Marc ; Heggen

Michael Engel

Sergiy Balanetskyy

Hans-Rainer Trebin

M Feuerbacher

Marc Heggen1

Michael Engel2

Sergiy Balanetskyy1

Hans-Rainer Trebin2

Michael Feuerbacher1

Structural variations in ϵ-type Al-Pd-(Mn, Fe) complex metallic alloy phases

Keywords: disclinations, phase transformations, plasticity of metals, TEM metadislocations, complex metallic alloys, tilings Transmission Electron Microscopy, Tiling, Metadislocation, Complex Metallic Alloys

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Several structurally related orthorhombic phases have been identified as ternary extensions of the Al 3 Pd phase in the Al-Pd-(Mn, Fe) alloy systems [1][2][3][4][5][6][7][8][START_REF] Yurechko | Phase Diagrams in Materials Science[END_REF]. They are observed in a composition range extending from the binary Al 3 Pd phase to about 5 % Mn and 10 % Fe [START_REF] Yurechko | Phase Diagrams in Materials Science[END_REF] and are termedphases. They are denoted l (l = 6, 16, 22, 28, 34 1 ) according to the index of the strong (0 0 l) diffraction spot which corresponds to the interplanar spacing of about 0.2 nm occurring in all of those phases [7][8][START_REF] Yurechko | Phase Diagrams in Materials Science[END_REF]. -type phases are also found in the alloy systems Al-Pd-(Rh, Re, Ru, Co, Ir) and Al-Rh-(Ru, Cu, Ni) [START_REF] Yurechko | Phase Diagrams in Materials Science[END_REF][START_REF] Balanetskyy | [END_REF].

In the literature, l -phases are betimes referred to using alternative notations: The 6 -and 28phases were denoted ' and , respectively [3][4][5][6]. Shramchenko and Denoyer [11] referred to the 6 -, 16 -, and 28 -phases as ', '_1, and '_3. On the other hand, Engel [12] referred to these phases as ', ' 1 , and ' 2 .

A particular feature of complex metallic -type phases is the occurrence of phase modulations and intermixing of structural elements. Modulated -type structures were observed in different alloy systems like Al 3 Pd, Al 72 Pd 25 Cr 3 , Al-Pd-Mn and Al-Pd-Fe [3,[START_REF] Balanetskyy | [END_REF][START_REF] Hiraga | The structures of decagonal and quasicrystalline phases with 1.2 nm and 1.6 nm periods, studied by high-resolution electron microscopy[END_REF]].

In the present paper, structural modifications of -type phases in Al-Pd-(Mn, Fe) are studied by 3 means of transmission electron microscopy (TEM). We describe local structural modifications related to the density and inclination of phason planes occurring at a phase boundary between 6and 28 -Al-Pd-Mn, where the phase boundary is built up by metadislocations. Furthermore structural alterations in the vicinity of metadislocations in -type Al-Pd-Mn and Al-Pd-Fe samples are studied. The presence of local unit cells corresponding to new monoclinic -type phases is reported. These phase modulations are discussed in terms of two tiling models. A generalized classification of -type structures is presented.

Structural description

The basic building blocks of all -phases are icosahedral-symmetric pseudo-Mackay clusters which are arranged in columns along the [0 1 0] direction [4]. The projection of the column centres on the (0 1 0) plane can be described in terms of so-called cluster tilings [1,4,[START_REF] Engel | [END_REF]. Two different tiles are used, a flattened hexagon H and a combination of a nine-edged banana-shaped polygon with a pentagon, termed phason line PL [3,5]. Figure 1 a shows the H and PL tiles, which are present in two different orientations, denoted H1, H2, PL1, and PL2. Figure 1 b andc depict the tiling representations of the phases 6 and 28 . These phases feature the orthorhombic space group Pnma and C2mm, respectively [4,24]. The 6 structure can be described by alternating rows of H1 and H2 tiles [4]. The 28 structure is characterised by additional PL1 tiles, which are aligned in rows along the [1 0 0] direction (figure 1 c). An equivalent version of the 28 structure can be constructed using PL2 tiles. All l phases have identical [1 0 0] and [0 1 0] lattice constants, which, in the case of the Al-Pd-(Mn, Fe) alloy system, are a = 2.35 and b = 1.66 nm [4,[START_REF] Yurechko | Phase Diagrams in Materials Science[END_REF]. The [0 0 1] lattice constants of the l phases differ. The lattice constants of the phases 6 and 1 The 34 phase was only observed in the alloy system Al-Pd-Mn [START_REF] Yurechko | Phase Diagrams in Materials Science[END_REF]. In the present work all translation vectors are expressed in terms of the basis of 6 . Furthermore, we will use the term -type phases not only for the phase sequence l (l = 6, 16, 22, 28, 34), but generalize it to all phases which can be described using the tiles H1, H2, PL1, and PL2 without creating gaps or overlaps. This includes also the so-called phase, which is structurally closely related to 6 . It consists of a parallel arrangement of only one type of H tiles [4].

A mixture of structural elements belonging to different -type phases is observed in the vicinity of metadislocations. These metadislocations are a new type of structural defect, which was first observed in the 6 -and 28 -phase [5,15]. They are always associated to a certain number of planar defects, so-called phason planes. A phason plane in the 6 -phase can be structurally described by a row of PL-tiles along the [1 0 0] direction in an H-tile matrix. Since, on the other hand, a row of PL tiles is a structural element of the 28 -phase (cf. figure 1 c), the presence of metadislocations leads to a mixture of structural features of the 6 -and the 28 -phase [5,16].

As an alternative to the cluster tiling involving H and PL tiles, other tilings can be used to describe the structure of -type phases. The rhomb tiling [3,12] offers a simpler geometrical description than the cluster tiling. The relation between these alternative tilings is depicted in figure 2 a. The cluster-tiles H and PL have an even number of vertices. Therefore, the vertices can be divided into two disjoint sets (i.e. the cluster tiling is bipartite). By connecting the nextneighbours of one of those sets the alternative tiling is obtained: H1/H2 are substituted by the obtuse rhombs OR1/OR2, PL1 is substituted by a -inflated hexagon H, and PL2 is substituted ). This substitution is area-preserving, reversible and can be applied to any cluster tiling representing an -type phase. The tiles B and H can be further decomposed into a combination of OR1, OR2, and an acute rhomb AR (figure 2 b). With this decomposition step, the cluster tiling can be substituted by a tiling using only obtuse and acute rhombs, OR1, OR2, and AR, and will then be referred to as a rhomb tiling [12]. The substitution of the tile H by OR1, OR2, and AR, however, is not reversible as the back transformation is ambiguous. For the classification of -type phases in this paper, representations in terms of the cluster tiling and the rhomb tiling will be used.

In -type phases local rearrangements of cluster centres are possible. These can be represented in the cluster tiling by an exchange and [0 0 1] displacement of PL1 and PL2 tiles (figure 2 c). This local movement of PL tiles, involving a correlated rearrangement of atomic clusters, is called a phason flip [3,17]. Phason flips can also be described using the tiles OR1, OR2, B, and H (figure 2 d) and the rhomb tiling (figure 2 e). In terms of the rhomb tiling, the distinction between PL1 and PL2 tiles is not possible (figure 2 e). Accordingly, the substitution of the second and third tiling of figure 2 to about 0.5 % strain. According to previous investigations on the plastic deformation properties of this material, the tests were performed in the ductile regime at a temperature of 700 °C. At temperatures below about 650°C the material is brittle [23]. The long specimen axis, corresponding to the compression direction, was oriented at an inclination of 45° with respect to the [0 1 0] direction in order to provide favourable conditions (i.e. a high Schmid factor) for metadislocation glide. The modes of metadislocation motion were discussed in detail by Feuerbacher and Heggen [22]. After mechanical testing the sample was quenched on a cold metal plate and prepared for TEM by standard techniques. The microstructural investigation was performed using a JEOL 4000 FX microscope and a Philips CM 200 FEG microscope equipped with a spherical-aberration corrector [18]. 6 -as well as 28 -phase regions, formed during plastic deformation, were observed. The phase boundary region between both phases was analysed. The phason plane density on the left hand side of the metadislocation is much higher than at the right hand side. In addition to the phason planes, a stacking fault [START_REF] Balanetskyy | [END_REF] is observed on the right hand side of figure 6. This type of defect was discussed by Klein et al. [15], but is not a matter of discussion in the present investigation.

The local structure of some selected areas in this sample region is analysed in figure 7 between those vectors is 108° and 72° rather than 90°. The lattice can therefore be considered a monoclinic rather than an orthorhombic lattice.

A similar structural study was made using an Al-Pd-Fe sample. Figure 8 shows a lattice fringe image in the vicinity of metadislocations (labelled 1 and 2), which are connected to -phase regions (labelled 3 and 4) [19]. The course of selected groups of phason-plane segments is indicated by curved lines. Again, they show a variable inclination with respect to the [1 0 0] direction, and a variable density of phason planes is observed. 12 the presence of metadislocations, phason planes are geometrically forced to circumvent defected areas (for example the ones labelled 8 in figure 6). This gives rise to various phason-plane inclinations with respect to the [1 0 0] direction. Furthermore, the introduction of additional phason planes by metadislocations (for example the ones labelled 3 in figure 6) causes a local variation of the phason plane density and to a local structural variation within the sequence of lphases (l = 6, 16, 22, 28, 34). The structural variations around the metadislocations are found in both samples investigated here, the plastically deformed single crystalline Al-Mn-Pd and cast polycrystalline Al-Mn-Fe sample. These observations are not dependent on selective processing of the samples.

The observation of phason plane inclinations with respect to the [1 0 0] direction leads a structural description in terms of a tiling model which is elaborated in the following subsection.

These local phason plane inclinations can be described in terms of monoclinic -type structures.

The discussion in terms of monoclinic structures is based on the structural investigation within the (0 1 0)-plane. Potential structural changes along the [0 1 0]-direction are not considered.

Monoclinic -type bulk phases have not been observed in the Al-Pd-(Mn, Fe) alloy system before.

An important characteristic regarding structural modifications connected to metadislocations intype phases is the mutual sense of the terms "defect" and "structural element". Phason planes associated with metadislocations are defects, but, on the other hand, they are elements of the ideal structure in all -type phases but 6 . In this sense, the presence of defects is always necessarily connected to structural variations of the surrounding material and to the local appearance of new monoclinic structures around the defects. Hiraga [START_REF] Hiraga | The structures of decagonal and quasicrystalline phases with 1.2 nm and 1.6 nm periods, studied by high-resolution electron microscopy[END_REF] studied decagonal and quasicrystalline phases by high-resolution electron microscopy in Al-Pd-(Cr, Co, Fe). He observed modulated -type structures, which can be classified according to our scheme. The areas shown in figure 11 of Hiraga's work can be identified as (3,1), (2,-1), (2,0), and (0,0) [START_REF] Hiraga | The structures of decagonal and quasicrystalline phases with 1.2 nm and 1.6 nm periods, studied by high-resolution electron microscopy[END_REF].

A comparable classification can be created in terms of the rhomb tiling. The unit cell is spanned by the basis vectors a and c, while b is the (fixed) basis vector perpendicular to the tiling plane.

The vectors a and c are given by integer linear combinations of the three vectors e i , i = 1, 2, 3, that span the rhomb tiles OR, AR (cf. figure 2). In contrast, the edges of the cluster tiling are described by a regular five-star basis. In this sense the rhomb tiling is a geometrically simpler description than the cluster tiling. The vector c connects two PL tiles in the [0 0 1] direction, while the vector a connects two next-neighbouring PL tiles along or inclined with respect ot the The parameters m, n can be determined by counting the relative densities of the tile types:

/ 1 c m = + + , (2) 2
/ 2 a n = + + , ( 3
) 2 tan n + = , (4)
#(H1) #(H2) #(PL) m + = , #(H1) #(H2) #(PL) n = . (5)
A direct consequence of these relations is the restriction n m .

It should be noted that the phases of the present classification scheme represent "simple" phases which can be built with the tiles H1, H2, PL1, and PL2 in the sense that they are described by two translation vectors connecting PL tiles. This results in structures which are characterised by a uniform distribution of straight phason lines. However, more complex periodic phases using H and PL tiles can be constructed, e.g. by the creation of supercells of the presented simple phases.

  irrational number of the golden mean and c( 6 )= 1.23 nm. Similar relations are found for other l phases: The factor relating their [0 0 1] lattice constant to that of 6 are + 1

  -shaped tile B (figure2 a

Furthermore, structural modificationsFigure 3

 3 Figure 3 shows an electron micrograph of a phase boundary between the 6 -(left-hand side) and

  a to c, which depict enlargements of the boxed regions of figure 6. Within areas of 10 to 20 nm in diameter, homogeneous local structures are observed, which can be matched by tilings. In figure 7 a, a space-filling tiling is made using PL and H tiles. The corresponding unit cell is spanned by the vectors (1 0 0) and (0 0 3+ ). This local structure can therefore be identified as 28 phase. The local structure in figures 7 b and c is characterised by inclined phason planes and different numbers of H1 and H2 tiles between the phason planes. The unit cell is spanned by the vectors (1 and (0 0 4+ ) (figure 7 b), and (1 0 1) and (0 0 6+ ) (figure 7 c), respectively. The angle

  Figures 9 a to c depict enlargements of the boxed regions in figure8, which are matched by cluster tilings. In figure9a the corresponding unit cell is spanned by the vectors (1 0 0) and (0 0 1+ ). This local structure can be identified as 16 phase. In figures 7 b and c the unit cell is spanned by the vectors (1 0 -1), (0 0 4+ ) and (1 0 -1), (0 0 2+ ), respectively. § 4. DiscussionMicrostructural investigationsIn this paper, a TEM study on structural modifications in complex metallic -type phases is presented. These structural modifications are connected to phason defects and metadislocations and occur on different length scales. For instance, figures 5 a and b show structural modifications, i.e. phason flips, on the scale of the lattice parameter. Single PL elements exhibit a modulated behaviour within the phason planes which are, on average, oriented along the [1 0 0] direction. Beyond that, figure3, 6, and 8 demonstrate that structural modulations exist on a larger some tens to hundreds of nanometres. They depend on the curvature and density of phason planes in these areas.On the unit-cell scale, local structural disordering was observed at an 6 -28 -phase boundary by means of high-resolution TEM. Various distances between phason planes along the [0 0 1] direction and the local occurrence of other unit cells (e.g. 34 ) were found. The average structure in the vicinity of the phase boundary has a larger separation between the phason planes than the perfect 28 -phase. Single PL elements are moved out of regularly stacked (0 0 1) planes by phason flips which results in a wavy course of the phason planes (figure5). In addition, the electron diffraction investigation reveals structural modulations in the direct vicinity of the phaseboundary region. The shifts of spots can be related to the presence of phason disorder[20,21].On a larger scale structural modifications are connected to the course of phason planes which are observed as dark lines in figure3. Metadislocations terminating the phason planes are not ideally stacked along [0 0 1] direction, but may be distant from the boundary along the [1 0 0] direction (see arrowhead in figure3). Accordingly, the density of phason planes varies locally and, on average, increases gradually from the 6 -to the 28 -phase region. As -type phases within a sequence l (l = 6,16,22, 28, 34) are distinguished by their [0 0 1] lattice constants, i.e. the distance between phason planes along the [0 0 1] direction, the variation of the phason plane distance due to the presence of metadislocations is associated with local structural variations. No sharp phase boundary but rather a transient region is observed between related -type phases. The intermediate region is characterized by a variation of the phason plane density leading to a continuous phase change and the presence of phason disorder. Structural modulations connected to the course of phason planes were furthermore investigated in the vicinity of metadislocations in an Al-Pd-Mn and an Al-Pd-Fe sample (figures 6 to 9). Due to

  In figure10, a systematic classification scheme for -type orthorhombic and monoclinic phases is presented. PL1 and PL2 tiles are depicted white, H1 and H2 tiles dark grey and light grey, respectively. The presented -type structures are characterized by different periodic arrangements of straight phason planes. They differ with respect to their inclination along the [1 0 0] direction and their spacing along the [0 0 1] direction. The central column of the figure represents the tilings of the orthorhombic -phases. From top to bottom we find 16 , 22 , 28 , 34 , and 40 . Their lattice vectors, expressed in terms of the basis of 6 (cf. §1; the basis vectors are shown in figure 2b), are given by (1 0 0) and (0 0 z+ ), z = 1, 2, 3, 4, 5,…. The first, second, fourth, and fifth column of figure 10 show -type tilings of monoclinic structures spanned by (1 0 -2), (1 0 -1), (1 0 1), and (1 0 2) translation vectors, respectively. Note that the structures in corresponding columns on the left and right hand side are related by a mirror operation. Tilings shown in the same row possess the same translation vector along the [0 0 1] direction; tilings in the same column possess the same inclination of phason planes. The tilings shown in figure 10 can be classified by two indices ( , ) m n according to the translation vectors (0 0 m+1+ ) and (1 0 n), with m = 0, 1, 2, 3,... and m n . The index n is confined corresponding to the latter relation in order to avoid overlap of the tiles. For reasons of presentation, the phases (3,-3), (4,-3), (4,-4), etc. were not shown in figure 10. Orthorhombic and monoclinic phases are classified by the indices 0 = n and 0 n , respectively. The 6 -phase and the -phase, consisting only of H tiles, can be expressed in terms of the present classification as the limit of an infinite distance between the phason planes, i.e. by the indices ( , 0) n and ( , ) n n ±for n , respectively.

  figure 7 and 9 can be assigned to the structures (3, 1) (figure 7 b and fig. 9 b), (4,1) (figure 7 c),

[ 1 0

 1 0] direction. In the { } i terms of the rhomb tiling is shown in figure11. OR1 and OR2 tiles are depicted white and AR tiles grey. The parameter m specifies the distance of the phason planes, and the parameter n the inclination of the phason planes with respect to the [0 0 1] direction. The lattice constants a, c, and their angle can be calculated as a function of m, n:

  the H1, H2, and PL tiles per unit cell is #(H1) m n = + , #(H2) m n = , #(PL) 2 = .

Figure 1 :

 1 Figure captions

Figure 2 :

 2 Figure 2: (a) The cluster tiles (light grey) can be substituted by the tiles OR1/OR2, H, and B.

Figure 3 :

 3 Figure 3: Transmission electron micrograph of a phase boundary in Al-Pd-Mn formed by

Figure 4 :

 4 Figure 4: Diffraction patterns of the sample areas a, b, and c of figure 3. The insets show

Figure 5 :

 5 Figure 5: High resolution micrographs of an Al-Pd-Mn sample in the 28 -phase region close to

Figure 6 :

 6 Figure 6: Lattice fringe image of an Al-Pd-Mn sample containing four metadislocations (1 to 4)

20 Figure 7 :

 207 Figure 7: Enlarged sections of the areas marked a, b, and c in figure 6. The local structure is

Figure 8 :

 8 Figure 8: Lattice-fringe image of an Al-Pd-Fe sample containing two metadislocations (1 and 2)

Figure 9 :

 9 Figure 9: Enlarged sections of the areas marked a, b, and c in figure 8. The local structure is

Figure 10 :

 10 Figure 10: Classification of -type phases using the cluster tiling. The phases are characterized by

Figure 11 :

 11 Figure 11: Classification of -type phases using the rhomb tiling. The tilings of figure 10 and 11

Figure 3 :Figure 5 :Figure 7 :Figure 9 :

 3579 Figure 3: Transmission electron micrograph of a phase boundary in Al-Pd-Mn formed by metadislocations between the 6-phase on the left and the 28-phase on the right hand side. Phason planes associated to the metadislocations can be observed as dark lines. One metadislocation is marked by an arrow. Diffraction patterns were taken at the sample areas a, b, and c. 529x385mm (72 x 72 DPI)

c and d leads to equal rhomb tilings in figure 2 e. § 2. Experimental details

	planes. A sample of approximately 5×2×2 mm 3 in size was uniaxially deformed in compression
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	An 6 -Al-Pd-Mn single crystal was grown by means of the Bridgman technique. The final
	composition of the ingot was Al 73.6 at.%, Pd 22.1 at.%, Mn 4.3 at.%. It was characterised
	metallurgically and oriented by Laue diffraction. The undeformed sample was investigated by
	means of TEM and shows the presence of pure 6 -phase with a low density of defects like phason
	6

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgments

We thank Dipl.-Ing. C. Thomas and M. Schmidt for the materials production. This work was supported by the Deutsche Forschungsgemeinschaft (PAK 36), and the 6 th Framework EU Network of Excellence "Complex Metallic Alloys" (Contract No. NMP3-CT-2005-500140).


Laijun ; Liu

Huiqing Fan
email: hqfan3@163.com

Lei Wang

Xiuli Chen

Pinyang Fang

induced Dielectric Relaxation and AC Conduction in CaCu3Ti4O12 Ceramics

Keywords: electronic properties, ferroelectrics, functional materials, grain boundary interfaces, hopping conduction, perovskites, titanium oxide Dielectric relaxation, Grain boundary, Impedance spectroscopy Dielectric relaxation, Grain boundary, Impedance spectroscopy PACS number(s): 77.22.Ch, 77.84.Dy

come

Introduction

Miniaturizing capacitor devices is demanding for higher dielectric permittivity materials progressively. In recent years, many researches have shown that the perovskite structure calcium copper titanate, CaCu 3 Ti 4 O 12 (CCTO), possesses a giant dielectric permittivity of 10 5 at room temperature, which is practically frequency independent between dc and 10 6 Hz. In addition, this material does not show ferroelectric transition in a wide temperature range (100-600 K), where the dielectric permittivity value remains constant, without suffering from limitations of commonly used ferroelectric materials. [1][2][3][4][5][6] The dielectric permittivity abruptly drops down to a value, ~100 when the temperature lowers below 100 K. 7,8 The dielectric response behavior of the material is characteristic of Debye-like relaxation with a single relaxation time. 9 A number of attempts to interpret this behavior in terms of intrinsic material properties 10,11 as well as an impurity and interface mediated phenomena 12 have been reported. One of the leading explanations for giant dielectric permittivity is the formation of nonconductive grain boundaries between conductive grains, i.e., grain-boundary layer capacitor behavior, common in doped perovskite titanates. [START_REF] Hench | Principles of Electronic Ceramics[END_REF] This model was supported by impedance spectroscopy. [START_REF] Fang | [END_REF][15][16] Recently, however, it is reported that the giant dielectric permittivity is associated with electrode polarization effects due to nonohmic contacts but not due to a grain boundary effect. 17 Generally, the electrode effects can be well separated by using different contacts and sample geometries. 18 However, few researchers attempted to separate electrode effect from grain boundary effect. Up to now it is still unclear that which effect is responsible for the giant dielectric permittivity in CCTO. This question may be resolved if the two responses, electrode relaxation and grain boundary relaxation, can be distinguished.

In this work, the measurement of dielectric and impedance spectroscopy of CCTO ceramics is reported in the temperature range from 298 K to 358 K at 0, 5, 10, 15, 20 and 40 V dc bias. A low frequency dielectric relaxation arising from electrode polarization was induced by dc bias. The electrode response and grain boundary response have been successfully distinguished from each other according to impedance spectroscopy and the frequency dependent ac conductivity with a dc bias.

Each component acts as a depletion layer to induce their own respective dielectric relaxations.

Experiment

Reagent grade oxides and nitrates, Ca(NO 3 ) 2 •4H 2 O, Cu(NO 3 ) 2 •3H 2 O and TiO 2 and sodium potassium chloride(NaCl : KCl = 44 : 56) were used as the starting materials. Appropriate amounts of the oxide and chloride powders were mixed together in the ball mill using zirconia balls as milling medium in ethanol for 12 h.

The dried powder mixtures were placed in covered high alumina crucibles, and then heated in a chamber furnace at 800 o C for 1 hour. On the completion of reaction, flux X-ray diffraction patterns were obtained using an automated diffractometer (XRD; X'Pert PRO MPD, Philips, Eindhoven, Netherlands) with Cu K 1 radiation. The surface morphologies of CCTO ceramics were observed by a scanning electron microscope (SEM; JSM-5610, JEOL, Tokyo, Japan). Both sides of the samples were polished and then brushed with silver conductive paste followed by heat treatment at 550 o C for 30 minutes in order to ensure good electrical contact. Electrical properties measurements were taken with an applied voltage of 500 mV using an Agilent 4294A impedance analyzer over the frequency range 100 Hz-13 MHz at room temperature and 100 Hz-100 kHz at selected temperatures between 298 K and 358 K at 0, 5, 10, 15, 20 and 40 V dc. The frequency dependence of the real part and imaginary part of dielectric permittivity of CCTO ceramics at 0 V, 5V, 10V, 15 V and 20 V dc bias at room temperature is shown in Fig. 3. The real part of dielectric permittivity with no dc bias is ~100,000 over a wide frequency range between 100 Hz and 7 MHz. And it enhances sharply with the increase of dc bias in low frequency (< 3 kHz), but an inverse change happens in high frequency (> 10 kHz, Fig. 3 inset). Two Debye-like relaxations present in the imaginary of dielectric permittivity, one is in 10 6 ~10 7 Hz associating with grain/domain response; the other is below 100 Hz associating with grain boundary response and/or electrode response. Admas et al. 16 found that the capacitance of grain boundary decreases with bias increasing. This is consistent with the expectation of a Schottky barrier model for the grain boundary with a reduction in potential barrier height in the forward bias direction and an increase in the width of depletion region. The dielectric-dc bias behavior of CCTO ceramics is consonant with this result in high frequency range. However, the low frequency dc-bias-field-induced dielectric relaxation is very interesting.

Results and discussion

Nonohmic electrode contacts always occur with many different types of materials due to their different electron work functions and electrode contact conditions and that surface roughness and wetting. The electrode polarization at the interface between the sample and electrode, like grain boundary response, may be one of the sources of the giant dielectric permittivity. In the following we will adopt the view that the change of dielectric permittivity in low frequency range is mainly arisen from external contacts. It is well known that thin depletion layers can arise, e.g., due to the formation of Schottky diodes at the electrode/bulk interface. 19 These thin layers of low conductivity act as high capacitance in parallel with a very large resistor, connected in series to the bulk, which can results in the erroneous detection is very large values of dielectric permittivity. Under the suitable conditions like application of a static bias across the sample, a pile of these low mobility charge carriers is collected around one of blocking electrode. This leads to a field gradient near the electrode and gives rise to a non-linear dielectric permittivity and loss tangent. Considering the blocking nature of the electrodes, two relaxation peaks should be observed, unfortunately, we can not find two complete peaks in Fig. 3. The reason is the measurement limit of our impedance analyzer. We can not obtain a high temperature and a wide range frequency. We will detect them by impedance spectroscopy.

C R C R C R C R j C R R C R R jX R Z + + + + + + = + = (1)
To further corroborate our findings, we also research the ac conductivity of CCTO ceramics. The bulk dc conductivity follows a variable-range-hopping mechanism under which the bulk ac conductivity also obeys a power law. the temperature is low and the lowest measurement frequency is still high. As seen from these lines, the conductivity of them increase rapidly starting at frequency f ~ 100 Hz, beyond which the can be described as

2 1 f f a t gb + + = ( 2
)
where gb is the dc grain boundary conductivity, a 1 , t and are three adjustable constants. In this formula, the gb +a 1 f t term describes the grain boundary conductivity relaxation. The conductivity of CCTO ceramics changes from the grain boundary response to the grain conductivity (domain conductivity) with the increase of frequency. The f 2 term describes approximatively the transition between the above two mentions relaxation behaviors.

A dc bias of 20 V is loaded on CCTO ceramics in order to find electrode relaxation clearly. Fig. 5b shows the frequency dependence on the ac conductivity at 298 K and 348 K. A long plateau presents in low frequency, associating with the dc electrode conductivity e . As the frequency f is increased to inflexion, also increases as a result of electrode conductivity relaxation as described by

s e f a 2 + = ( 3
)
where a 2 and s (0 < s 1) are two temperature dependent adjusting constants.

The a 2 f s term is an empirical expression representing the transport properties of polarons, electrons, and ions. 20 The solid lines are the fitting curves according to Eq. ( 2) in Fig. 5a. With an absence of dc bias, it is not clear that the conductivity changes from electrode relaxation to grain boundary relaxation. Maybe it happens in very low frequency.

) exp( 0 T k E B a = ( 4
)
where 0 is the prefactor, E a is the activation energy for the relaxation, k B is the Boltzmann constant, and T is the absolute temperature. Fig. 5a inset shows the plot of log versus 1/T, in which the solid line is fitted result using Eq. 4. Both obey the Arrhenius law with grain activation energies of 0.122 eV and grain boundary activation energies of 0.40 eV with an absence of dc bias.

The dot lines (1 and 3) are the fitting curves according to Eq. ( 3) and the solid lines (2 and 4) are the fitting curves according to Eq. ( 2) in Fig. 5b. The coefficients of fitting formulas are shown in Table I. The electrode conductivity increases from 1.10×10 -6 S/cm to 1.74×10 -4 S/cm as dc bias increases from 0 V to 20 V at 358 K.

Also, the grain boundary conductivity increases with the increases of temperature and dc bias (Fig. 4 and5). This reveals the strong electric field dependent nature of both electrode conductivity and grain boundary conductivity. Meanwhile, it is considered the conductivity is associated with electrode polarization effects and the development impedance spectroscopy and dialectic constant dependence of electrodes also show grain boundaries and electrode playing some important roles in giant dielectric constant. 16,17 In this work, our focus is the influence of grain boundaries and electrode on dielectric response.

Conclusions

CCTO ceramics fired silver electrodes were measured at different temperatures with a dc bias. The dependence of dielectric permittivity on dc bias clearly proves that the electrode causing an interfacial polarization process results in the detection of very strong dc-bias-field-induced dielectric relaxation.

  Suchnonohmic electrode contacts are well known to occur with many different types of materials due to their different electron work functions. The interface polarization Maxwell-Wagner relaxation, has been observed in many materials.

  from the powder particles by repeated washing with hot deionized water until no chloride ions were detected by silver nitrate. The obtained CCTO powder was pressed into pellets of 15 mm diameter and ~1 mm thickness by cold isostatic pressure method. The pellets were sintered at 1060 o C in air for 48 hours.

Figure 1

 1 Figure1shows typically the room-temperature XRD pattern for the CCTO

Z

  * plot of data obtained of CCTO ceramics with a dc bias of 0 V (a) and 10 V (b) at different temperatures are shown in Fig. 4. The resistances of grain and grain boundary decrease with the increase of temperature and dc bias. A large arc with nonzero intercept at high frequency (> 10 6 Hz) is shown in Fig. 4a and inset. Only a small section of the grain boundary arc can be observed at low temperature, and a steep incline at low frequency suggests that the resistance of grain boundary is much more than 10 5 •cm and its response frequency is less than 100 Hz. The grain boundary response and electrode response are still not distinguished. An impedance spectroscopy of CCTO ceramics with a dc bias of 10 V is shown in Fig. 4b and right inset. Both grain boundary resistance and grain resistance decrease with the increase of dc bias and temperature, which is consistent with a Schottky barrier response. The Fig. 4b shows the measuring impedance spectroscopy and calculating impedance spectroscopy according to an equivalent circuit, a series circuit of parallel RC elements. The impedance attributing to grain and grain boundary is calculated by Eq. 1 (Grain resistance R g = 135.27 , gain capacitance C g = 17.4 nF•cm, grain boundary resistance R gb = 5375.48 and gain boundary capacitance C gb = 80.49 nF•cm). Here shows a semicircular arc with its center lying underneath the abscissa. In comparison with Z* plot data, here the impedance changes from electrode relaxation to grain boundary relaxation as frequency increases beyond f = 19 kHz at 328 K with a dc bias of 10 V, as indicated by an arrow. The relaxation time of electrode and grain boundary is 0.955 ms and 0.026 ms with a dc bias of 10 V at 328 K, respectively. It is clear that the impedance responses from grain boundary and electrode are convoluted in the semicircular arc even if both measuring temperature and dc bias increase.

  This has been attributed to polaron relaxation, a phenomenon widely recognized in defective perovskite structure at low temperature. The frequency dependence on the ac conductivity at different temperatures is shown in Fig. 5a. The conductivity almost presents a step for T = 358 K at low frequency, as marked by ellipse, which is approximately equal to the dc electrode conductivity e . The step is very short because

  from negative value to positive value (from -5.75×10 -12 to 3.66×10 -12 S/cm), suggesting that the relationship between grain relaxation and grain boundary becomes closer with the increase of temperature. The activation energies of the aforementioned electrical responses from grains and grain boundaries are calculated. The activation energy is associated with the height of the potential-energy barrier restricting the motion of charge carriers. Conductivities of the grain (bulk), b , and grain boundary, gb , components are obtained and plotted against reciprocal temperature in Arrhenius format.

10 of

 10 Schottky barriers at the metal electrode/ceramic interface due to nonohmic contacts and grain boundaries effect.At recent, Zhu et al.21 claimed the existence of nanoscale disorder of Cu/Ca makes a remarkable effect on the dielectric properties of CCTO. A metalliclike polarizability in the surrounding insulating regions enhances dynamical electronic dielectric response. This mechanism is different from the "internal barrier layer capacitor" (IBLC) picture. It may be the origin of giant dielectric constant in CCTO single crystal and grains in CCTO ceramics. However, in CCTO polycrystals,

FIG. 4 .

 4 FIG. 4.Impedance complex plane plot obtained for CCTO ceramics with a dc bias

  Figure 1

TABLE I .

 I Conductivity and temperature-dependent adjusting constants of Eq.(2)

		and (3) in the fitting curves.
	FIG. 1.	XRD pattern of molten salt derived CCTO ceramics.
	FIG. 2.	SEM image of molten salt derived CCTO ceramics.
	FIG. 3.	Dielectric permittivity of CCTO ceramics as a function with different dc
		biases at room temperature. Debye-like relaxations present in low
		frequency with the increase of dc bias.

TABLE I Condition

 I

		gb	a 1	t	e	a 2	s
	Dc 0 V at 298K	-6.88	1.055	0.134	-5.75×10 -12
	Dc 0 V at 338K	-6.723	1.332	0.102	3.66×10 -12
	Dc 0 V at 358K	-5.96	1.016	0.107	2.54×10 -12
	Dc 20 V at 298K	-9.6	2.89	0.084	-7.32×10 -11 -4.76	0.005	0.643
	Dc 20 V at 348K	-6.9	1.478	0.095	-1.16×10 -12 -3.76	0.179	0.085

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgements

Page 10 of 23 http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters This work has been supported by the National Nature Science Foundation (50672075) and the Xi'an S&T Research Foundation (GG05015, GG06023), the SRFDP (20050699011) and EYTP and NCET Program of MOE, Doctorate Foundation (CX200704) and Science Creative Foundation (2006CR06) NPU of China.


Henry Proudhon

Warren J Poole
email: warren.poole@ubc.ca

Xiang Wang

Yves Jm Brechet

Y Bréchet

The role of internal stresses on the plastic deformation of the Al-Mg-Si-Cu alloy AA6111

Keywords: precipitation, strengthening mechanisms, stress-strain measurements, work-hardening

The role of internal stresses on the plastic deformation

of the Al-Mg-Si-Cu alloy AA6111

Introduction

A complete understanding of the plastic behaviour of two phase materials requires a multi-faceted approach as the presence of a second phase introduces a variety of new possible deformation mechanisms compared to single phase materials. At a general level, the second phase modifies the work hardening of material due to partitioning of stress and strain between the two phases. The modeling of these stresses for a wide range of volume fractions and particle shapes can be examined using by continuum models such as load transfer models, Eshelby and finite element method calculations [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF][START_REF] Clyne | An Introduction To Metal Matrix Composites[END_REF]. Alternatively, dislocation based models can be considered. Using the concept of geometrically necessary dislocations, the seminal work of Ashby examined the work hardening of Cu-SiO 2 single crystals [START_REF] Ashby | Strengthening Methods in Crystals[END_REF]. In this work, an intrinsic length scale, "the geometric slip distance", can be introduced to account for the effects of particle size, shape (plates vs. spheres) and volume fraction. In recent years, it has become possible to examine plasticity problems using discrete dislocation models; this seems to be promising in complex situations such as two phase systems e.g. see Cleveringa and Van Der Giessen [START_REF] Cleveringa | Comparison of discrete dislocation and continuum plasticity predictions for a composite material[END_REF], but much more work is needed here before commercially relevant systems can be explored.

Each approach has its limitations. For example, dislocation models have difficulty in predicting internal stresses such as observed during strain path changes (e.g. Bauschinger tests) while continuum models are inherently scale independent and cannot account for particle size effects. As a result, modelling approaches which incorporate aspects of both continuum and dislocation based approaches have been developed [START_REF] Embury | Plastic flow in dispersion hardened materials[END_REF].

For example, Brown and Stobbs developed a comprehensive framework to describe the work hardening behaviour for the Cu-SiO 2 single crystals system [START_REF] Brown | The work-hardening of copper-silica I. A model based on internal stresses, with no plastic relaxation[END_REF][START_REF] Brown | The work-hardening of copper-silica II. The role of plastic relaxation[END_REF]. Most recently, it has been shown that it is possible to extend continuum plasticity models by using higher order plasticity theories such as strain gradient plasticity which explicitly include a length scale [START_REF] Fleck | Strain gradient plasticity: Theory and experiment[END_REF].

It is also important to consider the possible interactions between plasticity and the second phase. First, for small particles (e.g. precipitates) which are sheared by dislocations, the increase in interfacial area can lead to the dissolution of the second phase [START_REF] Sargent | Dissolution of small precipitates[END_REF][START_REF] Gutierrez-Urrutia | The effect of coarse second-phase particles and fine precipitates on microstructure refinement and mechanical properties of severely deformed al alloy[END_REF]. Alternatively, deformation may lead to dynamic precipitation [START_REF] Deschamps | In-situ small-angle X-ray scattering study of dynamic precipitation in an Al-Zn-Mg-Cu alloy[END_REF][START_REF] Kelly | Guinier preston zones in an aluminium-silver alloy[END_REF]. Second, the build of internal stresses can lead to the intervention of alternative The internal stress in AA6111 3 processes such as particle fracture, interfacial decohesion or in the case of high aspect particles under compressive loading, elastic buckling and the formation of shear bands [START_REF] Tao | The mechanical behavior of directionally solidified Al-Ni in compression[END_REF]. In general, these processes depend critically on the stress-state, e.g. the magnitude and direction of the hydrostatic and deviatoric stresses.

Precipitation hardening aluminum alloys represent a particularly interesting and challenging two-phase system to examine. During precipitation, the volume fraction, size, shape and the composition of the second phase changes (i.e. a series of metastable precipitates is often observed). The early work of Byrne and co-workers [START_REF] Byrne | Precipitate hardening in an aluminum-copper alloy[END_REF] on Al-Cu single crystals found that there was a substantial change in macroscopic work hardening behaviour as a function of the precipitate state. A fundamental question that arose from these studies and the later work of Moan and Embury [START_REF] Moan | A study of the Bauschinger effect in aluminum-copper alloys[END_REF] was whether the enhanced work hardening behaviour observed in the Al-Cu samples (which have plate shaped precipitates) was best understood in the framework of additional storage of geometrically necessary dislocations (Ashby) or in terms of the development of long range back stress [START_REF] Brown | The work-hardening of copper-silica I. A model based on internal stresses, with no plastic relaxation[END_REF][START_REF] Atkinson | The work-hardening of copper-silica IV. The Bauschinger effect and plastic relaxation[END_REF][START_REF] Brown | The work hardening of fibrous composites with particular reference to the copper-tungsten system[END_REF]. It is not possible to differentiate these theories based on monotonic tests but the use of strain reversal, i.e. Bauschinger tests, is particularly useful for evaluating internal stresses [START_REF] Moan | A study of the Bauschinger effect in aluminum-copper alloys[END_REF][START_REF] Wilson | Reversible work hardening in alloys of cubic metals[END_REF][START_REF] Brown | Orowan's explanation of the Bauschinger effect[END_REF].

The current work examines the development of internal stresses as a function of precipitate state for the commercial aluminum alloy AA6111 using Bauschinger experiments. This system is a good candidate for detailed examination as it has recently been well characterized in terms of the nature of the precipitates, their size, shape and volume fraction [START_REF] Wang | Precipitation strengthening of the aluminum alloy AA6111[END_REF][START_REF] Esmaeili | On the precipitation-hardening behavior of the Al-Mg-Si-Cu alloy AA6111[END_REF][START_REF] Wang | The sequence of precipitation in the Al-Mg-Si-Cu alloy AA6111[END_REF] as well as the macroscopic work hardening behaviour [START_REF] Cheng | The influence of precipitation on the work hardening behavior of the aluminum alloys AA6111 and AA7030[END_REF] for a wide range of heat treatments. It is the objective of the current work to develop a comprehensive model framework for understanding the development of internal stresses in this alloy system for deformation at ambient temperature over a wide range of strains (i.e. strains from 0.01 to 0.6). its chemical composition was determined as shown in Table 1. For the tension-compression tests, axisymmetrical specimens were machined in the sheet thickness with a 20 mm gage length and a diameter ratio of 1.63 (the diameter is 9 mm in the grip section v.s. 5.5 mm in the gage length section). Specimens were all solution treated for 10 minutes in a salt bath at 560 . In order to study the effect of the precipitation structure, various ageing stages have been investigated. Based on previous work [START_REF] Poole | The shearable/non-shearable transition in Al-Mg-Si-Cu precipitation hardening alloys: Implications on the distribution of slip, work hardening and fracture[END_REF], 5 conditions have been selected from under-aged to massively over-aged, as shown in Table 2. For the sake of clarity, each condition will be hereafter identified by the combination of annealing time and temperature after quenching; e.g.

6H250 for 6 hours at 250 , see Table 2 for full details).

Mechanical tests were conducted on a computer controlled MTS servo-hydraulic machine. Tests have been conducted with a nominal strain rate of 10 -3 s -1 at room temperature. One specimen for each condition was tested in tension to failure to record the tensile properties. Then for each ageing condition, tension-compression tests were conducted in the following way: during the first straining phase, the specimen was deformed under tension to a certain amount of plastic strain. A clip-on extensometer was used to monitor the total strain, the target value having been determined from the tensile test. The strain was then reversed to apply at least 6% plastic strain in compression and at this point the specimen is unloaded. Tests were conducted on different samples with four different amounts of plastic strain in the tension loading phase (1%, 2%, 4% and 6%) to capture the evolution of the Bauschinger effect as a function of strain.

It will be shown that for the particular case of The internal stress in AA6111 5 the Bauschinger effect at larger strains (between 10% and 60%). With the method described above, the strain is limited to moderate values due to either necking or buckling of the sample. Therefore, additional specimens were processed in the following way: four 50 mm×100 mm×10 mm plates of the as received sheet were cut and prepared with the 7D250 condition. Three of these samples were rolled to 10, 20 and 40% reduction (the fourth sample was not rolled and was used to ensure there was no variation in the heat treatment due to the bulky geometry of the plate compared to the dogbone specimens). Specimens were then machined from the four plates, and both tension and compression tests were conducted as previously described.

The samples for transmission electron microscopy (TEM) were prepared as follows: mechanical polishing down to approximately 100 µm thickness and then jet polishing in a solution of 10% perchloric acid and 90% methanol at -35 . TEM observations were conducted using a PHILIPS CM12 transmission microscope operating at 120 kV.

Results

As shown by Poole and co-workers in the same alloy, the yield and work hardening behaviour depends critically on the heat treatment history of the material [START_REF] Poole | The shearable/non-shearable transition in Al-Mg-Si-Cu precipitation hardening alloys: Implications on the distribution of slip, work hardening and fracture[END_REF]. From the tensile behaviour, the yield stress values corresponding to 0.2% and 0.02% of permanent deformation σ 0.2% and σ 0.02% respectively, were determined and reported in Table 3. Tensile curves for a large set of ageing conditions in this material can be found elsewhere [START_REF] Poole | Modelling the stress-strain behaviour for aluminum alloy AA6111[END_REF].

Fig. 1 illustrates how the data recorded during the tensile testing was analysed. The process has three steps: (i) the true stress v.s. true strain curve is plotted; (ii) the compressive data is reversed and plotted as a function of cumulative strain; (iii) the elastic strain is removed and the true stress v.s. cumulative true strain can be compared to the monotonic behaviour. . Illustration of the stress-strain data processing in a three steps way in the case of 7H180 material with 2% forward strain: (a) true stress v.s. true strain curve; (b) reversed stress data plotted v.s. cumulative true strain (c) the elastic strain is removed and the curve is compared with the monotonic behavior.

F

Fig. 2 shows the forward-reverse behaviour obtained for different heat treatments after 1% plastic strain applied in the forward direction. Similar data have been obtained for forward strain values of 2%, 4% and 6%. One can readily observe that the reverse parts of the curves can be split in two; first a transient region where there is a very steep apparent work hardening rate, followed by a steady state one. The materials with different heat treatments show very different behaviours. For the under aged and peak aged material, the transient is rather short and the flow stress quickly comes back to the value reached during the forward straining. On the other hand for the over aged materials, the transient progressively becomes longer and more complicated. In particular for the materials aged at 250 , the transient part seems much more complicated with an inflexion in the reverse flow stress curve. While an in-depth investigation of the microscopic processes responsible of this inflexion is beyond the scope of this paper, it should be noted that this behaviour has been reported before [START_REF] Stoltz | Cyclic deformation and Bauschinger effect in Al-Cu-Mg alloys[END_REF][START_REF] Asaro | Elastic-plastic memory and kinematic-type hardening[END_REF] and was attributed to the recovery of the dislocation structures formed during the forward straining.

Bauschinger stress analysis

As previously suggested by Wilson [START_REF] Wilson | Reversible work hardening in alloys of cubic metals[END_REF] and followed by Atkinson et al. [START_REF] Atkinson | The work-hardening of copper-silica IV. The Bauschinger effect and plastic relaxation[END_REF], the evolution of the Bauschinger stress σ b as a function of the reverse plastic strain can be calculated by halving the difference between the monotonous and the reversed stress-strain data. The corresponding curves are shown in Fig. 3 evolution of the internal structure as the reverse strain develops. Fig. 3 presents a summary of the results of all Bauschinger tests. One can see that the results obtained for the 1% forward strain (see Fig. 2) are similar for the other values of 2%, 4% and 6%; however it is clear that for a given material, the general value of the Bauschinger stress increases as a function of the forward strain.

The difference in terms of the evolution of the Bauschinger stress as the reverse strain increases is significant. On the one hand, the under aged and peak aged materials (and to a lesser extend for the To further analyse these data, it is useful to choose a plastic strain offset to characterise the Bauschinger stress σ b . This is a rather important issue as the choice of the offset significantly affects the results.

Examination of the literature shows that there is no general agreement on this choice. In the work of Brown and coworkers on copper-silica, the internal stress is defined as being "permanent softening" and thus measured when the reverse curve becomes parallel to the tensile behaviour. This method does not seem suitable in our case since permanent softening is not readily observed. Alternatively, Lloyd chose to use a 3% plastic strain offset [START_REF] Lloyd | The Bauschinger effect in polycrystalline aluminium containing coarse particles[END_REF], while Moan and Embury focused on the transient behavior and have used very small offset values -between 0.001% and 0.1% plastic strain-to characterize the Bauschinger effect [START_REF] Moan | A study of the Bauschinger effect in aluminum-copper alloys[END_REF]. The internal stress in AA6111 choice of reverse strains has almost no practical effect, i.e. in all case the estimated stress is small, ranging from 1-8% of the yield stress. The value of 8% will be taken as a worst case estimate for the resolution of this technique. On the other hand, in the case of 6H250 or 7D250 the effect of the choice for reverse strains is more important, especially for low forward strains. By far the biggest difference in the estimate occurs when the reverse strain is increased from 0.5 to 1%. This is consistent with the data plotted in Fig. 3d and 3e which shows that the estimate of Bauschinger stress is changing rapidly in this range of the reverse strain. After 1% reverse strain the rate of change of Bauschinger stress is much lower in all cases.

It is well known that during the initial stage of the strain reversal, the dislocation structure is evolving in a complicated manner. For example, even single phase materials such as copper exhibit significant transient effects [START_REF] Pendersen | The Bauschinger effect in copper[END_REF]. Considering the strain resulting from the local motion of a dislocation sampling an array of point obstacles when the strain is reversed, Brown estimated that the transient strain would be approximately 1% plastic strain [START_REF] Brown | Orowan's explanation of the Bauschinger effect[END_REF]. This would be consistent with the present results which show that the region where the reverse stress is evolving the most rapidly also corresponds to approximately 1% plastic strain. As a result, the subsequent discussion will use the results obtained at 1% plastic reverse strain σ r 1% , as an estimate of the internal stress, see Eq. (1). While this is not completely satisfying from a theoretical point of view, we feel that in the absence of direct measurements for the stress in the second phase (by for example in situ X-ray diffraction), this represents a reasonable approach. The Bauschinger stress, measured with a 1% plastic strain offset σ b 1% , is thus used to estimate the value of the internal stress σ stored within the material at the point of reversal (where σ f0 denotes the flow stress reached at this point). From now on we shall use the term internal stress regardless whether we are refering to the experimental Bauschinger results or to the stress actually stored within the material.

F

Hence:

σ = σ b 1% = σ f 0 -|σ r 1% | 2 (1)
The measured internal stress increases in all cases with plastic strain as shown in Fig. 4a; but clearly, the five studied materials break into two groups. The under aged and peak aged materials show very little internal stress and the over aged materials show a significant level. The internal stress σ first increases with a very steep slope and then remains more or less stable, as shown by the upper dashed line on the figure.

Having characterised the internal stress, it is interesting to be able to estimate the fraction of the work hardening which can be attributed to the internal stress. This can be done by plotting the ratio of the internal stress to the level of work hardening: σ /(σ f 0 -σ Y 0.02% ) as shown in Fig. 4b. Again the five materials separate into two groups, underaged and peak aged materials have less than 20% of the work hardening which can be attributed to internal stress build-up while for the overaged materials, more than 50% of the work hardening (up to 86% for 7D250) can be attributed to the evolution of internal stress.

Large strain experiments with material aged 7D250

As shown by the results in Fig. 4, the material aged 7D250 exhibits a very large internal stress when it is plastically deformed. It is clear that the internal stress increases with the applied forward plastic strain but it is of interest to see if there is a limit to this increase. This cannot be answered from these experiments because we were limited to approximately 7-8% plastic strain due to either necking or buckling. Much higher values of strain can be achieved by cold rolling the material 1 . Considering the von Mises equivalent stress, rolling to 10%, 20% and 40% reduction lead to forward strain values of 0.12, 0.26 and 0.58. The experimental results for the Bauschinger stress as a function of reverse strain are presented in Fig. 3f and the corresponding results in terms of internal stress measured after 1% reverse plastic strain are summarized in Table 5 together with the other values measured with the tension-compression tests for 7D250.

It is clear that the increase of the internal stress observed before 10% strain does not continue. In fact, it is observed that after 10% plastic strain, the internal stress start decreasing. Fig. 5 presents some TEM micrographs of 7D250 material which provides evidence as to what is happening. Fig. 5a is a dark field image of a sample deformed 8% in tension. The dislocation structure is complex but one should observe that the precipitates are still intact and very straight. rolled specimens for 20% and 40% reduction. One should note in the latter two cases that the precipitates have fractured; this has actually been observed also for the 10% reduction cold rolled specimens.

Discussion

The results presented in the previous section have shown that depending on the thermal history of the material, very different behaviours are observed in terms of internal stress build-up. This section deals with analysing the results regarding the interaction of the dislocations with the precipitation structure. First we review the importance of the shearable/non-shearable transition and then the intervention of alternative processes is considered for the case of non-shearable particles.

The shearable/non-shearable transition

There is a general agreement about the precipitation sequence of strengthening alloys such as AA6111.

That sequence can be written as follow:

SSS → clusters/GP zones → β precursor of Q → equilibrium Q + Mg 2 Si
Here, SSS denotes the supersaturated solid solution; β are Mg The internal stress in AA6111 13

Table 6. Summary of the main information describing the precipitates structure in the 5 studied materials.

material nature of precipitates average radius precipitates are. . . observations and references on the precipitation sequence, see [START_REF] Wang | Precipitation strengthening of the aluminum alloy AA6111[END_REF]).

5M180 β + Q <1.2 nm shearable 7H180 β + Q 1.8 nm shearable 60D180 Q 3.3 nm mixed 6H250 Q ∼ 5
Table 6 provides a summary of the precipitates structure and the nature of the dislocations/precipitates interaction (based on monotonic work hardening and TEM studies) for the 5 different materials examined in this work. According to Table 6, one can correlate the observed behaviour for the internal stress build up during deformation with the nature of the dislocations/precipitates interactions. When the dislocations can pass through the precipitates (5M180 and 7H180), no Orowan loop is left around the precipitates and little or no internal stress is found within the material. As the size of precipitates increases, at some point it will become impossible for the dislocation to pass through. At this point, the dislocation will bow between the precipitates and then by-pass the precipitates, leaving one dislocation loop around the particle for each mobile dislocation. This mechanism is responsible for the internal stress measured in the cases of materials 6H250 and 7D250. In the case of 60D180, it is believed that the distribution is mixed -see [START_REF] Poole | The shearable/non-shearable transition in Al-Mg-Si-Cu precipitation hardening alloys: Implications on the distribution of slip, work hardening and fracture[END_REF] for high resolution TEM evidence that some precipitates are still sheared by the dislocations-so both mechanisms coexist which is consistent with the intermediate value of internal stress that was observed.

Deformation mechanisms at larger strains

The mechanical tests carried out on the rolled 7D250 samples have shown that the internal stress first increases with forward strain and then after approximately 10%, starts to decrease as a function of the amount of reduction. In other words, increasing the deformation in the specimen actually reduces the internal stress stored inside the material.

At first sight, this behavior seems inconsistent with the mechanisms we have just described, where the Bauschinger stress increases with ε p . But this can be understood by the fact that the storage of more and more loops around the particle correspondingly increases the axial stress on it. Thus, it is reasonable that at some point, the number of loops around the particle is large enough that the particle will fracture. This hypothesis is supported by TEM observation of 7D250 deformed by rolling shown previously on Fig. 5 where the fracture of the precipitates can readily be observed.

Experimental determination of the unrelaxed plastic strain

It is of interest to extend furter the analysis of the Bauschinger stress data. For this, only the over aged materials will be considered since they are the only ones to contain a population of non-shearable particles.

It has been shown for those materials that the internal stress first increases linearly (see Fig. 4a) with a slope Θ = d σ /dε. Although more data points would be needed to measure these slopes precisely, one can estimate (±200 MPa) the following values: Θ 60D180 = 1800 MPa, Θ 6H250 = 2200 MPa, Θ 7D250 = 2400

MPa (this would be consistent with a slight increase in the volume fraction of precipitates as a function of ageing time).

Those experimental values can now be used to calculate the unrelaxed plastic strain ε * p at the end of the forward loading for each experimental test. Because in the linear region (i.e. ε p ≤ 2%) the work hardening process is predominantly associated with storing more and more dislocation loops around the precipitates, the unrelaxed plastic strain is equal to the plastic strain:

ε * p = ε p for ε p ≤ 2% (2)
and one can now calculate the unrelaxed plastic strain ε * p for the data in Fig. 4a as: The internal stress in AA6111 15 The experimental values will be shown later on (see Fig. 6), together with the predictions achieved by the equation developed in section 5.1.

ε * p = σ Θ (3)

Modeling

5.1 A constitutive equation for the unrelaxed plastic strain, ε * p Brown and coworkers derived an expression for the internal stress starting with the unrelaxed shear plastic strain γ * p [START_REF] Brown | The work-hardening of copper-silica II. The role of plastic relaxation[END_REF]. For spherical particles of radius r, γ * p = n b/2 r where n is the number of dislocations responsible for the deformation, and b represents the magnitude of their Burgers' vector. We shall consider a similar approach, while taking into consideration the particular geometry and size of the precipitates.

In this case, the precipitates are lath shaped, aligned along the 100 directions and the dislocation glide along slip planes ({111} in FCC). Thus the precipitate length intercepting the slip planes l cos θ replaces the sphere diameter 2 r in the previous expression, with θ = 54.74°:

γ * p = n b l cos θ (4)
The actual deformation in the direction of interest must be related to the effective Burgers' vector in that direction: b sin θ. Introducing the Taylor factor M for multi-slip systems (γ = M ε) leads to the following expression:

ε * p = n b sin θ M l cos θ (5)
Now the rate of creation of these dislocation loops dn/dγ, as the shear strain γ operates, must be assessed. Because the loops will be stored along the axial direction of the precipitates, it should be proportional to l cos θ/(b sin θ), the number of slip planes intercepting the precipitate. However, this dislocation accumulation around a precipitate cannot increase without limit. Either the internal stress will be suficient to fracture the particles and provide new free surfaces where the loops can annihilate, or the stresses at the interface will trigger recovery mechanisms preventing further accumulation. The net effect is that the number of loops that can be stored must have an upper limit n * . That can be described easily if one pictures this effect as the maximum number of "available sites" n * . If n dislocations are stored, the proportion of sites unoccupied on which further storage is possible is (1 -n/n * ). Accounting of this effect amounts to multiply the "geometric storage rate" by this factor as expressed in Eq. ( 6). A similar approach as been recently used successfully to describe the storage of dislocations at grain boundaries [START_REF] Sinclair | A model for the grain size dependent work hardening of copper[END_REF]. Finally, one can write the rate of Orowan loops formation as:

dn dγ = l cos θ b sin θ 1 - n n * (6)
Using the Taylor factor and reordering give:

d(n/n * ) 1 -n/n * = M l cos θ n * b sin θ dε (7)
Integration of Eq. ( 7) leads to:

ln 1 - n n * = - M l cos θ ε n * b sin θ
Then n can be written:

n = n * 1 -exp - M l cos θ ε n * b sin θ (8)
and finally, combining with Eq. ( 5), the unrelaxed plastic strain comes to:

ε * p = n * b sin θ l cos θ 1 -exp - M l cos θ ε n * b sin θ (9)
From a physical point of view, n * is the maximum number of dislocation loops which can be stored around the precipitate. One should note that, if ε → 0, we have ε * p → ε p ; which is consistent with the experiments as presented in §4.3 (this behaviour is observed for ε p ≤ 2%). The predictions for the unrelaxed plastic strain are presented together with the experimental values in Fig. 6.

Modeling the internal stress

In order to model the internal stress, we will use a combined continuum/dislocation based approach.

As shown in Fig. 5, the precipitates can be considered as fibres oriented along the 100 directions in an aluminum matrix (note the precipitates are lath shaped but we shall consider a simpler view where their cross section is represented by a circle with an equivalent radius r 0 = 6.6 nm). The average initial precipitate length l 0 is approximately 300 nm, although there is some uncertainty here as it is difficult to measure in the TEM.

The internal stress σ is associated with the dislocations stored around the precipitates. According to this view, one can write: σ = α f σ ppt [START_REF] Gutierrez-Urrutia | The effect of coarse second-phase particles and fine precipitates on microstructure refinement and mechanical properties of severely deformed al alloy[END_REF] f being the volume fraction of the precipitates, σ ppt the average stress in the precipitates, and α a geometrical factor relative to the orientation of the precipitates with respect to the tensile axis. Because the experimental results have shown that it is important to account for the particle fracture, we shall adopt two different views depending wether or not the precipitate has been fractured or not:

• For unfractured precipitates, we use Eshelby's solution for ellipsoidal particles of high aspect ratio, i.e.

the stress is uniform within the inclusion. Then we have σ ppt = σ ppt = E ε * p , where E is the modulus of elasticity of the precipitates in the deformation direction. While we can now estimate ε * p from Eq. ( 9), determining the stiffness of the precipitates requires special attention and will be addressed in paragraph section 5.2.1.

• For fractured precipitates, we consider that the fracture process creates new free surfaces at the precipitates ends where the stress must now be zero. We then take the simplest possible view that the stress in the particles increases linearly from the end to the centre using a simple shear lag model which is presented in section 5.2.2. The internal stress in AA6111 19 is made using the Vienna ab initio simulation package (VASP) with Vanderbilt ultrasoft pseudopotentials and the generalized gradient approximation (GGA). The stress-strain approach is employed with the 0K lowest energy DFT (density functional theory) stoichiometry Al 3 Cu 2 Mg 9 Si 7 to give the elastic constants listed in Table 7.

Now, using the relations for hexagonal crystals (see for instance [START_REF] Hosford | The Mechanics Of Crystals And Textured Polycrystals[END_REF]), one can derive the stiffness coefficient S ij from the elastic constants. In particular, this leads to S 33 = 0.00848 GPa -1 . Then the Young modulus can be derived:

E = 1 S 33 = 118 GPa (11) 5.2.2
A simple plastic shear lag model for fractured precipitates. The Q phase precipitates are essentially perfect crystals, and as such will fail when the theoretical tensile strength σ u , is reached. We shall consider that the particles are brittle, so once the load reaches σ u , the particle fractures into two pieces of equal length. The experimental results suggest that the fracture stress of the precipitates should be within a 2500-3000 MPa range which is approximately E/40. This is within a factor of 4 theoretical tensile strength [START_REF] Kelly | Strong Solids[END_REF].

Once fractured, the ends of the particles are unbonded with the matrix and the stress transfer from the inclusion to the matrix is expected to be very different from the non-broken particles. We shall adopt a simple view where the stress follows a triangular evolution from zero at both ends and reaching its maximum in the middle of the precipitates as depicted on Fig. 7.

With this view the stress increases linearly with the following rate [START_REF] Lilholt | Hardening in two phase materials -II. plastic strain and mean stress hardening rate[END_REF]: τ being the shear stress at the interface (2τ = σ f ). Finally the average stress in the fibre is simply half the maximum stress:

dσ dx = 2 τ r = σ f r (12)
σ ppt = σ ppt 2 = l 4 σ f r (13)
According to this model, the breakup of precipitates will lower the maximum stress (and thus the average stress) and the internal stress calculated by Eq. (10) will decrease. The internal stress drop is highlighted by the hatched area on Fig. 7.

Of course, the fracture process is much more complex than the simple view we are developping here. First of all, the dislocation loops around the precipitates may not be perfectly equispaced, such that fracture may not occur at the centre of the precipitate. In addition, a precipitate will not be perfectly aligned with the 001 direction of the Al matrix and its geometrical parameter be different from the average ones.

To account for all these discrepancies, we have chosen to use a non unique fracture stress. A Gaussian distribution is used, centered on the mean value σ u and with a standard deviation σ u .

A consequence of using a Gaussian distribution for the fracture stress is the possibility to resolve readily the fraction of broken precipitates, which will be given exactly by the Gaussian cumulative distribution function. This is due to the fact that for a given stress σ = E ε * p all the precipitates with σ u ≤ σ will be broken. The total fraction of broken precipitates p 1 (the fraction of unbroken precipitates being The internal stress in AA6111 21 can be obtained by:

p 1 = 1 2 1 + erf E ε * p -σ u σ u √ 2 ( 14
)
where erf is the Gauss error function: erf (x) = 2/ √ π x 0 exp -t2 dt. After the first fracture, the fraction of precipitates broken i times (i > 1) among the ones broken i -1 times1 , is:

p i = 1 2   1 + erf    σ f r l 0 4 i -σ u σ u √ 2       (15)

5.2.3

Integration of the models. In order to fully describe the evolution of the internal stress carried out by the precipitates over a wide range of strain, we need to combine the models. The full model is constituted by a small computer program implementing the unrelaxed plastic strain expression stated by Eq. ( 9), the plastic shear lag model with the particle fracture. For completeness, the parameters used in the model have all been gathered in Table 8. Although all parameters have a physical meaning, some of them, namely n * and σ u are not precisely known and the values have been chosen to fit the experimental data. n * controls the rate of the elastic build up of dislocation loops and the maximum stress reached in the precipitates. A value of n * = 60 was chosen to fit the data presented on Fig. 6. Although no precise measure is available in the literature to check this value, it is worth noting that preliminary TEM observations counting the number of dislocation loops around the precipitates (in dark field contrast)

gave approximately 40 loops per precipitate after 8% tensile strain. Given the difficulty in making these measurements, we beleive this is in reasonable agreement with the maximum value of n * = 60. Now, with σ u E/40, the precipitate fracture occurs approximately for ε * p = 0.025 which according to Fig. 6 will happend slightly before a strain of 0.1. This is consistent with the experimental observations. In the end, a fracture stress variance of 170 MPa is chosen to fit the internal stress drop measured experimentally.

The matrix flow stress evolution is taken from the experimental tensile behavior. Eq. ( 14) and ( 15) the general form of the internal stress is given by:

σ = (1 -p 1 ) f E ε * p unbroken precipitates + f p 1 (1 -p 2 ) l 0 4 σ f r broken once + f p 1 p 2 l 0 8 σ f r broken twice + • • • (16)
The output of the simulation has been plotted in Fig. 8. One can see a very good agreement between the prediction by the model and the experimental values of internal stress measured by the Bauschinger tests. Fig. 8 shows that the initial internal stress carried out by the precipitates almost linearly increases with strain (initially ε * p = ε). This is followed by a regime where plastic relaxation occurs and then the precipitates start to fracture at a strain just below 0.1 resulting in a rapid drop of the internal stress. Due to increasing flow stress of the alloy σ f as the straining continues, most of the precipitates will fracture once more, further decreasing the stress borne by the precipitates. It is worth noting that with the current values of the parameters, a small fraction of the precipitates never fracture. Physically, this can be interpreted that some of the precipitates are oriented such that the load transfer is ineffective.

We are here standing at the limit of the continuum/discrete description of the deformation. The results of Fig. 8 show that adopting a physically based model taking into account the discrete nature of the dislocation/precipitates interaction is capable of describing micro-mechanical phenomena such as the internal stress build up and decrease, without having to explicitly account for the discrete view of the dislocation motion. However there could be a value for a discrete dislocation simulation to push further the physical description of those mechanisms by adopting a multiscale modeling framework. For example, discrete dislocation simulation could help to shape fundamental equations such as the rate of dislocation loop formation and provide physical input for the fitting parameters such as the fracture stress variance. Yet, some points remain unclear at the moment. One is that with α = 1, the internal stress contributed from the precipitates has the form of f E ε * p ; this implicitly supposes that all precipitates fully participate in the process. While this may be true in the case of spherical inclusion (such as the copper silica system), it is not true in our case since the precipitates are distributed with different orientations in the material with respect to the loading direction. There are several explanations possible to this issue. First one can argue that during deformation, the fibre-like precipitates tend to align with the loading direction. This will tend to lower down the orientation effect. Second, there are some uncertainties in the microstructural parameters values used in the model. Specifically, the Young modulus of the precipitates was evaluated from first principles and as such there may be some uncertainty in its value. Third, as previously discussed there is also a degree of uncertainty from the offset chosen to evaluate the internal stress from the Bauschinger tests. A direct measurement of the stress in the precipitates using for example, X-ray diffraction would be valuable.

Conclusions

The plastic deformation of a commercial precipitation strengthening aluminium alloy (AA6111) has been investigated for a large range of ageing conditions. The main results can be summarized as follows.

The development of internal stresses critically depends on the precipitates size and volume fraction. It has been shown that for the case of shearable particles, the development of long range internal stress is rather small and the work hardening is dominated by dislocation-dislocation interactions. On the other hand, for non-shearable precipitates, a significant fraction of the work hardening can be attributed to the development of internal stress. This is interpreted by long range elastic stresses produced by dislocation loops stored around the particles, and can be characterized by tension-compression test. It was found that plastic relaxation occurs after approximately 2% plastic strain. At higher strains, the dislocation loop storage is such that axial loading of the precipitates reaches a new regime where the precipitates fracture, reducing the internal stress in the material.

Following Brown and coworkers, the framework for modelling deformation in two phase materials has been revisited with the insight from the current experiments. First, a physically based expression for the unrelaxed plastic strain has been derived. This expression has been shown to accurately predict the plastic relaxation measured from Bauschinger experiments. To fully describe the internal stress evolution with the ongoing strain, a composite view of the material has been adopted. Within this view, a simple shear lag model has been used to compute the internal stress of the broken precipitates, allowing a good prediction of the internal stress development, up to strains of 0.6. This is, to the best of the authors knowledge, the first time that this regime is clearly identified and successfully modelled.

Figure 3 .

 3 Figure3. Bauschinger stress as a function of reverse plastic strain. For each ageing condition, 4 different forward strains were tested (1%, 2%, 4% and 6%); the curves for rolled samples are shown in the bottom right graph and have been annoted "CR".

Figure 4 .

 4 Figure 4. Internal stress measurements in AA6111, a) σ measured after 1% reverse strain, b) σ expressed as a percentage of the work hardening of the materials; symbols denote the same materials in both pictures.

Figure 5 .

 5 Figure 5. Observation of the microstructure of material 7D250 by TEM: a) dark field image of a sample deformed 8% in tension, b) bright field image after 20% cold rolling, c) bright field image after 40% cold rolling; arrows indicate examples of precipitates fracture.

Figure 6 .

 6 Figure 6. Prediction of the unrelaxed plastic strain by Eq. (9) (with n * = 60) compared to the values issued from the experiments and calculated with Eq. (3); as before, the error bars are based on a 200 MPa uncertainty in the measure of Θ.

5. 2 . 1 the

 21 elastic constants of the Q phase. The Q phase precipitates are lath shaped, perfect HCP crystals of Al 4 Cu 2 Mg 8 Si 7 , aligned along the 100 direction of the Al matrix. A critical piece of information needed is the elastic modulus of the crystal along its c axis. Unfortunately, no experimental values are known. As a result, we will use the first-principles calculation by Wang and Wolverton[START_REF] Wolverton | Crystal structure and stability of complex precipitate phases in Al-Cu-Mg-(Si) and Al-Zn-Mg alloys[END_REF]. The calculation

Figure 7 .

 7 Figure 7. Stress distribution in the simple shear lag model and illustration of the break up when the load reaches the tensile strength of the precipitate; x denotes the position along the precipitate length.

Figure 8 .

 8 Figure 8. Results of the internal stress model: a) internal stress predicted by the model, as a comparison, the calculation for f E ε and f E ε * p have also been plotted b) evolution of the fraction of broken precipitates.

  Illustration of the stress-strain data processing in a three steps way in the case of 7H180 material with 2% forward strain: (a) true stress v.s. true strain curve; (b) reversed stress data plotted v.s. cumulative true strain (c) the elastic strain is removed and the curve is compared with the monotonic behavior. 158x53mm (72 x 72 DPI) stress/strain curves, obtained by a Bauschinger test with a 1% forward plastic strain, for all materials. 145x99mm (72 x 72 DPI) Bauschinger stress as a function of reverse plastic strain. For each ageing condition, 4 different forward strains were tested (1%, 2%, 4% and 6%); the curves for rolled samples are shown in the bottom right graph and have been annoted "CR". 142x102mm (72 x 72 DPI) Internal stress measurements in AA6111, a) < > measured after 1\% reverse strain, b) < > expressed as a percentage of the work hardening of the materials; symbols denote the same materials in both pictures. 144x67mm (72 x 72 DPI) Observation of the microstructure of material 7D250 by TEM: a) dark field image of a sample deformed 8% in tension, b) bright field image after 20% cold rolling, c) bright field image after 40% cold rolling; arrows indicate examples of precipitates fracture. 130x56mm (72 x 72 DPI) Prediction of the unrelaxed plastic strain by Eq. (9) with n*=60 compared to the values issued from the experiments and calculated with Eq. (3); as before, the error bars are based on a 200 MPa uncertainty in the measure of . 71x66mm (72 x 72 DPI) Stress distribution in the simple shear lag model and illustration of the break up when the load reaches the tensile strength of the precipitate; x denotes the position along the precipitate length. 75x59mm (72 x 72 DPI) Results of the internal stress model: a) internal stress predicted by the model, as a comparison, the calculation for fE and fE *p have also been plotted b) evolution of the fraction of broken precipitates. 143x67mm (72 x 72 DPI)

Table 1 .

 1 Chemical composition of the AA6111 alloy (wt.%)

	Alloy	Mg	Si	Cu	Fe	Mn	Cr	Ti
	AA6111	0.8	0.6	0.7	0.25	0.2	0.05 0.06

Table 2 .

 2 Details of the five heat treatments applied to the specimens For long duration heat treatments, an air furnace was used rather than an oil bath; but to ensure a consistent heating rate in all experiments, the first hour was carried out in an oil bath like for the other treatments.

	Condition	label	solutionizing	quench ageing
	Under aged	5M180	10 min at 560	WQ	5 minutes at 180 in oil bath
	Peak aged Over aged 1 60D180 10 min at 560 7H180 10 min at 560	WQ WQ	7 hours at 180 in oil bath 1 hour at 250 in oil bath, then 60 days at 180 in air furnace a
	Over aged 2 Over aged 3	6H250 7D250	10 min at 560 10 min at 560	WQ WQ	6 hours at 250 in oil bath 1 hour at 250 in oil bath, then 7 days at 250 in air furnace a
	a

Table 3 .

 3 Yield stress measured with 0.02% and 0.2% plastic strain offsets for all investigated materials.

	Ageing	5M180 7H180	60D180 6H250 7D250
	σ Y 0.02% (MPa) σ Y 0.2% (MPa)	110 140	310 335	285 300	210 245	140 155

Table 4

 4

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presents the results showing the effect of choosing different reverse strains, i.e. 0.5, 1 and 2%, to estimate the Bauschinger stress from the experimental results. For the cases of 5M180 and 7H180, the

Table 4 .

 4 Measure of the Bauschinger stress (MPa) in the tensioncompression tests presented on Fig.3with three different plastic strain offsets.

	Forward strain 5M180 7H180 60D180 6H250 7D250
		plastic strain offset = 0.5%
	1%	6.5	10.7	31.3	45.8	37.9
	2%	8.6	19.8	49.8	50.3	42.4
	4%	10.1	26.0	53.7	47.3	46.4
	6%	11.0	26.5	47.7	44.6	50.6
		plastic strain offset = 1%
	1%	4.0	3.4	7.9	10.8	20.7
	2%	5.8	11.5	28.9	37.0	36.3
	4%	5.5	14.7	39.3	37.8	41.4
	6%	6.4	14.7	33.8	36.3	45.4
		plastic strain offset = 2%
	1%	2.1	1.0	4.4	7.3	14.7
	2%	5.1	5.7	9.9	16.9	28.2
	4%	4.2	6.1	23.7	26.6	34.9
	6%	4.2	7.3	22.6	29.6	36.9

Table 5 .

 5 Bauschinger stress measurements in material over aged 7 days at 250 , the tension-compression results have been put together with rolling-compression tests to achieve very large forward straining.

	Plastic strain (%)	Bauschinger stress σ b (MPa)
	1	20.7
	2	36.3
	4	41.4
	6	45.4
	12	31.7
	25.6	23.8
	58.7	16.6

  Cu 2 Mg 8 Si 7 ; and Mg 2 Si is a plate shaped equilibrium phase (for full details with TEM

			nm	mixed, mostly non-shearable
	7D250	Q	6.6 nm	non-shearable
	is likely to be Al 4

Table 7 .

 7 Elastic constants (unit is GPa) obtained by first-principles calculations for Al 3 Cu 2 Mg 9 Si 7 .

	C 11 = C 22	C 12	C 13	C 33	C 44 = C 55	C 66 = (C 11 -C 12 )/2
	147.3	36.8	31.0 128.3	36.4	55.5

  1 -p 1 ),

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Table 8 .

 8 Physical parameters used in the internal stress model.

	Parameter	Symbol	Value
	Volume fraction	f	0.019
	Young Modulus	E	118 GPa
	Orientation factor	α	1
	Burger's vector Maximum number of loops	b n *	0.286 nm 60
	Initial length Average radius	l 0 r	300 nm 6.6 nm
	Mean fracture stress	σ u	2900 MPa
	Fracture stress standard dev.	f σu	170 MPa

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

In Eq.[START_REF] Moan | A study of the Bauschinger effect in aluminum-copper alloys[END_REF] the fraction p i is relative. For instance, since to break twice, a precipitate must first break once, the actual fraction of precipitate broken twice is p 1 p

.

Acknowledgement

The authors greatfully acknowledge the help of C. Wolverton and Y. Wang for providing the elastic constants of the Q phase via first principle calculations. The authors are thankful to Pr. C. W. Sinclair for helpful discussion and comments on the present work. H. P. would like to thank the French ministry of foreign affairs for the Lavoisier fellowship during his work at the University of British Columbia.