Local versus energetic solutions in rate-independent brittle - PowerPoint PPT Presentation

Weierstrass Institute for Applied Analysis and Stochastics Local versus energetic solutions in rate-independent brittle delamination Marita Thomas (jointly with T. Roubíˇ cek & C. Panagiotopoulos) Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de

Rate-independent brittle delamination Crack initiation and growth along prescibed interface Γ C Ω + Modeling ansatz: Generalized Standard Materials Γ C � for delamination [Frémond82,87] . Ω − delamination variable z : [ 0 , T ] × Γ C → [ 0 , 1 ] volume fraction of active bonds DIMO 2013 · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de · Page 2 (23)

Rate-independent brittle delamination Crack initiation and growth along prescibed interface Γ C Ω + Modeling ansatz: Generalized Standard Materials Γ C � for delamination [Frémond82,87] . Ω − delamination variable z : [ 0 , T ] × Γ C → [ 0 , 1 ] volume fraction of active bonds Rate-independent system described by ( Q , E , R 1 ) Q : state space, R 1 : unidirectional, pos. 1-hom. dissipation potential DIMO 2013 · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de · Page 2 (23)

Rate-independent brittle delamination Crack initiation and growth along prescibed interface Γ C Ω + Modeling ansatz: Generalized Standard Materials Γ C � for delamination [Frémond82,87] . Ω − delamination variable z : [ 0 , T ] × Γ C → [ 0 , 1 ] volume fraction of active bonds Rate-independent system described by ( Q , E , R 1 ) Q : state space, R 1 : unidirectional, pos. 1-hom. dissipation potential Dissipation potential: � � a 1 | ˙ z | z ≤ 0 with a 1 > 0 if ˙ z ) = R 1 ( ˙ z ( x )) d s R 1 ( ˙ z ) : = R 1 ( ˙ ∞ else R ∞ Γ C Rate-independence ⇔ 1 -homogeneity: healing ∀ λ > 0 ∀ v : R 1 ( λ v ) = λ R 1 ( v ) R 1 ( 0 ) = 0 and forbidden 0 ˙ z DIMO 2013 · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de · Page 2 (23)

Rate-independent brittle delamination Crack initiation and growth along prescibed interface Γ C Ω + Modeling ansatz: Generalized Standard Materials Γ C � for delamination [Frémond82,87] . Ω − delamination variable z : [ 0 , T ] × Γ C → [ 0 , 1 ] volume fraction of active bonds Rate-independent system described by ( Q , E , R 1 ) Q : state space, R 1 : unidirectional, pos. 1-hom. dissipation potential Dissipation potential: � � a 1 | ˙ z | z ≤ 0 with a 1 > 0 if ˙ z ) = R 1 ( ˙ z ( x )) d s R 1 ( ˙ z ) : = R 1 ( ˙ ∞ else R ∞ Γ C Rate-independence ⇔ 1 -homogeneity: healing ∀ λ > 0 ∀ v : R 1 ( λ v ) = λ R 1 ( v ) R 1 ( 0 ) = 0 and forbidden ⇓ D 1 ( z 1 , z 2 ) = R 1 ( z 2 − z 1 ) Dissipation distance: 0 ˙ z DIMO 2013 · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de · Page 2 (23)

Rate-independent brittle delamination Crack initiation and growth along prescibed interface Γ C Ω + Modeling ansatz: Generalized Standard Materials Γ C � for delamination [Frémond82,87] . Ω − delamination variable z : [ 0 , T ] × Γ C → [ 0 , 1 ] volume fraction of active bonds Rate-independent system described by ( Q , E , R 1 ) Q : state space, R 1 : unidirectional, pos. 1-hom. dissipation potential DIMO 2013 · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de · Page 2 (23)

Rate-independent brittle delamination Crack initiation and growth along prescibed interface Γ C Ω + Modeling ansatz: Generalized Standard Materials Γ C � for delamination [Frémond82,87] . Ω − delamination variable z : [ 0 , T ] × Γ C → [ 0 , 1 ] volume fraction of active bonds Rate-independent system described by ( Q , E , R 1 ) Q : state space, R 1 : unidirectional, pos. 1-hom. dissipation potential � E ( t , u , z ) : = E bulk ( t , u )+ Γ C I [ z [[ u ]]= 0 ] ([[ u ( t )]] , z ( t ))+ I [ 0 , 1 ] ( z ) d x energy functional: displacement u : [ 0 , T ] × ( Ω + ∪ Ω − ) → R d , [[ u ]] : jump of u across Γ C z ( t )[[ u ( t )]] = 0 a.e. on Γ C brittle delamination: ∀ t ∈ [ 0 , T ] : DIMO 2013 · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de · Page 2 (23)

Rate-independent brittle delamination Crack initiation and growth along prescibed interface Γ C Ω + Modeling ansatz: Generalized Standard Materials Γ C � for delamination [Frémond82,87] . Ω − delamination variable z : [ 0 , T ] × Γ C → [ 0 , 1 ] volume fraction of active bonds Rate-independent system described by ( Q , E , R 1 ) Q : state space, R 1 : unidirectional, pos. 1-hom. dissipation potential � E ( t , u , z ) : = E bulk ( t , u )+ Γ C I [ z [[ u ]]= 0 ] ([[ u ( t )]] , z ( t ))+ I [ 0 , 1 ] ( z ) d x energy functional: displacement u : [ 0 , T ] × ( Ω + ∪ Ω − ) → R d , [[ u ]] : jump of u across Γ C z ( t )[[ u ( t )]] = 0 a.e. on Γ C brittle delamination: ∀ t ∈ [ 0 , T ] : Regularization of the brittle constraint by � � � � � = 0 allowed on Γ C adhesive contact: ∀ t ∈ [ 0 , T ] : z ( t ) u ( t ) � �� � penalized by energy term J k ([[ u ]] , z ) : = k 2 z | [[ u ]] | 2 DIMO 2013 · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de · Page 2 (23)

Goal: Model for brittle delamination displaying driving force for crack growth Γ − → brittle delamination Ansatz: Approximation adhesive contact This result is known for energetic solutions [Roubíˇ cek/Scardia/Zanini09] . DIMO 2013 · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de · Page 3 (23)

Goal: Model for brittle delamination displaying driving force for crack growth Γ − → brittle delamination Ansatz: Approximation adhesive contact This result is known for energetic solutions [Roubíˇ cek/Scardia/Zanini09] . But general drawback of energetic solutions: too early jumps Thus alternative notion of solution: local solutions DIMO 2013 · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de · Page 3 (23)

Goal: Model for brittle delamination displaying driving force for crack growth Γ − → brittle delamination Ansatz: Approximation adhesive contact This result is known for energetic solutions [Roubíˇ cek/Scardia/Zanini09] . But general drawback of energetic solutions: too early jumps Thus alternative notion of solution: local solutions Plan of the talk: 1. 2 notions of solution for rate-independent systems: energetic and local solutions 2. 1D-comparison of their behavior for adhesive contact 3. Alternative scaling for the local adhesive model Motivation as a stress-driven delamination model 4. Analytical results & challenges 5. Numerical experiment DIMO 2013 · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de · Page 3 (23)

1. Energetic & local solutions for rate-independent systems Definition [Mielke&Co] : q : [ 0 , T ] → Q is an energetic solution to ( Q , E , R 1 ) , if for all t ∈ [ 0 , T ] it holds ∂ t E ( · , q ( · )) ∈ L 1 (( 0 , T )) , E ( t , q ( t )) < ∞ and:  q ∈ Q : E ( t , q ( t )) ≤ E ( t , ˜ q )+ R 1 ( ˜ z − z ( t )) , ( S ) Stability : for all ˜  t � E ( t , q ( t ))+ Diss R 1 ( z , [ 0 , t ]) = E ( 0 , q ( 0 ))+ ∂ t E ( ξ , q ( ξ )) d ξ , ( E ) Energy balance :  0 where Diss R 1 ( z , [ s , t ]) : = sup all part. of [ s , t ] ∑ N j = 1 R 1 ( z ( ξ j ) − z ( ξ j − 1 )) . Results in mathematical literature: Rate-independent evolution of delamination (energetic solutions): e.g. [Koˇ cvara/Mielke/Roubíˇ cek06, Roubíˇ cek/Scardia/Zanini09, Freddi/Paroni/Roubíˇ cek/Zanini11,12, Mielke/Roubíˇ cek/Th12] Rate-dependent evolution of delamination: e.g. [Frémond82,87, Point88, Raous/Cangémi/Cocu99, Bonetti/Bonfanti/Rossi07,08,09,11,12] DIMO 2013 · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de · Page 4 (23)

1. Energetic & local solutions for rate-independent systems Definition [Mielke&Co] : q : [ 0 , T ] → Q is an energetic solution to ( Q , E , R 1 ) , if for all t ∈ [ 0 , T ] it holds ∂ t E ( · , q ( · )) ∈ L 1 (( 0 , T )) , E ( t , q ( t )) < ∞ and:  q ∈ Q : E ( t , q ( t )) ≤ E ( t , ˜ q )+ R 1 ( ˜ z − z ( t )) , ( S ) Stability : for all ˜  t � E ( t , q ( t ))+ Diss R 1 ( z , [ 0 , t ]) = E ( 0 , q ( 0 ))+ ∂ t E ( ξ , q ( ξ )) d ξ , ( E ) Energy balance :  0 where Diss R 1 ( z , [ s , t ]) : = sup all part. of [ s , t ] ∑ N j = 1 R 1 ( z ( ξ j ) − z ( ξ j − 1 )) . Definition [Mielke&Co] : q = ( u , z ) : [ 0 , T ] → U × Z is a local solution to ( Q , E , R 1 ) , if  For a.a. t ∈ [ 0 , T ] :     ( S u u ∈ U : E ( t , u ( t ) , z ( t )) ≤ E ( t , ˜ u , z ( t )) ,  loc ) Minimality : for all ˜     z ∈ Z : E ( t , u ( t ) , z ( t )) ≤ E ( t , u ( t ) , ˜ z )+ R 1 ( ˜ z − z ( t )) , ( S z loc ) Semistability : for all ˜   For all 0 ≤ t 1 < t 2 ≤ T :    t 2  �   E ( t 2 , q ( t 2 ))+ Diss R 1 ( z , [ t 1 , t 2 ]) ≤ E ( t 1 , q ( t 1 ))+ ∂ t E ( ξ , q ( ξ )) d ξ . ( E loc ) Energy ineq. :  t 1 DIMO 2013 · 13.09.2013 · Marita Thomas · Marita.Thomas@wias-berlin.de · Page 4 (23)