Arch. Metall. Mater. 62 (2017), 1,

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1 Arh. Metall. Mater. 6 (017), 1, DOI: /amm P.G. KOSSAKOWSKI* # EXPERIMENTAL DETERMINATION OF THE VOID VOLUME FRACTION FOR S35JR STEEL AT FAILURE IN THE RANGE OF HIGH STRESS TRIAXIALITIES This paper is onerned with the ritial void volume fration f F representing the size of mirodefets in a material at the time of failure. The parameter is one of the onstants of the Gurson-Tvergaard-Needleman (GTN) material model that need to be determined while modelling material failure proesses. In this paper, an original experimental method is proposed to determine the values of f F. The material studied was S35JR steel. After tensile tests, the void volume fration was measured at the frature surfae using an advaned tehnique of quantitative image analysis The material was subjeted to high initial stress triaxialities T 0 ranging from to The failure proesses in S35JR steel were analysed taking into aount the influene of the state of stress. Keywords: failure, ritial void volume fration f F, Gurson-Tvergaard-Needleman material model, S35JR steel, high stress triaxiality. 1. Introdution For many metals, failure is losely linked with the proesses that take plae in their mirostruture. The knowledge of the phenomena is essential to model the material failure. This is partiularly important in the ase of strutural steels used for engineering purposes as the information helps predit the time of failure of an entire struture. Mirostrutural observations of the failure proesses in many metalli materials reveal that there are three major mehanisms of frature: brittle frature, shear frature and dutile frature. Dutile frature is typial of basi strutural steels used in ivil engineering. The failure phenomena are due to the presene of mirostrutural defets in the form of voids, as shown shematially in Fig. 1. During deformation, voids form at inlusions and preipitates beause their stiffness is higher than that of the matrix. The failure of the material is thus dependent on the void growth, whih is the most important phenomenon observed during the damage proesses taking plae in the material mirostruture. The formation of mirodefets has influene on the material behaviour in the marosale ausing, for instane, a derease in the material strength (see the stress-strain σ(ε) urve in Fig ). Fig. 1. Formation of mirodefets in a material undergoing dutile frature (based on [1]) Fig.. Derease in the strength of the material due to the growth of mirostrutural defets desribed by the damage parameter d [] * KIELCE UNIVERSITY OF TECHNOLOGY, 7. TYSIĄCLECIA PAŃSTWA POLSKIEGO AL., KIELCE, POLAND # Corresponding author: kossak@tu.kiele.pl Unauthentiated

2 168 After the stress reahes a maximum, it begins to drop and neking is observed. Simultaneously, there is a signifiant rise in the void growth rate resulting in an inrease in the volume of mirodefets [3-4]. The prevailing state of stress also hanges, espeially when the stress triaxialities are high. These phenomena have been desribed by researhers looking for a measurable parameter to determine the relationship between the formation of mirodefets and the redution in stress σ in the material. Aording to the most general approah, the influene of the defet formation on the material strength an be desribed using the damage parameter d; it is written as a general formula: σ d = σ(1 d ). As an be seen, the values of stress determined by taking into aount mirodefets (σ d ) are muh lower than those determined using the assumptions of the ontinuum mehanis of solids (σ). The material fails when the damage defined by the damage parameter d reahes a ritial value, i.e. when d = 1. Sine the effet of mirodefets on the material strength at eah stage of failure is so lear, it should be taken into onsideration in many material models desribing the response and behaviour of a material undergoing various deformations until it fratures. For example, Kahanov haraterised this phenomenon using the damage parameter, whih is defined as the damaged area divided by the unaffeted area [5]. Some material models based on failure mehanis were developed later. An original approah was presented in 1977 by Gurson, who linked the material damage with the volume of mirodefets haraterised by the so-alled void volume fration f [6]. Aording to this model, a void exists physially in a unit ell of a material, as shown shematially in Fig. 3a. The void volume fration f is defined as f = V v /V, where V v is the volume of the voids and V is the volume of the material. Thus, the void volume fration f is desribed as randomly distributed avities (Fig. 3b). a) b) Fig. 3. (a) Mirodefet in a material unit ell; (b) randomly distributed avities representing the void volume fration f [4] The original Gurson model was developed by several authors. The major modifiation was made by Tvergaard and it involved introduing the oeffiients q i, whih define the plasti properties of the material [7]. Then, Tvergaard and Needleman redefined the funtion of void volume fration f as f * [8]. Now the Gurson yield funtion is ommonly defined as the Gurson- Tvergaard-Needleman (GNT) material model and it is expressed in the following form: e 0 q f 1 * osh q 3 m 0 * 1 q f 0 where: σ e effetive stress defined aording to the Huber- Mises-Henky strength hypothesis 1 e (1a) (with σ 1, σ, σ 3 being the prinipal stresses), σ 0 flow stress of the matrix material (yield stress), σ m hydrostati stress, f * modified void volume fration and q i Tvergaard oeffiients. The typial values of the Tvergaard oeffiients used for metalli materials are: q 1 = 1.5, q = 1.0 and q 3 = q 1 =.5 [7-9]. For years, the values were treated as onstants, whih an be easily found in the literature. Then, however, it was revealed that the Tvergaard oeffiients were dependent on the elasti and strength properties of the material, i.e. the modulus of elastiity E, the yield strength σ 0 and the strain hardening exponent N [10]. The values of the Tvergaard oeffiients for S35JR steel determined using this assumption are: q 1 = 1.90, q = 0.81 and q 3 = q 1 = 3.61 [4]. The modified void volume fration f * determines the influene of mirodefets on the stress state. While the material undergoes plasti deformation, its value hanges aording to the following funtion: f * f f 1/ q1 f f f F f f for for 3 f f f f where: f ritial void volume fration orresponding to the onset of void oalesene, f F ritial void volume fration orresponding to the material failure. Tvergaard and Needleman modified the original Gurson funtion of the void volume fration f in order to model the material failure for a ase when the void volume fration f is higher than the ritial void volume fration f until the stress apaity is lost. As an be seen from formula (), the modified void volume fration f * hanges depending on the relationship between the void volume fration f and the ritial void volume fration f, with the latter orresponding to the onset of void oalesene. In the range when f is higher than f, the modified void volume fration f * is a funtion of the Tvergaard oeffiient q 1 and the ritial void volume frations f and f F. Tvergaard and Needleman defined f F as the ritial void volume fration at the time of final failure, i.e. when f = f F [8]. The ritial void volume fration at the time of failure is one of the GTN parameters that need to be determined to desribe the material properties. Its value differs depending on the material struture and the state of stress. Generally, f F is determined by ombining experimental and numerial methods [11-15]; the other GTN parameters are defined in a similar way [3,16-18]. Another approah assumes that the ritial void volume fration f F is determined only from experimental results [19-0]. The value of f F is established on the basis of tensile tests measurements of the void volume fration at the time of failure. This (1) () Unauthentiated

3 169 proedure was used in this study to determine the ritial void volume fration at final failure f F for S35JR strutural steel on the basis of the mirostrutural analysis. In this paper, the problem is onsidered for two different sales: miro and maro. The analysis at the mirosale refers to the proess of formation of mirovoids due to deformation (GTN model). Sine the state of stress has a onsiderable effet on the dutile frature of the material studied, the experiments were onduted under spatial stress onditions. The analysis involved determining the range of high initial stress triaxialities for non-deformed speimens. This is a marosale problem beause the failure proesses are referred to stress triaxialities. It should be noted, however, that in this study the stress triaxiality was used as a parameter desribing the spatial state of stress observed in the analysed elements and, therefore, it should rather be treated in a general way. The first step of the study involved subjeting speimens made of S35JR steel to stati tension. The tests were performed on four types of speimens with a irumferential noth, illustrated in Fig. 4.. Properties of the material tested A ommon grade of strutural steel, i.e. S35JR, was onsidered in this study. It is a mild, low-arbon steel used in ivil engineering for the onstrution of buildings and non-building strutures, for instane, bridges. Its hemial omposition is haraterised by the maximum ontent of elements: C = 0.14%, Mn = 0.54%, Si = 0.17%, P = 0.016%, S = 0.06%, Cu = 0.9%, Cr = 0.1%, Ni = 0.1%, Mo = 0.03%, V = 0.00% and N = 0.01%. Large amounts of impurities are observed in the mirostruture of S35JR steel. The mean yield stress of S35JR steel obtained in tensile tests [4] was R 0. = 318 MPa, whih was muh more than the required value for this grade, R eh = 35 MPa, aording to the PN-EN standard [1]. The mean ultimate tensile stress was R m = 446 MPa, while the mean perentage elongation was A t = 33.9%. 3. Experimental approah to the determination of the ritial void volume fration f F for S35JR steel As the state of stress has a onsiderable effet on the failure proesses observed in metals, espeially S35JR steel [3,4], it needs to be analysed extensively. The omplexity of the problem was determined by the stress triaxiality parameter, T, defined as: T (3) where: σ m = (σ 1 + σ + σ 3 )/3 hydrostati stress, σ e effetive stress. The analysis dealt with high stress triaxialities determined for non-deformed speimens, i.e. those observed in the initial state; hene the index 0. The initial stress triaxialities T 0 onsidered in this study ranged from to Obviously, T may hange in time as a result of deformation. The hanges are dependent on the initial value of stress triaxiality, but this problem is not analysed here. m e Fig. 4. Geometry of the speimens subjeted to stati tension under high stress triaxialities The initial stress triaxialities T 0 determined for different noth radii R ranged from to 1.345, as shown in Table 1. TABLE 1 Initial stress triaxialities T 0 for the speimens tested Noth radius R [mm] Initial stress triaxiality T 0 = σ m /σ e Unauthentiated

4 170 The experimental proedure applied in this study was based on the original assumption of Tvergaard and Needleman, who defined the ritial void volume fration of a material at the time of failure f F. Its values were determined from the results of tensile tests onduted for U-nothed round speimens subjeted to tension until deohesion ourred. The frature surfaes shown in Fig. 5 were analysed in order to alulate the void volume fration at the time of failure, with the value orresponding to the parameter f F. The investigations onerning the ritial void volume fration f F were arried out using the methods of quantitative image analysis. The images of the frature surfaes were proessed to identify the areas overed by voids at the time of material deohesion. The brightest areas in Fig. 6 show the intervoid ligaments that formed before failure. To isolate the void-overed regions, it was neessary to onvert the photographs into two olour images. The greysale was applied as a riterion. The photographs were first binarised to selet the brightest areas representing the intervoid ligaments; then, the darker areas (voids) were removed. The image of the intervoid ligaments are shown in Fig. 7. Fig. 5. Frature surfae of a tensile speimen made of S35JR steel The values of f F were determined by analysing mirosopially the frature surfaes of all the speimens (Fig. 4) at magnifiations ranging from 500 to The view of a frature surfae in Fig. 6 indiates that S35JR steel underwent dutile frature. Fig. 7. Binarised frature surfae of S35JR steel From the images it was possible to alulate the surfae area of the white regions representing the voids. The result divided by the total surfae area of the images gave the void volume fration f F at the time of failure. As mentioned above, the analysis also involved determining the initial stress triaxialities (T 0 = ). 10 regions of frature surfaes on 30 speimens were measured for eah value of T 0 (eah speimen type). The outliers were rejeted using Chauvenet s riterion. 4. Results and disussion Fig. 6. Frature surfae of S35JR steel The analysis of the results onsisted in determining the ritial void volume fration f F at the time of failure for S35JR steel subjeted to high triaxial stresses. The values of f F obtained for different values of the parameter T 0 are provided in Table. The values of the parameter f F for all the analysed speimens are high; they range from to They are similar but slightly higher than the typial value, i.e. f F = 0.667, used for many metals, inluding the steel onsidered in this paper [3,17,18]. The obtained results are also higher than those Unauthentiated

5 171 reported in many studies onerning the GTN material model, i.e. f F = [8,1-14]. TABLE Critial void volume frations f F for different initial stress triaxialities T 0 Initial stress triaxiality T 0 = σ m /σ e Critial void volume fration f F Another issue is the relationship between the ritial void volume fration f F and the prevailing stress intensity. The values of f F are the lowest for the lowest initial stress triaxialities onsidered in this study, i.e. T 0 = 0.556, and they inrease with inreasing T 0. As shown in Fig. 8, two regions of f F (T 0 ) an be distinguished in the analysed range of the parameter T 0. In one region, T 0 < When T 0 = 0.739, there is a hange in f F (T 0 ). In the other region, i.e. when T 0 = , the relationship f F (T 0 ) is linear. Fig. 8. Critial Void Volume Fration f F versus Initial Stress Triaxiality T 0 urve for S35JR steel This relationship an be expressed as an approximation funtion: f F (T 0 ) = 0.1 T (4) It should be noted that the above relationship refers to the initial stress triaxialities, i.e. the state when the elements are not loaded. The proess of deformation auses the element to hange at the marosale. The naturally ourring hanges in the stress state are not the same as those reported for the material desribed by the GTN model at the mirosale. The values of f F are also expeted to be different for time-dependent stress triaxialities. This problem was not onsidered in this study; being omplex, it needs further investigation. 4. Conlusions The following onlusions an be drawn from the researh presented in this paper: the values of the ritial void volume fration f F ranged from to , depending on the prevailing state of stress; they were lose to and slightly higher than the typial value, i.e. f F = 0.667; there is a relationship between the parameter f F and the initial stress triaxility T 0 ; the higher the values of T 0, the higher the values of f F ; the linear relationship f F (T 0 ) for T 0 = an be represented by the approximation funtion f F (T 0 ) = 0.1 T , provided that the relationship was determined for initial stress triaxialities not taking into aount the hanges in the state of stress resulting from time-dependent deformations. REFERENCES [1] C. Ruggieri, Numerial investigation of onstraint effets on dutile frature in tensile speimens, J. Braz. So. Meh. Si. 6, (004). [] P.G. Kossakowski, W. Wiślik, Experimental determination and appliation of ritial void volume fration f for S35JR steel subjeted to multi-axial stress state, in: T. Łodygowski, J. Rakowski, P. Litewka (Eds.), Reent Advanes in Computational Mehanis, , CRC Press/Balkema, London, 014. [3] P.G. Kossakowski, An analysis of the load-arrying apaity of elements subjeted to omplex stress states with a fous on the mirostrutural failure, Arh. Civ. Meh. Eng. 10, (010). [4] P.G. Kossakowski, Mirostrutural failure riteria for S35JR steel subjeted to spatial stress states, Arh. Civ. Meh. Eng. 15, (015). [5] L.M. Kahanov, Time of the rupture proess under reep onditions, Izvestiia Akademii Nauk SSSR, Otdelenie Tekhniheskikh Nauk 8, 6-31 (1958). [6] A.L. Gurson, Continuum theory of dutile rupture by void nuleation and growth: Part I Yield riteria and flow rules for porous dutile media, J. Eng. Mater-T. ASME 99, -15 (1977). [7] V. Tvergaard, Influene of voids on shear band instabilities under plane strain onditions, Int. J. Frature 17, (1981). [8] V. Tvergaard, A. Needleman, Analysis of the up-one frature in a round tensile bar, Ata Metall. 3, (1984). [9] V. Tvergaard, Material failure by void growth to oalesene, Adv. Appl. Meh. 7, (1989). [10] J. Faleskog, X. Gao, C.F. Shih, Cell model for nonlinear frature analysis I. Miromehanis alibration, Int. J. Frature 89, (1998). [11] I.I. Cuesta, J.M. Alegre, R. Laalle, Determination of the Gurson Tvergaard damage model parameters for simulating small punh tests, Fatigue Frat. Eng. M. 33, (010). [1] Z.L. Zhang, C. Thaulow, J. Ødegård, A Complete Gurson model approah for dutile frature, Eng. Frat. Meh. 67, (000). Unauthentiated

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