nonlinear analysis: elastic-plastic material analysis

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Nonlinear Analysis:Elastic-Plastic Material Analysis


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Objectives

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

Section 3 – Nonlinear Analysis

Module 2 – Elastic-Plastic Materials


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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.



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


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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.



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


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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.



s

e

Monotonic stress strain curve in compression

Actual compression stress-strain curve following tensile work hardening

Tension stress-strain curve

Unload

Bauschinger effect


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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

31322123

22

21 ssssssssss eff

222222

61

21

xzyzxyzzxxzzyyyyxxeff ssssssssss




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Graphical Representation

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.



Two-dimensional representation of the von Mises yield criterion.


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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 seff.



ijijeff ssJ

233 2 s

131 Is ijij s

I1 = 1st stress invariantJ2= 2nd invariant of the

deviatoric stress tensor

deviatoric stress components

Alternate form of Von Mises yield criterion using indicial notation.


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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.

pij

eijij ddd eee

ijpij Sdd e

131 Is ijij s

ijijijij ss s 23

ijijij dd s




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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.




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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.



Force

Time

Monotonic Loading

Force

Time

Cyclic Loading


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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.



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


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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 (ET).

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



True

Stre

ss

True Strain

E

ET


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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.



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.


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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.




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Example ProblemSection 3 – Nonlinear Analysis


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.


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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.




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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.




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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.




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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.




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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.




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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.




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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



Elastic Results


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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.



nonlinear analysis: elastic-plastic material analysis

Documents

true stressstrain curves

strain values

tensile yield stress

lower yield stress

compressive yield stress

compression strain extremes

elasticplastic materialspage

logarithmic strain measures