Evolutionary -Convergence for a Delamination Model Thomas Frenzel, - PowerPoint PPT Presentation

General Framework & Theory Delamination Model Evolutionary Γ -Convergence for a Delamination Model Thomas Frenzel, Alexander Mielke Sept. 01, 2016 Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model (Static) Γ -Conv. Γ -Convergence ? 0 = D E ε ( u ε ) − → 0 = D E 0 ( u 0 ) Definition Let X be a Banach space, E ε : X → ( −∞ , ∞ ], we say E ε Γ Γ -converges (strongly or weakly) to E 0 and write E ε → E 0 if − ∀ u ε → u 0 : lim inf E ε ( u ε ) ≥ E 0 ( u 0 ) ∃ ˆ u ε → u 0 : lim E ε (ˆ u ε ) = E 0 ( u 0 ) Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model (Static) Γ -Conv. lower semi continuity − → u u Γ -limits are lower − → semi continuous. u u − → u u Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model (Static) Γ -Conv. choice of topology The choice of topology may affect the Γ -limit. Example Let a : [0 , 1] → R with 0 < a ≤ a ≤ a and denote its 1-periodic extension by a per . Then we define � 1 � x � u ( x ) 2 d x . E ε ( u ) = a per ε 0 Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model (Static) Γ -Conv. choice of topology � − 1 � 1 �� 1 � − 1 d x � With a arith = 0 a ( x ) d x and a harm = a ( x ) we 0 have the strong and the weak Γ -limit � 1 � 1 a arith u 2 d x a harm u 2 d x E s ( u ) = and E w ( u ) = 0 0 Microstructure is encoded in recovery sequence. For the weak topology we can choose u ε ( x ) = a per ( x /ε ) − 1 a harm u ( x ) . ˆ Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model (Static) Γ -Conv. choice of topology � − 1 � 1 �� 1 � − 1 d x � With a arith = 0 a ( x ) d x and a harm = a ( x ) we 0 have the strong and the weak Γ -limit � 1 � 1 a arith u 2 d x a harm u 2 d x E s ( u ) = and E w ( u ) = 0 0 Microstructure is encoded in recovery sequence. For the weak topology we can choose u ε ( x ) = a per ( x /ε ) − 1 a harm u ( x ) . ˆ Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model (Static) Γ -Conv. convergence of minimizers Theorem Let u ε be a minimizer of E ε . Then u ε → u 0 where u 0 is a minimizer of E 0 . In other words, Γ -convergence implies the convergence of solutions to the corresponding Euler-Lagrange-Equations, i.e., 0 = D E ε ( u ε ) − → 0 = D E 0 ( u 0 ) The sequence of solutions is a recovery sequence. Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Gradient Systems and Evol. Γ -Conv. Gradient Systems & Evolutionary Γ -Convergence microscopic system ε ց 0 macroscopic system u ε = A ε ( t , u ε ) ˙ u = A 0 ( t , u ) ˙ upscaling u 0 u 0 initial state − → ε time evolution ↓ ↓ upscaling u ε ( t ) = S ε ( t , u 0 u ( t ) = S 0 ( t , u 0 ) time t > 0 ε ) − → Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Gradient Systems and Evol. Γ -Conv. Gradient Systems & Evolutionary Γ -Convergence A gradient system ( X , E , R ) induces an evolution equation via u = D ξ R ∗ � � D ˙ u R ( u , ˙ u ) = − D u E ( t , u ) ⇐ ⇒ ˙ u , − D u E ( t , u ) Assume that E satisfies the chain rule, then � T � � � � � � E T , u ( T ) + J ( u , ˙ u ) ≤ E 0 , u (0) + ∂ t E t , u ( t ) d t 0 with � T � � + R ∗ � � J ( u , ˙ u ) := u ( t ) ˙ − D E ( t , u ( t )) d t . R 0 Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Gradient Systems and Evol. Γ -Conv. Gradient Systems & Evolutionary Γ -Convergence A gradient system ( X , E , R ) induces an evolution equation via u = D ξ R ∗ � � D ˙ u R ( u , ˙ u ) = − D u E ( t , u ) ⇐ ⇒ ˙ u , − D u E ( t , u ) Assume that E satisfies the chain rule, then � T � � � � � � E T , u ( T ) + J ( u , ˙ u ) ≤ E 0 , u (0) + ∂ t E t , u ( t ) d t 0 with � T � � + R ∗ � � J ( u , ˙ u ) := u ( t ) ˙ − D E ( t , u ( t )) d t . R 0 Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Gradient Systems and Evol. Γ -Conv. Gradient Systems & Evolutionary Γ -Convergence A gradient system ( X , E , R ) induces an evolution equation via u = D ξ R ∗ � � D ˙ u R ( u , ˙ u ) = − D u E ( t , u ) ⇐ ⇒ ˙ u , − D u E ( t , u ) Assume that E satisfies the chain rule, then � T � � � � � � E T , u ( T ) + J ( u , ˙ u ) = E 0 , u (0) + ∂ t E t , u ( t ) d t 0 with � T � � + R ∗ � � J ( u , ˙ u ) := u ( t ) ˙ − D E ( t , u ( t )) d t . R 0 Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Gradient Systems and Evol. Γ -Conv. Well-prepared evolutionary Γ -convergence � T � � � � � � E ε T , u ε ( T ) + J ε ( u ε , ˙ u ε ) ≤ E ε 0 , u ε (0) + 0 ∂ t E ε t , u ε ( t ) d t ↓ Γ- lim inf ↓ Γ- lim inf ↓ assptn. ↓ assptn. � T � � � � � � E 0 T , u 0 ( T ) + J 0 ( u 0 , ˙ u 0 ) ≤ E 0 0 , u 0 (0) + 0 ∂ t E 0 t , u 0 ( t ) d t Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Delamination Model f ( t ) Φ Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Delamination Model f ( t ) Φ � 1 1 � u 2 u ′ 2 + ε α Φ � E ε ( t , u ) = d x − f ( t ) u (0) , ε 0 � 1 1 u ′ 2 d x , R ( ˙ u ) = 2 ˙ 0 Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Delamination Model f ( t ) Φ 1 � � u ( t , y ) � u ( t , x ) = − u ( t , x ) + f ( t )(1 − x ) − ε α − 1 w ( x , y )Φ ′ ˙ d y ε 0 u ( t , 1) = 0 u (0 , x ) = u 0 ( x ) Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Apriori Estimates The EDB formulation � T � � � � � � E ε T , u ε ( T ) + J ε ( u ε , ˙ u ε ) ≤ E ε 0 , u ε (0) + ∂ t E ε t , u ε ( t ) d t 0 leads to bounds in 0 , T ; H 1 (0 , 1) H 1 � 0 , T ; H 1 (0 , 1) L ∞ � � � and resulting in convergence of u ε → u (up to a subsequence) in weakly in H 1 � 0 , T ; H 1 (0 , 1) 0 , T ; C β (0 , 1) � � � strongly in C and and u ε ( t ) ⇀ u ( t ) a.e. in H 1 (0 , 1) Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Γ -Convergence It is easy to verify that with respect to the weak H 1 (0 , 1) topologie we have � 1 1 � u ε � 2 u ′ 2 + ε α Φ E ε ( t , u ) = d x − f ( t ) u ε (0) ε ε 0 Γ − → � 1 1 2 u ′ 2 d x − f ( t ) u (0) . E 0 ( t , u ) = 0 Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Γ -Convergence The difficult part is the Γ -convergence (w.r.t. weak H 1 � 0 , T ; H 1 (0 , 1) � topology) of � T � � + R ∗ � � J ε ( u ε , ˙ u ε ) = R u ε ( t ) ˙ − D E ε ( t , u ε ( t )) d t 0 � 1 � T 1 2 + 1 ε + f ( t ) − µ ε ( u ε )) 2 d x d t u ′ 2( u ′ = 2 ˙ ε 0 0 � T � 1 1 2 + 1 � 2 d x d t u ′ � J 0 ( u , ˙ u ) = 2 ˙ − ˆ µ ( u 0 ) 0 2 0 0 Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Γ -Convergence The difficult part is the Γ -convergence (w.r.t. weak H 1 � 0 , T ; H 1 (0 , 1) � topology) of � T � � + R ∗ � � J ε ( u ε , ˙ u ε ) = R u ε ( t ) ˙ − D E ε ( t , u ε ( t )) d t 0 � 1 � T 1 2 + 1 ε + f ( t ) − µ ε ( u ε )) 2 d x d t u ′ 2( u ′ = 2 ˙ ε 0 0 � T � 1 1 2 + 1 � 2 d x d t u ′ � u ′ J 0 ( u , ˙ u ) = 2 ˙ 0 + f ( t ) − ˆ µ ( u 0 ) 0 2 0 0 Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Γ -Convergence The difficult part is the Γ -convergence (w.r.t. weak H 1 � 0 , T ; H 1 (0 , 1) � topology) of � T � � + R ∗ � � J ε ( u ε , ˙ u ε ) = R u ε ( t ) ˙ − D E ε ( t , u ε ( t )) d t 0 � 1 � T 1 2 + 1 ε + f ( t ) − µ ε ( u ε )) 2 d x d t u ′ 2( u ′ = 2 ˙ ε 0 0 � T � 1 1 2 + 1 � 2 d x d t u ′ � J 0 ( u , ˙ u ) = 2 ˙ D E 0 ( u 0 ) − ˆ µ ( u 0 ) 0 2 0 0 Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Investigation of Limit System We arrived at D R ( ˙ u ) = −DE 0 ( t , u ) + h ( t , u ) But we also have D R ( ˙ u ) = −DE 0 ( t , u ) ⇔ D R ( ˙ u ) = −DE 0 ( t , u ) + h ( t , u ) u ( t , x ) = − u ( t , x ) + f ( t )(1 − x ) ˙ Evolutionary Γ -Convergence for a Delamination Model

General Framework & Theory Delamination Model Outlook u ( t 1 ), u ( t 2 ) 3 2 1 x x 0 ( t 1 ) x 0 ( t 2 ) Fracture law x 0 ( t ) = const (Φ) f ( t ) 3 ˙ Evolutionary Γ -Convergence for a Delamination Model