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
any unload-reload cycles.

“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.
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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
Unload


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
reversed direction of loading.
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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
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

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
deformation when unloaded (i.e.
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
increment and the stress state under multiaxial loading.
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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
monotonic (non-cyclic) loads.
Kinematic hardening should be
used with cyclic loading
conditions to more accurately
predict the Bauschinger effect.
Monotonic Loading
Force
Time
Cyclic Loading
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
unidirectional loading.

Kinematic models are
recommended for cyclic loading.
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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

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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
high stress gradients.
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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
this problem because the load
does not cycle.

Midside nodes are used to help
capture the high stress/strain
gradients.

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
Maximum Applied Load
Module 2 – Elastic-Plastic Materials
Page 22

The onset of yielding starts at a
load level of 2,190 lb.

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.
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