Quasiconvexity at edges and corners and the nucleation of austenite in martensite

Transcription

1 Quasiconvexity at edges and corners and the nucleation of austenite in martensite Konstantinos Koumatos with John M. Ball and Hanuš Seiner Pattern Formation & Multiscale Phenomena in Materials 27/09/11

2 Experimental observations (Seiner) Specimen: Cu-Al-Ni single crystal rectangular bar of dimension mm in austenite edges approximately along (1, 0, 0), (0, 1, 0), (0, 0, 1), the principal directions of the cubic austenite.

3 Experimental procedure

4 alternative

5 Introduction Our work is also connected to the problem of necessary and sufficient conditions for strong local minimizers of E (y) := ϕ (x, y (x), Dy (x)) dx. Ω A classical problem in the calculus of variations - solved in the scalar case. In the vectorial case and Ω smooth, known necessary conditions are: satisfaction of the Euler-Lagrange equations positivity of the second variation quasiconvexity in the interior (Meyers, 1965) quasiconvexity at the boundary (Ball/Marsden, 1984)

6 Introduction Our work is also connected to the problem of necessary and sufficient conditions for strong local minimizers of E (y) := ϕ (x, y (x), Dy (x)) dx. Ω A classical problem in the calculus of variations - solved in the scalar case. In the vectorial case and Ω smooth, known necessary conditions are: satisfaction of the Euler-Lagrange equations positivity of the second variation quasiconvexity in the interior (Meyers, 1965) quasiconvexity at the boundary (Ball/Marsden, 1984)

7 Grabovsky and Mengesha (2009), provided a remarkable generalization of the Weierstrass theory showing that a strengthened version of the necessary conditions is sufficient.

8 Grabovsky and Mengesha (2009), provided a remarkable generalization of the Weierstrass theory showing that a strengthened version of the necessary conditions is sufficient. In our work, we introduce a simplified model and make use of a notion of quasiconvexity to explain why the nucleation cannot occur in the interior or the faces and edges, as well as show how quasiconvexity is lost at any given corner (allowing for the nucleation of austenite).

9 Grabovsky and Mengesha (2009), provided a remarkable generalization of the Weierstrass theory showing that a strengthened version of the necessary conditions is sufficient. In our work, we introduce a simplified model and make use of a notion of quasiconvexity to explain why the nucleation cannot occur in the interior or the faces and edges, as well as show how quasiconvexity is lost at any given corner (allowing for the nucleation of austenite). The result of Grabovsky and Mengesha, though maybe extendable to domains with edges and corners, works for smooth domains and is not directly applicable to our case.

10 We follow the approach to microstructure based on gradient Young measures (e.g. Ball/James, Kinderlehrer/Pedregal, Bhattacharya). These are families of probability measures ν = (ν x ) x Ω generated by sequences of gradients Dy j (assumed bounded in L ). Then, for all continuous f, f ( Dy j (x) ) νx, f = f (A) dν x (A) in L. M 3 3 In this approach, microstructures are idenified with gradient Young measure minimizers ν of an energy functional I θ (ν) = ν x, ϕ (, θ) dx, Ω = austenite at θ = θ c. Ω The underlying/macroscopic deformation gradient corresponds to the centre of mass of ν ν x = ν x, id = A dν x (A). M 3 3

11 We follow the approach to microstructure based on gradient Young measures (e.g. Ball/James, Kinderlehrer/Pedregal, Bhattacharya). These are families of probability measures ν = (ν x ) x Ω generated by sequences of gradients Dy j (assumed bounded in L ). Then, for all continuous f, f ( Dy j (x) ) νx, f = f (A) dν x (A) in L. M 3 3 In this approach, microstructures are idenified with gradient Young measure minimizers ν of an energy functional I θ (ν) = ν x, ϕ (, θ) dx, Ω = austenite at θ = θ c. Ω The underlying/macroscopic deformation gradient corresponds to the centre of mass of ν ν x = ν x, id = A dν x (A). M 3 3

12 Simplified model Let Ω describe the Cu-Al-Ni bar at austenite (θ = θ c ) and θ > θ c. The free-energy function ϕ : M+ 3 3 R {+ } is assumed lower semicontinuous (no coercivity or growth) s.t. δ, A SO (3) ϕ (A) = 0, A 6 i=1 SO (3) U i where δ > 0 and U i are the orthorhombic variants.

13 Cubic-to-orthorhombic variants α, β, γ > 0 and α γ β 0 0 β 0 0 U 1 = α+γ α γ U 2 = α+γ U 3 = U 5 = α γ 2 α+γ 2 0 α+γ 2 α γ 2 0 β 0 α γ 2 0 α+γ 2 α γ 2 α+γ 2 α γ 2 0 α+γ β U 4 = U 6 = γ α 2 α+γ 2 0 γ α α+γ 2 γ α 2 0 β 0 γ α 2 0 α+γ 2 γ α 2 α+γ 2 γ α 2 0 α+γ β

14 To make the problem more tractable we work with an energy that captures the essential behaviour of ϕ but becomes infinite off the wells - we invoke Γ-convergence. Let and ψ : M+ 3 3 (p > 3) - s.t. K := SO (3) 6 SO (3) U i i=1 R - lower semicontinuous and p-coercive ψ (A) 0 and ψ (A) = 0 A K. Define I k (ν) := ν x, kψ + ϕ dx. Ω

15 To make the problem more tractable we work with an energy that captures the essential behaviour of ϕ but becomes infinite off the wells - we invoke Γ-convergence. Let and ψ : M+ 3 3 (p > 3) - s.t. K := SO (3) 6 SO (3) U i i=1 R - lower semicontinuous and p-coercive ψ (A) 0 and ψ (A) = 0 A K. Define I k (ν) := ν x, kψ + ϕ dx. Ω

16 Claim: I k and Γ I w.r.t the weak- topology of L ( ( w Ω; C0 M 3 3 ) ) δ Ω ν x (SO (3)) dx, where I (ν) = +, ν x (SO (3)) = SO(3) dν x (A) is the volume fraction of austenite at x Ω. Note: Young measures lie in the closed unit ball of L w ( Ω; C0 ( M 3 3 ) ) = L 1 ( Ω; C 0 ( M 3 3 )) supp ν K otherwise.

17 For convenience we will write I (ν) = Ω ν x, W dx where the energy density W is given by δ, A SO (3) W (A) = 0, A 6 i=1 SO (3) U i +, otherwise. Denote the mechanically stabilized variant of martensite by U s so that the homogeneous gradient Young measure δ Us corresponds to a pure phase of U s.

18 For convenience we will write I (ν) = Ω ν x, W dx where the energy density W is given by δ, A SO (3) W (A) = 0, A 6 i=1 SO (3) U i +, otherwise. Denote the mechanically stabilized variant of martensite by U s so that the homogeneous gradient Young measure δ Us corresponds to a pure phase of U s.

19 Admissible measures Consider variations of δ Us which are localized in the interior, on faces, edges and at corners denoted by A i, A f, A e and A c. interior S face S edge S corner S We require: ν x = δ Us, x / S and y (x) = U s x, x S Ω (Dy (x) = ν x ). For faces, edges and corners S Ω is a free boundary.

20 In addition, we assume that det U s 1 and that det Dy (x) dx vol y (Ω) (C-N) Ω for the underlying deformations of admissible measures. (C-N) was introduced by Ciarlet & Nečas as a way to describe non-interpenetration of matter ensuring the a.e. injectivity of minimizers. Here, (C-N) guarantees that minimizers are homeomorphic and excludes a certain kind of self-contact of the boundary. Note: These conditions will only be relevant for faces and edges.

21 In addition, we assume that det U s 1 and that det Dy (x) dx vol y (Ω) (C-N) Ω for the underlying deformations of admissible measures. (C-N) was introduced by Ciarlet & Nečas as a way to describe non-interpenetration of matter ensuring the a.e. injectivity of minimizers. Here, (C-N) guarantees that minimizers are homeomorphic and excludes a certain kind of self-contact of the boundary. Note: These conditions will only be relevant for faces and edges.

22 For s = 1,..., 6, define M s = {e S 2 : U s e = max i {1,...,6} { U ie, 1e }} and M 1 s = {e S 2 : cof U s e = max i {1,...,6} { cof U ie, 1e }} Call vectors in M s and M 1 s, maximal directions for U s and Us 1. Main result: Let Ω be a parallelepiped (really, any convex polyhedron) with edges along directions in M s Us 1 1 If there exists ν A i A f A e A c such that I (ν) < I (δ Us ), then ν A c. M 1 s. 2 For any corner, there exists ν A c such that I (ν) < I (δ Us ). This says that austenite must and does nucleate at a corner.

23 For s = 1,..., 6, define M s = {e S 2 : U s e = max i {1,...,6} { U ie, 1e }} and M 1 s = {e S 2 : cof U s e = max i {1,...,6} { cof U ie, 1e }} Call vectors in M s and M 1 s, maximal directions for U s and Us 1. Main result: Let Ω be a parallelepiped (really, any convex polyhedron) with edges along directions in M s Us 1 1 If there exists ν A i A f A e A c such that I (ν) < I (δ Us ), then ν A c. M 1 s. 2 For any corner, there exists ν A c such that I (ν) < I (δ Us ). This says that austenite must and does nucleate at a corner.

24 Corners Quasiconvexity at U s at a corner: I (ν) I (δ Us ) ν A c ν x = δ Us S

25 Loss of quasiconvexity ν x = δ Us ν x = λδ Us + (1 λ) δ QUl ν x = δ R Then ν x (SO (3)) > 0 = I (ν) < 0 = I (δ Us ) proving part 2 of the result.

26 Interior Need to show: ν A i I (ν) I (δ Us ) ( ν x (SO (3)) = 0 a.e) ν x = δ Us S

27 Definition A function W : M 3 3 R {+ } is quasiconvex at F M 3 3 if for any homogeneous gradient Young measure µ with µ = F, W (F ) µ, W. Lemma: If W is quasiconvex at U s then I (ν) I (δ Us ) for all ν A i and nucleation cannot occur in the interior. Proof. Average the measure ν A i and note that the energy remains the same.

28 Definition A function W : M 3 3 R {+ } is quasiconvex at F M 3 3 if for any homogeneous gradient Young measure µ with µ = F, W (F ) µ, W. Lemma: If W is quasiconvex at U s then I (ν) I (δ Us ) for all ν A i and nucleation cannot occur in the interior. Proof. Average the measure ν A i and note that the energy remains the same.

29 Claim: W is quasiconvex at U s. Proof. We may assume that supp µ K and then can show that µ = U s = µ (SO (3)) = 0. This is precisely what we need to prove as then µ, W = 0 = W (U s ). Reminder: W (F ) < 0 F SO (3) Note: We show µ (SO (3)) = 0 using only that µ = U s NOT its homogeneity; we use this again for faces and edges. Also, (C-N) and det U s 1 is not needed and quasiconvexity in the interior is irrespective of the orientation of Ω.

30 Claim: W is quasiconvex at U s. Proof. We may assume that supp µ K and then can show that µ = U s = µ (SO (3)) = 0. This is precisely what we need to prove as then µ, W = 0 = W (U s ). Reminder: W (F ) < 0 F SO (3) Note: We show µ (SO (3)) = 0 using only that µ = U s NOT its homogeneity; we use this again for faces and edges. Also, (C-N) and det U s 1 is not needed and quasiconvexity in the interior is irrespective of the orientation of Ω.

31 Faces and Edges Need to show: I (ν) I (δ Us ) ν A f I (ν) I (δ Us ) ν A e ν x = δ Us ν x = δ Us S S Definition: quasiconvexity at faces (Ball/Marsden) quasiconvexity at edges

32 We involve the maximal directions for U s and U 1 s S y (S) e y r (t) = x 1 + te x 1 x 2 U s x 1 U s x 2 y (r (t))

33 S if e M s y (S) e y r (t) = x 1 + te x 1 x 2 U s x 1 U s x 2 y (r (t)) = U s r (t)

34 S if e M s y (S) e y x 1 x 2 U s x 1 U s x 2 ν x = Dy (x) = U s

35 S if e M s y (S) e y x 1 x 2 U s x 1 U s x 2 ν x = Dy (x) = U s = I (ν) I (δ Us )

36 The (C-N) condition excludes the following self-contact: S y (S) e y Then

37 S y (S) e y x 1 x 2 U s x 1 U s x 2 r (t) U s e

38 if U s e M 1 s (and det U s 1) S y (S) e y 1 x 1 x 2 U s x 1 U s x 2 y 1 (r (t)) = Us 1 r (t) r (t) U s e

39 if U s e M 1 s (and det U s 1) S y (S) e y 1 x 1 x 2 U s x 1 U s x 2

40 if U s e M 1 s (and det U s 1) S y (S) e y 1 x 1 x 2 U s x 1 U s x 2 ν x = Dy (x) = U s = I (ν) I (δ Us )

41 Figure: All edges our result applies to for s = 1 (lattice parameters obtained by Seiner).

42 Figure: Normals to all faces our result applies to for s = 1 (lattice parameters obtained by Seiner).

43 Remark on Grabosky & Mengesha s work The sufficiency proof is based on a decomposition lemma (I. Fonseca/S. Müller/P. Pedregal) splitting variations into a strong and a weak part. The strong part localizes and cannot lower the energy due to the quasiconvexity conditions, whereas, the weak part cannot lower the energy due to the positivity of the second variation. Departing from our singular energy and using the machinery of Grabovsky & Mengesha, one could hope to show that whenever ν is sufficiently close to δ Us with I (ν) < I (δ Us ) then ν must involve nucleation at a corner. Thank you.

44 Remark on Grabosky & Mengesha s work The sufficiency proof is based on a decomposition lemma (I. Fonseca/S. Müller/P. Pedregal) splitting variations into a strong and a weak part. The strong part localizes and cannot lower the energy due to the quasiconvexity conditions, whereas, the weak part cannot lower the energy due to the positivity of the second variation. Departing from our singular energy and using the machinery of Grabovsky & Mengesha, one could hope to show that whenever ν is sufficiently close to δ Us with I (ν) < I (δ Us ) then ν must involve nucleation at a corner. Thank you.

45

46

47 back