Crystal Plasticity Models for Hydro-forming of micro-tubes



Experimental and numerical investigation of localized thinning

in hydroforming of micro-tubes

W. Zhuang1, S. Wang2, J. Lin2*, D Balint2 and Ch. Hartl3

1. State Key Laboratory of Automotive Dynamic Simulation, Changchun 130022, PR China

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

3. Faculty of Automotive Systems Engineering and Production Engineering,

Köln D50679, Germany

Abstract

An experimental program has been carried out for hydroforming of stainless steel micro-tubes. Under careful control, it was found that failure takes place randomly, which is significantly different from observations of failure in hydroforming of macro-tubes, where failure loads and locations are predictable. This occurs because wall thinning of micro-tubes in forming processes is nonuniform, i.e. localized necking takes place randomly. To investigate the localized thinning mechanism, an integrated crystal plasticity finite element (CPFE) modeling system has been developed. In this paper, a simplified plane strain CPFE model is presented and used to investigate the localized thinning and failure features in hydroforming of micro-tubes. The crystal plasticity equations were implemented in the ABAQUS/Explicit FE code through a user-defined material subroutine, VUMAT. Single crystal, three-grain and polycrystal FE models were generated to study the localized thinning/necking mechanism and the effect of differing adjacent grain orientations, as well as the number of grains across the smallest specimen dimension, on the necking features. It has been confirmed from the analyses that the localized thinning observed in hydroforming of micro-tubes is significantly affected by the microstructure and grain orientations of the material.

Keywords: Hydroforming, micro-tubes, micro-mechanics modeling, crystal plasticity FE modeling.

* Corresponding author: Prof J Lin, jianguo.lin@imperial.ac.uk

1. Introduction

Hydroforming is a metal forming technology based on the application of pressurized liquid to generate defined workpiece shapes from tubular materials or sheet metals. The tube hydroforming technique has been widely used in automotive, aircraft and sanitary industries in recent years [1]. Compared with conventional punching and drawing processes, tube hydroforming has the advantage of part consolidation, weight reduction, improved structural strength and stiffness, lower tooling cost, fewer secondary operations, reduced dimensional variations, and reduced scrap. Hydroforming enables forming hollow complex-shaped components with integrated structures from single initial workpieces [2]. Modern electronic, telecommunication and medical technology products increasingly require tubular micro-components; the market for these products, and the micro-tubes they depend upon, is growing rapidly. Hydroforming is an efficient and time saving production method, and is capable of high productivity and accuracy in manufacturing micro-tubes.

Since an ordinary tubular component is generally much larger in its smallest dimension (i.e. thickness) compared with the grain size of the metal from which it is made, conventional macro-mechanics (continuum) FE modeling is effective for simulating typical hydroforming processes [3-5]. However, in hydroforming of micro-tubes, the material grain size can be on the order of the smallest part dimension. Thus, any given region within a micro-part may contain ten or fewer grains; when the number of grains across the smallest part dimension is sufficiently small, microstructural effects such as strain non-uniformity and localization, size effects [6] and twinning [7] may play an important role in the deformation.

To investigate the deformation features in hydroforming of micro-tubes, a hydroforming test facility has been designed and manufactured. Tests have been carried out for hydroforming of micro-tubes made of stainless steel having an initial blank outer diameter of 800 µm and a wall thickness of 40 µm. Fig. 1 shows some examples of hydroformed stainless steel micro-tubes having a final outer diameter of 1030 µm. Many tubes were formed using the same hydroforming parameters and it was found that rupture took place at different locations as a result of localized necking. Fig. 2 shows simulation results for a quarter section of a hydroformed micro-tube using a macro-mechanics modeling method (i.e. using conventional continuum plasticity theory); it can be observed that the wall thickness of the tube is uniform and no localized necking is predicted, unless large strain analyses together with imperfections (e.g. in the wall thickness) are used, which could give localized necking in the tube wall when using a conventional continuum plasticity model, e.g. for a power-law isotropic hardening material [8-9]. Localized necking and the subsequent fracture and rupture of the tube cannot be captured using conventional continuum plasticity without the introduction of imperfections [8-10], hence the need to account for the underlying crystal structure using crystal plasticity theory.

Due to the aforementioned differences in the deformation characteristics of hydroforming at the micro- and macro-scales, the workpiece of a micro-part cannot be regarded as a homogeneous continuum for process-simulation purposes. Crystal-plasticity (CP) theory, which assumes that homogenized crystallographic slip is the plastic deformation mechanism for crystalline materials, has attracted significant further attention in recent years because of its ability to relate the plastic deformation of micro-parts to their microstructures. CP models have been used successfully in the modeling of important phenomena, such as micro-crack initiation, crack propagation, fatigue, creep, etc. Crystal plasticity models based on a single-crystal yield function have been proposed for hydroforming by several investigators [11-17]. However, such models require sophisticated yield functions and specially designed algorithms so that non-uniqueness of stress can be avoided. Recently, Shi et al. [18] studied the localized necking of an aluminum tube under internal pressure. Localized necking was found to be associated with a surface instability resulting from nonuniform deformation. EBSD data was used to define the grain structure and orientations, with investigation into the effect of strain hardening and strain rate sensitivity on the predicted necking behavior. Grain distribution effects were explored by simulating a random grain structure with the same overall texture as the EBSD measurement, and a structure where cube-oriented grains were replaced by grains having a random orientation. This was sufficient to conclude that the spatial grain orientation distribution has a significant effect on necking in pressurized tubes [18]. In the present work, a systematic study of the effect of grain orientation distribution is the focus, with necking occurring naturally due to strain localization in grains having orientations locally favorable to crystallographic slip. An emphasis of the present research is on generating virtual microstructures in order to systematically study the effect of relative grain orientations (Schmid factors) and size (relative to the tube thickness) on the features of the localized thinning, and how this relates to the local state of stress in hydroforming of micro-tubes.

The overall objective of the present study was the development of an experimental method and an integrated, polycrystalline crystal plasticity finite element (CPFE) modeling technique for investigating the localized thinning features in the hydroforming of micro-tubes. Rate-dependent crystal plasticity theory is used to overcome the stress non-uniqueness. This theory has no yield function, and the numerical algorithms are stable and much more easily implemented in commercial FE software. Grains within the tube workpiece, their distributions and orientations are generated automatically using the previously developed VGRAIN software system. A set of crystal viscoplasticity model equations is implemented in the ABAQUS/Explicit FE code through a user-defined material subroutine, VUMAT. Single crystals, crystals having three grains contrived for studying orientation effects, and polycrystals have been generated and localized thinning analyzed using the simplified plane strain CPFE modeling technique.

2. Experimental program for hydroforming of micro-tubes

2.1. Test material, facility and procedures

Fig. 3a shows an example of a prototype micro-tube hydroforming test machine, which has been designed for investigating the feasibility of the hydroforming concept for the production of micro-tubes. This machine enables micro-hydroforming of tubular components having diameters between 0.2 and 1 mm. It is equipped with a spindle driven pressure intensifier which enables the application of up to 4000 bar internal pressure, the closing force being realized by a hydraulic drive, and the axial punches being moved by linear actuators with spindle gears.

Tests for the hydroforming of micro-tubes have been carried out using the machine. The length of the stainless steel tubular workpiece is 16 mm, and the outer diameters of the initial and formed tubes are 800 µm and 1030 µm, respectively. The experimental process is shown in Fig. 3b. At the beginning of the process the tube blank (800 µm) is placed into a die cavity (1030 µm) that corresponds to the final shape of the component. The dies are closed and the closing force applied while the tube is internally pressurized by a liquid to expand the component. Additionally the tube ends are axially compressed by sealing punches. The final tube is formed under the simultaneously controlled action of internal pressure and axial force. The forming pressure is applied linearly to a maximum value of 400 MPa to ensure the blank material is fully in contact with the die surface to form the desired part.

2.2. Examination of the features of tube failure

Roughly one hundred micro-tubes having the same initial and final geometry were formed using exactly the same hydroforming parameters. Despite carefully controlling the process, failure occurred in random locations for most of the formed micro-tubes; examples are shown in Fig. 1. Visual inspections showed that fracture took place in different locations, and in a random manner, before the die cavity was filled completely by the expanding tube. To investigate the random failure features, microstructure examination of the material was carried out to look at the relationship between grain size, as well as grain orientation distribution, and the failure mode of the material.

In order to examine the microstructure at the failure location, a formed micro-tube was cross-sectioned at the location of the rupture using a micro-wire cutting technique. By mounting, polishing and etching, the grains in a cross-section at the location of failure were revealed. Fig. 4 shows the sample, grains and grain boundaries at two locations in the cross section. First, it can be observed that the grain size is non-uniform in the vicinity of the failure. Fig 4c shows that the grain size is more uniform where no obvious localized necking has been observed; there is an average of approximately 2 grains across the thickness at that location. However, Fig. 4d shows some large grains close to the failure; in some areas, only one grain across the thickness of the material can be observed. In addition, localized necking is evident at the internal surface of the deformed tube, near the failure (Fig. 4d). Once localized necking occurs, failure takes place quickly. In the following sections, CPFE models have been created to study the localized necking features observed in the micrographs.

3. Crystal viscoplasticity constitution equations

Crystal plasticity theories are used to represent the flow of dislocations along slip systems in metallic crystals in terms of resolved shear strains. In particular, crystalline slip is assumed to obey Schmid’s law, i.e. the slipping rate [pic] in any particular slip system [pic] is related to the shear stress, [pic], acting on that slip system. The crystal plasticity theory used in this paper follows the pioneering work of Taylor [19], Hill and Rice [20] and Asaro [21]. The set of crystal viscoplasticity constitutive equations used are summarized below:

[pic] (1)

[pic] (2)

[pic] for an FCC crystal (3)

[pic] (4)

where [pic] is the average slip plane normal and [pic] is the average slip direction, with respect to the number of finite elements, Nel, within a single crystal grain (a polycrystal is simply an assemblage of single crystals with displacement compatibility at the common boundaries). The [pic] slip system is characterized by [pic] and [pic]. The number of slip systems and their orientations depends on the crystal lattice structure, e.g. an FCC crystal has 4 independent primary slip planes, and each has 3 independent slip directions (i.e. [pic]). At the beginning of deformation, [pic] for all α, and the slip plane normals [pic] and slip directions [pic] are the same for all of the elements within a single crystal grain; the initial grain orientation is defined by VGRAIN (introduced later) according to a probability distribution that characterizes the polycrystal grain orientation distribution. When plastic deformation occurs, [pic] and [pic] may have different values for the elements within a grain (in the finite strain formulation employed here; not the case when a small strain formulation is used). The use of [pic] and [pic] ensures that the same orientation is assigned to all elements within a grain.

Material strain hardening is specified based on the slip system strain hardness, [pic]. The self, [pic], and latent, [pic], hardening moduli are defined by Asaro [21] and Peirce et al. [22], which are directly related to accumulated shear strain [pic]:

[pic] (5)

[pic] (6)

where [pic] is the initial hardening modulus, [pic] is the initial shear strength, [pic] is the break through stress when plastic flow initiates and q is the 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 as one for isotropic hardening. [pic] is the fourth order stiffness tensor and the indices i, j, k and l take values between 1 and 3. In the initial state ([pic]), [pic], [pic], [pic] and [pic]. The material parameters for the equation set presented here are listed in Table 1 for 316L stainless steel, and Young’s modulus and Poisson’s ratio are 193 GPa and 0.34, respectively [23]. A high value of n is used here (Table 1) to reduce the viscoplastic effect of the material, as the hydroforming is carried out at low temperature.

|[pic] |[pic](s-1) |[pic] (MPa) |[pic] (MPa) |[pic] (MPa) |

|20.0 |0.001 |225 |330 |50 |

Table 1. The values of the material parameters for Eqs. (1) - (6).

4. Computational procedure for CPFE analysis

4.1. The numerical procedure

An integrated micro-mechanics CPFE modeling system contains three parts: microstructure generation and pre-processing, CPFE analysis and post-processing [24-25]. In the pre-processing stage, the so-called VGRAIN system introduced by Cao et al. [25] has been used to generate virtual grain structures according to parameters characterizing the microstructure. The grain structure within a defined region is generated according to the input values of average, maximum and minimum grain sizes. Orientations of grains are assigned according to a probability distribution either in a random form or with a chosen distribution (e.g. gamma). The generated virtual grains with their orientation information are input into ABAQUS/CAE for further pre-processing, e.g. adding boundary and loading conditions.

The CP material model is implemented in ABAQUS for explicit calculations via the user-defined material subroutine, VUMAT. The explicit implementation used here was adapted from the implicit algorithm developed by Huang [26] and compared to that of Harewood [23]. 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 in order that optimal processor parallelization can be achieved.

The accelerations and velocities at a particular point in time are assumed to be constant during a time increment and are used to solve for the next point in time, i.e. a forward Euler integration is carried out. To reduce dynamic effects, the ratio of the duration of the load to the fundamental natural period of the model is taken to be greater than five [27]; it has been found that by keeping the ratio of kinetic energy to the total internal strain energy < 5%, dynamic effects in the model are negligible [28-29].

4.2. Plane strain CPFE model

The geometry and the FE model of the cross-section of a micro-tube are shown in Fig. 5. In the CPFE model, a quarter section of the micro-tube was considered with symmetry boundary conditions along X = 0 and Y = 0. The minimum, average and maximum grain sizes of the material were taken to be 25, 30 and 40 µm, respectively, and 95% of the grains are within that range. Hence, there are between 1 and 2 grains across the thickness of the tube section on average. The grains and their orientations are generated using the VGRAIN system, and they are read into ABAQUS/CAE for further mesh generation, boundary conditions specification and loadings definition. The die is defined as a rigid part in ABAQUS/CAE. The maximum applied pressure is 400 MPa; this high pressure ensures that the workpiece is deformed sufficiently to come into complete contact with the die. 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, but the crystal plasticity equations above are valid for 3D analyses.

It is worth mentioning that large strain, polycrystal CPFE analyses require considerable CPU time. A 2D simplification (e.g. plane strain) reduces the computation time significantly and it is still possible to capture the interesting features, such as localized thinning, failure, etc., as observed in hydroforming of micro-tubes. The developed CPFE process modeling technique can be readily used for 3D hydroforming simulations if the 3D grain structures can be constructed effectively for the initial metal tube.

In order to investigate the relationship between localized thinning and microstructures of polycrystalline materials in hydroforming of micro-tubes, three FE models of a quarter cross-section of a tube were studied. The three models have the same initial geometry, but different crystal combinations: 1) A single crystal used to investigate the relationship between FCC slip systems and the loading direction; 2) A three-grain crystal comprising a small crystal section inserted symmetrically in the middle of the quarter FE model, having a different orientation relative to the grains adjacent to it, for studying the relationship between orientation mismatch and necking; And, 3) polycrystal structures for investigating the apparently random occurrence of local thinning for real hydroforming cases.

5. Computational results

5.1. Grain orientation effects of single crystals

To investigate the localized thinning features associated with the grain orientation and the applied load, a single crystal cross section model (single grain, uniform orientation) was employed. Fig. 6a shows a single FCC crystal with two sets of coordinate systems, the cubic crystal system and the global sample coordinate system, X-Y-Z. An assumption is made that the cubic crystal system may undergo only one rotation relative to the global system, around the Z axis; this is depicted in Fig. 6a where one of the potential slip systems, {111}, is also indicated for reference. Hence, the effect of arbitrary rotations is not considered; the rotation defined by the one angle ( is sufficient to illustrate the effect of grain orientation on necking. Fig. 6b illustrates the grain texture (in the global coordinate system) relative to a position on the tube specified by the angle (. It is indicated in Fig. 6b that the direction of the hoop stress at position (, [pic], the main driving force for forming the part, follows the tangent direction of the tube cross-section (thin tubes are considered here), whereas the crystal orientation remains unchanged over the tube cross section. The {111} slip system is indicated by the dotted lines, and the orientation of this slip system is specified relative to the [100] direction; the projection of the slip plane normal onto the X-Y plane, [pic], and the slip direction, [pic], are indicated. It should be noted that the angle between the hoop stress direction and the projection of the slip plane normal onto the X-Y plane is the same as that between the hoop stress direction and the slip plane normal itself, which is skewed in space relative to the X-Y plane; writing [pic], where [pic] is the normal to the slip plane and [pic] is the unit vector in the Z direction, it is clear that the scalar product between any vector in the X-Y plane with [pic], hence the angle between them, is equal to the scalar product between that vector and [pic]. Therefore, calculations of Schmid factor are correct when using the projection of the normal, as done here. Henceforth, [pic] is simply referred to as the normal to the slip plane. The Schmid factor for crystal orientation ( at location ( on the tube is [pic]. Thus, the location where plastic deformation occurs most readily on the {111} system, the location of maximum resolved shear stress, occurs where ( =(, which gives [pic]. The most difficult location to deform plastically is where [pic], which gives [pic]; in other words, the location most difficult to deform plastically is 45( from the location that is easiest to deform plastically. It should be noted that [pic] on the {111} system at a certain location does not imply plastic deformation does not occur; other slip systems out of the plane may be activated instead, but plastic deformation will in all cases be less than it is at a location where [pic] is greater on the {111} system. For example, if ( = 45(, the greatest plastic deformation will occur at θ = 45(, and the least at θ = 0( and θ = 90(.

Since the crystal orientation is constant over the tube cross-section, the direction of the hoop stress varies relative to the slip direction and it can be envisaged that necking will occur at the position where the resolved shear stress is maximum, which will be at the location where the [100] and/or [010] directions are perpendicular to the local direction of the hoop stress. The only case, for the quarter model, in which both [100] and [010] are perpendicular to the local hoop stress direction is ( = 0(; for positive (, the only location is that indicated by the [100] direction, and for negative (, the only location is that indicated by the [010] direction; in the case of a polycrystal, there may be many locations where necking occurs in the quarter model. The quarter model with symmetry conditions assumes a somewhat unnatural symmetry of the necking locations since for any ( > 0(, the [010] direction is effectively neglected and the slip in the other quadrants occurs by symmetric reflections of that in the first quadrant about the global coordinate axes. However, the necking locations are separated such that they will not appreciably interact, and in any case, the quarter model assumption has no influence on the qualitative behavior demonstrated in the simulations, which is the purpose of this study.

Numerical investigations were carried out with a grain orientation of ( = 60(. A single crystal was used for the tube cross-section and was analyzed using the loading and boundary conditions described above. Fig. 7 shows the deformed tube cross section superimposed with contours of accumulated plastic shear strain. It can be observed, as described above, that the thickness of the formed tube varies due to the variation in the direction of the hoop stress. The necking ratio, which is defined as the minimum thickness divided by the maximum thickness, is 0.74 for the simulation shown in Fig. 7. The thickness of the formed part for the single crystal structure gradually varies from the minimum thickness section (29.3 μm) at the position indicated by the crystal axis [100] to the maximum thickness section (39 μm), where the slip direction {111} (see Fig. 6a) and the hoop stress are perpendicular (hence the resolved shear stress on the {111} system is zero), which is the position 45o clockwise from the crystal axis [100]. The change in thickness for the formed tube is mainly due to the variation of the angle between the slip system and the local hoop stress direction. In addition, the hardening law of the material would affect the behavior, but different hardening laws are not explored in this study.

5.2 Effect of adjacent grain orientation on localized necking: a three grain model

A three grain model was created as shown in Fig. 8. The grain orientations are defined according to the convention shown in Fig. 6, and the tube dimensions are the same as those defined in Fig. 5. The two large grains, I and III, have the same size, and are sectors of the quarter model spanning 42o. Direction (1) represents the crystal axis [100], which is at α = 45o for both grains I and III. The small grain, grain II, is a sector of the quarter model spanning 6o. A series of CPFE analyses have been carried out while varying the small grain orientation (direction (1); crystal axis [100]) from α = 0o to 45o, while the orientations of the two large grains remain fixed at α = 45o.

With applied internal pressure, the tube blank is deformed to match the die shape. Fig. 9a shows the computational results for the hydroformed tubes with grain II orientations ([100] direction) of α = 0o, 20o and 30o relative to the global horizontal axis (see Fig. 8). When the crystal orientation for the small grain, grain II, is 0o, the plastic mismatch between it and the adjacent grains (I and III as shown in Fig. 8) is maximized and the ratio [pic], where [pic] is the thickness at the middle section of grain II and [pic] is the minimum thickness of the deformed tube section. The small grain is difficult to deform in the α = 0o orientation, because the Schmid factor (hence the resolved shear stress) is zero in the centre of the grain (α = 45o), and remains relatively small even at its outer extents since the grain size is small (6o span). The influence of this hard central grain causes the thinnest section to be located away from the interfaces between the small grain and the two large grains, i.e. the constraint of the small grain being difficult to deform plastically is felt by the neighboring grains until a sufficient distance away from that interface is reached (it is a compromise between the central grain, grain II, and the adjacent grains, as grains I and III would prefer to localize at the interfaces with the central grain, where their resolved shear stress is largest). However, as the orientation of the small grain was changed to 20o and 30o, the ratio [pic] reduced to 1.34 and 1.13, respectively. It can be observed from Fig. 9a that grain II deformed more as its orientation varied from 0o to 30o, demonstrating a reduction in the localization of plastic strain in the grains adjacent to grain II. Fig. 9b shows the variation of [pic] with grain II orientation; the ratio [pic] approaches 1 as the orientation of grain II varies from 0o to 45o.

5.3 Deformation effects for polycrystalline micro-tubes

Two cases were studied to investigate thinning for hydroforming of micro-tubes with polycrystalline material microstructures. In the two cases, the grain structures and grain orientations of the tube blank were identical, but the degree of deformation was different. This simulates the process of forming tubes with different final diameters from workpieces with the same initial diameter. The initial dimensions and grain structure of the material blank are shown in Fig. 5. For the first case (Fig. 10a), the radius of the die is 515 µm, thus, the ratio of final diameter to initial diameter, the deformation ratio, is 1.3. For the case shown in Fig. 10b, the radius of the die is 596 µm, hence the deformation ratio is 1.5.

Predicted localized thinning for the two cases is shown in Figs. 10a and 10b. It is clear that the wall thicknesses of the formed tubes are non-uniform and the amount of localized thinning increases with deformation dramatically and non-proportionally. The ratios of the maximum and minimum values of wall thickness in Fig. 10a to those in Fig. 10b are 0.97 (31.9/32.9) and 0.64 (13.7/21.4), respectively. It can be seen that the maximum value of the wall thickness remains approximately the same as the deformation progresses, but the minimum value of the wall thickness decreases sharply with further deformation. This indicates that once necking (localized thinning) takes place at a location, it progresses very quickly and leads to localized failure of the material.

Fig. 10c shows the progression of thinning with deformation, t/t0, where t0 is the initial thickness of the tube, for the three locations, A, B and C indicated in Figs. 10a and 10b, during the forming process. Location A represents the thinnest location and location C the thickest. It can be observed that during the initial tube expansion process, there is no obvious localized thinning taking place. However, when the deformation progresses to a certain extent (a deformation ratio above 1.05), more thinning occurs at location A. During the subsequent deformation, the localized thinning at A progresses very quickly and the thinning at location C progresses slowly. This indicates that failure must take place at location A. The other location of localized thinning, B, does not progress very much by the end of the deformation process as localization at A has taken over the deformation.

This localized thinning feature can be further demonstrated in Fig. 11 for deforming the tube to the die with the radius of 596 µm (deformation ratio: 1.5). It should be noticed that the deformed tube is not perfectly circular before the material finally contacted to the die surface. Even if the maximum pressure is reached, some particular areas of the tube surface may be difficult to touch the die surface due to the slip systems of the grains. It can be seen from Fig. 11a that once the necking takes place at specific locations, it progresses quickly and most of the deformation in the forming process would be carried by the locations. The thinnest location is at 0.28π from the x-axis. This localised necking feature can be further illustrated in Fig. 11b, where the thinnest location is indicated by the dotted line.

As discussed before, the necking location is related to the angle between the slip direction and the local direction of the hoop stress, but is also influenced by the orientation and size of neighboring grains. Thus, for a polycrystal case, the location and amount of localized necking are difficult to control in practice if there are only one or two grains through the thickness of the tube, as shown by the examples of localized failure in actual hydroforming of micro-tubes in Fig. 1; the location of failure is random and cannot be predicted without a detailed knowledge of the grain sizes and orientations. The experimentally observed random localized thinning and failure are also predicted by the CPFE analyses carried out in this study. Furthermore, the CPFE analyses confirm that conventional macro-mechanics (continuum) FE process modeling techniques (see Fig. 2) cannot be used to predict localized failure in hydroforming of micro-tubes unless imperfections are introduced [8-10].

5.4 Random thinning locations in polycrystal structures

Polycrystal structures and grain orientations are generated using the VGRAIN system automatically as mentioned before and shown in Fig. 5. To simulate the deformation and thinning behavior of hydroformed micro-tubes taken from the same piece of original material, the grain structures are generated twice using the same microstructure control parameters defined above, with grain orientations for both micro-tube models assigned randomly based on the orientation probability distribution within the VGRAIN system, characterized by the maximum, minimum and average grain sizes. Grain structures and grain orientations may be different between the two CPFE models, but they are within the ranges of the material specification. The simulation results for the hydroformed micro-tubes are shown in Fig. 12. It can be seen that the minimum and maximum values of the wall thickness of the formed tubes, shown in Figs. 12a (20.2 µm and 31.4 µm, respectively) and 12b (20.7 µm and 33.1 µm, respectively), are different for the two cases studied. The wall thickness of the two hydroformed micro-tubes having random grain orientations is non-uniform and difficult to predict. This is a result of the spatial variation of grain size, grain orientation and the constraints neighboring grains with different orientations apply to one another; this cannot be captured using conventional macro-mechanics (continuum) FE techniques (see Fig. 2). It can also be observed that localized thinning occurs at different locations due to the randomness of the grain orientations of the workpiece material, which is difficult to control in practice.

5. Conclusions

Tests on hydroforming of micro-tubes showed that premature failure takes place at random locations for materials having between 1 and 2 grains through the thickness of the initial tube workpiece. The random failure and localized thinning features observed in the experiments cannot be predicted using macro-mechanics (continuum) FE techniques. Effects can be generally understood and anticipated for single crystal tubes with known crystal structures through a geometric analysis of the Schmid factor, hence resolved shear stress, but this cannot be done for polycrystals. CPFE models have been developed, and these models can predict the important features of localized thinning due to grain structures and grain orientations. The research also shows that localized necking is related to the directions between the crystal slip systems and the local hoop stress direction. The thinnest location in the simulations is always in the section where the [100] crystal axis is perpendicular to the local hoop stress direction, and the thickest section is 45o from the location of the thinnest section for a single crystal structure. Thus, the location of localized thinning is easy to predict for a single crystal material but is difficult to predict for polycrystal cases without a detailed knowledge of the microstructure. However, the CPFE models developed in this study can be used to guide tube workpiece design by identifying the lower limit on the number of grains through the thickness such that an acceptable uniformity of thickness is achieved during forming, hence mitigating unpredictable failure, and a polycrystal grain distribution design that facilitates thickness uniformity. It was also demonstrated that adjacent grains with significant plastic mismatch compromise in determining the neck location, i.e. the grain that is easier to deform plastically may neck but not necessarily where the resolved shear stress is largest; neighboring grains that are difficult to deform plastically constrain those that are easy to deform, and thereby influence the neck location. Lastly, the simulations demonstrated that once necking takes place at the location that is most susceptible to plastic deformation by crystallographic slip, it progresses very quickly and leads to localized thinning, then failure; other potential areas of localized necking are subsequently irrelevant.

Acknowledgement

The authors thank the European Committee for their support on the FP7 project “Multiscale Modelling for Multilayered Surface Systems (M3-2S)”, Grant No: CP-FP 213600-2 M3-2S.

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11] Zamiri, A., Pourboghrat, F., Barlat, F. (2007) An effective computational algorithm for rate-independent crystal plasticity based on a single crystal yield surface with an application to tube hydroforming, Int. J. Plast., 23: 1126-1147.

12] Zamiri, A., Pourboghrat, F. (2009) A novel yield function for single crystals based on combined constraints optimization, Int. J. Plast., 26: 731-746.

13] Guan, Y., Pourboghrat, F., Barlat, F. (2006) Finite element modelling of tube hydroforming of polycrystalline aluminium alloy extrusions, Int. J. Plast., 22: 2366-2393.

14] Jansson, M., Nilsson, L., Simonsson, K. (2008) Tube hydroforming of aluminium extrusions using a conical die and extensive feeding, J. Mater. Process. Technol., 198: 14-21.

15] Korkolis, Y.P., Kyriakides, S. (2008) Inflation and burst of aluminium tubes, Part II: An advanced yield function including deformation-induced anisotropy, Int. J. Plast. 24: 1625-1637.

16] Nygards, M., Gudmundson, P. (2004) Numerical investigation of the effect of non-local plasticity on surface roughening in metals, Eur. J. Mech. A-Solid, 23: 753-762.

17] Houtte, P.V., Kanjarla, A.K., Van Bael, A., Seefeldt, M., Delannay, L. (2006) Multiscale modeling of the plastic anisotropy and deformation texture of polycrystalline materials, Eur. J. Mech. A-Solid, 25: 635-648.

18] Shi, Y., Wu, P.D., Lloyd, D.J., Embury, J.D. (2010) Crystal plasticity based analysis of localized necking in aluminum tube under internal pressure, Eur. J. Mech. A-Solid 29: 475-483.

19] Taylor, G.I. (1938) Plastic strains in metals, J. Inst. Met., 62: 307-324.

20] Hill, R., Rice, J.R. (1972) Constitutive analysis of elastic-plastic crystals at arbitrary strain, J. Mech. Phys. Solids, 20: 401–413.

21] Asaro, R.J. (1983) Crystal plasticity, J. Appl. Mech., 50: 921-934.

22] Peirce, D., Asaro, R.J. and Needleman, A. (1982) Material rate dependence and localized deformation in crystalline solids, Acta Metal., 31: 1951-1976.

23] Harewood, F. J., McHugh, P. E. (2007) Comparison of the implicit and explicit finite element methods using crystal plasticity, Comput. Mater. Sci. 39: 481-494.

24] Cao, J., Zhuang, W., Wang, S., Lin, J., (2009) Development of a VGRAIN system for CPFE analysis in micro-forming applications, Int. J. Adv. Manuf. Tech., 47: 981-991.

25] Cao, J., Zhuang, W., Wang, S., Ho, K.C., Zhang, N., Lin, J., Dean, T.A. (2009) An integrated crystal plasticity FE system for microforming simulation, J. Multiscale Model., 1: 107-124.

26] Huang, Y., (1991) A user-material subroutine incorporating single crystal plasticity in the ABAQUS finite element program, Harvard University Report, MECH 178.

27] Kutt, L.M., Pifko, A.B., Nardiello, J.A., Papazian, J.M. (1998) Slow-dynamic finite element simulation of manufacturing processes, Comput. Struct., 66: 1-17.

28] Chung, W.J., Cho, J.W., Belytschko, T. (1998) On the dynamic effects of explicit FEM in sheet metal forming analysis, Eng. Computations, 15: 750-776.

29] Choi, H.-H., Hwang, S.-M., Kang, Y.H., Kim, J., Kang, B.S. (2002) Comparison of implicit and explicit finite element methods for the hydroforming process of an automobile lower arm, Int. J. Adv. Manuf. Tech., 20: 407-413.

(a) Failure takes place randomly.

[pic]

(b) Close-up view of the failure.

Figure 1. Premature localized failure of formed micro-tubes.

Figure 2. Virtually formed micro-tube using a macro-mechanics modeling method.

(a) Micro-tube hydroforming machine.

(b) Micro-tube hydroforming tooling and test set-up.

Figure 3. Experimental setup for the hydroforming of micro-tubes.

[pic]

Figure 4. Failure of formed micro-tube: (a) Formed micro-tube with a crack, and (b), (c) and (d) the microstructure of a micro-tube.

Figure 5. Micro-mechanics (CPFE) model for tube hydroforming. Grains and grain boundaries are shown in the figure; different colors indicate different orientations.

[pic]

Figure 6. (a) FCC single crystal with one rotation relative to the micro-tube (global) coordinate system, about the Z-axis; (b) Cross section model of hydroforming of a micro-tube with the slip direction represented by dotted lines; ( gives the [100] direction, and ( a location on the tube in the global coordinate system.

Figure 7. Predicted localized necking for a single crystal structure with a grain orientation of 60( with respect to the sample (global) coordinate system.

Figure 8. A three-grain structure; the orientation of the large grains I and III are fixed and the orientation of grain II varies.

(a) Strain and necking variations as the orientation of grain II changes.

(b) Thinning behavior as a function of relative grain misorientation.

Figure 9. The dependence of localized necking behavior on the variation of the orientation of grain II, as depicted in Fig. 8

[pic]

[pic]

[pic]

Figure 10. Comparison of thinning features for deformation ratios of (a) 1.3 and (b) 1.5, and (c) necking development for the positions A, B and C.

a)

(b)

Figure 11. Thinning progress in forming the micro-tube to a deformation ratio of 1.5. (a) Field plots for the deformation history, and, (b) Normalized thinning profiles for different deformation ratios, where R is the radius of the tube at a certain deformation ratio.

[pic]

[pic]

Figure 12. Comparison of thinning features for microstructures generated twice by the VGRAIN system using the same control parameters.

-----------------------

1mm

[pic](MPa)

+374

+366

+358

+351

+343

+336

Hydraulic drive for tool closing and application of closing force

Guiding columns

Press frame

Top die

Bottom die

Connection for pressurizing media inlet

Spindle gear drives for sealing and application of axial force

Press slide

Top die

Initial tube

Sealing punch

Bottom die

1. Insertion of initial tube

2. Closing of top and bottom die

Internal pressure

Closing force

Hydroformed component

Media inlet

3. Hydroforming

4. Ejection of Hydroformed component

Axial force

(d)

(c)

(b)

(a)

1mm

Rigid Die

Workpiece

(a)

[001]

[010]

[100]

Z

X

Y

ðað

(a)

(b)

29.3 mðm

39 mðm

[100]

[010]

X

Y

60(

45(

Accumulated Shear Strain gð

Rigid Die

Workpiece

`!

a!

b!

48o

α

(a)

(b)

29.3 μm

39 μm

[100]

[010]

X

Y

60(

45(

Accumulated Shear Strain γ

Rigid Die

Workpiece

Ⅰ

Ⅱ

Ⅲ

48o

(1)

(1)

(1)

(2)

(2)

(2)

42o

0o

[pic]

[pic]

20o

[pic]

[pic]

30o

[pic]

[pic]

Max. Principal Strain

(degrees)

tII/tmin

A: 21.4(m

B: 28.4(m

C: 32.9(m

Max. Principal Strain

(a)

B: 25.2(m

A: 13.7(m

C: 31.9(m

Max. Principal Strain

(b)

A

B

C

t/t0 0

(c)

X

Y

Max. Principle Strain

Deformation Ratio:

1.0

1.1

1.2

1.3

1.4

1.5

0.28π

π/10

π/5

3π/10

2π/5

π/2

Arc Length / R

t/t0

Deformation Ratio:

1.0

1.3

1.5

1.2

31.4(m

20.2(m

(a)

Max. Principal Strain

20.7(m

33.1(m

(b)

Max. Principal Strain

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