Modelli Matematici. Gianni Gilardi. Cortona, giugno Transizione di fase solido solido in un sistema meccanico

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1 Modelli Matematici e Problemi Analitici per Materiali Speciali Cortona, giugno 2001 Transizione di fase solido solido in un sistema meccanico Gianni Gilardi Dipartimento di Matematica F. Casorati Università degli Studi di Pavia Via Ferrata 1, Pavia, Italy Tel: Fax: gilardi@dimat.unipv.it Home page: gilardi/

2 1 Generalities Joint work by: E. Bonetti, P. Colli, G.G., G. Schimperna (Pavia) W. Dreyer, J. Sprekels (WIAS Berlin) Motivation: Tin/lead solders solid solid phase change model Dreyer Müller, preprint 98 Different crystalline structures = coefficients depending on an order parameter Generalities 1

3 2 Generalities Γ u, Γ σ u = (u i ) χ domain in IR N (bdd, smooth, etc.) complementary parts of displacement order parameter Elasticity system for u with coefficients depending on χ and mixed boundary conditions coupled with Cahn Hilliard type equation for χ with forcing term depending on u and Neumann boundary conditions Initial conditions Generalities 2

4 3 Equations for u Equilibrium equation for the stress (σ ij ) σ ij x j = 0 Constitutive laws ( σ ij = C ijkl εkl ε ) kl C ijkl = C ijkl (χ) (usual symm & ellip) ε kl = 1 ( uk + u ) l 2 x l x k ε kl = ε kl(χ) (ε kl ) is the eigenstrain due to phase transition Boundary conditions u i = given σ ij n j = given on Γ u on Γ σ Equations for u 3

5 4 Equations for χ Cahn Hilliard type equations + div J = 0 (1) t J n = 0 on (2) J = M(χ) w (M elliptic) From (1 2) we deduce d χ dx = dt t dx = 0 constitutive equation for w (whence supplementary BC) initial condition for χ Equations for χ 4

6 5 Equations for χ w = w 1 + w 2 + w 3 2 χ w 1 = a kl x k x l a kl = a kl (χ) (elliptic) w 2 w 3 Hence = Ψ = n = 0 (Ψ is a double well potential) ( 1 ( εij ε ) ( ij Cijkl εkl ε ) ) kl 2 on w 3 = w 31 + w 32 where w 31 = σ kl ε kl w 32 = 1 2( εij ε ij ) C ijkl ( εkl ε kl) Equations for χ 5

7 6 All equations together σ ij x j = 0 σ ij = C ijkl (χ) ( ε kl ε kl(χ) ) ε kl = 1 ( uk + u ) l 2 x l x k t + div J = 0 J = M(χ) w w = w 1 + w 2 + w 31 + w 32 2 χ w 1 = a kl (χ) x k x l w 2 = Ψ w 31 = σ kl ε kl w 32 = 1 2( εij ε ij ) C ijkl boundary conditions as above initial condition for χ ( εkl ε kl) All equations together 6

8 7 Modified system Too difficult: no results. A significant paper: Garcke s abilitation thesis very general multiphase model: it contains the above problem but some quantities do not depend on χ. We modify in a different way, as shown below. Simplification/relaxation: M(χ) identity matrix (a kl (χ)) scalar a(χ) η ijkl := C ijkl(χ) constant w = µ χ + previous w, µ > 0 t Constraint: χ χ χ say 0 χ 1 Modified system 7

9 8 Model problem div ( k(χ) u y(χ) ) = 0 t w = 0 w χ t a( χ) χ + β(χ) + γ(χ) + z(χ) u + η u 2 initial and boundary conditions β maximal monotone in IR 2 and D(β) = [0, 1] k, y, a, γ, z Lipschitz functions of χ inf [0,1] k = k 0 > 0 and inf [0,1] a = a 0 > 0 η = k/ IR Model problem 8

10 9 Model problem A little more precise and better organized div ( k(χ) u y(χ) ) = 0 u = 0 on Γ u ( k( χ) u y(χ) ) n = 0 on Γ σ (1) t w = 0 w = χ t a( χ) χ + ξ + γ(χ) + z(χ) u + η u 2 ξ β(χ) n w = 0 n χ = 0 χ(0) = χ 0 on on (2) Model problem 9

11 10 Results Existence theorem. Assumptions as above. Moreover either (i) N = 1 or (ii) N = 2 and η = 0 sup a < a 0 := inf a χ 0 H 1 (), 0 χ 0 1, 0 < χ < 1 where χ is the mean value of χ 0. Then existence of a solution (u, χ, ξ, w) with u L (0, T ; H 1 ()) χ H 1 (0, T ; L 2 ()) L 2 (0, T ; H 2 ()) ξ L 2 (Q), Q = (0, T ) w L 2 (0, T ; H 2 ()) Results 10

12 11 Results Uniqueness means (u i, χ i, ξ i, w i ) solutions for i = 1, 2 then (u 1, χ 1 ) = (u 2, χ 2 ) Uniqueness theorem. (i) uniqueness if N = 1 (ii) uniqueness if N = 2 among the solutions such that u 4 dx dt < (1) Q (iii) uniqueness among all the solutions if N = 2, η = 0, and (1) holds for at least a solution. Results 11

13 12 Method for existence Fixed point of the map F = F 2 F 1 : F χ 1 F u 2 χ F 1 = find u from (1) with a given χ F 2 = find χ from (2) with a given u Schauder s theorem: Choice of the functional framework Continuity of F Relative compactness of its range V = { v H 1 () : v = 0 on Γ u } V = L 2 (0, T ; V ) (space for u) X = L 2 (0, T ; H 1 ()) (space for χ) K = K R = {χ X : χ [0, 1], χ X R} (R > 0 properly chosen) Method for existence 12

14 13 Study of F 1 Variational formulation of (1): V = { v H 1 } () : v = 0 on Γ u v 2 = v 2 Fix t and find u V such that ( k( χ) u y(χ) ) v = 0 v V Existence and uniqueness Basic estimate (v = u): k 0 u 2 u 1/2 sup y [0,1] where k 0 := inf [0,1] k, i.e. u 1 1/2 sup y k 0 [0,1] Conclusion for u = F 1 (χ) with χ K R : u L (0,T ;H 1 ()) Ĉ (1) where Ĉ is independent of χ and R. Study of F 1 13

15 14 Study of F 1 Continuity of F 1 : K R V: Assumptions: χ n, χ K R and χ n χ strongly in X Aim: u n := F 1 (χ n ) u := F 1 (χ) strongly in V Choose v = u n (t) u(t) and integrate over (0, T ): k 0 u n u 2 V = + Q Q k(χ n ) (u n u) 2 ( y( χ n ) y(χ) ) (u n u) Q ( k( χ) k(χ n ) ) u (u n u) c y(χ n ) y(χ) L2 (Q) + c u L p (Q) k( χ) k(χ n ) L q (Q) where c = 2T 1/2 1 Ĉ and p + 1 q = 1. Hence, a Meyers type result is needed: Zafran, J. Funct. Anal (general result) Study of F 1 14

16 15 Construction and study of F 2 F 2 : ( F 1 (K R ) ) Û K R, u χ Û = Ĉ { } v V : v L (0,T ;H 1 ()) Ĉ given by the basic estimate V = { v H 1 } () : v = 0 su Γ 0 V = L 2 (0, T ; V ) X := L 2 (0, T ; H 1 ()) K R = {χ X : χ [0, 1], χ X R} existence of χ uniqueness of χ χ K R for R large enough F 2 is continuous F 2 (Û) is relatively compact in X Construction and study of F 2 15

17 16 Existence of χ t w = 0 w χ t a( χ) χ + β(χ) + γ(χ) + z(χ) u + η u 2 n χ = n w = 0 χ(0) = χ 0 Abstract formulation: Trouble: H 1 () L 2 () H 1 () A : H 1 () H 1 () Aϕ, v = ϕ v χ + Aw = 0 on ϕ, v H 1 () w χ + a(χ) Aχ + β(χ) + γ(χ) + z(χ) u + η u 2 χ(0) = χ 0 we need η u 2 L 2 (Q) Hence, either 1D or η = 0. Existence of χ 16

18 17 Existence of χ Yosida regularization β ε as constraints are lost, take proper extensions of a, γ, z with a a 0 > 0 limit as ε 0 After a Galerkin approximation we have a solution (χ, w) = (χ ε, w ε ) to χ + Aw = 0 w = χ + a(χ) Aχ + β ε (χ) + γ(χ) + z(χ) u + η u 2 χ(0) = χ 0 Existence of χ 17

19 18 Existence of χ Main tool for a priori estimates: Then { X 0 := v H 1 () : X 0 := } v = 0 { } f H 1 () : f, 1 = 0 A X0 : X 0 iso X 0. Set N := {A X0 } 1 i.e., N solves generalized Neumann problems χ := 1 t = 0 χ 0 = 1 χ(t) t t at least formally. More precisely χ(t) χ and χ (t) belong to X 0 t Existence of χ 18

20 19 Existence of χ First a priori estimate: χ + Aw = 0 N (χ χ ) bad {}}{ w = χ + a(χ) Aχ + β ε (χ) + γ(χ) ( χ χ ) + z(χ) u + η u 2 and integrate the difference over (0, t). a(χ)a(χ), χ χ = χ ( a(χ)(χ χ ) ) a 0 χ Q 2 sup a χ 2 t Q t That is why we assume sup a < a 0. We get χ ε X L (0,T ;L 2 ()) C Choose here R > C and K = K R Existence of χ 19

21 20 Existence of χ Dealing with the limit The above estimate yields some weak convergence. However, due to the nonlinear terms, some stronger convergence is needed. This is given by further a priori estimates χ ε L (0,T ;H 1 ()) L 2 (0,T ;H 2 ()) c χ ε L2 (Q) c β ε (χ ε ) L 2 (Q) c w ε L2 (0,T ;H 2 ()) c Existence of χ 20

22 21 Existence of χ All this yields limit as ε 0 existence of χ Provided χ is unique, we have also: F 2 is well defined and estimates for F 2 (Û) relatively compact estimates for F 2 continuous Existence of χ 21

23 22 Uniqueness of χ (χ i, ξ i, w i ) = two solutions and χ := χ 1 χ 2, etc. } difference of χ i + Aw i = 0 difference of w i = χ i + a(χ i ) Aχ i + ξ i + γ(χ i ) + z(χ i ) u + η u 2 N χ χ and integrate the difference over (0, t). in the difference no η term worst terms: with a and z set a i := a(χ i ) set a i := a (χ i ) Uniqueness of χ 22

24 23 Uniqueness of χ Left hand side: 1 2 χ(t) 2 L 2 + ( 0) + bad = a 1 χ 2 + t 0 χ (a 1 χ) } {{ } bad a 1 χ χ 1 χ a 0 χ 2 L 2 c χ 1 L 4 χ L 4 χ L 2 a 0 2 χ 2 L 2 c χ 1 2 L 4 χ 2 L 4 χ 1 2 L 4 χ 2 L 4 GN 2D c χ 1 L χ 1 H 2 χ L 2 χ L 2 δ χ 2 L 2 + C δ χ 1 2 H 2 }{{} L 1 (0, T ) Gronwall χ 2 L 2 Uniqueness of χ 23

25 24 Uniqueness of χ... on the left hand side χ(t) 2 L 2 + t with some small coefficients 0 χ 2 L On the right hand side: bad = t 0 (GN 2D again) c δ t 0 t 0 + C δ t ( cz u L 2 χ 2 L 4 + Aχ 2, (a 2 a 1 )χ ) }{{} c ( u L 2 + Aχ 2 L 2) χ 2 L 4 ( u L 2 + Aχ 2 L 2 χ 2 L 2 0 ( u 2 H 1 + χ 2 2 H 2 ) }{{} L 1 (0, T ) Gronwall ) χ L 2 χ L 2 χ 2 L 2 Uniqueness of χ 24

26 25 Conclusion of existence Schauder s theorem applies. Hence: existence of a solution with u L (0, T ; H 1 ()) and (χ, ξ, w) according to our a priori estimates χ L (0, T ; H 1 ()) L 2 (0, T ; H 2 ()) since χ ε L (0,T ;H 1 ()) L 2 (0,T ;H 2 ()) c χ H 1 (0, T ; L 2 ()) since χ ε L 2 (Q) c ξ L 2 (Q) since β ε (χ ε ) L 2 (Q) c w L 2 (0, T ; H 2 ()) since w ε L 2 (0,T ;H 2 ()) c 2D with η 0 and 3D: open Conclusion of existence 25