Derivation of the model The PDE system Existence Long-time analysis Analysis of a model for adhesive contact with thermal effects Riccarda Rossi (Universit` a di Brescia) joint work with Elena Bonetti (Universit` a di Pavia), Giovanna Bonfanti (Universit` a di Brescia), Mathematical Models and Analytical Problems for Special Materials, Brescia, July 09–11 2009 Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis Fr´ emond’s modeling of adhesive contact Setting A viscoelastic body Ω ⊂ R 3 in adhesive contact with a rigid support on a (flat) part Γ Cont of its boundary ∂ Ω = Γ Dir ∪ Γ Neu ∪ Γ Cont . Related contributions .... Andrews, Cang´ emi, Chau, Cocou, Eck, Fern´ andez, Figuereido, Han, Jaruˇ sek, Klarbring, Krbec, Kuttler, Martins, Mu˜ noz-Rivera, Point, Racke, Raous, Shi, Shillor, Sofonea, Telega, Trabucho, Wright..... Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis Fr´ emond’s modeling of adhesive contact Setting A viscoelastic body Ω ⊂ R 3 in adhesive contact with a rigid support on a (flat) part Γ Cont of its boundary ∂ Ω = Γ Dir ∪ Γ Neu ∪ Γ Cont . Related contributions .... Andrews, Cang´ emi, Chau, Cocou, Eck, Fern´ andez, Figuereido, Han, Jaruˇ sek, Klarbring, Krbec, Kuttler, Martins, Mu˜ noz-Rivera, Point, Racke, Raous, Shi, Shillor, Sofonea, Telega, Trabucho, Wright..... Fr´ emond’s approach Based on [M. Fr´ emond, Non-smooth Thermomechanics , 2002] Account for microscopic motions in the macroscopic predictive theory Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis Fr´ emond’s modeling of adhesive contact Setting A viscoelastic body Ω ⊂ R 3 in adhesive contact with a rigid support on a (flat) part Γ Cont of its boundary ∂ Ω = Γ Dir ∪ Γ Neu ∪ Γ Cont . Related contributions .... Andrews, Cang´ emi, Chau, Cocou, Eck, Fern´ andez, Figuereido, Han, Jaruˇ sek, Klarbring, Krbec, Kuttler, Martins, Mu˜ noz-Rivera, Point, Racke, Raous, Shi, Shillor, Sofonea, Telega, Trabucho, Wright..... Fr´ emond’s approach Based on [M. Fr´ emond, Non-smooth Thermomechanics , 2002] Account for microscopic motions in the macroscopic predictive theory ◮ microscopic bonds are responsible for the adhesion, microscopic motions lead to rupture Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis Fr´ emond’s modeling of adhesive contact Setting A viscoelastic body Ω ⊂ R 3 in adhesive contact with a rigid support on a (flat) part Γ Cont of its boundary ∂ Ω = Γ Dir ∪ Γ Neu ∪ Γ Cont . Related contributions .... Andrews, Cang´ emi, Chau, Cocou, Eck, Fern´ andez, Figuereido, Han, Jaruˇ sek, Klarbring, Krbec, Kuttler, Martins, Mu˜ noz-Rivera, Point, Racke, Raous, Shi, Shillor, Sofonea, Telega, Trabucho, Wright..... Fr´ emond’s approach Based on [M. Fr´ emond, Non-smooth Thermomechanics , 2002] Account for microscopic motions in the macroscopic predictive theory ◮ microscopic bonds are responsible for the adhesion, microscopic motions lead to rupture ◮ account for the power of the microscopic motions in the power of the interior forces Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis State variables In the isothermal case [Bonetti-Bonfanti-R. ’07,’08] ◮ in the volume domain Ω: ◮ small deformation ( ε ( u ) symm. linear. strain tensor) (small perturbation assumption) thermal effects ( θ absolute temperature) ◮ ◮ on the contact surface Γ Cont : ◮ adhesion ( χ “phase parameter” related to the active bonds of the adhesion � “damage parameter”) ◮ effects of displacement ( u | Γ Cont trace of the displacement) thermal effects ( θ s absolute temperature) ◮ Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis State variables To account for thermal effects: [Bonetti-Bonfanti-R. preprint’08] ◮ in the volume domain Ω: ◮ small deformation ( ε ( u ) symm. linear. strain tensor) (small perturbation assumption) thermal effects ( θ absolute temperature) ◮ ◮ on the contact surface Γ Cont : ◮ adhesion ( χ “phase parameter” related to the active bonds of the adhesion � “damage parameter”) ◮ effects of displacement ( u | Γ Cont trace of the displacement) thermal effects ( θ s absolute temperature) ◮ Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis The equations for u and χ From the principle of virtual power (interior & exterior forces, no acceleration forces) ◮ momentum balance: 8 − div Σ = f in Ω × (0 , T ) , 8 Σ stress tensor > > 8 Σ n = R in Γ Cont × (0 , T ) , < < > R reaction on the contact surface < u = 0 in Γ Dir × (0 , T ) , f volume force , g traction > : > : > Σ n = g in Γ Neu × (0 , T ) , : Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis The equations for u and χ From the principle of virtual power (interior & exterior forces, no acceleration forces) ◮ momentum balance: 8 − div Σ = f in Ω × (0 , T ) , 8 Σ stress tensor > > 8 Σ n = R in Γ Cont × (0 , T ) , < < > R reaction on the contact surface < u = 0 in Γ Dir × (0 , T ) , f volume force , g traction > : > : > Σ n = g in Γ Neu × (0 , T ) , : ◮ equation for the microscopic motions: B − div H = 0 in Γ Cont × (0 , T ) , B interior microscopic work H · n s = 0 on ∂ Γ Cont × (0 , T ) , H microscopic work flux vector Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis The equations for θ and θ s Entropy balance for θ and θ s : ◮ for θ : 8 s internal entropy 8 s t + div Q = h in Ω × (0 , T ) , > > Q entropy flux vector < < ( Q · n = F on Γ Cont × (0 , T ) , h entropy source , Q · n = 0 on ∂ Ω \ Γ Cont × (0 , T ) , : > > F entropy flux through Γ Cont : Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis The equations for θ and θ s Entropy balance for θ and θ s : ◮ for θ : 8 s internal entropy 8 s t + div Q = h in Ω × (0 , T ) , > > Q entropy flux vector < < ( Q · n = F on Γ Cont × (0 , T ) , h entropy source , Q · n = 0 on ∂ Ω \ Γ Cont × (0 , T ) , : > > F entropy flux through Γ Cont : ◮ for θ s : ∂ t s s + div Q s = F in Ω × (0 , T ) , s s contact surface entropy Q s · n s = 0 on ∂ Γ Cont × (0 , T ) , Q s surface entropy flux vector Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis The equations for θ and θ s Entropy balance for θ and θ s : ◮ for θ : 8 s internal entropy 8 s t + div Q = h in Ω × (0 , T ) , > > Q entropy flux vector < < ( Q · n = F on Γ Cont × (0 , T ) , h entropy source , Q · n = 0 on ∂ Ω \ Γ Cont × (0 , T ) , : > > F entropy flux through Γ Cont : ◮ for θ s : ∂ t s s + div Q s = F in Ω × (0 , T ) , s s contact surface entropy Q s · n s = 0 on ∂ Γ Cont × (0 , T ) , Q s surface entropy flux vector Entropy balance : obtained by rescaling the internal energy balance (under small perturbation assumpt.): see [Bonetti-Fr´ emond’03, Bonetti-Colli-Fr´ emond’03, Bonetti’06, Bonetti-Colli-Fabrizio-Gilardi’06,’07,08, Bonetti-Rocca-Fr´ emond’07] Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis Constitutive laws Constitutive relations for Σ , R , s , Q , F , B , H , s s , Q s derive from the volume & surface free energies Ψ Ω = Ψ Ω ( u , θ ) , Ψ Γ Cont = Ψ Γ Cont ( u | Γ Cont , χ, θ s ) and the pseudo-potentials of dissipation Φ Ω = Φ Ω ( ∇ θ, ε ( u t )) , Φ Γ Cont = Φ Γ Cont ( ∇ θ s , χ t , θ | Γ Cont − θ s ) Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
Derivation of the model The PDE system Existence Long-time analysis Constitutive laws Constitutive relations for Σ , R , s , Q , F , B , H , s s , Q s derive from the volume & surface free energies Ψ Ω = Ψ Ω ( u , θ ) , Ψ Γ Cont = Ψ Γ Cont ( u | Γ Cont , χ, θ s ) and the pseudo-potentials of dissipation Φ Ω = Φ Ω ( ∇ θ, ε ( u t )) , Φ Γ Cont = Φ Γ Cont ( ∇ θ s , χ t , θ | Γ Cont − θ s ) The energy potentials include constraints on the variables for physical consistency: � nonsmooth (multivalued) operators in the constitutive eqns Riccarda Rossi Analysis of a model for adhesive contact with thermal effects
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