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

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[Estimation of Texture-Dependent Stress-Strain Curve and <i>r</i>-Value of Aluminum Alloy Sheet Using Deep Learning](https://mdr.nims.go.jp/datasets/8a323b45-aeeb-4558-afa7-4d350fea5c13)

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Estimation of Texture-Dependent Stress-Strain Curve and r-Value of Aluminum Alloy Sheet Using Deep LearningEstimation of Texture-Dependent Stress-Strain Curve and r-Value of AluminumAlloy Sheet Using Deep Learning+1Kohta Koenuma1, Akinori Yamanaka2,+2, Ikumu Watanabe3 and Toshihiko Kuwabara21Graduate School of Engineering, Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology,Tokyo 184-8588, Japan2Division of Mechanical Systems Engineering, Institute of Engineering, Tokyo University of Agriculture and Technology,Tokyo 184-8588, Japan3Research Center for Structural Materials, National Institute for Materials Science, Tsukuba 305-0047, JapanThe deformation of an aluminum alloy sheet is affected by its underlying crystallographic texture and has been extensively studied usingthe crystal plasticity finite element method (CPFEM). Numerical material test based on the CPFEM enables the quantitative estimation of thestress-strain curve and Lankford value (r-value), which depend upon the texture of aluminum alloy sheets. However, the application of CPFEM-based numerical material test to the optimization of aluminum alloy texture is computationally expensive. In this paper, we propose a method forrapidly estimating the stress-strain curves and r-values of aluminum alloy sheets using deep learning with a neural network. We train the neuralnetwork with the synthetic crystallographic texture and stress-strain curves calculated through the numerical material tests. To capture thefeatures of synthetic texture from a {111} pole-figure image, the neural network incorporates a convolution neural network. Using the trainedneural network, we estimate the uniaxial stress-strain curve and in-plane anisotropy of the r-value for various textures that contain Cube and Scomponents. The results indicate that the application of a neural network trained with the results of numerical material test is a promising methodfor rapidly estimating the deformation of aluminum alloy sheets. [doi:10.2320/matertrans.P-M2020853](Received March 12, 2020; Accepted July 20, 2020; Published November 6, 2020)Keywords: numerical analysis, tension test, texture, crystallite orientation, flow stress, deep learning1. IntroductionThe basic information needed to understand and predictthe deformation behavior of aluminum-alloy sheets duringplastic forming is the stress-strain curve and Lankford value(r-value), which strongly depend upon the crystallographictexture formed during the manufacturing process.1) There-fore, a method to estimate the texture-dependent stress-straincurves and r-values of aluminum alloy sheets is requiredfor their design and development. However, as the texture ofaluminum alloy sheets involves various types of preferredcrystal orientations, time-consuming material testing isrequired to experimentally clarify the relationship betweenthe texture and stress-strain curve.Numerical simulation based on the crystal plasticity finiteelement method (CPFEM) is effective for estimating thetexture-dependent stress-strain curves of aluminum alloys.2,3)The results of a benchmark test held at NUMISHEET 20184)demonstrated the potential of CPFEM in predicting thestress-strain curves and the in-plane anisotropy of r-valueof the 5000-series aluminum-alloy sheets. These resultsindicated the potential of CPFEM-based numerical simu-lation as a numerical biaxial tensile test5) for estimating thestress-strain curves of aluminum alloys as an alternative toexperimental material testing. In addition to the experimentalmaterial testing, an effective method to obtain the stress-strain curves, r-values, and other mechanical responses fromthe texture of aluminum alloy sheets would be ideal.For several years, researchers have explored machinelearning techniques efficiently to estimate the physicalresponses of a material. Although there are severalapproaches using machine learning, the neural networkapproach was extensively applied in the field of materialengineering in the latter half of the 1990s.6­8) Yoshitake et al.proposed the use of neural networks as an effective methodto estimate the fatigue crack growth rate from informationon the chemical composition and crystal grain-size ofnickel-base superalloy.6) Bhadeshia summarized the methodsthrough which neural networks have been applied inmaterials science and concluded that these were veryeffective tools for recognizing the material features andpredicting the material properties based on their features.8)However, it was difficult to handle large quantities ofcomplex data because the techniques related to machineleaning were immature and the computational performancewas poor.Recently, the drastic improvements in computationalperformance and the development of new machine learningalgorithms such as deep learning have facilitated theapplication of neural networks to complex classificationand nonlinear regression problems involving multidimen-sional information such as digital images.9­11) In the fieldof materials engineering, Adachi et al.12) proposed a methodfor estimating the stress-strain curves using microstructuralinformation such as the grain-size of steels and constructedan integrated software.13) As an example of the applicationof machine learning for aluminum alloys, Sheikh et al.14)developed a method to accurately estimate the flow stressdepending on the strain rate and temperature, in the coldplastic forming of A5083 aluminum alloy. Yuan et al.15)demonstrated that the random forest method16) can predict thetexture-dependent stress-strain curve and post-deformationtexture of the material using the stress-strain curve of coppercalculated through the viscoplastic self-consistent (VPSC)method as training data. Onoshima et al.17) showed that thestress-strain curve of A1145 aluminum alloy can be+1This Paper was Originally Published in Japanese in J. JSTP 61 (2020)48­55.+2Corresponding author, E-mail: a-yamana@cc.tuat.ac.jpMaterials Transactions, Vol. 61, No. 12 (2020) pp. 2276 to 2283©2020 The Japan Society for Technology of Plasticityhttps://doi.org/10.2320/matertrans.P-M2020853accurately reproduced through deep learning to estimate theparameters of crystal plasticity constitutive equations used fornumerical material test. However, to the best of the authors’knowledge, the application of deep learning with a neuralnetwork to estimate the stress-strain curves and r-valuesdirectly from the texture of aluminum alloy sheets has notbeen proposed.In this study, we propose a method to rapidly estimate thestress-strain curves and the in-plane anisotropy of r-valuefrom the texture of aluminum alloy sheets using deeplearning with a neural network. The proposed method isequivalent to the use of deep learning for regression analysisof the nonlinear relationship between the texture of aluminumalloy sheets and their stress-strain curves or r-values. Toperform this nonlinear regression, vast quantity of trainingdata are required for training the neural network. The trainingdata in this study includes the textures of aluminum alloysheets along with their corresponding stress-strain curvesand r-values. It is difficult to acquire this training data withina short period through experimental material testing.Therefore, we generate numerous stress-strain curves and r-values calculated in advance by numerical material test3) forapplication as training data for the neural network.Moreover, we describe the training of the neural networkto estimate the stress-strain curves and the in-planeanisotropy of r-value using various types of texture. Weverify the proposed method by demonstrating that ourestimation results agree well with the numerical material testresults which are assumed as the true values.2. Preparation of Training Data through NumericalMaterial TestTo prepare the training data for deep learning, weconducted several numerical material tests based on theCPFEM. These tests require the initial crystal orientationas the input data. In this section, we describe the appliedmethodology for the numerical material tests and thegeneration of the initial crystal orientation data includingthe preferred texture orientation of the aluminum alloy sheets.2.1 Numerical material test using the crystal plasticityfinite element methodThe numerical material tests used for generating trainingdata were based on the CPFEM with the homogenizationmethod, which has been previously described.3) For detailson the numerical material tests, see Ref. 3). In this section,we describe the work hardening model used in the numericalmaterial tests.The resolved shear stress ¸(¡) in slip system ¡ is given bythe following equation:¸ð¡Þ ¼ Pð¡Þij · ij; ð1Þwhere ·ij is the Cauchy stress tensor and Pð¡Þij is the Schmidtensor for slip system ¡ (¡ = 1, 2,+ , 12). For the shear-strain rate in slip system ¡, we use the following exponentialmodel proposed by Peirce et al.:18,19)_£ ð¡Þ ¼ _£0¸ð¡Þgð¡Þ¸ð¡Þgð¡Þ��������1m�1; ð2Þwhere _£0 and m are the reference shear strain rate and strainrate sensitivity index, respectively; g(¡) is the critical resolvedshear stress in slip system ¡, and its time evolution (i.e., workhardening) is expressed by the following equations:gð¡Þ ¼ ¸0 þZt_gð¡Þdt; ð3Þ_gð¡Þ ¼X¢hð¡¢Þj _£ ð¢Þj; ð4Þwhere ¸0 is the initial critical resolved shear stress; h(¡¢) isthe hardening coefficient matrix which represents thecontribution of slip system ¡ to the hardening of slip system¢, and is expressed by the following equation:20)hð¡¢Þ ¼ qhð£Þ þ ð1� qÞhð£Þ¤¡¢; ð5Þwhere ¤ij is Kronecker’s delta, and q is a coefficient thatrepresents the level of latent hardening. The value of h(£)is obtained from the following equation related to theaccumulated shear strain:hð£Þ ¼ h0nCfCð£ int þ £Þgn�1; ð6Þwhere h0, n, C, and £ int are the initial work hardeningcoefficient, hardening index, hardening coefficient, and initialshear strain, respectively, and £ is the accumulated shearstrain.To perform numerical material test, it is necessary todetermine the parameters in eqs. (2)­(6). Although this studydoes not target specific materials, we use the parametersdetermined based on the true stress-strain curves obtainedthrough the uniaxial tensile testing of A5182-O aluminumalloy sheet which is the test material in Ref. 3). The sheetincludes Cube-texture components; the other physicalproperty values and parameter values are as indicated inRef. 3).2.2 Generation of synthetic crystallographic texturesIn order to generate training data for the neural networkused for deep learning, we created synthetic crystallographictextures that include multiple preferred crystal orientationsand applied them as the basis for generating the initial crystalorientation data used in the numerical material tests. Thesynthetic textures, which are a type of “pseudo texture”generated by numerical calculation, were created as describedbelow.The crystallographic texture in aluminum-alloys includesseveral preferred orientations such as the Cube, Goss, S,and copper components.21) Although it would be desirable tocreate a synthetic texture that contains the various preferredorientations, instead of creating a synthetic texture thatincludes all the preferred orientations, we created a texturewith only two preferred orientations: the Cube texture, whichis a typical recrystallization texture of aluminum alloys, andthe S texture, which is a rolling texture, as well as randomtextures. The purpose of this study is to use deep learningto rapidly estimate the stress-strain curves and r-values ofaluminum alloy sheets from the texture information, anddemonstrate that the estimated stress-strain curves agree withthe numerical material test results. Hence, we validate theproposed method for a simple synthetic texture containing theCube and S textures.Estimation of Texture-Dependent Stress-Strain Curve and r-Value of Aluminum Alloy Sheet Using Deep Learning 2277For generating the synthetic texture using the above-mentioned procedure, it is assumed that the crystal-orientation distribution follows the 3D Gaussian probabilitydensity function f shown below, where the average valueis considered to be the ideal Cube and S orientations inBunge’s Euler angular space, i.e., (º1,º,º2) = (0, 0, 0) and(59°, 37°, 63°).fðº1; º; º2Þ ¼1ðffiffiffiffiffiffi2³pÞ3²3iexp � º21 þ º2 þ º222²2i� �; ð7Þwhere ²2i (i = Cube or S) is the variance that indicates thevariation in crystal orientation with respect to the ideal Cubeor S texture component.We generated various synthetic textures by varying thevolume fractions of the preferred orientations included inthe synthetic texture and the variance given by eq. (7). In thisstudy, the volume fractions of the Cube and S textures(denoted by VCube and VS, respectively) were varied inincrements of 10% in the range of 10 and 90%. When thesum of VCube and VS was less than 100%, the remainingorientations were randomly created using uniform randomnumbers. Based on Wu et al.,22) we changed the variancesof the Cube and S textures (denoted by ²2Cube and ²2S,respectively) every 3 deg2 in the range of 2­11 deg2.The initial crystal orientation data for the numericalmaterial tests were generated using eq. (7) according to thefollowing steps. Henceforth, we refer to this sequence ofsteps as “sampling.”Step 1 Generate three real numbers in the range of 0­1 usinguniform random numbers, and then convert them to realnumbers (a, b, c) that follow a normal distribution with amean = 0 and variance = 1 through the Box-Mullermethod.23)Step 2 Multiply the real numbers obtained in Step 1 by thestandard deviation ²i and 360° to obtain the random crystalorientation (º1A,ºA,º2A). The results are in degrees (°) (unit).Step 3 Compute the initial crystal orientation data (º1, º,º2)by adding the Euler angle of the ideal orientation to thecrystal orientation (º1A,ºA,º2A) obtained in Step 2. Forexample, for the S texture, we obtain a crystal orientationof (º1,º,º2) = (º1A + 59°, ºA + 37°, º2A + 63°).Step 4 Generate the initial crystal orientation data that obeysthe probability density function f expressed by eq. (7), byrepeating Steps 1­3 the same times as the desired number ofinitial crystal orientations.The initial crystal orientation data will differ slightlydepending on the random numbers used in Step 1 when usingthe above procedure to generate the initial crystal orientationdata, even if we do not change the values of volume fractionand variance used to generate the initial crystal orientationdata. Specifically, the greater is the variance ²2i in eq. (7), thegreater is the difference in the initial crystal orientationsgenerated by sampling, and the greater is the variation in theresults of the numerical material tests. Therefore, in thisstudy, the number of sampling is given by eq. (8), for aprobability density function f representing a given synthetictexture,Nsample ¼VCube²2Cube þ VS²2S2: ð8ÞFor example, when a synthetic texture is VCube = 0.5,²2Cube = 8 deg2, VS = 0.1, and ²2S = 5 deg2, the samplingnumber is 4 according to eq. (8). Therefore, for this synthetictexture, the numerical material test was performed four times,and all the results were included in the training data. Whenthis procedure was applied to all the volume fractions andvariances, a total of 1,468 initial crystal orientations weregenerated.In order to obtain the in-plane anisotropy of r-values forall the initial crystal orientations, three types of numericalmaterial tests were performed at tensile directions of 0°, 45°,and 90°, relative to the rolling direction (RD) of the sheet.Therefore, a total of 1,468 © 3 = 4,404 numerical materialtests were performed, and the results of all the calculationswere used as training data.3. Deep Learning with Neural NetworkThis section describes the neural network used in thisstudy as well as the deep learning method using the trainingdata generated by the method described in Section 2.3.1 Generation of {111} pole-figure image of synthetictextureIn this study, we entered the texture information into theneural network, and obtained the estimates of the stress-straincurves and in-plane anisotropy of the r-values as the output.The format for entering the texture information into theneural network is an important factor that determines theaccuracy of the stress-strain curve and r-value estimates. Themethod adopted in this study involved entering a {111} pole-figure image into the neural network. Recent neural network-based deep learning techniques for the image recognitionexhibit extremely powerful discrimination capabilities whenimages are used as the input.24) This suggests that the usageof image data representing textural features as input to theneural network is a suitable method for estimating the stress-strain curves with high accuracy.The {111} pole-figure images of the synthetic textures,which are the input to the neural network, were generated asfollows. First, for the initial crystal orientation data generatedusing the method described in Section 2.2, the positions ofthe poles on the {111} pole figures were determined. Further,to form the pole-figure image, the pole figure was dividedinto N © N subregions, where each subregion corresponds toone image pixel. The luminance value of each pixel was thenset according to the number of poles included in the pixel. Inthis study, each pixel luminance value L was set to 255n/a,where n is the number of poles contained in the pixel and ais a constant set to 10. In approximately 0.01% of thesynthetic texture images used as training data in this study,there were pixels whose luminance values exceeded 255.In these cases, the pixel luminance value was corrected to255. Using the above-mentioned method, we represented thedegree of orientation of the texture in terms of the individualpixel luminance, as depicted in Fig. 1(a).The {111} pole-figure images were formatted to PNG forthe ease of compression. In order to reduce the memoryrequired for deep learning, the image resolution was alwaysset to 2n © 2n pixels, i.e., a power of two on each axis.K. Koenuma, A. Yamanaka, I. Watanabe and T. Kuwabara2278Before the {111} pole-figure image was entered into theneural network, it was converted into a black-and-whiteinverted image, as shown in Fig. 1(b). This step wasperformed to render the image data similar to the format ofthe MNIST dataset,25) which includes a set of sample imagesoften used in deep-learning image recognition; it is noted thateven if the data were not converted into black-and-whiteinverted images, the results of training the neural networkwould not be affected.3.2 Neural network applied for deep learningNeural networks, which are mathematical models usedto implement deep learning, are inspired by the biologicalmechanism of the system of neurons in the brain.26) Figure 2displays the schematic of a simple neural network; the circlesin the figure indicate the modeled neurons. A neural networkcomprises an input layer that receives input data, hiddenlayers that extract the features of the data received by theinput layer, and an output layer that generates output data.Aweight is assigned to each input value xi for each neuron,and the weights are then are summed as shown in theequation below. This equation depicts the output of a neuronin the hidden layer shown in Fig. 2.zj ¼X3i¼1wðjÞi xi; ð9Þwhere wi( j) is the weight assigned to the input value xi tothe j th neuron. In neural network training, the weights wi( j)are determined such that the relationships between thetraining data inputs and outputs are reproduced. To determinethe appropriate weight wi( j) for multiple inputs, optimiza-tion algorithms such as the gradient descent are used toreduce the difference between the true output valuescontained in the training data and the neural-networkestimated outputs.27,28) In this study, Adam29) was used asthe optimization algorithm. A layer in which all the neuronsare connected to the preceding and following layers, asshown in Fig. 2, is called a fully connected layer. Varioustypes of neural-network connection methods have beenproposed, in addition to the fully connected layers used indeep learning.14,30,31)In this study, we supplemented the fully connected layerswith convolutional layers, pooling layers, and dropoutlayers.30) A neural network that uses convolutional layersis called a convolutional neural network (CNN). CNNcombines convolutional and pooling layers in succession tocompress the input image and capture the image features.30)The dropout layers assist in optimizing the weights andprevent overlearning by ignoring certain neurons duringneural-network training.31)In this study, the above-described layers were used toconstruct the neural network shown in Fig. 3. The role ofeach layer in our network is as follows:Layer 1 is a convolutional layer that receives a 2n © 2npixel {111} pole-figure image and converts it into 16 imagesof 25 © 25 pixels that capture the features of the input pole-figure image.Fig. 1 (a) Example of {111} pole figure of a synthetic crystallographictexture and (b) the corresponding image used for the input data to theneural network.Fig. 2 Schematic diagram of a neural network.Fig. 3 Neural network constructed in this study. This neural network is composed of two pairs of convolutional and pooling layers andtwo fully connected layers.Estimation of Texture-Dependent Stress-Strain Curve and r-Value of Aluminum Alloy Sheet Using Deep Learning 2279Layer 2 is a sum pooling layer that calculates the sum ofthe 2 © 2-pixel regions (kernels) in each of the 16 imagesgenerated by Layer 1 in order to compress them into 16 © 16-pixel images representing the luminance values. Layer 3 isa convolutional layer that converts the image compressedby Layer 2 into 32 images of 3 © 3 pixels to capture thefeatures of the pole-figure image. Layer 4 converts the {111}pole-figure input into a 32-dimensional feature value bycalculating the maximum value of each 3 © 3-pixel region(kernel) in each of the 32 images generated by theconvolution layer in Layer 3. Layer 5 improves theregression accuracy for nonlinear training data using themax pooling layer and a fully connected layer. Layer 6 isa fully connected layer for outputting the stress-strain curveor r-value corresponding to the feature value of the {111}pole-figure in the input. The stress and strain are normalizedto values between zero (minimum) and unity (maximum).The normalized stress values and their corresponding strainvalues are then output along with the maximum andminimum stress and strain values used for normalization.To estimate the in-plane anisotropy of r-value, the valueindicating the angle of the tensile direction relative to the RDis added to the 32-dimensional feature value generated bythe max pooling layer in Layer 4. This enables the estimationof the r-value for any tensile direction.Individual neural networks are trained separately toestimate the stress-strain curves and the in-plane anisotropyof r-value from the texture images.The greater the resolution of the {111} pole figures, thelonger it takes to train and apply the neural network. In thisstudy, the estimation accuracy was verified by training neuralnetworks under three conditions, namely n = 6, 7, and 8.It followed that no further improvement of the estimationaccuracy was observed when the neura network was trainedat resolutions more than n = 7. Therefore, we set theresolution of the {111} pole-figure images to n = 7, i.e.,128 © 128 pixels.3.3 Validation of the trained neural networkWe used the trained neural network to estimate the stress-strain curves and the in-plane anisotropy of r-values fromthe texture information, and then verified its accuracy bycomparing the results with the numerical material test results.As explained in Section 2.2, the initial crystal orientationdata generated during sampling can vary depending on therandom numbers, even if we do not change the values ofvolume fraction and variance used to generate the synthetictexture. Therefore, when performing numerical material teststo generate verification data, we used five initial crystalorientation data obtained by sampling five times, using theprobability density function f representing a given synthetictexture. When estimating using the trained neural network,we used the same probability density function f to generatethe fifty {111} pole-figure images used as inputs forestimating the stress-strain curves.4. Estimation Results Using Neural Network4.1 Estimation of the stress-strain curvesFigure 4 depicts the {111} pole figures representing thethree types of initial crystal orientation generated using VCubeset to 10%, 50%, and 90%, respectively. Here, ²2Cube, VS, and²2S were set to 5 deg2, 10%, and 5 deg2, respectively.Figure 5(a) displays the true stress-logarithmic plasticstrain curves calculated by numerical material tests andestimated by the trained neural network in the tensiledirection of RD using the initial crystal orientation shownin Fig. 4; the error bars in the figure indicate the varianceof the estimated value depending on the {111} pole figure-images input to the neural network. The flow stress calculatedby numerical material test decreases as VCube increases. Inthe numerical material tests, even the volume fraction andvariance in the Cube texture were identical to those in the Stexture, the flow stress varied due to minor differences in theinitial crystal orientation. Similar to the numerical materialtest results, in the stress-strain curves estimated by the trainedneural network, the flow stress decreased with the increasein VCube. At VCube values of 10% and 50%, the valuesestimated by the neural network agreed well with thosecalculated by numerical material tests. On the other hand,Fig. 4 {111} pole figure of the synthetic crystallographic texture used asthe input data for the numerical material test and the trained neuralnetwork. The volume fraction of the Cube texture, VCube, for each textureis (a) 10%, (b) 50%, and (c) 90%, respectively. The volume fraction of theS texture, and variances of the Cube and S textures are fixed at VS = 10%,²2S = 5 deg2, and ²2Cube = 5 deg2, respectively.Fig. 5 True stress-logarithmic plastic strain curves for the tensile directionof (a) RD and (b) TD calculated by numerical tensile tests and estimatedby the trained neural network using the different textures shown in Fig. 4.K. Koenuma, A. Yamanaka, I. Watanabe and T. Kuwabara2280when VCube = 90%, the neural network overestimates theflow stress compared to that calculated by the numericalmaterial test.Figure 5(b) depicts the estimated results of the true stress-logarithmic plastic strain curves, when the transversedirection (TD) is the tensile direction. Using the trainedneural network, it was possible to estimate the decrease inflow stress associated with the increase in VCube, similar tothe results in Fig. 5(a). However, the estimation accuracywhen VCube = 10% was less than that shown in Fig. 5(a).Figure 6 shows the {111} pole figures representing thethree types of initial crystal orientation generated when thevariance of the Cube texture (²2Cube) for each texture is 2 deg2,5 deg2, and 8 deg2, respectively. Here, VCube, VS, and ²2S wereset to 90%, 10% and 5 deg2, respectively.Figure 7(a) displays the true stress-logarithmic plasticstrain curves calculated by numerical material tests andestimated using the trained neural network in the tensiledirection of RD with the initial crystal orientation shown inFig. 6. The flow stress calculated by numerical material testdecreases as ²2Cube increases. On the other hand, the stress-strain curves estimated by the trained neural network exhibitsan opposite trend. However, when ²2Cube is 2 deg2 and 5 deg2,the results agree well with the numerical material test results.When ²2Cube is 8 deg2, the stress-strain curves estimated bythe neural network are greater than those obtained from thenumerical material tests.Figure 7(b) shows the estimated results of the true stress-logarithmic plastic strain curves when the tensile direction isTD. The flow stress calculated by numerical material testdecreases as ²2Cube increases, similar to the results shown inFig. 7(a). When ²2Cube is 8 deg2, the estimates by the neuralnetwork are greater than the numerical material test results.4.2 Estimation of in-plane r-valueThis section presents the results of using the trained neuralnetwork to estimate the in-plane anisotropy of the r-valuefrom the texture information. Table 1 lists the volumefraction and variance of the Cube and S texture componentsof the synthetic textures used for estimating the r-value.Figure 8 depicts the changes in the r-value associated withthe increase in the logarithmic plastic strain calculated bynumerical material test and estimated by the trained neuralnetwork. The values of r90 and r45 obtained by numericalmaterial test decrease with the increase in the logarithmicplastic strain. On the other hand, r0 increases with theincrease in the logarithmic plastic strain. For all the synthetictextures, when the logarithmic plastic strain is 0.05, thegreatest value is at r0, followed by r90 and r45. Comparingthe results of Cases 3, 4, and 5, as VCube increases, r45gradually approaches zero. Moreover, the higher the volumefraction of the S orientation relative to the volume fractionof the Cube orientation, the greater is the difference betweenr0 and r90.The above figures indicate that the estimates by the trainedneural network can grasp the tendency of results of thenumerical material tests. Furthermore, as the results of thenumerical materials tests are within the range of the r-valuevariation estimated by the trained neural network, it isestablished that the proposed method can be applied forestimating the anisotropy of r-values with the same accuracyas that of the numerical material tests.Fig. 6 {111} pole figure of the synthetic crystallographic texture used asthe input data for the numerical material test and the trained neuralnetwork. The variance of the Cube texture ²2Cube for each texture is (a)2 deg2, (b) 5 deg2, and (c) 8 deg2. The volume fraction of the S texture andvariances of the Cube and S textures are fixed at VCube = 90%, VS = 10%,and ²2S = 5 deg2, respectively.Fig. 7 True stress-logarithmic plastic strain curves for the tensile directionof (a) RD and (b) TD calculated by numerical tensile tests and estimatedby the trained neural network using the different textures shown in Fig. 6.Table 1 Volume fraction and variance of Cube and S textures used forgenerating synthetic crystallographic textures.Estimation of Texture-Dependent Stress-Strain Curve and r-Value of Aluminum Alloy Sheet Using Deep Learning 22815. Discussion5.1 Improvement of the stress-strain curve estimationaccuracyWe used the trained neural network to estimate the stress-strain curves based on the volume fraction and variance ofthe Cube texture. The results demonstrated that the stress-strain curves calculated through numerical material tests wereaccurately estimated. As mentioned in Section 1, performingnumerous numerical material tests and accumulating trainingdata is time consuming. However, if the neural networktraining data can be accumulated faster, for example, byusing parallel computation to increase the speed of numericalmaterial test, it should be possible to estimate the texture-dependent stress-strain curve within a short period. On theother hand, the estimation accuracy of the stress-strain curvesis clearly reduced with increasing VCube and ²2Cube. Hereafter,we discuss methods to improve this decrease in estimationaccuracy.In this study, we expressed textural differences byincreasing the pixel luminance value with a high degree ofintegration in the {111} pole-figure image. On the otherhand, because the luminance value in each pixel is expressedas a value between 0­255, the variance is relatively small,and differences in the texture cannot be clearly recognized.This may be the cause for the decrease in the estimationaccuracy of the neural network when the Cube texturecomponent is more than 80%. Therefore, the estimationaccuracy can be improved by modifying the methoddescribed in section 3.1 for calculating the pixel luminancevalue in the {111} pole-figure image, using a pole figureother than the {111} pole figure as the input data to theneural network, or a texture representation other than polefigures. In addition, when the variance of the texture isincreased, the estimation accuracy may be improved bygenerating more training data.5.2 Improvement of estimation accuracy of in-planeanisotropy of r-valueWe used a trained neural network to estimate the in-planeanisotropy of r-value for six types of synthetic texture. Theresults showed that for most textures, it was possible toestimate the r-values calculated by numerical material test.However, there were errors in the r-value estimated by theneural network. The causes for these errors and the solutionsare discussed below.Variations in the estimates of the r-values by the neuralnetwork were caused because only few r-values werecalculated by numerical material tests, rendering these valuesstrongly dependent on minor differences in the initial crystalorientation. As noted in Section 4.2, even if we do notchange the values of volume fraction and variance of theideal orientations constituting the synthetic texture, the initialcrystal orientation used as input information for thenumerical material tests will differ slightly depending onthe random numbers. Therefore, the numerical material testresults will vary depending on the difference. To solve thisproblem, it is necessary to increase the initial number ofcrystal orientations, in the numerical material test.As mentioned above, issues remain to be solved in futureresearch. From the results of this study, however, it isdemonstrated that the use of {111} pole figures as input datato a CNN-based neural network for capturing the texturalfeatures of aluminum alloy sheets is highly promising, andthat the stress-strain curves and the in-plane anisotropy ofr-value can be estimated with high accuracy. Althoughregression estimation by deep learning with a neural networkis essentially an interpolated estimate of the training data,it is advantageous from an engineering perspective, forexample, in reducing the number of material tests that wouldotherwise be required.6. ConclusionIn this study, we proposed the new method using deeplearning with the neural network to rapidly estimate thetexture-dependent stress-strain curves and r-values ofaluminum alloy sheets. The neural-network training datawere obtained through numerical material tests based onCPFEM, with synthetically generated texture data as theinput information. We stored the results of the numericalmaterial tests and used them as input for training the neuralnetwork, which could estimate with high accuracy the stress-strain curves and the in-plane anisotropy of r-value from the{111} pole figures of the textures.The challenge for future research is to improve theestimation accuracy of the neural network by improving thetraining data, for example, by supplementing the trainingdata with actual experimental data or numerical materialtest results based on a greater number of initial crystalorientations than those applied in this study.Fig. 8 Variation of r-value with logarithmic plastic strain calculated by thenumerical material tests and estimated by the trained neural network forcases 1­6 shown in Table 1. 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