Strojniški vestnik - Journal of Mechanical Engineering 64(2018)5, 339-348 © 2018 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2017.5137 Original Scientific Paper Received for review: 2017-12-08 Received revised form: 2018-02-17 Accepted for publication: 2018-03-13 Computational and Numerical Analysis of Ductile Damage Evolution under a Load-Unload Tensile Test in Dual-Phase Steel Juan Anduquia-Restrepo1 - Carlos Narvaez-Tovar2 - Rodolfo Rodriguez-Baracaldo1* 1 National University of Colombia, Department of Mechanical Engineering and Mechatronics, Group: Innovation in Manufacturing Processes and Materials Engineering, Colombia 2 National University of Colombia, Department of Mechanical Engineering and Mechatronics, Modelling Group and Numerical Methods in Engineering, Colombia Dual-phase (DP) sheet steels are used in the automotive industry. They have a microstructure that consists of a ferrite matrix with dispersed martensite islands giving a combination of good formability and high strength. However, they also exhibit ductile failure caused mainly by high strain incompatibility in both phases, which continues to be an issue of discussion among researchers. To capture the mechanical degradation of a DP sheet steel, this research focuses on the damage characterization using a continuum damage model and loading-unloading uniaxial tensile tests to quantify ductile failure without incurring expensive and difficult mechanical tests, which has the potential to provide an understanding of the identification of damage parameters in the metal-forming industry. By comparing experimental tests and computation simulations, the model presents minimum errors considering triaxiality as a constant value. Key words: continuum damage mechanics, dual-phase steels, load-unload test Highlights • Progressive deterioration of mechanical strength in a dual-phase steel is investigated by means of indirect measurement of elastic modulus. • The development of a hybrid calibration procedure between conventional experimental tests and numerical simulations to predict mechanical response is assessed. • A coupled isotropic J2 plasticity with hardening saturation and a ductile damage model is used. • Initial fracture takes place in a strain localization region. 0 INTRODUCTION DP steels are defined as low carbon steels that have formability, mechanical strength, hardenability, and toughness, enabling them to be used in the automotive industry for light-weight design [1]. The microstructure of DP steels is composed of two phases. Normally, hard martensite islands are embedded in a soft ferritic matrix. This influences the material's behaviour and, hence, its macroscale response because of the large plastic strains presented by the ferrite [2]. DP steels exhibit features during large plastic strains that make them distinct from other structural steels [3] to [5]. These features include the complex interaction of strain-hardening behaviour between the two phases, developing stress saturation effects during large strains, and dependence on failure modes by different states of stress, among others [3], [6] and [7]. Ductile failure can mainly occur via martensite cracking, martensite-ferrite interface decohesion, or both [3] and [8]. The effects may be considered through damage mechanics by three stages. The first stage is the microvoid nucleation during plastic flow. Next, the voids keep growing with continued plastic strain. Finally, the voids link to produce coalescence, and consequently, complete failure [7] to [9]. Therefore, a complete loss of load-carrying capability of the material occurs, which leads to collapse by ductile fracture [10] and [11]. Traditionally, the methods used to predict failure have been based on systematic and expensive testing of real models under laboratory conditions [12]. However, with the progressively growing knowledge about ductile failure mechanisms in steels, along with the development of computational power, it is becoming possible to define constitutive models that describe the internal behaviour in materials [13] and [14]. Continuum damage mechanics (CDM) has been a reliable tool to predict failure [15] and [16]. The conventional approach to CDM has focused on the interaction of the progressive deterioration of mechanical strength [17], which is formulated within the irreversible thermodynamics framework. This formulation, introduced by Lemaitre in [10], [17] and [18], presents a local damage indicator through an internal variable to describe the progressive deterioration of the ductile behaviour in steels up to fracture. The prediction of ductile damage, fracture, and forming limits in sheet metal forming processes *Corr. Author's Address: National University of Colombia, Department of Mechanical Engineering and Mechatronics, Bogota, Colombia, rodriguezba@unal.edu.co 339 requires the choice of a variable to identify and measure damage evolution; this is essential for obtaining a good understanding of nonlinear mechanical properties in new advanced steels, and it is mainly restricted by the complexity to be detected in experimental tests [19] and [20]. For practical industrial usage, the importance grows as one must assure a reliable identification of the material parameters using only simple conventional equipment to implement tests, such as uniaxial tension, hardening, among others [21] and [22]. The stiffness degradation by uniaxial tensile test with load-unload cycles is a recognized and effective method of assessing ductile damage processes on sheet metals through the calculation of reduced Young's modulus during the increases of the plastic strain. The ability to obtain associated small errors (about +5 %), makes it sufficiently reliable to be used in engineering applications [11] and [23]. Initially, this technique consisted of obtaining Young's modulus values only by the unloading ramp [10], [24] and [25]. However, various studies have reported the nonlinear of the unloading curve on some metals. For instance, DP steels present a denoted hysteretic loop of loading-unloading curves resulting from the strength ratio between the ferrite and martensite increases [1], [3], [26] and [27]. In contrast, the calculation of the stress-strain curve using extensometer reading for planar specimen tensile tests at the necking stage reduces the accuracy of damage measurement due to the nonuniform strain processes throughout the minimum cross-section [28]. In [28], it was considered necessary to apply an empirical method based on the geometrical relations of the specimen post-necking cross-section. Subsequently, [29] reported that the method proposed by [28] had difficulties in accurately calculating current cross-sectional areas. Instead of that, they proposed using load-displacement curves and necking evolution. In recent years, more accurate and reliable methodologies have been developed for the analysis of plastic instability through finite element analysis (FEA). For instance, [30] used an experimental-numerical method of digital image correlation (DIC) and FEA to obtain the local surface strain field. The results presented the high resolution of the measurements applicable to moderate plastic strain gradients. Cabezas et al. [31] used the Bridgmang-Zhang solution to study the large strain process of tensile sheets through a combined method applying an experimental stress-strain curve and FEA. Accordingly, the following work focuses on studying the damage behaviour in a DP590 steel by means of experimental tests and numerical simulations to assess the ductile fracture and the effect of the localization in strain on the damage evolution. For this, we have developed uniaxial tensile tests with load-unload cycles for obtaining the damage parameters through indirect measures of the elastic modulus degradation. The reason for this is that microdefects generate appreciable changes on the macro response of the material. In that manner, a fully coupled elastic-plastic-damage model using the theory of CDM has been implemented to predict the ductile damage behaviour observed during the tension tests. The formulation has been developed into an explicit integration scheme in which the simulation results are compared with the experimental results. 1 DUCTILE DAMAGE MODEL In this work, a version of Lemaitre's isotropic damage model is used without taking into account the strain rate [32]. This model establishes the hypothesis of strain equivalence to define constitutive behaviour between the damaged material represented by the tensor stress (o), and the virgin material represented by the effective tensor stress (cf). Both tensors are related by the damage parameter D, through Rabotnov's formulation [33]. CC = c / (1 - D), (1) Based on experimental observations, in [10] an indirect manner was postulated to measure the damage in ductile materials, through the degradation on elastic modulus while increasing plastic strain. The damage variable (Eq. (1)) is redefined as: a = i - E, E (2) where EE is the reduced elastic modulus and E is the elastic modulus in the ideally isotropic state. The evolution law for internal and observable variables can be obtained from the Helmholtz free energy, assuming that it follows a convex function [32]; derived into elastic, and plastic, ^p components as: y/ +yv = — i-C: s* :se (1 -D)+ Ra P 12 P + B exP(-bs ") (3) where p is the density, C the fourth order elasticity stiffness tensor, Rx and B are two material parameters of isotropic hardening, 8s the second order elastic strain tensor, and s p is the accumulated plastic strain. Considering that the energy release rate associated to the damage D, is the amount of energy available to initiate and propagate ductile damage [17], [18] and [32], this expression can be given by: Y = - CT„ 2 E (1 - D )2 2 _ (1 + v )+3(l - 2v ) Y CT V eq . (4) The expression from Eq. (4) inside the bracket can be contracted by a triaxiality factor: Rv = 2 (13 -3(1 - 2v ) ( Y sjp' (11) where j is the plastic consistency parameter, which obeys Kuhn-Tucker loading-unloading conditions. Y > 0; 0; /® = 0. (12) 1.1 Implementation of a Ductile Damage Model in FE Simulation In this section, we present the algorithm with a modified hardening law type saturation stress for the numerical integration of the elasto-plastic-damage Lemaitre's model inspired in the work proposed by Lee and Pourboghrat [37]. The procedure was used to compute the state variables of constitutive equations employing a predictor elastic/corrector plastic step. Furthermore, the J2 plasticity theory was coupled with the CDM criteria. Let [0, T] be the time interval of study and As, strain increment, be the required to update the variables at tn+1 [37] and [38]. Moreover, an, sSP and Dn at tn are known. The FE simulation was implemented using a fully explicit forward Euler integration scheme. Assuming additive rule, the strain increment, As, is defined in elastic increment Ase and plastic increment As p as, As=Ase + Asp. For elastic trial state, As p = 0, corresponding to the elastic Hooke's law coupled with the damage, which is computed from [37] a,nal =Gn + (j _ ) J + ^^ ). where, fftriaI is the elastic predictor, X and ^ are Lame's constants, and I is the identity matrix. Next, the yield surface is checked using Eq. (14) to evaluate whether the trial stress is within elastic domain [28]. The trial deviatoric part of stress tensor GdtnaI is defined by Eq. (15). [j (r ) 1 - D _trial _trial 1 r _trial a d = a — I a . d 3 (14) (15) If the yield condition &inal < 0 is satisfied, there is no plastic behaviour or damage evolution and the state variables are updated as trial values at t^i, using Eq. (15). = CT D , = D (16) Otherwise, the process is elastic-plastic and the plastic corrector step should be used to compute the plastic strain. Eq. (14) must satisfy the consistency condition 0 = 0 through the trial deviatoric stress oJnal for describing plastic flow, which requires that Od,n+i is on the expanded yield surface at the end of plastic step [37], expressed as: o. (17) ( \ trial ( \ a a V eq V e y n+1 where, q is the radial direction for the plastic correction [37], which must satisfy the hardening isotropic condition denoted by: (18) and Rn+i is the radius of the yield surface obtained at tn+i by Eqs. (4) and (ii), and Asp =7273 Ay , Rn+l = ^2/3 (1 - Dn+l )ay V2T3 Rv (l - Dn-7273 Ayan), (19) = where, a 2ES (1 - D ) (20) In this work a different hardening law was used with respect to the initial algorithm. The Voce type saturation law was adapted for DP steels in Eq. (20). Therefore, hardening modulus h = da /dep at n y,n ' n instance n is defined as: =Vyo ( - eXP (w . (2i) Being oSat and a^+i the material parameters. Thus, from Eq. (i4), can be represented by: o o. - 2^(1 - Dn )Ayq. (22) Taking Eqs. (13), (17) and (22), we obtained the next expression that leads to a second-order equation with respect to Ay, .2 where, a (A/) + b (A/) + c = 0, (23) a = a„h„, (24) b = anay,n-(1 - Dn )( + 3G), (25) * = (1 - D), (26) Note that the two roots computed of Eq. (23) should satisfy the following constrains: Ay = min(Ayt ), Ay > 0, i = 1,2. (27) Solving second-order equation, we obtained the plastic corrector (Ay), which is used to update the state variables at tn+i. Finally, when Dn+i reaches the damage critical condition Dc, the algorithm stops. a^ -2^(1 -Dn)Ayq, e^ = enp +V2T3 A/, D+i = Dn +V2T3 AawA/. (28) 2 EXPERIMENTAL RESULTS 2.1 Material A DP steel, DP590, with 3.4 mm thickness was used. The chemical composition was obtained with an optical emission spectrometer (OES) BAIRD SPECTROVAC equipment. The composition results were: 0,15 % C, 1.045 % Mn, 0.409 % Si, 0.037 % S and 0.05 % P. 900 -, 800700600500400300200100- \ t -□- 0°(RD) —0—45° (DD) ^^ 90° (TD) 00.00 Fig. 0.02 0.04 0.06 0.08 0.10 0.12 Engineering strain [-] 1. Engineering stress-strain curves According to the ASTM E8 standard on planar specimens [39], uniaxial tensile tests were performed q on planar specimens at 0° (rolling direction: RD), 45° (diagonal direction: DD) and 90° (transverse direction: TD). Quantitative analysis developed on curves (Fig. 1) shows an isotropic behaviour in the DP steel. The tests were conducted in a Shimadzu UH-500 universal test machine of 500 connecteded to a computer for control and data acquisition. All tests were developed with a 5 mm/min displacement rate at room temperature to avoid dynamic effects on the material response. Table 1. Load unload tensile test of the DP590 steel, standard deviations shown (S.D.) Properties Value S.D. Elastic E [GPa] 214.8 12.6 v [-] 0.3 - [MPa] 535.4 27.9 Plastic ou [MPa] 771.2 32.6 °Sat [MPa] 261.7 19.5 w [-] 48.0 5.8 2.2 Load-unload Tensile Tests 2.3 Damage Evaluation by Stiffness Degradation Load-unload tensile tests were performed to identify mechanical properties and damage parameters. An hourglass shape specimen was defined by the standard [40] to facilitate measurements of strain along the monitored length, and to assure the fracture at the centre of the specimen. For the tests, a minimum of 23 loops steps loading-unloading were performed with a crosshead speed of 5 mm/min at room temperature. The loading-unloading cycles were performed in steps of 0.5 mm/mm by controlling the strain, which was measured with an Epsilon 3542, extensometer. Three specimens were tested until fracture on the rolled direction. Fig. 2 presents stress-strain curves for loading-unloading tensile test. Fig. 2. Engineering load-unload curves According to the effect of strain-hardening and stress saturation that produces a retarded necking formation on DP steels [3]. The material plastic flow curve was obtained from the loading-unloading tensile test until before the onset of necking, considering the Considere criterion [41]. Table 1 summarizes elasto-plastic material parameters; hardening behaviour is obtained using Voce's law. To characterize the ductile damage of DP590, load-unload tensile tests computing the hysteretic closed loops are adopted in this work. The methodology discussed in [11] has been used as the guideline for the fitting process; this methodology identifies two conditions that influence the obtained damage measurements. The first condition is the variation from the elastic volume during the change of the plastic regime to elastic regime in the discharge due to elasto-viscoplastic material effects, extensometer response slightly disturbed by nonlinearity, the rigidity of the testing machine and clearances in its joints, slipping in clamps among others [10] and [11]. To treat this issue, a procedure to develop measurements under the range selection between 5 % and 80 % from ultimate load was chosen during unloading ramps. The second condition defines the stress-strain curve beyond necking to rectangular specimens [11] and [24]. For this study, the Scheider's solution was used as methodology from the finite element simulations [29]. Four steps are described: i) Definition of empirical expressions to true stressstrain measures in large uniform strains by a=F/A and s = ln(AL/ L0). Where F is the applied force, A is the current cross-section, AL is the elongation and L0 the initial length calibrated. t = t o- At ^L__-T* b = b — Ab Fig. 3. Illustration of necking of a rectangular specimen ii) Determination of a current cross-section (A) as function of: initial thickness (t0), initial width (b0), current thickness (t) and current width (b) (see Fig. 3), where Ab and At are changes of thickness and width, respectively. Calculate of a correction factor (fcorr) to flat tensile specimens, which was developed by Scheider et al. [29], f = J corr for s< 1.42s. for , (29) s> 1.42s, iv) = 1 = 0.22(s - 1.42su ) x (s- 0.78) +1 where su is the equivalent strain at maximum load. For this purpose, the value of su was 6.1 % and standard deviations of 0.4 %. Definition of effective stress under the Von Mises yield isotropic condition related with: F, A and fcorr as: O „. = Ff A J CO A (30) Fig. 4 shows details about necking zone through parameters A and fcorr. Reduction on current section area in terms of the equivalent strain follows a constant change (dashed line), while that the transition where material undergoes instability plastic by post-necking behaviour is clearly identified with e ~0.086 (solid line). A [m 0.00 0.02 0.04 0.06 0 08 0.10 0.12 Equivalent strain [-] Fig. 4. Behaviour A vs equivalent strain (dashed line) and fcorr vs equivalent strain (solid line) Later, using Hooke's law for the uniaxial state of stress and the hypothesis of strain equivalence, the corrected elastic modulus ( Ecorr ) may be defined as: (31) Fig. 5 shows the real degradation of the stiffness (Ew) in the function of the plastic strain. As can be seen, a decreasing of the real Young's modulus is triggered when plastic strain exceeds a threshold value of approximately 0.015. This value is particularly low in relation to the conventional sheet steels, where damage is mainly driven by excessive localization rather than the nucleation of microcavities [11] and [20]. Finally, the full deterioration process developed in DP takes place when Young's modulus reaches a critical value of Eon„ ~ 174.2 GPa. Fig. 5. Evolution of the reduction the real Young's modulus According to the corrected elastic modulus ( Ecorr ), the damage variable D from Eq. (1) is redefined as: D = 1 -- E (32) Fig. 6 shows D versus plastic strain behaviour for the material studied, a linear regression is used to determine damage resistance, S. It is also observed that the damage increases with the accumulation strain. Plastic strain [ Fig. 6. Linear regression procedure to obtain damage resistance The obtained equation by linear fit was employed to find the variation of the damage versus variation of the plastic strain (dD/dep), assuming regarding damage evolution in monotonic tensile loading that elastic strain is negligibly small to large plastic strains (ep ~ e) [23]. Eqs. (4) and (10) may be replaced in Eq. (11) and expressed in differential in Eq. (33): i: dD =- 2SE (1 - D )2 (33) In contrast, considering that the strain hardening saturates at the ultimate strength (au) [22], it can be assumed that the equivalent stress (aeq) is equal to au. Therefore, the damage resistance S can be written as: 5 = - 2E(1 - D)2 (dD / ds) (34) Critical damage DC is taken as the value just before which ductile fracture occurs. Table 2 summarizes the damage parameters identified for DP590. These parameters were entered in the material description for finite element simulations of the next section. to only in the transversal and normal directions. To ensure that the quasi-static condition is satisfied, the energy balance was monitored after every analysis. Density (p) used for the analysis was of 7850 kg/m3. Fig. 7 shows dimensions and boundary conditions on the specimen, where the geometry was reduced to one eighth due to the symmetry. The analysis was carried out until the structure reached a critical damage state, DC. The numerical simulation elongation response (AL) was monitored through a reference gauge half-length of 25 mm using a nodal displacement sensor (see Fig. 7b) to reproduce the experimental procedure. During the test, the specimen was elastically unloaded and reloaded intermittently; however, only the elasto-plastic loading curve is submitted. Table 2. Experimental damage parameters for DP590 steel Damage Value S [MPa] 1.4 Dc [-] 0.18 sP [-] 0.015 3 TENSILE TESTS SIMULATIONS Simulations were performed using finite element code ABAQUS/Explicit through a VUMAT subroutine to implement Lemaitre's model. The sample geometry was modelled using 3D eight-node brick elements with an integration point. The mesh dependency is investigated with four FE representations of the specimen with different sizes of the brick element. The minimum element size was defined with an aspect ratio between the total superficial area and the volume equal to 1 to avoid distortions and obtain values physically admissible in the onset of necking localization in the test [11] and [33] to [36]. For all calculations, a convergence test was developed using minimum size elements between 0.1 mm and 1 mm, quantitative comparison of all analyses indicated that the variation of solutions was less than 1 % when the chosen element size was in the interval range from 0.10 mm to 0.70 mm. Thus, the minimum mesh size was equal to 0.4 mm where the specimen was formed by 3852 elements. Later, the longitudinal displacement was imposed on the right-side end of the specimen by time step control. Boundary conditions were restricted in all directions on the left-side end, whereas from the right the conditions were restricted Fig. 7. a) Geometric illustration of the hourglass flat specimen (dimension in mm), b) FE mesh of the tensile test Fig. 8 shows the comparison of the results of the curves obtained from the simulation and experiment. It can be noted that the curve from the simulation follows a good agreement with the trend towards the centre; before the damage threshold is reached in the prelocalization, a strain value of 0.059 reaches the saturation stage given by a balance between the multiplication and annihilation of dislocations. Later, in the region on the curve where mechanical resistance is reduced, the simulation results show a good correlation with a lower tendency with respect to experimental points up to the critical damage at which failure occurs, where the numerical response of the damage model is marginally less rigid than the experimental results, reaching differences of about 12 %. In this case, the difference may be a result of constitutive formulations on the kinetic laws of damage evolution employed in this model, due to the fact that it considers the stress triaxiality from uniaxial tension as a constant value. Thus, the energy release rate associated with the damage gradually tends to be lower [20], [32] and [34]. 800 100- 0-f-,-,-,-,- 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Engineering strain [-] Fig. 8. Comparison of engineering stress-strain in tensile test loading-unloading The simulation reported a critical condition of damage in the internal material for a fracture strain value of 0.118. This value is lower than the experimental value due to the ability of the model to capture the initial fracture location under the criteria of prescribed damage. In contrast, with the experimental procedure developed, the mechanical response was monitored until fracture. 0 20-,- 0 18 0.00 0.02 0.04 0.06 0 08 0 10 0 12 0.14 Plastic strain [-] Fig. 9. Numerical comparison of the damage evolution versus plastic strain for DP590 steel Finally, Fig. 9 describes the damage evolution at the centre of the section that undergoes high levels in strain, which occurs by the nucleation, growth, and coalescence/linking of microcracks that control the interface decohesion of grain and phase boundaries [3] and [11]. Because of that, strength is reduced when damage increases proportionally by the accumulated strain. The value predicted numerically for critical damage is equal to 0.188. Low discrepancy is encountered with the experimental value reported. Despite the damage evolution results presenting good approximation using the Lemaitre's theory, slight nonlinear behaviour is also shown in Fig. 9. This is due to the fact that governing equations of the thermodynamic dissipation processes follow convex functions to describe the system evolution [17]. 4 CONCLUSIONS In this paper, an experimental methodology was employed to identify the mechanical properties and damage parameters for DP590 using loading-unloading cycles during tensile testing under a constant value of triaxiality. The damage was indirectly computed by the evolution of the elastic modulus. An hourglass specimen, along with an extensometer, was implemented in the material's parameters calibration. Of particular importance, the performance damage measurements under large plastic strains should be considered. The proposed procedure resulted in the avoidance on incorrect interpretations of the damage, due mainly to the effects of nonlinearities during loading-unloading cycles and the formation of the necking. Overall, the implemented model gives a good prediction of the loading-unloading uniaxial tensile tests. The average error obtained between computational and experimental results is minimal during performance. Therefore, it can support the damage model, providing good agreement for mechanical behaviour of the DP590 steel with experimental results under uniaxial stress state conditions. Finally, the ability to predict the mechanical response of the DP590 using CMD and simple mechanical tests provides a useful alternative to avoid time-consuming and expensive experimental designs to approximate the influence of internal defects on the integrity of sheet metal forming. However, the methodology should be improved considering variations on the triaxial state of stress to replicate more complex strain paths; this is essential to obtain several ductile fracture criteria and to take relevant information on the behaviour of the material; therefore, we consider including this concept for future works. 5 ACKNOWLEDGEMENTS The authors acknowledge the financial support received from the department of Vice-rector of Research of the National University of Colombia. M. Anduquia gratefully acknowledges the support of COLCIENCIAS through a doctoral fellowship. 6 BIBLIOGRAPHY [1] Dykeman, J., Hoydick, D., Link, T., Mitsuji, H. (2009). Material property and formability characterization of various types of high strength dual phase steel. SAE Technical Paper, p. 1-10, D0l:10.4271/2009-01-0794. [2] Kuziak, R., Kawalla, R., Waengler, S. (2008). Advanced high strength steels for automotive industry. 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