Anita Catapano

PhD Marco Montemurro
email: marco.montemurro@ensam.eu

A multi-scale approach for the optimum design of sandwich plates with honeycomb core. Part I: homogenisation of core properties

This work deals with the problem of the optimum design of a sandwich panel. The design process is based on a general two-level optimisation strategy involving dierent scales: the meso-scale for both the unit cell of the core and the constitutive layer of the laminated skins and the macro scale for the whole panel. Concerning the meso-scale of the honeycomb core, an appropriate model of the unit cell able to properly provide its eective elastic properties (to be used at the macro-scale) must be conceived. To this purpose, in this rst paper, we present the numerical homogenisation technique as well as the related nite element model of the unit cell which makes use of solid elements instead of the usual shell ones. A numerical study to determine the eective properties of the honeycomb along with a comparison with existing models and a sensitive analysis in terms of the geometric parameters of the unit cell have been conducted. Numerical results show that shell-based models are no longer adapted to evaluate the core properties, mostly in the context of an optimisation procedure where the parameters of the unit cell can get values that go beyond the limits imposed by a 2D model.

Sandwich structures are widely used in several elds: aviation, automotive, naval, construction industry. Their application, in fact, ranges from the most performing structures such as aircraft wings, helicopters rotor blades, racing yachts keels to home furnishings. This is because sandwich structures oer high capacity of stiness with a very low weight.

The dierence between a monolithic or laminated plate and a sandwich one concerns the presence of a low density cellular solid, i.e. the core, between two stier thin plates, that increases the geometric moment of inertia of the plate with a few weight increment.

However, when the choice of the designer falls on the use of a sandwich structure, for a given application, he must additionally dene the type of sandwich structure that has to be employed. We can, in fact, identify several types of sandwich structures according to the geometry and shape of the core: honeycomb, solid, foam, corrugated, truss, web cores.

The most important feature of the core is its relative density (ratio between the density of the cellular material and that of the material from which the cells walls are made) that can vary from 0.001 to, generally, 0.4, see [START_REF] Gibson | Cellular solids -Structure and properties[END_REF]. Almost any material can be used to build a cellular solid: polymers, metals, ceramics, composites, etc.. In aircraft and spacecraft applications sandwich panels are composed by glass or carbonber composite skins separated by aluminium or resin honeycombs, or by polymer foams.

In particular, the honeycomb cell size can be chosen to provide cores with dierent stiness and density properties. The result is a panel with very high bending stiness-to-weight and strength-to-weight ratios. A review on sandwich structures and their applications can be found in [START_REF] Gibson | Cellular solids -Structure and properties[END_REF][START_REF] Vautrin | Mechanics of sandwich structures[END_REF][START_REF] Vinson | The behavior of sandwich structures of isotropic and composite materials[END_REF].

The optimal design of sandwich structures is much more cumbersome than that of a classical monolithic structure. The diculties increase when the sandwich structure is made of composite skins and a honeycomb core. In this case we have to face, into the same design process, both the diculty of designing a laminated plate (concerning the skins) and the diculty of designing a complex 3D cellular core. Therefore, engineers always use some simplifying assumptions or rules to obtain, in an easier and faster way, a solution, e.g. the use of symmetric balanced stacks for the skins to ensure the membrane/bending elastic uncoupling along with the membrane orthotropy or the use of regular hexagonal geometry for the unit cell of the core. Such assumptions extremely shrink the solution space of potential congurations for the problem at hand. An alternative approach could consist in formulating the design problem of the sandwich panel as a constrained optimisation problem without introducing any simplifying hypothesis nor on the nature of the skins laminate neither on the geometry of the repetitive unit cell of the honeycomb core.

Obviously, this can be done on one hand by a rigorous mathematical formulation of the problem and on the other hand by the use of numerical techniques that must be able to explore the whole design space in order to nd the true global optimal conguration of the system. Moreover, unlike what is usually done in literature, our objective is twofold: on one hand we want to formulate and solve such a problem on dierent scales and on the other hand we want to include within the design process all possible parameters dening the structure (at each scale) as optimisation variables.

Therefore, in the framework of the design of a sandwich plate with honeycomb core and composite skins, the optimisation variables will be the material and the geometrical properties characterising both the skins and the core at each scale.

The design strategy proposed in this work is a numerical optimisation procedure that does not make any simplifying assumption in order to obtain a true global optimum conguration of the system. The design process is not submitted to restrictions, i.e. any parameter characterising our structure is an optimisation variable: thickness of the core, number of plies of skins, plies orientations and geometric parameters of the unit cell.

In order to face the design of the sandwich structure at both meso and macro scales, we used an optimisation strategy developed on two dierent levels: at the rst level we determine the optimal geometry of the unit cell together with the material and geometric parameters of the laminated skins (at this level the laminate representing each skin is modelled as an equivalent homogeneous anisotropic plate whose behaviour at the macroscale is described in terms of tensor invariants, i.e. the laminate polar parameters, see [START_REF] Vannucci | Plane anisotropy by the polar method[END_REF]). At the second level of the strategy we determine the optimal skins lay-up (the skin meso-scale) meeting the values of the material and geometrical parameters issued from the rst level of the strategy. The whole strategy is based on the use of the polar formalism [START_REF] Vannucci | Plane anisotropy by the polar method[END_REF][START_REF] Verchery | Les invariants des tenseurs d'ordre 4 du type de l'élasticité[END_REF] and on the Genetic Algorithm (GA) BIANCA [START_REF] Montemurro | A two-level procedure for the global optimum design of composite modular structures -application to the design of an aircraft wing. 2art 1: theoretical formulation[END_REF][START_REF] Montemurro | A two-level procedure for the global optimum design of composite modular structures -application to the design of an aircraft wing. 2art 2: numerical aspects and examples[END_REF]. For more details on the polar formalism and the related advantages the reader is addressed to Part II of the present work.

Concerning the model of the core, the rst level of the strategy involves two dierent scales:

• the meso-scale wherein the core is modelled via the repetitive unit cell characterised by some geometric variables;

• the macro-scale where the core is modelled as an homogeneous orthotropic solid whose mechanical response is described through the full set of elastic moduli whose values depend on the geometric parameters of the unit cell.

Therefore, the link between these two scales is represented by the homogenisation phase of the honeycomb core. To this purpose we decided to develop an accurate method to determine the material properties of the orthotropic core that will be assigned to the equivalent solid at the macro-scale.

A considerable number of analytical, numerical and experimental methods for predicting the eective elastic properties of the orthotropic honeycomb core as function of geometric and material characteristics of its unit cell can be found in the literature. The work of Gibson and Ashby [START_REF] Gibson | Cellular solids -Structure and properties[END_REF] represents the rst research study where the full set of elastic properties of a honeycomb core with constant wall thickness has been determined. Subsequently, this technique has been modied by Zhang and Ashby [START_REF] Zhang | The out-of-plane properties of honeycombs[END_REF] to include the double thickness walls to determine the out-of-plane properties. Another analytical approach is presented in [START_REF] Meraghni | Mechanical behaviour of cellular core for structural sandwich panels[END_REF] using the laminate theory. In 1997, Burton and Noor [START_REF] Burton | Assessment of continuum models for sandwich panel honeycomb cores[END_REF] present the full set of nine material properties derived analytically considering the inuence of the double thickness walls. Despite this last work represents a great step forward in the evaluation of the honeycomb structure properties it shows also a weakness: the out-of-plane shear properties of the honeycomb core are not univocally determined since the authors give only the lower and upper bounds for these properties. Other analytical approaches have been developed during the following years, for example [START_REF] Xu | Homogenized elastic properties of honeycomb sandwich with skin eect[END_REF] where the authors take into account also the skins eect or [START_REF] Schwingshackl | Determination of honeycomb material properties: existing theories and an alternative dynamic approach[END_REF] where some experimental results are used to develop an analytical method to determine the out-of-plane mechanical properties. Other researchers developed dierent numerical approaches based on the Finite Element (FE) technique to investigate the material properties of honeycomb structures. Grediac [START_REF] Grediac | A nite element study of the transverse shear in honeycomb cores[END_REF] was one of the rsts to determine the out-of-plane properties of the honeycomb through a FE approach. An analogous work, were the transverse shear stiness properties of honeycombs are determined through a FE method is presented in [START_REF] Shi | Equivalent transverse shear stiness of honeycomb cores[END_REF]. In [START_REF] Burlayenko | Eective elastic properties of foam-lled honeycomb cores of sandwich panels[END_REF] a displacement-based homogenisation technique is presented to evaluate the mechanical properties of foam-lled honeycomb cores. We can also nd other works where the eective properties of the sandwich plate are determined via FE static analyses conducted on the overall geometry of the honeycomb core instead of performing an homogenisation process on the unit cell, see for instance [START_REF] Sadowski | Eective properties for sandwich plates with aluminium foil honeycomb core and polymer foam lling -static and dynamic response[END_REF]. Further interesting works on purely experimental or mixed numerical/experimental techniques for the determination of the eective properties of the honeycomb core can be found in [START_REF] Foo | Mechanical properties of nomex material and nomex honeycomb structure[END_REF][START_REF] Karakoç | Experimental studies on mechanical properties of cellular structures using nomex honeycomb cores[END_REF].

A common weakness of the works about FE-based homogenisation techniques consists in the use of shell-like models for the unit cell of the honeycomb core. These models do not take into account the true geometry of the cell and, consequently, they are not able to properly estimate the inuence of the real 3D stress state on the eective core properties.

In particular, in this work, we need an adequate model of the honeycomb core, at the meso-scale, able to provide with a good level of accuracy the material properties of the core for any combination of design variables that (in the framework of the optimisation procedure) could give rise to a geometry of the unit cell with thick walls rather than thin ones. Therefore, before introducing the design problem of the sandwich structure and the related two-level optimisation strategy, we decided to develop an accurate 3D model, for the numerical homogenisation phase of the core, that has to be included within the rst level of the procedure and that represents the link between the meso and the macro-scale of the core.

In particular, concerning the evaluation of the eective material properties of the core, the skins eect, or skins inuence, is not taken into account during the homogenisation phase. In this work we adopt a conventional approach where the sandwich plate is modelled, at the macro-scale level, as a heterogeneous continuum composed by the union of three parts:

• two laminated skins (that a priori can be dierent) whose anisotropic mechanical behaviour is described through a set of tensor invariants concerning both membrane, bending and coupling stiness tensors (see Part II). Each skin is, hence, characterised by design variables of dierent nature (geometrical and material), i.e. its thickness (so the number of plies of the laminate) and its homogenised stiness tensors invariants (namely the laminate polar parameters, see Part II);

• the core whose mechanical response (at the macro-scale) is dened by the elastic moduli of the equivalent continuum that uniquely depend upon the geometrical design variables of the unit cell (dened at the meso-scale).

Neglecting the interaction between the skins and the core, at the meso-scale level of the core, corresponds to adopt the so-called free modulus model in the framework of the determination of the eective core properties, see [START_REF] Burlayenko | Eective elastic properties of foam-lled honeycomb cores of sandwich panels[END_REF][START_REF] Xu | Homogenized elastic properties of honeycomb sandwich with skin eect[END_REF].

Being this work rather long and considering the fact that it involves two dierent but linked main topics (the core homogenisation and the optimum design of the sandwich panel) we decided to divide its presentation into two parts. In this rst part, we will present the numerical homogenisation technique as well as the 3D FE model used to deal with the core homogenisation problem. In part II, we will present the formulation of the optimisation problem along with the two-level strategy and some numerical examples to prove its eectiveness.

The paper is organised as follows: the description of the core homogenisation problem is introduced in Section 2 and the FE model used for the numerical homogenisation is presented in Section 3. In Section 4 we present a numerical study to determine the eective in-plane and out-of-plane properties of the honeycomb along with a comparison with the existing analytical and numerical models and a sensitive analysis in terms of the geometric parameters of the unit cell. Finally, Section 5 ends the paper with some concluding remarks.

Homogenisation of core properties: problem description

In the last decades, several analytical, numerical and experimental techniques have been developed in order to determine the eective properties of honeycomb sandwich cores as function of geometric and material properties of the repetitive unit cell. Each method presents a certain level of sophistication. For example, analytical techniques, based on energy methods or homogenisation methods, make use of some simplifying assumptions to obtain the elasticity solution of the unit cell. To avoid the use of such assumptions, several studies were conducted to develop new experimental-based and numerical-based techniques for determining the eective core properties.

On one hand, experimental-based methods, see for instance [36], require a standardised procedure for the measurements. However, the main drawbacks of these procedures consist in the fact that they are very expensive in terms of both time and money and the obtained results are valid only for the particular material and geometry of the sample employed in the analysis.

On the other hand, numerical-based techniques, such as FE methods, do not make use of the simplifying assumptions used in analytical approaches and are not expensive. In addition, they can lead to realistic solutions of the elasticity problem in terms of stress and strain elds over the unit cell or within the whole structure of the honeycomb core.

As a consequence, in this work we have chosen a FE-based approach as a numerical homogenisation technique to determine the core properties. It should be noted that a special feature of the honeycomb is its repetitiveness, i.e. the periodicity in its shape. Therefore, at this stage, we will apply the homogenisation method at the meso-scale (the scale of the unit cell) and then, we will replace the actual cellular structure, at the macro-scale level, by an equivalent homogeneous anisotropic medium characterised by the elastic properties determined during the homogenisation phase. The proposed FE-based homogenisation technique leads us to include all the geometric parameters of the unit cell among the optimisation variables of the process without a great loss of computational time (about 11 seconds on a 2.50 GHz Dual Core processor for a single homogenisation analysis).

The basic assumptions made to evaluate the elastic response of our model and, hence, to determine the eective core properties are:

• linear, elastic behaviour for the material of the cell walls;

• perfect bonding for the wall-to-wall contact;

• the buckling of the cell walls is disregarded.

We recall that, as previously stated, since the aim of this rst paper does not consist in developing an equivalent homogeneous plate/solid model for the whole sandwich panel, we do not consider the skins inuence on the evaluation of the eective properties of the honeycomb core.

A scheme of the repetitive unit cell, used for the numerical homogenisation of the honeycomb core, is showed in Fig. 1.

The eective properties of the core are determined using the strain energy homogenisation technique of periodic media, see [START_REF] Barbero | Finite element analysis of composite materials[END_REF]. This technique makes use of the repetitive unit of the periodic structure to approximate its eective properties at the macro-scale level. The basic feature of the strain energy homogenisation technique consists in the assumption that the repetitive unit of the periodic structure and the corresponding unit volume of the homogeneous solid undergo the same deformation having, hence, the same strain energy.

In this case, the periodic structure is the honeycomb core whose repetitive unit cell has three planes of symmetry, thus we decided to exploit these symmetries using, in the homogenisation process, only an eighth of the repetitive unit cell as shown in Fig. 2.

The geometric characteristics of the repetitive unit used for the static analysis of the homogenisation model are shown in Fig. 3: t c is the thickness of the foil used to produce the honeycomb, l 1 is the length of the oblique side of the cell, l 2 represents the length of the horizontal sides of the repetitive unit while ϑ is the cell corrugation angle and h c is the height of the honeycomb core. We can now dene the related Representative Volume Element (RVE) of the honeycomb core whose volume is (see Fig. 2 and3):

V RVE = a 1 a 2 a 3 , (1)
with:

a 1 = 2l 2 + t c tan ( ϑ 2 ) + [ l 1 + t c tan ( ϑ 2 )] cos(ϑ) , a 2 = [ l 1 + t c tan ( ϑ 2 )] sin (ϑ) + t c , a 3 = h c 2 , ( 2
)
whereas the eective volume of the unit cell is:

V EFF = t c h c 2 [ l 1 + 2l 2 + 2t c tan ( θ 2
)
] .

(

) 3
The FE model for the homogenisation process has been created within the commercial FE code ANSYS . We used the 20-node solid element SOLID186 with three Degrees Of Freedom (DOFs) per node. The model along its structured mesh is illustrated in Fig. 4.

In this model the elements that do not belong to the unit cell geometry represent the second phase which has the properties of the so-called elastic air [START_REF]ANSYS Mechanical APDL Modeling and Meshing Guide[END_REF]. The second phase is introduced as a numerical artice to obtain the strain eld provided by the theory underlying the strain energy homogenisation technique, see [START_REF] Barbero | Finite element analysis of composite materials[END_REF] for more details. Unlike what is usually done in the literature, where the cell geometry is modelled with shell elements that show a geometric overlapping between the volumes built over the horizontal and oblique wall sides, in this work a 3D solid model is used to correctly represent the real geometry of the repetitive unit of the honeycomb structure. Our choice is due to the fact that, thanks to the 3D solid model, we can also take into account the eect of the full three-dimensional stress eld in the determination of the eective core properties. As we will show in the numerical study of Section 4, this eect cannot be neglected in any case, mostly when the geometrical conguration of the unit cell is far away from a classical conguration of regular hexagon with thin walls.

A further assumption, as specied in Section 2, concerns the elastic behaviour of the orthotropic homogeneous medium. The generalised Hooke's law for the RVE can be written as follows:

               σ 1 σ 2 σ 3 σ 4 σ 5 σ 6                =         C 11 C 12 C 13 0 0 0 C 21 C 22 C 23 0 0 0 C 31 C 32 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 55 0 0 0 0 0 0 C 66                        ε 1 ε 2 ε 3 ε 4 ε 5 ε 6                , ( 4
)
where σ and ε are, respectively, the stress and strain tensors expressed in Voigt's notation.

The link between tensorial and Voigt's notation for both tensors is:

               σ 1 σ 2 σ 3 σ 4 σ 5 σ 6                =                σ 11 σ 22 σ 33 σ 23 σ 13 σ 12                ,                ε 1 ε 2 ε 3 ε 4 ε 5 ε 6                =                ε 11 ε 22 ε 33 2ε 23 2ε 13 2ε 12                . ( 5
)
As σ and ε are referred to the equivalent homogeneous solid, each component of σ and ε is volume averaged.

The main objective of our homogenisation process is to determine all non-zero components of the stiness tensor C of Eq. ( 4). The expression of the components of C in terms of those of the stress and strain tensors is:

C αβ = σ α ε β with α, β = 1, 2, • • • , 6 and ε γ = 0 with γ = 1, 2, • • • , 6, γ ̸ = β . ( 6
)
We have to determine, now, the nine independent components of C.

In [START_REF] Barbero | Finite element analysis of composite materials[END_REF] the strain energy homogenisation technique is applied to the calculation of the elastic moduli of unidirectional laminae. In this work the author determines the ve components of the transversally isotropic ply stiness tensor by solving four dierent static analyses for the FE model of the RVE. In each one of these analyses the boundary conditions (BCs) are imposed in order to obtain a strain tensor having only one component dierent from zero. According to [START_REF] Barbero | Finite element analysis of composite materials[END_REF] we have determined the nine independent components of C for the honeycomb core through six static analyses on the FE model of Fig. 4. The corresponding BCs for each static analysis are resumed in Tables 1 and2.

These BCs are imposed in order to satisfy the symmetries of the RVE and to generate a strain eld in such a way that only one component of the tensor ε is dierent from zero for each static analysis. The relations giving the volume average strain components are:

ε 1 = u a 1 , ε 2 = u a 2 , ε 3 = u a 3 , ε 4 = u a 3 , ε 5 = u a 3 , ε 6 = u a 2 , ( 7
)
where u is the arbitrarily imposed displacement (see Tables 1 and2). Once the linear elastic problem, characterised by the boundary conditions of Tables 1 and2, is solved we can get the corresponding stress eld whose volume average value for the RVE is computed as follows:

σ α = 1 V RVE ∫ V RVE σ α (x 1 , x 2 , x 3 )dV , α = 1, 2, • • • , 6 . ( 8
)
Through the rsts three static analyses we are able to determine the components of the rsts three columns of the stiness tensor C while through the last three load cases we can determine the components belonging to the second half of its main diagonal, see Eq. ( 6). After calculating all nine independent components of C we can evaluate the eective elastic moduli of the honeycomb core in terms of the stiness tensor components using the well-known relationships reported in [START_REF] Jones | Mechanics of composite materials[END_REF].

As a nal remark, the equivalent density of the core is evaluated through the following relationship:

ρ RVE = ρt c [ l 1 + 2l 2 + 2t c tan ( θ 2
)] { 2l 2 + t c tan ( ϑ 2 ) + [ l 1 + t c tan ( ϑ 2 )] cos(ϑ) } {[ l 1 + t c tan ( ϑ 2 )] sin (ϑ) + t c } . ( 9
)
where ρ is the density of the material of the cell walls.

Numerical study

In order to prove the eectiveness of the proposed FE model we performed a numerical study by comparing our results with those obtained using the models of Burton and Noor [START_REF] Burton | Assessment of continuum models for sandwich panel honeycomb cores[END_REF] and Grediac [START_REF] Grediac | A nite element study of the transverse shear in honeycomb cores[END_REF].

The sandwich structure considered in this work is made of an aluminium honeycomb core and carbon composite skins, this last being a typical combination of materials employed in some real-world aerospace engineering applications, like those presented in [START_REF] Besant | Finite element modelling of low velocity impact of composite sandwich panels[END_REF][START_REF] Buitrago | Modelling of composite sandwich structures with honeycomb core subjected to high-velocity impact[END_REF][START_REF] Diamanti | Lamb waves for the non-destructive inspection of monolithic and sandwich composite beams[END_REF][START_REF] Taylor | Hypervelocity impact on carbon bre reinforced plastic / aluminium honeycomb: comparison with whipple bumper shields[END_REF]. The material properties of the aluminium alloy used for the honeycomb core as well as those of the elastic air (these last taken from [START_REF]ANSYS Mechanical APDL Modeling and Meshing Guide[END_REF]) are listed in Table 3. We rstly conducted the numerical tests on a reference honeycomb core whose unit cell sizes are given in Table 3 (taken from [START_REF]HexWeb honeycomb attributes and properties[END_REF]) while, secondly, we carried out a sensitivity analysis (in terms of all the geometric parameters of the unit cell) on the full set of the eective elastic properties.

Comparison with existing analytical and numerical models

The eective material properties for the reference unit cell (having the geometrical dimensions listed in Table 3) obtained using our FE model as well as those evaluated using the approaches of Burton and Noor [START_REF] Burton | Assessment of continuum models for sandwich panel honeycomb cores[END_REF] and Grediac [START_REF] Grediac | A nite element study of the transverse shear in honeycomb cores[END_REF] are listed in Table 4. In order to calculate with a good accuracy level the eective material properties of the core a convergence study in terms of mesh size has been carried out. In particular, in Fig. 5 we show the results of the convergence analysis for the equivalent elastic moduli in terms of number of divisions n div of the mesh along the cell wall thickness t c . As it can be seen the convergence is reached when n div is equal to four: this corresponds to a FE model having a total number of 52009 DOFs. However, when looking at the results of Fig. 5 one can notice that a FE model with only one division along the thickness direction t c is sucient to properly capture the results, being the maximum relative error of about 4.7% for the in-plane shear modulus G 12 which reduces from 0.67 MPa (one division) to 0.64 MPa (four divisions).

The main reason underlying this fact is that the average value of the stress components (involved into the evaluation of the equivalent elastic properties, see Eqs. ( 6)-( 8)) is slightly inuenced when passing from one to four divisions along t c . On the other hand the use of a 4 are compared by considering, in the framework of the models of Burton and Noor and Grediac, a unit cell having the same middle-surface as that of our FE model.

As it can be easily seen, the behaviour of the core at the macro-scale is orthotropic with the main orthotropy axes aligned with those of the Cartesian coordinate system of Fig. 2.

For this reference case, where we consider a regular hexagonal honeycomb cell with thin walls, the results given by our 3D solid model globally agree with those found by Burton and Noor and Grediac. In particular, concerning the evaluation of the three out-of-plane moduli E 3 , G 13 and G 23 , the relative dierence between our model and [11, 16] is very low: 1% for G 23 and about 2% for both G 13 and E 3 .

On the other hand, if we consider the in-plane moduli E 1 , E 2 , G 12 and the out-of-plane

Poisson ratios ν 13 and ν 23 the results obtained with our 3D FE-based model slightly dier from those provided by [11, 16]. The relative dierence ranges from 8% for E 1 to 23.5%

for G 12 . The main reason underlying the previous dierences is in the use of shell-based models and theories as done in [START_REF] Burton | Assessment of continuum models for sandwich panel honeycomb cores[END_REF][START_REF] Grediac | A nite element study of the transverse shear in honeycomb cores[END_REF] which are not able to properly evaluate the previous quantities. To understand this fact, let us consider the evaluation of the in-plane shear modulus G 12 which is calculated through the following relationship:

G 12 = C 66 = σ 6 /ε 6 . ( 10
)
From Eq. ( 10) we can see that G 12 depends only upon the average stress σ 6 and upon the imposed average strain ε 6 . This means that the value of the in-plane shear modulus depends on the accuracy of the numerical evaluation of the in-plane shear stress eld of the RVE, which is inuenced by the following aspects:

• σ 6 depends upon the correct evaluation of the normal and shear stress components in the oblique side (σ ηη , σ ξξ and σ ηξ ); indeed, all of these stresses are involved in the evaluation of σ 6 in the global frame of the RVE {O; x 1 , x 2 , x 3 }, see Figs. 6 and7;

• as shown in Figs. 6 and7, the previous stress components vary through the thickness of the oblique side and such variation is more pronounced at the intersection between the oblique and the horizontal walls;

• in the oblique face of the unit cell the normal out-of-plane stress σ ηη as well as the shear stress σ ηξ are non-negligible, being the order of magnitude of such components the same as the normal in-plane stress σ ξξ , see Figs. 6 and7;

• the shear stress σ 6 in the top and bottom horizontal sides of the unit cell varies through the thickness of the wall, as shown in Fig. 8.

The previous aspects cannot be correctly evaluated in the framework of a shell-based model such as those used in [START_REF] Burton | Assessment of continuum models for sandwich panel honeycomb cores[END_REF][START_REF] Grediac | A nite element study of the transverse shear in honeycomb cores[END_REF]. In particular the out-of-plane normal stress in the oblique wall is null for a shell model and the through-the-thickness variation of the shear stress can be properly evaluated only by higher order shell theories. Similar considerations can be repeated for the calculation of the rst three columns of the stiness tensor of the core and hence for the evaluation of the rest of the elastic properties which depend upon these quantities, see [START_REF] Jones | Mechanics of composite materials[END_REF].

Sensitivity analysis

Together with the determination of the eective elastic properties of the basic conguration having the geometry of Table 3, we also conducted a sensitivity analysis in which we change, one at a time, every geometric parameter of the unit cell by keeping constant the others.

When one of these parameters varies the rest get the values of the reference unit cell as reported in Table 3. For this reason such an analysis can be seen as an analysis of the mechanical response of the RVE, in terms of its eective material properties, in the neighbourhood of the reference conguration. Looking at the results shown in Figs. 9-13 we can deduce the following facts:

• when varying the parameter l 1 between 10t c and 100t c (Fig. 9) the trend of the in-plane elastic properties are in agreement with those provided by the model of Burton and Noor. Concerning the out-of-plane Young's modulus E 3 it is slightly underestimated by our 3D FE model when compared to that given by Burton and Noor and this gap increases when l 1 decreases. This phenomenon is due to the fact that when l 1 decreases it becomes of the same order of magnitude as the wall thickness t c . Under this condition the mechanical behaviour of the oblique side of the cell is no longer that of a thin plate, thus the model of Burton and Noor overestimates such modulus. Similar considerations could be done both for the shear moduli and the Poisson's ratios;

• when varying the parameter l 2 between 0.05l 1 and 0.5l 1 (Fig. 10) the trend of the results of our model globally agrees with that found using the model of Burton and

Noor; the relative dierence on the values of the dierent moduli found using the two models can be explained through the considerations previously done for the reference geometry of the unit cell. This relative dierence remains constant for all the elastic properties, with the exception of G 12 whose relative dierence strongly increases when l 2 decreases: this is due to the fact that when l 2 becomes of the same order of magnitude as t c , both the bottom and top horizontal sides of the unit cell cannot be modelled as thin plates, thus a shell-based model is no longer adapted to correctly capture such a phenomenon;

• when varying the parameter t c between 0.02l 2 and 0.1l 2 (Fig. 11) the dierence between the results of the present model and those issued from the model of Burton and Noor increases with t c . This phenomenon can be easily explained: when the wall thickness increases, the inuence of the true 3D geometry of the unit cell on the evaluation of the eective material properties becomes more and more important;

• when varying the parameter ϑ between 5 deg and 90 deg (Fig. 12) the trend of the results found with our model is practically the same as that provided by the model of Burton and Noor. The relative dierence between the values of the eective elastic properties found using the two models, increases when ϑ decreases, with the exception of G 13 and G 23 . This is due to the fact that when the corrugation angle of the cell decreases the unit cell becomes more and more at. In such a conguration the eect of the out-of-plane stresses σ 2 and σ 6 on the calculation of the eective core properties plays a crucial role, thus a shell-like model is not sucient to properly capture these phenomena;

• when varying the parameter h c between l 1 and 10l 1 (Fig. 13) we show that, actually,

the assumption underlying the model of Burton and Noor that the core properties do not vary with the core height is correct for seven elastic properties (E 1 , E 2 , E 3 , G 12 , ν 12 , ν 23 and ν 13 ). On the contrary, according to the model of Grediac [START_REF] Grediac | A nite element study of the transverse shear in honeycomb cores[END_REF], this assumption cannot be accepted when evaluating G 13 that varies with the parameter h c . Nevertheless, unlike the results provided by Grediac our model is able to capture also the weak variation of G 23 with the core thickness. Moreover, the relative dierence calculated on G 23 decreases when h c increases, ranging from 1% to 0,1%.

To be remarked that when varying each geometrical parameter of the cell, the curve describing the variation of the out-of-plane shear modulus G 13 always lies between the lower (LB) and upper (UB) bounds given by the model of Burton and Noor which are calculated using the strain energy associated rstly with a stress distribution satisfying the equilibrium (for the LB) and secondly with a strain eld allowing compatible deformation (for the UB). The reader is addressed to [START_REF] Burton | Assessment of continuum models for sandwich panel honeycomb cores[END_REF] for a deeper insight in the matter. As a conclusive remark of this section, we want to highlight the importance of using solid elements to build the FE model of the repetitive unit cell which are able to properly capture the inuence of the true 3D stress eld on the evaluation of the eective elastic properties of the honeycomb core.

Conclusions

The main aim of the present work is to deal with the problem of the optimum design of a sandwich panel composed of two laminated skins and a honeycomb core. In this rst part of the work we presented the numerical homogenisation technique as well as the related 3D FE model of the unit cell that will be used within the rst level of the optimisation procedure (see Part II). In particular, we need an adequate model of the honeycomb core, at the meso-scale, able to properly predict its equivalent material properties (at the macroscale) for any combination of design variables that, in the framework of the optimisation procedure, could give rise to a geometry of the unit cell which is far away from a classical shell-like geometry (i.e. a cell with thin walls).

To these purposes, rstly we demonstrated that shell-based models cannot correctly represent the true geometry of the unit cell of the honeycomb core and, consequently, they are not able to properly capture the inuence of the real 3D stress eld on the determination of the eective core properties. For these reasons, in this paper, we used a 3D solid FEbased model of the repetitive unit cell of the honeycomb core to predict its eective material properties in the most general case, i.e. by taking into account any combination of the geometric parameters of the unit cell.

The material properties determined using our model were, in addition, compared with those obtained using the models of Burton and Noor [START_REF] Burton | Assessment of continuum models for sandwich panel honeycomb cores[END_REF] and Grediac [START_REF] Grediac | A nite element study of the transverse shear in honeycomb cores[END_REF]. Numerical results show that, for a hexagonal unit cell with thin walls, the out-of-plane elastic moduli found by our model globally agree with those provided by shell-based analytical and numerical models. On the contrary, the in-plane elastic moduli found using our model show a relative dierence ranging from 8% to 23.5% (depending on the considered quantity) with respect to those provided by shell-based models. The main reason underlying such dierences is the inuence of the local out-of plane normal stress over the cell walls which is identically null in the framework of a shell-based model. Moreover, through an analysis on the local stress eld within the cell walls along with a sensitivity analysis, we proved that such models are no longer appropriate in the framework of an optimisation procedure that aims to take into account, among the design variables, the full set of the geometric parameters characterising the unit cell. In fact, for those congurations in which the thickness-to-side ratio of the cell walls goes beyond the limits imposed by a shell-like model, a 3D solid model of the unit cell is therefore necessary to properly describe the eect of the 3D stress eld on the evaluation of the full set of material properties of the equivalent solid that will be used at the macro-scale.

Part II of the present work will cover the formulation of the optimisation problem and provide a detailed description of the two-level optimisation strategy; numerical examples will be presented in order to demonstrate the eectiveness of the proposed approach.

Tables

1 st load case 2 nd load case 3 rd load case Nodes U 1 U 2 U 3 Nodes U 1 U 2 U 3 Nodes U 1 U 2 U 3
x 1 = 0 x 3 = 0 free free 0

x 3 = 0 free free 0

x 3 = 0 free free 0

x 3 = a 3 free free 0

x 3 = a 3 free free 0

x 3 = a 3 free free u

Table 1: Boundary conditions for the FE model of the RVE: 1 st , 2 nd and 3 rd static analyses.

4 th load case 5 th load case

6 th load case Nodes U 1 U 2 U 3 Nodes U 1 U 2 U 3 Nodes U 1 U 2 U 3
x 1 = 0 0 free free x 1 = 0 free 0 0

x 1 = 0 free 0 0

x 1 = a 1 0 free free x 1 = a 1 free 0 0

x 1 = a 1 free 0 0

x 2 = 0 x 3 = a 3 free free 0 Eective properties of the homogeneous orthotropic core vs. h c .

1 2 1 2 3

 12123 Sandwich Structures; Honeycomb; Homogenisation; Composite materials. Notations GA Genetic Algorithm FE Finite Element t c Thickness of the aluminium foil used to produce the honeycomb core l Length of the oblique sides of the hexagonal repetitive unit cell l Length of the horizontal sides of the hexagonal repetitive unit cell ϑ Corrugation angle of the hexagonal unit cell h c Height of the honeycomb core RVE Representative Volume Element for the honeycomb core V RV E Volume of the representative volume element (including the second phase, i.e. the elastic air) {O; x 1 , x 2 , x 3 } Global material frame of the representative volume element a Side length, along x 1 -axis, of the representative volume element a Side length, along x 2 -axis, of the representative volume element a Side length, along x 3 -axis, of the representative volume element DOFs Degrees Of Freedom σ α Volume averaged stress tensor components (Voigt's notation) ε β Volume averaged strain tensor components (Voigt's notation) C αβ Stiness tensor components of the homogenised core (Voigt's notation) σ αβ Volume averaged stress tensor components (tensorial notation) ε αβ Volume averaged strain tensor components (tensorial notation) BCs Boundary Conditions u Arbitrarily imposed displacement E i Young's moduli in the material frame {O; x 1 , x 2 , x 3 } G ij Shear moduli in the material frame {O; x 1 , x 2 , x 3 } ν ij Poisson's ratios in the material frame {O; x 1 , x 2 , x 3 } ρ RV E Eective (or relative) density of the honeycomb core ρ Density of the aluminium E Young's modulus of the aluminium ν Poisson's ratio of the aluminium E air Young's modulus of the elastic air ν air Poisson's ratio of the elastic air {O; ξ, η, z} Local frame dened on the oblique side of the hexagonal cell σ ξξ , σ ηη , σ ηξ Local stress tensor components expressed within the local frame

FE

  model of the RVE with four elements within the thickness lead us to properly describe the correct variation of the normal as well as the shear stress elds through the thickness (see Figs. 6-8). These considerations led us to use, in the present work, a FE model of the RVE characterised by four divisions along the cell thickness. It is worth noting that the results of Table

x 1 = a 1 u free free x 1 = a 1 0free free x 1 = a 1 x 2 = a 2 free 0 free x 2 = a 2 free u free x 2 = a 2

 11120222 free 0 free

x 2 = a 2 0 free 0 x 2 = a 2 free 0 free x 2

 2022

Figures

  Figures

Figure 1 :

 1 Figure 1: Honeycomb core structure (a) and the repetitive unit cell (b).

Figure 2 :

 2 Figure 2: The repetitive unit cell (a) and the overall sizes of the related RVE (b).

Figure 3 :

 3 Figure 3: Geometrical parameters of the unit cell.

Figure 4 :

 4 Figure 4: FE model of the RVE.

Figure 5 :

 5 Figure 5: Convergence analysis for the eective properties of the homogeneous orthotropic core in terms of number of divisions n div along the wall

Figure 6 :

 6 Figure 6: Variation of σ ηη (η) (a), σ ηξ (η) (b) and σ ξξ (η) (c) [MPa] within the oblique wall of the cell at the intersection with the top horizontal side, 6 th load case.

Figure 7 :

 7 Figure 7: Variation of σ ηη (η) (a), σ ηξ (η) (b) and σ ξξ (η) (c) [MPa] within the oblique wall of the cell at the intersection with the bottom horizontal side, 6 th load case.

Figure 8 :Figure 9 :

 89 Figure 8: Variation of σ 6 (η) [MPa] within the top horizontal wall (a) and the bottom horizontal wall (b) of the cell at the intersection with the oblique side, 6 th load case.

Figure 11 :Figure 12 :

 1112 Figure 11: Eective properties of the homogeneous orthotropic core vs. t c .

Table 2 :

 2 Boundary conditions for the FE model of the RVE: 4 th , 5 th and 6 th static analyses.

			Material properties
	Aluminium	E [MPa]	ν	ρ [Kg/mm 3 ]
			70000	0.33	2.7 × 10 -6
	Elastic air		E air [MPa]	ν air
			1 × 10 -3	0.0
	Geometrical parameters (reference values)
	l 1 [mm] l 2 [mm]	t c [mm]	θ [deg]	h c [mm]
	3.666	1.833	0.0635	60	20

Table 3 :

 3 Material and geometrical parameters of the unit cell.

	Properties	Present	Burton & Nooor [11] Grediac [16]
	E 1 [MPa]	0.884	0.815	0.815
	E 2 [MPa]	0.918	0.815	0.815
	E 3 [MPa]	1812.299	1848.185	1848.185
	G 12 [MPa]	0.640	0.489	0.489
	G 23 [MPa]	262.981	260.552	260.552
	G 13 [MPa]	390.833	156.331 (LB)	397.088
			434.254 (UB)
	ν 12	0.980	1.000	1.000
	ν 23	0.161 ×10 -3 0.145 ×10 -3	0.145 ×10 -3
	ν 13	0.167 ×10 -3 0.145 ×10 -3	0.145 ×10 -3
	ρ RVE [Kg/mm 3 ] 6.990 ×10 -8 7.117 ×10 -8

Table 4 :

 4 Eective material properties of the core for the reference geometry of the RVE.


T Couvant

L Legras

A Herbelin

A Musienko

G Ilevbare

D Delafosse

G Cailletaud

J Hickling

DEVELOPMENT OF UNDERSTANDING OF THE INTERACTION BETWEEN LOCALIZED DEFORMATION AND SCC OF AUSTENITIC STAINLESS STEELS EXPOSED TO PRIMARY PWR ENVIRONMENT

Understanding and quantification of interactions between localized deformation in materials and environmental-assisted cracking (EAC) could play an important role in maintaining the integrity of LWR components. Thus, a detailed understanding of strain localization during plastic deformation and of the underlying mechanisms is of great importance for the manufacturing and design of materials exposed to the environment of the primary circuit of PWRs. Thus, the crystal plasticity law of a 304L austenitic stainless steel has been identified, in order to quantify the effect of a change of strain path on the strain localization and increase the understanding of the contribution of the strain hardening and the strain incompatibilities on the mechanisms of initiation of SCC. Pre-deformation of the specimens used for SCC tests was evaluated using image correlation. Constant elongation rate tests and constant elongation tests were conducted for different levels of pre-deformations (0.07 <  < 0.18).

Examinations indicated the severe and deleterious effect of the strain localization due to a change of strain path on intergranular SCC susceptibility. Intergranular cracks initiated systematically in the low-deformed areas, where limited deformation (less than 1%) occurred during the exposure to the environment, and where high stress levels due to strain incompatibilities are expected.

INTRODUCTION

Understanding and quantification of interactions between localized deformation in materials and environmental-assisted cracking (EAC) could play an important role in maintaining the integrity of LWR components. A previous (1) review of the different types of identified strain localization-EAC interactions allowed key gaps to be identified. Thus, a detailed understanding of strain localization during plastic deformation and of the underlying mechanisms is important for the manufacture and design of materials exposed to the environment of the primary circuit of PWRs. Previous studies [START_REF] Couvant | Effect of strain path on SCC of AISI 304L stainless steel in PWR primary environment at 360[END_REF] have demonstrated the deleterious effect of cross tests on susceptibility of 304L to IGSCC in PWR primary water at 360°C. This result has been interpreted by the authors as a consequence of strain localization induced by orthogonal strain paths as defined by Schmitt (3). The objectives of this study were to correlate strain localization and selective oxidation using approaches based on quantitative and fundamental understanding, in order to evaluate the relevance of interactions between localized deformation and stress corrosion cracking (SCC) in austenitic alloys exposed to a PWR primary water. Thus, a wide range of techniques was deployed in this program to characterize the microstructures, to quantify strain localization and associated oxidation.

MATERIALS

Austenitic alloys tested in this work (304L and 316L) are 30-mm-thick plates. Chemical compositions and mechanical properties of the materials as manufactured are given in Table 1 and Table 2, respectively. Unlike 316L, 304L contains less than 5% of residual -ferrite and is subject to strain-induced martensite transformation: Ms = -133°C, Md 30 = -3°C, as predicted by Angel [START_REF] Angel | Formation of martensite in austenitic stainless steels[END_REF]. Concerning the 316L heat, Ms = -133°C, Md 30 = -40°C. The staking fault energy (SFE) of 304L is 24 mJ.m -2 and 33 mJ.m -2 for 316L [START_REF] Lacombe | Structures et diagrammes d'équilibre des diverses nuances d'acier inoxydables. Conséquences sur leurs traitements thermiques[END_REF]. Non-sensitized 304L and 316L were solution annealed at 1050°C and water quenched. The resultant microstructures, with no evidence of any carbide precipitates in the matrix or along the grain boundaries, were characterized using EBSD. Mean grain diameter is 27 µm for 304L and 90 µm for 316L. The distribution of misorientation angles in the range 6° <  < 59°, including twin boundaries ( > 54°).

Distributions are very similar for 304L and 316L. Thus, in these two materials, 30 to 50% of boundaries are twin boundaries. Misorientation angles in the range 37° <  < 54° correspond to about 30% of the boundaries. Inverse pole figures are given in Figure 1, at the same scale.

APPROACH

Specimens

Cross tests were used to promote strain localization (Figure 2). This design was optimized to guaranty strain localization in the cross-area of the specimen during prior-straining and second deformation in primary water and to avoid significant deleterious deformation in specimen fillets. The specimen is not symmetrical: only the front face is perfectly plane, designed to perform an easy mechanical polishing, material characterization and local strain evaluation. All characterizations and observations were done on 1 mm 2 at the center of the specimen's front face, prepared as described below:

 Three-step local polishing of each specimen, including mechanical polishing (0.04 µm finish) to remove superficial cold work, electropolishing (20°C, 19 V) in Struers A2 electrolyte A2 solution containing 73% ethanol, 10% butoxyethanol, 9% water, and 8% perchloric acid, and ionic polishing to remove the chemically affected superficial metal layer for relevant oxidation characterization.

 EBSD characterization at the center of each specimen (1 mm 2 ).

 Grid deposition (1 mm 2 , mesh size = 5 µm), by electronic micro lithography, at the center of the specimens using the following procedure: the center of the sample is coated by centrifugation with about 200 nm of a photosensitive polymer (poly methyl metacrylate). After heating (30 min at 135°C), the sample is locally irradiated (4 C/m 2 ) by a SEM beam (tension of acceleration = 30 kV, probe current = 3 nA, working distance = 30 mm), drawing a grid. The irradiated resin is then dissolved in a solution composed of methyl cetone (35%) and isopropylic alcohol (65%). A thin layer of gold is then deposited, first by cathodic pulverization, then by evaporation under vacuum. Residual resin is eliminated in a solution of ethyl acetate.

All cross specimens were finally prior deformed in ambient air, along X  axis.

Tensile tests

After pre-straining, a final machining of specimens is achieved by cutting off the arms used for presstraining to insure that deformation will be localized at the center of the specimen during the second loading at 360°C (Figure 2). Specimens were ultrasonically rinsed in ethanol and then in distilled water. Tests were carried out in Hastelloy (C-276) autoclaves. Specimens were insulated from the autoclave by oxidized Zircalloy to avoid galvanic coupling. Experiments were conducted under open circuit conditions. The environment was primary water (1000 ppm B as boric acid, 2 ppm Li as lithium hydroxide) at 360°C. The solution was previously de-aerated by evaporating 20% of the initial volume at 125°C, then hydrogen overpressure was introduced (30 cm 3 .kg -1 ) and controlled using a Pd-Ag probe. The chemistry was analyzed and validated before each test (B, Li, Cl -, SO 4 2-, F -). Pre-oxidation (5-7 days) in a primary water test environment was performed.

Image displacement analysis

Gridline images are collected before and after each step of deformation using scanning electron microscopy (SEM). All data are collected for the same working distance. Comparison of the grid before and after deformation reveals the material superficial displacement field used for evaluating deformation gradient tensor

F ~: 1 0 . ~x F x    (1)
where 0 x  is a vector in the initial configuration (before deformation), while 1 x  is the corresponding vector in the intermediate configuration (after priordeformation in air) or in the final configuration (after the deformation in the primary water). At each node of the grid, F ~ is evaluated using the four nearest nodes. Finally, Green Lagrange tensor E ~ is calculated at each node of the mesh using (2:

  I F F E T 2 1 ~  (2)
where T F ~is the transposed tensor of F ~.

Electronic microscopy

All specimens were examined by SEM after completion of testing in primary water, in order to identify sites of initiation. A dual-beam Helios Nanolab from FEI (SEM equipped with a Focused Ions Beam column) was used for preparing cross section thin foils on specifically deformed locations. This dual beam has a Schottky field emission gun column, Ga ion beam column, Pt and C gas injection system, carbon selective etching system, EDX detector and Autoprobe internal micromanipulator. TEM observations of specimens were performed using a FEI TECNAI G2 FEG STEM equipped with Gatan Imaging Filter, EDAX energy dispersive spectrometer and HAADF detector.

Finite element modeling

Crystal plasticity modeling is an appropriate approach to evaluate the heterogeneity of the continuous mechanical response in an aggregate, taking into account activation of slip systems. Therefore, such modeling is a relevant tool for comprehending SCC mechanisms, where strain hardening and strain incompatibilities play a significant role in the cracking mechanism. In the present modeling, the strain rate   was first decomposed into elastic e   and plastic p   parts (Equation 3).

p e         (3)
Considering that intragranular deformation is the main contribution, plastic strain rate p   in a grain is the sum of elementary shear rates s  extending over all active slip systems s (Equation 4):

       s s s s s s p m n n m 2 ~    (4)
where s n is normal vector to the slip plane and s m is a vector corresponding to slip direction. Twelve {1 1 1} 110 octahedral systems were assumed to be potentially active. Crystal plasticity models rely on the understanding that plastic deformation in crystals at low temperatures occurs by crystallographic slip following Schmidt's law. dislocations in active slip systems. Thus, flow rule s  is described by a phenomenological Norton-like viscoplastic relation (Equation 5), where K s is viscosity constant, n s is stress sensitivity, and s c  is the critical resolved shear stress on s:

s s s s n s s c s s s x x K x s           ) (  (5)
Resolved shear stress s  on slip system s is the projection of the stress tensor on orientation tensor s m ~(Equation 6):

s s m :    (6) s
x is the kinematic hardening, proportional to s  (Equations 7 and 8):

x s = c  s (7) s s s s d          (8)
where c is a constant.

  ) exp( 1 0 r cum r rs s c b h Q         (9)   t r r cum dt    (10)
where h is an interaction matrix, describing the self hardening (diagonal components) due to the interactions between coplanar slip systems, and the latent hardening (non-diagonal components) due to the interactions between secant slip systems. All components h rs of h are normalized by the selfhardening components. When strain hardening is isotropic, it is independent of the active slip system. Therefore, in this situation h sr = 1. Usually, selfhardening components are inferior to latent components. Both isotropic and kinematic hardenings are considered when modeling the behavior of the material. Nevertheless, in a first approach [START_REF] Cailletaud | A micromechanical approach to inelastic behaviour of metals[END_REF], it is assumed that h rs = 1. One free mesh has been produced, using Voronoi space subdivision, with 343 grains, 2174653 nodes, and 1599480 tetrahedral elements with 10 nodes (type c3d10), as shown in Figure 3. The finite element calculations were performed with the code Zebulon [START_REF] Besson | Object-oriented programming applied to the finite element method. Part II: application to material behaviors[END_REF].

RESULTS OF EXPERIMENTS

Strain localization after prior deformation

Results are given in Table 3. Prior mean equivalent strains E eq are within the range 0.065-0186. Figure 4 is an E eq mapping at the surface of specimen 1594-T112-06, considering an area of 250 µm × 250 µm. Strain is localized in slip bands oriented at ± 45° from the main axis load x  . Strain level in intense slip bands is typically two times higher than the mean deformation value. Between the intense slip bands, areas with only few percent of strain are observed. Figure 5 is an example of frequency of prior E eq observed at the surface of specimen 1594-T217-02. The E eq distribution is Gaussian.

Standard deviations of prior E

eq , SD(E eq ), measured on specimens are reported in Table 3, and plotted in Figure 6 against E eq . This figure shows the increase of SD(E eq ) with E eq . (11 is a fit of data plotted in Figure 6 that could be used to predict the Gaussian distribution of equivalent strain associated to a known mean value.

SD(E

eq ) = 0.35 E eq (11) Typically, the range of E eq , E xx , and E yy is two times higher than their mean value.

Figure 7 gives the evolution of the deformation as a function of the analyzed area, and shows that an analyze limited to 100 µm × 100 µm is sufficient for a good estimation of the mean, maximum, and minimum values of the different components of E. This means that an aggregate of tens of grains is sufficient to estimate with good precision the mean deformation of stainless steels and its heterogeneity.

Initiation of SCC in primary water

Results of constant elongation rate tests in primary water are presented in Table 4. Strain and strain rates are within the ranges 0.003 0.500 and 1.4 10 9 s 1 9.9 10 8 s 1 respectively. As expected, initiation of SCC was essentially intergranular.

The test on specimen 1594-T112-02 was interrupted for SEM examination after 270 hours of tension in a primary hydrogenated environment at 360°C. No initiation of SCC was noticed. Then, the specimen was tested again up to failure. Observations of the fracture surface indicated a preferential intergranular initiation in the prestrained area of the specimen (Figure 8). By contrast SCC propagation was essentially transgranular in the most deformed part of the specimen (Figure 9). This result indicates that IGSCC was enhanced by the change of strain path after prior deformation, while TGSCC was clearly promoted by the continuous deformation during testing.

Test on specimen 1594-T112-07 indicated an important susceptibility to IGSCC for a limited exposure to primary water (313 h). Prior deformation E xx and second deformation E yy levels were similar. By contrast, prior (prior E xx = 0.17) and second (second E yy < 0.01) deformations applied to specimen 1594-T112-08 were completely different. However, significant IGSCC initiation was observed on both specimens. Therefore, intergranular cracking resulted in the change of strain path, and large deformation during exposure to the environment is not required for IG cracking.

Figure 8 -IGSCC initiation followed by transition to TGSCC on fracture surface of specimen 1594-T112-2 after 1400 h of tension in primary environment.

Figure 9 -TGSCC propagation on fracture surface of specimen 1594-T112-2 after 1400 h of tension in primary environment.

Test on specimen 1594-T217-01 (316L) revealed a shallow intergranular initiation (Figure 10) appropriate for local strain evaluations. By constrast intergranular cracks were too opened in the case of specimen 1594-T217-02 (Figure 11). TGSCC was also noticed on this specimen.

No initiation was observed on specimen 1594-T217-10 after 577 h in primary water, for a prior deformation similar to those of specimen 1594-T217-01 and a limited second deformation (E yy = 0.003). Thus, time to initiation probably decreases when prior and second strain levels increase.

One test was conducted under constant displacement on specimen 1594-T112-03. Based on previous results [START_REF] Couvant | Investigations on the mechanisms of PWSCC of strain hardened austenitic stainless steels[END_REF], no relaxation was expected during the SCC test (1864 hours). A limited number of intergranular cracks were observed. Slip line traces are observed on the flanks of the cracks. This result suggests that IGSCC initiation is possible under constant load following a complex strain path when the second strain is limited.

Correlation between initiation and local deformation E yy in primary water is illustrated in Figure 12 and Figure 13. These figures indicate that SCC initiated at grain boundary where local strain gradients are significant. Therefore, an important effect of strain incompatibilities on intergranular cracking is expected. The cracked grain boundaries were in addition favorably oriented for cracking against loading axis.

Non-initiation (gray dots), IG-(red crosses) and TGSCC initiation (blue crosses) conditions are plotted against prior deformation E xx in ambient air, second deformation E yy in primary water in Figure 14. 0.3 mm² of the surface of the specimen 1594-T217-01 were analyzed, including 12182 measures of strain. 418 nodes were considered as representative sites for initiation. Most of the cracks were intergranular, confirming the appropriate design and loading used in this study to investigate IGSCC. No crack initiated where deformation was compressive during the exposure to primary water (Second E yy < 0). Most of the cracks initiated for 0.05 < prior-E xx < 0.1 and 0.01 < second-E yy < 0.04. It is confirmed that high levels of deformation during the exposure to the primary water are not necessary to initiate. No initiation

Figure 10 -IGSCC initiation on specimen 1594-T217-1 after 720 h of tension in primary environment.

Figure 11 -IGSCC initiation on specimen 1594-T217-2 after 1600 h of tension in primary environment.

Figure 12 -Deformation Eyy (in primary water) at the surface of specimen 1594-T217-1 (316L) after testing in a primary water (360°C).

Figure 13 -Deformation Eyy (in primary water) at the surface of specimen 1594-T217-1 (316L) after testing in a primary water (360°C).

Figure 14 -Preferential conditions for SCC initiation as a function of prior deformation in ambient air and second strain in primary water. Specimen 1594-T217-01 (316L) after testing in a primary water (360°C).

Oxide characterization

Several sites of sampling were chosen at the surface of specimens 1594-T112-06 and 15946T217-01. Each site of sampling was superposed with a local deformation map in order to correlate strain localization with local passive layer thickness. Figure 15 is an example of such correlation for the sites 1A and 1B. The level of prior deformation E xx in ambient air and second deformation E yy in primary water corresponding to the sampling for TEM examinations are reported in Table 5.

Figure 16 shows the details of the oxide formed in site 1A. In this figure, the metal-oxide interface is clearly below the grid-oxide interface, suggesting the anionic growth of the inner Cr-rich oxide layer in the metal. The thickness of the oxide below the grid does not seem affected by the presence of the grid. A preferential oxidation can be also noticed within pre-established shear bands from prior deformation in air. Spinels (magnetite) grew on the inner Cr-rich oxide layer and on the grid. Spinels are in epitaxy with the metal, despite the presence of the inner Cr-rich oxide layer.

Figure 17 is a STEM observation of site 1B, showing the possible differences of density between the outer and the inner part of the Cr-rich oxide layer. The outer part, at the interface with the spinels appears very regular and dense. This could be the result of the fine polishing including electrothen ionic polishing. By contrast, the inner part seems composed of small 'grains' growing inhomogeneously in the metal. The typical size of 'grains' observed in the inner oxide is similar to the fine dislocation structures in the underlaying metal. Detailed analyses were done in order to evaluate the effect of strain localization on oxide penetration and on metal-oxide interface. Figure 23 is an example of EFTEM Cr-map. Figure 24 is the associated binary map obtained after filtering. Image analysis of this binary map allows to quantify the thickness of the inner Cr-rich oxide layer (Figure 25). In this situation, the mean oxide penetration is 50 nm. Finally, results obtained after image analysis of inner oxide layers are summarized in Table 5.

Crack tip examination

Thin foils were removed (dual beam sampling) at the surface of specimens 1594-T112-06 (304L) and 1594-T217-01 (316L). Active intergranular crack tips were observed. Indeed, contrary to usually observed crack propagation tips, there is a high probability that all of the tips observed are active, because they correspond to the earlier stage of initiation, before any branching and screening by other cracks. Cracks usually initiated where gradient of deformations are observed. Figure 26 and Figure 27 clearly exhibits cracks on grain boundaries separating grains with clear different densities of dislocations. Crack depth did not exceed few microns. Cracks are usually opened (Figure 26), 200 nm typically, in good agreement 200 nm

Metal

with SEM observations at the surface (Figure 10). Opened cracks are filled with Fe oxide (Figure 28). A significant depletion of Fe is also observed up to 200 nm on the grain boundary, ahead of the tip. Similarly, a slight Cr depletion is observed on the grain boundary. Conversely, an important enrichment in Ni is observed.

RESULTS OF FEM

Identification of crystal plasticity law

The identification of the crystal plasticity law of the 304L is based on:

 Monotonic tensile tests at 20°C (  = 2 10 - 3 s -1 ) and at 360°C (  = 10 -6 s -1 ).  Monotonic shear tests at 20°C (  = 10 -3 s -1 ,  = 10 -1 s -1 ).

 Cyclic shear test at 20°C (2 cycles, /2 = 0.1,  = 10 -3 s -1 ).

 Cyclic hardening at 20°C (50 cycles,  =  0.024,  = 4 10 -3 s -1 ).

 o Loading ( 2 = 0.06, 360°C, 2 10 -3 s -1 ) of a secondary specimen removed at 90° from the pre-tension axis.

The elasto-viscoplastic constitutive law was identified by homogenization approach. The procedure of identification was the following:

 Isotropic elasticity was supposed, Young's modulus (E =185000 MPa) and Poisson's ratio ( = 0.3) were taken from ASTM Material Handbook.  Yield limit ( 0 ) was evaluated has the ratio of the macroscopic yield strength (170 MPa) and the mean value of the resolved shear stress (2.65).  If no explicit information is available for the interaction matrix the first estimation is h rs = 1.  For the creep power law, the value n = 7 was initially taken, in agreement with deformation-mechanism maps proposed by Frost and Ashby (8) for the same material.  High value of kinematic hardening (d) was chosen, in order to fit the first cyclic loading on kinematic hardening, while low value of isotropic hardening (b) was 200 nm chosen to fit the following cycles of loading on isotropic hardening.

Experimental data at room temperature (20°C) were used for the first identification. The final set of parameters is presented in Table 6. To appreciate the implications of the slip systems selected, the Taylor factor M has been calculated. M can be defined (12) as the sum of the slip on all crystallographic slip systems s, divided by the norm of inelastic strain tensor  p .

p s s M     (12)
Therefore, high M values are expected when the orientation is not favorable for gliding, or when several slip systems are required to accommodate the strain incompatibilities. Figure 31 indicates a severe impact from the change of strain path on the distribution of the Taylor factors: first, an important positive shift of M is observed, corresponding to the activation of a significant number of additional slip systems to accommodate the deformation; second, the distribution of M is more symmetric after a change of strain path. The asymmetry in the distribution of M indicates that most of the deformation implies one slip system (lower M values), while a limited part of the deformation involves multiple slips (higher M values).

Correlation between strain hardening and oxidation

Results obtained after image analysis of inner oxide layers are plotted in Figure 32. These results suggest an increase of the mean oxide penetration with the local cumulated equivalent deformation

E

. Typically, the maximum and the mean oxide penetrations increased by a factor 2 when 12% of axial deformation is reached. The increase of the mean value of the Cr-rich layer could indicate the decrease of the efficiency of passivity property of the oxide when strain hardening increases. This could result in the effect of the local nonhomogeneous diffusion in the metal on the microstructure of the oxide (grain size) and associated transport properties. Previous studies [START_REF] Couvant | Investigations on the mechanisms of PWSCC of strain hardened austenitic stainless steels[END_REF] already mentioned the increase of the Cr-rich oxide layer with pH in primary water, associated to an increase of the IGSCC susceptibility. Thus, both mechanical and chemical properties are susceptible to alter the properties of the passive layer formed at the surface of stainless steels exposed to primary environment. The alteration of these properties promotes SCC initiation when, in addition, local stress concentration can fracture the passive layer.

Correlation between loading and initiation

Most cracks identified in this study are intergranular. Nevertheless, shallow transgranular cracking also initiated. By contrast, following a simple monotonic strain path, pure trans-granular stress corrosion cracking is usually observed (2). Therefore, complex loading promotes intergranular initiation of SCC.

Strain is not homogeneous by nature. In this study, distribution of deformation levels were measured and calculated at the scale of the polycrystal. Local stress and especially intergranular stress are associated with strain heterogeneity. Strain heterogeneity at the scale of the polycrystal aggregate seems to control possible sites of cracking. Indeed, cracks initiate where the strain gradients are important. Therefore, intergranular stress due to strain incompatibilities seems to drive the stress corrosion cracking mechanism. Consequently, slip traces intersecting intergranular cracks are often observed on one grain adjacent to the crack. The typical range of angle of intersection is 45° to 90°. This observation also is in good agreement with previous observations of intergranular crack tip (9).

Assumed mechanism

Based on previous considerations, the following basic mechanism is assumed to describeboth initiation and propagation of SCC in stainless steels exposed to primary water:

1. Oxidation, leading to the formation of a superficial Cr-rich passive layer with a mean thickness of 50 nm. The maximum oxide penetration does not exceed 100 nm at 360°C, when 7 < pH 320°C < 8, if the material is not strain hardened. Nevertheless, the mean and maximum oxide penetrations increase for higher pH values and when the material is strain hardened.

2. When sufficient, local stresses lead to the fracture of the oxide or metal-oxide interface. Strain hardening, loading fluctuations, intergranular strain incompatibilies and irregular metal-oxide interfaces promote stress concentrations and, as a consequence, initiation or crack tip extension. Changes of strain path enhance stress concentrations on the grain boundaries.

CONCLUSION

Understanding quantification of interactions between localized deformation in materials and environmental-assisted cracking could play role in maintaining the integrity of LWR components. Thus, a detailed understanding of strain localization during plastic deformation and of the underlying mechanisms is important for the manufacture and design of materials exposed to the environment of the primary circuit of PWRs.

The objectives of this study was to evaluate the relevance of interactions between localized deformation and initiation of stress corrosion cracking in austenitic stainless steels exposed to a PWR primary environment and to correlate quantitatively strain localization and selective oxidation.

Experimental investigations and computations helped to quantify the effect of a change of strain path on strain localization and to increase understanding of the contribution of local strain and stress on SCC.

Works are in progress in order to evaluate strain localization effect on oxidation and IGSCC of Nibased alloys. Future works will focus on the coupling of identified mechanisms (deformation, oxidation, fracture) in order to model initiation.

Figure 1 -

 1 Figure 1 -Example of inverse pole figures of 304L (left) and 316L (right) at similar scales.

Figure 2 -

 2 Figure 2 -Front (left) and back (center) faces of the cross specimen for prior deformation in ambient air. Specimen after final machining for testing at 360°C along Y 

Figure 3 -

 3 Figure 3 -3D finite element mesh with 343 grains, 2174653 nodes, and 1599480 elements c3d10.

  distribution Frequency Mean value = 0.092888 Median value = 0.088738 SD = 0.037705 Analyzed surface = 0.0625 mm² Density = 40000 measures/mm²

Figure 4 -

 4 Figure 4 -Map of equivalent deformation evaluated at the surface of specimen 1594-T112-06 (304L). Analyzed area = 0.06 mm 2 .

Figure 5 -Figure 6 -

 56 Figure 5 -Evolution of min., mean, median, max. deformation Eeq and SD versus analyzed surface (40000 measures.mm -2 ) of specimen 1594-T217-01 (316L).

Figure 7 -

 7 Figure 7 -Evolution of min., mean, median, max. deformation Eeq and SD versus analyzed surface (40000 measures.mm -2 ) of specimen 1594-T217-01 (316L).

Figure 15 -

 15 Figure 15 -Strain localization (second strain Eyy) in site 1 at the surface of specimen 1594-T112-06.

Figure 16 -

 16 Figure 16 -Oxide observation (STEM, bright field) at the surface of specimen 1594-T112-06 (304L).

Figure 17 -

 17 Figure 17 -STEM on site 1B at the surface of specimen 1594-T112-06 (304L).

Figure 18

 18 Figure18indicates the position of the EDS scan presented in Figure19. Shear bands are crossing the metal-oxide interface, and 'lines' are continuing in the oxide, suggesting that shear bands were emerging before testing in primary water. So, the more pronounced oxide penetration along slip bands probably result in prior deformation. The origin of the x-axis in Figure19is in metal. No O is observed during the EDX scan up to 120 nm. Nevertheless, in this range of position, a decrease of Fe content is associated to an increase of Ni content and a stable Cr content. In addition, the increase of

Figure 18 -

 18 Figure 18 -Location of EDX profile on site 1A at the surface of specimen 1594-T112-06 (304L). STEM, HAADF detector.

Figure 20 Figure 19 -

 2019 Figure 20 indicates the position of the EDX scan presented in Figure 21. This last scan was realized perpendicularly to oxidized slip bands. On the most oxidized slip band, 50 nm ahead of the mean local metal-oxide interface, Fe content is 20% lower than Fe in the metal, 10 nm away. By contrast relative Cr content increases up to 17% and relative Ni content increases up to 56%. The Cr enrichment is observed approximately 6 nm away from the oxidized slip band. The Ni enrichment affects 10 nm, similarly to Fe depletion. The narrow oxidation of emerging slip bands is supposed to be a precursor of crack extension, limited to tens of nm, leading to initiation.

Figure 20 -Figure 21 -

 2021 Figure 20 -Location of EDX profile on site 2 at the surface of specimen 1594-T112-06 (304L). STEM, HAADF detector.

Figure 22 -

 22 Figure 22 -Porosity in inner oxide layer, at the interface with the metal. STEM on site 2 at the surface of specimen 1594-T112-06 (304L).

Figure 23 

 23 Figure 23  Cr-mapping of oxide layer at the surface of specimen 1594-T112-06 (304L), in site 1B.

Figure 24 .Figure 25 .

 2425 Figure 24. Binary Cr mapping of image presented in Figure 23.Specimen 1594-T112-06 (304L), in site 1B.

Figure 26 .

 26 Figure 26. IGSCC initiated between grains with apparent different levels of deformation. Sampling on specimen 1594-T217-01 (316L), in site 2 (Prior-Exx = 0, second Eyy = 0.03).

Figure 27 .

 27 Figure 27. IGSCC initiated between grains with apparent different levels of deformation. Sampling on specimen 1594-T217-01 (316L), in site 3 (Prior-Exx  0.1, second Eyy  0.03).

Figure 28 .

 28 Figure 28. EFTEM O, Fe, Cr and Ni maps on an intergranular crack initiated between grains with apparent different levels of deformation. Sampling on specimen 1594-T217-01 (316L), in site 2 (Prior-Exx = 0, second Eyy = 0.03).

  Creep test at 20°C under constant load (250 MPa, 216 h) and under trapezoidal load ( max = 250 MPa,  min /  max = 0.8, hold time = 3420 s, unload time = 90 s, reload time = 90 s, 201 h).  Creep test at 360°C (250 MPa, 6000h), including the preloading tension curve (  = 10 -3 s -1 ). Creep rate is very low (  = 1.7 10 -12 s -1 ).  Creep test at 360°C (400 MPa, 8000h), including the preloading tension curve (  = 10 -3 s -1 ). Creep rate is very low (  = 2.8 10 -12 s -1 ).  Non-isothermal (20°C-360°C) relaxation at 250MPa and 350MPa  Cross test (complex loading path) including: o Prior-tension  1 = 0.31, 20°C, 2 10 -3 s -1

2 .

 2 Figure29shows the significant gradient of Von Mises stress after a 2% pre-deformation (a), especially at grain boundaries, and the important increase of such a gradient after a change of strain path (b). Computations on the aggregates allow exhibiting the heterogeneity of the equivalent stress (Von Mises) and of the strain fields. Figure30

  Figure29shows the significant gradient of Von Mises stress after a 2% pre-deformation (a), especially at grain boundaries, and the important increase of such a gradient after a change of strain path (b). Computations on the aggregates allow exhibiting the heterogeneity of the equivalent stress (Von Mises) and of the strain fields. Figure 30 shows distribution of  xx at the end of pre-loading in direction x  (mean( xx ) = 0.02) and for an intermediate deformation (mean( xx ) = 0.01). More precisely, we can see the increase of the heterogeneity of  xx when mean deformation increases. For example,  xx is in the range 0.002-0.020 after 1% elongation, while  xx is in the range 0.005-0.040 after 2% elongation. By contrast, there is no significant evolution of stress  xx in the same range of deformation.

Figure 29 -

 29 Figure 29 -Von Mises stress field after 2% el. in x  direction

Figure 32 -

 32 Figure 32 -Evolution of Cr-rich oxide penetration (min., mean, max., SD) at the surface of 304L stainless steel as a function of local cumulated axial strain cum axial E

Table 1 -

 1 Chemical composition (wt. %).

	Material	Heat	C (%)	S (%)	P (%)	Si (%)	Mn (%)	Fe (%)	Ni (%)	Cr (%)	Mo (%)	N (%)	Al (%)
	316L	XY183	0.026	0.004	0.033	0.42	1.81 65.75	12.00	17.34	2.57 0.050	-
	AISI Requirement for 316L	< 0.03	< 0.03	< 0.045	< 1	< 2		10-13 16.5-18.5 2-2.5 < 0.11
	304L	T2575	0.026	0.002	0.027	0.52	1.49 68.68	9.45	19.23	0.24 0.064	0.033
	AISI Requirement for 304L	< 0.03	< 0.03	< 0.045	< 1	< 2		8-10	18-20		< 0.11

Table 2 -

 2 Mechanical properties in air at room temperature.

	Material	Heat	Ys0,2 (MPa)	UTS (MPa)	El. (%)	HV30
	316L	XY183	255	553	63
	304L	T2575	247	582	60	160
	RCCM-M220 requirement	> 170	> 500	> 45	126-179

Table 3 -

 3 Mean prior-deformation (in x  direction) in ambient air.

	Material	Specimen	Analyzed area (mm²) Mean(Exx) Mean(Eyy) Mean(Exy)	Mean( (s -1 ) E  ) xx	Mean(Eeq) SD(Eeq)	RES(Eeq) (mm²)
		1594-T112-02	0.01	0.120	-	-	10 -3	-	-	-
		1594-T112-07	0.12	0.110	-0.018	0.006	1.2 10 -3	0.136	0.050	-
	304L	1594-T112-03 1594-T112-04	0.50 0.04	0.115 0.069	-0.024 -0.047	-0.003 0.001	1.6 10 -3 1.0 10 -3	0.127 0.070	0.037 0.025	0.014 -
		1594-T112-06	0.06	0.087	-0.055	0.000	1.0 10 -3	0.088	0.038	0.016
		1594-T112-08	0.01	0.178	-0.089	0.016	1.0 10 -3	0.186	0.068	-
		1594-T217-01	0.50	0.066	-0.047	0.001	1.3 10 -3	0.073	0.038	0.103
		1594-T217-02	0.01	0.038	-0.029	0.000	1.1 10 -3	0.043	-	-
	316L	1594-T217-04 1594-T217-07	0.29 0.01	0.077 0.065	-0.047 -0.056	-0.005 -0.002	1.5 10 -3 1.1 10 3	0.085 0.075	0.037 0.028	0.045 -
		1594-T217-08	0.01	0.156	-0.069	0.013	1.5 10 3	0.159	0.044	-
		1594-T217-10	0.04	0.065	-0.039	0.005	1.0 10 -3	0.069	0.026	-

Table 4 -

 4 Results of tensile tests (in y 

	Material	Specimen	Mean Prior-Exx	Duration (h) Mean Eyy Mean yy E  (s -1 )	Remarks
							No initiation after 270 h. Initiation of
		1594-T112-02	0.120	1400	 0.500	 9.9 10 -8	IGSCC then TGSCC propagation up to
							failure.
	304L	1594-T112-06 1594-T112-07	0.070 0.100	519 313	0.040 0.110	2.1 10 -8 9.8 10 -8	Significant IGSCC initiation + shallow TGSCC initiation Significant IGSCC initiation + shallow TGSCC initiation
		1594-T112-08	0.170	430	0.006	3.9 10 -9	Significant IGSCC initiation + shallow TGSCC initiation
		1594-T112-03	0.111	1864	0.010	0	Few sites of IGSCC
		1594-T217-01	0.066	720	0.020	7.7 10 -9	Shallow IGSCC initiation
	316L	1594-T217-02	0.040	1606	0.270	4.7 10 -8	Significant IGSCC initiation + TGSCC
		1594-T217-10	0.060	577	0.003	1.4 10 -9

direction) in a primary PWR environment at 360°C.

Table 5 -

 5 Analysis of Cr-rich inner oxide penetrations at the surface of the specimens.

	Material	Site	Resolution (

nm/pixel) Local strain in the sample Analysed length (nm) Cr-rich inner oxide penetration (nm) Prior Exx Second Eyy Mean Median Min. Max. SD

		1A	2.82	0.10	0.02	1386	102	96	31	172	30
	304L	1A 1B	1.35 2.56	0.00 0.00	0.00 0.075	688 694	102 49	100 49	55 23	170 87	30 12
		2	1.41	0.02	0.07	679	81	82	54	101	9
		1G	5.88	0.02	0.10	1188	72	71	12	124	27
		1D	5.88	0.00	0.02	988	90	88	59	135	17
	316L	2G 2D	5.95 1.72	0.00 0.10	0.02 0.01	2077 890	46 73	48 74	6 40	101 105	25 14
		3G	5.88	0.10	0.01	2941	96	94	41	141	16
		3D	1.74	0.10	0.02	880	61	61	19	92	16

Table 6 -

 6 Material parameters after identification.