### Corporate Overview*4x3 PPT Version

```Nonlinear Analysis:
Elastic-Plastic Material Analysis
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Section 3 – Nonlinear Analysis
Objectives
Module 2 – Elastic-Plastic Materials
Page 2

The objectives of this module are to:

Provide an introduction to the elastic-plastic equations used in Autodesk
Simulation Multiphysics

Relate elastic-plastic material theory to the material parameters used in
Autodesk Simulation Multiphysics

Show how to set up and perform an analysis using elastic-plastic materials
in Autodesk Simulation Multiphysics
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Section 3 – Nonlinear Analysis
Module 2 – Elastic-Plastic Materials
Monotonic Stress-Strain Curves

Monotonic stress-strain curves
are obtained from a tensile test
that starts at zero load and
progresses to fracture without

“Engineering” stress-strain curves
are based on engineering stress
and strain measures.

“True” stress-strain curves are
based on true stress and
logarithmic strain measures.
Page 3
“True” stress-strain curves
should be used when performing
elastic-plastic finite element
analyses using Autodesk
Simulation Multiphysics.
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Section 3 – Nonlinear Analysis
Cyclic Stress-Strain Curves
Module 2 – Elastic-Plastic Materials
Page 4

Cyclic stress-strain curves are
obtained when the specimen is
cycled repetitively between tension
and compression stress or strain
values.

Strain controlled experiments cycle
between tension and compression
strain extremes to yield a cyclic
stress-strain curve as shown in the
figure.
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Cyclic stress-strain curves
are important to strain-life
fatigue life calculations.
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Section 3 – Nonlinear Analysis
Bauschinger Effect
Module 2 – Elastic-Plastic Materials
Page 5

The Bauschinger effect refers to a
decrease in the compressive yield
stress due to work hardening in
tension.
s
Tension stressstrain curve


It can also refer to a decrease in
the tensile yield stress due to
work hardening in compression.
Work hardening can be used to
increase the yield strength of a
material, but it does so at the cost
of a lower yield stress in the
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e
Actual compression
stress-strain curve
following tensile work
hardening
Monotonic stress
strain curve in
compression
Bauschinger effect
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Section 3 – Nonlinear Analysis
Yield Criteria
Module 2 – Elastic-Plastic Materials
Page 6
 The onset of yielding for ductile materials subjected to multi-axial
stress states can be predicted using the von Mises effective stress.
 This yield criterion can be written in several forms:

In terms of principal stress components
s eff  s  s  s  s 1s 2  s 2s 3  s 1s 3
2
1

2
2
2
3
In terms of Cartesian stress components
s eff


1
1 2
2
2
2
2
2 
s xx  s yy   s yy  s zz   s xx  s zz   s xy  s yz  s xz 


2
6

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Section 3 – Nonlinear Analysis
Graphical Representation
Module 2 – Elastic-Plastic Materials
Page 7

The elliptical curve shown in the figure
is the intersection of the 3-dimensional
von Mises yield surface with the s1, s2
principal stress plane (s3 = 0).

Note that the elliptical curve fits the
experimental data for the steel and
aluminum alloys (i.e. ductile materials).

Gray cast iron (brittle material) does
not exhibit significant plastic
deformation prior to fracture and the
von Mises criteria does not match the
experimental data.
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Two-dimensional representation of
the von Mises yield criterion.
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Section 3 – Nonlinear Analysis
Incompressibility
Module 2 – Elastic-Plastic Materials
Page 8



Ductile metals subjected to
moderate hydrostatic pressures
do not exhibit permanent
they do not yield).
The von Mises yield criterion is
consistent with this experimental
observation.
A hydrostatic stress state will give
a zero value for seff.
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Alternate form of Von
Mises yield criterion using
indicial notation.
s eff
3 ij
 3J 2 
sij s
2
deviatoric stress components
1
sij  s ij  I1
3
I1 = 1st stress invariant
J2= 2nd invariant of the
deviatoric stress tensor
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Section 3 – Nonlinear Analysis
Elements of Plasticity Theory
Module 2 – Elastic-Plastic Materials
Page 9
Key concepts of plasticity theory are:




The strain increment is decomposed into
elastic and plastic parts.
A yield surface is used to determine if the
material responds elastically or plastically.
Strain-hardening rules that determine the
shape and position of the yield surface in
the plastic region.
A plastic flow rule determines the
relationship between the plastic strain
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de ij  de  de
e
ij
p
ij
1
sij  s ij  I1
3
3
sij  ij sij  ij 
s   
2
dij  d s ij  ij 
de  d  Sij
p
ij
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Section 3 – Nonlinear Analysis
Isotropic & Kinematic Hardening
Module 2 – Elastic-Plastic Materials
Page 10

Isotropic hardening keeps the
center of the yield surface
stationary and accommodates
work hardening by allowing
the yield surface to get larger.

Kinematic hardening allows
the center of the yield surface
to move during work
hardening and keeps the size
of the yield surface constant.
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Section 3 – Nonlinear Analysis
Application of Hardening Rules
Module 2 – Elastic-Plastic Materials
Page 11


Isotropic or kinematic hardening
can be used when the system
being analyzed is subjected to
Kinematic hardening should be
conditions to more accurately
predict the Bauschinger effect.
Force
Time
Force
Time
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Autodesk Simulation Multiphysics Plasticity
Models

Autodesk Simulation Multiphysics
provides elastic-plastic material
models for isotropic or kinematic
hardening.

“True” stress-strain curves may be
approximated using a bilinear model
or entered directly.

Isotropic models should be used for

Kinematic models are
Section 3 – Nonlinear Analysis
Module 2 – Elastic-Plastic Materials
Page 12
List of elastic-plastic material
models found in Autodesk
Simulation Multiphysics.
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Section 3 – Nonlinear Analysis
Bilinear Models
Module 2 – Elastic-Plastic Materials
Page 13

“True” stress-strain curves can be
approximated using a bilinear
model.
ET
E


A bilinear model uses Young’s
Modulus (E) and a strainhardening modulus (ET).
True Strain
The Autodesk Simulation
Multiphysics material library
contains a strain-hardening
modulus for many metals.
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Section 3 – Nonlinear Analysis
Curve Models
Module 2 – Elastic-Plastic Materials
Page 14

Curve models allow an actual “true” stress-strain curve to be
entered and used.

Tabular data is often contained within the Autodesk Simulation
Multiphysics material models.
This image shows
the tabular data
found in Autodesk
Simulation
Multiphysics for
AISI 1020 cold
rolled steel.
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Tabular data
can be entered
manually or
imported from a
.csv file.
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Section 3 – Nonlinear Analysis
Analysis Type
Module 2 – Elastic-Plastic Materials
Page 15

An elastic-plastic material
model may be used with four
different Simulation analysis
types
MES with Nonlinear Material
Models
 Static Analysis with Nonlinear
Material Models
 Natural Frequency (Modal)
with Nonlinear Material Models
 MES Riks Analysis

Process for finding the nonlinear analysis
types supporting elastic-plastic material
models in Autodesk Simulation Multiphysics.
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Section 3 – Nonlinear Analysis
Example Problem
Module 2 – Elastic-Plastic Materials
Page 16

The response of a flat bar with a
hole at its center will be used to
demonstrate how to setup an
elastic-plastic material analysis.

There is a stress concentration at
the hole.

The objective is to determine how
the stress distribution changes
across the bar as it experiences
elastic-plastic deformation.
500 lb
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A fine mesh is used
where there will be
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Section 3 – Nonlinear Analysis
Elastic Response
Module 2 – Elastic-Plastic Materials
Page 17

The material is AISI 1020 cold
rolled steel and has a yield
strength of 50.8 ksi.

Based on the results of an elastic
analysis, the onset of yielding will
occur at a load of 2,190 lb.

The figure shows the stress
distribution at the onset of
yielding.
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Section 3 – Nonlinear Analysis
Analysis Type
Module 2 – Elastic-Plastic Materials
Page 18

The elastic-plastic
response will be computed
using the “Static Analysis
with Nonlinear Materials”
analysis type.

This must be set before
nonlinear materials will be
shown in the material
library.
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Section 3 – Nonlinear Analysis
Element Definition
Module 2 – Elastic-Plastic Materials
Page 19

The von Mises with Kinematic
Hardening or Isotropic Hardening
material model can be used for
does not cycle.

Midside nodes are used to help
capture the high stress/strain

The large displacement option is
used to account for geometry
changes as the plastic
deformation takes place.
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Section 3 – Nonlinear Analysis
Material Selection
Module 2 – Elastic-Plastic Materials
Page 20


AISI 1020 cold rolled steel is selected.
Since the bilinear stress-strain material model is being used, a
Strain Hardening Modulus appears in the properties.
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Section 3 – Nonlinear Analysis
Analysis Parameters
Module 2 – Elastic-Plastic Materials
Page 21
The load is applied in an increasing fashion in 20 load increments
(Capture rate). The duration is set to 1 second, but the problem does
not include any inertia effects.
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Section 3 – Nonlinear Analysis
Module 2 – Elastic-Plastic Materials
Page 22

The onset of yielding starts at a

The load is set to 5,000 lb. This
was determined after several
runs to be sufficient to let the
material yield completely across
the cross-section.
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Section 3 – Nonlinear Analysis
Results
Module 2 – Elastic-Plastic Materials
Page 23

These figures show the
difference between the stress
distribution computed using
both elastic and elastic-plastic
analysis types.

The high elastic stresses are
unrealistic because they do not
lie on the stress-strain curve of
the material.
Elastic Results
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Elastic-Plastic
Results
Education Community
Section 3 – Nonlinear Analysis
Summary
Module 2 – Elastic-Plastic Materials
Page 24

This module has provided an introduction to the elastic-plastic
constitutive equations used in Autodesk Simulation Multiphysics
software.

The difference between isotropic and kinematic hardening models
was discussed and related to when each should be used.

The material parameters required by Autodesk Simulation
Multiphysics for an elastic-plastic material were presented and
related to the theory.

The steps taken to set up an analysis that uses an elastic-plastic
material model were presented in the context of an example
problem.