Nonlinear Analysis:
Overview of Nonlinear 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
Section 3 – Nonlinear Analysis
Objectives

Module 1: Overview
Page 2
The objectives of this module are to:

Provide an overview of the nonlinear phenomena that may be encountered
in a displacement-based finite element analysis

Identify the four things that can make a problem nonlinear, and relate them
to the other modules in this section

Present the nonlinear material models contained within the Elasticity and
Plasticity material classes in Simulation

Explain the concept of a Mechanical Event Simulation that can include the
four things that make a problem nonlinear
© 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
What makes a problem Nonlinear ?
Section 3 – Nonlinear Analysis
Module 1: Overview
Page 3


If the behavior of a structural
system depends on its current
state, the force-displacement
relationship becomes
nonlinear.
Generalized
Force
nonlinear
There are four things that can
make a problem nonlinear.
linear
Material behavior
 Large Displacements/Rotations
 Stress Stiffening/Softening
 Intermittent boundary
conditions (contact)

© 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.
Generalized
Displacement
www.autodesk.com/edcommunity
Education Community
Section 3 – Nonlinear Analysis
Module 1: Overview
Nonlinear Phenomena Covered in Section 3
 The four things that can make a
problem nonlinear are covered in
the various modules in this
section.
 In most cases, more than one of
the four items is contained in the
example problem associated with
a specific module.
Large Displacements/Rotations
Impact
Riks Analysis
(Section 3, Module 5) (Section 3, Module 6)
Hyperelastic Material
Analysis
(Section 3, Module 3)
© 2011 Autodesk
Page 4
Nonlinear Materials
Polymers
Metals
Elastic-Plastic Analysis
(Section 3, Module 2)
Riks Analysis
(Section 3, Module 5)
Viscoelastic Material
Analysis
(Section 3, Module 4)
Hyperelastic Material
Analysis
(Section 3, Module 3)
Impact
(Section 3, Module 6)
Stress
Stiffening/Softening
Riks Analysis
(Section 3, Module 5)
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.
Intermittent Boundary
Conditions
Hyperelastic Material
Analysis
(Section 3, Module 3)
Impact
(Section 3, Module 6)
www.autodesk.com/edcommunity
Education Community
Section 3 – Nonlinear Analysis
Module 1: Overview
Material Behavior: Linear



Many materials demonstrate a
linear relationship over all or a
portion of their stress versus
strain response.
Metals exhibit this characteristic
prior to reaching a proportional
limit (sp).
Page 5
Stress
sy
sp
The proportional limit is close to
but different than the yield stress
(sy) that is defined at a strain
level of 0.002.
© 2011 Autodesk
Example stress-strain curve of
a metal
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.
s  E
0.002
www.autodesk.com/edcommunity
Strain
Education Community
Section 3 – Nonlinear Analysis
Material Behavior: Nonlinear


The response of many other
materials do not have linear
stress-strain relationships.
Module 1: Overview
Page 6
Uniaxial stress-stretch behavior of an
elastomer.
Stress, s
Tension
Examples include:
Metals stressed beyond the
proportional limit,
 Elastomers such as rubber which
demonstrate both nonlinear and
time dependent load-deformation
response, and
 Biological materials which also
exhibit nonlinear and time
dependent load-deformation
response.

© 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.
Stretch, l
Compression
www.autodesk.com/edcommunity
Education Community
Section 3 – Nonlinear Analysis
Module 1: Overview
Material Behavior: Elasticity

Page 7
Elastic materials have the
characteristic that they will return
to their original shape and stress
state when unloaded.
Elastic material models contained
in Simulation
Elasticity
Elastic

Elastic materials can be nonlinear
and/or time dependent.

Autodesk Simulation Multiphysics
contains many material models
that can be used to model elastic
materials.
© 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.
Isotropic
Orthotropic
Variable tangent
Curve
Curve with cut-off
Thermoelastic
Duncan-Chang Soil
Foam
 Blatz-Ko
 Hyperfoam
www.autodesk.com/edcommunity
Hyperelastic






Mooney-Rivlin
Aruda-Boyce
Ogden
Yeoh
Neo-Hookian
Van der Waals
Viscoelastic
 Thermal creep
 Viscoelastic
versions of the
hyperelastic
models
Education Community
Section 3 – Nonlinear Analysis
Module 1: Overview
Material Behavior: Plasticity



Page 8
Plastic materials have the
characteristic that they do not
return to their original shape and
stress state when unloaded.
Autodesk Simulation Multiphysics
also contains several material
models that can be used to
model the plastic response of
materials.
Plastic material models contained
in Simulation
Plasticity
Viscoplastic
Plastic
Von Mises yield function
with isotropic and
kinematic hardening
Plastic
Thermal creep
Druker-Prager
yield function
Plasticity material models can be
used to determine the “bent”
shape of a metal loaded beyond
the proportional limit.
© 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
Section 3 – Nonlinear Analysis
Large Displacement/Rotations
Module 1: Overview
Page 9

Small displacement analysis:
 The displacement–induced deformations
are small.
 The change in the spatial orientation
(rotation) is small.
Deformed configurations
of a fishing pole
dy

Large displacement analysis:
 The displacement–induced deformations
can be finite.
 The spatial orientation change may no
longer be infinitesimal.
© 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.
dx
q
The material at the end of the
pole undergoes large
displacements and rotations.
www.autodesk.com/edcommunity
Education Community
Section 3 – Nonlinear Analysis
Large Displacements/Rotations: Measures



Small displacement analyses can be
performed using infinitesimal strain
and Cauchy stress measures.
Large Displacements/Rotations require
the use of finite deformation stress and
deformation measures (e.g. Green’s
strain and the 2nd Piola-Kirchoff stress)
The infinitesimal strain equations can
not handle large rotations; Green’s
strain equations can.
© 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.
Module 1: Overview
Page 10
Infinitesimal Strain Equations
 xx 
u
v
w
;  yy  ;  zz 
x
y
z
1  u
v 
1  v
w 
 xy    ;  yz    
2  y x 
2  z y 
1  u w 


2  z x 
 xz  
Only good for infinitesimal
displacements and rotations.
www.autodesk.com/edcommunity
Education Community
Section 3 – Nonlinear Analysis
Large Displacements/Rotations: Loads


Pressures or tractions specified
relative to surface orientations
can have a big effect when large
displacements/rotations are
encountered.
In the two figures, the pressure
always remains normal to the
surface of the part as it deforms.
Module 1: Overview
Page 11
Undeformed
Configuration
Pressure
Surface
Deformed
Configuration
Surface
© 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
Pressure
Education Community
Section 3 – Nonlinear Analysis
Stress Stiffening/Softening
Module 1: Overview
Page 12
 The stress level in a system always
affects its stiffness.
The increased bending stiffness
of a high pressure hose is due to
stress stiffening.
 Sometimes the effect is small and it can
be neglected. In other cases it has a
significant effect.
 The increase in the frequency of a
guitar string is an example of stress
stiffening.
An unpressurized hose is
easily rolled up.
 The decrease in the compression
stiffness of a column is an example of
stress softening.
© 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
Section 3 – Nonlinear Analysis
Intermittent Boundary Conditions
Module 1: Overview
Page 13
 Boundary conditions provide
constraints that mechanical systems
must satisfy during an event.
Contact between the O-ring and
the gland walls causes an
intermittent boundary condition.
 In many cases the boundary conditions
are constant with time.
O-ring
 In other cases the boundary conditions
are time dependent and can depend
on interactions among various
components in the system.
 Contact between components is a type
of intermittent boundary condition.
© 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.
Prescribed displacements are a
time-dependent boundary
condition that can be used in
nonlinear simulations.
www.autodesk.com/edcommunity
Education Community
Section 3 – Nonlinear Analysis
What else is true about Nonlinear Analysis ?
Module 1: Overview
Page 14




The solution of nonlinear problems
requires incremental techniques and
iterations within each increment to ensure
equilibrium at the end of each increment.
An incremental solution is performed by
incrementing the applied loads until a
certain level of loading is reached.
A pure incremental technique can cause
error accumulation from one increment to
the next leading to an incorrect solution.
Equilibrium iterations should be
performed to force the solution to lie on
the equilibrium path within a preset
tolerance.
© 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.
The Newton-Raphson Method is the
fundamental solution method used
in a nonlinear analysis
Slope = KT
A 2
Fext
error  Runb
Rint
B
1
u1
u u
uA 2
www.autodesk.com/edcommunity
u
Education Community
Section 3 – Nonlinear Analysis
Time and Pseudo Time
Module 1: Overview
Page 15

Load curves are used in nonlinear analyses to define how the load
varies.

In a dynamic analysis or time dependent material analysis, the load
curves define how the load varies with time.

In a static analysis the load curve is also defined as a function of
time.
Time is a parameter or “psuedo time” used to define the variation of the load
during the event.
 “Psuedo time” is simply a parameter, and does not create a true timedependent response.

© 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
Section 3 – Nonlinear Analysis
Mechanical Event Simulation (MES)
Module 1: Overview
Page 16


Mechanical Event Simulation (MES)
combines kinematic, rigid, and flexiblebody dynamics and nonlinear stress
analysis capabilities.
MES can simultaneously analyze
mechanical systems experiencing large
deformations, nonlinear material
properties, kinematic motion, and
forces caused by that motion and then
predict the resulting stresses.
© 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.
A Mechanical Event Simulation
(MES) can simultaneously include
the four things that make a
problem nonlinear.




Material behavior
Large Displacements/Rotations
Stress Stiffening/Softening
Intermittent boundary
conditions (contact)
www.autodesk.com/edcommunity
Education Community
Some characteristics of a MES problem
Section 3 – Nonlinear Analysis
Module 1: Overview
Page 17






Large scale motion and/or deformations with or without
change in the positions and directions of loads and/or
constraints
Acceleration/inertia
Damping
Contact or impact
Nonlinear material behavior (such as plastic deformation due
to exceeding the material yield strength)
Loads and results are time-dependent, providing many
instantaneous results “snapshots” over a user-defined period
of time
© 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
Section 3 – Nonlinear Analysis
Module Summary

Module 1: Overview
Page 18
The four types of nonlinearities that can be encountered in a finite
element analysis include:
Nonlinear material behavior
 Large displacements/rotations
 Stress stiffening/softening
 Intermittent boundary conditions (contact)


Each of these nonlinearities can be modeled and analyzed using
Autodesk Simulation Multiphysics software.

The other modules in this section provide details on how to model
and perform an analysis that contains one or more of these
nonlinearities.
© 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