ZHUANG Weimin , WANG Shiwen, BALINT Daniel, and LIN Jianguo

1 State Key Laboratory of Automotive Dynamic Simulation, Jilin University, Changchun 130022, China

2 Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK

1 Introduction

Hydro-forming is a metal forming technology based on the application of pressurized liquid media to generate defined workpiece shapes from tubular materials or sheet metals. Tube hydro-forming has been widely used in automotive and aircraft industries in recent years[1]. Compared with conventional punching and drawing processes,tube hydro-forming has the advantage of part consolidation,weight reduction, improved structural strength and stiffness, fewer secondary operations, reduced dimensional variations, and reduced scrap. Hydro-forming provides the possibility to form hollow complex-shaped components with integrated structures from single initial workpieces[2].Currently, modern products for electronic, telecommunication and medical technology applications require tubular micro-components for an ever-growing market. Microhydroforming is an efficient and time saving production method that offers the required productivity and accuracy for the manufacturing of these components.

Since the tube is much larger than the grain size of the metal from which it is made, conventional macromechanics finite element(FE) is generally effective in modeling hydroforming processes[3–6]. In hydro-forming of a micro-tube, for example, the outer diameters of the initial and the formed tube are 800 μm and 1 030 μm, respectively, the wall thickness is 40 μm and the average grain size is about 30 μm. Cracking occurs as shown in Fig. 1 for the formed parts due to the localized thinning of the material in the hydro-forming process. Many tubes were formed using the same hydro-forming parameters and it was found that cracking takes place at different locations.The random nature of the cracking in hydro-forming of micro-tubes cannot be captured by using conventional macro-mechanics process modeling techniques.

Fig. 1. Examples of hydro-formed micro-tubes with random cracking observed.

In micro-forming, the grain size can be comparable to the smallest dimension of the part. Thus any region within a micro-part may contain fewer than ten grains, compared with hundreds or thousands in macro-forming. Due to the differences in the deformation characteristics at the micro and macro levels of the forming process, the workpiece for a micro part can no longer be regarded as a homogeneous continuum for process simulation purposes[7]. Crystal plasticity(CP) theory, which assumes that crystalline slip is the dominant deformation mechanism in crystalline materials,has attracted significant attention due to its ability to relate the plastic behavior of micro-parts to their microstructures.The main objective of this paper is to develop an integrated crystal plasticity finite element(CPFE) polycrystal modeling technique for capturing the localized thinning features for the hydroforming of micro-tubes.

A set of crystal viscoplastic constitutive equations is introduced in section 2. The FE model and the integrated numerical process, including the virtual grain generation facility, are presented in section 3. Computational results,which include localized thinning features for different microstructures of the material and under different deformation conditions, are detailed in section 3. Finally,conclusions are given in section 4.

2 Numerical Procedures for Micromechanics Hydro-forming Simulation

2.1 Crystal viscoplasticity constitutive equations

Crystal plasticity theories are used to represent the flow of dislocations along slip systems in metallic crystals in terms of resolved shear strains[8]. In particular, the crystalline slip is assumed to obey Schmidt’s law, i.e., the slipping rateαγ˙ in any particular slip system α is related to the shear stress, τα, on the same plane. The crystal plasticity theory used in this paper follows the pioneering work of TAYLOR[9], HILL, et al[10], and ASARO[11]. The set of crystal viscoplasticity constitutive equations used in the simulations is summarized below:

where Cijklis the fourth order stiffness tensor and the indices i, j, k and l take values between 1 and 3.is an average slip plane normal andis an average normal direction, and

The αth slip system is defined by a combination ofandThe number of slip systems and their orientations depend on the crystal lattice, e.g., an face centered cubic(FCC) crystal has 4 slip planes and each slip plane has 3 slip directions, i.e., α=1, 2,…, 12. In an FE analysis, each grain is divided into a certain number of elements (Nel). At the beginning of the deformation, τα=0, and the slip plane normaland slip directionare the same for all the elements within a grain, which takes its initial grain orientation from the virtual GRAIN (VGRAIN) software, which will be introduced later, according to a probability distribution embedded within the system. When plastic deformation occurs,andmay have different values in the different elements within a grain. The use ofandensures that the same orientation is assigned to all the elements within a grain.

Material strain hardening is specified based on slip system strain hardness, gα. The self, hαα, and latent, hαβ, hardening moduli are defined by ASARO[11]and PEIRCE, et al[12], which is directly related to the accumulated shear strain γ. They are defined by

where h0is the initial hardening modulus, g0is the initial shear strength, gsis the break through stress when plastic flow initiates and q is a hardening factor. In Taylor’s isotropic hardening assumption, the self and latent hardening rates are assumed to be the same. Hence, the value of the hardening factor, q, is taken to be equal to one. In the initial state (t=0), σij=0, γα=0, gα=g0and εkl=0. The material constants used with the equation set for 316L stainless steel are listed in Table. Young’s modulus and Poisson’s ratio are 193 GPa and 0.34, respectively[13]. The high value of n used here is chosen to reduce the viscoplastic effect of the material, as the deformation is at low temperature.

Table. Values of material constants in Eqs. (1) and (2)

The CP material model is implemented in ABAQUS via the user-defined subroutine VUMAT. This implementation was based on an implicit algorithm developed by HUANG[14]and further developed by HAREWOOD, et al[13]. In explicit finite element calculation procedures, the task can be split up easily and solved by a number of processors. Hence, the VUMAT can be constructed with a vectorized interface. This means that when a simulation is carried out using multiple processors, the analysis data can be split up into blocks and solved independently. Thus,vectorization can be preserved in the writing of the subroutine so that optimal processor parallelization can be achieved.

2.2 Integrated micro-mechanics forming simulation system

(1) Voronoi diagram. Significant research has been carried out which has demonstrated that Voronoi tessellation can be used to generate polygonal grain-structures[15]. Voronoi tessellation divides a region into convex polygons or cells, which fills the space without overlap. Voronoi tessellations can be constructed in either two- or threedimensions. For the CPFE applications in this paper, 2D Voronoi tessellations are used. Firstly, N nuclei (or seeds)are generated in a planar (x-y) area. A Cartesian-coordinate system is chosen, with nucleation points being created in the area by generating x- and y-coordinates independently from pseudo-random numbers distributed evenly between zero and unity. After the first point has been specified, each subsequent random point is accepted only if it is greater than a minimum allowable distance from any existing point, until N nuclei are seeded; a more-detailed description of the method can be found in Ref. [16]. The distribution of the seeds can be approximated by the oneparameter gamma distribution:

(2) Definition of initial grain orientations. Another important feature in virtual microstructure generation is the assignment of the initial orientations of every grain within the model. Crystals, with their inherent directions of slip(i.e., slip planes), are usually oriented randomly. To simulate the crystal deformation in micro-mechanics modeling,the grain orientations need to be defined based on their intrinsic physical features. Two angles (θ =[0, 2π] and ψ =[0, π]), which are relative to global coordinates, are used to define the initial grain-orientations and are shown in Fig. 2[17].

Fig. 2. Relation between spherical coordinates and the global coordinates

An orientation matrix, g, expressed in terms of the spherical coordinates of the sample directions, e.g., x= RD(rolling direction), y = TD (transverse direction) and z=ND (normal direction), in the coordinate system of the crystal directions, can be obtained as follows:

In the VGRAIN software package, two angles θ and ψ can be assigned according to probability theory. These angles describe the components of the initial orientation for each grain. In an FE analysis, each grain is divided into a number of elements and the determination of grain orientations in the subsequent increments will be discussed in a later section. The crystal orientations can also be represented by pole figures, which are commonly used and accepted in materials science.

(3) Numerical procedures. A VGRAIN system, which is described in detail by CAO, et al[15], has been developed and used to generate virtual grain structures according to the physical parameters of a material. The grain structure within a defined region is generated according to the input values of average, maximum and minimum grain sizes(Fig. 3). Orientations of grains are assigned according to a probability distribution either in a random form or with a prescribed distribution. The generated grain structures together with grain orientations are input to commercially available FE codes, such as ABAQUS, where further FE pre-processing, such as meshing, boundary and loading conditions, can be carried out. In this work, the generated virtual grain distributions with their orientation information are transferred into ABAQUS/CAE for further preprocessing. A flow diagram for the overall integrated CPFE modeling system is shown in Fig. 4.

Fig. 3. Main interface of the VGRAIN system for generating virtual grain structures using a Gamma distribution based on physical parameters

Fig. 4. Integrated numerical procedure for micro-mechanics modeling

2.3 FE model for micro-mechanics hydro-forming process modeling

The geometry (with dimensions) and the FE model of the cross-section of a micro-tube are shown in Fig. 5.

Fig. 5. Micro-mechanics model, with grains and grain boundaries, for tube hydro-forming

In the CPFE model, a quarter section of the micro-tube is considered. The minimum, average and maximum grain sizes of the material are 25, 30 and 40 μm, respectively,and 95% of the grains are within that range. Hence there are about 1–2 grains across the thickness of the tube section on average. The grains and their orientations are generated by using the VGRAIN system, which are read into ABAQUS/CAE for further mesh generation, boundary and loading definitions. The die is defined in ABAQUS/CAE as well. The maximum applied loading pressure is 400 MPa; this high pressure ensures the workpiece is deformed to the die completely. A friction coefficient of 0.1 is used when the workpiece and the die are in contact during the forming process. For simplicity, a 2D plane strain CPFE analysis was carried out here.

It is worth mentioning that CPFE analyses require considerable computer CPU time. 2D CPFE analyses could reduce the computational time significantly while still allowing the interesting features, such as localized thinning,failure, etc, in hydro-forming of micro-tubes to be captured.The fully-developed CPFE process modeling technique can be readily used for 3D hydro-forming simulations, if the 3D grain structures can be constructed effectively for the initial metal tubes.

3 Computational Results

3.1 Unpredictable thinning in polycrystals with large grains

Polycrystalline structures and grain orientations are generated using the VGRAIN system automatically as previously mentioned. To simulate the deformation and thinning behavior of two hydro-formed micro-tubes, which are taken from the same piece of the material, the grain structures are generated twice using the same microstructure control parameters defined above, and the orientations of both are assigned randomly based on the embedded probability theories within the VGRAIN system. This indicates that the microstructures and grain orientations may be different between the two CPFE models, although they are within the range of the material specification. The results of the virtually hydro-formed micro-tubes are shown in Fig. 6. It can be observed that the minimum and maximum values of the wall thickness of the formed tubes, shown in Fig. 6(a) (20.2 μm and 31.4 μm) and Fig. 6(b) (20.7 μm and 33.1 μm), are different for the two cases studied. The wall thicknesses of the two hydro-formed micro-tubes with random grain orientations are not uniform and are difficult to predict. This is due to the variation in grain size and grain orientation, and the relationship between grains and their neighbors. This complicated relationship and the localized thinning features cannot be captured using conventional macro-mechanics FE techniques. It can also be observed that localized thinning occurs at different locations. This is mainly due to the grain orientations of the workpiece material, which are also difficult to control in practice.

3.2 Deformation effects for polycrystalline microstructures

To investigate the deformation effects in thinning of hydro-formed micro-tubes with polycrystal material microstructures, two cases were studied. The grain structures and grain orientations of the workpiece for the two cases are identical, but the deformation ratios are different. This simulates the process of forming the tubes with different diameters from the same workpiece. In the first case (Fig.7(a)), the radii of the die and the outside of the tube are 515 μm and 400 μm, respectively, thus, the ratio of the deformation is 1.3. For the other case shown in Fig. 7(b),the radius of the die is 596 μm and the deformation ratio is 1.5.

Fig. 6. Predicted thinning features of the part with two microstructures generated with the same control parameters

The predicted localized thinning features for the two cases are shown in Fig. 7. It can be clearly seen that the wall thickness of the formed tube is not uniform and the amount of the localized thinning increases with the increase in deformation dramatically and non-proportionally.The ratios of the two maximum and minimum values of wall thickness shown in Figs. 7(a) and 7(b) are 0.97(31.9/32.9 for maximum values) and 0.64 (13.7/21.4 for minimum values), respectively. It can also be seen that the maximum values of the wall thickness for the two cases are almost the same as the deformation progresses, but the minimum values of the wall thickness of the tubes decreases sharply with the increase in deformation. This indicates that once localized necking takes place at a location,it would progress very quickly and lead to the localized failure of the material. The necking position is related to the angle between the directions of the slip systems and the hoop stress, which is also governed by the orientations and sizes of the neighboring grains. Hence, for a polycrystal case, the position and the amount of localized necking are difficult to control in practice if there are only one or two grains through the thickness of the tubular part. One example of localized failure in hydro-forming of micro-tubes is shown in Fig. 1. The location of failure is random and cannot be predicted. This experimentally observed random localized thinning/failure feature has been reproduced by the CPFE analysis carried out in this work. Furthermore,the CPFE analysis results confirm that traditional macromechanics FE process modeling techniques cannot be used to predict the localized failure in hydro-forming of microtubes.

Fig. 7. Comparison of thinning features for the deformation ratios of 1.3 and 1.5

4 Conclusions

(1) Traditional macro-mechanics FE techniques can only be used for process simulation in forming of macrocomponents. However, in hydro-forming of micro-tubes, if the ratio of the wall thickness of the micro-tube and the grain size of the material is low, CPFE analysis must be used. Otherwise, the important localized thinning features resulting from the microstructure variation and grain orientations of the material cannot be captured.

(2) It has been demonstrated that the location of localized thinning cannot be predicted for polycrystalline cases,which occurs randomly. This occurs because the grain size distribution and individual grain orientations are not fixed.

(3) Once necking takes place at a preferred location, it progresses very quickly and leads to localized failure as observed in the experiments.

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