static tensile strength tests for specimens with a circular cross-section, according to PN-EN10002-1 (2004). The nominal dia- meter of the specimens was φ = 10mm, the length of the measuring ba- se l0 = 50mm, and the primary cross-sectional area S0 = 78.5mm 2. The average values obtained during the tests were as follows: the yield stress R0.2 = 318MPa, the tensile strength Rm = 446MPa, and the displacement percentage A5 = 33.9%. Using the averaged σ(ε) curve, it was possible to determine the nominal normal stress σ and the longitudinal strain ε. Theparameters of theGTNmodel for S235JRsteelwere establishedon the basis ofmicrostructure analysis, numerical modelling of tensile strength tests, 598 P.G. Kossakowski and results of the analysis performedbefore byKossakowski (2009, 2010), and Kossakowski and Trąmpczyński (2011). The initial void volume fractionwas determined during themicrostructure analysis as f0 = 0.0017 = 0.17%. The Tvergaard parameters were determi- ned according to the results obtained by Faleskog et al. (1998). For the ratio R0.2/E = 0.00155 and strain-hardening exponent N = 0.195, the values of Tvergaard parameters were established as q1 =1.91, q2 =0.79 and q3 =3.65. The other parameters of the GTNmodel were defined numerically, basing on the studies byKossakowski (2009, 2010), andKossakowski andTrąmpczyński (2011). The numerical calculations were performed using a program based on the finite element method, Abaqus Explicit version 6.10. The elements were modelled as axially symmetrical components using standard elements (Aba- qus, 2010). The averaged tensile strength curve was approximated by the equation ε=                σ E for σ<σ0 ε1−ε0 σ1−σ0 (σ−σ0)+ε0 for σ0 ¬σ¬σ1 ε0+ σ01 E ( σ σ01 )1/N for σ>σ1 (5.1) where: ε – strain, ε0 – yield strain, ε1 – initial hardening strain, σ – stress, σ0 – yield stress, σ1 – initial hardening stress, σ01 – initial stress at the beginning of the nonlinear part of the approximation curve, E – modulus of elasticity, N – strain-hardening exponent. Theparameters of theGTNmodelwere determinedbasing on the σ(ε) cu- rves obtained through numerical analysis and experiments. At each iteration step, the values of the GTN model parameters were changed within certa- in limits. The optimization criterion was the convergence of the σ(ε) values obtained numerically and experimentally. The critical void volume fraction, fc, above which the nominal material strength decreased, was determined as 0.06, basing on the determined value of f0, as suggested by Richelsen and Tvergaard (1994). The value of the void volume fraction corresponding to the complete loss of material strength fF was assumedas 0.667. Thevolume fraction of thenucleated voids fN was 0.04, the average nucleation strain of inclusion-related voids εN was 0.3, and the standard deviation of the strain sN was 0.05. All parameters of the GTN model are summarized in Table 1. Simulation of ductile fracture of S235JR steel... 599 Fig. 7. The approximation of the true stress-strain σ(ε) curve (a) and nominal stress-strain σ(ε) curves (b) of S235JR steel Table 1.Microstructural parameters of the GTNmodel of S235JR steel f0 fc fF q1 q2 q3 εN fN sN 0.0017 0.06 0.667 1.91 0.79 3.65 0.3 0.04 0.05 6. Numerical simulation of ductile fracture of S235JR steel using computational cells with microstructurally-based length scales Thenumerical analysiswas also performedusingAbaqusExplicit version 6.10. The elements modelled were the same as those used during the tensile strength tests. The specimens with a circular cross-section were subjected to quasi-static tension and a displacement control increase with the strain rate ε̇ = 0.1s−1. The modelling was performed for axially symmetrical elements. Because of the symmetry, themodellingwas conducted only for half-specimens (Fig.8). A sharp notch with a depth of R = 0.05mm was modelled in the middle of the sample length. During numerical simulations, two approaches based on computational cells with microstructurally-based length scales were applied. Twomodels were used during the numerical simulations. For the first one, referred as GTN model, the damaged Gurson-Tvergaard-Needleman material model was assumed for thewhole numericalmodel. The dimensions of the cell localised near the crack plane were equal to D×D/2, where D was equal to the characteristic length lc = 250µm determined during microstructural examinations. The second model, referred as Cell model, was based on the assumptions given in the previous section. The elements adjacent to the crack plane were modelled by making use of the GTN material model. Their dimensions were equal to D×D/2. The value of Dwas equal to themean spacing between the 600 P.G. Kossakowski Fig. 8. Geometry of numerical models used during simulations large voids D=100µm. The simulations were also conducted for cell dimen- sions D = 250µm, based on the characteristic length lc. The elastic-plastic material model was assumed for the rest of the numerical model according to Fig.4. Thenumerical simulations of ductile fracture ofS235JR steelwerebasedon the force-elongation F(l) curves. The analysis began with the determination of the coefficients of approximation equation (5.1). The yield stress was esta- blishedaccording to thevalues obtainedduring the tests, i.e. σ0 =318MPa. It was found that the strain-hardening exponent N hadavery significant impact on the force F and elongation l determined numerically. The optimization condition was the convergence of the force-elongation F(l) values obtained numerically and experimentally using themean square error criterion. For all analysed cases, the best agreement of the results was achieved for the strain- hardening exponent N =0.195. Force-elongation curves F(l) determined du- ring experiments and simulations by using GTN and Cell models are shown in Figs.9-11. For comparison, the curves determined assuming N =0.19 and N = 0.20 are also shown. Others constants of approximation equation (5.1) were determined as follows: yield strain ε0 = 0.002, initial hardening strain ε1 =0.015, initial hardening stress σ1 =333MPa, initial stress at the begin- ning of the nonlinear part of the approximation curve σ01 = 198MPa, and modulus of elasticity E=205GPa. Simulation of ductile fracture of S235JR steel... 601 It should be noted that the tensile strength curves obtained by applying GTN model were consistent with the experimental results in the first range, until the maximum force F was reached. In the range from the maximum force to failure, the forces F determined during the experiments were higher than those found from numerical simula- tions usingGTNmodel. Therewas a sharp decrease in the force corresponding to elongation equal to ∼ 8.0mm, followed by the element failure after the maximum force was reached (Fig.9). The visible softening was noticed when GTN modelwas used for the whole numerical model. The simulated failure of elements was noticed for elongation equal to 11.7-12.0mm, depending on the strain-hardening exponent N. Fig. 9. Force-elongation F(l) curves determined during experiments and numerical simulations by using GTN modelwith D=250µm Fig. 10. Force-elongation F(l) curves determined during experiments and numerical simulations by usingCell model with D=100µm 602 P.G. Kossakowski Good agreement of the results in the first range of strength curveswas also foundwhenCellmodelwasused.For cell dimensions D=100µm, the forces F established by numerical simulations were higher than those determined in experiments in the final part of force-elongation F(l) curves (Fig.10). Fig. 11. Force-elongation F(l) curves determined during experiments and numerical simulations by usingCell model with D=250µm In thewhole range of strength curves the best agreement of resultswas no- ticed for dimension D = 250µm (Fig.11). The simulated failure of elements was noticed for elongation equal to 12.3-13.7mm and 11.7-12.2mm, depen- ding on the strain-hardening exponent N for dimension D = 100µm and D=250µm, respectively. 7. Discussion As shown in the previous section, it was possible to numerically simulate duc- tile fracture and failure of elements made of S235JR steel subjected to static tension taking into consideration effects of microstructural defects. It should be noted that the tensile strength curves obtained by applying bothGTN and Cell model were more or less consistent with the experimental results. In all cases, the best agreement was achieved by assuming a strain- hardeningexponent N =0.195. In thefirst rangeuntil themaximumforcewas reached, the strength curves obtained by applying bothGTN andCell models were fully consistent with the experimental results. However, in the further part of the tensile strength curves, differences were noticed depending on the model assumed.The final parts of simulated curves, in the range from thema- ximum force to failure, weremuch consistentwith the experimental data using Simulation of ductile fracture of S235JR steel... 603 Cell model in comparison toGTN model. It is clearly visible in Fig.12, where experimental and simulated strength curves for the strain-hardening exponent N = 0.195 are presented. The softening effect was noticed for elements mo- delled usingGTN model, due to significant reduction of the strength through the assuming the voided material for the entire numerical model. A different situation was observed whenCell model was used. The values of forces in the final part of tensile curves were much consistent with the experimental data comparing to GTN model. During simulations performed using Cell model, there was no softening effect. Fig. 12. Force-elongation F(l) curves determined during experiments and numerical simulations by usingGTN andCell models (N =0.195) When usingCell model, the process zone at the fracture surface is model- led as a layer of the material containing a pre-existing population of similar sized voids, containing a single void of some initial volume f0. It is close to real situation, because for S235JR steel the fracture mechanism displays a macroscopically planar fracture process zone of one or two void spacings in thickness (Kossakowski andTrąmpczyński, 2011).Outside the fractureprocess zone, the voids exhibit little or no growth. Considering the above, it can be concluded that applying Cell model provides simulation of the ductile failure process close to a real situation in relation to the microstructure of S235JR steel. As can be seen in Fig.12, the cell dimension D has impact on results of simulation, revealing in the final part of the force-elongation curves. For the dimension D = 250µm, values of the force were close to mean experimental data. When using the mesh based on the cell dimension D = 100µm, the results obtainedwere higher due to a very small process zone containing voids leading to a smooth reduction of the material strength. It can be concluded 604 P.G. Kossakowski that the modelling of ductile fracture of S235JR using Cell model should be performed based on the cell dimension D=250µm rather than D=100µm. When using Cell model, it should be noticed that the process zone needs to be assumed a priori in the region of potential failure. Unfortunately, for structural elements with complicated geometry, it may be difficult to predict precisely. The process zone determined incorrectly can lead to false results. Therefore, it seems that practical application of Cell model method may be limited. 8. Conclusions The following conclusions have been drawn basing on the results of numerical simulations of ductile fracture of S235JR steel using computational cells with microstructurally-based length scales: • The numerical analysis showed that by applying theGurson-Tvergaard- Needlemanmodel, which takes into account the real microstructural pa- rameters for S235JR, it is possible to simulate failure of elements sub- jected to static tension resulting from ductile fracture. • In order to simulate ductile fracture of S235JR steel, the averaged tensile strength curve approximated by equation (5.1) is proposed. The best agreement of numerical and experimental results was noticed for the strain-hardening exponent N = 0.195 and yield stress obtained during strength tests, i.e. σ0 =318MPa. • The tensile strength curves obtained by applyingGTNmodelwere consi- stentwith the experimental results in the first range, until themaximum force was reached. However, in the range from themaximum force to fa- ilure, the forces F determined during the experiments were higher than those calculated during numerical simulations. ApplyingGTNmodel for whole numerical models, leads to softening followed by amuchmore re- duction of the material strength in comparison to void growth noticed during the experiments. • When using Cell model, the process zone is modelled as a layer of the material containing a pre-existing population of similar sized voids, con- taining a single void of some initial volume f0. It is close to a real situation, because for S235JR steel the fracture mechanism displays a macroscopically planar fracture process zone of one or two void spacings in thickness, while outside the fracture process zone, the voids exhibit little or no growth. The force-elongation curves obtained during numeri- cal simulations were more or less consistent with the experimental data Simulation of ductile fracture of S235JR steel... 605 in the whole range, from the beginning up to failure. Considering the above, it can be concluded that applyingCell model provides simulation of the ductile failure process close to a real situation in relation to the microstructure of S235JR steel. • The results of simulationsweremuchmore consistentwith strength tests when using Cell models with cell dimensions D = 250µm rather than D = 100µm. It was followed by a very small process zone when used cell dimensions D = 100µm interpreted as the mean spacing between the voids nucleated from large inclusions, leading to a smooth reduc- tion of the material strength. When using cell dimensions D = 250µm interpreted as the characteristic length lc defined by the dimensions of plateaus and valleys formed from coalescing inclusion colonies leading to fracture, the results of simulations were much more consistent with the experimental data. • Application of Cell model requires determination and modelling of the process zone a priori. In many cases, it may be difficult to predict it precisely, but the process zone determined incorrectly can lead to fal- se results. Taking the above into consideration, it seems that practical application ofCell model method is limited. References 1. ABAQUS 6.10 Analysis User’s Manual, Dassault Systèmes, 2010 2. 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Xia L., Shih C.F., 1995a, Ductile crack growth – I. A numerical study using computational cells with microstructurally-based length scales, Journal of the Mechanics and Physics of Solids, 43, 2, 233-259 27. Xia L., Shih C.F., 1995b, Ductile crack growth – II. Void nucleation and geometry effects on macroscopic fracture behavior, Journal of the Mechanics and Physics of Solids, 43, 12, 1953-1981 Symulacja pękania ciągliwego stali S235JR z zastosowaniem elementów zdefiniowanych na podstawie mikrostrukturalnych skal długości Streszczenie W artykule przedstawiono wyniki modelowania numerycznego pękania ciągliwe- go oraz zniszczenia elementów wykonanych ze stali S235JR poddanych statycznemu rozciąganiu. W analizie oparto się na modelu materiałowym Gursona-Tvergaarda- Needlemana (GTN), uwzględniającym strukturę materiałową. Parametrymateriało- we modelu GTN stali S235JR przyjęto w oparciu o jej własności mikrostrukturalne i wytrzymałościowe. Celem badań było opracowanie i weryfikacja procedury umoż- liwiającej przeprowadzanie obliczeń inżynierskich rozciąganych elementów konstruk- cyjnych przy zastosowaniumodelu GTN. Podczas symulacji numerycznych uwzględ- niono efekt wielkości siatki, stosując metody oparte na mikrostrukturalnych skalach długości. Uzyskane wyniki ujawniają wysoką wrażliwość krzywych wytrzymałościo- wychw zakresie od obciążeniamaksymalnegodo zniszczenia, w zależności odmetody oraz wielkości siatki zastosowanej w obliczeniach. Manuscript received April 19, 2011; accepted for print September 16, 2011