event there charts) - university of toronto t-space · 2020. 4. 7. · the ramberg-osgood equation...
TRANSCRIPT
This manuscript has been reproaiced from the microfilm master. UMI films the
text directly from the original or copy submitted. Thus, some thcsis and
dissertation copies are in typewriter face, while others may be from any type of
cornputer pflnter.
The quality of this reproduction is dependent upon the quality of the copy
submitted. Broken or indistinct print, colored or poor quality illustrations and
photographs, pnnt bbedthrough, substandard margins, and impmper alignment
can adversely affect repmdudion.
In the unlikely event that the author did not send UMI a complete manuscript and
there are missing pages, aiese will be noted. Also, if unauthorized copyright
matenal had ta be removed, a note will indicate the deletion.
Ovenize materials (e.g., maps, dawings, charts) are reproduced by secüoning
the original, beginning at the upper Mt-hand corner and conünuing from left to
right in equal sections with small overîaps.
Photographs included in the original manuscript have b e n reproduced
xerographically in this copy. Higher quality 6' x 9' black and Wite photographic
prints are available for any photographs or illustrations appeanng in this copy for
an additional charge. Contact UMI directly to order.
Bell & Howell Information and Leaming 300 North Zeeb Road, Ann Arbor, MI 481061346 USA
NOTE TO USERS
This reproduction is the best copy available
URlI
Simulation the Tube Drawing Process using the Finite Element Method
A thesis submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
Zulfikar H. A. Kassam
Department of Metallurgy and Materials Science Faculty of Applied Science and Engineering
UniversiSr of Toronto
May 1998
O Zulfikar H. A. Kassam
National Library I*I of Canada Bibliothèque nationale du Canada
Acquisitions and Acquisitions et Bibliographic Services services bibliographiques
395 Wellington Street 395, rue Wellington ûîtawaON K1AON4 Ottawa ON K 1 A ON4 Canada Canada
The author has granted a non- exclusive licence allowing the National Library of Canada to reproduce, loan, distribute or sel! copies of this thesis in rnicroform, paper or electronic formats.
The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts fiom it may be printed or otherwise reproduced without the author's permission.
Your hlo Votre reference
Our fi& Notre relerence
L'auteur a accordé une licence non exclusive permettant à la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la forme de microfiche/filrn, de reproduction sur papier ou sur format électronique.
L'auteur conserve la propriété du droit d'auteur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation.
ABSTRACT
The power law and the Ramberg-Osgood constitutive equations are commonly used for describing material behavior. The Ramberg-Osgood equation is the more popular one as it is capable of describing behavior of several materials over the entire range of deformation. Previous research revealed, however, that the Ramberg-Osgood equation is not capable of describing the behavior of materials which exhibit sudden changes.
One of the goals of this research was to develop a more accurate and efficient way of describing materials behavior. This goal has been achieved as a new equation, called the Alpha constitutive equation. has been developed. This equation is very accurate and efficient as well due to the ability of one equation being able to describe the behavior of any material - even materials that exhibit strain softening.
The second phase of this research focused on developing a finite element program to simulate the tube drawing process which involves large plastic deformation and complicated b o u n d q conditions. A specialized code to simulate the tube drawing in the presence of a mandrel has been developed. This finite element program has the ability to accept materiais data in the form of a modified Rarnberg-Osgood equation as well as the Alpha equation. A special technique was devised and irnplemented in the fmite element program to avoid instability that tends to occur at the die exit.
The finite element simulation results achieved in terms of the prediction of the drawing force compares well with the results obtained by Ontario Hydro Technologies. In addition. experiments conducted at Ontario Hydro revealed the formation of shear cracks on the outer surface of the tube. The finite element simulation results are consistent with this observation as the results indicate that maximum shear stress does exist a t the outer walls of the tube which is responsible for the formation of shear cracks.
The FEM program developed has the added advantage of having the capability to determine the stresses under unsteady state (transient) conditions. In addition, the residual stresses in the rnaterid can be determined.
This research work is far from over - it is only the beginning of a very promising step to reach the goal of being able to use mathematicaîly simulations to accurately simulate tube forming processes.
1 am greatly indebted to my supervisor Prof. Zhirui Wang for his guidance and support throughout the course of this work and to Ontario Hydro Technologies for their financial assistance during this research. I would also iike to thank Dr. Edward Ho of Ontario Hydro Technologies for his help and advise. 1 would also like to thank him for carrying out the tube drawing experiments on the Hydraulic Drawbench Test Facility (HDTF). In addition. 1 would like to thank Prof. G. Bendzsak for taking the time to go through this work and suggest a number of improvements. I am also grateful to Prof. S. A. Meguid from the
Mechanical Engineering Department for checking the technical correctness of this work.
Findy. 1 would like to thank my Parents Hussein and Farida Kassam for their support throughout my academic career. as well as my Wife Yasmin for her support and patience through the course of this work.
APPROACE TO SOLVE METAL FORMING PROBLEMS 1
1. PROBLEMS ENCOlJNTERED DURING FORMING 1
1. 1. ADDRESSING FORMING PROBLEMS 2
1. 2. IMPORTANCE OF STUDYING DEFORMATION MECHANICS 3
1. 3. SI7JDYING DEFORMATION MECHANICS OF FORMING PROCESSES - TRADITIONAL METPODS VS. PROCESS MODELS 3
1.4. ANALYTICAL TECHNIQUES 1.4. 1. LimitationsOfAnalyticaITechniques
1.5. NUMERlCAL TECHNIQUE - FINITE ELEMENT METHOD 7
fdTeRATURE SURVEY AND OBJECTIVES
2. 1. DEmXoPMENT OF FINITE ELEMENT METHOD
2. 2. USING NON-LINEAR FEA IN METAL FORMING PROCESSES
2. 3. CURRENT AREAS OF RESEARCH 2. 3. 1. Modeling of Boundary Conditions 2.3. 2. Modeling Behavior of Anisotropic Materials 2. 3. 3. Modeling of Kinematic Hardening 2. 3. 4. Prediction and Effect of Damage Formation 2. 3. 5. Microstructure Prediction 2. 3. 6. Testing and Evaluation of Models 2. 3. 7. Numerical Simulation of Forming Processes
REVIEW OF CONSTITUTIVE EQUATIONS
3.2. THE CONVENTIONAL RAMBERG-OSGOOD EQUATION 3.2. 1. Problems with the Conventional Ramberg-Osgood Eqn.
3.3. NEW APPROACH FOR THE RAMBERG-OSGOOD EQN.
ELASTIC AXXSYMMETRXC STRESS ANALYSIS
4. 2. DISPLACEMENT FUNCTIONS
4. 3. STRAIN 4. 3. 1 . Initial Strain (Thermal Strain)
4. 4. ELASIICITY MATRIX 4. 4. 1. Orthotropic Materials 4. 4. 2. Isotropie Materials
4. 5. ELEMENT STIFFNESS MATRU(
4.6. LX)ADING CONDITION 4. 6. 1 . Extemal Nodal Forces 4.6. 2. Dlstributed Body Forces 4. 6. 3. Traction Forces (pressure loading) 4. 6. 4. Forces due to Initial Strain (change in temperature)
FEM FOR ELASTIC - LARGE PLASTIC DEFORMATION 57
5. 2. FINITE DEFORMATIONS 59 5. 2. 1. Strain-Displacement Relationships: Green and Almansi Strain Tensors 62
5. 3. DIFFERENT STRESS MEASURES 68
5. 4. FINITE ELEMENT FORMULATIONS 71
5. 5. UPDATED iAGRANGIAN FORMULATION 5. 5. 1. Lhearization of Equilfbrium Equations 5. 5. 2. Determination of Stiffness Matrix
PLASTIC STRESS-STRAIN RELATfONSlCiIPS 84
6. 2. ERRORS INVOLVED IN ELASTIC-PLASIIC ANALYSIS 85
6.3. FLOW FWLE FOR ISOTROPIC MATERIALS: PRANDTL - REUSS EQUATIONS 86
6.4. THE PUSSITC POTENTIAL (YIELD) FUNCïiON
6. 5. mxlW R U E FOR AN ANISOTROPIC MATERiAL
6.6. THE ELASIIC-PLASI'IC CONSTITUTIVE MAT= 6. 6. 1. Elastic-Plastic Constitutive Matrix for Isotropic Materials 6. 6. 2. Elastic-Plastic Constitutive Matrix for Anisotropic Materials 6. 6. 3. Instantaneous Plastic Modulus. H' (or H,)
6.7 . N E W N - W H S O N I'IERATIVE SCHEME 6. 7. 1. Convergence Criteria for Newton-Raphson lterations
6 .8 . INTEGRAIION OF FLOW RULES TO CALCULATE STRESS INCREMENTS 6. 8. 1. Crossing the Yield Surface 6. 8. 2. Forward-Euler technique
EXPERlMENTAL SETUP AND PROCEDURE RESULTS AND DISCUSSION FOR THE CONSTiTWïWE EQUATIONS 117
7. 1. MATERIALS. EXPERIMENTAL SETUP & PROCEDURE 1 17
7. 2. RESULTS AND DISCUSSION 118
7.3. THE ALPHA CONSrrrUTIVE EQUATION IS INTRODUCED 120
7.4. THE ABILITY OF ALPHA CONSTITUTIVE EQUATION IN DESCRlBING BEHAVIOR OF MATEFWKS 127 7. 4. 1. The Ability of Alpha Constitutive Equation in Describing Strain
Softening in Zr-2.5wtYoNb Pressure Tube Material 132
7.5. USING a-EQUATION IN FINITE ELEMENT ANALYSIS 134
7.6. CONCLUDING REMARKS 134
ACCURACY OF THE FEM PROGRAM DEVELOPED 137
8.1. TESllNG THE FINITE ELEMENT ANALYSE CODE FOR ACCURACY IN ELASTIC SIMULATIONS 137
8.2. TESTING FOR ACCURACY IN SIMULATING ELASTIC- PLASTIC MATERIALS 141
8.3. TESITNG FOR ACCURACY IN SIMULATING E W I C - PLASTIC MATERLALS 'MAT OBEY THE a EQUATION 146
CF1).wrET& 9
FINITE ELEMENT ANALTSIS TO SIMULATE TUBE DRAWING PROCESS
CHOOSiNG THE APPROPRIATE FINITE ELEMENT APPROACH
FEM TECHNIQUE FOR LARGE PLASTIC DEFORMATION
FEM TECHNIQUE FOR NON-LINEAR HARDENING MATERIALS
CHOICE OF STRESS STATE
CHOICE OF FRICTION MODEL 9.5. 1. FRICTION LAYER TECHNIQUE 9.5.2. ALTERNATlVE METHOD FOR SIMULATINC FRICTION CONDlTIONS
TUBE DIMENSIONS
MESH DESIGN
BOUNDARY CONDITIONS
TUBE DRAWING EXPEFüMENTS
GRAPHICAL PLUE OF VARIABLES
RADML AND AXIAL DISPLACEMENTS
RADIAL, AXIAL, SHEAR AND CIRCUMFERENTiAL STRAINS (MATERIAL DEFORMATION)
APPLIED LOCAL STRESS P A m R N
WSIDUAL STRESSES
RESIDUAL STRAINS
EFFECT OF FRICTION
Figure 3.1 Cornparison between the effectiveness of the powcr law and Rarnberg- 30 Osgood equation.
Figure 3.2 Comparison between the three parameters method and the ln- ln plot 32
Figure 3.3 Behavior of modified Zr-2.5 wt% Nb along the circumferenlial direction 33 being describeû by two Ramberg-Osgood equations evaluated using 1 n- 1 n plot rnethod.
Figure 4.1 , Element of an axisymmetric solid 14.11 37 Figure 4.2 Strain and stresses involved in the analysis üf axisymmetric solids. 4 1 Figure 5.1 Motion and deformation of a body in a Cartesirin coordinate systcm (adripted 58
Fipure 5.2 Figure 6.3 Figure 6.4
Figure 7.1
Figure 7.2 1 behavior of aluminum A356. 1
Configuration of body at time t and time t + t. 64 One-dimensional stress-strain relationship with linear hardening. 102 (a) Full newton-Raphson iteration scheme and (b) Modified Newton- 1 03
Figure 7.3 1 Experimental data points and the Rarnberg Osgood equation describing the 1 120
Raphson iteration scheme. Experimental data points and the Ramberg Osgood equation describing the behavior of aluminum 6O6 1. Experimental data points and the Ramberg Osgood equation descnbing the
1 behavior of 70130 brass. 1
119
1 20
Figure 7.4
Figure 7.5
Figure 7.6 Figure 7.7
The above diagram shows the diffe~nce between the definition of (a) a, used by Ramberg-Osgood where a, is constant and (b) a sued by Alpha constitutive equation where a is a variable {a=f(e)} and is evaluated at many points on the stress strain-curve. Experimental data points and the Alpha constitutive equation descnbing the bchavior of 70/30 brass. Plots of a versus total tnie strain (e) for brass, copper and steel. Experimental data points and the Ramberg Osgood equation describing the behavior of 1018 steel and copper. Experimental data points and the Alpha constitutive equation describing the behavior of modified Zr-2.5wt%Nb CANDU pressure tube material in compression along the axial, redid and circumferential directions of the
viii
124
127
128 129
133
Figure 8.1
Fisure 8.2
Figure 8.3
Figure 8.4
The above is the Cylindrical pressure vesse1 that was used for elastic finite element simulation of assess the accuracy of finite element. a=O.Sm, b= 1 .ûm, and p= 1ûûMpa. Finite element mesh used to analyze stresses in the cylindrical pressure vessel subiected to an interna1 pressure of 100Mpa. Variation of the circumferential stress component, 000, with distance from the center (radial distance) Variation of the radial stress component, orr, with distance from the center (radial dis tance)
138
138
139
140
1 Figure 8.5 1 Variation of the axial stress component. ozz. with distance from the cenw (
1 Figure 8.6 1 Theoretical elastic-plastic stress-strain curve is compand to the results of 1 1 1 QUAD (our p r o m ) and MSA (Commercial promarn) 1 1 Figure 8.7 1 Theoretical elastic-linear hardening stress-strain curve is compared to the 1
results of QUAD and NISA. Figure 8.8 Theoretical Rarnberg-Osgood stress-strain curve is compared to the results 143
of QUAD and NISA for a material that exhibits a very low rate of
- -
~ i ~ u i e 8.9 Theoreti=al Ram berg-Osgood stress-strain curve is compared to the results 143 of QUAD and NISA for a material that exhibits a high rate of hardenin~.
Figure 8.10 Alpha constitutive equation (solid line) describing the [ensile behavior of 1 44 70/30 brass and the QUAD FEM simulations results (triangles) exhibits
Figure 8.1 1
Figure 8.12
Figure 9.1
Figure 9.2
Figure 9.3 Figure 9.4
Figure 10. la
Figure IO. 1 b
Figure 10.2a
very good correlation. The Alpha constitutive equation curves describing the behavior of copper 145 and FEM simulations results showing good agreement when the yield stress used in the one indicating the dcviation from linearity. The Alpha constitutive equation curves describing the behavior of method 145 Zr-Z.Swt%Nb CANDU pressure tube material and the FEM simulation results showing good agreement when the yield stress used in thc one indicating the deviation from linearity. Shows the 3-dimensional (A) and 2 dimensional simplification of the 156 element which is in contact with the die interface. The abovc diagram shows the setup used by Lu and Wright for trip drawing 159 (top) and the setup we used for tube drawing boitom. The above diagram shows the setup of the drawing operation 168 Schematic Diagram illustrating general layout of Hydraulic Drawbench Test 1 173 Facility Thc radial displacement as a function of radial position (along the thickness 177 of the tube) is shown at different axial positions (along the draw direction). The above results are for U=0.05 at steady state. The radial displacement as a function of radial position (along the thickness 178 of the iube) is shown at different axial positions (along the draw dircction). The above results are for U=O.1 at steady state. The axial displacement as a function of radial position (along the thickness 179 of the tube) is shown at different axial positions (along the dmw direction).
1 1 The above resulrs are for U=0.05 at steady state. 1
I Figure 10.2b I The axial displacement as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). 1
1 1 The above results are for U=O.l at steady state. 1 Figure 10.3a
Figure 10.3b
Figure 10.4a
Figure 10.4b
The resultani speedfdrawing speed as a function of radial position (along the 182 thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for U=0.05 at steady state. The resultant speeddrawing speed as a function of radial position (along the 183 thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for U=O. 1 at steady state. The direction of flow as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for U=0.05 at steady state. The direction of flow as a function of radial position (along the itiickness of 185 the tube) is shown at different axial positions (along the draw direction). The above results are for U a . 1 at steady state.
1 Figure 10.1 la 1 ~heabove diagram shows the axial strain contours at steady state conditions 1 199 1
Figure 10.5a
Figure 10.5b
Figure 10.6a
Figure 10.6b
Figure 10.7a
Figure 10.7b
Figure 10.8a
Figure 10.8b
Figure 10.9a Figure 10.9b Figure 10 .9~ Figure 10.10a
Figure IO.lOb
1 Figure 10.1 1 b 1 The above diagram shows the axial strain contours at steady state conditions 1 200 1 1 Figure lO.l2a 1 The above diagram shows the radiai strain contours at sready state 1 201 1
The radial strain as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw dircction). The abovc results are for Ud.05 at steady state. The radiai strain as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for U=O. 1 at steady state. The axial strain as a function of radiai position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The
. above results are for Ud.05 at steady state. The axial sirain as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for Ud.1 at steady state. The shear strain as a function of radial position (along the thickness of the tube) is shown at different axiaI positions (along the draw direction). The abovc results are for U=O.OS at steady state. Thc shear smin as a function of radial position (along the thickness of the tube) is shown at different axiaI positions (along the draw direction). The abovc results are for U=O. 1 at steady state. The circumferentiai strain as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results arc for U=0.05 at steady state. The circumferentiai strain as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for U=O. 1 at steady state. The meshes showing the progression of deformation. The mesh shown at steady state for U=0.05. The mesh shown at steady state for Ud. 1. The above diagram shows the radial strain contours at steady state conditions for U=0.05 The above diagram shows the radial strain contours at steady state
1 Figure 10.12b 1 The above diagram shows the shear suain contours at steady state 1 202 1
186
187
, 188
1 89
190
191
192
193
194 195 196 197
198
Figure 10.13a
L
Figure 10.13b
conditions for U d . 1 The above diagram shows the circumferential strain contours at steady state
Figure lO.I4a
Figure 10.14b
Figure IO.15a
Figure 10.15b
203 conditions for U=O.OS The above diagram shows the circumferential strain contours at steady state 204
The above diagram shows the equivalent suain contours at steady state conditions for U=0.05 The above diagram shows the equivalent strain contours at steady state conditions for U=O. 1 The above diagram shows the radial strain contours at steady state conditions for U=0.05 The above diagram shows the radial strain contours at steady state conditions for U=O. 1
205
206
207
208
A
1 Figure 10.l6a
1 Figure 10.16b
1 Figure 10.l7a
1 Figure 10.17b
1 Figure IO.18a
1 Figure 10.18b
1 Figure 10.19a
1 Figure 10.19b
Figure 10.20a
Figure 10.20b
I Figure 10.2 1 a
I Figure 10.2 1 b
I Figure 10.22a
1 Figure 10.22b
I Figure 1 O.23a
I Figure 1 0 3 b
Figure 10.26
The above diagram shows the axial strain contours at steady state conditions 1 209
The above diagram shows the axial strain contours at steady state conditions 210 for U=O. 1
above results are for U9 .1 at steady state. 1
The above diagram shows the shear stress contours at steady stale 21 1 conditions for U=0.05 The above diagram shows the radial stress contours at steady state 212 conditions for U=O. 1 The above diagram shows the circumferential stress contours at steady state 213 conditions for U=0.05 The above diagram shows the circumferential stress contours at steady state 214 conditions for Ud .2 The above diagram shows the equivalent stress contours at steady state 215 conditions for U=0.05 The above diagram shows the equi valent stress contours at steady staic conditions for U=0.2 The radial stress as a function of radial position (along thc thickness of the tube) is s h o w at different axial positions (along the draw direction). Thc above results are for U=0.05 at steady state. The radia1 stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The
216
222
223
Thc axiaI stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). Thc above results are for Ud.05 at stcady state. The axial stress as a function of radial position (along the thickness of rhe tube) is shown at different axial positions (along the draw direction). The above results are for U=O. 1 at steady state. The shear stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). Thc above results are for U=0.05 at steady state. The shear stress as a function of radial position (along the thickness of thc tube) is shown at different axial positions (along the draw direction). The above results are for U=O. 1 at stcady state. The circurnferential stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for U=0.05 at steady state. The circumferential stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for U=O. 1 at steady state. Residual Stresses for U=0.05 Residual Stresses for U=O. 1 Residual Stresses for Ud.05 Residual Stresses for U=O. 1 Finite element calculation showing the effect of friction on drawing force compared to the expenmental resul t
224
225
226
227
228
229
232 23 3 234 235 236
Metal Fomina and the Finite Elernent Method 1 p-pp-p p. - - - -
T E n Approach to 801ve M e t a l Forming Problema
1. PROBLEMS ENCOUNTERED DURING FORMING
Meta1 forming operations are of prime importance to the manufac-
turing industry . During metal forming operations. an initially simple
part. for example. a billet or a sheet blank. is plastically deformed by
using tools and dies in order to obtain the desired final configuration.
A common example of metal fonning operations is the tube draw-
ing process. The tube drawing process is frequently a finishing operation
conducted on an extruded tube in order to enhance the mechanical
properties and produce the desired shape and surface finish. The tube
drawing process is camed out at room temperature. Often. the deforma-
tion that needs be imparted to the tube is more than 10V0 reduction in
area and. sometimes. even as high as 60% reduction in area. Because of
the relatively poor flow properties of metals at room temperature as
compared to hot working temperatures a t which extrusions are per-
fomed, the material may form voids and eventually crack if attempts are
made to impart high amounts of deformation in one single step. In order
to elude this problem. deformation is usually imparted by performing a
series of draws, usually three to four draws. in order to obtain the final
product. The tube is sometimes annealed in between drawing operations
Metal Formina and the Finite Element Method 2
in order to improve the drawability of the material. In any case. during
each drawing operation. large amounts of plastic deformation are im-
parted to the material. Therefore. depending upon the drawability of the
material. voids may still form. During service. these voids may grow and
link up to form a microcrack. Many microcracks do in t um link up to
form a major crack that eventually propagates causing catastrophic
failure.
1. 1. ADDRESSING FORMING PROBLEMS
The formation of voids depends on the strain and stress state in
the material during the forming operation which in turn is influenced by
parameters such as die angle. reduction in area, etc.. used in the drawing
operation. Knowledge of the effect of these parameters on the strain and
stress states in the material during the drawing operation could be used
to predict the conditions under which defects would form. Consequently.
it would possible to determine the conditions (parameters) that are
required to produce defect free tubes. The main goal would therefore be to
study the deformation mechanics i.e.. the forces, stresses. deformations
and strains associated with the forming process. and determine the
parameters under which the strain and the stress states are ideally
suited to produce defect free tubes with desired mechanical properties.
Metal F o m i n ~ and the Finite EIernenf Method 3
1.2. IMPORTANCE OF STUDYING DEFORMATION MECHANICS
As mentioned above, the most important aspect to be considered
during metal forming operations are the deformation mechanics. The
deformation mechanics depend on the material properties of the work-
piece. geometry of the workpiece. the die geometry (such as die angle and
die length). and the processing conditions such as drawing speed and the
friction at the interface between the die and the workpiece.
Information about the deformation mechanics is required in order
to design the dies and equipment appropriately. and to predict the effect
of die design (e-g. die angle. reduction achieved. etc.) and processing
parameters (e.g. friction condition at the die/workpiece interface. draw-
ing speed. etc.) on the stress experienced by the workpiece. Calculation of
stresses in the workpiece allows one to predict the occurrence of defects.
Therefore, with the help of this information, the die design and process-
ing parameters can be altered to prevent the occurrence of defects.
1.3. STUDYiNG DEFORMATION MECHANICS OF FORMING
PROCESSES - TRADITIONAL METHODS VS. PROCESS
MODEM
In order to study the deformation mechanics for the pwpose of
assessing the manufacturability of a product. two methods may be used;
(1) traditional method and (2) process models.
The traditional method involves iterative trials of different proc-
esses before a suitable process is decided upon, followed by iterative
trials on the selected process with different die designs and process
Meta1 for min^ and the Finite Element Method 4
parameters. This is done to ensure the manufacturability of the product
under the given conditions and to assess the resulting quality of the
product obtained. However. this approach is not only expensive and time
consuming but also the final choice of die design process conditions is
based upon pure judgment of the personnel in charge and may not
necessarily yield optimum combination of mechanical properties of the
resulting product. That is, there may be other combinations of die design
and processing conditions that may yield better mechanicd properties
but may have been overlooked for two reason: (1) lack of scientific
understanding of the process and (2) financial constraints do not allow
for adequate trials to be carried out until the mechanical properties of
the end product are optimized.
On the other hand. process or mathematical models rnay be used
to simulate the process using different die geometry and process parame-
ters. The advantage of using such models is that. the optimum process-
ing conditions can be obtained a t a lower cost and in a relatively short
period of time a s compared to the traditional method. The optirnization
process is facilitated by the scientific understanding of the foming
process that is associated with this approach.
Having established the relative importance of process models. the
next step is to determine the different types of models available to the
designer. In general, the available models or techniques can be further
classified into two categories; (1) analytical techniques and (2) numerical
techniques.
Metal Formina and the Finite Element Method 5
1.4. ANALVTICAL TECHNIQUES
There are a number of analytical techniques which can be used to
determine the stress distribution and the force or stress required to
obtain a certain degree of deformation in metal forming processes. The
methods most commonly used are the slab method. the slip-line field
methc J, and the upper- and lower-bound techniques.
In the slab methoci, the workpiece being defomed is discretized into
a number of slabs. For each slab, simplifying assumptions are made
with respect to the forces/stresses acting on the slab. The theory as-
sumes that the material is rigid-perfectly plastic (no strain hardening).
isotropic and homogeneous. Considering the equilibrium of forces acting
on the slab. one can calculate the load required to obtain a certain
degree of deformation and. also. to detemine. approximately. the stress
distribution.
The slip-lviefield method is used for problems where plane strain
conditions exist. The slip-line field theory allows the determination of
stresses in a plasticaiiy defomed body even when the defornation is not
uniform throughout the body ( 1.11. From the stress distributions.
velocity fields can be calculated through plasticity equations. However,
the results obtained from the slip-line field theos. do not cornelate very
well with the experimental results [1.2]. The theory assumes that the
material is ngid-perfectly plastic (no strain hardening). isotropic and
homogeneous.
Upper- und lower-bound techniques have also been used to analyze
metal forming processes. An upper-bound solution provides an overesti-
Metal for min^ and the Finite Efement Method 6 --
mation of the required deformation force while the lower-bound solution
provides an underestimation of the force. The degree of agreement
between the upper- and lower-bound predictions is an indication of how
close the prediction is to the exact value. From a practical viewpoint. the
upper-bound technique is more important than the lower-bound tech-
nique since calculations based on the former technique will always result
in an overestimation of the load that the die will have to withstand. In
the upper-bound technique a kinematically admissible velocity field is
constmcted. Information leading to a good selection of velocity fields
cornes from experirnental evidence and experience. This method. with
experience. can deliver a fast and relatively accurate prediction of loads
and velocity distributions. The application of this method is. however.
restricted to materials which can be considered to be isotropic. homoge-
neous. and rigid-perfectly plastic.
1. 4. 1. Limitations Of Andytical Techniques
The analytical methods mentioned above provide a quick and
simple way of predicting forming loads. overall geometry changes of
deforming workpieces, and an approximate detemination of optimum
process conditions. However, there are a number of shortcomings associ-
ated with these methods. First and foremost. these methods can only be
used in cases where the material is isotropic. This fact severely limits the
application of these methods since. in actual fact. many materials are
either anisotropic to begin with. or attain highly anisotropic properties
as deformation progresses. Secondly. the analytical techniques assume
Meta1 Forming and the Finite Element Method 7
that the matenal is rigid-perfectly plastic. This assumption implies that
the flow stress of the material is a constant regardless of the amount of
total deformation experienced by the material or the rate of deformation
(strain rate). In reality. the flow stress for most materials depend upon
the total deformation and the deformation (strain) rate. In addition. the
analytical methods can only be used if the die and workpiece geometry is
simple and the boundaxy conditions are simple.
1.5. NUMERICAL TECHNIQUE - FINITE ELEMENT METHOD
On the other hand. a numerical technique called the finite element
method overcomes al1 the above mentioned shortcomings associated with
the analytical techniques. The FEM simulation provides accurate in for-
mation with regards to the forces. stresses. deformations and strains in
the workpiece. This provides a more realistic insight into the deformation
mechanics during forming. thereby. enabling a more scientific approach
to be adopted in metal forming operations. Damage criteria can then be
used to predict the formation of localized defects. This feature is ex-
tremely useful since it allows one to alter the die design and/or process
parameters to ensure that the stress state attained during the forming
process does not initiate voids in workpiece leading to catastrophic
failure during service: this consideration is extremely important if the
component is to be subjected to cyclic loads during s e ~ c e . Also. the
finite element method offers large economic savings through minimiza-
tion of development lead times. the improvement of design. and the
manufacture of higher quality and more reliable components. Moreover.
Meta1 Forming and the Finite Element Method 8
the finite element has the capability to determine. or at least predict to a
reasonable degree of accuracy. the degree of strain hardening the mate-
rial has experienced as a result of the forming process. Knowledge of the
arnount of strain hardening a material has expenenced enables the
prediction of the yield stress of the final product.
Metal Formin2 and the Finite Element Method 9
REFERENCES
G . E. Dieter, Mechanical Metallurgy, McCraw-Hill, New York
(1986).
S. Kobayashi. S-I Oh and T. Altan, Metal Forming and the Finite
Element Method, Oxford Univ, FYess, New York (1989).
O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method,
4th Edition, McGraw-Hill, London (1989) 22, 34-36.
R. D. Cook, D. S. Malkus and M. E. Plesha, Concepts and
Applications of Finite Element Analysis, 3rd Edition, John Wiley
& Sons, New York (1989) 18.
W. F. Hosford and R. M. Caddell, Metal Forming: Mechanics and
Metallur~. Prentice Hall, Englewood Cliffs, N. J. ( 1993).
E. M. Mielnik. Metalworking Science and Engineering, McGraw-
Hill. New York ( 199 1).
Finite Elettient Form trlations for Lar~e Elastic- Plastic Deforma tion 1 O
2. 1. DEVELOPMENT OF FINITE ELEMENT METHOD
Linear elastic finite element melhod (FEM) was developed first. It
was used to solve some of structural engineering problems where elastic
analysis was adequate. The application of the finite element method to
engineering problems increased tremendously since the advent of non-
linear FEM which allowed the incorporation of elastic and plastic
deformation. By using non-linear finite element analysis (FEA). engi-
neering problems that involved plastic deformations could now be
analyzed. Included in this category are buckling and post-buckling
analysis. as well as metal forrning operations.
The earliest paper on non-linear finite element analysis was
written by Turner et al. (2.11. In order to cater for geometric non-linearity
(large deformations and large rotations), 'incremental' procedures which
involved updating of nodal coordinates were adopted by Turner et al. [2.1]
and Argyris [2.2.2.3]. A sirnilar approach was used by Zienluewicz [2.4]
and Marcal 12.51 to cater for material non-linearity.
Utilization of non-linear finite element analysis to solve plasticity
problems required the developrnent of a structural tangent stiffness
rnatrix that would relate increments in load to the increments in dis-
Finite Element Formulations for Large Elastic-Plastic Deformation 11
placement. This task was accomplished by various workers
[2.4,2.6.2.7.2.8]. At the same time. it was realized that the incremental
(or forward Euler) approach which was commonly used resulted in an
unquantifiable build up of error. In order to obviate Ulis problem. Mallet
and Marcal [2.9] and Oden [2.10] used the full Newton-Raphson iteration
scheme. A modified Newton-Raphson iteration scheme was also devel-
oped and recommended by Zienkiewicz [2.4] amongst others.
2. 2. USING NON-LINEAR FINITE ELEMENT ANALYSIS IN
METAL FORMING PROCESSES
Application of non-linear finite element analysis to solve problems
involving large-strain plasticity (especially metal forming operations) has
gradually increased over the last 20 years, but widespread industrial
recognition of the technique's value has been lacking until recently
12.111. The employrnent of the finite element method has greatly en-
hanced the analysis capability for metal forming research. The finite
element method (FEM) is now being used by many industrial firms for
the purpose of analyzing metal forming operations.
2.3. CURRENT AREAS OF RESEARCH
Presently. research is being conducted for the purpose of developing
codes that can accurately simulate large strain plastic deformations. as
in metal forming analysis. Success has however been limited. The most
commonly sirnulated processes are upsetting [2.13] and sheet drawing
Finite Element Formulations for Large Elastic-PIastic Defornation 12
12.14. 2.151. There has been some work in simulating metal forming
processes such as rolling. extrusion and stamping: these processes
involve large plastic deformation and large temperature changes. To
tackle such problems. thermo-mechanical coupled finite element analysis
have been developed [2.16.2.17]. The finite element method has also been
used to simulate superplastic forming 12.181. As mentioned. success has
been limited because of the problems faced by researchers. These prob-
lems are now discussed.
2.3. 1. M o d e h g of Boundary Conâitions
Even today. boundary conditions. including die-workpiece contact
problems and accurate representation of friction conditions. present
many problems for finite element simulations 12.1 1.2.191.
According to Berry [2.19]. the real missing link in the application
of finite element methods to metal forrning processes has been a general.
automatic algorithm for treating the complicated contact that occurs
during forming including workpiece entxy and exit from the die.
Furthemore. accurate modeling of friction conditions have not
been successful. Friction remains one of the most difficult aspects to
incorporate properly into a finite element model. and phenornena such as
localized lubricant breakdown have not been considered at al1 [2.11].
Nevertheless. some progress has been made towards improving the
techniques used to model friction conditions. Attempts have been made
to incorporate velocity-dependent tenns in iterative solutions [2.20.2.2 11
or Vary surface restraint on a purely empirical basis 12.221.
Finite Elenlent Formulations for Large Elastic-Plastic Deformation 13
2. 3. 2. Modeling Behavior of Anisotropic Matericils
Anisotropic material behavior. which includes initial anisotropy.
induced anisotropy and anisotropic hardening. is another area that has
not received a great deal of attention [2.11]. According to Dogui and
Sidoroff [2.23]. the theoretical aspects of large-strain anisotropic elas to-
plasticity finite element formulation are not yet fully understood. The
best founded formulation is quite complicated as it requires two different
rotations: the Jaumann rotation which is needed for the elastic part
cannot be used for the plastic part because it may lead to unreasonable
behavior (2.231.
2. 3. 3. Modeiing of Kinematic Hardehg
Bauschinger effect is exhibited by many engineering materials and
is extremely important as it influences the generation of residual stresses
during forming operations. In order to incorporate the Bauschinger effect
into finite element cornputer codes. various kinematic hardening models
have been used. Those models which are considered valid for finite strain
are in active current use. However. it appears from research findings
presented that they involve huge errors 12.241. Therefore. there is a need
for algorithms which can accurately sirnulate kinematic hardening along
with anisotropic hardening.
Finite Element Formulations for Large Elastic-Plastic Deformation 14
2. 3. 4. Rediction and Effect of Damage Formation
Meta1 forming operations involve large amounts of plastic strain
and. therefore. depending upon the stress state that exists within the
workpiece. there is a possibility of nucleation and growth of voids and
cracks. As a consequence of this phenornenon. calkd damage. the
mechanical properties of the material deteriorate. Hence. the material's
ability to resist catastrophic failure during subsequent forming processes
or service diminishes tremendously. Failures are usually ductile in
nature and are a consequence of plastic instability which occurs as a
result of void and crack formation. Material damage is responsible for
phenornena such as central bursts in extrusion. alligator cracks in
rolling. and shear cracks in tube drawing. Modeling of damage effects
requires one to incorporate the effect of damage on the mechanical and
physical properties of the material and. more importantly. utilize a
fracture criterion to predict catastrophic failure.
2. 3. 4. 1. Change in elastic properties
Extensive plastic deformation may affect the apparent elastic
modulus and Poisson's ratio by virtue of one of these three mechanisms:
(1) nucleation and growth of voids, (ii) nucleation and growth of cracks,
and (iii) change in material texture. Several studies have been devoted to
the determination of effect of the presence cavities and planar cracks on
the apparent elastic moduli. Examples of such studies are ref. [2.25] and
[2.26].
Finite Elemen t Formulations for Large Elastic-Plastic Deforniation 15
2. 3. 4. 2. Effect on plastic deformation behauior
In continuum plasticity theory. there are two elements of the
constitutive mode1 that are used at the macroscopic level i2.271: (i) the
true stress-straui cuwe and (ii) the yield surface. Both of these are
affected by plastic deformation and darnage accumulation.
For a typical stress-strain curve it is obsenred that. after exceeding
the limit of proportionality, the material strain-hardens continuously up
to a maximum stress ou~s. known as the ultimate tensile strength of the
material. Thereafter, due to an advanced evolution of damage in localized
regions as a consequence of localization of deformation into shear bands
and/or occurrence of necking, the load decreases with increasing defor-
mation until fracture occurs. This phenornenon. also called plastic
instability , has an important influence on the redistribution of stresses
and strains within the whole structure and directly influences the
occurrence of failure. Modeling this phenornena is extremely important
but. unfortunately, not a trivial task. Metallographic work has been done
on some metals in order to determine the damage pattern that leads to
final failure of the material [2.28].
The amount of porosity in the material also influences the yield
surface. Some theoretical and numerical analysis of solids containing a
spherical void network has been conducted in order to quanti@ the
influence of the voids on the yield surface i2.291.
Finite Element Formulations for Large Elastic-Plastic Deformat ion 16
2. 3. 4. 3. Modeling of damage effects
A mathematical formulation has been presented by Mathur and
Dawson [2.30] for the purpose of analyzing damage accumulation by the
nucleation and growth of voids. In their study two different approaches
were introduced. The numerical predictions that resulted from calcula-
tions showed good agreement with results reported on sheet drawing
experiments. The effect of die angle geometry on the accumulation of
material damage was also studied.
2. 3. 5. Microstructure Prediction
Finite element analysis results can be coupled with metallurgical
knowledge for the purpose of predicting anisotropy and texture evolution.
General Electric have devised an expert system to accomplish this task
[2.3 11. However, a substantial amount of work needs to be done in this
2. 3. 6. Testhg and Evaluation of Modele
According to Jain [2.32]. the development of proper. applicable.
and reliable materials data and constitutive models for materials has
been largely neglected or ignored. Consequently, the testing, validation
and widespread implementation of process modeling has not progressed
at a satisfactory rate. Jain (2.321 goes to great lengths to emphasize the
importance of proper, valid. accurate and applicable data for input in
fuiite element analysis. After dl. the validity of the results depends very
much on the accuracy of the input data.
Finite Element Formulations for Large Elastic-Plastic Deformation 17
As mentioned previously . it is extremely important that materials
data input into the finite element simulations be accurate because the
final result can only be as accurate as the input data regardless of the
accuracy of the simulation. Although accurate information about
materials behavior is readily available, there is a need to represent the
stress-strain behavior of a material in the form of a constitutive equa-
tion relating the strain to the stress in order for the data to be useable
by the finite element method. Previous research 12.331 has, however.
indicated that the currently used constitutive equations cannot describe
material behavior accurately over the entire strain range and hence are
deemed inadequate for usage in finite element calculations. Nevertheless.
the practice of using these equations has prevailed because there has
been no alternative approach.
Keeping this in mind. prelirninary research conducted during the
Master's thesis 12.341 was focused on testing zirconium-niobium CANDU
pressure tube material and determining whether the existing constitutive
equation is capable of describing materials behavior accurately. During
this research. it was found that the currently used power law as well as
the Ramberg-Osgood equation could not describe materials behavior
accurately . The power law equation has two great shortcomings: (i) the power
law could only describe material behavior in the fully developed plastic
region and. hence. the behavior of the material in the transition region
between elastic and plastic could not be modeled, (ii) the behavior in the
Finite Elemegt Formulations for Large Elastic-Plastic Deformatiorz 18
fully developed plastic region could not be modeled accurately when the
strain hardening behavior was cornplicated.
The Ramberg-Osgood equation on the other hand, a t least theo-
retically, could describe the deformation behavior of the material over the
elastic region, the transition region between elastic and plastic as well as
the fully developed plastic region. From the previous research 12.33,
2.341. it was found that this was not the case in practice. The problem is
that the equation in its original f o m was developed by Ramberg and
Osgood who based their calculations on heat treated steel - therefore.
this equation was capable of describing the behavior of steel very accu-
rately. However. zirconium strain hardens in a different way and hence
these equations were incapable of describing it's behavior accurately. It
was noted that the strain hardening parameter that determines the
behavior the transition region. namely a, needs to be a variable and not
be a constant a s previously proposed. This observation is important as
al1 material will behave differently in the transition region depending
upon the slip systems that are activated and the interaction between
them. The strain hardening behavior in the fully plastic region will also
be different depending upon the material, but this has already been
considered by Ramberg and Osgood in the hardening parameter n. In the
previous study 12.33, 2341, the a was denoted as variable instead of a
constant and. as a result, a much better curve-fit was obtained that
facilitated the description of materials behavior quite accurately.
Finite Element Forniula tions for Large Elastic-Plastic Defornia tion 19 - --
One problem faced. however. was that zirconium exhibited a very
complicated strain hardening behavior in certain directions. especially
the circumferential direction where the strain hardening behavior
changes considerably as the material deforms. TWo Ramberg-Osgood
equations were required to describe material behavior accurately. one
which was valid in the elastic region. transition region and small plastic
strains region and the second equation valid for higher plastic strains.
The use of two equations instead of one was cumbersome but unfortu-
nately could not be avoided. This implied that the Raxnberg-Osgood
equation did not have universal applicability.
2. 3. 7. Numerical Simulation of Forming Rocesses
The subject matter is not trivial by any means by virtue of the fact
that there are many factors that combine together to make the problem
vexy cornplex. As mentioned. there has been considerable amount of work
done on simulating upsetting tests where the friction conditions and the
contact issues between the die and the workpiece can be resolved easily.
However, there has been limited arnount of work done to sirnulate the
bar drawing operation because of the complexity involved in simulating
the friction conditions as well as the contact problems. Sirnilar problems
are faced during simulation of the tube sinking process. The tube draw-
ing process. however. is even more complicated by the presence of the
mandrel for which the boundary conditions as well as the contact
problems need to be resolved.
Finite Elenlent Formulations for Large Elastic-Plastic Deformation 20
2.4. OBJECTIVES
Based on the above background analysis, the main objective is to
simulate the tube drawing process. and at the same time. tackle some of
the important problems that research scientists face. Specifically. the
following objectives are established:
(1) To develop a constitutive equation form that cari describe the
behavior of most materials by a single equation. including materi-
als that demonstrate strain softening. The main goal of this study
is to enable the stress-strain response of al1 materials to be de-
scribed accurately by means of a constitutive equation that can be
easily incorporated in FEM codes; input of accurate materials data
into FEM prograrns should yield more accurate results.
(II) To simulate the tube drawing process with the presence of two
contact surfaces namely the die wall and the mandrel. A similar
study has not been found in literature. Correspondhg finite ele-
ment formulations for the cylindrical coordinate system will be ex-
plicitly developed together with formuations of the method to deal
with the complex contact problems including the method used to
relieve stress at die exit. Furtherrnore. the new constitutive equa-
tion developed will be incorporated into the finite element code as
tested for accuracy. Finally, the finite element simulation results
will be compared with experimental results to determine the accu-
racy of the developed finite element program.
Finite Element Formirla fions for Large Elastic-Plastic Deforma fion 21
REFERENCES
M. J. Turner. E. H. Dill. H. C. Martin and R. J. Melosh. "Large
deflection of structures subject to heating and external load." J .
Aem. Sci. 27 (1960) 97- 106.
J. H. Argyris. Recent Advances in Matrix Methods of Structural
Analvsis, Pergamon Press ( 1964).
J. H. Argyns. "Continua and discontinua." Proc. Con$ MatrDc
Methods in Struct. Mech. Air Force Inst. of Tech.. Wright Patter-
son Air Force Base, Ohio (Oct. 1965).
O. C. Zienkiewicz. The Finite Element Method in Engineering
Science, McGraw-Hill, London ( 197 1).
P. V. Marcal. "Finite element analysis with material non-
linearities-theory and practice." Recent Advances in Mat& Meth-
ods of Structural Analysis & Design, ed. R. H. Gallangher et al..
The University of Alabama Press (1971) 257-282.
P. V. Marcal and 1. P. King. "Elastic-plastic analysis of two-
dimensional stress systems by the finite element method." Int. J .
Mech Sci. 9 (1967) 143-155.
Y. Yamada. N. Yoshimura and T. Sukarai, "Plastic stress-strain
matrix and its application for the solution of elasto-plastic
problerns by the finite element method." Int. J. Mech. Sci. 10
(1968) 343-354.
Finite Element Formulations for Large Elastic-Plastic Defornzation 22
I2.81 0. C. Zienkiewicz. S. Valliapan and 1. P. King. Elasto-plastic
solutions of engineering problems - Initial stress finite element
approach." Int. J. Num. Meth. Eng. 1 (1969) 75- 100.
[2.9] R. H. Mallet and P. V. Marcal. "Finite element analysis of non-
linear structures." J. of Struct. Diu. 94 (1968) 208 1-2 105.
I2.101 J. T. Oden. "Numerical formulation of non-linear elasticity
problems." J. of Struct. Diu. 93 (1967).
[S. 1 11 P. Hartley. 1. Pillinger and C. E. N. Sturgess. "European develop-
ments in simulating forming processes using three-dimensional
anaiysis." JOM 43 (10) (1991) 12.
12.121 R. Duggirala. "Using the finite element method in metal forming
processes." JOM 42 (2) (1990) 24.
I2.131 L. M. Taylor and E. B. Becker. "Some computational aspects of
large deformation. rate independent plasticity problems." Comp.
M e t h Appl. Mech & Eng. 41 (1 983) 25 1 -277.
I2.141 S. C-Y. Lu. E. J. Appleby. R. S. Rao. M. L. Devenpeck. P. K.
Wright and 0. Richmond, "A numericai solution of strip drawing
employing measured die-interface bounday conditions obtained
with transparent sapphire dies." Numerical Methods in Indusbial
FormUig Frocesses. Swansea (1982) 735-746.
[2.15] S. C-Y. Lu and P. K. Wright. "Finite element modeling of plane
strain strip drawing with interface friction," J. of Eng. for Ind.
110 (1988) 101-1 10.
Finite Element Formulations for Large Elastic-Plastic Defontration 23
12.161 J. van der Lugt and J. Huetink. 'Thermal mechanically coupled
finite element analysis in metal forming processes." Comp. M e t k
Appl. Mech. & Eng. 54 (1986) 145-160.
12.171 O. C. Zieiikiewicz. "Flow formulation for numerical solution of
forming processes." Numencal Anafysis of Forming Processes, Ed.
J. F. T. Pittman. O. C. Zienkiewicz, R. D. Wood and J. M. Aiex-
ander, John Wiley & Sons (1984) 1-44.
12.181 N. Rebelo, "FEA of forming," Machine Design 60 ( 13) ( 1988) 12 1 -
123.
I2.191 D. T. Berry. "Stamping out forming problems with FEA." Mech.
Eng. 110 (1988) 58-62.
I2.201 P. Chabrand. Y. Pinto and M. Raous. "Numerical modeling of
friction for metal forming processes." Modeling of Metalforming
Processes. ed. J. L. Chenot and E. Onate, Kluwer, Dordrecht, the
Netherlands ( 1988) 93-99.
12.211 J. E. Jackson, Jr. et al., "Some numencal aspects of frictional
modeling in material forming processes." mction and Materials
Characterizations, ed. 1. Haque et al.. ASME, New York (1988) 39-
46.
12.221 I . Pillinger. P. Hartley and C. E. N. Sturgess. "Modeling of fric-
tional tool surfaces in finite element metalforming analyses."
Modeling of Metalforming Processes, ed. J. L. Chenot and E.
Onate, Kluwer. Dordrecht, the Netherlands (1988) 85-92.
12.231 A. Dogui and F. Sidoroff. "Large strain formulation of anisotropic
elasto-plasticity for metal forming." Computational Methods for
Finite Element Formulations for Larae Elastic-Plastic Deformation 24
Predicting Material Processin. Defects. ed . M. Predeleanu . Elsevier.
Amsterdam ( 1987) 8 1-92.
12.241 E. H. Lee. "Finite deformation effects in plasticity analysis. "
Numerlcal Analysis of Forming Rocesses, Ed. J. F. T. Pittman. O.
C. Zienkiewicz. R. D. Wood and J. M. Alexander. John Wiley &
Sons (1984) 385.
(2.251 B. Budiansky and R. J. O'Connell, "Elastic moduli of a cracked
solid," Int. J. Solids & Stmct. 12 (1976) 8 1-96.
12.261 M. Hlavacek. "Effective elastic properties of materials with high
concentration of aligned spheroidal pores." Int. J. Sotids & Struct.
22 (1986) 3 15-332.
12.271 M. Predeleanu. "Finite strain plasticity analysis of damage effects
in metal forming processes. " Cornpututional Methods for Predicting
M a t e r a Processhg Defects. ed. M. Predeleanu. Elsevier. Arns ter-
dam (1987) 295-307.
12.281 A. Pineau, "Review of fracture rnicromechanics and a local
approach to predicting crack resistance in low strength steels."
Advances in Fracture Mechanics. vol. 2. ed. D. Francois, Per-
gamon Press (198 1) 553-577.
(2.29) A. L. Gurson. "Continuum theory of ductile rupture by void
nucleation and growth: Part I Yield criteria and flow rules for po-
rous ductile media. J. Eng. Mater. & Tech. (1 977) 2- 15.
(2.301 K. K. Mathur and P. R. Dawson. "Darnage evolution modeling in
bu& foming processes." Computationd Methods for PTedicting Ma-
Finite Element Fomulations for L a r ~ e Elastic-Plastic Defornation 25
terial Processing Defects. ed. M. Predeleanu. Elsevier. Amsterdam
(1987) 25 1-262.
K. J . Meltsner. "A me tallurgical expert system for interpreting
FEA." J O M 43 (10) (1991) 15.
S. C. Jain. "Recognizing the need for materials data: The missing
link in process modeling." JOM 43 (10) (199 1) 6.
2. H. A. Kassam. 2. Wang and E.T.C. Ho. "Constitutive Equa-
tions for a Modified Zr-2.5 wt?40 Nb Pressure Tube Material," Mat.
Sci. & Eng. A158 (1992) 185- 194.
2. H. A. Kassam. "Deformation Behavior of Zr-2.5wt%Nb Alloy."
M a s ter's Thesis ( 1 992).
Finite! Elernent Formulations for L a r ~ e Elastic-Plastic Deformafion 26
3. 1. INTRODUCTION
Most finite element programs use uniaxial stress-strain curve data
as a basis for representing material behavior in three dimensions.
Therefore, it is extrernely important that accurate data be input into the
finite element simulations as the results obtained are only as accurate
as the input data. Frequently, in commercial programs, the stress-strain
behavior data is requested in the form of the power law or the Ramberg-
Osgood equation 13.1 1.
The Rarnberg-Osgood equation, because of its simplicity. elegance and
effectiveness in describing stress-strain relations for a number of
materials (especially steel), has become very popular. RiLEM (Reunion
Internationale des Laboratones d'Essais et de Recherches sur les Materi-
aux et les Constructions). France, uses the Ramberg-Osgood eqn. as a
standard basis to describe stress-strain relations of engineering materials
under monotonic and uniaxial tensile loading 13.2, 3.3. 3.41.
Unfortunately. previous research (Kassam et al. 13.51) on Zr-
2.5wtYoNb CANDU pressure tube material showed that the conventional
Rarnberg-Osgood equation cannot describe the stress-strain behavior in
Finife Elemenf Formulations for Larae Elastic-Plastic Defornation 27
al1 instances, especially. when a rnaterial exhibits complicated strain
hardening behavior or demonstrates strain softening.
Therefore. it is instructive to briefly review the Ramberg-Osgood
equation not only because of its importance, but also due to the fact
that the basis of the Rarnberg-Osgood equation instigated a new consti-
tutive equation.
3.2. THE CONVENTIONAL RAMBERG-0SG00D EQUATION
The Ramberg-Osgood equation in its original form is given below
i3.11:
where E is the elastic constant and K and N are the parameters which
have to be determined from experirnental data. The dimensionless form of
the Ramberg-Osgood equation. however, is more popular and is written
as:
where
Finite Element Formulations for Large Elastic-Plastic Deformation 28
1 - ml a, =
ml (3.41
and 0, is the secant yield strength that is defmed to be equal to the
stress at the intersection of a line through the origin, having a slope
equal to mlE (Ocm, cl). with the experirnentally obtained uniaxial
stress-strain curve. The constant ml is usually chosen to be 0.7 so that
the secant yield strength is close to the 0.2% yield stress. Ramberg and
Osgood 13.11 found this to be true for steels. Since rn, is usually chosen
to be equal to 0.7, it implies that a, = 3/7. Substituting this a, value
in equation (2) yields the following form of the Ramberg-Osgood equa-
tion:
The parameter N can be evaluated by using the following relationship
proposed by Ramberg and Osgood (3.1):
0 1 log - 0 2
Finite Element Formulations for Large Elastic-Plastic Deformafion 29
where m2 c m be chosen to be equal to 0.85 and a2 is the secant yield
stress obtained for this m, value. The above method of calculating the
parameters for the Ramberg-Osgood equation is known as the three
parameters rnethod
3. 2. 1. Roblems with the Conventional Ramberg-Osgood
wnm During previous research 13.51, the parameters for the power law
equation and the Ramberg-Osgood equation. along the radial direction of
the Zr-2.5 wtVo Nb pressure tube were calculated and are given below:
power law eqn. : a = (1 .28~10~) E
7 5
Rambeig-Osgood eqn. : E = - + E
Finite Element Formulations for Large Elastic-Plastic Deformntion 30
O Experimental data points
---- Power law eqn.
- Thrcc Paramcters Melhod
O 1 2 3 4 5 6 7 8
True Strain (%)
Figure 3.1 Cornparison between the effectiveness of the Power Iaw and
the Ramberg-Osgood equation.
In order to compare the effectiveness of the above power law and
the Rambeg-Osgood equations in descnbing the behavior of the material
in the radial direction. the experimental data points were plotted to-
gether with the above equations in Fig. 3.1. I t was noted that the power
law equation adequately describes the behavior of the material in the
fully developed plastic region only . The Ramberg-Osgood equation
obtained can satisfactorily describe the behavior of the material in the
elastic and transition region between elastic and plastic. However. it
does not adequately descnbe the material behavior in the fully developed
plastic region. This problem arose due to the fact that zirconium is a
Finite Element Fomldations for Larae Elastic-Plastic Deformation 31 -- - - - -. - - - - -- -- - - - - - - - -
highly anisotropic material and the grain orientation with respect to the
applied stress direction determines not only the stress level at which
plastic deformation will commence but also the strain hardening behav-
ior. Consequently. the overall strain hardening behavior is affected and.
therefore. the strain hardening behavior exhibited at the early stages of
plastic deformation will change considerably as deformation progresses.
Bearing this in mind, a different approach was taken during this based
upon the ideas of Ramberg and Osgood, in order to obtain a better curve-
fit. Another fact that affects the usability of this equation is that the a,
value is based upon steels. Many materials exhibit strain hardening
behavior that is quite different from steel.
3.3. NEW APPROACH FOR THE RAIlbBERG-OSGOOD EQN.
The new approach for the Rarnberg-Osgood equation that was
proposed in previous work i3.51 utilizes the Ramberg-Osgood equation
given in eqn. (3.2). but uses a new technique for determining the parame-
ters a, and Ai in order to obtain a better curve fit. The first fundamental
concept that lay behind this is the fact that the a, parameter had to be a
variable that would be dependent on the material properties. Secondly. it
was important to realize that the early stages of plastic deformation
should not dictate what would happen as deformation progresses as
material behavior can change considerably. Therefore, d u ~ g this
research, a technique was developed which would calculate the strain
Finite Element Formulations for L a w e Elastic-Plastic Deformation 32
hardening coefficient N based on deformation behavior over a larger
strain range. The new method that was developed for obtaining the
parameters will be described briefly and one can refer to ref. 13.51 for
greater detail.
By rearranging equation (3.2) and taking natural logs on both
sides. the following equation is obtained:
Therefore. a graph of ln((&/&,) - (o/o,) ) {y-axis) versus ln(o/o,) {x-axis)
will veld a straight line if the stress-strain curve can be described by the
Ramberg-Osgood equation. The slope of the line. N . and the y-intercept.
ln a,. are calculated by conducting linear regression analysis on the ln-
ln data. In this thesis. this is referred to as the Ln-ln plot method.
L L V V I , -0 - w -- --
1 O00 -
Three parameters Method
O 1 2 3 4 5 6 7 8
True Sfrain (S)
Figure 3.2 Cornparison between the three parameters method and the In-ln plot method
Finite Element Formulations for Large Elastic- Plastic Deformation 33
Thereafter. the In-ln plot rnethod was used for calculating the
parameters of the Ramberg-Osgood equation for many materials and it
proved to be more successful than the three parameters method 13.51.
(Fig. 3.2). In spite of the great improvement that resulted from this new
technique. it was found that when the material showed complicated
strain hardening behavior (which was due to the change in the underly-
ing micro-mechanisms by which plastic deformation took place), a single
Rarnberg-Osgood could not adequately describe material behavior over
the entire deformation range. In such cases. this new technique (ln-ln
plot method) was extended to obtain two Rarnberg-Osgood equations.
each equation being valid in a certain range of deformation as shown in
Figure 3.3.
1400 I
True Strain (%}
Figure 3.3 Behavior of modified Zr-2.5 wt% Nb dong the circumferential direction being described by two Rarnberg-Osgood equations evaluated using In-ln plot method.
Finite Elernezz t Formulations for Large Elasf ic-Plastic Defortnation 34
The necessity to establish two Rarnbeg-Osgood equations can also
anse for other cases as well. e.g.. in the case where high material ductil-
ity necessitates that the equation be valid over large ranges of deforma-
tion (>25% strain). In fact, a nurnber of materials do not obey the
Ramberg-Osgood equation over large ranges of deformation strain.
Therefore. two Ramberg-Osgood equations can be used to describe
material behavior more accurately if a single equation seems inadequate.
However. exceptions to this rule are materials that exhibit strain sof-
tening. In this instance. not even two equations can describe material
behavior. This shortcoming in the conventional and modified Ramberg-
Osgood constitutive equations is attributed to the fact that these
equations can only describe rnonotonic strain hardening behavior.
In the present study, tests conducted on aluminum 6061 and
aluminum A356 indicated that a single Rarnberg-Osgood equation can
describe the behavior of these materials very accurately. Problems have.
however, been encountered in ?0/30 brass where even two Ramberg-
Osgood equations seem to be incapable of describing the material
behavior accurately. These will be discussed later.
The next obvious step would have been to establish three Ramberg-
Osgood equations. each of which would only be valid in a certain range
of deformation. However. it becomes rather cumbersome to develop a
number of Rarnberg-Osgood equations, each one being valid only over a
small range of deformation. Furthemore. as mentioned earuer, the
Ramberg-Osgood equation cannot descnbe behavior of materials that
exhibit strain-softening .
Finite Element Fonn~rlations for Large Elastic-Plastic Deformation 35
Therefore, one of the prirnary goals of this research was to explore
the possibility of developing a new equation that could describe the
behavior of any material accurately using a single equation so as to
avoid the inconvenience of developing several equations which describe
the behavior of any given single material. Also, it was critical that the
newly developed equation should adequately describe strain softening
behavior.
This work effort led to the development of a new f o m of constitu-
tive equation which is referred to as the Alpha constitutive equation The
nomenclature for this equation will become apparent when the basis of
this equation is described in the results and discussion section.
Finite Elenien t Forrni~lations for Large Elastic-Plasf ic Defornation 36
REFERENCES
13.11 W. Rambeg. and W. R. Osgood, "Description of stress-strain
curves by three parameters." Tech. Note 902. Nationnl Advisory
Cornmittee of Aeronautics. Washington. D. C. ( 1 943).
(3.21 Kato. B.. , 'Tension testing of metallic structural materials for
determining stress strain relations under rnonotonic and uniaxial
tensile loading." RILEM draft recommendation. MateriaLs & Struc-
tures. Vol. 23 (1990) pp. 35-46.
13.33 Kato. B.. Aoki, H.. and Yamanouchi. H.. 1990. "Standardized
mathematical expression for stress-strain relations of structural
s tee1 under monotonic and uniaxial tension loading." RiLEM
Tech. Comm.. Materials & Shuchves, Vol. 23 (1990) pp. 47-58.
13.41 Martino. A. D.. Landolfo, R.. and Mazzolani. F. M.. "The use of
the Rarnberg-Osgood law for materials of round-house type,"
RlLEM Tech. Comm.. Materials & Shichlres. Vol. 23 (1990) pp.
59-67.
13.53 Kassam. 2. H. A., Wang, 2.. and Ho. E.T.C.. "Constitutive
equations for a rnodified Zr-2.5 wt% Nb pressure tube material,"
Matends Science & Engineering, Vol. A1 58 (1992) pp. 185- 194.
Zulfikar H. A. Kassam Finite Element Analysis for Tube Draruing 37
It has been observed that while the theoretical foundations for conduct-
h g finite element analysis on elastic and plastic axisymmetric problems
has been well established. al1 this information has never been compiled
in a single literature. i.e.. al1 the information is hagmented making it
very difficult to understand even for those familiar with the basics of
finite element method. In addition. many equations are only stated in a
cryltic forrn making it ves, difficult to understand. Furthemore, this
theory is advanced that it is not taught even in mechanical or materials
graduate schools in North Arnerica or Europe with the exception of a
handful of Universities who are very actively involved in this research.
Therefore, it seem appropriate that the fundamental theory be
presented here in a systematic and simple manner so that persons
wishing to dwelve in further research in this aspect can use this idorma-
tion. Aiso. al1 the equations are presented in a detailed manner so that
the subject matter is easier to understand and appreciate.
Zulfikar H. A. Kassarn Finite Element Analysis for Tube Drawing 38
4. 1. INTRODUCTION
The axisyrnmetric case can be simplified to a two-dimensional
problem. By syrnmetry. two components of displacement. that in the r
(radial)-direction and z (axial)-direction. define completely the state of
strain and. hence. the state of the stress. The displacements in the r
and z - directions are denoted by u and v .
Figure 4.1. Element of an axisymmetric solid [4.1].
respectively. Figure 4.1 shows a triangular element and the volume of
revolution it encompasses: al1 integrations over the volume of an element
have to take this aspect into consideration. The triangular element has
been use here. but the principles are general and can be applied for other
element types.
Zitlfikar H. A. Kassani Finite Element Analysis for Tube Drawing 39
In spite of the fact that axîsymmetric problems are basically 2D
problems. four components of strain have be considered as compared to
three components in the usual 2D plane stress and plane strain prob-
lems. This is due to the fact that any radial displacement automatically
induces a strain in the circumferential direction, and as the stresses in
this direction are not zero. this fourth component of strain and of the
associated stress has to be considered. Therein lies the main difference
between the axisyrnmetric case and plane stress/plane strain problems.
4. 2. DISPIACENIENT FUNCTIONS
A typical triangular element with the nodes i. j. k numbered in an
anticlockwise order is shown in Figure 4.1. Node i has nodal coordinates
ri and zi , u i and u i are the displacements of node i in the r and z
directions. respectively. etc. The displacements within a triangular
element can be given by
where
ai = rjZk - rkZj
aj = r k Z i - riZk
ak = riZj - r j Z i
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 40
The equations for vertical displacement v are also sirnilar as given below:
24 = det
The displacements can therefore be written as
where
1 ri Zi
1 9 2J 1 rk Zk
The chosen displacement function automatically guarantees
= 2 . (area of triangle ijk ) (4. lc)
continuity of displacements with adjacent elements because the dis-
placements v a q linearly along any side of the triangle and. with identical
Zulfikar H. A. Kassam Finife Element Analysis for Tube Drawinp 41 - -- --
displacement imposed at the nodes. the same displacement will clearly
exist al1 along an interface.
Ngure 4.2. Strains and stresses involved in the analysis of axisymmet-
ric solids,
4. 3. STRAIN
As mentioned earlier. four components of strain have to be consid-
ered in this case. These strains and the associated stresses are shown in
Fig. 4.2.
Zulfikar H . A. Kassam Finite Element Analvsis for Tube Drawina 42
Therefore, in the axisyrnmetric case with triangular element, [BI is given
by
and {u) is given by
Using the relations for the triangular element given in eqn. (4.3). the [BI
matrix as defined in equation (4.5) is given by
Zulfïkar H . A. Kassam Finite Element Analusis for Tube Drawinrr 43
It is clear from above that in the axisymmetric case the [BI matrix
involves the r and z t e m s and hence the strain is no longer a constant
within the element as in the plane stress or strain case but a function of
position within the element. This variation is due to the tenn which
is a function of r and z and. hence, depends on the position within the
element. If the imposed nodal displacements are such that u is propor-
tional to r then indeed al1 the strains within the element will be con-
stant.
4. 3. 1. Initial Strain (Thermal Strain)
In this case the initial strain vector is given by:
In general. the initial strain may depend on the position within the
element. However, to simplify matters. the initial strain is usually
defined by an average value which is constant throughout the element.
Zulfikar H. A. Kassntn Finife Eiement Analvsis for Tube Drawina 44 - -- --
In the case of an isotropic material, the thermal ~trains introduced
due to an increase AT in temperature of an element with a coefficient of
thermal expansion a is given by
It should be noted that no shear strains are caused by dilatation (change
in volume) due to change in temperature. In the case of orthotropic
material. the initial strain vector is given by
4.4. ELASTICITY MATRIX
The elasticity matrk [De] relates the strains {E} to the stresses {a) in
the forrn
Zulfikar H. A. Kassam Finife Element Annlvsis for Tube Drawina 45
4.4. 1. Orthotropic Materials
The [De] matrix for orthotropic materials is given by
where
Zulfikar H. A. Kassam Finite Element Annlysis /or Tube Drawing 46
D14 = O (4.12d)
and
The definition for Poisson's ratio is:
Zulfikar H . A. Kassam Finite Element Analysis for Tube Drawing 47 . . -- - - -
strain in the '2' direction "12 = ' strain in the '1' direction (direction of applied stress)
The shear modulus is equal to:
4. 4. 2. Isotropie Materiale
For the specid case of isotropie materiais. El = % = E3 = E. v12 =
v13 = vZ3 = v and G12 = G. In this case. therefore, it can be shown that
the [De] matrix is given by
Zulfikar H . A. Kassam Finite Element Analysis for Tube Drawinn 48
4.5. ELEMENT STIFFNESS MATRIX
There are various ways in which an element stiffness matrix can
be derived. One of the methods that can be used is the UQnQfiOnd
method. The variational method can be used to derive the stiffness
matrix for any elastic problem. The first step of the variational method is
to express the potential energy of the element in terms of the nodal
displacements.
where Lf is the potential energy, Ue is the elastic strain energy and W is
the work done. The elastic strain energy U, is given by 14.21
where the integral is evaluated over the volume of the element. dV, and
the work done is given by
where, as before, {u} is the displacement vector and {fi is the resultant
nodal force vector. Having determined al1 the terms related to the poten-
tial energy the next step is to rninimize the potential energy by differenti-
Zulfikar H. A. Kassnm Finite Element Analysiç for Tube Drnwing 49
athg the potential energy 17 with respect to the displacement lu), and
equating it to zero, or
The above statement implies that for equilibrium to be ensured the total
potential energy must be stationary for variations of admissible dis-
placements [4.1]. In stable elastic situations the total potential energy is
not only stationary but is a minimum. Thus the finite element method
seeks such a minimum within the constraint of an assumed displace-
ment pattern.
The greater the number of degrees of freedorn. the more accurate
the solution will be thereby ensuring complete equilibrium. provided
that the tme displacement can. in the limit. be approximated by the
displacement functions. The necessary convergence conditions for the
finite element process could thus be derived.
Substituting equations (4.15) and (4.16) into eqn. (4.14) and
differentiating the potential energy with respect to the displacements
according to eqn. (4.17) results in the following expression
(j. I B I ~ D ~ I I B I ~ V ) lui - 10 = 0
Zulfikar H. A. Kassam Finite EJemenf Analysis for Tube Drawing 50
Rearranging yields the following equation
where
and [LI is known as the element shmess ma^.
Now that the element stiffness matrix [k] has been evaluated, al1
these matrices can be assembled to forrn the global stiffness matrix.
Thereafter. the global stiffness matrix has to be reduced to take into
account the boundary conditions. The reduced stiffness matrix can then
be inverted and pre-multiplied by the resultant of al1 the loading vectors
to solve for displacements. This information c m then be used to calcu-
late strains and stresses.
As denved before. the element stiffness matrur for the general
elastic case is defined by the following relationship
Zulîïkar H. A. Kassam Finite Element Analysis for Tube Drazuina 51
vol
In the axisymmetric case. dV = 2 x r dr dz.. Therefore,
In the axisymmetric case. since the [BI matrix depends on the r
and z. the integration of the above equation is not straight forward. The
stiffness matrix has to be evaluated by numerical
second option also exists: that of evaluating [B
(one point numerical integration)
integration. However. a
1 at a centroidal point
Therefore. in this case, the elernent stiffness matrix is given by
where A is the area of the triangle. I t has been stated by Zienkiewicz and
Taylor [4.1] that if the numerical integration is of such an order that the
Zulfikar H . A. Kassam Finite Element Analysis for Tube Drawinn 52
volume of the element is exactly determined by it. then in the limit of
subdivision. the solution will converge to the exact answer.
In this case, since the volume of the body of revolution is given
exactly by the product of the area and the path swept around by its
centroid. the solution will converge to the exact answer as the element
size is reduced. The 'one-point integration approach' described above.
surprisingly. yields more accurate results as compared to the exact
integration. This is because the exact integration yields ln(ri / r k ) terms
which cannot be evaluated accurately since to the ratio (ri /rk ) ap-
proaches unity as the distance from the centroid increases and/or the
element size is very small.
4.6. LOADING CONDITION
4. 6. 1. Externat Nodai Forces
In the case of plane stress and plane strain problems the assignîng
of the extemal loads is straightforward. However. in axisymmetric
problems. one has to be careful. This is because the nodal forces repre-
sent a combined effect of the force acting along the whole circumference
of the circle swept by the node. Therefore. if f, represents the radial
component of the force per unit length of the circumference of the node
which is at a distance r from the center. the extemal nodal force f & ~ is
equal to
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 53
fenf = z x r f , (4.26)
Similarly . in the axial direction.
where jz represents the axial component of the force per unit length of
the circumference of the node which is at a distance r from the center.
Therefore, for an element. the external nodal forces Ifenf} is given
by
where f is the force acting on the node. the first subscript denotes the
direction in which the force is acting and the second subscript denotes
the node on which this force is acting; ri is the radial coordinate of
node i, etc.
Zulfiknr H . A. Kassam Finite Element Analysis for Tube Drawing 54
4. 6. 2. Distributed Body Forces
For the axîsyrnmetric case the extemal distributed body force
vector {fdbf ) is given by
where b, is the distnbuted body force per unit mass in the r -direction, b,
is the distributed body force per unit mass in the z -direction. p is the
density. A is the area of the element and is the average radius of the
element and is given by
The basic principle behind this formula is force = body force/unit
volume multiplied by the volume of the element. The resultant force is
divided by 3 so that al1 the three nodes share an equal arnount of force.
4. 6. 3. Traction Forces (pressure loadirmg)
The forces due to traction on the edge ij , f a is given by
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 55
where t , and t, are the boundary stresses, i.e., traction (force per unit
surface area of the boundary edge). in the r and z directions. respec-
tively. l ÿ is the length of side ij and r is the radial coordinate of the
nodes.
4. 6. 4. Forces due to Initial Strain (change in tempera-
ture)
The forces introduced due to change in temperature (fEo } are given
Zu[F'kar H. A. Kassam Finite Element Analysis for Tube Drawing 56
REFERENCES
[4.1] 0. C . Zienkiewicz and R. L. Taylor. The Finite Element Method.
Fourth Edition, McGraw-Hill, London (1 989) 22.34-36.79.
[4.2] S. A. Meguid. The Finite Element Method in Mechanical Engi-
neering. University of Toronto Press. Toronto ( 1989).
Zulfikar W. A. Kassant Finite Element Analvsis for Tube D r a w i n ~ 57
Finite Element formulation^ for Elaatic - Large Plamtic Deformation
S. 1. INTRODUrnON
The strains involved in rnetal forming problems are large and.
therefore, in order solve the problem using the finite element method.
elastic-plastic stress analysis is required. Many equations which were
valid for elastic analysis are no longer valid when plasticity is incorpo-
rated. For example. the infinitesimal definition of strain (geometric
linearity). and the linear relationship between stress and strain (material
linearity). which are both valid for (infinitesimal) elastic deformations
are no longer valid when considering elastic-plastic deformation. In
short. plasticity involves geometric non-linearity and material non-
linearity.
Geometric non-linearity stems down from the fact that plasticity
involves finite strains and large rotations such that the infinitesimal
definition of strain will yield a non-zero value of strain for rigid body
rotations; this result is obviously incorrect and, therefore. a new defini-
tion of strain has to be used which makes use of higher order terms. The
popular definitions of strains are the Green-Lagrange sbain tensor and
the Almansi-Eulerian straui tensor. Moreover. in problems involving finite
plastic deformation the correct measure of stress needs to be utilized.
This aspect will be described in more detail in the following sections.
Zulfikar H . A. Kassam Finite Element Analvsis for Tube D r a w i n ~ 58
Also, since the finite deformations are involved, there is a need to
conduct incremental analysis. i.e.. using incremental load steps and
updating the mesh after each load step.
ATION [+At
CONFlGURATION ATTIME t
Figure 5.1. Motion and deformation of a body in the Cartesian coordi-
nate system (adapted from ref. [5.3]).
In addition. when considering plasticity. it is well known that the
material behavior. i.e.. the relationship between stress and strain. is
Zulfikar H. A. Kassam Finile Element Analusis for Tube Drawina 59
non-linear. In order to account for this non-linear behavior iterations
are required within each load step until equilibrium is established.
Special techniques are required in order to ensure that the loading path
does not deviate from the yield surface as a result of non-linear material
behavior . As a consequence of material and geometric non-linearity. the
stiffness matrix will also be different as compared to the stiffness matrix
which was valid in the elastic case. in that. extra tems are required. Al1
these aspects will be described in detail in the subsequent sections.
At this point it should be noted that the displacement functions
are valid for both elastic and elastic-plastic cases since the displacement
function depends only on the type of element used. i.e.. whether the
element is 3-noded triangular. 4-noded quadrilateral. etc.
5 .2 . FINITE DEFORMATIONS
Since metal forrning problems involve large displacements and
large strains. unlike in elastic deformation. the position of the body in
space changes as deformation progresses. For example. consider the
motion and deformation. of a body in a Cartesian coordinate system.
Note that the general concepts introduced here are equally valid in the
curvilinear (cylindrical. spherical) coordinate systems although the
specific equations may be different. The configurations of a body at time
O. time t and time t + A t are given in Figure 5.1. In general, for any
problem involving large deformations, the airn is to determine the
Zulfikar H. A. Kassarn Finite Elenient Analysis for Tube Drawing 60
equilibrium positions of the body at discrete intervals in time. i.e., a t
time 0, At, SAt, .. . . . t, t +At, where At is the tirne increment. Assume
that the solution for the kinematic and static variables for al1 the time
steps from time O to time t. inclusive. have been solved, and the solution
for time t + A t is required. I t should be acknowledged that the solution
procedure for the next equilibrium position is typical and the process is
therefore applied repetitively until the complete solution is obtained.
Nornencla ture
It is useful a t this stage to explain the notations which will be
employed. The coordinates describing the configuration of the body a t
O t tirne O are O xl . %, O x3. at time t are xl . 3. ' x3. and at time t +At
are t +At X1
t +At %*
t +At x3. where the left superscripts refer to the
configuration of the body at the stated tirne. and the right subscripts
refer to the coordinate axes.
The notation for the displacements of the body is similar to the
notation for the coordinates: at Ume t the displacements with respect to
t the position of the body at tirne O are ul. u2. ' u3, and at tirne t +At
the displacements with respect to the position of the body a t time O are
t +At t +At u17 %* ' u3. Therefore
(5. la)
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawina 6 1
t + d t x n = O ri + t + ~ t YI (n = 12.3) (5. lb)
The increments in displacements from time t to time t +At are given by
During deformation. the surface area. volume and density of the
body are changing continuously and at time O. t. t +At. these quantities
are denoted by O A. A. t +At A, V, V. '+" V. and O p. ' p. t +At P.
respectively. Note that the mass will obviously remain constant.
In the case of applied forces. stresses and strains. a similar nota-
tion is adopted; the le3 superscript indicates the configuration in which
the quantity occurs. In addition. a kft subscript is included to indicate
with respect to which configuration the quantity was measured. For
example. the traction [boundary stress) and the body force per unit mass
t +At t +At at time t + A t , but measured in configuration t , are &. t bn*
respectively. where n = 1.2.3.
Let us now consider the notation used for stresses. Since the
Cauchy stresses always refer to the configuration in which they occur.
t +dl - t +Al the Cauchy stress tensor at time t + A t is denoted by ou = t + d t ou .-
The second Piola-Kirchhoff stress tensor corresponding to the configura-
tion at time t +At but measured in configuration at time t is denoted by
Zulfikar H. A. Kdssarn Finife Elenlent Analvsis for Tube Drawina 62
t +At S . The different stress measures. including the Cauchy and second t ij
Piola-Kirchhoff stress tensor will be described in detail in a later section.
Considering the strains. the Cauchy's infinitesimal strain tensor
e.. ; the referred to the configuration a t time t +At is denoted by
Green strain tensor using the displacements from the configuration a t
time t to the configuration at time t +At. and referred with respect to the
t +At configuration at time t is denoted by . Note that the configura-
tion at time O may also be taken as reference. The Almansi strain tensor
always refers to the configuration in which it occurs and. hence. the
t + A t - t +At tensor at time t +At is denoted by €0 = t + ~ t Eij .'
5.2. 1. Strain-Displacement Relationships: Green and Al-
mansi Strain Tensors
Since the strains involved in plasticity are large. the strain-
displacement relationships have to be reconsidered. Le.. the iinear
relationships between displacements and strains given in equation (3.4)
(for axisymrnetric case) are no longer valid. If these definitions of strain
are used to calculate strains in a body which expenences a large amount
of deformation or a large rotation. the result yielded will be inaccurate or
even incorrect. It can easily be shown that if a body is subjected to a
large rotation without straining, then the linear strain-displacement
relationship will give a non-zero value of strain. This is obviously incor-
rect since for a rigid-body rotation the strain must be equal to zero by
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 63
definition. Therefore. it seems appropriate at this stage to consider a
definition of strain where the calculated strain is zero when a body
undergoes rigid-body rotation. For this reason one has to use another
definition for strain when tackling problems involving finite strains. For
the sake of convenience. the following equivalent notations are being
introduced,
t t Position coordinates X I ,X2,X3 = xi. x2. x3.
- t + A t Position coordinates xl .x2,x3 = t +At X2.
t +At X1 * x3
- t +At 1 Length L = 1 : L =
- t +Al Displacement y, - r, - t x, = r, -X, (n=1.2 ,3)
- - Cauchy's infinitesimal strain tensor eÿ - ey
There are two types of strain tensors that are frequently used in
t +At E - . , which is a the finite strain theory: (i) the Green's sbain tensor, y
strain tensor defined in Lagrangian coordinates and (ii) the Ahunsi's
t +At shdn tensor, , . which is a strain tensor defined in Eulerian
coordinates.
Zulfkar H. A. Kassam Finite Element Analysis for T~dbe Drnwing 64
P'( xi, X2, x3)
CONFIGURATION ATTIME t+At
P( Xi, X2, X3)
Q( x I +dx 1, x2+dx2,
Figure 5.2 Configuration of body at tirne t and tirne t + At.
The Green's strain tensor in the Cartesian coordinate system is
given by
Zulfihr H. A. Kassam Finite Element Analysis for Tub2 Drawing 65
t +At t +At E.. ) , that is the Green's strain tensor e u ) = { + { Y
has a linear part (equal to the infinitesimal definition of strain) and a
non-linear part.
The Almansi's strain tensor in the Cartesian coordinate system is
given by
t +AC t +At t +At or ( +At Ey ) = { +At 11 - { +At I n 1 9 that is the Almansi's strain
tensor also has a linear part (equal to the infinitesimal definition of
strain) and a non-iinear part. From the above definitions of the Green's
and Alrnansi's strain tensors it is clear that for small deformation one
needs only to consider the linear part since the non-linear part will be
very small. Therefore. for infinitesimal strains. since the linear part is
the predominant part and the non-linear part negligible. the distinction
between Green's (Lagrangian) and Alrnansi's (Eulerian) strain tensors
disappears since it is immaterial whether the derivatives of the displace-
ments are calculated at the position of a point before or d e r deforma-
tion.
Zulfikar H. A. Kassam Finite Element Analysis for Tube D r a w i n ~ 66
The Green's strain tensor is related to the Aimansi's strain tensor
by the following relationship:
where. in general.
and
final volume Vf Jacobian = J = initial
For 2-D plane stress case
Zulfikar H. A. Kassarn Finite Element Analysis for Tube Drawing 67
and for the plane strain case.
The Green's s M n tensor in the cylindrical coordinate system
(axisymmetric case) also has a linear part (equal to the infinitesimal
t +At definition of strain) and a non-linear part. i.e.. ( E~ } = gy 4 +
E - - ) . The Green's strain tensor is given by f+"t y n
d u - - dR
Zulfikar H. A. Kassam Finite Eletnent Analysis for Tube Drawing 68
and the Almansi's strain tensor in the cylindrical coordinate system
(axisymmetric case) is given by
In this case. i.e.. axisymmetric case.
S. 3. DIFFERENT STRESS MEASURES
In finite defomation analysis. the strains and rotations expen-
enced are large and. hence, the usual linear definitions of strain are no
longer accurate. Consequently . two different types of strain rneasures
t +At were introduced; (i) Green strain tensor measures strain with
reference to the original (undeformed) configuration of the solid and (ii)
t +At Almansi strain tensor , measures strain based on the current
Zulfikar H. A. Kassam Finite Elemenf Analysis for Tube Drawing 69 - - - - - - -
configuration of the solid. One can appreciate the need to modw the
usual definitions of stresses in the solid since. for a certain applied force.
the calculated stresses will depend upon the geometry of the body. For
example. the usual definition of normal stress in elastic problerns is
(force acting on the deformed geometry) / (original undeformed area) : this
definition of stress is called the second Pioh-Kirchoff stress, and the
stress tensor is denoted by [SI. However. in actual fact. the stress has to
be calculated based on the force/final area: this definition of stress is
analogous to the definition of true stress and is called the Cauchy stress,
[al. The second Piola-Kirchoff stress is accurate enough to be used in
cases where there is infinitesimal amount of deformation, i.e., the
strains and the rotations are small. since the deformed area is approxi-
mately equal to the undeformed area. For large strains and rotations,
however, the Cauchy stress provides a more accurate definition of stress.
The relationship between these two stress measures will be given shortly.
At this stage. it is worthwhile rnentioning that there are two other types
of stresses which also are important when dealing with finite deforma-
tions; these are the Jirst Pioh-Kirch08 stress [Pl and the Kirchhoff or
nominaI stress. [r]. The relationships between al1 these stresses can be
derived based on the principle of virtual work 15.1).
The Cauchy and the second Piola-Kirchoff stress are related
through the equivalent virtual work concepts in that
Zuljïkar W. A. Kassam Finite Efement Analysis for Tube Drawina 70
where Ui is virtual work done when there is an increment [6E] in the
Green's strain at a second Piola-Kirchoff stress level of [SI and the
resulting Cauchy stress is [al; furthemore, since a small increment in
strain is being considered. the Cauchy's infinitesimal strain tensor. [lie 1 ,
is deemed appropriate. The integration is carried out over initial volume
V and final volume Vf. respectiveiy.
Since
Making use of the above relationship. eqn (5.22) can be manipulated to
obtain
where
[.cl = J [al (5.16)
is known as the Kirchhoff or nominal stress. By manipulating eqn. (5.15)
it can be shown that
Zulfikar H. A. Kassam Finife Element Analvsis for Tube Drawina 71
There is one more stress measure that needs to be introduced: this
is the first Pioh-Kirchoff stress tensor, [Pl. The relationship between this
stress measure and the Cauchy and second Piola-Kirchoff stress tensor is
given by
5. 4. FINITE ELEMENT FORMULATIONS
In the finite element analysis of metal-forming problems. there are
various aspects that have to be taken into consideration i5.21.
1. The material satisfies a constitutive rate equation which c m be of
complicated form and may change considerably during plastic
straining (non-linear material behavior) .
2. The rate of stress used in the constitutive equation is not an
ordinary rate. but a CO-rotational (Jaumann) rate. Altematively.
the Cauchy stress may be used.
ZuZfikar H . A. Kassam Finite Element Analysis for Tube Drawing 72
3. The constitutive rate equation is formulated with respect to the
previous (Lagrangian) or current (Eulerian) configuration of the
material a t any tirne.
4. Analysis up to large strains is desired.
5. Large material rotations may be involved.
The above mentioned conditions for non-linear analyses entai1 the
u tilization of a n incremental formulation. Le.. an approach where the
deformation is applied in a finite number of steps.
5 .5 . UPDATED LAGRANGIAN FORNIULATION
The updated Lagrangian finite element formulation shall be used
for solving problems involving large elastic-plastic deformation. The
formulation described here is valid for non-linear material behavior. large
displacements and large strains.
Based on the principle of virtual displacements. the following non-
linear equilibrium equation can be derived.
on-linear equilibrium equation for updated Lagrangian
Zulfikar Fi. A. Kassam Finite Element Analysis for Tube Drawing 73
The above is a non-linear equilibrium equation in the incremental
displacement un .
1 Non-linear part of strain qy = 5 ukmi ukJ
The constitutive relation between stress and strain increments is now
given by
The above equation related the Lagrangian strains to the second Piola-
Kirchoff stress, and is used in finite element analysis to calculate the
stresses after the strain has been deterrnined. Since the second Piola-
Kirchoff stress is based upon the material configuration before the
application of the force at time t, this stress measure cannot be used if
an accurate result is desired. Instead. the Cauchy stress needs to be
calculated based upon the Second-Piola Kirchhoff stress. The equation
Zulfïkar H. A. Kassam Finite Element Analvsis for Tube Drawina 74
that relates these two stress measures has been given previously in eqn.
(5.17) which is once again presented here:
The Cauchy stress is the stress in the element in the deformed configura-
tion. The only problem with the Cauchy stress is that the coordinates of
the element rotate with element rotation. Therefore, in deformations
where large rotations occur. each element may end up with a different
local coordinate system. Hence, it is necessary to resort to a stress
system which is invariant to rotation namely the Jaumann stress rate.
The Jaumann stress rate can be calculated from the Cauchy stress rate
by the following equation:
where D/Dt denotes the tirne derivative and L$, are the Cartesian compo-
nents of the spin tensor
Zupkar H . A. Kassam Finite Element Analysis for Tube Drawing 75
5. 5. 1. Linearization of Equilibtlum Equations
The solution for equation (5.20) cannot be calculated directly since
they are non-linear in the displacement increments. Approximate solu-
tions can be obtained by assuming that in equation (5.24) ey = eÿ .
This means that. in addition to using 4 E~ = eu . the incremental
constitutive relation employed is
Taking into account the above mentioned linearization procedure.
the equation (5.20) for the U.L. formulation may be written as I5.31:
Linearized equilibrium equation for updated Lagrangian
Equation (5.26) is an equilibrium equation for the updated Lagrangian
formulation which is linearized in the incremental displacement and
shall be used as the basis for isoparametric finite element analysis. Since
the procedure for assembling the elemental stlffness matrices to form the
global stiffness mat* is standard. attention will be focused on the
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 76
derivation of the element stifhess matrix followed by the development of
particular equations for the case of a triangular axisymmetric element.
5. 5. 2. Determination of StUhess Ma-
Differentiating each t e m in equation (5.26) with respect to the
displacement. the following equation is obtained for a single element:
where
t t In equations (5.28) to (5.30) the B and BNL are the linear and
non-linear strain-displacement transformation matrices. respectively.
and the elements of the incremental material property matrix D ~ P is
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 77
the elastic-plastic constitutive mat* corresponding to. and defined
t with respect to the configuration at tirne t. a is a matrix of Cauchy
t A stresses and a is a vector of Cauchy stresses. both in the configura-
tion at time t. and +At F is the resultant nodal force vector.
The above quantities for an axisymmetric triangular element have
been evaluated. The displacement function for the triangular element is
as follows:
and
Zul/ikar H. A. Kassam Finite Elenrent Analysis for Tube Drawing 78
= R& - Ri& bj = Zk * Z, 9 = Ri * Rk (5.34)
a k = - b k = & - q ck = Rj - Ri
24 = det = 2 . (area of triangle ijk ) (5.35)
Zulfikar H . A. Kassani Finite Elemenf Analysis for Tube Drnwing 79
Linear straindisplruiement tranqfonnation matrir
Since
t { t e l = l t BL I M
> and {u} =
Therefore,
where u is the displacement from time t to time t +At.
Zulj ïhr H . A. Kassam Finite Element Analysis for Tube Drawing 80
Non- Linear strain-displacement tranqfonnation ma&
Zulfikar H. A. Kassam Finite Element Analysis for Tube Dtawing 81
Zulfikar H. A. Kassarn Finite Element Analysis for Tube Drawing 82
REFERENCES
M. A. Crisfield. Non-linear Finite Element Anal~sis of Solids and
Structures, John Wiley & Sons. Chichester. UK. (199 1).
J. C. Nagtegaal and F. E. Veldpaus, "On the Implementation of
Finite Strain Plasticity Equations in a Numerical Model," in Nu-
merical Analysis of Forming Processes. Edited by J. F. T. Pittman.
O. C. Zienkiewicz, R. D. Wood and J. M. Alexander, John Wiley
& Sons Ltd., (1984) 351-371.
K. J. Bathe, E. Ramm and E. L. Wilson, "Finite Eiement For-
mulations for Large Deformation Dynamic Analysis." Intl. J. for
N u m M e t h vi Engineering 9 (1975) 353-386.
H. D. Hibbitt, P. V. Marcal and J. R. Rice, "Finite Element
Formulation for Problems of Large Strain and Large Displace-
ments," Int. J. Solids Struct. 6 (1970) 1069- 1086.
J. T. Oden, Finite Elements on Nonlinear Continua, McGraw-
Hill, New York (1972).
S. Yaghmai and E. P. Popov. "Incremental Analysis of Large
Deflections of Shells of Revolution," Int. J. Solids Struct. 7 (197 1 )
1375- 1393.
Zulfïkar H . A . Kassam Finite Elernent Analysis for Tube Drazuing 83
15.71 S. Kobayashi. S. Oh and T. Altan. Metal Forming and the Finite
Element Method. Oxford University Press. New York. U.S.A.
(1989) 3-4.174-175.
15.81 Y. Shimazeki and E. G . Thompson. "Elasto-Viscoplastic Flow
with Special Attention to Boundary Conditions." Intl. J. for Num.
Metk inEngUleerVlg 17 (1981) 97.
15-91 J. C. Nagtegaal and J. E. De Jong. "Some Computational Aspects
of Elastic-Plastic Large Strain Analysis." Intl. J. for Num. Meth. in
Engineering 17(1981) 15.
Zulfikar H . A. Kassam Finite Element Analvsis for Tube Drawina 84
6. 1. INTRODUCTION
The main differences between elastic and plastic deformation in
metal-based materials are:
(il
(iii)
elastic deformation is linear while plastic deformation is generally
non-linear.
elastic strains are recoverable after the loading is removed. i.e.. the
original state of strain is retained after the load is removed; on the
other hand, plastic strains are not recoverable. i.e.. plastic defor-
mation results in permanent deformation.
the elastic strains are uniquely related to the stresses through
Hooke's law and the elastic strains can be evaluated for any given
stress state without any regard to the way in which the stress state
was obtained; however, in plastic deformation, the strains are not
uniquely determined by the stress state but depend on the history
of loading or how the stress state was reached.
Owing of the dependence of plastic strains on the loading (stress)
path, the plastic sb.& increments have to be calculated throughout the
loading (stress) history and summed up together to detexmine the total
strain. However, in the case that the stresses are applied proportionally.
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 85
i.e.. al1 the stresses increase in the same ratio. then the plastic strain
state is independent of the stress history and depends only on the final
stress state. The former case of non-proportional loading seems to be
more common.
Since large amounts of deformation are involved in metal forming
operations. each step (increment) needs to be as large as possible. i.e.,
need finite strain increments. so as to minimize the number of steps
required to obtain a solution but at the same time ensuring that accu-
racy is maintained. Since material behavior in the plastic region is
generally non-linear, the utilization of a single tangent stiffness matrix
reflecting behavior at the start of an increment will result in large error
in the obtained solution. A number of iterations within each increment
are therefore required in order to follow the stress-strain trajectoxy more
closely. The Newton-Raphson method is commonly used to perform these
iterations.
6. 2. ERRORS INVOLVED IN ELASTIC-PLASTIC ANALYSIS
One important aspect to appreciate is that in spite of the fact that
the numencal approach adopted involves the utilization of strain incre-
ments and a number of iterations within each increment, it is inevitable
that solution procedures used to calculate increments in stresses based
on iterative or incremental strain will lead to some emor [6.2]. This error
does not relate to a lack of equilibrium but to the errors introduced
during integration of the flow niles. i.e.. calculation of incremental
stresses from iterative or incremental strains using a tangent (linear)
Zulfikar H . A. Kassam Finite Element Analilsis for Tube Drawing 86
stiffness matrix that does not follow the stress-strain trajectory in the
exact manner. In plasticity it is well known that the strain path may
have a significant influence on material behavior if the loading is not
proportional (proportional loading is when increments in strain are
always proportional to the stress state). However. most analysts assume
a linear strain path within an increment. Consequently. inability to
follow the stress-strain trajectory in the exact manner introduces some
errors not only because of the fact that the constitutive laws are not
exactly followed because of discrete increments but also due to the fact
that the exact loading path cannot be followed. In addition, usage of
tangential constitutive matrix (forward Euler scheme) to calculate stress
increments also results in deviation of the stress state from the yield
surface. In this research. however. the error is going to be minimize by
using the f o m d Euler scheme combined with subincrementation and a
special technique to retum the stress back to the yield surface within
each subincrement. This is the only way to ensure that the material
behavior is sirnulated as closely as possible.
6.3. FLOW RULE FOR ISOTROPIC MATERIALS: PRANDTL - REUSS EQUATIONS
In 1870 Saint-Venant [6.4] proposed that the principal axes of
strain increment coincided with the principal stress axes. The general
three-dimensional equations relating the totd straui Uicrement, &O, to
the deviatoric stresses. q'. were given by Levy [6.5] and independently by
Zulfikar N. A. Kassam Finitéi Elernent Analysis for Tube Drawing 87
von Mises [6.6]. These equations. known as the Levy-Mises equations. are
given by:
where oÿ' are the deviatoric stresses which are given by
cU is a non-negative proportionality term which reflects the material
behavior in the plastic region and. therefore, may Vary throughout the
loading history. In the above equations the total strain increments are
assumed to be equal to the plastic strain increments. i.e.. the elastic
strains are ignored. Thus these equations can only be applied in cases
where the plastic deformation is very large compared to elastic deforma-
tion such that the elastic deformation can be ignored. An alternative
- -
form of the Levy-Mises equations was proposed by Prandtl [6.7] and
Reuss [6.8] who isolated the elastic strains. The resulting equations
therefore correlate the plastic strain increments to the deuiatoric stresses.
These equations are as follows:
Equation (6.3) can be expanded to obtain the plastic strain increments
explicitly. These equations in the expanded f o m are
Zuljïkar H. A. Kassam Finife Elemenf Analysis for Tube Drawing 89
The t e m dA. must reflect material behavior and is given by
where GP is the effective incremental strain and 5 is effective stress.
This relationship is usually determined during uniaxial testing. The
terms are defined as follows:
The plastic stress-strain relationships can therefore be written as fol-
lows:
Zulfikar H. A. Kassam Finite Element Analvsis for Tube D r a w i n ~ 90
or in general
The dEP / 5 ratio should reflect the material behavior in the plastic
region. By multiplying the numerator and the denorninator by d a the
following expression is obtained:
Zwlfikar H. A. Kassanr Finife Elenrent Analysis for Tube Drawing 91
where
Therefore the general plastic stress-strain relationship (for isotropic
materials) is given by
6. 4. THE PLASTIC POTENTIAL (YIELD) FUNCTION
The equation for plastic flow in isotropic material, i.e., eqn. (6.3)
could also be denved by making use of the plastic potentiaL The concept of
plastic potential is based upon the hypothesis that there exists a plastic
potential function f that is scalar function of stress, i.e.. f = f (ag ), from
which the ratio of the components of the plastic strain increment. de{ ,
c m be obtained by partially differentiating f (q ) with respect to the
stresses 09 . Thus
Zulfiknr 13. A. Kassarn Finite Element Analysis for Tube Drawing 92
where d  . as rnentioned previously, is a non-negative proportionality
constant that reflects material behavior.
The plastic potential is associated with the yield function. For the
von-Mises yield critenon (isotropie matenal), the plastic potential
func tion. f (ou ) . is given by
Differentiating the plastic potential function w.r.t. stress yields:
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 93
where oy ' are the components of the deviatoric stress tensor. Therefore
the plastic stress strain relationship is given by
which is the same as equation (6.1).
6 5 FLOW RULE FOR AN ANISOTROPIC MATERIAL
For an anisotropic material the hypothesis is made that there
exists a plastic potential f (ou ) so that the incremental strains may be
derived by partially differentiating f (ou ) with respect to og i6.91.
where F, G. H, L, M, and N are constants which characterize the anisot-
ropy of the matenal. These constants are related to the yield stresses in
x. y. and z -directions. i.e.. to X, Y, and 2. respectively. by the following
relationships :
Zult3ar H. A. Kassam Finitr Element Analysis for Tube Drawing 94
Solving the above equations simultaneously yields
As mentioned before. the incremental strains may be derived by partially
dmerentiating f (ou ) with respect to ag and multiplying by CU' where dA'
serves the same purpose for anisotropic materials as does cU for iso-
tropic materials; dA' = (dEP / 5 ) . For exarnple. d f / d a , = [G (o, - a,
) + H (O, - ayy )) / [F + G + H 1. The relationship between incremental
plastic strain and the stress is. therefore. given by
Zulfikar H. A. Kassnm Finite Element Analilsis for Tube Drawina 95
Note that during plastic deformation of a material the state of
anisotropy changes. Usually the change in anisotropy is negligible
compared with the initial state of anisotropy. In cases where the change
in anisotropy is significant. al1 the anisotropic parameters F, G. Hl L, M.
and N and the proportionality terni &' . d l of which reflect material
behavior. must be re-evaluated. If the state of anisotropy remains
constant, then the yield stresses must increase proportionally as the
matenal strain hardens and it then follows that the anisotropic parame-
ters must decrease in proportion.
As mentioned before cUf = (dEP / 5 ) where 5 is the equivalent
stress and &P is the equivalent plastic strain increment. The equiva-
lent stress for anisotropic material is given by
Zuiflluzr H. A. Kassarn Finite Element Analysis for Tube Drawing 96
If the anisotropy of the material is negligible then F = G = H and L = M =
N = 3F. then the above equation reduces to equation (6.7), the equiva-
lent yield stress for isotropie materials. The equivalent plastic strain
increment is given by
6. 6. THE ELASTIC-PLASTIC CONSTITUTIVE MATRIX
(da) = [ D ~ ~ ] {dE} (6.22)
where
1 [D ep 1 = [D - S
dF ID {s} {z} ID el (6.23)
Zttl /kar H . A. Kassam Finite Element Analysis for Tube Drawing 97
and S, as defined in eqn (6.35) is given by
AU the abow equatbns are valid for isotropie and anisotropic materialS.
Specific equations for each type of material will now be derived.
6. 6. 1 Elastic-Plastic Constitutive Matrix for Isotropic
Materials
2 1 {Z} = 3 [a, - 5 (O* + q,y l
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 98
Implying that
Elastic-Plastic Constitutive Matrix for Anisotropic
Mat erials
Zulfikar H. A. Kassam Finite Element Analvsis for Tube Drawing 99
Irnplying that
H' is the instantaneous plastic modulus of the material in a rnultiaxial
stress state and is generally defined as
For anisotropic materials. however, instantaneous plastic modulus of
the material. H' . will depend on the magnitude and direction of the
strain. Therefore, to cater for this difference, the following relationship is
proposed:
Zulfikar W . A. Kassam Finite Element Analysis for Tube Drawina 100
where the instantaneous plastic modulus of the material is denoted by H
foiiowed by a subscript which describes the modulus in a particular
direction. Note that for isotropic materials, the modulus will have the
same value in al1 the directions. Therefore,
and. therefore. the plastic matrix. [D 1 . for anisotropic materials is
given by
Zulfikar H . A. Kussam Finite Element Analysis /or Tube Drawing 101
6. 6. 3. Instantaneous Plastic Modulw. 8' (or H,)
The instantaneous plastic modulus. H' or H, (H followed by
subscripts describing the direction of applied strain). is not simply equal
to the tangential instantaneous modulus. Et . because application of
stress beyond the elastic limft causes not only plastic strain but also
elastic strain. On the other hand the instantaneous plastic modulus. H'.
by the definition provided in equation (6.52). is supposed to encornpass
the change in effective stress as a function of plastic strain only and not
the total strain. A relationship between H' and Et must therefore be
derived. Consider a simple case of an elastic-linear strain hardening
material (see Fig. 6.3). Note that the relationship developed here is
applicable in al1 cases, i.e.. also in cases where the strain hardening is
non-linear as is the case in material which obey the power law or the
Ramberg-Osgood equation. Refemng to Fig. 6.3, an increment in stress
causes elastic and plastic strains. The elastic strain. dP. is given by
do/E . The total strain. &'. is given by &/Et . The plastic strain. &p. is
therefore given by
Zulfïkar H . A. Kassam Finite Elemenf Analysis for Tube Drawing 102
1 Figure 6.3 One-dimensional stress-strain relationship with linear
hardening .
6.7 . NEWTON-RAPHSON ITERATIVE SCHEME
In order to cater for non-linear material behavior it was previously
mentioned that an iterative scheme is required for each step to converge.
The Newton-Raphson iterative scheme is most commonly used to cany
out this task. There are mainly two different types of Newton-Raphson
(N-R) schemes; these are the (a) full Newton-Raphson method and (b)
modified Newton-Raphson method. The difference between these two
scheme is illustrated graphically in Figs. 6.4(a)&(b).
Zulfikar H . A. Kassam Finife Elernent Analvsis for Tube Drawina 103
Force
.- Displacemen
Force
Figure 6.4 (a) Full Newton-Raphson iteration scheme and (b) Modified
Newton-Raphson'iteration scheme.
Zuifikar H . A. Knssarn Finite Element Analysis for Tube Drawing 104
From the figure it is clear that the full N-R method involves the recalcu-
lation of the stiffness matrix after each iteration while the modified N-R
calculates the stiffness matrix only at the start of the increment. i.e.. at
the start of the first iteration only and. hence. the latter approach is
simpler to execute from a programming point of view. However. from the
figure. it is clear that the full N-R method converges faster (quadratic
convergence) than the modified N-R method which does not converge
quadratically and the procedure often diverges I6.141. Consequently. the
line search procedure has to be used in conjunction with the modified N-
R method in order to ensure convergence. This iteration algorithm is
particularly suitable for structures exhibiting extreme material non-
linearity such as strain softening. On the other hand. the full N-R
method is more effective for geometrically non-linear problems than the
modified N-R method. Based on the fact that elastic-plastic problems are
geometrically non-linear and the convergence rate for the full N-R
method is faster, this technique shall be used.
The N-R method is now explained further with reference to Figure
6.4(a). It should be noted that any prescribed values of displacement
must be incorporated at the first iteration. Non-homogeneous boundary
conditions, i.e.. nodal points where the displacements have been pre-
scribed to be non-zero. for each increment can be taken care of at this
stage by multiplying the particular column in the stiffness mattix
corresponding to this node by the non-zero prescnbed displacement to
form a force vector (AFnh} (nit denotes non-homogeneous component of
the force vector) which is then subtracted from the resultant extemal
Zulfikar H . A. Kassani Finite Element Analysis for Tube Drawing 105
force vector (AF} to yield the effectiw force vector {me$. The incremental
force {AF,d = (6~(')) is then applied to the solid. and by using the stiff-
ness matrix evaluated a t the start of the iteration, the iterative nodal
displacements ( 6 ~ ( ' ) } is calculated. Thereafter the iterative strain is
calculated followed by calculation of the corresponding iterative stress
increment by integrating the flow niles; this concept will explained in
detail in the next section. Based upon the calculated iterative stress
values, the applied elernental force vector {dl)'} (note that small f
denotes that the calculation is for a single element) is evaluated by using
the following equation
The result will be an applied ehmental force wctor (dl)'} for each element
with each component of the vector corresponding to a nodal point that
rnakes up the elernent. A global applied force vector ( d l ) * ) iç then
calculated by summing up the applied elemental force vectors. The
residual force vector force (y~) will now be the resultant extemal force
vector. ( 6 ~ ( ~ ) ) . for the next iteration and is calculated by subtracting the
applied global force vector (AF(l)*} from the previous resultant eirtemal
force vector (AF} (not the effective force vector). Le.,
Zulfiknr H. A. Kassarn Finite Element Analvsis for Tube D r a w i n ~ 106
The iterative force vector { 6 ~ ( ~ ) } is then applied to the solid using the
stiffness matrix re-evaluated a t the start of the present iteration and this
time setting any prescribed components of nodal displacement (including
non-homogenous ones) to be equal to zero. The iterative displacement
D SU(^)} is calculated by multiplying the inverse of the stiffness matrix with
the residual force vector (6~(~)}. The incremental displacements are then
calculated by summing the iterative displacements. The incremental
displacements are then used to determine the incremental strains
followed by integrating the flow rules in order to calculate the incre-
mental stresses. Once again. based upon the recalculated incremental
stress values, the applied elemental force vector {d2)*} is now calculated
using equation (6.35). A global applied force vector [d2)') is then calcu-
lated by summing up the applied elemental force vectors {d2)*}. The
residual force vector (VI will now be the resultant extemczi force vector,
{ 6 ~ ( ~ ) ) , for the next iteration and is calculated by subtracting the applied
global force vector (AF(2)*] from the resultant extemal force vector (6F(2)} ,
Zulfiknr H. A. Kassam Finite Elment Analysis for Tube Drawing 107
{w) = {d3)} = { - IAF(~)*} (6.37)
This procedure i s repeated until the values of the residual forces {y} or
iterative displacements (SU} are smaller than some required convergence
criteria (as described below) in order to obtain a converging solution.
6. 7. 1. Convergence Criteria for Newton-Raphson Itera-
tions
When using incremental/iterative solution algorithms. a conver-
gence criteria must be defhed in order to assess whether equilibrium has
been achieved. Selection of an appropriate convergence cnteria is ex-
tremely important. The commonly used criteria are the (i) Euclidean
residual nom. (ii) Euclidean displacement nom, and (iii) work nom.
One may use more than one criterion for the purpose of ensuring that
equilibrium has been established. In such a case al1 the chosen criteria
must be fulfilled before concluding that equilibrium has been estab-
lished. In addition to choosing a criterion, or a set of criteria, it is
important to make a correct choice for the tolerance. An excessively tight
tolerance may result in unnecessary iterations and consequently a waste
of computer resources, while a slack tolerance may provide inaccurate
answer. Therefore. an effective convergence cntenon (criteria) together
with a realistic tolerance is a precondition for accurate and economic
solutions [6.14, Sec. 2.3- 141.
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drazuina 208
Assigning tolerance values is very much a matter of experience. In
general, sensitive geometrically non-linear problems require tight conver-
gence criteria in order to maintain the solution on the correct equilib-
rium path, whereas a slack tolerance is usually more effective with
problem that predominantly exhibit material non-linearity and. in such
a case, high local residual forces may have to be tolerated.
6. 7. 1 . 1 . Eucliclean residual nom
The Euclidean residual norm yv is defined as the n o m (root mean
square) of the residual forces ( y ) as a percentage of the norm of the
incremental effective extemal force vector LW,^ and is written as
Note that the components of the above vectors do not include the
constrained nodes. For problems involving predominantly geometric non-
linearity. a tolerance of y . < 0.1 is suggested. In the case where plasticity
predominates. a more flexible tolerance of 1.0 < y, < 5.0 is suggested by
Zul'kar H. A. Kassam Finite Element Analysis for Tube Drawing 109
6. 7. 1 . 2. Euclidean displacement nom
The Euclidean displacement norm yu is defined as the norm of the
iterative displacernent {6~(*)} as a percentage of the norm of the total
incremental displacement {AU) and is written as
l 1 {SU"$ i 1 Yu = I I {AU} 1 1
x 100
The above critenon is a physical measure of the nodal displacements
during the current iteration. A tolerance of 0.1 < yu < 1 .O is suggested.
6. 7. 1. 3. Work n o m
The Euclidean work norm is defined as the work done by the
residual forces on the current iteration as a percentage of the work done
by the effective forces on the first iteration, i.e..
A value of 1 .OE-6 c < LOE-3 is suggested.
Zulfikar H. A. Kassanl Finite Elemenf Analusis for Tube Drawin~ 110
6.8. INTEGRATION OF FLOW RULES TO CALCULATE
The elastic-plastic constitutive matrices, [D ep 1, derived in earlier
sections can be used during the iteration process in order to obtain a
converging solution, i.e., a solution of incremental displacements for
given increment in applied forces and certain boundary conditions. The
change in displacement a t each iteration can then be used to calculate
the iterative strains. However. the resulting change in stresses cannot be
calculated directly from change in strain using the elastic- plastic consti-
tutive matrices since the rnaterial behavior is not linear and, further-
more, the increments are finite and not infinitesimal which rules out the
possibility of assuming linear relation over the range (that is possible
when the strain increment is infinitesimal). Utilization of the elastic-
plastic constitutive matrices which depict matenal behavior a t the start
of an increment will therefore result in the unsafe drift from the yield
surface. i.e.. caiculated value of stress increment will be larger than the
real value.
In order to overcome these problems, the following schemes can be
used:
(1) fonuard-Euler scheme with a technique to retum the stress statr to
the yeld surface
(2) backward-Euler scheme
(3) SubUzcrements
(4) mean n o d method
Zulfiknr H. A. Knssam Finite Element Analysis for Tube Drawing 111
In each case. the aim is to update the stresses at a Gauss point
given (a) the old stresses. strain and equivalent plastic strains and (b)
the new strains. For al1 procedures. the first step is to use the elastic
stress-strain relationship to update the stresses. If these updated
stresses are found to lie within the yield surface then the material is
assumed to have remained elastic. In this case. therefore. integration of
rate equations in not necessary as the relationship between stress and
strain is linear in this region. However. if the elastic stresses are outside
the yield surface. the first step is to determine the point (or stress state)
at which these elastic stresses cross the yield surface. One of the above
mentioned integration procedures is then used to calculate the stress
increment in the plastic regime. The determination of the point of
crossing of the yield surface is described first. followed by the forward-
Euler scheme combined with sub-incrementation to integrate the rate
equations.
6. 8. 1. Crossing the Yield Surface
Many. but not all. of the integration procedures require the
location of the intersection of the elastic stress vector with the yield
surface. This necessitates the utilization of the yield function denoted by
F and previously defmed as
Zulfikar H . A. Kassam Finite Element Analysis for Tube Druwing 112
In this case. if the stress state at the beginning of the increment 1s given
by ' ag , the calculated elastic stress increment for the current step is
given by
where {dgÿ ] is the strain increment. In order to determine the crossing of
the yield surface it is required that
where a is the parameter to be deternined. aA is the stress at the point
where the stress crosses the yield surface. A is a point on the yield
surface where the stress crosses the yield and the stresses ' ag at time t
are such that
e i.e.. below the yield surface. while, with a = 1. the stress (t cg + da ÿ )
is such that
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 113
a has to be determined.
6. 8. 2. Fornard-Euler technique
A predictor method is required to calculate the increment in stress
while a corrector method is required to retum the stress to the yield
surface. The standard predictor [6.11] is the forward-Euler procedure
which is given by
with al1 the calculation done with respect to the stress state at point A
and constitutive laws which are valid at point A. The above equation
implies that the elastic-plastic constitutive matrix [ D ~ ~ ] is used to
calculate the stress increment due to plastic strain amounting to (1-a )
Zulfikar H. A. Kassarn Finite Element Analilsis for Tube Drawing 114 -- -
To yield more accurate results it is recommended that the strain
increment be sub-divided into smaller. equal increments. This technique
is cal1ed sub-incrementation In this technique. each increment is subdi-
vided into many subincrements and in each small subincrement. the
elastic-plastic constitutive matrix may be considered to be linear. This
technique ensures that the point C is not very far from the yield surface
thereby allowing the stress-strain trajectory to be followed more closely
and. hence. minirnizes the error. In fact. this is the technique that was
adopted.
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 115
REFERENCES
[6.1] M. A. Crisfield, Non-linear Finite Element Analysis of Solids and
Structures. John Wiley & Sons. Chichester (199 1).
[6.2] J. H. Argyns, L. E. Vaz and K. J. William. "Improved Solution
Methods for Rate Problems." Comp. M e t h in Appl. Mech &
Engng. 16 (1978) 23 1-277.
[6.3] D. M. Tracey and C. E. Freese. ""Adaptive Load Incrementation in
Elasto-Plastic Finite Element Analysis." Comp. & Stmct. 13
(198 1) 45-53.
16.41 B. Saint-Venant, Compt. Rend. 70 (1870) 473-480.
[6.6] R. von Mises. Goemiiger Nuchr. Math Phys. (1913) 582-592.
16.71 L. Prandtl. Proc. of the I st Int. Congress on Appl. Mech. (1925) 43-
54.
[6.8] E. Reuss. 2. Angew. Math Mech. 10 (1930) 266-274.
Zulf ikarif . A. Kassani Finite ElernentAnalysisforTube Drawing 116
16.91 R Hill. The Mathematical Theory of Plasticiw. Odord Univ.
Press. London (1950) 25.
(6.101 T-R. Hsu. The Finite Element Method in Themomechanics. Allen
& Unwin, London (1 986).
16.1 11 D. R. J. Owen and E. Hinton, Finite Elements in Plasticitv -
Theory and Practice. Pineridge Press. Swansea ( 1980).
[6.12] M. Ortiz and J. C. Sirno. "An Analysis of a New Class of Integra-
tion Algorithms for Elasto-Plastic Constitutive Relations. Int. J.
Num M e t h Engng.. 23 (1986) 353-366.
(6.131 G . W. Rowe. C. E. N. Sturgess. P. Hartley and 1. Pillinger. Finite
Elernent Plas ticity and Metalforming Analysis , Cambridge Univer-
sity Press. New York (1991) 93.
(6.14) LUSAS manual, 2.3-10
Zulfïkar H . A. Kassam Finite Element Analvsis for Tube Drawina 117
Experimental 8etup and Procedure, Remultm and Dimcu88ion for the Conmtitutive Equationm
As mentioned previously the Rarnberg-Osgood equation is inadequate for
describing the behavior of al1 materials, i.e., it has its limitations and
therefore lacks universal applicability. Therefore. testing was conducted
on a number of different materials and efforts were directed towards
formulating a universal constitutive equation that can accurately
descnbe mechanical behavior of al1 materials. This aspect is extremely
important for the integration of these material properties into finite
element calculations.
The materials tested and the new constitutive equation developed
to accurately descnbe material behavior are now presented.
7. 1. MATE-, EXPERXMENTAL SETUP & PROCEDURE
Uniaxial tensile test-s have been conducted on aluminum 6061,
aluminum A356. 70/30 brass. copper and 1018 steel using the 810
Materials Testing System (MT'S). The tensile specimens were machined
according to ASTM standards, i.e.. cylindrical cross-section with 2" gage
length. Al1 the specimens were annealed before mechanical testing. The
load was rneasured by a 100W load cell. and the strain by an extensome-
Zulfikar H . A. Kassam Finite Element Analysis for Tube Drawina 118
ter. The output analog voltage signals obtained from the MT'S signal
conditioners. corresponding to the load and strain. were digitized using
the WB-800 data acquisition card that was interfaced with an IBM
compatible computer system: the computer recorded and displayed the
incoming signals a t discrete intervals (0.5s). Al1 the tests are conducted
at an average strain rate of 1x10-4s- in order to minimize or virtually
eliminate strain rate effects. The tests were conducted until final failure.
However. attention was focused on the material behavior up to the
commencement of necking (point of maximum load).
Overall systematic error from the measurements for load and
extension through the data acquisition system was less that 0.5%. The
overall random error associated with the experiment. including those due
to slight misalignment. non-uniform or non-homogeneous properties.
etc.. is less than 3%.
Data on Zr-2.5Nb sarnples were obtained from previous work 17.11.
7. 2. RESULTS AND DISCUSSION
Tests conducted on aluminum 6061 and aluminum A356 indicate
that a single Ramberg-Osgood equation c m describe the behavior of
these materials very accurately (see Figs. 7.1 and 7.2). Problems were
encountered. however. in 70/30 brass where even two Rarnberg-Osgood
equations were incapable of describing the stress-strain response accu-
rately over a large deformation range (see Fig. 7.3). There is a significant
strain range between 3-696 strain over which neither of the two equa-
tions described materials' behavior accurately. The method of obtaining
Zulfikar H . A. Kassam Finite Element Analysis for Tube Drazuing 119
two Ramberg-Osgood equations have been descrfbed in detail previously
[7.1].
In addition to the inability of describing materials behavior over
large ranges of deformation. the Ramberg-Osgood equation cannot
describe the behavior of materials that exhibit strain-softening.
Therefore. efforts were directed to search for a different equation
that could describe the behavior of any material accurately using a
single equation so as to avoid the inconvenience of developing several
equations to describe behavior of a single material. The possibility of
using this equation to describe strain softening is also studied.
- AL.UMINUM 606 1 CURVE RAMBERG OSGOOD EQUATiON &82 GPa. a1 =3 16 MPa. a=0.305, n= 19.9
True Strain (%)
Figure 7.1 Experimentd data points and the Rarnberg Osgood equation describing the behavior of aluminum 606 1.
Zulfïkar H . A . Kassam Finite Element Analvsis for Tube Drawina 120
- ALUMINUM A356 CURVE RAMBERG OSGOOD EQUATlON E=82 GPa, a l=194 MPa, ~0.440. n= i 2.8
. . . 0.5 1 .O 1.5
True Striain (9%)
Figure 7.2 Experimental data points and the Ramberg-Osgood equation describing the behavior of aluminum A356
1 O 15
True Strain (%)
Figure 7.3 Experimental data points and the Rarnberg-Osgood equation describing the behavior of 70/30 brass.
Zulfikar Fi. A. Kassam Finite Element Analysis for Ttrbe D r a w i n ~ 121
The result of this extensive search is a new form of constitutive equa-
tion. and we cal1 it the Alpha constitutive equation The name for this
equation will become apparent as the basis of this equation is described
in the following section.
7. 3. The Alpha Constitutive Equation
The idea of this new equation cornes from the parameter a, that is
used in the Ramberg-Osgood equation. To understand the significance of
the parameter a, . the modified Rarnberg-Osgood equation needs to be
understood and. therefore. is described here briefly.
where q is the secant yield strength and e, is the elastic strain at this
stress value defined as follows:
E~ =a, /E
At a = al. the above equation reduces to
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawinn --- -
By the definition given in eqn. 7.2. it is apparent that E I represents the
elastic strain at the stress value of 01 while e at a= al represents the
total strain at that stress value. The is denoted as E as it would
reflect the fact that it represents the elastic strain. but only at this
particular stress value alone - up so far. however. the E represents the
elastic strain at a= 01 . The above equation c m . therefore, be written as
e since the total strain E is the sum of the elastic strain. E , and the
plastic strain. E P. By manipulating the above equation. it can easily be
shown that
a, = E P / ~ = constant (7.5)
i.e.. a, is the ratio of the plastic strain to the elastic strain at the stress
value of al . Therefore. according to the Rarnberg-Osgood equation. a, is
a constant and the value of a, will depend upon the behavior of the
material in the transition region between elastic and the fully developed
plastic region. This observation indicates that the Rarnberg-Osgood
equation emphasizes the transition region and assumes that the strain
hardening behavior exhibited by the material in the early stages of
Zulfikur H . A. Kassam Finite Element Analysis for Tube Drawing 123
deformation will persist throughout. In actual fact. strain hardening
behavior exhibited at the early stages of deformation does not persist
over the entire range of deformation due to the cornplicated interactions
on a microscopic scale. It is due to this reason that the Rarnberg-Osgood
equation cannot describe material behavior over large ranges of defoma-
tion, i.e., above 5% strain. Furthermore, the behavior of materials that
experience a change in the predominant plastic deformation mechanism
in the fully developed plastic region cannot be described by the Ramberg-
Osgood equation for the same reason.
In order to elirninate this problem. the Alpha Constitutive Equation is
developed. In this denvation for the Alpha constitutive equution, a parame-
ter a is used. the definition for which is similar to a, given in eqn. ( 10).
However. unlike the Ramberg-Osgood equation where a, is a constant
and is evaluated as o = 0 1 , the value of a is determined at many
(arbitrary) points on the experirnental stress-strain curve. This implies
that a is a variable. The variable a can be a function of the total strain.
i.e.. a = f ( ~ ) . or altematively. a can be described in terms of the stress.
Le.. a = f(o). For now, consider a = f ( ~ ) . Therefore. a can be define as
follows
a = E P / E ~ =
The daerence between the a, definition of Rarnberg-Osgood equation
and this new definition of a is illustrated in Fig. 7.4.
Zulfikar H . A. Kassam Finite Element Analysis for Tube Drawing 124
Strain
a evaluated at these points
a ( ~ ) = e P i / ~ e ~ and is evaluated at many points implying that a is
1 a variable I
Strain
Figure 7.4 The above diagram shows the difference between the defini- tion of (a) a, used by Ramberg-Osgood where a, is constant. m,=0.7 and (b) a used by Alpha constitutive equation where a is a variable (a=f(~)) and is evaluated at many points on the stress-strain curve.
Zulfikar H. A. Kassam Finite Elernent Analysis for Tube Drawing 1 25
The a value at any point dong the stress-strain curve can be
evaluated by the following expression: for a (a,€) pair of values on the
experirnental stress-strain curve,
By using eqns. (1 1) and (13). the total strain E can therefore be written
as
It has been aiready noted previously that a = f(a ) and. hence. it is more
convenient to remange the above equation into the following form:
The above equation is what we define as the Alpha constitutive equation
This equation, although having a simple form. relates the true stress, a
Zulfikar H . A. Kassam Finite Elenient Analvsis for Tube Drawina 126
to the total true strain, E. via the elastic modulus. E. and the parameter,
Therefore. the experimental results can be used to calculate the
elastic modulus, E, and the variable a as a function of the total strain:
the functional relationship between a and E c m be conveniently repre-
sented by a polynomial equation. The calculation of this polynomial
function is s traightforward. For example. twenty or more (preferably
equidistant) points can be chosen from the experimental stress strain
curve. For each of these points one can determine the a value by using
eqn. (12). A graph of a (y-axis) versus total strain. E. (x-axis) can be
plotted. There are many programs on the commercially available that
enable the calculation of the best fit polynomial - there is no limit to the
order of the polynomial that can be used. However. one would try to
minimize the order of the polynomial for the sake of convenience. but at
the sarne time ensuring that the r2 value is greater than 0.999 if possi-
ble. The r2 value is a measure of the accuracy of the curve-fit and experi-
ence obtained during this research shows that this value of r2 can be
achieved and is indeed necessaxy in order to be able to descnbe the
material behavior accurately. E and a = f(e ) have to be determined to
obtain a constitutive equation to describe material behavior as per eqn.
(15).
Zulfikar H . A. Kassam Finite Elenent Analysis for Tube Drawing 127
7. 4. THE ABILITY OF ALPHA CONSTITUTIVE EQUATION IN
DESCRIBING BEHAVIOR OF MATERIALS
In order to illustrate the effectiveness of the Alpha constitutiue
equation to describe material behavior. the experimental data points
depicting the stress-strain response of 70/30 brass in tension are plotted
together with the Alpha constitutive equation (see Fig. 7.5) in the form of
eqn. (15). i.e.. a = EE / ( 1 +a). For 70/30 brass. E is equal to 72 GPa and
a is a function of the total strain (%) and is determined from the plot
shown in Fig. 7.6 and is found to be the following:
1 O 15
True Strain (%)
Figure 7.5 Expenmental data points and the Alpha constitutive equa- tion describing the behavior of 70/30 brass.
Zulfikar H. A. Kassant Finite Element Analysis for Tube Drawing 128
70130 BRASS 70
True Strain (%)
Figure 6 Plots of a versus total strain. E, for brass, copper and steel.
Figure 7.6 Plots of a versus total tnie strain (e) for brass. copper and
steel.
As can observe from Fig. 7.5. the curve-fit is very good (?=0.9998). and
much better than the one shown in Fig. 7.3 where even two Rarnberg-
Osgood equations could not describe material behavior accurately. This
result is indeed very encouraging as the Alphn constitutiue equation has
demonstrated the capability of describing material behavior over a vev
large range of deformation.
Zulfikar H . A. Kassani Finite Element Analysis for Tube Drawing 129
&*ad-'
FOR COPPER
10 15 20
True Strain (9%)
Ngure 7.7 Experimental data points and the Alpha constitutive equa-
tion describing the behavior of 1018 steel and copper.
To further test the ability of the Alpha constitutiue equation. the
experimental data points and the corresponding Alpha curve-fits for
copper and 10 18 steel are shown in Fig. 7.7. The parameters a and E
for these materials are tabulated in Table 7.1 and Table 7.2, respectively.
The Alpha equation describes the behavior of copper very accurately over
the entire strain range. For 1018 steel, however, which exhibits a 'knee'
at the onset of yielding corresponding to the lower yield point, a simple
polynomial relationship between a and the true strain is not adequate
to descnbe this complex behavior. A simple and straightfomard method
Zulfikar H . A. Kassam Finite Elenient Analysis for Tube Drawing 130
Zulfikar H . A. Kassam Finite EIement Analusis for Tube D r a w i n ~ 131
TABLE 7. 2 Elastic modulus of tested materials
ELASTIC MODULUS
(GPa)
Modified Zr-2.5Nb (Axial tension) 1 95
BRASS (tension)
COPPER (tension)
10 18 STEEL (tension) Modified Zr-2.5Nb (Circumferential compression)
Modified Zr-2.5Nb axial cornoression)
Modified Zr-2.5Nb (Radial compression) 1 109
72
94
200
115
96
of overcoming this problem is to create a database of the relationship of
a and true strain over this very small range where the polynomial
equation cannot describe material behavior. In spite of this shortcorn-
ing. the Alpha constitutiw equation obtained by deriving a polynomial
relationship between a and tme strain has demonstrated that it can
descnbe the behavior of such type of steel over more than 90% of the
s train range.
From the above results. it is clear that a single Alpha constitutiw
eqmtion possesses the ability to accurately describe material behavior
over very large ranges of deformation. This observation at least solves one
of the problems that has been encountered with the Ramberg-Osgood
equation where. frequently. more than one equation is required to
describe material behavior accurately.
Zulfikar H . A. Kassam Finite Elernent Analysis for Tube Drawing 132
7.4. 1 The Ability of Alpha Constitutive Equation in De-
scribing Strain Softening in Zr-2.SwtohNb Pressure Tube Material
Another major concem regarding the employment of the currently
available constitutive equations is the inability of these constitutive
equations in describing behavior of materials that exhibit strain soften-
ing.
In order to investigate whether the Alpha constitutiw equation has
the ability to overcome this deficiency that is present in the currently
used constitutive equations. the data from experimental stress-strain
results of modified Zr-2.5wtOhNb CANDU pressure tube (7.1) is used t o
obtain the corresponding Alpha constitutive equatiom. The relationship
between a and the total tme strain (96) is then is also tabulated in Table
Zrilfikar W . A. Kassam Finite Element Analysis for Tube Drawing 133
O EXPERIMENTAL DATA POINTS IN AXIAL COMPRESSION
. ALPHA CONSTlTUTIVE EQUATION IN AXIAL COMPRESSION
A EXPERIMENTAL DATA POINTS IN CIRCUMFERENTIAL COMPRESSION
------- ALPHA CON!jTiTüTIVE EQUATION IN CIRCUMFERENTIAL COMPRESSION
a EXPERIMENTAL DATA POINTS IN RADIAL COMPRESSlON
- 9 - ALPHA CONSTITUTIVE EQUATION IN RADIAL COMPRESSION
3 4 5 True Strain (%)
Figure 7.8 Experimental data points and the Alpha constitutive equa- tion describing the behavior of modlfied Zr-2.5wtVoNb CANDU pressure tube material in compression along the axial. radial and circumferential directions of the tube.
To determine the effectiveness of these Alpha constitutive equatims.
the experimental data points obtained for modified Zr-2.5wt%Nb CANDU
pressure tube material in compression are plotted in Fig. 7.8 along with
the corresponding constitutive relations. As can be seen from this figure.
the Alpha constitutive equntion can describe the behavior of this material
very accurately in al1 the directions.
It is important to note that the behavior along the circumferential
direction c m also be described by a single Alpha constitutive relation.
This observation 1s very important as previous research (7.1) revealed
that the material behavior in this direction could not be described by a
Zulfikur H. A. Kassam Finite Element Analysis for Tube Drawing 1 34
single Ramberg-Osgood equation. This was due to the complicated
behavior exhibited in this direction as a result of the change in the
predominant mechanism responsible for plastic deformation (1).
Of more striking importance is the fact that the Alpha constitutive
eqUQtiOn can describe the strain softening behavior exhibited by this
matenal in compression dong the axial direction of the tube.
7.5. USMG a-EQUATION IN FIMTE ELEMENT ANALYSIS
Implementing the a-equation in finite element analysis is very
simple. The most important consideration is that the hardening rate,
H*=da/de, needs to be evaluated. This parameter can easily be determined
by numerical means.
7.6. CONCLUDING REMARgS
From the above discussion it is clear that a newly developed
constitutive equation. the Alphn constitutive equation, can describe
behavior of different types of matenal over large range of deformation.
e.g. 70/30 brass and copper. In addition, it can also describe behavior of
materials which demonstrate complicated hardening behavior, such as
modified Zr-2.5wt%Nb CANDU pressure tube material dong the circum-
ferential direction which exhibited a sharp change in the strain harden-
ing behavior. Furthexmore. the Alpha constitutive w o n proved capable
of describing strain softening as in modified Zr-2.5wt%Nb CANDU
Zulfikar N. A. Kassam Finite Element Analysis for Tube Drawing 1 35
pressure tube material in compression dong the axial direction. This
particular aspect gives the Alpha constitutive equation a definite advantage
over the other constitutive relations. Since the Alpha constitutive equntion
has proved to be more accurate in describing materials behavior than the
currently used constitutive equations. it could be used in finite element
cornputer programs in order to provide more accurate input of material
behavior for the simulations.
Zulfïkar H. A. Kassam Finite Elenaent Analysis for Tube Drawing 136
REFERENCES
I7.1) Kassarn. 2. H. A., Wang, Z., and Ho. E.T.C.. "Constitutive
equations for a modified Zr-2.5 wt% Nb pressure tube materiai,"
Materbls Science & Engineering, Vol. A1 58 (1 992) pp. 185- 194.
Zulfikar H. A. Kassaln Finite Element Analysis for Tube Drawinp 137
One of the primary concern during the development of the program
was the validity and accuracy of the results obtained from the finite
element simulations.
8.1. TESTXNG THE FINITE ELEMENT ANALYSIS CODE FOR
ACCURACY IN ELASTIC SIMULATIONS
In order to test the accuracy of the program, the stress distribution
in a thick walled cylindrical pressure vesse1 subjected to intemal pres-
sure has been calculated. The geometry of the cylinder used is shown in
Figure 8.1 and the mesh design for finite element calculations is shown
in Figure 8.2. The analytical solutions for stresses in the radial direction.
circumferential direction, and axial directions were calculated as a
function of radial distance; there is no variation along the axial direction
due to absence of any stress in this direction and there is no variation
along the circumferential direction due to the being axisymmetric.
It would be interesting to compare the analytical results to those
obtained using the finite element prograrn that is developed this re-
search; we cal1 it QUAD. For further checking, the solution for stress
distribution has also been obtained using a commercially available finite
element package called EMRCNISA.
Zulfikar H. A. Kassam Finite Element Analysis for Tube Druwing 138
Figure 8.1 The above is the cylindrical pressure vessel that was used for elastic finite element simulation to assess the accuracy of finite element method. a=0.5m. b=l.Om. and p=100MPa.
UNIFORM PRESSURE p = 100 MPa
Figure 8.2 Finite element mesh used to analyze stresses in the cylindri- cal pressure vessel subjected to an intemal pressure of 100MPa.
As mentioned, the finite element mesh used to obtain the solution
is shown in Figure 8.2. The inner wall of the tube has a radius of 0.5m
Zulfikar H . A . Kassam Finite Element Analvsis for Tube Drawirig 139 - --
and the outer wall has a radius of lm. Uniform pressure of 100 MPa is
applied at the inner wall. The finite element mesh consists of 100
elements. The cylinder at the bottom edge is constrained from moving
dong the z-direction (axial direction), hence. u,. which represents the
displacement along the z-direction. is equal to zero. The analytical
result. the finite element solution obtained by the program QUAD. and
the finite element solution from commercially available program called
EMRCNISA are compared in the graph below.
0.6 0.7 0.8 0.9
RADIAL DISTANCE (ml
Figure 8.3 Variation of the circumferential stress component. a,,, with distance from the center (radial distance)
The lines represent the distribution of stresses obtained by using
the elastic analytical solutions. The different syrnbols denote the results
Zulfiicar H. A. Kassarn Finite Element Analysis for Tube D r a w i n ~ 140
- THEORETICAL
O QUAD
x EMRCNISA
0.7 0.8
RADiAL DISTANCE (m)
Figure 8.4 Variation of the radial stress component, O,, with distance from the center (radial distance)
--- TlIEOREllCAL O QUADa,
I EMRCNISA a p
RADIAL DISTANCE (ml
Figure 8.5 Variation of the axial stress component, a,, with distance from the center (radial distance)
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawina 141
obtained by the computer program developed QUAD and the results
obtained by using the commercially available program.
From the graphs show- in Figures 8.3 to 8.5 it is observed that the
result obtained from finite element computer program QUAD and
EMRCNISA are in excellent agreement with the analytical solutions
except a t the edges. The edge effects exist in finite elernent analysis when
there is a stress gradient a t the edge or free surface. The edge effects are
noticeable in the graphs 8.3 and 8.4 where there is a stress gradient a t
the edge. In the axial direction, however, where the stress is supposed to
be equal to zero and therefore no stress gradients exists. Consequently.
it should be expected that the results obtained at the edge for the axial
stress should not be influenced by the edge. The QUAD results are
consistent with this. However, the results obtained from EMRCNISA do
indicate a pronounced edge effect. This provides some indication of the
validity and the accuracy of the finite element program has been devel-
oped. Note that al1 the calculations mentioned above are based on elastic
analysis .
8.2. TESTING FOR ACCURACY IN SIMULATING ELASTIC-
PLASTIC MATERIALS
Well. how about the ability of the program to simulate plâstic
deformation behavior in metallic materials? To test this, finite element
simulations were conducted using different type of materials behavior
that are commonly observed. The actual (theoretical) material behavior is
compared with that obtained from finite element analysis using both
QUAD and EMRCNISA. The simulation involved a single axisymmetric
element being subjected to incremental displacement boundaxy condi-
Zulfiknr H. A. Kassarn Finite Element Analysis for Tube Dmwing 142
tions that would eventually cause considerable amount of plastic defor-
mation. The results from different types of materials behavior are shown
in the graphs below.
S train
Figure 8.6 Theoretical elastic-plastic stress-strain curve is compared to the results of QUAD (our program) and NISA (commercial program)
1 I I 1 I 0.0 1 0.02 0.03 0.04 0.05 O.
Strain
Zulftkar H. A. Kassam Finite Element Amlysis for Tube Drawing 143
Figure 8.7 Theore tical elas tic-linear hardening stress-strain curve is compared to the results of QUAD and NlSA
O . O E + O O ~ O
r 0.0 1
I
0.02 1 I
0.03 0.04 0.
Strain
Figure 8.8 Theoretical Ramberg-Osgood stress-strain cuwe is compared to the results of QUAD and NISA for a material that exhibits a very low rate of hardening.
Strain
Figure 8.9 Theoretical Ramberg-Osgood stress-strain curve is compared to the results of QUAD and NISA for a material that exhibits a high rate of hardening.
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 144
The graphs above indicate that the program (QUAD) gives vexy good
agreement with the theoretical results in simulating plastic deformation.
The results are rnuch better than that obtained from EMRCNISA. This
observation justifies the implementation of this finite element method
program for calculation of stresses in metal forming operations where the
strains are usually very high.
350 :
A QUAD FEM RESULTS
O 2 3 4 5 6 7 8 9 10 1 1 I
True Strain (t)
Figure 8.10 Alpha constitutive equation (solid line) describing the tensile behavior of 70/30 brass and the QUAD FEM simulations results (triangles) exhibit very good correlation.
Zulfikur H . A. Kassarn Finite Element Analysis for Tube Drazoing 145
FOR COPPER FEM RESULT. YS=40MPn
True Strain (%)
Figure 8.11 The Alpha constitutive equation c w e s descnbing the behavior of copper and the FEM simulation results showing good agree- ment when the yield stress used in the one indicating the deviation from linearity .
- ALPHA CONSTlTUllVE EQUATiON IN ClRCUMFEREKnAL COMPRESSION
O FEM RESULT. YS=880MPa
0 FEM RESULT. YS=700MPa
O ! h n I I 8 I 1 I 1
O 1 2 3 4 5 6 7
True Strain (%)
Figure 8.12 The Alpha constitutive equation cuves descnbing the behavior of modified Zr-2.5wt%Nb CANDU pressure tube material and the FEM simulation results showing good agreement when the yield stress used in the one indicating the deviation from linearity.
Zulfiknr H. A. Kassam Finite Element Analysis for Tube Drazving 146
8. S. TESTING FOR ACCURACY M $-TING ELASTIC-
PLASTIC MATERIALS THAT OBEY THE a EQUATION
It is important to determine the accuracy of the result obtained for
the a-constitutive equation. Therefore, FEM simulations were conducted
using three different materials, namely brass, copper and Zr-2. 5wtohNb
alloy. The FEM simulation results are compaxed to the original data. It is
observed that there is excellent agreement between the actual data and
the FEM simulation results provided that the yield stress used is the one
indicating deviation from linearity. It is therefore concluded that the a-
equation can be integrated into FEM programs and can provide accurate
representation of materiaîs behavior.
ZulJikizr H. A. Kassam Finite Element Analysis for Tube Drawing 147
Finite Element Analyaia to Simulate the I Tube Drawing Procesa
9.1. CHOOSING THE APPROPRIATE FINITE ELEMENT
APPROACH
Having embarked upon this project, there were many factors to be
considered. This is due to the fact that there are many different methods
and many different techniques that are available. Each method seems to
have its advantages and disadvantages. Final decision of the method
that will be used had to be based upon a number of criteria. The method
chosen should be such that it cm:
accurately simulate large plastic deformation (including large
strains and rotations) and calculate the deformation, velocity,
strain. force and stress fields in the workpiece.
simulate behavior of non-linear plastic deformation.
calculate stresses that are independent of rotation of the material.
sirnulate transient (non-steady s tate) and steady state problems.
determine the effect of plastic defoxmation path upon the final
stress state.
detexmine the residual stresses in the material &ter the forming
operation.
Zulfikar H. A. Kassam Finite Element Annlysis for Tube Drawing 148
determine accurately the die entry and die exit conditions.
4 accurately simulate the friction boundary conditions at the inter-
face between the die and the workpiece.
9.2. FEM TECHNIQUE FOR LARGE PLASTIC DEFORMATION
In order to satisfy these criteria. research was conducted to deter-
mine the appropriate methods and techniques that could yield the
required result. For accurate simulation of large plastic deformation. the
updated Lagranginn scheme is chosen. The updated Lagrangian method
can simulate problems that involve large plastic deformations. It can
also trace the plastic deformation history of the material and, hence. c m
determine the effect of the deformation path on the subsequent stress
states in the materiai and also determine the residual stresses in the
material. It can also solve non-steady state (transient) problems. The
above rnentioned reasons j u s t e the choice of the updated Lagrangian
method for simulation of metal foming problerns.
9.3. FEM TECHNIQUE FOR NON-LINEAR HARDENING
MATERIALS
In order to sirnulate behavior of non-linearly hardening materials.
the modified Ramberg-Osgood constitutive equation is incorporated into
the finite element code. The modified Ramberg-Osgood equation has a
major advantage over the conventional Ramberg equation in that the
conventional equations assumes that the plastic deformation parameter
Zulfikur H . A. Kussam Finite Element Analysiç for Tube Druwing 149
a, is has a constant value of 3/7 while the modified equation allows the
user to specify the a, value based upon the material. In addition. the
Alpha Constituive Equation was also programmed into the Finite
Element code as well. thereby. providing flexibility in the means by which
the constitutive equation is described.
9.4. CHOICE OF STRESS STATE
The other important factor that is to be considered is the signifi-
cant effect of finite deformation and rotation of the material during
deformation on the resulting stress state. The Piola-Kirchoff stress (like
engineering stress) is not adequate as the deformation is substantial and
this method ignores the effect of change in shape of the material on the
stress state. Cauchy stress (analogous to true stress) is more accurate as
it takes into account the change in shape of the material on the result-
ing stress state. However. in this method. the calculated stress is local in
nature in that the stress axis rotates with the material. The problem
with this type of approach is that stress needs to be calculated for each
element but each element does not rotate the same amount. Somehow, a
stress value has to be determined that in not local but instead global in
nature. This requires that the stress state does not v q with rotation of
the material. This can be achieved by using the spin invariant Jaumann
stress. The incorporation of this calculation into finite element codes is
not trivial. However, this is extremely necessary in order to ensure
accuracy of the calculated result.
Zulfikar H . A. Kassarn Finite Element Analysis for Tube Drawing 150 -- -
9. 5. CHOICE OF FRICTION MODEL
Friction plays a vital role during a forming process. However. i t
presents many problems for finite element simulations 19.1.9.21. Hartley
et al 19.11 admits that friction remains one of the most difficult aspects
to incorporate properly into a finite element model. According to Berry
19.21. the real missing link in the application of finite element methods
to metal forming processes has been a general, automatic algorithm for
treating the complicated contact that occurs during forming including
workpiece entry and exit from the die. There have been a nurnber of
recommendations as to the ways in which friction can be modeled. It is
therefore very important to study the available techniques of simulating
friction conditions between the die and the interface and assess its
capability for accurate simulation of friction conditions.
9. 5. 1. FRICTION LAYER TECHNIQUE
Different methods 19.4. 9.51 have been proposed in order to cater
for the friction boundary conditions a t the interface between the die and
the workpiece using the shear friction model. One of the methods of
taking into account the friction b o u n d q conditions has been proposed
by Hartley, Sturgess and Rowe I9.41. In this method, a layer of 'friction'
elements are created a t the interface between the die and the workpiece.
These friction elements have a stiffness which is equal to the stiffness of
the workpiece modified by a function of the interface shear factor.
Friction. therefore. is artificially imposed upon the interface nodes by the
so called friction elements by virtue of the shear force associated with
Zulfkar H . A. Kassam Finite Element Analysis for Tube Drawing 151
the deformation of these friction elements. The authors claim that there
is excellent agreement between the results obtained by finite element
simulation using this frictional elements approach and the results
obtained from ring compression tests conducted on aluminum.
In order to evaluate the capability of this technique, it was pro-
grarnmed into the finite element code at a very early stage of develop-
ment. This is due to the fact that this approach is very simple and can
easily be integrated into the finite element code. An attempt was made to
simulate a compression test on a cylindrical specimen using axisyrnmet-
rical elements (assuming conditions do not Vary with the angle. 8).
However. this method proved to be unsatisfactory. The reason are
explained below.
I t is found that at rn (shear friction coefficient) values as low as
0.1. the friction elements deform much more than the workpiece ele-
ments because of having a lower stiffness. This effect was significant
when the material was subjected to overall strains in excess of 20%. The
friction elements experienced very high amounts of deformation due to
the fact that they were not only less stiff than the b u k but also near the
surface where deformation is usually higher. Consequently, the shape of
the friction elements distorted appreciably making them unsuitable for
the purpose of transferring shear force to the adjacent elements in the
workpiece.
Furthemore. the extremely high strains experienced by these
friction elements made it very difficult for the finite element solution to
converge and. frequently. caused instability. i. e.. the solution diverges.
Zulfikar H . A. Kassam Finite Element Analysis for Tube D r a w i n ~ 152
I t was also noted that this approach is only valid in the case where
there is relatively little displacement between the die and the workpiece.
Therefore. this method cannot handle metal forming operations such as
tube drawing or strip drawing where there is a significant amount of
relative displacement between the die and the workpiece.
9. 5. 2. ALTERNATIVE METHOD FOR SIlldULATING
FRICTION CONDITIONS
An alternative technique for simulating friction conditions i s the
one proposed by Lu and Wright 19.51. This technique will be discussed in
length as detailed research was conducted to assess the validity of the
approach. It was thereafter incorporated into f i i te element code to check
its performance and accuracy. Important modifications were made to this
method in order to ensure reliability and accuracy of the results ob-
tained. All details will be discussed in this chapter.
There are two methods of sirnulating friction conditions in finite
element simulations. These are the interjùce velocity model and the
interface fiction mode2.
In the interface velocity model. as the name implies. the velocity at
the boundaries and contact region are known. For example, in the case
of the of wire drawing or solid bar drawing. the velocities at the center
line and the velocity at the interface between the die and the workpiece
must be known. together with the drawing speed. In the case of the tube
drawing process, the velocities at the interface between the die and the
workpiece and between the mandrel and the workpiece must be known as
Zulfiknr H. A. Kassarn Finite Element Analysis for Tube D r a w i n ~ 153
well as the drawing speed. This type of approach has the advantage in
that no knowledge of friction boundary conditions is required because
the effect of the friction is embedded in the information of the velocity
profile. i.e., the velocity at the interface is a direct consequence of the
friction conditions existing at the interface. In fact. finite element
simulation can be carried out with knowledge of the interface velocities
to determine the friction conditions that exist at the interface! This is
exactly opposite to the interface friction model whereby knowledge of the
friction conditions at the contact regions are required and the finite
element solution will allow one to determine the velocities at every point
in the body including the contact region.
There are a few disadvantages with regards to the interface velocity
model in that a number of experiments are required for each case study
in order to detemine the velocities at the contact regions. Every time a
parameter such as the drawing speed, die angle. friction condition.
material properties. etc.. are changed. new expertments need to be
conducted to measure the velocity profile at the interface. Each experi-
mental result is therefore valid for the specific parameters employed in
the forming process. This type of approach is therefore very expensive. In
addition. since rnost of the data is derived from experirnents. the finite
element modeling used is limited to calculation of strains and stresses in
the body. including the detexmination of residual strains in the body.
The finite element modeling, therefore, loses its predictive capability in
that in cannot allow one to determine the strain and stress field if any of
the parameter in the forming process changes. Herein lies most of the
Zulfikar H. A. Kassam Finite Element Analvsis for Tube D r a w i n ~ 154
benefit yielded by using finite element method which in the case of the
interface velocity model cannot be exploited. Nevertheless. the interface
velocity model is useful for conduction accurate simulations which may
be used for verification of the finite element model.
The interface friction model. as mentioned earlier. requires knowl-
edge of the fnction conditions that exist at the interface. While accurate
knowledge of the friction conditions at the contact points cannot be
determined easily, approximate values can be obtained from literature.
These Mction coefficient will depend upon the surfaces that corne into
contact with each other and the lubrication condition at the interface.
The method has a significant advantage over the interface velocity model
in that finite element modeling can be conducted predict the feasibility of
manufacturing a part should any or al! of the parameters in the process
change or be changed. This gives the interface friction model full predic-
tive capability.
The interface friction model is incorporated into finite element
methods through specification of the traction (forces) that result as a
consequence of the presence of the friction between the interface. Al1
parameters such as die angle. die length. material properties of work-
piece. draw reduction. etc. can be changed to examine the effect it has on
the process and the product.
The friction is modeled by considering the relationships between
the normal and the tangential components of the surface traction that
result from the presence of friction. This method is based on the general
linear boundary condition method that was proposed by Lu and Wright
Zulfikar H . A. Kassam Finite EIement Analusis for Tube Drawina 155 ppppp - --
I9.5). This method was designed to model friction conditions on non-
aligned surfaces. Another area of concem was the feasibility of obtaining
accurate solutions from this approach.
There are many advantages associated with implementing the
general linear boundary condition method. Based on the physics of the
friction model. a relationship between the traction components can be
determined and incorporated into the finite element code in a very
general marner. This method is based upon controlling the displace-
ments of the nodes that are in contact with the die in accordance with
the geometry of the contact surface. In addition. a relationship between
the forces in the two principal (global) directions is formulated based
upon the friction model to be implement.
Zulfikar H . A. Kassam Finite Element Analysis for Tube Drawing 156
AXIAL DIRECTION, z 4
IE INTERFACE
4-NODED OF INTER
ELEMENT .EST
DIE-WORKPIECE INTERFACE
RADIAL DIREXTION, r
z
(BI Figure 9.1 shows the 3-dimensional (A) and 2-dimensional simplifica- tion of the elernent which is in contact with the die interface.
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 157
Figure 9.1 shows the 3-dimensional as well as the 2-dimensional
diagrams to demonstrate the conditions that exit at the interface. I t is
apparent that the nodes in contact with the boundary. namely nodes A
and B. can only move dong the interface between the workpiece and the
die. Therefore. for each of these nodes. a displacement boundary condi-
tion has to be imposed such that these nodes can only move dong the
boundary. This entails the imposition of a boundary condition in which
the displacements in the radial and axial directions are constrained so
as to confirm with the boundary. Let the displacement in the radial
direction be termed y, and the displacement in the axial direction be
termed v,, where n is the node of interest. Consequently. a relationship
can be formulated to confirm with the boundary conditions. This will be
in the fonn of:
- y, cosa = v, sina
or on rearranging the above equation,
u,, cosa + v,, sîna = O
The above boundary condition has to be superimposed upon the nodes
that are in contact with the die-workpiece interface. In addition. a force
(traction) boundary condition has to be applied to account for the
friction boundary conditions that exist a t the interface.
Zulfikar H . A. Kassam Finite Element Analusis for Tube D r a w i n ~ 158
For coulomb friction. the FNom and the Fm,, forces are related
in the following manner.
where p is the coefficient of friction.
Upon resolving the forces. one obtains the following equations:
F,,,, = -(Fr cosa + F, sina)
FTANCEM = Fr sina - F, cosa
Substituting equations (9.4) and (9.5) into (9.3) and rearranging. the
following equation is obtained:
Fr (sina + pcosa) - F, (cosa - psina) = O (9.6)
Al1 the equations shown above represent the physics of the prob-
lem. The main concem that remains is the method that needs to be used
in order to integrate the above equations into the finite element method.
Zulfïkar Fi. A. Kassam Finite Element Analysis for Tube Drawing 159
Figure 9.2 The above diagram shows the setup used by Lu and Wright for strip drawing (top) and the setup used for tube drawing (bottom)
The finite element simulation is broken down into many step in
order to simulate plastic deformation. Similarly. the displacement and
force boundary conditions represented above will be appiied incremen-
Z u l f h r H. A. Kassam Finite Element Analysis for Tube Drawing 160
tally. Thus. a more accurate representation of the equation requires the
specification of the incremental nature of the problem.
The solution procedure adopted in this study resembles closely to
the one adopted Lu and Wright who studied strip drawing. The finite
element simulation begins at the drawing end of the tube. labeled DE.
which at the beginning is actually outside the die. This implies that this
section is considered to have been drawn already. The solution proce-
dure ends when steady state conditions are achieved. The length of the
tube is designed such that the steady state conditions are achieved well
before the other end reaches the die entrance. This is an important
consideration in order to avoid any perturbances affecting the solution.
When the solution procedure begins. it is assurned that the whole
material is homogeneous. This implies that the material properties are
the same at every point. This approach overlooks the fact that the
material which is experiencing deformation under the die and the
material that has been already drawn may have different mechanical
properties compared to the undeformed material that has to be drawn.
Nevertheless. this seems to be a reasonable assumption bearing in mind
that the tube will be drawn a sufficient distance in order to achieve
steady state condition and. therefore. produce the necessary information
about the residual stresses as well as the mechanical properties in the
material after the drawing process.
It is important to note that in the tube drawing process there are
two boundaries. The first one is the one between the die walls and the
outer walls of the tube and the second one is between the mandrel and
Zulfikar H. A. Kassam Finite Element Analilsis for Tube Drawina 161
the inner wall of the tube. The presence of two boundaries does make the
problem more cornplex.
The essential aspect of al1 problems described by differential
equations is the boundary conditions that exist. There are two ways by
which boundary conditions at these two interfaces can be specified. The
first method is by determining the velocity field a t the interface by
experimental techniques and using this information to spec@ the
displacements at the nodal points that corne into contact with the
boundary. This method is indeed very powerful. However, the major
drawback of this technique is that it is only valid for a certain set of
conditions such as material property. friction coefficient between the die
surfaces and the workpiece, the die geometry. etc. Therefore. if any of the
conditions change. then the data will be of no use. This is a significant
drawback as it does not allow this analysis to be predictive. which in fact
is the most important goal. It is the predictive capabilities that will
provide the vision to foresee problems before they occur and Save time
and cost of actually performing the tests in the laboratory without any
indication of the end result. In addition, determination of velocity fields
is possible for strip drawing process where a grid can be etched ont0 the
side surfaces prior to the drawing process, and this gnd pattern analyzed
when steady state conditions are attained. In the tube drawing process.
however, because of the geometry being a tube. it is not possible to use
this technique.
Another option is to use the interface boundary conditions that
depict the friction conditions existing at the interface. The way in which
ZulJikar H. A. Kassarn Finite Element Analysis for Tube Drawing 162
the friction conditions can be simulated in the finite element method is
quite cornplex. However, an attempt will be made to explain the tech-
nique as clearly as possible.
Regardless of the friction mode1 implemented. the conditions at
the drawing end of the tube need to specified. A displacement Au,
(displacement dong the axial direction) is specified to be some value that
would result in sufficient displacement but at the same time not yield
very high strain in a single increment: large increments in strain in a
single increment may cause instability and the solution would not
converge. At the same. the displacement in the radial direction, Au,. is
defined to be zero; incorporating this condition is important as it avoids
and unnecessary distortion of the elements at the drawing end which
does occur in the absence of this additional b o u n d q condition. The
reason for this distortion is that the force is concentrated in the few
nodes at the drawing end of the tube.
The boundary conditions that are applicable on the inner and
outer surfaces of the tube are now described. The tube surfaces c m be
separated into two different classes. There are some regions which are
free, i.e.. they are not in contact with another surface. Theses regions are
easy to deal with as the forces applied on these regions is equal to zero.
However. of more importance are the regions that corne into direct
contact with the die walls or the mandrel. In addition, it must be noted
that the workpiece moves as the solution procedure progresses. Therefore
the regions that are in contact with the die walls and the mandrel with
Zulfikar H . A. Kassam Finite Element Analysis for Tube Drawing 163
continuously change as the simulation progresses. Therefore. there is a
need to have a procedure built into the program that will keep track of
the nodes that in contact with the surfaces a t any given time. Specific
conditions need to be set up to determine the moment in time that the
node first cornes into contact with surfaces and at which point the
contact is lost. In short. a procure is needed to continuously update the
location of the workpiece with respect to the die so as to ensure that the
friction boundary conditions are applied at the proper locations.
Once a node comes into contact with the die. the boundary
conditions of the node change from a traction free (force = O) condition
to one where the friction boundary conditions are applicable. Similarly.
when a node loses contact with the surface, the boundary conditions
change back to traction free state where no forces are applied. Very few
finite element techniques are available for simulating sliding contact
boundary conditions of which the most appealing was the one by Lu and
Wright.
Detemaining The Contact Points
Since the friction boundary conditions need to be applied at the
surfaces which corne into contact with the die wdls or the rnandrel, it is
necessary to detemine the moment a t which the workpiece surface
establish this contact. Therefore. it is important to establish a mecha-
nism by which the point at which the surface node of the workpiece first
comes into with die wdls or mandrel and the point a t which they lose
contact with these surfaces.
Zurflkar H . A. Kassam Finite Element Analysis for Tube Drawing 164 -- - -
The method used by Lu and Wright is simple. The method they
propose is as follows: on the die-entry side the X-coordinate of nodes
such as P on the surface ABCD are monitored at each step. When the X-
coordinate. X(P). of any such node exceeds the X-coordinate of a speci-
fied point M. X(M). the node P is considered to be eligible for contact. The
point M is chosen upstrearn from the idealized strip-die intersection to
allow for possible bulging of the strip material. When X(P) is greater than
X(M). the Y-coordinate of P is also checked and compared with the Y-
coordinate of P', the vertical projection of P on the die surface. If X(P) is
greater than or equal to X(M) and Y(P) greater than or equal to Y(P'). the
node is considered to be on the die surface and the boundary condition is
changed from a traction free to a specified traction (friction) rnodel.
Possible penetration of the node into the die is also monitored.
When the penetration is less than 5% of the corresponding drawing
increment. it is ignored. Otherwise. the increment is repeated with a
smaller displacement.
A similar technique was used in this research with slight varia-
tions. Instead of using an imaginary point M beyond which contact is
not recognized even if contact has occurred. the equation of the lines
that make up the die surface and the mandrel is established. This
program also requires one to spec* the nodes that make up the outer
and inner surfaces; these are the ones that will come into contact with
the die surface and the mandrel. respectively. After each iteration. the
program checks to establish whether the surface nodes have come into
contact with the die or the mandrel. This procedure is more accurate as
Zultïkar H . A. Kassarn Finite Element Analusis for Tube Drawina 165 - . - - - - - - - -- - -
it does not wait for d l the iterations of the present increment to finish
before applying the boundary conditions. Therefore. there is virtually no
penetration of the die or the mandrel resulting in a more accurate
solution.
The die exit conditions can be accounted for by determining the
coordinate of the point at the die exit (which is fixed) and compare it to
the nodal coordinates in order to establish whether the node has exit the
die.
One important aspect to note is that the stresses developed in the
workpiece as a consequence of massive deformation during the drawing
process must be relieved to a certain extent as the matenal leaves the
die. This process has to be artificially achieved. Initially. the technique
proposed by Lu and Wright was used. This however caused instability as
the force was reîieved in one step. Modifications were made so that the
stress is relieved incrementally over a short period of tirne to alleviate the
problem of instability (lack of convergence).
The friction condition is incorporated into the finite element
solution as described here. The basis of this technique is to establish two
important aspects; (1) the constraint in the displacement. u. of the nodes
which corne into contact with the surface and (2) the friction force. f.
generated at this surface is applied to the corresponding node.
The relationships are as follows:
aul + b b = c
AL+ BA= C
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 166
where a. b A and B are constants depending on die geometry and friction
conditions. and 0 is the angle of the die surface line beginning at the exit
and joining to the entrance; in the cartesian coordinate system. the
angle is negative. e.g.. for a 15 degree semi-cone angle. the 0 value is -75
degrees. The subscripts 1 and 2 refer to the two directions which in this
case is r (radial) and z (axial).
If. for example. if a friction boundaxy condition is to be specified at
node P. then the equations that need to integrated into the finite ele-
ment code are as follows:
%P., + bh~= c (9. Sa)
AL,,+ B ! P = C (9.8b)
The equations in a finite element analysis calculation is in the form of
the small letter designate local conditions while the capital letters
designate global conditions. The above equations imply that the local
conditions at the nodal point need to be integrated into the global
system of equations. The resulting equations in the global mat* is as
follows : (missing)
The equation for the force boundaxy condition is as follows:
Zulfikar H. A. Kassam Finite Element Analysis for Tube Drawing 167
K, is the i j entry of the global stiffhess mat& and j = 1.2.. ... . n. In
addition, the node in contact with the surface has to constrained to
move dong the contact surface by implementing the following equations:
F2p1 = c (9.1 la)
& ~ - 1 . 2 ~ - 1 = a (9.1 1b)
&P- I.DP = b (9.1 1 ~ )
% P - L . ~ = 0 (9.11d)
where j = 1. 2. ....... n and j is not equal to 2P-1 or 2P. n is the total
degrees of freedom.
For the coulomb friction case,
a = sin 0
B = -sin 0 + pcos 8
C = O
9.6. TUBE DIMENSIONS
The tube being drawn was chosen to have the same dimension as
one of the experiments conducted at Ontario Hydro Technologies. This
was done so as to be able to compare the finite element simulation
Zulfikar H. A. Kassam Finite Elment Analysis for Tube Drawinl~ 168
results to the experimental results conducted at Ontario Hydro Tech-
nologies.
Figure 9.3 The above diagram shows the setup of the drawing operation
Zulfiknr H . A. Knssarn Finite Element Analysis for Tube Drawing 169
The inner radius of the tube prior to the drawing operation is
11.05mm, and the outer diarneter is 12.84mm. During the drawing
operation. the tube thickness and the diarneter are reduced such that the
final inner radius of the tube is 10.67mrn and the outer radius is
1 1.94mm. The die semi-cone angle is 1 5".
9.7. MESH DESIGN
The mesh consisted of 390 axisyrnmetric 4-noded quadrilateral
elements. There were 432 nodes in total. Each node had two degrees of
freedom. one along the radial direction and one along the axial direction.
9.8. BOUNDARY CONDITIONS
Displacement boundary conditions are imposed on the nodes at
the drawing end of the tube in order to mode1 the drawing process.
Friction b o u n d q conditions are imposed upon the nodes that are in
contact with the die and the nodes that are in contact with the mandrel.
The general linear boundary condition method is used to incorporate the
friction conditions. This method is explained in detail earlier in the
chapter. The coefficient of friction. p. is assumed to be constant over the
entire contact surface.
9.9. TUBE DRAWING EXPERIMENTS
A hydrawlic drawbench test facility (HDTF) has been set up at Ontario
Hydro Research Division (OHRD) . The drawbench was custom-designed
Zulfikar H. A. Kassam Finite Etement Analusis for Tube Drawina 170
and manufactured by ASTec Inc. of Baltimore. Maryland to meet techni-
cal requirements specified by Ontario Hydro Research Division (OHRD)
personnel. The load capacity of the hydraulic drawbench is about 135kN
(30.000 lbs). The range of lengths of scaled-down tubes that can be cold
drawn is 15-230 cm (6"-90"). The stroke of the actuator ram is equipped
with an MTS Tempsonics LDT position Sensor (Mode1 TT' SCRU09000)
position encoder to monitor or track the position of the draw in any
given experiment. The drawbench has two built-in load cells to monitor
and/or record the draw force and the mandrel force. The speed of the
actuator arm can be manually adjusted to give draw speeds in the range
O - 50 mm/s. A schematic diagrarn illustrating the general layout of the
HDTF is presented in Figure 9.4.
Data Acquisition System
The HDTF has a dedicated data acquisition system. This system consists
of a IBM compatible personal cornputer coupled to a Hewlett Packard
mode1 HP3421A Data Acquisition Control Unit. A custom designed
software has been developed in-house to acquire and process experirnen-
tal test data obtained during a given draw experiment. Such parameters
as draw force. mandrel force. position of the rnoving work peice. and
temperature can be monitored and recorded in a given draw test. The
software has been developed to display the engineering values of the draw
parameters of interest in real time to facilitate a more interactive test
environment. The test data are also automatically logged to a file which
Zulfikar H. A. Kassam Finite Element Anal~s is for Tube Drawing 171
can be retrieved for further post-test processing. The processed data can
be plotted graphically as well.
Awilury Equipment
In order to cary out the draw experiments, the following auxilliary
equipment is needed:
a) Die and mandrel sets:
b) Mandrel holder and load train assembly;
C ) Load train assembly to grip the workplace or tube specimen.
Cold Drawing of Aluminum Alloy 6061 Tubes
The starting material is an alluminum alioy 6061 seamless tube received
in the T6511 temper condition. The dimensions of the aluminum alloy
tube for the given experirnent is shown in Figure 9.3. Pnor to performing
cold draw experiments. the tube samples were "pointed" (swaging down
the leading or front end of the tube samples for gripping purposes) and
then given a heat treatrnent of 4.0 hours at 400 OC in a resistance air
fmace . The lubncant used was an extreme pressure (EP). molidenum-
disulphide based grease.
The elastic modulus of this material is 70 GPa and the Poisson's
ratio is 0.345. The yield stress of the material is 235 MPa. The stress-
strain response of the material up to fracture can be described by the
modified Rarnberg-Osgood equation with the following parameters:
Secant yield strength. o, = 235 MPa
Zulfikar H . A. Kassarn Finite Elernent Analysis for Tube Drawing 172
Plasticity parameter. a, = 0.275
Strain Hardening parameter, N = 28.7
Zirlfikar H. A. Kassam Finite Element Analysis for Tube Drawing 173
Figure 9.4. Schematic Diagram illustrating general layout of Hydraulic Drawbench Test Facility
Zulfikar N. A. Kassam Finite Element Analysis for Tube Drawinn 174
REFERENCES
19.11 P. Hartley. 1. Piliinger and C. E. N. Sturgess, "European develop-
ments in simulating forming processes using three-dimensional
analysis." JOM 43 (10) (1991) 12.
I9.21 D. T. Berry. "Starnping out forming problems with FEA," Mech
Eng. 110 (1988) 58-62.
19-31 J.F.T. Pitmann, R.D. Wood, J .M. Alexander and O.C.
Zienkiewicz. Numerical Methods in Industrial Forming Processes.
Pineridge Press. Swansea. U.K.. 1982.
19.41 P. Hartley. C. E. N. Sturgess and G . W. Rowe. "Friction in Finite
Element Analyses of Metalforming Processes," Int. J. Mech. Sci.
21 (1979) 301-31 1.
19.51 S C-Y. Lu and Wright. 'Finite Element Modeling of Plane-Strain
Strip Drawing with Interface Friction." Journal of Engineering for
Indusiq 110 (1988) 101-110.
ZuIfikar H. A. Kassam Tube Drawing Results 175
Tube Drawing Results
10.1 GRAPHICAL PLOTS OF VARIABLES
The tube drawing setup is shown in Figures 9.3 and 9.4. The z-direc-
tion is the axial direction. Variables such as displacement. strain and stress
are plotted as a function of radial position (along the thickness of the tube).
at different axial positions (along the tube drawing direction). This will pro-
vide information with respect to the change in the instanteneous displace-
ment. i.e.. velocity, strain and stress patterns as the tube is drawn out of
the die. The axial positions are absolute with the reference point origina-
tion at the end of the tube that is opposite to the drawn end of the tube
when the drawing process commenced according to the finite element simu-
lation. The radial position has a reference point along the axis of the cylin-
der. The simulation carried out is axisymmetric and. therefore. the results
are the same along the circumferential direction regarless of the exact an-
gular position. It should be emphasized that the displacements plotted here
are not absolute displacements but in fact instantaneous displacements in
a certain time increment at steady state. This implies that these instanta-
neous displacements reflect the velocity profile of the flow behavior. The
axial positions plotted have the following significance: Table 12.1. Data Plotted
al Position, A nlficance - 10 mm Initial die entrance No
12.0 mm Further stage of deformation Yes 12.84 mm Die curves to become straight Yes 13.0 mm Straight region of the die Yes 13.5 mm Die exit point Yes 14.0 mm Post deformation stage Yes
Zulfikar H. A. Kassariz Tu br Dra w i n ~ Results 176
The apparent position where the tube is in contact with the die is a t 10mm.
This is before deformation commences. As deformation progresses and the
drawing operation reaches a steady state. i.e., a state variable such as
displacement (velocity) is constant at any position within the die. the ap-
parent point of contact is between 11.0 to 11.5 mm. Therefore the results
are plotted at A=lZ.Omm in order to give an idea of how deformation
progresses at the early stages. Results plotted at A=12.84mm represent
the conditions at point of curvature of the die back to the straight position.
This is the position where almost al1 the deformation has already been
irnparted to the material. Meanwhile, results at A= 13.0 indicate the changes
that may occur due to change in the geometry of the die from being con-
verging to parallel. The results at A = 14.0mm indicate the conditions after
the material has exited the die.
10.2. RADIAI, AND AXIAL DISPLACEMENTS
The radial and axial (instanteneous) displacements are of interest as
they provide an indication of the flow profile duMg the drawing process.
Al2 displacements rnentioned here are instanteneous displacements although
the word instanteneous is not speciJcally mentioned in al1 cases. The radial
displacement as function of radial position (through the thickness of the
tube), at different axial positions (along the tube drawing direction) is plot-
ted in the Figure 10. la. The results plotted represent conditions at steady
state. The radial displacements are naturally negative (indicating material
movement towards the centre of the die) as a result of contact with the
converging dies. In the ideal case. plane sections would remain plane. Please
note that the ideal case is only hypothetical and is used here for cornpari-
son purposes only.
ZulfZkar W. A. Kassam Tube Dra w inlp Results 177
11.0 11.5 12.0
Radial Position (mm)
Figure 10.la The radial displacement as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=0.05 at steady state.
Zulf2kar H. A. Kassam Tube Drawing Results 178
Radial Position (mm)
Figure 10.lb The radial displacement as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for ~=0 .1 at steady state.
Zulfikar H. A. Kassam Tube Drawing Results 179
1 0.5 11.0 11.5 12.0 12.5
Radial Position (mm)
Figure 10.2a The axial displacement as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=0.05 at steady state.
Zulfrkar H. A. Kassam Tube Drawing Results 180
In the ideal case where plane sections would remain plane. the de-
formation would be uniform across the thickness of the tube and conse-
quently the instantaneous radial displacement (radial velocity) as a func-
tion of the radial position would be a straight line. However. as observed in
the graph. it is clear that the line is far from being straight except during
the early stages of deformation (A= 12.0mm). This indicates that the defor-
mation during the early stages is reasonably uniform except in the region
close to the contact surface of the outer tube section. This is to be expected
because it is very close to the first contact point between the die and the
workpiece and the effect of friction has not spread deep into the thickness
direction of the tube. As the material is drawn further into the die and
approaches the straight section (A= l2.84mm), the radial displacement line
becomes curved significantly. This indicates that as deformation progresses.
the deformation through the thickness of the tube becomes highly non-
uniform. The instantaneous radial displacement in the negative direction
initially increases but thereafter decrease considerably as the material ex-
its the die. Le.. the instantaneous radial displacement (radial velocity) tends
to zero. This is due to the fact that as the material exits the converging part
of the die. there are virtually no forces acting in the radial direction. The
tendency of the radial displacement to go to zero implies that the material,
after exiting the converging section of the die. tends to move only dong the
axial direction as would Se expected.
In order to verify that the change in the radial velocity as deforma-
tion progresses is attributed to friction. the result for radial velocity for a
higher value of friction coefficientrn (p=O. 1) is plotted in Figure 10.1 b. On
comparing 10.1 a with 10. l b . it is observed that the radial velocity profile a t
axial position 12.0mm is the same since a t l2.Omm deformation has just
Zulfikar H. A. Kassam Tube Drawinn Results 181
Radial Position (mm)
Figure 10.2b The axial displacement as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=0.1 at steady state.
Zulfikar H. A. Kassam Tube Drawing ResuZts 182
m..
Radial Position (mm)
Figure 10.3a The resultant speeddrawing speed as a function of radial position (along the thickness of the tube) is shown at different axial positions (dong the draw direction). The above results are for p=0.05 at steady state.
Zulfikar H. A. Kassarn Tube Dra w i n ~ Results 183
Radial Position (mm)
Figure 10.3b The resultant speed/drawing speed as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=O. 1 at steady state.
Zulfikar H. A. Kassam Tube Drawina Results 184
11.0 11.5 12.0
Radial Position (mm)
Figure 10.4a The direction of flow as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=0.05 at steady state.
Zulfikar H. A. Kassam Tube Drawinp Results 185
Radial Position (mm)
Figure 10.4b The direction of flow as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for u=0.1 at steady state.
Zulfikur H. A. Kassam Tube Drawinn Results 186
(O. 00)
(O. 1 O)
(0.20)
(0.3 O)
(0.40)
Radial Position (mm)
Figure 10.5a The radial strain as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for y=0.05 at steady state.
Zulfikar H. A. Kassarn Tube Drawing Results 187
(0.00)
(O. 1 O)
(0.20)
(0.30)
(0.40)
Radial Position (mm)
Figure 10.5b The radial strain as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=O. 1 at steady state.
Zulfiùar H. A. Kassam Tube Drawing Results 188
Radial Position (mm)
Figure 10.6a The axial strain as a function of radial position (along the thickness of the tube) is shown at different axial positions (dong the draw direction). The above results are for p=0.05 at steady state.
Zulfrkar H. A. Kassarn Tube Drawina Results 189
Radial Position (mm)
Figure 10.6b The axial strain as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=O. l at steady state.
Zulflkar H. A. Kassarn Tube Drawing Results 190
0.05 - a
(0.00) -
(0.05) -
(0.10) -
(O. 15) -
Radial Position (mm)
Figure 10.7a The shear strain as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=0.05 at steady state.
Zulfikar H. A. Kassam Tube Drawinp Results 191
J
0.15 -
0.10 -
0.05 - 4
(0.00) - .
(0.05) -
(O. 1 O) - d
1
Radial Position (mm)
Figure 10.7b The shear strain as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for y=0.1 at steady state.
Zulfrkar H. A. Kassam Tube Drawina Results 192
Radial Position (mm)
Figure 10.8a The circumferential strain as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=0.05 at steady state.
Zulfikar H. A. Kassam Tube Drawing Results 193
Radial Position (mm)
Figure 10.8b The circumferential strain as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=0.1 at steady state.
Zulfikar H. A. Kassan~ Tube Drawing Results 1 94
Figure 1 O.9a The meshes showing the progression of deformation
Zulfikur H. A. Kassum Tube Drawing Rcsults 195
Figure 10.9b The mesh shown at steady state for p-0.05.
Zurfikar H. A. Kassam Tube Drawing Results - 196
Figure 10.9~ The mesh shown at steady state for p-O. 1.
Zulfikar H. A. Kassam Tube Drawing Results 197
9 m w b , C V m a ) F d o o o r ~ ~ y c y
o o o o o o o o o
Figure [email protected] The above diagram shows the radial strain contours at steady state conditions for p=0.05.
Zulfikar H. A, Kassam Tube Drawinn Resrrlts 198
Figure la. lob The above diagrani shows the radial strain contours at steady state conditions for p=0.1.
Zulfikar H. A. Kassani Tube Drawing Results 199
Figure W. 1 la The above steady state conditions for
diagram p=0.05.
shows the axial strain contours at
Zulfiknr H. A. Kassam Tube Druwing Results 200
Figure 1Q.llb The above diagram shows the axial strain contours at steady state conditions for p=0.1.
Zulfikar H. A. Kassam Tube Drawing Results 20 1
Figure le. 12a The above diagram shows the shear strain contours at steady state conditions for p=0.05.
Zurfikar H. A. Kassam Tube Dra wing Resuirs 202
Figure 1Q.12b The above diagrarn shows the shear strain contours at steady state conditions for p=O. 1.
Zulfikar H. A. Kassam Tube Drawing Results 203
Figure 10.13a The above diagram shows the circumferential strain contours at steady state conditions for p=0.05.
Zuljikar H. A. Kassam Tube Druwing Results 204
Figure 10.13b The above diagrarn shows the circuderential strain contours at steady state conditions for k=O. 1.
Zulfikur H. A. Kassam Tube Drawin~ Resulis 205
Figure la. 14a The above diagram shows the equivalent strain contours at steady state conditions for p=0.05.
Zulfikar H. A. Kassam Tube Drawin~ Results 206
Figure 18.14b The above diag~-am shows the equivalent strain contours - at steady state conditions for p=0.1.
ZuIfîkar H. A. Kassanl Tube Drawing Results 207
~igurc 143.lEia The above diagrarn shows the radial stress contours at steady state conditions for p=0.05.
Zulfkar H. A. Kussam Tube Druwing Resul~s 208
Figure 14.15b The above diagram shows the radial stress contours at steady state conditions for p=O. 1.
Zulfikar H. A. Kassam Tube Drawina Resu1t.s 209
Figure 18.16a The above diagram shows the axial strain contours at steady state conditions for p=0.05.
Zulfikur H. A. Kussam Tube Dru winf Results 210
Figure 10.16b The above diagram shows the axial strain contours at steady state conditions for p=O. 1.
Zulfikcr r H. A. Kassatn Tube Drawin~ Resulfs 21 1
Figure 1Q.17a The above diagram shows the shear stress contours at steady state conditions for p=0.05.
Zulfikar H. A. Kassam Tube Drawing Results 212
Figure 10.17b The above diagram shows the shear stress contours at steady state conditions for p=0.1.
Zulfikar H. A. Kassam Tube Drawing Resulrs 2 13
Figure 1Q.18a The above diagram shows the circumferential stress contours at steady state conditions for p=0.05.
Zulfkar H. A. Kusscim Tube Dru wing Resulfs 2 14
Figure lQ.18b The above diagram shows the circumferential stress contours at steady state conditions for p=0.2.
Zulfikar H. A. Kassarn Tube Drawirta Results SIS
Figure 10.19a The above diagram shows the equivalent stress contours at steady state conditions for y=0.05.
Zurfikar H. A. Kassani Tube Druwing Results 216
m o o o o o a o o > 0 0 0 0 0 0 0 0 a O b C O U 3 W C ' , C U r O
Figure [email protected] I h e above diagram shows the equivalent stress contoürs at steady state conditions for p=0.2.
Zulfikar H. A. Kassarn Tube Drawing Results 2 17
begun due to the contact established between the workpiece and the die
walls. However, the profile shows an appreciable change at axial positions
l2.84mm and 13 .O mm where deformation has progressed substantially.
This indicates that the friction conditions strongly influence the radial ve-
locity profile.
The instantaneous axial displacement as a function of position dong
the thickness of the tube is shown at different axial positions in Figure
10.2. If plane sections remained plane. then the axialvelocity would be the
sarne across the thickness of the tube. However, the non-uniform velocity
profile shown in Figure 10.2, even at axial position of 12.0mm. indicates
that the friction conditions have affected axial velocily even at very early
stages of deformation implying that there is a npple effect from the heavily
deformed part that has affected the flow pattern in the adjacent material
due to the workpiece being a continuum body. Nevertheless. in the pres-
ence of friction. one would expect that the instantaneous axial displace-
ment (axial velocity) would be a maximum close to the inner surface of the
tube and minimum at the outer surface of the tube because of the friction
conditions that prevail at the contact surfaces. It should be noted that the
effect of the friction forces is much more severe at the outer surface due to
the converging nature of the die while much less frictional forces are en-
countered between the workpiece and the mandrel in spite of the fact that
the sarne friction coefficient was used between workpiece and die and be-
tween workpiece and mandrel. From the results. however. it is observed
that during the early of deformation (A=12.0mm) of deformation. the mini-
mum axial displacement is not at the outer surface but instead quite a
significant distance away from the surface. This indicates that the material
near the outer surface, in spite of the prevailing friction conditions. moves
Zulfikar H. A. Kassanl Tube Drawinn Results 2 18
faster along the drawing direction than the adjacent material. This behav-
ior is evident even when the ratio of the resultant speed/drawing speed is
plotted as a function of radial position a t different axial positions (Figure
10.3). This phenornenon is only possible if the material rotation occurs in
this outer region that is close to the outer surface which results in speed-
ing up the flow of material adjacent to the surface.
To prove this fact. the direction of the flow of the material is plotted
a s a function of radial position a t various axial positions (Figure 10.4). In
the ideal case. the direction of flow should increase monotonically from -
90" to -75'. The results. however. show that this is far from the case thereby
indicating material rotation close to the outer surface.
As the matenal exits the die (refer to Figure 10.2). the axial displace-
ment is maximum close to the inner surface and a minimum at the outer
surface indicating smooth flow of the materiai. The resultant speed/draw-
ing speed (Figure 10.3) also indicates that the material flow has become
smooth a t the die curvature point (axial position of 12.84mm) and onwards
a s the ratio monotonically decreases from the inner edge to the outer edge.
Correspondingly, the direction of flow changes monotonically from -90" to
-75". One stnking thing to note is that the resultant speed/drawing speed
ratio (Figure 10.3) is very close to 1 at axial position 12.84mm at the inner
edge of the tube while at the outer edge it is 0.91 for p=0.05 and 0.87 for
p=0.1. This implies that velocity profile a t the inner edge is not affected
much at the inner egde while the friction conditions affect the outer edge
quite significantly. Moreover. the different ratio values observed for the two
friction coefficients indicates that the friction coefficient also affects the
flow velociw on the outer edge significantly. After the material exits the die,
the axial displacement reaches a constant value of 1 x 10-1 mm. which is
Zulfikar H. A. Kassarn Tube Drawin~ Results 219
equal to the displacement at the drawing end. This implies that after the
material exits the die. the flow velocity of the material is equal to the draw-
ing speed. This fact is evident when the ratio of the resultant speed/draw-
ing speed is plotted: the ratio tends to 1 as the material exits the die. Fur-
themore. the fact that the effect of friction at the i m e r edge that is in
contact with the mandrel has much less contact with the mandrel bas
much less effect on the flow profile a s opposed to the contact with the
converging die walls indicates the importance of the angle of contact sur-
face: the mandrel contact surface is in the direction of the drawing force
and hence has a minimum effect on the flow pattern. The die surface makes
a 15" angle with the draw direction and. hence. exerts a significant effect
due to the converging nature of the die. This implies that the die angle is a
significant parameter and. consequently, a smaller die angle would sub-
stantially reduce the effect on the velocity profile thereby lowering the strains
and stresses the workpiece material is subjected to.
10.3. RADXAL, AXIAL, SHEAR AND CIRCUMFERENTIAL
STRAINS (MATERIAL D E F O U T I O N )
During the tube drawing process, the material is being compressed
along the radial direction by the converging die and the mandrel resulting
in a reduction in thickness of the tube. The compressive forces along the
radial direction translate to negative radial strains as shown in Figure 10.5.
The material experiences the greatest compressive radial strain at the in-
ner surface and the least compressive radial strain (upto 0.27 or 27% strain)
at the outer surface. The radial strain increases as the deformation
progresses from A= 12 .Omm to A= l2.84mm. The radial strains thereafter
does not change much as it is about to exit the die (A=13.0mm). However.
Zulfikar H. A. Kassarii Tube Drawinn Results 220
upon exit (A= 14.0mm). the radial strains are relieved partially a s the con-
straining effect from the die and the mandrel is relieved.
The maximum reçu1 tant compressive radial s train a t axial position
of 14mm is about 15% and is more uniform across the thickness of the
tube. Furthermore. as a result of the compressive strains dong the radial
direction and the tensile nature of the drawing force. the material expen-
ences tensile axial strains (Figure 10.6). The material closest to the inner
surface experiences the maximum tensile strain while the material at the
outer surface expenences the minimum tensile strain. The tensile strain
increases as deformation progresses. The maximum tensile strain is as
high as 29%. Upon exiting the die (A=14.0) the strains are relieved to some
extent, and the maximum tensile strain at 14mm is about 19%.
The shear strain pattern is shown in Figure 10.7. The maximum
shear strain is about 14%. The shear strains. with the exception at 12.0mm.
indicate a maximum value at the inner and outer surfaces. This is to be
expected at that is where contact occurs with rnandrel and die, respec-
tively . One interesting feature to note is that the shear strain pattern gen-
erally indicates that high shear strains develop at the inner and outer sur-
face during deformation. On the same token it has been noted by Ontario
Hydro researchers that shear cracks appear frequently at the outer surface
of the tube. This indicates that the shear strain developed during deforma-
tion may be a cntical pararneter that influences the appearance of this
crack.
The circumferentîal strain pattern is shown in Figure 10.8. The maxi-
mum compressive circurnferential strain is less than 4% strain. The cir-
cumferential strain is much less than the radial strain. This is due to the
fact that the change in the diarneter of the tube is reflected in the circum-
ferential strain while the change in tube thickness is reflected in the radial
strain; the change in thickness is quite significant while the change is di-
ameter of the tube as result of the drawing operation is comparatively mucli
smaller. The circumferential strain as a function of radial position at van-
ous positions indicate that the circumferential strain profile is more or less
uniform across the thickness of the tube except during the early stages of
deformation. As deformation progresses. the circurnferential strain increases
until it reaches the die curvature point whereby it starts decreasing. The
maximum compressive circumferential strain at 14.0mm is approximately
1.5%.
For reference purposes. the strain contours for al1 the strains are
s h o w in Figures 10.10 - 10.13. Meanwhile, Fig. 10.9 shows the deformed
meshes at different stages.
The strains plotted above indicate the amount of deformation the
material has undergone. This information in turn gives us an idea with
regards to how much the material has hardened in the process. It should
be noted that corresponding to the degree of deformation that is shown in
the plot of effective or equivalent strain. there is a material von-Mises stress
that reflects the material's strength (the material strain hardens during the
deformation process). The constitutive equation of the material determines
the relationship between the effective strain and the material strength. As
shown in Figure 10.14. the effective strain is a maximum close to the inner
surface of the tube. while the lowest effective strain is a t the outer surface
of the tube.
This implies that the material close to the inner surface of the tube
must have undergone large amounts of deformation and, hence, higher
Zulfikar H. A. Kassam Tube Drawing Results 222
Radial Position (mm)
Figure 10.20a The radial stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=0.05 at steady state.
Zulfikar H. A. Kassam Tube Drawing Results 223
Radial Position (mm)
Figure 10.20b The radial stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=O. 1 at steady state.
Zulftkar H. A. Kassam Tube Drawing Results 224
Radial Position (mm)
Figure 10.21a The axial stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=0.05 at staedy state.
Zulfikar H. A. Kassam Tube Drawing Results 225
Radial Position (mm)
Figure 10.21b The axial stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for u=O. 1 at staedv state.
Zu lfikar H. A. Kassam Tzbe Drawing Results 226
11.0 11.5 12.0 12.5
Radial Position (mm)
Figure10.22a The shear stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=0.05 at steady state.
Zu ffikar H. A. Kassarn Tube Drawing Results 227
Radial Position (mm)
FigurelO.22b The shear stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=O. l at steady state.
Zuffrkar H. A. Kassam Tube Drawing Results 228
Radial Position (mm)
Figure 10.23a The circumferential stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for u=0.05 at steadv state.
Zulfikar H. A. Kassam Tube Drawing Results 229
10.5 11.0 11.5 12.0 12.5
Radial Position (mm)
Figure 10.23b The circumferential stress as a function of radial position (along the thickness of the tube) is shown at different axial positions (along the draw direction). The above results are for p=O. l at steady state.
amount of strain hardening which translates to higher strength in the
material in this region. On the other hand. the material close to the outer
surface expenences lower effective strain and must have a correspondingly
lower strength.
10.4. APPLIED LOCAL STRESS PATTERN
Contour and graphical plots of the applied stresses are shown in
Figures 10.15 to 10.23. The maximum applied radial stress (Figure 10.15)
is calculated by the finite element based upon the incremental strain and
the corresponding strain hardening rate calculated from the effective stress-
strain curve. The contour results indicate that the maximum compressive
applied radial stress is a t the outer surface (-460MPa compressive). The
radial stress initially increases considerably at the outer edge during the
early deformation stages as is shown Fig. 10.20. This radial applied stress
starts to decrease as the material approaches the curve point in the die as
the constraining effect from the die and the mandrel are no longer present.
The radial applied stress tends to vary between zero and lOOMPa at 14.0mm
with the maximum radial stress found at the inner surface.
The applied axial stress pattern (Figure 10.2 1) is highly non-uniform
varying from O-200MPa at the inner surface to 880 MPa a t the outer sur-
face. High tensile circumferential stresses develop at the outer surface upon
exiting the die due to change in the boundary conditions.
The shear stress (Fig. 10.23) reaches a maximum value of 200MPa
during deformation. Overall. the shear stress is a maximum in the region.
The shear stress at the tube surfaces after exiting the die tends to zero as
would be expected.
The circumferential stress during the early stages of deformation is
negative (upto 500 MPa compressive) a s would be expected. However, upon
exiting the die. the forces acting on the free surfaces are relieved and. as a
result. high circumferential stresses build up at the outer surface. This
seems to be very much in contrast to radial stresses which tend to go zero
as the material exits the die. According to the contour plot shown in Figure
10.18, however, these high tensile stresses relax as the workpiece is pulled
further away from the die.
The highest equivalent applied stress (Figure 10.19) is at the outer
surface of the tube which is actually the weaker point in the material (low
effective strain corresponds to lower strength of the materiai). Since the
material in this region has low strength while experiences the highest stress,
it is apparent that this region will be susceptible to failure if the maximum
strength of the material is exceeded.
10.5. RESIDUAL STRESSES
The residual stresses for friction conditions p=0.05 and ~=0.1 are
plotted in Figures 10.24a and 1 O.24b. respectively. These are actually
stresses plotted at 15mm: this is the position beyond which the effect of
the die diminishes completely and virtually no change in strains and stresses
beyond this point.
From these figures. it is observed that the residual radial stress is
very small [<GOMPa absolute value). The axial stresses are very small at the
inner region of the tube while it is very high at the outer regions of the tube
(upto and above 500MPa) which have in close contact with the die w d s .
Note that the axial stresses are not exactly residual stresses as a drawing
force is still being applied under steady state conditions.
11 11.5
Radial Position (mm)
Figure 18.24a Residual Stresses for p=0.05
(O. IO) ; 1 1
11 11.5
Radial Position (mm)
Figure 10.25a Residual Strains for p=0.05
0 RADIAL STRAIK
O AXl AL STRAI N
A SHEAR STRAlN
rn CIRCUhl STRAIN
1 0.5 11 11.5
Radial Position (mm)
Figure 10.25b Residual Strains for p=O. 1
Zulfikar H. A. Kassam Tube Drawinr: Results 236
Graph of Drawing Force versus Friction
Cl Drawing Force
Friction coefficient (p)
Figure 10.26 Finite element calculation showing the effect of friction on drawing force compared to the experimental result.
Zulfikar H. A. Kusstrrrt Trilw Drnwin~ Results 237
The circumferential residual stresses are low at the inner region of
the tube while being high at the outer region of the tube (max. 250MPa).
This value of residual stress seems rather high and ominous because the
tensile circumferential 1-esidual stress can result in premature failure if the
tube is subjected to high circumferential stresses at the outer edge. Luck-
ily. however. these tube are subjected to interna1 pressure where the cir-
cumferential stress is much higher in the inside region as compared to the
outside region of the tube as shown in Figure 9.3.
The shear residual stresses are close to zero at the outer surfaces as
expected. and is less than 100 MPa in the interior region.
10. 6. RESIDUAL STRAINS
The residual strains are plotted in Figures 10.25(a) and (b). The re-
sidual strain in the inner regions of the tube is virtually zero. but increases
substantially in the outer regions of the tcbe upto alrnost 10%. T h e
axial strains may not be referred to as residual strain due to the drawing
force being present.
The shear and circumferential strains on the other hand are virtu-
ally zero.
10. 7. EFFECT OF FRICTION
Friction. as would be expected. has an effect on the stress and strain
distributions altliough the dfeci may not be as striking as one would have
thought. The hicher friction conditions basically result in încreased re-
sidual radial and axial sti-ain (compare Fig. 10.25a and 10.25b).
More impoi-tantly. friction has a profound effect on the drawing force
as tabulated below.
Zulfika r H. A. KUSM I 1 1 Tirhe Dra rving Results 238
Friction coefficient
0.05
o. 1
0.2
Drawing force
16.6 k.N
19.5 kN
27.3 kN
The expenments conducted at Ontario Hydro indicated that the drawing
force was above 10 kN. The results are shown in figure 10.26. In compar-
ing these calculations i t is evident that the actual existing fnction condi-
tions may be less than the one that were used in these simulations. Inter-
polation conducted on the FEM result indicate that the drawing force in-
creases significantly as the friction coefficient increases and the relation-
ship is described by a second order polynomial.
Zulfikar H. A. Kussmi Trihl~ DI-ctrvina Results 239
Conclusions
A new equation. i-ekn-ed to as the Alpha constitutive equation, has
been developed. The novel Seature of this equation is that the param-
eter alpha is a variable defined as the ratio of plastic strain to elastic
strain. This keeps an accurate track of materiais' changing property
while deformaiion progresses. Consequently. the Alpha equation has
been shown to acrui'ately describe the behavior of al1 tested materi-
als. The Alpha constitutive equations for brass. copper, steel and Zr-
2.5Nb alloy have been developed.
The Alpha constitutive rquation has also been shown to describe strain
softening behavior wllich is a unique feature of this equation. No other
constitutive equation 1x1s claimed to be able to describe strain soften-
ing behavior.
Equations periairii~ir: sprcifically to the finite element simulation of
large plas tic de formation in the cylindrical coordinate system have
been workcbd out and presented in detail.
A finite elenient program to simulate the tube drawing process has
been developed. This program is unique based on the fact that is
tailor maclr to handlr tube drawing processes with the presence of
die wall ancl mandi-el while contact friction at these two interfaces is
handled bv Coulomb friction.
The technique of hancllinfi friction conditions is similar to that adopted
by Lu and \VI-ight \vit11 a veiv important distinction. The technique for
relieving stresses was modified in such a way as to avoid instability.
This was achieved by relieving the stress in a step-wise manner in-
stead or a single step.
The finite element code was venfied by using a number of simula-
tions - it has been shown that the code developed in this rtsearch is
more accurate than a conimercial package.
The finite element code incorporates the Alpha equation - this has
been done l'or the firsi lime ever. The results yielded indicate that the
Alpha constitutive equation can be integrated into fmite element and
has the ability to accurately simulate materials behavior.
The finitr c-lenieni <*nclr was further venfied through tube drawing
experimcnis coiiductzcl at Ontario Hydro.
- The esprrimentally niaasured drawing force is close to the draw-
ing foi-cc. clalru lai et! fi-oni finite element simulations
- Durine tube drawing expenments. shear cracks were found at
the o u t w surlace o f the tube. FEM simulations conducted indi-
cate t hat the sheai- strain is a maximum a t the outer surface which
is consisieni wit l i the experimental results.