![Page 1: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/1.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Formulations variationnelles utilisant lesbipotentiels
Marius Buliga1 Gery de Saxce2 Claude Vallee3
1 ”Simion Stoilow” Institute of Mathematics of the RomanianAcademy, Bucharest
2 LML (UMR CNRS 8107), Villeneuve d’Ascq3 LMS (UMR CNRS 6610), Poitiers
10eme Colloque Franco-Roumain de Mathematiques AppliqueesPoitiers, 26-31 aout 2010
August 24, 2010
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 2: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/2.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Plan de l’expose
Standard materials
Implicit standard materials. Bipotentials
Applications of bipotentials in mechanics
The construction problem
Blurred constitutive laws
The model of Berga & de Saxce
Conclusion
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 3: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/3.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Standard materials
I Standard Material (Halphen & Nguyen Quoc Son, 1975)uses Moreau work in Convex Analysis
I Φ – convex dissipation potential defining a constitutive law by:y ∈ ∂Φ(x) OR x ∈ ∂Φ∗(y) OR Φ(x) + Φ∗(y) = 〈x , y〉
I y is a stress variable, x a strain rate variable, and 〈x , y〉denotes the duality product between them.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 4: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/4.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Standard materials
I Standard Material (Halphen & Nguyen Quoc Son, 1975)uses Moreau work in Convex Analysis
I Φ – convex dissipation potential defining a constitutive law by:y ∈ ∂Φ(x) OR x ∈ ∂Φ∗(y) OR Φ(x) + Φ∗(y) = 〈x , y〉
I y is a stress variable, x a strain rate variable, and 〈x , y〉denotes the duality product between them.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 5: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/5.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Standard materials
I Standard Material (Halphen & Nguyen Quoc Son, 1975)uses Moreau work in Convex Analysis
I Φ – convex dissipation potential defining a constitutive law by:y ∈ ∂Φ(x) OR x ∈ ∂Φ∗(y) OR Φ(x) + Φ∗(y) = 〈x , y〉
I y is a stress variable, x a strain rate variable, and 〈x , y〉denotes the duality product between them.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 6: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/6.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Standard materials
I X and Y are two spaces with duality 〈·, ·〉 → R
I Φ : X → R ∪ +∞ convex, proper, lsc function
I Φ∗ : X → R ∪ +∞ is the Fenchel-Moreau conjugate:
Φ∗(y) = sup〈x , y〉 − Φ(x) | x ∈ X
I ∂Φ is the subdifferential of Φ:
∂Φ(x) = y ∈ X ∗ | Φ(ξ)− Φ(x) ≥ 〈ξ − x , y〉
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 7: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/7.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Standard materials
I X and Y are two spaces with duality 〈·, ·〉 → R
I Φ : X → R ∪ +∞ convex, proper, lsc function
I Φ∗ : X → R ∪ +∞ is the Fenchel-Moreau conjugate:
Φ∗(y) = sup〈x , y〉 − Φ(x) | x ∈ X
I ∂Φ is the subdifferential of Φ:
∂Φ(x) = y ∈ X ∗ | Φ(ξ)− Φ(x) ≥ 〈ξ − x , y〉
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 8: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/8.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Standard materials
I X and Y are two spaces with duality 〈·, ·〉 → R
I Φ : X → R ∪ +∞ convex, proper, lsc function
I Φ∗ : X → R ∪ +∞ is the Fenchel-Moreau conjugate:
Φ∗(y) = sup〈x , y〉 − Φ(x) | x ∈ X
I ∂Φ is the subdifferential of Φ:
∂Φ(x) = y ∈ X ∗ | Φ(ξ)− Φ(x) ≥ 〈ξ − x , y〉
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 9: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/9.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Standard materials
I X and Y are two spaces with duality 〈·, ·〉 → R
I Φ : X → R ∪ +∞ convex, proper, lsc function
I Φ∗ : X → R ∪ +∞ is the Fenchel-Moreau conjugate:
Φ∗(y) = sup〈x , y〉 − Φ(x) | x ∈ X
I ∂Φ is the subdifferential of Φ:
∂Φ(x) = y ∈ X ∗ | Φ(ξ)− Φ(x) ≥ 〈ξ − x , y〉
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 10: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/10.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Standard materials
I Fenchel inequality: Φ(x) + Φ∗(y) ≥ 〈x , y〉
y ∈ ∂Φ(x) ⇐⇒ x ∈ ∂Φ∗(y) ⇐⇒ Φ(x) + Φ∗(y) = 〈x , y〉
II Remark that if we denote b(x , y) = Φ(x) + Φ∗(y) then theFenchel inequality becomes: b(x , y) ≥ 〈x , y〉
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 11: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/11.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Standard materials
I Fenchel inequality: Φ(x) + Φ∗(y) ≥ 〈x , y〉
y ∈ ∂Φ(x) ⇐⇒ x ∈ ∂Φ∗(y) ⇐⇒ Φ(x) + Φ∗(y) = 〈x , y〉
II Remark that if we denote b(x , y) = Φ(x) + Φ∗(y) then theFenchel inequality becomes: b(x , y) ≥ 〈x , y〉
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 12: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/12.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Standard materials
I Fenchel inequality: Φ(x) + Φ∗(y) ≥ 〈x , y〉
y ∈ ∂Φ(x) ⇐⇒ x ∈ ∂Φ∗(y) ⇐⇒ Φ(x) + Φ∗(y) = 〈x , y〉
II Remark that if we denote b(x , y) = Φ(x) + Φ∗(y) then theFenchel inequality becomes: b(x , y) ≥ 〈x , y〉
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 13: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/13.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Implicit standard materials. Bipotentials
I b = b(x,y) is a dissipation bipotential defining a implicitconstitutive law
y ∈ ∂b(·, y)(x)
I b : X × Y → R ∪ +∞ is bi-convex, bi-lsc b(x , y) ≥ 〈x , y〉I y ∈ ∂b(·, y)(x) ⇐⇒ x ∈ ∂b(x , ·)(y) ⇐⇒ b(x , y) = 〈x , y〉
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 14: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/14.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Implicit standard materials. Bipotentials
I b = b(x,y) is a dissipation bipotential defining a implicitconstitutive law
y ∈ ∂b(·, y)(x)
I b : X × Y → R ∪ +∞ is bi-convex, bi-lsc b(x , y) ≥ 〈x , y〉
I y ∈ ∂b(·, y)(x) ⇐⇒ x ∈ ∂b(x , ·)(y) ⇐⇒ b(x , y) = 〈x , y〉
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 15: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/15.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Implicit standard materials. Bipotentials
I b = b(x,y) is a dissipation bipotential defining a implicitconstitutive law
y ∈ ∂b(·, y)(x)
I b : X × Y → R ∪ +∞ is bi-convex, bi-lsc b(x , y) ≥ 〈x , y〉I y ∈ ∂b(·, y)(x) ⇐⇒ x ∈ ∂b(x , ·)(y) ⇐⇒ b(x , y) = 〈x , y〉
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 16: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/16.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Implicit standard materials. Bipotentials
I b : X × Y → R ∪ +∞ is bi-convex, bi-lsc b(x , y) ≥ 〈x , y〉
I y ∈ ∂b(·, y)(x) ⇐⇒ x ∈ ∂b(x , ·)(y) ⇐⇒ b(x , y) = 〈x , y〉
I particular case: separable bipotential b(x , y) = Φ(x) + Φ∗(y)
I funny example: Cauchy bipotential b(x , y) = ‖x‖‖y‖
y = λx , λ ≥ 0 ⇐⇒ x = µy , µ ≥ 0 ⇐⇒ ‖x‖‖y‖ = 〈x , y〉I (Cauchy-Bunyakovsky-Schwarz inequality)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 17: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/17.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Implicit standard materials. Bipotentials
I b : X × Y → R ∪ +∞ is bi-convex, bi-lsc b(x , y) ≥ 〈x , y〉
I y ∈ ∂b(·, y)(x) ⇐⇒ x ∈ ∂b(x , ·)(y) ⇐⇒ b(x , y) = 〈x , y〉
I particular case: separable bipotential b(x , y) = Φ(x) + Φ∗(y)
I funny example: Cauchy bipotential b(x , y) = ‖x‖‖y‖
y = λx , λ ≥ 0 ⇐⇒ x = µy , µ ≥ 0 ⇐⇒ ‖x‖‖y‖ = 〈x , y〉I (Cauchy-Bunyakovsky-Schwarz inequality)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 18: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/18.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Implicit standard materials. Bipotentials
I b : X × Y → R ∪ +∞ is bi-convex, bi-lsc b(x , y) ≥ 〈x , y〉
I y ∈ ∂b(·, y)(x) ⇐⇒ x ∈ ∂b(x , ·)(y) ⇐⇒ b(x , y) = 〈x , y〉
I particular case: separable bipotential b(x , y) = Φ(x) + Φ∗(y)
I funny example: Cauchy bipotential b(x , y) = ‖x‖‖y‖
y = λx , λ ≥ 0 ⇐⇒ x = µy , µ ≥ 0 ⇐⇒ ‖x‖‖y‖ = 〈x , y〉I (Cauchy-Bunyakovsky-Schwarz inequality)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 19: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/19.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Implicit standard materials. Bipotentials
I b : X × Y → R ∪ +∞ is bi-convex, bi-lsc b(x , y) ≥ 〈x , y〉
I y ∈ ∂b(·, y)(x) ⇐⇒ x ∈ ∂b(x , ·)(y) ⇐⇒ b(x , y) = 〈x , y〉
I particular case: separable bipotential b(x , y) = Φ(x) + Φ∗(y)
I funny example: Cauchy bipotential b(x , y) = ‖x‖‖y‖
y = λx , λ ≥ 0 ⇐⇒ x = µy , µ ≥ 0 ⇐⇒ ‖x‖‖y‖ = 〈x , y〉I (Cauchy-Bunyakovsky-Schwarz inequality)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 20: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/20.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Implicit standard materials. Bipotentials
I b : X × Y → R ∪ +∞ is bi-convex, bi-lsc b(x , y) ≥ 〈x , y〉
I y ∈ ∂b(·, y)(x) ⇐⇒ x ∈ ∂b(x , ·)(y) ⇐⇒ b(x , y) = 〈x , y〉
I particular case: separable bipotential b(x , y) = Φ(x) + Φ∗(y)
I funny example: Cauchy bipotential b(x , y) = ‖x‖‖y‖
y = λx , λ ≥ 0 ⇐⇒ x = µy , µ ≥ 0 ⇐⇒ ‖x‖‖y‖ = 〈x , y〉I (Cauchy-Bunyakovsky-Schwarz inequality)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 21: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/21.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Applications in mechanics
I contact with friction - de Saxce & Feng (1991)
b(v , f ) =
µfn‖vt‖ if f ∈ Kµ, vn ≤ 0+∞ otherwise
I non-associated Drucker-Prager model - Berga & de Saxce(1994)
b(εp, σ) =
C1ε
pm + C2(σm − c
tan φ)‖εp‖ if σ ∈ K , εp ∈ K ′
+∞ otherwise
εpm = tr εp , σm = tr σ , C1 =
c
tanφ, C2 = kd(tan θ − tanφ)
I cam-clay - de Saxce (1995), coaxial laws - Vallee et al. (1997)Lemaitre plastic ductile damage law - Bodoville (1999)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 22: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/22.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Applications in mechanics
I contact with friction - de Saxce & Feng (1991)
b(v , f ) =
µfn‖vt‖ if f ∈ Kµ, vn ≤ 0+∞ otherwise
I non-associated Drucker-Prager model - Berga & de Saxce(1994)
b(εp, σ) =
C1ε
pm + C2(σm − c
tan φ)‖εp‖ if σ ∈ K , εp ∈ K ′
+∞ otherwise
εpm = tr εp , σm = tr σ , C1 =
c
tanφ, C2 = kd(tan θ − tanφ)
I cam-clay - de Saxce (1995), coaxial laws - Vallee et al. (1997)Lemaitre plastic ductile damage law - Bodoville (1999)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 23: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/23.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Applications in mechanics
I contact with friction - de Saxce & Feng (1991)
b(v , f ) =
µfn‖vt‖ if f ∈ Kµ, vn ≤ 0+∞ otherwise
I non-associated Drucker-Prager model - Berga & de Saxce(1994)
b(εp, σ) =
C1ε
pm + C2(σm − c
tan φ)‖εp‖ if σ ∈ K , εp ∈ K ′
+∞ otherwise
εpm = tr εp , σm = tr σ , C1 =
c
tanφ, C2 = kd(tan θ − tanφ)
I cam-clay - de Saxce (1995), coaxial laws - Vallee et al. (1997)Lemaitre plastic ductile damage law - Bodoville (1999)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 24: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/24.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
I MΦ = (x , y) | y ∈ ∂Φ(x) maximal cyclically monotone
I Rockafellar theorem: The following are equivalent: -M ⊂ X × Y maximal cyclically monotone - ∃Φ convex, lscM = MΦ
I for b bipotential let Mb = (x , y) | b(x , y) = 〈x , y〉I if b(x , y) = Φ(x) + Φ∗(y) then Mb = MΦ
I PROBLEM: Given M ⊂ X × Y , find b bipotential s.t.M = Mb
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 25: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/25.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
I MΦ = (x , y) | y ∈ ∂Φ(x) maximal cyclically monotone
I Rockafellar theorem: The following are equivalent: -M ⊂ X × Y maximal cyclically monotone - ∃Φ convex, lscM = MΦ
I for b bipotential let Mb = (x , y) | b(x , y) = 〈x , y〉I if b(x , y) = Φ(x) + Φ∗(y) then Mb = MΦ
I PROBLEM: Given M ⊂ X × Y , find b bipotential s.t.M = Mb
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 26: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/26.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
I MΦ = (x , y) | y ∈ ∂Φ(x) maximal cyclically monotone
I Rockafellar theorem: The following are equivalent: -M ⊂ X × Y maximal cyclically monotone - ∃Φ convex, lscM = MΦ
I for b bipotential let Mb = (x , y) | b(x , y) = 〈x , y〉
I if b(x , y) = Φ(x) + Φ∗(y) then Mb = MΦ
I PROBLEM: Given M ⊂ X × Y , find b bipotential s.t.M = Mb
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 27: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/27.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
I MΦ = (x , y) | y ∈ ∂Φ(x) maximal cyclically monotone
I Rockafellar theorem: The following are equivalent: -M ⊂ X × Y maximal cyclically monotone - ∃Φ convex, lscM = MΦ
I for b bipotential let Mb = (x , y) | b(x , y) = 〈x , y〉I if b(x , y) = Φ(x) + Φ∗(y) then Mb = MΦ
I PROBLEM: Given M ⊂ X × Y , find b bipotential s.t.M = Mb
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 28: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/28.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
I MΦ = (x , y) | y ∈ ∂Φ(x) maximal cyclically monotone
I Rockafellar theorem: The following are equivalent: -M ⊂ X × Y maximal cyclically monotone - ∃Φ convex, lscM = MΦ
I for b bipotential let Mb = (x , y) | b(x , y) = 〈x , y〉I if b(x , y) = Φ(x) + Φ∗(y) then Mb = MΦ
I PROBLEM: Given M ⊂ X × Y , find b bipotential s.t.M = Mb
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 29: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/29.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
M. Buliga, G. de Saxce, C. Vallee: Existence and construction ofbipotentials for graphs of multivalued laws, J. Convex Analysis15(1) (2008) 87-104.
I Cover M with cyclically monotone graphs Mλ
M ⊂⋃λ∈Λ
Mλ
each Mλ gives Φλ convex
I define b(x , y) = infλ∈Λ
(Φλ(x) + Φ∗λ(y))
I Theorem: M = Mb and b bipotential if the familyΦλ | λ ∈ Λ satisfies an implicit convexity inequality.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 30: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/30.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
M. Buliga, G. de Saxce, C. Vallee: Existence and construction ofbipotentials for graphs of multivalued laws, J. Convex Analysis15(1) (2008) 87-104.
I Cover M with cyclically monotone graphs Mλ
M ⊂⋃λ∈Λ
Mλ
each Mλ gives Φλ convex
I define b(x , y) = infλ∈Λ
(Φλ(x) + Φ∗λ(y))
I Theorem: M = Mb and b bipotential if the familyΦλ | λ ∈ Λ satisfies an implicit convexity inequality.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 31: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/31.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
M. Buliga, G. de Saxce, C. Vallee: Existence and construction ofbipotentials for graphs of multivalued laws, J. Convex Analysis15(1) (2008) 87-104.
I Cover M with cyclically monotone graphs Mλ
M ⊂⋃λ∈Λ
Mλ
each Mλ gives Φλ convex
I define b(x , y) = infλ∈Λ
(Φλ(x) + Φ∗λ(y))
I Theorem: M = Mb and b bipotential if the familyΦλ | λ ∈ Λ satisfies an implicit convexity inequality.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 32: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/32.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
Example: the Cauchy bipotential b(x , y) = ‖x‖‖y‖M(b) = (x , y) : x = λy , λ ≥ 0 ∪ (0, y) : y ∈ Rn
‖x‖‖y‖ = infλ∈[0,+∞]
(Φλ(x) + Φ∗λ(y))
I for λ ∈ (0,+∞) take Φλ(x) =λ
2‖x‖2
I for λ = 0 take Φλ(x) = 0,
I for λ = 0 take Φ∗λ(y) = 0.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 33: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/33.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
Example: the Cauchy bipotential b(x , y) = ‖x‖‖y‖M(b) = (x , y) : x = λy , λ ≥ 0 ∪ (0, y) : y ∈ Rn
‖x‖‖y‖ = infλ∈[0,+∞]
(Φλ(x) + Φ∗λ(y))
I for λ ∈ (0,+∞) take Φλ(x) =λ
2‖x‖2
I for λ = 0 take Φλ(x) = 0,
I for λ = 0 take Φ∗λ(y) = 0.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 34: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/34.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
Example: the Cauchy bipotential b(x , y) = ‖x‖‖y‖M(b) = (x , y) : x = λy , λ ≥ 0 ∪ (0, y) : y ∈ Rn
‖x‖‖y‖ = infλ∈[0,+∞]
(Φλ(x) + Φ∗λ(y))
I for λ ∈ (0,+∞) take Φλ(x) =λ
2‖x‖2
I for λ = 0 take Φλ(x) = 0,
I for λ = 0 take Φ∗λ(y) = 0.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 35: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/35.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
M. Buliga, G. de Saxce, C. Vallee: Non maximal cyclicallymonotone graphs and construction of a bipotential for theCoulomb’s dry friction law, J. Convex Analysis 17(1) (2010)
I if M is made by several pieces which are not maximalcyclically monotone, like in the case of Coulomb friction law,then apply (a slight generalization of) the previous resultcombined with the following one.
I define b(x , y) = max (Φ1(x) + Φ∗1(y),Φ2(x) + Φ∗2(y))
I Theorem: b bipotential if and only if Φ1, Φ2 satisfy acondition expressed in terms of inf-convolutions.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 36: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/36.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
M. Buliga, G. de Saxce, C. Vallee: Non maximal cyclicallymonotone graphs and construction of a bipotential for theCoulomb’s dry friction law, J. Convex Analysis 17(1) (2010)
I if M is made by several pieces which are not maximalcyclically monotone, like in the case of Coulomb friction law,then apply (a slight generalization of) the previous resultcombined with the following one.
I define b(x , y) = max (Φ1(x) + Φ∗1(y),Φ2(x) + Φ∗2(y))
I Theorem: b bipotential if and only if Φ1, Φ2 satisfy acondition expressed in terms of inf-convolutions.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 37: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/37.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
M. Buliga, G. de Saxce, C. Vallee: Non maximal cyclicallymonotone graphs and construction of a bipotential for theCoulomb’s dry friction law, J. Convex Analysis 17(1) (2010)
I if M is made by several pieces which are not maximalcyclically monotone, like in the case of Coulomb friction law,then apply (a slight generalization of) the previous resultcombined with the following one.
I define b(x , y) = max (Φ1(x) + Φ∗1(y),Φ2(x) + Φ∗2(y))
I Theorem: b bipotential if and only if Φ1, Φ2 satisfy acondition expressed in terms of inf-convolutions.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 38: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/38.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
M. Buliga, G. de Saxce, C. Vallee: Bipotentials for non monotonemultivalued operators: fundamental results and applications, ActaApplicandae Mathematicae (2009), DOI10.1007/s10440-009-9488-3.
I if M is only maximal monotone, not cyclically monotone thenit admits a globally convex bipotential,
I related with the theory of selfdual lagrangians
I related with minimax problems.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 39: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/39.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
M. Buliga, G. de Saxce, C. Vallee: Bipotentials for non monotonemultivalued operators: fundamental results and applications, ActaApplicandae Mathematicae (2009), DOI10.1007/s10440-009-9488-3.
I if M is only maximal monotone, not cyclically monotone thenit admits a globally convex bipotential,
I related with the theory of selfdual lagrangians
I related with minimax problems.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 40: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/40.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The construction problem
M. Buliga, G. de Saxce, C. Vallee: Bipotentials for non monotonemultivalued operators: fundamental results and applications, ActaApplicandae Mathematicae (2009), DOI10.1007/s10440-009-9488-3.
I if M is only maximal monotone, not cyclically monotone thenit admits a globally convex bipotential,
I related with the theory of selfdual lagrangians
I related with minimax problems.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 41: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/41.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Blurred constitutive laws
G. de Saxce, M. Buliga, C. Vallee, Blurred constitutive laws andbipotential convex covers, to appear in Mathematics andMechanics of Solids
I (merci Michel Jean)
I Theorem: Let b be the bipotential which models Coulombfriction and ε > 0. Then the blurred Coulomb friction law
distance(y , ∂b(·, y)(x)) ≤ ε
can be expressed as an implicit constitutive law with the helpof a bipotential (and we construct it).
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 42: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/42.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Blurred constitutive laws
G. de Saxce, M. Buliga, C. Vallee, Blurred constitutive laws andbipotential convex covers, to appear in Mathematics andMechanics of Solids
I (merci Michel Jean)
I Theorem: Let b be the bipotential which models Coulombfriction and ε > 0. Then the blurred Coulomb friction law
distance(y , ∂b(·, y)(x)) ≤ ε
can be expressed as an implicit constitutive law with the helpof a bipotential (and we construct it).
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 43: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/43.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Blurred constitutive laws
M. Buliga, G. de Saxce, C. Vallee, Blurred maximal cyclicallymonotone graphs and bipotentials, in revision at Journal ofMathematical Analysis and Applications
Theorem: Let φ : X → R ∪ +∞ be a convex, lsc, proper
function and ε > 0. If for any y ∈ Y the set⋃
‖y−y‖≤ε
∂φ∗(y) is
convex then the problem
distance(y , ∂φ(x)) ≤ ε
can be expressed as an implicit constitutive law with the help ofthe bipotential
b(x , y) = φ(x) + inf‖a‖≤ε
[φ∗(y − a) + 〈x , a〉]Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 44: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/44.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
A. Berga, G. de Saxce, Elastoplastic finite element analysis of soilproblems with implicit standard material constitutive laws, Rev.Eur. des Elements Finis 3(3) (1994), 411-456
”One of the advantages of the new formulation is to extend the classical
Calculus of Variations to non associated constitutive laws. In the
theoretical frame of the Implicit Standard Materials, a new functional,
called bifunctional, is introduced, depending on both the displacement
and stress field.”
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 45: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/45.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
A. Berga, G. de Saxce, Elastoplastic finite element analysis of soilproblems with implicit standard material constitutive laws, Rev.Eur. des Elements Finis 3(3) (1994), 411-456
”The exact solution of the Boundary Value Problem corresponds to the
simultaneous minimization of the bifunctional, firstly with respect to
kinematically admissible displacement fields, when the stress field is equal
with the exact one, and secondly with respect to statically admissible
stress fields, when the displacement field is the exact one. The two
minimization problems are the direct extension of the dual variational
principles of displacements and stresses.”
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 46: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/46.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
non-associated Drucker-Prager model
I ε = D(u) =1
2
(∇u +∇uT
), ε = εe + εp
I σ = Sεe , be(εe , σ) = 1
2〈εe ,Sεe〉+ 1
2〈S−1σ, σ〉
I εp ∈ ∂bp(εp, ·)(σ)
bp(εp, σ) =
C1εpm + C2(σm − c
tan φ)‖εp‖ if σ ∈ K , εp ∈ K ′
+∞ otherwise
I u is CA(u0) : u = u0 on ∂0Ωσ is SA(fv , fs) : div σ + fv = 0 in Ω, σ · n = fs on ∂1Ω∂Ω = ∂1Ω ∪ ∂2Ω (disjoint union, ...)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 47: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/47.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
non-associated Drucker-Prager model
I ε = D(u) =1
2
(∇u +∇uT
), ε = εe + εp
I σ = Sεe , be(εe , σ) = 1
2〈εe ,Sεe〉+ 1
2〈S−1σ, σ〉
I εp ∈ ∂bp(εp, ·)(σ)
bp(εp, σ) =
C1εpm + C2(σm − c
tan φ)‖εp‖ if σ ∈ K , εp ∈ K ′
+∞ otherwise
I u is CA(u0) : u = u0 on ∂0Ωσ is SA(fv , fs) : div σ + fv = 0 in Ω, σ · n = fs on ∂1Ω∂Ω = ∂1Ω ∪ ∂2Ω (disjoint union, ...)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 48: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/48.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
non-associated Drucker-Prager model
I ε = D(u) =1
2
(∇u +∇uT
), ε = εe + εp
I σ = Sεe , be(εe , σ) = 1
2〈εe ,Sεe〉+ 1
2〈S−1σ, σ〉
I εp ∈ ∂bp(εp, ·)(σ)
bp(εp, σ) =
C1εpm + C2(σm − c
tan φ)‖εp‖ if σ ∈ K , εp ∈ K ′
+∞ otherwise
I u is CA(u0) : u = u0 on ∂0Ωσ is SA(fv , fs) : div σ + fv = 0 in Ω, σ · n = fs on ∂1Ω∂Ω = ∂1Ω ∪ ∂2Ω (disjoint union, ...)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 49: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/49.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
non-associated Drucker-Prager model
I ε = D(u) =1
2
(∇u +∇uT
), ε = εe + εp
I σ = Sεe , be(εe , σ) = 1
2〈εe ,Sεe〉+ 1
2〈S−1σ, σ〉
I εp ∈ ∂bp(εp, ·)(σ)
bp(εp, σ) =
C1εpm + C2(σm − c
tan φ)‖εp‖ if σ ∈ K , εp ∈ K ′
+∞ otherwise
I u is CA(u0) : u = u0 on ∂0Ωσ is SA(fv , fs) : div σ + fv = 0 in Ω, σ · n = fs on ∂1Ω∂Ω = ∂1Ω ∪ ∂2Ω (disjoint union, ...)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 50: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/50.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
I discretized in time problem: knowing the fields ofdisplacement, deformation (elastic and plastic), stress, theincrements of the imposed boundary conditions and volumeforce, find the increments ∆u, ∆σ, ∆εe , ∆εp
I ∆ε = D(∆u) , ∆ε = ∆εe + ∆εp
I ∆σ = S∆εe
∆εp ∈ ∂∆bp(∆εp, ·)(∆σ)
∆bp(∆εp,∆σ) = (∆t)bp(1
∆t∆εp, σ0 + ∆σ)− 〈∆εp, σ0〉
I ∆u is CA(∆u0) , ∆σ is SA(∆fv , δfs)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 51: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/51.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
I discretized in time problem: knowing the fields ofdisplacement, deformation (elastic and plastic), stress, theincrements of the imposed boundary conditions and volumeforce, find the increments ∆u, ∆σ, ∆εe , ∆εp
I ∆ε = D(∆u) , ∆ε = ∆εe + ∆εp
I ∆σ = S∆εe
∆εp ∈ ∂∆bp(∆εp, ·)(∆σ)
∆bp(∆εp,∆σ) = (∆t)bp(1
∆t∆εp, σ0 + ∆σ)− 〈∆εp, σ0〉
I ∆u is CA(∆u0) , ∆σ is SA(∆fv , δfs)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 52: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/52.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
I discretized in time problem: knowing the fields ofdisplacement, deformation (elastic and plastic), stress, theincrements of the imposed boundary conditions and volumeforce, find the increments ∆u, ∆σ, ∆εe , ∆εp
I ∆ε = D(∆u) , ∆ε = ∆εe + ∆εp
I ∆σ = S∆εe
∆εp ∈ ∂∆bp(∆εp, ·)(∆σ)
∆bp(∆εp,∆σ) = (∆t)bp(1
∆t∆εp, σ0 + ∆σ)− 〈∆εp, σ0〉
I ∆u is CA(∆u0) , ∆σ is SA(∆fv , δfs)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 53: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/53.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
I discretized in time problem: knowing the fields ofdisplacement, deformation (elastic and plastic), stress, theincrements of the imposed boundary conditions and volumeforce, find the increments ∆u, ∆σ, ∆εe , ∆εp
I ∆ε = D(∆u) , ∆ε = ∆εe + ∆εp
I ∆σ = S∆εe
∆εp ∈ ∂∆bp(∆εp, ·)(∆σ)
∆bp(∆εp,∆σ) = (∆t)bp(1
∆t∆εp, σ0 + ∆σ)− 〈∆εp, σ0〉
I ∆u is CA(∆u0) , ∆σ is SA(∆fv , δfs)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 54: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/54.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
discretized in time problem (Pdisc): knowing the fields ofdisplacement, deformation (elastic and plastic), stress, theincrements of the imposed boundary conditions and volume force,find the increments u, σ, εe , εp
ε = D(u) =1
2
(∇u +∇uT
), ε = εe + εp
σ = Sεe , be(εe , σ) = 1
2〈εe ,Sεe〉+ 1
2〈S−1σ, σ〉
εp ∈ ∂bp(εp, ·)(σ) (with a bp computed from the old plastic
bipotential and the input fields)u is CA(u0) , σ is SA(fv , fs)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 55: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/55.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
I define the inf-convolution:b(ε, σ) = inf be(ε
e , σ) + bp(εp, σ) : εe + εp = ε
(due to the nice form of be this is like a Moreau-Yosidaregularization w.r.t. ε, so b is smooth in ε)
I Proposition: If bp is a bipotential then (Pdisc) is equivalentwith the following problem (P):
ε = D(u) =1
2
(∇u +∇uT
), u is CA(u0) , σ is SA(fv , fs)
σ ∈ ∂b(·, σ)(ε)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 56: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/56.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce
I define the inf-convolution:b(ε, σ) = inf be(ε
e , σ) + bp(εp, σ) : εe + εp = ε
(due to the nice form of be this is like a Moreau-Yosidaregularization w.r.t. ε, so b is smooth in ε)
I Proposition: If bp is a bipotential then (Pdisc) is equivalentwith the following problem (P):
ε = D(u) =1
2
(∇u +∇uT
), u is CA(u0) , σ is SA(fv , fs)
σ ∈ ∂b(·, σ)(ε)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 57: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/57.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
b(ε, σ) = inf be(εe , σ) + bp(ε
p, σ) : εe + εp = εProperties of b:
I b(ε, σ) ≥ 〈ε, σ〉
I b(·, σ) is convex and smooth
I b is not a bipotential!
I b(ε, ·) is lsc but not convex!Nevertheless (Pdisc) is equivalent with (P) because bp is abipotential.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
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Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
b(ε, σ) = inf be(εe , σ) + bp(ε
p, σ) : εe + εp = εProperties of b:
I b(ε, σ) ≥ 〈ε, σ〉I b(·, σ) is convex and smooth
I b is not a bipotential!
I b(ε, ·) is lsc but not convex!Nevertheless (Pdisc) is equivalent with (P) because bp is abipotential.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 59: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/59.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
b(ε, σ) = inf be(εe , σ) + bp(ε
p, σ) : εe + εp = εProperties of b:
I b(ε, σ) ≥ 〈ε, σ〉I b(·, σ) is convex and smooth
I b is not a bipotential!
I b(ε, ·) is lsc but not convex!Nevertheless (Pdisc) is equivalent with (P) because bp is abipotential.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 60: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/60.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
b(ε, σ) = inf be(εe , σ) + bp(ε
p, σ) : εe + εp = εProperties of b:
I b(ε, σ) ≥ 〈ε, σ〉I b(·, σ) is convex and smooth
I b is not a bipotential!
I b(ε, ·) is lsc but not convex!Nevertheless (Pdisc) is equivalent with (P) because bp is abipotential.
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 61: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/61.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
Problem (P):
I ε = D(u) =1
2
(∇u +∇uT
), u is CA(u0) , σ is SA(fv , fs)
σ ∈ ∂b(·, σ)(ε)
I b(ε, σ) ≥ 〈ε, σ〉I b(·, σ) is convex and smooth (∂b(·, σ)(ε) is univalued)
I b(ε, ·) is lsc but not convex
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 62: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/62.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
Problem (P):
I ε = D(u) =1
2
(∇u +∇uT
), u is CA(u0) , σ is SA(fv , fs)
σ ∈ ∂b(·, σ)(ε)
I b(ε, σ) ≥ 〈ε, σ〉
I b(·, σ) is convex and smooth (∂b(·, σ)(ε) is univalued)
I b(ε, ·) is lsc but not convex
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 63: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/63.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
Problem (P):
I ε = D(u) =1
2
(∇u +∇uT
), u is CA(u0) , σ is SA(fv , fs)
σ ∈ ∂b(·, σ)(ε)
I b(ε, σ) ≥ 〈ε, σ〉I b(·, σ) is convex and smooth (∂b(·, σ)(ε) is univalued)
I b(ε, ·) is lsc but not convex
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 64: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/64.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
Problem (P):
I ε = D(u) =1
2
(∇u +∇uT
), u is CA(u0) , σ is SA(fv , fs)
σ ∈ ∂b(·, σ)(ε)
I b(ε, σ) ≥ 〈ε, σ〉I b(·, σ) is convex and smooth (∂b(·, σ)(ε) is univalued)
I b(ε, ·) is lsc but not convex
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 65: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/65.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
Problem (P):
ε = D(u) =1
2
(∇u +∇uT
), u is CA(0) , σ is SA(0, 0)
σ ∈ ∂b(·, σ)(ε)
I b(ε, σ) ≥ 〈ε, σ〉
I b(·, σ) is convex and ∂b(·, σ)(ε) is univalued
I b(ε, ·) is lsc but not convex
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 66: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/66.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
Problem (P):
ε = D(u) =1
2
(∇u +∇uT
), u is CA(0) , σ is SA(0, 0)
σ ∈ ∂b(·, σ)(ε)
I b(ε, σ) ≥ 〈ε, σ〉I b(·, σ) is convex and ∂b(·, σ)(ε) is univalued
I b(ε, ·) is lsc but not convex
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 67: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/67.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
Problem (P):
ε = D(u) =1
2
(∇u +∇uT
), u is CA(0) , σ is SA(0, 0)
σ ∈ ∂b(·, σ)(ε)
I b(ε, σ) ≥ 〈ε, σ〉I b(·, σ) is convex and ∂b(·, σ)(ε) is univalued
I b(ε, ·) is lsc but not convex
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 68: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/68.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
Bifunctional: B(ε, σ) =
∫Ω
b(ε, σ) dx
Algorithm: the solution (u, σ) of (P) is the limit of the sequence
0. initialize u0 ∈ CA(0), σ0 ∈ SA(0, 0)
1. given un, σn
a. (global step) find un+1 such that
B(D(un+1), σn) ≤ B(D(v), σn) ∀v ∈ CA(0)
b (local step) take σn+1 ∈ ∂b(·, σn)(D(un+1)) (integration byparts shows that σn ∈ SA(0, 0) for any n, in a weak sense)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 69: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/69.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
Bifunctional: B(ε, σ) =
∫Ω
b(ε, σ) dx
Algorithm: the solution (u, σ) of (P) is the limit of the sequence
0. initialize u0 ∈ CA(0), σ0 ∈ SA(0, 0)
1. given un, σn
a. (global step) find un+1 such that
B(D(un+1), σn) ≤ B(D(v), σn) ∀v ∈ CA(0)
b (local step) take σn+1 ∈ ∂b(·, σn)(D(un+1)) (integration byparts shows that σn ∈ SA(0, 0) for any n, in a weak sense)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 70: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/70.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
Bifunctional: B(ε, σ) =
∫Ω
b(ε, σ) dx
Algorithm: the solution (u, σ) of (P) is the limit of the sequence
0. initialize u0 ∈ CA(0), σ0 ∈ SA(0, 0)
1. given un, σn
a. (global step) find un+1 such that
B(D(un+1), σn) ≤ B(D(v), σn) ∀v ∈ CA(0)
b (local step) take σn+1 ∈ ∂b(·, σn)(D(un+1)) (integration byparts shows that σn ∈ SA(0, 0) for any n, in a weak sense)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 71: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/71.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
Bifunctional: B(ε, σ) =
∫Ω
b(ε, σ) dx
Algorithm: the solution (u, σ) of (P) is the limit of the sequence
0. initialize u0 ∈ CA(0), σ0 ∈ SA(0, 0)
1. given un, σn
a. (global step) find un+1 such that
B(D(un+1), σn) ≤ B(D(v), σn) ∀v ∈ CA(0)
b (local step) take σn+1 ∈ ∂b(·, σn)(D(un+1)) (integration byparts shows that σn ∈ SA(0, 0) for any n, in a weak sense)
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 72: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/72.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
I Variational formulation: a la NayrolesX is the space of ε, for example X = L2(Ω,Mn×n
sym (R))Y is the space of σ, for example Y = L2(Ω,Mn×n
sym (R))
they are in duality by 〈ε, σ〉1 =
∫Ω
ε : σ dx
I U is the space of u, let’s take U = W 1,2(Ω, Rn)
I P is the space of f = (fv , fs) (pairs of volume force, surfaceforce) seen in duality with U by〈u, f 〉2 =
∫Ω u · fv dx +
∫∂0Ω
u · fs
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 73: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/73.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
I Variational formulation: a la NayrolesX is the space of ε, for example X = L2(Ω,Mn×n
sym (R))Y is the space of σ, for example Y = L2(Ω,Mn×n
sym (R))
they are in duality by 〈ε, σ〉1 =
∫Ω
ε : σ dx
I U is the space of u, let’s take U = W 1,2(Ω, Rn)
I P is the space of f = (fv , fs) (pairs of volume force, surfaceforce) seen in duality with U by〈u, f 〉2 =
∫Ω u · fv dx +
∫∂0Ω
u · fs
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 74: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/74.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
I Variational formulation: a la NayrolesX is the space of ε, for example X = L2(Ω,Mn×n
sym (R))Y is the space of σ, for example Y = L2(Ω,Mn×n
sym (R))
they are in duality by 〈ε, σ〉1 =
∫Ω
ε : σ dx
I U is the space of u, let’s take U = W 1,2(Ω, Rn)
I P is the space of f = (fv , fs) (pairs of volume force, surfaceforce) seen in duality with U by〈u, f 〉2 =
∫Ω u · fv dx +
∫∂0Ω
u · fs
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 75: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/75.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
I Variational formulation: a la NayrolesD : U → X linear and continuous, D(u) = 1
2
(∇u +∇uT
)
I U0 ⊂ U is the space CA(0), X0 = D(U0)
I Y0 ⊂ Y is the space SA(0,0), defined as the space of allσ ∈ Y such that:
∀u ∈ U0 〈σ,D(u)〉 = 0
I Integrate the relation σ ∈ ∂b(·, σ)(D(u)) from (P) to get:u ∈ U0 , σ ∈ Y0 , ∀ε ∈ X B(D(u), σ) ≤ B(ε, σ)− 〈ε, σ〉1
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 76: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/76.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
I Variational formulation: a la NayrolesD : U → X linear and continuous, D(u) = 1
2
(∇u +∇uT
)I U0 ⊂ U is the space CA(0), X0 = D(U0)
I Y0 ⊂ Y is the space SA(0,0), defined as the space of allσ ∈ Y such that:
∀u ∈ U0 〈σ,D(u)〉 = 0
I Integrate the relation σ ∈ ∂b(·, σ)(D(u)) from (P) to get:u ∈ U0 , σ ∈ Y0 , ∀ε ∈ X B(D(u), σ) ≤ B(ε, σ)− 〈ε, σ〉1
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 77: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/77.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
I Variational formulation: a la NayrolesD : U → X linear and continuous, D(u) = 1
2
(∇u +∇uT
)I U0 ⊂ U is the space CA(0), X0 = D(U0)
I Y0 ⊂ Y is the space SA(0,0), defined as the space of allσ ∈ Y such that:
∀u ∈ U0 〈σ,D(u)〉 = 0
I Integrate the relation σ ∈ ∂b(·, σ)(D(u)) from (P) to get:u ∈ U0 , σ ∈ Y0 , ∀ε ∈ X B(D(u), σ) ≤ B(ε, σ)− 〈ε, σ〉1
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 78: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/78.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
I Variational formulation: a la NayrolesD : U → X linear and continuous, D(u) = 1
2
(∇u +∇uT
)I U0 ⊂ U is the space CA(0), X0 = D(U0)
I Y0 ⊂ Y is the space SA(0,0), defined as the space of allσ ∈ Y such that:
∀u ∈ U0 〈σ,D(u)〉 = 0
I Integrate the relation σ ∈ ∂b(·, σ)(D(u)) from (P) to get:u ∈ U0 , σ ∈ Y0 , ∀ε ∈ X B(D(u), σ) ≤ B(ε, σ)− 〈ε, σ〉1
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 79: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/79.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
I Variational formulation: a la Nayroles
u ∈ U0 , σ ∈ Y0 , ∀ε ∈ X B(D(u), σ) ≤ B(ε, σ)− 〈ε, σ〉1
I Remark that:∀v ∈ U0 B(D(u), σ) ≤ B(D(v), σ) (that is 1a.)
I σ ∈ ∂1B(·, σ)(D(u)) (that is 1b.). Indeed, that means
∀ε ∈ X B(ε, σ) ≥ B(D(u), σ) + 〈ε− D(u), σ〉1
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 80: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/80.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
I Variational formulation: a la Nayroles
u ∈ U0 , σ ∈ Y0 , ∀ε ∈ X B(D(u), σ) ≤ B(ε, σ)− 〈ε, σ〉1
I Remark that:∀v ∈ U0 B(D(u), σ) ≤ B(D(v), σ) (that is 1a.)
I σ ∈ ∂1B(·, σ)(D(u)) (that is 1b.). Indeed, that means
∀ε ∈ X B(ε, σ) ≥ B(D(u), σ) + 〈ε− D(u), σ〉1
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 81: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/81.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
The model of Berga & de Saxce (revised)
I Variational formulation: a la Nayroles
u ∈ U0 , σ ∈ Y0 , ∀ε ∈ X B(D(u), σ) ≤ B(ε, σ)− 〈ε, σ〉1
I Remark that:∀v ∈ U0 B(D(u), σ) ≤ B(D(v), σ) (that is 1a.)
I σ ∈ ∂1B(·, σ)(D(u)) (that is 1b.). Indeed, that means
∀ε ∈ X B(ε, σ) ≥ B(D(u), σ) + 〈ε− D(u), σ〉1
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 82: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/82.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Conclusion
I Variational formulation:
We have a fixed point problem in σ ∈ Y0:
σ ∈ ∂1B(·, σ)(D(U0))
I alternatively:D(U0) ∩ ∂1 (B(·, σ))∗ (σ) 6= ∅
I (remark that (B(·, σ))∗ (σ) has an integral expression as asum of be and (bp(·, σ))∗ (σ))
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 83: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/83.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Conclusion
I Variational formulation:
We have a fixed point problem in σ ∈ Y0:
σ ∈ ∂1B(·, σ)(D(U0))
I alternatively:D(U0) ∩ ∂1 (B(·, σ))∗ (σ) 6= ∅
I (remark that (B(·, σ))∗ (σ) has an integral expression as asum of be and (bp(·, σ))∗ (σ))
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels
![Page 84: Formulations variationnelles utilisant les bipotentiels · 2010-09-01 · Plan de l’expos´e Standard materials Implicit standard materials. Bipotentials Applications of bipotentials](https://reader033.vdocuments.us/reader033/viewer/2022042018/5e76526dbf38df4df41d2166/html5/thumbnails/84.jpg)
Plan de l’exposeStandard materials
Implicit standard materials. BipotentialsApplications of bipotentials in mechanics
The construction problemBlurred constitutive laws
The model of Berga & de SaxceConclusion
Conclusion
I Variational formulation:
We have a fixed point problem in σ ∈ Y0:
σ ∈ ∂1B(·, σ)(D(U0))
I alternatively:D(U0) ∩ ∂1 (B(·, σ))∗ (σ) 6= ∅
I (remark that (B(·, σ))∗ (σ) has an integral expression as asum of be and (bp(·, σ))∗ (σ))
Marius Buliga1 Gery de Saxce2 Claude Vallee3 1 ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest 2 LML (UMR CNRS 8107), Villeneuve d’Ascq 3 LMS (UMR CNRS 6610), Poitiers 10eme Colloque Franco-Roumain de Mathematiques Appliquees Poitiers, 26-31 aout 2010Formulations variationnelles utilisant les bipotentiels