Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 4, pp. 995-1006, Warsaw 2014 STRESS MODIFIED CRITICAL STRAIN CRITERION FOR S235JR STEEL AT LOW INITIAL STRESS TRIAXIALITY Paweł Grzegorz Kossakowski Kielce University of Technology, Faculty of Civil Engineering and Architecture, Kielce, Poland e-mail: kossak@tu.kielce.pl Ductile fracture of low carbon structural steel (S235JR) at low initial stress triaxiality has been predicted using a method based on the Stress Modified Critical Strain (SMCS) crite- rion and the Gurson-Tvergaard-Needleman (GTN)material model. The influence of micro- defects on the material strength has been taken into account. The investigations, including tensile tests, have been conducted for standard cylindrical unnotched tensile specimens at low triaxial stresses. An advanced finite elementmethod has been used to determine several SMCSmodel parameters. Keywords: StressModified Critical Strain (SMCS) criterion, Gurson-Tvergaard-Needleman (GTN) material model, numerical simulations, S235JR steel 1. Introduction Assessment of the load-carrying capacity of structural elements that are damaged or overloaded is a common problem encountered in engineering practice. In either case, if the capacity of elements is exceeded, the structure, especially whenmade of steel, is no longer safe. In extreme cases, the structuremay operate in a pre-failure state, leading finally to a catastrophe. The load- -carrying capacity analysis becomes more complicated when structural elements are plastically deformed or otherwise damaged. Risk assessment is absolutely necessary for structures showing signs of failure, i.e., visible deformations or cracks. A thorough analysis of the load-carrying capacity of structural elements made of steel in a pre-failure condition is definitely difficult,mainly because of the high complexity of the problem. Some structural elements are able to operate without failure even though they are beyond the elastic limit. In such a case, we cannot applymethods based on the classic strength hypotheses; we can only use assumptions andmethods of the theory of plasticity. Serious engineering problems may also be attributable to cracks. Cracks affect the load- -carrying capacity of single elements, and consequently, the entire structure. It can be difficult to assess stresses around a crack and determine their influence on the load-carrying capacity of an element and on the real fracture toughness of the material, i.e., steel. The knowledge of the type of fracture is crucial.Whenbrittle or plastic fracture occurs,we can use the classicmethods of fracture mechanics. In the case of transition or elastic-plastic fracture, however, wemay find it highly problematic to correctly assess the fracture toughness. Other important problems that should be analyzed by structural engineers are the influence of defects of structural elements on the stability of whole structures as well as the influence of defects of welds on the load-carrying capacity of welded connections. Even though many procedures have been developed and fracture mechanics provides some well-defined criteria, the analysis of the load-carrying capacity of an element containing a crack is likely to be affected by various parameters, the choice of which is no longer obvious. It is essential to consider the criteria for the application of linear and nonlinear fracture mechanics. Theyare dependenton the size and shape of theplastic zone,which canbemeasuredusingmany 996 P.G. Kossakowski acceptable methods. In the case of full plasticity, accurate assessment of the fracture toughness can be difficult because of the problemswithmeasurement of the critical value of the J-integral. Othermaterial constants can alsobeused for the analysis in a fairlywide range (minimum,mean or probable values), and theymay significantly affect the results. Hence the need to systematize and unify the methodology for assessing the strength and safety of mechanical systems and building structures. Many components used in construction are made of steel. Crack initiation in steel is related to thematerial microstructure, thus, ductile fracture is a basicmechanism of the failure.Micro- -cracks occur as a result of micro-defects (voids) present in the material structure nucleating on inclusions and second-phase particles in the material matrix. The nucleation, growth and coalescence of voids are the consecutive steps in the micro-damage evolution. The void growth is responsible for the development of localized plastic deformations, being the most important stage leading to material failure. Ductile fracture has been analyzed bymany researchers, but many problems are still neces- sary to solve. For instance, experimental methods are used to analyze the coalescence of voids. To find outmore about ductile fracture, we need to carry outmore extensive research and deve- lopmore effectivemethods for determining the load-carrying capacity and predicting the failure time. Today, one of the methods used for predicting ductile fracture as a function of multi-axial stresses and strains in the plastic range is the Stress Modified Critical Strain (SMCS) model. The model seems to be a very useful method from the practical point of view in comparison with the conventional fracture toughness methods. To determine the ductility of a material, the SMCS model uses quantitative results, whereas in a traditional Charpy impact test, for example, assessment is made on the basis of qualitative data. The SMCS model may be used also in analyses of structural elements with no visible cracks, when there is no crack tip yielding. Engineers applying this model deal with multi-axial, especially triaxial, stresses, and strains in the plastic range. In traditional methods, analysis is based on themaximum plastic strains. The SMCS method combines experimental testing with numerical analysis, with the latter being particularly important here. Selecting the bestmodel to assess the failure of a givenmetal material is the most significant decision that a structural engineer needs to make. Phenomena connected with ductile fracture may be analyzed and simulated using damage material models, taking into account the influence of microstrucural defects on the material strength. One of the first damage material models to take into consideration the relationship between the particular stages of failure and the strength of material was the Gurson material model (1977) developed for porousmedia. In theGursonyield function, being amodified formof the Huber-Mises-Hencky hypothesis, the strength of a material is affected by an increase in the void volume fraction.TheoriginalGursonmodel has undergonevariousmodifications.Themain ones were proposed by Tvergaard (1981), Tvergaard and Needleman (1984), and Needleman and Tvergaard (1984). The resulting Gurson-Tvergaard-Needleman (GTN) material model is now one of the basic damage models recommended for use in analysis of ductile fracture. The GTNmaterial modelmay be helpful inmodelling and simulating the stresses and strains during material failure, as they are parameters that are necessary to determine the failure criteria based on the other models, such as the SMCSmodel. In this study, ductile fracture in structural steel was predicted by means of a method that is a combination of the SMCS and GTN models. The mechanisms of ductile crack initiation weremodelled using the spatial distributions of stresses and strains at themicroscale. Although this approach focuses on fracture behaviour, which is a particularly complex issue, the model is anticipated to become a useful tool for solving various engineering problems. It is important that ductile fracture should be analyzed by referring to the microstructural aspects and that failure in the particular steel elements should be predicted on the basis of quantitative data. Stress Modified Critical Strain criterion for S235JR steel... 997 Much of the previous research on the SMCSmethod has focused onmechanical engineering applications and on steels that are not common in construction. This study has been conducted for low-carbon S235JR steel, which is a basic steel used in the construction industry in Poland and other European countries. S235JR steel is applied to produce a number of structural ele- ments, especially for construction applications. To predict and prevent failures and collapses of steel structures in construction, engineers need reliable methods for predicting failure processes in thematerial and the element under load conditions. It is thus vital to develop failure criteria for structural steels used in construction, including S235JR steel. Thispaperanalyzes theapplicability of theSMCSmodel to engineeringproblemswith regard to S235JRsteel. The studyhasbeen conducted for cylindrical unnotched specimens at low initial stress triaxiality σm/σe =1/3, where σm and σe denote the hydrostatic stress and the effective stress, respectively. The investigations have been partially based on the results presented by Kossakowski (2012a,b). They involved performing a microstructural analysis, tensile tests and an advanced finite element analysis by means of the GTN model. The fundamental purpose of the study, however, is to determine the SMCScriterion at low initial stress triaxiality for S235JR steel. 2. The SMCS model and the length scale parameter The SMCSmodel is one of themethods used for the assessment ofmaterial failure based on the relationship between the spatial plastic strain and the stress state at failure. The fundamental assumptions of the SMCS model were prepared by McClintock (1968) and Rice and Tracey (1969), who focused on the void growth phenomenon leading to the material failure. They noticed that the void growth is due to changes in the strain and stress state, defined by two key parameters, the effective strain εple and the stress triaxiality σm/σe. The SMCSmodel and method were further improved by Hancock andMackenzie (1976), Hancock and Brown (1983), Johnson andCook (1985), Marino et al. (1985), Panontin and Sheppard (1995), Bandstra et al. (2004), Benzerga et al. (2004), Kanvinde and Deierlein (2004, 2006) and Chi et al. (2006). Asmentioned before, ductile fracture is a complex phenomenon, dependent onmany factors such as anisotropy of the void distribution, spacing and shape, void nucleation, changes and evolution in void shapes, void-to-void interactions, primary void nucleation and the nucleation andgrowth of secondaryvoids.As themodels of these phenomenaare too complex andadvanced to use in common engineering practice, it is necessary to propose a simple and accuratemethod to simulate ductile fracture to use in engineering assessments. Assuming that ductile fracture is largely dependent on void growth, and that void nucleation does not affect the ductile fracture process significantly, the void coalescence may be modelled and defined by the critical void ratio only, neglecting the void-to-void interactions or void loca- lization. The void growth phenomenon has been analyzed by many researchers, including Berg (1962), Rice andTracey (1969), McClintock (1968), Thomason (1968), and Brown and Embury (1973). According to the SMCS model applied in this study, prediction of ductile failure requires considering the relationshipbetween stress triaxiality andplastic strainonly.Fromthese assump- tions, it is clear that we can apply the SMCS failure criterion to low carbon steels, commonly used in the construction industry. Ductile crack initiation in these steels is mainly due to the growthandcoalescence of voids responsible for thedevelopment of localized plastic deformations. The SMCS fracture criterion enables us to evaluate the initiation of ductile fracture depen- ding on the stress and strain, definedbymulti-axial stresses and plastic strains. The relationship between instantaneous stresses and strains at fracture initiation can be used to determine the critical plastic strain εplc , being a critical parameter, as a function of stress triaxiality σm/σe. 998 P.G. Kossakowski According to the SMCSmodel, the critical plastic strain εplc is calculated as εplc =αexp ( − 3 2 σm σe ) (2.1) where α is the toughness parameter, σm – hydrostatic stress, σe – effective stress. The toughness parameter α is a constant for thematerial tested, generally determined thro- ugh experiments. This parameter is established by combining the experimental and numerical methods. The time to failure is calculated from the results of the strength test, while the critical plastic strain εplc is obtained through numerical simulations of the stress and strain state at failure. The toughness parameter α can be calculated as follows α= εplc exp ( −3 2 σm σe ) (2.2) During thenumerical simulationsperformedbymeansof thematerial damagemodels, several problems have been encountered. When ductile fracture was simulated with a finite element method, i.e., in the nonlinear range, the mesh-size effects revealed the softening of the final part of the strength curve. This phenomenon is strictly connected with the fracture processes occurring at the micro-scale. According to the fracture criterion, two conditions must be satisfied to initiate a crack. The critical stress and the plastic strain must be exceeded over the critical volume of the material, which, in a two-dimensional analysis, is defined by the characteristic length lc. Thus, the SMCS fracture criterion can be written as εpl >εplc over r > lc (2.3) where εpl is the plastic strain, εplc – critical plastic strain, r – length, lc – characteristic length. Fracture condition (2.3) is shown schematically in Fig. 1, where the region of high stress and plastic strain is presented. The fundamental parameter is the difference between the effective strain εple and the critical plastic strain ε pl c , marked by contours to indicate the subsequent loading steps in the region near the crack tip. Fig. 1. Predicting ductile crack initiation defined by the characteristic length lc (based onKanvinde andDeierlein, 2006) Crack formation ispredicted for aprogressively increasingdistanceuntil the fracture criterion is satisfied over the characteristic length lc. The first loading step when the fracture criterion is satisfied over the characteristic length lc corresponding to fracture initiation is presented as a solid line. Stress Modified Critical Strain criterion for S235JR steel... 999 As can be seen, the characteristic length lc is a basic parameter determining the crack initiation process, which is particularly important in numerical simulations of ductile fracture. The characteristic length lc defines the propermesh size in the region prone to crack initiation and, accordingly, failure.Theparameter is vital in themodellingof regionswithhigh stress-strain gradients, for instance, the area near the crack tip.TheSMCS fracture criterion is satisfiedwhen the size of the finite elements is exceeded over the critical volume of the material, represented in two dimensions by the characteristic length lc. Summing up, the mesh size should be at least equal to the characteristic length lc. The theory and methods used to determine the characteristic length lc are presented further in the paper. 3. The GTN material model Asmentioned before, theGursonmaterialmodel for porousmedia is one of thematerial damage models that take into account the influence of microstrucural defects on the material strength. TheGurson yield criterion replaced the Huber-Mises-Hencky yield criterion. It is assumed that the proportion of voids in the plastic potential function is dependent on the void volume frac- tion f, defining the influence of microfailure (pores, voids) on thematerial strength. According to the GTNmodel, failure loads related to ductile fracture can be determined by analyzing some microstructural parameters and plastic properties of the material. As shown in many studies conducted for alloys and structural steels, theGTNmodel provides goodagreement between the predictions and the experimental data. To numericallymodel the load limit for any element subjected to any load, we need to apply an appropriate procedure and take the effects of micro-failure into account. The parameters of thematerial microstructure can be determined by analyzing the actual data rather than matching the parameters to the tensile test results. Accordingly, the load-carrying capacity can be calculated numerically by analyzing the stress state, changes in the microstructure and, finally, the safety reserves of the particular elements. The GTN yield criterion modified by Tvergaard (1981), Tvergaard and Needleman (1984) and Needleman and Tvergaard (1984) is written as Φ= (σe σ0 )2 +2q1f ∗cosh ( −q2 3σm 2σ0 ) − (1+q3f∗2)= 0 (3.1) where σe is the effective stress, σ0 – yield stress, σm – hydrostatic stress, f ∗ – modified void volume fraction, qi – Tvergaard’s parameters. As can be seen, the material porosity is dependent on themodified void volume fraction f∗ f∗ =            f for f ¬ fc fc+ fF −fc fF −fc (f−fc) for fc σ1 (4.1) where ε is the 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. The strength response of S235JR steel described bymodel (4.1) has been used to determine theGTNmodel parameters. First, the fundamental GTNparameter, initial porosity defined by the void volume fraction, has been determined throughmicrostructural analysis as f0 =0.0017. Like in the work by Faleskog et al. (1998), Tvergaard’s parameters have been established on the basis of the strength properties obtained experimentally. The values of Tvergaard’s para- meters are: q1 =1.91, q2 =0.79 and q3 =3.65 for R0.2/E =0.00155 and the strain-hardening exponent N =0.195. The other parameters of the GTN model have been determined by combining the experi- mental and numerical methods, as described byKossakowski (2010, 2012a,b). The results of the tensile tests have beenmodelled numerically using a program based on the Finite ElementMe- thod, Abaqus Explicit version 6.10. TheGTNmaterial model parameters have been established numerically on the basis of the σ(ε) curves obtained experimentally. The values of the GTN parameters have been changed iteratively within certain limits using the optimization criterion based on the convergence of the σ(ε) strength curves obtained numerically and experimentally. All the GTNmaterial model parameters for S235JR steel are summarized in Table 1. Table 1.Microstructural parameters of theGTNmodel for S235JR steel (Kossakowski, 2012a) 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 1002 P.G. Kossakowski 4.3. The SMCS criterion for S235JR steel at low initial stress traxiality The analysis of the SMCS criterion for S235JR steel at low initial stress triaxiality has been determinedusing the results of the tensile tests obtained for unnotched specimenswith a circular cross-section. The minimum initial stress triaxiality σm/σe is 1/3. The key parameters of the SMCS model, i.e., the critical strain and stress triaxiality, have been determined by using the strength curves obtainedduringexperimentswith the spatial stress-straindistributions at failure modelled numerically bymeans of the GTNmaterial model. The force-elongation F(l) curves obtained experimentally and numerically are shown in Fig. 3. The time to failure is the fundamental parameter required to establish the spatial stress- -strain distributions. Fig. 3. Force-elongation F(l) curves obtained during the experiments and numerical simulations for S235JR steel at low initial stress triaxiality The stress-strain state has been determined by numerically simulating the tensile test results using the Abaqus Explicit version 6.10, a program based on the Finite Element Method. The main idea is to couple the SMCS andGTNmodels to predict fracture in the structural steel on the basis of the failure criterion (2.3). The distributions of stress and strain at material failure have been identified numerically using theGTNmaterial model, which enabled us to determine the SMCS failure criterion for S235JR steel at low initial stress triaxiality. In the numerical simulations, axially symmetrical elements, representing specimens with a circular cross-section, have been subjected to static tension with strain increasing at a rate of 10−2 s−1. The modelling has been performed on 1/4 specimens because of the specimen symmetry. For this approach, it is possible to block horizontal displacements along the specimen axis (left edge of the model) and vertical displacements in the crack tip region (lower edge of the model), as shown in Fig. 4. Crack formation is initiated by producing a sharp notch (R=0.05mm)at the bottomof the numerical model, in the middle of the gauge length. The process zone has been modelled using microstructural length scales suitable for S235JR steel. The dimensions of the finite elements localized near the crack plane are D×D/2, where D corresponds to the characteristic length lc =250µm (Fig. 4). There are 120 finite elements in the failure zone and 484 finite elements in the other regions. Thenumber of nodeswas 670. TheGTNmaterialmodel developed for porous media has been adapted to suit the whole numerical model using the parameters provided in Table 1. The effect of the FEM mesh density has been analyzed for an approach in which the size of the FEMmesh corresponds to the distance between primary voids in the material structure. The FEM model has been constructed in the same way as the fundamental FEM model based on the 250× 250µm mesh in the fracture zone. In the model analyzed here, the size of the Stress Modified Critical Strain criterion for S235JR steel... 1003 Fig. 4. Geometry of the numerical models used for the simulations FEM mesh has been established on the basis of the average distance between primary voids, which is 100µm (Kossakowski, 2012a). The size of themesh in the failure zone is 100×100µm. The values of and the relationships between the stress triaxiality ratio σm/σe and the effective strain εple in the failure plane are almost identical as those observed in the fundamental model based on a 250µmmesh. It is important to note that there is no considerable difference between the toughness parameter α for an FEM mesh with a greater density and that obtained for a 250×250µm mesh in the failure zone, because it is only 2.7% lower. It can be concluded that the density of the FEM mesh with finite elements ranging 100-250µm in the failure zone has no significant effect. Taking account of this and good convergence of the results of tensile test simulations with a mesh based on the length scale lc, i.e. a 250× 250µm mesh in the failure zone, we assume that this mesh size is fundamental. The results presented below concern the approach with the 250×250µm FEMmesh density in the failure zone. Failure has been identified numerically on the basis of the force-elongation curves shown in Fig. 3, taking into consideration the time to failure observed during the experiments. The corresponding stress-strain state has been determined using the GTN material model, i.e., by analyzing the influence of the micro-defects on the material strength. Figure 5 shows the di- stributions of the effective strain εple and the stress triaxiality σm/σe determined at a load corresponding to that at the fracture initiation. As can be seen from Fig. 5, the most intensive distributions of the effective strain εple and the stress triaxiality σm/σe at failure are observed in the central part of the elements, along the axis of symmetry. This region is the most prone to crack initiation. Therefore, the critical stress-strain distributions in this region have been used to determine the SMCS failure criterion for S235JR steel. The critical SMCSparameters have been based on the stress triaxiality σm/σe and effective strain εple curves determined at fracture initiation at thenodes of thefinite elements along the line from the centre of the cross-section of the fracture region up to the side, as shown in Fig. 6. It can be seen that the stress triaxiality changes significantly along the line going from the centre of the minimum cross-section to the side. At the central point, σm/σe =1.123, while at the end σm/σe = 0.318. The effective strain distribution is similar, but the differences at the central and end points are smaller. At the central point, the effective strain εple was 0.997, while at the end εple was 0.760. 1004 P.G. Kossakowski Fig. 5. Distributions of the effective strain εple (a) and the stress triaxiality σm/σe (b) at failure Fig. 6. Stress triaxiality σm/σe and the effective strain ε pl e curves The distributions and the critical values in the region prone to crack initiation and failure have been determined as the critical plastic strain, εplc = 0.997, and the corresponding stress triaxiality, σm/σe = 1.123. According to formula (2.2), the toughness parameter α has been calculated as α= εplc exp ( −3 2 σm σe ) =5.374 (4.2) Thevalue of the toughness parameter,α=5.374,maybe treated as thematerial constant for S235JR steel at low initial stress triaxiality, σm/σe =1/3, in the static range. Themethod and the criterion presented above can be applied to solve practical engineering problems. According to the SMCScriterion, the failure of S235JR steel can be predicted using the following condition εpl >εplc =5.374exp ( − 3 2 σm σe ) (4.3) It should be noted that the SMCS criterion, (4.3), may be applied also to steel grades with metallurgical content and strength properties comparable with those of S235JR steel in the case of low initial stress triaxiality. Stress Modified Critical Strain criterion for S235JR steel... 1005 5. Conclusions The SMCS model applied to predict ductile fracture in steels seems superior to both the co- nventional fracture toughness methods and the more sophisticated assessment techniques. The SMCS model, based on the distributions of multi-axial, especially triaxial, stresses and plastic strains can also be used to analyze the behaviour of structural elements with no visible cracks, or no crack tip yielding. Themodel and the failure criterion seemwell-suited for awide range of engineering problems, especially the prediction of failure and the assessment of the load-carrying capacity of structures. The conclusions can be summarized as follows: • TheSMCScriterion is an easy anduniversal tool todealwith various engineeringproblems related to the strength and load-carrying capacity of steel structural elements. It can be used to predict ductile crack initiation when cracks are not visible, when there is no crack tip yielding, in the case of large-scale yielding and in flaw-free geometries. • By combining the SMCScriterionwith theGTNmodel,we can determine the stress-strain state at failure.When the strength of amaterial is analyzed, the influence ofmicro-defects needs to be taken into account. • The toughness parameter for S235JR steel is: α=5.374. This value can be treated as the material constant to be used in the fracture condition when the material is under static triaxial stress. 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