Intro Stabilité Localisation Conclusions

Matthieu Mazière1

Description

Niveau: Supérieur
Intro Stabilité Localisation Conclusions STRAIN LOCALISATION AND OTHER INSTABILITIES Matthieu Mazière1, S. Forest1 1Mines ParisTech 17. Mars 2010

• homogeneous deformation


• general bifurcation


• localisation conclusions


• strain localisation


• stability finite strain


• finite strain


• formulation mechanical problem

Sujets

• Finite strain theory

Informations

• Publié par: chaeh
• Publié le: 01 mars 2010
• Nombre de lectures: 21
• Langue: Français
• Poids de l'ouvrage: 4 Mo

Extrait

Intro Stabilité Localisation Conclusions
STRAIN LOCALISATION AND OTHER INSTABILITIES
1 1Matthieu Mazière , S. Forest
1Mines ParisTech
17. Mars 2010Intro Stabilité Localisation Conclusions
OUTLINE
INTRODUCTION - NON HOMOGENEOUS DEFORMATIONS
FINITE STRAIN UNIQUENESS AND STABILITY
Finite strain formulation
Mechanical problem
Uniqueness and stability criteria
Application
STRAIN LOCALISATION
General bifurcation modes in elastoplastic materials
Compatible modes
Stability / ellipticity of the boundary value problem
Linear perturbation method
CONCLUSIONS
Bilan
Autres localisationsIntro Stabilité Localisation Conclusions
OUTLINE
INTRODUCTION - NON HOMOGENEOUS DEFORMATIONS
FINITE STRAIN UNIQUENESS AND STABILITY
Finite strain formulation
Mechanical problem
Uniqueness and stability criteria
Application
STRAIN LOCALISATION
General bifurcation modes in elastoplastic materials
Compatible modes
Stability / ellipticity of the boundary value problem
Linear perturbation method
CONCLUSIONS
Bilan
Autres localisationsNecking
Localisation
Intro Stabilité Localisation Conclusions
NON HOMOGENEOUS DEFORMATION PHENOMENA...
...AND UNEXPECTED
BucklingLocalisation
Intro Stabilité Localisation Conclusions
NON HOMOGENEOUS DEFORMATION PHENOMENA...
...AND UNEXPECTED
Buckling
NeckingIntro Stabilité Localisation Conclusions
NON HOMOGENEOUS DEFORMATION PHENOMENA...
...AND UNEXPECTED
Buckling
Necking
Localisationv(x) = A cos(!x) +B sin(!x)
v(0) = 0
CL :
v(L) = 0

A = 0
A cos(!L) +B sin(!L) = 0

1 0 A 0
=
cos(!L) sin(!L) B 0Fx
f g =int | {z }
v(x)z
MM = v(x)Ffz det(M) = 0! Compression
00M = EI v (x)fz Gz det(M) = 0! Buckling
00 2v (x) +! v(x) = 0
s ! sin(!L)!!L = k
2F EIGzavec,! = F =c1 2EIGz L
Intro Stabilité Localisation Conclusions
BUCKLING - LIMIT LOAD
6v(x) = A cos(!x) +B sin(!x)
v(0) = 0
CL :
v(L) = 0

A = 0
A cos(!L) +B sin(!L) = 0

1 0 A 0
=
cos(!L) sin(!L) B 0
| {z }
M
det(M) = 0! Compression
det(M) = 0! Buckling
! sin(!L)!!L = k
2 EIGz
F =c1 2L
Intro Stabilité Localisation Conclusions
BUCKLING - LIMIT LOAD

Fx
f g =int v(x)z
M = v(x)Ffz
00M = EI v (x)fz Gz
00 2v (x) +! v(x) = 0
s
F
avec,! =
EIGz
6Intro Stabilité Localisation Conclusions
BUCKLING - LIMIT LOAD
v(x) = A cos(!x) +B sin(!x)
v(0) = 0
CL :
v(L) = 0

A = 0
A cos(!L) +B sin(!L) = 0

1 0 A 0
=
cos(!L) sin(!L) B 0Fx
f g = | {z }int v(x)z
MM = v(x)Ffz det(M) = 0! Compression
00M = EI v (x)fz Gz det(M) = 0! Buckling
00 2v (x) +! v(x) = 0
s ! sin(!L)!!L = k
2F EIGzavec,! = F =c1 2EI LGz
6Intro Stabilité Localisation Conclusions
NECKING
Elasticity
F F
1
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