Step
00
Table of Contents
1. Introduction on Nonlinear Analysis
2. Presentation of procedure and options in Nonlinear Analysis
3. Nonlinear Geometry
4. Nonlinear material(1)-Elastoplasticity
5. Nonlinear material(2)-Hyperelasticity
6. Nonlinear Contacts
7. Conclusion and advice for a better use of Nonlinear Analysis
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Nonlinear Static Analysis
1
Step
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Nonlinear Static Analysis
2
Step
Introduction on Nonlinear Analysis
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Nonlinear Static Analysis
3
Step
01
What is Nonlinear Analysis ?
Most of the physical phenomena are nonlinear
 When nonlineary can be neglected, Analysis can be performed using linear Analysis
(more effective)
 When nonlineary cannot be neglected, Nonlinear analysis should be performed
What is it???
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Nonlinear Static Analysis
4
Step
02
What is Nonlinear Analysis ?
3 causes of Nonlinearity
Geometric Nonlinearity
Material Nonlinearity
When an object is subjected
When the relation between
When the contact of an
to excessive deformation or
Stress and Strain isn’t elastic,
object with another is
the load direction
Nonlinear Elasto-Plastic
changing.
ischanging
Theory has to be used.
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Contact Nonlinearity
Nonlinear Static Analysis
5
Step
03
What is Nonlinear Analysis ?
What is linear Analysis?
F
K
Following Hook’s Law
Linear relation
F=Kδ
K
δ
u(=δ)
Result of linear analysis?
F
A force of 1kg create a deformation of 1mm
A force of 10kg create a deformation of 10mm
As K is always constant, F value can be obtained easily
 When Stiffness is constant, Hook’s Law can be linearized.
 In most structures, Constant Stiffness is taken as an assumption.
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Why??
Nonlinear Static Analysis
6
Step
04
Linear Analysis VS Nonlinear Analysis
What is Nonlinear Analysis?
F
F
K3
K
K2
K=Constant
(a) Linear Analysis
K≠Constant
K1
u
(b) Nonlinear Analysis
u
 When Load increases, stiffness changes.
 The relation Load-displacement is a nonlinear function.
 In linear Analysis, K is constant, and so displacement U can be obtained simply if Load F is known.
In other words, if the slope of the curve is known, behavior of the solid can be determined with
only one calculation. In Nonlinear Analysis, the slope of the curve is always changing, thus a
new calculation with updated slope value and several calculation steps are required.
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Nonlinear Static Analysis
7
Step
05
Linear Analysis VS Nonlinear Analysis
Nonlinear Analysis Examples
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Nonlinear Static Analysis
8
Step
06
Reasons to use Nonlinear Analysis
In which circumstances is nonlinear analysis required?
When more accurate data are necessary
When position where contact happens is changing
When large deformation is susceptible to happen
When stress level approaches yielding point of material
In order to determine precisely buckling load
When an unusual big displacement is observed
When hyperelastic material like rubber is used
When deformation gradually increases due to a constant load which is
applied for a long time
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Nonlinear Static Analysis
9
Step
07
Numerical Analysis s Methodology of Nonlinear Analysis
Numerical Analysis Methodology of Nonlinear Analysis
Linear Analysis
F
 In linear analysis, response of a structure submitted to a
Linear
load can be determined using the linear equation F=KU.
Nonlinear
Nonlinear Analysis
 F=H(U) is not a linear equation, so F≠KU
 In nonlinear analysis, the load can be divided in several
load steps and the equation ΔF=[Kx][ΔU] can be used
for each load step to find displacement.
u

When applying a load in nonlinear analysis, this load can de divided in
smaller load steps and then displacement can be calculated with several
iterations.
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Nonlinear Static Analysis
10
Step
08
Numerical Analysis s Methodology of Nonlinear Analysis
Numerical Analysis Methodology of Nonlinear Analysis
 Incremental Method
F
 Newton-Raphson
F
error occurs
ΔF
ΔF
ΔF
ΔF
Improvement
ΔF
u
Error is reduced by
a supplementary
iteration at each
load step
ΔF
u
 Stiffness is updated at each load step.
 Stiffness is updated at each load step.
 Problem: error accumulated at each load step
 Error is reduced by adding an internal stiffness
will create some big error at the end of the
iteration for each load step.
analysis.
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Nonlinear Static Analysis
11
Step
09
Numerical Analysis s Methodology of Nonlinear Analysis
Newton-Raphson Method
F
[KT][ΔU] = {F1} - {FIx}
iteration
 In linear analysis, when F(Ext load) = F(Int load) and
F2
solution can be found (convergence) then it
verify the equation F=KU.
 In nonlinear analysis, when {F1}-{FIx} is comprised
ΔF
between a certain error tolerance, solution can
F1
ΔF
FIx
be determined and load step converge.
error
error
error
 There are three convergence criteria based on Load,
displacement and work.
Error is reduced progressively
F
Load
displacement
u
surface = work
u
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Nonlinear Static Analysis
12
Step
10
Numerical Analysis s Methodology of Nonlinear Analysis
Convergence Criteria / Error Tolerance
F
 Usually, two convergence criteria are
combined together to obtain satisfying
②Load
results.
①displacement ① displacement+③work or ② load + ③ work
(used a lot in practice)
 Usually, displacement criteria is used for
systems which are not very sensible to
Surface = ③Work
F
load variation.
Use of load criteria
u
Use of displacement criteria
ΔF
In this case, tolerance criteria is converging
with ΔF, but is not regarding to Δu.
Δu
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u
Nonlinear Static Analysis
13
Step
11
Numerical Analysis s Methodology of Nonlinear Analysis
Numerical Analysis Methodology of Nonlinear Analysis
 Full Newton - Raphson  Modified Newton - Raphson  Initial Stiffness Method
F
F
F
u
u
u

Update stiffness each time

Update Stiffness at each Load step

Keep value of initial Stiffness

It takes time to calculate stiffness

Number of increments is increased

Number of increments is increased

If there is no convergence problem,

If there is no convergence problem, this
this method is faster than simple
method is faster than simple Newton-
Newton-Raphson
Raphson

In midas NFX, results can be obtained using any of these iterative methods
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Nonlinear Static Analysis
14
Step
12
Numerical Analysis s Methodology of Nonlinear Analysis
What is Buckling?

All the structures have a stable equilibrium state.

When a load is applied and it create a large deformation of the structure, it is said to be in an unstable equilibrium state.

Such instability is not due to the material but to the geometrical shape of the structure.

By performing buckling analysis, it is possible to determine the buckling load (Maximum Load at which buckling will
occur) and the buckling Mode (Deformed Shape due to buckling) of the structure.
Axial Load

When a structure is submitted to an external
load, the equilibrium state external load=
Neutral equilibrium
F
state
internal load simply doesn’t apply. Structure is
said to be in a stable equilibrium state.
Buckling Load(Fcr)
Unstable
equilibrium state
u
Stable
equilibrium state
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Linear Buckling Analysis vs Nonlinear buckling Analysis?
Nonlinear Static Analysis
15
Step
13
Numerical Analysis s Methodology of Nonlinear Analysis
Linear Buckling VS Nonlinear Buckling
F
Limitations of Linear Buckling Analysis
Linear Buckling
 It can be dangerous if buckling load is overestimated.
Nonlinear Buckling
It cannot determine behavior after buckling
 Material is supposed elastic and so nonlinear material
Real structure
behavior
u
Advantage of Nonlinear Buckling Analysis
※ In case of nonlinear buckling
F
behavior is not considered.
 Possibility to calculate the real buckling load.
From this point, tangential
stiffness is either 0 or negative
-> No convergence
 Nonlinear material behavior can be considered.
The use of Newton-Raphson Method to estimate structural behavior
Snap-through
after buckling is very difficult. In this case, another method has to
be used.
Arc-length, Displacement control Method
u
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Nonlinear Static Analysis
16
Step
14
Numerical Analysis s Methodology of Nonlinear Analysis
Arc-length Method
F
F

ΔP
u
u
 Calculation of results of a load step could be negative or null Stiffness .
 Usually used when Nonlinear Buckling occurs in snap-through shape.
 midas NFX provides Crisfield(CRIS), Riks(RIKS), Modified Riks(MRIKS) Methods.
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Nonlinear Static Analysis
17
Step
15
Numerical Analysis s Methodology of Nonlinear Analysis
Arc-length Method
• Max No. of Increments
 In function of the nonlinearity, Arc-Length Method can converge
faster than the number of increments, but it can also diverge. In order
to account for this case, sufficient number of increments has to be set.
• Load contribution scale factor
 When load contribution Scale Factor is “1”, Load and displacement
are unknown. When it is equal to “0”, only displacement is unknown.
Default parameter is “0”.
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Nonlinear Static Analysis
18
Step
16
Numerical Analysis s Methodology of Nonlinear Analysis
Displacement Control Method
 In most of the problems, structural behavior is determined using Load control.
 In case of Nonlinear Buckling, it is difficult to determine the load after buckling, so it is
better to use displacement control.
 Applied Load (Buckling load) can be determined by the constraint force.
Force (Constraint)
Imposed displacement
1.2
1
0.8
0.6
0.4
0.2
0
0
Constraint
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50
100
150
200
250
300
Displacement
Nonlinear Static Analysis
19
Step
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Nonlinear Static Analysis
20
Step
Presentation of procedure and
options in Nonlinear Analysis
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Nonlinear Static Analysis
21
Step
01
Presentation of procedure and options in Nonlinear Analysis
Analysis Procedure
 Linear Analysis Procedure
 Nonlinear Analysis procedure
Preparation of Geometric
Model
Preparation of Geometric
Model
Material Assignment
Material Assignment
Apply Element Properties
Apply Element Properties
Mesh Preparation
Mesh Preparation
Assignment of Boundary
Conditions
Assignment of Boundary
Conditions
Insert Loading Condition
Insert Loading Condition
Create Analysis Case and
Perform Analysis
Create Analysis Case and
Perform Analysis
Verify the Results
Verify the Results
Material Nonlinearity
Assignment
Nonlinear Contact
Assignment
Activate Geometric
Nonlinearity Option

As the process of linear and nonlinear Analysis are the same, it is a good idea to Train first by performing linear analysis
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Nonlinear Static Analysis
22
Step
02
Presentation of procedure and options in Nonlinear Analysis
Analysis Option
In nonlinear analysis, different methods have to be used in order to find a different solution from linear analysis.
After creating the analysis case, diverse options can be selected.
1. Method to Create Analysis Case
2. Support Geometric Nonlinearity
3. Number of Increments, set
convergence criteria
4. Intermediate Output Request
5. Advanced Nonlinear Parameters
6. Use of Subcases
7. Use of Restart feature
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Nonlinear
Analysis
Options
Nonlinear Static Analysis
23
Step
03
Presentation of procedure and options in Nonlinear Analysis
Method to Create Analysis Case
Analysis & Results >> Analysis case >> General
1
or
2
1
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Nonlinear Static Analysis
24
Step
04
Presentation of procedure and options in Nonlinear Analysis
Method to Consider Geometric Nonlinearity
2
Check it to consider Geometric Nonlinearity
1
Click on Nonlinear Static (Required)
and select Subcase control button

When it is not obvious to consider large deformation, the best
way is to check it first and then evaluate the deformation results.
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Nonlinear Static Analysis
25
Step
05
Presentation of procedure and options in Nonlinear Analysis
Number of Increments, Convergence
Criteria Settings
F
• Determine number of
increments (number of ΔF)
ΔF
ΔF
ΔF
Ex) For a Load of 100N divided in 20
increments, Load will be 5N for each
step.
Select a maximum of 2 convergence criteria
Usually Load + Work Convergence criteria are
used
• Usually, displacement criteria is used for systems
which are not very sensible to load variation.
•
•
With 3 Convergence Criteria, Convergence is
quite difficult to obtain.
With 1 convergence criteria, Convergence is
easy but it is difficult to obtain rational results.
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u
F
Load
Disp.
Surface = Work
u
Nonlinear Static Analysis
26
Step
06
Presentation of procedure and options in Nonlinear Analysis
Intermediate Output Request
F
ΔF/2 = bisecting Increment
ΔF = Load Step
Bisecting increment: When it
cannot converge in one increment,
the increment is divided in 2 and
calculation is performed on each.
u
• Every Increment : Output of all increments results (including bisected
increments).
• Every Non-bisecting Increment : Output of all increments results except
bisected increments.
• Last Increment : Export only the result from the last increment.
• Every N Non-bisecting Increment : Export the results every N Load step.
Nonlinear analysis doesn’t provide only 1 result like linear analysis but
provide a result for every increment (even for bisected increments). In
order to set the appropriate output request , Intermediate output
request has to be chosen in the analysis control window.
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Nonlinear Static Analysis
27
Step
07
Presentation of procedure and options in Nonlinear Analysis
Advance Nonlinear Parameters - 1
 If you Check off “Use Default Settings”, you can set manually
the parameters for the Stiffness update scheme
 Stiffness update scheme can be changed. If
you use default settings, Midas NFX will choose
automatically the method to use to update the
stiffness.
 SEMI : Stiffness Update is done after 1 iteration to take account
of new load on each load increment.
 ITER : Stiffness will be updated for each step.
Number of iterations before Stiffness Update = 1 : NewtonRaphson is performed.
Number of iterations before Stiffness Update = Max No. of
Iterations per Increment : Newton-Raphson is performed.
Number of iterations before Stiffness Update > Max No. of
Iterations per Increment : Stiffness is not updated
 When Bisection happens, it is due to the fact that either real
Number of iteration per increment is higher than the Max No.
of Iterations per Increment or it is because solution diverges.
The default setting is "5 times".
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Nonlinear Static Analysis
28
Step
08
Presentation of procedure and options in Nonlinear Analysis
Advance Nonlinear Parameters - 2
• Terminate Analysis on failed convergence
 When convergence fail, analysis will be terminated. If this
option is not checked, analysis will continue even if it doesn't
converge.
• Max No. of iterations per increment
 It sets the maximum number of iterations at each increment. If
it doesn’t converge after this number of increments, the load step
will be bisected and analysis will be performed again.
Ex: if 10N Load doesn’t converge, analysis will be performed again
on a 5N bisected load step.
• Max. Bisection Level
 It sets the maximum number of bisection possible of 1 load step.
Ex) If the Maximum bisection level is 5, a load step of 10N can be
divided 5 times : 5N, 2.5N, 1.25N, 0.625N, 0.3125N
• Enable Line Search
 If the solution of the nonlinear analysis has the particularity to
converge with some oscillations, line search method can help to
get convergence.
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Nonlinear Static Analysis
29
Step
09
Presentation of procedure and options in Nonlinear Analysis
Method to use Subcases (Load Step) -1
 Subcases are sets composed of Loads and boundary conditions applied to the Analysis problem.
 Each Analysis Subcase results are linked with the results of the previous Subcase.
Analysis & Results >> Analysis Case >> General
Subcase Setting
• Boundary conditions, Loads and Contacts can be
[ << ] : Inactivate all the Sets
assigned to each Subcases by Drag-and-Drop.
[ >> ] : Activate all conditions for all
sets in all subcases.
 Subcase Control/Subcase Output Results
have to be defined for each Subcase.
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Nonlinear Static Analysis
30
Step
10
Presentation of procedure and options in Nonlinear Analysis
Method to use Subcases (Load Step) -2
F
Load2
Subcase ②
Load1
Subcase ①
Subcase ③
Time (case)
Subcase① : Load1 Applied
Subcase② : Load 2 is applied in addition to Load1
Subcase③ : All the Loads are removed
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Nonlinear Static Analysis
31
Step
11
Presentation of procedure and options in Nonlinear Analysis
Method to use Subcases (Load Step) -3
Example1
1
2
Example2
1
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2
Nonlinear Static Analysis
32
Step
12
Presentation of procedure and options in Nonlinear Analysis
Method to use Restart feature-1
When the solution of the nonlinear analysis is submitted to highly nonlinear environment,
convergence becomes difficult and it may happen that the analysis stop before the end.
In this case, convergence can be obtained by changing the analysis parameters (number of increments,
convergence criteria,…), but analysis has to be performed again and it can be a very time-consuming
process.
Use of restart function to begin the analysis again from the point where it stopped
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Nonlinear Static Analysis
33
Step
13
Presentation of procedure and options in Nonlinear Analysis
Method to use Restart feature-2
1
2
4
5
3
6
During this 3 subcases, if analysis didn’t
converge after the 2nd subcase, analysis
4
have to be run only from the 2nd subcase.
If 3 subcases exists, 3 restart files will be
created.
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Nonlinear Static Analysis
34
Step
14
Analysis method of Nonlinear Analysis Results
Equivalent Stress
Uncheck Nodal Average
to see Equivalent stress
results.
Right click
5
1
2
3
Check SOLID STRS Equivalent stress

4
When a nonlinear analysis using Material nonlinearity is performed
(Hyper elastic model), Equivalent stresses must be checked. When
material nonlinearity is not considered, von-mises stresses have to
be checked.
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Nonlinear Static Analysis
35
Step
15
Analysis method of Nonlinear Analysis Results
Effective Plastic Strain
3
Right click
1
2
4
Check SOLID STRS EFFECTIVE PLASTIC

4
When equivalent stresses are superior to the yield strength
of the material, effective plastic strain are created.
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Nonlinear Static Analysis
36
Step
Geometric Nonlinearity
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Nonlinear Static Analysis
37
Step
01
Geometric Nonlinearity
Geometric Nonlinearity
 Occurrence of large displacement/large rotation in the structure
 Occurrence of large strains
 Excessive deformation increases, regardless of the material properties and the stiffness
changes
 Dynamic loads can be applied (direction of the load is changing in function of the
structural deformation)
M
linear
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M
nonlinear
Nonlinear Static Analysis
38
Step
02
Analysis Theory of Geometric Nonlinearity
Large Displacement/ Large Rotation
When the load applied to a structure create some large deformation or rotation, the
stiffness of the Structure changes. If the elements rotate, the stiffness of the total structure
changes.
Supposing two Mesh sets submitted to different axial loads:
Horizontal stiffness
(a)
(b) horizontal and vertical stiffness
* Usually, when Geometric nonlinearity is considered, it means
that large deformation, large rotation and large strain are
present, but it is important to note that large
deformation/rotation don’t always create large strain!
Large displacement
Large rotation
ex)
Large Strain
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Nonlinear Static Analysis
39
Step
03
Analysis Theory of Geometric Nonlinearity
Large Strain
When an object submitted to loads is deformed and when the surface or the area of the
elements change too much, the stiffness of the object will also change.
In the examples below, the stiffness along the axial load will progressively change:
(a)

(b)
When Material Nonlinearity is considered, usually large displacement is also
present. This is why it is better to consider the geometric linearity each time
stress-strain curve is nonlinear.
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Nonlinear Static Analysis
40
Step
04
Analysis Theory of Geometric Nonlinearity
Follower force
 Definition: Force which can change direction and application position according to
structural deformation.
* Why is Follower force nonlinear ?
The force direction determines structure’s deformation
 Then, the structure deformation determines force direction
Thrust Force=W
(a)
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(b)
Nonlinear Static Analysis
41
Step
05
Analysis Theory of Geometric Nonlinearity
Follower force
 Apply follower load in midas NFX:
1
2
※ Starting node Direction of the load
kept.
3
4
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Nonlinear Static Analysis
42
Step
06
Geometric Nonlinearity
Analysis method for geometric nonlinearity
 When the mesh deformation is too large, some error can happen, this is why it is
better to think about the mesh size according to the deformation.
 Analysis convergence is more easy for 1st order element rather than for 2nd order
element.
 If the solution of the nonlinear analysis has the particularity to converge with some
oscillations, line search method can help to get convergence.
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Nonlinear Static Analysis
43
Step
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Nonlinear Static Analysis
44
Step
Material Nonlinearity(1)
– Elastoplasticity
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Nonlinear Static Analysis
45
Step
01
Material Nonlinearity
Material Nonlinearity
 In linear Analysis, Deformation of material is considered elastic
 In order to consider plastic deformation of materials, Nonlinear analysis
with nonlinear material have to be used.
Types of nonlinear material models
s
s
s
sY
E
e
Elasto-plastic model
e
Hyperelastic (Nonlinear elastic) model
Creep model
time
 It is important to select the right material model
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Nonlinear Static Analysis
46
Step
02
Elastoplastic model
Properties of Elasto Plastic Model
σ
 Uniaxial tensile test stress-strain graph
Ultima
te
stress
Yield
stress
rupture
 Have to Consider the characteristics
due to cyclic loading(Stress-strain
behavior under repeated loading
alters the properties of the material)
Loading
Unloading
Elastic
Module
perfect deformati Necking
on
plasticity
hardening
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ε
* If there is no cyclic loading, is it necessary
to consider the hardening model?
Nonlinear Static Analysis
47
Step
03
Elastoplastic model– Yield criterion
Yield criterion
 TRESCA Model (maximum shear stress theory)  Von-Mises Model (Torsional energy theory)
maximum shear stress > Simple tensile test yield
value /2
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Tri-axial Stress State shape deformation energy >
deformation energy of Simple tensile test yield value
Nonlinear Static Analysis
48
Step
04
Elastoplastic model– Yield criterion
Shape deformation energy
In triaxial stress state, the cause of yielding is the torsion which causes
shape deformation, it is not the isotropic volumetric pressure.
y
y
x
z
=
y
x
z
Tri-axial stress state
Isotropic volumetric pressure
Doesn’t cause yielding
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+
x
z
Shape deformation occurring due to torsion
Cause yielding
Nonlinear Static Analysis
49
Step
05
Elastoplastic model– Hardening model
Hardening model
 Isotropic hardening model
 Kinematic hardening model
σ
σ
sY
2sY
ε
ε
sY
 Yield stress increase of the
same ration for tension and
compression
 Yield stress domain is
constant
 Plastic deformation moves
along with the center of yield
stress domain
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Nonlinear Static Analysis
50
Step
06
Elastoplastic model– Hardening model
3D stress hardening model
Hardening model for 3D stress state stress domain is represented by a surface
 Isotropic Hardening model
σ2
 Kinematic Hardening model
σ2
σ1
Initial yield surface
 Surface of the stress domain
increases
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σ1
Initial yield surface
 Center of stress domain
surface moves
(surface area is maintained)
Nonlinear Static Analysis
51
Step
07
Elastoplastic model– Hardening model
Bauschinger effect
The Bauschinger effect refers to a property of materials where the material's stress/strain characteristics ch
ange as a result of the microscopic stress distribution of the material. For example, an increase in tensile yi
eld strength occurs at the expense of compressive yield strength.
Isotropic Hardening model and Kinematic hardening model are inaccurate, so we use a
combination of the 2 models  center can move and yield surface increase at the same time
σ2
σ
σ1
ε
Initial yield surface
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Nonlinear Static Analysis
52
Step
08
Stress-strain curve input method
Engineering stress VS True stress
 Materials stress-strain curve Obtained through tensile tests does not take into account the changes in the
area. (engineering stress- nominal strain curve)
 It is better to replace the input by the true stress – true strain curve which considers the change in surface.
L0
ΔL
Undeformed
A0
L
Deformed
True stress, True Strain calculation equations:
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A
* this equation is valid only for 1D problem
Nonlinear Static Analysis
53
Step
09
Stress-strain curve input method
Engineering stress VS True stress
Plastic Strain  Total Strain  (
e True  ln(1  e eng )
Eng. Strain
(mm/mm)
Eng. Stress
(MPa)
True Strain
(mm/mm)
True Stress
(MPa)
0.00000
0.00
0.00000
0.00
0.00112
264.70
0.00112
0.00400
264.70
0.00837
Yield Stress
)
Young ' s Modulus
Plastic Strain
(mm/mm)
True Stress
(MPa)
265.00
0.00000
265.00
0.00399
265.76
0.00287
265.76
276.14
0.00834
278.45
0.00722
278.45
0.01811
332.96
0.01795
338.99
0.01683
338.99
0.03170
383.16
0.03121
395.31
0.03009
395.31
0.04574
414.51
0.04472
433.47
0.04361
433.47
0.06505
439.14
0.06302
467.71
0.06190
467.71
0.08273
451.17
0.07949
488.50
0.07837
488.50
0.10447
458.31
0.09937
506.19
0.09825
506.19
0.12521
460.50
0.11797
518.16
0.11685
518.16
s True  s eng (1  e eng )
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Nonlinear Static Analysis
54
Step
10
Stress-strain curve input method
Perfectly Plastic Model
 Inclination of the curve in plastic range is 0.
 Beginning of plastic deformation is defined by Yield Stress
σ
ε
Perfectly Plastic Model
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Nonlinear Static Analysis
55
Step
11
Stress-strain curve input method
Bi-Linear Model
 Inclination of the curve in plastic range is
defined by a linear curve.
 Plastic Hardening Curve is defined by stressstrain plastic hardening function.
σ
ε
Bi-linear Model
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Nonlinear Static Analysis
56
Step
12
Stress-strain curve input method
Stress –strain function, Plastic Hardening function
Stress-Strain function
 Plastic Hardening function (Defined as the plastic
part only of the stress –strain curve)

In the first line, stress and strain are null.

In the second line, Elastic strain and yield stress
are entered.

In the third line, Plastic strain and Plastic stress
are entered

In the first line, enter 0 for the strain and yield
stress.

In the second line, enter the plastic strain and
plastic stress.
* Value of the strain should be calculated correctly
form the yield point, which is the starting point of
the curve.
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Nonlinear Static Analysis
57
Step
13
Stress-strain curve input method
Multi-Linear Model
 Hardening interval of the curve is composed of several segments.
 Curve is defined from real test data from experiment
σ
ε
Multi-linear Model
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Copy and Paste
from Excel
Nonlinear Static Analysis
58
Step
Material Nonlinearity(2) –
Hyper Elasticity
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Nonlinear Static Analysis
59
Step
01
Properties of Rubber materials
Rubber Material
 Incompressibility appears because of the complex link between multiple polymer chains.
 Differently from metallic materials, rubber materials can be submitted to large deformation due to
nonlinear elastic forces.
 Because of their elastic resilience and vibration damping properties, rubber materials are energyabsorbent, excellent dust removal, dust-proof and soundproof.
Physical
properties
of rubber
material
hyperelasticity
viscoelasticity
Strain energy function, W
Creep effect
ε
Tertiary
Creep
Primary
Creep
Steady State
Creep
time

What is viscoelasticity?
Phenomenon in which the physical
properties of the material depend on
Hysteresis effect
 Vibration
damping
the time.
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Nonlinear Static Analysis
60
Step
02
Hyper elastic models
What is an Hyper elastic material?
 Material loaded and submitted to a deformation of 500% can recover its original shape after unloading.
 As a nonlinear material, nonlinear analysis have to be used, but principle of superposition can still be
used as for linear analysis.
s
s
Elasticity
K
e
Permanent deformation
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e
Nonlinear Static Analysis
61
Step
03
Hyper elastic models
What are the properties of Hyper elastic materials?
 Stress can be derivated from the strain energy density function.
 Stress can be assumed from integration of the strain.
 Midas NFX provide Mooney-Rivlin, Polynomial, Ogden, Blatz-Ko models of the strain energy density
function W.
 The best way to approximate the strain energy density function is to use stress-strain experimental
data.
s
Real Behavior
Strain energy density
function used in midas
NFX
e
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Nonlinear Static Analysis
62
Step
04
Theory of hyper elastic models
Strain energy density function (W)
 Polynominal
Na
Shape deformation
Volumetric change
i+j=1

Nd
i=1
, ( Aij ,Di : Material constants)
More higher-order model can better express the stress-strain experimental data but requires
more material constants
 Mooney-Rivlin Model
W( J1 , J2 , J3 ) = A10(J1 – 3) + A01(J2 – 3) + D(J3 – 1)2
 It is the case of Na=1 in the polynomial equation.
 It is the most widely used model. (the model is rather convenient than accurate)
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Nonlinear Static Analysis
63
Step
05
Theory of hyper elastic models
Strain energy (W)
 Ogden Model
Na
i+j=1
μi
αi
Nd
i=1
, ( αi, μi ,Di : Material constants)
 It is difficult to express clearly the behavior of the material by using energy function
expressed with principal strains (J1,J2,J3).
 Material constants are not directly expressing the physical properties of the material,
whole function is expressing it.
 Blatz-Ko Model
μ
2
I2
I3
 For the foam materials in which incompressible nature is not strong, the material
constants are reduced to the initial shear stiffness μ only.
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Nonlinear Static Analysis
64
Step
06
Theory of hyper elastic models
Strain energy density function (W)
 Definitions ( J1 , J2 , J3 )
* J1 , J2 , J3 = 1,2,3 axis strain invariants
L1
λ1L1
λ2L2
L2
λ3L3
L3
Undeformed
Deformed
λ1 · λ2 · λ3 = 1  incompressible
J1 = λ12 + λ22 + λ32
J2 = λ12 · λ22 + λ22 · λ32 + λ32 · λ12
J3 = λ12 · λ22 · λ32
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Nonlinear Static Analysis
65
Step
07
Hyper elastic Material Assignment
Calculation of material constants
 For Hyper –elastic material, a range of test have to be performed to obtain all material
constants (for elasto-plastic materials, only uni-axial tensile test is necessary).
1. Uni-axial tensile test
2. Bi-axial tensile test
3. Simple shear test
4. Pure shear test
5. Volume change test
Eng. stress / Eng. strain measured through the following
experiment  Use the least squares method to
determine the material constants.
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Nonlinear Static Analysis
66
Step
08
Hyper elastic Material Assignment
Calculation of material constants
※ 시편 길이비 10:1
Uni-axial tensile test
Simple shear test
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Bi-axial tensile test– can be replaced by Pure shear test– pure shear stress calculation is
simple compression experiment
achieved using tensile experiment results and
rotation at 45˚
Volume change test– 취구는 크게 만들고 재료는 작은 재료를 넣는
압축력은 강성이 크기때문에 취구의 연성이 결과에 반
Nonlinear Static Analysis
67
Step
09
Hyper elastic Material Assignment
Calculation of material constants using stress-strain data-1
④ Select the type of experiment
① Select Hyper Elastic Tab
⑥ Click Add
⑤ Enter experiment data
② Assign Model Type
⑦ enter Poisson's ratio or
volumetric deformation
experimental data
⑨ Click Add
⑧ Select Type and order of calculation
③ Click Evaluate experimental data
⑨ Click Fit to Test Data…
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Nonlinear Static Analysis
68
Step
10
Hyper elastic Material Assignment
Calculation of material constants using stress-strain data-2
⑩ Click on close after verifying stability limit information of the material
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Nonlinear Static Analysis
69
Step
11
Hyper elastic Material Assignment
Calculation of material constants using stress-strain data-3
shape deformation
material constants
calculated
Mooney-Rivlin Model
W = A10( J1 – 3) + A01( J2 - 3 ) + D( J3 –
1 )2
Volume change material
constant calculated
⑪ Click Save and close
⑪ Click OK
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Nonlinear Static Analysis
70
Step
12
Phenomenon that cannot be expressed with Hyper elastic material
Hysteresis Effect
 When a rubber material composed of complex chains of polymers is loaded and
unload some loss of energy due to friction can occur.
Rubber
molecular
structure
s
loading
Theory
unloading
Real shape
Energy loss
e
 Stress softening (Mullin`s effect) : Phenomenon
which describe the stabilization of the stress-strain
curve from the initial shape to a stabilized shape
after several cyclic loading.
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Nonlinear Static Analysis
71
Step
13
Phenomenon that cannot be expressed with Hyper elastic material
Precautions to take for Hyper elastic Analysis
① Local slope can have a negative value, so the low order model is more stable.
s
Low order model
After Curve fitting, if such case happen, it is
High order model
better to consider the low-order model
(If presence of a negative slope in the curve)
e
② For Elasto plastic material, Von Mises stresses are usually investigated, whereas for
Hyper elastic materials, compressive stresses only are investigated.
A10 ,A01 << D1
,
WD <<
WH
In other words, stresses which are causing shape deformation are relatively smaller than
stresses causing volume change.
③ For complex rubber hyper-elastic materials , only nonlinear elasticity is considered
( Viscosity and Hysteresis are not included)
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Nonlinear Static Analysis
72
Step
Contact Nonlinearity
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Nonlinear Static Analysis
73
Step
01
Contact Nonlinearity
What is the reason to use contacts?
Load application

Load influence sent and
received through the nodal
points are connected to
adjacent elements.
In other words, if the nodes are not connected together, they cannot transfer the effect
of loads.
Most of the time, real models are composed of more than 2 bodies and it is quite
difficult to connect the nodes.
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How to do ?
Nonlinear Static Analysis
74
Step
02
Outline of Contact Nonlinearity
Why is contact necessary?
No contact
Penetration occurs
between the
elements
No mutual influence because the
nodes are not connected
Contact
Elements in
contact
Contact creation
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Nonlinear Static Analysis
75
Step
03
Outline of Contact Nonlinearity
What makes Nonlinearity phenomenon?
100N...200N...500N
1. Contact boundary and contact stresses
cannot be guessed in advance.
 Research of the contact boundary is a part of the
analysis process.
 As the degree of the contact stress or the
contact area depends on the size of the load,
the stiffness value change nonlinearly.
Contact Stress
Contact
Force
2. Rapid changes occur in the contact force
 Before the body touch each other contact
force is null and rapidly increases at the
moment when the contact occurs.
Penetration
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Nonlinear Static Analysis
76
Step
04
Contact Nonlinearity
Classification according to the contact surface properties
 Flexible-flexible contact
 Body 1 and Body 2 have similar stiffness values
and can all change shape.
Body 1
Body 2
 Most general condition of contact
 Flexible-Rigid contact
 When the stiffness of one of the body is much greater than the other, this
body can be defined as rigid.
 Stiffness matrix cannot be calculated numerically.
ex)  107

107

 Steel


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



1

1 
In this situation, considering this body as
rigid can help to reach convergence.
Ex) Iron (rigid) in contact with rubber
(flexible).
Rubber
Nonlinear Static Analysis
77
Step
05
The penalty method
The penalty method
When contact happens between 2 bodies, a small penetration first occurs, then some springs are created to link nodes which
are violating contact condition and surface of contact. A load (F=KΔδ) is then applied in the opposite direction to reduce
progressively the penetration. This is the process described below:
1.
Research of nodes/segments which infringe the contact condition .
Load application
Body 1
spring
Potential contact nodes
Penetration (Δδ)
Body 2
Nodes violating contact condition
2. Input a contact force to the nodes/segments which infringe the contact condition.
Contact force (Pushing out force)
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Nonlinear Static Analysis
78
Step
06
The Penalty Method
Contact Pair - Master, Slave
 Contact constraints
Nodes from Slave contact are not able to penetrate Master contact surface.
Nodes from Master contact are able to penetrate Slave contact surface.
For dense mesh, there is no real difference, but for coarse mesh, a big difference
can happen in function of Master-slave choice.
 Selection of principal contact face (Master) and intermediate nodes (Slave)
Rigid surface is usually set as Master surface.
Convex surface is usually set as Slave surface.
Surface mesh more densely is usually set as Slave surface.
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Master
Slave
Slave
Master
Nonlinear Static Analysis
79
Step
07
Contact Force
Contact Force ?
 When penetration occurs, an external pushing force is applied. This force
is called Contact Force.
 The larger the penetration, the bigger is the contact force.
 Contact Force (spring in compression)
FC  Kn g
Penalty parameter (Kn) : Contact Stiffness
- Dependent on the stiffness of the material
- The larger is Kn, the smaller is the penetration (in order to obtain same contact force)
g<0
FC
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Nonlinear Static Analysis
80
Step
08
Contact Force
Contact Stiffness Parameter
 Contact stiffness is expressed as the product
of a constant and the material stiffness.
Kn  SF  E
SF  1.0
Elastic module of flexible body
With high contact stiffness, solution will be more accurate.
 A high contact stiffness is used to limit penetration.
 As we use a high contact stiffness, the model is
submitted to vibrations and it may cause a convergence
problem.
Load
Contac
t Force
< when relatively high contact stiffness is applied, it causes vibrations in the model>
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Nonlinear Static Analysis
81
Step
09
Contact Force
Friction Force
Friction force
mFC
Body 1
Friction force doesn’t increase
Cause shear stress
Friction force
Kt
Relative
displacement
Stick region,
Just before slipping happens a small elastic
deformation happen. If the load is released in
this region, it will come back to the initial state.
Slip region, Kt·s doesn’t exist, only μFc is
applied
Contact force
Ff  Kt s
 FC
Stick (Kt: Tangent
Slip
stiffness)
In other words, the tangential stiffness determines the
status of the Stick
There is a relation with the shear strength of the material.
If Kt is high, the behavior of the contact surface is
almost rigid.
Friction
force
mFC
Kt
Relative
displacement
If Kt is small, the relative displacement depends linearly
of the friction force.
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Nonlinear Static Analysis
82
Step
10
Contact Force
Horizontal Stiffness
 Expressed as the product of a constant and
the material stiffness.
Kt  SF  E
SF  0.5
Elastic module of flexible body
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Nonlinear Static Analysis
83
Step
11
Types of contact
Types of contact
Vertical
Behavior
Horizontal
Behavior
Welded Contact No separation No separation
Linear
Contact
Sliding Contact No separation Sliding occurs
Contact
Rough Contact
Separation
No Sliding
General
Contact
Separation
Sliding occurs
(Friction)
Noninear
Contact
*Linear contact can be applied in nonlinear analysis,, but
nonlinear contacts can not be used in Linear Analysis.
*If friction is used, shear stress will happen before sliding in
movement direction.
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Normal Behavior
Shear Behavior
Nonlinear Static Analysis
84
Step
12
Types of contact
Types of contact
 Surface – Surface Contact
 User don’t need to define separately Master contact and Slave contact surfaces, because contact
occurs indifferently in the 2 directions.
 It is perfect when sliding and friction in multiple directions are considered.
 Surface – node Contact
 In the case when contact occurs between nodes of slave contact surface and Master surface, nodes
of the slave contact surface will always penetrate the Master contact surface.
 If unreasonable Slave and Master contact surface are chosen, it can results in incorrect results or in
convergence error.
 In other words, Slave and Master contact surfaces have to be chosen well, with good
understanding of the penalty method defined in the previous slides.
 In the penalty Method…
Master contact surface is usually chosen as the surface flat or with relatively high
stiffness, whereas, Slave contact surface is usually chosen as the surface convex or
with relatively low stiffness. Moreover, Surface meshed more densely has to be
defined as the slave surface.
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Nonlinear Static Analysis
85
Step
13
Method of contact creation
Auto Contact
 When contact is found within the scope of the
search, contact is automatically defined.
 When search distance is set to “Auto”, search
distance is determined in function of the mesh size.
 The Master contact surface and the slave contact
area are determined randomly.
ex) when search distance is set to 20mm
Contact surface is search between
meshes within 20mm of distance.
 If an excessively large Searching Distance is used, meshes
which are not in contact will be included in the contact, so
an appropriate Searching Distance should be used.
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Nonlinear Static Analysis
86
Step
14
Method of contact creation
Manual Contact
 Master contact and Slave contact surfaces are
defined manually.
 By selecting only the areas where contact is
expected to happen, duration of analysis can be
reduced.
Expected area of contact
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Nonlinear Static Analysis
87
Step
15
Method of contact creation
Self Contact
 When contact occur between external surface of elements of one mesh set,
self-contact is used.
 Self contact is efficient when contact points are numerous and when it is
difficult to estimate the contact points.
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Nonlinear Static Analysis
88
Step
16
Contact Algorithm
Contact Parameters
 Contact tolerance
Constant value used for the calculation of the contact search distance.
(For welded contact and bi-linear sliding contact, contact happen when
Master and Slave contact surfaces are within defined search distance. In
case of general contact or rough contact, contact happens when Master
and Slave contact surfaces arrive within the search distance).
 Master Contact Extension Ratio
Contact surface search distance is defined as the product of Master
contact surface mesh size and this value. Defaults value is 0.005 and can
be modified to increase or decrease the contact search area
 Remove Initial Penetration by Adjusting Slave Nodes
At the beginning of the analysis, if the slave contact nodes are already
penetrating the master contact surface, position of the slave nodes is
changed in order to suppress automatically the penetration.
 Structural Nonlinear Analysis without Geometric NonlinearityMax. Search Distance
When Geometric Nonlinearity is not considered, this option defines
the search distance from the Master contact surface to find the
slave contact surface.
•
Normal Failure force: When using breaking weld
contact, contact will be separated if the contact
force is superior to the Normal Failure Force.
•
Shear Failure Force: When using breaking weld
contact, contact will be separated if the shear
contact force is superior to the Shear Failure Force.
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Nonlinear Static Analysis
89
Step
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Nonlinear Static Analysis
90
Step
Conclusion and advice for a better
use of Nonlinear Analysis
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Nonlinear Static Analysis
91
Step
01
Conclusion and advice for a better use of Nonlinear Analysis
Principal useful tips
① Perform Linear Analysis Case before running Nonlinear Analysis
 By performing linear analysis, you can judge whether or not to perform Nonlinear Analysis (depending on the
deformation).
 If stresses or displacement are too large in linear analysis, design of the product has to be reviewed.
 By performing linear analysis, areas where large stresses or deformation happen can be estimated and smaller mesh
can be refined in these areas.
 By performing linear analysis, areas where contacts happen can be estimated in order to create the most appropriate
and effective contact surfaces.
② Parameter settings required for Nonlinear Analysis
 Load increments are one of the most important of the parameter settings. It cannot work properly for example if
the plastic state is reached since the first increment. Increment number has to be set in order to get reasonable
results (It is still possible to get difficult convergence)
 Convergence criteria selection, Geometric nonlinearity option, Line search Method, Subcase activation, Contact
parameters
 Use of Restart feature(Convergence-related parameters can be applied and adjusting to save a lot of time)
 Understand the principle of special nonlinear analysis (geometry, contact, material)

Ex 1) Choice of hyper elastic material to estimate properties of rubber materials
Ex 2) Use of Arc-length method for nonlinear buckling analysis
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Nonlinear Static Analysis
92
Step
02
Conclusion and advice for a better use of Nonlinear Analysis
Principal useful tips
③ Finite element modelling for nonlinear analysis
With appropriate modelling, analysis speed and convergence will increase substantially. Nonlinear analysis consumes a
lot of time if we compare with linear analysis and a simplification of the model is also necessary. In order to create
the best model for Nonlinear analysis, the points below have to be considered:
1)
Use symmetry condition when possible
 Cannot be used for nonlinear buckling analysis or dynamic analysis
2)
Use Beam, shell, or flat idealized elements when possible
3)
Minimize the use of nonlinear material model
 Use nonlinear material model only for element which are subjected to plastic deformation
4)
Create dense and smooth mesh in areas of high strain
 Make even denser mesh in areas of contact

5)
Evaluate the zones where large deformation will happen and make appropriate mesh to fit deformed shape.
When possible, use rotation or extraction feature to create Hexa or Penta element meshes
 Results will be much more accurate than tetrahedral elements.
Pay attention to the Mesh quality (Aspect Ratio, Skew Angle) , because it can create convergence problems in
bad mesh quality areas.
 Simplify the fillets and holes which are not important for your analysis
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Nonlinear Static Analysis
93