Size effects from grain statistics in ultra-thin metal sheets

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1 Journal of Materials Processing Technology 174 (2006) Size effects from grain statistics in ultra-thin metal sheets T. Fülöp a,b, W.A.M. Brekelmans b, M.G.D. Geers b a Netherlands Institute for Metals Research (NIMR), Rotterdamseweg 137, P.O.Box 5008, 2600 GA, The Netherlands b Materials Technology, Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB, The Netherlands Received 20 April 2003; received in revised form 6 January 2005; accepted 6 January 2006 Abstract This paper deals with the simulation of the mechanical response of ultra-thin ductile metal sheets. The basic theory used for the analysis is rate-dependent single crystal plasticity. The associated algorithms are implemented into a Finite Element code and a procedure for the realization of the discretization is proposed. A uniaxial tensile test and a three-point bending test are computationally evaluated. Supported by simulations, the effect of the number of surface grains over the total number of grains is investigated. The numerical results demonstrate the reliability of the procedure and its capability to predict the variation of the mechanical overall properties with changing grain size or sheet thickness Elsevier B.V. All rights reserved. Keywords: Size effects; Micro-forming; Miniaturization; Crystal plasticity; MEMS 1. Introduction The forming of parts having relatively small dimensions plays an increasingly important role in the manufacturing industry. Many of the recent electronics devices contain sheet metal components of aluminum or copper, with a thickness typically in the range of m. Miniature parts also appear in chemical micro-reactors or medical devices. Thanks to a high accuracy and production rate, metal forming is still a leading technology in the fabrication of many micro-parts. However, several factors influence the application of forming in the small size range. The limitations are related to accuracy demands imposed on the forming tools, to material handling, to positioning and most importantly to the change of the mechanical properties of the formed part itself. The latter not only influences the quality of the final product, but affects also the manufacturing conditions (e.g. reaction forces and deformed shape). Furthermore, thin metal films are used in several applications for micro-electromechanical systems (MEMS), where their mechanical response is of primary importance. The change in the overall mechanical behavior of ultra-thin sheets due to a reduction of the dimensions is governed by socalled size effects. In this paper, the attention will be given exclusively to those size effects that result from the changes in Corresponding author. Tel.: ; fax: address: m.g.d.geers@tue.nl (M.G.D. Geers). grain statistics. This becomes manifest by two different phenomena [1]. The first cause for the occurrence of a size effect is the change of the number of the grains in the thickness direction of the sheet (i.e. a geometrical change of the grain statistics). If only a few grains are remaining over the thickness, the overall mechanical response will be heavily influenced by the orientation of the individual grains. This will result in an increase of the scatter on the manufacturing parameters like applied forces and geometrical results, considered over a number of experimental realizations. The scatter also clearly applies to the fracture behavior, as thin sheets tend to fail in a brittle mode. The second effect of size reduction is the relative increase of the free surface of the material. This results in a relative decrease of the forming forces. An opposite effect can be expected from the presence of constraining boundary layers obstructing deformation which aspect is not investigated here. The occurrence of grain size effects makes the application of a traditional process design methodology difficult, or even impossible. Therefore, the design of micro-forming operations is still largely performed through empirical techniques. In parallel, several theoretical and numerical methods have been developed to create a link between the microstructure and the macroscopic response of crystalline metals. However, most of these methods address the conventional size ranges and their applicability to small parts is not fully addressed. The present study focuses on the validity of micromechanical methods applied to the deformation of ultra-thin sheets. The material under consideration is commercially available pure /$ see front matter 2006 Elsevier B.V. All rights reserved. doi: /j.jmatprotec

2 234 T. Fülöp et al. / Journal of Materials Processing Technology 174 (2006) aluminum (FCC lattice). First, a presentation of the modeling strategy will be given, after which the analysis of the deformation of ultra-thin sheets is assessed from the point of view of size effects resulting from grain statistics. 2. Analysis of the crystalline behavior The analysis of the mechanical behavior of metal parts consisting of a polycrystalline microstructure can be performed in globally two different ways. The first possibility is the use of statistical or averaging theories. The associated homogenization approaches render a macroscopic approximation of the underlying physical processes, which is typically relevant for a huge collection of grains in the characteristic dimensions of the component examined. For a limited number of grains however, a more precise procedure is obtained through the local use of single crystal plasticity models combined with finite element calculations. The applicability on the analysis of ultra-thin sheet metal forming will be examined in the rest of this paper. In the present section, the applied single crystal plasticity theory is shortly reviewed. The objective of single crystal plasticity is to create a constitutive law for individual FCC crystals, which, in the present paper, will be implemented into a finite element code where the interaction with neighboring grains can be taken into account. The constitutive law will be formulated in the context of a nonlinear kinematics framework (finite strains and rotations). The conventional multiplicative elastic plastic decomposition [2] is adopted as point of departure: F = F e F p, (1) where F is the deformation gradient tensor. The elastic part F e of this tensor contains the elastic lattice distortion and the lattice rotation. The lattice distortion gives rise to stresses, here represented by the second Piola Kirchhoff stress tensor T. The tensor F p describes the plastic deformation which is the net effect of the crystallographic slip, keeping the lattice undistorted and unrotated. Consistently, the plastic deformation is assumed to be volume preserving, so det(f p ) = 1. With the introduction of the elastic Green Lagrange strain tensor E e which is work conjugate to the stress tensor T, the elastic constitutive equation can be written as: T = 4 C : E e, (2) with E e = 1 2 (FeT F e I), (3) where 4 C is the fourth order elasticity tensor. It can be specified for FCC metals in terms of three independent stiffness parameters, C 11, C 12 and C 44 [3]. The evolution of F p can be expressed in terms of the slip rates on the slip systems using the plastic part L p of the velocity gradient tensor: Ḟ p = L p F p, with L p = α γ α S α 0. (4) In this equation γ α is the slip rate on slip system α and S α 0 = m0 αnα 0 is the non-symmetric Schmid tensor for that system, representing the spatial orientation of α in the initial, unloaded configuration. The unit vectors m0 α and nα 0 correspond to the slip direction and the slip plane normal on α, respectively. A viscoplastic flow rule for metallic materials is assumed to be formulated according to [4], γ α τ α 1/m = γ 0 sgn(τ α ), (5) S α where the resolved shear stress τ α is defined by τ α = T : S α 0, γ 0 denotes a reference slip rate parameter, m the strain rate sensitivity parameter and S α is the slip resistance. The quantification of the parameters γ 0 and m for a specific material will be addressed in the next section. The evolution of the slip resistance S α is assumed to be governed by the slip rates on all slip systems according to Ṡ α = β h αβ γ β. (6) The moduli h αβ determine the strain hardening on slip system α due to cross-slip on system β and are derived from the hardening law selected. In the following, a Kocks-type hardening expression [5] is adopted: ( ) a h αβ = h 0 1 Sα q αβ, (7) S s where h 0,S s and a are material self-hardening parameters and q αβ introduces latent hardening. In the following, q αβ is taken equal to 1.4 when the slip systems α and β are non-coplanar and equal to 1.0 otherwise. 3. Finite element model For the multi-crystalline analysis, the single crystal approach was implemented into a finite element code. The basic equations presented in the previous section were locally used at integration point level. Three-dimensional simulations were performed on models representing materials obtained from different processing conditions. The processing history affects the microstructural grain orientations and grain geometries. In the following, the most important features and their numerical description are summarized. In order to perform a finite element simulation of a forming process of thin sheets with a grain size comparable to the thickness, the individual grains need to be discretized. For this reason, a good approximation of the real grain structure is necessary. There are three different ways to obtain such an approximation. The first method consists of a detailed measurement of the shape and size of the real grains and subsequently the construction of an appropriate finite element mesh. While this method might be generally used for multi-crystals (i.e. polycrystalline materials composed of a small number of grains with one or two grain layers in the sheet thickness direction [6]), it is less suitable for the analysis of forming operations on real materials due to the possibly large total number of grains and the presence of

3 T. Fülöp et al. / Journal of Materials Processing Technology 174 (2006) Fig. 1. Example of the grain structure and orientations in (a) undeformed and (b) deformed state after 50% pre-straining in longitudinal direction. volume grains which cannot be characterized by conventional experimental techniques [7]. The second way to obtain a realistic grain structure is the generation by grain growth simulation. Several models are available to perform such simulations, aiming to predict of the grain structure created by solidification from a liquid. However, only a few grain growth models are applicable in 3D, sharing the drawback that they are very complex and time consuming [8]. The remaining part of this section applies a third approach of microstructure generation, the application of Voronoi tessellation. Mathematically formulated, a number of spatial control points is defined, after which a Voronoi polygon (polyhedron in 3D) is attributed to each control point as the collection of points in a plane (in a volume) which are closer to the considered control point than to the other control points. Physically, a Voronoi structure can be identified with simple homogeneous crystal growth using the following assumptions: all crystal nuclei appear at the same time and no new nucleus is created during the process; all nuclei are fixed in space and contribute equally; all nuclei grow radially with an equal rate and they stop growing when they mutually touch. The control points determining the Voronoi structure are the so-called Poisson points of the tessellation and their positions can be used to define the full granular structure. Their inter-distance is related to the mean grain size and their location can be generated randomly. It can be shown that the Voronoi pattern based on random control points realistically approximates the real grain structure of metallic materials [9]. Several algorithms were developed in the last years to generate the Voronoi polygons/polyhedra starting from known control points. Most of them are designed for planar structures, but some can also be applied in 3D [10]. Additional attention should be paid to the finite element mesh generation. The obviously most natural choice is the direct meshing of the Voronoi polygons/polyhedra. However, this process becomes difficult in three-dimensional cases, as currently no robust method is available which, for structures composed of general polyhedra, generates a reasonable (limited) number of elements. It is important to recall at this point that an iterative crystal plasticity formulation has to be solved in each integration point which limits the maximal resolution of the mesh. In the following, another approach will be used, based on the simple generation of a three-dimensional cubic mesh. The material properties to be assigned to each integration point are retrieved by searching the grain in which that integration point is located. The advantage of this approach is its simplicity, however, an important drawback is that the generated mesh is not able to describe the grain boundaries in detail (their influence is in fact smeared over the size of an element). Two representative examples are shown in Fig. 1. Where the Voronoi patterns and the associated finite element meshes are illustrated. In both cases a rectangular specimen of 100 m thick sheet is considered with an average grain size of 300 m. The finite element discretization is composed of 2268 eight-node continuum elements. Fig. 1a presents a specimen without any pre-deformation. Physically, such a configuration can be obtained after full recrystallization. In the numerical representation, all of the grains were created from randomly distributed control points. In contrast to Fig. 1a, Fig. 1b represents the grain structure of a sheet sample pre-deformed by uniaxial tension. It is also obtained from a random Voronoi pattern in an adapted (compressed) geometry followed by an appropriate additional distortion. In the following, the effects of sheet thickness and microstructure on the forming process will be studied for ultra-thin sheets through numerical investigations. To avoid any additional geometrical effects that are not directly related to the change in the microstructural properties, the similarity law proposed by Kals [1] has been respected during the creation of the finite element meshes. According to the requirement of geometrical similarity,

4 236 T. Fülöp et al. / Journal of Materials Processing Technology 174 (2006) Table 1 Geometries used for the numerical investigations Dimensions, in mm (length width thickness) Number of grains over the thickness Surface grains over volume grains ratio all dimensions were uniformly scaled. Table 1 lists the length, width and thickness of the models used for the simulations. The mean grain size which is set by the tessellation was kept constant, i.e. 100 m. Table 1 also gives the number of the grains in the thickness direction, showing that in the case of a sheet thickness of 100 m there is only one grain over the thickness, corresponding to the multi-crystalline limit. The thickness of 500 m leads to about five grains over the thickness, approaching more the polycrystalline case. As for the models presented in Fig. 1, the finite element mesh prepared for these geometries also comprises 2268 three-dimensional cubic continuum elements for all sheet thicknesses. With eight integration points per element, these meshes are reasonably able to describe the behavior of the grain structure defined. Another important aspect of the simulation of microforming is the characterization of the orientation of individual grains. In order to study the effects of the crystallographic orientation on the mechanical properties, for each geometry ten different grain samples have been considered, five of them without and five with pre-deformation. The pre-deformation was fixed to 50%, according to Fig. 1b. For each grain structure, five different orientation patterns were defined. For the undeformed cases, the orientations were selected randomly. In the case of the pre-deformed samples, a rolling texture was used, obtained from simple Taylor simulations [11] on a material with a random initial distribution. Fig. 1 also presents the pole figures for the random and rolled orientations, where stereographic projections are used. The orientations were attributed by assuming that initially each grain is homogeneous, so that the orientation is constant for the whole grain. The material parameters were specified using realistic values for aluminum. The different constants, suitable for a quantitative simulation in the context of single crystal plasticity are adopted from [12] and listed in Table 2. Table 2 Material parameters for aluminum Elastic constant, C 11 Elastic constant, C 12 Elastic constant, C 44 Initial flow stress, S 0 Saturation stress, S s Initial hardening, h MPa MPa MPa MPa 61.8 MPa 58.4 MPa s 1 Reference slip rate, γ 0 Hardening exponent, a 2.25 Strain rate sensitivity, m Hardening parameter, q 1.4 or Analysis Fig. 2. Flow curves for different sheet thicknesses. Two different deformation modes have been investigated, uniaxial tension and three-point bending, using the finite element modeling approach as outlined in the previous section. For both cases, rate type boundary conditions were specified. The results for uniaxial loading with a nominal strain rate of 0.01 s 1 are presented in Fig. 2, where the stress strain responses are compared for different sheet thicknesses. The deformation range is up to 30% which is the common region of uniform deformation (no necking) for aluminum alloys. Average curves of the numerical investigations are plotted, for a given sheet thickness along with the obtained scatter. The difference between the average response of undeformed and pre-deformed configurations appears to be very small, such that they cannot be distinguished in the figure. The figure highlights the influences of the sheet thickness on the mechanical response for a constant grain size. The first evident geometrical effect illustrated by Fig. 2 is the reduction of the flow stress with reduction of the sheet thickness, which corresponds to experimental observations [13]. This phenomenon can be explained by the increase of the relative contribution of surface grains, which are less constrained than the volume grains. This less constrained situation results in the decrease of the stress. This effect becomes more important with increasing deformation, implying that the hardening becomes less effective, also observed experimentally [1]. So in case of uniaxial tension, a smaller thickness at constant grain size leads to a relative decrease in the resistance to plastic flow. A different effect of the microstructure dependence, which is also shown in Fig. 2, is the reduced predictability of the tensile response for thinner sheets. The figure presents the scatter of the tensile stresses obtained for different initial textures. The microstructure dependence becomes manifest in the increase of the mutual differences between the calculated stress values for smaller thicknesses. Predictability may therefore decrease considerably, if the precise grain orientations are not taken into account. This consequence of miniaturization can be explained by the (partial) loss of the polycrystalline properties in the sheet. In the case of the thick materials, there are still enough grains in a cross section to maintain the polycrystalline behavior. In the case of ultra-thin sheets on the contrary, the properties in a given

5 T. Fülöp et al. / Journal of Materials Processing Technology 174 (2006) Fig. 3. Ratio of the average roughness R a to the current sheet thickness t for different initial thicknesses. cross section are increasingly dominated by individual grains. Larger scatter can be expected when the initial orientations are more dispersed. This explains the observation already noted that the scatter is larger for initially random orientations and much smaller, even negligible when the simulations are performed on pre-deformed configurations. The relative importance of the individual grains becomes manifest in the variations of the deformation of different cross sections showing inhomogeneous thickness reductions. Additionally, the surface grains experience a reduced rotational resistance. The effect of these two mechanisms can be illustrated through the evaluation of the surface roughness for different levels of deformation. Fig. 3 gives the results for all thicknesses by presenting the relative average roughness, calculated by R a = 1 N N (z i z), (8) i=1 where N is the number of finite element nodes on the surface, z i is the out of plane coordinate of node i and z indicates the mean position of the surface. As expected, the surface roughness obtained for different sheet thicknesses increases with the loading, and this in a more pronounced manner for thinner sheets. The occurrence of locally smaller cross sections triggers an instability more rapidly and therefore leads to a decrease of the formability of thinner sheets, also observed experimentally [14]. From a macroscopic point of view, assuming homogeneous material behavior, the deformations in a pure tensile configuration would be uniform. However, in most practical applications the loading conditions will introduce a gradient of strain in the material. To examine the effect of this gradient for thin sheets, three-point bending of samples with different thicknesses is investigated. The initial geometries are similar to those used for the tensile tests. Three thicknesses are considered with different grain structures and orientations. The bending is realized by a prescribed constant transverse velocity at the central part of the samples while the two extremities of the model are clamped. Fig. 4 visualizes the response by considering the normalized bending force, F/w 0 t 0, in dependence of the normalized central displacement, f/t 0. Here, F is the applied force, f is the associated displacement, t 0 and w 0 are the initial sheet thickness and Fig. 4. Normalized bending force F/w 0 t 0 versus the normalized punch displacement, f/t 0 for bending of thin sheets. width of each of the specimens, respectively. The response averaging procedure was used again, however, it should be noted that the scatter in the different responses was substantially larger than for the tensile tests. The increase of the scatter is especially apparent for the case for thinner sheets, which can be explained by similar arguments as for uniaxial tension. For bending, the loaded volume is smaller than the total volume of the specimen, giving more pronounced effects of the individual grains. The results demonstrate that the normalized bending force decreases considerably with decreasing sheet thickness. This effect can be closely connected to the apparent decrease of the macroscopic flow stress, and was also observed experimentally [15]. 5. Conclusions The aim of this work was a crystal plasticity based analysis of the forming of ultra-thin sheets and the verification of the predictive capabilities of the applied finite element simulation. For this purpose, a rate-dependent slip model was implemented with a Kocks-type strain hardening expression. Simulations of uniaxial tension and three-point bending tests have been carried out. The results of the analyses illustrate the dependence of the calculated results on the initial grain orientations. On the other hand, the presence of various size effects was observed. 1. The development of the flow stress during deformation depends on the number of grains in the thickness direction. Both the initial yield limit and the hardening decrease with decreasing thickness. These effects can be observed for uniaxial tension and even more pronounced for bending, and can be explained by the increasing importance of the contribution of the surface grains. 2. Due to the different orientations of the grains located in the sheet plane, the deformation is no longer uniform even under homogeneous loading conditions (tension). The occurring spatial heterogeneity results in an increase of the surface roughness, introducing local imperfections. This effect is more important for thinner sheets and leads to a decrease of the formability.

6 238 T. Fülöp et al. / Journal of Materials Processing Technology 174 (2006) An additional effect of the microstructure dependence is the substantially decreased predictability of the mechanical response for thinner sheets, if individual grains and orientations are not explicitly taken account. This phenomenon becomes manifest in the increase of the deviations between the calculated responses for smaller thicknesses. The lack of predictability is more pronounced for random initial textures than for well-developed rolling textures. Acknowledgments This research was carried out under the project number MC in the framework of the Strategic Research Programme of the Netherlands Institute for Metals Research ( References [1] R.T.A. Kals, Fundamentals on the miniaturization of sheet metal working processes, Ph.D. thesis, Erlangen-Nürnberg, Germany, [2] E.H. Lee, Elastic plastic deformation at finite strains, J. Appl. Mech. Am. Soc. Mech. Eng. 36 (1969) 1 6. [3] L. Anand, M. Kothari, A computational procedure for rate independent crystal plasticity, J. Mech. Phys. Solids 44 (1996) [4] S.R. Kalidindi, C.A. Bronkhorst, L. Anand, Texture evolution in FCC metals, J. Mech. Phys. Solids 40 (1992) [5] D. Peirce, R.J. Asaro, A. Needleman, Overview 21 an analysis of nonuniform and localized deformation in ductile single crystals, Acta Metall. 30 (1982) [6] J. Harder, A crystallographic model for the study of local deformation processes in polycrystals, Int. J. Plasticity 15 (1999) [7] L. Margulies, G. Winther, H.F. Poulsen, In situ measurement of grain rotation during deformation of polycrystals, Science 291 (2001) [8] M. Jessel, P. Bons, L. Evans, T. Barr, K. Stüwe, Elle: the numerical simulation of metamorphic and deformation microstructures, Comp. Geosci. 27 (2001) [9] M.S. Wu, J. Guo, Analysis of a sector crack in a three-dimensional Voronoi polycrystal with microstructural stresses, Trans. ASME 67 (2000) [10] S. Kumar, S.K. Kurtz, V.K. Agarwala, Micro-stress distribution within polycrystalline aggregate, Acta Mech. 114 (1996) [11] A.H. van den Boogaard, Thermally enhanced forming of aluminum sheet, Ph.D. thesis, University of Twente, The Netherlands, [12] A. Kumar, P.R. Dawson, Computational modeling of FCC deformation textures over Rodrigue s space, Acta Mater. 48 (2000) [13] U. Engel, R. Eckstein, Microforming from basic research to its realization, J. Mater. Proc. Technol (2002) [14] T.A. Kals, R. Eckstein, Miniaturization in sheet metal working, J. Mater. Proc. Technol. 103 (2000) [15] L.V. Raulea, A.M. Goijaerts, L.E. Govaert, F.P.T. Baaijens, Size effects in the processing of thin metal sheets, J. Mater. Proc. Technol. 115 (2001)