Thiebaud Richeton
email: thiebaud.richeton@univ-lorraine.fr

Stéphane Berbenni

Effects of heterogeneous elasticity coupled to plasticity on stresses and lattice rotations in bicrystals: A Field Dislocation Mechanics viewpoint

Keywords: Bicrystals, Incompatibilities, Internal stresses, Lattice rotations, Field Dislocation Mechanics, Jump conditions

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Many studies coupling experiments and theoretical investigations (e.g., [START_REF] Gemperlova | Compatibility stresses in deformed bicrystals[END_REF][START_REF] Hirth | The influence of grain boundaries on mechanical properties[END_REF][START_REF] Liang | GND accumulation in bi-crystal deformation: Crystal plasticity analysis and comparison with experiments[END_REF][START_REF] Mayama | Geometrically necessary dislocation structure organization in FCC bicrystal subjected to cyclic plasticity[END_REF][START_REF] Ohashi | X-ray microdiffraction and strain gradient crystal plasticity studies of geometrically necessary dislocations near a Ni bicrystal grain boundary[END_REF][START_REF] Rey | Slip heterogeneities in deformed aluminium bicrystals[END_REF]1982;[START_REF] Saada | Planar dislocation arrays and crystal plasticity[END_REF][START_REF] Sun | Mesoscale investigation of the deformation field of an aluminium bicrystal[END_REF][START_REF] Sutton | Interfaces in Crystalline Solids[END_REF][START_REF] Vehoff | The effect of hydrogen and segregation on fatigue crack nucleation at defined grain boundaries in nickel bicrystals[END_REF][START_REF] Acharya | Constitutive analysis of finite deformation field dislocation mechanics[END_REF] have been carried out on bicrystals in order to get a better understanding of the specific role of grain boundaries during the deformation of polycrystals. At a material surface of discontinuity (grain, twin or phase boundary), incompatibility stresses can develop due to material elastic anisotropy. Much theoretical work (e.g., [START_REF] Gemperlova | Compatibility stresses in deformed bicrystals[END_REF]Qamar and Husain,1989;Tewary et al., 1988), including Finite Element simulations (e.g., [START_REF] Schick | Stress concentrations at grain boundaries due to anisotropic elastic material behavior[END_REF][START_REF] Vehoff | Fatigue crack nucleation at interfaces[END_REF], has been devoted to estimate these elastic incompatibility stresses in order to predict crack initiation and propagation path. When plasticity enters into play, plastic strain incompatibilities add to the picture and yield additional stresses and lattice rotations. Among these fields, it may actually be distinguished between those uniquely due to a difference in plastic distortions and those arising from a coupling between elastic and plastic heterogeneities. Many models make an assumption of homogeneous isotropic elasticity or of pure rigid viscoplastic behavior (no elasticity) and thus overlook the contribution of both elastic and elastic-plastic coupling incompatibilities. The present paper aims notably at underlying the potentially significant role of coupling incompatibilities. For this purpose, explicit closed-form solutions of stress and lattice misorientation fields are derived for two semi-infinite crystals undergoing uniform plastic distortions in the general context of heterogeneous anisotropic elasticity. The derivation is performed by means of the concepts of the elastic theory of continuously distributed dislocations (ECDD) initiated by [START_REF] Kröner | Kontinuumstheorie der Versetzungen und Eigenspannungen[END_REF][START_REF] Kröner | Continuum theory of defects in physics of defects[END_REF] and [START_REF] Willis | Second order effects of dislocations in anisotropic crystals[END_REF]. It deals with the Nye's tensor α [START_REF] Nye | Some geometrical relations in dislocated crystals[END_REF] which reflects only the so-called geometrically necessary dislocations (GND) densities following [START_REF] Ashby | The deformation of plastically non-homogeneous materials[END_REF], also referred to as "excess" or "polar" dislocation densities in the literature. The Nye's tensor α is a continuous rendition of the collective arrangements of dislocations and the associated lattice incompatibility. This theory was rethought some years ago by [START_REF] Acharya | A model of crystal plasticity based on the theory of continuously distributed dislocations[END_REF] (see also [START_REF] Acharya | Constitutive analysis of finite deformation field dislocation mechanics[END_REF][START_REF] Acharya | Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model mesoscopic field dislocation mechanics: Part I[END_REF] in considering the permanent deformation arising due to dislocation motion. In the Field Dislocation Mechanics (FDM) theory of [START_REF] Acharya | A model of crystal plasticity based on the theory of continuously distributed dislocations[END_REF], the elastic distortions are uniquely decomposed into compatible and incompatible parts whereas the evolution equation for the GNDs [START_REF] Mura | Continuous distribution of moving dislocations[END_REF] is added within an efficient numerical set-up.

Incompatibility problems constitute a long-standing issue of materials mechanics (see the pioneering works of [START_REF] Bilby | Bristol Conference Report on Defects in Crystalline Solids[END_REF][START_REF] Frank | Symposium on The Plastic Deformation of Crystalline Solids[END_REF][START_REF] Indenbom | Internal stresses in crystals[END_REF][START_REF] Kröner | Kontinuumstheorie der Versetzungen und Eigenspannungen[END_REF][START_REF] Willis | Second order effects of dislocations in anisotropic crystals[END_REF]. Different techniques have been envisaged to address these issues and determine stresses and lattice rotations: Green functions [START_REF] Berveiller | Généralisation du problème de l'inclusion et application à quelques problèmes d'élastoplasticité des matériaux hétérogènes[END_REF][START_REF] Berveiller | The problem of two plastic and heterogeneous inclusions in an anisotropic medium[END_REF][START_REF] Kröner | Modified Green functions in the theory of heterogeneous and/or anisotropic linearly elastic media[END_REF], stress potential functions for plane problems [START_REF] Kröner | Continuum theory of defects in physics of defects[END_REF][START_REF] Berveiller | Contribution à l'étude du comportement plastique et des textures de déformation des polycristaux métalliques[END_REF] or Fourier Transforms [START_REF] Berbenni | Intra-granular plastic slip heterogeneities: discrete vs. mean field approaches[END_REF][START_REF] Rey | The elastic field of periodic dislocation networks[END_REF][START_REF] Saada | Elastic field of dislocation networks and grain boundaries[END_REF]. In the case of bicrystals, [START_REF] Gemperlova | Compatibility stresses in deformed bicrystals[END_REF] imagine the total stresses as a sum of the applied and compatibility stresses and the total strains as a sum of elastic strains in relation with applied stresses, elastic strains in relation with compatibility stresses and plastic strains. Then, they consider boundary conditions for stresses and compatibility conditions for strains at the interface to find out the stresses. For threedimensional heterogeneous elasticity problems, [START_REF] Michelitsch | Stress functions and internal stresses in linear threedimensional anisotropic elasticity[END_REF] have extended the Kröner stress function method. It is also noteworthy that some homogenization schemes for laminates microstructures were derived to calculate the stresses using elastic stress concentration and influence tensors (e.g., Franciosi andBerbenni, 2007, 2008;[START_REF] Stupkiewicz | Modelling of laminated microstructures in stressinduced martensitic transformations[END_REF]. Netherveless, to the authors' knowledge, no explicit closed-form formulas of stresses have been given for bicrystals with completely anisotropic elasticity and heterogeneous plasticity. In addition, the context of the present paper is different from the preceding approaches since it aims at directly applying the FDM theory [START_REF] Acharya | A model of crystal plasticity based on the theory of continuously distributed dislocations[END_REF] in its static form.

The paper outline is as follows. In Section 2, notation conventions are settled. The static FDM theory is recalled in Section 3 and formulated for heterogeneous elasticity and plasticity with respect to a reference homogeneous medium. Conventional continuity conditions at surfaces of discontinuity are also recalled in this section, in addition to the expression of the surface-dislocation density tensor and the jump condition ensuring the conservation of the Burgers vector. Section 4 deals with the issue of an infinite bicrystal with planar boundary undergoing piecewise uniform plastic distortions and elastic properties. The static FDM theory is accordingly used to derive analytical expressions of stress and lattice rotation fields in the general case of heterogeneous anisotropic elasticity. The validations of these explicit formulae are checked in Section 5 from Crystal Plasticity Finite Element simulations on a bicrystal with periodic boundary conditions assuming each crystal has a cubic symmetry (fcc metals). The contribution of the different sources of incompatibilities is further analyzed. In Section 5, analytical expressions of residual stresses and lattice misorientations are also used to perform comparisons with results derived from the isotropic approximation. Finally, concluding remarks follow.

Notations

A bold symbol denotes a tensor and ⊗ v w the tensorial product of tensors v and w . We denote Av the action of the second-order tensor A on the vector v , producing a vector. The symbol AB represents tensor multiplication of the second-order tensors A and B and a ":" denotes the contracted product between two tensors. The operation × is the tensorial cross product, defined as:

( ) ( ) T T , ∀ × = × c A v c A c v ,
the symbol ∀ being shorthand for "for all" and the superscript T denoting the transpose of a matrix. If A and B are second-order tensors, C a fourth-order tensor and v a vector, we have in rectangular Cartesian coordinates

(1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) T , , : grad curl ji ij ij j i ij jl il ilmn mn il i l il ljk ik j il ljk ij k il ijk j k i A A v A B C B v A A v v w = = = = = =∈ × =∈ × = ∈ A Av AB C A v A A v v w
where jkl ∈ is a component of the third-order alternating Levi-Civita tensor.

Besides, the following reduction convention is adopted concerning the writing of the Hooke's law in matrix notation

(2)

σ ε σ ε σ ε σ ε σ ε σ ε ⎡ ⎡ ⎤ ⎡ ⎤ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣
σ ε σ ε ε ε ε ε ⎤ ⎡ ⎤ ⎡ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎦ ⎣ ⎦ 33 12 31 23 σ σ σ σ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
where ij σ are the symmetric Cauchy stresses, e ij ε the elastic strains, ij c the elastic stiffnesses and ij s the elastic compliances.

FDM static theory for heterogeneous elasticity and plasticity

Field equations

The initial static elastic theory of continuously distributed dislocations (ECDD) as initiated by [START_REF] Kröner | Kontinuumstheorie der Versetzungen und Eigenspannungen[END_REF][START_REF] Kröner | Continuum theory of defects in physics of defects[END_REF], [START_REF] Indenbom | Internal stresses in crystals[END_REF], [START_REF] Willis | Second order effects of dislocations in anisotropic crystals[END_REF], and reworked by [START_REF] Acharya | A model of crystal plasticity based on the theory of continuously distributed dislocations[END_REF] includes the following set of field equations [START_REF] Acharya | A model of crystal plasticity based on the theory of continuously distributed dislocations[END_REF][START_REF] Acharya | Constitutive analysis of finite deformation field dislocation mechanics[END_REF][START_REF] Acharya | Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model mesoscopic field dislocation mechanics: Part I[END_REF] (3)

⊥ ⊥ ⊥ ⊥ = = + = + = + = = = =- = u β β β β β β β β β β β α β β σ C ε (e) (f div 0 ) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ = σ
where u is the displacement field and β the total distortion field (i.e., the displacement gradient) which decomposes into its elastic e β and plastic part p β (Eq.(3)a). Due to lattice incompatibility, the elastic and plastic distortion tensors are not gradients and thus not curl free. They have compatible ( e β , p β ) and incompatible parts ( e ⊥ β , p ⊥ β ) (Eq.(3)b). While the compatible part is a gradient and thus curl free (Eq.(3)c), the incompatible part results from geometrically necessary dislocations (GNDs) distribution α , and is solution to the incompatibility equation (Eq.(3)d) (see for instance [START_REF] Acharya | A model of crystal plasticity based on the theory of continuously distributed dislocations[END_REF][START_REF] Acharya | Constitutive analysis of finite deformation field dislocation mechanics[END_REF] for a mathematical definition of second-order tensor fields A and ⊥ A and the associated unique orthogonal decomposition

⊥ = +

A A A ). As a consequence of the Stokes-Helmholtz decomposition of the elastic and plastic distortions (Eq.(3)d, see also [START_REF] Acharya | Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model mesoscopic field dislocation mechanics: Part I[END_REF] 1 for a more complete description), the incompatible elastic distortion must offset exactly the incompatible plastic distortion to ensure the continuity of matter (i.e. β is curl free, Eq.(3)a) (4) The stress tensor σ is obtained from the tensor of elastic stiffnesses C (Eqs.( 2) and (3)e) and satisfies the static equilibrium equation without body forces (Eq.(3)f). The elastic and plastic strain tensors ( e ε , p ε ) are the symmetric parts of the elastic and plastic distortion tensors ( e β , p β ), respectively.

Navier-type equation and incompatibility sources

The fluctuations of elastic moduli Let us anticipate that in the case of a bicrystal, the reference medium will be chosen later to be one of both crystals. Due to the space uniformity of p0 β in infinite media, its curl is zero so that p0 β is compatible (its incompatible part p0 Eqs.( 5) and ( 7) can be introduced in Eqs.

(3)a-f to determine the Navier-type equation ( 8)

( ) ( ) ( ) ( ) ( ) ( ) 0 p 0 p p div : grad 0 ⊥ ⎡ ⎤ + - - - = ⎣ ⎦ C δC x u x β δβ x δβ x which reads (9) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 p 0 p p0 p p div grad div div div : grad div : div : div : 0 ⊥ ⊥ - - + - - - = C u x C δβ x C δβ x δC x u x δC x β δ C x δβ x δC x δβ x
Eq.( 9) can be recast as (10)

( ) ( ) ( ) 0 div grad 0 + = C u x f x where ( ) ( ) ( ) (c) (i) = + f x x f
f x is the total fictive body force due to incompatibility sources in the material.

( )

(c)
f x is a fictive body force containing the compatible total distortion ( grad u ) and the compatible parts of ( )

p β x (11) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 p p 0 p (c) div : grad div div : div : = - - - x δC x u x C δβ x δC x β δC x f δβ x ( ) (i)
f x is a fictive body force containing the incompatible part of ( )

p β x (12) ( ) ( ) ( ) ( ) ( ) 0 p ( ) p i div div : ⊥ ⊥ = - - x C δβ x δC x δβ f x
Eqs.( 11) and ( 12) show that the general case where plastic and elastic properties are both heterogeneous give rise to coupling incompatibility effects through

( ) ( ) ( ) p div : - δC x δβ x .
The effects of these elastic-plastic coupling incompatibilities on stresses and elastic rotations will be analyzed and discussed in Section 4 through analytical derivations in the case of an infinite bicrystal with planar boundary and piecewise uniform elastic and plastic properties.

Interestingly, two particular cases can be derived from Eqs.( 11),( 12). The first case corresponds to homogeneous plastic fields. This case yields ( )

(i) 0 = f x
and following

Eqs.(4), ( 7) and (11) (13)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (c) p0 e div : grad div : ⎡ ⎤ = = - = ⎣ ⎦ f x x δC x u x β C x f δ β x
The second case corresponds to homogeneous elastic properties and yields from Eqs.( 6),( 11) and (12) (14)

( ) ( ) ( ) ( ) (c) (i) 0 p div = + =- f x x x C f δβ f x

Burgers vector conservation at surfaces of discontinuity

Let us now consider a fixed plane interface I separating two crystals (crystal I and crystal II) in the current configuration of a continuous medium, with unit normal vector n oriented from crystal I to crystal II (Fig. 1). The limit of continuous fields approaching the interface I from crystal I is assigned the superscript I , and that of fields approaching I from crystal II the superscript II . Hence, the jump of a field g at a material surface of discontinuity is denoted [ ] II I = g g g . Conventional continuum mechanics requires continuity of both the traction vector σn and the displacement u across the perfectly bonded interface (15

) [ ] [ ] 0 ; 0 = = σ n u
A consequence of the displacement continuity is Hadamard's kinematic compatibility condition (Hadamard, 1903). For small displacements and in the absence of interface motion, it states that ( 16)

[ ] [ ] 0 grad × = × = n n u β
Thus, for any unit vector t lying in the interface I, Eq.( 16) can also be written in the form (17

) [ ] [ ] grad , 0 I ∀ ∈ = = u t β t t
Hadamard's condition reflects the "tangential continuity" of the total distortion β across the interface seen as a material surface of discontinuity. Surfaces of discontinuity such as grain or sub-grain boundaries are conventionally described in terms of surface-dislocations belonging to the interface [START_REF] Bilby | Bristol Conference Report on Defects in Crystalline Solids[END_REF][START_REF] Bullough | Continuous distributions of dislocations: surface dislocations and the crystallography of Martensitic transformations[END_REF][START_REF] Christian | Dislocations and lattice transformations[END_REF]. The components of the surfacedislocation density tensor S α in the interface I are expressed as a non-dimensional ratio of a Burgers vector length over a segment length. The surface-dislocations are confined to the interface ( )

S 0 = α n
and form a two-dimensional array. For small displacements, they allow accommodation of a discontinuity of the elastic distortion tensor e β across the interface [START_REF] Beausir | Spatial correlation in grain misorientation distribution[END_REF][START_REF] Bilby | Bristol Conference Report on Defects in Crystalline Solids[END_REF][START_REF] Bullough | Continuous distributions of dislocations: surface dislocations and the crystallography of Martensitic transformations[END_REF][START_REF] Christian | Dislocations and lattice transformations[END_REF][START_REF] Sutton | Interfaces in Crystalline Solids[END_REF]

) (18) ( ) S e S e
, or equivalently

I ⎡ ⎤ ⎡ ⎤ ∀ ∈ × = = - × ⎣ ⎦ ⎣ ⎦ t α n t β t α β n
Eq.( 18) can be thought as generalized Frank-Bilby relations [START_REF] Bilby | Bristol Conference Report on Defects in Crystalline Solids[END_REF][START_REF] Frank | Symposium on The Plastic Deformation of Crystalline Solids[END_REF].

Using the standard elastic-plastic decomposition of the total distortion (Eq.(3)(a)) and the Hadamard's condition (Eq.( 16)), the surface-dislocations density tensor can also be formulated as

( )

S p S p , or equivalently I ⎡ ⎤ ⎡ ⎤ ∀ ∈ × = - = × ⎣ ⎦ ⎣ ⎦ t α n t β t α β n
Eqs.( 18) and ( 19) state that any jump in the elastic (or plastic) distortion tensor between the two lattices can be accommodated by an appropriate singular surface-dislocation distribution supported by the interface. Some recent analyses [START_REF] Acharya | Jump condition for GND evolution as a constraint on slip transmission at grain boundaries[END_REF][START_REF] Acharya | New perspectives in plasticity theory: dislocation nucleation, waves, and partial continuity of plastic strain rate[END_REF]Mach, 2009;[START_REF] Beausir | Spatial correlation in grain misorientation distribution[END_REF][START_REF] Mach | Continuity in the plastic strain rate and its influence on texture evolution[END_REF]Puri et al., 2011;[START_REF] Richeton | Continuity constraints at interfaces and their consequences on the work hardening of metal-matrix composites[END_REF][START_REF] Taupin | Size effects on the hardening of channeltype microstructures: a Field Dislocation Mechanics-based approach[END_REF] suggest that a more consistent description of the boundary area can be obtained if continuity of the incompatible elastic (or plastic) displacement across the interface is postulated, i.e. if there is conservation of the Burgers vector content across the interface (according to [START_REF] Kröner | Continuum theory of defects in physics of defects[END_REF], the true Burgers vector b of all dislocation lines threading a surface S of the deformed material configuration, surrounded by a contour C is defined for small displacements as ). In such continuous modeling, the boundary area is seen as an extended region supporting a continuous density of bulk GNDs, and the singular description of the incompatibility between crystals through surface-dislocations is surrendered. Hence, the surface-dislocation density tensor is zero and Eqs.( 18) and ( 19) reduce to (Mach, 2009;[START_REF] Beausir | Spatial correlation in grain misorientation distribution[END_REF][START_REF] Mach | Continuity in the plastic strain rate and its influence on texture evolution[END_REF]Puri et al., 2011[START_REF] Richeton | Continuity constraints at interfaces and their consequences on the work hardening of metal-matrix composites[END_REF] (20) e e p p 0 0 , or equivalently 0 0

I ⎧ ⎧ ⎡ ⎤ ⎡ ⎤ = ×= ⎪⎣ ⎦ ⎪⎣ ⎦ ∀ ∈ ⎨ ⎨ ⎡ ⎤ ⎡ ⎤ = ×= ⎪ ⎪ ⎣ ⎦ ⎣ ⎦ ⎩ ⎩ β t β n t β t β n

Analytical derivations for infinite bicrystal with planar boundary 4.1. Piecewise uniform plastic distortions

Here, we follow the static theory of continuously distributed dislocations described in Section 3. The objective is to give the stress fields and lattice misorientations (i.e., elastic rotation jumps) in both crystals of an infinite medium assuming uniform plastic distortion and elastic constants in each crystal. The static configuration is represented in Fig. 2 where the infinite grain boundary plane spreads in the (O,x 1 ,x 3 ) plane. To be consistent with the Frank-Bilby procedure [START_REF] Frank | Symposium on The Plastic Deformation of Crystalline Solids[END_REF][START_REF] Bilby | Bristol Conference Report on Defects in Crystalline Solids[END_REF], crystals I and II are separated by the discontinuity interface S with unit normal n directed toward crystal II from crystal I along the (O,x 2 ) direction. It is noteworthy that the present calculations would also work for any permutation in the choice of the coordinate axis along the normal to the interface. In consistency with Section 3, the superscript I is used for fields referring to crystal I and II for fields referring to crystal II. The static problem of Fig. 2 is invariant along (O,x 1 ) and (O,x 3 ). As a result, for any field g referring to the bicrystal properties, we have (21

) ,1 ,3 0 = = g g
The linear theory of small perturbations is considered and the set of field equations set out in Section 3 (Eq.(3)a-f) is solved. In the following, the reference homogeneous infinite medium is chosen as crystal I. Linear elasticity is considered with piecewise uniform elastic moduli

(22) ( ) I 2 ijkl ijkl ijkl C C C H x ⎡ ⎤ = + ⎣ ⎦ where ( ) 2 H x is the Heaviside unit step function along (O,x 2 ) (23) ( ) 2 2 2 1 if 0 0 if 0 x H x x > ⎧ = ⎨ < ⎩
The total volume of the bicrystal is denoted V . The bicrystal is submitted to tractions on its external boundary s = T Σn , where Σ is assumed macro homogeneous and s n stands for the outward normal of the external boundary. As previously mentioned, piecewise uniform spatial fluctuations of the plastic distortion components is considered (24) ( )

I 2 p p p ij ij ij H x ⎡ ⎤ = + ⎣ ⎦ β β β
Such assumption is actually equivalent as considering an infinite planar surface-dislocation distribution S α (and a nil density of polar dislocations α in the bulk). From Eqs.( 18) and ( 19), the surface-dislocations density tensor S α reads (25)

⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ - - ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = - = - ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ - - ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ α β β β β β β β β β β β β
The full development of the compatibility conditions applied to tensors e β and p β (Eq.(3)c) is given in Appendix A. Because of the invariance along (O,x 1 ) and (O,x 3 ) (Eq.( 21)), we have ,1 3)) reduce to (26)

( ) ( ) ,2 ,2 2, 0 e p ij ij j β β ∀ ≠ = =
These equations actually apply to regions where fields are smooth, i.e. for the present situation, respectively in crystal I and in crystal II. At surfaces of discontinuity, they have corresponding jump conditions ( 0

e p ⎡ ⎤ ⎡ ⎤ × = × = ⎣ ⎦ ⎣ ⎦ β n β n , cf.
Eq.( 16)). At the interface between crystal I and crystal II, these jump conditions read (27) 2, 0

e p ij ij j β β ⎡ ⎤ ⎡ ⎤ ∀ ≠ = = ⎣ ⎦ ⎣ ⎦
As consequence of Eqs.( 26) and ( 27

( ) ' ' 2, e p p p ij ij ij ij ij ij j K K β β β β ⊥ ∀ ≠ = - - - =
Because for 2 j ≠ , p ij β is spatially uniform and because p β is assumed piecewise uniform (Eq.( 24)), Eq.( 29) can finally be written (30)

( ) 2 2, e p ij ij ij j K H x β β ⎡ ⎤ ∀ ≠ = -⎣ ⎦
where ij K is another spatially uniform function.

Stresses 4.1.1.1 Heterogeneous anisotropic elasticity

Because of the invariance along (O,x 1 ) and (O,x 3 ) (Eq.( 21)), stress equilibrium (Eq.( 3

)f) reduces to 2,2 0 i = σ
. Besides, self-equilibrium of stresses reads

(31) 1 Σ ij ij V dV V σ = ∫
⎧ = = + + + + + ⎪ = = + + + + + ⎨ ⎪ = + = + + + + + ⎩ ε β σ σ σ ε β σ σ σ ε β β σ σ σ
From Eq.( 30), the elastic strains 11 e ε , 33 e ε , 31 e ε are uniform piecewise functions. Since the elastic compliances are also uniform piecewise functions, Eq.( 33) shows that the stress components are uniform piecewise functions too and can be written

(34) ( ) I 2 ij ij ij H x ⎡ ⎤ = + ⎣ ⎦ σ σ σ with [ ] 2 0 i = σ
from Eq.( 32). Furthermore, assuming that the volume fraction of each crystal in the infinite medium is equal (i.e., I

I I V V V V =
), the macroscopic stress reads from Eq.( 31) (35) ( )

I I I 1 Σ 2 ij ij ij = + σ σ
Substituting the expressions of the elastic distortions given by Eq.(30) into the system Eq.( 33), we have

(36) ( ) ( ) ( ) ( ) ( ) ( )
Σ Σ Σ Σ Σ Σ 2 Σ Σ Σ p p p K H x s s s s s s K H x s s s s s s K K H x s s s s s s ⎧ ⎡ ⎤ - = + + + + + ⎣ ⎦ ⎪ ⎪ ⎡ ⎤ - = + + + + + ⎨ ⎣ ⎦ ⎪ ⎡ ⎤ + - = + + + + + ⎪ ⎣ ⎦ ⎩ ε σ σ σ ε σ σ σ ε σ σ σ
It is now possible to derive the expressions of the constant functions ij K by writing the former equations into crystal I ( 2 0 x < ) using Eq.( 35)

(37) ( ) ( ) ( ) (
) ( )( )

I
2 Σ Σ Σ Σ Σ Σ 2 Σ Σ Σ Σ Σ Σ K s s s s s s s s s K s s s s s s s s s K K s s s σ σ σ σ σ σ σ σ σ = - + + + + + + + + = - + + + + + + + + + =- + + ( ) ( )
Σ Σ Σ Σ Σ Σ 2 p p p K s s s s s s K s s s s s s K K s s s s ε σ σ σ ε σ σ σ ε σ σ σ ⎡ ⎤ = + + + + + + ⎣ ⎦ ⎡ ⎤ = + + + + + + ⎣ ⎦ ⎡ ⎤ + = + + + + ⎣ ⎦ ( ) II II 22 45 12 56 23 Σ Σ Σ s s ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ + + ⎩
By summing the two expressions of each constant function and by introducing the notation

I I I ij ij ij s s s = + %
, the system writes also

(39) [ ] [ ] [ ] ( ) ( ) [ ] [ ] [ ] ( ) (
ε σ σ σ ε σ σ σ ε σ σ σ ⎧ = + + + + + = + + + + + = + + + + + ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
and from Eqs.( 30) and ( 39) they also read

(41) [ ] [ ] [ ] ( ) ( ) ( ) [ ] [ ] [ ] ( ) ( ) (
)

II
ε ε σ σ σ ε ε σ σ σ ε ⎡ ⎤ = - + + + + + + + + + ⎣ ⎦ ⎡ ⎤ = - + + + + + + + + + ⎣ ⎦ = % % % % % % [ ] [ ] [ ] ( ) ( ) (
)

II
1 2 Σ Σ Σ Σ Σ Σ 2 p s s s s s s s s s ε σ σ σ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎡ ⎤ - + + + + + + + + + ⎪ ⎣ ⎦ ⎩ % % %
The combination of Eqs.( 40) and ( 41) leads then to the following linear system

(42) [ ] [ ] [ ] ( )( ) [ ] [ ] [ ] ( )(
)

II
Σ Σ Σ 2 Σ Σ Σ Σ Σ Σ 2 Σ Σ Σ p p s s s s s s s s s s s s s s s s s s s s s ⎡ ⎤ + + =- + + + + + + ⎣ ⎦ ⎡ ⎤ + + =- + + + + + + ⎣ ⎦ + + = σ σ σ ε σ σ σ ε σ σ σ % % % % % % % % % [ ] [ ] [ ] ( )
2 Σ Σ Σ 2 Σ Σ Σ p s s s s s s ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎡ ⎤ - + + + + + + ⎣ ⎦ ⎩ ε
The expressions of the stress components in crystal II follow directly from the resolution of this linear system of 3 equations with 3 unknowns. Then the stresses in crystal I are given from Eq.( 35) by ( 43)

I I I 2Σ ij ij ij = - σ σ
Let us now adopt the following notations

(44) [ ] [ ] [ ] ( ) [ ] [ ] [ ] ( ) [ ] [ ] [ ] ( )
Σ Σ Σ 2 Σ Σ Σ Σ Σ Σ 2 Σ Σ Σ 2 2 Σ Σ Σ 2 Σ Σ Σ p p p s s s s s s s s s s s s s s s s s s ⎧ ⎡ ⎤ ⎡ ⎤ = + + + - + + ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ = + + + - + + ⎨ ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ = + + + - + + ⎣ ⎦ ⎣ ⎦ ⎩ ε ε ε ε ε ε
In the general case of piecewise uniform anisotropic elasticity with non zero macroscopic stress, we have then

(45) ( ) ( ) ( ) ( ) ( )
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ - - - + - ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ = + + - - ⎡ ⎤ ⎡ ⎤ - - + - + ⎣ ⎦ ⎣ ⎦ = ε ε ε σ ε ε σ % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % ( ) ( ) ( ) ( ) *
⎡ ⎤ - ⎣ ⎦ + + - - ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ - + - + - ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ = + + - - ε ε ε ε σ % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 13 15 35 s s s ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ % % % It is
Δ Δ ΔΔ ij ij ij ij ij = Σ + + + s ε s ε σ σ σ σ with - [ ] ( ) p ij ij σ ⎡ ⎤ Σ = = = ⎣ ⎦ s ε 0 ,
p p Δ Δ Δ Δ ij ij ij ij ij = -Σ + + s ε s ε σ σ σ σ
, internal stress term due to couplings between elastic and plastic incompatibilities ( ) p Δ Δ s ε . ( )

Heterogeneous isotropic elasticity

II II II 2 1 E = + μ ν
), the system of Eq.( 42) reads

I I I 1 1 2 2 Σ Σ Σ 1 1 2 2 Σ Σ Σ 1 1 2 p p E E E E E E E E E E E E E E E E ν ν ν ν ν σ σ ε ν ν ν ν ν σ σ ε σ ε μ μ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎡ ⎤ + - + =- + - + - ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎣ ⎦ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎡ ⎤ - + + + =- + - - + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎣ ⎦ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ + =- ⎜ ⎟ ⎝ ⎠ 31 31 I 2 Σ p μ ⎧ ⎪ ⎪ ⎪ ⎪ ⎡ ⎤+ ⎣ ⎦ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
and its resolution leads to the following solutions

2 1 2 1 1 1 1 1 1 Σ Σ Σ 2 1 1 p p E E E E E E E E E E E E E E E E E ⎛ ⎞ - - ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ - ⎛ ⎞ ⎜ ⎟ ⎡ ⎤ ⎡ ⎤ + + + - + - + + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎣ ⎦ ⎣ ⎦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ = ⎛ ⎞ ⎛ ⎞ + - + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ + = ν ν ν ν ν ν ν ν ν ε ε μ μ μ σ ν ν ν ν σ ( ) (
2 1 2 1 1 1 1 1 1 Σ Σ Σ 2 1 1 2 2 p p p E E E E E E E E E E E E E ⎛ ⎞ - - ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ - ⎛ ⎞ ⎜ ⎟ ⎡ ⎤ ⎡ ⎤ + + - + - + + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎣ ⎦ ⎣ ⎦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ + - + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎡ ⎤ = - + ⎣ ⎦ + ν ν ν ν ν ν ν ε ε μ μ μ ν ν μ μ μ σ ε μ μ II 31 I I I Σ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ μ μ 4
Σ 1 Σ 1 Σ p p p p p μ σ ε ν ε ν μ σ ε ν ε ν σ μ ε ⎡ ⎤ ⎡ ⎤ = - + ⎣ ⎦ ⎣ ⎦ - ⎡ ⎤ ⎡ ⎤ = - + ⎣ ⎦ ⎣ ⎦ - ⎡ ⎤ = ⎧ ⎪ ⎪ ⎪ ⎨ -⎣ ⎦ ⎪ ⎪ ⎪ ⎩
Finally, using Eq.( 35) we have

(50) ( ) ( ) ( ) ( ) ( ) 11
Σ sgn 1 Σ sgn 1 Σ sgn p p p p p x x x μ σ ε ν ε ν μ σ ε ν ε ν σ μ ε ⎧ ⎡ ⎤ ⎡ ⎤ = - + ⎣ ⎦ ⎣ ⎦ ⎪ - ⎪ ⎪ ⎨ ⎡ ⎤ ⎡ ⎤ = - + ⎣ ⎦ ⎣ ⎦ ⎪ - ⎪ ⎡ ⎤ = - ⎪ ⎣ ⎦ ⎩ where ( ) 2 sgn 1 x = when 2 0
x > (i.e. in crystal II following Fig. 2) and

( )

2 sgn 1 x = -when 2 0
x < (i.e. in crystal I).

Comparison with other works and techniques

To the authors' knowledge, the explicit stress formula obtained in the general context of heterogeneous anisotropic elasticity with plastic strains (Eq.( 45)) had never been formulated in a closed-form solution before. However, the explicit expression in the particular case of isotropic elasticity (Eq.( 50)) was already derived by other techniques: Fourier transforms [START_REF] Rey | The elastic field of periodic dislocation networks[END_REF], second order Beltrami stress function (see [START_REF] Kröner | Continuum theory of defects in physics of defects[END_REF] for plane problems in the real space [START_REF] Berveiller | Contribution à l'étude du comportement plastique et des textures de déformation des polycristaux métalliques[END_REF][START_REF] Rey | Slip heterogeneities in deformed aluminium bicrystals[END_REF] or from consideration of boundary conditions for stresses and compatibility conditions for strains at the interface [START_REF] Gemperlova | Compatibility stresses in deformed bicrystals[END_REF]. Note however that the present calculations do not require the assumption of plastic incompressibility (i.e. our calculations are also valid in the case of dilatant plasticity) in contrast with the calculations performed by [START_REF] Berveiller | Contribution à l'étude du comportement plastique et des textures de déformation des polycristaux métalliques[END_REF]. Furthermore, it is noteworthy that the linear system of equations (Eq.( 42)) giving the full solutions was actually written by [START_REF] Gemperlova | Compatibility stresses in deformed bicrystals[END_REF] for the most general case. It can be noted however that the form is somewhat different than in Eq.( 42) and that there is a typographic error in the published expression. A correct expression can be found in the book of Sutton and Ballufi (1995, p.707). [START_REF] Gemperlova | Compatibility stresses in deformed bicrystals[END_REF] also formulated the solution in the case of heterogeneous isotropic elasticity and found expressions equivalent to Eq.( 48) (Sutton and Ballufi, 1995). The most noticeable difference between the approach of [START_REF] Gemperlova | Compatibility stresses in deformed bicrystals[END_REF] and the present one is that they consider boundary conditions for stresses and compatibility conditions for strains at the interface whereas we consider stress equilibrium and the compatibility conditions for e β and p β in the bulk (Eq.( 28)). Hence, the proof is given from Eq.( 30) that elastic strains and stresses are piecewise uniform functions whereas it is an implicit assumption in the work of [START_REF] Gemperlova | Compatibility stresses in deformed bicrystals[END_REF].

Besides, Eq.( 45) can also be derived in terms of elastic stress concentration and influence tensors following the homogenization techniques described in [START_REF] Stupkiewicz | Modelling of laminated microstructures in stressinduced martensitic transformations[END_REF], and in Franciosi andBerbenni (2007, 2008). Thus, after some algebra manipulations on these tensors, we can recast the stresses as follows (51)

B B F B B F σ ε σ ε ⎧ ⎡ ⎤ = + + ⎪ ⎣ ⎦ ⎨ ⎡ ⎤ = + - ⎪ ⎣ ⎦ ⎩
where P, P' denote plane components 11, 33, 13 (corresponding resp. to 1, 3, 5 in the reduction convention settled by Eq.( 2)), and A denotes anti-plane components 22, 12, 23 (resp. 2, 4, 6 in the reduction convention) and

⎛ ⎞ ⎡ ⎤ ⎣ ⎦ ⎜ ⎟ ⎜ ⎟ ⎡ ⎤ ⎡ ⎤ = ⎣ ⎦ ⎣ ⎦ ⎜ ⎟ ⎜ ⎟ ⎡ ⎤ ⎣ ⎦ ⎝ ⎠ ε ε ε ε . I ' PP B (resp. II
B B F I B B F σ ε σ ε ⎧ ⎡ ⎤ = + + ⎣ ⎦ ⎪ ⎨ ⎡ ⎤ = - - - ⎪ ⎣ ⎦ ⎩
The expressions for the components of II ' PP B , II PA B and ' PP F are obtained from the explicit Eq.( 45) and are given in Appendix B. Using the general homogenization technique recently developed in [START_REF] Stupkiewicz | Modelling of laminated microstructures in stressinduced martensitic transformations[END_REF], Franciosi andBerbenni (2007, 2008), the explicit formulas for bicrystals with general elastic anisotropy are given by Eq.( 53) together with Eqs.(B.1), (B.2) and (B.3). These explicit formulas were not reported in the literature to our knowledge.

Elastic energy density

The volume density of elastic energy is computed as (54) 1 2

e ij ij = ϕ σ ε
where the elastic strains e ij ε can be deduced from the Hooke's law (Eq.( 2)). Thus

(55) 1 2 ij ijkl kl s = ϕ σ σ
where ij σ are given by Eqs.( 43) and ( 45) for both crystals. In addition, the equivalent elastic rotation vector e Ω is introduced such that (57) e e , ∀ = × l ω l Ω l In the following, only an analytical formula for its jump, ⎡ ⎤ ⎣ ⎦ e Ω , is derived, which allows giving further insights on the lattice misorientation angle between two crystals, assuming their initial orientation is known. From Eq.( 25), the jump of the tangential elastic distortions is known (58) 2,

e p ij ij j ⎡ ⎤ ⎡ ⎤ ∀ ≠ = - ⎣ ⎦ ⎣ ⎦ β β
The normal components can be deduced from the jump of elastic strains which are known from the Hooke's law (Eq.( 2)). In particular, we have (59)

⎧ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = - = + ⎪ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎨ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = - = + ⎪ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎩ β ε β ε β β ε β ε β
Then the jumps of lattice rotations are accordingly derived

(60) ( ) ( ) 1 2 3 2 3 2 3 2 1 3 3 1 1 3 3 1 2 1 2 2 1 Ω 1 Ω 2 Ω e e
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = - = + ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = = - ⎨ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = - = - + ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎩ ω ε β ω β β ω ε β with (61) ( ) ( ) ( ) ( ) [ ] [ ] [ ] [ ] [ ] [ ] ( ) ( ) ( ) ( ) [ ] [ ] [ ] [ ] [ ]
⎡ ⎤ = -Σ + -Σ + -Σ ⎣ ⎦ + + + + Σ + Σ + Σ ⎡ ⎤ = -Σ + -Σ + -Σ ⎣ ⎦ + + + + Σ + Σ + Σ ε σ σ σ σ σ σ ε σ σ σ σ σ σ 3 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
In comparison with the stresses, the present solutions are only independent of compliance

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = + + ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ s β s β with - ( ) Δ p Ω Ω 0 e e i i ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = = ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎛ ⎞ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = - + ⎜ ⎟ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ s β s β
, term due to elastic-plastic coupling incompatibilities ( )

p Δ Δ s β .
We can notice in Eq.( 60) that only

1 Ω e ⎡ ⎤ ⎣ ⎦ and 3 Ω e ⎡ ⎤ ⎣ ⎦ depend on the three sources of incompatibilities.

2 Ω e ⎡ ⎤ ⎣ ⎦ is independent of elastic constants and results only from plastic incompatibilities (see in Appendix C the full expressions of the spin vector jump decomposition).

It is interesting to relate Eq.( 60) to the well-known relation established by [START_REF] Kröner | Continuum theory of defects in physics of defects[END_REF] which expresses the gradient of elastic rotation vector in function of elastic strains and Nye's tensor α where I is the unit tensor. Indeed, this equation is expressed in the bulk but in the limit of a vanishing thin planar interface of discontinuity with unit normal n , it becomes (see the proof of this argument in Appendix D)

(64) ( ) ( ) ( ) T T e S S 1 tr 2 ⎡ ⎤ ⎡ ⎤ ⊗ = - × - + ⎣ ⎦ ⎣ ⎦ e Ω n ε n α α I
The result of Eq.( 60) is then exactly retrieved from Eq.( 64) when the former is applied to the configuration of Fig. 2 (the expression of S α being given in Eq.( 25)).

Heterogeneous isotropic elasticity

In the case of heterogeneous isotropic elasticity, many compliance tensor components vanish (65) ( )

1 2 3 2 3 2 3 1 1 3 3 1 2 2 1 1 1 Ω Σ 2 μ 1 Ω 2 1 1 Ω Σ 2 μ e p e p p e p ⎧ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = + ⎪ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = - ⎨ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎛ ⎞ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = - + ⎪ ⎜ ⎟ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎪ ⎝ ⎠ ⎩ β β β β
and the solutions do not depend anymore on coupling incompatibilities.

Homogeneous isotropic elasticity

In the case of homogeneous isotropic elasticity, Eq.( 60) further simplifies into (66) ( )

1 2 3 2 3 1 1 3 3 2 1 Ω 1 Ω 2 Ω e p e p p e p ⎧ ⎡ ⎤ ⎡ ⎤ = ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = - ⎨ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ = - ⎣ ⎦ ⎣ ⎦ ⎩ β β β β
All the components are then independent of elastic constants.

Consequence of plastic distortion tangential continuity

If the tangential continuity of the plastic distortion, as described in Section 3.3 (Eq.( 20)), is assumed to be satisfied at the interface of Fig. 2

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = = = = = = ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ β β β β β β
In the case of piecewise uniform plastic distortions, it conducts to (68)

Δ Δ Ω Ω ij ij ij e e i i ⎧ = Σ + ⎪ ⎨ ⎡ ⎤ ⎡ ⎤ = ⎪ ⎣ ⎦ ⎣ ⎦ ⎩ s s σ σ
which means that jump of internal stresses and lattice misorientations should be related to elastic incompatibilities only. As a consequence, stresses and lattice misorientations should only develop in an elastic way and thus should recover to zero when the macroscopic uniform stress Σ is unloaded to zero (no residual stress and no residual lattice misorientation). In addition we should have in particular

2 Ω 0 e ⎡ ⎤ = ⎣ ⎦ .
In homogeneous elasticity, the tangential continuity of the plastic distortion leads to ( 69)

Ω 0 ij ij e i = Σ ⎧ ⎪ ⎨ ⎡ ⎤ = ⎪⎣ ⎦ ⎩ σ
Hence, it introduces the continuity of both stresses and lattice rotations. [START_REF] Mach | Continuity in the plastic strain rate and its influence on texture evolution[END_REF] already showed that the tangential jump condition of Eq.( 20) can lead to the continuity of lattice rotations but in the case of a pure viscoplasticity. This result is indeed retrieved from Eqs.( 60) and ( 67) if jumps of elastic strains are neglected. In their paper, [START_REF] Mach | Continuity in the plastic strain rate and its influence on texture evolution[END_REF] noticed also that in the presence of elastic strains, rotations are no more continuous. Concerning the prediction of texture evolution, it thus appears that an assumption of homogeneous elasticity may actually be equivalent to the hypothesis of a pure viscoplasticity (also true without considering the tangential continuity of the plastic distortion, cf. Eq.( 66) and Eq.( 60) without elastic strains).

Finally, it should be recalled here that Eqs.( 68) and ( 69) were established in the peculiar case of piecewise uniform plastic distortions. Hence, the tangential continuity of the plastic distortion actually induces the tangential uniformity of plastic distortions, which further leads to a development of stresses and lattice misorientations due only to elastic heterogeneous properties. In the more realistic case of tangentially continuous but not necessarily uniform plastic distortions (which is not the scope of the present paper), stress and lattice misorientation evolutions should be very much related to the limiting values of the tangential plastic distortions at the interface as seen for instance from numerical simulations in [START_REF] Richeton | Continuity constraints at interfaces and their consequences on the work hardening of metal-matrix composites[END_REF] and [START_REF] Taupin | Size effects on the hardening of channeltype microstructures: a Field Dislocation Mechanics-based approach[END_REF].

Application to bicrystals deformation

The first part of this section aims at checking the correctness of the analytical formulas established in Section 4 from Crystal Plasticity Finite Element simulations and then at showing the relative contribution of the different sources of incompatibilities. The second part of the section deals with elastic/plastic bicrystals and displays the values of residual stresses and residual lattice misorientations for different orientations of the elastic crystal. Systematic comparisons with the isotropic approximations are also performed.

Finite element modelling of a bicrystal with infinite planar interface

5.1.1 Description of the crystal plasticity model A 3D finite element mesh of 20-node brick elements with reduced integration (C3D20R) is used for the discretization a bicrystal bar (10 x 100 x 10 elements, see Fig. 3). This bicrystal bar is designed to mimic the configuration of Fig. 2. Periodic boundary conditions are indeed imposed along the lateral sides. The two crystals are thus separated by an infinite planar interface with normal along (O,x 2 ) (Fig. 3). Fixed velocities are prescribed on the upper and bottom face (Fig. 3). Besides, a simplified single crystal plasticity model inspired by the work of [START_REF] Peirce | Material rate dpendence and localized deformation in crystalline solids[END_REF] is implemented at small strains within a Finite Element commercial software (ABAQUS/Standard). For the sake of further analyses'clarity, our model assumes that plasticity results solely from the possible activation of a single slip system (hence, there is no consideration of latent hardening). The plastic velocity gradient tensor p L is then conventionally expressed as (70

) p γ = ⊗ L s m &
where s and m are the slip direction vector and glide plane normal, respectively. The plastic slip rate on the single slip system γ& is given by the Orowan's law,

m d bV γ ρ = & . m ρ
is the density of mobile dislocations, which is here assumed to remain constant. b is the magnitude of the Burgers vector and d V is the averaged collective dislocation velocity on the slip system. d V is assumed to follow the power law relationship (71) ( )

0 sgn n d h V V τ τ σ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠
where :

τ = ⊗
s m σ is the resolved shear stress on the glide plane, with reference velocity 0 V and stress exponent n as material parameters.

( )

sgn 1 τ = when 0 τ > and
( )

sgn 1 τ = - when 0 τ < .
The reference shear stress h σ reflects short range obstacle overcoming for dislocations. In this simplified model, the reference shear stress is allocated a stationary value, meaning that no isotropic hardening is considered. The implementation of this model is made through the use of the rate tangent modulus method developed by [START_REF] Peirce | A tangent modulus method for rate dependent solids[END_REF]. A linear interpolation within the time increment Δt is thus employed to define the increment of plastic shear strain Δγ (72)

( ) Δ Δ Δ 1 t t t t + ⎡ ⎤ = - + ⎣ ⎦ γ χ γ χ γ & &
where the time integration parameter χ ranges from 0 to 1 with χ = 0 corresponding to the Euler forward explicit integration scheme. Here, a classical value of 0.5 is chosen (see [START_REF] Peirce | A tangent modulus method for rate dependent solids[END_REF]. Cubic symmetry elastic stiffnesses for copper are given in Table 1 and are considered to run the simulation. Other material parameters are given in Table 2. On the upper and bottom face, the following material velocities are prescribed (73

) 2 1 3 2 v and v v 0 along L x L = ± Γ = = = ± &
where Γ & =10 -3 s -1 is the initial applied macroscopic strain rate and L =50 µm is the initial length along (O,x 2 ) of each crystal. It should be noted that this loading test does not correspond to a pure tensile test since all the components of the macroscopic stress tensor are non zero. Indeed, constraining the cross-sectional contraction of the material by prescribing zero transverse velocities at the sample ends ( 1 3 v v 0 = = ) leads to transverse reaction forces. The component 22 Σ remains however the prevailing one close to a pure uniaxial tensile load. The crystals orientations are indicated in Table 2. Crystal I has the crystallographic direction [ ] 123 along the tensile axis, which corresponds to a typical single slip orientation for a fcc single crystal [START_REF] Takeuchi | Work hardening of copper single crystals with multiple glide orientations[END_REF], therefore justifying our model's simplifying assumption (Eq.( 70)). Plasticity in crystal I thus solely results from dislocation glide on the system 101 111 ⎡ ⎤ ⎣ ⎦ . The threshold shear stress h σ is affected by a very huge value in crystal II, so that the behavior of crystal II remains purely elastic during the small range of tensile strains (< 0.2 %) investigated in the FE simulation. This consideration is justified by the chosen orientation for crystal II (Table 2). In the elastic domain, crystal II exhibits much lower resolved shear stresses on its slip systems than the shear stress on system 101 111 ⎡ ⎤ ⎣ ⎦ in crystal I. As a consequence, crystal II should normally be in a plastic state much later on than crystal I.

Validation of the explicit closed-from solutions

The infinite planar interface under study thus separates a crystal undergoing elasticity and single slip plasticity from a crystal of the same composition but with a different orientation and which undergoes only elasticity. Fig. 4 exhibits a comparison between the stresses in crystal II provided by the FE simulation ( num ij σ ) and those directly derived from the expressions of Eq.( 45) ( theo ij σ ). It is noteworthy that the former formulas are static whereas the FE calculations are a consequence of an incremental procedure based on an evolving viscoplastic strain as defined in Eqs.( 70) to (72). theo ij σ is actually computed from Eq.( 45) with the overall stress Σ and the jumps of plastic strain jump p ⎡ ⎤ ⎣ ⎦ ε taken from the FE calculations and updated after each time increment. In the same way, Fig. 5 shows a comparison between the lattice misorientations provided by the FE simulation ( e num i ⎡ ⎤ ⎣ ⎦ Ω )

and those directly derived from the analytical formulas ( e theo i ⎡ ⎤ ⎣ ⎦ Ω computed from Eq.( 60)

with Σ and the jump of plastic distortions p ⎡ ⎤ ⎣ ⎦ β as inputs). For both cases, it can be seen that the FE solutions match perfectly the analytical ones (the differences being within the numerical precision of the computer) thus giving a proof of the correctness of the formulas established in Section 4.

Relative contribution of the different sources of incompatibilities

From the evolution of Σ and p ⎡ ⎤ ⎣ ⎦ β provided by the FE simulation, our analytical developments enable to quantify the contribution of the different incompatibility sources on stresses and lattice misorientations as exposed in Eqs.( 46) and ( 62). Examples of these decompositions are given in crystal II for 11 σ and 1 Ω e ⎡ ⎤ ⎣ ⎦ in Figs. 6 and7, taking the plastic crystal (crystal I) as the reference medium. For the present elastic/plastic bicrystal, the major contribution arises from the terms due to plastic incompatibilities alone

( p Δ ij ε σ , p Δ Ω e i ⎡ ⎤ ⎣ ⎦ β
). However, it can be observed that the other terms (

Δ ij s σ , p Δ Δ ij s ε σ , Δ Ω e i ⎡ ⎤ ⎣ ⎦ s , p Δ Δ Ω e i ⎡ ⎤ ⎣ ⎦ s β
), which would be absent in homogeneous elasticity, still represent a significant part. It can notably be inferred that the coupling term between elastic and plastic incompatibilities might exhibit huge contributions for some peculiar configurations. For instance, in Fig. 7, contribution of coupling incompatibilities on

1 Ω e ⎡ ⎤ ⎣ ⎦ is quite consequent and opposite in sign with respect to the one due to plastic incompatibilities alone. Therefore, heterogeneous elasticity may influence strongly on texture developments in the course of plastic deformation in polycrystalline aggregates. This aspect is often neglected in the crystallographic texture developments simulated by polycrystalline models (Self-Consistent models, Crystal Plasticity Finite Element models or FDM models restricted to pure viscoplastic behaviour or elastic-viscoplastic behaviour with homogeneous isotropic elasticity).

Analytical predictions of residual stresses and lattice misorientations

The same bicrystal than in Section 5.1 is used as a starting configuration. While the plastic crystal (crystal I) remains fixed, the elastic one (crystal II) is subsequently authorized to rotate around (O,x 2 ) by an angle θ . Indeed, in case of a tensile test, the resolved shear stresses on slip systems remain unchanged when the crystal is rotated around the tensile axis. It can thus be assumed that crystal II undergoes pure elastic distortions for any value of θ , providing that the tensile strain remains low enough. Then, assuming a given low value of plastic slip (results are shown for γ = 0.01) for the system

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = = = ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ε ε ε ε ε ε in
Eq.( 44)). In the case 0 = Σ , we have precisely ⎣ ⎦ in function of θ for γ = 0.01. Results derived from the general consideration of heterogeneous anisotropic elasticity are compared with solutions derived from isotropic approximations (the isotropic average elastic constants are computed from a Voigt average [START_REF] Hirth | Theory of dislocations[END_REF], see Table 1) which involves the dying out of the coupling terms in Eq.( 74). Results are also displayed for two fcc materials, Cu which strongly anisotropic (anisotropic coefficient ( )

(74) p p p p Δ ΔΔ Δ Δ Δ Ω Ω Ω ij ij ij e e e i i i ⎧ = + ⎪ ⎨ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = + ⎪ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎩ ε s ε β s β σ σ σ
44 11 12 a 2 3.26 c c c = = -
), and Al which is quite less ( a 1.22 = , see Table 1). The difference between Cu and Al in stresses and in the density of stored elastic energy (Figs. 8 and9) arises solely because of different values of elastic coefficients in crystal I. Isotropic assumption leads however to the same solutions for Cu and Al concerning the lattice misorientation vector, since the former do not depend on elastic constants in this case (Eq.( 66)). In the general heterogeneous anisotropic case, lattice misorientations do depend on elastic constants, as it is illustrated in Fig. 10 where a consequent difference between Cu and Al is observed. In the same way, concerning stresses and stored elastic energy density, Figs. 8 and9 show that the discrepancy between isotropic and anisotropic solutions is much more pronounced for Cu than for Al. From the insets of Figs. 8, 9 and 10, it is seen that the relative error made by the isotropic approximation is strongly dependent on the orientation of crystal II and can reach values above 40% in Cu. Strong discrepancies can occur for both stresses and lattice misorientations. For 1% slip, the rotation of the elastic crystal around the tensile axis can lead in Cu to a stress variation reaching 50 MPa (Fig. 8) and a lattice rotation variation reaching 0.1° (Fig. 10).

As a consequence, considering elastic anisotropy is crucial not only to predict stresses with good accuracy but also to predict correct texture evolution. Besides, several crystals may display higher elastic anisotropy than pure Cu (e.g., a 8.91 = for β-brass or a 12.5 = for a Cu-14wt.% Al-4.2wt.% Ni alloy, see Suezawa and Sumini, 1976). This study also illustrates the role that elastic anisotropy may have on strain path dependence in polycrystals. In the course of a Bauschinger test for instance, fields due to pure elastic incompatibilities will recover to zero when 0 = Σ [START_REF] Yaguchi | Grain boundary contribution to the Bauschinger effect in Beta-Brass bicrystals[END_REF] but fields due to elastic-plastic coupling incompatibilities will remain in addition to fields due to pure plastic incompatibilities. It must be noted that elastic incompatibilities may nevertheless influence the Bauschinger effect by contributing to the selection of slip systems and affecting their activity around the grain boundary [START_REF] Vehoff | The effect of hydrogen and segregation on fatigue crack nucleation at defined grain boundaries in nickel bicrystals[END_REF][START_REF] Yaguchi | Grain boundary contribution to the Bauschinger effect in Beta-Brass bicrystals[END_REF]. Furthermore, in fatigue tests, elastic incompatibilities alone can induce some plasticity near the grain boundary [START_REF] Vehoff | The effect of hydrogen and segregation on fatigue crack nucleation at defined grain boundaries in nickel bicrystals[END_REF][START_REF] Yaguchi | Grain boundary contribution to the Bauschinger effect in Beta-Brass bicrystals[END_REF]. As a result of cumulative plastic strain, plastic and coupling incompatibilities also develop and provoke stress concentrations which may finally lead to crack initiation.

Conclusions

The present paper considered the issue of bicrystal elasto-plasticity assuming an infinite planar grain boundary. The bicrystal is considered as a heterogeneous elasticplastic solid undergoing piecewise uniform plastic distortions (where plasticity is not restricted to be incompressible) and elastic properties. Explicit closed-form solutions of stress and lattice misorientation fields in the general context of heterogeneous anisotropic elasticity (with and without plastic distortion tangential continuity) were derived by use of the concepts of Field Dislocation Mechanics in the context of Frank-Bilby interfacial dislocation. The obtained expressions were validated by Crystal Plasticity Finite Element simulations on a bicrystal with periodic boundary conditions. Analytical solutions made it possible to quantify the contributions of the different incompatibility sources, namely the contribution arising from elastic incompatibilities alone, that due to plastic incompatibilities alone and the one due couplings between elastic and plastic incompatibilities. Applications were performed on infinite elastic/plastic bicrystals for different orientations of the elastic crystal. Values of residual stresses and lattice misorientations were compared with the isotropic approximations. In most situations, plastic incompatibilities had the most prominent effects but elastic and even more coupling incompatibilities showed significant contributions depending on the material elastic constants, the relative orientation of both crystals and the external loading. However, it must be recalled here that the study was conducted in single slip plasticity only as it was restricted to small value of plastic slip. With increasing plastic slip, allowing for multiple slip systems activity and latent hardening will strongly affect the evolution of Σ and p ⎡ ⎤ ⎣ ⎦ β .

The validity of the obtained analytical expressions will still remain under the assumptions considered (i.e. linear anisotropic elasticity and homogeneous plastic distortion within each crystal) but the relative part of the different incompatibility terms of σ and e ⎡ ⎤ ⎣ ⎦ Ω may evolve differently from the results exposed in Figs.6 and 7. In the same way, plastic accommodation on different slip systems could modify the discrepancy between the anisotropic solutions and the isotropic approximations (Figs.8, 9, 10) with increasing plastic slip.

Only plastic and elastic-plastic coupling incompatibilities do contribute directly to residual fields. Under load however, all incompatibility types may affect the selection of slip systems and therefore the distribution of plastic strains. Because of elastic-plastic coupling incompatibilities, the necessity to consider elastic anisotropy in models for accurate prediction of residual textures is also demonstrated. Actually, it has been shown in the present paper that an assumption of homogeneous elasticity restricts to the hypothesis of pure viscoplasticity. Finally, as discussed by [START_REF] Gemperlova | Compatibility stresses in deformed bicrystals[END_REF], it must be underlined that the expressions of stresses and lattice misorientations obtained in an infinite bicrystal should be valid in a finite bicrystal only in a thin region adjacent to the interface and far from the finite boundaries and edges like free surfaces. In polycrystalline aggregates, these field expressions might give some correct estimates close to grain boundaries and far from triple junctions. In addition, it should be mentioned that they remain valid if the interface considered is a twin or a phase boundary. Therefore, the derived formulas are expected to be also of some use in dealing with twinning or multiphase alloys deformation and fatigue issues. 53) are first formulated (with the same notations as in the main text)

β β β β β β β β β β β β β β β β β β β β β β β β β β ⎛ - - ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ - - - - - - - - - - - = 22,
β β β β β β β β β β ⎛ ⎞ ⎛ ⎞ ⎜ - ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ - - - ⎝ ⎠ - ⎠ ⎝ Appendix B.
(B.1) ( ) ( ) ( ) ( ) ( ) ( ) (
⎤ - + - ⎣ ⎦ ⎡ ⎤ - + - + - ⎣ ⎦ = - ⎡ ⎤ - + - + - ⎣ ⎦ = - % % % % % % % % % % % % % % % % % % % % % % % % % % % % ( ) ( ) ( ) ( ) ( ) (
⎤ + - + - ⎣ ⎦ ⎡ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎤ - + - + - ⎣ ⎦ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = ⎪ ⎩ = =- % % % % % % % % % % % % % % % % %
B B D ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ - - + - + - = = ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ = % % % % % % % % % % % %
Lastly, from Eq.( 45), the non zero stress influence tensor components

F F F F F F F F F F F F F F F F s s s D s s s D s s s D s s s s D s s s s D s s s s D ⎧ - = ⎪ ⎪ ⎪ - = - = = = = ⎨ - = = - = = = = - = = = = % % % % % % % % % % % % % % % % % % % % % ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Appendix C. Full explicit expressions of the spin vector jump decomposition The starting point is the expression of the surface-dislocations density tensor S α given for small displacements by Eq.( 18). Decomposing the elastic distortion into its symmetric and skew-symmetric parts leads to (D.1) Taking the transposed of this equation, we finally retrieve the expression of Eq.(64). To the authors' knowledge, this expression and the attached proof was not given in the literature. Eq.( 46)).

(C.1) [ ] ( ) [ ] ( ) [ ] ( ) [ ] [ ] [ ] ( ) ( ) [ ] ( ) [ ]
⎧ ⎡ ⎤ ⎡ ⎤ = + + + ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = - ⎨ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ = - + + + ⎣ ⎦ ⎣ ⎦ ⎪ ⎩ β ε ε ε β β ε ε ε σ σ σ β β β σ σ σ β (C.

Figures captions

Cubic stiffnesses

  .(3)a-b,[START_REF] Acharya | Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model mesoscopic field dislocation mechanics: Part I[END_REF] explicitly used the following unique Stokes-Helmholtz decomposition for the elastic distortion tensor: where u is the total displacement field,u z is a vector field whose gradient forms the compatible part of e β (denoted e ) β and χ is the incompatible part of e β (denoted e ) ⊥ β such that curl = α χ to be consistent with Eq.(3)d.

  resulting term when all sources of incompatibilities are set to zero (in this case, one rightly retrieves that the stress equals the macroscopic uniform stress), -

  In the case of piecewise uniform isotropic elasticity (i.e heterogeneous elasticity) characterized for crystal I (resp. crystal II) by Young modulus I

  .1.1.3 Homogeneous isotropic elasticityIn the case of homogeneous and isotropic elasticity now characterized by uniform Young

  the elastic stress concentration tensors for crystal I (resp. crystal II), and' PP Fis the non zero part of the elastic stress influence tensor. From overall stress equilibrium (Eq.(35)), the elastic stress concentration tensors verify

  anisotropic elasticityThe lattice rotations are linked to the elastic rotation e ω , defined as the skew-symmetric part of the elastic distortion tensor

Figs

  Figs. 8, 9 and 10 display respectively the evolution of residual stress components 11 σ , 33 σ ,

  Full explicit component expressions of the elastic concentration and influence tensors obtained from Eq.(45) Following Eq.(45), the elastic stress concentration tensor components of

  Figur unit v

Figure 3 .Figure 4 .

 34 Figure 3. Sketch of the bicrystal bar used in the FE simulation.

Figure 5 .

 5 Figure 5. Comparison between the lattice misorientations provided by the FE simulation ( e num i ⎡ ⎤ ⎣ ⎦ Ω ) and those directly derived from the analytical formulas ( e theo i ⎡ ⎤ ⎣ ⎦ Ω).

Figure 6 .

 6 Figure 6. Contribution of the different incompatibility source terms for 11 σ in crystal II (cf.

Figure 7 .

 7 Figure 7. Contribution of the different incompatibility source terms for

Figure 8 .

 8 Figure 8. Evolution of the residual stress components 11 σ , 33 σ , 31 σ in crystal II with the angle θ for γ = 0.01. Crystal I remains fixed while crystal II can rotate around (O,x 2 ) by the angle θ . Inset: relative error made by the isotropic approximation for Cu.

  )

	⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩	11 33 13 K K K	= = +	11 33 p p ⎡ ⎤ 31 2 1 2 1 K ⎣ ⎦ ⎡ ⎤ 31 p + + ⎣ ⎦ ⎡ ⎤ = ⎣ ⎦ ε ε ε	II 11 II 11 15 s σ σ 1 13 11 2 s s + ( σ + + 11 s 13 33 s II +	II 33 II 33 σ σ 35 33 + + II s σ	II 31 II 31 σ σ II 55 31 15 35 s s s + σ	I 11 11 I 13 11 Σ Σ s s + + s 13 33 I I 33 33 Σ Σ s + + s 15 31 I I 35 31 Σ Σ s ) ( 36 23 16 23 Σ Σ s s + + + 34 12 14 12 Σ Σ s s + + 23 22 12 22 Σ Σ s s + + + % % % % % % ) I I I 25 22 45 12 56 23 15 11 35 33 55 31 Σ Σ Σ Σ Σ Σ s s s s s s + + + + + + % % %
	From Eq.(33), the unknown elastic distortion components in crystal II are expressed as (40) ( II II II II II II II II II II 11 11 11 13 33 15 31 12 22 14 12 16 23 Σ Σ Σ e s s s s s s ) ( II II II II II II II II II II 33 13 11 33 33 35 31 23 22 34 12 36 23 Σ Σ Σ e s s s s s s ) ( ) II II II II II II II II II II 31 15 11 35 33 55 31 25 22 45 12 56 23 2 Σ Σ Σ e s s s s s s

  interesting to notice that these solutions are independent of compliances 22

								s , 24 s , 26 s ,
	44 s , 46 s , 66 s and of plastic strains 12 p ε , 22 p ε , 23 p ε . It can be seen that the stresses contain
	different sources of incompatibilities as highlighted in the general static FDM theory
	considered in Section 3.2. If one writes	ij s %	=	2	ij s	I	ij s ⎡ ⎤ + ⎣ ⎦	and considers crystal I as the
	reference medium, it is indeed possible to decompose the stress field into the following
	way
	(46)						p	p

  )

			I			2		I					I
	II		11		33 55	35		13	15 35	13 55	15	13 35	33 15
	1111
			I			2		I					I
	II		13		33 55	35		33	15 35	13 55	35	13 35	33 15
	1133
	I 15 15 35 s s II ( 2 33 55 35 I I 11 13 55 13 II 3311 2 s s s s B ⎡	I 35 11 55 s s D	15 s 15 D ) ( 2 I 15 s 35 13 55 s s 13 15 55 I s s s s	11 35 s s 13 35 s s	)	33 15 s s
	3333 B II	2	I 13 s	(	15 35 s s	13 55 s s	)	I 33 s	(	11 55 D s s	15 s	2	) ( I 35 s	13 15 s s	11 35 s s	)
	II 3313 B

  )

				I					I			2		I
		II		15	15 35	13 55		35	11 55	15			55	13 15	11 35
		3331
				I					I					I				2
	II	II		11	13 35	33 15		13	13 15	11 35		15		11 33		13
	1311	3111
	II	II	⎡	I 13	13 35	33	15		I 33 s	13 15 s s	11 35 s s		I 35 s		11 33 s s		13 s	2
	1333	3133								D
	II 1313 B	II 1331 B	II 3113 B		II 3131 B	2	I 15 s	(	13 35 s s	33 15 s s	)	I 35 s	(	13 15 s s D	11 35 s s	)	I 55 s	(	11 33 s s	13 s	2	)

Table 1 .

 1 Values of elastic stiffnesses for Cu and Al.

Table 2 .

 2 Material parameters used in the FE simulation.

Acknowledgements

TR is grateful to the French Agence nationale de la Recherche (ANR) for financial support under contract "PHIRCILE" (ANR 2010 JCJC 0914 01).

Appendix A. Development of the compatibility conditions

The full development of the compatibility conditions (


Harikrishnan Jayaprakash

Liwei Guo

Shengdong Wang

Dr Christian Bruneau

Cédric Fischmeister
email: cedric.fischmeister@univ-rennes1.fr

M Harikrishnan Jayaprakash

M Liwei Guo

Dr M Shengdong Wang

Acceptorless and base-free dehydrogenation of alcohols mediated by a dipyridylamine-Ir(III) catalyst

Keywords: dehydrogenation, alcohol, acceptorless, base-free, iridium

Several dipyridylamine-Ir(III) and dipyridylamine-Ru(II) complexes have been evaluated in the acceptorless dehydrogenation of alcohols in the absence of base additives. Iridium catalysts were found superior to ruthenium complexes and the nature of the bridging nitrogen in dipyridylamine ligands was also evidenced as a key parameter. Catalytic reactions were conducted in toluene but more sustainable solvents such as anisole and pcymene were found suitable for this transformation.

Introduction

The catalytic dehydrogenation of alcohols [1] often quoted as oxidant free oxidation is for many reasons a very appealing oxidation process. Following the pioneering studies made by Robinson, [2] this reaction was quickly identified as a valuable route for the production of hydrogen. [3][4][5] The dehydrogenation of alcohols for the production of carbonyl compounds is more recent and has found numerous interesting applications in tandem and hydrogen-borrowing reactions for the synthesis of organic compounds [6] but also as a catalytic tool for the production of (bio)-alcohols via the Guerbet reaction. [7] The oxidation of alcohols through acceptorless dehydrogenation is a clean process as dihydrogen is the unique coproduct of the reaction. This reaction proceeds therefore with high atom economy unlike many stoichiometric oxidants (Cr and Mn compounds, Dess-Martin, Swern reagents..) used in organic synthesis and it does not require any oxidant (O2, H2O2, RCO3H,…) when compared to other catalytic oxidation processes. A number of transition metal catalysts are known to promote the acceptorless dehydrogenation of alcohols [8][START_REF] Fujita | Iridium catalysts operating under base[END_REF][10][START_REF] Van Buijtenen | Ruthenium and osmium catalysts operating under base-free conditions[END_REF][START_REF] Wang | Rhodium catalyst[END_REF][13][START_REF] Song | Iron catalysts[END_REF][15] leading to ketone from secondary alcohols whereas primary alcohols do not always deliver the expected aldehydes and rather furnish ester derivatives. Among the catalyst reported so far, those operating under base-and additive-free conditions [START_REF] Fujita | Iridium catalysts operating under base[END_REF][START_REF] Van Buijtenen | Ruthenium and osmium catalysts operating under base-free conditions[END_REF]13,15] are the most attractive as they generate less wastes and they tolerate reagents bearing base-sensitive functional groups. A survey of the catalysts operating under base-free conditions shows that bifunctional catalysts bearing basic ligands such as pincer, carboxylate and other chelating ligands are employed. Alternatively, Kanai reported an original process for the room temperature acceptorless dehydrogenation of alcohols using a hybrid ternary catalytic system. [16] Recently, we have been interested in the synthesis of catalysts bearing chelating dipyridylamine ligands [17,18] in a number of catalytic transformations. 2,2-Dipyridylamine is a non-expensive commercially available compound and 2,2'-dipyridylamine derived ligands are easily accessible using simple synthetic procedures. We have developed a number of dpa-based catalysts that showed very good performances in hydrogenation reactions [19] and base-free dehydrogenation of formic acid [20] and nitrogen heterocycles. [21] These previous studies clearly demonstrated the high efficiency of this type of catalysts in hydrogenation and dehydrogenation processes and prompted us to investigate their potential in the acceptorless and base-free dehydrogenation of secondary and primary alcohols.

Results and Discussion

The iridium-chloro catalysts Ir1-3 and the zwitterionic iridium complexes Ir4-5 were first evaluated as catalysts for the dehydrogenation of the model substrate 1-phenylethanol (Figure 1). These catalysts were first screened in the base-assisted dehydrogenation of 1-phenylethanol at 130 °C in toluene (Scheme 1). This first set of experiments revealed the modest activity of these catalysts in particular for Ir1, Ir4 and Ir5 and demonstrated that Ir-Cl complexes Ir2 and Ir3 were more efficient than their Ir-OSO3 counterparts. This result contrasts with our previous studies on the dehydrogenation reactions of formic acid and N-heterocycles for which Ir-OSO3 catalysts surpassed their Ir-Cl counterparts. [20,21] The nature of the reagent may of course have a strong influence on the catalytic pathway but other parameters should also be considered as key parameters. Indeed, previous studies on formic acid and Nheterocycle dehydrogenations were conducted in water where ligand exchange (OSO3/H2O) was likely proceeding whereas decoordination of the -OSO3 ligand might be more difficult in toluene. These results also emphasized the deleterious influence of a bridging N-H vs N-R in the dipyridylamine ligand (Ir1 and Ir5) hence also contrasting with the dehydrogenation of formic acid where the N-H moiety was necessary to ensure the best catalytic performances. [20] Scheme 1 Dehydrogenation of 1-phenylethanol (Conversion determined by gas chromatography)

Another parameter to consider to rationalize these results is the reaction pH. Whereas the dehydrogenation of formic acid was performed under acidic conditions, [20] the dehydrogenation of 1-phenylethanol is conducted under alkaline media likely leading to the deprotonation of the bridging N-H hence transforming the neutral dipyridylamine ligand into an anionic dipyridylaminate [22] and consequently to a new catalytic species. However, as reported hereafter in the base-free dehydrogenation study, this hypothesis may not be the unique answer to the poor activity of Ir1 and Ir4 since these two complexes presented again poor performances under base-free conditions (vide infra) Having identified Ir2 and Ir3 as the most potent catalysts for the dehydrogenation of 1-phenylethanol, we turned our attention to the nature of the base. As depicted in Table 1, none of the inorganic bases tested led to better results than Cs2CO3. The same conclusion was obtained when Ir3 was evaluated with the same inorganic bases (see ESI). As emphasized in the introduction, base-free dehydrogenation of alcohols is strongly desired as these conditions guarantee the integrity of base-sensitive functional groups while minimizing wastes production. The dehydrogenation of 1-phenylethanol was attempted under base-free conditions and considering the issues raised here above, i.e. deprotonation of bridging N-H in dipyridylamine ligand, all the catalysts were again evaluated under these pH neutral conditions. Our investigations demonstrated that the base-free dehydrogenation of 1-phenylethanol could be achieved with high conversion provided a higher reaction temperature and a longer reaction time were implemented. All reactions were conducted under a flow of argon as a simple open system failed to furnish good results (see ESI). This experimental observation is consistent with a gas (H2) release from the reaction medium. Of note, Ir2 led to full conversion of 1-phenylethanol in refluxing toluene for 24 h (Table 2). It should be noted that despite the neutral condition, Ir1, and Ir4 bearing the NH-dipyridylamine ligand were yet inefficient as the zwitterionic Bn-dipyridylamine Ir5. These results confirmed that zwitterionic complexes were not suitable for this transformation and that the NH-dipyridylamine ligand was also unsuitable. In the latter case this result can no longer be explained by the deprotonation of the NHdipyridylamine ligand but may be related to the lower electron density at the metal in Ir1 vs Ir2 and Ir3, which was identified as an important parameter for the dehydrogenation of formic acid. [20] This result provides some hints for a possible reaction mechanism in which the basicity of the dipyridylamine ligand would be a key parameter. Entry Catalyst Conversion (%) [b] 1 Ir1 7 2 Ir2 100 (94) [c] (97) [d] 3 Further tests were performed with Ir2 in order to identify the optimized reaction conditions (Table 3). This series of experiments demonstrated that 2 mol% of catalyst was the lowest catalyst loading acceptable to ensure high conversion in a reasonable amount of time and that a 1 M concentration of substrate was also an important requirement (Table 3, entries 4-6). Finally, greener and more sustainable solvents [23] such as anisole and p-cymene were evaluated and found to be suitable for this transformation whereas the reaction conducted in cyclopentylmethyl ether (CPME) led to poor conversion of 1phenylethanol (Table 3, entries 7-9). [a] Reaction conditions: 1-phenylethanol (40 mg, 0.33 mmol), Ir2, solvent, 140 °C (oil bath temperature), 24 h, argon flow. [b] Determined by gas chromatography. [c] CPME (110 °C) [d] Anisole [e] p-cymene

With respect to other transition metals, iridium-based complexes are in general the most performing catalysts in dehydrogenation of alcohols. However due to the high cost of iridium metal, less expensive transition metals should also be amenable to perform this reaction. A few examples of ruthenium catalysts promoting the dehydrogenation of alcohols under base-free conditions have been reported. [START_REF] Van Buijtenen | Ruthenium and osmium catalysts operating under base-free conditions[END_REF] They require in general high temperature (up to 165 °C) and long reaction times (up to 5 days) to reach high conversions. The ruthenium catalysts Ru1-3 were evaluated in the base-free and base-assisted dehydrogenation of 1-phenylethanol (Figure 2). Under both type of experimental conditions, these catalysts displayed poor performances (see ESI). These results are in agreement with our previous studies on hydrogenation and dehydrogenation reactions where iridium catalysts always surpassed ruthenium catalysts. The scope and limitations of the base-free dehydrogenation of alcohols with Ir2 was then investigated using a range of secondary alcohols. As depicted in Figure 3, 1-phenylethanol derivatives bearing an electron-donating or an electronwithdrawing group at the para-position were efficiently transformed into the corresponding alcohols (Figure 3, 2a-2d) whereas steric hindrance at the ortho-position was problematic only with an electron-withdrawing substituent. Steric hindrance on the aliphatic side chain also showed some limitations upon moving from an isopropyl to a very bulky t-butyl substituent (Figure 3, 2g-2i). To the best of our knowledge, the synthesis of compound 2i by dehydrogenation of alcohol is unprecedented. Despite full conversion of the hydroxyl-precursor, the synthesis of 2h was hampered by side reactions thus leading to a moderate yield of 55%. Notably, aliphatic secondary alcohols were efficiently converted to the corresponding ketones ((Figure 3, 2k-2l). The challenging dehydrogenation of primary alcohols was also investigated. In this case, 2m was obtained in a modest 50% yield whereas cinnamyl alcohol was fully converted but with poor selectivity as 2 products resulting from partial (2nb) of full reduction (2nc) of the expected product (2na) were detected by GC/MS leading to the isolation of 2na in a modest 45% yield. Based on the experimental results obtained during the screening of the various catalysts implemented in this study, a possible reaction mechanism is presented in Figure 4. As reported earlier in this manuscript, the activity of the catalyst was drastically improved when the dipyridylamine ligand was made of a bridging tertiary amine vs secondary amine. This prompted us to propose a mechanism in which the basicity of the bridging nitrogen was key parameter. Just like other bifunctional catalysts based on pincer ligands, the bridging nitrogen would play the role of a base abstracting the acidic proton of the alcohol thus leading to the possible intermediate b. Subsequent -hydride elimination would release the ketone product leading to c and hydrogen extrusion would then regenerate Ir2. Another mechanism involving the decoordination of one of the pyridine ring that would act as a basic centre is also conceivable. Further experimental and theoretical studies will be necessary to shed light on the mechanism

Conclusions

In conclusion, we have implemented an iridium catalyst for the acceptorless and base-free dehydrogenation of alcohols. This catalyst displays high efficiency in a number of alcohol dehydrogenation reactions performed under experimental conditions similar to those reported until now. However, by contrast to many of the recently reported catalysts based on complex ligand architectures, this catalyst is easily prepared from accessible compounds. Of note, the nature of dipyridylamine ligand, particularly the substitution pattern at the bridging nitrogen turned to be a key parameter, as previously observed in the dehydrogenation of formic acid. We will take advantage of the tunability of the dipyridylamine ligand to further improve the catalyst performances in organic media and in water.

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Figure 1 .

 1 Figure 1. Ir-dpa complexes evaluated in the dehydrogenation of 1-phenylethanol

Figure 2

 2 Figure 2 Ruthenium complexes

Figure 3

 3 Figure 3 Reaction scope with isolated yields (Reaction condition: substrate (0.33 mmol), Catalyst (2 mol%), argon flow. [a] refluxing toluene (140 o C, oil bath temperature), 24 h; [b] refluxing toluene, 17 h; [c] anisole, 150 o C, 24 h)

Figure 4

 4 Figure 4 Proposed mechanism

Table 1 .

 1 Dehydrogenation of 1-phenylethanol with Ir2. Screening of bases

	[a]
	Entry	Base	Conversion (%) [b]
	1	Cs2CO3	45
	2	Na2CO3	29
	3	K2CO3	20
	4	KOAC	20
	5	NaOAc	11
	6	K3PO4	No conversion
	[a] Reaction conditions: 1-phenylethanol (40 mg, 0.33 mmol), Ir2 (4 mg,
	0.006 mmol, 2 mol%), base (15 mol%), toluene (3 mL), 130 °C (oil
	bath), 17 h, argon flow. [b] Determined by gas chromatography. [c] A
	mixture of acetophenone 2 and 1,3-diphenylbutan-1-one was obtained
	in a 34/66 ratio as determined by gas chromatography.

Table 2 .

 2 Base-free dehydrogenation of 1-phenylethanol

[a]

Table 3 .

 3 Experimental condition optimisation

			[a]
	Entry	Catalyst (mol%)	[1] (mol.L -1 )	t (h)	Conversion (%) [b]
	1	2	0.1	24	100
	2	1	0.1	24	85
	3	0.5	0.1	24	54
	4	2	0.1	17	69
	5	2	0.01	17	33
	6	2	1	17	99
	7 [c]	2	0.1	17	12
	8 [d]	2	0.1	17	92
	9 [e]	2	0.1	17	75

Acknowledgements

The authors acknowledge the China Scholarship Council for grants to SW and LG.


Shengdong Wang

Dr Haiyun Huang

Christian Bruneau

A S Wang

Dr C Fischmeister
email: cedric.fischmeister@univ-rennes1.fr

Formic acid as hydrogen source for the iridium-catalyzed reductive amination of levulinic acid and 2-formylbenzoic acid

Formic acid as a hydrogen source for the iridium-catalyzed reductive amination of levulinic acid and 2-formylbenzoic acid

Introduction

Green and sustainable catalytic processes for the transformation of biomass-derived platform compounds has been recognized as a key technology for tackling the problems resulting from the massive use of fossil resources.

1 Levulinic acid (LA) that is produced from carbohydrates or biomass wastes is one of the most valuable platform chemicals with multiple applications. 1g, 2 For instance, levulinic acid is the precursor of γ-valerolactone, a compound that can be used as liquid fuel or food additive 3 and reaction media in numerous transformations. 4 LA is also a precursor of succinic acid via oxidation reactions. 5 Pyrrolidone derivatives are important structural units in organic chemistry which find applications in various domains such as surfactants, intermediates for inks, fibers and drugs.

6

Recently, N-substituted 5-methyl-2pyrrolidones were identified as biosourced substitutes for the extensively used NMP (N-methylpyrrolidone) solvent in the ruthenium catalyzed C-H bond functionalization reaction. 7 Despite the availability of synthetic methods including intramolecular hydroamination, 8 hydroamination-cyclization, 9

N-Alkylation 10 and intramolecular hydroamidation, 11 there is an ongoing interest for the development of greener synthesis of N-heterocyclic compounds in particular those employing renewable resources and environmentally friendly and safe conditions. Since the early report by Frank in 1947, 12 the synthesis of N-substituted 5-methyl-2-pyrrolidones by reductive amination of levulinic acid has been investigated with various heterogeneous 13 and homogeneous 14 catalysts using diverse hydrogen sources. In 2017, we initiated a project based on dipyridylamine-containing organometallic catalysts.

A first generation of ruthenium and iridium complexes enabling the reduction of levulinic acid into γ-valerolactone in the presence of triethylamine was reported. It is noteworthy that two examples of reductive amination of levulinic acid by formic acid and a two-fold excess of benzylamine or o-aniline were described, highlighting some selectivity issues. 15a The same year, we reported the synthesis of second generation catalysts with enhanced performances 15b that were used for the base-free synthesis of a broad scope of N-heterocycles by reductive amination of stoichiometric amounts of levulinic acid and various amines including very bulky derivatives using molecular hydrogen as hydrogen source. 7 In particular, an iridium-dipyridylamine catalyst operated at 110 °C under low hydrogen pressure (5 bar). More recently, we disclosed a third generation of catalyst incorporating an electron-enriched dipyridylamine ligand as very efficient for the dehydrogenation of formic acid 15c and the reversible hydrogenation/dehydrogenation of quinoline derivatives. These results prompted us to investigate the synthesis of Nheterocycles by reductive amination of levulinic acid with amines using formic acid, a hydrogen source less studied than dihydrogen. In 2011, Fu 14a prepared several pyrrolidone derivatives by reductive amination of levulinic acid catalyzed by [RuCl 2 (p-cymene)] 2 and t-Bu 3 PHBF 4 using formic acid. This catalyst showed good performances with alkylamines but much lower activity with arylamines. At the same period, several amines were successfully transformed into pyrrolidones at 130 °C in water with an Au-based heterogeneous catalyst using FA as reductant. 13a In 2013, Xiao reported an iridium catalyst operating at pH 3.5 using a mixture of formic acid and sodium formate 14b while García 13c described Ru-NPs catalyst operating at 120 °C, the latter suffering from selectivity issues. Coordination polymers were also employed for the synthesis one compound (5-methyl-2pyrrolidone) from levulinic acid and ammonium formate at 120 °C, N-heterocycles being synthesized by a one-pot synthesis This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins employing ketones, levulinic acid and ammonium formate.

13d

Most recently, Burtoloso 14i reported the reductive amination of LA using a homogeneous iron catalyst in water. In this study, several pyrrolidone derivatives were obtained in good yields but a very high reaction temperature was necessary, typically 180 °C hence lessening the green character of the process. It must also be mentioned that metal-free transformations were reported by Xiao 14c and Andrioletti 14d but these processes required either undesirable organic solvents and bases or harsh conditions of temperature and pressure. Apart from the impressive results obtained by Han with a Pt/P-TiO 2 heterogeneous catalyst operating at ambient conditions of pressure and temperature 13m and by Sun using AuPd nanoparticles at 1 bar of hydrogen pressure and 85 °C, 13n the latter suffering from some scope limitations, it is still necessary to improve the sustainability of catalytic methods for the synthesis of pyrrolidone derivatives. This can be tackled by the implementation of modern synthetic tools such as flow synthesis [START_REF] Batista | a) odr uez-adr n, uente-an a o[END_REF] or by the design and implementation of more efficient catalysts and catalytic process. Herein, we describe the efficient base-free reductive amination of levulinic acid and 2-fromylbenzoic acid under mild conditions in water using tailor-made catalysts with enhanced performances. We have recently reported that dipyridylamine (dpa)-based Ir1 complex (Figure 1) was a very effective catalyst for hydrogenation of levulinic acid using formic acid as hydrogen source [15b] and we could further extend the scope of this catalyst to the reductive amination of levulinic acid under hydrogen pressure. [7] The introduction of the electrondonating dimethylamino-group to the dpa ligand (Figure 1, Ir2-Ir4) resulted in the disclosure of Ir3 as a very efficient catalyst for the dehydrogenation of aqueous or neat formic acid.

[15c]

These results prompted us to investigate the potential of these catalysts in the aqueous phase reductive amination of levulinic acid using formic acid as source of hydrogen.

Results and discussion

We initiated our investigations with the reductive amination of levulinic acid with aniline as a model reaction. Several experimental parameters were screened and optimized. As depicted in Table 1, we first explored the reductive amination of levulinic acid with our previously reported Iridium complexes (Table 1, entries 1-4). This catalyst screening revealed the superiority of the electron rich catalyst Ir3, which enabled full and selective conversion of LA into the corresponding pyrrolidone as detected by 1 H NMR of the crude reaction mixture. Further studies of the reaction parameters identified 60 °C and 0.05 mol% as the optimal temperature and catalyst loading, respectively. Interestingly, when the reaction was conducted at a lower temperature, the aminoacid intermediate 1a' was detected giving some clues about the catalytic mechanism (Table 1, entries 7-8). It is also worth mentioning that low reaction temperature prevented the formation of -valerolactone as a side-product resulting from direct reduction of levulinic acid.

H NMR analysis

With the standard conditions in hands, we performed the reductive amination of levulinic acid with a number of commercially available amines. A broad scope of aniline derivatives with various steric and electronic demands was first evaluated. All these amines furnished the corresponding pyrrolidones in good to excellent yields (Table 2,entries 2,3,7, 8). It is noteworthy that the catalyst tolerated very bulky aniline derivatives allowing the synthesis of 1d and 1e in good yields. This is to the best of our knowledge, the first synthesis of these compounds using FA as hydrogen source. For comparison, they were previously obtained in similar yields under 5 bar of hydrogen pressure at 110 °C with 0.1 mol% of Ir1. 7 Another important feature is depicted in Having demonstrated the applicability of the catalysts to a number of amine using pure levulinic and formic acid, we turned our attention to a one-pot process starting from glucose. This one-pot process is an interesting approach considering that LA and FA can be obtained from the acidic treatment of glucose but their separation is difficult. 17 The treatment of glucose with sulphuric acid at 170 °C led to a reaction mixture that was simply filtrated at room temperature in order to remove insoluble materials (humins).

To the filtrate were directly added the iridium catalyst Ir3 and aniline (Scheme 1). After 16 h at 60 °C, the expected pyrrolidone 1a was obtained in a modest but competitive 23 % yield since the one-pot process was performed without pH neutralization of the intermediate filtrate.

14a Scheme 1. One pot synthesis of 1a from glucose

The generality of this catalytic reaction was further extended to the reaction of 2-formylbenzoic acid with amines aiming at preparing N-arylisoindolinone derivatives (Table 3). These compounds display interesting biological properties and therefore represent an attractive target in organic synthesis. Thus, when reacted with aliphatic (Table 3, entry 1) or aromatic amines (Table 3, entries 2-6), 2-formylbenzoic acid was converted into N-subsituted isoindolinones in good to excellent yields. Here too, electron-donating (Table 3, entry 3), electron-withdrawing (Table 3, entry 4) and bulky amines (Table 3, entries 5, 6) were efficiently transformed into the desired compounds. b Isolated yield Following our investigations on the efficiency and scope of the reaction, we turned our attention to the reaction pathways. As generally accepted, the reaction mechanism first involves the formation of an imine and further reduction to the corresponding amine, which in turn reacts in an intramolecular fashion with the carboxylic acid. As observed in the initial optimization of the reaction conditions, an amine was detected as a side-product upon decreasing the reaction temperature to 40 °C. This reaction at 40 o C in the presence of 0.05 mol% of Ir3 (Scheme 2, 1) was repeated and produced the expected mixture of 1a':1a in a 75:25 ratio. After solvent removal, this crude reaction mixture was left stirring for an additional hour resulting in an inverted 21/79 ratio of amine/pyrrolidone. This result demonstrates that the reaction mechanism follows the generally proposed mechanism. To confirm this transient amine intermediate, the reaction was performed with 6-oxo-heptanoic acid and 4-chloroaniline since, as observed in the initial screening, long chain derivatives are less reactive. As anticipated, the intermediate amine 3a' was isolated in good yield without detection of the 7-membered cyclic product (Scheme 2).

Reaction intermediates

On the basis of the above results and considering that an iridium hydride complex can be obtained upon reacting Ir3 with formic acid, 15 a plausible reaction pathway is proposed in Scheme 3. It must be mentioned that according to our previous studies, 15 the reaction likely proceeds by transfer hydrogenation with FA but also by hydrogenation resulting from the fast dehydrogenation of formic acid into carbon dioxide and dihydrogen. In both cases the key catalytic step is the reduction of the imine A by the metal-hydride species M leading to the amino-acid B which upon cyclisation delivers the desired product.

Scheme 3. Reaction pathway of reductive amination of LA.

Conclusions

In summary, we have achieved the reductive amination of levulinic acid into a number of pyrrolidone derivatives with direct extension to the synthesis of piperidones and Narylisoindolinones. Unlike most of the reported catalytic processes, our catalyst could efficiently involve very bulky amines. Most of all, this catalytic method makes use of the easily accessible and safe to handle formic acid as hydrogen source and a biosourced substrate hence ensuring a safe and sustainable process, the single co-product of the reaction being carbon dioxide. The green and sustainable features of this process are further strengthened by the mild reaction conditions and the use of an aqueous media. Nevertheless, efforts will be necessary to improve further the catalytic performances since we are using a rather expensive noble metal. The search for catalysts based on more accessible transition metals is currently underway in our group.

Experimental Section

Levulinic acid (98%), amines and 2,2'-dipyridylamine (98%) were purchased from Sigma-Aldrich. Water was HPLC grade and used as received. Ir1, Ir2, Ir3, Ir4 were synthesized according to the previous synthetic procedures. 15

  15d

Figure 1 .

 1 Figure 1. Iridium complexes used in this study.

Table 1 .

 1 Ir-catalyzed reductive amination of Levulinic acid.

	a

b

The yield of A and B was determined by 1

Table 2

 2 The reductive amination protocol was extended to the synthesis of 6membered lactams (piperidone derivatives) which are also valuable compounds in organic chemistry. Three derivatives were thus prepared from 5-oxo-hexanoic acid. However, it was necessary to increase the reaction temperature to 100 °C to reach high yields(Table 2, entries 13-15)

	, entries 7

with two aliphatic amines including a linear (Table 2, entry 9) and a cyclic amine (Table 2, entry 10) was also successful, yielding the desired products in good yields (76-83%). Likewise, two benzyl amine derivatives led to the corresponding Please do not adjust margins Please do not adjust margins pyrrolidones in good yields (Table 2, 1k, 1l). b isolated yield. c 100 o C (oil bath).

Table 3

 3

	. Ir3-catalyzed reductive amination of 2-formylbenzoic acid: Substrate scope a
	Entry Substrate	Product	Yield b (%)
	1	2a	93
	2	2b	96
	3	2c	90

  NMR spectra were recorded on a Bruker Avance I 300 MHz or Avance III 400 MHz spectrometers. General procedure for reductive amination of LA: Levulinic acid (2 mmol) or formylbenzoic acid (2 mmol), formic acid (4 mmol), amine (2.2 mmol), catalyst (0.01-0.05 mol %) and 2 mL of water were added to a heavy walled Schlenk tube equipped with a Teflon screw cap. The mixture was stirred at the appropriate temperature for the desired time. Volatile compounds were removed under vacuum. The crude mixtures were analysed by

1 H NMR. and purified by column

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 3

Acknowledgements

The authors acknowledge the China Scholarship Council for a grant to SW.

Conflicts of interest

There are no conflicts to declare.