Available online at www.sciencedirect.com
Procedia Engineering 10 (2011) 1663–1669
ICM11
Towards a distribution-based formulation of FEM for micro-scale components J. Lütjensa*, P. Bobrovb, M. Hunkela, and W. Wosniokb a
Stiftung Institut für Werkstofftechnik, Bremen, Germany b
Institut für Statistik, Universität Bremen, Germany
Elsevier use only: Received date here; revised date here; accepted date here
Abstract Increasing demands for reliable production processes of micro-scale components in large quantities has promoted the research on micro cold forming processes. The need to accelerate product design by the use of simulation has revealed limitations of the classical finite element method (FEM) in the sub-millimeter domain where the local behavior may be dominated by one or a few grains only. As a consequence, average material characteristic values obtained from macroscopic specimens are no longer sufficient to describe the local work-piece behavior. Instead, an uncertainty of the material parameters is encountered, which can be characterized mathematically by probability distributions. The research activities presented here focus on the feasibility of FEM simulations based on distributed material parameters. In order to reduce the computational load, pre-calculation of a sufficient number of load cases in representative volume elements (RVE) for usage in a material data base is proposed. Work-piece simulations are then performed as a multi-scale approach with different levels of interactions between adjacent RVEs. Material characterizations of DC01 sheets and their application to numerical simulation of bulge-tests are presented. © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of ICM11
Keywords: Distribution-based FEM simulation; bulge-test; pole figure; feature size
1. Introduction Classical finite-element method (FEM) simulations of work-pieces with dimensions in the micrometer range suffer from the fact that local variations in the microstructure such as individual grains are usually not regarded. Instead, material properties gained in macroscopic experiments incorporating a large number of grains are used. However, since there may be only a few or even one single grain dominating the local behavior, this can produce erroneous solutions, particularly if the failure of a part is initiated by a local flaw [1]. Local variations can include
*
Corresponding author: Jörn Lütjens, tel.: +49-421-218-5407; fax: +49-421-218-5333. E-mail address:
[email protected].
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of ICM11 doi:10.1016/j.proeng.2011.04.278
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grain size or form, grain orientation, grain boundary effects such as segregations, surface defects, dislocation density, and other effects. One way to overcome this deficiency is to regard the statistical variations of the material properties by performing Monte-Carlo simulation runs. However, it can be estimated that this requires too large a number of repetitions to be manageable [2]. In order to obtain a statistical measure of the failure probability without excessive calculation load, the collaborative research center 747 "Mikrokaltumformen - Prozesse, Charakterisierung, Optimierung" funded by the Deutsche Forschungsgemeinschaft DFG attempts to reformulate the classical FEM. The new formulation will operate on probability distributions of material properties as opposed to average values in the classical formulation [2]. The interaction of the distributed state variables via homogenization methods and representative volume elements (RVE) is to yield a distribution of resulting properties such as strains or stresses, which can be interpreted as a failure probability. Mathematical details of this on-going work are to be elaborated in the future. For the present time, the validity of the statistical approaches is being assessed by comparison with standard Monte-Carlo calculations. 2. Methods The work presented here covers a central aspect of the distributed properties, the local orientations. To this end, experimentally determined distributions are being compared to distributions in an FEM simulation. The experiment chosen is a hydraulic cupping test, aka bulge-test. This test, which is often used to determine the forming limit diagram of sheet metals, offers a spatially resolved measurement of local strains via a camera based system [8]. Furthermore, the microstructure of the material was investigated by standard metallography as well as by XRD-pole figure measurements. The details of the bulge-test set-up have been reproduced using the FEM software Abaqus. By assigning local material orientations conforming to the texture parameters of the material, distributions of local strains are calculated and compared to the experiments. 2.1. Bulge-test description Fig. 1 depicts the experimental set-up for the bulge-test. The specimen is fixed on the pressure vessel by a drawing die. By the regulating valve the internal pressure is raised to a maximum value of 1 MPa. The pressure gauge measures the actual pressure in the cavity. The time to reach the maximum pressure was 15 s. The die diameter was 6.4 mm. A total of 39 samples were measured. Since the stress tensor at the clamped circular edge is different from the stress tensor in the center of the sample (more towards uni-axial loading), only the inner disk up to 2/3 of the total radius is included in the analysis. We consider this part of the specimens to by relatively unaffected by the boundary conditions, an estimate that has been verified by simulation studies. Optical deformation measurement system specimen
drawing die
P
pressure vessel
gas
pressure gauge
Fig. 1. Experimental set-up for the bulge-test. Courtesy of BIAS, Bremen.
Regulating valve
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2.2. Simulation details The simulations are carried out using the software Abaqus. A total of 1526 shell elements with quadratic basis functions are used to model the specimen. The fixation including drawing die as well as blank holder are modeled as rigid body shells. A force of 50 N is applied to the blank holder in order to prevent the specimen from being drawn in. A friction coefficient of 0.2 is used for the contact between sheet and holder but the simulation is set up so that no noticeable slip occurs. The pressure applied to the bottom of the sheet is set to 1 MPa. An individual orientation is assigned to each element such that their Euler angle distribution conforms to the general texture. The distribution of orientations also converges to a superposition of canonical Gaussian distribution on SO(3) by the central limit theorem [3, 4]. The orientations can be calculated using the method of small incremental rotations [5] for full 3D rotations. This method requires a principal orientation matrix and two parameters, a and b, to describe a texture component. The assignment of in-plane orientations to shell elements via the method of small rotations breaks down to a Box-Muller-method [6]. This procedure also produces a spherical Gaussian distribution, but here one angle T and one parameter b is used only. The texture parameters have been extracted from Table 1. 3. Experimental results 3.1. Material characterization The material under investigation was a 1.0330 DC01 (Armco) steel with a sheet thickness of 50 m. The average grain size was determined as 8.6 m in rolling direction and 6.1 m in normal direction. Details can be found in [7]. In Fig. 2, a pole figure measured by XRD of the sheet surface is shown. The material exhibits a typical rolling texture with parameters and proportions given in Table 1. These parameters have been estimated by numerical approximation of the measured and calculated pole figures. 95% of the local orientations employ an angle of under 19° between the local (100)-axis and the principal (100)-axis. Table 1 Texture components identified in the measured pole figures of DC01 and corresponding parameters for the method of small incremental rotations. Large values of these parameters describe wider orientation distribution functions. proportion
a
b
{111}(112)
0.6
0.3
0.4
{211}(011)
0.2
0.3
0.2
{100}(011)
0.2
0.5
0.3
Rolling direction
principal orientation
a
b
Fig. 2. XRD pole figure (110) of DC01 sheet at the surface (a) and simulated pole figure according to texture parameters in Table 1 (b). Rolling direction: vertical.
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major strain H [%] Fig. 3. Results of the bulge-test on DC01 as-delivered (top) and after heat treatment 870 °C / 30 min (bottom). Left: Plots of the measured major strains, BIAS Bremen. Right: Histogram of the major strains. Rolling direction: horizontal.
3.2. Bulge-test results After 15 s, a pressure of 1 MPa is measured. Fig. 3 shows in the upper row the measured distributions of the major strains for one sample (as-delivered) as well as a histogram of the same data for the entire series. The average major strain is 0.217 %. The distribution is roughly Gaussian with a standard deviation of 0.12 % (absolute value of strain). Almost the entire specimen at this stage is deformed elasto-plastically. We observe a characteristic diameter of 200 m for regions of approximately constant major strain values. This value has been calculated by determining the length d for which the feature size function S(d) = /Vtot reaches 90 % of its total growth. Here, are the standard deviations within local circles of diameter d, averaged over the entire sample. In Fig. 3, the same situation is also depicted for a DC01 sheet after heat treatment for 30 min at 850 °C. The heat treatment has an effect on the mechanical properties (by reducing the sheet thickness by 20 % and the elastic limit to about 130 MPa) as well as on the microstructure (by shifting the average grain size to 26 m and by reducing the dislocation density) [7]. The coarser grains should be responsible for a coarser distribution pattern in the measured strains, and in fact an enlargement of the feature size to 250 m is observed. In the histogram, a larger standard deviation of 0.26 % as opposed to 0.12 % is observed for the heat treated sample. This can be explained by the non-
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Rolling direction
linearity of the stress-strain-curve, which lets large strains grow even larger. The remarkably higher average strain can be explained partly by the reduced sheet thickness and partly by the higher ductility after heat treatment.
ļ Fig. 4. Plot of the measured in-plane strains of DC01 as-delivered in rolling (left) and in normal direction (right). One Pixel corresponds to 82 m. Rolling direction is horizontal in this plot.
An interesting feature is evident when plotting the strains in x- and y-direction separately, as in Fig. 4 (DC01specimen as-delivered). Both average levels as well as distribution patterns of the two quantities are different. The average strain in rolling direction (x) is 0.198 % as opposed to 0.189 % in the y-direction; the standard deviations are 0.143 % and 0.122 %, respectively. More obvious, regions of similar x-strain appear to be aligned in x-direction whereas no preferred orientation of the y-strains is observed. These differences must be attributed to the material texture. 4. Simulation results In Fig. 5, results from one simulation run are depicted. The local variation of the effective strains is best viewed by plotting one strain component, in this case Hyy. The deformation of the metal sheet under pressure from below is exaggerated by a factor of 4. As could be expected, higher stresses with more uniaxial components are found near the edge of the disk. This is due to the fact that a local bending in combination with compressive stresses from a slight inward movement acts on the material. For this reason, only the inner disks were evaluated statistically. The distribution of the major strains has a similar form compared to the distribution of the measured strains. The average strain is 0.104 %, which is between the measured values for the two material variations. The standard deviation of the major strains 0.19 % is also within the range of the measured values. This was not to be expected because of the missing implementation of feature or grain size in the simulation. In the model, each element was assigned a distinct local orientation. The element size of approximately 20 m is also by a factor of 10 smaller than the measured feature size in the bulge-test plots. Future investigations will show whether the different standard deviations are caused by the different feature scales. The anisotropy of the experimental strain distributions could not be reproduced in the simulation. The average inplane strains in and normal to the rolling direction differ by only 1 % whereas the difference in the experiment was 4.5 %. The standard deviations do not show any significant difference. Fig. 5 also shows a comparison of the histograms of the strains from simulation and experiment.
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experiment as-delivered
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major strain H [%] Fig. 5. Simulation results. Left: Plot of the strains Hyy for one simulation run. Deformation scale factor 4. The entire sheet inside the drawing die is shown. Neighbor elements experience different strains due to their local orientations. Right: Histogram of the major strain distributions for simulation and experiment.
5. Conclusions Although a number of simplifications have been made for the simulation model and parameters, many characteristic aspects of the experiment could be reproduced by the simulation. Comparisons of the average strains, of the standard deviations, or of the underlying strain distributions validate the chosen approach. Particularly the modeling of the texture by the method of small incremental rotations based on pole figure measurements appears to yield satisfactory results. One aspect that could not be reproduced is the anisotropic properties of the material. This could be due to the lack of a feature size aspect in the model, since the manifestation of anisotropies may depend on the coherency size of the anisotropic feature. This is a topic for future investigations. Several factors in the simulation can still be improved. As stated above, the regions of common orientations and the tessellation of the volume should be decoupled. Effects of the grain size can then be investigated more thoroughly. Even more important seems the pattern of the grain forms as a carrier of anisotropic properties. Finally, an open question is whether the use of shell elements is sufficient to catch all aspects of the material. The presented model of small rotations constitutes part of the plans to create a distribution-based FEM method [2] that permits to study failure probabilities and other stochastical properties without massive use of Monte-Carlo simulation runs as in this work. The current work attempts to provide fundamental knowledge to support this approach. Acknowledgements The authors gratefully acknowledge the financial support by DFG (German Research Foundation) for Subproject B2 "Verteilungsbasierte Simulation" within the SFB 747 (Collaborative Research Center "Mikrokaltumformen Prozesse, Charakterisierung, Optimierung)". The authors wish to thank Hanna Wielage from BIAS Bremen as well as Jérémy Epp from IWT Bremen for providing the experimental data. References [1] Lütjens J, Bobrov P, Hunkel M, Texture-related distributions of elastic properties in thin rolled sheets, Proc. 15 th International Metallurgy & Materials Congress, Istanbul, 2010 (to be published) [2] Bobrov P, Lütjens J, Montalvo Urquizo J, Wosniok W, Hunkel M, Schmidt A, Timm J, Zur verteilungsbasierten Modellierung von Mikrowerkstoffen, In: 4. Kolloquium Mikroproduktion, BIAS-Verlag, Bremen, 2009 [3] Parthasarathy, The central limit theorem for the rotation group, Theory of Probabilities and its Application, 9, 273 –282, 1964
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[4] Borovkov M, Savyolova T, Approximation of the class of canonical normal distributions by means of the method of random rotations in the texture analysis, Industrial Laboratory 68 16–21, 2002 [5] Bobrov P, Montalvo Urquizo J, Schmidt A, Wosniok W, Mechanic-Stochastic Model for the Simulation of Polycrystals, Report 10-03, ZeTeM, Univ. Bremen, 2010 [6] Press WH, Teukolsky SA, Vetterling WT, Flannery BP, Numerical Recipes in C - The Art of Scientific Computing, Cambridge Press, ISBN 0-521-43108-5, 1992 [7] Köhler B, Bomas H, Hunkel M, Lutjens J, Zoch HW, Yield strength behaviour of carbon steel microsheets after cold forming and after annealing, Scripta Materialia, Vol. 62, Issue 8, April 2010, 548-551, ISSN 1359-6462, DOI: 10.1016/j.scriptamat.2009.12.039. [8] Vollertsen F, Hu Z, Wielage H, Blaurock L, Fracture Limits of Metal Foils in Micro Forming, 36th International MATADOR Conference, eds.: Hinduja S, Li L., 49-52, Springer Verlag London, 2010
