Modeling and analysis of a nonlinear PDE-system for phase - PowerPoint PPT Presentation

Weierstrass Institute for Applied Analysis and Stochastics DFG Research Center MATHEON Modeling and analysis of a nonlinear PDE-system for phase separation and damage Christiane Kraus (joint work with Christian Heinemann, WIAS, Elena Bonetti, Antonio Segatti, Pavia) Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de

Outline 1. Introduction 2. Modeling of a system for phase separation and damage processes 3. Numerical simulations 4. Existence results for the introduced system ⊲ different chemical energy densities ⊲ different elastic energy densities Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 2

Motivation Morphology in solder joints ⊲ Phase separation and coarsening Solder ball After solidification 3h. 300h. ⊲ Crack initiation and propagation along the phase boundary Tin-lead solders in electronic devices (mobile phones, PCs, micro-chips,...) Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 3

Literature Phase separation and coarsening ⊲ Phase field models of Cahn-Hilliard type with elasticity Analysis: Carrive/Miranville/Pietrus 00, Garcke 00, Miranville 00/01, Bonetti/Colli/Dreyer/Gilardi/Schimperna/Sprekels 02, Pawłow/Zaja ¸czkowski 08/09/10 Damage ⊲ Elliptic system/differential inclusion Engineering: Marigo 93, Frémond/Nedjar 96, Bourdin 00, Miehe 07, Ubachs/Schreurs/Geers 06, Hakim/Karma 09 Analysis: Mielke/Roubíˇ cek 03/10, Bonetti/Schimperna/Segatti 05, Thomas 10, Knees/Rossi/Zanini 11, Rocca/Rossi 12 Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 4

Literature Phase separation and coarsening ⊲ Phase field models of Cahn-Hilliard type with elasticity Analysis: Carrive/Miranville/Pietrus 00, Garcke 00, Miranville 00/01, Bonetti/Colli/Dreyer/Gilardi/Schimperna/Sprekels 02, Pawłow/Zaja ¸czkowski 08/09/10 ⇓ Aim ⊲ Developing of a phase field model for phase separation and damage processes ⊲ Analytical properties, numerical simulations ⇑ Damage ⊲ Elliptic system/differential inclusion Engineering: Marigo 93, Frémond/Nedjar 96, Bourdin 00, Miehe 07, Ubachs/Schreurs/Geers 06, Hakim/Karma 09 Analysis: Mielke/Roubíˇ cek 03/10, Bonetti/Schimperna/Segatti 05, Thomas 10, Knees/Rossi/Zanini 11, Rocca/Rossi 12 Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 4

New phase field model - energy Two components c + , c − , c + + c − = 1 ⇒ c = c + − c − = Variables c : concentration, u : displacement field z : damage, 0 ≤ z ≤ 1 Free energy � 1 2 | ∇ c | 2 + ψ ( c )+ 1 � 2 | ∇ z | 2 + h ( z )+ W ( c , e ( u ) , z ) � E ( c , u , z ) = ˆ d x Ω ψ ( c ) Chemical energy density ψ : polynomial or logarithmic type Elastic energy density ∇ u +( ∇ u ) T � e ( u ) = 1 � c c − c + 2 W 1 ( c , e ( u )) = 1 2 ( e ( u ) − e ∗ ( c )) : C ( c )( e ( u ) − e ∗ ( c )) e ∗ ( c ) : eigenstrain, C ( c ) : stiffness tensor (sym., pos. def.), W ( c , e ( u ) , z ) = ( g ( z )+ δ ) W 1 ( c , e ( u )) , δ > 0 , g : monotonically increasing with g ( 0 ) = 0 , g ∈ C 1 ([ 0 , 1 ]) . Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 5

New phase field model - energy and dissipation Free energy � 1 2 | ∇ c | 2 + ψ ( c )+ 1 � 2 | ∇ z | 2 + h ( z )+ W ( c , e ( u ) , z ) � E ( c , u , z ) = ˆ d x Ω Dissipation potential � β � 2 | ∂ t z | 2 − α∂ t z � R ( ∂ t z ) = ˆ d x Ω Constraints for the damage variable 0 ≤ z ≤ 1 , ( z = 0 completely damaged, z = 1 undamaged) ∂ t z ≤ 0 unidirectional process R ( ∂ t z ) � E ( c , u , z ) : = ˆ E ( c , u , z )+ Ω I [ 0 , 1 ] ( z ) d x � R ( ∂ t z ) : = R ( ∂ t z )+ Ω I ( − ∞ , 0 ] ( ∂ t z ) d x ˆ ∂ t z Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 6

New phase field model - evolution system � � 1 2 | ∇ c | 2 + ψ ( c )+ 1 2 | ∇ z | 2 + h ( z )+ W ( c , e ( u ) , z )+ I [ 0 , 1 ] ( z ) � E ( c , u , z ) = d x Free energy Ω � β � � R ( ∂ t z ) = 2 | ∂ t z | 2 − α∂ t z + I ( − ∞ , 0 ] ( ∂ t z ) d x Dissipation potential Ω Evolution system (ES) in classical formulation Evolution law for the mass concentration ∂ t c = − div J , J = − M ∇ w w = D c E ( c , u , z ) = −△ c + ψ ′ ( c )+ W , c ( c , e ( u ) , z ) Quasistatic balance of forces 0 = D u E ( c , u , z ) = div W , e ( c , e ( u ) , z ) � � Evolution law for the damage variable 0 ∈ D z E ( c , u , z )+ ∂ R ( ∂ t z ) ⇒ 0 = −△ z + h ′ ( z )+ W , z ( c , e ( u ) , z )+ r + β∂ t z − α + s with ⇐ r ∈ ∂ I [ 0 , 1 ] ( z ) , s ∈ ∂ I ( − ∞ , 0 ] ( ∂ t z ) . Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 7

New phase field model - initial-boundary conditions Assumptions Ω ⊂ R N bounded Lipschitz domain, D ⊂ ∂ Ω with H n − 1 ( D ) > 0 . Initial-boundary conditions (IBC) u ( t ) = b ( t ) on D × ( 0 , T ) , Dirichlet conditions M ∇ w · ν = 0 on ∂ Ω × ( 0 , T ) , Neumann conditions ∇ c · ν = 0 on ∂ Ω × ( 0 , T ) , ∇ z · ν = 0 on ∂ Ω × ( 0 , T ) , W , e · ν = 0 on ∂ ( Ω \ D ) × ( 0 , T ) , c ( 0 ) = c 0 , Initial conditions with 0 ≤ z 0 ≤ 1 . z ( 0 ) = z 0 Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 8

New phase field model - growth assumptions Assumptions: ⊲ ψ ∈ C 1 ( R ) , h ∈ C 1 ([ 0 , 1 ]) , W 1 ∈ C 1 ( R × R N × N ) , W 1 ( c , e ) = W 1 ( c , ( e ) t ) . Growth assumptions for W 1 : ⊲ | W 1 ( c , e ) | ≤ C ( | c | 2 + | e | 2 + 1 ) , ⊲ η | e 1 − e 2 | 2 ≤ ( W 1 , e ( c , e 1 ) − W 1 , e ( c , e 2 )) : ( e 1 − e 2 ) , ⊲ | W 1 , e ( c , e 1 + e 2 ) | ≤ C ( W 1 ( c , e 1 , z )+ | e 2 | + 1 ) , ⊲ | W 1 , c ( c , e ) | ≤ C ( | c | 2 + | e | 2 + 1 ) . Growth assumption for ψ : ⊲ | ψ ′ ( c ) | ≤ C ( | c | 2 ∗ / 2 + 1 ) Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 9

New phase field model - weak formulation Weak formulation q = ( c , w , u , z ) is a weak solution of (ES) with (IBC) if ⊲ c ∈ L ∞ ( 0 , T ; H 1 ( Ω )) , w ∈ L 2 ( 0 , T ; H 1 ( Ω )) , u ∈ L ∞ ( 0 , T ; H 1 ( Ω ; R N )) , z ∈ L ∞ ( 0 , T ; H 1 ( Ω )) ∩ H 1 ( 0 , T ; L 2 ( Ω )) , 0 ≤ z ≤ 1 a.e. and ∂ t z ≤ 0 a.e. ⊲ for all ξ ∈ L 2 ( 0 , T ; H 1 ( Ω )) , ∂ t ξ ∈ L 2 ( Ω T ) and ξ ( T ) = 0 : � � ( c − c 0 ) ∂ t ξ d x d s = − M ∇ w · ∇ ξ d x d s Ω T Ω T ⊲ for all ξ ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) and a.e. t ∈ ( 0 , T ) : � � � � Ω w ξ d x = ∇ c · ∇ ξ + ψ ′ ( c ) ξ + W , c ( c , e ( u ) , z ) ξ d x Ω ⊲ for all ξ ∈ H 1 ( Ω , R N ) with ξ = 0 on D and a.e. t ∈ ( 0 , T ) : � Ω W , e ( c , e ( u ) , z ) : e ( ξ ) d x = 0 Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 10

New phase field model - weak formulation Weak formulation (continued) ⊲ Variational inequality − ( Ω ) ∩ L ∞ ( Ω ) and a.e. t ∈ ( 0 , T ) : for all ξ ∈ H 1 � � � ∇ z · ∇ ξ + h ′ ( z ) ξ + W , z ( c , e ( u ) , z ) ξ − α ξ + β∂ t z ξ d x + � r , ξ � , 0 ≤ (1) Ω where r ∈ ∂ I H 1 + ( Ω ) ∩ L ∞ ( Ω ) ( z ) . ⊲ Energy estimate for a.e. t ∈ ( 0 , T ) : � t � � − α∂ t z + β | ∂ t z | 2 + M ∇ w · ∇ w � E ( c ( t ) , u ( t ) , z ( t ))+ d x d s Ω 0 � t � Ω W , e ( c , e ( u ) , z ) : e ( ∂ t b ) d x d s ≤ E ( c ( 0 ) , u ( 0 ) , z ( 0 ))+ (2) 0 Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 11

New phase field model - weak formulation Weak formulation (continued) ⊲ Variational inequality − ( Ω ) ∩ L ∞ ( Ω ) and a.e. t ∈ ( 0 , T ) : for all ξ ∈ H 1 � � � ∇ z · ∇ ξ + h ′ ( z ) ξ + W , z ( c , e ( u ) , z ) ξ − α ξ + β∂ t z ξ d x + � r , ξ � , 0 ≤ (1) Ω where r ∈ ∂ I H 1 + ( Ω ) ∩ L ∞ ( Ω ) ( z ) . ⊲ Energy estimate for a.e. t ∈ ( 0 , T ) : � t � � − α∂ t z + β | ∂ t z | 2 + M ∇ w · ∇ w � E ( c ( t ) , u ( t ) , z ( t ))+ d x d s Ω 0 � t � Ω W , e ( c , e ( u ) , z ) : e ( ∂ t b ) d x d s ≤ E ( c ( 0 ) , u ( 0 ) , z ( 0 ))+ (2) 0 Theorem (Heinemann & K.) For smooth solutions, (1) and (2) are equivalent to 0 ∈ ∂ R ( ∂ t z )+ ∂ z E ( c , u , z ) . Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 11

Main result - existence Notion of weak solutions ⊲ Cahn-Hilliard system with elasticity ⊲ variational inequality for z ⊲ energy estimate Assumptions ⊲ Initial and boundary conditions ⊲ Conditions for ψ , h and W Theorem (Heinemann & K.) Existence of weak solutions Let b ∈ W 1 , 1 ([ 0 , T ] ; W 1 , ∞ ( Ω ; R N )) , c 0 ∈ H 1 ( Ω ) and z 0 ∈ H 1 ( Ω ) . Then there exists a weak solution ( c , u , w , z ) of (ES) with the previous initial and bound- ary conditions. Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 12

Numerical simulations (Müller, WIAS) ⊲ loading proportional to the time t ⊲ initial damage seed ⊲ different thermal expansions of the phases Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 13

Numerical simulations Damage field is constant in time Phase separation and damage · Levico, 11. September 2013 · C. Kraus · Page 14