IMPLEMENTATION OF NON-LOCAL THEORY INTO THE...
144
IMPLEMENTATION OF NON-LOCAL THEORY INTO THE UPPER- AND LOWER-BOUND DAMAGE-BASED MATERIAL MODEL by Michael D. Landry B.Sc.E., University of New Brunswick, Fredericton, New Brunswick, 2008 B.Sc., Cape Breton University, Sydney, Nova Scotia, 2006 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering In the Graduate Academic Unit of Mechanical Engineering Supervisor: Z. T. Chen, PhD, Department of Mechanical Engineering Examining Board: J. Hall, PhD, Department of Mechanical Engineering, Chair A. Simoneau, PhD, Department of Mechanical Engineering Y. Ni, PhD, Department of Chemical Engineering This thesis is accepted by the Dean of Graduate Studies THE UNIVERSITY OF NEW BRUNSWICK December, 2009 ©Michael D. Landry, 2010
Transcript of IMPLEMENTATION OF NON-LOCAL THEORY INTO THE...
IMPLEMENTATION OF NON-LOCAL THEORY INTO THE UPPER- AND
LOWER-BOUND DAMAGE-BASED MATERIAL MODEL
by
Michael D. Landry
B.Sc.E., University of New Brunswick, Fredericton, New Brunswick, 2008
B.Sc., Cape Breton University, Sydney, Nova Scotia, 2006
A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Engineering
In the Graduate Academic Unit of Mechanical Engineering
Supervisor: Z. T. Chen, PhD, Department of Mechanical Engineering
Examining Board: J. Hall, PhD, Department of Mechanical Engineering, Chair
A. Simoneau, PhD, Department of Mechanical Engineering
Y. Ni, PhD, Department of Chemical Engineering
This thesis is accepted by the
Dean of Graduate Studies
THE UNIVERSITY OF NEW BRUNSWICK
December, 2009
©Michael D. Landry, 2010
ii
Abstract
The combined upper- and lower-bound material model is an excellent tool for
designers to accurately predict the behaviour of advanced high strength steels in
complicated loading scenarios. The combined model is composed of the Gurson
upper-bound and the Sun and Wang lower-bound plasticity models. It is well known
that the upper-bound model suffers from mesh sensitivity. Due to the similarities in
derivation between the upper- and lower-bound models, the lower-bound was tested
and showed a similar loss of mesh objectivity. The non-local theory was implemented
into the dual bound model as a regularization technique in an attempt to resolve, or
reduce the behaviour of mesh sensitivity. Non-local theory is a known solution to
similar problems in the upper-bound and the results show that the method works well
for the lower-bound too. The implementation was tested using a plane strain tension
test and a straight tube hydroforming model. In the plane strain tension test, the
porosity band width was regulated with respect to mesh size through the use of the
non-local theory. In the hydroforming simulation, variation in burst pressure and
corner fill expansion due to mesh sensitivity was reduced when the non-local code
was implemented.
iii
Acknowledgements
I would like to thank Dr. Chen for giving me the opportunity to work in such a
great research group and to Auto21 for providing the financial support necessary
throughout my stay. Financial support was also provided by the Natural Sciences and
Engineering Research Council of Canada and is greatly appreciated. I am very
thankful for the help and guidance of the faculty and staff of the Mechanical
Engineering Department on a wide range of subjects too broad to mention here. Their
help has allowed me to accomplish much more than I thought was possible. Working
in the windowless lab of H-13 was made much easier by the great company provided
by Cliff, Hossein, and Xiaobo. Finally I would like to thank my wife, Jill, for her
continued support and the support of our family.
iv
Table of Contents
Abstract ............................................................................................................................... ii
Acknowledgements ............................................................................................................ iii
Table of Contents ............................................................................................................... iv
List of Tables .................................................................................................................... vii
List of Figures .................................................................................................................... ix
Nomenclature ................................................................................................................... xiv
Chapter 1: Introduction ....................................................................................................... 1
1.1. Motivation ................................................................................................................ 2
1.2. Rationale for software selection ............................................................................... 3
1.3. Objectives of the work ............................................................................................. 4
Chapter 2: The Upper- and Lower-Bound Damage-based Material Model ....................... 6
2.1. Upper-Bound Model ................................................................................................ 6
2.2. Lower-Bound Model .............................................................................................. 10
2.3. Damage Development and Modelling ................................................................... 13
2.3.1. Void Nucleation .............................................................................................. 13
2.3.2. Void Growth ................................................................................................... 16
2.3.3. Void Coalescence............................................................................................ 17
2.4. Implementation ...................................................................................................... 18
2.5. Verification ............................................................................................................ 21
v
Chapter 3: Non-Local Theory ........................................................................................... 23
3.1. Introduction ............................................................................................................ 23
3.1.1. Non-Local Integral .......................................................................................... 23
3.1.2. Explicit Gradient ............................................................................................. 25
3.1.3. Implicit Gradient ............................................................................................. 26
3.2. Implementation ...................................................................................................... 27
3.3. Verification ............................................................................................................ 34
Chapter 4: Mesh Sensitivity .............................................................................................. 36
4.1. Mesh Sensitivity in the Upper-Bound Approach ................................................... 41
4.2. Upper-Bound Non-Local Solution ......................................................................... 43
4.3. Mesh Sensitivity in the Lower-Bound Approach .................................................. 45
4.4. Lower-Bound Non-Local Solution ........................................................................ 47
4.5. Characteristic Length ............................................................................................. 49
Chapter 5: Straight Tube Hydroforming ........................................................................... 52
5.1. Introduction ............................................................................................................ 52
5.2. Experiment Dimensions and Modelling Approach................................................ 58
5.3. Results and Validation ........................................................................................... 67
5.3.1. Porous Zone .................................................................................................... 67
5.3.2. Burst Pressure and Corner Fill Expansion ...................................................... 71
5.3.3. Mesh Objective Material Model Study ........................................................... 76
vi
Chapter 6: Conclusions and Recommendations ............................................................... 80
6.1. Conclusions ............................................................................................................ 81
6.2. Discussion and Recommendations ........................................................................ 83
References ......................................................................................................................... 85
Appendix A ....................................................................................................................... 93
Appendix B ..................................................................................................................... 116
Appendix C ..................................................................................................................... 122
Curriculum Vitae
vii
List of Tables
Table 1 Demonstration material parameters ..................................................................... 40
Table 2 Mechanical properties of the demonstration material (Bardelcik, 2006) ............ 41
Table 3 Porous band width for the local upper-bound solution ........................................ 42
Table 4 Porous band width for the non-local upper-bound solution ................................ 44
Table 5 Porous band width for the local lower-bound solution ........................................ 46
Table 6 Porous band width for the non-local lower-bound solution ................................ 48
Table 7 Convergence study of the output time frequency in the hydroforming model .... 65
Table 8 Coefficient of friction for contact between the tube, die and null mesh
(Bardelcik 2006) ....................................................................................................66
Table 9 Tubular hydroforming tangential mesh discretizations and aspect ratios. ........... 67
Table 10 Porosity zone dimensions for upper- and lower-bound local models ................ 69
Table 11 Porosity zone dimensions for upper- and lower-bound non-local models ........ 70
Table 12 Characteristic lengths used for the non-local calculations ................................. 71
viii
Table 13 Burst pressure and corner fill expansion for upper- and lower-bound local
models ................................................................................................................... 73
Table 14 Burst pressure and corner fill expansion for upper- and lower-bound non-
local models .......................................................................................................... 75
Table 15 Von Mises equivalent stress at various pressures for both upper- and lower-
bound solutions ..................................................................................................... 77
Table 16 Burst pressure for von Mises at 90% UTS and the dual bound model .............. 79
ix
List of Figures
Figure 1 Transformation of a material containing voids (a) into either a spherical unit
cell (b) or a cylindrical one (c) where a is the radius of the void and b is the
cell radius (Gurson, 1977)....................................................................................... 8
Figure 2 The influence of porosity and triaxiality on the Gurson yield criterion
(Worswick and Pelletier, 1998) ............................................................................ 10
Figure 3 A comparison of the dual bound model and the extended stress based
forming limit curve (see Bardelcik, 2006) for both stress and strain based
nucleation (Butcher, 2009) .................................................................................... 12
Figure 4 Experimental powder metallurgy tension and compression samples against
the Gurson upper-bound and Sun and Wang lower-bound (Sun and Wang,
1989) ..................................................................................................................... 12
Figure 5 Al2O3 particles reinforcing aluminum 6061 showing (left) debonding and
(right) particle cracking under horizontal loading (Kanetake et al., 1995) .......... 14
Figure 6 A cubic unit-cell with central spherical void: The void undergoes dilatational
growth until the void touches the cell wall (Thomason, 1998)............................. 17
Figure 7 Interactions between the yield function and plastic multiplier allow
regulation of the porosity as a substitute for non-localizing the strain
increment............................................................................................................... 28
x
Figure 8 Information is recorded into record files until the end of the current time step
at which point the non-local porosity for the entire model is calculated and
returned for the next step. ..................................................................................... 30
Figure 9 Schematic of the Pre-Model Process which includes the element counter and
insertion of non-local porosity .............................................................................. 32
Figure 10 Schematic of the Pre-Model Process which includes the element counter
and insertion of non-local porosity ....................................................................... 33
Figure 11 Verification of the upper-bound non-local addition precedent to element
failure, from left to right: local banding, non-local banding with a negligible
characteristic length, and non-local banding with an exaggerated characteristic
length..................................................................................................................... 35
Figure 12 Test sample used to demonstrate mesh sensitivity, the grey element has 1%
higher initial porosity in comparison to the other elements. ................................. 38
Figure 13 Velocity profile used for the plane strain tension tests..................................... 38
Figure 14 Porous band width versus output frequency: □ 0.5mm mesh at the 2.3x10-2
fringe, ◊ 0.5mm mesh at the 2.6x10-2
fringe, ○ 0.5mm at the 2.9x10-2
fringe, □
0.25mm mesh at the 2.3x10-2
fringe, ◊ 0.25mm mesh at the 2.6x10-2
fringe. ...... 39
xi
Figure 15 Mesh sensitivity demonstrated in the upper-bound model through the use of
a 0.5mm, 0.33mm and 0.25mm element size mesh under horizontal tensile
loading................................................................................................................... 43
Figure 16 Mesh sensitivity regularization in the upper-bound model through the use
of a non-local model with characteristic lengths of 0.19, 0.17, and 0.15 mm
for the coarse, medium and fine meshes ............................................................... 43
Figure 17 Mesh sensitivity regularization in the upper-bound model through the use
of a non-local model with characteristic lengths of 0.2125, 0.1913, and 0.1750
mm for the coarse, medium and fine meshes ........................................................ 45
Figure 18 Mesh sensitivity demonstrated in the lower-bound model through the use of
a 0.5mm, 0.33mm and 0.25mm element size mesh under horizontal tensile
loading. Point A clearly shows lower concentrations of porosity at smaller
element sizes, whereas point B shows the opposite behaviour. ............................ 46
Figure 19 Mesh sensitivity regulated in the lower-bound model through the use of a
variety of characteristic lengths: 0.16, 0.1575, and 0.1450 mm for the coarse,
medium, and fine meshes, respectively. Point C is a distinct porous band
which displays consistency in dimension and magnitude across all meshes. ....... 47
Figure 20 Mesh sensitivity regulated in the lower-bound model through the use of a
second set of characteristic lengths: 0.19, 0.1650, and 0.15 mm for the 0.5mm,
0.33mm, and 0.25mm element size samples, respectively. Point D is a region
xii
of high porosity, and E a region of moderate porosity on the upper and lower
edges of the sample. .............................................................................................. 49
Figure 21 Regions of effective characteristic lengths for various element sizes .............. 51
Figure 22 Characteristic length sets based on uniform porosity distribution: ◊ Gurson
Group 1, □ Gurson Group 2, ○ Sun and Wang Group 1, and Δ Sun and Wang
Group 2 ................................................................................................................. 51
Figure 23 Free forming or bulge forming allows the material to deform without an
exterior die under pressure from an internal liquid (Goodarzi et al., 2005) ......... 54
Figure 24 The effect of different anisotropy parameters can be significant on
hydroforming operations Carleer et al. (2000). .................................................... 55
Figure 25 Failure mechanisms in tubular hydroforming include (from left to right)
buckling, wrinkling, bursting, and folding............................................................ 56
Figure 26 Corner fill expansion is the percentage that the tube expands into the die.
The distance the pipe moves (A) is divided by the original distance from the
pipe to the die corner (B) to obtain the percentage. .............................................. 58
Figure 27 The straight tube hydroforming die used by Bardelcik (2006) ........................ 59
Figure 28 The square cross-section of the straight tube hydroforming die (Bardelcik,
2006) ..................................................................................................................... 60
xiii
Figure 29 Internal pressure loading schedule ................................................................... 60
Figure 30 One eighth, 2-dimensional sliver model for straight tube hydroforming with
8-node brick elements ........................................................................................... 62
Figure 31 Secondary coordinate system for boundary conditions on the 45 degree
symmetry plane, the null mesh (brown) is clearly visible under the tube
elements (red)........................................................................................................ 63
Figure 32 Regions of various boundary conditions: the green (1) and red (2) sets are
constrained from displacement in the tangential and normal directions and
from rotation in the radial direction; the blue (3) set is constrained from
displacement in the normal direction .................................................................... 65
Figure 33 Porous zone width for the upper-bound model, top: 30 element perimeter
local model, bottom: 150 element perimeter local model ..................................... 68
xiv
Nomenclature
ija = yield surface normal
A = distance between formed tube and die wall
ib = Sun and Wang parameters
B = distance between tube and die wall before forming
c = a function of characteristic length
e
ijklC = elastic constitutive moduli
d = plastic multiplier
kl = total strain increment
ij = Kronecker delta
E = elastic or Young‟s modulus
= local strain
= non-local strain
M = effective plastic strain rate in the matrix
N = mean nucleation strain
f = void volume fraction or porosity
*f = effective porosity for the Gurson-Tvergaard-Needleman model
cf = critical porosity value
ff = final porosity value indicating complete loss of load carrying
capacity
growthf = void volume fraction growth rate of change
xv
localf = local void volume fraction, used in contrast to non-local porosity
nucleationf = void nucleation rate of change
Nf = void volume fraction at nucleation
*
uf = ultimate porosity
0f = initial void volume fraction
G = shear moduli
K = bulk moduli
fl = characteristic length.
n = unit normal to the surface of the body, strain hardening exponent
subn = number of sub-increments
= upper-bound model parameter
iq = Gurson fitting parameters
s = a comparable location to x
NS = standard deviation of the second phase Gauss distribution
= flow stress in the matrix,
= matrix flow stress rate
e = von Mises equivalent stress,
hyd = hydrostatic stress on the unit cell
hyd = macroscopic effective stress rate
y = initial yield stress,
N = critical nucleation stress value
xvi
T
hyd = hydrostatic component of the trial elastic stress tensor
T
ij = trial elastic stress tensor
0
ij = elastic stress tensor at the beginning of the increment
= weighting function
= flow potential for yield criteria
max = a user defined parameter
( )T
ij = the yield function evaluated using the total strain increment
v = Poisson‟s ratio
EW = element width
x = the point of interest in non-local integral equation
= neighbourhood submitted to non-local averaging
1
Chapter 1: Introduction
Industrial production often presents complicated problems which are
necessary to simulate in advance to predict a possible solution. Simulation also
provides a manner to improve production techniques and in turn reduce financial cost.
In recent years, hydroforming, the process of metal forming using a pressurized fluid
to conform stock material to the cross-section of a die, has been introduced into a
variety of industrial processes for producing products, including structural parts in the
automotive industry. Hydroforming has been introduced to increase production times
and reduce weight, thereby increasing vehicle efficiency (Mortimer, 2001).
Hydroforming complex parts often requires bending the stock tube before
hydroforming. This introduces changes in the strain path (Baradari, 2005) and
therefore requires capable prediction techniques.
Continuum damage-based material models such as the Gurson (1977) upper-
bound model and the Sun and Wang (1989) lower-bound model have advanced the
capabilities of modeling prediction of ductile fracture, providing more accuracy in
modeling metal forming operations like hydroforming. However, their solutions are
not accurate enough in some aspects of complex forming operations. In response to
these concerns a variety of “add-ons” or outright changes have been made to the
upper- and lower-bound models to address shortfalls in their design (Tvergaard, 1981;
Needleman, 1987; Sun and Wang, 1995; Aliabadi, 2001). These enriched models
2
become increasingly accurate and are therefore desirable when modeling an
expensive forming operation.
The dual bound model, proposed by Butcher (2006), is used to provide a
forming limit band for various sheet metal forming operations which may contain
high strain gradients. A common issue when dealing with processes that include a
high gradient of strain is mesh size and orientation dependency (César de Sá et al.,
2006), also known as mesh sensitivity. Researchers have focused on implementing
non-local or gradient theory into material models to address this problem (Reusch et
al., 2003; Tvergaard and Needleman, 1995; Leblond et al., 1994; Geers et al., 2003).
These high strain gradients are therefore of concern because mesh sensitivity may be
present in both the Gurson upper-bound and the Sun and Wang lower-bound models
which comprise the dual bound model. In this chapter, the motivation behind the
present work is introduced. In addition, the overall objectives of the study are
presented.
1.1. Motivation
Legislative restrictions and public demand for less expensive, more fuel
efficient, safer automobiles have forced the use of hydroforming in many modern
manufacturing plants in lieu of traditional stamp and weld methods.
Research into producing a more reliable material model for the simulation of
hydroforming of automotive parts has resulted in the development of a dual bound
3
model in which both the Gurson (1977) and the Sun and Wang (1989) yield criteria
are used to predict material failure and formability (Butcher, 2006).
The current behaviour of the dual bound model when simulating straight tube
hydroforming operations is unacceptable in that it does not display realistic void
volume fraction or porosity concentration and therefore is not indicative of the true
failure pressure. This unrealistic concentration of void volume fraction is a symptom
of mesh sensitivity, which is present in the combined dual bound model and will be
demonstrated later in this thesis.
1.2. Rationale for software selection
Selection of appropriate software for the execution of this research depended
on a variety of factors. Since this is first and foremost, an extension of existing
research efforts, compatibility with the material model available at the Applied
Mechanics and Manufacturing Laboratory (AMML) of UNB is of paramount
importance. Secondly, the study is aimed at, and funded directly by the automotive
industry and the Government of Canada. Prudency dictates the utilization of similar
tools to those used in the industry to reduce the implementation problems that may
occur in cross platform adaptation. These two areas support the use of LSTC‟s LS-
Dyna. LS-Dyna is currently utilized by AMML for their dual bound material model,
noted in Butcher (2006), and is a widely used program by industry for simulating a
variety of forming processes of automobile structures (Livermore Software
4
Technology Corporation, 2009; Livermore Software Technology Corporation, 2002) .
Other companies currently working with or who have worked with LS-Dyna in the
past include Toyota, General Motors, Ford, DaimlerChrysler, and Delphi Automotive
Systems (Livermore Software Technology Corporation, 2002). In addition to the
widespread use of LS-Dyna, it was important that the study used the package, partly
due to the large investment in this environment that had already taken place, and also
because of the direction of the research group in the near future.
1.3. Objectives of the work
Several results were desired at the onset of this project. The first objective was
to demonstrate mesh sensitivity in both the Gurson (1977) and Sun and Wang (1989)
models. A variety of methods were used in the literature and the one chosen will be
discussed further in Chapter 2. After the mesh sensitivity was confirmed in both
upper- and lower-bound models, the selection and implementation of a regularizing
mechanism within the confines of the user defined material subroutine in LS-Dyna
was considered. The effectiveness of this mechanism was then demonstrated through
the analysis of porosity distribution in a plane strain tension test and, finally, through
studying the differences in burst pressure and corner fill expansion between the
regularized and non-regularized simulations in straight tube hydroforming.
The remainder of the thesis is organized as follows: The upper- and lower-
bound material model is examined; a description of non-local theory is introduced
and how the theory was implemented; mesh sensitivity and solutions are displayed;
5
the straight tube hydroforming discussion is posted; and finally, the conclusions and
recommendations are presented.
6
Chapter 2: The Upper- and Lower-Bound Damage-based Material
Model
Material properties are of the highest priority when building a realistic
simulation of a metal forming operation. The upper-bound, damage-based material
model proposed by Gurson (1977) has been widely accepted to predict ductile
fracture in various forming processes (Baradari, 2005; Brunet et al., 1996; Medo et
al., 2009; Nahshon and Xue, 2009). However, the Gurson model is designed to
overestimate the formability of the material because of its upper-bound nature.
Recently, Butcher (2006) presented the idea of using a lower-bound and upper-bound
combined damage-based model to create a formability band. By running both
simulations consecutively, a solution can be created that not only gives a reasonable
prediction, but also a bounded region for which the material will most likely behave.
In this chapter, the bounds of the model are described along with the treatment of the
damage process. Finally, the implementation of the model in the finite element
package and methods for verification are discussed.
2.1. Upper-Bound Model
The Gurson model (1977) and the following Sun and Wang (1989) model are
often referred to as damage-based yield criteria. The Gurson based yield criterion
7
(Gurson, 1977) performs quite effectively as a micromechanical damage-based
model, especially when the contributions of the modifications presented in recent
literature are considered (Tvergaard, 1981; Tvergaard and Needleman, 1984). Due to
the physical basis and good prediction characteristics, the Gurson model is a popular,
well proven model (Butcher, 2006; Tvergaard, 1981; Needleman, 1987; Baradari,
2005; Sun and Wang, 1989). Together with the Sun and Wang (1989) model,
damage-based yield criteria incorporate the effects of damage or void volume
fraction on the material and are therefore suited to describing the ductile failure of
materials which have failure mechanisms governed by porosity development initiated
by the debonding or cracking of second phase particles distributed in the material.
The Gurson model (Gurson, 1977), based on a PhD dissertation from Brown
University, has a micromechanical basis for its derivation. In this model, the material
is represented by a collection of unit cells, each with a singular void, centered in the
rigid-plastic, incompressible, homogeneous von Mises material. Micromechanical
analysis was conducted representing the macroscopic mechanical behaviour of the
material by the plastic solution of the unit cell under equivalent loading. The Gurson
criterion is considered an upper-bound yield locus. The material model finds its
upper-bound nature by constructing the microscopic velocity field in the matrix
material as the solution to allow dilation in the unit cell while maintaining
incompressibility of the matrix material (Gurson, 1977). This velocity field is
required to maintain compatibility and coexistence with the surrounding unit cells and
their prescribed macroscopic rate of deformation (Gurson, 1977). Two different unit
8
cells were proposed, one is cylindrical, and the other, spherical. The cells are
presented in Figure 1. In this work, the spherical model is used.
Figure 1 Transformation of a material containing voids (a) into either a spherical unit cell (b) or a
cylindrical one (c) where a is the radius of the void and b is the cell radius (Gurson, 1977)
For a spherical void, the Gurson yield criterion is described as
22
2
32 cosh 1 0
2
hyde f f
( 1 )
where e is the von Mises equivalent stress, indicates the flow stress in the
matrix, hyd refers to the hydrostatic stress on the unit cell and f is the void volume
fraction.
A variety of modifications have been made to the original Gurson model
including the one added by Tvergaard in 1981 (Tvergaard, 1981). Tvergaard noticed
3
1
2 b
a
R
1
3
2
a R
b
θ
(a) (b) (c)
9
that bifurcation occurred at small load values and the strains were twice that predicted
by a similar numerical process. He introduced a series of fitting parameters labelled
iq where 1q =1.5,
2q =1 and 3q = 2
1q . The introduction of these parameters can be seen
in Equation 2
22
1 2 32
32 cosh 1 0
2
hyde fq q q f
( 2 )
If void coalescence is predicted by a critical void volume fraction as suggested
by Tvergaard and Needleman (1984), then f is replaced in the equation above by
*f . This new variable, known as the effective porosity, includes the accelerating
effect of void coalescence on void growth rate. With the use of this addition, the
criterion is referred to as the Gurson-Tvergaard-Needleman (GTN) yield criterion. A
more detailed explanation of this addition is included in Section 2.3.3.
The load carry capacity is displayed by the behaviour of the model under
various porosities. When the triaxiality changes and the porosity varies, the behaviour
of the yield function changes accordingly, as can be seen in Figure 2. It is worth
noting that the Gurson model will reduce to the von Mises model for a zero void
volume fraction material as shown in the diagram.
10
Figure 2 The influence of porosity and triaxiality on the Gurson yield criterion (Worswick and
Pelletier, 1998)
2.2. Lower-Bound Model
The Sun and Wang model describes a lower-bound solution similar to the
spherical unit cell described by Gurson (1977). To generate the lower-bound model,
Sun and Wang constructed a stress field which satisfies the boundary condition and
the static equilibrium of the unit cell instead of the velocity field that Gurson used
(Sun and Wang, 1989). This statically admissible stress field forms the principle
difference between the two models. The Sun and Wang yield function reads
12
22
2 2
3
3cosh
20
31 sinh
2
hyd
e
hyd
b f
b
b f
( 3 )
1.0
0.5
0.0
σe
/ σ
σhyd / σ
0.0 2.0 4.0 6.0
f = 0.1 0.025 0.01 0.0001
von Mises
11
where
112 ln
2b f
2 1 (1 ln )b f f
120 0
2 2 213
2
3 3coth sinh
2 2
mt mtbb f
b
and
0 0.65 lnmt f .
If the original Gurson model is desired, it is possible to use the Sun and Wang
model by simply setting 1b =2,
2b =1+ 2f , and 3b = 0. Both Gurson (1977) and Sun
and Wang (1995) introduced the nucleation and growth into the models where the
rate of porosity is defined as the sum of the nucleation and growth porosity rates
which are related to the effective and hydrostatic plastic strain rate, respectively.
Further reading on the topic is presented in Section 2.3.
Since, according to Sun and Wang (1995), the Gurson model overestimates
formability and the Sun and Wang model is shown to be an accurate lower-bound
(Sun and Wang, 1995), it is practical to approach industrial forming operations using
both models. This technique, proposed by Butcher (2006), can be used to define a
band of formability in which the material will reliably fail. This dual bound approach
also produces an alternative to more complicated modelling techniques. Figures 3 and
4 show the potential of using dual bound modelling for failure prediction in tube
hydroforming of advanced high strength steel.
12
Figure 3 A comparison of the dual bound model and the extended stress based forming limit curve
(see Bardelcik, 2006) for both stress and strain based nucleation (Butcher, 2009)
Figure 4 Experimental powder metallurgy tension and compression samples against the Gurson upper-
bound and Sun and Wang lower-bound (Sun and Wang, 1989)
Experiment XSFLC GT: stress SW: stress GT: strain SW strain
Effe
ctiv
e St
ress
σe
(kg/
mm
2)
6
5
4
3
2
1
0 0.1 0.2 0.3
Porosity f
Gurson
Sun and Wang
Tension Compression
0 66.5 133
Bu
rst
Pre
ssu
re (
MP
a)
End-feed load (kN)
150
130
110
90
70
50
13
2.3. Damage Development and Modelling
Ductile fracture is caused by void damage development in the material once
loading is applied to a critical level. Damage is defined as the void volume fraction in
a material. Plastic failure due to damage development in a porous material is
categorized into three major phases, none of which are required to occur
homogeneously across the entire medium. The premiere phase is void nucleation,
followed by void growth and eventually coalescence.
2.3.1. Void Nucleation
Void nucleation can occur by two different events and indicates the beginning
of the plastic failure process in ductile materials. Nucleation can be seen to occur
either through inclusion fracture or the matrix material debonding from the inclusion
surface (Faleskog and Shih, 1997), as shown in Figure 5 (Kanetake et al., 1995).
14
Figure 5 Al2O3 particles reinforcing aluminum 6061 showing (left) debonding and (right) particle
cracking under horizontal loading (Kanetake et al., 1995)
Void nucleation can be predicted using either a strain or stress-based mechanism.
Strain controlled models usually follow the same form:
nucleation Mf A ( 4 )
where M is the effective plastic strain rate in the matrix and
2
2
1exp
2 N
NM
N
N
SS
fA
( 5 )
20 μm 20 μm
15
In addition to the effective plastic strain rate, N and
NS are the average and
standard deviation of the nucleation strain, and Nf is the void nucleating particle
volume fraction (Picart et al., 1992).
Stress based void nucleation follows a similar pattern but substitutes the
nucleation strain rate with effective matrix stress rate, , and hydrostatic stress rate,
hyd (Needleman, 1987):
nucleation hydf B ( 6 )
Similar to the strain based nucleation, the coefficient is defined as
2
1exp
22
hyd NN
N yN y
fB
SS
( 7 )
where y is the initial yield stress, is the flow stress, hyd is the hydrostatic
component of the stress tensor, and N and
NS are the average and standard
deviation of the nucleation stress. These approaches have become more prominent
among the nucleation schemes since Gurson‟s dissertation was released (Picart et al.,
1992; Worswick and Pelletier, 1998; Chu and Needleman, 1980).
Void nucleation in the present model is selected to be strain controlled to
provide a direct relationship between the void-damage-induced material softening and
strain localization mechanisms. Strain controlled nucleation is also representative of
16
the behaviour of some materials, for example DP600 (Butcher et al., 2009), the
material to be used later in the thesis.
2.3.2. Void Growth
Following void nucleation is void growth, the second phase in void damage
development. Void growth has been investigated for a variety of void shapes and
dimensions. The premier work on the subject is attributed to McClintock (1968) and
his solution for a through thickness hole of circular or elliptical shape in an infinite
plate. Following that work, Rice and Tracy presented the solution for a three
dimensional spherical void in an infinite block of matrix material (1969).
Void growth rate is commonly implemented into continuum damage material
models via the following relation
3(1 ) p
growth hydf f ( 8 )
where p
hyd is the hydrostatic component of the plastic strain rate (Butcher et al.,
2009). This relation works because the matrix material is assumed to be a plastic
incompressible solid, therefore all void growth is directly related to bulk material
volume change.
17
2.3.3. Void Coalescence
Void coalescence is the third and final stage in the process of failure in a
porous medium. Coalescence is the rapid linkage of the adjacent voids separating a
localized area within the material. This process usually escalates quickly into final
catastrophic material failure on a global scale. Void coalescence can occur through
three different mechanisms: direct impingement of adjacent voids, necking of the
ligament separating voids, or shearing of the ligament among the voids (Baradari,
2005). The first of the three mechanisms is quite unlikely due to the unrealistically
large reduction in density, seen in Figure 6, and this can be confirmed through the use
of a scanning electron microscope (SEM) (Thomason, 1998).
Figure 6 A cubic unit-cell with central spherical void: The void undergoes dilatational growth until the
void touches the cell wall (Thomason, 1998).
Other mechanisms of ligament necking and shear are considered to be more
likely than dilatational mechanisms due to the high porosity levels required to match
fracture strains (Thomason, 1998). Despite the advantages in modelling the physical
phenomenon of ligament necking and shear, techniques which do so, like the plastic
limit load criterion for inter-void ligament necking, require microstructure parameters
Vf0 = 0.01 Vf = 0.05 Vf = 0.1 Vf = 0.15 Vf = 0.2 Vf = 0.25 VfU = 0.52
18
and have only recently been adapted to deal with low triaxiality and shear effects
(Butcher and Chen, 2009).
In this work a traditional approach to post-coalescence behaviour was taken.
In this approach, implemented by Tvergaard and Needleman (1984) to the Gurson
yield model, a critical porosity value, cf is used to indicate a point at which void
coalescence starts, leading to an accelerated failure and material softening, as follows
* *
f
if
( ) if
c
uc c c
c
f f f
f f ff f f f f
f f
( 9 )
where *f is an effective porosity term which replaces porosity in the yield criterion
expressed by Eq. (1), ff is the final porosity value indicating complete loss of load
carrying capacity, and *
uf is called the ultimate porosity and is usually defined as
1/1q where
1q is the fitting coefficient from Tvergaard‟s (1981) addition to the
Gurson model.
2.4. Implementation
Butcher‟s dual bound material model is executed in a very similar manner to
that which the original author used the model (2006). Both the upper- and lower-
19
bound models are introduced to the LS-Dyna v970 explicit code through a user
defined material subroutine. This scalarized code is capable of executing both models
through Worswick and Pelletier‟s sub-increment elastic predictor-normal corrector
scheme (1998). This approach provides a robust integration scheme capable of
handling high strain rate loading in terms of both significant triaxiality and large
strain rates (Worswick and Pelletier, 1998).
The integration scheme was implemented for the use of both the upper- and
lower-bounds prior to the author‟s involvement in the project. The scheme is capable
of dealing with a wide range of loading scenarios, including impulse loading where
the hydrostatic stress may become larger than the flow stress by an order of
magnitude or more. The subroutine can effectively handle this sort of loading despite
the Gurson equations becoming unstable (Worswick and Pelletier, 1998). As such, the
stability of the scheme is more than enough for tension and hydroforming simulations
(Butcher, 2006)
The scheme begins with calculating a trial elastic stress,
0T e
ij ij ijkl klC . ( 10 )
These trial stresses are evaluated with the use of the yield function to
determine if the increment is elastic or if it is elastic-plastic. The later case, elastic-
plastic, would be indicated by a flow potential larger than unity ( ( ) 1T
ij ).
20
In order to use the integration scheme in high strain rate applications,
Worswick and Pelletier recommend dividing the total strain increment into sub-
increments with the number of increments determined from the following equation:
max
( )T
ij
subn
.
( 11 )
In this statement max is a user defined parameter and ( )T
ij is the yield function
evaluated using the total strain increment.
Continuing, if the magnitude of the yield function evaluated at the trial elastic
stress, indicates that the stress state is outside of the yield surface ( ( ) 1T
ij ), it is
returned using a plastic multiplier, d , and the yield surface normal, ija . The
corrected stress state is then
T
ij ij ija d ( 12 )
The plastic multiplier cannot be obtained directly but is found using a Newton-
Raphson iteration scheme until the yield criterion is satisfied within a tolerance
acceptable to the user. The iteration scheme is defined as follows
21
1( ) ( ) /( )
I I I dd d d
d d
( 13 )
The second necessary component to the normal corrector equation (Equation 12) is
the yield surface normal, ija . According to Butcher (2006), this value is calculated
using
3( ) sinh
2
hydT T
ij ij ij hyd ij
Ga K f
( 14 )
where 1
2
q for the upper-bound and
1 ln
2 8
f for the lower-bound model. G, K
and ij represent the shear modulus, bulk modulus and Kronecker delta, respectively.
Livermore Software Technology Corporation‟s LS-Dyna v.970 and the user
defined material subroutine were executed using an Intel Xeon 64bit processor
operating the open source CentOS 5.0 operating system (Community Enterprise
Operating System Development Team, 2009).
2.5. Verification
Verification that the material model behaves as expected can be done in
several ways. Both the upper- and lower-bound models used here have been tested
22
extensively through previous research efforts over the past 4 years in AMML at UNB
(Butcher, 2006; Butcher and Chen, 2009). This makes them a very reliable base on
which to introduce the regularization techniques to be mentioned later in the thesis.
Each of the upper- and lower-bounds is comprised of formulations that revert to
easily checked material models. By constraining void volume fraction to be
negligible, the Gurson model (1977) displays the behaviour of the von Mises model
as was mentioned earlier in the upper-bound section of this chapter. The lower-bound
model is also designed to be quite accommodating for verification. The model
displays the same results as the upper-bound when the following settings are used for
the b parameters:
1 2b , 2
2 1b f , and 3 0b ( 15 )
23
Chapter 3: Non-Local Theory
3.1. Introduction
Due to the local derivation of the Gurson (1977) and Sun and Wang (1989)
models, it is important to consider the effect of mesh sensitivity whenever using these
criteria. It is often desirable to remove the mesh sensitivity effects, especially when
stress localization causes artificially high stress regions (for example: around the end
of a crack tip). Damage-induced strain softening behaviour in the material model also
leads to orientation dependency in the areas of high strain gradients (César de Sá et
al., 2006). Localization around these stress concentrators can be addressed by the
addition of a regularization mechanism. These mechanisms come in a variety of
forms; however the three we will focus on include the non-local integral type model,
the explicit gradient type model and the implicit gradient type model.
3.1.1. Non-Local Integral
Non-local formulations of elasticity can be traced to the work of Eringen and
Edelen (1972) where the contemporary non-local theory was implemented through
averaging model constituents over a spatial neighbourhood defined by a distance
constant know as the characteristic length (Peerlings et al., 2001). The first practical
24
application of non-local theory to a continuum damage model was implemented by
Bazant and Pijaudier-Cabot (1987) followed by application to the creep problem by
Saanouni et al. (1989; Brunet et al., 2004). Non-local integral style approaches follow
very similar formulation in most sources of literature:
1( ) ( ) ( ; ) ( )
( ; ) ( )localdf x df x s s x d s
s x d s
( 16 )
where x is the point of interest, s is a comparable location in the neighbour hood ,
and is a weighting function described as follows:
2
3/2 2 2
1( ; ) exp
(2 ) 2f f
x ss x
l l
( 17 )
In the equation above, fl is the characteristic length and the distance function
x s can be either the Euclidean distance or geodetic distance (Polizzotto, 2001).
Euclidean models and geodetic models approach distance by two separate
philosophies. Euclidean models calculate the shortest distance between 2 points in
space by subtracting their coordinates and therefore do not compensate for possible
absences of material between the two points in question. On the other hand, geodetic
models calculate the shortest distance between two points without intersecting the
25
boundary surface of the body (Polizzotto, 2001). They can therefore compensate for
absences of material, providing a more mechanically sound relation between the
points of interest. Since there is an increased computational expense in calculating the
geodetic function, and the likelihood of small details dividing the part within the
range of any significant weighting is low, the Euclidean function was selected as
sufficient.
Multiple controlling factors contribute to the non-local regularization
including weight function. The weight function chosen for this work was a simple
Gaussian distribution. This weight function was chosen due to its unbounded range
and incorporation in several other works (Polizzotto, 2001; Brunet et al., 2005;
Stromberg and Ristinmaa, 1996). The Gaussian distribution is computationally more
expensive for the same reasons that it is more realistic; all elements in the body
influence each other and therefore must be accounted for in the calculation, if even
just marginally weighted.
3.1.2. Explicit Gradient
Another option for regularizing mesh sensitivity is the use of an explicit
gradient model. If the non-local equation is applied to a relatively smooth field, one
can use the Taylor series expansion about the localdf term of Eq. (16) in terms of x to
provide an equation in terms of the gradient of ( )localdf x (Brunet et al., 2005). The
following equation is found after neglecting terms of fourth order and higher
(Peerlings et al., 2001):
26
2
2( ) ( ) ( )4
f
local local
ldf x df x df x
( 18 )
This Taylor series provides the link between the explicit gradient approach
and the traditional non-local integral model. It is important to note that the link
however is not rigidly enforced, as the convergence of the Taylor series expansion
isn‟t guaranteed for highly localized deformations (Peerlings et al., 2001).
Abandoning these terms in the expansion causes the disparity between the explicit
gradient model and a truly integral style non-local one (Peerlings et al., 2001).
3.1.3. Implicit Gradient
Peerlings et al. (2001) formulated yet another gradient model by use of the
Laplacian on the Taylor series expansion from the explicit formulation. This model
uses the following Helmholtz equation
2c ( 19 )
27
when applied to the local von Mises equivalent strain . In this equation, c is equal
to
2
4
fl, and represents the non-local equivalent strain. Due to the nature of
Helmholtz equations, boundary conditions must be satisfied including the following
one which states that flow of regularized strain across the boundaries of the object‟s
surface must be negligible, i.e.
0n
( 20 )
where n is the unit normal to the surface of the body. It is important to note that the
terms neglected during truncation of the Taylor series expansion of the implicit
formula are different than those omitted for that of the explicit formula. In this
difference of terms lies the defining nonlocal character which allows interaction
between finitely separated points as opposed to the explicit model (Peerlings et al.,
2001).
3.2. Implementation
The implementation of the non-local code depends on an important
relationship between the strain and porosity. This relationship is in part, due to the
integration scheme mentioned in Section 2.4. In the integration scheme, the porosity
is used to evaluate the yield function (Equation 1 or 3), which is then used to derive
the plastic multiplier, d , which subsequently moves the stress state back to the yield
28
surface. The stress state is then returned to the solver to determine the next strain
increment (see Figure 7). By this relation, limiting the localization of porosity can
limit strain localization. The approach of applying non-local theory to only the
damage parameter was introduced by Pijaudier-Cabot and Bazant (1987). This
relationship is fundamental to the remainder of the implementation process.
Figure 7 Interactions between the yield function and plastic multiplier allow regulation of the porosity
as a substitute for non-localizing the strain increment.
Limitations on the user material subroutine‟s access to strain increments,
coordinates and other important information created a unique scenario for calculating
Plastic
Multiplier
Yield
Function
Porosity
Stress State Strain
Increment
Void
Nucleation
and Growth
29
the averaging scheme. The finite element package requires the stress state for each
element to be calculated by the material routine and returned to the solver before the
solver releases the strain increments to the material routine about the next element in
the time step. This forces each element‟s stress state and porosity to be evaluated
independent of the other elements in the body unless information is carried between
the elements and time steps by a method outside of the solver, i.e. a mechanism in the
user defined material routine. The transfer of the porosities between time steps in the
model provides an ideal location for the change between local and non-local void
volume fraction to occur (Figure 8). Therefore, by changing the local porosity to the
non-local porosity, localization of the strain can be avoided.
Evaluation of the non-local porosity only occurs at the end of each time step.
This process allows all the local porosities to be known in advance, eliminating the
problem of evaluating each element independent of the others as was mentioned
above. This implementation can be seen in the flow chart in Figure 8.
30
Figure 8 Information is recorded into record files until the end of the current time step at which point
the non-local porosity for the entire model is calculated and returned for the next step.
Modification of the original dual bound material model for the purpose of
non-local calculation requires several additions. The changes can be summarized in
three separate ways: changes to the inputs of the material model, a pre-model process,
and finally a post-model process. These changes are documented in the following
paragraphs.
The original inputs for the user defined material subroutine were not sufficient
to calculate the non-local porosity, so additions were necessary. Four significant
additions to the input of the material model were made: the internal element
identifications, the internal part and node identifications, the node coordinate
locations, and the dimensions of the previous arrays for access requirements. Internal
element identifications were used throughout the program to monitor transfers to and
Previous Time Steps
T= n-1
Element 1
T= n-1
Element 2
T= n-1
Element 3
T= n
Element 1
T= n
Element 2
T= n
Element 3
Non-Local Evaluation
of All Elements
Local Record
Non-Local Record
31
from recording files and also calculate the total number of elements in the body to be
processed. Internal part and node identifications were required to access the node
coordinates and the dimensions of these arrays were subsequently necessary to read
the arrays themselves.
The pre- and post-model processes are present in the code through the use of
two “if” statements, positioned at the beginning and the end of the material
evaluation. Both “if” statements are activated by the declaration of a characteristic
length in the material input card of the LS-Dyna card deck. The first “if” statement,
called the pre-model process and seen in Figure 9, contains an element counter to
determine the total number of elements in the body. In addition to the element
counter, if the time is not equal to zero, the “if” statement overwrites the recorded
porosity from the previous time step with the recalculated non-local porosity from the
non-local calculator described below.
32
Figure 9 Schematic of the Pre-Model Process which includes the element counter and insertion of
non-local porosity
The second “if” statement, called the post-model process is located at the end
of the material model routine and calls two separate files (see Figure 10). The first file
is an element locator routine, which is called at every time step for every element.
The second file is the non-local calculator which is called only after the last element
of the time step is solved by the material routine.
Pre-Model Process
Characteristic Length > 0
Element Counter
Is t=0?
Replace f
with fnon-local
End “if” Statement
Yes No
33
Figure 10 Schematic of the Pre-Model Process which includes the element counter and insertion of
non-local porosity
The element locator file, as mentioned above, is run after each element and
contains various elements. The element locator file has a series of tasks which begin
with the detection of the element type and the extraction of the proper number of
element coordinates from LS-Dyna‟s reference array. These coordinates are used to
find the center of the element for the non-local averaging scheme. The volume or area
of the element, dependent of element type, is then calculated in an external file and
returned to be archived with the element location and porosity to a record file for the
non-local calculator. This completes the duties of the element locator file.
The final component to the non-local addition is the non-local calculator itself.
This routine is called at the end of the time step by the last element in the body. The
coordinates, volume or area, and the porosity of all of the elements solved in that time
step are retrieved from a record file. A loop is then started over all the elements. For
Post-Model Process
Characteristic Length > 0
Call
Non-local Calculator
Call
Element Locator
End “if” Statement
Yes
No
Is this the last element?
34
each element, the distance between all other elements and itself is calculated. These
distances are used in conjunction with the area or volume information to evaluate the
non-local porosity. A check is done to ensure that the porosity only increases and
finally the non-local void volume fraction is written to a record file to be inserted
during the next time step.
3.3. Verification
To verify that the non-local code works, several simple behaviours can be
observed. The nature of an averaging scheme with an adjustable neighbourhood
allows the comparison of the local results with the non-local results at a
neighbourhood of negligible size. Since the neighbourhood is too small to influence
the results, the same solution should be present for both the local and non-local codes.
In contrast, adjusting the characteristic length (and subsequently, the neighbourhood)
to one much larger than the element size should result in uniform distribution of the
void volume fraction at any given time step. Evidence of these two behaviours can be
seen in Figure 11 where a specimen of material is loaded using a uniaxial tensile
displacement. Additionally, the width of the porosity bands should increase with
increasing characteristic length. This behaviour was observed in all the trials
throughout the study.
35
Figure 11 Verification of the upper-bound non-local addition precedent to element failure, from left to
right: local banding, non-local banding with a negligible characteristic length, and non-local banding
with an exaggerated characteristic length.
f = 0.05
f = 0.02
36
Chapter 4: Mesh Sensitivity
In order to obtain correct results for a finite element simulation, it is important
that the simulation remain objective to any imaginary variables or other constructions
non-existent in the real world problem. One artificial construction, produced to
employ the finite element method, is the mesh used to divide the body into discrete
volumes. Usually, an analyst may increase the number of elements used in the mesh
until the consistent results are obtained. This is called a convergence study. However,
in some scenarios, including using a mesh sensitive material model, this approach
does not produce the desired effect.
It is well known that classical damage models introduce mesh sensitivity by
allowing void development to soften the material when strain is introduced (Cesar de
Sa et al., 2006; Tvergaard and Needleman, 1995). This mesh sensitivity is a result of
the governing equations loosing ellipticity (Baaser and Tvergaard, 2003) and
allowing strain to lose continuity and localize (Engelen et al., 2003).
In order to detect this strain localization, porosity is analyzed. Porosity is
chosen for several reasons. Firstly, the void volume fraction can be directly related to
the strain through the nucleation and growth schemes of voids presented above, and
secondly, non-localizing the porosity instead of the strain increment provides valid
solution (Pijaudier-Cabot and Bazant, 1987) and would be less computationally
expensive to extract from the existing user material subroutine. For these reasons, the
porosity will be the subject of the analysis.
37
In order to detect the mesh sensitivity using porosity, many different tests
have been developed (Reusch et al., 2003; Baaser and Tvergaard, 2003; Drabek and
Bohm, 2005; Tvergaard and Needleman, 1995; Ramaswamy and Aravas, 1998). Most
of these tests are quite similar, comprising of a single rectangular two dimensional
specimen with a prescribed uniaxial displacement. In accordance with this
convention, the test used in this work follows closely with that presented by others,
but mostly that used by Reusch et al. (2003).
The test consists of a two dimensional rectangular patch of material 10 mm
wide and 12 mm tall, seen in Figure 12. Boundary conditions are placed on the left
side in the x-direction, and rotation is also restricted in the y- and z-direction. A
displacement is prescribed to the right side of the mesh in the x-direction of 3 mm
using the velocity profile described in Figure 13. This velocity profile was chosen to
produce a continuous acceleration profile and therefore reduce the influence of
inertial effects. The patch is meshed using a plane strain shell element (type 13)
included with LS-Dyna (Livermore Software Technology Corporation, 2003). To
induce localized damage banding, a single element has been introduced with a 1%
prescribed increase in initial void volume fraction over the rest of the sample; this
element is shaded grey in Figure 12. This prescription is applied in the UMAT during
an initialization phase.
38
Figure 12 Test sample used to demonstrate mesh sensitivity, the grey element has 1% higher initial
porosity in comparison to the other elements.
0
0.5
1
1.5
2
2.5
3
3.5
0 0.5 1 1.5 2
Figure 13 Velocity profile used for the plane strain tension tests
x
y 10.0
12.0
u = 3.0 mm
Time (second)
Vel
oci
ty (
mm
/sec
on
d)
0 0.5 1 1.5 2
3.5
3
2.5
2
1.5
1
0.5
0
39
Simulation output frequency was determined using a band width convergence
study. Each simulation was directed to write an output file every 5 milliseconds to
capture banding. A convergence study was conducted on the coarse (0.5mm) and fine
(0.25mm) mesh to ensure the 5 millisecond output time step presents realistic banding
behaviour. Output frequencies of 5, 7.5, 10, 50 and 100 milliseconds were tested and
convergence was demonstrated as can be seen in the figure below. Porosity band
width was measured in millimetres for each mesh at set fringe levels: 2.3 x 10-2
, 2.6 x
10-2
, and 2.9 x 10-2
void volume fraction. Both the 50 and 100 millisecond output
frequency did not display banding before failure and therefore are not present in
Figure 14.
0
0.5
1
1.5
2
2.5
3
0.005 0.006 0.007 0.008 0.009 0.01
Figure 14 Porous band width versus output frequency: □ 0.5mm mesh at the 2.3x10-2
fringe, ◊ 0.5mm
mesh at the 2.6x10-2
fringe, ○ 0.5mm at the 2.9x10-2
fringe, ■ 0.25mm mesh at the 2.3x10-2
fringe, ◊
0.25mm mesh at the 2.6x10-2
fringe.
Output Frequency (seconds)
0.005 0.006 0.007 0.008 0.009 0.01
Ban
d W
idth
(m
m)
3
2.5
2
1.5
1
0.5
0
40
The demonstration material for the plane strain tension test was based on a
real dual phase steel, DP600. DP600 was chosen to provide a realistic basis for the
material to be used in the demonstrations. It is a material with well-known properties
to the AMML and has been used before for hydroforming simulations (Butcher,
2006; Butcher and Chen, 2009). DP600, however, is not conducive to demonstrating
porous banding and mesh sensitivity, due to its void evolution behaviour. Since real
materials differ in void behaviour, it is valid to modify the void parameters of DP600
to display the porous banding. The modification can be justified because mesh
sensitivity is a result of the strain softening mechanism in the damage-based material
model. Therefore, the material parameters cannot control whether or not mesh
sensitivity occurs, but can only influence how prominent the effect is in the
simulation. It is important to note that the plastic strain-effective stress curve used to
describe the behaviour of DP600 was also used with the demonstration material,
unmodified in any way. The volume fraction of void nucleating particles was set to
5.5%, which is the percentage of martensite in DP600, and initial porosity was set to
0.07%, as experimentally determined by Winkler et al. (2008). Ten percent of the
nucleation strain was chosen as its standard deviation, sN (Butcher et al., 2009)
whereas the nucleation strain itself was chosen to be 0.2 (Table 1) (Butcher, 2006).
Table 1 Demonstration material parameters
f0 fN fc ff εN sN
(% εN)
0.0007 0.055 0.05 0.07 0.20 10
41
The mechanical properties of DP600 used in the demonstration model are
listed in Table 2.
Table 2 Mechanical properties of the demonstration material (Bardelcik, 2006)
E (GPa) y (MPa) K (MPa) n
206 0.30 413.54 795.8 0.115
Three separate mesh sizes were created to test for mesh sensitivity. Mesh
discretizations for the plane strain tensile test included a 20, 30 and 40 element wide
sample. These samples, in order to maintain the dimensions in Figure 12, were
prescribed 24, 36, and 48 elements vertically. These selections correspond to square
elements with sides of 0.5 mm, 0.33 mm and 0.25 mm.
4.1. Mesh Sensitivity in the Upper-Bound Approach
The loss of mesh objectivity in the upper-bound model is a well known
phenomenon (Leblond et al., 1994; Tvergaard and Needleman, 1995; Tvergaard and
Needleman, 1997); however, submitting the model to the test provides verification
that the approach is capable of detecting the spurious mesh sensitivity of the upper-
bound material. The non-local theory and the Gurson (1977) material model were
addressed by Leblond, Perrin, and Devaux (1994) with the first integrated non-local
42
Gurson model. Tvergaard and Needleman (1995) also produced a non-local version
of the Gurson model. Since then, a variety of non-local and gradient enhanced Gurson
models have been developed, attempting to limit the strain localization resulting from
the effects of the strain softening mechanism adhered to in damage-based material
models. The production of a localized result on a variety of meshes was therefore
expected from the upper-bound approach and, as mentioned above, was used to verify
that the test was capable of displaying mesh sensitivity.
Three separate mesh discretizations were constructed with equal element
horizontal and vertical side lengths of 0.5 mm, 0.33 mm and 0.25 mm to demonstrate
the mesh sensitivity. Each mesh size produced remarkably different band sizes in the
time step immediately preceding the first element‟s failure. The results from this
series can be seen in Table 3 where the band widths are displayed at various porosity
fringe levels and Figure 15.
Table 3 Porous band width for the local upper-bound solution
Element Size
(mm)
Band Width at Indicated Porosity Fringe Level (mm)
0.023 0.026 0.029
0.5
0.33
0.25
2.59
1.85
0.78
1.73
1.18
0.39
1.00
0.66
-
43
Figure 15 Mesh sensitivity demonstrated in the upper-bound model through the use of a 0.5mm,
0.33mm and 0.25mm element size mesh under horizontal tensile loading
4.2. Upper-Bound Non-Local Solution
The upper-bound non-local solution behaved in the expected manner when
contrasted with the upper-bound local solution. The band width became much more
consistent under regularization as can be seen in Figure 16. By implementing the non-
local code, a significant change in porosity localization was seen.
Figure 16 Mesh sensitivity regularization in the upper-bound model through the use of a non-local
model with characteristic lengths of 0.19, 0.17, and 0.15 mm for the coarse, medium and fine meshes
f = 0.05
f = 0.05
f = 0.02
f = 0.02
44
Mesh regularization can be performed at various band widths aside from those
shown above. Another set of characteristic lengths were used to moderate the
localized porosity bands to a significantly different width. Characteristic lengths were
determined by trial-and-error. Band widths for both sets of characteristic lengths can
be referenced in Table 4, where the width at a variety of porosity fringe levels is
recorded and shows effective regularization by the non-local method. The porosity
fringe levels of the second trial can be seen in Figure 17. These results confirm that
the process is capable of both detecting and regularizing void volume fraction
localization.
Table 4 Porous band width for the non-local upper-bound solution
Characteristic
Length
(mm)
Element Size
and
(mm)
Band Width at Indicated Porosity Fringe Level
(mm)
0.023 0.026 0.029 0.032 0.035
0.1900
0.1700
0.1500
0.5
0.33
0.25
3.19
-
-
2.22
2.17
2.37
1.52
1.55
1.33
0.82
0.87
0.75
-
-
-
0.2125
0.1913
0.1750
0.5
0.33
0.25
-
-
-
-
-
-
-
-
-
2.22
2.51
2.45
1.23
1.35
1.35
45
Figure 17 Mesh sensitivity regularization in the upper-bound model through the use of a non-local
model with characteristic lengths of 0.2125, 0.1913, and 0.1750 mm for the coarse, medium and fine
meshes
4.3. Mesh Sensitivity in the Lower-Bound Approach
The lower-bound approach to determining mesh sensitivity followed the same
procedure that was performed for the upper-bound model in Section 4.1. Since the
Sun and Wang (1989) model is a relatively unknown or at least much less used model
in comparison to the Gurson (1977) model, its mesh sensitivity had not yet been
explored. The tensile test described above was performed again with the previously
mentioned 20, 30 and 40 element wide meshes. Loading was also prescribed in terms
of a horizontal displacement control. All of these factors were identical to those
prescribed to the upper-bound model during the previous tests.
The results of the simulations varied in some ways from those found by the
upper-bound model. It is interesting to note the general width of the localized porosity
bands was wider than those discovered through the upper-bound solution. This is due
to the softer nature of the lower-bound solution and its propensity to promote a larger,
more distributed porous zone. Uneven banding was found across the mesh sizes as
f = 0.05
f = 0.02
46
can be seen in Figure 18. Significant differences in the magnitude and spatial area of
the porosity could be seen in the zones of low porosity (point A in Figure 18)
indicating that the band width in the coarse mesh was much larger than that of the
fine. Also, the porosity gradient between the low (point A) and high (point B)
porosity zones, became much larger as the mesh was refined due to the increase in
spatial area of the zones and difference in void volume fraction. Measurements of the
porosity bands can be seen in Table 5 at various fringe levels.
Figure 18 Mesh sensitivity demonstrated in the lower-bound model through the use of a 0.5mm,
0.33mm and 0.25mm element size mesh under horizontal tensile loading. Point A clearly shows lower
concentrations of porosity at smaller element sizes, whereas point B shows the opposite behaviour.
Table 5 Porous band width for the local lower-bound solution
Element Size
(mm)
Band Width at Indicated Porosity Fringe Level (mm)
0.023 0.026 0.029 0.032
0.5
0.33
0.25
-
-
3.32
-
2.90
2.45
3.25
2.01
1.82
2.03
0.84
1.25
f = 0.05
f = 0.02
A
B
47
4.4. Lower-Bound Non-Local Solution
Non-local processing of the lower-bound solution was completed with two
separate sets of characteristic lengths similar to that which occurred with the upper-
bound. A trial-and-error approach to finding the characteristic lengths was used.
Characteristic lengths were determined by matching the porosity distribution across
the element sizes. Lengths below a quarter of the element dimension were ineffective
in altering band formation. Two separate characteristic length sets were used to
ensure localization can be regulated at a desired width, consistent with that of the
actual material. The first set of lengths can be seen in Figure 19. A table containing
the width of the porosity bands measured at various porosity fringe levels can be seen
in Table 6.
Figure 19 Mesh sensitivity regulated in the lower-bound model through the use of a variety of
characteristic lengths: 0.16, 0.1575, and 0.1450 mm for the coarse, medium, and fine meshes,
respectively. Point C is a distinct porous band which displays consistency in dimension and magnitude
across all meshes.
f = 0.05
f = 0.02
C
48
Table 6 Porous band width for the non-local lower-bound solution
Characteristic
Length
(mm)
Element Size
and
(mm)
Band Width at Indicated Porosity Fringe Level
(mm)
0.029 0.032 0.035 0.038
0.1600
0.1575
0.1450
0.5
0.33
0.25
3.68
3.79
3.78
2.53
2.67
2.59
1.44
1.52
1.49
-
-
-
0.1900
0.1650
0.1500
0.5
0.33
0.25
-
-
-
-
-
-
-
-
-
0.79
0.99
0.95
All three meshes displayed consistent gradients between the low and high
porosity regions once the non-local code had been utilized. Also to note is the band
width of the diagonal band (see point C on Figure 19) between the highest region, and
the moderate areas on the upper edge. The second set of characteristic lengths
displayed similarly positive results. Bands can be compared by examining the results
using a second set of slightly different characteristic lengths as shown in Figure 20
and above in Table 6.
49
Figure 20 Mesh sensitivity regulated in the lower-bound model through the use of a second set of
characteristic lengths: 0.19, 0.1650, and 0.15 mm for the 0.5mm, 0.33mm, and 0.25mm element size
samples, respectively. Point D is a region of high porosity, and E a region of moderate porosity on the
upper and lower edges of the sample.
This second set displays slightly more variation in the regions of high porosity
(point D in Figure 20) due to the trial and error approach to characteristic length
selection. This is contrasted by consistent diagonal band width and related porosity
gradients in the region between the highest porosity (point D), and the moderate areas
(point E in Figure 20) on the upper and lower edges.
4.5. Characteristic Length
Much research is left to be done concerning characteristic lengths in the non-
local theory. Often parameters are guided by intuition because a definition of a
correct non-local formulation has yet to be achieved (Jirasek and Marfia, 2005). Some
authors consider it a material parameter (Drabek and Bohm, 2005), and others believe
it is dependent on loading or environmental factors (Baaser and Tvergaard, 2003).
f = 0.05
f = 0.02 D
E
50
Various meshes studied to produce the uniform band widths that were seen in
the upper- and lower-bound non-local solution sections provided some insight into
the range of the characteristic length that would provide reasonable results. A plot of
the characteristic lengths and their effectiveness can be seen in Figure 21. Some
conclusions that could be drawn from this figure include the ineffectiveness of the
characteristic length at a range less than a quarter of the dimension of the element size
as was mentioned above. Certain characteristic lengths become ineffective with non-
comparable mesh sizes. A lower limit was expected as the weighting of the points
separated by more than 4 times the characteristic length would be negligible,
effectively rendering the calculation a local one. In addition, it is also evident that the
effective upper limit of the characteristic length increases with the increase in number
of elements in the model. These two behaviours combine to create the valid zones
seen in Figure 21.
Characteristic lengths that provided similar porosity distributions showed
distinct behaviours. A plot of the characteristic lengths that produced similar porosity
distributions shows an upward trend when the element size was increased (Figure 22).
Also, similar shaped curves are present in all groups to the exception of Sun and
Wang Group 2 curve. This group required a smaller length for the medium element
size sample than the Gurson Group 1 which shared the characteristic length values of
both the smallest and largest element sizes. This could be attributed to the difference
in porosity zone development between the two material models.
51
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Effe
ctiv
e Z
on
e in
Un
its
of
(Ch
arac
teri
stic
Le
ngt
h /
Ele
me
nt
Size
)
1mm 0.5mm 0.25mm
Element Size
Figure 21 Regions of effective characteristic lengths for various element sizes
Figure 22 Characteristic length sets based on uniform porosity distribution: ◊ Gurson Group 1, □
Gurson Group 2, ○ Sun and Wang Group 1, and Δ Sun and Wang Group 2
0.15
0.14
0.17
0.16
0.18
0.19
0.20
0.21
0.22
0.22 0.27 0.32 0.37 0.42 0.47 0.52
Element Size (mm)
Ch
arac
teri
stic
Len
gth
(m
m)
0.18
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Effe
ctiv
e Zo
ne
in U
nit
s o
f (C
har
acte
rist
ic L
engt
h/
Elem
ent
Size
)
1mm 0.5mm 0.25mm
Element Size
52
Chapter 5: Straight Tube Hydroforming
5.1. Introduction
Hydroforming is becoming a common occurrence in the production of
automobiles. The procedure is a metal forming process that uses pressurized fluid to
deform the blank into a die cavity. This technology has had large advances in the past
years, both in the field and in the laboratory. An excellent example of this is Opel‟s
Astra. In this vehicle, a 30% decrease in the weight of the sub-frame assembly was
achieved through hydroforming (Mortimer, 2001). Nowadays, hydroforming
technology has been used in many automotive applications including exhaust
systems, drive shafts, axle assemblies, substructures, seat frames and other
components.
Although implemented relatively recently in the automotive industry,
hydroforming is not a new technique. It can be traced back to World War II when
German manufacturers used it to produce aircraft components (Zhang, 1999).
Tubular hydroforming specifically, can be attributed to the work of Grey et al. when
advances in copper fittings were needed for domestic sanitation (1939).
In first half of the 20th
century, hydroforming was just beginning; future works
would develop a variety of methods. Research in both thick and thin walled cylinders
continued throughout the 1950‟s and 1960‟s with the first report of a formal process
for forming parts reported by Fuchs in April of 1966 (Koc and Altan, 2001). The
53
1970‟s saw more research into pressure media with investigations into various
materials and wall thicknesses. The 1970‟s also saw the first attempt to incorporate
Hill‟s theory of anisotropy into modelling of hydroforming tubes (Koc and Altan,
2001). End feed, a method of applying force to the ends of the work piece, was
introduced in the 1980‟s and allowed the introduction of more advanced concepts and
methods like segmented dies for axisymmetric parts (Dohmann and Hartl, 1997). All
these activities prompted a variety of conferences and seminars. A collection of these
gatherings has been active since at least 1996 in the US and 1997 in Germany (Zhang,
1999).
Hydroforming is a versatile procedure and can be presented in tube, flat sheet,
hydromechanical deep drawing and other formats. Tube hydroforming can be
subdivided into straight tube and prebent tube hydroforming with straight tube
hydroforming being possible with and without an external die.
The simplest form of tube hydroforming comes in the form of straight tube
hydroforming without a die. Also called bulge testing, burst testing, bulge forming of
tubes, free forming or liquid bulge forming (Koc and Altan, 2001), straight tube die-
less hydroforming is used to predict material behaviour (Asnafi, 1999; Asnafi and
Skogsgårdh, 2000; Tirosh et al., 1996). The method has been used since the 1970‟s to
investigate the different fluid media possible to apply the pressure while
hydroforming (Koc and Altan, 2001). This method has maintained a large grip on the
research community as an effective tool. A diagram of the free-form hydroforming
process can be seen in Figure 23.
54
Figure 23 Free forming or bulge forming allows the material to deform without an exterior die under
pressure from an internal liquid (Goodarzi et al., 2005)
Carleer et al. (2000) investigated the effect of different material parameters
and constructed an analytical model to rank the suitability of steels to hydroforming
using bulge forming. These material parameters can have significant effects. For
example, the effect of normal anisotropy parameter of sheet metal on hydroforming
can be seen in Figure 24.
55
Figure 24 The effect of different anisotropy parameters can be significant on hydroforming operations
Carleer et al. (2000).
Straight tube hydroforming is characterized by the use of an exterior die but is
otherwise similar to bulge forming. Hydroforming with a die provides not only a
research tool but also a method for producing products. An excellent example is the
Mercedes Benz produced cam shaft with a 20 second cycle time (Zhang, 1999). The
process adds tribological effects between the tube and die and more complex loading
paths to the otherwise similar bulge hydroforming.
A variety of investigations in failure due to bursting in both un-restrained and
square die tubular hydroforming have been performed (Chow and Yang, 2002;
Manabe and Amino, 2002). Failure in straight tube hydroforming is categorized into 4
major groups: wrinkling, buckling, folding and bursting (Figure 25). The prediction
of these defects in straight tube hydroforming was studied using Hill‟s quadratic yield
criterion with a power law hardening rule by Kim and Kim (2002). The result of the
R = 1 R = 3
56
model was compared with experimental results with varying success. The wrinkling
limit was predicted accurately; however the bursting limit overestimated the
capabilities of the material. The work provides insight into how various processes can
be used to predict these failures. Once the failure phenomenon is understood for
straight tube hydroforming, more complicated hydroforming processes can be
analyzed.
Figure 25 Failure mechanisms in tubular hydroforming include (from left to right) buckling,
wrinkling, bursting, and folding.
The corner fill expansion (CFE) is often used as a measure of formability
when evaluating straight tube hydroforming. It is the percentage of decrease in the
distance from the corner of a square die to the surface of the tube (see Figure 26).
Corner fill expansion is calculated as follows (Bardelcik, 2006):
57
%CFE 100 10016.14 mm
A A
B ( 21 )
It is worth noting that the value of 16.14 mm for B was selected based on the
straight tube hydroforming process studied in the present work. Geometry for the
remainder of the setup can be seen in Section 5.2.
Corner fill expansion has a variety of influences. Kridli et al. (2003)
investigated the effect of the strain hardening coefficient (n) on corner fill expansion.
They discovered materials with higher strain hardening coefficients formed to a
higher corner fill expansion because of their higher strain capability at any given
interior tube pressure. Bardelcik (2006) investigated the effects of increasing friction
on the corner fill expansion of pre-bent hydroformed DP600 tubes. Results showed
similar CFE for the corner located on the inside of the bend for both raised and
nominal friction levels. The corner located on the outside of the bend behaved in a
different manner by displaying a decrease in CFE for the raised friction simulation.
Both the strain hardening coefficient and friction can change the corner fill expansion
at appreciable levels if not monitored and controlled properly.
58
Figure 26 Corner fill expansion is the percentage that the tube expands into the die. The distance the
pipe moves (A) is divided by the original distance from the pipe to the die corner (B) to obtain the
percentage.
5.2. Experiment Dimensions and Modelling Approach
The straight tube hydroforming simulations were designed to predict the
results presented in Bardelcik (2006). These experiments were composed of a single
straight tube of DP600, placed within a square cross-section die, and filled with a
pressurized fluid until the tube ruptured.
The DP600 tube placed inside the hydroforming die is replicated in the finite
element model through several dimensions. The real tubes that were processed in the
hydroforming die were 1016 mm (40 inches) long and had a radius of 38.1 mm (1.5
inches). The tubes were manufactured by Stelco and “tubed” by Nova Tube Ontario
using roll forming (Bardelcik, 2006). The average thickness of the tube walls were
1.85 mm. These dimensions were replicated in the finite element model to every
detail except the length which was shortened to an arbitrary small amount, 1.963 mm.
This dimension was used to simplify the modelling procedure and is largely
59
irrelevant, aside from being of a small value, due to the use of a pressure surface to
load the tube in the simulation, and not a loading scheme that would be dependent on
the length, for example: a series of point loads.
The dimensions of the die were also reproduced in the cross-section. The
dimensions of the experimental die can be found in Figure 27 as they appear in
Bardelcik (2006). This die allows a 0.5 mm (0.02 inch) diametrical clearance between
the outer surface of the tube and the die wall. The dimensions of the finite element die
model are designed to reproduce the center, square, section of the die which is 431.8
mm (17 inches) long and is described by the dimensions present in Figure 28.
Figure 27 The straight tube hydroforming die used by Bardelcik (2006)
431.8mm [17”]
50.8mm [2”]
292.1mm [11.5”]
60
Figure 28 The square cross-section of the straight tube hydroforming die (Bardelcik, 2006)
The pressure schedule for the experiments followed a simple linear pattern.
The loading sequence was ranged from a minimum pressure of 0 MPa to a maximum
pressure of 153.7 MPa (22,293 psi). This is replicated in the simulation by increasing
the pressure to the experimental maximum (153.7 MPa) over 56 seconds as can be
seen in Figure 29.
0
153.7
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56
Time (seconds)
Tub
e P
ress
uri
zati
on
Figure 29 Internal pressure loading schedule
76.701mm
3.125mm
Symmetry Plane
Time (seconds)
Tub
e P
ress
ure
61
The modelling of a hydroforming process can be simplified in many ways,
especially, if the process is straight tube hydroforming, which is the case in this study.
Because of the computational expense of evaluating the non-local code, all efforts
were made to reduce the number of elements to the minimum amount possible. A
look at other hydroforming works show the 80,000 brick element, 1/16th
symmetry
model to be a popular approach (Butcher, 2006; Butcher et al., 2009; Bardelcik,
2006). This model would take extremely long periods to calculate in the current non-
local environment, upwards of 10 weeks by initial approximations. In order to reduce
this computation time, the model was treated as a two-dimensional sliver of the tube
as can be seen in Figure 30. The disadvantage of treating the simulation as a two
dimensional process (eliminating the possibility of studying end-feed) is negated
when the reduced number of computations in the smaller model is considered. By
simplifying the simulating of straight tube hydroforming, the benefit to the model
efficiency is prevalent.
62
Figure 30 One eighth, 2-dimensional sliver model for straight tube hydroforming with 8-node brick
elements
The sliver model is composed of three separate entities, the tube, the die and a
pressure surface. The tube is made of 8-node, constant stress, brick elements and was
assigned the dual bound material model with DP600 material characteristics. The die
consists of Belytschko-Tsay (Hallquist, 1998) shell elements and was assigned a built
in rigid material supplied with LS-Dyna (Livermore Software Technology
Corporation, 2003). In order to apply load to the interior surface of the tube, a null
mesh layer is necessary. This layer consists of fully integrated shell elements with a
power law plasticity material routine. The thickness of both shell layers for the die
and the inner pressure is 0.1 micrometers. The thickness is irrelevant for the die
because only one side will be used and it is a rigid body. However, for the null mesh,
63
the very small thickness effectively eliminates any influence on the tube‟s behaviour
(Dyment, 2004).
Boundary conditions reflect the requirements of a 1/8th
symmetrical sliver of
the tube. Two separate symmetry planes were defined, one vertical and one at a 45
degree angle. The first set of boundary conditions were assigned to a coordinate
system along the 45 degree plane. The coordinate system was oriented so the x-axis
was radial, z-axis was tangential and y-axis was normal to the 2-dimensional
representation of the tube. Restrictions on displacement were set in the y-, and z-
directions with rotations restricted in the x-direction. The coordinate system at the
single node constraint locations is illustrated in Figure 31.
Figure 31 Secondary coordinate system for boundary conditions on the 45 degree symmetry plane, the
null mesh (brown) is clearly visible under the tube elements (red)
X
Y
Z
64
A second set of boundary conditions was placed in conjunction with the
vertical symmetry plane. This set of boundary conditions was defined with respect to
the global coordinate system and as such was restricted from displacement in the x-
and z-directions and from rotations in the y-direction.
A third set of boundary conditions was placed on the remaining nodes (all
nodes except for the nodes with constraints already applied to them) to maintain
symmetry conditions in the z-direction. This set of nodes was only restricted from
displacement in the z-axis. Each set is displayed by a different colour in Figure 32:
the first set indicated in green, the second in red and the third in blue. Recording of
the output data occurred every 5 milliseconds as was the case for the plane strain
tensile test. This output frequency was validated through the use of a convergence
study. The results of this study may be found in Table 7.
65
Figure 32 Regions of various boundary conditions: the green (1) and red (2) sets are constrained from
displacement in the tangential and normal directions and from rotation in the radial direction; the blue
(3) set is constrained from displacement in the normal direction
Table 7 Convergence study of the output time frequency in the hydroforming model
Number of
Tangential
Elements
Zone Width at the Indicated Output Frequency (mm)
0.005 seconds 0.01 seconds 0.05 seconds 0.1 seconds
30
150
10.90
6.91
10.78
6.80
10.84
6.66
10.77
6.86
The contact algorithms for the model use a penalty based approach that comes
packaged in the solver. Tube and die interactions were controlled by the
*CONTACT_SURFACE_TO_SURFACE card with parameters discovered
experimentally in Bardelcik (2006). The coefficient of friction between the tube and
(1)
(2)
(3)
66
the die was determined using a twist compression test (Bardelcik, 2006) and the
values for both surface pairs are shown in Table 8. The coefficient of friction between
the null mesh and the tube was defined to be a very small value to reduce any
influence the mesh would have on the tube‟s behaviour.
Table 8 Coefficient of friction for contact between the tube, die and null mesh (Bardelcik 2006)
Contact Surfaces Contact Algorithm Coefficient of
Friction
Tube – Die
Tube – Null Mesh
SURFACE_TO_SURFACE
AUTOMATIC_SURFACE_TO_SURFACE
0.035
0.0001
Applying pressure to the inside surface of the tube was accomplished through
an interior null mesh and a pre-defined pressure schedule. A shell mesh with
negligible thickness (0.1 micrometers) was loaded using a shell mesh loading scheme
and the pressure schedule indicated in Figure 29. Kinematic energy was compared to
the internal energy during the simulations to ensure the simulation performed as
expected.
Mesh discretization for the hydroforming procedure was defined by varying
the number of elements in the tangential direction. The number of elements varies
from 30 to 150 in the 1/8th
cross section model and was limited by computation
efficiency. These dimensions provide element widths between 0.97, and 0.19
millimetres. The number of elements in the radial direction was restricted to 4
67
elements, each 0.46 mm tall, representing the convention of the other 80,000 node
hydroforming models. Mesh discretizations can be seen in Table 9.
Table 9 Tubular hydroforming tangential mesh discretizations and aspect ratios.
Number of
Tangential
Elements
Total Number of
Elements
Element Width
(mm)
Element Aspect
Ratio
(Width/Height)
30
40
60
100
150
120
160
240
400
600
0.973
0.730
0.487
0.292
0.195
2.10
1.58
1.05
0.631
0.421
5.3. Results and Validation
5.3.1. Porous Zone
The straight tube hydroforming model was evaluated and the zone of porous
concentration was compared. The porous zone is a defined region of porosity
concentration that develops in the simulations at failure. Simulations of straight tube
hydroforming were performed for both the upper- and lower-bound models using the
local and non-local versions of the code. Both of the local versions (the upper- and
68
lower-bounds) displayed mesh sensitivity when the porosity of the model was
displayed on a similar scale and can be measured through the dimensions of the
porous concentration zone. Figure 33 displays the upper-bound model at imminent
failure where the differences in porosity zone width are evident. The zone width and
height was measured at the 0.148 % porosity contour line for the lower-bound and
0.104 % for the upper-bound at the tube‟s inner surface. These values were used
because they describe the lowest value that can decisively define the porous zone
when 10 fringe levels are used at a fixed scale between 0.5% and 0.06% porosity.
Notice that the height of the porous zone was only marginally affected as the mesh
discretization in the radial direction was held constant across the various meshes
(only a 0.12 mm difference between the finest and coarsest mesh is present).
Figure 33 Porous zone width for the upper-bound model, top: 30 element perimeter local model,
bottom: 150 element perimeter local model
The width of the porous zone changed significantly when the number of
elements was increased for both the upper- and lower-bounds. It‟s also clear from
Porous Zone Width
f = 0.0006
f = 0.005
Porous Zone Width
69
Table 10 that the amount that the zone width was reduced was quite similar between
the models (about 36%) for similar decreases in element width for both models. This
36% decrease in zone width over the entire set of meshes was produced by an 80%
reduction in element width. It is noted that a large change in element width is
necessary to produce a small change in porous zone width, which was not the case for
the plane strain tensile test.
Table 10 Porosity zone dimensions for upper- and lower-bound local models
Number of
Tangential
Elements
Initial
Element
Width EW
(mm)
Upper-Bound Porous
Zone
Lower-Bound Porous
Zone
Height
(mm)
Width
(mm)
Width
(mm)
30
40
60
100
150
0.973
0.730
0.487
0.292
0.195
1.37
-
-
-
1.49
10.9
9.37
8.13
7.27
6.91
10.8
9.55
7.96
7.15
6.86
The non-local code was implemented and the characteristic length, fl , was
determined from moderating the porous zone width in a similar fashion that porosity
band width was used in Chapter 4. The widths of the zones are described in Table 11
and were determined by using the local 30 element zone width as a reference. A
70
maximum deviation of 9.8 percent was obtained through trial and error for both the
upper- and lower-bounds.
Table 11 Porosity zone dimensions for upper- and lower-bound non-local models
Number of
Tangential
Elements
Initial Element
Width EW (mm)
Upper-Bound
Porous Zone
Width
Lower-Bound
Porous Zone
Width
30 (Local)
40
60
100
150
0.973
0.730
0.487
0.292
0.195
10.9
11.4
10.9
10.1
9.83
10.8
10.2
9.92
10.2
9.83
The characteristic lengths, determined through the process, decreased as the
element width was reduced and the number of elements was subsequently increased.
A comparison of the characteristic lengths and element widths are presented in Table
12. These relative values do not remain constant over all the meshes but rather
increase with the number of elements.
71
Table 12 Characteristic lengths used for the non-local calculations
Number of
Tangential
Elements
Upper-Bound Porosity Zone Lower-Bound Porosity Zone
Characteristic
Length
fl (mm)
f El W
Characteristic
Length
fl (mm)
f El W
30
40
60
100
150
Local
0.225
0.198
0.168
0.140
-
0.308
0.407
0.575
0.718
Local
0.225
0.204
0.178
0.140
-
0.308
0.419
0.610
0.718
5.3.2. Burst Pressure and Corner Fill Expansion
With the characteristic lengths determined, a comparison of the local and non-
local models can be prepared in terms of burst pressure and corner fill expansion. The
corner fill expansion (CFE) and burst pressure was predicted for both the local upper-
and lower-bounds and can be found in Table 13. In a similar fashion to the porous
zone distribution, the predicted burst pressure and corner fill expansion represent
good approximations with a smaller mesh discretization. However, when the number
of elements increases, the deviation between the CFE and burst pressure predictions
from the experimental values increases too.
72
An experimental value for the burst pressure of DP600 for straight tube
hydroforming was determined by Bardelcik (2006). The value was determined at
100% of the burst pressure, representing the pressure inside the tube immediately
before failure. A comparison of the 30 element wide mesh and the experimental
values shows a 2.6 MPa difference in burst pressure for the upper-bound, and a 4
MPa difference for the lower-bound. This difference is increased to 12.4 MPa for the
upper-bound and 12.6 MPa for the lower-bound when the experiment is compared to
the 150 element wide mesh. This variation represents a 17.8 and 18 percent deviation
respectively.
Corner fill expansion for the straight tube hydroforming of DP600 was also
determined by Bardelcik (2006). The value was determined at 90% of the burst
pressure‟s value. Corner fill expansion is simulated closely using the 30 element wide
mesh with a 2 percent difference using the upper-bound model from the experimental
result and negligible difference using the lower-bound. This error expanded to 19
percent and 16.5 percent for the upper- and lower-bounds, respectively when the 150
element wide mesh was employed. All of the values for corner fill expansion can be
found in Table 13.
73
Table 13 Burst pressure and corner fill expansion for upper- and lower-bound local models
Number of
Tangential
Elements
Upper-Bound Lower-Bound
Burst
Pressure
(MPa) [psi]
90 %
Interrupted
CFE
(mm) [%]
Burst
Pressure
(MPa) [psi]
90 %
Interrupted
CFE (mm) [%]
30
40
60
100
150
67.0 [9713]
63.2 [9164]
59.9 [8682]
57.9 [8392]
57.2 [8299]
7.57 [46.9]
6.97 [43.1]
6.52 [40.4]
6.28 [38.9]
6.03 [37.3]
65.6 [9516]
62.1 [9000]
58.8 [8521]
57.0 [8260]
57.0 [8268]
7.40 [45.9]
6.93 [42.9]
6.43 [39.9]
6.14 [38.0]
6.03 [37.4]
Experiment 69.6 [10,092] 7.42 [46] 69.6 [10,092] 7.42 [46]
Implementation of the non-local code allowed for some decrease in error in
both the burst pressure and corner fill expansion predictions. Since the non-local
models were calibrated with respect to the porous zone width of the 30 element mesh,
comparisons of the deviation between the experiment and the 30 element mesh for
both the CFE and burst pressure will be identical to that of the local models.
74
Burst pressures for the upper- and lower-bound non-local models using a
larger number of elements seem slightly better than those associated with the local
versions. The upper-bound now reports a 17.1 percent error with respect to the
experimental values for burst pressure, and the lower-bound reports a 17.5 percent
error. This is a smaller amount of error than the local version of the code reported
(17.8 for the upper-bound and 18 percent for the lower) for burst pressure. Each mesh
size also reported the same or better values than the local models across both
categories.
Corner fill expansion was also improved across all categories. CFE was
presented with a 15 percent error from the worst upper-bound mesh and a maximum
of 16.5 percent error was reported from the lower-bound when compared to the
experimental datum. These values are much less than the 19 and 16.5 percent errors
reported in the local versions of the upper- and lower-bound codes. Table 14 contains
the burst pressure and corner fill expansion for the upper- and lower-bound non-local
models.
75
Table 14 Burst pressure and corner fill expansion for upper- and lower-bound non-local models
Number of
Tangential
Elements
Upper-Bound Lower-Bound
Burst
Pressure
(MPa) [psi]
90 %
Interrupted
CFE
(mm) [%]
Burst
Pressure
(MPa) [psi]
90 %
Interrupted
CFE (mm) [%]
30 (Local)
40
60
100
150
67.0 [9713]
63.2 [9173]
60.0 [8704]
58.1 [8429]
57.7 [8369]
7.57 [46.9]
7.07 [43.8]
6.60 [40.9]
6.31 [39.1]
6.25 [38.7]
65.6 [9516]
62.8 [9109]
59.5 [8636]
57.8 [8379]
57.4 [8320]
7.40 [45.9]
7.05 [43.7]
6.56 [40.6]
6.28 [38.9]
6.21 [38.4]
Experiment 69.6 [10,092] 7.42 [46] 69.6 [10,092] 7.42 [46]
The variations are higher for corner fill expansion and burst pressure than the
variation between the porous zone width in the 30 element local mesh, and the other
non-local meshes. Since the non-local meshes have fitted porous zone widths through
the adjustment of the characteristic length, an investigation into why the magnitude of
error is larger in the burst pressure and CFE than the porous zone width is found in
the following section.
76
5.3.3. Mesh Objective Material Model Study
Burst pressure and 90% corner fill expansion both showed signs of
improvement when the non-local code was implemented, but the change was less
than the fitted porosity distribution variation. This difference can be studied using a
von Mises plasticity model. The von Mises plasticity model does not contain a strain
softening mechanism, due to the lack of a strain softening mechanism like void
nucleation and growth, and is therefore not susceptible to mesh sensitivity. In order to
examine the error found in the burst pressure and corner fill expansion predictions, a
small series of simulations were performed. A von Mises plasticity model can be
attained by setting the porosity in the Gurson (1977) upper-bound model to 0 and
restricting any void nucleation from occurring. By using the alternative model, a
distinction can be made between mesh sensitivity and any other source of error, like
discretization error.
A relevant point of comparison was needed between the two simulations to
determine if the error present in the dual bound model is a result of mesh sensitivity
or other sources. To determine this point, the von Mises equivalent stress was output
for both the upper- and lower-bound simulations just before failure. They show
different values which indicate that the calculation of the von Mises equivalent stress
is dependent on the solution provided from the material model (see Table 15). This is
expected from the formulation of the von Mises equivalent stress seen below (Chen,
2008):
77
2 2 2 2 2 2
11 22 22 33 33 11 12 23 31
16
2e ( 22 )
where e is the von Mises equivalent stress, and ij represents the
components of the stress tensor.
Table 15 Von Mises equivalent stress at various pressures for both upper- and lower-bound solutions
Internal
Pressure
Number of
Tangential
Elements
von Mises Equivalent Stress at Indicated Pressure
Upper-Bound Local
Solution (MPa) [psi]
Lower-Bound Local
Solution (MPa) [psi]
30 Burst
150 Burst
150 Burst
30
30
150
718.7 (@67.0 [9713])
696.9 (@57.2 [8298])
718.7 (@57.2 [8298])
713.8 (@65.6 [9516])
693.1 (@56.3 [8168])
713.8 (@56.3 [8168])
A comparable internal pressure in the von Mises plasticity model can be
determined through the von Mises equivalent stress. By examining Table 15, it is
evident that the equivalent tensile strength at failure predicted by the dual bound
model is approximately 90% of the ultimate tensile strength (UTS = 795.8MPa) of
DP600 (Bardelcik, 2006). It is important to note that using 90%UTS instead of
100%UTS is a justified approach because the objective of the exercise is to compare
the results with the amount of error found in the von Mises plasticity model at the
78
final loading capacity predicted by the dual bound model, not the final loading
capacity predicted by the von Mises plasticity model. The 90%UTS value can then be
used as a yield criterion to determine the appropriate internal pressures from the von
Mises plasticity model.
By determining the pressure at the interior of the tube when the equivalent
von Mises stress (the maximum value of all elements in the body) matches 90% of
the ultimate tensile strength of DP600 (716 MPa), the difference in pressures from the
various mesh sizes will show the error not attributed to mesh sensitivity (see Table
16). These pressures produced by the von Mises plasticity model should be similar to
the burst pressures found by the dual bound model, and when examining Table 16, it
is evident that they are very close indeed.
The spread in the final pressure between the meshes of the von Mises model,
and the spread in burst pressure between the dual bound local and non-local models‟
meshes is the final product of this study. Spread in pressure between the 30 element
wide and 150 element wide meshes are listed in Table 16. This variation in pressure
provides a solution to the question of why the burst pressure and corner fill expansion
were not corrected as thoroughly as expected through the use of the non-local routine.
Spread in the non-local versions of the dual bound model is distinctly smaller than the
spread in their local counterparts. This reduction in variation is a result of the non-
local method eliminating or at least reducing the mesh sensitivity in the solution. Also
to note is the similar variation in the von Mises plasticity model when compared to
the non-local model. This similarity in spread indicates that the variation remaining
79
between the meshes of the dual bound model is not caused by mesh sensitivity but
rather sources of error that must be evident in both models, like discretization error.
Table 16 Burst pressure for von Mises at 90% UTS and the dual bound model
Number of
Tangential
Elements
Pressure (MPa) [psi]
von Mises
(90% UTS)
Local
Upper-
Bound
Non-Local
Upper-
Bound
Local
Lower-
Bound
Non-Local
Lower-
Bound
30
40
60
100
150
64.5 [9355]
61.8 [8957]
57.6 [8360]
56.3 [8161]
56.3 [8160]
67.0 [9713]
63.2 [9164]
59.9 [8682]
57.9 [8392]
57.2 [8299]
67.0 [9713]
63.2 [9173]
60.0 [8704]
58.1 [8429]
57.7 [8369]
65.6 [9516]
62.1 [9000]
58.8 [8521]
57.0 [8260]
57.0 [8268]
65.6 [9516]
62.8 [9109]
59.5 [8636]
57.8 [8379]
57.4 [8320]
30 - 150
Spread
8.2 9.8 9.3 8.6 8.2
80
Chapter 6: Conclusions and Recommendations
The dual-bound material model has been successfully reformatted to produce
non-local solutions for problems suffering from damage-induced ductile failure and
mesh sensitivity. The loss of mesh objectivity was clearly demonstrated in both the
upper- and lower-bound models through the use of two separate loading scenarios.
The non-local theory was invoked in the dual bound model as a regularization
technique in an attempt to resolve the problem. The implementation was tested using
a plane strain tension test and a straight tube hydroforming model. A straight tube
hydroforming model was designed which produces reasonable predictions with
outstanding computational performance when compared with experimental results
and a larger, 80,000 element, 3 dimensional model, respectively. In the plane strain
tension test, the porosity band width was regularized with respect to mesh size
through the use of the non-local theory. In the hydroforming simulation, variation in
burst pressure and corner fill expansion due to mesh sensitivity was reduced when the
non-local code was implemented. The effectiveness of the non-local implementation
was also confirmed through a comparison with experimental data and a mesh
objective material model.
81
6.1. Conclusions
The following conclusions can be made concerning the work presented in this thesis:
1. Although the mesh sensitivity of the upper-bound is well known (Leblond,
1994; Reusch et al. 2003), the dependence of the lower-bound or Sun and
Wang (1989) model on mesh size is an original finding of this work.
2. Non-local theory is capable of regularizing the mesh sensitivity in the lower-
bound Sun and Wang (1989) material model.
3. The local lower-bound model predicts a geometrically larger porosity zone
than the porosity distribution predicted by the local upper-bound model.
4. Non-local regularization may be directly implemented into a user-defined
material subroutine in LS-Dyna without using the built-in non-local feature,
therefore allowing the theory to be applied to user-defined material models.
5. The finite element model of straight tube hydroforming predicts burst pressure
and corner fill expansion reasonably well.
82
6. Characteristic lengths below a quarter of the element size are too small to
weigh the closest external element with any significance, resulting in a near
local solution.
7. When the size of the elements in a model decreased, an increased
characteristic length is required to obtain a similar porosity distribution to that
which was found in the larger element size model at any given time.
8. The smallest characteristic length that provides uniform porosity distribution
at any given time increases when the size of the elements in the model
decreases.
83
6.2. Discussion and Recommendations
The following recommendations are made for future work:
1. Varying the material properties for the hydroforming and plane strain tensile
test would provide more insight into the possible dependency of the
characteristic length on material parameters.
2. Reprogramming the dual-bound material model to take advantage of the
parallel processing power now available in the AMML would significantly
reduce computation times.
3. It would be advantageous to perform an experimental analysis on porous zone
distribution in DP600 during hydroforming and other metal forming
operations. This data could be used to validate the predictions in this thesis
and provide a foundation for a more in depth study of the characteristic length
measure.
4. Some improvements in performance may be possible by changing the non-
local weight function to a bounded weight function, reducing redundancy in
the data record method found in the non-local framework code, and finding a
more efficient manner to calculate the volume of elements, also found in the
non-local framework code.
84
5. The effect of mesh sensitivity on hydroforming in the radial direction should
be investigated further. Hydroforming simulations could also include pre-bent
tubes and end feed if the simulation‟s efficiency was significantly increased,
as this would require a larger, more comprehensive model.
6. Additions could be made to the material model including the new plastic limit
load criterion and shear modifications on void coalescence (Butcher and
Chen, 2009). This advance of the model into more modern approaches may
provide a better physical foundation to the model and more accurate
predictions.
85
References
Aliabadi M., 2001. Nonlinear fracture and damage mechanics. WIT Press, Chap. 8.
Asnafi N., 1999. Analytical modelling of tube hydroforming. Thin-Walled Structures
34, 295-330.
Asnafi N., Skogsgårdh A., 2000. Theoretical and experimental analysis of stroke-
controlled tube hydroforming. Materials Science and Engineering A279, 95-110.
Baaser H., Tvergaard V., 2003. A new algorithmic approach treating nonlocal effects
at
finite rate-independent deformation using the Rousselier damage model.
Computer Methods in Applied Mechanics and Engineering 192, 107-124.
Baradari J., 2005. PhD Thesis: Damage in hydroforming of pre-bent aluminum alloy
tubes. University of Waterloo, Canada.
Bardelcik A., 2006. MASc Thesis: Effect of pre-bending and hydroforming
parameters on the formability of advanced high strength steel. University of
Waterloo, Canada.
Bazant Z., Pijaudier-Cabot G., 1987. Nonlocal damage theory. Journal of
Engineering Mechanics 113, No. 10.
Brunet M., Sabourin F., Mquil-Touchal S., 1996. Prediction of necking and failure in
3 D. sheet forming analysis using damage variable. Journal De Physique IV vol.
6, no. 6.
86
Brunet M., Morestin F.,Walter H., 2004. Damage identification for anisotropic sheet-
metals using a non-local damage model. International Journal of Damage
Mechanics 13, 35-57.
Brunet M., Morestin F.,Walter-Leberre H., 2005. Failure analysis of anisotropic
sheet-metals using a non-local plastic damage model. Journal of Materials
Processing Technology 170, 457-470.
Butcher C.,2006. MScE Thesis: A combined upper and lower bound approach for
finite element modeling of ductile fracture. University of New Brunswick,
Canada.
Butcher C., 2009. Personal Communication.
Butcher C., Chen Z., 2009. A void coalescence model for combined tension and
shear. Modelling and Simulation in Materials Science and Engineering 17, 1-15.
Butcher C., Chen Z., Bardelcik A., Worswick M., 2009. Damage-based finite-element
modeling of tube hydroforming. International Journal of Fracture 155, 55-65.
Carleer B., van der Kevie G., de Winter L., van Veldhuizen B., 2000. Analysis of the
effect of material properties on the hydroforming process of tubes. Journal of
Materials Processing Technology 104, 158-166.
César de Sá J., Areias P., Zheng C., 2006. Damage modeling in metal forming
problems using an implicit non-local gradient model. Computational Methods in
Applied Mechanics and Engineering 195, 6646-6660.
Chen J., Yuan H., 2002. A micro-mechanical damage model based on gradient
plasticity: algorithms and applications. International Journal for Numerical
Methods in Engineering 54, 399-420.
87
Chen Z., 2008. ME6373 Mechanics of continua. In-class lecture, University of New
Brunswick, November 5th
.
Chow C., Yang X., 2002. Bursting for fixed tubular and restrained hydroforming.
Journal of Materials Processing Technology 130-131, 107-114.
Chu C., Needleman A., 1980. Void nucleation effects in biaxially stretched sheets.
Journal of Engineering Materials and Technology 102, 249-256.
Community Enterprise Operating System Development Team [online]. [Accessed
October, 2009]. Available from World Wide Web: <http://www.centos.org>
Dohmann F., Hartl Ch., 1997. Tube hydroforming - research and practical
application. Journal of Materials Processing Technology 71, 174-186.
Drabek T., Bohm H., 2005. Damage models for studying ductile matrix failure in
composites. Computational Materials Science 32, 329-336.
Dyment J., 2004. MASc Thesis Effect of the bending process on hydroformability of
steel tubes. University of Waterloo, Canada.
Engelen R., Geers M., Baaijens F., 2003. Nonlocal implicit gradient-enhanced elasto-
plasticity for the modelling of softening behaviour. International Journal of
Plasticity 19, 403-433.
Eringen A., Edelen D.,1972. On nonlocal elasticity. International Journal of
Engineering Science 10, 233-248.
Faleskog J.,Shih C.,1997. Micromechanics of coalescence-I synergistic effects of
elasticity, plastic yielding and multi-size-scale voids. Journal of the Mechanics
and Physics of Solids 45, 21-50.
88
Geers M., Ubachs R., Engelen R., 2003. Strongly non-local gradient-enhanced finite
strain elastoplasticity. International Journal for Numerical Methods in
Engineering 56, 2039-2068.
Goodarzi M., Kuboki T., Murata M., 2005. Deformation analysis for the shear
bending process of circular tubes, Journal of Materials Processing Technology
162-163, 492-497.
Grey J., Devereaux A., Parker W., June 1939. Apparatus for making wrought metal
T's, US Patent 2,203,868.
Gurson A., 1977. Continuum theory of ductile rupture by void nucleation and growth:
Part 1- Yield criteria and flow rules for porous ductile media. Journal of
Engineering Materials and Technology 99, 2-15.
Hallquist J., 1998. LS-DYNA Theoretical Manual. Livermore Software Technology
Corporation.
Jiresek M., Marfia S., 2005. Non-local damage model based on displacement
averaging. International Journal for Numerical Methods in Engineering 63, 77-
102.
Kanetake N., Nomura M., Choh T., 1995. Continuous observation of microstructural
degradation during tensile loading of particle reinforced aluminum composites.
Material Science and Technology 11, 1246-1252.
Kim S., Kim Y., 2002. Analytical study for tube hydroforming. Journal of Materials
Processing Technology 128, 232-239.
Koc M., Altan T., 2001. An overall review of the tube hydroforming (THF)
technology. Journal of Materials Processing Technology 108, 384-393.
89
Kridli G., Bao L., Mallick P., Tian Y., 2003. Investigation of thickness variation and
corner filling in tube hydroforming. Journal of Materials Processing Technology
133, 287-296.
Leblond J., Perrin G., Devaux J., 1994. Bifurcation effects in ductile metals with
nonlocal damage. Transactions of the American Society of Mechanical Engineers
61, 236-242.
Livermore Software Technology Corporation [online]. [Accessed October, 2009].
Available from World Wide Web: <http://www.lstc.com/lsdyna.htm>
Livermore Software Technology Corporation, 2002. Program Table of Contents . 7th
International LS-DYNA User‟s Conference.
Livermore Software Technology Corporation, 2003. LS-DYNA Keyword User‟s
Manual: Nonlinear Dynamic Analysis of Structures.
Manabe K., Amino M., 2002. Effects of process parameters and material properties
on deformation process in tube hydroforming. Journal of Materials Processing
Technology 123, 285-291.
McClintock F.,1968. A Criterion for ductile fracture of growth of holes. Journal of
Applied Mechanics 35, 363-371.
Mediavilla J., Peerlings R., Geers M., 2006. An integrated continuous-discontinuous
approach towards damage engineering in sheet metal forming processes.
Engineering Fracture Mechanics 73, 895-916
Medo B., Rakin M., Gubeljak N., Sedmak A., 2009. Application of complete Gurson
model for prediction of ductile fracture in welded steel joints. Key Engineering
Materials 399, 13-20.
90
Mortimer J., 2001. BMW breaks new ground with hydroforming. Assembly
Automation 21, 317-320.
Nahshon K., Xue Z., 2009. A modified Gurson model and its application to punch-out
experiements. Engineering Fracture Mechanics 76, 997-1009.
Needleman A., 1987. A continuum model for void nucleation by inclusion
debonding. Journal of Applied Mechanics 54, 525-531.
Peerlings R., Geers M., de Borst R., Brekelmans W., 2001. A critical comparison of
nonlocal and gradient-enhanced softening continua. International Journal of
Solids and Structures 38, 7723-7746.
Picart P., Oudin J., Bennani B., 1992. Finite element simulation of void nucleation
growth and coalescence in isotropic standard elasto-plasticity: application to cold
forging. Journal of Materials Processing Technology 32, 179-188.
Polizzotto C., 2001. Nonlocal elasticity and related variational parameters.
International Journal of Solids and Structures 38, 7359-7380.
Ramaswamy S., Aravas N., 1998. Finite element implementation of gradient
plasticity models Part I: Gradient-dependent yield functions. Computer Methods
in Applied Mechanics and Engineering 163, 11-32.
Reusch F., Svendsen B., Klingbeil D., 2003. A non-local extension of the Gurson-
based ductile damage modeling. Computational Materials Science 26, 219-229.
Rice J., Tracey D., 1969. On the ductile enlargement of voids in triaxial stress fields.
Journal of the Mechanics and Physics of Solids 17, 201-217.
91
Saanouni K., Chaboche J., Lesne P., 1989. On the creep crack growth prediction by
a non-local damage formulation. European Journal of Mechanic Solids 8, 437-
459.
Shima S., Oyane M., 1976. Plasticity theory for porous metals. International Journal
of Mechanical Sciences 18, 85-291.
Stromberg L., Ristinmaa M., 1996.FE-formulation of a nonlocal plasticity theory.
Computer Methods in Applied Mechanics and Engineering 136, 127-144.
Sun Y., Wang D., 1989. A lower bound approach to the yield loci of porous
materials. Acta Mechanica Sinica 5, 237-243.
Sun Y., Wang D., 1995. Analysis of shear localization in porous materials based on a
lower bound approach. International Journal of Fracture 71, 71-83.
Thomason P., 1998. A view on ductile-fracture modelling, Fatigue and Fracture of
Engineering Materials and Structures 21, 1105-1122.
Tirosh J., Neuberger A., Shirizly A., 1996. On tube expansion by internal fluid
pressure with additional compressive stress. International Journal of Mechanical
Sciences 38, No. 8-9, 839-85.
Tvergaard V., 1981. Influence of voids on shear band instabilities under plane strain
conditions. International Journal of Fracture 17, 389-407.
Tvergaard V., Needleman A., 1984. Analysis of the cup-cone fracture in a round
tensile test bar. Acta Metallurgica 32, 157-169.
Tvergaard V., Needleman A., 1995. Effects of nonlocal damage in porous plastic
solids. International Journal of Solids and Structures 32, No.8/9, 1063-1077.
92
Tvergaard V., Needleman A., 1997. Nonlocal effects on localization in a void-sheet.
International Journal of Solids and Structures 34 No.18, 2221-2238.
Winkler S., Thompson A., Salisbury C., Worswick M., Van Riemsdijk I., Mayer R.,
2008. Strain rate and temperature effects on the formability and damage of
advanced high-strength steels. Metallurgical and Materials Transactions A 39,
1350-1358.
Worswick M., Pelletier P., 1998. Numerical simulation of ductile fracture during high
strain rate deformation. European Physical Journal of Applied Physics 4, 257-
267.
Xue L., 2008. Constitutive modeling of void shearing effect in ductile fracture of
porous materials. Engineering Fracture Mechanics 75, 3343-3366.
Yuan H., Chen J., Krompholz K., Wittmann F., 2003. Investigations of size effects in
tensile tests based on a nonlocal micro-mechanical damage model. Computational
Materials Science 26, 230-243.
Zhang S., 1999. Developments in hydroforming. Journal of Materials Processing
Technology 91, 236-244.
93
Appendix A
User Material FORTRAN Code Modifications
94
A.1 Modifications to ‘dyn21.f’
In the „urmathn‟ subroutine (Line 3101):
46 call umat46 (cm(mx+1),eps,sig,hsv,dt1,capa,'brick',tt,i,
. a(islcnt(16)),a(n8))
was changed to:
46 call umat46 (cm(mx+1),eps,sig,hsv,dt1,capa,'brick',tt,i,
. a(islcnt(16)),a(n8),lft,llt,nnm1+i,a(lc1h),r_mem(dm_x),9)
go to 60
A.2 Modifications to ‘umat46_3d.f’
All subroutines with “_g3d” correspond to the upper-bound model and have “_s3d”,
lower-bound counterparts. Both terms may be considered replaceable, as identical
changes were made to both.
In the „umat46‟ subroutine (Line 1):
subroutine umat46 (cm,eps,sig,hisv,dt1,capa,etype,time,n,crv,npc)
was changed to:
subroutine umat46 (cm,eps,sig,hisv,dt1,capa,etype,time,n,crv,npc,
. lft,llt,inl,ixsnl,xnl,knl)
In the „umat46‟ subroutine (Lines 72,75,109,112, 153, and 156):
call umat_g3d (cm,eps,sig,hisv,dt1,capa,etype,time,n,crv,npc)
was changed to:
call umat_g3d (cm,eps,sig,hisv,dt1,capa,etype,time,n,crv,npc
95
1 lft,llt,inl,ixsnl,xnl,knl)
A.3 Modifications to ‘umat_g3d.f’ and ‘umat_s3d.f’
Identical changes were made to both the umat_g3d.f and umat_s3d.f files, to brevity,
only the umat_g3d.f changes are listed here.
In the „umat_g3d‟ subroutine (Line 1):
subroutine umat_g3d (cm,eps,sig,hisv,dt1,capa,etype,time,n,
. crv,npc)
was changed to:
subroutine umat_g3d (cm,eps,sig,hisv,dt1,capa,etype,time,n,
. crv,npc,lft,llt,inl,ixsnl,xnl,knl)
In the „umat_g3d‟ subroutine, the following was added (Line 299):
c
c *******************************************
c Non-Local MDL Aug27 08
c *******************************************
c Nonlocal Common Block
common/nonlocal/BrickVol,Ar,nel
integer pc,period
c
if(cm(47).gt.0)then
c
c
if(time.eq.0)then
if(inl.gt.nel)then
nel=inl
endif
endif
c
c Open the nonlocal data file
c
if(time.ne.0)then
c
96
open(10,file='fnlrecord.dat',status='unknown',access='append',
1 form='unformatted')
c
c Find the correct record:
c
do 2 in=1,(nel-(inl-1))
backspace(10)
2 continue
c
1 read(10) inlr,fnl
c
if(inlr.ne.inl)then
goto 1
endif
close(10)
c
c Overwrite the Porosity with the Non-Local Porosity
c
hisv(2)=fnl
c end if
end if
end if
c
c End Non-Local MDL Aug27 08
c *******************************************
c
In the „umat_g3d‟ subroutine, the following was added (Line 401):
c Increases Initial Porosity for Selected Element (Shear Band Test)
if(inl.eq.cm(46))then
hisv(2)=1.01*cm(28)
else
hisv(2)=cm(28)
endif
c
In the „umat_g3d‟ subroutine, the following was added (Line 941):
c
c **************************************************
c NON-LOCAL IMPLEMENTATION - MDL April 27th 09
c **************************************************
c
c elemloc stores the local porosity, volume and element location
97
c to frecord.dat file for nonlocal processing at the end of the
c time-step.
c
c nlocalpor calculates non-local porosity for all elements in
c this time step and saves the values to frecord.dat to be used in
c the next time step.
c
if(cm(47).gt.0)then
c
call elemloc (cm(45),time,llt,lft,etype,inl,ixsnl,xnl,knl,hisv(2))
c
if(inl.eq.nel)then
c
call nlocalpor (cm,eps,sig,hisv,dt1,capa,etype,
1 time,n,crv,npc,lft,llt,inl,ixsnl,xnl,knl)
end if
c
end if
c
c---------------------------------------------------------------------
c
A.4 Creation of ‘elemloc.f’
The file was created to find the center of each element, location and area or volume at
the end of processing the stresses. It extracts coordinates from arrays handed from the
solver and calls the proper area or volume routine. This information is recorded to a
record file at the end of the routine.
c FILE CREATED AUGUST 07 2008 -MDL
c
c This file was created to calculate the center of an element and
c return it's coordinates based on the coordinates of the nodes
c that comprise the element.
c
subroutine elemloc (eltype,time,llt,lft,etype,inl,ixsnl,xnl,
. knl,f)
c
c
c
common/nonlocal/BrickVol,Ar,nel
c
98
c Declaration of Variables
c
dimension ixsnl(knl,*),xnl(3,*)
c
character*(*) etype
c
real A1(3),A2(3),A3(3),A4(3),A5(3),A6(3),A7(3)
real A8(3)
real*4 xcoord
real*4 ycoord
real*4 zcoord
real location,loc1,loc2,loc3
c
integer*4 nodeID
integer nbofnodes
integer row
integer col
c
c Detect the element type and set number of nodes that
c correspond
c
c
if(eltype.le.0.5)then
nbofnodes=8
elseif(eltype.gt.0.5)then
nbofnodes=4
else ! etype is a 'beam '
nbofnodes=2
print *,'WARNING: Beam type elements are not supported.'
endif
c
c Initialize coordinates to zero
c
xcoord=0
ycoord=0
zcoord=0
c
c Average the nodal coordinates to give element location
c
do 70 m=1,nbofnodes
if(eltype.le.0.5)then
nodeID=ixsnl((m+1),inl)
elseif(eltype.gt.0.5)then
loc1=(inl*5)
loc2=loc1-4+m
loc3 =loc2/9
location=loc3+1
row=floor(location)
99
if(row.eq.location)then
row=row-1
col=9
else
col=nint((location-row)*9)
endif
nodeID=ixsnl(col,row)
endif
xcoord=(xnl(1,nodeID)+xcoord)
ycoord=(xnl(2,nodeID)+ycoord)
zcoord=(xnl(3,nodeID)+zcoord)
70 continue
xcoord=xcoord/nbofnodes
ycoord=ycoord/nbofnodes
zcoord=zcoord/nbofnodes
c
c
c Detect the element type and calculate area or volume
c of the element
c
if(eltype.le.0.5)then
A1=(/xnl(1,ixsnl(2,inl)),xnl(2,ixsnl(2,inl)),xnl(3,ixsnl(2,inl))/)
A2=(/xnl(1,ixsnl(3,inl)),xnl(2,ixsnl(3,inl)),xnl(3,ixsnl(3,inl))/)
A3=(/xnl(1,ixsnl(4,inl)),xnl(2,ixsnl(4,inl)),xnl(3,ixsnl(4,inl))/)
A4=(/xnl(1,ixsnl(5,inl)),xnl(2,ixsnl(5,inl)),xnl(3,ixsnl(5,inl))/)
A5=(/xnl(1,ixsnl(6,inl)),xnl(2,ixsnl(6,inl)),xnl(3,ixsnl(6,inl))/)
A6=(/xnl(1,ixsnl(7,inl)),xnl(2,ixsnl(7,inl)),xnl(3,ixsnl(7,inl))/)
A7=(/xnl(1,ixsnl(8,inl)),xnl(2,ixsnl(8,inl)),xnl(3,ixsnl(8,inl))/)
A8=(/xnl(1,ixsnl(9,inl)),xnl(2,ixsnl(9,inl)),xnl(3,ixsnl(9,inl))/)
c
elseif(eltype.gt.0.5)then
loc1=(inl*5)
loc2=loc1-3
loc3 =loc2/9
location=loc3+1
row=floor(location)
if(row.eq.location)then
row=row-1
col=9
else
col=nint((location-row)*9)
endif
nodeID=ixsnl(col,row)
A1=(/xnl(1,nodeID),xnl(2,nodeID),xnl(3,nodeID)/)
c
loc1=(inl*5)
loc2=loc1-2
loc3 =loc2/9
100
location=loc3+1
row=floor(location)
if(row.eq.location)then
row=row-1
col=9
else
col=nint((location-row)*9)
endif
nodeID=ixsnl(col,row)
A2=(/xnl(1,nodeID),xnl(2,nodeID),xnl(3,nodeID)/)
c
loc1=(inl*5)
loc2=loc1-1
loc3 =loc2/9
location=loc3+1
row=floor(location)
if(row.eq.location)then
row=row-1
col=9
else
col=nint((location-row)*9)
endif
nodeID=ixsnl(col,row)
A3=(/xnl(1,nodeID),xnl(2,nodeID),xnl(3,nodeID)/)
c
loc1=(inl*5)
loc2=loc1
loc3 =loc2/9
location=loc3+1
row=floor(location)
if(row.eq.location)then
row=row-1
col=9
else
col=nint((location-row)*9)
endif
nodeID=ixsnl(col,row)
A4=(/xnl(1,nodeID),xnl(2,nodeID),xnl(3,nodeID)/)
endif
c
if(eltype.le.0.5)then
c
call hexahedron (A1,A2,A3,A4,A5,A6,A7,A8)
c
elseif(eltype.gt.0.5)then
call area (A1,A2,A3,A4)
c
else ! etype is a 'beam '
101
print *,'Non-Local Processing is not capable of beam type'
endif
c
c Write location, volume and porosity to the recording file.
c
open(10,file='frecord.dat',status='unknown',access='append',
1 form='unformatted')
c
if(inl.eq.1)then !Rewinds file at beginning of elements
rewind(10)
c
end if
c
write(10) inl,f,xcoord,ycoord,zcoord,BrickVol,Ar
c
close(10)
c
c
return
end
A.5 Creation of ‘nlocalpor.f’
This routine reads all of the element location, and area or volume information stored
by „elemloc.f‟ and uses the data to calculate the non-local porosity. The new porosity
is stored to be used in the following time-step.
c
c----------------------------------------------------------------
c NON-LOCAL POROSITY CALCULATION
c Integral Style Non-Local Porosity
c
c FILE CREATED AUGUST 07 2008 -MDL
c Modified April 29th 2009 - MDL
c
c----------------------------------------------------------------
c
c This file opens a reference file, frecord.dat and calculates
c the non-local porosities for an entire time step then records
c the values to fnlrecord.dat file to use in the next time step.
c
subroutine nlocalpor (cm,eps,sig,hisv,dt1,capa,etype,
. time,n,crv,npc,lft,llt,inl,ixsnl,xnl,knl)
c
102
c
common/gurs1/
1 pnew(128),cc(128),scale(128),aj2(128),ak(128),davg(128),
2 sj2(128)
common/gurmat/g0,xtype,qs,qh,qn,qc,qm,xjnk1,
1 bk,cp,tmelt,tref,eref,rho0,limit,hard_xp,q1,q2,q3,
2 toler,xiter,phimax,ntype,xhyd,enucl,fnucl,snucl,f0,
3 fc,ff,sin,xjnk4,t0,cint,s1,s2,s3,gamma,gamma1,
4 xjnk5,xlhist,delt,refrate,refstr,xpstr
equivalence (qs,qaa),(qh,qbb),(qn,qcc),(qc,qdd),(qm,qee),
1 (xjnk1,qer)
equivalence (qs,phi0),(qh,phi1),(qh,phi2),(qn,phi3),(qc,phi4),
1 (qm,phi5),(xjnk1,q)
common/fstar/f(128)
common/nonlocal/BrickVol,Ar,nel
c
c Declaration of Variables
c
dimension ixsnl(knl,*),xnl(3,*)
dimension cm(*),cmat(48)
equivalence (g0,cmat)
character*(*) etype
dimension eps(*),sig(*),hisv(*),sig0(6),hisv0(30)
dimension crv(101,2,*)
dimension npc(*)
c
c
c VARIABLES
c
c fnonloc - Nonlocal Porosity
c x,y,zcoord - x,y,z coordinates of element
c fary - array of local porosities of all elements
c x,y,zcoary - array of x,y,z coordinates of all elements
c ielidary - array of internal element id's
c A1-8 - nodal coordinates for the current element
c A - First sum of non-local equation
c B - Second sum of non-local equation
c
real A
real B
real Bi
real fnonloc(nel)
real fnonlocprev(nel)
real fary(nel),xcoary(nel),ycoary(nel),zcoary(nel)
real BrickVolary(nel),Arary(nel)
real D1,D2,D3
real xminuss(nel)
real domoverdenom
103
real length,lengthc,twolengthtwo
c
integer ielidary(nel)
integer in
integer inprev(nel)
integer inp
c
c
c Characteristic Length
c
length=cm(47)
lengthc=(length*length)*length
twolengthtwo=2*length*length
c
c Read coordinates, volume and local porosities of all elements
c
open(10,file='frecord.dat',status='unknown',access='append',
1 form='unformatted')
c
rewind(10)
c
do 10 in=1,nel
read(10) ielidary(in),fary(in),xcoary(in),ycoary(in),
1 zcoary(in),BrickVolary(in),Arary(in)
c
10 continue
c
close(10)
c
c
c Loop over every element in the Part
c
c
do 90 inp=1,nel
c
c Calculate distance between current element and all others
c
do 30 in=1,nel
D1=xcoary(inp)-xcoary(in)
D2=ycoary(inp)-ycoary(in)
D3=zcoary(inp)-zcoary(in)
xminuss(in)=(sqrt((D1*D1)+(D2*D2)+(D3*D3)))
30 continue
c
c Select Incremental Element
c
if(cm(45).le.0.5)then
c
104
dOmega=BrickVolary(inp)
c
elseif(cm(45).gt.0.5)then
c
dOmega=Arary(inp)
c
endif
c
c Evaluate dOmega devided by (2*pi)^(3/2) and l^3
c
domoverdenom=(dOmega/((lengthc)*(2*3.14159265358979323)**(3/2)))
c
c
c Evaluate A
c
A=0
do 40 in=1,nel
A=((exp((-xminuss(in))/(twolengthtwo))*domoverdenom)+A)
c
40 continue
c
c Evaluate B
c
B=0
Bi=0
do 50 in=1,nel
Bi=(fary(in)*(exp((-xminuss(in))/(twolengthtwo))*domoverdenom))
B=Bi+B
c
50 continue
c
c Check for invalid division
c
if(A.eq.0.0)then
print *,'A used to be :',A
A=0.00000000001
print *,'A set to 0.00000000001'
print *,'WARNING: Non-local porosity may not be valid'
print *,' for element :',inp
end if
if(B.eq.0.0)then
print *,'B used to be :',B
B=0.00000000001
print *,'B set to 0.00000000001'
print *,'WARNING: Non-local porosity may not be valid'
print *,' for element :',inp
end if
c
105
c Evaluate the non-local porosity
c
fnonloc(inp)=(B/A)
c
c
90 continue
c
c Write non-local porosities of all elements
c
open(11,file='fnlrecord.dat',status='unknown',access='append',
1 form='unformatted')
c
rewind(11)
c
if(time.gt.0)then
do 97 in=1,nel
read(11) inprev(in),fnonlocprev(in)
97 continue
rewind(11)
else
do 98 in=1,nel
fnonlocprev(in)=0
98 continue
end if
c
do 99 in=1,nel
c
c
c Check to ensure porosity can only increase
c
if(fnonlocprev(in).gt.fnonloc(in))then
write(11) in,fnonlocprev(in)
else
write(11) in,fnonloc(in)
end if
99 continue
c
close(11)
c
return
end
A.6 Creation of ‘hexahedron.f’
The „hexahedron.f‟ routine calculates the volume of a hexahedron which represents
the constant stress 8 node brick element used in the solver.
106
c
c HEXAHEDRON - This function generates the volume of a
c hexahedron based on eight 3 dimensional Cartesian coordinates.
c This program generates various tetrahedrons from triangular
c surfaces between the nodes and a central temporary node xc. This
c approach only works if the shape is approximately cuboid.
c
subroutine hexahedron (A1,A2,A3,A4,A5,A6,A7,A8)
c
c Check tolerance at end of file for crazy shaped elements
c
c CREATED BY MICHAEL LANDRY
c AUGUST 19TH 2008
c
c VARIABLES
c
c A1-A8 - Node coordinates for the brick element
c
c
c INDEXING VARIABLES
c
c idx - counter for triangular faces
c
c
c Common Blocks
c
common/nonlocal/BrickVol,Ar,nel
c
c Declaration of Variables
c
real A1(3),A2(3),A3(3),A4(3),A5(3),A6(3),A7(3)
real A8(3)
real V1(3),V2(3),Va(3)
real L1,L2,La
real Nx,Ny,Nz
real N(3)
real R1(3),R2(3),R3(3)
real DisV(3)
real D
real Area
real BrickVol
real Vola
real xc(3)
integer idx
c
c
c For loop to cycle through all the faces (12 triangles)
c and both possible combinations (2 per square face)
107
c
BrickVol=0
do 1 idx=1,24
c
c
if(idx.eq.1)then
R1=A1
R2=A2
R3=A4
end if
if (idx.eq.2)then
R1=A3
R2=A4
R3=A2
end if
if (idx.eq.3)then
R1=A2
R2=A6
R3=A3
end if
if (idx.eq.4)then
R1=A7
R2=A3
R3=A6
end if
if (idx.eq.5) then
R1=A6
R2=A5
R3=A7
end if
if (idx.eq.6)then
R1=A8
R2=A7
R3=A5
end if
if (idx.eq.7)then
R1=A5
R2=A1
R3=A8
end if
if (idx.eq.8)then
R1=A4
R2=A8
R3=A1
end if
if (idx.eq.9)then
R1=A5
R2=A6
108
R3=A1
end if
if (idx.eq.10)then
R1=A2
R2=A1
R3=A6
end if
if (idx.eq.11)then
R1=A4
R2=A3
R3=A8
end if
if (idx.eq.12)then
R1=A7
R2=A8
R3=A3
end if
if (idx.eq.13)then
R1=A2
R2=A1
R3=A3
end if
if (idx.eq.14)then
R1=A4
R2=A1
R3=A3
end if
if (idx.eq.15)then
R1=A6
R2=A2
R3=A7
end if
if (idx.eq.16)then
R1=A3
R2=A2
R3=A7
end if
if (idx.eq.17)then
R1=A5
R2=A6
R3=A8
end if
if (idx.eq.18)then
R1=A7
R2=A6
R3=A8
end if
if (idx.eq.19)then
109
R1=A1
R2=A5
R3=A4
end if
if (idx.eq.20)then
R1=A8
R2=A5
R3=A4
end if
if (idx.eq.21)then
R1=A6
R2=A5
R3=A2
end if
if (idx.eq.22)then
R1=A1
R2=A5
R3=A2
end if
if (idx.eq.23)then
R1=A3
R2=A4
R3=A7
end if
if (idx.eq.24)then
R1=A8
R2=A4
R3=A7
end if
c
c
c Calculate central temporary node
c
xc=(A1+A2+A3+A4+A5+A6+A7+A8)/8
c
c
c Create vectors with the edges of the triangle bases
c
V1 = (/(R2(1)-R1(1)),(R2(2)-R1(2)),(R2(3)-R1(3))/)
V2 = (/(R3(1)-R1(1)),(R3(2)-R1(2)),(R3(3)-R1(3))/)
c
c
c Calculate normal to the base
c
Nx = (V1(2)*V2(3))-(V1(3)*V2(2))
Ny = (V1(3)*V2(1))-(V1(1)*V2(3))
Nz = (V1(1)*V2(2))-(V1(2)*V2(1))
N = (/Nx,Ny,Nz/)/(sqrt((Nx*Nx)+(Ny*Ny)+(Nz*Nz)))
110
c
c
c Distance from central point to base
c
DisV = (/(R1(1)-xc(1)),(R1(2)-xc(2)),(R1(3)-xc(3))/)
c
D = abs(DisV(1)*N(1)+DisV(2)*N(2)+DisV(3)*N(3))
c
c
c Calculate Area of the Base
c
c Finding the Length of two vectors, which form 2 sides of the
c triangle
c
L1 = sqrt((V1(1)*V1(1))+(V1(2)*V1(2))+(V1(3)*V1(3)))
L2 = sqrt((V2(1)*V2(1))+(V2(2)*V2(2))+(V2(3)*V2(3)))
c
c
c Length of third side of triangle V1,V2,Va
c
Va= V1-V2
La = sqrt((Va(1)*Va(1))+(Va(2)*Va(2))+(Va(3)*Va(3)))
c
c
c Area of the triangles and subsequently the base using Heron's
c formula. First the lengths are sorted by length
c
if ((L1.ge.L2).ge.La)then
a=L1
b=L2
c=La
elseif((L1.ge.La).ge.L2)then
a=L1
b=La
c=L2
elseif((L2.ge.L1).ge.La)then
a=L2
b=L1
c=La
elseif((L2.ge.La).ge.L1)then
a=L2
b=La
c=L1
elseif((La.ge.L1).ge.L2)then
a=La
b=L1
c=L2
else
111
a=La
b=L2
c=L1
endif
c
c Using stable version of Herons formula:
c
Area= 0.25*sqrt((a+(b+c))*(c-(a-b))*(c+(a-b))*(a+(b-c)))
c
c
BrickVol=(((0.333333333)*Area*D)+BrickVol)
c
if(idx.eq.12)then
Vola=BrickVol
end if
c
1 continue
c
BrickVol=BrickVol/2
c
if (abs(Vola-BrickVol).ge.(0.05*BrickVol))then
print *,'WARNING: Element Shape Affected Volume Calculations'
print *,'Vola=',vola
print *,'BrickVol=',Brickvol
end if
c
return
end
A.7 Creation of ‘area.f’
The „area.f‟ routine calculates the area of a parallelogram which represents the plane
strain 4 node shell element used in the solver.
subroutine area(A1,A2,A3,A4)
c
c AREA - This function generates the area of a plane bounded by
c four 3 dimensional Cartesian coordinates. This program assumes
c 2 separate triangles that make up the area of the
c plane. Adding the area of these triangles produces
c the area of the plane.
c
c CREATED BY MICHAEL LANDRY , AUGUST 8TH 2008
112
c
c WARNING: ALL 4 NODES MUST BE ON THE SAME PLANE!!!
c
c VARIABLES
c
c A1-A4 - Node coordinates surrounding the area
c V1-V3 - Vectors defining a plane
c Va,Vb - Vectors defining the opposite edge of the triangles
c L1-L3 - Lengths of the V1-V3 vectors
c La,Lb - Lengths of the L1-L3 vectors
c aa1,bb1,cc1 - Lengths of triangle 1 (longest to shortest)
c aa2,bb2,cc2 - Lengths of triangle 2 (longest to shortest)
c Ar - Current sum of the area of a base triangles
c
c INDEXING VARIABLES
c
c none
c
c Common Blocks
c
common/nonlocal/BrickVol,Ar,nel
c
c Declaration of Variables
c
real A1(3),A2(3),A3(3),A4(3)
real V1(3),V2(3),V3(3),Va(3),Vb(3)
real R1(3),R2(3),R3(3)
real L1,L2,L3,La,Lb,RL1,RL2,RL3
real V1dotV2,V2dotV3,V3dotV1
real Theta12,Theta23,Theta31
real aa1,aa2,bb1,bb2,cc1,cc2
c Calculate Area of the Base
c Make 2 triangles, calculate area, and add to get base.
c Finding the Length of three vectors, which form 4 sides of two
c triangles (one is shared)
c
V1 = (/(A2(1)-A1(1)),(A2(2)-A1(2)),(A2(3)-A1(3))/)
V2 = (/(A3(1)-A1(1)),(A3(2)-A1(2)),(A3(3)-A1(3))/)
V3 = (/(A4(1)-A1(1)),(A4(2)-A1(2)),(A4(3)-A1(3))/)
c
L1 = sqrt((V1(1)**2)+(V1(2)**2)+(V1(3)**2))
L2 = sqrt((V2(1)**2)+(V2(2)**2)+(V2(3)**2))
L3 = sqrt((V3(1)**2)+(V3(2)**2)+(V3(3)**2))
c
c Determine which vector is shared between the two triangles
c Find the angles between the vectors
c
V1dotV2=V1(1)*V2(1)+V1(2)*V2(2)+V1(3)*V2(3)
113
V2dotV3=V2(1)*V3(1)+V2(2)*V3(2)+V2(3)*V3(3)
V3dotV1=V3(1)*V1(1)+V3(2)*V1(2)+V3(3)*V1(3)
c
Theta12=abs(acos(V1dotV2/(L1*L2)))
Theta23=abs(acos(V2dotV3/(L2*L3)))
Theta31=abs(acos(V3dotV1/(L3*L1)))
c
if (Theta12.eq.max(Theta12,Theta23,Theta31))then
R1=V2
RL1=L2
R2=V3
RL2=L3
R3=V1
RL3=L1
c
V1=R1
L1=RL1
V2=R2
L2=RL2
V3=R3
L3=RL3
elseif (Theta23.eq.max(Theta12,Theta23,Theta31))then
R1=V3
RL1=L3
R2=V1
RL2=L1
R3=V2
RL3=L2
c
V1=R1
L1=RL1
V2=R2
L2=RL2
V3=R3
L3=RL3
elseif (Theta31.eq.max(Theta12,Theta23,Theta31))then
end if
c
c Length of third side of triangle V1,V2,Va & V2,V3,Vb
c
Va= V1-V2
Vb= V3-V2
c
La = sqrt((Va(1)**2)+(Va(2)**2)+(Va(3)**2))
Lb = sqrt((Vb(1)**2)+(Vb(2)**2)+(Vb(3)**2))
c
c Area of the triangles and subsequently the base using Heron's
c formula.
114
c First the lengths are sorted by length
if (L1.ge.L2.ge.La)then
aa1=L1
bb1=L2
cc1=La
elseif(L1.ge.La.ge.L2)then
aa1=L1
bb1=La
cc1=L2
elseif(L2.ge.L1.ge.La)then
aa1=L2
bb1=L1
cc1=La
elseif(L2.ge.La.ge.L1)then
aa1=L2
bb1=La
cc1=L1
elseif(La.ge.L1.ge.L2)then
aa1=La
bb1=L1
cc1=L2
else
aa1=La
bb1=L2
cc1=L1
end if
c
if (L2.ge.L3.ge.Lb)then
aa2=L2
bb2=L3
cc2=Lb
elseif(L2.ge.Lb.ge.L3)then
aa2=L2
bb2=Lb
cc2=L3
elseif(L3.ge.L2.ge.Lb)then
aa2=L3
bb2=L2
cc2=Lb
elseif(L3.ge.Lb.ge.L2)then
aa2=L3
bb2=Lb
cc2=L2
elseif(Lb.ge.L2.ge.L3)then
aa2=Lb
bb2=L2
cc2=L3
else
115
aa2=Lb
bb2=L3
cc2=L2
end if
c
c Using stable version of Herons formula:
c
Ar= 0.25*sqrt((aa1+(bb1+cc1))*(cc1-(aa1-bb1))*(cc1+
1 (aa1-bb1))*(aa1+(bb1-cc1)))
Ar= 0.25*sqrt((aa1+(bb1+cc1))*(cc1-(aa1-bb1))*(cc1+
1 (aa1-bb1))*(aa1+(bb1-cc1)))+Ar
c
c
return
end
116
Appendix B
Plane Strain Tension Test LS-Dyna Card Deck
117
B.1 Input Deck ‘1mmShear.in’
This is a typical input deck for the plane strain tension test. The deck below describes
a 1mm element size which wasn‟t used for the final experiments due to the
excessively coarse mesh it produces. Long lists of data in the following code have
been reduced for brevity.
$
*KEYWORD
$
$ Element 46 should be raised 1% Porosity
*TITLE
C:\FEM-MODEL\1mmShear
$
$
$ User supplied input:
$
*KEYWORD
$
*TITLE
1mm Shear Band Test, Coarse Mesh
$
$ User supplied input:
$
$---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8
$
$ N-mm-ms-g-MPa
$
$ Optional Control Cards that have been modified.
$
$---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8
$
*CONTROL_TERMINATION
$ Time (in millisec) that the simulation stops.
$ ENDTIM ENDCYC DTMIN ENDNEG ENDMAS
2.000,0,0.6,0,0
$
$
$
*CONTROL_TIMESTEP
0 0.90 0 0 0 0 1 0
$
$
*CONTROL_OUTPUT
$ npopt neecho nrefup iaccop opifs ipnint ikedit
0,0,0,0,2,1000,0
$
$
*CONTROL_ENERGY
$ Used to allow hourglass energy to be calculated and stored in the
$ GLSTAT and MATSUM ASCII files
$ HGEN RWEN SLNTEN RYLEN
2
$
$*CONTROL_HOURGLASS
$6
$
$
*CONTROL_SHELL
$ BWC (warping stiffness) changed from 2 to 1. Recommended for B-T elements
$ ISTUPD (shell thickness change option) changed to 1 to allow membrane strains
$ to affect the shell thickness.
118
$ WRPANG ESORT IRNXX ISTUPD THEORY BWC MITER PROJ
20.000 0 -1 1 13 1 1 0
$
$ DATABASE
$
*DATABASE_EXTENT_BINARY
$ NEIPH NEIPS MAXINT STRFLG SIGFLG EPSFLG RLTFLG ENGFLG
25 25 3 1 1 1 1 1
$ CMPFLG IEVERP BEAMIP DCOMP SHGE STSSZ
0 0 0 2 0 0
$
$
*DATABASE_BINARY_D3PLOT
$dt,lcdt
0.005
$
*DATABASE_NODAL_FORCE_GROUP
3
$
*DATABASE_NODFOR
0.005
$
*DATABASE_MATSUM
0.005
$
*DATABASE_GLSTAT
0.005
$
$---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8
$
$ Parts
$
$---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8
$
*PART
Tensile bar
$ PID SECID MID EOSID HGID GRAV ADPOPT TMID
1 1 1 0 0 0 0 0
$
$$---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8
$
$
*INCLUDE
DP600.insert
$
$
$ Plane Strain (x-y) Shell Element
*SECTION_SHELL
$ SID ELFORM SHRF NIP PROPT QR/IRID ICOMP
1 13 0.83 1.0 0.0 0.0 0
$ T1 T2 T3 T4 NLOC
1.0000 1.0000 1.0000 1.0000 0
$
$---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8
$
*SET_NODE_LIST
$ Nodes along y axis
1
133,122,111,100,89,78,67,56
45,34,23,12,1
$
$
*SET_NODE_LIST
$ Nodes with velocity boundary condition
$ SetID
2,
143,132,121,110,99,88,77,66
55,44,33,22,11
$
$
*SET_NODE_LIST
119
$ Nodes for Nodal Force Database File (Node List 1)
3
133,122,111,100,89,78,67,56
45,34,23,12,1
$
$---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8
$
$ Set Boundary Conditions
$
$---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8
$
$
$ Boundary condition for y-symmetry
*BOUNDARY_SPC_SET
1 0 1 0 0 0 1 1
$
$ Boundary condition for displacement
*BOUNDARY_PRESCRIBED_MOTION_SET
$ NSID DOF VAD LCID SF VID DEATH BIRTH
2 1 0 2 3 0
$
$ velocity of deformation curve
$
*INCLUDE
velocity_notch_4_0.txt
$
$---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8
$
$ End user defined input
$
$---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8
$
*NODE
1,0,0,0
2,1,0,0
…
143,10,12,0
$
*ELEMENT_SHELL
$ eid pid n1 n2 n3 n4
1,1,1,2,13,12
2,1,2,3,14,13
…
120,1,131,132,143,142
$
*END
B.2 Input Deck ‘DP600.in’
This is a material input deck for the plane strain tension test. The deck below
describes a “DP600 like” material, and should not be confused with the true material
parameters used for the tubular hydroforming. The DP600 file name was retained to
integrate multiple materials into the card deck with relative ease. Material parameters
120
included in this sample for may vary from the actual values used in the simulations.
It‟s recommended to consult the tables in the proper section of the thesis body for the
true values. Long lists of data in the following code have been reduced for brevity.
$ $ Material model for DP600 K=795.8 n=0.115 $ *MAT_USER_DEFINED_MATERIAL_MODELS $ MID RO MT LMC NHV IORTHO IBULK IG 1,7.887E-03,46,48,20,0,9,1 $ IVECT IFAIL 0 1 $ G0 XTYPE QS LCID 80232.56 7 413.54 1 $ K CP TMELT TREF EREF RHO0 LIMIT HARD 164285.71 1 0.115 $ Q1 Q2 Q3 TOLER NITER PHIMAX NTYPE XHYD 1.00 1.00 1.000 0.0001 50 1 1 0.35 $ ENUCL FNUCL SNUCL F0 FC FF B 0.200 0.0550 0.100 0.0007 0.050 0.070 1 $ T0 CINT S1 S2 S3 GAMMA GAMMA1 $ XLHIST DELT REFRATE REFSTR ElType ShearEl CHARLEN PERIOD 0,0,0,0,1,46,0,1 $ $ $ *DEFINE_CURVE $ lcid sidr scla sclo offa offo $ i i f f f f 1 0 1 1 0 0 $ abscissa ordinate $ f f 0,413.54 0.001,447.39
… 0.6,943.21 $ *END
B.3 Input Deck ‘velocity_notch_4_0.in’
This is the file used to control displacement during the tension test. Long lists of data
in the following code have been reduced for brevity.
$
$ Velocity boundary condition
$
$ 1 mm travel 1 mm/msec peak velocity VS time curve
$
*DEFINE_CURVE
$ LCID SIDR SFA SFO OFFA OFFO DATTYP
2 0 1 1 0 0 0
$ XAXIS-A1 YAXIS-O1
0.000, 0.00000
0.029, 0.00201
121
…
2.000, 0.00000
$
*END
122
Appendix C
Tube Hydroforming LS-Dyna Card Deck
123
C.1 Input Deck ‘3dsliver.in’
This is a typical input deck for the straight tube hydroforming test. Long lists of data
in the following code have been reduced for brevity.
*KEYWORD 18250000 $ *TITLE 3D_SLIVER_HYDROFORM:DP600 $ $ User supplied input: $ 0 $ $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ $ Title: 2d Straight Tube Hydroforming Model $ Date: June 24, 2009 $ Base units of: mm, msec, grams gives other units of N, MPa $ Author: Unknown $ Modified By: M. Landry $ $ Optional Control Cards that have been modified. $ $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ *CONTROL_TERMINATION $ Time (in msec) that the simulation stops after. $ ENDTIM ENDCYC DTMIN ENDNEG ENDMAS 40.0 0 $ *CONTROL_HOURGLASS $ Change the default hourglass control to a Flanagan-Belytschko with exact volum $ integration... recommended for large deformations $ IHQ QH
6 0.100 $ *CONTROL_SHELL $ BWC (warping stiffness) changed from 2 to 1. Recommended for B-T elements $ ISTUPD (shell thickness change option) changed to 1 to allow membrane strains $ to affect the shell thickness. $ WRPANG ESORT IRNXX ISTUPD THEORY BWC MITER PROJ 20.000 0 -1 1 2 1 1 0 $ *CONTROL_CONTACT $ Allow the Shell thickness to be considered in surface to surface and node to $ surface type contacts. $ SLSFAC RWPNAL ISLCHK SHLTHK PENOPT THKCHG ORIEN ENMASS 0.100 0.000 2 1 1 1 1 $ USRSTR USRFAC NSBCS INTERM XPENE SSTHK ECDT TIEDPRJ 0 0 10 0 4.000 $ Card 3 $ Card 4 1 $
*CONTROL_ENERGY $ Used to allow hourglass energy to be calculated and stored in the $ GLSTAT and MATSUM ASCII files $ HGEN RWEN SLNTEN RYLEN 2 2 2 1 $ $ $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ $ ASCII and LS TAURUS output $ $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ *DATABASE_RCFORC $ DT 0.005 $ *DATABASE_RBDOUT
124
$ DT
0.005 $ *DATABASE_GLSTAT $ DT 0.005 $ *DATABASE_MATSUM $ DT 0.005 $ *DATABASE_BINARY_D3PLOT $ DT/CYCL LCDT NOBEAM 0.005 $ *DATABASE_EXTENT_BINARY $ NEIPH NEIPS MAXINT STRFLG SIGFLG EPSFLG RLTFLG ENGFLG 20 0 7 1 1 1 1 1 $ CMPFLG IEVERP BEAMIP DCOMP SHGE STSSZ N3THDT 0 0 0 2 2 1 2 $ $
$---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ Parts $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ *PART tube (steel) $ PID SECID MID EOSID HGID GRAV ADPOPT 1 1 1 0 $ *PART tube (shell mesh used for HF) $ PID SECID MID EOSID HGID GRAV ADPOPT 102 102 102 0 $ *PART Hydroforming Die $ PID SECID MID EOSID HGID GRAV ADPOPT 100 100 100 0 $ $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8
$ Materials and Sections $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ $ Used for shell mesh for pressure application *MAT_POWER_LAW_PLASTICITY $ MID RO E PR K n SRC SRP 102 7.887E-03 207000 0.29 795.8 0.115 0 0 $ SIGY VP $ 294 0 $ $ DP600 material model *INCLUDE DP600.insert $ $ $ Hydroforming Die Material *MAT_RIGID $ MID RO E PR N COUPLE M ALIAS 100, 7.8869E-03, 208000., 0.3 $ CMO CON1 CON2 1.0 7.0 7.0
$LCO or A1 A2 A3 V1 V2 V3 $ $ Defines Element Formulation *SECTION_SOLID $ SID ELFORM 1 1 $ $ *SECTION_SHELL $ sid elform shrf nip propt qr/irid icomp 102 16 0.0 7 1 0 0 $ t1 t2 t3 t4 nloc 0.0001 0.0001 0.0001 0.0001 0.0 $ $ *SECTION_SHELL $ SID ELFORM SHRF NIP PROPT QR/IRID ICOMP
125
100 2 0.83 3.0 0.0 0.0 0
$ T1 T2 T3 T4 NLOC 0.0001 0.0001 0.0001 0.0001 0 $ $ $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ 1/8th Symmetry plane with constraints based on node sets within the mesh file $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ *DEFINE_COORDINATE_SYSTEM 88,0,0,0,1,1,0 1,1,1 $ *BOUNDARY_SPC_SET 2,88,0,1,1,1,0,0 $ $ Other Boundary Conditions $ *BOUNDARY_SPC_SET 3,0,1,0,1,0,1,0 $ *BOUNDARY_SPC_SET
4,0,0,0,1,0,0,0 $ $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ Contact Definitions $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ $ $ $ *CONTACT_SURFACE_TO_SURFACE $ $ Tube to Die $ $ SSID MSID SSTYP MSTYP SBOXID MBOXID SPR MPR 1 100 3 3 $ FS FD DC VC VDC PENCHK BT DT 0.035 0.035 1.0 238.8 10 1 0.0 $ SFS SFM SST MST SFST SFMT FSF VSF $
$ *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE $ $ Tube to Null Shell Mesh $ $ SSID MSID SSTYP MSTYP SBOXID MBOXID SPR MPR 1 102 3 3 $ FS FD DC VC VDC PENCHK BT DT 0.0001 0.0001 1.0 238.8 10 1 0.0 $ SFS SFM SST MST SFST SFMT FSF VSF $ $ $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ Pressure Application $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ *LOAD_SHELL_SET $ EID LCID SF AT 1 1103 1 $
$ Ordinate Scale Factor for Converting psi to MPa *DEFINE_CURVE $ LCID SIDR SFA SFO OFFA OFFO DATTYP 1103 0 2.0 -0.006895 0 0 0 $ $ XAXIS-A1 YAXIS-O1 0.0, 0.0 2.0, 0.0 30.0, 22293.0 100.0,22293.0 $ $ *INCLUDE 40x6.in $ $ *END
Curriculum Vitae
Candidate Information:
Michael Landry
537 Windsor Street, Apartment B.
Fredericton, New Brunswick
E3B 4G2
Email: [email protected]
Education:
Bachelor of Science in Engineering 2008
Mechanical Engineering
University of New Brunswick, Fredericton, NB
Bachelor of Science 2006
Cape Breton University, Sydney, NS
Diploma in Engineering 2006
Cape Breton University, Sydney, NS
Conferences:
Landry M., 2009. Mesh sensitivity and non-local theory applied to the dual-bound
material model. Mechanical Engineering Graduate Students Conference 5,
University of New Brunswick, Fredericton, N.B.
Butcher C., Landry M., Chen Z., 2009. Poster: Microstructure Based Material Models,
APMA-AUTO21 Annual Conference and Exhibition, Hamilton, Ont.
Publications:
Landry M., Chen Z., August 13, 2009. Mesh Sensitivity in Sun and Wang‟s Damage
Based Material Model. Meccanica, Under review.
Landry M., Chen Z., Xia W., April 16, 2009. An Approximate Lower Bound Damage-
Based Yield Criterion for Porous Ductile Sheet Metals. Journal of Applied Mechanics,
Under review.
