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 microtubes 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
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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 macromechanics (continuum) FE modeling is effective for simulating typical hydroforming processes [35]. 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
2
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
3
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 userdefined 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 microtubes. 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.
4
In order to examine the microstructure at the failure location, a formed micro-tube was crosssectioned 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   in any particular slip system  is related to the shear stress,
  , 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:
   mi ij s j
(1)

   a sgn      g 
g    h   ,

n
(2)
  1, 2...12
for an FCC crystal
(3)



 ij  Cijkl   kl   klp   Cijkl   kl   0.5   sk ml  sl mk  

where mi 
1
N el
(4)


Nel
 mi ,k is the average slip plane normal and s j 
k 1
1
N el
N el
 s
k 1
j
,k
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
5
common boundaries). The  slip system is characterized by mi and s j . 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.   1, 2,...12 ). At
the beginning of deformation,    0 for all α, and the slip plane normals mi and slip directions
sj 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, mi and sj 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 mi and s j 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, g  . The self, h ,
and latent, h , hardening moduli are defined by Asaro [21] and Peirce et al. [22], which are
directly related to accumulated shear strain  :
h
2

h( )  h0sech h0


 qh   
 g s  g0 
 =

(5)
12
   
(6)
 1
where h0 is the initial hardening modulus, g 0 is the initial shear strength, g s 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. Cijkl is the fourth order stiffness tensor
and the indices i, j, k and l take values between 1 and 3. In the initial state ( t  0 ),  ij  0 ,    0 ,
g   g 0 and  kl  0 . 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.
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n
a (s-1)
h0 (MPa)
g s (MPa)
g 0 (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
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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 crosssection 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
8
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}<110>, 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 ,   , 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}<110> 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, , and the slip direction, , 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 , where is
the normal to the slip plane and is the unit vector in the Z direction, it is clear that the scalar
product between any vector in the X-Y plane with , hence the angle between them, is equal to the
scalar product between that vector and . Therefore, calculations of Schmid factor are correct when
using the projection of the normal, as done here. Henceforth, is simply referred to as the normal to
the slip plane. The Schmid factor for crystal orientation  at location  on the tube is . Thus, the
location where plastic deformation occurs most readily on the {111}<110> system, the location of
maximum resolved shear stress, occurs where  =, which gives     max    2 . The most
difficult location to deform plastically is where , which gives ; 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 on the {111}<110> 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 t a is greater on the {111}<110> 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
9
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 m) at the position indicated by the crystal axis [100] to
the maximum thickness section (39 m), where the slip direction {111}<110> (see Fig. 6a) and the
hoop stress are perpendicular (hence the resolved shear stress on the {111}<110> 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.
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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 tII / tmin  1.56 , where tII is the thickness at the middle
section of grain II and tmin 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 tII / tmin 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 tII / tmin with grain II orientation; the
ratio tII / tmin 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),
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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
12
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 nonuniform 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]
13
crystal axis is perpendicular to the local hoop stress direction, and the thickest section is 45 o 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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16
(a) Failure takes place randomly.
1mm
(b) Close-up view of the failure.
Figure 1. Premature localized failure of formed micro-tubes.
17
 e (MPa)
+374
+366
+358
+351
+343
+336
Figure 2. Virtually formed micro-tube using a macro-mechanics modeling method.
18
Hydraulic drive for tool closing
and application of closing force
Guiding columns
Press frame
Press slide
Top die
Bottom die
Connection for pressurizing
media inlet
Spindle gear drives for sealing
and application of axial force
(a) Micro-tube hydroforming machine.
Top die
Initial tube
Sealing punch
Bottom die
2. Closing of top and bottom die
1. Insertion of initial tube
Internal pressure
Closing force
Hydroformed
component
Media inlet
Axial force
3. Hydroforming
4. Ejection of Hydroformed component
(b) Micro-tube hydroforming tooling and test set-up.
Figure 3. Experimental setup for the hydroforming of micro-tubes.
19
(b)
(c)
(a)
1mm
(d)
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.
20
Rigid Die
Workpiece
Figure 5. Micro-mechanics (CPFE) model for tube hydroforming. Grains and grain boundaries are
shown in the figure; different colors indicate different orientations.
21
[001]
Z

X
[010]
(a)
Y
[100]
(a)
(b)
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.
22
[100]
29.3 m
Accumulated
Shear Strain 
45
Y
39 m
[010]
60
X
Figure 7. Predicted localized necking for a single crystal structure with a grain orientation of 60
with respect to the sample (global) coordinate system.
23
Rigid
Die
(1)
(2)
Ⅰ
(2)
(1)
Ⅱ
(2)
(1)
Ⅲ
42o
48o
Workpiece
Figure 8. A three-grain structure; the orientation of the large grains I and III are fixed and the
orientation of grain II varies.
24
0o
tmin
tII
tmin
20o
tII
o
tmin
30
tII
Max. Principal Strain
(a) Strain and necking variations as the orientation of grain II changes.
1.6
1.5
t II/t min
1.4
tII/tmin
1.3
1.2
1.1
1
0.9
0
10
20
30
40
50
Grain II Orientation (degrees)
(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
25
B: 28.4m
A: 21.4m
Max. Principal Strain
C: 32.9m
(a)
B: 25.2m
A: 13.7m
Max. Principal Strain
C: 31.9m
(b)
1.2
1
C
t/t00.6
B
t/t。
0.8
0
0.4
A
0.2
0
1
1.1
1.2
(c)
1.3
1.4
1.5
1.6
Deformation Ratio
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.
26
Y
Max. Principle Strain
0.28π
Deformation Ratio: 1.0 1.1 1.2 1.3 1.4 1.5
X
(a)
1.2
Deformation Ratio: 1.0
1
t/t0
0.8
1.2
0.6
1.3
0.4
1.5
0.2
0
π/10
18
3π/10
π/5
36
54
Arc Length / R
2π/5
72
π/2
90
(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.
27
31.4m
20.2m
Max. Principal Strain
(a)
20.7m
Max. Principal Strain
33.1m
(b)
Figure 12. Comparison of thinning features for microstructures generated twice by the VGRAIN
system using the same control parameters.
28