Mechanical-Nonlin 13.0 App5A Chaboche

8/12/2019 Mechanical-Nonlin 13.0 App5A Chaboche

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Customer Training Material

Appendix 5A

Nonlinear Kinematic

Hardening with

Structural Nonlinearities

A5A-1ANSYS, Inc. Proprietary

2010 ANSYS, Inc. All rights reserved.Release 13.0

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ANSYS Mechanical Nonlinear Kinematic Hardening (Chaboche)

Customer Training MaterialA. Review of Linear Kinematic Hardening

For linear kinematic hardening, the yield surface translates as a rigid

body during plastic flow.

(kinematic hardening is a form of anisotropic hardening)

The elastic region is equal to twice the initial yield stress. This is called

the Bauschinger effect.

1

2y'

Subsequent

Yield Surface

Initial Yield

Surface



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32


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Customer Training Material... Review of Linear Kinematic Hardening

The yield criterion can therefore be expressed as:

0:2

== yF

s

s

where s is the deviatoric stress, y is the uniaxial yield stress, andis the back stress (location of the center of the yield surface).

Note in the previous plot that the center of the yield surface has. , ,

2y. The back stress is linearlyrelated to plastic strain via:

plC = 2



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Bilinear kinematic hardening (BKIN) is an example of linear kinematic

hardening.

,

relationship of back stress and equivalent plastic strain.

Can be used for cyclic loading since it includes the Bauschinger

effect (elastic region equal to twice the initial yield stress).

However, linear kinematic hardening is recommended for situations-

strain).

Because there is only one plastic slope (tangent modulus), this is not

representative of true metals as the hardening is constant.



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A note on multilinear kinematic hardening:

Multilinear kinematic hardening is different than BKIN, and it uses the

Besselin a.k.a. subla er or overla model. It characterizes multilinear

behavior as a series of elasto-perfectly plastic subvolumes, each ofwhich yields at different points, so no back stress a is used.



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Customer Training MaterialB. Nonlinear Kinematic Hardening

Nonlinear kinematic hardeningis similar to linear kinematic

hardening except for the fact that the evolution law has a nonlinear

term (the recall term {}): iiplii C =

32

where pl is equivalent plastic strain while is accumulated plasticstrain.

The yield criterion is expressed as:

3

where Ris a constant definin the ield stress similar to linear

:2 == ss



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kinematic hardening.


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Customer Training Material... Nonlinear Kinematic Hardening

The yield surface can be described graphically as shown below:

The current yield surface shifts in principal stress space

ere s a m ng y e sur ace, as exp a ne n e nex s e. n o er

words, the behavior approaches perfectly plastic, unlike linear kinematic

hardening, which does not change slope.

1 Limiting Yield

R

Surface

Limiting

value of {}



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Current Yield

Surface


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Customer Training Material... Nonlinear Kinematic Hardening

Nonlinear kinematic hardening has the following characteristics:

Nonlinear kinematic hardening does not have a linear relationship

between hardenin and lastic strain.

The nonlinear kinematic hardening term is associated with the translation

of yield surface. A non-zero value of results in a limiting value of .s means a , un e near nema c ar en ng, e y e sur ace

cannot translate forever in principal stress space. The translation is

limited within a specific region.

The constant R(yield stress), which describes the size of the elastic

domain, is added on the response. If a limiting value of exists, then a

limitin ield surface will also exist.

Nonlinear kinematic hardening is suitable for large strains and cyclic

loading, as it can simulate the Bauschinger effect. It can model



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w u .


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Customer Training MaterialC. Chaboche Model

The Chaboche Kinematic Hardening modelis an example of

nonlinear kinematic hardening. As noted previously, the yield

function is

( ) ( ) 0:23 == RF ss

1 Limiting Yield

Surface

RC/

32

Current Yield

m ng

value of {}



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Surface


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Customer Training Material... Chaboche Model

The back stress is a superposition of up to five kinematic models:

in

pln

TdC

C &&&& 12

+==

where:i

ii dTC3 11 ==

n s e num er o nema c mo e s o use

is the back stress (location of the center of the yield surface).Ci is constant that is proportional to hardening modulus

plis equivalent plastic strain

i is rate of decrease of hardening modulus is accumulated lastic strain. is temperature



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Note that if n=1 and 1=0, Chaboche will reduce to Bilinear Kinematic (nolimiting value for 1).


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Customer Training Material... Chaboche Model

This figure breaks down the Chaboche model parameters and how

they are related to each other:

n is 3 the number of kinematic models combined to ether.

R is the yield stress (constant value)

Values 1 3 are the backstresses calculated from the

previous equation. Constants

C1 C3 and 1 3 are

+R

R

associated with these values.

R describes the yield surface

whereasdescribes the 2

1 2 3

shifting of the center of the

yield surface.

1

3

R=160

C =80000 =2000



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o e a 3= , so ere s nolimiting surface for .

C2=10000, 2=200C3=2500, 3=0


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Customer Training MaterialD. Defining the material input

To define a Chaboche model, from the Engineering Data Tool Box

Expand the Plasticity folder to expose Chaboche Kinematic Hardening

The Chaboche model will then appear



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to enter appropriate values


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Customer Training Material... Defining the material input

Up to five kinematic hardening models can be defined under Chaboche

via Engineering data GUI.

command object under the geometry branch for the applicable part(s).



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Customer Training Material... Defining the material input

As with the other metal plasticity models, Chaboche can also be

defined with temperature dependence properties in a tabular format



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Customer Training MaterialE. Ratchetting & Shakedown

Ratchetting occurs under a nonsymmetricstress-controlled loading

There is a progressive increase in strain at each cycle.

Shakedown occurs under a nonsymmetricstress-controlled loading

.



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Customer Training Material... Ratchetting & Shakedown

0.06

0.07

1500

2000

Ratchetting

Ratchetting modeled

with Chaboche using:

n=1

R=980,

=

Plasticstrain

0.01

0.02

0.03

0.04

.

Stress(M

Pa)

-500

0

500

1000

= 400

Time

0 2 4 6 8 10

0.00

.

Plastic strain

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

-1500

-1000

t

strain

0 004

0.006

.

(MPa)

500

1000

1500Shakedown Loading

Controlled Stress

Unsymmetry

Plas

ti

0.000

0.002

.

Stres

s

-1500

-1000

-500

0

Shakedown modeled with

Chaboche using:

n=2



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Time

0 2 4 6 8 10

Plastic strain

0.000 0.002 0.004 0.006 0.008 =

C1=224000 , C2=200001=400,20000, 2=0


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Considerations for Ratchetting:

Ratchetting is the accumulation of plastic strain under an unsymmetric

stress-controlled cyclic loading.

Linear kinematic models cannot capture ratchetting, as shown below

Bilinear Kinematic Hardening Multilinear Kinematic Hardenin



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Considerations for Ratchetting (contd):

On the other hand, a single nonlinear kinematic model(n=1) for the

Chaboche model can capture ratchetting, as shown below.

1000

1500

In the figure on the left,the blue lines indicate a

500

sequence. Note that no

ratchetting occurs, and

it is a stable cycle.

-500

0

- 1. 50E -0 3 - 1. 00 E- 03 - 5. 00 E- 04 0. 00 E+ 00 5 .00 E- 04 1. 00 E- 03 1 .5 0E -0 3 2 .0 0E -0 3 2 .5 0E -0 3 3 .0 0E -03

Unsym

Symm

The red lines indicate a

nonsymmetric loading

se uence. Because the

-1000

plastic slopes aredifferent (due to value of

back stresses), plastic



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- s ra ns con nue o

accumulate.


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Considerations for Ratchetting (contd):

Ratchetting occurs due to the fact that the initial slope in compression

- -

unsymmetric, C-D is nearer to the limiting yield surface, so its slope ismore asymptotic.

C

A

B



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Considerations for Shakedown (contd):

One way to model shakedown is with at least two kinematic models in

. . One kinematic model will have i0, which will provide a ratchetting

effect, as shown previously.

On the other hand, another model will have i=0 to provide a stabilizationeffect. Recall that i=0 is equivalent to bilinear kinematic hardening, so

Together, the two models will provide ratchetting with stabilization after a

certain number of cycles. This is called shakedown.



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Customer Training MaterialF. Determining Chaboche Parameters

WB-Mechanical does not have a curve fitting routine for reading in

cyclic test data and determining Chaboche parameters directly.

modulus) and (rate of decrease of hardening) can be derived perprocedures outlined on 1st and 4th references on last slide. - , ,

symmetrically loaded at different strain amplitudes.



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Customer Training Material... Determining Chaboche Parameters

For each set of test data:

Determine kapproximately

from the elastic domain.

pl

k is usually half the elastic

domain.

strain range pl Determine the stressrange Plotting (/2-k) against

( pl/2) to estimate theasymptotic valuecorresponding to C/

C/



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Customer Training Material... Determining Chaboche Parameters

Using an appropriate hyperbolic

expression, fit the results to solve for

C1 and 1 in relation to /2-k.

=

2tanh

2 1

1

1

plCk

Data from three

separate tests

Note: Curve fitting procedures often benefit from reasonable initial

values. Since C1 is the initial hardening modulus, the slope after the



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y e s ress can e a en as an es ma e o 1 an roug e re a on

C1/1 determined in previous step, an initial value of 1 can be obtained.


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Customer Training MaterialReferences

Lemaitre, J. and Chaboche, J.L., Mechanics of Solid Materials,

Cambridge University Press, 1990.

, . .,

Cyclic Viscoplasticity, International Journal of Plasticity, Vol. 5, pp.247-302, 1989.

Chaboche, J.L., On Some Modifications of Kinematic Hardening to

Improve the Description of Ratchetting Effects, International Journal

of Plasticity, Vol. 7, pp. 661-678, 1991.

Chaboche Nonlinear Kinematic Hardening Model, Sheldon Imaoka,

Memo Number STI0805A, May 4, 2008



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Mechanical-Nonlin 13.0 App5A Chaboche

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