In this paper, we study liquid crystal nematodynamics (i.e., the hydrodynamics of nematic liquid crystals). In previous work, we investigated periodic mesomorphic media [START_REF] Chechkin | Nematic liquid crystals. Existence and uniqueness of periodic solutions to Ericksen-Leslie equations[END_REF] and homogenization of micro inhomogeneous nematic liquid crystals (periodic in [START_REF] Ratiu | Homogenization of the equations of the dynamics of nematic liquid crystals with inhomogeneous density[END_REF] and random in [START_REF] Chechkin | Nematodynamics and random homogenization[END_REF]).

The hydrodynamic theory of liquid crystals, due to Ericksen and Leslie, was developed in the 1960's [START_REF] Ericksen | Conservation laws for liquid crystals[END_REF][START_REF] Ericksen | Hydrostatic theory of liquid crystals[END_REF][START_REF] Leslie | Some constitutive equations for anisotropic fluids[END_REF][START_REF] Leslie | Some constitutive equations for liquid crystals[END_REF]; see also [START_REF] Ericksen | Continuum theory of nematic liquid crystals[END_REF][START_REF] Leslie | Continuum theory for nematic liquid crystals[END_REF]. The subject of our research is the Ericksen-Leslie system describing the dynamics of nematic liquid crystals

u -μΔu = -∇p -∂ ∂xj ∂F ∂nx j • (∇n) + F + f, div u = 0, J n -2qn + h = g + G, n = 1, (1)
where summation on repeated indices is understood, n xj := ∂ ∂xj n, and ∂F ∂n xj

• (∇n) := ∂F ∂n k, xj ∇n k (see also [1, formulas (3.90), (3.99), (3.100), (5.2)], [11, page 90], or [START_REF] Lin | Existence of solutions for the Ericksen-Leslie system[END_REF]). Here, u is the Eulerian, or spatial velocity vector field, n = (n 1 , n 2 , n 3 ) is the director field, the constant μ > 0 is the viscosity coefficient, the constant J > 0 is the moment of inertia of the molecule, F(x, t) and G(x, t) are given external forces, and ˙:= ∂ ∂t + u • ∇ is the material derivative. The terms f and g correspond to the dissipative part of the stress tensor and the dissipative part of the intrinsic body force, respectively, and they depend on u, n, and their derivatives. The function F(n, ∇n) is the Oseen-Zöcher-Frank free energy and is defined by

F(n, ∇n) := K 1 n • curl n + 1 2 K 11 (div n) 2 + K 22 (n • curl n) 2 + K 33 n × curl n 2 . ( 2
)
The molecular field h is defined by

h := ∂F ∂n - ∂ ∂x j ∂F ∂n xj . ( 3
)
The pressure p and the Lagrange multiplier 2q are determined, respectively, by the conditions div u = 0 and n = 1. We are interested in the non-dissipative case, i.e., g = 0, f = 0. Since the liquid crystal is nematic, we necessarily have K 1 = 0. For simplicity, we study the one-constant approximation, i.e., we assume

K 11 = K 22 = K 33 =: K > 0.
Thus, using n = 1, the Oseen-Zöcher-Frank free energy [START_REF] Chechkin | Nematic liquid crystals. Existence and uniqueness of periodic solutions to Ericksen-Leslie equations[END_REF] becomes

F = K 11 2 (div n) 2 + K 22 -K 33 2 (n • curl n) 2 + K 33 2 n 2 curl n 2 = K 2 (div n) 2 + K 2 (n j,xi n j,xi -n i,xj n j,xi ), (4)
where the subscript x j means partial derivative relative to the spatial coordinate x j and summation on repeated indices is understood, irrespective of their position. Hence,

∂F ∂n = 0, ∂F ∂n i,xj = K(div n)δ ij + Kn i,xj -Kn j,xi . (5)
Lemma 1.1. The non-dissipative (i.e., f = 0, g = 0) nematic (i.e., K 1 = 0) Ericksen-Leslie equations in the one constant approximation (i.e., K := K 11 = K 22 = K 33 > 0) are

u -μΔu = -∇p -K n xj • (∇n) xj + F, div u = 0, J n + J ṅ 2 -h • n + G • n n + h = G, n = 1, (6)
where the molecular field has the expression

h = ∂F ∂n - ∂ ∂x j ∂F ∂n xj = -KΔn. ( 7
)
and

p := p + K 2 (div n) 2 -n i,xj n j,xi . Proof. Formula (5) implies 1 K ∂ ∂x j ∂F ∂n xj • (∇n) = (div n)δ ij + n i,xj -n j,xi ∇n i xj = n k,x k xj ∇n j + n k,x k ∇n j,xj + n i,xj ∇n i xj -n j,xj xi ∇n i -n j,xi ∇n i,xj = n i,xj ∇n i xj + 1 2 ∇ (div n) 2 -n i,xj n j,xi
and hence the first equation in (1) becomes the first equation in [START_REF] Ericksen | Hydrostatic theory of liquid crystals[END_REF]. In addition, (3) and ( 5) yield

h i = -K(n k,x k xj δ ij + n i,xj xj -n j,xixj ) = -KΔn i .
which is [START_REF] Leslie | Some constitutive equations for anisotropic fluids[END_REF]. Next, taking the dot product of the second equation in [START_REF] De Gennes | The physics of liquid crystals[END_REF] with n yields

2q = J n • n + h • n -G • n = -J ṅ 2 + h • n -G • n (8)
since n • ṅ = 0. Therefore, the second equation in (1) becomes

0 = J n -2qn + h -G = J n + J ṅ 2 -h • n + G • n n + h -G
which is the second equation in [START_REF] Ericksen | Hydrostatic theory of liquid crystals[END_REF].

We rewrite the Ericksen-Leslie equations (6) as a system of three equations of first order by introducing a new vector field ν. This vector field appeared for the first time in [START_REF] Gay-Balmaz | The geometric structure of complex fluids[END_REF], motivated by the geometric and variational structure of the conservative version of the Ericksen-Leslie equations (i.e., μ = 0,

f = g = 0, F = G = 0). Theorem 1.1. If (u, n) is a solution of Ericksen-Leslie equations (6), n = 1, define the vector field ν := n × ṅ. Then, (u, ν, n) is a solution of the system u -μΔu = -∇p -(Kn k,xj ∇n k ) xj + F, div u = 0, ( 9
) J ν = -K(Δn) × n + n × G, ( 10
) ṅ = ν × n, (11)
with initial conditions n 0 and ν 0 satisfying n 0 = 1 and n 0 • ν 0 = 0. Conversely, if (u, ν, n) is a solution of the system (9)-( 11) with initial conditions n 0 and ν 0 satisfying n 0 = 1 and n 0 • ν 0 = 0, then (u, n) is a solution of the Ericksen-Leslie equations [START_REF] Ericksen | Hydrostatic theory of liquid crystals[END_REF].

Proof. Suppose that (u, n) is a solution of Ericksen-Leslie equations ( 6) and define the vector field ν := n × ṅ. Then, ν × n = (n × ṅ) × n = ṅ since n = 1 and n • ṅ = 0, which gives [START_REF] Chandrasekhar | Liquid crystals[END_REF]. The first equation in [START_REF] Ericksen | Hydrostatic theory of liquid crystals[END_REF] coincides with [START_REF] Ericksen | Continuum theory of nematic liquid crystals[END_REF]. Finally,

J ν = J(n × ṅ) ˙= n × J n (6) = -n × h + n × G (7) = -K(Δn) × n + n × G
which is [START_REF] Leslie | Continuum theory for nematic liquid crystals[END_REF]. Since (u, n) is a solution of (6), we have n = 1. In addition, n • ν = n • (n × ṅ) = 0. Therefore, the initial conditions of the system ( 9)-( 11) necessarily satisfy n 0 = 1 and n 0 • ν 0 = 0, as stated.

Conversely, suppose that (u, ν, n) is a solution of the system ( 9)- [START_REF] Chandrasekhar | Liquid crystals[END_REF] with initial conditions satisfying

n 0 = 1 and n 0 • ν 0 = 0. Then ( n 2 ) ˙= 2n • ṅ (11) = 2n • (ν × n) = 0 and (n • ν) ˙= ṅ • ν + n • ν = (ν × n) • n + n • 1 J (-KΔn × n + n × G) = 0
by [START_REF] Leslie | Continuum theory for nematic liquid crystals[END_REF] and [START_REF] Chandrasekhar | Liquid crystals[END_REF], which implies n = 1 and n • ν = 0.

We have

n × ṅ (11) = n × (ν × n) = ν since n = 1 and n • ν = 0. Conversely, if ν = n × ṅ, then ν × n = (n × ṅ) × n = ṅ, since n = 1.
The first equation in [START_REF] Ericksen | Hydrostatic theory of liquid crystals[END_REF] coincides with [START_REF] Ericksen | Continuum theory of nematic liquid crystals[END_REF]. Finally, by ( 7), [START_REF] Chandrasekhar | Liquid crystals[END_REF], n = 1, and ν • n = 0, we get

J n + J ṅ 2 -h • n + G • n n + h = J ν × n + Jν × ṅ + J ṅ 2 + KΔn • n + G • n n -KΔn (10) = -K((Δn) × n) × n + (n × G) × n + Jν × (ν × n) + J ν 2 + KΔn • n + G • n n -KΔn = KΔn -K(n • Δn)n + G -(G • n)n -J ν 2 n + J ν 2 n + K(Δn • n)n + (G • n)n -KΔn = G
which is the second equation in [START_REF] Ericksen | Hydrostatic theory of liquid crystals[END_REF].

In terms of ν, by ( 8) and [START_REF] Chandrasekhar | Liquid crystals[END_REF], we have 2q

= h • n -J ν 2 -G • n.
Because of Theorem 1.1, we work in the rest of the paper exclusively with the Ericksen-Leslie equations expressed as ( 9)- [START_REF] Chandrasekhar | Liquid crystals[END_REF] with initial conditions (u 0 , ν 0 , n 0 ) satisfying n 0 = 1 and n 0 • ν 0 = 0.

Since the structure of the full Ericksen-Leslie system is sufficiently complicated, simplified models have been introduced in order to get some reasonable results. Most of the known results are obtained under the assumption J = 0. For the incompressible model in [START_REF] Lin | Nonlinear theory of defects in nematic liquid crystal: phase transition and flow phenomena[END_REF], Lin introduced a simplification of the general Ericksen-Leslie system that keeps many of the mathematical difficulties of the original system by using a Ginzburg-Landau approximation to relax the nonlinear constraint. Namely, instead of the restriction n = 1, the penalty term 1 2 ( n 2 -1) 2 was added to the free energy functional. In [START_REF] Lin | Nonparabolic dissipative system modeling the ow of liquid crystals[END_REF], Lin and Liu showed the global existence of weak solutions and smooth solutions for that approximation. In [START_REF] Shkoller | Well-posedness and global attractors for liquid crystals on Riemannian manifolds[END_REF], a very simple proof of local well-posedness for this coupled system was provided using a contraction mapping argument. It was proved that this system is globally well-posed and has compact global attractors in 2D. Recently, Hong [START_REF] Hong | Global existence of solutions of the simplified Ericksen-Leslie system in dimension two[END_REF] and Lin-Liu-Wang [START_REF] Lin | Liquid crystal flows in two dimensions[END_REF] showed, independently, the global existence of weak solutions of an incompressible model in two dimensional space. Moreover, in [START_REF] Lin | Liquid crystal flows in two dimensions[END_REF], the regularity of solutions, except for a countable set of singularities whose projection on the time axis is a finite set, has been obtained (see also [START_REF] Hong | Global existence of solutions of the liquid crystal ow for the Oseen-Frank model in R 2[END_REF]). In [START_REF] Wang | Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data[END_REF], Wang established a global well-posedness theory for the incompressible liquid crystals for rough initial data. A simplified Ericksen-Leslie system for two-dimensional compressible flow was considered in [START_REF] Jiang | Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions[END_REF]. Hieber, Nesensohn, Prüss, and Schade analyzed in [START_REF] Hieber | Dynamics of nematic liquid crystal flows: the quasilinear approach[END_REF] the simplified system as a parabolic evolutional equation in an L p -L q -setting and studied the system near an equilibrium.

In the present paper we focus on the system (9)- [START_REF] Chandrasekhar | Liquid crystals[END_REF] in the hyperbolic case J > 0 and prove an existence and uniqueness theorem for two-dimensional periodic media as well as for the problem in a bounded domain of the plane. We also prove the finite propagation speed of waves in such media. Some results in this paper were announced in [START_REF] Ratiu | Existence and uniqueness theorems in two-dimensional nematodynamics. Finite speed of propagation[END_REF].

Two-Dimensional Solution in a Periodic Domain

Consider a liquid crystal flow in R 3 . The flow is called two-dimensional if all unknowns in the Ericksen-Leslie system are independent of the third coordinate x 3 ; so we can suppose that they are all defined on a plane (x 1 , x 2 ). Let Q T := (0, T ) × T, where T := R 2 /Z 2 is the two-dimensional flat torus. We study the system (9)-( 11) in Q T with initial conditions

u(0, x) = u 0 , ν(0, x) = ν 0 , n(0, x) = n 0 , ( 12
)
satisfying n 0 = 1 and n 0 • ν 0 = 0. Here u, ν, n are unknown vector fields, p is an unknown scalar function, and J, K, μ are fixed strictly positive numbers. The material derivative

f t + 3 i=1 u i f xi turns into f t + 2 i=1 u i f xi .
It is natural to consider a flat motion, i.e., u : T → R 2 × {u 3 = 0}. Note that the vector fields n, ν are always three-dimensional, even if they are defined on a flat two-dimensional domain; in particular the director field is not necessarily tangential to the plane {x 3 = 0}.

Notations and Definitions

Throughout the paper we use the following notations: 3 ), or a vector field with values in R 3 ;

• ḟ := ∂f ∂t + u • ∇f = f t + u j f xj is the material time derivative of f ; • a bold letter b denotes a 3-dimensional vector b = (b 1 , b 2 , b
• standard summation convention is used on repeated indices, independent on their position, e.g., 9)-( 11) hold almost everywhere.

a i b i := i a i b i ; • L 2 (T) := v : T → R 3 | T v 2 d 2 x < ∞ ; • W m 2 (T) is the Sobolev space of functions on T having m distributional derivatives in L 2 (T); • Sol(T) := {v : T → R 3 | v ∈ C ∞ (T), div v = 0}; • Sol(Q T ) := {v ∈ C ∞ (Q T ) | v(t, •) ∈ Sol(T), ∀t ∈ (0, T )}; • Sol 2 (T) is the closure of Sol(T) in the norm L 2 (T); • Sol m 2 (T) is the closure of Sol(T) in the norm W m 2 (T). Definition 2.1. A quadruple (u, ν, n, ∇p ) is a strong solution of problem (9)-(12) in the domain Q T if (i) u is a time-dependent vector field in L 2 ((0, T ); Sol 3 2 (T)), u t ∈ L 2 (Q T ); (ii) ν is a vector field in L ∞ ((0, T ); W 2 2 (T)), ν t ∈ L ∞ ((0, T ); L 2 (T)); (iii) n is a vector field in L ∞ ((0, T ); W 3 2 (T)), n t ∈ L ∞ ((0, T ); W 1 2 (T)); (iv) ∇p ∈ L 2 (Q T ); (v) u, n, ν satisfy the initial conditions (12), i.e., (u, n, ν) → (u 0 , n 0 , ν 0 ) in L 2 (T) as t → 0; (vi) Equations (
The first goal of the paper is to prove existence of strong solutions to the problem ( 9)- [START_REF] Lin | Existence of solutions for the Ericksen-Leslie system[END_REF].

Theorem 2.1. Let F = 0, G = 0. Suppose u 0 ∈ Sol 2 2 (T), Δν 0 ∈ L 2 (T), Δn 0 ∈ W 1 2 (T).
Then there is a T > 0 such that the solution to problem (9)-( 12) (as given in Definition 2.1) does exist.

The proof of this theorem is given in the next subsections.

Galerkin-Type Approximations

We begin the proof with a classical approximation method.

Select two sequences of subspaces

E 1 ⊂ E 2 ⊂ • • • and F 1 ⊂ F 2 ⊂ • • • such that ∪ k∈N E k is dense in Sol 2 2 (T) and ∪ k∈N F k is dense in W 2 2 (T) (and, consequently, in W 1 2 (T)). Since the inclusions W 1 2 (T) → L 2 (T)
and Sol 1 2 (T) → Sol 2 (T) are both compact, the operator Δ : Sol(T) → Sol(T) is a symmetric operator that extends to a self-adjoint operator on Sol 2 (T) and L 2 (T). Fix bases of eigenfunctions of Δ in Sol 2 (T) and L 2 (T) and denote by E k and F k the linear span of the first k eigenfunctions in Sol 2 (T) and L 2 (T), respectively. The subspaces E k and F k consist of smooth functions and thus ∪ k∈N E k and ∪ k∈N F k are dense in Sol 2 2 (T) and W 2 2 (T), respectively.

Proposition 2.1. Let (u 0k , ν 0,k , n 0,k ) ∈ E k × F k × F k be an approximation of the initial data (u 0 , ν 0 , n 0 ) for fixed k.
Then, for some T > 0, there exists a solution

(u k , ν k , n k ) ⊂ C 0 ((0, T ); E k × F k × F k ) of the problem (u k,t , ω) = -(u l k u k,x l , ω) + μ(Δu k , ω) -K(Δn k • ∇n k , ω), ( 13
)
J(ν k,t , ζ) + (Ju l k ν k,x l , ζ) + K(Δn k × n k , ζ) = 0, ( 14
) (n k,t , ψ) + (u l k n k,x l , ψ) -(ν k × n k , ψ) = 0, ( 15
) (u k , ν k , n k )| t=0 = (u 0k , ν 0k , n 0k ), ( 16
)
where the identities above hold for all

ω ∈ E k , ζ, ψ ∈ F k , and (u, v) = T u • v dx 1 dx 2 is the L 2 (T)-inner product.
Indeed, the system ( 13)-( 16) could be regarded as a Cauchy problem for the ordinary differential equation X t = f (X) in 3k-dimensional space with continuous right-hand side. Due to the Cauchy-Peano theorem, there exists some small T 0 > 0 such that this problem has a solution for |t| < T 0 =: T . Remark 2.1. If u 3 0k = 0, then for any t < T 0 , we have

u 3 k (t) = 0.
The next step is to get a uniform estimate on (u k , ν k , n k ) in some appropriate norm.

Energy Conservation

We need the following identity.

Lemma 2.1. For all t ∈ (0, T ) we have

T u k (t) 2 + J ν k (t) 2 + K ∇n k (t) 2 dx 1 dx 2 + 2μ t 0 T ∇u k 2 dx 1 dx 2 dt = T u 0k 2 + ν 0k 2 + ∇n 0k 2 dx 1 dx 2 .
Proof. In this proof we use (u, ν, n) instead of (u k , ν k , n k ), in order to simplify notation. In Eqs. ( 13)- [START_REF] Lin | Nonparabolic dissipative system modeling the ow of liquid crystals[END_REF] we substitute (ω, ζ, ψ) = (u, ν, -KΔn). Since Δ : F k → F k , this substitution is allowed. Taking the integral over the interval (0, t) in (13) and using periodicity of all functions, we obtain

T 1 2 u 2 t 0 dx 1 dx 2 = t 0 T -μ ∇u 2 + Kn s xj n s xi u i xj dx 1 dx 2 dt. ( 17
)
The sum of the integrals over the interval (0, t) of the relations ( 14) and ( 15) reads

0 = t 0 T J(ν • ν t + u l ν i x l ν i ) + K (Δn × n) • ν + (ν × n) • Δn -n i t n i xsxs -u j n i xj n i xsxs dx 1 dx 2 dt = T 1 2 J ν 2 + Kn i xs n i xs dx 1 dx 2 t 0 + t 0 T Ku j xs n i xs n i xj dx 1 dx 2 dt. ( 18
)
Taking the sum of ( 17) and ( 18) we obtain

1 2 T u 2 + J ν 2 + K ∇n 2 dx 1 dx 2 t 0 + μ t 0 T ∇u 2 dx 1 dx 2 dt = 0,
which proves the stated identity.

Corollary 1. Problem (13)-( 16) has a solution for every T > 0.

Proof. Reconsider our problem as an ordinary differential equation X t = f (X) with continuous right hand side. The value T 0 obtained from the Cauchy-Peano theorem depends only on the function f and on the norm of the initial data. Since f doesn't depend on t and for any t ∈ (0, T 0 ) the solution

X(t) = (u k (t), ν k (t), n k (t)) is bounded with X(0) = (u k (0), ν k (0), n k (0)
), the Cauchy-Peano theorem guarantees the existence of the solution on the interval (t, t + T 0 ) for any t < T 0 , and, consequently, on the interval (0, 2T 0 ). Repeating the procedure, this proves the existence of the solution for any t ∈ (0, NT 0 ), where N is an arbitrary natural number.

Estimates on Higher Derivatives

Unfortunately, the results of Lemma 2.1 are not sufficient to prove the convergence of (u k , ν k , n k ) to the solution of ( 9)- [START_REF] Lin | Existence of solutions for the Ericksen-Leslie system[END_REF]. We need more precise estimates.

Theorem 2.2.

There exists T > 0 and C > 0, depending only on the initial data, and constants J, K, μ > 0, such that

u k L2((0,T );W 3 2 ) , u k L∞((0,T );W 2 2 ) , ν k L∞((0,T );W 2 2 ) , n k L∞((0,T );W 3 2 ) ≤ C, u k,t L2(QT ) , ν k,t L2(QT ) , ∇n k,t L2(QT ) ≤ C.
Proof. We begin by proving the first set of inequalities. In equation [START_REF] Lin | Nonlinear theory of defects in nematic liquid crystal: phase transition and flow phenomena[END_REF], set ζ = -Δν and integrate over the domain (0, T ) using Green's identities. We have

J 2 T ∇ν 2 dx 1 dx 2 T 0 = QT - J 2 u j x k ν i xj ν i x k -K(Δn × n x k ) • ν x k -K(Δn x k × n) • ν x k dx 1 dx 2 dt. ( 19
)
The integral of ( 15) with ψ = KΔ 2 n can be written as

K 2 T n x k xj • n x k xj dx 1 dx 2 T 0 = K QT u j n i x k xj Δn i x k + u j x k n i xj Δn i x k -(ν × n x k + ν x k × n) • Δn x k dx 1 dx 2 dt = K QT -2u j x k n xj x k • Δn -Δu j n xj • Δn + (ν x k × n x k ) • Δn + (n × ν x k ) • Δn x k dx 1 dx 2 dt. ( 20
)
To estimate the second derivatives of u, we set ω = -Δu in [START_REF] Gay-Balmaz | The geometric structure of complex fluids[END_REF] and integrate over (0, T ):

1 2 T ∇u 2 dx 1 dx 2 T 0 + μ QT Δu 2 dx 1 dx 2 dt (13) = QT -u j x k u xj • u x k + Kn xi • ΔnΔu i dx 1 dx 2 dt. ( 21
)
The sum of ( 19)-( 21) is

1 2 T J ∇ν 2 + K 2 j,k=1 n xj x k 2 + ∇u 2 dx 1 dx 2 T 0 + μ QT Δu 2 dx 1 dx 2 dt = QT -Ju j x k ν xj • ν x k -2Ku j x k n xj x k • Δn + K(ν x k × n x k ) • Δn -u j x k u xj • u x k dx 1 dx 2 dt.
We estimate the integral on the right-hand side:

1 2 T J ∇ν 2 + K 2 j,k=1 n xj x k 2 + ∇u 2 dx 1 dx 2 T 0 + μ QT Δu 2 dx 1 dx 2 dt ≤ C 1 T 0 esssup ∇u(t) ∇ν(t) 2 L2(T) + 2 i,j=1 n xixj (t) 2 L2(T) + ∇u 2 L2(T) + esssup ∇n(t) ∇ν(t) L2(T) Δn(t) L2(T) dt. ( 22
)
Since ij u xixj 2 L2(T) = Δu 2 L2(T) (see [24, III, §8]), we have estimated u in the L 2 ((0, T ); W 2 2 (T))- norm.
We need to estimate esssup ∇u(t) and esssup ∇n(t) in terms of higher derivatives. Unfortunately, the W 2 2 -norm is not enough, so we repeat the previous procedure for (ω, ζ, ψ) = (Δ 2 u, Δ 2 ν, -KΔ 3 n). We use below the identity ∇(Δu) 2 := Δu x k • Δu x k . From ( 13)-( 15), we have

1 2 T J Δν 2 + Δu 2 + K ∇(Δn) 2 dx 1 dx 2 t 0 + Qt μ ∇(Δu) 2 dx 1 dx 2 dt ≤ - Qt u j x k (2Jν xj x k • Δν + 2u xj x k • Δu + KΔn xj • Δn x k + 2Kn xixj x k • Δn xi ) + Δu j (Jν xj • Δν + u xj • Δu + Kn xj x k • Δn x k ) + Ku j x k xi n xj x k • Δn xi + KΔu j x k n xj • Δn x k -K(Δn x k • n xj + n xj x k • Δn)Δu j x k + K(Δ 2 n × n + 2(Δn x k × n x k )) • Δν + K(Δν × n) • Δ 2 n -K(2ν xj x k × n xj + 2ν xj × n xj x k + ν x k × Δn + ν × Δn x k ) • Δn x k dx 1 dx 2 dt ≤ C T 0 esssup ∇u(t) ν(t) 2 W 2 2 (T) + u(t) 2 W 2 2 (T) + n(t) 2 W 3 2 (T) + u W 2 4 (T) + ν W 1 4 (T) ν(t) W 1 4 (T) + u(t) W 1 4 (T) + n(t) W 2 4 (T) ν(t) W 2 2 (T) + u(t) W 2 2 (T) + n(t) W 3 2 (T) + u W 3 2 (T) n(t) 2 W 2 4 (T) + esssup ∇n(t) ||ν(t)|| W 2 2 (T) n(t) W 3 2 (T) + esssup ν(t) n(t) 2 W 3 2 (T) dt. ( 23
)
Since for any periodic f ∈ W 1 2 ([0, 1] 2 ), we have

f Lp ≤ C( ∇f L2 + f L2 ) for any p < ∞, (24)
and any periodic g

∈ W 1 p ([0, 1] 2 ), max g ≤ C( ∇g Lp + g L2 ) for p > 2, (25)
we conclude from Lemma 2.1, ( 22), [START_REF] Jiang | Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions[END_REF], and (23) that

I (t) ≤ I (0) + C 4 t 0 I (t) 2 dt, as t < T 1 , (26)
where

I (t) := u(t) 2 W 2 2 (T) + J ν(t) 2 W 2 2 (T) + K n(t) 2 W 3
2 (T) + 1, and the time T 1 depends only on initial data, and the positive constants J, K, μ. Indeed, due to the Cauchy inequality and ( 24), ( 25)

T 0 esssup ∇u (I (t) -1)dt ≤ ε T 0 ∇(Δu(t)) 2 L2(T) dt + C ε T 0 I 2 (t)dt
for any ε > 0 and some C ε > 0. The same inequality holds for the terms containing u W 2 4 (T) or u W 3 2 (T) . The rest of the terms can be estimated with C

t 0 I 3 2 dt ≤ C t 0 I 2 dt.
If ε is sufficiently small, we get [START_REF] Mikhailov | Partial differential equations[END_REF].

Next, we need the following simple lemma, a kind of Gronwall-Bellman inequality.

Lemma 2.2. Let Y be a measurable function on R. Suppose that for almost all t we have

0 ≤ Y (t) ≤ Y (0) + k t 0 Y (s) 2 ds.
Then

Y (t) ≤ Y (0) 1 -ktY (0)
.

Proof. Define the function X(t) by the integral equation

X(t) = Y (0) + ε + k t 0 X(s) 2 ds ⇐⇒ X = kX 2 , X(0) = Y (0) + ε, i.e., X(t) = Y (0) + ε 1 -kt(Y (0) + ε) .
The difference W = Y -X satisfies the inequality

W (t) ≤ -ε + k t 0 (X(s) + Y (s))W (s)ds.
Since Y and X are measurable, the function

f (t) = -ε + t 0 (X(s) + Y (s))W (s)ds
is continuous. Suppose t 0 is the least zero of f . Since X + Y > 0, the condition f (t 0 ) = 0 means that W is non-negative on a positive measure subset of (0, t 0 ). But this is in contradiction with f (t) < 0 as t < t 0 ; consequently f (t) < 0 for all t > 0.

Letting ε → 0, yields the statement in the lemma.

We continue the proof of the theorem. From ( 26), we conclude that for any t < min{T 1 , (C 4 I (0)) -1 /2} =: T 2 we have

I (t) ≤ C 5 , (27)
where C 5 depends only on K, μ, J, I (0). This proves the first set of three inequalities in the statement of the theorem. Using Lemma 2.2 we can now estimate the time-derivative u t . Set ω = u t in (13) and rewrite the resulting identity as

QT 2 u t 2 dx 1 dx 2 dt + 1 2 μ T ∇u 2 dx 1 dx 2 t 0 = QT 2 -u j u xj • u t + Δn • n xj u j t dx
u t 2 + μ Qt ∇u 2 dx 1 dx 2 t 0 ≤ u j u xj 2 L2(Qt) + Δn • ∇n 2 L2(Qt) ≤ C 6 .
The same type of inequalities can be obtained in a similar fashion for ν t and ∇n t . This proves the second set of inequalities in the statement of the theorem.

Convergence of the Approximations

In this subsection we finish the proof of Theorem 2.1.

Theorem 2.2 provides the existence of measurable functions u, ν, n and a subsequence of (u k , ν k , n k ) such that

u k u weakly in L 2 ((0, T ); Sol 2 2 (T)), u k,t u t weakly in Sol 2 (Q T ), ν k * ν *-weakly in L ∞ ((0, T ); W 2 2 (T)), n k * n *-weakly in L ∞ ((0, T ); W 3 2 (T)), u k,t u t , ν k,t ν t , ∇n k,t ∇n t weakly in L 2 (Q T ).
Moreover, due to standard embedding theorems,

u k → u strongly in Sol 2 (Q T ), ν k → ν strongly in L 2 (Q T ),
and

∇n k → ∇n strongly in L 2 (Q T ). Fix ω ∈ ∪ k C 1 (0, T ; E k ), ζ(t), ψ(t) ∈ ∪ k C 1 (0, T ; F k
) and integrate ( 13)-( 15) over (0, T ). Passing to the limits as k → ∞ we have 9)- [START_REF] Chandrasekhar | Liquid crystals[END_REF].

⎧ ⎪ ⎨ ⎪ ⎩ QT ( u -μΔu -KΔn i ∇n i ) • ωdx 1 dx 2 dt = 0, QT J K ν • ζ + (Δn × n) • ζ dx 1 dx 2 dt = 0, QT ( ṅ • ψ -(ν × n) • ψ) dx 1 dx 2 dt = 0. (28) Since ∪ k C 1 ((0, T ); E k ) is dense in L 2 ((0, T ); Sol 2 (T)) and ∪ k C 1 ((0, T ); F k ) is dense in L 2 (Q T ), these equations imply (
If ∇p is the Hodge projection of u -μΔu -Δn∇n ∈ L 2 (Q T ) on the orthogonal complement of L 2 ((0, T ); Sol 2 (T)), then ∇p ∈ L 2 (Q T ).

Finally, we check the initial conditions [START_REF] Lin | Existence of solutions for the Ericksen-Leslie system[END_REF].

Fix φ ∈ Sol 2 (T) and consider the family of functions

f k (t) = (u k (t), φ) L2(T) . Since f k (t) = (u k,t (t), φ) L2(T) (u t , φ) L2(T) weakly in L 2 (0, T ), it follows that f k tends to (u, φ) L2(T) in C(0, T ). Thus, u(t, •) tends to lim k u 0k = u 0 weakly in L 2 (T). Also T (u(t, x) -u(0, x)) 2 dx 1 dx 2 = T t 0 u t dt 2 dx 1 dx 2 ≤ t T t 0 u 2 t dtdx 1 dx 2 .
Since u t ∈ L 2 (Q t ), the function u(t, x) is continuous in L 2 (T)-norm with respect to t. Consequently, u 0 is both a weak and a strong limit.

The weak limits of the other variables to their respective initial conditions are checked in the same way. This proves Theorem 2.1.

Uniqueness

Theorem 2.3. Suppose that (u 1 , ν 1 , n 1 , p 1 ) and (u 2 , ν 2 , n 2 , p 2 ) are solutions of the problem (9)-( 12) in the domain Q T . Then, for some 0

< T 0 ≤ T (u 2 , ν 2 , n 2 , ∇p 2 ) = (u 1 , ν 1 , n 1 , ∇p 1 )
almost everywhere in Q T0 .

Proof. First of all, every solution of the problem satisfies identities (28) for all 28). With this substitution, for any τ < T , the identities (28) give

ω ∈ W 1 2 (Q T ), ζ ∈ W 1 2 (Q t ), ψ ∈ L 2 (0, T ; W 2 2 ). Denote w = u 1 -u 2 , f = ν 1 -ν 2 , g = n 1 -n 2 and set (ω, ζ, ψ) = (w, f, -KΔg) in (
Qτ w t • w + u i 2 w xi • w + w i u 1,xi • w + μ ∇w 2 + KΔn 1 • g xj w i + KΔg • n 2,xj w j dx 1 dx 2 dt = 0, ( 29
) Qτ J(f t • f + u i 2 f xi • f + w i ν 1,xi • f) + K(Δg × n 1 + Δn 2 × g) • f) dx 1 dx 2 dt = 0, ( 30
) -K Qτ g t • Δg + u 1 i g xi • Δg + w i n 2,xi • Δg -(ν 1 × g + f × n 2 ) • Δg) dx 1 dx 2 dt = 0. (31)
Next, we rewrite these identities as

T 1 2 w(τ ) 2 d 2 x + Qτ μ ∇w 2 d 2 xdt = Qτ -w i u 1,xi • w -K(Δn 1 • g xj w i + Δg • n 2,xj w j dx 1 dx 2 dt, ( 32
) J 2 T f(τ ) 2 dx 1 dx 2 = - Qτ Jw i ν 1,xi • f + K(Δg × n 2 + Δn 1 × g) • f dx 1 dx 2 dt, ( 33
) K 2 T ∇g(τ ) 2 dx 1 dx 2 = K Qτ w i n 2,xi • Δg -u i 1,x k g xi • g x k + (ν 1,x k × g) • g x k -(f × n 2 ) • Δg) dx 1 dx 2 dt. ( 34
)
Adding the identities (32)-(34), we get 1 2 T ( w(τ

) 2 + J f(τ ) 2 + K ∇g(τ ) 2 ) dx 1 dx 2 + μ Qτ ∇w(t) 2 dx 1 dx 2 dt = - Qτ w j u i 1,xj w i + KΔn 1 • g xj w i + Jw i ν 1,xi • f + K(Δn 1 × g) • f + Ku i 1,x k g xi • g x k -K(ν 1,x k × g) • g x k dx 1 dx 2 dt.
Due to the embedding theorems and the Hölder inequalities we get 1 2 T ( w(τ

) 2 + J f(τ ) 2 + K ∇g(τ ) 2 ) dx 1 dx 2 + μ Qτ ∇w(t) 2 dx 1 dx 2 dt ≤ C τ 0 f L2(T) ∇ν 1 Lq(T) + ∇g L2(T) Δn 1 Lq(T) w Lp(T) + g L2(T) ∇ν 1 Lq(T) + ∇f L2(T) Δn 1 Lq(T) g Lp(T) + ∇u 1 L∞(T) w 2 L2(T) + ∇g 2 L2(T) dt ≤ C(δ)τ 1 2 esssup t Δn 1 (t) Lq(T) + ∇ν 1 (t) Lq(T) + ∇u 1 (t) L2((0,T );L∞(T)) esssup t w(t) 2 L2(T) + f(t) 2 L2(T) + ∇g(t) 2 L2(T) + ∇w(t) 2 L2(Qτ )
for some q > 2,

p -1 = 1/2 -q -1 .
Taking the τ -esssup of the left hand side and comparing it to the second factor on the right hand side, shows that for τ sufficiently small we have

(w, f, ∇g) = (0, 0, 0). Consequently, (u 1 , ν 1 ) = (u 2 , ν 2 ). However, ġ = -w i n 2,xi + ν 1 × g + f × n 2 . Since f = 0 and w = 0, we have d dt g(t) = ν 1 (t) × g(t)
with initial condition g(0) = 0. This implies that g = 0, i.e., n 1 = n 2 . Since ∇p i is the projection of u -μΔu -Δn∇n, we have ∇p 1 = ∇p 2 .

Theorem 2.3 is proved.

Liquid Crystal in the Presence of External Forces

Theorem 2.1 can be easily extended if F, G = 0.

Theorem 2.4. Suppose u 0 ∈ Sol 2 2 (T), ν 0 ∈ W 2 2 (T), n 0 ∈ W 3 2 (T) and F ∈ L 2 ((0, T ); W 1 2 (T)), G ∈ L 1 ((0, T ); W 2 2 (T)); F 3 = 0.
Then there exists some 0 < T 0 < T such that the solution (as in Definition 2.1) of problem (9)-( 11) exists and is unique in Q T0 .

Liquid Crystal Flow in Bounded Domains

Let Ω be a bounded domain in R 2 and consider nematic liquid crystal flow in the cylinder Ω × R which does not depend on the third coordinate.

Since all functions in the Ericksen-Leslie system depend only on the points (x 1 , x 2 ) ∈ Ω, we are studying Eqs. ( 9)- [START_REF] Chandrasekhar | Liquid crystals[END_REF] in domain (0, T ) × Ω with initial conditions [START_REF] Lin | Existence of solutions for the Ericksen-Leslie system[END_REF] and additional boundary conditions

u ∂Ω = 0, n -n 1 ∂Ω = 0, ν| ∂Ω = 0 for any t > 0, (35)
where n 1 is a given vector field on Ω × R.

Condition u ∂Ω = 0 means that the domain has impenetrable boundary and that the fluid moves without slipping; nn 1 ∂Ω = 0 describes the director position at the boundary. The third condition comes from the original Ericksen-Leslie system and means that ṅ = 0 at the boundary.

The Definition of the Solution and the Existence Theorem

We begin by introducing some notations.

In this section we let Q T := (0, T ) × Ω,

• Sol (Ω) := {v : Ω → R 3 such as v ∈ C ∞ 0 Ω), div v = 0}; • Sol (Q T ) := {v ∈ C ∞ (Q T ) : ∀t v(t, •) ∈ • Sol (Ω)}; • Sol m 2 (Ω) is the closure of • Sol (Ω) in the norm W m 2 (Ω) • W m 2 (Ω)
is the subspace of W m 2 (Ω) with zero trace (see, for instance [START_REF] Sobolev | Some applications of functional analysis in mathematical physics, 3rd edn[END_REF][START_REF] Mikhailov | Partial differential equations[END_REF]). The definition of a solution of the Ericksen-Leslie equations is quite similar to the one in Definition 2.1, with some changes because of the boundary. Definition 3.1. The quadruple (u, ν, n, ∇p ) is a strong solution of problem ( 9)-( 12), (35

) in domain Q T if • u is a vector field in L 2 ((0, T ); • Sol 1 2 (Ω)) ∩ L 2 ((0, T ); W 2 2 (Ω)), u t ∈ L 2 (Q T ); • ν is a vector field in L ∞ ((0, T ); • W 1 2 (Ω)), ν t ∈ L ∞ ((0, T ); L 2 (Ω)); • n -n 1 is a vector field in L ∞ ((0, T ); • W 1 2 (Ω)) ∩ L 2 ((0, T ); W 2 2 (Ω))
, where n 1 is a given constant vector field in (x, t),

n t ∈ L ∞ ((0, T ); W 1 2 (Ω)); • ∇p ∈ L 2 (Q T );
• u, n, ν satisfy initial conditions [START_REF] Lin | Existence of solutions for the Ericksen-Leslie system[END_REF], i.e., (u, n, ν) → (u 0 , n 0 , ν 0 ) in L 2 (Ω) as t → 0; • Equations ( 9)- [START_REF] Chandrasekhar | Liquid crystals[END_REF] hold almost everywhere.

In this section we suppose the third component of the director to be equal to zero. Then we have n = (cos θ, sin θ, 0), ν = (0, 0, ν), where θ is a new unknown function. The Ericksen-Leslie system becomes

u -μΔu = -∇ p + K 2 ∇θ 2 -KΔθ∇θ, div u = 0, ( 36
) J ν = -KΔθ, (37) θ = ν (38)
with boundary and initial conditions

u ∂Ω = 0, θ -θ 1 ∂Ω = 0, ν| ∂Ω = 0 for any t > 0, (39)
u(0, x) = u 0 (x), ν(0, x) = ν 0 (x), θ(0, x) = θ 0 (x). ( 40
) Theorem 3.1. Assume that Ω is a domain with C 2 -boundary. Let θ 0 ∈ W 3 2 (Ω), ν 0 ∈ W 2 2 (Ω), u 0 ∈ • Sol 1 2 (Ω) ∩ W 2 2
(Ω); Δu 0 ∂Ω = 0 and assume that for some d > 0 we have

θ 0 (x) = θ 1 ≡ const, ν 0 (x) = 0 if dist(x, ∂Ω) < d.
Then the solution of (36)-( 40) exists for some T > 0 and is unique.

The proof proceeds along the same lines as that of Theorem 2.1. We point out the necessary modifications.

Instead of ( 36)-(38) we consider the regularized system, where equation ( 36) is replaced with

u -μΔu + εΔ 2 u = -∇P -Δθ∇θ (41)
with additional boundary conditions Δu ∂Ω = 0; where ε is a small parameter, P = p + K 2 ∇θ 2 . Also, we replace θ with θ 1 + θ, where θ is a new unknown function. As in the proof of Theorem 2.1, select two sequences of subspaces

E 1 ⊂ E 2 ⊂ • • • and F 1 ⊂ F 2 ⊂ • • • such that ∪ k E k is dense in • Sol 1 2 (Ω) ∩ W 2 2 (Ω) and ∪ k F k is dense in W 2 2 (Ω)∩ • W 1 2 (Ω).
It is still useful to choose E k and F k to be the linear span of the first k eigenfunctions of the Laplacian in the corresponding spaces.

Define the finite-dimensional solution (u k , ν k , θk ) ⊂ C 0 ((0, T ); E k ×F k ×F k ) by the system of ordinary differential equations

(u k,t , ω) = -(u l k u k,x l , ω) + (μΔu k -εΔ 2 u k , ω) -K(Δ( θk + θ 1 ) • ∇( θk + θ 1 ), ω), ( 42
) J(ν k,t , ζ) = -(Ju l k ν k,x l k, ζ) -K(Δ( θk + θ 1 ), ζ), ( 43
) ( θk,t , ψ) = -(u l k ( θk + θ 1 ) x l , ψ) + (ν k , ψ), ( 44
) (u k , ν k , θ k )| t=0 = (u 0k , ν 0k , θ 0k ). ( 45
)
The identities (42)-( 45) hold for all ω ∈ E k , ζ, ψ ∈ F k . (For simplicity we use the notation θ k = θk + θ 1 ). The solution of this finite-dimensional problem, obviously, exists for some T > 0.

The result of Sect. 2.3 still holds.

Lemma 3.1. For all t > 0 we have

Ω u k (t) 2 + J ν k (t) 2 + K ∇θ k (t) 2 dx 1 dx 2 t 0 + 2μ t 0 Ω ∇u k (s) 2 dx 1 dx 2 ds + 2ε t 0 Ω Δu k (s) 2 dx 1 dx 2 ds = 0.
Proof. Repeat the procedure used in Lemma 2.1.

We need to check that the boundary integrals are equal to zero. Due to the condition u k | ∂Ω = 0, all terms containing u i k vanish. The only term not containing u k is

T 0 ∂Ω θ k,t θ k,xi N i dS
derived from (44), where (N 1 , N 2 ) is an exterior normal vector field. Since θ is a smooth vector field and θ 1 does not depend on t, θ t | ∂Ω = 0, it follows that this term vanishes. The Lemma is proved.

As in section 2.3, Lemma 3.1 guarantees that the solution of the problem (42)-( 45) exists for all T > 0.

Next, we estimate higher derivatives.

Lemma 3.2.

There exists T > 0 and C > 0 depending on ε, the initial and boundary data, but not depending on k, such that u k L2((0,T );W 3 2 (Ω)) , ∇ν k L∞((0,T );L2(Ω)) , θ k L∞((0,T );W 2 2 (Ω)) ≤ C. Note that T > 0 and C > 0 depend on ε, which won't be sufficient to prove Theorem 3.1. Lemma 3.3 below gives the necessary uniform estimates.

Proof. In this proof we use (u, ν, θ) instead of (u k , ν k , θk ).

Consider Eqs. ( 42)-(44) and substitute (ω, ζ, ψ) = (Δu, Δν, Δ 2 θ). First, we note that (44) could be rewritten as

1 2 Ω (Δ θ) 2 dx 1 dx 2 T 0 = T 0 ∂Ω ( θ -ν)Δθ x k N k -Δθ( θ -ν) x k N k dSdt + QT -2u j x k θ xj x k Δθ -Δu j θ xj Δθ + ΔνΔθ dx 1 dx 2 dt.
The first term of the boundary integral is equal to zero, since θ(t, x) = 0 and ν(t, x) = 0 if x ∈ ∂Ω.

Recall that F k is defined to be the linear span of the first k eigenfunctions {v m | m = 0, . . . , k} of the Laplacian, i.e.,

Δv m = λv m , v m | ∂Ω = 0, v m ∈ • W 1 2 (Ω); then Δv m ∈ • W 1 2 (Ω) and thus Δ θ ∈ • W 1 2 (Ω)
. This shows that the second term in the boundary integral also vanishes.

Since Δθ = Δ θ, we have an equation similar to [START_REF] Wang | Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data[END_REF]. The analogues of ( 19) and ( 21) are obtained in the same way and, consequently, we can prove an analogue of inequality ( 22 We also need inequality (11.8) from [24, Chapter III]

∀p > 1 ∃C ∀v ∈ W 2 p (Ω), v| ∂Ω = 0 v W 2 p (Ω) ≤ C( Δv Lp(Ω) + v L2(Ω) )
and the Sobolev embedding theorems to estimate u W 2 p (Ω) (and, consequently, max ∇u ) in terms of Δu W 1 2 (Ω) and lower derivatives. Finally, we can obtain an inequality similar to [START_REF] Mikhailov | Partial differential equations[END_REF] for for some 1 < α ≤ ∞, and that u is bounded by a constant m > 0. Assume also that ∇n 0 and ν 0 vanish for xx 0 < r. Then ∇n and ν are equal to zero for xx 0 < r -(m + max{1, K/J})t, M t α-1 α < 1 2 .

I (t) := u(t) 2 W 1 2 (Ω) + J ν(t) 2 W 1 2 (Ω) + K n(t) 2
In the case of a bounded domain Ω, we impose the additional assumption u ∂Ω = 0, ν(x, t) = 0, n(x, t) t = 0 if (x, t) ∈ {x ∈ ∂Ω | xx 0 < rt(m + max{1, K/J})}.

Proof. Let x 0 = 0. Taking ζ = νϕ and ψ = -KΔnϕ in (28) we get which proves the statement.

J 2 ν 2 ϕdx
Remark 4.1. The proof of Theorem 4.1 is independent of the existence and uniqueness proof and result. Moreover, in the proof we can suppose (u, ν, n) to satisfy only [START_REF] Leslie | Continuum theory for nematic liquid crystals[END_REF] and [START_REF] Chandrasekhar | Liquid crystals[END_REF] but not [START_REF] Ericksen | Continuum theory of nematic liquid crystals[END_REF].

Fig. 1 .

 1 Fig. 1. The structure of smectic (left), nematic (center) and cholesteric (right) liquid crystals

2 L2

 2 ((0,t);W 3 2 (Ω)) + 1, and, consequently, (27). This proves Lemma 3.2.

Theorem 4 . 1 .

 41 Consider the Eqs.[START_REF] Leslie | Continuum theory for nematic liquid crystals[END_REF],[START_REF] Chandrasekhar | Liquid crystals[END_REF], suppose thatw ij := u j xi + u i xj satisfy esssup x |w ij (x, t)| Lα(0,T ) ≤ M/2

  1 dx 2 dt. Since u j u xj and Δn • n xj are uniformly bounded in L 2 (Q t ) [which follows from (27) and standard embedding theorems], we conclude

	Qt

  Next, we need Lemma 8.1 of[24, Chapter III] to estimate u W 2 2 (Ω) in terms of Δu L2(Ω) and u L2(Ω) , namely,

				):
	1 2	T	J ∇ν 2 + K	2 j,k=1	θ xj x k	2 + ∇u 2 dx 1 dx 2	T 0
			+ μ
			T				2
		≤ C 1	esssup ∇u(t)	∇ν(t) 2 L2(Ω) +	θ xixj (t) 2 L2(Ω) + ∇u 2 L2(Ω)
			0				i,j=1
			+ ∇θ(t) L4(Ω) Δu(t) L4(Ω) Δθ(t) L2(Ω) dt.	(46)
					u W 2

QT Δu 2 dx 1 dx 2 dt + ε QT ∇(Δu) 2 dx 1 dx 2 dt 2 (Ω) ≤ C( Δu L2(Ω) + u L2(Ω) ).

  + u j ϕ xj ) ν 2 -K(Δn × n) • νϕ dx 1 dx 2 dt, + u j ϕ xj ) ∇n 2 -K(ν × n) • Δnϕ -Ku j x k n xj • n x k ϕ -K ṅ • n xj xj dx 1 dx 2 dt. + ∇n 2 dx 1 dx 2 dt ≤ 0.

	1 dx 2 (ϕ t K t 0 = J 2 2 ∇n 2 ϕdx 1 dx 2 t 0 = K 2 (ϕ t Add the previous two identities and rewrite the result as
	J 2	ν 2 ϕdx 1 dx 2	t 0	+		K 2	∇n 2 ϕdx 1 dx 2	t 0	≤	φ	J 2	ν 2 +	K 2	∇n 2
		+ K max
	and estimate
			φ	J 2K	ν 2 +	1 2	∇n u i x i x		J 2K	ν 2 +	1 2	∇n 2
	+ ν 2 Consequently, |φ ( x + m t)| 2
		J 2K	ν(t)φ	1 2 2 2 +	1 2	φ	1 2 ∇n(t) 2 ≤ max
												α-1 α	esssup t	φ	1 2 ∇n 2 2 ,

i,j,x

w ij ∇n 2 ϕ + K ν ∇n ∇ϕ dx 1 dx 2 dt. Denote ϕ = φ( x +m t), where φ(x) ∈ C 1 (R), φ = 0 for x > r, φ ≤ 0. Let m := m+max{1, K/J} 2 + ν ∇n ∇ϕ dx 1 dx 2 dt ≤ φ ( x + m t) m + i,j,x w ij ∇n 2 ϕdx 1 dx 2 dt ≤ Mt

Acknowledgments. GAC was partially supported by RFBR Grant 15-01-07920. TSR was partially supported by the NCCR SwissMAP Grant of the Swiss NSF. All authors were partially supported by the Government grant of the Russian Federation under the Resolution No. 220 "On measures designed to attract leading scientists to Russian institutions of higher education" according to the Agreement No. 11.G34.31.0054, signed by the Ministry of Education and Science of the Russian Federation, the leading scientist, and Lomonosov Moscow State University (on the basis of which the present research is organized).

Proof of Theorem 3.1. Due to Lemma 3.2, the sequences u k , ν k , θ k are weakly-precompact in L 2 ((0, T );

2 (Ω)), L ∞ ((0, T );

• W 1 2 (Ω)), and L ∞ ((0, T ); W 2 2 (Ω)). From the embedding theorems, u k u ε weakly in L 2 ((0, T );

This proves existence of solutions of the regularized problem (37), ( 38), (41) for any ε > 0. To prove Theorem 3.1, it is sufficient to obtain uniform estimates for higher derivatives of (u ε , ν ε , θ ε ). Lemma 3.3. For sufficiently small T 1 > 0, not depending on ε, there is a constant C > 0, also not depending on ε, such that

Proof. Consider, instead of (37) and (38), the equations

where χ(x) is a smooth function,

Repeat the Galerkin procedure for the problem (39), (40), (41), (49), (50) as presented in Sect. 2. Since all nonlinear terms in (49), (50) vanish on the boundary, we can prove the analogue of Theorem 2.2.

Proposition 3.1.

There exists a strong solution (ũ, ν, θ) of the problem (39), (40), (41), (49), (50). The solution is unique and satisfies inequalities (47), (48), where the constants C and T depend on d and don't depend on ε.

The proof is identical to that of Theorem 2.2 and the argument in Sect. 2.6, so it won't be repeated here.

Let us prove that, for some T 1 < T not depending on ε, the two solutions coincide, i.e.

Define constant T 1 > 0 by

Suppose t 0 ∈ K . Then the solution (u ε , ν ε , θ ε ) of the regularized problem exists in some interval (0, t 0 + δ). Indeed, let us consider the Eqs. (37), (38), (41) with standard boundary conditions and initial conditions (u ε , ν ε , θ ε )(t 0 ) = (ũ, ν, θ)(t 0 ). Thus, in the open interval (t 0 , t 0 + δ) the new problem has a solution that is a continuation of (u ε , ν ε , θ ε ).

Next, we show that (u ε , ν ε , θ ε ) and (ũ, ν, θ) coincide on some open interval. Consider Eqs. (37), (38) in the domain Q t0 . Since u ε ∈ L 2 ((0, t 0 ); W 1 ∞ (Ω)), we can apply Theorem 4.1 (all we need from this theorem is the existence of the triple (u, ν, n) satisfying Eqs. [START_REF] Leslie | Continuum theory for nematic liquid crystals[END_REF], [START_REF] Chandrasekhar | Liquid crystals[END_REF] and this fact does not depend on any statement below); thus θ ε = θ 1 , ν ε = 0 in an α-neighborhood of ∂Ω if 0 < t < T for some constant T ≤ t 0 . This constant does not depend on ε since u ε = ũ for any t < t 0 and ũ L2((0,t0);W 1 ∞ (Ω)) is also ε-independent by Proposition 3.1. Due to the definition of T 1 , we have α > d 3 and T = t 0 . Moreover, since u ε ∈ L 2 ((0, t 0 + δ); W 1 ∞ (Ω)) and θ ε t=t0 = θ 1 , ν ε t=t0 = 0 in the α-neighborhood of ∂Ω, there exists a constant δ ε depending on ε such that

Now it is easy to check that the solution (u ε , ν ε , θ ε ) satisfies the Eqs. ( 41), ( 49), (50) in Q T1 , and hence coincides with (ũ, ν, θ) in some neighborhood of t = t 0 . Consequently, K is open.

We now show that K is closed, i.e., if

This is impossible since we have inequalities (47), (48) for the solution (ũ, ν, θ). Hence t 0 ∈ K .

Therefore, K is both open and closed in [0, T 1 ] which implies that K = [0, T 1 ] and concludes the proof of the lemma.

Continuing the proof of Theorem 3.1, select a subsequence ε k such that (u, ν, n) is a weak limit of (u

, and L ∞ ((0, T ); W 2 2 (Ω)). The vector fields (u, ν, n) are the first three terms of the solution. The term ∇p is the Hodge projection of u -μΔu -Δn∇n ∈ L 2 (Q T ) on the orthogonal complement of L 2 (0, T ; Sol 2 (Ω)).

As in Sect. 2.5, one checks that the initial conditions hold. The proof of uniqueness is identical to that in Sect. 2.6.

Remark 3.1. As can be seen from the proof, all that is needed is that n 1 is piecewise constant on the boundary ∂Ω.

The same result holds if external forces are present.

Theorem 3.2. Suppose Ω, n 0 , ν 0 , u 0 , n 1 satisfy the conditions of Theorem 3.1. Assume also that F ∈ L 2 ((0, T );

Then the solution exists and is unique for some T > 0. Remark 3.2. If G = (G 1 , G 2 , 0), the director field cannot be represented as n = (cos θ, sin θ, 0). Remark 3.3. The analogues of the Theorems 3.1 and 3.2 hold also in the case of the director field n = (cos θ sin φ 0 , sin θ sin φ 0 , cos φ 0 ), ν = (0, 0, ν 3 ), where φ 0 is a constant angle and θ, ν 3 are new unknown functions.

Finite Propagation Speed

In this section we consider the strong solutions of the Ericksen-Leslie equation both in the periodic (see Definition 2.1) and the bounded domain case (see Definition 3.1). For simplicity in this section we will use fdx 1 dx 2 instead of T fdx 1 dx 2 or Ω fdx 1 dx 2 depending on the nature of the domain.


Morgan Bertin
email: bertin@lmt.ens-cachan.fr

Chaowei Du
email: c.du@tue.nl

Johan P M Hoefnagels
email: j.p.m.hoefnagels@tue.nl

François Hild
email: hild@lmt.ens-cachan.fr

François Hild

Crystal plasticity parameter identification with 3D measurements and Integrated Digital Image Correlation

Keywords: Crystal plasticity, Digital image correlation, Finite-element

published or not. The documents may come   L'archive ouverte pluridisciplinaire

analysis; In-stu test; Optical microscopy

Introduction

The mechanical properties of materials often originate from physical and multi-scale phenomena that are due to complex and heterogenous microstructures. One of the first method to consider heterogeneous media for computing macromechanical responses is based on homogenization [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF]. In order to solve this problem micromechanical frameworks have been developed for elastic materials [START_REF] Hashin | A variational approach to the theory of the elastic behaviour of multiphase materials[END_REF] and in elastoplasticity [START_REF] Hill | Continuum micro-mechanics of elastoplastic polycrystals[END_REF]. Due to complex behaviors and the difficulty to observe and measure at the microscale the main difficulty remains the identification of the constitutive parameters.

The lattice structure (e.g., FCC, BCC) is considered to derive the constitutive equations for single crystals [START_REF] Franciosi | Multislip in f.c.c. crystals a theoretical approach compared with experimental data[END_REF][START_REF] Cuitino | Computational modelling of single crystals[END_REF][START_REF] Tabourot | Generalised constitutive laws for f.c.c. single crystals[END_REF]. The present study aims at coupling experimental and numerical results and provides a compromise between a reasonable computation time and a sufficiently small scale to achieve the observation of the micromechanical behavior in a material with real engineering applications. Under the same assumptions, Raabe et al. [START_REF] Raabe | Micromechanical and macromechanical effects in grain scale polycrystal plasticity experimentation and simulation[END_REF] characterized the micromechanical behavior in polycrystalline aluminum with the measurement of the displacement on the observe surface. The authors demonstrated by 2-dimensional Finite Element analyses that the micromechanical response obtained from the crystal plasticity law was closer to the experimental observation and was able to predict the stress and strain heterogeneities. However, the underlying microstructure is a key feature and 3D orientation mapping has been performed by Musienko et al. [8] to compare experimental and numerical strains also on polycrystalline aluminum. Using a 3D analysis, Martins et al. [START_REF] Martins | Full 3d spatially resolved mapping of residual strain in a 316L austenitic stainless steel weld specimen[END_REF] focused on the residual strain measured in a weld zone of 316L stainless steel.

Digital Image Correlation enables full displacement and strain fields to be measured and provides insight into the micromechanics of solids. Imaging systems capable of higher magnifications allow the heterogeneity of the micromechanical response to be evaluated. A first approach [START_REF] Hoc | A procedure for identifying the plastic behavior of single crystals from the local response of polycrystals[END_REF] using full field measurements obtained from SEM acquisitions has led to the identification of a single crystal plasticity law. Dmitrieva et al. [START_REF] Dmitrieva | Lamination microstructure in shear deformed copper single crystals[END_REF] studied shear bands in a single copper crystal and Tasan et al. [START_REF] Tasan | Microstructural banding effects clarified through micrographic digital image correlation[END_REF] characterized micromechanical banding effects.

Other techniques may be used, e.g., Raabe et al. [START_REF] Raabe | Grain-scale micromechanics of polycrystal surfaces during plastic straining[END_REF] acquired pictures with a confocal microscope.

The combination of both experimental and numerical aspects has been investigated by Héripré et al. [START_REF] Héripré | Coupling between experimental measurements and polycrystal finite element calculations for micromechanical study of metallic materials[END_REF] while applying the Finite Element Method Updating technique (FEMU [START_REF] Kavanagh | Finite element applications in the characterization of elastic solids[END_REF]). Tasan et al. [START_REF] Tasan | Integrated experimental-simulation analysis of stress and strain partitioning in multiphase alloys[END_REF] proposed an integrated experimental-numerical approach to investigate stress-strain partitioning in multiphase alloys, although, the mechanical properties of the material have been obtained with a separated technique, i.e., nano indentation tests. Guery et al. [START_REF] Guery | Identification of crystal plasticity law parameters using kinematic measurements in polycrystals[END_REF] have identified constitutive parameters of the law proposed by Méric and Cailletaud [START_REF] Méric | Single crystal modeling for structural calculations: Part 2 -finite element implementation[END_REF] using full field measurement, SEM acquisitions and FEMU-UF. The results provide an internal check of the constitutive model with respect to the experiment with the identified parameters. However, several difficulties remain (e.g., unknown underlying microstructure) and the specimen design is still one of the biggest challenges because it is hard to build and to test mechanically.

The objective of this study is to perform an investigation at the microscale and to identify the material parameters of a crystal plasticity model with Integrated Digital Image Correlation (IDIC [START_REF] Roux | Digital image mechanical identification (DIMI)[END_REF][START_REF] Neggers | Timeresolved integrated digital image correlation[END_REF][START_REF] Bertin | Optimization of a biaxial tensile specimen geometry for the identification of constitutive parameters based upon full field measurements[END_REF][START_REF] Bertin | Integrated digital image correlation applied to elastoplastic identification in a biaxial experiment[END_REF]). Figure 1 shows the designed identification procedure. Initially, the crystallographic features, i.e., phase, grain (size, number and orientation) and the geometry of the studied volume are assessed and lead to the design of a representative 3D numerical model. Secondly, the out-of-plane displacement holds essential information regarding the active slip systems. Therefore an experimental method is introduced that utilizes confocal microscopy combined with a 3D surface global DIC formulation to quantify full three-dimensional displacement field of the specimen surface and use these 3D data to extract boundary conditions and identify crystal plasticity parameters. Because of the relatively height noise level computations are run with stabilized boundary conditions. Finally, the sought parameters are identified with integrated-DIC that combines all experimental data, i.e., the surface height profiles, constitutive model, and crystallographic features.

The paper is organized as follows. Section 2 introduces the experimental and numerical tools utilized herein. Section 3 shows the experimental results prior to the identification discussed in Section 4.

Coupled experimental/numerical procedure

The objectives are i) to observe a crystal-specific micromechanical behavior under confocal microscopy, ii) to quantitatively measure the 3D surface dis-placement field at the grain scale with DIC, iii) to simulate the mechanical response with a representative model of the microstructure and the chosen crystal plasticity law, and iv) to couple experimental and numerical results in a single numerical framework to identify (as much as possible) the material and kinematic unknowns. The experimental methodology relies on the measurement of surface height profiles of a micro-specimen (µ-specimen) within a macroscopic sample that is mechanically loaded in tension.

Specimen

To demonstrate the potential of the proposed technique a relevant engineering material is chosen for this study, namely, 1-mm thick as-received IF-steel sheet. The latter possesses a very low carbon content and Table 1 gathers its chemical composition. IF steel is widely used, e.g., in the automotive industry, for its deep drawability properties [START_REF] Lee | Microstructures and mechanical properties of ultra low carbon interstitial free steel severely deformed by a multi-stack accumulative roll bonding process[END_REF].

The desired mechanical properties of IF-steels are, a) low yield stress, b) high hardening modulus resulting in stress levels after the material has been plastically deformed, c) high Lankford coefficients to improve deep-drawability capabilities and d) high fracture strain. The ferritic microstructure is body center cubic (BCC). The average grain size is of the order of 10 µm.

The preparation of the specimen requires four steps: i. From the 1-mm thick sheet, a large specimen is cut (50 mm × 10 mm) and ground down to 0.3 mm thickness and polished on both sides, to yield a very flat specimen with the two surfaces parallel to each other.

ii. A large surface area of the large specimen is investigated using Orientation Imaging Microscopy (OIM, see Figure 2(a,b)). The location of the µspecimen is chosen to contain two grains that are larger than 15 µm having a clean grain boundary and a misorientation angle less than 1 • within the grain so that both crystal and structure are relatively clean (Figure 2(c)).

iii. The µ-specimen is milled using Focused Ion Beam technique (Figure 2(d)).

The obtained µ-specimen is 12 µm long and 4 µm wide with an equilateral triangular basis. Its length and depth are a compromise between a size consistent with the resolution of confocal microscopy and the milling speed of the FIB process.

iv. The last step is platinum deposition to create a random pattern for DIC purposes. The latter is obtained by depositing 20 nm high and 0.8 µm in diameter semi-spherical dots at specific locations. Their size has been adapted with respect to the confocal microscope features.

Figure 2 shows secondary electron images related to steps (iii) and (iv).

Figure 2(d) shows the underlying microstructure and the grain boundary by observation from the side. Furthermore, it validates the assumption that the apex line of the µ-specimen is entirely free. Figure 2(e) shows the pattern obtained after platinum deposition. The latter has also been deposited on the border of the µ-specimen to perform DIC on a larger area.

Table 2 gathers the lattice orientations of the two grains identified from the OIM analysis.

Experimental protocol

Figure 3 shows the experimental apparatus within the confocal microscope, i.e., the large specimen clamped in the tensile stage. Before starting the experiment, the µ-specimen is successively, i) positioned and aligned inside the tensile stage, ii) aligned with the translation and rotation stage, and iii) positioned in the center of the field of view of the sensor. During the experiment the large specimen is subjected to a velocity equal to 0.5 µm.s -1 and at approximatively every 5 µm stopped while the image acquisition procedure is performed. Because the early stage of plastic strain and its corresponding kinematic hardening behavior are being sought, three cycles corresponding to 1 %, 2 % and 3 % global strain are applied without compression.

Since the µ-specimen can never be perfectly centered on the microscale it moves in the field of view when strained. Therefore, an in-house routine was written to maintain the µ-sample inside the field of view by correcting the rigid body motions using DIC and controlling the motorized (x, y)-sample accordingly.

The first image of the experiment corresponds to the reference, then for each load step, a first image is captured to evaluate the rigid body motions.

The latter ones are applied as a correction by moving the motorized (x, y)-stage and finally a second image is captured. In the z-direction, the contraction of the large specimen will move the µ-sample out of the depth of field. An auto-frontal procedure is used to address this issue.

Height profile images are captured in confocal profilometry mode by a Sen-sofar Plµ2300 Optical Confocal Microscope using a CCD camera (definition: 557 × 557 pixels) with a Nikon EPI 150× objective lens, resulting in a square field of view of 61 × 61 µm 2 . This technique allows for a very high resolution in height (estimated to be 9 nm for this particular specimen configuration) whereas the in-plane resolution is limited by the diffraction limit of the blue light wavelength. However, a careful inspection of the height profiles revealed some artifacts caused by, i) the lens defects and dust particles, and ii) the CCD pixel positions. Both errors were identified to be systematic because they remain at the same location on the images as long as the imaging parameters are fixed and the same lens is used.

Figure 4(a) shows an image with the systematic errors previously listed. The correction method consists of an acquisition under the identical parameters of 400 images at 400 random locations on a flat reference surface. All images are averaged, i.e., added and normalized by the total number of images, and a plane is fitted and subtracted to this average profile to obtain a mean height level of zero mean with the corresponding root mean square error in the height level as small as possible. This process allows only the systematic error to be kept (Figure 4(b)) and by subtraction from all the experimental images the biases are almost completely removed without corrupting the experimental data. This procedure helps the DIC computation that needs reliable height levels over time. The loads are also measured and the standard resolution is evaluated as

γ F = 1 N.

3D Digital Image Correlation

Global DIC [START_REF] Besnard | Finite-element" displacement fields analysis from digital images: Application to Portevin-Le Châtelier bands[END_REF] relies on the registration of an image f in the reference configuration and a series of pictures g in the deformed configurations. The specific approach followed herein adds to the conventional 2D approach the out-of-plane displacement as an offset to the height conservation [START_REF] Han | Measuring nanoscale stress intensity factors with an atomic force microscope[END_REF][START_REF] Neggers | Direct stress-strain measurements from bulged membranes using topography image correlation[END_REF][START_REF] Bergers | On-wafer time-dependent high reproducibility nano-force tensile testing[END_REF].

The registration problem consists of minimizing the sum of squared differences between the deformed image corrected by the measured 3D displacements u(x, t) = u(x, t)e x + v(x, t)e y + w(x, t)e z and the reference image (written for each time t independently) over the Region of Interest

χ 2 f (t) = 1 2γ 2 f N Ω Ω g (x + u(x, t)e x + v(x, t)e y , t) -f (x, t) -w(x, t)e z ) 2 (2.1)
with respect to the parameterization of the sought displacement fields u(x, t), where x = xe x + ye y is any considered pixel, Ω the Region of Interest (ROI),

N Ω its area in terms of number of pixels. The term γ f represents the standard deviation (expressed in height level) of noise (assumed additive white Gaussian) that affects each image independently (including the reference one, which is responsible for the factor of 2). Ten images have been acquired initially and the standard deviation is evaluated to be γ f = 9 nm. The minimization of χ 2 f is achieved by successive linearizations and corrections, using a modified Newton

Raphson scheme

[M ]{δu} = {b (i) } (2.2)
where [M ] is the DIC matrix, {δu} the vector gathering all increments of measured displacement amplitudes, and {b (i) } the residual column vector at itera-tion i. The DIC matrix reads

M ij = Ω (∇f (x) • ψ i (x)) (∇f (x) • ψ j (x)) (2.3)
and the right hand side term

b i = Ω (∇f (x) • ψ i (x)) (f (x) -g(x + ũ(x))) (2.4)
where ũ(x) is the current estimate of the displacement field and the image gradient is enriched with the variations of height levels, namely, ∇f = ∂f ∂x e x + ∂f ∂y e y -e z . A consequence of the addition of the out-of-plane displacement in the functional is that the problem has more degrees of freedom that might affect the minimization.

The integration of mechanical and kinematic constraints into the 3D-DIC problem, namely Integrated-DIC (IDIC), is performed by choosing as unknowns the sought parameters associated with a constitutive law or boundary conditions. The sensitivity fields S U (x) [START_REF] Lecompte | Mixed numerial-experimental technique for orthotropic parameter identification using biaxial tensile tests on cruciform specimens[END_REF] are chosen as a basis of functions for the displacement field

[S U ] = ∂{u} ∂{p} (2.5)
where {u} is the computed nodal displacement field, {p} is the vector gathering the unknown parameters, i.e., sought material parameters or boundary conditions. Furthermore, specific features of the loading history may or may not be sensitive. Treating an image series is equivalent to summing over the corresponding contributions at each time step: χ2 f = 1/N t Nt t=1 χ 2 f (t) to probe the sensitivities over the whole time domain. The solution at convergence gives an evaluation of the model error compared to the noise level. This indicator is useful because it allows the user to validate or not the investigated constitutive law(s) [START_REF] Bertin | Optimization of a biaxial tensile specimen geometry for the identification of constitutive parameters based upon full field measurements[END_REF].

Last, if the same mesh is used, and the DIC matrix has been computed IDIC simply consists of projecting the nodal displacement field onto the sensitivity fields [START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF]. Thus, the sought parameters are identified by iteratively solving the linear system until convergence

{δp} = 1 2γ 2 f [M ] -1 IDIC [S U ] t {b} (2.6)
where

[M ] IDIC = 1/(2γ 2 f )[S U ] t [M ][S U ]
is the kinematic Hessian. The covariance matrix of the identified parameters is written as [START_REF] Bertin | A basis and method for removal and replacement of doubtful height observations in topographic measurements[END_REF][START_REF] Bertin | Optimization of a biaxial tensile specimen geometry for the identification of constitutive parameters based upon full field measurements[END_REF] [

C I p ] = {δp} ⊗ {δp} = [M ] -1 IDIC (2.7)
where • is the mean value of •. Another useful indicator is the correlation matrix (no index summation used), where i refers to the row number and j the column number of the sought parameter

(Cor p ) ij = (C p ) ij (C p ) ii (C p ) jj (2.8)

Crystal plasticity model

To demonstrate the potential of the proposed technique to identify parameters of a crystal plasticity model, the Méric and Cailletaud [START_REF] Méric | Single crystal modeling for structural calculations: Part 2 -finite element implementation[END_REF] model is chosen.

It is based on the decomposition of the total strain into an elastic and plastic strain. The elastic behavior assumes cubic symmetry in agreement with the BCC lattice. Using the Kelvin-Voigt notations, Hooke's fourth-order tensor [C] depends on three elastic constants of the material, namely C 11 , C 12 and C 44 .

The plastic deformation initiates from crystallographic slip in the crystal lattice and satisfies incompressibility. A slip system (θ) is defined by a slip plane n θ 0 and a slip direction on the plane s θ 0 in the initial configuration of the lattice (• 0 ). In the case of a BCC crystallographic lattice, 24 slip systems are considered, namely 12 principal slip systems from 110 and 12 secondary systems from 112 . The gradient tensor F θ = ∇ t 0 ⊗ x, where ⊗ is the dyadic product, corresponds to the lattice strain

F θ = I + γ θ s θ 0 ⊗ n θ 0 (2.9)
where γ θ is the applied shear on the θ th slip system, I is the second order identity tensor. The linear elasticity law relates the second Piola-Kirchhoff stress tensor

S = J e F -1 e σ (F t e ) -1
where σ is the stress tensor and the Green-Lagrange strain

tensor E e = 1/2(F t e F e -I); S = C : E e , where • : • is the double inner product.

Since an elastoviscoplastic framework is assumed, the plastic strain rate is driven by the resolved shear stress on each slip system θ. The velocity gradient tensor reads

L θ = Ḟ θ F θ -1 = γθ s θ 0 ⊗ n θ 0 (2.10)
where γθ the resolved shear stress rate on the θ th slip system. The orientation of the lattice is important with respect to the loading direction. Thus, for Θ slip systems the velocity gradient tensor reads

L = Θ θ=1 γθ s θ 0 ⊗ n θ 0 = Θ θ=1 γθ P θ 0 (2.11)
where P θ 0 = s θ 0 ⊗ n θ 0 is the so-called non-symmetric Schmid tensor. The decomposition of the deformation gradient rate tensor becomes

L = L e + F e L p F -1 e (2.12)
where L e and L p are the elastic and plastic gradient rate tensors. To complete the constitutive description of the crystalline material, the plastic slip rates (2.10) rely on the resolved shear stress. The relationship between the resolved shear stress τ θ and the shear rate γθ reads

γθ = |τ θ -cx θ | -τ 0 K n η θ (2.13)
where τ 0 is the critical resolved shear stress, c is the hardening coefficient, (K, n)

are material constants related to Norton law, η θ = sign(τ θ -cx θ ) and where [H θκ ] is the interaction matrix, δ θκ is the Kronecher operator and H θκ = c(η κ -dx κ )δ θκ the hardening moduli of the interaction matrix. Table 3 gathers all model parameters and their initial values chosen based on the IF literature [START_REF] Franciosi | Glide mechanisms in b.c.c. crystals: An investigation of the case of α-iron through multislip and latent hardening tests[END_REF][START_REF] Tjahjanto | Modelling of the effects of grain orientation on transformation-induced plasticity in multiphase carbon steels[END_REF].

C++ implementation

One of the main challenges is the coupling between experimental and numerical data in a single framework. An in-house C++ code dealing with all the facets of the problem from the acquired height profile images to the identification of parameters while computing the mechanical responses has been designed. Regarding the FE computations (performed in Code Aster [START_REF] David | Comportements élastoviscoplastiques mono et polycristallins[END_REF]), a C++ routine has been written to automatically generate the input files with all the necessary data, i.e., the microstructure, the boundary conditions, the mesh, and the constitutive parameters. Simulations are automatically started and the results are sent back to the main code to solve the identification problem iteratively.

This main code deals, with i) the surface profile images in height levels, ii) the 2D mesh (in pixels) related to the global DIC formulation, iii) the 3D mesh (in mm) used to solve the finite element problem, and iv) the 2D mesh (in pixels)

where the sensitivity fields are projected on the DIC matrix to solve the Integrated DIC problem. As proposed in Ref. [START_REF] Bertin | Optimization of a biaxial tensile specimen geometry for the identification of constitutive parameters based upon full field measurements[END_REF], the parameterization is based on a log scale of the ratio between the current value of the parameter and its initial guess. It is defined such that the sought parameters in the new setting {q} are expressed from the initial basis as {q} = log{p/p 0 }, where {p} is the vector gathering the values of the parameters and {p 0 } their initial values.

As a key feature, the measurement, the computation, and the identification steps rely on a 3D surface formulation where the out-of-plane displacements are also used to identify the micromechanical behavior of crystals. The simulations are performed on the measured 3D microstructures with their lattice orientation. Furthermore, the applied boundary conditions are obtained from the displacements measured via DIC. The numerical simulations being 3D, the displacement of the triangular boundaries are obtained assuming a linear interpolation between the edge L + and L -(i.e., intersection between the measured surface (S + and S -) and the triangular boundary surface) and the apex (P + and P -) where its displacement is equal to the mean value of the corresponding edge (respectively L + and L -).

To conclude, a regularization method [START_REF] Gras | Identification of a set of macroscopic elastic parameters in a 3d woven composite: Uncertainty analysis and regularization[END_REF] is implemented to prevent meaningless identification when the influences of material parameters on the observables are weak. Therefore, a functional (χ R ) is associated with the identification functional ( χf ) to enable for the changes of the sought parameters only if their corresponding sensitivities are higher than a specific bound. χ 2 R is a convex function reaching its minimum equal to zero and reads

χ 2 R = {p}[C R p ] -1 {p} (2.16)
where [C R p ] is the covariance matrix of the material parameters. The regularized functional is written as χ 2 f = χ2 f +χ 2 R because both functionals are dimensionless and normalized to one. Finally, the regularization of the linear system reads

([M ] IDIC + λ * [I]) {δp} = {b} + λ * ({p 0 } -{p}) (2.17)
where λ * is the regularization parameter whose choice is performed to cancel out the influence of noise [START_REF] Gras | Identification of a set of macroscopic elastic parameters in a 3d woven composite: Uncertainty analysis and regularization[END_REF].

Experimental results

320 profile images are acquired and 8-bit encoded such that the total range is used over the entire images set. The ratio between mm and pixels is equal to 110 nm/pixel, and the ratio between mm and height levels is constant and equal to 6.5 nm/(digital level). Figure 6 shows the reference image and the last image of the experiment after the application of the bias correction.

Figure 7 shows a Backscattered Electron (BSE) image of the µ-specimen and its surrounding area where slip bands are revealed due to Electron Channelling Contrast. Furthermore plasticity has occurred during the experiment, as expected. The signal possesses a notable amount of noise, which is challenging for further investigations. Figure 8(b) also shows the prescribed displacement at the boundary of the macroscopic specimen. The ratio between the two displacements (of the order of 5,000) is constant, as expected. The distance between the two grips is 32 mm and the µ-specimen length 12 µm is equal to 3.8 × 10 -4 . However, the ratio experimentally observed is equal to 2 × 10 -4 and corresponds to a strain of the µ-specimen of 1.6%. This results shows that the microscopic strain is not equal to the global macroscopic strain. It originates from microscopic heterogeneities or small discontinuities of the macroscopic geometry.

3D-DIC measurements

Figure 9 shows the digital level residual fields reconstructed at the last time step of the experiment and the last iteration of the DIC procedure. The latter is obtained by computing the difference between each pixel of the reference image and the corrected deformed image [START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF] and decreases by about 16 %. The gain originates from the vertical displacement determination that affects the entire ROI.

The main issue (see Figure 9) is that the signal to noise ratio is very small. To overcome this challenge a new strategy is proposed, i) piecewise-linear time steps are assumed for each loading phase and for each considered surfaces (S + and S -) for the components u Y and u Z (see Figure 10 in red), ii) the displacements of the apex (lines L + and L -) of both boundary surfaces are identified with integrated 3D-DIC, and iii) the measured macroscopic loads are adapted to fit the section of the microscopic specimen and some material parameters are identified based on the minimization of the combination of the loads and image residuals. The u X component is not considered because of its larger noise level.

With this pre-processing strategy, two objectives have already been fulfilled, namely, i) the displacement fields have been measured over time and ii) the micromechanical plastic behavior and its evolution have been observed. The remaining objectives to achieve are i) the coupling between experimental and numerical solutions (with the selected crystal plasticity law), ii) the identification of the displacement of the apex of the boundary triangles and iii) the identification of the constitutive parameters.

Boundary conditions and material parameter identification

The identification is investigated with the 3D model and the measured microstructure. First, the in-depth displacements at the constrained boundaries are assessed. Four unknowns, namely λ + Y , and λ + Z , λ - Y , and λ - Z are applied as an amplitude correction to u + Y and u + Z at point P + (apex of S + ), and to u - Y and u - Z at point P -(apex of S -, see Figure 11).

Sensitivity analysis

A sensitivity analysis is carried out and the uncertainty is evaluated with the covariance matrix [C p ] IDIC of the unknowns [START_REF] Bertin | Integrated digital image correlation applied to elastoplastic identification in a biaxial experiment[END_REF]. A 2% perturbation on each parameter is applied to compute the sensitivity fields. The vector of kinematic and material parameters reads The high level of anti-correlation between the parameters (c, d) will impact their identifications.

{p} = {λ + Y , λ + Z , λ - Y , λ - Z , τ 0 , c, d}.

Identification

The underlying boundary conditions are now determined with Integrated-DIC by solving iteratively Equation (2.17). Figure 13 Table 4 gathers the measured values and compares the dimensionless residuals between the initial 3D-DIC solution, with more kinematic freedom and the integrated DIC solution, with only few degrees of freedom associated with boundary conditions. Therefore, the small residual increase is regarded as a validation of the integrated approach.

The monitored macroscopic loads are scaled to the specimen section and a combined image and load identification scheme is used, namely IF-DIC. The global cost function reads

χ2 = N f 1 + N f χ2 f + 1 1 + N f χ 2 F (4.1)
where the weight of the two functionals originates from a Bayesian foundation [START_REF] Bertin | Optimization of a biaxial tensile specimen geometry for the identification of constitutive parameters based upon full field measurements[END_REF] to account in a fair manner for both sources of information. χ 2 F is the load residual

χ 2 F = 1 γ 2 F N t {F m -F c } t {F m -F c } (4.2)
where N t is the number of time step, {F m } and {F c } the vectors gathering the measured and computed loads, respectively, and γ F = 5.97 × 10 -6 N is the standard load uncertainty [START_REF] Bergers | On-wafer time-dependent high reproducibility nano-force tensile testing[END_REF]. The same choice of regularization methodology is made and the global functional reads χ 2 = χ2 + χ 2 R . As a first step, only the material parameters are sought, while the boundary conditions are the previously assessed ones. Figures 14(a [START_REF] Bergers | On-wafer time-dependent high reproducibility nano-force tensile testing[END_REF] to provide relevant data to perform a precise identification with accurate measurements of the load history at the scale of the µ-specimen. Such a change of the methodology would, however, come at the expense of a more complex experimental procedure.

Conclusion

The investigation and the characterization of the micromechanical behavior of a micro-beam made of two grains have been achieved. On the experimental side, two new procedures have been proposed, i) the design of a triangular prism shape micro-specimen with a known microstructure embedded in a macrospecimen and ii) the use of a global DIC algorithm to measure the 3D surface displacement field. On the identification side, a novel strategy has been designed to overcome the encountered high noise level on the height profile images. The latter has prevented direct approaches such as Finite Element Method Updating to succeed because the initial 3D global DIC measurement is too noisy. The choice was made to use an integrated approach to work directly on the images but the scatter on the boundary conditions prevented a direct evaluation of the sensitivity fields.

Based on this limitation, a two-step identification strategy was adapted, i) estimation of the boundary conditions and ii) determination of the constitu-tive parameters. The boundary conditions were assumed to be piece-wise linear with time. First, the identification via integrated-DIC was performed only with the height profile images to assess the underlying boundary conditions. Subsequently, the macroscopic loads were adapted to the specimen section and both sources of information were combined to assess the sought constitutive parameters, which describe the kinematic hardening relationship in the crystal plasticity material model.

The concept proposed herein is very general and can be applied to any micromechanical experiment aiming to characterize a constitutive law describing a micro-mechanical behavior. While optical profilometry brings the advantage of 3D displacement fields, other imaging systems can also be use such as scanning electron microscopy or regular optical microscopy depending on the sought models and the observation scale. The technique gives new insights into the micro-mechanical behavior with the combination of images and loads based on a methodology that can cope with high noise levels.

Figure 5 (

 5 Figure 5(a) shows the finite element model of the microstructure where the two

Figure 8 (

 8 Figure 8(a) shows the u Y component of the displacement field obtained with the 3D-DIC algorithm at the last time step. The corresponding mesh is also

Figure

  12(a) shows the kinematic Hessian for the entire loading history. The most influent parameters are λ + Y and λ - Y whereas the material parameters exhibit the smallest influences. This result shows that accurate identification of the boundary conditions is of paramount importance prior to any analysis of the material parameters. Figure12(b) shows the corresponding correlation matrix.

  (a) shows the change of the four kinematic parameters and Figure 13(b) shows that of the functional χ f . The converged solution is reached in 10 iterations. Furthermore, three kinematic parameters, namely λ - Y , λ + Z , and λ - Z converged to a close solution with respect to their initial values. However, λ + Y varies by about 20 %, which means that a significant shear strain occurs on the subsurface of the µ-specimen during the experiment.

  ) and14(b) show the changes of the material parameters and residuals. The global residual (χ) decreases by about 1 % and stabilizes after 4 iterations. Furthermore, while the load residual (χ F ) is significantly minimized the kinematic residual remains approximately constant.

Figure 14 (

 14 Figure 14(c) shows the experimental, initial, and final computed load histories.

FiguresFigure 1 : 2 Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :

 1223456 Figures

Figure 7 :

 7 Figure 7: Backscattered electron (BSE) image of a larger view at the end of the

Figure 8 :

 8 Figure 8: (a) Component u Y of the displacement field at the last time step of the experiment evaluated via DIC. (b) Displacement amplitude (u + Y(t) -u - Y (t))in the longitudinal direction and prescribed displacement with an adapted scale

Figure 9 : 1 digital2Figure 10 :Figure 11 :

 911011 Figure 9: (a) Residual field at the last time step of the experiment for the first

Figure 12 :Figure 13 :Figure 14 :

 121314 Figure 12: (a) Kinematic Hessian [M ] IDIC and (b) correlation matrix of the kinematic Hessian [M ] IDIC , both for the entire loading history

Table 5

 5

	gathers the constitutive parameters at convergence. Up to 50 %
	changes are observed with respect to the initial guess. The load residual has
	been decreased by 25 %, which proves that it is sensitive to the sought param-
	eters. However, its final value is still large, which is an indication of a model
	error associated with the constitutive law. Last, no changes in the kinematic pa-
	rameters are obtained when a new identification is performed with the assessed
	20

Table 1 :

 1 Chemical composition of the studied IF-steel

	Composition	C	Mn	Al	Ti	Nb	Si	S	P	N(ppm)	Fe
	(wt %)	0.0022 0.09 0.015 0.033 0.008 0.003 0.007 0.012	28	bal.

Table 2 :

 2 Euler angles of the two grains measured via OIM and maximum misorientation angle (δ) within each grain Euler angle ( • ) misorientation ( • )

	Denomination	ψ	θ	φ	δ
	Grain 1 (red)	281 8	47	0.70
	Grain 2 (blue) 211 45 150	0.76

Table 3 :

 3 All parameters of the Meric and Cailletaud crystal plasticity model and their initial values chosen based on the IF literature[START_REF] Franciosi | Glide mechanisms in b.c.c. crystals: An investigation of the case of α-iron through multislip and latent hardening tests[END_REF][START_REF] Tjahjanto | Modelling of the effects of grain orientation on transformation-induced plasticity in multiphase carbon steels[END_REF]

	Mechanism		Elasticity		Viscosity		Plasticity
	Model parameter	C 11	C 12	C 44	K	n	τ 0	c	d
	Unit	GPa	GPa GPa MPa (-) MPa GPa	(-)
	Initial value	233.5 135.5 118	12	10	40	40	1500
				37

Table 4 :

 4 Identified values of the kinematic parameters

	Name χ f	λ + Y	λ + Z	λ -Y	λ -Z
	DIC	3.7	-	-	-	-
	IDIC	4.5 0.81 1.02 0.98 1.02

Table 5 :

 5 Identified parameters of the crystal plasticity model

	Name	χ	χ f	χF	τ0 (MPa) c (GPa) d (-)
	Initial value 4.64 4.46	78	40	40	1600
	IF-DIC	4.52 4.47	58	21	44	1140

Acknowledgment MB thanks Professor Marc G.D. Geers for giving him the opportunity to join the Mechanics of Material research group to perform the experiments reported herein. The authors acknowledge the help of Dr. Hugo Leclerc in the implementation of the 3D-DIC code. This work was supported by the Région Ile de France and BPI France (DICCIT and THERMOFLUIDE-RT projects).