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Akinori Yamanaka, Ryunosuke Kamijyo, Kohta Koenuma, [Ikumu Watanabe](https://orcid.org/0000-0002-7693-1675), Toshihiko Kuwabara

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[Deep neural network approach to estimate biaxial stress-strain curves of sheet metals](https://mdr.nims.go.jp/datasets/16102982-584f-4d35-899b-24333ca94956)

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Deep neural network approach to estimate biaxial stress-strain curves of sheet metalsMaterials and Design 195 (2020) 108970Contents lists available at ScienceDirectMaterials and Designj ourna l homepage: www.e lsev ie r .com/ locate /matdesDeep neural network approach to estimate biaxial stress-straincurves of sheet metalsAkinori Yamanaka a,⁎, Ryunosuke Kamijyo b, Kohta Koenumab, Ikumu Watanabe c, Toshihiko Kuwabara aa Division of Advanced Mechanical Systems Engineering, Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16, Naka-cho, Koganei-shi, Tokyo 184-8588, Japanb Department of Mechanical Systems Engineering, Graduate School of Engineering, Tokyo University of Agriculture and Technology, 2-24-16, Naka-cho, Koganei-shi, Tokyo 184-8588, Japanc Research Center for Structural Materials, National Institute of Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, JapanH I G H L I G H T S G R A P H I C A L A B S T R A C T• DNNs were used to estimate biaxialstress-strain curves of aluminum alloysheets.• Pole figure images and 3D orientationmaps were explored as input data.• DNNs were as accurate as numericalbiaxial tensile tests, but much faster.• A new approach to virtual datageneration for material modeling wasdemonstrated.⁎ Corresponding author.E-mail address: a-yamana@cc.tuat.ac.jp (A. Yamanakahttps://doi.org/10.1016/j.matdes.2020.1089700264-1275/© 2020 The Authors. Published by Elsevier Ltda b s t r a c ta r t i c l e i n f oArticle history:Received 20 May 2020Received in revised form 1 July 2020Accepted 9 July 2020Available online 15 July 2020To improve the accuracy of a sheet metal forming simulation, the constitutive model is calibrated using resultsfrom multiaxial material testing. However, multiaxial material testing is time-consuming and requires special-ized equipment. This study proposes two different deep neural network (DNN) approaches, a two- and three-dimensional convolutional neural network (DNN-2D and DNN-3D), to efficiently estimate biaxial stress-straincurves of aluminum alloy sheets from a digital image representing the sample's crystallographic texture. DNN-2D is designed to estimate biaxial stress-strain curves from a digital image of {111} pole figure, while DNN-3Destimates the curves from a 3D image of the texture. The twoDNNswere trained using synthetic texture datasetsand the corresponding biaxial stress-strain curves obtained fromcrystal plasticity-based numerical biaxial tensiletests. The accuracy of the two trained DNNs was examined by comparing the results from that of the numericalbiaxial tensile tests. It was observed that both the DNNs could estimate biaxial stress-strain curves with high ac-curacy. Though DNN-3D provides with a better estimation than DNN-2D, it displays lower computational effi-ciency. Thus, the two DNNs and their training procedures offer a new and efficient method to provide virtualdata for material modeling.© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license(http://creativecommons.org/licenses/by/4.0/).Keywords:Deep neural networkMaterial modelingMultiaxial material testingAluminum alloy sheets).. This is an open access article under1. IntroductionCommercial finite element simulation software is widely used in in-dustry to simulate sheet metal forming process [1,2]. Many phenome-nological constitutive models based on yield functions have beendeveloped and incorporated into commercial finite element softwarethe CC BY license (http://creativecommons.org/licenses/by/4.0/).http://crossmark.crossref.org/dialog/?doi=10.1016/j.matdes.2020.108970&domain=pdfhttp://creativecommons.org/licenses/by/4.0/https://doi.org/10.1016/j.matdes.2020.108970mailto:a-yamana@cc.tuat.ac.jphttps://doi.org/10.1016/j.matdes.2020.108970http://creativecommons.org/licenses/by/4.0/http://www.sciencedirect.com/science/journal/www.elsevier.com/locate/matdesFig. 1. {111} pole figure of crystallographic texture in a 5182-O aluminum alloy sheet(a) measured using EBSD and (b) illustrated as a density plot.2 A. Yamanaka et al. / Materials and Design 195 (2020) 108970over the past decades [3–8]. The precise prediction of defects (e.g. frac-ture and springback) using sheet metal forming simulations relies onthe accuracy of the yield function, the constitutivemodel, and its param-eters (i.e. material model which should reproduce the actual plastic de-formation behavior). Thematerialmodel can be calibrated using variousmultiaxial material testingmethods [9] that measures the plastic defor-mation behavior of sheet metals under multiaxial stress conditions likethe hydraulic bulge [10], stack compression [11], biaxial tensile test(cruciform sample) [12–14], and multiaxial tube expansion tests [15].The calibrations are generally performed based on the contour ofequal plastic work and the direction of incremental plastic strain ratesmeasured by multiaxial material tests. Therefore, acquisition of experi-mental data bymultiaxial material tests is key to obtain highly accuratesheet metal forming simulation results [16–18].The crystal plasticity finite elementmethod (CPFEM) is another pop-ular method for simulating the plastic deformation behavior of sheetmetals during metal forming process [19,20]. Previous studies of sheetmetal forming simulations using CPFEM have investigated deep draw-ing and spherical punch forming [21–26]. However, CPFEM-basedsheet metal forming simulations have not been widely adopted in theindustry due to the high computational demand of themethod, despitethe use of high-performance computers.Crystal plasticity-based simulations can also be used in virtual mate-rial testing [27] and virtual laboratory [28]. In virtual material testing, amaterial model is calibrated using the results from crystal plasticity-based simulations instead of time-consuming experimental multiaxialmaterial tests. Hence, virtual material testing has been widely appliedto calibrate material models of sheet metals [27,29–36]. However, cal-culating the contour of equal plastic work and the yield locus using vir-tualmaterial testing requires the use of user-developed source codes forcrystal plasticity simulations, thus involving a high computational load.Machine-learning is an effective way to estimate the mechanical re-sponse of materials from their microstructural information. Manymachine-learning algorithms have been proposed, where artificial neu-ral networks (ANNs) have gained popularity in thefield ofmaterials sci-ence since the 1990s [37–39]. An early review article by Bhadeshia [39]published in 1999 had already anticipated that ANNwill be a promisingmachine-learning tool for estimating material properties. In fact, ANNshave since been appliedwidely for the estimation of mechanical behav-ior for variousmaterials [40–45]. Yang et al. [43] proposed a newmeth-odology to predict the stress-strain curve of binary composites usingANN and principal component analysis. Janab et al. [44] used a geneticalgorithm and ANN to predict the rate-dependent tensile flow behaviorof AA5182-O aluminum alloy sheets. More recently, Ali et al. [45]employed ANN to estimate the stress-strain curve and texture evolutionof AA6063-T6 aluminum alloy under non-proportional loadingconditions.We have previously proposed a deep neural network (DNN)-basedmethodology for the estimation of uniaxial stress-strain curves andthe anisotropy of Lankford value (r-value) of aluminum alloy sheetsusing crystallographic texture data [46]. In the previous study, DNNwas trainedwith a large training dataset generated using crystal plastic-ity finite element simulations of uniaxial tensile testing, referred to asnumerical material tests [32]. The trained DNN successfully estimatedthe uniaxial stress-strain curve and anisotropic evolution of r-valuewith the same accuracy as numerical material tests [46]. However, tothe best of the author's knowledge, machine-learning-basedmethodol-ogy has not yet been used to estimate biaxial stress-strain curves ofsheet metals.The purpose of this study is to propose a new DNN approach to effi-ciently estimate biaxial stress-strain curves of sheet metals from theirunderlying microstructural features. Two DNNs were developed; oneto estimate biaxial stress-strain curves from a digital image of {111}pole figure (DNN-2D) and another to estimate biaxial stress-straincurves from a three-dimensional (3D) image representing the crystallo-graphic texture in a voxelized Euler angle space (DNN-3D). Both theDNNs were trained using synthetic crystallographic texture datasetscontaining typical preferred texture components in aluminum alloysheets, namely Cube, Goss, S, Brass, and Copper-components. Biaxialstress-strain curves, used in the training dataset, were generated viaCPFEM based numerical biaxial tensile tests. The two trained DNNswere validated by comparing the estimated biaxial stress-strain curveswith those obtained from numerical biaxial tensile tests. To facilitatefurther implementation of this proposed DNN approach, the trainedDNNs, training parameters, and training datasets are made availablefor free download at https://github.com/Yamanaka-Lab-TUAT/DNN-NMT [47].2. Materials and methods2.1. Experimental tensile testingThe numerical biaxial tensile tests, used to generate the trainingdataset, were validated using experimental uniaxial and biaxial tensiletesting of a 5182-O aluminum alloy sheet (initial thickness = 1.0mm). The crystallographic texture of the sheet was measured usingelectron backscattered diffraction (EBSD). The number of crystal orien-tations obtained from the EBSDmeasurement was 1,009,957. The {111}pole figure of themeasured texture is given in Fig. 1. The pole figure in-dicated that the texture in the 5182-O aluminum alloy sheet included aweak {001}〈100〉 Cube-component.The true stress-true strain curve obtained from the uniaxial tensiletest was used to identify the parameters in the crystal plasticity consti-tutive equation used in the numerical biaxial tensile tests. The uniaxialtensile samples were prepared in accordance with the Japan IndustrialStandards (JIS 13 B-type specimen). The rolling direction (RD) of thesheet was set parallel to the tensile direction in the uniaxial tensiletest. The uniaxial tensile tests were performed using the AutographAG-Xplus 100 kN instrument (SHIMADZU Co.). A fixed equivalent plas-tic strain rate in the order of 5.0 × 10−4 s−1 was used during uniaxialtensile testing under a quasi-static condition. Two specimens eachwere tested for RD and transverse direction (TD) of the sheet.The biaxial tensile tests were performed on cruciform samples usinga servo-controlled testing machine. The detailed specifications of thesamples and testing machine have been previously reported [14,15].Tensile forceswere applied along RD and TDof the sheet and various ra-tios of the true stress components along the RD and TD (σ11: σ22 = 4:1,2:1, 4:3, 1:1, 3:4, 1:2, and 1:4, where σ11 and σ22 correspond to the truestress along RD and TD, respectively) on a linear stress pathwere inves-tigated. A fixed equivalent strain rate in the order of 5.0 × 10−4 s−1 wasused during biaxial tensile testing. Two specimens were tested for eachstress ratio and the two stress-strain curves thus obtained from the bi-axial tensile tests were averaged. The averaged biaxial stress-straincurves were compared to those calculated from the numerical biaxialtensile tests.https://github.com/Yamanaka-Lab-TUAT/DNN-NMThttps://github.com/Yamanaka-Lab-TUAT/DNN-NMT3A. Yamanaka et al. / Materials and Design 195 (2020) 1089702.2. Numerical biaxial tensile testingThe constitutive equation based on the crystal plasticity theory[48–51] and the finite element model used in the numerical biaxialtensile tests are presented in this section.2.2.1. Crystal plasticity constitutive equationThe crystal plasticity constitutive equation proposed by Peirce et al.[48] was used in our study. Multiplicative decomposition of a deforma-tion gradient tensor (Fij) in the Cartesian coordinate system gives thefollowing equation:Fij ¼ F�ik Fpkj ð1Þwhere Fik∗ and Fkjp are the elastic and the plastic components of the defor-mation gradient tensor, respectively. The velocity gradient tensor (Lij) isdefined as:Lij ¼ _Fik F−1kj ð2Þwhere F:ik is the rate of the deformation gradient tensor. The velocitygradient tensor in Eq. (2) can be divided into two tensors as:Lij ¼ Dij þWij ð3Þwhere Dij is the deformation rate tensor and Wij is the continuum spintensor. These tensors are comprised of elastic and plastic components:Dij ¼ Deij þ Dpij ð4ÞWij ¼ W�ij þWpij ð5Þwhere Dije and Dijp are the elastic and plastic components of the defor-mation rate tensor, respectively and Wij∗ and Wijp are the elasticand plastic components of the spin tensor, respectively. The plasticcomponents of the deformation rate and the spin tensors aredescribed as:Dpij ¼Xnslipα¼1P αð Þij_γ αð Þ ð6ÞWpij ¼Xnslipα¼1ω αð Þij_γ αð Þ ð7Þwhere nslip is the number of slip systems in a crystal and γ:αð Þ is theplastic shear strain rate for the αth slip system. As 12 slip systems areinvolved in a face-centered cubic crystal, we have nslip = 12 and αcan vary from 1 to 12. We have used the following equation proposedby Pan et al. [49] to calculate the plastic shear strain rate:_γ að Þ ¼ _γ0τ að Þg að Þ����τ að Þg að Þ����1m−1ð8Þwhere γ0:is the reference shear strain rate, τ(α) is the resolved shearstress for αth slip system, m is the strain rate sensitivity parameter,and g(α) is the critical resolved shear stress (CRSS) for αth slip system.Further, Pij(α) and ωij(α) in Eqs. (6) and (7) are given as follows:P αð Þij ¼ 12s� αð Þi m� αð Þj þm� αð Þi s� αð Þj� �ð9Þω αð Þij ¼ 12s� αð Þi m� αð Þj −m� αð Þi s� αð Þj� �ð10Þwhere si∗(α) (i=1, 2, 3) is a unit vector in the slip direction andmi∗(α) is aunit vector normal to the slip plane in the deformed configuration. Thecrystal rotation was described as follows:s� αð Þi ¼ F�ijs0 αð Þj ð11Þm� αð Þi ¼ m0 αð Þj F�−1ji ð12Þwhere mi0(α) is a unit vector in the direction of initial slip and si0(α) is aunit vector normal to the slip plane.The constitutive equation for finite deformation is given as:σ∇ij ¼ CeijklDkl−Xnslipα¼1R αð Þij_γ αð Þ ð13ÞFurther,R αð Þij ¼ ω αð Þik σkj−σ ikωαð Þkj þ CeijklPαð Þkl ð14Þwhere∇σ ij is the Jaumann rate of the Cauchy stress tensor and Cijkle is theelastic modulus tensor.The evolution of CRSS was used to evaluate the strain-hardening ofthe material by employing the following equation:g að Þ ¼ τ0 þZt_g að Þdt ð15Þwhere t denotes time and τ0 is the initial CRSS. g:αð Þ is the rate of CRSS,and is defined as:_g αð Þ ¼Xnslipβ¼1h αβð Þ _γ βð Þ��� ��� ð16Þwhere h(αβ) is the hardening coefficient matrix. This term is further de-fined as:h αβð Þ ¼ q αβð Þ dτ γð Þdγþ 1−q αβð Þ� � dτ γð Þdγδαβ ð17Þwhere q(αβ) is a matrix describing the level of latent-hardening and δαβis the Dirac delta function. The relationship between the shear stressand the accumulated plastic shear strain (γ) is given as:τ γð Þ ¼ τ0 þ h0 C γint þ γð Þf gn0 ð18Þwhere h0 is the initial hardening coefficient, n’ is the hardening index, Cis the hardening constant, and γint is the initial plastic shear strain. Theparameters used to describe the strain-hardening behavior of the testsample included τ0, h0, n’, C, and γint. These parameters were calibratedby fitting the uniaxial stress-strain curve for RD (i.e. σ11: σ22= 1:0) cal-culated from the numerical biaxial tensile test to the reference experi-mental uniaxial tensile test data.The mathematical homogenization method [52,53] was used to cal-culate the biaxial stress-strain curves of aluminumalloy sheets based onthe underlying crystallographic texture of the samples using numericalbiaxial tensile tests. The homogenization method derives the governingequations for two-scale boundary value problems (BVPs) in bothmicro-scopic and macroscopic length-scales. The microscale BVP was solvedusing finite element method and the microscale mechanical behaviorof themicrostructurewas analyzed in specificmacroscopic stress states.Further, themacroscopicmechanical behaviorwas evaluated at each in-tegration point of themacroscale finite elementmodel using themicro-scale BVP solutions. Thismicro-macro coupling schemehas beenwidelyapplied in the finite element modeling of elastoplastic materials[53–56]. A detailed formulation of the two-scale finite element simula-tion using the crystal plasticity constitutive equation is described inSupplementary Material and was based on a previously reportedmethod [21].TDRD{111}Fig. 3. {111} pole figure of the initial 1000 crystal orientations used as the input data in thenumerical biaxial tensile test (σ11: σ22 = 1:0), which was performed to calibrate thestrain-hardening parameters of the crystal plasticity constitutive equation. The initialcrystal orientations were sampled from the results of EBSD measurement.Table 1Material constants for 5182-O aluminum alloy and parameters used in thisstudy.Elastic constants [GPa](Voigt notation)C11 = 108.2C12 = 61.3C44 = 28.5Reference shear rate, γ:0 [s−1] 0.5Strain rate sensitivity factor, m 0.0054 A. Yamanaka et al. / Materials and Design 195 (2020) 1089702.2.2. Finite element model for numerical biaxial tensile testThe finite element model used for the numerical biaxial tensile testsis shown in Fig. 2. Two length-scales were defined, where the macro-scopic scale was denoted by xi (i = 1, 2, 3) and the microscopic scaleby yi (i = 1, 2, 3). The macroscale coordinate axes for i = 1, 2 and 3were defined parallel to the RD, TD, and normal direction (ND) of thesheet, respectively. The finite element (FE) model for the macroscale(macro-FE model) consisted of a single hexagonal isoparametric ele-ment with eight integration points. The FE model for the microscale(micro-FE model) included a representative volume element for thecrystallographic texture of a sheet andwas described by a cubic domain.Themicro-FEmodel was divided by 125 elements based on the same fi-nite element type (isoparametric elementwith eight integration points)used for the macro-FE model. Thus, crystallographic texture consistingof 1000 crystal orientations was described in the micro-FE model. Theinitial 1000 crystal orientations were sampled from the EBSD measure-ment results (1,009,957 crystal orientations) based on the STATmethod[57]. Fig. 3 shows the {111} pole figure of the 1000 crystal orientationsused as the input data for the numerical biaxial tensile test (σ11: σ22= 1:0), which was performed to calibrate the strain-hardening param-eters of the crystal plasticity constitutive equation. The same initial crys-tal orientation dataset was assigned to all integration points in themacro-FE model.The material constants for 5182-O aluminum alloy and the parame-ters used in this study are listed in Table 1 [32]. A fixed strain sensitivityfactor (m) of 0.005 was chosen because the strain sensitivity of alumi-num alloy is generally low.The strain-hardening parameters were calibrated using the numeri-cal biaxial test (σ11: σ22= 1:0) and the nodal velocity along the RDwasapplied to the nodes of the macro-FE model to obtain a nominal strainrate (5.0 × 10−4 s−1) similar to the experimental uniaxial tensile test.Further, the nodal forces in the numerical biaxial tensile tests were ap-plied to the nodes of themacro-FEmodel along the RD and TDusing thesame algorithm as the experimental biaxial tensile test to ensure a con-stant true stress ratio. The plastic strain rate during biaxial tensile defor-mation was set in the order of 5.0 × 10−4 s−1. The same seven linearstress paths evaluated in the experimental biaxial tensile tests werealso evaluated using the numerical biaxial tensile tests (i.e. σ11: σ22 =4:1, 2:1, 4:3, 1:1, 3:4, 1:2 and 1:4). The macroscopic nominal stresswas calculated by dividing the integrated value of the nodal forces onthe surface of the macro-FE model with the initial cross-sectional areaof the macro-FE model. The macroscopic nominal strain was calculatedby dividing the change in the side length of the macro-FE model afterdeformation by the initial length.2.2.3. ValidationAs stated in the previous section, the strain-hardening parameters inEq. (18) were identified by fitting the true stress-true strain curveFig. 2. The macro- and microscale finite element mobtained from the numerical biaxial tensile test (σ11: σ22 = 1:0) tothe experimental uniaxial stress-strain curve for the RD of the 5182-Oaluminum alloy sheet. In Fig. 4 (a), the calculated stress-strain curve iscompared with the experimental result. From the fitting of the curves,we identified the strain-hardening parameters which are listed inTable 2. Furthermore, the true stress-true strain curve for TDwas calcu-lated using numerical biaxial tensile test (σ11: σ22 = 0:1) based on theidentified parameters, and the results are compared with the experi-mentally obtained curve as shown in Fig. 4(b).Biaxial true stress-logarithmic plastic strain curves were calculatedusing the calibrated numerical biaxial tensile tests. The calculatedstress-strain curves displayed high correlation with the experimentalresults at true stress ratios of σ11: σ22 = 4:1, 3:4, 1:2, and 1:4 (seeFig. 5). However, the numerical biaxial tensile tests slightlyoverestimated the true stress at the other stress ratios. Overall, the nu-merical biaxial tensile tests provided a reasonably accurate predictionof biaxial tensile testing. Therefore, numerical biaxial tensile tests withthe parameters listed in Table 2were used to generate the DNN trainingodels for the numerical biaxial tensile tests.Fig. 4. True stress-true strain curves obtained from experiment (uniaxial tensile test) and numerical biaxial tensile test for the (a) RD and (b) TD of 5182-O aluminum alloy at a true stressratio of σ11:σ22 = 1:0.Table 2Strain-hardening parameters calibrated based on the numerical biaxial tensiletest at a true stress ratio of σ11: σ22 = 1:0.Initial CRSS, τ0 [MPa] 45Latent-hardening matrix, q(αβ) 1.0Initial hardening coefficient, h0 [MPa] 115Hardening index, n’ 0.24Hardening constant, C 17Initial plastic shear strain, γint 0.15A. Yamanaka et al. / Materials and Design 195 (2020) 108970and validation datasets. Themain goal of this study was to demonstratethat DNNs can estimate biaxial stress-strain curves based on crystallo-graphic texture with same accuracy as the numerical biaxial tensileFig. 5. Biaxial true stress-logarithmic plastic strain curves at seven true stress ratios calculatedtensile tests.tests. Therefore, it was assumed that the strain-hardening behaviorand the corresponding parameters (see Table 2) were not affected bychanges in crystallographic texture.2.2.4. Training, validation, and test datasetsThe development of a DNN machine-learning algorithm requirestraining, validation, and test datasets. During training, the DNN iden-tifies patterns in a training dataset that includes input and outputdata. Here, the input data was comprised of true stress ratios and digitalimages of the synthetic crystallographic texture, while the output datawas the biaxial true stress-logarithmic plastic strain curves calculatedusing the numerical biaxial tensile test. The validation dataset wasused to optimize the weights and biases of the DNN, after which thetest dataset was used for testing the performance of the trained DNN.from experiments (biaxial tensile testing with a cruciform sample) and numerical biaxialTable 3Bunge Euler angle of the ideal orientations of the different preferred texture components.Component Euler angles (ϕ 1, ϕ, ϕ 2; °)Cube {001}〈100〉 (0, 0, 0)Goss {110}〈001〉 (0, 45, 0)S {123}〈634〉 (59, 37, 63), (27, 58, 18), and (53, 75, 34)Brass {110}〈112〉 (35, 45, 0) and (55, 90, 45)Copper {112}〈111〉 (90, 35, 45) and (39, 66, 27)6 A. Yamanaka et al. / Materials and Design 195 (2020) 108970It is important to note that the DNN was not further optimized duringthis independent testing step.2.3. Crystallographic texture2.3.1. Synthetic crystallographic textureThe synthetic crystallographic texture was based on five preferredtexture components generally observed in aluminum alloys, namelyCube {001} 〈100〉, Goss {110}〈001〉, S {123}〈634〉, Brass {110} 〈112〉,and Copper {112} 〈111〉. Cube and Goss are recrystallization texturecomponents, whereas S, Brass, and Copper are deformation texturecomponents. The Bunge Euler angles (ϕ1, ϕ, and ϕ 2) of the ideal orien-tations of each texture component are given in Table 3.The synthetic texture was modeled using a three-dimensionalGaussian distribution function based on the ideal orientations given inTable 3. The Gaussian distribution function is expressed as:f ϕ1;ϕ;ϕ2ð Þ ¼ 1ffiffiffiffiffiffi2πp� �3ζ3iexp −ϕ21 þ ϕ2 þ ϕ222ζ2i !ð19Þwhere ζi2 (i=Cube, S, Goss, Brass, and Copper) denotes the variance ofeach preferred texture component in the synthetic texture with respectto the ideal orientation. The synthetic texture was determined usingEq. (19) as follows:Step 1: The volume fraction (Vi) and variance (ζi2) (i=Cube, S, Goss,Copper, and Brass) of each texture component in the synthetic tex-ture were determined. The number of crystal orientations of eachtexture component (Nori(i)) in the synthetic texture was calculated asNori(i) = 1000Vi.Step 2: Three random real numbers (a, b, c) between 0 and 1 weregenerated from a Gaussian distribution with mean = 0 andvariance = 1.Step3:The real numbers generated inStep2weremultipliedby the stan-dard deviation (ζi) to determine the random orientation (ϕ’1, ϕ’, ϕ’2; °).Step 4: The ideal orientation of the texture component was added tothe randomorientation obtained in Step 3 to give the synthetic textureorientation (ϕ 1,ϕ,ϕ 2) (e.g. orientation angle (ϕ1,ϕ,ϕ 2)= (ϕ’1+59°,ϕ’+ 37°, ϕ’2 + 63°) for the S-component).Fig. 6. Schematic diagram of the procedure for crStep 5: Steps 2 to 4 were repeated for all the crystal orientations.Various synthetic textures were generated by changing the volumefraction of the preferred texture component in the synthetic texture(Vi) in 10% increments from 0% to 100%. If the sum of the volume frac-tions of the preferred texture components was less than 100%, the re-mainder was attributed to a random component. The variance (ζi2) inStep 1 was changed every 5 deg2 from 5 to 15 deg2, as described in apreviously reported method byWu et al. [58]. A total of 5944 synthetictextures were generated for the training and validation datasets.For the test dataset, synthetic texturewas generated by changing thevolume fraction of the preferred texture components (Vi) in 10% incre-ments from0% to 60%,while the variance (ζi2)was changed every 5 deg2from 5 to 15 deg2. A total of 252 synthetic textures were generated forthe test dataset.2.3.2. Pole figure of synthetic crystallographic textureDNN-2Dwas developed to estimate biaxial stress-strain curves froma digital image of the {111} pole figure of a synthetic texture. The digitalimage of a {111} pole figure was generated as follows (see Fig. 6):Step 1: The position of the pole (Q(x, y)) for a crystal orientation inthe synthetic texture was determined via stereographic projection.Step 2: The projection plane was divided into N2 = 2n × 2n sub-domains, where the number of sub-domains corresponded to theresolution of the digital image of the {111} pole figure.Step 3: The sub-domain, Q(i, j), containing the pole, Q(x, y), was de-termined, where i (i = 1–2n) and j (j = 1–2n) denote the indices ofthe sub-domain.Step 4: Steps 1, 2, and 3were repeated for all the crystal orientationsin the synthetic texture.Step 5: The number of poles in each sub-domain was calculated anddenoted asM(i, j). The luminance of each pixel in the digital image ofthe {111} pole figure was calculated as L(i, j) = 255 M(i, j)/a. Thevalue of a is fixed, where a value of 10 was used in this study.A small proportion (0.01%) of the calculated pole figures containedpixels with a luminance value above 255. The luminance of these pixelswas corrected to 255.Fig. 7 shows the digital images of {111} pole figures produced by theabove procedure, in which the synthetic texture consisted of a singlepreferred texture component. The gray scale represents the integrationdegree of the texture component. High resolution pole figure imagescan generate a large computational load during DNN training and esti-mation of biaxial stress-strain curves using the trained DNN. Therefore,the effect of pole figure image resolution on estimation accuracy wasevaluated by trial-and-error. A preliminary investigation indicatedthat a {111} pole figure image resolution of 128 × 128 pixels (i.e. n =7) was most suitable.The digital images of the {111} pole figures were converted viamonochrome inversion during the training of DNN-2D, which followthe Modified National Institute of Standards and Technology (MNIST)eating a digital image of a {111} pole figure.Fig. 7. Digital images generated from {111} pole figures in which the synthetic texture consisted of a single preferred texture component, namely (a) Cube, (b) S, (c) Goss, (d) Brass, and(e) Copper. The pole figures were generated at a constant volume fraction (Vi = 100%) and variance (ζi2 = 10 deg2), where i = Cube, S, Goss, Brass, and Copper.7A. Yamanaka et al. / Materials and Design 195 (2020) 108970dataset [59]. The 5944 pole figure digital images generated for all thesynthetic textures were saved in portable network graphics (PNG)format.2.3.3. 3D orientation map of synthetic crystallographic textureThe 3D images of the synthetic texture used as the input data forDNN-3D were created using the following procedure:Step 1: The position of crystal orientations in the synthetic texturewas calculated in a 3D Euler angle space ranging 0° ≦ ϕ1 ≦ 360°, 0°≦ ϕ ≦ 180°, and 0° ≦ ϕ 2 ≦ 360° (see Fig. 8(a)).Step 2: The 3D Euler angle space was divided into Nx × Ny × Nzvoxels.Step 3: The orientation densities (ρi) of the ith (i=1, 2,…,Nx ×Ny ×Nz) voxelwere calculated asρi= ni / b, where ni is the number of ori-entations in the ith voxel and b is a constant for regularizing ρi be-tween 0 and 1. A matrix of orientation density (ρi) was used totrain DNN-3D.The number of voxels in the 3DEuler angle spacewasNx ×Ny ×Nz=32×16×32, thus the total number of voxelswas equal to the resolutionof the digital image of {111} pole figure (27 × 27). A b-value of 40 wasused to ensure that the synthetic texture was only observed when theorientation density (ρi) was larger than 1, thus the orientation densitywas correct to 1. An example of a 3D image of a synthetic texture com-prising a single Goss component with a variance of ζGoss2 = 10 deg2 isshown in Fig. 8(b). The 3D image data of a synthetic texture is hereafterreferred to as 3D orientation map.2.3.4. Biaxial true stress-logarithmic plastic strain curvesThe training and validation datasets contained biaxial true stress-logarithmic plastic strain curves calculated using the numerical biaxialFig. 8. (a) 3D plot of a synthetic texture containing a single Goss component (ζGoss2 = 10 deg2texture with the distribution of orientation density (ρi) in the voxelized Euler angle space.tensile tests based on synthetic texture. A total of 53,496 stress-straincurves were generated by performing the numerical biaxial tensiletests at 9 true stress ratios, namely σ11: σ22 = 1:0, 4:1, 2:1, 4:3, 1:1,3:4, 1:2, 1:4, and 0:1, based on 5944 synthetic textures. The test datacontained biaxial true stress-logarithmic plastic strain curves calculatedusing the numerical biaxial tensile tests based on 252 synthetic textures,thus producing a total of 2268 stress-strain curves (252 textures × 9true stress ratios).The dimensions of output data from the trained DNNs was reducedby processing the biaxial stress-strain curves as follows:Step 1: Non-dimensional true stress-logarithmic plastic straincurves were calculated by normalizing the true stress-logarithmicplastic strain curves obtained from the numerical biaxial tensiletests by their maximum values (σmax and εmaxp ).Step 2: The non-dimensional true stresses (σ i; i = 1, 2, …, ndiv) atequal intervals between 0.5 and 1.0 and the corresponding non-dimensional logarithmic plastic strain (εp0; i=1, 2,…, ndiv) were cal-culated. Here, ndiv is a constant.Thus, the biaxial stress-strain curves were presented as a numericalsequence including σ i, εpi (i = 1, 2, …, ndiv), σmax and εmaxp , where ndivwas set to 50 in this study.2.4. Deep learning method2.4.1. Architecture of DNN-2DDNN-2Dwas developed to estimate biaxial stress-strain curves froma digital image of the {111} pole figure representing synthetic texture(see Fig. 9). The six-layered DNN-2D was based on a similar DNN re-ported by Koenuma et al. [46], and consisted of convolution, maxpooling, and fully connected layers. A convolution layer and a max) in the 3D Euler angle space. (b) The corresponding 3D orientation map of the synthetic8 A. Yamanaka et al. / Materials and Design 195 (2020) 108970pooling layerwere used in the first layer to receive the input image (128× 128 pixels). A filter size of 7 × 7 and the stride value of 3 was used inthe convolution layer, while a filter size of 3 × 3 was used in the maxpooling layer. Thus, 16 images of 14 × 14pixelswere obtained to extractthe features of the input {111} pole figure. The second layer convertedthese 16 images to 32 images of 6 × 6 pixels using a convolution layerwith a filter size of 5 × 5 and a stride value of 2. The third layer com-pressed the 32 images into 64 images of 1 × 1 pixel using a convolutionlayer with a filter size of 5 × 5 and stride value of 1, and a max poolinglayerwith a filter size of 2 × 2. This layer further captured the features ofthe {111} pole figure image. The fourth layer combined the 64 imagesand corresponding true stress ratio (i.e. σ11: σ22) with a fully connectedlayer of 512 units. The nine true stress ratios (σ11: σ22 = 1:0, 4:1, 2:1,4:3, 1:1, 3:4, 1:2, 1:4, and 0:1) were converted to real numbers (0,0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, and 1, respectively). The neuralnetwork included two-branches to output the non-dimensional loga-rithmic plastic strains (εpi ; i = 1, 2, …, ndiv), maximum true stress(σmax) and logarithmic plastic strain (εmaxp ) values in the output layer.The fifth layer was a fully connected layer with 512 units to improvethe regression accuracy of the non-linear training data, whereas thesixth layer was a fully connected layer to output the normalized loga-rithmic plastic strain (εpi ), maximum true stress (σmax), and maximumlogarithmic plastic strain (εmaxp ) for RD and TD.In DNN-2D and DNN-3D, the convolution was followed by batchnormalization [60,61] in the first, second and third layers. Further,the exponential linear unit (ELU) [62] was used as an activationfunction for all layers except for the output layer. The mean squarederror (MSE) of the loss function was applied in DNN-2D and DNN-3D.DNN-2Dwas constructed and trained in theNeuralNetwork Consoledeveloped by Sony Network Communications Inc. [63]. The detailedspecifications of the layers used in DNN-2D are given in the web refer-ence [64] and the trained DNN-2D, the training parameters, and theFig. 9. Schematic illustration of DNN-2D for estimating biaxial stress-strain curvesdatasets used for training and validation are freely available at https://github.com/Yamanaka-Lab-TUAT/DNN-NMT [47].2.4.2. Architecture of DNN-3DDNN-3Dwas developed to estimate biaxial stress-strain curves fromthe 3D orientation map of a synthetic texture (see Fig. 10). The DNNconsisted of six layers and was constructed based on Keras [65] frame-work. The first layer consisted of a convolution layer and a max poolinglayer to receive the input data (3D orientation map). The convolutionlayer had a filter size of 7 × 7 × 7 and a stride value of 1, while themax pooling layer had a filter size of 2 × 1 × 1. The input data was con-verted to 16 voxel datasets of 14 × 12 × 14. The second layer convertedthe 16 voxel datasets to 32 voxel datasets of 6 × 5 × 6 using a convolu-tion layer with a filter size of 5 × 5 × 5 and a stride value of 2. The thirdlayer further compressed the 16 voxel datasets to 64 voxel datasets of 1× 1 × 1 using a convolution layer with a filter size of 5 × 5 × 5 and astride value of 1, and a max pooling layer with a filter size of 2 × 1 ×2. The fourth, fifth and sixth layers applied were same as thoseemployed in DNN-2D.If a larger number of intermediate layers were to be included, the es-timation accuracy of the biaxial stress-strain curves is expected to im-prove. However, it can result in overtraining due to increase in thenumber of optimized weights and bias. A preliminary investigation ofthe effect of the number of intermediate layers on the estimation accu-racy was conducted by including 1, 2, and 3 intermediate layers. The es-timation accuracy did not improve when 3 intermediate layers wereused, thus 2 intermediate layers (i.e., fourth and fifth layers) were cho-sen. The number of units in the intermediate layers were evaluated inthe range 256 to 1024, thus confirming that 512 units was suitable.2.4.3. Training and validation of DNN-2D and DNN-3DDNN-2D and DNN-3D were trained using a mini-batch-basedtraining scheme based on the Adam optimization algorithm [66].from a digital image of the {111} pole figure representing a synthetic texture.https://github.com/Yamanaka-Lab-TUAT/DNN-NMThttps://github.com/Yamanaka-Lab-TUAT/DNN-NMTFig. 10. Schematic illustration of DNN-3D for estimating biaxial stress-strain curves from a 3D orientation map of a synthetic texture.9A. Yamanaka et al. / Materials and Design 195 (2020) 108970A mini-batch size of 256 was chosen based on a preliminary evaluationof various mini-batch sizes (64, 128 and 256).The resulting curves exhibited variations in the loss function (MSE)during the training and validation of the DNNs (Fig. 11). Training wasterminated at 6000 epochs for DNN-2D and 3000 epochs for DNN-3D,where a full day was required for 5000 epochs using a graphic process-ing unit (NVIDIA TITAN V). The optimized weights and biases used forthe estimation of the biaxial stress-strain curves were determinedafter 1060 epochs and 1330 epochs during the validation of DNN-2Dand DNN-3D, respectively.Fig. 11. Variation in the loss function (MSE) during the training and validation of (a) DNN-2D arespectively. (For interpretation of the references to colour in this figure legend, the reader is r3. Results3.1. Synthetic textures for testing the trained DNNsTable 4 shows the volume fraction (Vi) and the variance (ζi2) of thepreferred texture components in three synthetic textures (Textures A,B, and C) used to test the trained DNNs. The three synthetic textureswere not included in the training or validation datasets. The volumefraction and the variance of Textures A, B, and C were determinedbased on the texture of previously reported aluminum alloy sheetsnd (b) DNN-3D, where the red and blue lines represent the training and validation curves,eferred to the web version of this article.)Table 4Volume fraction (Vi) and the variance (ζi2) of the preferred texture components in the synthetic textures (A, B, and C) used to test the trained DNN-2D and DNN-3D.Cube S Goss Brass Copper RandomVcube ζcube2 VS ζS2 VGoss ζGoss2 VBrass ζBrass2 VCopper ζCopper2 VRandomTexture A 6% 11 deg2 29% 7 deg2 5% 7 deg2 15% 7 deg2 15% 7 deg2 30%Texture B 5% 14 deg2 41% 9 deg2 4% 6 deg2 22% 9 deg2 16% 11 deg2 12%Texture C 18% 13 deg2 8% 8 deg2 10% 12 deg2 2% 13 deg2 4% 5 deg2 58%10 A. Yamanaka et al. / Materials and Design 195 (2020) 108970[67]. Examples of the digital images of the {111} pole figures and the 3Dorientation maps for Textures A, B, and C are shown in Figs. 12 and 13,respectively. The major texture component in Textures A and B wasthe S-component, while Texture C contained a large volume fractionof the Cube-component and random texture.The trained DNN-2D and DNN-3Dwere tested by comparing the es-timated biaxial stress-strain curves with those calculated using the nu-merical biaxial tensile tests. The crystal orientations in the synthetictextures were dependent on the random numbers which were used togenerate the synthetic textures. Thus, five sets of synthetic textureswere generated based on the volume fraction and the variance listedin Table 4. The five generated synthetic textures were used as initialcrystal orientations in the numerical biaxial tensile tests. The biaxialstress-strain curves obtained from the numerical biaxial tensile testswere used as the reference data for the trained DNN estimations.The accuracy of the biaxial stress-strain curves estimated by thetrained DNNs was evaluated based on 50 {111} pole figures and 3D ori-entation maps of Textures A, B, and C generated using the volume frac-tions and variances listed in Table 4. The pole figures and the 3Dorientation maps were used as input data for the trained DNNs and bi-axial stress-strain curves were estimated. The estimated mean biaxialstress-strain curve was compared with the numerical biaxial tensiletest results.Fig. 12. Example of digital images of {111} poleFig. 13. Example of 3D orientation maps3.2. Estimation of biaxial stress-strain curves using the trained DNNsThe biaxial true stress-logarithm plastic strain curves for Texture Aestimated using the trained DNN-2D are given in Fig. 14. The trainedDNN-2D provided an accurate estimation in comparison to the biaxialstress-strain curves calculated from the numerical biaxial tensile testsfor all values of the stress ratios. Further, as shown in Fig. 15, the biaxialtrue stress-logarithmplastic strain curveswere estimated by the trainedDNN-3D with the same accuracy as the trained DNN-2D.The results obtained from the trained DNN-2D and DNN-3D werequantitatively compared based on the contours of equal plastic workin the stress space. This approach was first introduced by Hill andHutchinson [68] and has been widely applied to evaluate the work-hardening behavior of sheet metals subjected to biaxial loading. Thecomparison between the work contours calculated from the biaxialstress-strain curves estimated by the trained DNNs and the numericalbiaxial tensile tests for Textures A, B, and C is illustrated in Fig. 16. Al-though the estimated work contours deviated slightly from the refer-ence data at stress ratios of σ11: σ22 = 4:1 and 1:2, the results of theDNNs were in good agreement with the numerical biaxial tensile testresults.The estimation capability of the trained DNNs was evaluated basedon the root mean squared error (RMSE) calculated as follows:figure for Textures (a) A, (b) B, and (c) C.for Textures (a) A, (b) B, and (c) C.Fig. 14. Biaxial true stress-logarithmic plastic strain curves for Texture A calculated using the numerical biaxial tensile tests (black) and estimated by the trained DNN-2D (red). (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)11A. Yamanaka et al. / Materials and Design 195 (2020) 108970RMSE j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1NtestXNtesti¼1dij� �2vuut ð20Þwhere j (j=1, 2,…, 9) is the index number of nine true stress ratios, viz.σ11: σ22 = 1:0, 4:1, 2:1, 4:3, 1:1, 3:4, 1:2, 1:4, and 0:1. As shown inFig. 17(a), dij is the difference between the stress points consistingthe work contours calculated from the DNN and numerical biaxialtensile test results for the jth stress path; i (i = 1, 2, 3, …, Ntest)denotes the ith test dataset and Ntest is the number of the test datasets(i.e. Ntest = 252).The RMSE for ε0p = 0.01, 0.02, 0.03, and 0.04 is illustrated in Fig. 17(b) and (c), which indicates that the RMSE increased with increase inε0p. Further, the RMSE for all stress ratios was less than 7 MPa, whichcorresponded to 3% of the maximum estimated true stress value. Fur-thermore, with the exception of σ11: σ22 = 1:4 and σ11: σ22 = 1:2,when ε0p = 0.01, the RMSE of the work contours estimated by thetrained DNN-3D were smaller than DNN-2D. These findings demon-strated that the trained DNN-3D provided a more accurate estimationthan DNN-2D.3.3. Computational efficiency of DNN-2D and DNN-3DIn the previous section, we showed that the trained DNN-3D can es-timate the biaxial stress-strain curves and the work contour with ahigher accuracy than DNN-2D for most of the stress ratios. This sectioncompares the computational efficiency of DNN-2D and DNN-3D, whichis an important aspect in the practical use of DNNs.To compare the computational efficiency in the training of DNN-2Dand DNN-3D, we measured the training time per epoch for DNN-2Dusing the profiling function of Neural Network Console [69]. On theother hand, the training time per epoch for DNN-3D was evaluated byaveraging the training time required for 5 epochs, which was obtainedby using Keras's callback function. The result elucidates that the trainingefficiency of DNN-2D (training time per epoch: 13.45 s) is better thanthat of the DNN-3D (training time per epoch: 23.43 s).We further investigated the computational efficiency in the esti-mation of biaxial stress-strain curves using DNN-2D and DNN-3Dby measuring the time spent for generating the curves for nine stressratios using 50 sets of synthetic texture A. The result clearly showsthat the estimation of biaxial stress-strain curves using DNN-2DFig. 15. Biaxial true stress-logarithmic plastic strain curves for Texture A calculated using the numerical biaxial tensile tests (black) and estimated by the trained DNN-3D (red). (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)12 A. Yamanaka et al. / Materials and Design 195 (2020) 108970(required time: 8.347 s) is faster than that of DNN-3D (requiredtime: 12.560 s).In summary, DNN-3D can estimate the biaxial stress-strain curvewith a higher accuracy, though its computational efficiency is lowerthan that of DNN-2D. This decline in computational efficiency was ex-pected since DNN-3D needs to perform the feature extraction of the3D orientation map using the 3D convolution neural network.4. DiscussionTo calculate the biaxial stress-strain curves for nine true stress ratios,the numerical biaxial tensile tests requires at least one hour with paral-lel computing usingmultiple CPUs. On the other hand, the trainedDNNsprovided an accurate estimation of biaxial stress-strain curves for 50synthetic textures in under a minute. The trained DNNs are computa-tionally efficient tools for predicting biaxial tensile deformation behav-ior of aluminum alloy sheets without the use of any user-developedsource codes for crystal plasticity simulations. Therefore, DNN is apromising method for generating virtual data in the material modelingof sheet metals.The biaxial stress-strain curves estimated by the trained DNNscorrelate well with the numerical biaxial tensile test results. How-ever, as shown in Fig. 17(b) and (c), the RMSE at the stress ratio ofσ11: σ22 = 1:4 is higher than the RMSE at other stress ratios. To iden-tify the origin of high RMSE, we examined the relationship betweenthe synthetic textures in the test datasets and the RMSE. Table 5shows the five synthetic textures for which we found the five highestvalues of di8 (see Eq. (20)) at the reference plastic strain of ε0p = 0.04.It is clearly observed that the high di8 correlates with a relatively highvolume fraction of Goss-component (Vgoss ≥ 50%). To further demon-strate the correlation between di8 and Vgoss, all biaxial stress-straincurves at the stress ratio of σ11: σ22 = 1:4 in the training datasetsare shown in Fig. 18; we also show the five biaxial stress-straincurves in the test datasets corresponding to the synthetic textureslisted in Table 5. The five stress-strain curves for the TD (black solidlines in Fig. 18), which the trained DNNs could not accurately esti-mate, lies in the region with low training data density, resulting inhigh di8 and consequently a high RMSE. Therefore, the high RMSEat the stress ratio of σ11:σ22 = 1:4 shown in Fig. 17 can be reducedby increasing the training data density.Fig. 16. Contours of equal plastic work calculated from the biaxial true stress-logarithmic plastic strain curves estimated by the trained DNNs and calculated using the numerical biaxialtensile tests for (a) Texture A, (b) Texture B, and (c) Texture C,where ε0p is the reference plastic strain and represents the logarithmic plastic strain in the uniaxial tensile state in RD (plasticwork per unit volume).13A. Yamanaka et al. / Materials and Design 195 (2020) 108970The accuracy of the DNNs can be further improved. In this study, the{111} pole figure images were used as input data for DNN-2D, but thestereographic projection used to plot the pole figures may yield thesame pole on a pole figure even if we use different crystal orientations.It was thus difficult for the DNN-2D to distinguish slight differences inthe synthetic textures. The estimation accuracy of DNN-2D is expectedto improve if multiple pole figures were used as for input images forDNN-2D, i.e. complementing the {111} pole figure with {100} and{110} pole figures.It was assumed that changes in crystallographic texture did not af-fect the strain-hardening behavior of aluminum alloy sheets. On thecontrary, the stress-strain curve of an aluminum alloy sheet isdependent on crystallographic texture, crystal grain size, precipitates,and othermicrostructures [70]. In particular, the dislocation density sig-nificantly affects stress-strain curve of aluminum alloy sheets. The esti-mation of biaxial stress-strain curves, which depends on such variousmicrostructural factors, using a DNN approach requires sufficientlylarge database of microstructural data and stress-strain curves of vari-ous aluminumalloy sheets. Although someprevious studies have devel-oped databases for specific aluminum alloys [71,72], an appropriatedatabase of the biaxial tensile deformation behavior of aluminumalloy sheets has not yet been developed. The development of a databasecontaining experimental multiaxial stress test results is a major issuehindering the further development of the proposed DNN approach.Fig. 17. (a) Schematic representation of dij used for calculating the RMSE (Eq. (20)). RMSE of the contours of equal plasticwork estimated by the trained (b) DNN-2D and (c) DNN-3Dwithreference to the numerical biaxial tensile tests at different reference plastic strains (ε0p).Table 5Volume fraction (Vi) and variance (ζi2) of the five synthetic textures for which highest di8 at the reference plastic strain of ε0p= 0.04were observed in the test datasets. The texture ID cor-responds to the index i of di8.Texture ID Cube S Goss Brass Copper RandomVcube ζcube2 VS ζS2 VGoss ζGoss2 VBrass ζBrass2 VCopper ζCopper2 VRandom51 10% 9 deg2 10% 13 deg2 60% 8 deg2 10% 13 deg2 10% 8 deg2 0%63 10% 10 deg2 10% 10 deg2 50% 6 deg2 10% 13 deg2 10% 14 deg2 10%134 20% 6 deg2 10% 7 deg2 50% 10 deg2 10% 10 deg2 10% 7 deg2 0%3 10% 5 deg2 10% 11 deg2 50% 12 deg2 10% 14 deg2 20% 13 deg2 0%90 10% 12 deg2 10% 10 deg2 50% 13 deg2 10% 5 deg2 10% 14 deg2 10%Fig. 18. Biaxial true stress-logarithmic plastic strain curves in the training datasets (redand blue lines); the opacity of the curves shows the training data density. Yellow andblack solid lines show the biaxial stress-strain curves for RD and TD in the test datasetscorresponding to the synthetic textures listed in Table 5. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version ofthis article.)14 A. Yamanaka et al. / Materials and Design 195 (2020) 108970Database development is a time-consuming task, but transfer learning[73–75] is a promising methodology for re-training a pre-trained DNNwith a small experimental dataset. Nonetheless, it should be emphasizedthat the DNNs developed in this study will contribute to future work.5. ConclusionsThe validity of DNN approach for efficient material modeling of sheetmetals was demonstrated by performing numerical experiments. Specif-ically, twoDNNs (DNN-2D andDNN-3D)were developed to estimate thebiaxial true stress-logarithmic plastic strain curves of aluminum alloysheets from images of crystallographic texture based on preferential tex-ture components. The input image data included {111} pole figure im-ages for DNN-2D and 3D orientation maps for DNN-3D. Training,validation and testing datasets were generated using numerical biaxialtensile tests based on CPFEM. The numerical biaxial tensile test was ex-perimentally validated based on uniaxial and biaxial tensile tests of5182-O aluminum alloy sheet. The biaxial stress-strain curves estimatedby the trained DNNs highly correlated with those calculated by the15A. Yamanaka et al. / Materials and Design 195 (2020) 108970numerical biaxial tensile tests. The precision of DNN-2D and DNN-3Dwas compared based on work contours, which indicated that the DNN-3D is more accurate than DNN-2D. However, the computational effi-ciency of DNN-2D was found to be higher than that of DNN-3D.This study has demonstrated that the proposed DNNs and othermachine-learning procedures offer a new approach for the generationof virtual data aimed at material modeling of sheet metals. The applica-tion of proposed DNN based approach to real sheet metals relies on thedevelopment of a multiaxial material test database and the future workaims to retrain the DNNs with an improved database.Data availabilityThe data that supports the results of this study are available from thecorresponding author upon reasonable request.Declaration of Competing InterestThe authors declare that they have no known competing financialinterests or personal relationships that could have appeared to influ-ence the work reported in this paper.AcknowledgementThe authors would like to thank Prof. Hiroshi Utsunomiya of OsakaUniversity for the EBSDmeasurements of 5182-O aluminumalloy sheet.This research was financially supported by JSPS Grant-in-Aid for Scien-tific Research (KAKENHI) (B) Grant Number JP17H03425 andJP20H02476.Appendix A. Supplementary dataSupplementary data to this article can be found online at https://doi.org/10.1016/j.matdes.2020.108970.References[1] E. Tekkaya, State-of-the-art of simulation of sheet metal forming, J. Mater. Proc.Technol. 103 (2000) 14–22.[2] M.A. Ablat, A. Qattawi, Numerical simulation of sheet metal forming: a review, Int. J.Adv. Manuf. Technol. 89 (2017) 1235–1250.[3] R. Hill, A theory of the yielding and plastic flow of anisotropic metals, Proc. RoyalSoc. Lond. A 193 (1948) 281–297.[4] F. Barlat, R.C. Becker, Y. Hayashida, Y. Maeda, M. Yanagawa, K. Chung, J.C. Brem, D.J.Lege, K.Matsui, S.J. Murtha, S. Hattori, Yielding description for solution strengthenedaluminum alloys, Int. J. Plasticity 13 (1997) 385–401.[5] F. Yoshida, T. Uemori, A model of large-strain cyclic plasticity describing theBauschinger effect and workhardening stagnation, Int. J. Plasticity 18 (2002)661–686.[6] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.H. Choi,E. Chu, Plane stress yield function for aluminum alloy sheets – part 1: theory, Int. J.Plasticity 19 (2003) 1297–1319.[7] D. Banabic, H. Aretz, D.S. Comsa, L. Paraianu, An improved analytical description oforthotropy in metallic sheets, Int. J. Plasticity 21 (2005) 493–512.[8] H. Vegter, A.H. Van den Boogaad, A plane stress yield function for anisotropic sheetmaterial by interpolation of biaxial stress states, Int. J. Plasticity 22 (2006) 557–580.[9] T. Kuwabara, Advances in experiments on metal sheets and tubes in support of con-stitutive modeling and forming simulations, Int. J. Plasticity 23 (2007) 385–419.[10] International Organization for Standardization, Metallic Materials\\Sheet andStrip\\Determination of Biaxial Stress-strain Curve byMeans of Bulge TestWith Op-tical Measuring Systems (ISO Standard No. 16808:2014), https://www.iso.org/stan-dard/57777.html 2014 (accessed on 24 June 2020).[11] L.M. Alves, C.V. Nielsen, P.A.F. Martins, Revisiting the fundamentals and capabilitiesof the stack compression test, Exp. Mech. 51 (2011) 1565–1572.[12] A. Makinde, L. Thibodeau, K.W. Neale, Development of an apparatus for biaxial test-ing using cruciform specimens, Exp. Mech. 32 (1992) 138–144.[13] T. Kuwabara, S. Ikeda, K. Kuroda, Measurement and analysis of differential workhardening in cold-rolled steel sheet under biaxial tension, J. Mater. Proc. Tech.80–81 (1998) 517–523.[14] International Organization for Standardization, Metallic Materials – Sheet and Strip– Biaxial Tensile Testing Method Using a Cruciform Test Piece (ISO Standard No.16842:2014), https://www.iso.org/standard/57812.html 2014 (accessed on 24June 2020).[15] T. Kubabara, F. Sugawara, Multiaxial tube expansion test method for measurementof sheet metal deformation behavior under biaxial tension for a large strain range,Int. J. Plasticity 45 (2013) 103–118.[16] T. Kuwabara, A. Van Vael, E. Iizuka, Measurement and analysis yield locus and workhardening characteristics of steel sheets with different r-values, Acta Mater. 50(2002) 3717–3729.[17] T. Kuwabara, M. Kuroda, V. Tvergaard, K. Nomura, Measurement of subsequent yieldsurface of prestrained sheet metal, Acta Mater. 48 (2000) 2071–2079.[18] D. Yanaga, T. Kuwabara, N. Uema, M. Asano, Material modeling of 6000 series alumi-num alloy sheets with different density cube textures and effect on the accuracy offinite element simulation, Int. J. Solids Struct. 49 (2012) 3488–3495.[19] F. Roters, P. Eisenlohr, L. Hantcherli, D.D. Tjahjant, T.R. Bieler, D. Raabe, Overview ofconstitutive laws, kinematics, homogenization and multiscale methods in crystalplasticity-element modeling: theory, experiments, applications, Int. J. Plasticity 58(2010) 1152–1211.[20] F. Roters, P. Eisenlohr, T.R. Bieler, D. Raabe, Crystal Plasticity Finite Element Methodsin Materials Science and Engineering, Wiley-VCH Verlag Gmbh & Co., Weinheim,2010.[21] E. Nakamachi, N.N. Tam, H. Morimoto, Multi-scale finite element analyses of sheetmetals by using SEM-EBSD measured crystallographic RVE models, Int. J. Plasticity23 (2007) 450–489.[22] T.J. Barrett, M. Knezevic, Deep drawing simulations using the finite element methodembedding a multi-level crystal plasticity constitutive law: experimental verifica-tion and sensitivity analysis, Comput. Methods Appl. Mech. Eng. 354 (2019)245–270.[23] N. Guo, J.Wang, C.Y. Sun, Y.F. Zhang,M.W. Fu, Analysis of size dependent earing evo-lution in micro deep drawing of TWIP steel by using crystal plasticity modeling, Int.J. Mech. Sci. 165 (2020) 105200.[24] Y. Shi, H. Jin, P.D. Wu, Analysis of cup earing for AA3104-H19 aluminum alloy sheet,Eur. J. Mech. A Solids 69 (2018) 1–11.[25] D. Raabe, Y. Wang, F. Roters, Crystal plasticity simulation study on the influence oftexture on earing in steel, Comput. Mater. Sci. 34 (2005) 221–234.[26] I. Tikhovskiy, D. Raabe, F. Roters, Simulation of earing during deep drawing of an Al-3% Mg alloy (AA 5754) using a texture component crystal plasticity FEM, J. Mater.Proc. Tech. 183 (2007) 169–175.[27] M. Kraska, M. Doig, D. Tikhomirov, D. Raabe, F. Roters, Virtual material testing forstamping simulations based on polycrystal plasticity, Comput. Mater. Sci. 46(2009) 383–392.[28] J. Gawad, D. Banabic, A. Van Bael, D. Comsa, M. Gologanu, P. Eyckens, P. Van Houtte,D. Roose, An evolving plane stress yield criterion based on crystal plasticity virtualexperiments, Int. J. Plasticity 75 (2015) 141–169.[29] F. Roters, M. Diehl, P. Shanthraj, P. Eisenlohr, C. Reuber, S.L. Wong, T. Maiti, A.Ebrahimi, T. Hochrainer, H.-O. Fabritius, S. Nikolov, M. Friák, N. Fujita, N. Grilli,K.G.F. Janssens, N. Jia, P.J.J. Kok, D. Ma, F. Meier, E. Werner, M. Sticker, D.Weygand, D. Raabe, DAMASK-the Dusseldorf Advanced Material Simulation Kitfor modeling multi-physics crystal plasticity, thermal, and damage phenomenafrom the single crystal up to the component scale, Comput. Mater. Sci. 158 (2019)420–478.[30] F. Han, M. Diehl, F. Roters, D. Raabe, Using spectral-based representative volume el-ement crystal plasticity simulations to predict yield surface evolution during largescale forming simulations, J. Mater. Proc. Technol. 277 (2020) 116449.[31] B.H. Frodal, L.E.B. Dælhi, T. Børvik, O.S. Hopperstad, Modelling and simulation ofductile failure in textured aluminum alloys subjected to compression-tensionload, Int. J. Plasticity 118 (2019) 36–69.[32] K. Hashimoto, A. Yamanaka, J. Kawaguchi, T. Sakurai, T. Kuwabara, Biaxial tensile de-formation simulation of 5000 series aluminum alloy sheet using crystal plasticity fi-nite element method based on homogenization method and its experimentalvalidation, J. Jpn. Inst. Light Met. 65 (2015) 196–203 (in Japanese).[33] A. Yamanaka, K. Hashimoto, J. Kawaguchi, T. Sakurai, T. Kuwabara, Material model-ling and forming simulation of 5182 aluminum alloy sheets using numerical biaxialtensile test based on homogenized crystal plasticity finite element method, J. Jpn.Inst. Light Met. 65 (2015) 561–567 (in Japanese).[34] W. Liu, B.K. Chen, Y. Pang, A. Najafzadeh, A 3D phenomenological yield functionwith both in and out-of-plane mechanical anisotropy using full-field crystal plastic-ity spectral method for modelling sheet metal forming of strong textured aluminumalloy, Int. J. Solid Struct. 193-194 (2020) 117–133.[35] K. Zhang, B. Holmedal, T. Manik, A. Saai, Assessment of advanced Taylor models, theTaylor factor and yield-surface exponent for FCC metals, Int. J. Plasticity 114 (2019)144–160.[36] S. Coppieters, T. Hakoyama, P. Eychens, H. Nakano, A. van Bael, D. Debruyne, T.Kuwabara, On the synergy between physical virtual sheet metal testing: calibrationof anisotropic yield functions using a microstructure-based plasticity model, Int. J.Mater. Form. 12 (2019) 741–759.[37] S. Yoshitake, V. Narayan, H. Harada, H.K.D.H. Bhadeshia, D.J.C. Mackay, Estimation ofthe γ and γ’ lattice parameters in nickel-based superalloys using neural networkanalysis, ISIJ Int. 38 (1998) 495–502.[38] H. Fujii, D.J.C. Mackay, H.K.D.H. Bhadeshia, Bayesian neural network analysis of fa-tigue crack growth rate in nickel base superalloys, ISIJ Int. 36 (1996) 1373–1382.[39] H.K.D.H. Bhadeshia, Neural networks in materials science, ISIJ Int. 39 (1999)966–979.[40] N. Haghdadi, A. Zarei-Hanzaki, A.R. Khalesian, H.R. Abedi, Artificial neural networkmodeling to predict the hot deformation behavior of an A356 aluminum alloy,Mater. Design 49 (2013) 286–391.[41] G. Allegri, Modelling fatigue delamination growth in fiber-reinforced composites:power-law equations or artificial neural networks? Mater. Design 155 (2018)59–70.https://doi.org/10.1016/j.matdes.2020.108970https://doi.org/10.1016/j.matdes.2020.108970http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0005http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0005http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0010http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0010http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0015http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0015http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0020http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0020http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0020http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0025http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0025http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0025http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0030http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0030http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0030http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0035http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0035http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0040http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0040http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0045http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0045https://www.iso.org/standard/57777.htmlhttps://www.iso.org/standard/57777.htmlhttp://refhub.elsevier.com/S0264-1275(20)30504-9/rf0055http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0055http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0060http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0060http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0065http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0065http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0065https://www.iso.org/standard/57812.htmlhttp://refhub.elsevier.com/S0264-1275(20)30504-9/rf0075http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0075http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0075http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0080http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0080http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0080http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0085http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0085http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0090http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0090http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0090http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0095http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0095http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0095http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0095http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0100http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0100http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0100http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0105http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0105http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0105http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0110http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0110http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0110http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0110http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0115http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0115http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0115http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0120http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0120http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0125http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0125http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0130http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0130http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0130http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0135http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0135http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0135http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0140http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0140http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0140http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0145http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0145http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0145http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0145http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0145http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0145http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0145http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0150http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0150http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0150http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0155http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0155http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0155http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0160http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0160http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0160http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0160http://refhub.elsevier.com/S0264-1275(20)30504-9/rf01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A. Yamanaka et al. / Materials and Design 195 (2020) 108970[42] G. Liu, L. Wang, Y. Yi, L. Sun, L. Shi, S. Ma, Inverse identification of graphite damageproperties under complex stress states, Mater. Design 183 (2019) 108135.[43] C. Yang, Y. Kim, S. Ryu, G.X. Gu, Prediction of composite microstructure stress-straincurves using convolution neural network, Mater. Design. 189 (2020) 108509.[44] A. Jenab, I.S. Sarraf, D.E. Green, T. Rahmaan, M.J. Worswick, The use of genetic algo-rithm and neural network to predict rate-dependent tensile flow behaviour ofAA5182-O sheets, Mater. Design. 94 (2016) 262–273.[45] U. Ali, W. Muhammad, A. Brahme, O. Skiba, K. Inal, Application of artificial neuralnetwork in micromechanics for polycrystalline metals, Int. J. Plasticity 120 (2019)205–219.[46] K. Koenuma, A. Yamanaka, I. Watanabe, T. Kuwabara, Estimation of texture-dependent stress-strain curve and r-value of aluminum alloy sheet using deeplearning, Mater. Trans. (2020) (in print).[47] Yamanaka Laboratory, Tokyo University of Agriculture and Technology, Deep NeuralNetwork-based Numerical Material Test Project, https://github.com/Yamanaka-Lab-TUAT/DNN-NMT 2020. (Accessed 24 June 2020).[48] D. Peirce, R.J. Asaro, A. Needleman, An analysis of nonuniform and localized defor-mation in ductile single crystals, Acta Metall. 30 (1983) 1087–1119.[49] J. Pan, J.R. Rice, Rate sensitivity of plastic flow and implications for yield-surface ver-tices, Int. J. Solid Struct. 19 (1983) 973–987.[50] U.F. Kocks, H. Mecking, The physics and phenomenology of strain hardening, Prog.Mater. Sci. 48 (2003) 171–273.[51] D. Pierce, C.F. Shih, A. Needleman, A tangent modulus method for rate dependentsolids, Comp. Structs. 18 (1984) 875–887.[52] J.M. Guedes, N. Kikuchi, Preprocessing and postprocessing formaterials based on thehomogenization method with adaptive finite element methods, Comput. MethodsAppl. Mech. Eng. 83 (1990) 143–198.[53] K. Terada, N. Kikuchi, A class of general algorithms for multi-scale analysis of hetero-geneous media, Comput. Methods Appl. Mech. Eng. 190 (2001) 5427–5464.[54] K. Terada, K. Matsui, M. Akiyama, T. Kuboki, Numerical re-examination of themicro-scale mechanism of the Bauschinger effect in carbon steels, Comput. Mater. Sci. 31(2004) 67–83.[55] I.Watanabe, K. Terada, M. Akiyama, Two-scale analysis for deformation-induced an-isotropy of polycrystalline metals, Comput. Mater. Sci. 32 (2005) 240–250.[56] I. Watanabe, K. Terada, E.A. de Souza Neto, D. Peric, Characterization of macroscopictensile strength of polycrystalline metals with two-scale finite element analysis, J.Mech. Phys. Solid. 56 (2008) 1105–1125.[57] L.S. Tóth, P. van Houtte, Discretization techniques for orientation distribution func-tions, Textur. Microstruct. 19 (1992) 229–244.[58] P.D. Wu, S.R. MacEwan, D.J. Lloyd, K.W. Neale, Effect of cube texture on sheet metalformability, Mater. Sci. Eng. A364 (2004) 182–187.[59] Y. LeCun, L. Bottou, Y. Bengio, P. Haffner, Gradient-based learning applied to docu-ment recognition, Proc. IEEE 86 (1998) 2278–2324.[60] I. Goodfellow, Y. Bengio, A. Courville, Deep Learning, MIT Press, Cambridge, MA,2016.[61] S. Ioffe, C. Szegedy, Batch normalization: accelerating deep network training by re-ducing internal covariate shift, Proc. ICML 32 (2015) 448–456.[62] D.A. Clevert, T. Unterthiner, S. Hochreiter, Fast and accurate deep network learningby exponential linear units (ELUs), in: Y. Bengio, Y. LeCun (Eds.),4th InternationalConference on Learning Representations, ICLR2016, San Juan, Puerto Rico, May 2-4, 2016, Conference Track Proceedings 2016, pp. 1–14https://arxiv.org/abs/1511.07289.[63] Sony, Neural Network Console, https://dl.sony.com/ 2020 (accessed on 24 June2020).[64] Sony, Neural Network Console: Layer Reference, https://support.dl.sony.com/docs/layer_reference/ 2020 (accessed on 24 June 2020).[65] Keras, Python Deep Learning Library, https://keras.io/ 2020 (accessed on 24 June2020).[66] P.D. Kingma, J. Ba, Adam: a method for stochastic optimization, in: Y. Bengio, Y.LeCun (Eds.),3rd International Conference on Learning Representations, ICLR2015,San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings 2015, pp. 1–15-http://arxiv.org/abs/1412.6980.[67] X.H. Zeng, M. Ahmad, O. Engler, Texture gradient, average texture, and plastic an-isotropy in various Al–Li sheet alloys, Mater. Sci. Technol. 10 (1944) 581–591.[68] R. Hill, J.W. Hutchinson, Differential hardening in sheet metal under biaxial loading:a theoretical framework, J. Appl. Mech. 59 (1992) S1–S9.[69] Sony, Neural Network Console: Profiling Function, https://support.dl.sony.com/docs/tutorial-profiling-the-training-processing-time/ 2020 (accessed on 24 June2020).[70] T. Kuwabara, T. Mori, M. Asano, T. Hakoyama, F. Barlat, Material modeling of 6016-Oand 6016-T4 aluminum alloy sheets and application to hole expansion forming sim-ulation, Int. J. Plasticity 93 (2017) 164–186.[71] M. Dixit, R.S. Mishra, K.K. Sankaran, Structure-property correlations in Al 7050 andAl 7055 high-strength aluminum alloys, Mater. Sci. Eng. A478 (2008) 163–172.[72] J.K. Paik, K.J. Kim, J.H. Lee, B.G. Jung, S.J. Kim, Test database of the mechanical prop-erties of mild, high- tensile and stainless steel and aluminium alloy associated withcold temperatures and strain rates, Ships Offshore Struct. 12 (2017) 5230–5256.[73] X. Li, Y. Zhang, H. Zhao, C. Burkhart, L.C. Brinson, W. Chen, A transfer learning ap-proach for microstructure reconstruction and structure-property predictions, Sci.Rep. 8 (2018) 13461.[74] S. Wu, Y. Kondo, M. Kakimoto, B. Yang, H. Yamada, I. Kawajima, G. Lambard, K.Hongo, Y. Xu, J. Shiomi, C. Schick, J. Morikawa, R. Yoshida, Machine-learning-assisted discovery of polymers with high thermal conductivity using a moleculardesign algorithm, npj Comput. Mater. 5 (2019) 66.[75] H. Yamada, C. Liu, S. Wu, Y. Koyama, S. Ju, J. Shimoi, J. Morikawa, R. Yoshida,Predicting material properties with little data using shotgun transfer learning, ACSCent. Sci. 5 (2019) 1717–1730.http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0210http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0210http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0215http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0215http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0220http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0220http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0220http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0225http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0225http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0225http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0230http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0230http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0230https://github.com/Yamanaka-Lab-TUAT/DNN-NMThttps://github.com/Yamanaka-Lab-TUAT/DNN-NMThttp://refhub.elsevier.com/S0264-1275(20)30504-9/rf0240http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0240http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0245http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0245http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0250http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0250http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0255http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0255http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0260http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0260http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0260http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0265http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0265http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0270http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0270http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0270http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0275http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0275http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0280http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0280http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0280http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0285http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0285http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0290http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0290http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0295http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0295http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0300http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0300http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0305http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0305https://arxiv.org/abs/1511.07289https://arxiv.org/abs/1511.07289https://dl.sony.com/https://support.dl.sony.com/docs/layer_reference/https://support.dl.sony.com/docs/layer_reference/https://keras.io/http://arxiv.org/abs/1412.6980http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0335http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0335http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0340http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0340https://support.dl.sony.com/docs/tutorial-profiling-the-training-processing-time/https://support.dl.sony.com/docs/tutorial-profiling-the-training-processing-time/http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0350http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0350http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0350http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0355http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0355http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0360http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0360http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0360http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0365http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0365http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0365http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0370http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0370http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0370http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0370http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0375http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0375http://refhub.elsevier.com/S0264-1275(20)30504-9/rf0375 Deep neural network approach to estimate biaxial stress-�strain curves of sheet metals 1. Introduction 2. Materials and methods 2.1. Experimental tensile testing 2.2. Numerical biaxial tensile testing 2.2.1. Crystal plasticity constitutive equation 2.2.2. Finite element model for numerical biaxial tensile test 2.2.3. Validation 2.2.4. Training, validation, and test datasets 2.3. Crystallographic texture 2.3.1. Synthetic crystallographic texture 2.3.2. Pole figure of synthetic crystallographic texture 2.3.3. 3D orientation map of synthetic crystallographic texture 2.3.4. Biaxial true stress-logarithmic plastic strain curves 2.4. Deep learning method 2.4.1. Architecture of DNN-2D 2.4.2. Architecture of DNN-3D 2.4.3. Training and validation of DNN-2D and DNN-3D 3. Results 3.1. Synthetic textures for testing the trained DNNs 3.2. Estimation of biaxial stress-strain curves using the trained DNNs 3.3. Computational efficiency of DNN-2D and DNN-3D 4. Discussion 5. Conclusions Data availability Declaration of Competing Interest section26 Acknowledgement Appendix A. Supplementary data References