Modeling of Sheet Metals with Coarse Texture via Crystal Plasticity

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1 Modeling of Sheet Metals with Coarse Texture via Crystal Plasticity B. Klusemann 1, A.F. Knorr 2, H. Vehoff 2 and B. Svendsen 1 1 Material Mechanics, RWTH Aachen University, Aachen, Germany 2 Chair of Material Science, University of Saarland, Saarbrücken, Germany benjamin.klusemann@rwth-aachen.de, tel: ABSTRACT In this contribution experimental and theoretical investigations of sheet metal mesocrystals with coarse texture are performed. One focus of this work is on size effects due to a lack of statistical homogeneity. The overall mechanical response is then strongly influenced by the orientation of the individual grains. For this purpose a crystal-plasticity-based finite-element model is developed for each grain, the grain morphology, and the specimen as a whole. The crystal plasticity model itself is ratedependent and accounts for local dissipative hardening effects. This model is applied to simulate the thin sheet metal specimens with coarse texture subjected to tension loading at room temperature. Investigations are done for body-centered-cubic Fe-3%Si and face-centered-cubic Ni samples. Comparison of simulation results to experiment are given. INTRODUCTION The relation between microstructure, material properties and mechanical response is a basic issue of research in material science and material mechanics. From the modeling point of view, a common concept used to account for the effect of the microstructure on the material behavior is that of a representative volume element (RVE). This concept is based on the assumption of scale separation between the microstructural and macrostructural lengthscale. If the characteristic size of the system (e.g., sheet thickness) approaches that of the microstructure (e.g., grain size), however, such scale separation is no longer given and one must resort to other means of representing the effect of microstructural heterogeneity on the system behavior. As the macrostructural lengthscale approaches the microstructural one, the degree of material heterogeneity increases, and the local microstructural behavior may deviate significantly from the average macrostructural behavior [e.g., 10, 17]. In this case, the model has to account for the microstructural details such as orientation details of the grain structure [e.g., 18] or phase distribution [e.g., 23, 13]. In the extreme case, the microstructural and macrostructural lengthscales are of the same order of magnitude, and one must resort to numerical modeling of the microstructure with the help of, e.g., the finite-element method [e.g., 4, 28, 19, 14]. In the case of polycrystalline materials, for example, such finite-element models are often constructed with the help of, e.g., optical and / or EBSD data on the grain morphology. In specimens with more than one grain over the thickness, the common method of projecting the two-dimensional EBSD information uniformly in the third dimension will generally lead to incorrect results [e.g., 26]. If the specimen is one grain thick, however, such an optical- / SEM- / EBSD-based approach should be reasonable. For such a specimen a number of size effects are expected to influence its mechanical properties. These effects have been known for years and are still the subject of active research [e.g., 5, 6, 9, 12]. The overall mechanical response is strongly influenced by the orientation of the individual grains if the number of grains over the thickness is fairly small [5]. In the case of thin sheets the mechanical properties in a given cross section are increasingly dominated by each individual grain as reported in [7]. Due to the different orientations of the grains located in the sheet plane, the deformation is no longer uniform even under homogeneous loading conditions. This heterogeneity and the size-dependence of deformation give rise to size effects [e.g., 8]. To understand and predict the behavior of such specimens correctly, simulation and experiment have to be compared locally, e.g., within individual grains in a polycrystalline specimen. For this purpose, detailed local experimental information is necessary [7]. The purpose of the current work is the investigation and modeling of so called oligocrystals which are specimens consisting of one or more coarse-textured layers over their thickness. As example a body-centered cubic Fe-3%Si and a face-centered cubic pure Ni sheet metal sample are investigated. The Fe-3%Si sample has been investigated experimentally by [7, 8]. These samples are grown in such a way that there is only one grain over the thickness, and grain boundaries are perpendicular to the sample surface. The modeling is carried out with the help of crystal-plasticity and the finite-element method [CPFEM: e.g., 18, 5, 19].

2 Related previous experimental and modeling work to oligocrystals includes for example that of [21], who investigated grain interaction in an Al oligocrystal with columnar grains subject to plane strain channel die extrusion. In addition, [28] examined plastic localization and surface roughening in an Al oligocrystal. Statistical size effects also relevant to the current work have been investigated by [5] in ultra-thin ( mm) Al oligocrystals. On the other hand, [24] investigated the behavior of a Cu oligocrystal characterized by multiple grains over the thickness during plane strain compression. The discrepancy between experiment and simulation results noted by them was attributed among other things to a lack of information about the grain morphology in the thickness direction. The paper is structured in the following fashion. First the single-crystal model is given. After this experimental and simulation results for an Fe-3%Si and an pure Ni oligocrystal are presented. The work ends with a summary. SINGLE-CRYSTAL MODEL Lets a,n a, andt a := n a s a represent the glide direction, glide plane normal, and direction transverse to the glide direction in the glide plane, of thea th glide system, respectively. As usual,(s a,t a,n a ) represent an orthonormal system assumed constant with respect to the intermediate local configuration as determined by the inelastic local deformationf P. As usual, the evolution of the intermediate local configuration andf P is modeled by the large-deformation form F P = L P F P = a γ as a F T Pn a (1) in the case of glide-based large-deformation crystal plasticity. Here, γ 1,γ 2,... represent the glide-system shears whose evolution is modeled here via the power-law form τ γ a = γ a 0 0 τ m dir(τa a d ) (2) in terms of the Schmid τ a := s a Mn a and Mandel M stresses. Here, dir(τ a ) = τ a / τ a is the shear-rate direction, γ 0 represents a characteristic glide shear-rate, m 0 is the strain-rate exponent, and τ d a is the dissipative slip resistance whose evolution is modeled by the interaction form τ d a = b hd ab γ b (3) [e.g., 1]. Here, the saturation model h d ab = q ab h 0 (1 τd b /τds 0 )n 0, a,b = 1,2,..., (4) [e.g., 2] is assumed for the components of the hardening matrix, with q ab = 1.0(1.4) fora = b(a b) the components of the matrix of hardening-rate ratios [e.g., 1] for the bcc case,h 0 the initial hardening rate,τ0 ds the saturation value ofτd b, andn 0 the hardening rate exponent. For the fcc case the components of the matrix of hardening-rate ratiosq ab are given byq ab = 1.0(1.4) forasame glide plane asb(a other glide plane asb). The current model is completed by the linear elastic constitutive relation S E = C E E E (5) for the elastic second Piola-Kirchhoff stress S E determining the MandelM S E and Kirchhoff K = F E S E F T E (6) stresses, the former in the context of small elastic strain. Here, C E is the constant elasticity tensor, E E = 1 2 (FT E F E I) represents the elastic Green strain, andf E := FF 1 P is the elastic local deformation as usual. The algorithmic formulation of the model combines explicit update at the integration-point level combined with implicit update at the finite-element / structural level for satisfaction of the boundary conditions. The resulting mixed algorithm has been implemented into the commercial program ABAQUS via the user material (UMAT) and user element (UEL) interfaces. The simulations to be discussed below were all carried out in ABAQUS/Standard. To ensure reliable and robust numerical results, adaptive time-step-size control is employed for the explicit update of inelastic model quantities used at the integration-point level. In particular, this latter is based on the magnitude of the inelastic velocity gradient L P related to the corresponding approximation of the algorithmic flow rule for F P. The critical value of this parameter for stability is determined empirically via one-element tests. For more details, the interested reader is referred to [11]. RESULTS FOR Fe-3%Si For the experimental investigation of [7], a test sample of approximately 5mm width and 15 mm length was laser-cut from a larger Fe-3%Si sheet of thickness 1 mm consisting of a single layer of grains having a mean diameter of about 2 mm. The

3 specimen was subject to simple tension under quasi-static loading conditions (10 3 s 1 ). During the test, sample geometry, grain morphology, and the local lattice orientation were measured at selected total strain states (0%, 1.5%, 4%, 10%, 19.5%) in the tension direction. The experimental results from [7] with respect to the orientation gradient and change in specimen shape and grain morphology are shown in Figure 1. The orientation gradient [e.g., 25, 8] is a measure of the local (maximum) mismatch between the orientation of a given point and that of its neighbors. More precisely, this is a measure of the change in lattice orientation between two neighboring (regularly-spaced) measurement points in the plane of the EBSD measurements. To calculate these, letr i andr i+1 represent the orientation of two adjacent points in either thei = x ori = y direction in the plane. Then ( θ i := min arccos 1 Q G 2 (R i QR i+1 1)) (7) c represents the orientation gradient at i modulo crystal symmetry transformations, i.e., elements of the crystal symmetry group G c (here cubic). The values of θ x and θ y determine in turn the measure θ = max{ θ x, θ y } (8) of maximal local orientation gradient and so the OG mapping. In the experimental case,r i and R i+1 are determined directly from the EBSD data. In the model case, R i and R i+1 in (7) and so (8) are determined by the spatial distribution of the elastic local rotationr E. Figure 1: Orientation gradient (OG) θ during tensile loading of Fe-3%Si oligocrystal determined experimentally at (from top to bottom) 1.5%, 4%, 10%, and 19.5%, total strain [7]. The OG results are superimposed on the current specimen geometry and grain morphology in contour form (red line used as approximation of shape change). Points in the specimen where EBSD data was not obtained or too poor to determine the OG distribution are shown in black. The 3D model specimen in Figure 2 was obtained from the 2D experimental information of the undeformed sample via direct extrusion of the specimen shape and grain morphology into the third dimension. This represents a possible source of discrepancy between the experimental and simulation results to be discussed below. Transition regions on either end of the actual specimen consisting of elastic isotropic material have been introduced in order to transmit the tension boundary conditions more accurately to the more complex specimen boundary [29]. As input data for the simulation the measured EBSD data is used as initial orientation. The orientation in every grain is assumed to be homogeneous. Room-temperature values for the material parameter in the single-crystal model assumed for the simulations are shown in Table 1 which are taken from [14]. Here it is assumed that slip in Fe-3%Si occurs in 111 direction on the{110} and{112} planes. Further it is also assumed that the material parameters for {110} and {112} systems are equal and that these systems do not interact. To model this, the corresponding coupling terms in the hardening matrixq ab are set to zero. In the following the deformation behavior and the evolution of the orientation gradient between experiment and simulation are investigated. For sake of comparison and to understand the influence of the assumed hardening law better simulations have

4 Figure 2: FE-model of the Fe-3%Si tensile specimen. Individual grains are numbered for reference in the sequel. Table 1: Material parameter values assumed for bcc Fe-3%Si. In particular, the elastic constant values are from [20], and the other paramters have been determined in [14]. Glide-system parameter values are assumed to be equal for both glide-system families{110} and{112} considered in this work. c E11 [GPa] c E12 [GPa] c E44 [GPa] τ d 0 [MPa] γ 0 [s 1 ] m 0 h 0 [MPa] τ ds 0 [MPa] n been carried out neglecting all hardening. In the initial stages of loading where little or no hardening can have occurred, this is not unreasonable and allows a check of the initial conditions of the model independent of the hardening modeling. hardening excluded hardening included Figure 3: Comparison of experimental (red thin line) and simulation (black thick line) results of Fe-3%Si oligocrystal for the specimen geometry and grain morphology at a) 1.5%, b) 4%, c) 10%, d) 19.5%, total strain in the tension direction for simulations neglecting (left) and including (right) hardening. First we turn to a comparison of experimental and simulation results for change in specimen shape and grain morphology as shown in Figure 3. As shown by the comparison of grain boundary motion for the results where hardening is neglected, generally the simulation underestimates the amount of grain deformation in the grains to the left of grains 13 and 14, and overestimates it in grains 13, 14 and 15. Discrepancies such as those seen in grains 13 and 14 are significantly enhanced by incipient specimen-level deformation localization and shear-band formation, in particular in the case of ideal viscoplasticity. It

5 is interesting to note that grain 14 had already the highest Schmid factor at the start of the deformation [14]. Comparing the simulation results including hardening to the results neglecting hardening suggests that, up to 4%, little or no deviation between the simulation results is visible. These results agree as well with experiment up to this point. After this point, however, the effect of including hardening becomes quite apparent. In particular, hardening results in a reduction in the prediction of the amount of grain deformation, something particularly apparent in the grains to the left of grain 13, but also in grains 13, 14 and 15. The grain boundary morphology is also well predicted, even at large deformation only a deviation in the contraction of grain 15 and 16 is observed. hardening excluded hardening included OGM_max e e e e e e e e e e e e e+00 Figure 4: Comparison of experimental results from [7] and modeling results for the orientation gradient (OG) θ of Fe-3%Si oligocrystal at total deformation states of (from top to bottom) 1.5%, 4%, 10%, and 19.5% in the tension direction. The OG results are superimposed on the current specimen geometry and grain morphology in contour form. Points in the specimen where EBSD data was not obtained or too poor to determine the OG distribution are shown in black. Next we turn to the investigation of the orientation gradient. The experimental results shown in Fig. 1 as well as the simulation results in Fig. 4 are only depicting the orientation gradient inside each grain due to the fact that the initial orientation gradient over the grain boundaries (misorientation) is much larger than the orientation gradient during loading. Again the simulations are performed for neglecting and including hardening. Consider first the simulation results neglecting hardening. As for the deformation results a localization of the gradient can be observed for grain 13, 14 and 15. It can be seen from this results that the simulation neglecting hardening cannot predict the correct tendency for the OG in the experiment. Again up to 4% deformation a similar orientation gradient can be observed for the simulations neglecting and including hardening, however, afterwards no correlation is anymore visible. In contrast the simulation results including hardening predicts for example the band-like distribution of high OG at the boundary between grains 1 and 4 (and perhaps grain 5 as well where data is missing) as seen in the experiment. As well, the homogeneous lattice orientation, i.e., lack of an OG, in the middle of grains 4 and 8 in the experiment is also seen in grains 4, 5 and 8 in the model. Further, the development of higher OGs in grain 9 near its boundary with grain 13, as well as in grains 13 and 15 near their common boundary, is present in the model results. On the other hand, the OG band in grain 15 parallel to its boundary with grain 16 is missing, as is the OG in grain 17 near its boundary with grain 15. In addition, the experimental and model OG distributions in grain 16, especially near the boundary with grain 14, are different. Then again, the development of OG bands in grain 11 near its boundaries with grains 8 and 12, although much more diffuse than in the experiment, is present. In summary, the simulations including hardening of the bcc Fe-3%Si sample were able to predict the experimental results with respect to the deformation behavior and the orientation gradient evolution quite good.

6 RESULTS FOR Ni An investigation of a face-centered cubic metal was done on an 99.99%-pure nickel sample. A test sample of approximately 18 mm width and 50 mm length was cut by spark erosion from a larger sheet of 2 mm thickness. After the annealing process (first 24h at 1350C then further 24h at 1425C) the average specimen thickness is reduced to 0.5 mm and grains have a mean diameter of about 1 mm, which implies the achieved single grain layer condition. Due to the annealing process also width and length of the test sample have been reduced to 16.5 mm and 48.5 mm, the thickness varied linear from 0.3mm to 0.6mm from left to right. The specimen was subject to simple tension under quasi-static loading conditions (10 3 s 1 ). During the test, the sample geometry, grain morphology, and the local lattice orientation were measured at certain total strain states (0%, 1%, 2%, 4%, 6%) in the tension direction. Figure 5: Model of the pure Ni tensile specimen. The 3D model specimen of the pure Ni sample in Figure 5 was obtained via extrusion of the specimen shape and grain morphology into the third dimension from the 2D experimental information of the undeformed Ni sample. Transition regions on either end of the actual specimen consisting of elastic isotropic material have been introduced in order to transmit the tension boundary conditions more accurately. As input data for the simulation the measured EBSD data is used as initial orientation with the orientation in every grain assumed to be homogeneous. In the case of face-centered cubic pure nickel it is known that the deformation occurs on 4{111} planes in 3< 110 > directions. The material parameter for these slip systems are identified on experimental data from [22] for[111] single crystal tensile data for room-temperature and quasi-static loading conditions ( ε 0 = s 1 ). For the model identification a single crystal in the simulation is rotated into the [111] direction and therefore the axes of the crystallographic system have to be rotated by the euler angles{φ 1 = cos 1 3 ; Φ = π 4 ; φ 2 = 0}. The tensile load is applied into x-direction which results initially in 6 active slip systems. The identification is done using LS-OPT in conjunction with ABAQUS by fitting the stress-strain curves. The optimization techniques rely on response surface methodology (RSM) [15], a mathematical method for constructing smooth approximations of functions in a design space. The approximations are based on results calculated at numerous points in the multi-dimensional design space. In this study, the material parameters are the design variables, and the model together with the data determines the objective function of the corresponding optimization problem. The shear-rate sensitivity,m 0 is assumed to be 20 and reference shear rate γ 0 to be 10 3 s 1 for the current case of quasi-static loading conditions which is in accordance to [27]. The identified parameters are shown in Table 2. Here it has to be noted that the material parameters are only fitted for crystal orientation of[111]. However, it is known that Ni shows significant different stress-strain curves for different orientations [e.g., 3, 16] depending on the fact whether the crystal is showing single or double slip. Of course, this represents one possible source of discrepancy between the experimental and simulation results. Table 2: Identified hardening parameter values for pure nickel based on the experimental data from [22] for an [111] single crystal. c E11 [GPa] c E12 [GPa] c E44 [GPa] τ d 0 [MPa] γ 0 [s 1 ] m 0 h 0 [MPa] τ ds 0 [MPa] n

7 first Piola-Kirchhoff stress [MPa] simulation experiment strain [%] Figure 6: Comparison of experimental and simulation results for first Piola-Kirchhoff stress (P = F A 0 witha 0 =1.544mm) in loading direction over strain ( l l 0 withl 0 =49.5mm) for pure Ni oligocrystal. During the experiment the applied displacement and corresponding reaction force were continuously recorded. The results are shown in Fig. 6. The reference area used to calculate the first Piola-Kirchhoff stress is the initially smallest cross-section of the specimen given by A 0 =1.544mm. The simulation results were obtained in the same way. However, in contrast to the experiment the specimen was continuously loaded in the simulation. It can be observed that the initial yield point in the experiment is lower compared to the results in the simulation which might be related to the fact that the material parameters used in the simulation are obtained from [111] single crystal data. After 4% total strain the simulation underestimates the required force for the specified displacement. This might due to the evolution of geometrically necessary dislocations (GNDs) in the specimen which strengthen the material. The influence of including these GNDs in the simulation for additional hardening is on-going research and will be studied with help of the experimental obtained orientation gradient. Next we investigate the development of the specimen geometry and grain morphology. Exemplarily the results for 4% total strain are compared between experiment and simulation in Fig. 7a). In general, it can be observed that the simulation is able to predict the specimen geometry and grain morphology for 4% total strain quite well. However, from these results it is very difficult to see where the main activity occurs. Therefore the total slip of the {111} glide system family is shown in Fig. 7b). It can be observed that the main deformation occurred in grain D (for labeling see Fig. 5). Furthermore a relative high activity can be seen in grain A, B and at the boundary of grain D to grain C. This is related to the fact that in this region the specimen is slightly smaller in width and thickness compared to regions more on the right of the specimen. Fig. 7c) shows a micrograph from light microscopy after the shear band is visible which occurs at 6% total strain (cp. Fig. 6). Although in the simulation failure is not considered, the total slip indicates where the highest deformation is present. This can be seen as an indicator where a shear band would be most likely start. From the simulation results it would be most likely that a shear band could occur in the region of grain D, C and E. In the experiment the shear band was observed in grain D and C which at least for grain D this could be anticipated by the simulation. In summary, these first exemplary simulation results of the face-centered pure Ni sample show that the simulation was able to predict the general behavior correctly, however, certain deviations occur which might be related to the fact that only[111] single crystal data were considered for the material parameter identification. Further a failure model has to be included to be able to predict the shear band. SUMMARY & OUTLOOK The current work has focused on the modeling and simulation of the behavior of two thin metal sheets, one consisting of a single layer of large grains of Fe-3%Si (bcc) and one consisting of a single layer of large grains of pure Ni (fcc). Since such material are highly heterogeneous, they are modeled with the help of single-crystal plasticity for each grain in the specimen and the finite-element method for the grain morphology and specimen as a whole. The single-crystal model is rate-dependent and

8 a) AccSlip b) e e e e e e e e e e+00 c) Figure 7: a) Comparison of experimental (red line) and simulation (black line) results for the specimen geometry and grain morphology at 4% total strain in the tension direction. b) Total slip for {111} glide system family at 6% total strain projected on deformed geometry. c) Micrograph from light microscopy of the specimen after shear band occur at 6% total strain accounts for (local) dissipative hardening effects. The predictions of the model are compared with experimental results of thin sheets of Fe-3%Si and pure Ni loaded incrementally in tension. For the Fe-3%Si sample the specimen geometry and grain morphology and the development of the orientation gradient were analyzed. Two modeling cases were examined and compared with each other. In the first case, all hardening was neglected, resulting in ideal viscoplastic behavior of the grains. Initially, reasonable agreement is obtained; but as one can imagine, further loading and increasing deformation leads to significant hardening. As such, neglecting all hardening results in overestimate of the deformation in favorably oriented grains and to corresponding mismatch with experiment. Including hardening leads to quite good agreement. For the pure Nickel sample the stress-strain curve were compared between simulation and experiment which showed some deviations. Further, the specimen geometry and grain morphology were exemplary investigated as well as the strain field. In general the simulation showed the same tendency as obtained in the experiment. However, certain deviations occur which might be related to the fact that only [111] single crystal data were considered for the material parameter identification. Besides dissipative hardening, the effects of additional strengthening due to grain size and misorientation distributions, as well as that of additional hardening due to GND development in the specimen, on the deformation behavior will be investigated in the future. Further a failure model has to be included to be able to predict the occurrence of the shear band in the experiment which will be on-going work. Due to the fact that with Fe-3%Si and pure Nickel representatives of body-centered-cubic and face-centered-cubic materials

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