Elastic-Plastic Results - Autodesk

badwaterreflectiveΠολεοδομικά Έργα

29 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

78 εμφανίσεις

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Nonlinear Analysis:

Elastic
-
Plastic Material
Analysis

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Objectives


The objectives of this module are to:



P
rovide an introduction to the elastic
-
plastic equations used in Autodesk
Simulation Multiphysics



R
elate elastic
-
plastic material theory to the material parameters used in
Autodesk
Simulation Multiphysics



S
how how to set up and perform an analysis using elastic
-
plastic materials
in Autodesk
Simulation Multiphysics

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 2

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 3

“True” stress
-
strain curves
should be used when performing
elastic
-
plastic finite element
analyses using Autodesk
Simulation Multiphysics.

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Cyclic Stress
-
Strain Curves


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.


Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 4

Cyclic stress
-
strain curves
are important to strain
-
life
fatigue life calculations.

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Bauschinger

Effect


The
Bauschinger

effect refers to a
decrease in the compressive yield
stress due to work hardening in
tension.



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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 5

s

e

Monotonic stress
strain curve in
compression

Actual compression
stress
-
strain curve
following tensile work
hardening

Tension stress
-
strain curve

Unload

Bauschinger

effect

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Yield Criteria


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





In terms of Cartesian stress components




3
1
3
2
2
1
2
3
2
2
2
1
s
s
s
s
s
s
s
s
s
s






eff























2
2
2
2
2
2
6
1
2
1
xz
yz
xy
zz
xx
zz
yy
yy
xx
eff
s
s
s
s
s
s
s
s
s
s
Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 6

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Graphical Representation


The elliptical curve shown in the figure
is the intersection of the 3
-
dimensional
von
Mises

yield surface with the
s
1
,
s
2

principal
stress plane (
s
3

= 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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 7

Two
-
dimensional representation of
the von
Mises

yield criterion.

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Incompressibility


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
s
eff
.


Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 8

ij
ij
eff
s
s
J
2
3
3
2


s
1
3
1
I
s
ij
ij


s
I
1

= 1
st

stress invariant

J
2
= 2
nd

invariant of the
deviatoric stress tensor

deviatoric

stress components

Alternate form of Von
Mises

yield criterion using
indicial notation.

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Elements of Plasticity Theory

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 multi
-
axial loading.


p
ij
e
ij
ij
d
d
d
e
e
e


ij
p
ij
S
d
d



e
1
3
1
I
s
ij
ij


s






ij
ij
ij
ij
s
s



s



2
3


ij
ij
ij
d
d

s




Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 9

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Isotropic & Kinematic Hardening


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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 10

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Application of Hardening Rules


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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 11

Force

Time

Monotonic Loading

Force

Time

Cyclic Loading

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 12

List of elastic
-
plastic material
models found in Autodesk
Simulation Multiphysics.

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Bilinear Models


“True” stress
-
strain curves can be
approximated using a bilinear
model.



A bilinear model uses Young’s
Modulus (E) and a strain
-
hardening modulus (E
T
).



The Autodesk Simulation
Multiphysics material library
contains a strain
-
hardening
modulus for many metals.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 13

True Strain

E

E
T

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Curve Models


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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 14

This image shows
the tabular data
found in Autodesk
Simulation
Multiphysics for
AISI 1020 cold
rolled steel.

Tabular data
can be entered
manually or
imported from a
.
csv

file.

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Analysis Type


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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 15

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Example Problem

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 16

500 lb

A fine mesh is used
where there will be
high stress gradients.


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.


© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Elastic Response


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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 17

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Analysis Type


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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 18

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Element Definition


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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 19

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Material Selection


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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 20

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Analysis Parameters

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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 21

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Maximum Applied Load


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.

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 22

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Results


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
-
Plastic
Results

Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 23

Elastic Results

© 2011 Autodesk

Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunity

Education Community

Summary


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.



Section 3


Nonlinear Analysis

Module 2


Elastic
-
Plastic Materials

Page 24