See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/301764266
A study of material Non-linearity during deformation using FEM sofware
Thesis · April 2015
DOI: 10.13140/RG.2.1.4049.4961
CITATIONS
READS
0
3,989
1 author:
Emayavaramban Elango
Ecole Centrale de Nantes
2 PUBLICATIONS 2 CITATIONS
SEE PROFILE
Some of the authors of this publication are also working on these related projects:
Numerical Modelling of Nonlinear materials View project
All content following this page was uploaded by Emayavaramban Elango on 01 May 2016.
The user has requested enhancement of the downloaded file.
A STUDY OF MATERIAL NON-LINEARITY
DURING DEFORMATION USING FEM
SOFTWARE
(Project Term January-April, 2015)
CAPSTONE PROJECT REPORT
Submitted by
Emayavaramban E
Registration Number: 11103698
Milan Chhetri
Registration Number: 11109647
Akash Dan
Registration Number: 11113741
Sonu Yadav
Registration Number: 11102819
Project Group Number: MERGCO135
Course Code: MEC 494
Under the Guidance of
Mr. AKASH SAINI – ASSISTANT PROFESSOR
School Of Mechanical Engineering
Lovely Professional University, Jalandhar
PUNJAB
A Study of material non-linearity during deformation using FEM software
DECLARATION
We hereby declare that the project work entitled “A Study of material nonlinearity during deformation using FEM software” is an authentic record of our
own work carried out as requirements of Capstone Project for the award of B.Tech
(Hons.) degree in Mechanical Engineering from Lovely Professional University,
Phagwara, under the guidance of Akash Saini, during January to April 2015.
All the information furnished in this capstone project report is based on our own
intensive work and is genuine.
Project Group Number: MERGCO135
Name of Student 1: Emayavaramban E
Registration Number: 11103698
Name of Student 2: Akash Dan
Registration Number: 11113741
Name of Student 3: Milan Chhetri
Registration Number: 11109647
Name of Student 4: Sonu Yadav
Registration Number: 11102819
(Signature of Student 1)
Date:
(Signature of Student 2)
Date:
(Signature of Student 3)
Date:
(Signature of Student 4)
Date:
i
A Study of material non-linearity during deformation using FEM software
CERTIFICATE
This is to certify that the declaration statement made by this group of students is
correct to the best of my knowledge and belief. They have completed this Capstone
Project under my guidance and supervision. The present work is the result of their
original investigation, effort and study. No part of the work has ever been
submitted for any other degree at any University. The Capstone Project is fit for
the submission and partial fulfillment of the conditions for the award of B.Tech
(Hons.) degree in Mechanical Engineering from Lovely Professional University,
Phagwara.
Signature and Name of the Mentor: Akash Saini
UID: 16899
Designation: Assistant Professor
School of Mechanical Engineering,
Lovely Professional University,
Phagwara, Punjab.
Date:
ii
A Study of material non-linearity during deformation using FEM software
ACKNOWLEDGEMENT
First and foremost, we would like to thank our Parents for encouraging and
providing
constant
motivation.
Besides,
we
take
immense
pleasure
in
thanking Assistant Professor Mr. Akash Saini for having permitted us in carrying
out this project. He not only provided his whole hearted support but also acted as a
guru, friend and what not. He inspired and corrected us as and when required. These
mere words can’t describe his generosity. We also wish to express our deep sense of
gratitude to all the PAC Committee Members for the approval of this project. Last
but not the least, we would like to express our heartfelt thanks to our friends &
classmates for their help and support.
With Regards
Emayavaramban E
Akash Dan
Milan Chhetri
Sonu Yadav
iii
A Study of material non-linearity during deformation using FEM software
TABLE OF CONTENTS
Declaration .................................................................................................................... i
Certificate .....................................................................................................................ii
Acknowledgement ......................................................................................................iii
List of tables ................................................................................................................ vi
List of figures .............................................................................................................vii
Symbols and notation .................................................................................................. ix
Introduction .........................................................................................1-1
Literature Review................................................................................2-2
Theory .................................................................................................3-7
3.1.1
Origin of Plasticity .............................................................................. 3-7
3.1.2
Concept of Continuum Plasticity ........................................................ 3-9
3.2
Rate – Independent Plasticity ...................................................................3-10
3.2.1
Yield Criterion .................................................................................. 3-10
3.2.2
Flow Rule .......................................................................................... 3-10
3.2.3
Hardening Rule ................................................................................. 3-11
3.3
An elaboration for the types of Hardening Models: .................................3-11
3.3.1
Isotropic hardening ........................................................................... 3-11
3.3.2
Kinematic hardening: ........................................................................ 3-12
3.4
Yield Surfaces ..........................................................................................3-21
3.4.1
Famous Yield Surfaces: .................................................................... 3-21
Scope of the study .............................................................................4-25
Objective ...........................................................................................5-26
Research methodology ......................................................................6-27
Complete work plan with timeline ....................................................7-29
Expected outcome .............................................................................8-33
Research and experimental work done .............................................9-34
iv
A Study of material non-linearity during deformation using FEM software
9.1
Material selection .....................................................................................9-34
9.2
Element Selection:....................................................................................9-37
9.3
Nonlinear material model .........................................................................9-38
9.3.1
Bilinear kinematic hardening ............................................................ 9-38
9.3.2
Multi-linear Kinematic Hardening Constant .................................... 9-39
9.3.3
Bilinear isotropic Hardening ............................................................. 9-41
9.3.4
Multi linear Isotropic Hardening ...................................................... 9-42
9.3.5
Nonlinear Kinematic Hardening Constants (TB,CHABOCHE) ...... 9-43
9.4
Loading conditions ...................................................................................9-44
9.4.1
LC_1 Ramped loading and unloading .............................................. 9-44
9.4.2
LC_2 to LC_6 Cyclic loading conditions ......................................... 9-44
9.5
Solution method and test details...............................................................9-46
9.5.1
Static and Quasi-static analysis:........................................................ 9-46
9.5.2
Implicit dynamics.............................................................................. 9-46
9.5.3
Explicit dynamics.............................................................................. 9-47
9.5.4
Test details ........................................................................................ 9-47
9.6
Application based study ...........................................................................9-48
Result and discussions.................................................................10-49
Conclusion...................................................................................11-67
References .................................................................................................................. 68
Appendix I ................................................................................................................. 70
v
A Study of material non-linearity during deformation using FEM software
LIST OF TABLES
TABLE 7-1 TEAM MEMBERS LIST ....................................................................................................... 7-29
TABLE 7-2 GANTT CHART ................................................................................................................. 7-32
TABLE 9-1MATERIAL SPECIFICATION FOR 9NI-4CO-0.20C .............................................................. 9-35
TABLE 9-2 CALCULATED MATERIAL PROPERTIES .............................................................................. 9-36
TABLE 9-3MATERIAL PARAMETER FOR BKIN .................................................................................. 9-39
TABLE 9-4 BISO MAT DATA.............................................................................................................. 9-41
TABLE 9-5 INPUT DEFINITION FOR MISO .......................................................................................... 9-42
TABLE 9-6CHABOCHE MATERIAL PARAMETER INPUTS ...................................................................... 9-43
TABLE 9-7 LOADING CONDITION PARAMETERS VALUE ..................................................................... 9-45
TABLE 9-8 TEST DETAILS .................................................................................................................. 9-47
vi
A Study of material non-linearity during deformation using FEM software
LIST OF FIGURES
FIGURE 3-1PLASTIC DEFORMATION ..................................................................................................... 3-7
FIGURE 3-2.: A SINGLE CRYSTAL CONTAINING SLIP PLANE WITH NORMAL ‘N’, SLIP DIRECTION ‘S’, AND
LOADED IN DIRECTION ‘T’. ....................................................................................................... 3-8
FIGURE 3-3 SIMPLE MODELS OF ELASTIC AND PLASTIC DEFORMATION ............................................... 3-9
FIGURE 3-4CASE OF ISOTROPIC HARDENING SHOWING EXPANSION IN YIELD SURFACE ALONG WITH
UNIAXIAL STRESS-STRAIN CURVE. .......................................................................................... 3-12
FIGURE 3-5REVERSE LOADING WITH ISOTROPIC HARDENING SHOWING (A) YIELD SURFACE (B) STRESSSTRAIN CURVE........................................................................................................................ 3-13
FIGURE 3-6 KINEMATIC HARDENING SHOWING (A) THE TRANSLATION, |X| OF THE YIELD SURFACE WITH
PLASTIC STRAIN, (B) THE RESULTING STRESS–STRAIN CURVE WITH SHIFTED YIELD STRESS IN
COMPRESSION—THE BAUSCHINGER EFFECT .......................................................................... 3-14
FIGURE 3-7 BAUSCHINGER EFFECT.................................................................................................... 3-19
FIGURE 3-8(A) EDGE DISLOCATION AND (B) DISLOCATION PILE-UPS ON THE GRAIN BOUNDARIES .... 3-19
FIGURE 3-9 SURFACES ON WHICH INVARIANTS 𝐼1, 𝐽2, 𝐽3 ARE PLOTTED IN PRINCIPAL STRESS SPACE 3-21
FIGURE 3-10 THE VON MISES YIELD SURFACES IN PRINCIPAL STRESS COORDINATES CIRCUMSCRIBES A
CYLINDER WITH RADIUS AROUND THE HYDROSTATIC AXIS. ALSO SHOWN IS TRESCA’S
HEXAGONAL YIELD SURFACE. ................................................................................................ 3-22
FIGURE 3-11 THE VON MISES YIELD SURFACE FOR CONDITIONS OF PLANE STRESS ............................ 3-23
FIGURE 3-12 VON MISES YIELD SURFACE FOR PLANE STRESS AND THE CORRESPONDING STRESS–STRAIN
CURVE .................................................................................................................................... 3-24
FIGURE 3-13 VIEW OF DRUCKER–PRAGER YIELD SURFACE IN 3D SPACE OF PRINCIPAL STRESSES .... 3-24
FIGURE 6-1 FLOW CHART OF THE APDL CODING .............................................................................. 6-28
FIGURE 9-1: TYPICAL TENSILE STRESS-STRAIN CURVES FOR 9NI-4CO-0.20C STEEL PLATES AT
VARIOUS TEMPERATURES. ...................................................................................................... 9-35
FIGURE 9-2: TYPICAL COMPRESSIVE STRESS STRAIN CURVES AND COMPRESSIVE TANGENT MODULUS
CURVES FOR 9NI-4CO-0.20C STEEL PLATE AT VARIOUS TEMPERATURES. ............................. 9-36
FIGURE 9-3 LINK 180 ....................................................................................................................... 9-37
FIGURE 9-4 TBPLOT FOR BKIN ...................................................................................................... 9-38
FIGURE 9-5 TBPLOT FOR MKIN...................................................................................................... 9-39
FIGURE 9-6 MATERIAL INPUT FOR BISO ........................................................................................... 9-41
FIGURE 9-7 MATERIAL PLOT FOR MISO ........................................................................................... 9-42
FIGURE 9-8 LC_1 .............................................................................................................................. 9-44
FIGURE 9-9 LC_6 .............................................................................................................................. 9-45
FIGURE 9-10 LC_2 ............................................................................................................................ 9-45
FIGURE 9-11 LC_3 ............................................................................................................................ 9-45
FIGURE 9-12 LC_4 ............................................................................................................................ 9-46
FIGURE 9-13 LC_5 ............................................................................................................................ 9-46
FIGURE 9-14 MESHED GEOMETRY FOR TEST_24 AND TEST_25 ...................................................... 9-48
vii
A Study of material non-linearity during deformation using FEM software
FIGURE 10-1 TEST 1 (LOADING CONDITION 1) ................................................................................. 10-49
FIGURE 10-2 TEST_ 1 PLASTIC STRAIN OVER TIME .......................................................................... 10-49
FIGURE 10-3 STRESS - STRAIN ......................................................................................................... 10-49
FIGURE 10-4 TEST_1 STRAIN ENERGY ALONG DISPLACEMENT ........................................................ 10-50
FIGURE 10-5 TEST _ 2 MISO STRAIN VS UY.................................................................................... 10-51
FIGURE 10-6 MISO WITH CYCLIC LOADING .................................................................................... 10-51
FIGURE 10-7 TEST _3 STRAIN VS DISPLACEMENT ............................................................................ 10-52
FIGURE 10-8 TEST_3 STRAIN ENERGY OVER TIME ........................................................................... 10-52
FIGURE 10-9 STRAIN ENERGY PLOT FOR TEST_3 ............................................................................. 10-52
FIGURE 10-10 COMPARISON OF BISO WITH BKIN ......................................................................... 10-53
FIGURE 10-11 DISPLACEMENT LOAD FOR TEST_4 ......................................................................... 10-54
FIGURE 10-12 TEST_4 STRESS VS STRAIN ...................................................................................... 10-54
FIGURE 10-13 TEST_6 DISPLACEMENT PLOT .................................................................................. 10-54
FIGURE 10-14 TEST_6 STRESS VS STRAIN ...................................................................................... 10-54
FIGURE 10-15 DISPLACEMENT LOAD FOR TEST_8 ......................................................................... 10-54
FIGURE 10-16 TEST_8 STRESS VS STRAIN ...................................................................................... 10-54
FIGURE 10-17 TEST_4 STRAIN ENERGY OVER TIME........................................................................ 10-55
FIGURE 10-18 TEST_10 STRESS -STRAIN ....................................................................................... 10-56
FIGURE 10-19 TEST_11 PLASTIC STRAIN INCREMENT (MKIN) ...................................................... 10-57
FIGURE 10-20 TEST_4 PLASTIC STRAIN INCREMENT (BKIN) ......................................................... 10-57
FIGURE 10-21 TEST_12 ELASTIC STRESS ....................................................................................... 10-57
FIGURE 10-22 TEST_6 ELASTIC STRESS ......................................................................................... 10-57
FIGURE 10-23 TEST_12 STRAIN ENERGY ....................................................................................... 10-58
FIGURE 10-24 TEST_12 TOTAL STRESS AND TOTAL STRAIN ......................................................... 10-58
FIGURE 10-25 TEST_12 LC_4 ........................................................................................................ 10-59
FIGURE 10-26 TEST _ 12 PLASTIC STRAIN, ELASTIC STRAIN, TOTAL STRAIN .................................. 10-59
FIGURE 10-27TEST_14 LC_5 ........................................................................................................ 10-60
FIGURE 10-28 TEST_14 PLASTIC STRAIN........................................................................................ 10-60
FIGURE 10-29 TEST_14 STRAIN ENERGY ....................................................................................... 10-61
FIGURE 10-30 CHABOCHE MODEL WITH DIFFERENT Γ VALUE .......................................................... 10-62
FIGURE 10-31 TEST_19 PLASTIC STRAIN ....................................................................................... 10-62
FIGURE 10-32 ELASTIC STRAIN AND DISPLACEMENT FOR TEST_20 ............................................... 10-63
FIGURE 10-33 COMPARISON OF BKIN AND CHABOCHE FOR LC_5 ............................................. 10-63
FIGURE 10-34 PARTIALLY DEFORMED SHEET METAL (TEST_24) .................................................... 10-64
FIGURE 10-35 PLASTICALLY DEFORMED SHEET METAL (TEST_24) ................................................ 10-64
FIGURE 10-36 ENERGY SUMMERY (TEST_24) WITH PLASTICITY MODEL ....................................... 10-65
FIGURE 10-37 ENERGY SUMMERY (TEST_25) WITHOUT PLASTICITY MODEL ................................. 10-65
viii
A Study of material non-linearity during deformation using FEM software
SYMBOLS AND NOTATION
Symbols
Ci
K
n
α
γi
σ
𝜎𝑖
𝜎𝑖𝑗
𝜎𝑦
𝜎′
𝜎𝑒
𝜎̅
εpl̇
[K]
{x}
[M]
{ẍ }
[C]
{𝑥̇ }
λ
Q
𝐼1
𝐽2 , 𝐽3
𝑆𝑠𝑦
𝑆𝑦𝑐
𝑆𝑦𝑡
∈̇
𝑝̇
U
k
P
x
Name
Material Constant
Yield stress of materials
Number of kinematic models
Back Stress
Material Constant (TB,CHABOCHE)
Stress
Principle Stress
Stress tensor
Yield stress
Deviatoric Stress
Equivalent Stress
Effective Equivalent stress
Accumulated equivalent plastic strain
Stiffness matrix
Displacement matrix
Mass matrix
Acceleration matrix
Damper matrix
Velocity matrix
Plastic multiplier
Function of stress to term the plastic potential
First principal invariant of the Cauchy Stress
Second and third principal invariants
Shear yield strength.
The uniaxial yield stresses in compression.
The uniaxial yield stresses tension.
The rate of effective plastic strain.
The accumulated plastic strain rate
Strain Energy
Stiffness of the material
Force Applied
Displacement
ix
A Study of material non-linearity during deformation using FEM software
INTRODUCTION
As per today most of the structural elements which are a part of buildings, bridges and so
on face an ever increasing threat of failure mainly due to cyclic loading. Just more than a
score ago during the industrial revolution, the world saw a complete transformation. That
era witnessed development from skyscrapers, radio towers to suspension bridges. Though
these developments brought a complete change but it also brought along with it the
dangers of failures and loss of lives, because of the factors which were not taken into
account at that time such as reverse and repetitive loading.
The famous example of greatest failure of engineering can be of the famous suspension
bridge established in U.S, due to constant cyclic loading the bridge succumbed to failure
and resulted in the loss of many lives.
Not until the theories presented by Tresca in 1864, the world understood these grave
problems. Tresca postulated that the plastic deformations taking place in these structural
elements mainly occurred due to Crystal slip occurring in the microstructures. Though it
considered the plastic deformation but still it didn’t dealt with the real time problems
which occurred due to cyclic loading.
Hitherto 1956, it was Prager who first introduced cyclic loading for these structural
elements with the help of Bauschinger effect and proposed kinematics hardening model.
With the successive improvisation in the hardening models in the subsequent years,
definitely the damage to life and property was reduced. But the only problem it posed was
cost and time it took for analysis.
With the advent of simulation softwares for analysis proposes such as Ansys by Boeing
Company and Hyperworks by Altair engineering, a dramatic change occurred. Now these
software's provided solutions for even the complex and most challenging problems. Thus
it solved the problem for analysis time frame and also provided accurate results.
Now in our present work we demonstrate the analysis of these materials and try to
comprehend their nonlinear behavior and provide subtle solutions to the analysis.
1-1
A Study of material non-linearity during deformation using FEM software
LITERATURE REVIEW
Structural members are frequently subjected to cyclic loading resulting in hysteresis
behavior which is very important in examining the dynamic response of these members
against repeated loading such as earthquake, thermo-mechanical loading of pressure
vessels, and dynamic cyclic loading of shaft and wind motion of structures. In most cases,
the failure takes place in these members due to such kind of loading which results in low
cycle or high cycle fatigue of the materials through the loss of structural integrity.
Tresca in 1864 provided the earliest yield criterion in case of Isotropic materials which
was proposed by him on the basis of his observation that plastic strains appear by
crystallographic gliding under acting shear stresses. According to this criterion the
material passes from elastic to a plastic state when the maximum shear stress 𝜏𝑚𝑎𝑥
reaches a critical value. In the general case, the criterion may be written as follows:
Max {|{|𝜎1 − 𝜎2 |, |𝜎2 − 𝜎3 |, |𝜎3 − 𝜎1 |} = 𝜎0
Around that time, It was conjectured whether the plasticity in material is affected or not
by hydrostatic stresses. This criterion was first proposed independently by Huber and
Mises and then later further improvised by Hencky. The observation was based on the
fact that plasticity in materials is completely independent of Hydrostatic stresses and so it
leads to the conclusion that elastic energy of distortion influences the transition from
elastic to a plastic state comes naturally. Though as according to the criterion the name
should have been Huber–Mises–Hencky Yield Criterion but for simplicity it was
known as the well-known Mises Yield Criterion.
This criterion can be formulated as follows:
The material passes from elastic to a plastic state when the elastic energy of distortion
reaches a critical value that is independent of the type of the stress state.
Hill(1948) proposed the Incremental flow theory of plasticity which describes the
yielding of the material by work hardening models, a yield surface, and a flow law and is
one of the most commonly used methods in the modelling of elastic-plastic loading of
structures. This model was based on Isotropic hardening.
2-2
A Study of material non-linearity during deformation using FEM software
Hill’s criterion was based on Anisotropy and was a generalization of the Huber–Mises–
Hencky Yield Criterion. The material on which he performed the experiment was
supposed to have anisotropy with three orthogonally symmetrical planes.
General Hill’s 1948 Criterion is of the form:
2
2
2
𝐹(𝜎22 − 𝜎33 )2 + 𝐺(𝜎33 − 𝜎11 )2 + 𝐻(𝜎11 − 𝜎22 )2 + 2𝐿𝜎23
+ 2𝑀𝜎31
+ 2𝑁𝜎12
=1
Where F, G, H, L, M & N are constants which are determined experimentally and 𝜎𝑖𝑗 are
the stresses.
Hill assumed that equal hardening occurs in all directions in yield surface and so this
model considers that the plastic flow occurs in all directions. However, in cyclic
loadings this model cannot predict the experimental evidence of the Bauschinger effect,
which is significant due to stress reversals.
Drucker and Prager (1952) introduced the DP Model which pertained to pressure
sensitive materials such as rock, soil and concrete. In simple words, it is a pressuredependent model for determining whether a material has failed or undergone plastic
yielding. The friction angle, cohesion and plastic dilation are essential for the Drucker–
Prager type material models. Parameters in Drucker–Prager (DP) type plasticity model
are related to friction angle and cohesion govern the yielding and hardening criteria, while
the parameter related to plastic dilation determines the flow rule.
The DP model has paved the way for easier implementation for numerical simulations of
concrete materials and structures.
What sets Drucker-Prager plasticity model different from typical metal plasticity models
is that it contains a dependence on hydrostatic pressure. This means that if there is some
hydrostatic tension, the yield strength would be smaller.
Prager in 1956 overcame the difficulty of cyclic loading and was successfully able to
demonstrate the Bauschinger Effect by suggesting a Kinematic Hardening model which
was later improvised by Ziegler in 1959. As a matter of fact, Prager was the first to
introduce the word Kinematic Hardening. The kinematic model proposed by him assumed
a single yield surface such that during the process of plastic loading the yield surface
translates in the stress space and its shape and size remains unchanged.
He formulated that if the initial yield surface is described as:
2-3
A Study of material non-linearity during deformation using FEM software
𝐹 = 𝑓(𝜎) − 𝑘 = 0
Then due to kinematic hardening in the process of plastic deformation, the subsequent
yield surface takes the form as:
𝑓(𝜎 − 𝛼) − 𝑘 = 0
Where α is the tensorial hardening parameter called the back stress, that represents the
center of the yield surface and ‘k’ is a material constant which represents the size of the
yield surface
One of the drawbacks of Prager’s kinematic Hardening model as predicted by Ziegler was
that it does not give consistent results for three dimensional and two dimensional cases.
The reason being that the yield function assumes different shapes for one, two, or threedimensional cases.
Ziegler observed that in Prager’s model during loading, as the yield surface translates
towards positive direction of loading, it moves in the negative 2-and 3-directions,
although no load applied in these directions, causing what is known as transverse
softening which is not at all desirable.
Ziegler (1959) overcame the shortcomings of Prager’s Model and instead of assuming
that the yield surface moves along the normal direction, he assumed that the movement
takes place in the radial direction. According to Ziegler, his model is consistent with the
results in all the three dimensions and also does not show any transverse hardening or
softening effect.
Armstrong and Fredrick(1966) initiated the non-linear kinematic hardening rule which
presented rules on how to simulate the ratcheting phenomena using the idea of strain
hardening and a recovery term in their equation. They accomplished this by adding a
“recovery term” to the Prager’s evolution law thereby making the accumulation of plastic
strain possible. The hardening rule is of the following form:
𝑑𝛼 =
2
2
𝐵𝑑𝜀 𝑝 − 𝛾𝛼 √ 𝑑𝜀 𝑝 𝑑𝜀 𝑝
3
3
Hoffman (1967) added a linear combination of the normal stresses σ11, σ22 and σ33, to
make the yield surface nonsymmetrical with respect to the origin. It was done so as to
2-4
A Study of material non-linearity during deformation using FEM software
tackle the disadvantage of the Hill criterion as it does not allow modeling of materials
with different values of the yield stress in tension and in compression.
Chaboche et al. (1979) and Chaboche (1986) proposed a decomposing hardening rule in
which the back stress is decomposed into several components where each of the
components, individually involved according to AF RULE (Armstrong & Fredrick).
Form of decomposed hardening rule:
𝑑𝛼 = 𝛴𝛼𝑖 ,
𝑑𝛼𝑖 =
2
2
𝐵𝑖 𝑑𝜀 𝑝 − 𝐵𝑖 𝛼𝑖 √ 𝑑𝜀 𝑝 𝑑𝜀 𝑝
3
3
Where i=1, 2, 3….
Lately, a lot of modifications have been made in their decomposed models in order to
improve the uniaxial and multi axial ratcheting.
Bari and Hassan (2000), provided with an exemplary work in ratcheting. They divided
the uniaxial strain controlled hysteresis curve into segments and related a number of the
material constants to each segment. However, some of the parameters where determined
by trial and error in order to produce a good fit to the uniaxial hysteresis curve.
Mahbadi and Eslami in 2002 and 2006 worked upon the cyclic loading of beams and
thick vessels in accordance with the Prager and AF kinematic hardening model. Their
work includes the cyclic torsion of a shaft in the elastic-plastic zone using the Prager
kinematic hardening model via the finite element formulation. The stress-strain and
residual shear stress and strain in the shaft subjected to cyclic loading is obtained in the
developed finite element code
Yoshida and Uemori (2006) have proposed an advanced constitutive model of largestrain cyclic plasticity which can describe cyclic plasticity characteristics, such as the
Bauschinger effect and cyclic hardening characteristics, as well as the anisotropy of sheet
metals, thus called ‘Yoshida and Uemori model’. The test was carried out as biaxial
2-5
A Study of material non-linearity during deformation using FEM software
tension experiments, under proportional and nonproportional loadings, conducted on 980
MPa HSS. By using this model, the accurate prediction of springback becomes possible.
This model is considered as the best model for sheet metal operations among the existing
Kinematic Hardening models. It is so because it includes seven parameters of cyclic
plasticity, and each parameter has a cyclic definition. No artificial mathematical
parameters are included. Furthermore, Young’s modulus depending on plastic strain is
introduced in the model to describe stress-strain response more accurately after stress
reversal.
2-6
A Study of material non-linearity during deformation using FEM software
THEORY
3.1
Plasticity
According to Hooke’s law, all
materials behave elastically when
deformed below the yield point.
That is to say, the materials revert
to their original shapes as soon as
on the removal of the load.
Beyond yield point, materials
starts losing its elasticity and
begin to plastic deform. This
phenomenon
of
Figure 3-1plastic deformation
materials
deforming plastically and leading to permanent deformation after a certain point is known
as plasticity.
3.1.1
Origin of Plasticity
Crystal Slip is considered as the genesis of plasticity. Metals are usually crystalline in
nature, in which atoms are arranged in an orderly manner. The grain boundaries of these
materials mark regions of different crystallographic orientations.
So during plastic deformation, crystal slip occurs resulting in different crystal planes of
atoms to move relative to another.
Few important features of Plastic Slip:


1
There is no volume change: incompressibility condition of plasticity1.
It is a shearing process.
Not all plastic deformations are incompressible. If a material is porous, it can well
shrink in size and lead to volume change.
3-7
A Study of material non-linearity during deformation using FEM software
Plastic deformation in crystals normally occurs by the movement of the line defects
known as dislocations. Crystal Slip occurs due to dislocations running through the crystal
and at the edges. Each dislocation contributes just one Burger’s vector of relative
displacement, but with many such dislocations, the displacements become large.
Slip tends to occur preferentially on certain crystal planes and in certain specific crystal
directions. The combination of a
slip plane and a slip direction is
called a slip system (figure 3-2).
These tend to be the most densely
packed planes and the directions in
which the atoms are packed closest
together.
Slip will take place on the slip system, that is, the crystal will yield, when shear stress
reaches the CRSS. This is known
as Schmid’s law. The resolved
shear stress is of the form:
Figure 3-2.: A single crystal containing slip plane
with normal ‘n’, slip direction ‘s’, and loaded in
direction ‘t’.
𝜏 = 𝜎 cos ∅ cos 𝜆 = 𝜎(𝑡. 𝑛)(𝑡. 𝑠)
Material non-linearity is associated with the inelastic behavior of a component or system.
This type of nonlinearity arises when the material exhibits non-linear stress-strain
relationship.
This figure represents a structure which exhibits a softening behavior after yielding. In
linear elastic analysis modulus of elasticity defined the stress-strain relationship. But in
the case of non-linear material analysis, the modulus of elasticity is only the first
definition point of an overall behavior. Inelastic behavior may be characterized by a
force-deformation (F-D) relationship, also known as a backbone curve, which measures
strength against translational or rotational deformation.
Plasticity is divided into sub parts –

Micro plasticity

Continuum plasticity
3-8
A Study of material non-linearity during deformation using FEM software
Figure 3-3 Simple models of elastic and plastic deformation
Micro plasticity-It is a local phenomenon in metals which occurs for stress values where
the metal is globally in the elastic domain while some local areas are in the plastic
domain.
Continuum plasticity-It is a branch of mechanics that deals with the analysis of the
kinematic and the mechanical behavior of material modeled as a continuous mass rather
than as discrete particles.
3.1.2
Concept of Continuum Plasticity
Materials, such as solids, liquids and gases, are composed of molecules separated by
empty space. On a microscopic Scale, materials have cracks and discontinuities.
However, certain physical phenomena can be modeled assuming the materials exist as a
continuum, meaning the matter in the body is continuously distributed and fills the entire
region of space it occupies. A continuum is a body that can be continually sub-divided
into infinitesimal elements with Properties being those of the bulk material.
3-9
A Study of material non-linearity during deformation using FEM software
3.2 RATE – INDEPENDENT PLASTICITY
It is an irreversible straining process which occurs in a material once a certain level of
stress is reached. Different types of material behavior pertaining to it are as follows:

Bi linear Isotropic Hardening

Multi Linear Isotropic Hardening

Multi Linear Kinematic Hardening
To have an understanding of Rate-Independent plasticity, we must understand about the
following ingredients:
3.2.1
Yield Criterion
The yield criterion determines the stress level at which yielding is initiated. The most
often used yield criterion is the Von Mises Yield Criterion.
The Von Mises yield function is given as:
1
2
3
𝑓 = 𝜎𝑒 − 𝜎𝑦 = ( 𝜎 ′ : 𝜎 ′ ) − 𝜎𝑦
2
This equation satisfies the yield criterion, which is given by
f<0
:
Elastic deformation
f=0
:
Plastic deformation.
Since, we are considered with the plastic deformation. So let us know about the
assumptions that are made:

The yield is independent of Hydrostatic Stresses.

Yield in polycrystalline metals tends to be isotropic.

Yield stresses measured in compression have the same magnitude as yield stresses
measured in tension.
3.2.2
Flow Rule
The flow rule determines the direction of plastic straining and is given as:
{𝑑𝜀 𝑝𝑙 = 𝜆 {
3-10
𝜕𝑄
}}
𝜕𝜎
A Study of material non-linearity during deformation using FEM software
λ= plastic multiplier (which determines the amount of plastic straining)
Q = function of stress termed the plastic potential (which determines the direction of
plastic straining)
If “Q” is the yield function (as is normally assumed), the flow rule is termed associative
and the plastic strains occur in a direction normal to the yield surface.
3.2.3
Hardening Rule
The hardening rule describes the changing of the yield surface with progressive yielding,
so that the conditions (i.e. stress states) for subsequent yielding can be established.
Types of Hardening Rules
a) Kinematic hardening assumes that the yield surface remains constant in size and
the surface translates in stress space with progressive yielding.
b) Work (or isotropic) hardening and kinematic hardening. In work hardening, the
yield surface remains centered about its initial centerline and expand in size as the
plastic strain develops. For materials with isotropic plastic behavior this is termed
isotropic hardening.
3.3 AN ELABORATION FOR THE TYPES OF HARDENING MODELS:
3.3.1
Isotropic hardening
When metals are deformed plastically, they harden and thus the stress required for further
plastic deformation increases, which is a function of accumulated plastic strain ‘p’, which
can produced as:
𝑝 = ∫ 𝑑𝑝 = ∫ 𝑝̇ 𝑑𝑡
Isotropic Hardening generally occurs when the yield surface expands uniformly in all
directions in the stress space.
In the below fig., we observe that the loading is in the 2-direction, so the load point
moves in the 𝜎2 direction from zero until it meets the initial yield surface at 𝜎2 = 𝜎𝑦 .
Yield occurs at this point. In order for hardening to take place, and for the load point to
stay on the yield surface, the yield surface must expand as 𝜎2 increases (figure 3-4). The
3-11
A Study of material non-linearity during deformation using FEM software
amount of expansion is often taken to be a function of accumulated plastic strain, p, and
for this case, the yield function is of the form:
𝑓(𝜎, 𝑝) = 𝜎𝑒 − 𝜎𝑦 (𝑝) = 0
Figure 3-4Case of Isotropic Hardening showing expansion in yield surface along with
uniaxial stress-strain curve.
3.3.2
Kinematic hardening:
When the load increases monotonically, hardening is assumed to be Isotropic. But for the
case of cyclic loading, Isotropic does not seem to be appropriate.
This can be explained with the help of a figure as follows:
At a strain of 𝜀𝑖 , corresponding to load point (1) as shown, the load is reversed so that the
material behaves elastically (the stress is now lower than the yield stress) and linear
stress–strain behavior results up until load point (2). At this point, the load point is again
on the expanded yield surface, and any further increase in load results in plastic
deformation. We observe that isotropic hardening leads to a very large elastic region
(Figure 3-4), on reversed loading, which is often not what would be seen in experiments.
According to experimental analysis, a smaller elastic region is expected and this results
from what is known as the Bauschinger Effect and the Kinematic Hardening.
In Kinematic Hardening, generally the yield surface translates in stress space rather than
expanding. Explanation can be described as follows:
3-12
A Study of material non-linearity during deformation using FEM software
Figure 3-5Reverse Loading with Isotropic Hardening showing (a) yield surface (b)
stress-strain curve
The stress increases until the yield stress,𝜎𝑦 is achieved. With continued loading, the
material deforms plastically and the yield surface translates. When load point (1) is
achieved, the load is reversed so that the material deforms elastically until point (2) is
achieved when the load point is again in contact with the yield surface. The elastic region
is much smaller than that of isotropic hardening.
In fact, for the kinematic hardening,
The elastic region is of size 2𝜎𝑦 whereas for the isotropic hardening, it is 2(𝜎𝑦 + 𝑟).
In the case of plastic flow with kinematic hardening, note that the consistency condition
still holds; the load point must always lie on the yield surface during plastic flow. In
addition, normality still holds; the increment in plastic strain has direction normal to the
tangent to the yield surface at the load point.
The yield function describing the yield surface must now also depend on the location of
the surface in stress space. Consider the initial yield surface shown in the below figure 36. Under applied loading and plastic deformation, the surface translates to the new
3-13
A Study of material non-linearity during deformation using FEM software
location shown such that the initial center point has been translated by |x|. We now need
to determine the stresses relative to the new yield surface center to check for yield.
In the absence of kinematic hardening, the yield function written in terms of tensor
stresses is
′
3
𝑓 = 𝜎𝑒 − 𝜎𝑦 = ( 𝜎 ′ : 𝜎 ′ ) − 𝜎𝑦
2
With kinematic hardening, however, it is:
1/2
3 ′
′
′
′
𝑓 = ( (𝜎 − 𝑥 ) ∶ (𝜎 − 𝑥 )) − 𝜎𝑦
2
Figure 3-6 Kinematic hardening showing (a) the translation, |x| of the yield surface with
plastic strain, (b) The resulting stress–strain curve with shifted yield stress in
compression—the Bauschinger effect
KINEMATIC HARDENING MODELS:
Prager Rule
It was introduced by Prager (1956) and describes the translation of the yield surface.
According to this model, the simulation of plastic response of materials is linearly related
with the plastic strain. The equation proposed by Prager to describe the evolution of the
back-stress is
𝛼̇ ij = 𝑐 ̇ ∈ "ij
3-14
A Study of material non-linearity during deformation using FEM software
Where c is a constant derived from a simple monotonic uniaxial curve and
∈̇ "ij is the rate of effective plastic strain.
Armstrong and Frederick
This model was proposed by Armstrong and Frederick (1966), in which it simulates the
multi-axial Bauschinger effect that is actually the movement of the yield surface in the
stress space. As compared to the previously existing models, this one predicts
Bauschinger effect for example, the uniaxial cyclic loading test. Through experimental
results, it was found out that Armstrong & Frederick predictions were more accurate than
Prager is and Mises models for cyclic axial loading and torsion tension of a thin tube tests
on annealed copper. This model also proposed some advancement in terms of simplicity
for computer programs. Although the subroutine for calculating strain increments from 10
stress and stress increments were more complex than the ones for Prager Model, however,
there was improvement in results and better correlation with experiments.
Armstrong and Frederick model (1966) is based on the assumption that the most recent
part of the strain history of a material dictates the mechanical behavior. Its kinematic
hardening rule was predicted by the expression
𝛼̇ 𝑖𝑗 =
2
𝐶1 ∈̇"𝑖𝑗 − 𝐶2 𝛼𝑖𝑗 𝑝̇
3
Where 𝑝̇ is the accumulated plastic strain rate given as 𝑝̇ = √2/3 ∈̇𝑖𝑗 ∈̇𝑖𝑗 .
The constants C1 and C2 are determined from uniaxial tests.
Chaboche
Proposed by Chaboche and his co-workers (1979, 1991), this model is based on a
decomposition of non-linear kinematic hardening rule proposed by Armstrong and
Frederick. This decomposition is mainly significant in better describing the three critical
segments of a stable hysteresis curve. These three segments are:

The initial modulus when yielding starts,

The nonlinear transition of the hysteresis curve after yielding starts until the curve
becomes linear again,

The linear segment of the curve in the range of higher strain.
3-15
A Study of material non-linearity during deformation using FEM software
To improve the ratcheting prediction in the hysteresis loop, Chaboche et al. (1979),
initially proposed three decompositions of the kinematic hardening rule, corresponding to
the above three segments of the hysteresis curve. Using this decomposition, the ratcheting
prediction improved as compared to the A-F model. In the same work, Chaboche (1986)
analyzed three models to describe kinematic hardening behavior. The first model that was
studied uses independent multiyield surfaces as proposed by Mroz (1967). This model is
useful in generalizing the linear kinematic hardening rule. It also enables the description
of:

The nonlinearity of stress-strain loops, under cyclically stable conditions,

The Bauschinger effect, and

The cyclic hardening and softening of materials with asymptotic plastic
shakedown.
The shortcoming of this model is its inability to describe ratcheting under asymmetric
loading conditions.
The second type of models used only two surfaces, namely the yield and the bounding
surfaces, to describe the material. The Dafalias-Popov (1976) model was chosen under
this category, as it shows the following differences against the Mroz (1967) model:

It uses two surfaces whereas Mroz (1967) uses a large number of surfaces

In terms of the general transition rule for the yield surface, the Mroz formulation
had an advantage over this model

This model gives a function to describe a continuous variation of the plastic
models, thus enabling description of a smooth elastic-plastic transition.
In the Mroz (1967) model, the number of variables needed for the description of
ratcheting is very high and for cyclic stabilized conditions no ratcheting occurs. In the
two-surface model, the updating procedure to describe a smooth elastic-plastic transition
and simulate ratcheting effects leads to inconsistencies under complex loading conditions.
The nonlinear kinematic hardening rule is an intermediate approach of the models that
uses differential equations that govern the kinematic variables. The varying hardening
modulus can be derived directly based on these equations, whereas in the case of the
Mroz (1967) model, non-linearity of kinematic hardening was introduced by the field of
hardening moduli associated with several concentric surfaces. In the case of the Dafalias
3-16
A Study of material non-linearity during deformation using FEM software
and Popov (1976) model, it was done by continuously varying then hardening modulus,
from which the translation rule of the yield surface is deduced. It was later found that this
model tends to greatly over-predict ratcheting in the case of normal monotonic and
reverse cyclic conditions. To overcome these pitfalls, Chaboche (1991) introduced a
fourth decomposition of the kinematic hardening rule based on a threshold. This fourth
rule simulates a constant linear hardening with in a threshold value and becomes
nonlinear beyond this value. With the use of this fourth decomposition, the overprediction of ratcheting is reduced and there is an improvement in the hysteresis curve.
This is because, with in the threshold, the recall term is ignored and linear hardening
occurs as it did without the fourth rule. Beyond the threshold the recall term makes the
hardening non-linear again and reduces the ratcheting at a higher rate to avoid overprediction.
Voyiadjis and Kattan
Voyiadjis and Kattan (1990) proposed a cyclic theory of plasticity for finite deformation
in the Eulerian reference system. A new kinematic hardening rule is proposed, based on
the experimental observations made by Phillips et al. (1973, 1974, 1979, and 1985). This
model is shown to be more in line with experimental observations than the Tseng-Lee
model (1983), which is obtained as a special case.
Voyiadjis and Kattan model uses the minimum distance between the yield surface and the
bounding surface as a key parameter. Once this distance reaches a critical value, the
direction of motion of the yield surface in the vicinity on the bounding surface is changed
and the Tseng-Lee model (1983) is used to ensure tangency of the two surfaces at the
stress point. This model predicts a curved path for the motion of the yield surface in the
interior of the bounding surface. On the other hand, Tseng-Lee (1983) assumes that the
center of the yield surface moves in a straight line. Voyiadjis and Kattan model has been
proven to give good results that conform to experimental observations.
Voyiadjis and Sivakumar
A robust kinematic hardening rule is proposed by Voyiadjis and Sivakumar (1991,1994)
to appropriately blend the deviatoric stress rate rule and the Tseng-Lee rule in order to
satisfy both the experimental observations made by Phillips et al. (1974, 1975, 1977,
1979, 1985) and the nesting of the yield surface to the limit surface. In this model, an
3-17
A Study of material non-linearity during deformation using FEM software
additional parameter is introduced to reflect the dependency of the plastic modulus on the
angle between the deviatoric stress rate tensor and the direction of the limit back stress
relative to the yield backstress. This model was tested for uniaxial (or proportional) and
non-proportional (multiaxial) loading conditions. The results obtained were than
compared with experimental results, and their correlation was proven to be very accurate.
Voyiadjis and Basuroychowdhary
Voyiadjis and Basuroychowdhary (1998) proposed a two-surface plasticity model using a
nonlinear kinematic hardening rule to predict the non-linear behavior of metals under
monotonic and non-proportional loadings. The model is based on Frederick and
Armstrong (1966), Chaboche (1989, 1991), Voyiadjis and Kattan, and Voyiadjis and
Sivakumar (1991, 1994) models. The stress rate is incorporated in the evaluation equation
of back-stress through the addition of a new term. The new term creates an influence of
the stress rate on the movement of the yield surface, as proposed by Phillips et al. (1974,
1975). The evolution equation of backstress is given as four components of the type
NLK-T (Non-Linear Kinematic with Threshold).
Baushinger effect
If one takes a fresh sample and loads it in tension into the plastic range, and then unloads
it and continues on into compression, one finds that the yield stress in compression is not
the same as the yield strength in tension, as it would have been if the specimen had not
first been loaded in tension. In fact the yield point in this case will be significantly less
than the corresponding yield stress in tension. This reduction in yield stress is known as
the Bauschinger effect. The effect is illustrated in above figure shown. The solid line
depicts the response of a real material. The dotted lines are two extreme cases which are
used in plasticity models; the first is the isotropic hardening model, in which the yield
stress in tension and compression are maintained equal, the second being kinematic
hardening, in which the total elastic range is maintained constant throughout the
deformation. The presence of the Bauschinger effect complicates any plasticity theory.
However, it is not an issue provided there are no reversals of stress in the problem under
study.
3-18
A Study of material non-linearity during deformation using FEM software
Figure 3-7 Bauschinger effect
Physical Nature of the Bauschinger Effect
A proper understanding of the physical origin of Bauschinger Effect helps in acquiring
more refined plasticity models and ultimately improves the simulation results. At room
temperature, the main source for the Bauschinger effect as in the case of metal plasticity
is a dislocation structure. During deformation, dislocations move, activating slip on the
energetically favorable slip systems, and thus resulting in increasing the density of the
dislocations gradually. Dislocations overlap; accumulate at obstacles producing
dislocation tangles and pile-ups. This increases the resistance to further dislocation
motions and causes a hardening of the metal.
Figure 3-8(a) Edge dislocation and (b) dislocation pile-ups on the grain
boundaries
3-19
A Study of material non-linearity during deformation using FEM software
The Bauschinger effect can be generally ascribed to long-range effects, such as internal
stresses due to dislocation interactions (Figure 3-10), dislocation pile-ups at grain
boundaries or Orowan loops around strong precipitates, and to short-range effects, such as
directionality of mobile dislocations in their resistance to motion or annihilation of the
dislocations during the reverse loading.
The primary driving force of the Bauschinger effect can be explained by the motion of the
less stable dislocation structures such as pile-ups. Pile-up occurs as a cluster of
dislocations is unable to move past the barrier. As accumulated dislocations generate
microscopic back-stresses, they will assist the movement of dislocations in the reverse
direction and the yield strength becomes lower. This occurs directly after the change of
load direction or during unloading and takes place simultaneously with elastic
deformation. With this microscopic mechanism one can explain such macroscopic
phenomena as the transient softening, the early re-plastification and the reduction of the
Young’s modulus.
Another mechanism is, when the strain direction is reversed, dislocations of the opposite
sign can be produced from the same source that produced the slip-causing dislocations in
the initial direction. Dislocations with opposite signs can attract and annihilate each other.
Since strain hardening is related to an increased dislocation density, reducing the number
of dislocations reduces strength. The work hardening stagnation can be explained by the
partial disintegration of the performed dislocation cell structures and the subsequent
resumption of work hardening to the formation of new dislocation structures. The socalled cross effect during orthogonal loading is referred to the fact that the dislocation
structures which developed during pre-loading in a given direction act as obstacles to slip
on systems activated in the orthogonal direction after the change of loading direction.
Other mechanisms beside the crystallographic slip can also macroscopically contribute to
the Bauschinger effect. Twinning is crucial particularly for the metals with hexagonal
close-packed lattice such as magnesium or zircon. During the cold forming of the
magnesium alloys the twinning under compression can occur, which leads to the essential
reduction of the yield strength. Other factors which contribute to such material behavior
on the macroscopic level could be a change of the crystallographic texture during plastic
deformation, stress induced phase transformation or porosity evolution.
3-20
A Study of material non-linearity during deformation using FEM software
3.4 YIELD SURFACES
A yield surface is defined as a five
dimensional
surface
in
the
six-
dimensional stress space. It is usually
convex and the state of stress is elastic
from the inside. When the stress state
lies on the surface, the material is said
to have reached its yield point and the
material is said to have become plastic.
Further deformation of the material
causes the stress state to remain on the
yield surface, even though the shape
and size of the surface may change as
the plastic deformation evolves. This is
because stress states that lie outside the
yield surface are non‐permissible in rate‐
independent plasticity, though not in
Figure 3-9 Surfaces on which invariants
𝐼1 , 𝐽2 , 𝐽3 are plotted in principal stress space
some models of viscoplasticity.
The yield surface is usually expressed in terms of a three‐dimensional principal stress
space (𝜎1 , 𝜎2 , 𝜎3), a two‐ or three‐dimensional space spanned by stress invariants
(𝐼1 , 𝐽2 , 𝐽3 ) or a version of the three‐dimensional Haigh–Westergaard stress space. Thus we
may write the equation of the yield surface in the forms:
3.4.1

𝑓(𝜎1 , 𝜎2 , 𝜎3 ) = 0

𝑓(𝐼1 , 𝐽2 , 𝐽3 ) = 0
Famous Yield Surfaces:
Tresca’s Yield Surface:
Henri Tresca proposed the Tresca Yield Criterion. It is also known by the name
Maximum Shear Stress Theory or the Tresca-Guest Theory.
It is expressed as:
3-21
A Study of material non-linearity during deformation using FEM software
1
1
𝑀𝑎𝑥(|𝜎1 −𝜎2 |, |𝜎2 − 𝜎3 |, |𝜎3 − 𝜎1 |) = 𝑆𝑠𝑦 = 𝑆𝑦
2
2
Where Ssy is the shear yield strength and Sy is the tensile yield strength.
Tresca yield surface is actually a three dimensional space of principal stresses. It is a
prism of six sides and having infinite length. This means that the material remains elastic
when all three principal stresses are roughly equivalent, no matter how much it is
compressed or stretched. It is a cross section of the prism along the 𝜎1 , 𝜎2 plane.
Figure 3-10 The von Mises yield surfaces in principal stress coordinates circumscribes a
cylinder with radius around the hydrostatic axis. Also shown is Tresca’s hexagonal yield
surface.
Von Mises Yield Surface:
Von Mises yield surface is a circular cylinder of infinite length whose axis is inclined at
equal angles to the three principal stresses and is located in the three dimensional space of
principal stresses.
3-22
A Study of material non-linearity during deformation using FEM software
A cross section of the von Mises cylinder on the plane of 𝜎1 , 𝜎2 produces the elliptical
shape of the yield surface.
It is expressed in the following form:
(𝜎1 − 𝜎2 )2 + (𝜎2 − 𝜎3 )2 + (𝜎3 − 𝜎1 )2 = 2𝑆𝑦2
It is apparent from this that hydrostatic stress has no effect on yield according to the von
Mises criterion. Even infinite, but equal, principal stresses σ1, σ2, and σ3 will never cause
yield, since 𝜎𝑒 remains zero.
Figure 3-11 the von Mises yield surface for conditions of plane stress
Figure 3.14 shows the von Mises yield surface for conditions of plane stress, showing
the increment in plastic strain, 𝑑𝜺𝒑 , in a direction normal to the tangent to the surface.
Von Mises yielding is based on difference of normal stress, but independent of
hydrostatic stress. This figure 3.10 shows von Mises yield surface for plane stress and the
corresponding stress–strain curve obtained for uniaxial straining in the 2-direction.
3-23
A Study of material non-linearity during deformation using FEM software
Figure 3-12 Von Mises yield surface for plane stress and the corresponding stress–strain
curve
Drucker–Prager yield surface:
Drucker–Prager yield surface is a regular cone in the three dimensional space of principal
stresses. The Drucker‐Prager yield criterion is also commonly expressed in terms of the
material cohesion and friction angle. It provides provisions for handling materials with
differing tensile and compressive yield strengths. It is most often used for concrete where
both normal and shear stresses can determine failure. The Drucker–Prager yield criterion
may be expressed as:
𝑚−1
𝑚 + 1 (𝜎1 − 𝜎2 )2 + (𝜎2 − 𝜎3 )2 + (𝜎3 − 𝜎1 )2
(
) (𝜎1 + 𝜎2 +𝜎3 ) + (
)√
= 𝑆𝑦𝑐
2
2
2
Where 𝑚 =
𝑆𝑦𝑐
𝑆𝑦𝑡
Figure 3-13 View of Drucker–Prager yield surface in 3D space of principal stresses
3-24
A Study of material non-linearity during deformation using FEM software
SCOPE OF THE STUDY
The main scope of the study is that we learned the non-linearity of material under
different load conditions. We also learned the Mechanical APDL of the Ansys FEM
package. We came to understand that Isotropic hardening is not useful in situations where
components are subjected to cyclic loading.
Isotropic hardening does not account for Bauschinger effect and predicts that after a few
cycles,
the
material
(solid)
just
hardens
until
it
responds
elastically.
To fix this, alternative laws i.e. kinematic hardening laws have been introduced. As per
these hardening laws, the material softens in compression and thus can correctly model
cyclic behavior and Bauschinger effect.
In the case of isotropic hardening, if you plastically deform a solid, then unload it, then
try to reload it again, you will find that its yield stress (or elastic limit) would have
increased compared to what it was in the first cycle.
Again, when the solid is unloaded and reloaded, yield stress (or elastic limit) further
increases (as long as it is reloaded past its previously reached maximum stress). This
continues until a stage (or a cycle) is reached that the solid deforms elastically throughout
(that is, if the cycles of load are always to the same level, then after just one cycle your
specimen on subsequent cycles will just be loading and unloading along the elastic line of
the stress strain curve). This is isotropic hardening.
Essentially, isotropic hardening just means if you load something in tension past yield,
when you unload it, then load it in compression, it will not yield in compression until it
reaches the level past yield that you reached when loading it in tension. In other words if
the yield stress in tension increases due to hardening the compression yield stress grows
the same amount even though you might not have been loading the specimen in
compression. It is a type of hardening used in mathematical models for finite element
analysis to describe plasticity though it is not absolutely correct for real materials.
4-25
A Study of material non-linearity during deformation using FEM software
OBJECTIVE

To identify very essential and most used material nonlinear properties form the
ocean of material model.

To choose the material that possess material nonlinear property and that is very
much used in the industry.

To understand the working of the commercially available FEM software
packages like ANSYS.

To understand the core concepts of the material nonlinearity like yield surface,
ratcheting etc.

To find a way to input data of these material data in to the software packages.

Solve all the models by proper solver the suites the analysis.

To get the results of the study and plot them as graph.

To compare the results of BISO and MISO model, under the category of isotropic
hardening. After compression, to point out the notable differences between. And
to do energy analysis on these model more in-depth comparison.

To study the kinematic hardening by using BKIN, MKIN and CHABOCHE
models. And to points out differences in their results

To use the previous studies to analysis a metal forming process and compare it
with the
linear material analysis and to point out the importance of the nonlinear
material modeling.
5-26
A Study of material non-linearity during deformation using FEM software
RESEARCH METHODOLOGY
Material nonlinearity is associated with the inelastic behavior of a component or
system. Inelastic behavior may be characterized by a force-deformation relationship,
which measures strength against translational or rotational deformation.
We used Ansys Mechanical APDL to plot curves under different load conditions, which
is a FEM software package of ANSYS. We used APDL as solver, as coding can easily be
done for analysis and we are familiar with work environment and material models in
Ansys APDL and the result provided by it are very accurate than other FEM software
packages and is also efficient in time.
We used different loading conditions on our material from LC 1 to LC 6 which has been
described in section 9.4.
ANSYS Workbench was also used for the purpose of Explicit Dynamics. We performed
analysis of the sheet metal bending operation and comprehended the comparison between
the material linearity and non-linearity processes. The results also paved the way for
understanding the immense application of the Material non linearity in sheet metal
bending operations.
We used 9Ni-4Co-0.20C steel as our material for analysis because it was suitable with
our loading conditions and has the desired conditions required for analysis. We used
Element as LINK180 which is a 3-D Truss (spar) element in Ansys of because of features
such as:

Uses only one element between pins.

No bending of the element is considered.

3 DOF’s: UX, UY, UZ (in 3D).

Few Real Constants: Cross-Sectional Area, Added Mass.

Plasticity, creep, rotation, large deflection, and large strain capabilities are
included.
We used two hardening models Kinematic hardening and Isotropic hardening for the
purpose of analysis. Kinematic hardening was used for cyclic loading as such type of
loading does not produce favorable results in isotropic hardening.
6-27
A Study of material non-linearity during deformation using FEM software
Figure 6-1 Flow chart of the APDL coding
This is so because, in kinematic hardening Bauschinger effect takes place which is very
favorable and is an accurate condition for analysis of non-linearity. We have produced
results using both the models and compared them on the basis of different loading
conditions with a detailed explanation.
As a non-material model we used BISO, MISO, BKIN, MKIN and CHABOCHE
Material Model for analysis in APDL. We use these material model as it easily satisfies
the Yield criterion. In these material models, we have plotted different curves under
different loading conditions.
6-28
A Study of material non-linearity during deformation using FEM software
COMPLETE WORK PLAN WITH TIMELINE
We divided our team into a group of four where each individual was assigned with
individual task.
Table 7-1 Team members list
Name
Task assigned
Emayavaramban E
Ansys APDL coding
Milan Chhetri
Material modelling
Akash Dan
Analysis of Material Properties
Sonu Yadav
Theory of Plasticity
After the division of individual tasks, we started Preparing on our individual topic
to get information and to know as much as possible.
Daily Routine for assessment of work performed by each individual:

A general meeting of one hour during the weekdays.

Sharing what we learnt. Each individual explains other members of the
team with his work study.

After that, we acquainted ourselves with Ansys APDL coding with the
help of Emayavaramban.
Study Approach:
1. Analysis and comparison of hardening method - isotropic hardening and kinematic
hardening with the help of stress-strain graph under cyclic loading, linear loading and
time dependent loading in order to have different curves under graph which will help
to sort out all the hardening process in detail.
2. First we went through BISO (Bilinear Isotropic Hardening) which uses von mises
yield criterion. And we obtained stress-Strain curve in which the initial slope of the
curve is taken as the elastic modulus of the material. At the specified yield stress, the
curve continues along the second slope defined by the tangent modulus. The tangent
modulus cannot be less than zero nor greater than the elastic modulus. In BISO we got
bilinear curve. Subsequently, We got different curves for different loading.
7-29
A Study of material non-linearity during deformation using FEM software
3. After that, we did analysis for MISO (Multi-linear isotropic hardening) under
different loading conditions. The MISO is similar to BISO except that a multi-linear
curve is used instead of bilinear curve. We plotted graph for non-cyclic loading in
MISO. We used MISO even for those which do not support MKIN (Multi-linear
kinematic hardening). We used MISO for large strain cycling where kinematic
hardening could exaggerate the Bauschinger effect. We got linear total stress-total
strain curve, starting at the origin, with positive stress and strain values. The curve
was continuous from the origin through 100max stress-strain points. The slope of the
first segment of the curve must correspond to the elastic modulus of the material and
no segment slope should be larger. No segment can have a slope less than zero. The
slope of the stress-strain curve is assumed to be zero beyond the last user-defined
stress-strain data point. We specified different temperature-dependent stress-strain
curves in order for comparison.
4. After obtaining the different Stress-Strain curves under different loading conditions,
we analyzed BKIN (Bilinear kinematic Hardening). In BKIN the total stress range is
equal to twice the yield stress, so that Bauschinger effect is included. As BKIN is only
restricted to material which satisfies Von Mises yield criteria. We defined the material
behavior by a bilinear total stress-total strain curve starting at the origin and with
positive stress and strain values. In the plot the initial slope of the curve is taken as the
elastic modulus of the material. At the specified yield stress, the curve continues
along the second slope defined by the tangent modulus .The tangent modulus cannot
be less than zero nor greater than elastic modulus. We came to know that Rice’s
Hardening Rule is applied for BKIN which states that it take relaxation with
temperature increase into account.
5. After obtaining different plot for BKIN we analyze for MKIN Multi-linear kinematic
hardening). We used MKIN and KINH in order to plot different curves. We came to
know that both MKIN and KINH used Besseling Model which is called sub-layer or
overlay model. The material response is represented by multiple layers of perfectly
plastic material; the total response is obtained by weighted average behavior of all
layers and the Individuals weights are derived from the uniaxial stress-strain curve.
The uniaxial behavior is described by a piece wise linear “total stress-total strain
curve” starting at the origin with positive stress and strain values. We obtained the
plot where the slope of the first segment of the curve must correspond to the elastic
modulus of the material and no segment slope should be larger. The slope of the
7-30
A Study of material non-linearity during deformation using FEM software
stress-strain curve is assumed to be zero beyond the last user-defined stress-strain data
point. We used KINH because layers are scaled (Rice’s model which is which does
take relaxation with temperature increase into account), providing better
representations. As KINH allows using 40 temperature-dependent stress-strain curves
we used it to plot different curves under different loadings. We defined more one
stress-strain curve for temperature-dependent properties, and then each curve
contained the same number of points. The assumptions are that the corresponding
points on the different stress-strain curves represent the temperature dependent yield
behavior of a particular sub layer. The MKIN curve is continuous from the origin with
a maximum of five total stress total strain points. The slope of the first segment of the
curve must correspond to the elastic modulus of the material and no segment slope
should be larger. We came to know that MKIN can also be used in conjunction with
the TBOPT option (TB, MKIN,,,, TBOPT). TBOPT has the following three valid
arguments:
1. No stress relaxation with temperature increase (this is not recommended
for non-isothermal problems);also produces thermal ratcheting
2. Recalculate total plastic strain using new weight factors of the sub-volume.
3. Scale layer plastic strains to keep total plastic strain constants; agrees with
Rice’s model (TB, BKIN with TBOPT=1).Produces stable stress-strain
cycles.
6. After getting the different plot for MKIN we analyze the Non-linear kinematic
hardening CHABOCHE. We use CHABOCHE model for simulating the cyclic
behavior of materials. Like the BKIN & MKIN options, we can also use this model to
simulate monotonic hardening and the Bauschinger effect. We can also superpose up
to five kinematic hardening models and an isotropic hardening model to simulate the
complicated cyclic plastic behavior of materials, such as cyclic hardening or
softening, and ratcheting or shakedown. We plotted different curves for
CHABOCHE.
7-31
A Study of material non-linearity during deformation using FEM software
Table 7-2 Gantt chart
Process Weeks
1
2
3
1) Team Formation and mentor selection
2) Project Pre –Research
3) Team and project approval
4) Understanding the research methodology
5) Literature review
6) Planning the research work
7) Division of works
8) Ansys Trial study
9) Modelling BISO and MISO models
10) Processing of the data of BISO and MISO
11) Working with BKIN and MKIN
12) Analysis of results of BKIN and MKIN
13) Analysis using Chaboche model
14) Analysis on application based studies
15) Report and presentation preparation
7-32
4
5
6
7
8
9
10
11
11+
A Study of material non-linearity during deformation using FEM software
EXPECTED OUTCOME
On this research we are expecting to identify the similarities, pros and cons of bilinear
and multi linear isotropic hardening models, we are also expecting results that are similar
to experimental results. During energy analysis the simulated results should match the
analytical results.
Kinematic Hardening should include Bauschinger effect in to it. During cyclic
loading it demonstrates the behavior of the kinematic yield surface translation. We could
also able to identify the similarities and differences in BKIN and MKIN models.
The Chaboche model is nonlinear kinematic model which includes ratcheting. At the end
we should able to demonstrate the differences between Chaboche method and MKIN
model.
We should be able to apply all the above mentioned material model to a practical
application based problem and solve. And also to describe the importance and
significance of adding nonlinear material model to the study.
8-33
A Study of material non-linearity during deformation using FEM software
RESEARCH AND EXPERIMENTAL WORK DONE
9.1 MATERIAL SELECTION
The material we selected for Non Linearity Analysis is 9Ni-4Co-0.20C Steel. It is
intermediate alloy steel categorized under 9Ni-4Co series of steels and has alloy content
substantially higher than that of Low alloy steels but lower than the stainless steels.
The high Chromium content owns to the fact that it provides improved Oxidation
Resistance while Nickel addition to non-secondary hardening steels lowers the transition
temperature and improves low-temperature toughness.
Typical composition of 9Ni-4Co-0.20C (percent by weight) includes:
C – 0.20; Mn – 0.25; Fe – Balance; Si – 0.1; Cr – 0.75; Ni – 9.0; Co – 4.5; Mo – 1;
V – 0.08; P – 0.01
9Ni-4Co-0.20C Steel is an alloy which possesses the following properties:



Excellent Fracture Toughness
Excellent Weldability
High Hardenability when heat treated to 190 to 210 ksi ultimate tensile
strength.
Few other important characteristics of this alloy:


The alloy does not require any pre-heat or post-heat treatment while it is being
welded in the heat-treated condition. Even the hardening in certain sections of
the alloy can be at the least be 8 mm thick.
The alloy can retain its microstructural changes even up to 900° F, which is
approximately 100° F below the tempering temperature.
The heat treatment process for the material can be summarized as follows:






Normalizing for 1hr at around 1650°F per inch of cross-section.
Cooling in air up to room temperature.
Reheating to around 1520°F for 1hr per inch of cross-section.
Quenching in oil or water.
After Quenching, maintaining it at a temperature of around -100° F for 2hrs.
Performing double tempering at 1035° F for 2 hrs.
9-34
A Study of material non-linearity during deformation using FEM software
Table 9-1Material Specification for 9Ni-4Co-0.20C
Specification
Form
AMS 6523
Sheet, Strip & Plate
As for the selection of this material against the other alloys belonging to the category of
intermediate alloy steels such as 9Ni-4Co-0.30C and 5Cr-Mo-V, we draw out the
drawbacks of the other materials:

In 5Cr-Mo-V alloy steels, distortion is observed during heat treatment.

In 9Ni-4Co-0.30C steel, pre-heat and post-heat treatment is required prior to
welding.
Figure 9-1: Typical Tensile Stress-Strain Curves for 9Ni-4Co-0.20C steel plates at
various temperatures.
9-35
A Study of material non-linearity during deformation using FEM software
Figure 9-2: Typical Compressive Stress Strain Curves and Compressive Tangent
Modulus Curves for 9Ni-4Co-0.20C steel plate at various temperatures.
Table 9-2 calculated material properties
Properties (SI units)
RT
700
900
Poison ratio
0.3
0.3
0.3
Modulus of elasticity
1.94E+11 1.72E+11
1.59E+11
Tangent Modulus
4.88E+10 5.21E+10
5.42E+10
Fracture point
1.42E+09 1.17E+09
1.02E+09
Yield Strength
1.12E+09 8.63E+08
6.34E+08
CHABOCHE (C)
1.46E+11 1.24E+11
1.37E+11
Back Stress
2.92E+08 3.15E+08
3.81E+08
CHABOCHE (γ)
500.87
358.39
9-36
393.42
A Study of material non-linearity during deformation using FEM software
9.2 ELEMENT SELECTION:
The element we chose for analysis is
LINK180. LINK180 is a 3-D Truss (spar)
element (Figure 9-3). By truss elements we
mean that it is a subset of beam-type elements
which can’t carry moments and thus have no
bending DOF’s. These are most commonly
Figure 9-3 LINK 180
known as “Two-Force members” as these can only carry axial loads.
It has found its usage in many engineering applications such as in modelling trusses,
sagging cables, links, springs, and so on.
Desired properties of LINK 180

Every node in a truss model is a ball and socket (or spherical) joint.

Uses only one element between pins.

As in a pin-jointed structure, no bending of the element is considered.

A Uniaxial Tension Compression Element with 3 DOF’s: UX, UY, UZ (in
3D).

Material Props: Modulus, Density

Real Constants: Cross-Sectional Area, Added Mass.

Plasticity, creep, rotation, large deflection, and large strain capabilities are
included.

Isotropic hardening plasticity, kinematic hardening plasticity, Hill anisotropic
plasticity, Chaboche nonlinear hardening plasticity is supported.
Element is defined by ET, ITYPE, Ename command, In this we also have to specify the
element number which is later used to identify different properties of the element s during
meshing. The R, Nset, R1 is to real constrain. In this case it is the cross section of the link
element. The element requires two nodes to be defined. The element x-axis is oriented
along the length of the element from node I toward node J. Element loads are described
in Nodal Loading.
9-37
A Study of material non-linearity during deformation using FEM software
LINK180: Important Assumptions and Restrictions

The spar element assumes a straight bar, axially loaded at its ends and of uniform
properties from end to end.

The length of the spar must be greater than zero, so nodes I and J must not be
coincident.

The cross-sectional area must be greater than zero.

The displacement shape function implies a uniform stress in the spar.

Stress stiffening is always included in geometrically nonlinear analysis
(NLGEOM, ON). Pre-stress effects can be activated by the PSTRES command.

To simulate the tension-/compression-only options, a nonlinear iterative solution
approach is necessary.
9.3 NONLINEAR MATERIAL MODEL
9.3.1
Bilinear kinematic hardening
We include BKIN in order to assume that
total stress range is equal to twice the yield stress,
so that Bauschinger effect is included. It is
restricted to material which satisfies Von Mises
yield criteria. The material behavior is described
by a bilinear total stress-total strain curve starting
at the origin and with positive stress and strain
values. The initial slope of the curve is taken as
the elastic modulus of the material. At the
specified yield stress (c1), the curve continues
Figure 9-4 TBPLOT for BKIN
along the second slope defined by the tangent
modulus, c2 .The tangent modulus cannot be less than zero nor greater than elastic
modulus.
Initialize the stress-strain table with TB,BKIN. For each stress-strain curve, define
the temperature [TBTEMP], then define C1 and C2 [TBDATA]. You can define up to six
temperature-dependent stress-strain curves (NTEMP=6 maximum on the TB command)
in this manner.
9-38
A Study of material non-linearity during deformation using FEM software
Table 9-3Material parameter for BKIN
Constant
Meaning
C1
Yield Stress (Force/Area)
C2
Tangent Modulus (Force/Area)
BKIN can be used with TBOPT option. In this case, TBOPT takes two arguments
.For TB,BKIN,,,,,0, there is no stress relaxation with an increase in temperature. This
option is not recommended for non-isothermal problems. For TB,BKIN,,,,1, Rice’s
hardening rule is applied (which does take relaxation with temperature increase into
account).
9.3.2
Multi-linear Kinematic Hardening Constant
We can use KINH & MKIN to
model metal plasticity behavior under
cyclic loading.
The two model use Besseling Model
which is called sub layer or overlay
model.
The
represented
material
by
perfectly plastic
response
is
layers
of
multiple
material;
the
total
response is obtained by weighted average
behavior
of
all
layers.
Individuals
weights are derived from the uniaxial
stress-strain curve .The uniaxial behavior
Figure 9-5 TBPLOT for MKIN
is described by a piece wise linear “total stress-total strain curve” starting at the origin
with positive stress and strain values. The slope of the first segment of the curve must
corresponds to the elastic modulus of the material and no segment slope should be larger.
The slope of the stress-strain curve is assumed to be zero beyond the last user-defined
stress-strain data point.
The KINH is recommended because layers are scaled (Rice’s model), providing
better representations. The KINH option allows us to define up to 40 temperaturedependent stress-strain curve. If we define more one stress-strain curve for temperature9-39
A Study of material non-linearity during deformation using FEM software
dependent properties, then each curve should contain the same number of points. The
assumptions is that the corresponding points on the different stress-strain curves represent
the temperature dependent yield behavior of a particular sub layer.
For stress vs. total strain input, initialize the stress-strain table with TB,KINH .For
stress
vs.
plastic
strain
input,
initialize the
stress-strain
table
with
either
TB,KINH,,,,PLASTIC or TB,PLASTIC,,,,KINH. Input the temperature of the first curve
with the TBTEMP, then input stress and strain values using the TBPT. Input the
remaining temperatures and stress-strain values using the same sequence (TBTEMP
followed By TBPT).
The MKIN curve is continuous from the origin with a maximum of five total stress total
strain points. The slope of the first segment of the curve must correspond to the elastic
modulus of the material and no segment slope should be larger.
In MKIN we can define up to 5 temperature dependent stress-strain curves. We can use
only 5 points for each stress-strain curve & each stress-strain curve must have the same
set of strain values.
Initialize the stress-strain table with TB,MKIN, followed by a special form of the
TBTEMP command (TB-TEMP,,STRAIN) to indicate that strains are defined next. The
constants (C1-C5), entered on the next TBDATA command, are the five corresponding
strain values (the origin strain is not input). The temperature of the first curve is then
input with TBTEMP, followed by the TBDATA command with the constants C1-C5
representing the five stresses corresponding to the strains at that temperature. You can
define up to five temperature-dependent stress strain curves (NTEMP=5 max on the TB
command) with the TBTEMP command.
MKIN can also be used in conjunction with the TBOPT option (TB,MKIN,,,,TBOPT).
TBOPT has the following three valid arguments:
0–
No stress relaxation with temperature increase ((this is not recommended
for non-isothermal problems) also produces thermal ratcheting.
1–
Recalculate total plastic strain using new weight factors of the sub volume.
9-40
A Study of material non-linearity during deformation using FEM software
2–
Scale layer plastic strains to keep total plastic strain constants; agrees with
Rice’s model (TB,BKIN with TBOPT=1).Produces stable stress-strain
cycles.
9.3.3
Bilinear isotropic Hardening
This option (TB,BISO) uses the
Von Mises yield criteria coupled with an
isotropic work hardening assumptions. The
material behavior is described by a bilinear
stress-strain curve starting at the origin with
positive stress and strain values. The initial
slope of the curve is taken as the elastic
modulus of the material. At the specified
yield stress (C1), the curve continues along
the second slope defined by the tangent
modulus C2.The tangent modulus cannot be
less than zero nor greater than the elastic
Figure 9-6 Material input for BISO
modulus.
Initialize the stress-strain table with TB,BISO. For each stress-strain curve, define
the temperature [TBTEMP], then define C1 & C2 [TBDATA]. Define up to six
temperature-dependent stress strain curves (NTEMP=6).
Table 9-4 BISO mat data
Constant
Meaning
C1
Yield Stress (Force/Area)
C2
Tangent Modulus (Force/Area)
9-41
A Study of material non-linearity during deformation using FEM software
9.3.4
Multi linear Isotropic Hardening
The option (TB,MISO) is similar
to BISO except that a multi-linear curve
is used instead of bilinear curve. It can be
used for non-cyclic load histories or for
those elements that do not support the
multi-linear kinematic hardening option
(MKIN). This option may be preferred
for large strain cycling where kinematic
hardening
could
Bauschinger
exaggerate
effect.
The
the
uniaxial
behavior is described by apiece-wise
linear total stress-total strain curve, starting
Figure 9-7 Material Plot for MISO
at the origin, with positive stress and strain
values. The curve is continuous from the origin through 100max stress-strain points. The
slope of the first segment of the curve must correspond to the elastic modulus of the
material and no segment slope should be larger. No segment can have a slope less than
zero. The slope of the stress-strain curve is assumed to be zero beyond the last userdefined stress-strain data point.
You can specify up to 20 temperature-dependent stress-strain curves. For stress vs. total
strain input, initialize the curves with TB,MISO. For stress vs. plastic strain input,
initialize the curves with TB,PLASTIC,,,,,MISO .Input the temperature for the first curve
[TBTEMP], followed by up to 100 stress-strain points (the origin stress strain point is not
input)[TBPT]. Define up to 20 temperature-dependents stress-strain curves (NTEMP=20,
maximum on the TB command) in this manner .The constants X,Y) entered on the TBPT
command are:
Table 9-5 Input definition for MISO
Constant
Meaning
X
Strain value (Dimensionless)
Y
Corresponding stress value (force/area)
9-42
A Study of material non-linearity during deformation using FEM software
9.3.5
Nonlinear Kinematic Hardening Constants (TB,CHABOCHE)
This option (TB,CHABOCHE) uses the Chaboche model for simulating the cyclic
behavior of materials. Like the BKIN & MKIN options, you can use this model to
simulate monotonic hardening and the Bauschinger effect. You can also superpose up to
five kinematic hardening models and an isotropic hardening model to simulate the
complicated cyclic plastic behavior of materials, such as cyclic hardening or softening,
and ratcheting or shakedown.
The Chaboche model implemented is
𝑛
𝑛
1
𝑖
2
1 𝑑𝐶𝑖
𝛼̇ = ∑ 𝛼𝑖=̇
∑ 𝐶𝑖 𝜀 𝑝𝑙̇ − 𝛾𝑖 𝛼𝑖 𝜀̇ 𝑝𝑙 +
𝜃̇ 𝛼𝑖
3
𝐶𝑖 𝑑𝜃
The yield function is:
𝑓(𝜎, 𝜀 𝑝𝑙 ) = 𝜎 − 𝑘 = 0
εpl̇ =accumulated equivalent plastic strain2. Initialize
the
data
table
with
TB,CHABOCHE for each set of data , define the temperature [TBTEMP], then define C1
through Cm [TBDATA] , where m=1 +2 NPTS .the maximum number of constants ,m is
11, which correspond to 5 kinematic models [NPTS=5 on the TB command]. The default
value form is 3, which corresponds to one kinetic model [NPTS=1]. You can define up to
1000 temperature-dependent constants ([NTEMP X m ≤ 1000] maximum on the TB
command) in this manner. The constant C1 through C (1+2NPTS) are:
Table 9-6Chaboche material parameter inputs
2
Constant
Meaning3
C1
K4=yield stress
C2
C1=material constant for first kinematic model
C3
𝛾1=material constant for first kinematic model
C4
C2=Material constant for second kinematic model
C5
𝛾2=Material constant for second kinematic model
3
a dot located above any of these quantities indicates the first derivative of the quantity with respect to time
K, and all C and 𝛾 values in the right column are material constants in CHABOCHE model.
4
Define k using BISO,MISO, or NLISO ,through the TB command.
9-43
A Study of material non-linearity during deformation using FEM software
C(2NPTS)
𝐶𝑁𝑃𝑇𝑆 =Material constant for last kinematic model
C(1+2NPTS)
𝛾𝑁𝑃𝑇𝑆 =Material constant for last kinematic model
9.4 LOADING CONDITIONS
To effectively study all the aspects of the material nonlinearity, various loading
conditions are used. Most of the loading conditions are cyclic. To make the cyclic loading
process easy we have defined some parameters and looping condition and also we divided
the loading condition into six categories (LC_1 to LC_6).
The parameters for loading conditions are

Amplitude (AMP)

Amplitude increment (AMPINC)

Mean (MEAN)

Mean increment (MEANINC)

Time increment (TINC)

Over all solution time (TFIN)
Figure 9-8 LC_1
9.4.1
LC_1 Ramped loading and unloading
Ramped loading (figure 9-8) is specifically applied for the isotropic hardening
because it cannot effectively show the cyclic loading. Here overall 0.2 s is used to load
and unload with the displacement of 0.003 m. this loading condition is designated as
LC_1.
9.4.2
LC_2 to LC_6 Cyclic loading conditions
We have applied five different kinds of cyclic loading for kinematic hardening
process. This loading condition is applied by a looping condition. The loop condition
terminates when T(time) reaches TFIN. Each time the time is incremented by the TINC
parameter. LC_2 (figure 9-10) in a constant amplitude cyclic load. Here the mean and
amplitude are set to constant. This is load is a very basic cyclic loading. LC_3 (figure 911) is increased amplitude load where AMPINC parameter is given some non-zero and
positive value. This load extends the yield surface at every cycle.
9-44
A Study of material non-linearity during deformation using FEM software
LC_4 (figure 9-12) is very much
similar to the previous one only difference is
that this one is not symmetric to x axis. In
this loading condition MEAN set to non-zero
value. LC_5 is a very important loading for
the Chaboche model. Because it shows the
ratcheting if the material. In LC_5 (figure 913) MEAN is increased by the parameter
Figure 9-9 LC_6
MEANINC. LC_6 (figure 9-9)is a unique
conditon. It has two differrent cycling loading first with a zero mean and constant loand
and suddenly the mean is creased and also the amplitude is incresed. This condition is
made to simulted to rapid change in the yield surface.
Table 9-7 Loading condition parameters value
Parameters
LC_2
LC_3
LC_4
LC_5
LC_6
Amplitude (AMP)
0.003
0.003
0.003
0.003
0.003
0.005
Amplitude increment (AMPINC)
0
0.001
0.001
0
0
0
Mean (MEAN)
0
0
0.002
0
0
0.002
Mean increment (MEANINC)
0
0
0
0.001
0
0
Time increment (TINC)
0.2
0.2
0.2
0.2
0.2
0.2
Over all solution time (TFIN)
2
2
2
2
1.5
3
Figure 9-11 LC_3
Figure 9-10 LC_2
9-45
A Study of material non-linearity during deformation using FEM software
Figure 9-12 LC_4
Figure 9-13 LC_5
9.5 SOLUTION METHOD AND TEST DETAILS
We have used different king of solution to complete this project. First we see what are
types of simulation are available.
9.5.1
Static and Quasi-static analysis:
By the name static we can guess that it is time independent analysis. Time is not
considered a parameter. So, the velocity and acceleration is also not considers since that
are the function of time. Quasi-static load means the load is applied so slowly that the
structure deforms also very slowly (very low strain rate) and therefore the inertia force is
very small and can be ignored. We have used this method to solve the research based
problems.
[𝐾] ∙ {𝑥} = {𝐹}
9.5.2
Implicit dynamics
Implicit analysis requires a numerical solver to invert the stiffness matrix once or even
several times over the course of a load/time step. This matrix inversion is an expensive
operation, especially for large models. So this model no very efficient for our study. For
material nonlinearity we used implicit method
[𝑀]{𝑥̈ } + [𝐶]{𝑥̇ } + [𝐾]{𝑥} = {𝐹}
{𝑥} = [𝐾]−1 ({𝐹} − [𝑀]{𝑥̈ } − [𝐶]{𝑥̇ })
9-46
A Study of material non-linearity during deformation using FEM software
9.5.3
Explicit dynamics
In explicit dynamic analysis, nodal accelerations are solved directly (not iteratively) as
the inverse of the diagonal mass matrix times the net nodal force vector where net nodal
force includes contributions from exterior sources (body forces, applied pressure, contact,
etc.). Explicit analysis handles nonlinearities with relative ease as compared to implicit
analysis. This would include treatment of contact and material nonlinearities.
[𝑀]{𝑥̈ } + [𝐶]{𝑥̇ } + [𝐾]{𝑥} = {𝐹}
{𝑥̈ } = [𝑀]−1 ({𝐹} − [𝐶]{𝑥̇ } − [𝐾]{𝑥})
9.5.4
Test details
We have created series of test to simulate all the aspects of above methods, loading
condition, Material model. Which are mentioned in the table given below
Table 9-8 Test details
TEST NO
Non-linear Material model
Loading Condition
Solution method
TEST_1
BISO
LC_1
Quasi-static
TEST_2
MISO
LC_1
Quasi-static
TEST_3
MISO
LC_2
Quasi-static
TEST_4
BKIN
LC_2
Quasi-static
TEST_5
BKIN
LC_1
Quasi-static
TEST_6
BKIN
LC_3
Quasi-static
TEST_7
BKIN
LC_4
Quasi-static
TEST_8
BKIN
LC_5
Quasi-static
TEST_9
BKIN
LC_6
Quasi-static
TEST_10
MKIN
LC_1
Quasi-static
TEST_11
MKIN
LC_2
Quasi-static
TEST_12
MKIN
LC_3
Quasi-static
TEST_13
MKIN
LC_4
Quasi-static
TEST_14
MKIN
LC_5
Quasi-static
TEST_15
MKIN
LC_6
Quasi-static
9-47
A Study of material non-linearity during deformation using FEM software
TEST_16
CHABOCHE(C3=700)
LC_1
Quasi-static
TEST_17
CHABOCHE(C3=300)
LC_1
Quasi-static
TEST_18
CHABOCHE(C3=0)
LC_1
Quasi-static
TEST_19
CHABOCHE
LC_2
Quasi-static
TEST_20
CHABOCHE
LC_3
Quasi-static
TEST_21
CHABOCHE
LC_4
Quasi-static
TEST_22
CHABOCHE
LC_5
Quasi-static
TEST_23
CHABOCHE
LC_6
Quasi-static
TEST_24
BKIN
LC_1
Explicit dynamics
TEST_25
--
LC_1
Explicit dynamics
9.6 APPLICATION BASED STUDY
TEST_24 and TEST_25 are the application based study. They have done this study to test
the practical application of nonlinear material modelling in real life. We chose sheet metal
bending operation. The study is ANSYS workbench with explicit dynamics model. The
meshed geometry is shown below.
Figure 9-14 Meshed geometry for TEST_24 and TEST_25
9-48
A Study of material non-linearity during deformation using FEM software
RESULT AND DISCUSSIONS
Isotropic model is very easy
and
requires
much
lesser
computational power to solve.
However
isotropic
hardening
lacks Bauschinger effect. Since
the increase in yield surface, this
model is not convenient to show
in cyclic loading we have applied
a ramped loading and unloading
to show the Decrement from
plastic region. A Displacement of
Figure 10-1 Test 1 (loading condition 1)
3E-003 m is imposed and removed.
As explained before the bilinear model is very easy way for modelling plasticity but it
lacks accuracy. The bilinear material input is explained in the chapter and the figure
captures the Stress – Strain graph of the BISO model
Figure 10-3 Stress - strain
Figure 10-2 Test_ 1 Plastic strain over time
The figure 10-3 shows the total Strain vs Total Stress plot, this graph is very essential to
understand the complex behavior of plasticity. Here till the yield point the slope is
constant which is equal to modulus of elasticity after reaching the plastic region the slop
reduces and maintains the constant value, which is the property of the BISO model.
Figure 10-2 shows the plastic strain over time. At beginning there no plastic stain because
of elastic deformation and after the yield point, Plastic strain is directly proportional to
10-49
A Study of material non-linearity during deformation using FEM software
the displacement. During the reversal load the material reaches the elastic region which in
turn produces straight line denoting no increase in plastic strain.
Figure 10-4 Test_1 Strain energy along displacement
The plot of strain energy with the displacement figure 10-4 gives a parabolic curve which
can the strain energy equation.
𝑥1
𝑈 = ∫ 𝑃𝑑𝑥 =
0
1
𝑃𝑥
2 1 1
The above equation is to find Strain energy if we apply force as loading. If we have
displacement as a loading condition following formula will suit it much better.
𝑥1
𝑈 = ∫ 𝑘𝑥 𝑑𝑥 =
0
1
𝑘𝑥1 2
2
The above equation is a quadratic equation. Quadratic equation gives parabola when we
plot them. That explains why the stain energy curve for displacement gives a parabolic
result.
10-50
A Study of material non-linearity during deformation using FEM software
Unlike
bilinear
hardening,
MISO
isotropic
(multi
linear isotropic hardening)
requires True stress strain
curve. We need to list the
values of stress and strain
points in the TB,MISO
command. This model is
much more accurate than the
previous
nonlinear
Figure 10-5 test _ 2 MISO Strain vs Uy
because
plastic
of
its
region
which traces the actual
material property.
It is very important to add the true stress-train strain value. The figure 10-5 shows the
graph between strain and displacement, unlike the previous one the plastic region is
curved which is made possible by adding multiple points into the data. Again during
unloading it simply follows the slops of the initial graph, which is a practically proved
one.
MISO model is not very
effective
for
cyclic
plastic
loading but we have done an
analysis to satisfy our curiosity.
At a short glace it won’t make
any
sense
but
it
is
the
representation of the increase in
yield surface during subsequent
loading. In this model the
elastic region tends to increase
Figure 10-6 MISO with Cyclic loading
and the elastic energy absorbing
capacity also increase in each loading. Figure 10-7 shows the graph between strain and
displacement.
10-51
A Study of material non-linearity during deformation using FEM software
Figure 10-7 Test _3 strain vs displacement
Figure 10-8 test_3 strain energy over time
Figure 10-9 Strain energy plot for Test_3
10-52
A Study of material non-linearity during deformation using FEM software
Now we talk about kinematic hardening model. It is appropriate to discuss BKIN
model first. Similar to BISO model this is also easy to model and analysis. Engineers
choose BKIN over MKIN because of its simplicity but as always it lacks accuracy. As
BKIN model is for cyclic loading. We tested it in a various loading condition (LC_1 to
LC_6). For simplify the understanding of the BKIN model first study was made with
LC_1. On applying the ramped loading the material tends to deform elastically followed
by plastic deformation and elastic deformation by compression. Till the point of yield
strength during compress the material behave similar to BISO model. But here the
compressive yield occurs much sooner the previous model. Because during the kinematic
hardening the yield surface tends to shift, which demonstrates Bauschinger effect.
Figure 10-10 Comparison of BISO with BKIN
10-53
A Study of material non-linearity during deformation using FEM software
Figure 10-11 Displacement Load for TEST_4
Figure 10-12 TEST_4 stress vs strain
Figure 10-13 TEST_6 displacement plot
Figure 10-14 TEST_6 stress vs strain
Figure 10-15 Displacement load for TEST_8
Figure 10-16 TEST_8 stress vs strain
10-54
A Study of material non-linearity during deformation using FEM software
Fig 10-11 to 10-16 shows the different loading condition and its effect Strain – strain
curve on the BKIN model. In Fig-10-12 exactly shows the translation of the yield surface
if the yield surface enlarges then the stress – strain curve would be different like to fig 106. On every cycle the hysteresis loop follows the same path. Area under the curve gives
us the strain energy of loading. Fig-10-14 shows hysteresis loop for load with increased
amplitude with constant mean. Here the loop is angled. The loop is extending on each and
every cycle because of the increase displacement load. The increased loop area says it
need more energy on every cycle to plastically deform the material. The model takes
Bauschinger effect in a very simplest way. It net a very accurate way for describing the
kinematic hardening. Because it shows only lesser value of energy dissipated then the
actual material dissipation. This model also cannot describe the ratcheting or memory
effect. The loop is angled which does not happen in real material. Designers use this
model because of its simplicity as explained before. As shown it does not support
Figure 10-17 TEST_4 strain energy over time
ratcheting effect. The curve again is angles and not very accurate. The figure also
explains very well that the yield surface shifts in same direction during all of the loading.
Here energy absorbed will be equally increasing in all the loops.
10-55
A Study of material non-linearity during deformation using FEM software
The MKIN model is very similar to MISO in terms of inputting but there are lots of
variations in output of both the model. MKIN is a extended form of the BKIN model. It is
similar to the piecewise linear kinematic hardening rule. This material modelling is also
known as Besseling model. Here the plastic stress strain curve is loaded to define the
plastic deformation. This model approximates the experimental results better than the
BKIN model. Still energy dissipated will be less than the experimental study.
Figure 10-18 TEST_10 Stress -Strain
The figure 10-18 shows the stress – strain on a ramped load we can notice that
during the reversed loading the yielding occurs much sooner than
MISO model also explains the yield surface translation. During yielding the curve is
smooth and changing which is achieved by the inputting 20 data points. Again the area
under this curve gives the strain energy of the load. When we check the plastic strain
increment of both the model we can easily identify them. Over all plastic stain increment
is much less than the MKIN model. This is because of the inaccurate input data for the
BKIN model. This also explains why the strain energy dissipation is more for the MKIN
model.
10-56
A Study of material non-linearity during deformation using FEM software
Figure 10-19 TEST_11 Plastic strain
increment (MKIN)
Figure 10-20 TEST_4 Plastic strain
increment (BKIN)
During constant amplitude cyclic loading the results matches with BKIN model in
most aspects except the curved line of the stress strain curve during the plastic
deformation. When we compare the increased amplitude load we can find much
difference in the solution. First of all the curve is not angled and increment is not also in
angles and it agrees the experiments results much better.
Figure 10-22 TEST_6 Elastic stress
Figure 10-21 TEST_12 Elastic stress
Even the elastic strain over time can also easily differentiate the BKIN and MKIN
models. In BKIN the lines are straight as the time passes the plastic region tends to
increase. In MKIN model it is can be seen as curved region.
10-57
A Study of material non-linearity during deformation using FEM software
Figure 10-24 TEST_12 Total Stress and Total Strain
During the load with
increases amplitude the true
functionality of this model
can be understood. the fig
10-24
shows the property
of the stress strain curve
during incresed amplitude f
load. Wich clearly shows
the translation of the yield
surface unlike in MISO
model. On each and every
cyclic
energy
Figure 10-23 TEST_12 Strain energy
dissipation
increases on each cycle because of the incresed load.
10-58
A Study of material non-linearity during deformation using FEM software
The MKIN model appears to be strange
when we apply asymmetric displacement
load. The elastic stain values fluctuate with
a constant range but the plastic strain
increases
as
the
load
increases
in
asymmetric way. The overall strain also
follows the same path as the plastic strain.
As we can notice that the mean of the
plastic and total strain is not equal to zero.
But the mean of elastic strain equal to
Figure 10-25 TEST_12 LC_4
zero.
Figure 10-26 TEST _ 12 plastic strain, elastic strain, total strain
10-59
A Study of material non-linearity during deformation using FEM software
In the fifth loading condition (LC_5)
the load is cyclic. For time up to 2 sec
it maintains smaller amplitude value
after that both the amplitude and
mean of the load in increased to test
the condition. The results are shown.
First the hysteresis loop stays in a
common place as the load shifts the
loop also moves to new place and
Figure 10-27TEST_14 LC_5
continue to loop at the location. Now we
analysis the strain energy curve. First the energy increase follows the same oscillation as
the load but the energy curve is second order curve. As the load shifts the energy as
increases rapidly with larger magnitude which shows larger energy dissipation.
Figure 10-28 TEST_14 plastic strain
10-60
A Study of material non-linearity during deformation using FEM software
Figure 10-29 TEST_14 Strain energy
In most aspects the multilinker and nonlinear hardening rule are same. In
nonlinear analysis we use Besseling function where we don’t need back stress but in the
nonlinear analysis there is an evolution term which gives the nonlinear property to this
yield function. This model has two important material inputs Ci and γ. Where Ci is the
initial tangent modulus of the material. The second parameter γ controls the rate at which
the hardening module decreases with increase in plastic strain. If γ is set to zero the
nonlinear analysis act like a linear model similar to the BKIN model. The Chaboche
model was proposed by decomposing the Armstrong and Frederick. The Chaboche model
can combine up to 5 nonlinear model in single analysis. The figure 10-30 shows the
importance the parameter γ. It is already said that when γ = 0 the model behaves like a
linear model which is BKIN model. When we have a non-zero value for γ the ratcheting
property can be seen and it defines the curvature for the stress – strain curve.
10-61
A Study of material non-linearity during deformation using FEM software
Figure 10-30 Chaboche model with different γ value
From the graph we can notice one
thing the parameter γ not only
affect the slop it also indirectly
affect the energy dissipated on the
loading we can clear identify by
the larger area formed by the
curve having higher value of γ.
When we have the cyclic loading
for the Chaboche model clearly is
different from the MKIN. In
Figure 10-31 TEST_19 Plastic strain
MKIN the curve will overlap but in
Chaboche due to the ratcheting. It tends to shift on each cycle this because of the back
stress.
10-62
A Study of material non-linearity during deformation using FEM software
Clearly MKIN and Chaboche
model shows very loss difference
for LC_3 and LC_4.
When we have increased mean
in load we can get the ratcheting
on the material. The figure 10-33
shows the difference between the
results of material with BKIN
and CHABOCHE models. The
Figure 10-32 Elastic strain and Displacement for
TEST_20
graph clear tells us there are lots
of variation in both models.
BKIN model clearly doesn’t
support ratcheting. The whole curve is angled and there is no strain accumulation. But on
the Chaboche model the strain accumulates. When we input three or more material
models in Chaboche it even supports Shakedown.
Figure 10-33 Comparison of BKIN and CHABOCHE for LC_5
10-63
A Study of material non-linearity during deformation using FEM software
TEST_24 is done to show the practical application and functionality for of modelling
material nonlinearity. Sheet metal bending operation is analyzed using ansys workbench.
Here the sheet is a deformable body, the punch and die are rigid bodies. The punch is
displaced over a distance to bend the sheet metal. Figure 10-34 shows the partially
deformed sheet metal and the plastic strain starts to form in the middle of the plate.
Figure 10-34 partially deformed sheet metal (TEST_24)
The figure 10-35 shows a completely bend sheet metal. Here the punch is drawn back still
the sheet metal retain its deformed shape, which shows plastic deformation. This result
cannot be achieved by linear material models.
Figure 10-35 plastically deformed sheet metal (TEST_24)
10-64
A Study of material non-linearity during deformation using FEM software
Figure 10-36 Energy Summery (TEST_24) with plasticity model
Figure 10-37 Energy summery (TEST_25) without plasticity model
10-65
A Study of material non-linearity during deformation using FEM software
When we analyze the overall energy of the system we can easily understand the plastic
deformation. The figure 10-36 Shows the energy summery of the test. First the internal
energy begins with zero value as the material tests to bend the system begins the store
more energy as a form of strain mean while the energy is also dissipates as plastic
deformation. At the end when we unload the object the internal energy remain in the peak
value. Which shows the energy is permanently stored in the material. Which is a plastic
deformation.
The nest graph shows the energy summery when there is no nonlinear material property is
defined (figure 10-37). Here the energy tends to peak very similar to the previous test but
when we unload the system the energy returns back to the original state. That shows
elastic deformation. To exactly predict the deformation and failure of the material which
under goes plastic deformation it is very essential to define a non-linear material model in
it.
10-66
A Study of material non-linearity during deformation using FEM software
CONCLUSION
This study shows the basic nonlinear material models. There are many material models
are present and are created every year. The report shows basic differences between
models as follows. The BISO is very simply and easy to model isotropic hardening. But it
lacks accuracy. In other hand MISO is more accurate because it traces the deformation
along the strain- strain curve. Both the models cannot predict Bauschinger effect and
strain softening. These studies are done by simple ramped loading and unloading. The
BKIN model includes Bauschinger effect and it is a kinematic hardening model. Similar
to BISO, BKIN is easy but inaccurate. Multi linear kinematic hardening is different than
BKIN, and it uses the Besseling model. It characterizes multi linear behavior as a series
of elasto‐perfectly plastic ‘sub volumes,’ each of which yields at different points, so no
back stress is used. MKIN model cannot show ratcheting and Shakedown property of the
material. Armstrong and Frederick developed the nonlinear material model. Chaboche
model is the advanced form Armstrong and Frederick model. It used up to five material
model together. Chaboche model is more accurate in predicting the solution. It is not very
popular because it is hard to calibrate the material with the Chaboche material parameters.
Chaboche model is good in predicting ratcheting and shakedown effect. This paper
doesn’t discuss about this. One more aspect of this paper is also to discuss the practical
application of the nonlinear material models. A sheet metal bending operation is
discussed in brief. The paper also describes about the energy interaction during the sheet
bending with and without nonlinear material models. Though this paper described most
part of the rate-independent plastic model. the paper fails to describe about the rate
dependent plasticity, creep, gasket model, hyper elasticity etc. more steady are to be done.
11-67
A Study of material non-linearity during deformation using FEM software
REFERENCES
[Online]. - http://homepages.engineering.auckland.ac.nz.
A Comparative Analysis of Linear and Nonlinear Kineamtic Hardening Rules in
Computational Elastoplasticity [Journal] / auth. Angelis Fabio De // Technische
Mechanik. - 2012. - pp. 164-173.
Advanced Mechanics of Solids [Book] / auth. Srinath L S.
An Application of Finte Element Method and Design of Experiments in the
Optimization of Sheet Metal Blanking Process [Journal] / auth. Al-Momani Emad and
Ibrahim Rawabdeh // Jordan Journal of Mechanical and Industrial Engineering. - mar
2008. - Vol. 2. - pp. 53-63.
Application of the Finite Element method in cold forging process [Journal] / auth.
Cristina Maria Oliveira Lima Roque and Sergio Tonini Button // J. of teh Braz. Soc,
Mechanical Sciences. - 2000. - 2 : Vol. XXII. - pp. 189--202.
Bilinear Isotropic hardening behavior [Report] / auth. Ranganathan Raghavendar,
Bradly Verdant and Ranny Zhao.
Chaboche Nonlinear Kinematic Hardening model [Journal] / auth. Imaoka Sheldon. 2008.
Characterization of Material Parameters [Conference] / auth. S.M.Humayun Kabir,
Tae-In Yeo and Sang-Ho Kim // World Congress on Engineering. - London : [s.n.], 2009.
Computational Inelasticity [Book] / auth. J.C.Simo and T.J.R.Hughes. - [s.l.] : Springer.
Consideration of material behaviour in the numerical solution of cyclic thermal an
mechanical loading using kinematic hardening [Conference] / auth. Rust Wilhelm,
Clemens Groth and Günter Müller // ANSYS Conference. - Houston, PA : [s.n.], 1994.
Elaticity in Engineering Mechanics [Book Section] / auth. Arthur P. Boresi, Ken P.
Chong and James D. Lee.
Introduction to Computational Plasticity [Book] / auth. Fionn Dunne and Nik
Petrinic. - [s.l.] : Oxford University Press.
LS-DYNA Support [Online]. - http://www.dynasupport.com/.
68
A Study of material non-linearity during deformation using FEM software
Mechanical APDL help file [Book] / auth. Ansys.
Non-linear Finite Element Modeling of the Titanium [Journal] / auth. Alexey I.
Borovkov and Denis V. Shevchenko .
Nonlinear Static - 1D Plasticity- Various Forms of Isotropic Hardening [Journal] /
auth. L.Yaw Louie. - 2012.
Nonlonear analysis using regular Yield surface [Conference] / auth. G.Asteris
Panagiotis // 15th ASCE Engineering Mechanics Conference. - New York : [s.n.], 2002.
NUMERICAL
MODELING
OF
ELASTO-VISCOPLASTIC
CHABOCHE
CONSTITUTIVE EQUATIONS USING MSC.MARC [Journal] / auth. Andrzej
Ambroziak.
Numerical Simulation of Sheet Metal Forming for High Strenght Steels [Report] /
auth. Arwidson Claes.
Parameter-refreshed Chaboche model [Journal] / auth. Budaházy Viktor and László
Dunai // periodica polytechnica. - 2015.
Researchgate [Online]. - http://www.researchgate.net/.
Some Aspects for the Simulation of a Non-Linear Problem with Plasticity and
Contact [Report] / auth. Gaertner Eduardo Luís and Marcos Giovani Dropa de Bortoli.
Theory of Plasticity [Book] / auth. J.Chakrabarty.
Wikipedia [Online]. - www.en.wikipedia.com.
69
A Study of material non-linearity during deformation using FEM software
APPENDIX I
APDL code for TEST_23 with Chaboche material model
/CLEAR !****************TITLE*********************
/TITLE, TEST_23_CHOB_LC_6
/UNITS, SI
/PREP7
!**************DEFINING PARAMETERS***********
A=0.02
L=0.2
!***************ELEMENT DEFINITION**************
ET,1,LINK180
!***********REAL CONTRAINS AND GEOMENTRY**************
R,1,A
N,1,
N,2,,L
E,1,2
!*******************MATERIAL INPUT**************
MP,EX,1,194499E6
MP,PRXY,1,0.3
MP,DENS,7833
TB,CHAB,1
! CHABOCHE TABLE
TBDATA,1,1.12E9,1.46E11,500.8754
D,1,ALL,0
!******************SOLUTION ROUTINE
FINISH
/SOL
!ANTYPE,TRANS
OUTRES,ALL,ALL
NSUB,30,300,10
AMP=0.003
70
A Study of material non-linearity during deformation using FEM software
!*********************LC__6__******************
AMPINC=0
MEAN=0
MEANINC=0
TINC=0.2
TFIN=1.5
*DO,T,0.1,TFIN,TINC
D,2,UY,MEAN+(AMP*0.5)
TIME,T
SOLVE
D,2,UY,MEAN-(AMP*0.5)
TIME,T+0.1
SOLVE
AMP=AMP+AMPINC
MEAN=MEAN+MEANINC
*ENDDO
AMP=0.005
AMPINC=0
MEAN=0.002
MEANINC=0
TINC=0.2
TFIN=3
*DO,T,1.5,TFIN,TINC
D,2,UY,MEAN+(AMP*0.5)
TIME,T
SOLVE
D,2,UY,MEAN-(AMP*0.5)
TIME,T+0.1
SOLVE
AMP=AMP+AMPINC
MEAN=MEAN+MEANINC
*ENDDO
71
View publication stats