Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Constraints: irreversibility, unilateral contact.. ◮ Admissible values for z : � z ∈ [0 , 1] ◮ z ( x ) = 1: at x ∈ Γ C adhesive completely sound & fully effective ◮ 0 < z ( x ) < 1: at x ∈ Γ C a fraction of the molecular links is broken ◮ z ( x ) = 0: at x ∈ Γ C surface is completely debonded ◮ θ is the absolute temperature: � θ > 0 ◮ Damaging of the glue is a unidirectional process: z ≤ 0 . (irreversibility) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Constraints: irreversibility, unilateral contact.. ◮ Admissible values for z : � z ∈ [0 , 1] ◮ z ( x ) = 1: at x ∈ Γ C adhesive completely sound & fully effective ◮ 0 < z ( x ) < 1: at x ∈ Γ C a fraction of the molecular links is broken ◮ z ( x ) = 0: at x ∈ Γ C surface is completely debonded ◮ θ is the absolute temperature: � θ > 0 ◮ Damaging of the glue is a unidirectional process: z ≤ 0 . (irreversibility) ◮ No interpenetration between Ω 1 and Ω 2 : � unilateral contact conditions Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Unilateral (frictionless) Signorini contact • Signorini conditions in complementarity form: ˆ ˆ ˜ ˜ u · ν ≥ 0 on Γ C × (0 , T ) (no interpenetration) (Sign 1 ) σ | Γ C ν · ν ≥ 0 on Γ C × (0 , T ) (Sign 2 ) |{z} traction stress on Γ C ˆ ˆ ˜ ˜ σ | Γ C ν · u = 0 on Γ C × (0 , T ) (Sign 3 ) σ | Γ C ν · t = 0 on Γ C × (0 , T ) ∀ t s.t. ν · t = 0 (Sign 4 ) (Sign 2 ) & (Sign 3 ) & (Sign 4 ) yield ◮ [ [ u ] ] · ν > 0 ⇒ σ | Γ C ν = 0 (no reaction) ◮ [ [ u ] ] · ν = 0 ⇒ σ | Γ C ν = λν , λ ≥ 0 (reaction is triggered) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for u Momentum equilibrium equation ̺ .. u − div ( σ ) = F in Ω × (0 , T ) . Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for u Momentum equilibrium equation ̺ .. u − div ( σ ) = F in Ω × (0 , T ) . Ansatz of generalized standard solids: ◮ inertial effects Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for u Momentum equilibrium equation ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E θ = F in Ω × (0 , T ) . (eq u ) Ansatz of generalized standard solids: ◮ inertial effects ◮ stress σ features viscous response of material (Kelvin-Voigt rheology) ` ´ σ = D e ( . u ) + C e ( u ) − E θ | {z } viscosity C , D 4th-order positive definite and symmetric tensors matrix of thermal expansion coefficients E Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for u Momentum equilibrium equation ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E θ = F in Ω × (0 , T ) . (eq u ) Ansatz of generalized standard solids: ◮ inertial effects ◮ stress σ features viscous response of material (Kelvin-Voigt rheology) ` ´ σ = D e ( . u ) + C e ( u ) − E θ | {z } viscosity C , D 4th-order positive definite and symmetric tensors matrix of thermal expansion coefficients E ◮ F applied bulk force Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for u Momentum equilibrium equation ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E θ = F in Ω × (0 , T ) . (eq u ) Ansatz of generalized standard solids: ◮ inertial effects ◮ stress σ features viscous response of material (Kelvin-Voigt rheology) ` ´ σ = D e ( . u ) + C e ( u ) − E θ | {z } viscosity C , D 4th-order positive definite and symmetric tensors matrix of thermal expansion coefficients E ◮ F applied bulk force + boundary conditions on ∂ Ω = Γ D ∪ Γ N : u = 0 on Γ D × (0 , T ) σ n = f on Γ N × (0 , T ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for u Momentum equilibrium equation ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E θ = F in Ω × (0 , T ) . (eq u ) Ansatz of generalized standard solids: ◮ inertial effects ◮ stress σ features viscous response of material (Kelvin-Voigt rheology) ` ´ σ = D e ( . u ) + C e ( u ) − E θ | {z } viscosity C , D 4th-order positive definite and symmetric tensors matrix of thermal expansion coefficients E ◮ F applied bulk force + boundary conditions on ∂ Ω = Γ D ∪ Γ N : u = 0 on Γ D × (0 , T ) σ n = f on Γ N × (0 , T ) + complementarity problem on Γ C encompassing adhesion variable z in Signorini contact Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for θ Heat equation c v ( θ ) . θ + div ( j ) = D e ( . u ): e ( . u ) + θ CE : e ( . u ) + G in Ω × (0 , T ) It balances heat flux & rate of heat production due to dissipation: Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for θ Heat equation c v ( θ ) . θ + div ( j ) = D e ( . u ): e ( . u ) + θ CE : e ( . u ) + G in Ω × (0 , T ) It balances heat flux & rate of heat production due to dissipation: ◮ c v ( θ ) heat capacity Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for θ Heat equation ` ´ c v ( θ ) . θ + div − K ( e ( u ) , θ ) ∇ θ = D e ( . u ): e ( . u )+ θ CE : e ( . u )+ G in Ω × (0 , T ) (eq θ ) It balances heat flux & rate of heat production due to dissipation: ◮ c v ( θ ) heat capacity ◮ j heat flux, given by Fourier’s law in an anisotropic medium j = − K ( e ( u ) , θ ) ∇ θ, K ( e ( u ) , θ ) pos. def. matrix heat conduction coefficients Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for θ Heat equation ` ´ c v ( θ ) . θ − div K ( e ( u ) , θ ) ∇ θ = D e ( . u ): e ( . u ) + θ CE : e ( . u ) + G in Ω × (0 , T ) (eq θ ) It balances heat flux & rate of heat production due to dissipation: ◮ c v ( θ ) heat capacity ◮ j heat flux, given by Fourier’s law in an anisotropic medium j = − K ( e ( u ) , θ ) ∇ θ, K ( e ( u ) , θ ) pos. def. matrix heat conduction coefficients ◮ D e ( . u ): e ( . u ) viscous dissipation potential in the bulk Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for θ Heat equation ` ´ c v ( θ ) . θ − div K ( e ( u ) , θ ) ∇ θ = D e ( . u ): e ( . u ) + θ CE : e ( . u ) + G in Ω × (0 , T ) (eq θ ) It balances heat flux & rate of heat production due to dissipation: ◮ c v ( θ ) heat capacity ◮ j heat flux, given by Fourier’s law in an anisotropic medium j = − K ( e ( u ) , θ ) ∇ θ, K ( e ( u ) , θ ) pos. def. matrix heat conduction coefficients ◮ D e ( . u ): e ( . u ) viscous dissipation potential in the bulk ◮ G external heat source Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for θ Heat equation ` ´ c v ( θ ) . θ − div K ( e ( u ) , θ ) ∇ θ = D e ( . u ): e ( . u ) + θ CE : e ( . u ) + G in Ω × (0 , T ) . (eq θ ) + Neumann boundary conditions on ∂ Ω: K ( e ( u ) , θ ) ∇ θ · n = g on ∂ Ω × (0 , T ) , g external heat source Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for θ Heat equation ` ´ c v ( θ ) . θ − div K ( e ( u ) , θ ) ∇ θ = D e ( . u ): e ( . u ) + θ CE : e ( . u ) + G in Ω × (0 , T ) . (eq θ ) + Neumann boundary conditions on ∂ Ω: K ( e ( u ) , θ ) ∇ θ · n = g on ∂ Ω × (0 , T ) , g external heat source + conditions Γ C featuring dissipation rate on Γ C 1 ` ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ K ( e ( u ) , θ ) ∇ θ | + Γ C + K ( e ( u ) , θ ) ∇ θ | − · ν + η ( u , z ) θ = 0 on Γ C × (0 , T ) , (T1) Γ C 2 ˆ ˆ ˜ ˜ K ( e ( u ) , θ ) ∇ θ · ν = ζ ( . z ) on Γ C × (0 , T ) (T2) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for θ Heat equation ` ´ c v ( θ ) . θ − div K ( e ( u ) , θ ) ∇ θ = D e ( . u ): e ( . u ) + θ CE : e ( . u ) + G in Ω × (0 , T ) . (eq θ ) + Neumann boundary conditions on ∂ Ω: K ( e ( u ) , θ ) ∇ θ · n = g on ∂ Ω × (0 , T ) , g external heat source + conditions Γ C featuring dissipation rate on Γ C 1 ` ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ K ( e ( u ) , θ ) ∇ θ | + Γ C + K ( e ( u ) , θ ) ∇ θ | − · ν + η ( u , z ) θ = 0 on Γ C × (0 , T ) , (T1) Γ C 2 ˆ ˆ ˜ ˜ K ( e ( u ) , θ ) ∇ θ · ν = ζ ( . z ) on Γ C × (0 , T ) (T2) ◮ (T1): transient condition on Γ C with η ([ [ u ] ] , z ) heat transfer coefficient � heat convection [ [ θ ] ] jump of temperature across Γ C ◮ (T2) balances normal jump of heat flux j = − K ∇ θ with dissipation rate ζ ( . z ) on Γ C Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem (Frictionless) unilateral contact in the adhesive case • Complementarity problem with σ = D e ( . u ) + C ( e ( u ) − E θ ) and z Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem (Frictionless) unilateral contact in the adhesive case • Complementarity problem with σ = D e ( . u ) + C ( e ( u ) − E θ ) and z ˆ ˆ ˜ ˜ u · ν ≥ 0 on Γ C × (0 , T ) (no interpenetration) (Sign 1 ) ` ˆ ˆ ˜ ˜´ (Sign new σ | Γ C ν + κ z u · ν ≥ 0 on Γ C × (0 , T ) ) 2 ` ˆ ˆ ˜ ˜´ ˆ ˆ ˜ ˜ (Sign new σ | Γ C ν + κ z u · u = 0 on Γ C × (0 , T ) ) 3 ` ˆ ˆ ˜ ˜´ (Sign new σ | Γ C ν + κ z u · t = 0 on Γ C × (0 , T ) ∀ t s.t. ν · t = 0 ) 4 Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem (Frictionless) unilateral contact in the adhesive case • Complementarity problem with σ = D e ( . u ) + C ( e ( u ) − E θ ) and z ˆ ˆ ˜ ˜ u · ν ≥ 0 on Γ C × (0 , T ) (no interpenetration) (Sign 1 ) ` ˆ ˆ ˜ ˜´ (Sign new σ | Γ C ν + κ z u · ν ≥ 0 on Γ C × (0 , T ) ) 2 ` ˆ ˆ ˜ ˜´ ˆ ˆ ˜ ˜ (Sign new σ | Γ C ν + κ z u · u = 0 on Γ C × (0 , T ) ) 3 ` ˆ ˆ ˜ ˜´ (Sign new σ | Γ C ν + κ z u · t = 0 on Γ C × (0 , T ) ∀ t s.t. ν · t = 0 ) 4 When z = 0 (Sign new ) & (Sign new ) & (Sign new ) reduce to Signorini conditions 2 3 4 ◮ [ [ u ] ] · ν > 0 ⇒ σ | Γ C ν = 0 (no reaction) ◮ [ [ u ] ] · ν = 0 ⇒ σ | Γ C ν = λν , λ ≥ 0 (reaction is triggered) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem (Frictionless) unilateral contact in the adhesive case • Complementarity problem with σ = D e ( . u ) + C ( e ( u ) − E θ ) and z ˆ ˆ ˜ ˜ u · ν ≥ 0 on Γ C × (0 , T ) (no interpenetration) (Sign 1 ) ` ˆ ˆ ˜ ˜´ (Sign new σ | Γ C ν + κ z u · ν ≥ 0 on Γ C × (0 , T ) ) 2 ` ˆ ˆ ˜ ˜´ ˆ ˆ ˜ ˜ (Sign new σ | Γ C ν + κ z u · u = 0 on Γ C × (0 , T ) ) 3 ` ˆ ˆ ˜ ˜´ (Sign new σ | Γ C ν + κ z u · t = 0 on Γ C × (0 , T ) ∀ t s.t. ν · t = 0 ) 4 When z > 0 (adhesion active) (Sign new ) & (Sign new ) & (Sign new ) yield 2 3 4 ˆ ˆ ˜ ˜ σ | Γ C ν = λν − κ z u ν, λ ≥ 0 even for λ = 0 there’s a reaction ∼ κ z [ [ u ] ] counteracting separation � this is the elastic response of the adhesive Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem (Frictionless) unilateral contact in the adhesive case • Complementarity problem with σ = D e ( . u ) + C ( e ( u ) − E θ ) and z ˆ ˆ ˜ ˜ u · ν ≥ 0 on Γ C × (0 , T ) (no interpenetration) (Sign 1 ) ` ˆ ˆ ˜ ˜´ (Sign new σ | Γ C ν + κ z u · ν ≥ 0 on Γ C × (0 , T ) ) 2 ` ˆ ˆ ˜ ˜´ ˆ ˆ ˜ ˜ (Sign new σ | Γ C ν + κ z u · u = 0 on Γ C × (0 , T ) ) 3 ` ˆ ˆ ˜ ˜´ (Sign new σ | Γ C ν + κ z u · t = 0 on Γ C × (0 , T ) ∀ t s.t. ν · t = 0 ) 4 Equivalently formulated as differential inclusion ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ σ | Γ C ν + κ z u + ∂ I C u ∋ 0 on Γ C × (0 , T ) , with C = C ( x ) = { v ∈ R d : v · ν ( x ) ≥ 0 } for a.a. x ∈ Γ C and ∂ I C convex analysis subdifferential of the indicator function I C . Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem General contact conditions on Γ C Signorini contact can be replaced by ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ σ | Γ C ν + κ z u + ∂ I C u ∋ 0 on Γ C × (0 , T ) with C = C ( x ) closed cone for a.a. x ∈ Γ C . Examples ◮ (Signorini) unilateral contact, no interpenetration C = C ( x ) = { v ∈ R d ; v · ν ( x ) ≥ 0 } for a.a. x ∈ Γ C ◮ tangential slip along Γ C C = C ( x ) = { v ∈ R d ; v · ν ( x ) = 0 } for a.a. x ∈ Γ C ◮ very simplified model: C = C ( x ) linear subspace of R d Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for z Flow rule for z ˛ 2 − a 0 ∋ 0 ˛ ˜˛ ˛ˆ ˆ ˜ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ on Γ C × (0 , T ) (eq z ) u It’s a balance law between dissipation and stored energy Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Equation for z Flow rule for z ˛ 2 − a 0 ∋ 0 ˛ ˜˛ ˛ˆ ˆ ˜ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ on Γ C × (0 , T ) (eq z ) u It’s a balance law between dissipation and stored energy ◮ ζ ( . z ) dissipation potential on Γ C , enforces irreversibility ◮ I [0 , 1] ( z ) � constraint z ∈ [0 , 1] ˛ 2 ∼ elastic response of the adhesive ˛ ˛ 1 2 κ ˛ [ [ u ] ] ◮ ◮ a 0 (phenomenological specific) stored energy by disintegrating the adhesive. Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Rate-dependent vs. rate-independent evolution for z ˛ 2 − a 0 ∋ 0 ˛ ˛ˆ ˆ ˜ ˜˛ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ u on Γ C × (0 , T ) (eq z ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Rate-dependent vs. rate-independent evolution for z ˛ 2 − a 0 ∋ 0 ˛ ˛ˆ ˆ ˜ ˜˛ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ u on Γ C × (0 , T ) (eq z ) Viscous models ζ = ζ ( . z ) has superlinear growth at infinity. In particular, z ) = 1 z | 2 + I ( −∞ , 0] ( . ζ ( . 2 | . z ) (gradient flow case) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Rate-dependent vs. rate-independent evolution for z ˛ 2 − a 0 ∋ 0 ˛ ˛ˆ ˆ ˜ ˜˛ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ u on Γ C × (0 , T ) (eq z ) Viscous models ζ = ζ ( . z ) has superlinear growth at infinity. In particular, z ) = 1 z | 2 + I ( −∞ , 0] ( . ζ ( . 2 | . z ) (gradient flow case) Rate-independent models, our choice ζ = ζ ( . z ) has linear growth at infinity: 1-positively homogeneous ζ ( λ v ) = λζ ( v ) ∀ λ ≥ 0 In particular, ζ ( . z ) = a 1 | . z | + I ( −∞ , 0] ( . z ) with a 1 (phenomenological specific) dissipated energy by disintegrating the adhesive. Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Rate-dependent vs. rate-independent evolution for z ˛ 2 − a 0 ∋ 0 ˛ ˛ˆ ˆ ˜ ˜˛ z ) − a 1 + ∂ I [0 , 1] ( z ) + 1 ∂ I ( −∞ , 0] ( . 2 κ u on Γ C × (0 , T ) (eq z ) Viscous models ζ = ζ ( . z ) has superlinear growth at infinity. In particular, z ) = 1 z | 2 + I ( −∞ , 0] ( . ζ ( . 2 | . z ) (gradient flow case) Rate-independent models, our choice ζ = ζ ( . z ) has linear growth at infinity: 1-positively homogeneous ζ ( λ v ) = λζ ( v ) ∀ λ ≥ 0 In particular, ζ ( . z ) = a 1 | . z | + I ( −∞ , 0] ( . z ) with a 1 (phenomenological specific) dissipated energy by disintegrating the adhesive. Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Rate-independent evolutions ˛ 2 − a 0 ∋ 0 ˛ ˜˛ ˛ˆ ˆ ˜ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ on Γ C × (0 , T ) (eq z ) u Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Rate-independent evolutions ˛ 2 − a 0 ∋ 0 ˛ ˜˛ ˛ˆ ˆ ˜ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ on Γ C × (0 , T ) (eq z ) u Features ◮ invariance under time-rescaling : ζ is 1-homogeneous ⇒ ∂ζ is 0-homogeneous Hence z is solution of (eq z ) if and only if z ◦ α is solution of (eq z ) for every strictly increasing reparametrization α . ◮ Typical of activated systems: z responds to the activation energy in a rate-independent way possibly with hysteresis effects Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Rate-independent evolutions ˛ 2 − a 0 ∋ 0 ˛ ˜˛ ˛ˆ ˆ ˜ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ on Γ C × (0 , T ) (eq z ) u Mathematical difficulties ◮ ζ does NOT grow superlinearly at ∞ � no “good” estimates for . z standard regularity of t �→ z ( t ) is ONLY BV ◮ z may have jumps!!! � weak formulations Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Rate-independent evolutions ˛ 2 − a 0 ∋ 0 ˛ ˜˛ ˛ˆ ˆ ˜ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ on Γ C × (0 , T ) (eq z ) u Mathematical difficulties ◮ ζ does NOT grow superlinearly at ∞ � no “good” estimates for . z standard regularity of t �→ z ( t ) is ONLY BV ◮ z may have jumps!!! � weak formulations Theory of energetic solutions [Mielke et al.] Weak, derivative-free formulations, based on ◮ energetic balance (energy identity) ◮ stability conditions ◮ enforcing irreversibility Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem The complete PDE system: viscous vs. rate-independent behaviour ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E θ = F in Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem The complete PDE system: viscous vs. rate-independent behaviour ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E θ = F in Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), ` ´ c v ( θ ) . = D e ( . u ): e ( . u ) + θ CE : e ( . θ − div K ( e ( u ) , θ ) ∇ θ u ) + G in Ω × (0 , T ) , + Neu. b.c. on ∂ Ω × (0 , T ), Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem The complete PDE system: viscous vs. rate-independent behaviour ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E θ = F in Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), ` ´ c v ( θ ) . = D e ( . u ): e ( . u ) + θ CE : e ( . θ − div K ( e ( u ) , θ ) ∇ θ u ) + G in Ω × (0 , T ) , + Neu. b.c. on ∂ Ω × (0 , T ), ˆ ˆ ˜ ˜ D e ( . u ) + ( C e ( u ) − E θ ) ν = 0 on Γ C × (0 , T ) , ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E θ )) | Γ C ν + κ z u + ∂ I C u ∋ 0 on Γ C × (0 , T ) , | {z } σ | Γ C Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem The complete PDE system: viscous vs. rate-independent behaviour ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E θ = F in Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), ` ´ c v ( θ ) . = D e ( . u ): e ( . u ) + θ CE : e ( . θ − div K ( e ( u ) , θ ) ∇ θ u ) + G in Ω × (0 , T ) , + Neu. b.c. on ∂ Ω × (0 , T ), ˆ ˆ ˜ ˜ D e ( . u ) + ( C e ( u ) − E θ ) ν = 0 on Γ C × (0 , T ) , ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E θ )) | Γ C ν + κ z u + ∂ I C u ∋ 0 on Γ C × (0 , T ) , | {z } σ | Γ C ˛ 2 − a 0 ∋ 0 ˛ ˛ˆ ˆ ˜ ˜˛ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ u on Γ C × (0 , T ) , Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem The complete PDE system: viscous vs. rate-independent behaviour ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E θ = F in Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), ` ´ c v ( θ ) . = D e ( . u ): e ( . u ) + θ CE : e ( . θ − div K ( e ( u ) , θ ) ∇ θ u ) + G in Ω × (0 , T ) , + Neu. b.c. on ∂ Ω × (0 , T ), ˆ ˆ ˜ ˜ D e ( . u ) + ( C e ( u ) − E θ ) ν = 0 on Γ C × (0 , T ) , ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E θ )) | Γ C ν + κ z u + ∂ I C u ∋ 0 on Γ C × (0 , T ) , | {z } σ | Γ C ˛ 2 − a 0 ∋ 0 ˛ ˛ˆ ˆ ˜ ˜˛ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ u on Γ C × (0 , T ) , 1 ` ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ K ( e ( u ) , θ ) ∇ θ | + Γ C + K ( e ( u ) , θ ) ∇ θ | − · ν + η ( u , z ) θ = 0 on Γ C × (0 , T ) , Γ C 2 ˆ ˆ ˜ ˜ K ( e ( u ) , θ ) ∇ θ · ν = ζ ( . z ) on Γ C × (0 , T ) . Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem The complete PDE system: viscous vs. rate-independent behaviour ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E θ = F in Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), ` ´ c v ( θ ) . = D e ( . u ): e ( . u ) + θ CE : e ( . θ − div K ( e ( u ) , θ ) ∇ θ u ) + G in Ω × (0 , T ) , + Neu. b.c. on ∂ Ω × (0 , T ), ˆ ˆ ˜ ˜ D e ( . u ) + ( C e ( u ) − E θ ) ν = 0 on Γ C × (0 , T ) , ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E θ )) | Γ C ν + κ z u + ∂ I C u ∋ 0 on Γ C × (0 , T ) , | {z } σ | Γ C ˛ 2 − a 0 ∋ 0 ˛ ˛ˆ ˆ ˜ ˜˛ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ u on Γ C × (0 , T ) , 1 ` ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ K ( e ( u ) , θ ) ∇ θ | + Γ C + K ( e ( u ) , θ ) ∇ θ | − · ν + η ( u , z ) θ = 0 on Γ C × (0 , T ) , Γ C 2 ˆ ˆ ˜ ˜ K ( e ( u ) , θ ) ∇ θ · ν = ζ ( . z ) on Γ C × (0 , T ) . Not fully rate-independent model: viscosity -driven equations for u and ϑ coupled with rate-independent evolution for z . Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem The complete PDE system: viscous vs. rate-independent behaviour ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E θ = F in Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), ` ´ c v ( θ ) . = D e ( . u ): e ( . u ) + θ CE : e ( . θ − div K ( e ( u ) , θ ) ∇ θ u ) + G in Ω × (0 , T ) , + Neu. b.c. on ∂ Ω × (0 , T ), ˆ ˆ ˜ ˜ D e ( . u ) + ( C e ( u ) − E θ ) ν = 0 on Γ C × (0 , T ) , ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E θ )) | Γ C ν + κ z u + ∂ I C u ∋ 0 on Γ C × (0 , T ) , | {z } σ | Γ C ˛ 2 − a 0 ∋ 0 ˛ ˛ˆ ˆ ˜ ˜˛ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ u on Γ C × (0 , T ) , 1 ` ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ K ( e ( u ) , θ ) ∇ θ | + Γ C + K ( e ( u ) , θ ) ∇ θ | − · ν + η ( u , z ) θ = 0 on Γ C × (0 , T ) , Γ C 2 ˆ ˆ ˜ ˜ K ( e ( u ) , θ ) ∇ θ · ν = ζ ( . z ) on Γ C × (0 , T ) . Not fully rate-independent model: viscosity -driven equations for u and ϑ coupled with rate-independent evolution for z . General theory for rate-independent evolutions coupled with viscous evolutions: [Roub´ ıˇ cek’09,’10] Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Mathematical difficulties (I) ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E θ = F in Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), ` ´ c v ( θ ) . θ − div K ( e ( u ) , θ ) ∇ θ = D e ( . u ): e ( . u ) + θ CE : e ( . u ) + G in Ω × (0 , T ) , + Neu. b.c. on ∂ Ω × (0 , T ), ˆ ˆ ˜ ˜ D e ( . u ) + ( C e ( u ) − E θ ) ν = 0 on Γ C × (0 , T ) , ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E θ )) | Γ C ν + κ z u + ∂ I C u ∋ 0 on Γ C × (0 , T ) , ˛ 2 − a 0 ∋ 0 ˛ ˜˛ ˛ˆ ˆ ˜ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ u on Γ C × (0 , T ) , 1 ` ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ K ( e ( u ) , θ ) ∇ θ | + Γ C + K ( e ( u ) , θ ) ∇ θ | − · ν + η ( u , z ) θ = 0 on Γ C × (0 , T ) , Γ C 2 ˆ ˆ ˜ ˜ K ( e ( u ) , θ ) ∇ θ · ν = ζ ( . z ) on Γ C × (0 , T ) . ♦ (quadratic) coupling terms between (eq u ) and (eq θ ) � only L 1 -estimates for r.h.s. of (eq θ ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Enthalpy reformulation Only L 1 estimates for the r.h.s. of ` ´ c v ( θ ) . = D e ( . u ): e ( . u ) + θ CE : e ( . θ − div K ( e ( u ) , θ ) ∇ θ u ) + G in Ω × (0 , T ) ⇒ Boccardo-Gallou¨ et techniques + suitable growth conditions on c v , e.g. c 0 ( θ + 1) ω 0 ≤ c v ( θ ) ≤ c 1 ( θ + 1) ω 1 Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Enthalpy reformulation Only L 1 estimates for the r.h.s. of ` ´ c v ( θ ) . = D e ( . u ): e ( . u ) + θ CE : e ( . θ − div K ( e ( u ) , θ ) ∇ θ u ) + G in Ω × (0 , T ) ⇒ Boccardo-Gallou¨ et techniques + suitable growth conditions on c v , e.g. c 0 ( θ + 1) ω 0 ≤ c v ( θ ) ≤ c 1 ( θ + 1) ω 1 To combine this with time-discretization, enthalpy re-formulation R ϑ w = w ( θ ) = 0 c v ( r ) d r , θ = Θ( w ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Enthalpy reformulation Only L 1 estimates for the r.h.s. of ` ´ c v ( θ ) . = D e ( . u ): e ( . u ) + θ CE : e ( . θ − div K ( e ( u ) , θ ) ∇ θ u ) + G in Ω × (0 , T ) ⇒ Boccardo-Gallou¨ et techniques + suitable growth conditions on c v , e.g. c 0 ( θ + 1) ω 0 ≤ c v ( θ ) ≤ c 1 ( θ + 1) ω 1 To combine this with time-discretization, enthalpy re-formulation R ϑ w = w ( θ ) = 0 c v ( r ) d r , θ = Θ( w ) hence ` ´ w − div . K ( e ( u ) , w ) ∇ w = D e ( . u ): e ( . u )+Θ( w ) CE : e ( . u )+ G in Ω × (0 , T ) (eq w ) with θ ( w 1 /ω 1 − 1) ≤ Θ( w ) ≤ C 2 θ ( w 1 /ω 0 − 1) C 1 Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Mathematical difficulties (II) ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E Θ( w ) = F Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), ` ´ w − div . K ( e ( u ) , w ) ∇ w = D e ( . u ): e ( . u ) + Θ( w ) CE : e ( . u ) + G Ω × (0 , T ) , + Neu. b.c. on ∂ Ω × (0 , T ), ˆ ˆ ˜ ˜ D e ( . u ) + ( C e ( u ) − E Θ( w )) ν = 0 Γ C × (0 , T ) , ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z u + ∂ I C u ∋ 0 Γ C × (0 , T ) , ˛ 2 − a 0 ∋ 0 ˛ ˜˛ ˛ˆ ˆ ˜ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ Γ C × (0 , T ) , u 1 ` ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ K ( e ( u ) , w ) ∇ w | + Γ C + K ( e ( u ) , w ) ∇ w | − · ν + η ( u , z ) Θ( w ) = 0 Γ C × (0 , T ) , Γ C 2 ˆ ˆ ˜ ˜ K ( e ( u ) , w ) ∇ w · ν = ζ ( . z ) Γ C × (0 , T ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Mathematical difficulties (II) ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E Θ( w ) = F Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), ` ´ w − div . K ( e ( u ) , w ) ∇ w = D e ( . u ): e ( . u ) + Θ( w ) CE : e ( . u ) + G Ω × (0 , T ) + Neu. b.c. on ∂ Ω × (0 , T ), ˆ ˆ ˜ ˜ D e ( . u ) + ( C e ( u ) − E Θ( w )) ν = 0 Γ C × (0 , T ) , ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z u + ∂ I C u ∋ 0 Γ C × (0 , T ) , ˛ 2 − a 0 ∋ 0 ˛ ˜˛ ˛ˆ ˆ ˜ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ Γ C × (0 , T ) , u 1 ` ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ K ( e ( u ) , w ) ∇ w | + Γ C + K ( e ( u ) , w ) ∇ w | − · ν + η ( u , z ) Θ( w ) = 0 Γ C × (0 , T ) , Γ C 2 ˆ ˆ ˜ ˜ K ( e ( u ) , w ) ∇ w · ν = ζ ( . z ) Γ C × (0 , T ) ♦ coupling terms between (eq u ), (eq w ) and (eq z ) involve traces of u and w on Γ C � need of sufficient regularity of u and w to control u | Γ C and w | Γ C Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Mathematical difficulties (III) ` ` ´´ ̺ .. D e ( . u − div u ) + C e ( u ) − E Θ( w ) = F Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), ` ´ = D e ( . u ): e ( . u ) + Θ( w ) CE : e ( . w − div . K ( e ( u ) , w ) ∇ w u ) + G Ω × (0 , T ) , + Neu. b.c. on ∂ Ω × (0 , T ), ˆ ˆ ˜ ˜ D e ( . u ) + ( C e ( u ) − E Θ( w )) ν = 0 Γ C × (0 , T ) , ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z u + ∂ I C u ∋ 0 Γ C × (0 , T ) , ˛ 2 − a 0 ∋ 0 ˛ˆ ˛ ˆ ˜ ˜˛ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ u Γ C × (0 , T ) , 1 ` ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ K ( e ( u ) , w ) ∇ w | + Γ C + K ( e ( u ) , w ) ∇ w | − · ν + η ( u , z ) Θ( w ) = 0 Γ C × (0 , T ) , Γ C 2 ˆ ˆ ˜ ˜ K ( e ( u ) , w ) ∇ w · ν = ζ ( . z ) Γ C × (0 , T ) ♦ rate-independent evolution for z � lack of regularity of t �→ z ( t ), z may have jumps, . z need not be well-defined! Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the equation for z (I) ˛ 2 − a 0 ∋ 0 ˛ ˜˛ ˛ˆ ˆ ˜ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ u Weak, derivative-free formulation � semi-stability condition : ` ´ ` ´ ` ´ z ∈ L ∞ (Γ C ) : ∀ ˜ Φ u ( t ) , z ( t ) ≤ Φ u ( t ) , ˜ z + R ˜ z − z ( t ) for a.a. t ∈ (0 , T ) (S) with ◮ dissipation potential Z 8 Z < a 1 | ˜ z − z | d S if ˜ z ≤ z a.e. in Γ C , ` R z − z ) := ˜ ζ (˜ z − z ) d S = Γ C : Γ C + ∞ otherwise. ◮ stored energy functional Z Z “ κ ” 1 ˛ 2 + I [0 , 1] ( z ) − a 0 z ˆ ˆ ˜ ˜ ˛ˆ ˛ ˆ ˜ ˜˛ Φ( u , z ) := 2 C e ( u ): e ( u ) d x + I C ( u )+ 2 z u d S Ω Γ C Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the equation for z (I) ˛ 2 − a 0 ∋ 0 ˛ ˜˛ ˛ˆ ˆ ˜ z ) + ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . 2 κ u Weak, derivative-free formulation � semi-stability condition : ` ´ ` ´ ` ´ z ∈ L ∞ (Γ C ) : ∀ ˜ Φ u ( t ) , z ( t ) ≤ Φ u ( t ) , ˜ z + R ˜ z − z ( t ) for a.a. t ∈ (0 , T ) (S) with ◮ dissipation potential Z 8 Z < a 1 | ˜ z − z | d S if ˜ z ≤ z a.e. in Γ C , ` R ˜ z − z ) := ζ (˜ z − z ) d S = Γ C : Γ C + ∞ otherwise. ◮ stored energy functional Z Z “ κ ” 1 ˛ 2 + I [0 , 1] ( z ) − a 0 z ˆ ˆ ˜ ˜ ˛ˆ ˛ ˆ ˜ ˜˛ Φ( u , z ) := 2 C e ( u ): e ( u ) d x + I C ( u )+ 2 z u d S Ω Γ C Remark: (S) only a semi -stability condition ( u ( t ) is fixed)! This reflects the fact that PDE system NOT FULLY RATE-INDEPENDENT , u has viscosity-driven evolution! Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the equation for z (II) ˛ 2 − a 0 ˛ ˛ˆ ˆ ˜ ˜˛ ∂ I [0 , 1] ( z ) + 1 ∂ζ ( . z ) + 2 κ u ∋ 0 on Γ C × (0 , T ) (eq z ) | {z } = ∂ z Φ( u , z ) implies ` ´ ` ´ ` ´ z ∈ L ∞ (Γ C ) : ∀ ˜ Φ u ( t ) , z ( t ) ≤ Φ u ( t ) , ˜ z + R z − z ( t ) ˜ for a.a. t ∈ (0 , T ) (S) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the equation for z (II) Proof: • use that ∂ζ ( . z ) ⊂ ∂ζ (0) (1-homogeneity of ζ ) z ∈ L ∞ (Γ C ) and test • fix ˜ “ ” ˛ 2 − a 0 ˛ˆ ˛ ˆ ˜ ˜˛ ∂ I [0 , 1] ( z ( t )) + 1 − 2 κ u ( t ) ∈ ∂ζ ( . z ( t )) ⊂ ∂ζ (0) by ˜ z − z ( t ) | {z } = − ∂ z Φ( u ( t ) , z ( t )) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the equation for z (II) Proof: • use that ∂ζ ( . z ) ⊂ ∂ζ (0) (1-homogeneity of ζ ) z ∈ L ∞ (Γ C ) and test • fix ˜ “ ” ˛ 2 − a 0 ˛ ˛ˆ ˆ ˜ ˜˛ ∂ I [0 , 1] ( z ( t )) + 1 − 2 κ u ( t ) ∈ ∂ζ ( . z ( t )) ⊂ ∂ζ (0) by ˜ z − z ( t ) | {z } = − ∂ z Φ( u ( t ) , z ( t )) • Hence Z Z ζ (˜ z − z ( t )) − ζ (0) ≥ �− ∂ z Φ( u , z ( t )) , ˜ z − z ( t ) � Γ C Γ C | {z } | {z } ` = 0 = R z − z ( t )) ˜ Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the equation for z (II) Proof: • use that ∂ζ ( . z ) ⊂ ∂ζ (0) (1-homogeneity of ζ ) z ∈ L ∞ (Γ C ) and test • fix ˜ “ ” ˛ 2 − a 0 ˛ ˛ˆ ˆ ˜ ˜˛ ∂ I [0 , 1] ( z ( t )) + 1 − 2 κ u ( t ) ∈ ∂ζ ( . z ( t )) ⊂ ∂ζ (0) by ˜ z − z ( t ) | {z } = − ∂ z Φ( u ( t ) , z ( t )) • Hence Z Z ζ (˜ z − z ( t )) − ζ (0) ≥ �− ∂ z Φ( u , z ( t )) , ˜ z − z ( t ) � Γ C Γ C | {z } | {z } ` = 0 = R z − z ( t )) ˜ Φ( u , · ) convex ≥ Φ( u ( t ) , z ( t )) − Φ( u ( t ) , ˜ z ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the equation for z (II) Proof: • use that ∂ζ ( . z ) ⊂ ∂ζ (0) (1-homogeneity of ζ ) z ∈ L ∞ (Γ C ) and test • fix ˜ “ ” ˛ 2 − a 0 ˛ˆ ˛ ˆ ˜ ˜˛ ∂ I [0 , 1] ( z ( t )) + 1 − 2 κ u ( t ) ∈ ∂ζ ( . z ( t )) ⊂ ∂ζ (0) by ˜ z − z ( t ) | {z } = − ∂ z Φ( u ( t ) , z ( t )) • Hence Z Z ζ (˜ z − z ( t )) − ζ (0) ≥ �− ∂ z Φ( u , z ( t )) , ˜ z − z ( t ) � Γ C Γ C | {z } | {z } ` = 0 = R z − z ( t )) ˜ Φ( u , · ) convex ≥ Φ( u ( t ) , z ( t )) − Φ( u ( t ) , ˜ z ) Remark: if t �→ z ( t ) absolutely continuous ( no jumps ): semi-stability condition (S) (+ energy identity) ⇒ (eq z ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the equation for u From ̺ .. D e ( . 8 ` ` ´´ u − div u ) + C e ( u ) − E Θ( w ) = F in Ω × (0 , T ) , > > < + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), [ D e ( . [ u ) + ( C e ( u ) − E Θ( w ))] ] ν = 0 on Γ C × (0 , T ) , > ( D e ( . > ` ´ : u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z [ [ u ] ] + ∂ I C [ [ u ] ] ∋ 0 on Γ C × (0 , T ) , Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the equation for u From ̺ .. D e ( . 8 ` ` ´´ u − div u ) + C e ( u ) − E Θ( w ) = F in Ω × (0 , T ) , > > < + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), [ D e ( . [ u ) + ( C e ( u ) − E Θ( w ))] ] ν = 0 on Γ C × (0 , T ) , > ( D e ( . > ` ´ : u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z [ [ u ] ] + ∂ I C [ [ u ] ] ∋ 0 on Γ C × (0 , T ) , to 8 [ [ u ] ] ∈ C on Γ C × (0 , T ), > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > : Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the equation for u From ̺ .. D e ( . 8 ` ` ´´ u − div u ) + C e ( u ) − E Θ( w ) = F in Ω × (0 , T ) , > > < + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), [ D e ( . [ u ) + ( C e ( u ) − E Θ( w ))] ] ν = 0 on Γ C × (0 , T ) , > ( D e ( . > ` ´ : u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z [ [ u ] ] + ∂ I C [ [ u ] ] ∋ 0 on Γ C × (0 , T ) , to 8 [ [ u ] ] ∈ C on Γ C × (0 , T ), > > > > > Z > > ` ´ > > ̺ u ( T ) · . v ( T ) − . u ( T ) d x > > > Ω > > Z T > Z > > ` ` ´ > + D e ( . u ) + C e ( u ) − E Θ( w ) : e ( v − u ) d x d t > > < 0 Ω Z T Z T Z Z ` . ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ > − ̺ . u · v − . u d x d t + κ z u · v − u d S d t > > > > 0 Ω 0 Γ C > > > Z T Z T Z Z Z > > ` ´ > ≥ ̺ u 0 · . v (0) − u (0) d x + F · ( v − u ) d x d t + f · ( v − u ) d S d t > > > > Ω 0 Ω 0 Γ N > > > > > : for all test func. v s.t. v = 0 on Γ D × (0 , T ) and [ [ v ] ] ∈ C on Γ C × (0 , T ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the enthalpy equation From = D e ( . u ): e ( . u ) + Θ( w ) B : e ( . . ` ´ 8 w − div K ( e ( u ) , w ) ∇ w u ) + G Ω × (0 , T ) , > > < + Neu. b.c. on ∂ Ω × (0 , T ) ` ´ K ( e ( u ) , w ) ∇ w | + Γ C + K ( e ( u ) , w ) ∇ w | − 1 · ν + η ([ [ u ] ] , z )[ [Θ( w )] ] = 0 Γ C × (0 , T ) , > 2 ] · ν = ζ ( . Γ C > : [ [ K ( e ( u ) , w ) ∇ w ] z ) Γ C × (0 , T ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the enthalpy equation From = D e ( . u ): e ( . u ) + Θ( w ) B : e ( . . ` ´ 8 w − div K ( e ( u ) , w ) ∇ w u ) + G Ω × (0 , T ) , > > < + Neu. b.c. on ∂ Ω × (0 , T ) ` ´ K ( e ( u ) , w ) ∇ w | + Γ C + K ( e ( u ) , w ) ∇ w | − 1 · ν + η ([ [ u ] ] , z )[ [Θ( w )] ] = 0 Γ C × (0 , T ) , > 2 ] · ν = ζ ( . Γ C > : [ [ K ( e ( u ) , w ) ∇ w ] z ) Γ C × (0 , T ) to Z T Z Z 8 > w ( T ) v ( T ) d x + K ( e ( u ) , w ) ∇ w · ∇ v − w . v d x d t > > > Ω 0 Ω > > Z T > Z > > ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ˆ ˆ ˜ ˜ > + η ( u , z ) Θ( w ) v d S d t > > > > 0 Γ C > > < Z T Z T v | + Γ C + v | − Z Z ` ´ Γ C D e ( . u ): e ( . u ) + Θ( w ) C : E e ( . = u ) v d x d t − h z ( d S d t ) > 2 > 0 Ω 0 Γ C > > Z T Z T > Z Z Z > > > + Gv d x d t + gv d S d t + w 0 v (0) d x > > > > 0 Ω 0 ∂ Ω Ω > > for all test functions v , > > : with h z measure induced by dissipation ζ Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the adhesive contact PDE system Find a triple ( u , z , w ) with u ∈ W 1 , 2 (0 , T ; W 1 , 2 Γ D (Ω; R d )) ∩ W 1 , ∞ (0 , T ; L 2 (Ω; R d )) , z ∈ L ∞ (Γ C × (0 , T )) ∩ BV ([0 , T ]; L 1 (Γ C )) , w ∈ L r (0 , T ; W 1 , r (Ω \ Γ C )) ∩ L ∞ (0 , T ; L 1 (Ω)) ∩ BV ([0 , T ]; W 1 , r ′ (Ω \ Γ C ) ∗ ) ∀ 1 ≤ r < d +2 d +1 , Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the adhesive contact PDE system Find a triple ( u , z , w ) with u ∈ W 1 , 2 (0 , T ; W 1 , 2 Γ D (Ω; R d )) ∩ W 1 , ∞ (0 , T ; L 2 (Ω; R d )) , z ∈ L ∞ (Γ C × (0 , T )) ∩ BV ([0 , T ]; L 1 (Γ C )) , w ∈ L r (0 , T ; W 1 , r (Ω \ Γ C )) ∩ L ∞ (0 , T ; L 1 (Ω)) ∩ BV ([0 , T ]; W 1 , r ′ (Ω \ Γ C ) ∗ ) ∀ 1 ≤ r < d +2 d +1 , fulfilling ◮ semi-stability ◮ weak formulation of the momentum equation ◮ weak formulation of the enthalpy equation Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Weak formulation of the adhesive contact PDE system Find a triple ( u , z , w ) with u ∈ W 1 , 2 (0 , T ; W 1 , 2 Γ D (Ω; R d )) ∩ W 1 , ∞ (0 , T ; L 2 (Ω; R d )) , z ∈ L ∞ (Γ C × (0 , T )) ∩ BV ([0 , T ]; L 1 (Γ C )) , w ∈ L r (0 , T ; W 1 , r (Ω \ Γ C )) ∩ L ∞ (0 , T ; L 1 (Ω)) ∩ BV ([0 , T ]; W 1 , r ′ (Ω \ Γ C ) ∗ ) ∀ 1 ≤ r < d +2 d +1 , fulfilling ◮ semi-stability ◮ weak formulation of the momentum equation ◮ weak formulation of the enthalpy equation ◮ total energy inequality Z Z 1 u ( T ) | 2 d x + Φ ` ´ ̺ | . u ( T ) , z ( T ) + w ( T ) d x 2 Ω Ω Z Z ≤ 1 u (0) | 2 d x + Φ ` ´ ̺ | . u (0) , z (0) + w (0) d x 2 Ω Ω Z T Z T Z T Z T Z Z Z Z + F · . u d x d t + f · . u d S d t + G d x d t + g d S d t 0 Ω 0 Γ N 0 Ω 0 ∂ Ω Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Existence theorem (I) Under conditions on the data c v , K , η + conditions on the initial data ( u 0 , . u 0 , z 0 , w 0 ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Existence theorem (I) Under conditions on the data c v , K , η + conditions on the initial data ( u 0 , . u 0 , z 0 , w 0 ) if ̺ = 0 and C = C ( x ) general closed cone in R d Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Existence theorem (I) Under conditions on the data c v , K , η + conditions on the initial data ( u 0 , . u 0 , z 0 , w 0 ) if ̺ = 0 and C = C ( x ) general closed cone in R d then the Cauchy problem for the weak formulation ◮ semi-stability ◮ weak formulation of the momentum equation ◮ weak formulation of the enthalpy equation ◮ total energy inequality Z ` ´ Φ u ( T ) , z ( T ) + w ( T ) d x Ω Z T Z Z ` ´ ≤ Φ u (0) , z (0) + w (0) d x + F · . u d x d t Ω 0 Ω Z T Z T Z T Z Z Z + f · . u d S d t + G d x d t + g d S d t 0 Γ N 0 Ω 0 ∂ Ω Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Existence theorem (I) Under conditions on the data c v , K , η + conditions on the initial data ( u 0 , . u 0 , z 0 , w 0 ) if ̺ = 0 and C = C ( x ) general closed cone in R d then the Cauchy problem for the weak formulation has a solution ( u , w , z ). Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Existence theorem (II) Under conditions on the data c v , K , η + conditions on the initial data ( u 0 , . u 0 , z 0 , w 0 ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Existence theorem (II) Under conditions on the data c v , K , η + conditions on the initial data ( u 0 , . u 0 , z 0 , w 0 ) if ̺ > 0 and C = C ( x ) linear subspace in R d Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Existence theorem (II) Under conditions on the data c v , K , η + conditions on the initial data ( u 0 , . u 0 , z 0 , w 0 ) if ̺ > 0 and C = C ( x ) linear subspace in R d then the Cauchy problem for the weak formulation ◮ semi-stability ◮ weak formulation of the momentum equation ◮ weak formulation of the enthalpy equation ◮ total energy inequality Z Z 1 u ( T ) | 2 d x + Φ ` ´ ̺ | . u ( T ) , z ( T ) + w ( T ) d x 2 Ω Ω Z Z ≤ 1 u (0) | 2 d x + Φ ` ´ ̺ | . u (0) , z (0) + w (0) d x 2 Ω Ω Z T Z T Z T Z T Z Z Z Z + F · . u d x d t + f · . u d S d t + G d x d t + g d S d t 0 Ω 0 Γ N 0 Ω 0 ∂ Ω Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Existence theorem (II) Under conditions on the data c v , K , η + conditions on the initial data ( u 0 , . u 0 , z 0 , w 0 ) if ̺ > 0 and C = C ( x ) linear subspace in R d then the Cauchy problem for the weak formulation has a solution ( u , w , z ). Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Outline of the proof (I) ♦ Approximation via ε - Yosida regularization of the constraint [ [ u ] ] ∈ C on Γ C , i.e. ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z u + ∂ I C u ∋ 0 on Γ C × (0 , T ) replaced by ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z u + ( ∂ I C ) ε u ∋ 0 on Γ C × (0 , T ) . Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Outline of the proof (I) ♦ Approximation via ε - Yosida regularization of the constraint [ [ u ] ] ∈ C on Γ C , i.e. ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z u + ∂ I C u ∋ 0 on Γ C × (0 , T ) replaced by ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ ( D e ( . u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z u + ( ∂ I C ) ε u ∋ 0 on Γ C × (0 , T ) . Penalization In the case of (frictionless) Signorini contact C = C ( x ) = { v ∈ R d ; v · ν ( x ) ≥ 0 } for a.a. x ∈ Γ C ]) = − 1 ] · ν ) − ν , hence approximation reduces to then ( ∂ I C ) ε ([ [ u ] ε ([ [ u ] penalization − 1 ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ · ν ) − ν ∋ 0 on Γ C × (0 , T ) ( D e ( . u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z u ε ( u Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Outline of the proof (II) ♦ Approximation of the ε - Yosida regularized problem via semi-implicit time discretization : τ > 0 time-step � partition { t 0 = 0 < t 1 < . . . < t k < . . . < t K τ = T } Time-discrete problem: find { ( u k ετ , w k ετ , z k ετ ) } K τ k =1 fulfilling ̺ D 2 t uk “ D t uk e ( uk ετ ) − E Θ( wk ˛ e ( uk ˛ γ − 2 e ( uk ” = Fk ετ − div D e ` ´ + C ` ετ ) ´ + τ ˛ ετ ) ˛ ετ ) in Ω , ετ τ + Dir. b.c. on Γ D + Neu. b.c. on Γ N ´ 1 2 −√ τ D t wk K ( wk ετ , e ( uk ετ )) ∇ wk D t uk D t uk + Θ( wk D t uk + Gk ετ − div ` ` ´ ` ´ : e ` ´ ετ ) E : C e ` ´ in Ω , D e ετ ετ ετ ετ τ 2 + Neu. b.c. on ∂ Ω, κ ∂ζ ( D t zk ετ ) + ∂ I [0 , 1]( zk uk ˛ 2 − a 0 + τα zk ˛ ˜˛ ετ ) + ˛ˆ ˆ ˜ ετ ∋ 0 on Γ C , ετ 2 D e ( D t uk ετ ) + C ( e ( uk ετ ) − Θ( wk ˛ e ( uk ˛ γ − 2 e ( uk ˛ ˛ ˆ ˆ ετ ) E ) + τ ετ ) ετ ) ˜ ˜ ν = 0 on Γ C , κ zk uk uk h D e ( D t uk ετ ) + C ( e ( uk ετ ) − Θ( wk ˛ e ( uk ˛ γ − 2 e ( uk i ˆ ˆ ˜ ˜ + ( ∂ I C ) ε ( ˆ ˆ ˜ ˜ ) + ετ ) E ) + τ ˛ ετ ) ˛ ετ ) ν ετ ετ ετ on Γ C , ˛ 2 ´ µ 2 − 1 ˆ + τβ ` uk uk ˛ ˛ˆ ˆ ˜ ˜˛ ˆ ˜ ˜ 1+ = 0 ετ ετ 1 K ( wk ετ , e ( uk ετ )) ∇ wk ετ | + + K ( wk ετ , e ( uk ετ )) ∇ wk ετ |− u k − 1 , zk Θ( wk ` ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ · ν + η ( ετ ) ετ ) = 0 on Γ C , Γ C Γ C ετ 2 K ( wk ετ , e ( uk ετ )) ∇ wk ν = − ζ ( D t zk ˆ ˆ ˜ ˜ ετ ) on Γ C . ετ Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Outline of the proof (III) ♦ A priori estimates ♦ Passage to the limit as τ ↓ 0 ♦ Passage to the limit as ε ↓ 0 ⇒ Existence of a solution to the weak formulation Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Outline of the proof (III) ♦ A priori estimates ♦ Passage to the limit as τ ↓ 0 ♦ Passage to the limit as ε ↓ 0 ⇒ Existence of a solution to the weak formulation formally shown on the PDE system ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E Θ( w ) = F in Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), ` ´ w − div . K ( e ( u ) , w ) ∇ w = D e ( . u ): e ( . u ) + Θ( w ) CE : e ( . u ) + G in Ω × (0 , T ) + Neu. b.c. on ∂ Ω × (0 , T ), ˆ ˆ ˜ ˜ D e ( . u ) + ( C e ( u ) − E Θ( w )) ν = 0 on Γ C × (0 , T ) , ( D e ( . ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z u + ∂ I C u ∋ 0 on Γ C × (0 , T ) , ˛ 2 − a 0 ∋ 0 z ) + ∂ I [0 , 1] ( z ) + 1 ˛ ˛ˆ ˆ ˜ ˜˛ ∂ζ ( . 2 κ u on Γ C × (0 , T ) , 1 K ( e ( u ) , w ) ∇ w | + Γ C + K ( e ( u ) , w ) ∇ w | − ` ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ · ν + η ( u , z ) Θ( w ) = 0 on Γ C × (0 , T ) , Γ C 2 ˆ ˆ ˜ ˜ K ( e ( u ) , w ) ∇ w · ν = ζ ( . z ) on Γ C × (0 , T ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Outline of the proof (III) ♦ A priori estimates ♦ Passage to the limit ⇒ Existence of a solution to the weak formulation formally shown on the PDE system ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E Θ( w ) = F in Ω × (0 , T ) , + Dir. b.c. on Γ D × (0 , T ) + Neu. b.c. on Γ N × (0 , T ), ` ´ w − div . K ( e ( u ) , w ) ∇ w = D e ( . u ): e ( . u ) + Θ( w ) CE : e ( . u ) + G in Ω × (0 , T ) + Neu. b.c. on ∂ Ω × (0 , T ), ˆ ˆ ˜ ˜ D e ( . u ) + ( C e ( u ) − E Θ( w )) ν = 0 on Γ C × (0 , T ) , ( D e ( . ˆ ˆ ˜ ˜ `ˆ ˆ ˜ ˜´ u ) + ( C e ( u ) − E Θ( w ))) | Γ C ν + κ z u + ∂ I C u ∋ 0 on Γ C × (0 , T ) , ˛ 2 − a 0 ∋ 0 z ) + ∂ I [0 , 1] ( z ) + 1 ˛ ˛ˆ ˆ ˜ ˜˛ ∂ζ ( . 2 κ u on Γ C × (0 , T ) , 1 K ( e ( u ) , w ) ∇ w | + Γ C + K ( e ( u ) , w ) ∇ w | − ` ´ ˆ ˆ ˜ ˜ ˆ ˆ ˜ ˜ · ν + η ( u , z ) Θ( w ) = 0 on Γ C × (0 , T ) , Γ C 2 ˆ ˆ ˜ ˜ K ( e ( u ) , w ) ∇ w · ν = ζ ( . z ) on Γ C × (0 , T ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Basic a priori estimates (I) First estimate ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E Θ( w ) = F × . u ` ´ + w − div . K ( e ( u ) , w ) ∇ w = D e ( . u ): e ( . u ) + Θ( w ) CE : e ( . u ) + G × 1 ˛ 2 − a 0 ∋ 0 × . ˛ ˛ˆ ˆ ˜ ˜˛ z ) + ∂ I [0 , 1] ( z ) + 1 + ∂ζ ( . 2 κ u z = total energy balance: Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Basic a priori estimates (I) First estimate ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E Θ( w ) = F × . u ` ´ + w − div . K ( e ( u ) , w ) ∇ w = D e ( . u ): e ( . u ) + Θ( w ) CE : e ( . u ) + G × 1 ˛ 2 − a 0 ∋ 0 × . ˛ ˛ˆ ˆ ˜ ˜˛ z ) + ∂ I [0 , 1] ( z ) + 1 + ∂ζ ( . 2 κ u z = total energy balance: Z Z 1 u ( T ) | 2 d x + Φ ` ´ ̺ | . u ( T ) , z ( T ) + w ( T ) d x 2 Ω Ω Z Z =1 u (0) | 2 d x + Φ ` ´ ̺ | . u (0) , z (0) + w (0) d x 2 Ω Ω Z T Z T Z T Z T Z Z Z Z + F · . u d x d t + f · . u d S d t + G d x d t + g d S d t 0 Ω 0 Γ N 0 Ω 0 ∂ Ω Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Basic a priori estimates (I) First estimate ` ` ´´ ̺ .. u − div D e ( . u ) + C e ( u ) − E Θ( w ) = F × . u ` ´ + w − div . K ( e ( u ) , w ) ∇ w = D e ( . u ): e ( . u ) + Θ( w ) CE : e ( . u ) + G × 1 ˛ 2 − a 0 ∋ 0 × . ˛ ˛ˆ ˆ ˜ ˜˛ z ) + ∂ I [0 , 1] ( z ) + 1 + ∂ζ ( . 2 κ u z = total energy balance: Z Z 1 u ( T ) | 2 d x + Φ ` ´ ̺ | . u ( T ) , z ( T ) + w ( T ) d x 2 Ω Ω Z Z =1 u (0) | 2 d x + Φ ` ´ ̺ | . u (0) , z (0) + w (0) d x 2 Ω Ω Z T Z T Z T Z T Z Z Z Z + F · . u d x d t + f · . u d S d t + G d x d t + g d S d t 0 Ω 0 Γ N 0 Ω 0 ∂ Ω ⇒ “Energy” a-priori estimates on u , w , z Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Basic a priori estimates (II) ◮ Boccardo-Gallou¨ et estimates on the enthalpy equation & interpolation ⇒ bounds for ∇ w ◮ . w estimated by comparison in the enthalpy equation Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Basic a priori estimates (II) ◮ Boccardo-Gallou¨ et estimates on the enthalpy equation & interpolation ⇒ bounds for ∇ w ◮ . w estimated by comparison in the enthalpy equation ◮ By comparison in the momentum equilibrium equation Z T Z T ˛ ˛ Z Z ˛ ˛ `ˆ ˆ ˜ ˜´ ̺ .. uv + ∂ I C u v ˛ ≤ C for all test functions v ˛ ˛ ˛ 0 Ω 0 Γ C ⇒ inertial term and subdifferential term CANNOT be estimated separately . Hence we distinguish cases ̺ = 0 & general C vs. ̺ > 0 & linear C Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Basic a priori estimates (II) ◮ Boccardo-Gallou¨ et estimates on the enthalpy equation & interpolation ⇒ bounds for ∇ w ◮ . w estimated by comparison in the enthalpy equation ◮ By comparison in the momentum equilibrium equation Z T Z T ˛ ˛ Z Z ˛ ˛ `ˆ ˆ ˜ ˜´ ̺ .. uv + ∂ I C u v ˛ ≤ C for all test functions v ˛ ˛ ˛ 0 Ω 0 Γ C ⇒ inertial term and subdifferential term CANNOT be estimated separately . Hence we distinguish cases ̺ = 0 & general C vs. ̺ > 0 & linear C Compactness theorems: strongly/weakly converging (sub)sequences of approx. solutions ( u n , w n , z n ) → ( u , w , z ) Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Passage to the limit: step (I) Momentum equation & Semi-stability condition & Total energy inequality ◮ By strong-weak convergences we deduce that ( u , w , z ) fulfils weak momentum equation . Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Passage to the limit: step (I) Momentum equation & Semi-stability condition & Total energy inequality ◮ By strong-weak convergences we deduce that ( u , w , z ) fulfils weak momentum equation . ◮ Arguing by ◮ lower semicontinuity ◮ recovery sequence trick we obtain that ( u , w , z ) fulfils semi-stability condition . Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
Modeling PDE system: mathematical difficulties Weak formulation Analysis and existence results Open problem Passage to the limit: step (I) Momentum equation & Semi-stability condition & Total energy inequality ◮ By strong-weak convergences we deduce that ( u , w , z ) fulfils weak momentum equation . ◮ Arguing by ◮ lower semicontinuity ◮ recovery sequence trick we obtain that ( u , w , z ) fulfils semi-stability condition . ◮ Lower semicontinuity argument ⇒ total energy inequality . Riccarda Rossi Analysis of a rate-independent model for adhesive contact with thermal effects
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