Enhancement of void growth model for the anisotropic ductile metal (tt)

ABSTRACT The aim of work presented in this dissertation was to perform the improvement of the existing void growth-based damage models used for the ductile fracture analysis and prediction of sheet metals, which are subjected plastic deformation The original metal material is usually containing the second phase particles or/and inclusions Once the metallic material under deformation lead to the nucleation, growth and coalescence of micro-voids that it is root of ductile damage in industrial and civil products The first objective of this work was enhancement of N L Dung micro-void growth model to predict ductile fracture behavior of sheet aluminum alloys, typical for civil structures with anisotropic properties and their implementation in user-defined material subroutine (VUMAT) The explicit finite element code has been chosen for implementation of new material models Constitutive model with anisotropic yield criterion, damage growth and failure mechanism has been developed and implemented into ABAQUS/Explicit software The second important aspect of this dissertation was performance of tensile experiments in three different orientations of materials for identification of mechanical behavior of high strength sheet aluminum alloys AA6061-T6 The results from these tests allowed derivation of material constants for constitutive models and help to a better understanding of anisotropic material behavior The tensile tests were also used to validate the accuracy and applied capability of enhanced constitutive material models The constitutive models were developed within the general framework of ductile damage mechanics Coupling of the quadratic yield function Hill48 with damage model based on micro-mechanical and continuum damage mechanics (CDM) theories has been chosen to suit the anisotropic behavior of sheet material The validation of the constitutive models has been performed by numerical simulations of tensile and Nakajima tests The micro-crack and fracture initiation, crack path, damage criteria and forming limit diagram (FLD) of aluminum alloy AA6061-T6 are predicted using these constitutive models INTRODUCTION 1.1 The research motivation The fracture phenomenon can be observed any everywhere in our daily lives The phenomenon of ductile fracture is usually happening in metallic forming process under plastic deformation Today, although high quality design and manufacturing processes can result in robust, strong products, the causation of fail cannot be avoided in some cases However, this will not be an issue if processes such as damage elimination or healing are included in the application Therefore, the prediction and characterization of micro-crack initiation, fracture propagation and the final failure of the material is of such importance that it has become a special field in materials science Recently, damage modeling and predicting of metal material and their alloys are becoming more and more important as an object of research in recent years Until now, phenomenological continuum damage mechanics combined with finite element method has mostly been used for numerical modeling of a material damage Two phenomenological approaches are usually used The first approach is based on theory of continuum damage mechanics (CDM) In this method, damage variable is modeled by a scalar variable integrated with a suitable yield function [1, 2] One of the disadvantages of classical continuum mechanics is the impossibility of predicting the micro-crack initiation The second approach is based on the micro-mechanical theory This method using a yield function containg a porousity (void volume fraction) that descibed softening phenomenon of matrix material Therefore, this theory-based damage model is also known as porous ductile material model [3-5] The primary advantage of this approach is its micro-crack and fracture predictability via porousity of matrix material Recently with the rapid development of advanced high strength steel and aluminum alloy sheets, suitable material models are required for accurately describing their anisotropic behavior Therefore, improving the accuracy of ductile fracture prediction for various metals by using various fracture predicted models is still needed to continue 1.2 The research objectives An investigation of ductile fracture of sheet metallic material and their alloy using N L Dung micro-void growth models [5, 6] was performed in this dissertation The following specific objectives had to be achieved in this work: Understanding the mechanism of microscopic ductile fracture of metallic materials and their alloys Improving the original damage models for predicting ductile fracture and shear damage of anisotropic sheet metals Developing the user-defined material subroutine (VUMAT) for both porous ductile material and continuum damage mechanics (CDM) theory-based models Conducting the experiments to determine the mechanical behavior of material and calibrate the material parameters for the constitutive models Applying the damage models to predict the ductile fracture of isotropic and anisotropic metals 1.3 Research methodology An approach based on theoretical framework of ductile fracture together with experimental observation is applied to this dissertation The ductile damage models of N L Dung [5, 6] are first enhanced to anisotropic sheet metal and modified for shear damage by using the classical CDM and micromechanical ductile damage theories After this enhancement the damage models are written in Fortran program language as the user material subroutines (VUMAT) for the Abaqus/Explicit finite element package using the numerical algorithms [7, 8] During this process the code is frequently verified from various aspects in such a way that it works along with the existing capability of Abaqus/Explicit software without any errors Once the VUMAT subroutines are successfully developed, the damage models would be verifed via predicting ductile fracture of practical application (tensile test, deep drawing…) 1.4 The contributions of dissertation An enhancement of N L Dung void growth model [5, 6] for anisotropic metal using quadratic yield criterion Hill48 is proposed in this work This approach can be applied to the various yield criteria This work is also modified the N L Dung model [5, 6] for ductile fracture prediction under pure and simple shear loading states The micro-crack and fracture criteria are formed by relationship between equivalent plastic fracture strain and stress triaxiality They help to reduce computational time once they associated with CDM theory The forming limit diagram (FLD) of aluminum alloy sheet AA6061-T6 is suggested in this dissertation The VUMAT subroutines can be used as the sourced codes for implementing a new material model and developing current work in future 1.5 Dissertation outline The outline of this dissertation is as follows An introduction to ductile fracture mechanism of metallic material that occurs due to the nucleation, growth and coalescence of voids is presented in chapter A literature review of the existing porous ductile material models is also shown in this chapter Chapter details the enhancing ductile fracture criterion and porous ductile material model (N L Dung models) for anisotropic material The numerically implemented procedure using the stress integrating algorithms to seek element stress and state variables of damage models are represented in chapter The experiments and an optimized tool are conducted to identify the input material parameters of damage models would be outlined in chapter In chapter 6, the finite element calculations for tensile tests and deep drawing process are performed The numerical simulations of prediction of ductile fracture in anisotropic metals are also compared to experimental results A ductile fracture criterion of the aluminum alloy is also proposed based on finite element analysis Finally, in chapter conclusions are made based on the results gained in previous chapters and some discussions are presented for future work In addition, the appendixes and cited documents are also included in this dissertation DUCTILE FRACTURE OF METALLIC MATERIAL Ductile fracture of metallic material is due to heterogeneous microscopic structures that decline the mechanical properties of the material The metallic material is usually containing the resource of microscopic damage such as distributed micro-voids which might be process induced during loading are more tending to crack or failure The Figure 2.1 shows the formation process of ductile fracture due to micro-void nucleation, growth and coalescence in the metal sheet under uniaxial tension [9] In the recent time of several decades, the experimental studies and analytical models of void nucleation, growth and coalescence have been conducted by many researchers [3, 10-16] The original void growth models are not feasible for ductile fracture prediction under the range of low and negative stress triaxialities [17-19], i.e., they cannot be used to predict ductile fracture in the cases of compressed and pure shearing loads To improve the predictive ability of the GTN-like model, Xue [18], Nahshon and Hutchinson [19] have proposed the void coalescence models due to the relatively shearing and rotational voids during plastic deformation In this work, the N L Dung porous ductile model [5] would be associated with Nahshon and Hutchinson [19] shear damage criterion In addition, a modification of the N L Dung ductile fracture criterion [6] for shear damage using CDM theory is also performed in this work Force (N) Plastic strain evolution Void nucleation, growth and coalescence a) Displacement (mm) b) Figure 2.1 Ductile fracture mechanism of metallic material: a) specimen, b) the process of void nucleation, growth and coalescence versus plastic strain evolution [9] DUCTILE FRACTURE MODELLING 3.1 3.1.1 The continuum damage mechanics (CDM) model The constitutive equations The yield criterion in stress space is expressed in the following form:  ( e , D , f )= e − (1 − D  ) f =0 (3.1) where  is softening exponent and  e denotes equivalent stress, For isotropic matrix material e =e = M ises (3.2) s ij : s ij For anisotropic matrix material, Hill48 equivalent stress is used as follow e =e H ill =  ij : H ijkl :  kl (3.3) The constant tensor H ijkl for Hill48 material is represented by 6x6 matrix as in the material principal coordinate system The damage variable D is considered as internal variable corresponding to a material degradation  D  1, D =0 where corresponds to a virgin material while D = corresponds to fully damaged material  D = D crit p f  dD =1 (3.4) D crit is critical value of damage variable,  p f is equivalent plastic fracture strain of matrix material, d D denotes the evolution rate of damage variable For the ellipsoidal void, N L Dung [5] proposed a critical accumulated damage variable as follow (assuming      ), dD g   (1 − n )  +  +    (1 − n )  −   3 = sinh    cosh   − n   ( )  f   f    −  −   p +  d f  Where  i ( i = 1, 2, ) denote the principal stress components d p (3.5) the Hill48 equivalent plastic strain rate: d p = d p ( Hill 48 ) ( d  ij : H ijkl : d  ij ) −1 = −1 Where H ijkl is the pseudo-inverse matrix of Hill48 material (3.6) 3.1.2 An extension of the void growth model for shear damage As a drawback of void growth-based damage model under simple or pure shear load, the past studies indicated that there is no void growth under shear loading states because of zero stress triaxiality but the voids are still rotated under this condition In an earlier study by McClintock [20] for void growth in shear bands, the fracture due to void growth in the longitudinal direction of the shear bands and the void shear is given by   ln  = r   ln +  + dam age due to relatively shearing and rotational void  (1 + n ) sinh (1 − n )  m  (3.7) f dam age due to void grow th McClintock [20] introduced a damage criterion including both void growth anf relatively shearing and rotational as Eq.(3.7) This work modifed the N L Dung damage accumulate criterion and proposed a shear damage variable as follow, ln + N (  D = s   ln   − g r D crit   Ratio of p  ) (3.8) p f  dD g / 2r0 calculated through initial VVF The damage criterion for general loading case is written D = D + g ( ) D g s (3.9) Where g (  ) is Lode weighted function Complete damage occurs once damage variable reaches to unity i.e., 3.2 D =1 The porous ductile model In order to consider anisotropic aspect of sheet material, the original Dung model [5] will be associated with ewuivalent stress Hill48 as follow,  D ung − H ill 48   H ill 48 = e   f     + fq1 cosh      (1 − n )  m   − − q2 f  f  =0 (3.10) The clossed yield surface of the Dung-Hill48 model is shown in Figure 3.1 Figure 3.1 The yield surface presentation of the Dung-Hill48 model in normalized principal stress space NUMERICAL IMPLEMENTATION OF THE DUCTILE DAMAGE MODELS This chapter describes the implementation of the constitutive models in chapter into FEM code of ABAQUS/Explicit software via the VUMAT subroutines For this dissertation, the VUMAT subroutines are written by program language Fortran 90 and integrated with FEM code of ABAQUS/Explicit software package version 6.14-3 4.1 Numerical implementation of CDM model The Hill48 yield criterion would be implemented by using “cutting-plane” algorithm [7] The flowchart of stress integration algorithm of CDM model is given in appendix 4.2 Numerical implementation of the porous ductile model A numerical algorithm for pressure-dependent plasticity models [8] is applied to this dissertation The flowchart of stress integration algorithm of porous ductile model is shown in appendix 4.3 Verification of user-defined material subroutine (VUMAT) To verify the accuracy of the implementation against known values and element distortion, the VUMAT subroutines would be verified via single element, tensile specimen and deep drawing process Results of tensile test: Figure 4.1 shows an identical crack path obtaining by experiment and numerical simulation of uniaxial tensile specimen Figure 4.2 shows a comparison of force-displacement curve between the experiment and numerical simulations The numerical results show a good agreement with experimental data when using the CDM-Hill48 model with fitted parameter set, whereas a larger displacement amount is archived by Dung-Hill48 model comparing with that of experiment This may be due to effect of hardening exponent in Dung-Hill48 model during plastic deformation process a) b) c) Figure 4.1 Comparison of the crack path (a) experiment [21], (b) CDM-Hill48 model, (c) Dung-Hill48 model 10 The critical damage parameter of due to void growth (D ) g crit and softening exponent (  ) would be calibrated Table 5.4 Initial guess values and constrains for optimization process Parameter D crit β Upper limit 3.0 3.0 Initial guess 2.0 2.0 Lower limit 0.1 1.0 g g The Table 5.4 shows boundary condition and initial values of D crit and  The best-fit material parameters that archived from optimization process are given in Table 5.5 and the optimal force-displacement curves are shown in Figure 5.2 Table 5.5 The best-fit material parameters for CDM model Parameter D crit β CDM-Mises 2.65 1.25 CDM-Hill48 1.32 1.96 g Figure 5.2 The best-fit force-displacement curve using CDM model 5.2.3 Porous ductile model The Table 5.6 shows boundary condition and initial values of input pparameters for the porous ductile model The best-fit material parameters that archived 14 after optimization process are given in Table 5.7 and the optimum forcedisplacement curves are shown in Figure 5.3 Table 5.6 Initial guess values and constrains for optimization process q1 q2 fF fC f0 εN sN fN Upper limit 2.0 4.0 0.25 0.1 0.0020 0.65 0.15 0.07 Initial guess 1.5 2.25 0.15 0.06 0.0018 0.3 0.1 0.05 Lower limit 1.0 1.5 0.101 0.02 0.0014 0.085 0.05 0.03 Table 5.7 The optimal values for Dung-Hill48 model Parameter q1 q2 fF fC f0 N sN fN Value 1.321 2.582 0.14 0.087 0.0016 0.115 0.054 0.0515 Figure 5.3 The best-fit force – displacement curve using of Dung-Hill48 model DUCTILE FRACTURE PREDICTION OF AA6061-T6 ALUMINUM ALLOY 6.1 6.1.1 The tensile tests Geometries, mesh and boundary conditions The R-notched specimens are cut from thin sheet that its nominal thickness of mm All experimental data is obtained using tensile testing machine Testometric M500-30AT The initial mesh size at analysis zone is 0.5 mm x 0.5 mm The eight-node brick element type with reduced integration and hourglass 15 control (C3D8R) has been used The left-hand side of specimen is fixed wheareas the tensile load is applied on the right-hand side of the specimen 6.1.2 Ductility prediction The predicted values of ductility and corresponding errors are presented in Table 6.1 The relative error is calculated by eq (6.1) as below, D uctility ( exp ) − D uctility ( sim ) Error ( % ) =  100 (6.1) D uctility ( exp ) Where the subscripts • ( ex p ) and • ( sim ) imply the experimental and predicted ductility, respectively Table 6.1 The ductility predictions of the R-notched specimen Ductility (%) Specimen Error (%) Experiment CDMHill48 DungHill48 CDMHill48 DungHill48 Dog-bone R6 14.77 2.67 14.66 2.80 14.63 2.65 0.74 4.87 0.97 0.79 R3 2.18 2.24 2.22 2.75 1.54 R1.5 Average 1.99 1.66 1.98 16.88 6.31 0.70 1.00 The maximum error values of predicted ductility by CDM-Hill48 model is 16.88 % and by Dung-Hill48 model is 1.54 % The average error values of 6.31 % and of 1.00 % are obtained by CDM-Hill48 model and Dung-Hill48 model, respectively The detail results are given in Table 6.1 Crack initiation and propagation prediction The micro-crack initiation locations are determined by extracting VVF along minimum section of the specimens and are given in Figure 6.1 It predicted that in the case of dog-bone and R6 notched specimens micro-crack initiated at center, whereas in the case of R3, R1.5 and shear specimens micro-crack initiates near periphery 16 Figure 6.1 Micro-crack location of R-notched specimens The predicted fracture initiation locations using CDM-Hill48 model are identical to those of Dung-Hill48 model For all R-notched and shear specimens, the fracture propagates along minimum section The fracture initiates at center of the dog-bone and R6 specimens, whereas fracture initiates at periphery of R1.5 and shear specimens For the R3 specimen, there is small deference between predicted result by CDM-Hill48 model and Dung-Hill48 model Fracture occurs at periphery before when predicted by CDM-Hill48 while that the fracture initiates at very close to periphery when using DungHill48 model The summary of predicted result of fracture initiation locations is given in Table 6.2 Table 6.2 Summary of fracture initiation location prediction Specimen Dog-bone R6 R3 R1.5 Shear CDM-Hill48 Center Center Periphery Periphery Dung-Hill48 Center Center Periphery Near periphery Periphery Periphery 6.1.3 Ductile fracture strain prediction The predicted micro-crack and fracture strains that using as the ductile fracture criteria of the AA6061-T6 aluminum alloy sheet are described in Figure 6.2 17 Figure 6.2 Equivalent plastic fracture strain as a function of average stress triaxiality 6.2 Forming limit diagram (FLD) prediction Forming limit points identified from earliest failure elements in each specimen using the combined Dung-Hill48 model are plotted in Figure 6.3 The same procedure is applied to simulation results with the GTN model using the same parameter set to determine limiting strains (contour plots of this case are not shown) Based on these points forming limit diagrams of the material are obtained for the two models The two curves are just a small distance apart resulting from a mild anisotropy of the material In particular, the GTN isotropic material model underestimates the forming limits in most cases except for in proximity of the equi-biaxial stretching mode Equivalent plastic strains at the forming limits are computed and plotted against corresponding average triaxiality values in Figure 6.4 Representation of these states are located at the zone between micro-crack and macro-crack loci Hence, the micro-crack condition ( f = f C ) can be applied as a conservative measure of the formability of the material Also, the micro-crack condition ( f = f F ) serves as an upper bound 18 Figure 6.3 The forming limit diagram of AA6061-T6 aluminum alloy Figure 6.4 The equivalent plastic fracture strain of AA6061-T6 aluminum alloy CONCLUSIONS AND FUTURE WORK 7.1 The overall conclusions This research has focused on improving and applying the ductile damage models that can predict the plastic failures of the anisotropic metallic sheet This has required a combination of high quality experimental data obtaining by 19 tensile tests and the material parameter calibration of the damage models The principal conclusions have been summarized in detail below: The original ductile damage models were enhanced for anisotropic matrix material The applicability of both continuum damage mechanics theory-based model (CDM-Hill48 model) and porous ductile theory-based model (DungHill48 model) were examined The user-defined material subroutines (VUMAT) in ABAQUS/Explicit were successfully developed for the numerical computation These VUMAT subroutines can be usefully sourced code for implementation of various material models in the future The series of experiment are also conducted for identifying the input material parameters and validating the proposed damage models The FEM simulations were performed on uniaxial tension, deep drawing and Nakajima tests The numerical results show that the response of forcedisplacement curve obtaining by Hill48 matrix material-based damage models is more suited to experimental data than those of the von Mises matrix material assumption This reflects the correctness of the enhanced models comparing with original damage models Using critical values of the void volume fraction, the predictions of micro- and fracture initiation of the material are performed to obtain the equivalent plastic strains at fracture In that sense, the porous plasticity model is employed to establish a ductile fracture criterion of the material via the construction of a plastic strain – triaxiality relation Such relation can later be used as an alternative fracture criterion based on CDM theory The FLD of AA6061-T6 sheet is predicted using the Dung-Hill48 model The predicted result sits above the one obtained with GTN model for most deformation mode except for the neighborhood of the equi-biaxial stretching state, the different apart reflects anisotropic assumption of matrix material in 20 Dung-Hill48 model The prediction of forming limits are consistent with the established fracture plastic strain - triaxiality relation that obtained by tension and Nakajima test when using both damage models (CDM-Hill48 and DungHill48) 7.2 The recommends for future work In this work, to save computational time, a consistent mesh size is used for calibrating and validating the ductile fracture analyses However, in finite element method, the mesh dependence is well known problem for damage analyses due to softening behavior of matrix material Therefore, it would be adequate evaluation if a particular mesh size which gives accurate results is investigated The aluminum alloys are known as the crystalline structure material so that a further damage investigation based on grain structure analyses at microscopic level is important This work marks the first attempt to incorporate sheet metal anisotropy into a porous plasticity model to predict ductile fracture and forming limits of the aluminum alloy As such, a simplest extension of the J2 flow theory, the Hill’s quadratic plasticity model, is chosen Nevertheless, it is rather well-known that the aluminum alloy considered obeys non-quadratic yield function Therefore, the present approach should be applied to more sophisticated yield functions The effect of strain rate and temperature in warm and hot forming process should be studied The tests should be carried out on specimens under lager range of stress triaxialities to archive full locus of fracture strain for AA6061-T6 aluminum alloy 21 Appendix 1: Flow chart of stress integrated algorithm of CDM-Hill48 model 22 Appendix 2: Flow chart of stress integrated algorithm of Dung-Hill48 model 23 References [1] J Lemaitre, "A continuous damage mechanics model for ductile fracture," Journal of engineering materials and technology, vol 107, no 1, pp 83-89, 1985 [2] L Xue, "Damage accumulation and fracture initiation in uncracked ductile solids subject to triaxial loading," International Journal of Solids and Structures, vol 44, no 16, pp 5163-5181, 8/1/ 2007 [3] A L Gurson, "Continuum Theory of Ductile Rupture by 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10// 1952 26 List of published papers [1] H H Nguyen, T N Nguyen, and H C Vu Ductile fracture prediction and forming assessment of AA6061-T6 aluminum alloy sheets International Journal of Fracture, 2018, 209.1-2: 143-162 [2] H H Nguyen, T N Nguyen, and H C Vu, "Forming limit curve determination of AA6061-T6 aluminum alloy sheet,"Journal of Science and Technology Development, vol 20, No.K2-2017, 2017, p 51-59 [3] H H Nguyen, T N Nguyen, and H C Vu Forming Limit Diagram Prediction of AA6061-T6 Sheet Using a Microscopic Void Growth Model In: International Conference on Advanced Engineering Theory and Applications Springer, Cham, 2017 p 1026-1036 [4] H H Nguyen, T N Nguyen, and H C Vu "Ductile Fracture Prediction and Formability Assessment of AA6061-T6 Sheets using a Porous Plasticity Model" in Proceedings of the 11th Southeast ASEAN Technical University Consortium Symposium – SEATUC-11, March 13 and 14, 2017 ed 268, Ly Thuong Kiet, TP Ho Chi Minh, Viet Nam: Conference Proceedings by CD-ROM with International Standard Serial Number, ISSN: 2186-7631 [5] H H Nguyen, T N Nguyen, and H C Vu, "Implementation and Application of Dung Model to Analyze Ductile Fracture of Metallic Material" in in AETA 2015: Recent Advances in Electrical Engineering and Related Sciences, Springer, Cham, 2016, p 903-913 [6] H H Nguyen, T N Nguyen, and H C Vu, "Application of the Dung Microscopic Damage Model to Predict Ductile Fracture of the Deep Drawn Aluminum Alloy Sheets" in AETA 2015: Recent Advances in Electrical Engineering and Related Sciences, Springer, Cham, 2016, p 891-901 27 [7] H H Nguyen, T N Nguyen, and H C Vu, "Ductile Fracture Prediction of Docol DP600 Steel by the Numerical Simulations," in Proceedings of the 4rd International Conference on Engineering Mechanics and Automation – ICEMA4, August 25 and 26, 2016 ed 18, Hoang Quoc Viet, Ha Noi, Viet Nam: Publishing House for Science and Technology, 2016, p 318-325 [8] H H Nguyen, T N Nguyen, and H C Vu, "Phân tích động lực học tăng trưởng lỗ hổng vi mô vật liệu đàn dẻo," in Tuyển tập Hội nghị Cơ học toàn quốc, Hà Nội, Việt Nam, 2014, p 175-180 [9] H H Nguyen, T N Nguyen, and H C Vu, "The Dung Void Growth Model for Shear Failure," Journal of Science and Technology, vol 52(2C), p 214-255, 2014 [10] H H Nguyen, T N Nguyen, and H C Vu, "Ductile Fracture Analysis of API X65 Steel by Modified Gurson Model in ABAQUS," in Proceedings of the 3rd International Conference on Engineering Mechanics and Automation - ICEMA3, October 15, 2014 ed 18, Hoang Quoc Viet, Ha Noi, Viet Nam: Publishing House for Science and Technology, 2014, p 427-432 28 ... extension of the void growth model for shear damage As a drawback of void growth- based damage model under simple or pure shear load, the past studies indicated that there is no void growth under... the mechanical behavior of material and calibrate the material parameters for the constitutive models Applying the damage models to predict the ductile fracture of isotropic and anisotropic metals... Understanding the mechanism of microscopic ductile fracture of metallic materials and their alloys Improving the original damage models for predicting ductile fracture and shear damage of anisotropic