ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 ·...

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ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon, France [email protected] ATHENS Course MP06 Nonlinear Computational Mechanics March 16 to 20, 2009

Transcript of ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 ·...

Page 1: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ELASTO-PLASTICITY

Jean-Louis Chaboche

ONERA, 29 av. de la Division Leclerc92320 Châtillon, France

[email protected]

ATHENS Course MP06

Nonlinear Computational MechanicsMarch 16 to 20, 2009

Page 2: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �

Comportements non linéaires

moteur fusée

A 380

essai au solde l’A 380

moteur avion civil

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �

ConstructionGénie Civil

Automobile

talus

collecteur d’échappement

arrachement

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �

Pièces soumises à des sollicitations thermomécaniques sévères

rocketengineaubes de

turbine

collecteurs d’échappement

chambres de combustion

Page 5: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �

� GENERAL INTRODUCTION

� ELEMENTARY NOTIONS

� ELASTIC LIMIT CRITERIA

� FORMULATION OF RATE INDEPENDENT PLASTICITY

� PRANDT-REUSS INCREMENTAL PLASTICITY

� PRAGER’S LINEAR KINEMATIC HARDENING

� NOTION OF GENERALIZED STANDARD MODELS

� NON-LINEAR KINEMATIC HARDENING

Introduction and Elasto-plasticity

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �

Structural analysis with non-linearities

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Page 7: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity 2

Various scales in metallic materials and structures

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity :

<<

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Page 9: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity =

� GENERAL INTRODUCTION

� ELEMENTARY NOTIONS

� ELASTIC LIMIT CRITERIA

� FORMULATION OF RATE INDEPENDENT PLASTICITY

� PRANDT-REUSS INCREMENTAL PLASTICITY

� PRAGER’S LINEAR KINEMATIC HARDENING

� NOTION OF GENERALIZED STANDARD MODELS

� NON-LINEAR KINEMATIC HARDENING

Introduction and Elasto-plasticity

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity 34

Notations>σσσσ

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intrinsic : indicial :σσσσ

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�?

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��

��� ====�

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity 33

Invariants

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3 σσσσσσσσσσσσ −−−−====

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εεεεεεεεεεεε ========

Page 12: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity 3�

Characteristic tests in tension

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σσσσ

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σσσσ

σσσσ

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;�������

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity 3�

Multiaxial tests

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traction –

compression –

internal

pressure

traction –

compression –

torsion

Page 14: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity 3�

Notion of plastic strain

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σσσσ

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity 3�

�� εεεεεεεεεεεε ++++====���� ������ 0

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Notion of plastic strain

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity 3�

Notion of hardening

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity 32

Bauschinger effects

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity 3:

Rheological elementary models

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εεεεηηηησσσσ �====

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εεεεηηηησσσσ �====

�� σσσσσσσσσσσσ ≤≤≤≤≤≤≤≤−−−−

�����

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity 3=

Parallel and series assemblies

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σσσσ

F�σσσσ

εεεε

σσσσ�σσσσ

�σσσσ−−−−

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �4

σσσσ

F

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σσσσ)

F

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;�������Viscosity effects

Page 21: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �3

� GENERAL INTRODUCTION

� ELEMENTARY NOTIONS

� ELASTIC LIMIT CRITERIA

� FORMULATION OF RATE INDEPENDENT PLASTICITY

� PRANDT-REUSS INCREMENTAL PLASTICITY

� PRAGER’S LINEAR KINEMATIC HARDENING

� NOTION OF GENERALIZED STANDARD MODELS

� NON-LINEAR KINEMATIC HARDENING

Introduction and Elasto-plasticity

Page 22: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

��������������

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3F ==== (((( )))) �A3

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Elastic limit criterion = yield criterion

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3C σσσσσσσσσσσσσσσσ ========

Page 23: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

4�555�1 �D ≤≤≤≤−−−−==== σσσσ

/ � �����0

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&�����

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3σσσσ �σσσσ

von Mises and Tresca

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====>�����3

4 ==== (((( )))) 411�

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��D ====−−−−−−−−−−−− σσσσσσσσ

Page 24: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

(((( )))) 4�555�1�� ����D ≤≤≤≤−−−−==== σσσσφφφφ

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3σσσσ �σσσσ

von Mises, Tresca and Edelman-Drucker

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(((( )))) 4�555�1�� ����D ≤≤≤≤−−−−==== σσσσφφφφ

Page 25: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

Edelman-Drucker

������ ��������

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Page 26: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

�������� �

;��� ����� ��������

4�555�1��&� �D ≤≤≤≤−−−−++++==== σσσσαααασσσσ

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/ � �

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3σσσσ

�σσσσ�

Drucker – Prager

��&� σσσσ

Page 27: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �2

K����� � � ��C�

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Anisotropic criteria

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Page 28: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �:

.���1����3=�=��0

(((( )))) (((( )))) ��

� ≤≤≤≤−−−−−−−−==== �555�1C�C

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An isotropic non-symmetric criterion

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-300

-200

-100

0

100

200

300

-300 -200 -100 0 100 200 300

Barlat-Cazacu

Drucker

von Mises

Page 29: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �=

� GENERAL INTRODUCTION

� ELEMENTARY NOTIONS

� ELASTIC LIMIT CRITERIA

� FORMULATION OF RATE INDEPENDENT PLASTICITY

� PRANDT-REUSS INCREMENTAL PLASTICITY

� PRAGER’S LINEAR KINEMATIC HARDENING

� NOTION OF GENERALIZED STANDARD MODELS

� NON-LINEAR KINEMATIC HARDENING

Introduction and Elasto-plasticity

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �4

� ������ �� ����� �

� � �������� ���� � ��� �

� � ����� ����

� ;������� ����

� � �������� � ��� �

Rate-independent incremental plasticity theory

Context of theory:

Main ingredients :

� ������ ���� ���������

� �� � ���� ���������

� � 8��� ������ �����������

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �3

Hardening

;�������

σσσσ

εεεε

(�� ����� (�����

�������

������������

��8���������

����

������

���;�

������������ 0 σσσσ�

combination of :isotropic hardeningkinematic hardeningothers…

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

initial

��

�σσσσ

3σσσσ �σσσσ

����� ====

Isotropic hardening

��

��

��(

δ δ δ δ �(�

δ �(M�M�

σσσσ

�εεεε

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

�%O εεεε==== (�������3=�=

Kinematic hardening (linear)

O

initial

�σσσσ

3σσσσ �σσσσ

σσσσ

�εεεε

O

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

4 <<<<

4 ====

������

� � ��� 8�� ����������������� ��������

σσσσ∂∂∂∂∂∂∂∂====

40

<<<<∂∂∂∂∂∂∂∂ σσσσσσσσ

σσσσ

������

σσσσ�

4 <<<< �

40

≥≥≥≥∂∂∂∂∂∂∂∂ σσσσσσσσ

����4 ==== ������� � 8

4 >>>>

� ������� ����

Loading-unloading condition

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

&;� � 8�� �;� ������ ����������� � ��� ����� 8�; �;� ������

������������ ������� � ������ �;�� �;� � 8�� � �����

8�; ��� �;�� ����7�������������� �

40� ≥≥≥≥−−−− εεεεσσσσσσσσ �

4 ====

�εεεε�

σσσσ

Principle of maximal plastic work

Pσσσσ σσσσ�

σσσσ�εεεε�

Pσσσσ

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

% ��������8�; �;� ��?��.����� ��(������

� � λλλλσσσσ

λλλλεεεε ��� ====∂∂∂∂∂∂∂∂

====

����� �;� ����� � � �/�?��

����� ����� �����7��������������� ���;� ����� � �

���������� �;� ;��� ����� ���������0

������

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Concequency : Normality rule

4C

&�C

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∂∂∂∂∂∂∂∂====��������

����

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∂∂∂∂∂∂∂∂

σσσσσσσσ

4 <<<<

4 ====

������

σσσσ∂∂∂∂∂∂∂∂====

�σσσσ

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �2

��� ����� 8�; �;� ;������� �������������� �;� ��������������

4!0!

0

>

>

====∂∂∂∂∂∂∂∂

++++∂∂∂∂∂∂∂∂

==== ��� σσσσσσσσ

4 ====�

Consistency condition

4 <<<<������

σσσσσσσσ�

(((( )))) 4!� > ====σσσσ

(((( )))) 4�!!�� >> ====++++++++ σσσσσσσσ

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �:

Combination of the different rules

� ����� 0 � � λλλλσσσσ

λλλλεεεε ��� ====∂∂∂∂∂∂∂∂

====

;������� 0 (((( ))))>11 +�F+ σσσσλλλλ�� ====

� �������� 0 4+

F0�+

+

0

1

11

1

====∂∂∂∂∂∂∂∂

++++====∂∂∂∂∂∂∂∂

++++∂∂∂∂∂∂∂∂

==== λλλλσσσσσσσσσσσσ

�����

σσσσσσσσ

σσσσλλλλ �

� 0�;

3

+

F

0

1

1

====

∂∂∂∂∂∂∂∂

−−−−

∂∂∂∂∂∂∂∂

====1

�+

1F;

∂∂∂∂∂∂∂∂−−−−====

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�0�� �F;

3

� σσσσεεεε �� ====

� ;������� 8�; �� ������� � 8

������������

<<<<≥≥≥≥

====4 4

4 3� �F

�?�F?? ====

;������� ������� �����

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �=

4 ====

σσσσ∂∂∂∂∂∂∂∂====

σσσσ

4 <<<<������

4 ====

σσσσ∂∂∂∂∂∂∂∂==== �

σσσσ

4 <<<<������

4� ====�

� λλλλεεεε �� ====

�� λλλλσσσσ

λλλλεεεε ��� ====∂∂∂∂∂∂∂∂

====

Associated plasticity : Non-Associated plasticity :

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �4

Non-associated plasticity : example of soils mechanics

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����� �

(((( )))) 4+� 1 ≤≤≤≤σσσσ)����� � �� 0

� ������� �3�� ��������� � 8

(((( )))) 4&�� ====εεεε�

�7���/�� ���� ��� �� 0

7���0�

�εεεε�σσσσ∂∂∂∂

∂∂∂∂

4

�C

σσσσ&��3 ====

1

�J�4

��J�4

(������ � 8� ������ 0 (((( )))) 4+�� 1 ====σσσσ

σσσσλλλλλλλλεεεε

∂∂∂∂∂∂∂∂

========�

�� ��� 4 ==== 40

≥≥≥≥

∂∂∂∂∂∂∂∂ σσσσσσσσ

� ���

4� ====εεεε� 4 <<<< 40

<<<<

∂∂∂∂∂∂∂∂ σσσσσσσσ

� �

F������� ����� 0 � ���? 7�;�/ ��� 5�5�5555+1 ====�

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �3

Elasto-plastic tangent operator (non-associated flow)

� 8����� ��0

� ����� ��;� ������ � �� 0

(((( ))))�0 εεεεεεεεσσσσ −−−−ΛΛΛΛ====F 1�M����80

εεεελλλλ �� 00;

3ΛΛΛΛ======== �00;; � ΛΛΛΛ++++====

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�0;

3

� σσσσεεεε �� ====

;������� �����

σσσσ∂∂∂∂∂∂∂∂====

σσσσ∂∂∂∂∂∂∂∂====�

(((( )))) ====−−−−−−−−ΛΛΛΛ====∂∂∂∂∂∂∂∂

++++∂∂∂∂∂∂∂∂

==== λλλλεεεεεεεελλλλσσσσσσσσ

�������

1

1 ;00+

F0

4;�0000 � ====−−−−ΛΛΛΛ−−−−ΛΛΛΛ==== λλλλλλλλεεεε ���

% �������� 0

(((( )))) (((( ))))εεεεεεεελλλλεεεεεεεεεεεεσσσσ ������� 0,0�0,;

30�000

� −−−−ΛΛΛΛ====ΛΛΛΛ−−−−ΛΛΛΛ====−−−−ΛΛΛΛ====

(((( )))) εεεεεεεε �� 00�0;

30 ΛΛΛΛΛΛΛΛ−−−−ΛΛΛΛ==== ⊗⊗⊗⊗

εεεεσσσσ �� 0�

==== (((( )))) ΛΛΛΛΛΛΛΛ−−−−ΛΛΛΛ==== ⊗⊗⊗⊗ 0�0;

3

� � ====

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

� ����� 0

� �������� 0

��������

εεεελλλλ �� 0��

�====

1�1�1�>1�> ��� εεεεµµµµεεεε �� ====ΛΛΛΛ

Perfectly plastic behavior (associated plasticity)

� � λλλλσσσσ

λλλλεεεε ��� ====∂∂∂∂∂∂∂∂

====

40�0

========∂∂∂∂∂∂∂∂

==== σσσσσσσσσσσσ

���

���������� ��������� ������������������������������������ ��������������� �����

������� ���������� �������������������������������

4�00�00���00�0�� ====ΛΛΛΛ−−−−ΛΛΛΛ====−−−−ΛΛΛΛ==== λλλλεεεεεεεεεεεεσσσσ �����

� �� �� �������� ��� / � ���������� � 0 �� >1�>�11�>>1� δδδδδδδδδδδδδδδδµµµµδδδδλδλδλδλδ ++++++++====ΛΛΛΛ

µµµµ��� 1�>1�> ====ΛΛΛΛ

�00�

00�

ΛΛΛΛ

ΛΛΛΛ====

εεεελλλλ

��

�D

>>

��

σσσσ====

���C σσσσσσσσ −−−−====

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

�� �;��������

������������

����

����

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−−−−−−−−====

�3

�3

3

��

σσσσ

Flow direction associated with von Mises criterion

σσσσσσσσ

σσσσ ∂∂∂∂∂∂∂∂

∂∂∂∂

∂∂∂∂====

∂∂∂∂∂∂∂∂

====�

0�

�D

σσσσσσσσ ====�D

�D

>>

��

σσσσ====

��D σσσσσσσσ −−−−====

>

1�

1�

�D>

��

σσσσσσσσ

∂∂∂∂∂∂∂∂

∂∂∂∂

∂∂∂∂====

(((( )))) 1�>>1�>�1�>1�

>

1�

3

3�

�δδδδδδδδδδδδδδδδδδδδδδδδ

σσσσ++++++++========

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σσσσ====

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��

�3

�3

3

� σσσσ����������������

����

����

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−−−−−−−−====

Page 44: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

� GENERAL INTRODUCTION

� ELEMENTARY NOTIONS

� ELASTIC LIMIT CRITERIA

� FORMULATION OF RATE INDEPENDENT PLASTICITY

� PRANDT-REUSS INCREMENTAL PLASTICITY

� PRAGER’S LINEAR KINEMATIC HARDENING

� NOTION OF GENERALIZED STANDARD MODELS

� NON-LINEAR KINEMATIC HARDENING

Introduction and Elasto-plasticity

Page 45: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

initial

��

�σσσσ

3σσσσ �σσσσ

����� ====

Isotropic hardening

��

��

��(

δ δ δ δ �(�

δ �(M�M�

σσσσ

�εεεε

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

Incremental Prandtl-Reuss plasticity/ � �������<���� �� �� ;������� <������ ����� ��������

����� ====

���������� ������������ 0

4�&�1� �D ≤≤≤≤−−−−−−−−==== σσσσ

� �������� 0 4&�&�B1����B�0� ====−−−−−−−−==== ���� σσσσ

�D

�D �

� �

σσσσσσσσσσσσ

σσσσ====

∂∂∂∂

∂∂∂∂====

∂∂∂∂∂∂∂∂

====

�&�&�B10�� �F���B�

3� ��� −−−−==== σσσσεεεε

��������������������

λλλλεεεεεεεε ���� ======== ��0

��

&�&�B10�� �F���B�

3��� −−−−==== σσσσλλλλ

�� εεεεεεεεεεεε ��� ++++====

3��&�))

3� σσσσννννσσσσ

ννννεεεε ��� −−−−++++

====1

����� ====

�Dσσσσ

���B�;� ====33σσσσ====

�33εεεε====

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �2

Ramberg-Osgood or Hollomon equation of tensile curve

����� ������� � 80

4�&�1� �D ====−−−−−−−−==== σσσσ

3�D �Q1 ++++====σσσσ

3�Q1 εεεεσσσσ ++++====

============−−−−

3

� �

Q���B�;

3�D

Q

1

Q−−−−

������������

������������

−−−−====

σσσσ

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σσσσ

3�4

�44�����

��4

344

�44

45� 35�354

�εεεε�

�(�

4

����� ====

1

����

�3�,��� �4L%

���������� 0

�(�3�:1 ====��5� ====

�(����Q ====

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �:

Isotropic hardening and cyclic loading

σσσσ

εεεε

����������������

����������������

� � �������� ���/�

��� �����������

�����������������

������������� ����������������������

����������� ������� �

���7���� ��7�������� ������� �

��������� ����������������������

���7���� ��7����G ��� ����� ������ H����� ���� ���

�����������������

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �=

Prandtl-Reuss plasticity at variable temperature

σσσσ

)?���� 0��� ����������������� �������� ���������� 0

�&�1

1

������

&

�&�1�D ====σσσσ

������

���&�1 ++++

4�&�1� �D ≤≤≤≤−−−−−−−−==== σσσσ �&�&�B10�� �F���B�

3� ��� −−−−==== σσσσεεεε

��������������������

� ����

��� �

Page 50: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �4

� GENERAL INTRODUCTION

� ELEMENTARY NOTIONS

� ELASTIC LIMIT CRITERIA

� FORMULATION OF RATE INDEPENDENT PLASTICITY

� PRANDT-REUSS INCREMENTAL PLASTICITY

� PRAGER’S LINEAR KINEMATIC HARDENING

� NOTION OF GENERALIZED STANDARD MODELS

� NON-LINEAR KINEMATIC HARDENING

Introduction and Elasto-plasticity

Page 51: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �3

�%O εεεε==== (�������3=�=

Kinematic hardening (linear)

O

initial

�σσσσ

3σσσσ �σσσσ

σσσσ

�εεεε

O

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ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

�44��(�

�σσσσ

3σσσσ

�44��(�

�� �

� ����

����

�σσσσ

3σσσσ

344��(�

3

��� � � �����

����

+� %�3�

3

3 −−−−−−−− � 3%���

3 −−−−

������������������� ��������!�"�������������� � ��� � �����5��3=:�

Example of kinematic hardening in some stainless steels

Page 53: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

Incremental plasticity with Linear Kinematic Hardening/ � �������<���1����� ;������� <������ ����� ��������

�%

�O εεεε====

���������� ������������ 0

4�&�1�O�C ≤≤≤≤−−−−−−−−==== σσσσ

� �������� 0

4&�&�B1%�

�0�

� ====−−−−−−−−==== ���� εεεεσσσσ

�O�C

BO�

� �

−−−−−−−−

====∂∂∂∂∂∂∂∂

====σσσσσσσσ

�&�&�B10�� �F%

3� ��� −−−−==== σσσσεεεε

��������������������

&�&�B10�� �F%

3��� −−−−==== σσσσλλλλ

���

�� ====εεεε

1

4

33σσσσ

�33εεεε

1�3333�

�%OO εεεε========

λλλλεεεεεεεε ���� ======== ��0

��

Page 54: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

� GENERAL INTRODUCTION

� ELEMENTARY NOTIONS

� ELASTIC LIMIT CRITERIA

� FORMULATION OF RATE INDEPENDENT PLASTICITY

� PRANDT-REUSS INCREMENTAL PLASTICITY

� PRAGER’S LINEAR KINEMATIC HARDENING

� NOTION OF GENERALIZED STANDARD MODELS

� NON-LINEAR KINEMATIC HARDENING

Introduction and Elasto-plasticity

Page 55: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

Thermodynamic approach – Internal variables

���

�!00 ααααεεεεσσσσαααα

ααααψψψψρρρρεεεε

εεεεψψψψρρρρψψψψρρρρ ����� ++++====

∂∂∂∂∂∂∂∂

++++∂∂∂∂∂∂∂∂

====

� � � �� ������/���7����� 7���/�7��������������;�� ���� �� ��� ������

������������ �����������������������

� �;� �� �� � ������� /���7���� �� 8� � �;� �?���� �� � � ��

������ � ������

��� �� ααααεεεερψρψρψρψρψρψρψρψ ====

� /���� �� ������� �;� �� ��� ������ ����� �;� �� ��� � �� ���� 0

ψψψψ

ψψψψ

��!

ααααψψψψρρρρ

εεεεψψψψρρρρσσσσ

∂∂∂∂∂∂∂∂

====∂∂∂∂∂∂∂∂

====

� 8� �������;�� �;��� ������/���7���������� ���� � ���������� �;�

8; �� ���� ;�� ��

Page 56: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

Definition of intrinsic dissipation

(((( )))) (((( ))))�����

!0000 ααααεεεεσσσσεεεεσσσσεεεεσσσσψψψψεεεεσσσσ ������� ++++−−−−++++====−−−−====ΦΦΦΦ

(((( )))) (((( ))))��

� �N!�K ααααεεεεσσσσ −−−−========

� �������� ������ ��� �;� � ������ 7��8��� �;� ��� ����� ��;�����

������ ��� �;� ��� ����� �� ��� ������

� �� � �������� ���;��� �������� � ������

� �7/ ���� �;��� � � ������ ��������� ������ 7�;�/ ��

� &;� '������N�� *��������� ������������������� ������������ ������ 0

� ���� � (��� ��C�� R������ N�� ��3

�;�������� � C����3 ���S��

K

ΦΦΦΦ�

NK!0 ���

���� ====−−−−====ΦΦΦΦ ααααεεεεσσσσ

Page 57: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �2

� ������0��?����������8 �1 ������� ����� ������������� 0

F���0����� � ������� �;� �?��

�?��� ����� �;� � �������

Plastic flow – Generalised normality

�0 εεεεσσσσ �� ====ΦΦΦΦ

NK!0 ���

���� ====−−−−====ΦΦΦΦ ααααεεεεσσσσ

� )?���� �� �?�������� ��������� R������� �������� 8����

8� ��� � ��?����;� ������ ������ ��0

� S� ;�/��� ��?���������������� �;� � ������� R�8� ������0

� � λλλλσσσσ

λλλλεεεε ��� ====∂∂∂∂∂∂∂∂

====K

N

∂∂∂∂∂∂∂∂

==== λλλλ��

NK �

/���;�� ���σσσσ∂∂∂∂∂∂∂∂$

�0 εεεεσσσσ �

�!$ ∂∂∂∂∂∂∂∂

(((( )))) (((( ))))����

� !� !0!�$ σσσσλλλλααααεεεεσσσσσσσσ ��� −−−−−−−−====

4�!�� � ≤≤≤≤σσσσ

4�!�� � ≤≤≤≤σσσσ

� ��

�!

∂∂∂∂∂∂∂∂

−−−−==== λλλλαααα ��

� " ����� ��������� � 8��� ��;������� �/ ��� � 0

Page 58: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �:

Example : linear kinematic hardening in 1D

������F

3)

3��� εεεεεεεεεεεεεεεεψψψψ ++++====

� ������������ � 8����

� ������ ��� �;� ������ ��� �� ���;� ����� ����8�;

� ������/���7������������������� �;� ����� ������� �F��

� ������ ������ ��J���� ����� ������ T �� ��� ������ 0�

�εεεε

4O�O�� � ====−−−−−−−−==== σσσσσσσσσσσσ�O��

� −−−−==== σσσσεεεε �����

�εεεε

(((( )))) (((( )))) ������OO εεεεεεεεσσσσεεεεεεεεσσσσεεεεσσσσεεεεσσσσψψψψεεεεσσσσ ��������� −−−−====++++−−−−++++====−−−−====ΦΦΦΦ

(((( )))) ��OO �� ���� σσσσσσσσεεεεσσσσ ====−−−−====−−−−====ΦΦΦΦ

� *� ��� ������ � �������� 8�; ��/������������ � 8��F�54 εεεε

σσσσ)

F

�σσσσ

Page 59: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �=

Example : linear isotropic hardening in 1D

������F

3)

3��� εεεεεεεεεεεεεεεεψψψψ ++++====� $��� ������ 8����

� ������� ������ ��J���� ����� ������ T �� ��� ������ 0�

(((( )))) ����� �� �������� σσσσσσσσεεεεσσσσεεεεσσσσψψψψεεεεσσσσ ====−−−−====−−−−−−−−====−−−−====ΦΦΦΦ

�)εεεε

εεεεψψψψσσσσ ====

∂∂∂∂∂∂∂∂

====

� &;� ������� /���7���������� ����� ��� ����� ������������ ��αααα � �!

� -��� ����� � 8���� 4O�O�� � ====−−−−−−−−==== σσσσσσσσσσσσ

� *� ��� ������ ���� � 7� �������� ��������� � � ������ ����������F�54 �

�F�

� ====∂∂∂∂∂∂∂∂

====ψψψψ

Page 60: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �4

Hardening state variables consistent with GSM

� -��� ����� � 8�; � 7��� 1����� <�� �� �� ;�������

� ��� ����������/���7������� ����� 8�; ���

4�O���O�� � ====−−−−−−−−−−−−==== σσσσσσσσσσσσ

��

O

εεεε

σσσσλλλλλλλλαααα ���� ====

∂∂∂∂∂∂∂∂

====∂∂∂∂∂∂∂∂

−−−−====� Q����� ;������� 0

� �� �� �� ;������� 0

� ���������/���7���� ��1����� ;������� �����������/��� � �����������

��

� ���� ========

∂∂∂∂∂∂∂∂

−−−−==== λλλλλλλλ

�εεεε

Oαααα

�αααα

Page 61: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �3

� GENERAL INTRODUCTION

� ELEMENTARY NOTIONS

� ELASTIC LIMIT CRITERIA

� FORMULATION OF RATE INDEPENDENT PLASTICITY

� PRANDT-REUSS INCREMENTAL PLASTICITY

� PRAGER’S LINEAR KINEMATIC HARDENING

� NOTION OF GENERALIZED STANDARD MODELS

� NON-LINEAR KINEMATIC HARDENING

Introduction and Elasto-plasticity

Page 62: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

Non-Linear Kinematic Hardening

ααααααααψψψψ 0%�

3� ====

O

σσσσ

�εεεε

� ������� ���/��

�.�

��� ααααεεεεαααα −−−−====αααααααα

ψψψψ%

�O

� ====∂∂∂∂

∂∂∂∂====

���������� D������� 0

�� ������� ���/��

σσσσ

�εεεε

(((( ))))�� 0% ααααααααψψψψ ====O ���� � �� ����� �

Page 63: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

!���� ���$������1��3=��

�O.%O�

���� −−−−==== εεεε

;�������

�(������

�����

��� /���

���� ��� ����� � �� ��� � ������� � 0

�� O.%O εεεεεεεε ��� −−−−====

(((( ))))(((( ))))�

�.�?�.

%O

.

%O εεεεεεεενννννννν −−−−−−−−����

����

��������

−−−−++++====

Non-Linear Kinematic Hardening

��0

�� εεεεεεεε ��� ====

O

σσσσ

�εεεε

%A.

Page 64: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

���7���� ������0

(((( ))))(((( ))))�

�.�?�.

%O

.

%O εεεεεεεενννννννν −−−−−−−−����

����

��������

−−−−++++====

������������

������������

∆∆∆∆====

�.&��;

.

%O

εεεε

1�

O

�++++==== ∆∆∆∆σσσσ∆∆∆∆

���������0

Non-Linear Kinematic Hardening

O

σσσσ

�εεεε

%A.

O

σσσσ∆∆∆∆

Page 65: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

�U��A�εεεε∆∆∆∆

σσσσ∆∆∆∆

"���"�%�=4

&!�+

+��2=��% 7���!���

���"%.�3�

Cyclic curves : Non-Linear Kinematic Hardening

Page 66: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity ��

����������������

−−−−==== −−−− ��444

3 �3�4Oεεεε

����������������

−−−−==== −−−− ��44

� �3344Oεεεε

O�J���44�ε ε ε ε �

O�J�O3<�O�

<�O�

1�J�344

σσσσσσσσ

344

�44

�44

434 � �εεεε

34��44) �

==== )�����J���44

∞∞∞∞

����====

OO

�O.%O �

��� −−−−==== εεεε

Multi - Kinematic Hardening %;�7 �;� � � ���������3=:3

Page 67: ELASTO-PLASTICITYmms2.ensmp.fr/.../archives-transparents/JLC-Plasticity.pdf · 2014-01-14 · ELASTO-PLASTICITY Jean-Louis Chaboche ONERA, 29 av. de la Division Leclerc 92320 Châtillon,

ATHENS – Course MP06 – 16 – 20 March 2009 Elasto-plasticity �2

General anisotropic hardening

(((( )))) (((( )))) 4�&�1�O0�0O ≤≤≤≤−−−−−−−−−−−−−−−−==== σσσσσσσσ O

�� �� �� ;������� 0

������� ���

Q����� ;������� 0

��������� ��

!�� �� �� ;������� 0

�� ��� ��

.�� �� ��� ;������� 5�5�5

���; ���1 ���� ��

O

� O �

��� �� �� 0�� ��� �1������ �� ��

��� �� ���

B33σσσσ

B��σσσσ

B��σσσσ

4