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The Bipotential Approach

Géry de Saxcé. The Bipotential Approach. Laboratory of Mechanics of Lille UMR CNRS 8107. University of Lille 1 France. Mechanics of contact. Plasticity of Soils. Cyclic Plasticity of metals. OUTLINE. The bipotential. one-to-one normality law: POTENTIAL. multivalued

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The Bipotential Approach

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  1. Géry de Saxcé The Bipotential Approach Laboratory of Mechanics of Lille UMR CNRS 8107 University of Lille 1 France

  2. Mechanics of contact Plasticity of Soils Cyclic Plasticity of metals OUTLINE The bipotential

  3. one-to-one normality law: POTENTIAL multivalued normality law: SURPOTENTIAL non-associated law: BIPOTENTIAL CONSTITUTIVE LAWS topological linear space duality product dual space of X

  4. FENCHEL’S INEQUALITY EXTREMAL COUPLE multivalued normality law (or subnormality law) constitutive law inverse law convex SURPOTENTIAL

  5. Good properties of The normality law Plastic Limit Analysis: bound theorems Calculus of Variations: existence of functionals Step-by-step computation: the tangent stiffness matrix is symetric and positive-definite STANDARD MATERIALS Materials admitting a superpotential

  6. THE CLASSICAL FORMULATION:two functions of one variable load function: plastic potential (Melan): such that : BIPOTENTIAL FORMULATION:one function of two variables NON-ASSOCIATED PLASTICITY deviatoric stress yield surface hydrostatic stress

  7. CORNER STONE INEQUALITY BIPOTENTIAL b bBI-CONVEX

  8. IMPLICIT SUBNORMALITY LAW constitutive law inverse law EXTREMAL COUPLE

  9. Halphen & Nguyen Quoc Son (1975) STANDARD MATERIAL the bipotential is separable IMPLICIT STANDARD MATERIAL material admitting a bipotential

  10. The bipotential Mechanics of contact

  11. sticking sliding dual cone no contact complete contact law: de Saxcé & Feng (1998) predictor predictor/corrector scheme by projection onto Coulomb’s cone corrector UNILATERAL CONTACT LAW WITH COULOMB’s DRY FRICTION

  12. De Saxcé & J. Fortin (2000) Discret Element code MULTICOR Discharge of a silo (1600 particles)

  13. time-space average spatial average De Saxcé, J. Fortin & O. Millet (2004) Discharge of a silo (1580 particles)

  14. Plasticity of Soils The bipotential Mechanics of contact

  15. indicatory function (well potential) 0 K implicit subnormality law …ADMITS A BIPOTENTIAL coupling term vertex term kinematical condition plasticity criterion NON-ASSOCIATED DRUCKER-PRAGER MODEL… dilatancy angle friction angle deviatoric stress yield surface cohesion hydrostatic stress

  16. compatibility: load step equilibrium: constitutive law: de Saxcé & Berga (1994) Implicit function theorem: local tangent matrix: symmetric coupling matrix positive definite hessian matrix Structural stiffness matrix: MODIFIED NEWTON TANGENT METHOD load displacement

  17. de Saxcé & Berga (1994) Collapse mechanism associated law ( ) non associated law ( ) gain in computation time> 30 %

  18. Cyclic Plasticity of metals The bipotential Mechanics of contact Plasticity of Soils

  19. limit surface stresss yield surface back-stressX generalized yielding law current yield stressR …ADMITS A BIPOTENTIAL indicatory function (well potential) coupling terms kinematic hardening isotropic hardening plasticity criterion 0 K THE NON LINEAR KINEMATICAL HARDENING RULE… traction limit yield stress X plastic strain stabilized cycle

  20. material bipotential contact bipotential external action potential BIFUNCTIONAL

  21. for the solutions of the boundary value problem: BIFUNCTIONAL CORNER STONE INEQUALITY OF THE BIPOTENTIAL + PRINCIPLE OF VIRTUAL POWER

  22. A POSTERIORI ESTIMATOR OF THE FINITE ELEMENT MESH ERROR bifunctional de Saxcé & Hjiaj (1999)

  23. > variable repeated loads : Plastic Shakedown current load domain load 2 reference load domain load 1 incremental collapse ratchet plastic fatigue accommodation shakedown load shakedown PLASTIC SHAKEDOWN ANALYSIS >proportional loads: Plastic Limit Analysis

  24. constant traction alternated torsion De Saxcé, Bouby, Tritsch (2003) alternated torsion constant traction Lemaitre, Chaboche (1988) plane stress SHAKEDOWN OF A SAMPLE IN TRACTION/TORSION plane strain

  25. CONCLUSIONS : the bipotential - provides a natural extension of the calculus of variations • is a theoretical tools (for instance extending the bound theorems • of Plasticity to the materials admitting a bipotential) • is a constructive method to propose new, fast and robust algorithms OPEN QUESTIONS Existence, uniqueness and construction of the bipotential CONCLUSIONS OTHER BIPOTENTIALS Cam-clay (de Saxcé, 1995) Coaxial laws (Vallée et al., 1997) Lemaitre plastic ductile damage law (Bodovillé, 1999)

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