The ultimate ductile fracture strain determination method for the specimen of circular cross-section has been presented by FEM method. The state of stress in individual locations of tensile tested specimen in successive process phases has been determined unequivocally with the stress triaxiality k. It has been demonstrated that the plane specimen’s fracture strain value in the fracture location varies and depends on the state of stress, which is present in the final specimen’s tension phase. The ductile fracture strain values in various fracture locations for steel, copper and aluminum specimen have been experimentally determined and compared. The simple and practical method to determine this strain has been proposed.U članku je data metoda proračuna određivanja plastičnoga loma uzoraka s okruglim prijesekom sa MKE metodom. Stanje naprezanja u pojedinačnim mjestima vlačnog pokusnog uzorka u pojedinim fazama procesa su bile određene koeficientom troosnog naprezanja k. Dokazano je, vrijednosti naprezanja pri lomu plošnih uzoraka u oblasti prijeloma se mjenjaju u ovisnosti od stanja naprezanja koje je pokazano u završnoj fazi vučenog uzorka. Vrijednosti deformacije pri plastičnom razaranju u raznim mjestima prijeloma za čelik, bakar i aluminij su bile eksperimentalne proračunate i uspoređene. Određena je jednostavna i praktička metoda pri proračunu tih deformacija

S. Kut

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S. KUTA SIMPLE METHOD TO DETERMINE DUCTILEFRACTURE STRAIN IN A TENSILE TEST OF PLANE SPECIMEN’SReceived – Prispjelo: 2009-11-05Accepted – Prihva}eno: 2009-12-18Original Scientific Paper – Izvorni znanstveni radINTRODUCTIONIn several practical cases, the ultimate ductile frac-ture strain determined with tensile test is accepted as amaterial plasticity measure 1. In this case, the plastic-ity has to be defined as an ability of a material to accom-modate high permanent strains until fracture appearswhere this strain reaches certain value called ultimatefracture strain p. The strain value until fracture dependsnot only on the material type, but also on other severalfactors, as: strain speed, strain history, material startingstructure, temperature, specimen geometry, etc. It is im-possible to account for all factors in a single mathemati-cal description, due to a complexity of phenomena andan insufficient state of the art, mainly for phenomenapresent during a plastic strain. Several experiments 2-5have demonstrated that the material fracture processstrongly depends on the hydrostatic stress. This conclu-sion has been independently induced based on experi-ments 6-8.Recently, several different fracture criteria, includ-ing the state of hydrostatic stress, have been developed8-10. However, the practical application of above cri-teria to forecast the fracture during the metal formingprocess has been feasible thanks the numerical comput-ing methods, which enable to determine the material’sstate of stress during the plastic forming process. Cur-rently, the ductile fracture criteria are commonly usedwhen simulating various plastic processing processes10-13. However, the practical application of the crite-ria requires the experimental determination of the duc-tile fracture strain p value for a given material. Thisstrain is usually determined based on the tensile test, butthe determination method indeed is not so obvious, andin several cases even doubtful.In most cases, the tensile test is performed againstcircular or rectangular cross-section specimens. Con-sidering that the ductile fracture strain p around thefracture zone equals the equivalent strain z in this zone,it can be calculated using the equation:    p z    23122232 (1)For circular cross-section specimen (Figure 1a), thestrain components in direction 1 and 2 are calculated us-ing the equation:  1 210   lndd(2)METALURGIJA 49 (2010) 4, 295-299 295The ultimate ductile fracture strain determination method for the specimen of circular cross-section has beenpresented by FEM method. The state of stress in individual locations of tensile tested specimen in successiveprocess phases has been determined unequivocally with the stress triaxiality k. It has been demonstrated thatthe plane specimen’s fracture strain value in the fracture location varies and depends on the state of stress,which is present in the final specimen’s tension phase. The ductile fracture strain values in various fracture loca-tions for steel, copper and aluminum specimen have been experimentally determined and compared. The sim-ple and practical method to determine this strain has been proposed.Key words: tensile test, ductile fracture strain, stress triaxiality, finite element method (FEM)Jednostavna metoda odredjivanja plasti~noga prijeloma i ~vrsto}e plo{nih uzoraka. U ~lanku jedata metoda prora~una odre|ivanja plasti~noga loma uzoraka s okruglim prijesekom sa MKE metodom. Stanjenaprezanja u pojedina~nim mjestima vla~nog pokusnog uzorka u pojedinim fazama procesa su bile odre|enekoeficientom troosnog naprezanja k. Dokazano je, vrijednosti naprezanja pri lomu plo{nih uzoraka u oblastiprijeloma se mjenjaju u ovisnosti od stanja naprezanja koje je pokazano u zavr{noj fazi vu~enog uzorka. Vrijed-nosti deformacije pri plasti~nom razaranju u raznim mjestima prijeloma za ~elik, bakar i aluminij su bile eksperi-mentalne prora~unate i uspore|ene. Odre|ena je jednostavna i prakti~ka metoda pri prora~unu tihdeformacija.Klju~ne rije~i: vlak, plasti~ko razaranje, troosno stanje naprezanja, metoda kona~nih elemenata (MKE)ISSN 0543-5846METABK 49(4) 295-299 (2010)UDC – UDK 669.14-418:539.37:620.17=111S. Kut, Faculty of Mechanical Engineering and Aeronautics, RzeszówUniversity of Technology, Rzeszów, PolandThe strain component in direction 3 is calculated us-ing the constant volume condition:    1 2 30 2    	 (3)If (2) and (3) are substituted to (1) and transformed,the equation to determine the ductile fracture strain pfor circular cross-section specimen is achieved.p 210lndd(4)If a tensile tested specimen is plane (Figure 1b), thestrain components in directions 1 and 2 are different andmay be calculated using the equation:1 lnbb10(5)2 lngg10(6)The strain component in direction 3 is calculated us-ing the equation:     1 2 3 1 20    	 ( ) (7)If (5), (6) and (7) are substituted to (1), the equationto determine the ductile fracture strain p for rectangularcross-section specimen is achieved:    p    	  2312221 22( ) (8)The equation (8) has been derived provided that thespecimen’s cross-section shape is not changed after thestrain. Actually, the specimen’s cross-section shape af-ter the tensile failure differs significantly from the start-ing shape (Figure 2a). After the tensile failure, the planespecimen’s cross-section has a shape of a saddle (Figure2b), and it means that the ductile fracture strain value isnot identical within the cross-section, but varies signifi-cantly. That’s why the calculation of the ductile fracturestrain p for plane specimen is not so obvious, as for thecircular cross-section specimen.The lack of reference data how to proceed in thiscase has been a basis to perform the experiments, in or-der to develop the method to specify the ductile fracturestrain p for rectangular cross-section specimens.EXPERIMENTAL WORKThe static tensile test has been performed using UTS100 tensile testing machine. The plane sheet metal spec-imens have been tested made of the following material:steel, copper, and aluminum 5 251. The mechanicalcharacteristics and the strain hardening curve parame-ters for tested materials, achieved based on the tensiletest, have been presented in Table 1. In order to deter-mine material constants K and n, the specimen elonga-tion has been measured using the extensometer alongthe section l0 = 80 mm. Then the strain hardening curvep = f() has been plotted for the points below maximumtension force. The stress p for individual strain harden-ing curve points has been calculated as the ratio of theforce to the variable specimen cross-section, calculatedbased on the constant volume condition. The logarith-mic strain for individual strain hardening curve pointshas been calculated from the equation  = ln(l/l0), where:l0 = 80 mm, l – the length of section after specimen elon-gation. The strain hardening curve points calculated thisway have been approximated with an equation p = Kn.The measuring bases to indicate the measuring zonehave been marked on the specimen surface. L, P – Lat-eral, S – Middle (Figure 3). The zone width and speci-men thickness have been measured in these locationsbefore and after the specimen tensile failure. The geo-metrical values have been measured using thetoolmaker’s microscope with an accuracy of 0,01 mm.The average specimen thickness g1 after the tensile fail-ure within individual areas (L, P, and S, C) has been de-termined as follows:1) the specimen thickness has been measured afterthe tensile failure in examined areas, with an in-terval of approx. 0,5 mm on specimen width,2) thickness g1 has been calculated as an arithmeti-cal mean of measured thickness values within in-dividual areas.The measured values of geometrical parameters inindividual locations before and after the specimen ten-296 METALURGIJA 49 (2010) 4, 295-299S. KUT: A SIMPLE METHOD TO DETERMINE DUCTILE FRACTURE STRAIN IN A TENSILE TEST OF PLANE SPECIMEN’SFigure 1 Typical tensile specimens: a) a round specimen,b) a flat specimenFigure 2 Cross-sectional area in the neck at fracture:a) before fracture, b) after fractureTable 1 Mechanical properties of materials testedMaterialYieldstressRe /MPaUltimatestrengthRm /MPaStrain har-dening co-efficientStrain har-dening ex-ponentK /MPa nsteel 399 447 591 0,072copper 97 217 389 0,262aluminium 65 175 378 0,31sile failure, for individual materials have been stated inTable 2. Then for all measured zones, the strain value 1and strain value 2 have been calculated using (5) and(6), and then the ductile fracture strain values in an indi-vidual zones have been calculated using (8) (Table 2).The analysis of the fracture strain value p for indi-vidual materials demonstrates that, for all cases, thehighest strain appears in the middle part of specimen,and the lowest in the lateral part. The experiments per-formed show that the more plastic material is (in thiscase - copper), the higher fracture strain value p differ-ence. For aluminum and copper specimen, both thestrain 1 in direction 1 (specimen width) and the strain 2in direction 2 (specimen thickness) achieve the highestvalue in zone S, its middle part. Whereas for the steelspecimen, the highest strain 2 appears in direction 2(specimen thickness) in zone S, and in direction 1 (spec-imen width) strain 1 is slightly lower than in its middlepart (Table 2).It is also supposed that the differences betweenstrains in the middle and lateral part of the specimen willincrease as the specimen width is increased. The meanstrain value measured for an entire specimen is inaccu-rate and depends mainly on its geometry. The followingquestion arises: where the fracture strain for the planespecimens should be measured?FEM NUMERICAL SIMULATIONTo answer the question as referred to the above, thesteel specimen tension process numerical simulation hasbeen performed using MSC Marc Mentat software,which enables solving non-linear and contact problems.FEM simulation’s geometrical model has been createdbased on the experimental model. The purpose of thenumerical simulation in this case is neither detailedanalysis of stresses and strains nor determining their val-ues. The purpose of the simulation is to indicate the area,where the state of stress on the tensioned specimen is theclosest to uniaxial tension, within an entire strain rangeup to specimen tensile failure. Therefore the specimentension process has been analyzed in the plane stresscondition. The elastic-plastic material model withnon-linear strain hardening has been adopted, describedby the following equation 14:      EK n( )( )00(9)The material parameters for elastic strain have beenas follows: E = 210000 MPa, = 0,3. The strain harden-ing parameters K, n are presented in Table 1. In order tocreate FEM grid of deformable sheet metal, Class 4Type 3 elements has been used – plane-stress quadrilat-eral 15. The start point of necking has been determinedbased on Hill’s equation in form of 16:*( )n1(10)where: * - critical strain for the onset of local necking, 3 1, n - strain hardening exponent.The tension simulations have been performed for anentire specimen, placed in the measuring area of anextensometer (II) holding the griping area of the speci-men, right at the tensile testing machine grips. Such a pur-poseful placement of the extensometer (II) enabled the in-troduction of the movement boundary condition for thespecimen modeled as in the experiment. This also en-abled eliminating the machine structure susceptibility er-rors. The boundary condition has been also introduced fornodes placed at the ends of modeled specimen in the mea-suring area of an extensometer (II). The node movementstowards the specimen axis have been forced in the bound-ary condition. The node movement perpendicularly to-wards the specimen has been disallowed.METALURGIJA 49 (2010) 3, 295-299 297S. KUT: A SIMPLE METHOD TO DETERMINE DUCTILE FRACTURE STRAIN IN A TENSILE TEST OF PLANE SPECIMEN’SFigure 3 Tensile specimen with marked control linesTable 2 Specimen’s geometry and average strain va-lues in analyzed regionsMate-rialDesignation ofparametrMeasuring zonesL, PlateralSmiddleC totalSteelvalues ofgeome-trical pa-rametersmmgo 3,55 3,55 3,55bo 2,1 2 14,92g1 2,34 1,7 2,2b1 1,47 1,56 10,97plasticstrain1 -0,357 -0,248 -0,3072 -0,417 -0,736 -0,564p 0,774 1,024 0,884Coppervalues ofgeome-trical pa-rametersmmgo 3,55 3,55 3,55bo 2,1 2,1 14,85g1 1,27 0,98 1,12b1 1,51 1,48 10,05plasticstrain1 -0,32 -0,35 -0,392 -1,028 -1,278 -1,154p 1,416 1,724 1,606Aluminiumvalues ofgeome-trical pa-rametersmmgo 3,55 3,55 3,55bo 2,1 2,25 14,78g1 2,1 1,75 1,95b1 1,81 1,93 11,75plasticstrain1 -0,148 -0,153 -0,2292 -0,525 -0,707 -0,599p 0,707 0,918 0,855Due to such an assumed boundary condition, the lo-cal necking appears exactly halfway the length oftensioned specimen.The tension simulation has been performed until spec-imen tensile failure, and it corresponds to extensometr(II) displacement, which was 15,33 mm for the steel spec-imen. The tensile force curves have been prepared andcompared (Figure 4) in order to validate the FEM simula-tion. The limit value of ductile fracture strain depends onthe present state of stress. In the mechanical & mathemat-ical modeling approach, non-dimensional stresstriaxiality k = m/H, where m is a mean normal stress, His an equivalent stress, is the very important parameter,which unequivocally specifies the plane state of stress(Figure 5). If this factor is known, it is possible to deter-mine the state of stress in any point of strained object,e.g.: if k = 0 – this is a simple shear (Figure 5.c), k = 0,66 –it is a biaxial regular tension (Figure 5.e), k = - 0,33 – it isan uniaxial compression (Figure 5.b), etc.In considered case we determine the strain for thetensile test, so that k factor value is 0,33. As seen in FEMcalculations, the uniaxial state of stress is present in aninitial tension phase and lasts until the neck is created,and then once Rm limit is exceeded, the states of stress inindividual zones differ significantly (Figure 6).The state of stress in the lateral zone L changesslightly in the biaxial tension direction, reaching k =0,36 in its final phase. Whereas the state of stress in themiddle zone S changes significantly in the simple sheardirection, reaching k = 0,106 in its final phase.The state of stress factor k distribution in the initialand final phase of the specimen tensile test has been pre-sented on Figure 7. The k factor value is explicitly dif-ferent in the middle and lateral part of the specimen un-der test.CONCLUSIONS1. The experiments performed show that the frac-ture strain in the tensile test for plane specimen298 METALURGIJA 49 (2010) 4, 295-299S. KUT: A SIMPLE METHOD TO DETERMINE DUCTILE FRACTURE STRAIN IN A TENSILE TEST OF PLANE SPECIMEN’SFigure 4 Comparison of the tensile force-experimentaland numerical results (steel specimen)Figure 5 The k factor values for an individual plane stresscases: a) biaxial compression, b) uniaxial com-pression, c) simple shear, d) uniaxial tension, e)biaxial tensionFigure 6 Comparison of stress triaxiality in different spe-cimen zone: L, P – lateral, S - middleFigure 7 Distribution of stress triaxiality k: a) initial phaseof tensile test (3-item), b) final phase of tensiletest (20-item)must be determined in L or P zone, as the state ofstress in these zones is the closest to the uniaxialtension for all tensile test.2. The calculation of the ductile fracture strain foran entire cross-section C is highly inaccurate andthe error mostly depends on the specimen dimen-sions.3. The presented method of the ductile fracturestrain determination is simple and can be per-formed during the conventional tensile test, oncethe base line is marked on the specimen surface.REFERENCES1 Czichos, H., Saito, T., Smith L.: Springer Handbook of Ma-terials Measurement Methods. Springer-Verlag New York,2006, pp. 302-307.2 Bao, Y., Wierzbicki, T.: On fracture locus in the equivalentstrain and stress triaxiality space. Int. J. Mech. Sci. 46(2004), 81-98.3 Mohr, D., Henn, S.: Calibration of stress-triaxiality depen-dent crack formation criteria: A new hybrid experimen-tal-numerical method. Exp.Mech. 47 (2007), 805-820.4 Oh, C.-K., et al.: Development of stress-modified fracturestrain for ductile failure of API X65 steel, Int. J. Fract. 143(2007), 119-133.5 Kim, J., et al.: Modeling of void growth in ductile solids:Modeling of void growth in ductile solids: effects of stresstriaxiality and initial porosity. Eng. Fract. Mech. 71 (2004),379–400.6 Bao, Y.: Dependence of ductile crack formation in tensiletests on stress triaxiality, stress and strain ratios. Eng. Fract.Mech. 72 (2005), 505–522.7 Zhu, H., et al.: Investigation of fracture mechanism of 6063aluminum alloy under different stress states. Int. J. Fract.146 (2007), 159–172.8 Zhao, Z., et at.: An improved ductile fracture criterion forfine-blanking process. J. Shanghai Univ. (Sci.), 13 (2008) 6,702-706.9 Oyane, M., et al.: Criteria for ductile fracture and their ap-plications. J. Mech. Working Techn. 4 (1980), 65-81.10 Hambli, R., Reszka, M.: Fracture criteria identificationusing an inverse technique method and blanking experi-ment. Int. J. Mech. Sci. 44 (2002), 1349-1361.11 KUT, S.: The method of ductile fracture modeling and pre-dicting the shape of blanks. Progressive Technologies andMaterials. OWPRz, Rzeszów, 2007, 15-25.12 Thipprakmas, S., et al.: An investigation of step taper-sha-ped punch in piercing process using finie element method. J.Mat. Proc. Techn. 197 (2008), 132-139.13 Yu, S., at al.: Ductile fracture modeling of initiation andpropagation in sheet-metal blanking processes, J. Mat. Proc.Techn. 187-188 (2007), 169-172.14 Chen, W.F., Han D. J.: Plasticity for Structural Engineers.Springer-Verlag New York, 1988, 12.15 MSC Software, MSC.Marc Volume B, Element Library,Version 2007.16 MSC Software, MSC.Marc Volume A, Theory and UserInformation, Version 2007.Note: The responsible translator English language isG. Rêbisz, Rzeszów, Poland.METALURGIJA 49 (2010) 3, 295-299 299S. KUT: A SIMPLE METHOD TO DETERMINE DUCTILE FRACTURE STRAIN IN A TENSILE TEST OF PLANE SPECIMEN’S

Jednostavna metoda odredjivanja plastičnoga prijeloma i čvrstoće plošnih uzoraka

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