Homogenization for a nonlinear ferroelectric model Aida Timofte Institute of Mathematics “Simion Stoilow” of the Romanian Academy Bucharest, 29.10.2008

Modeling and existence result for ferroelectric materials Ferroelectrics: spontaneous polarization, which can be reoriented by the application of an electric field. Derived property: piezoelectricity (the ability to develop an electrical charge proportional to a mechanical stress). Applications: electronics and optics (piezoelectric materials for medical ultrasound imaging and actuators, electro-optic materials for data storage and displays).

Basic quantities in the theory: u : Ω → Rd, elastic displacement field, D : Rd → Rd, electric displacement field, q : Ω → Rdq , internal variables (for instance, remanent strain rem or remanent polarization Prem). Stored-energy functional: E E(t, u, D, q) =

Z





Z

1 W (x, e(u), D, q) + α(x, ∇q) dx + |D|2 dx Ω Rd \Ω 20

−h`(t), (u, D)i, W Helmholtz free energy and e(u) infinitesimal strain tensor o n 1 T d×d d×d T e(u) = (∇u + ∇u ) ∈ Rsym := σ ∈ R σ =σ . 2

The external loading `(t): h`(t), (u, D)i =

Z Rd

+

Eext (t, x)·D(x) dx +

Z ΓNeu

Z Ω

fvol (t, x)·u(x) dx

fsurf(t, x)·u(x) da(x),

Eext , fvol , and fsurf, are applied external fields. Dissipation functional: R R(q) ˙ =

Z Ω

R(x, q(x)) ˙ dx,

where R(x, ·) : Rdq → [0, ∞) convex function homogeneous of degree 1 ( ⇐⇒ rate-independence).

Energetic formulation: based on energies only. Advantage: avoids derivatives of E and of the solution (u, D, Q). Under suitable smoothness and convexity assumptions the energetic formulation is equivalent to the differential formulation. Mielke & Theil & Levitas ’99; Mielke, Theil & Levitas ’02; Mielke & Theil ’04; Mainik & Mielke ’05; Mielke ’05 ; Francfort & Mielke ’06; Mielke & Timofte ’06. We have shown: Existence: The energetic formulation (S) & (E) has solutions for suitable initial data, if the constitutive functions W and R satisfy reasonable continuity and convexity assumptions. Uniqueness: under much stronger conditions.

Functions spaces: (u, D, Q) ∈ Y := F × Q, (u, D) ∈ F := H1 Γ

Dir

d L2 div (R ) :=

n

d (Ω; Rd) × L2 div (R ),

o ψ ∈ L2(Rd; Rd) div ψ = 0 ,

q ∈ Q := H1(Ω; Rdq ).

Definition: We call (u, D, q) an energetic solution of the problem associated with E and R, if the stability condition (S) and the energy balance (E) hold for every t ∈ [0, T ]: c q) c qb ; b D, b + R(q−q(t)) b b D, (S): E(t, u(t), D(t), q(t)) ≤ E(t, u, for all u,

(E): E(t, u(t), {z D(t), q(t))} + | present energy

Z t

R(q(s)) ˙ ds

|0 {z } dissipatedZ energy t

= E(0, u(0), {z D(0), q(0))} − | initial energy

˙ h`(s), (u(s), D(s))i ds .

| 0 {z } reduced work of external forces

Existence theorem (A. Mielke, A.T. ’06) Suppose that the constitutive functions W and R satisfy reasonable continuity and convexity assumptions. Then for each stable initial data (u0, D0, q0) the energetic problem (S) & (E) has a solution (u, D, q) : [0, T ] → Y, with (u(0), D(0), q(0)) = (u0, D0 , q0). Moreover, we have (u, D, q) ∈ L∞([0, T ]; Y).

Assumptions (

R : Ω × Rdq → [0, ∞), R ∈ C0(Ω × Rdq ), cR |V | ≤ R(x, V ) ≤ CR |V | for all x ∈ Ω, V ∈ Rdq ,

for some fixed constants cR , CR > 0, R(x, ·) : Rdq → [0, ∞) is 1-homogeneous and convex for every x ∈ Ω. (

d dq W : Ω × Rd×d sym × R × R → [0, ∞] α : Ω × Rdq ×d → [0, ∞]

are Caratheodory functions.

Coercivity and convexity assumptions: W (x, ε, D, q) + α(x, V ) ≥ c(|ε|2 + |D|2 + |q|2 + |V |2) − C d dq dq ×d, for every (x, ε, D, q, V ) ∈ Ω × Rd×d sym × R × R × R for some fixed constants c, C > 0,   W (x, ·, ·, q) : Rd×d × Rd → [0, ∞] is convex for every (x, q) ∈ Ω × Rdq , sym  α(x, ·) : Rdq ×d → [0, ∞] is convex for every x ∈ Ω.  1 ` ∈ C [0, T ], (H1 Γ

Dir

 d ∗ 2 d ∗ (Ω; R )) × Ldiv (R ) .

ε problem Ω ⊂ Rd, connected open bounded set, with Lipschitz boundary Γ; Y ⊂ Rd, unit periodicity cell; ε > 0. Assume W, α, R, to be Y -periodic in the first argument. Energy functional: Eε (t, u, D, q) =

Z Ω





W ( xε , e(u), D, q) + α( xε , ∇q) dx +

−h`(t), (u, D)i, Dissipation functional: Z



 x ˙ = R ε , q(x) ˙ dx. Rε(q) Ω

Z

1 |D|2 dx Rd \Ω 20

Energetic formulation We call (uε, Dε, qε) an energetic solution of the problem associated with Eε and Rε, if for every t ∈ [0, T ] the stability condition (Sε) and the energy balance (Eε) hold: c q) b D, b + Rε(q−q b ε(t)) (Sε) : Eε(t, uε(t), Dε(t), qε(t)) ≤ Eε(t, u, c qb ; b , D, for all u

(Eε) : Eε (t, uε(t), Dε(t), qε(t)) +

Z t 0

Rε(q˙ε(s)) ds

= Eε(0, uε(0), Dε(0), qε(0)) −

Z t 0

˙ h`(s), (uε(s), Dε(s))i ds.

Existence for (Sε ) & (Eε ) 0 0 ε ε For all ε > 0 and stable (u0 ε , Dε , qε ), the energetic problem (S ) & (E ) has a solution (uε, Dε, qε) : [0, T ] → Y, with 0, q 0 ). , D (uε(0), Dε(0), qε(0)) = (u0 ε ε ε

Moreover, we have (uε, Dε, qε) ∈ L∞([0, T ]; Y).

Two-scale homogenized problem (ε → 0)

H := H1ΓDir (Ω)d × L2(Ω; H1av (Y ))d, Z o 1 1 f (y) dy = 0 , Hav(Y ) := f ∈ Hper (Y ) Y ( ) Z d × Y ) := ψ ∈ L2(Rd × Y )d div ψ = 0 , L2 ( R divx ψ(x, y) dy = 0 , y Div Y n

Q := H1(Ω)dq × L2(Ω; H1av (Y ))dq , Z := H × L2Div (Rd × Y ) × Q. d × Y ), Q = (q , Q ) ∈ Q, define ( R For U = (u0, U1) ∈ H, D ∈ L2 0 1 Div

E(t, U, D, Q) =

where D(x) :=

Z Z



Ω Y

Z Y



W (y, e ˆ(U ), D, q0) + α(y, ∇xq0 + ∇y Q1) dy dx Z

Z

1 + |D|2 dy dx − h`(t), (u0, D)i, Rd\Ω Y 20

D(x, y) dy ∀x ∈ Rd and eˆ(U ) := ex(u0) + ey (U1 ). ˙ = R(Q)

Z Z Ω Y

R(y, q˙0 (x)) dy dx.

Energetic formulation of the two-scale homogenized problem: (S)& (E) For all t ∈ [0, T ]: e , Q) e, D e + R (Q e − Q(t)) (S) : E(t, U (t), D(t), Q(t)) ≤ E(t, U e , Q) e,D e ∈ Z, for every (U

(E) : E(t, U (t), D(t), Q(t)) +

Z t 0

˙ R(Q(s)) ds = E(0, U (0), D(0), Q(0)) −

Z t 0

˙ h`(s), (u0(s), D(s))i ds.

Claim: (S)& (E) is the two-scale homogenized problem for (Sε )& (Eε).

Main Theorem Let (uε, Dε, qε) : [0, T ] → Y be a solution for (Sε ) & (Eε ). Assume Eε (0, uε(0), Dε(0), qε(0)) → E(0, U 0, D0, Q0). for some Z 0 = (U 0 , D0, Q0) ∈ Z. Then there exists a subsequence (uε0 , Dε0 , qε0 )ε0 such that w2c

∀t ∈ [0, T ] : (uε0 (t), Dε0 (t), qε0 (t)) * Z(t) = (U (t), D(t), Q(t)) in Z, where Z : [0, T ] → Z is a solution of (S)& (E), with the initial condition Z(0) = Z 0 .

Two-scale convergence method Nguetseng 1989 (SIAM J. Math. Anal.); Allaire 1992 (SIAM J. Math. Anal.); Cioranescu & Donato 1999 (Oxford Lecture Series Math. Appl., 17); Lukkassen, Nguetseng & Wall 2002 (Int. J. Pure and Appl. Math.). Ω ⊂ Rd, bounded open set; Y = [0, 1)d, reference cell; Cper (Y ): subspace of C(Rd) of Y -periodic functions; Definition (two-scale convergence) A sequence {vε} in L2(Ω) two-scale converges to V ∈ L2(Ω × Y ), if 

Z

x lim vε(x)ψ x, ε→0 Ω ε



dx =

Z Z Ω Y

V (x, y)ψ(x, y) dy dx.

for every ψ = ψ(x, y) from L2(Ω; Cper (Y )). ts

We then write vε * V .

Weak and strong two-scale convergence Periodic unfolding operator Tε Cioranescu, Damlamian & Griso 2002   



x Tε : L2(Ω) → L2(Rd × Y ); Tεv(x, y) = vex ε + εy . ε Relation with two-scale convergence Assume Ω has Lipschitz boundary. Let a bounded sequence (vε)ε in L2(Ω) and let V ∈ L2(Ω × Y ). Then ts

vε * V ⇐⇒ Tε vε|Ω×Y * V (weakly) in L2(Ω × Y ). Definition Let {vε} ⊂ L2(Ω) and V ∈ L2(Ω × Y ). We say that {vε} w2 (w2): weakly two-scale converges to V (write vε * V ), if Tεvε * V (weakly) in L2(Rd × Y ). s2

(s2): strongly two-scale converges to V (write vε −→ V ), if Tεvε → V (strongly) in L2(Rd × Y ).

Proposition If {vε} ⊂ L2(Ω) is bounded, then ts

w2

vε * V ⇐⇒ vε * V. Two-scale convergence of gradients ∞ 1 H1 per (Y ) is the closure of Cper (Y ) in H (Y ).

R o 1 f ∈ Hper (Y ) Y f (y) dy = 0 . Notation: For v ∈ L2(Ω) define Ev ∈ L2(Ω × Y ) by Ev(x, y) = v(x).

H1 av(Y ) :=

n

Theorem Assume |∂Ω| = 0. Let vε * v0 weakly in H1 (Ω). Then s2

vε −→ Ev0, and there is a subsequence {vε0 } and V1 = V1 (x, y) in L2(Ω; H1 av(Y )), such that w2

∇vε0 * E∇v0 + ∇y V1.

Definition Let (U, D, Q) ∈ Z, with U = (u0, U1 ) and Q = (q0 , Q1). A sequence (uε, Dε, qε)ε in Y is called (w2c): weakly two-scale cross-convergent to (U, D, Q), if w2

w2

∇uε * ∇u0 + ∇y U1,

uε * Eu0, w2

qε * Eq0,

w2

D ε * D,

w2

∇qε * ∇q0 + ∇y Q1.

w2c

We write (uε, Dε, qε) * (U, D, Q). (s2c): strongly two-scale cross-convergent to (U, D, Q), if s2

uε −→ Eu0,

s2

∇uε −→ ∇u0 + ∇y U1 ,

s2

qε −→ Eq0, s2c

s2

∇qε −→ ∇q0 + ∇y Q1.

We write (uε, Dε, qε) −−→ (U, D, Q).

s2

Dε −→ D,

Theorem 1 (two-scale Γ-convergence of Rε) Let (qε)ε a bounded sequence in H1(Ω)dq and Q = (q0 , Q1) ∈ Q. w2

(i) If qε * Eq0 , then lim inf Rε(qε ) ≥ R(Q). ε→0

s2

(ii) If qε −→ Eq0, then lim Rε(qε) = R(Q). ε→0

Theorem 2 (two-scale Γ-convergence of Eε ) Let (uε, Dε, qε)ε a bounded sequence in Y and (U, D, Q) ∈ Z. w2c

(i) If (uε, Dε, qε) * (U, D, Q), then lim inf Eε(t, uε, Dε, qε) ≥ E(t, U, D, Q). ε→0

s2c

(ii) If (uε, Dε, qε) −−→ (U, D, Q), then lim Eε(t, uε, Dε, qε) = E(t, U, D, Q). ε→0

Proposition 1 (joint recovery sequence) w2c

Let a stable sequence (uε, Dε, qε)ε ⊂ Y, with (uε, Dε, qε)ε * (U, D, Q) ∈ e , Q) e ∈ Z, there is a joint recovery sequence Z. Assume for each (Ue , D f , qe ) ⊂ Y, such that e ε, D (u ε ε ε h

i f e ε, Dε, qeε) + Rε(qeε−qε) − Eε(t, uε, Dε, qε) lim sup Eε(t, u ε→0 e , Q) e,D e + R (Q e − Q) − E(t, U, D, Q). ≤ E(t, U

Then (U, D, Q) is stable (satisfies (S)). Proposition 2 For every (U, D, Q) ∈ Z, there exists a sequence (uε, Dε, qε)ε in Y, such that s2c

(uε, Dε, qε) −−→ (U, D, Q)

Lemma 1 Let (uε)ε and (vε)ε be bounded sequences in Lp(Ω), respectively Lq (Ω), where 1p + 1q = 1. Then  

w2



Z

Z

Z

uε * U ∈ Lp(Ω × Y )  ⇒ uε(x)vε(x) dx → U (x, y)V (x, y) dy dx. s2  v * q Ω Ω Y V ∈ L (Ω × Y ) ε Lemma 2 s2

Let (vε)ε be a bounded sequence in Lp(Ω), with vε * V ∈ Lp(Ω × Y ). x ) for x ∈ Rd. Then (Y ) and f (x) = f ( Consider f ∈ L∞ ε per ε s2

fεvε * f V, where f (x, y) := f (y) for every (x, y) ∈ Ω × Y .

Periodic unfolding and other related operators Periodic unfolding operator   



x Tεv(x, y) = v ε + εy . ε

Tε : Lp(Rd) → Lp(Rd × Y );

Z

1 U (ξ, y) dξ. PεU (x, y) = d ε ε([ xε ]+Y )

Pε : Lp(Rd × Y ) → Lp(Rd × Y ); Periodic folding operator p

d

p

d

Fε : L (R × Y ) → L (R );



n o FεU (x) = PεU x, xε .

Notations o d Ωε := int ε(λ + y) λ ∈ Z , y ∈ Y, ε(λ + int(Y )) ∩ Ω 6= ∅ , n o p p d d Lex(Ω) := v ∈ L (R ) v ≡ 0 a.e. on R \Ω , n o p p d d Lex(Ω × Y ) := U ∈ L (R × Y ) U ≡ 0 a.e. on (R \Ω) × Y . n

Properties: p

p

(i) Tε is an isometry and Tε (Lex(Ω)) ⊂ Lex(Ωε × Y ). p

p

(ii) kPεk ≤ 1 and Pε(Lex(Ω × Y )) ⊂ Lex(Ωε × Y ). For every U ∈ Lp(Rd × Y ), we have lim kPε U − U kLp(Rd ×Y ) = 0.

ε→0 p

p

(iii) Fε(Lex (Ω × Y )) ⊂ Lex (Ωε). (iv) We have TεFε = Pε, FεTε = idLp(Rd) , and PεTε = Tε . (v) The adjoint of Tε : Lp(Rd) → Lp(Rd × Y ) is the operator Fε : Lq (Rd × Y ) → Lq (Rd), where 1p + 1q = 1.

Proposition Let (uε)ε be a bounded family in Lp(Ω) with p ∈ (1, ∞), and U ∈ Lp(Ω × Y ). Then the following statements are equivalent: (ii) Tεuε * U in Lp(Ω×Y ), Ω×Y

ts

(i) uε * U,

w2

(iii) uε * U in Lp(Ω×Y ).

Proposition Let ρ : Rd × Rm → [0, ∞) be measurable and Y -periodic in the first argument. Then Z





Z

x , v(x) dx = ρ ρ(y, Tε v(x, y)) dx dy, d d ε R R ×Y p

for every v ∈ Lp(Rd). If ρ(·, 0) ≡ 0 and v ∈ Lex(Ω), then Z





Z

x , v(x) dx = ρ ρ(y, Tε v(x, y)) dx dy. ε Ω Ωε ×Y The integrals are finite if |ρ(y, z)| ≤ C(1 + |z|p ). In particular, we have Z Ω

v(x) dx =

Z Ωε ×Y

Tεv(x, y) dx dy

for every v ∈ Lpex(Ω).

References [1] A. Mielke. Evolution in rate-independent systems. In Handbook of Differential Equations II, Evolutionary Equations. Elsevier B.V., 2, 461–559, 2005. [2] A. Mielke and A. M. Timofte. An energetic material model for time-dependent ferroelectric behaviour: Existence and uniqueness. Math. Meth. Appl. Sci., 29 (2006), 1393–1410. [3] A. Mielke and A. M. Timofte. Two-scale homogenization for evolutionary variational inequalities via the energetic formulation. SIAM J. Math. Anal., 39 (2007), 462–668. [4] A. Timofte. Homogenization for a nonlinear ferroelectric model. Asymptotic Analysis, in press.