Xavier Iturrioz √1

Rodrigo Alvear-Perez √1

Nadia De Mota √

Christel Franchet ✦2

Fabrice Guillier ✦

Vincent Leroux ∫

Hubert Dabire

Melissande Le Jouan √

Hadjila Chabane

Romain Gerbier √

Alain Berdeaux

Jean-Luc Galzi ✦

Marcel Hibert

Catherine Llorens-Cortes √
email: c.llorens-cortes@college-de-france.fr

Identification and Characterization of E339-3D6, the First Nonpeptidic Apelin Receptor Agonist

Apelin plays a prominent role in body fluid and cardiovascular homeostasis. To explore further upstream the role played by this peptide, non-peptidic agonists and antagonists of the apelin receptor are required. To identify such compounds which do not exist to date, we used an original Fluorescence Resonance Energy Transfer-based assay to screen a GPCR-focused library of fluorescent compounds on the human EGFP-tagged apelin receptor. This led to isolate E339-3D6 that displayed a 90 nmol/L affinity, behaved as a partial agonist with regard to cAMP production and as a full agonist with regard to apelin receptor internalization.

Finally, E339-3D6 induced vasorelaxation of rat aorta precontracted with noradrenaline and potently inhibited systemic vasopressin release in water-deprived mice when intracerebroventricularly injected. This compound represents the first non-peptidic agonist of the apelin receptor, the optimization of which will allow to develop a new generation of vasodilator and aquaretic agents.

INTRODUCTION

Apelin, is a bioactive peptide isolated from bovine stomach extracts and identified as the endogenous ligand of the human orphan G protein-coupled receptor (GPCR), APJ [START_REF] O'dowd | A human gene that shows identity with the gene encoding the angiotensin receptor is located on chromosome 11[END_REF][START_REF] Tatemoto | Isolation and characterization of a novel endogenous peptide ligand for the human APJ receptor[END_REF].

Apelin derives from a single 77-amino acid precursor, proapelin which has a fully conserved C-terminal 17-amino acid sequence in all species studied, apelin 17 (K17F), including the pyroglutamyl form of apelin 13 (pE13F) (see Table S1 for sequences) [START_REF] Tatemoto | Isolation and characterization of a novel endogenous peptide ligand for the human APJ receptor[END_REF][START_REF] Habata | Apelin, the natural ligand of the orphan receptor APJ, is abundantly secreted in the colostrum[END_REF][START_REF] Lee | Characterization of apelin, the ligand for the APJ receptor[END_REF]. Both peptides (K17F and pE13F) naturally occur in rat brain and plasma [START_REF] De Mota | Apelin, a potent diuretic neuropeptide counteracting vasopressin actions through inhibition of vasopressin neuron activity and vasopressin release[END_REF]. They exhibit a strong inhibitory activity on forskolin-induced cAMP production in cells expressing the human [START_REF] Habata | Apelin, the natural ligand of the orphan receptor APJ, is abundantly secreted in the colostrum[END_REF][START_REF] Medhurst | Pharmacological and immunohistochemical characterization of the APJ receptor and its endogenous ligand apelin[END_REF] or the rat apelin receptor [START_REF] De Mota | Cloning, pharmacological characterization and brain distribution of the rat apelin receptor[END_REF]. These peptides promote phosphorylation of ERKs, Akt and p70 S6 kinase [START_REF] Masri | Apelin (65-77) activates p70 S6 kinase and is mitogenic for umbilical endothelial cells[END_REF]. They are also highly potent inducers of apelin receptor internalization [START_REF] El Messari | Functional dissociation of apelin receptor signaling and endocytosis: implications for the effects of apelin on arterial blood pressure[END_REF][START_REF] Reaux | Physiological role of a novel neuropeptide, apelin, and its receptor in the rat brain[END_REF].

Apelin and its receptor are both widely distributed in the brain [START_REF] Lee | Characterization of apelin, the ligand for the APJ receptor[END_REF][START_REF] De Mota | Cloning, pharmacological characterization and brain distribution of the rat apelin receptor[END_REF][START_REF] O'carroll | Distribution of mRNA encoding B78/apj, the rat homologue of the human APJ receptor, and its endogenous ligand apelin in brain and peripheral tissues[END_REF]12) but are particularly highly expressed in the supraoptic (SON) and paraventricular (PVN) hypothalamic nuclei. Dual labeling studies demonstrate that within these two nuclei, apelin and its receptor co-localize with arginine vasopressin (AVP) in magnocellular neurons [START_REF] De Mota | Apelin, a potent diuretic neuropeptide counteracting vasopressin actions through inhibition of vasopressin neuron activity and vasopressin release[END_REF][START_REF] Reaux | Physiological role of a novel neuropeptide, apelin, and its receptor in the rat brain[END_REF][START_REF] O'carroll | Regulation of rat APJ receptor messenger ribonucleic acid expression in magnocellular neurones of the paraventricular and supraopric nuclei by osmotic stimuli[END_REF][START_REF] Goazigo | Dehydration-induced cross-regulation of apelin and vasopressin immunoreactivity levels in magnocellular hypothalamic neurons[END_REF]. In lactating rats, characterized by increases in both synthesis and release of AVP, central injection of K17F inhibits the phasic firing pattern of AVP neurons, thereby resulting in decreased AVP release in the blood circulation, and increased aqueous diuresis [START_REF] De Mota | Apelin, a potent diuretic neuropeptide counteracting vasopressin actions through inhibition of vasopressin neuron activity and vasopressin release[END_REF].

Moreover, after water deprivation, endogenous levels of AVP and apelin are conversely regulated to optimize systemic AVP release necessary to avoid additional water loss at the kidney level [START_REF] De Mota | Apelin, a potent diuretic neuropeptide counteracting vasopressin actions through inhibition of vasopressin neuron activity and vasopressin release[END_REF][START_REF] Goazigo | Dehydration-induced cross-regulation of apelin and vasopressin immunoreactivity levels in magnocellular hypothalamic neurons[END_REF]. Recently, we also showed that such opposite regulation of plasma apelin and AVP levels by osmotic stimuli exists in humans suggesting that apelin, like AVP, may participate in the maintenance of body fluid homeostasis not only in rodents but also in humans [START_REF] Azizi | Reciprocal regulation of plasma apelin and vasopressin by osmotic stimuli[END_REF].

Apelin and its receptor are also present in the cardiovascular system, i.e. heart, kidney and vessels [START_REF] Japp | The apelin-APJ system in heart failure: pathophysiologic relevance and therapeutic potential[END_REF]. Systemically administered apelin decreases arterial blood pressure (BP) [START_REF] Lee | Characterization of apelin, the ligand for the APJ receptor[END_REF][START_REF] El Messari | Functional dissociation of apelin receptor signaling and endocytosis: implications for the effects of apelin on arterial blood pressure[END_REF][START_REF] Reaux | Physiological role of a novel neuropeptide, apelin, and its receptor in the rat brain[END_REF][START_REF] Tatemoto | The novel peptide apelin lowers blood pressure via a nitric oxidedependent mechanism[END_REF], via a nitric oxide (NO)-dependent mechanism [START_REF] Tatemoto | The novel peptide apelin lowers blood pressure via a nitric oxidedependent mechanism[END_REF]. Consistent with these data, apelin receptor-deficient mice display an exaggerated pressor response to systemic angiotensin II (AngII), suggesting a counter-regulatory effect of apelin on AngII [START_REF] Ishida | Regulatory roles for APJ, a seven-transmembrane receptor related to angiotensin-type 1 receptor in blood pressure in vivo[END_REF]. Finally, apelin improves cardiac contractility and reduces cardiac loading [START_REF] Ashley | The endogenous peptide apelin potently improves cardiac contractility and reduces cardiac loading in vivo[END_REF](20)[START_REF] Szokodi | Apelin, the novel endogenous ligand of the orphan receptor APJ, regulates cardiac contractility[END_REF], suggesting a role for apelin in the regulation of cardiovascular functions. In agreement with this hypothesis, when apelin-deficient mice are subjected to chronic pressure overload by surgical constriction of the aorta, they develop severe and progressive heart failure [START_REF] Kuba | Impaired heart contractility in Apelin gene-deficient mice associated with aging and pressure overload[END_REF].

Given the broad array of physiological actions of apelin, its receptor represents a new interesting target for therapeutic research and drug design. In this context, the development of a nonpeptidic apelin receptor agonist by favouring diuresis and improving the contractile performance of the myocardium whilst reducing peripheral resistances could be particularly useful for the treatment of heart failure.

In order to identify an agonist or antagonist of the apelin receptor, we used in the present study a novel Fluorescence Resonance Energy Transfer (FRET)-based assay to screen a library of fluorescent (lissamine-tagged) nonpeptidic compounds on the human N-terminal EGFP-tagged apelin receptor, stably expressed in HEK-293 cells. This screening led to isolate several hits whose affinities (9 x 10 -8 to 10 -6 M) have been confirmed in classical binding experiments using radiolabelled pE13F. The most potent hit, E339-3D6 (23) displayed an affinity of 9 x 10 -8 mol/L, behaved as a partial agonist with regard to cAMP production and as a full agonist with regard to apelin receptor internalization. E339-3D6 could be therefore considered as a biased agonist. Using molecular modeling, we provided evidence that the binding moieties of E339-3D6 and pE13F overlapped within the receptor binding cavity. This novel ligand represents an unique and interesting lead compound with agonist properties on the apelin receptor, inducing ex vivo a concentration-dependent vasorelaxation of rat aorta precontracted with noradrenaline (NA) and decreasing in vivo systemic AVP release in waterdeprived mice, both effects being comparable to those produced by apelin.

MATERIALS AND METHODS

Drugs and Antibodies. Apelin 17 (K17F) and the pyroglutamyl form of apelin 13 (pE13F) were synthesized by NeoMPS (Strasbourg, France), see Table S1 for amino acid sequences.

Animals. Male Swiss mice (18-20 g) and adult male Wistar Kyoto rats (300-350 g) were maintained under 12 h light/dark cycle with free access to food and water and were obtained from Charles River Laboratories (L'Arbresle, France) and from Janvier (Le Genest-St-Isle, France). All animal experiments were carried out in accordance with current institutional guidelines for the care and use of experimental animals.

Chemistry.

The solid phase synthesis of the screened fluorescent compound library has been previously described [START_REF] Hibert | Collection of traceable compounds and uses thereof[END_REF]. Solid phase re-synthesis of the best hit, E339-3D6 ( 23) and of lissamineundecanoic-apelin-13 were described in the online data supplement.

Screening by FRET.

A library of 800 fluorescent compounds (lissamine-tagged) were screened by FRET on HEK 293 cells stably expressing the full length EGFP-tagged human apelin receptor or the EGFP-∆ 16 human apelin receptor (See online data supplement for details).

Membrane Preparations and Radioligand Binding Experiments. Membranes from CHO

or HEK 293 cells stably expressing respectively, the rat apelin receptor-EGFP and the EGFP-∆ 16 human apelin receptor, were prepared as previously described [START_REF] Iturrioz | Histidine 450 plays a critical role in catalysis and, with Ca2+, contributes to the substrate specificity of aminopeptidase A[END_REF]. Membrane preparations (5 µg/assay) were incubated for 1 h at 20°C with 2 x 10 -10 mol/L [ 125 I]pE13F in binding buffer alone or in presence of K17F, pE13F or E339-3D6 at various concentrations.

Reaction was stopped and filtered on Whatman GF/C filters. After washing radioactivity was counted. Saturation-binding curves were obtained by incubating membrane proteins with [ 125 I]pE13F at different concentrations (See online data supplement for details). cAMP Assay. The cAMP assay was performed as previously described [START_REF] De Mota | Cloning, pharmacological characterization and brain distribution of the rat apelin receptor[END_REF] (online data supplement for details).

Internalization Assay. CHO cells stably expressing the rat apelin receptor-EGFP or AT1 A -EGFP were seeded at 20 % confluency on glass coverslips coated with polylysine (WI; 0.01 %) (Sigma-Aldrich, St Quentin, France). Internalization was performed by incubating the cells at 37 °C for various times with 10 -7 mol/L K17F or lissamine-apelin 13 or with different concentrations of E339-3D6 as previously described [START_REF] El Messari | Functional dissociation of apelin receptor signaling and endocytosis: implications for the effects of apelin on arterial blood pressure[END_REF]. Cells were then mounted in Mowiol for confocal microscopic analysis (See online data supplement for details).

Quantification of Internalization by Digital Image Analysis.

Quantification of the extent of ligand-induced rat apelin receptor-EGFP internalization was performed by confocal microscopy coupled to digital image analysis as previously described (9, 26) (See Supplementary Methods for details).

Aortic Rings Preparation and Isometric Tension

Recording. According to [START_REF] Puybasset | Coronary and systemic hemodynamic effects of sustained inhibition of nitric oxide synthesis in conscious dogs. Evidence for cross talk between nitric oxide and cyclooxygenase in coronary vessels[END_REF][START_REF] Trochu | Beta 3-adrenoceptor stimulation induces vasorelaxation mediated essentially by endothelium-derived nitric oxide in rat thoracic aorta[END_REF], rat aortic rings were equilibrated in physiological salt solution for 120 min under a resting tension of 2 g. After checking the integrity of the endothelium, cumulative concentrationresponse curves to K17F (10 -12 to 10 -6 mol/L), E339-3D6 (10 -12 to 10 -6 mol/L) or Ach (10 -10 to 10 -4 mol/L) were constructed after precontraction with NA (3 x 10 -6 mol/L) in presence or in absence of endothelium. Each concentration of the drug was added every 15 min (See online data supplement for details).

Intracerebroventricular Injections in Mice and AVP Radioimmunoassay. K17F (1 µg)

and E339-3D6 (from 0.03 to 2 µg) were administrated by i.c.v. route in conscious mice with free access to water or deprived of water for 24 h as previously described [START_REF] De Mota | Apelin, a potent diuretic neuropeptide counteracting vasopressin actions through inhibition of vasopressin neuron activity and vasopressin release[END_REF]. Animals were killed 1 min after the injection, and trunk blood (0.5-1 mL ) was collected in chilled tubes containing 50 µL of 0.3 mol/L EDTA pH 7.4. AVP concentrations were determined as previously described (5) from 0.2 mL of plasma by using a specific vasopressin-[Arg 8 ] RIA kit (Biovalley, Marne la Vallée, France).

Data and Statistical Analysis.

Values are given as means ± standard error (S.E.M). One-way ANOVA or ANOVA for repeated measures followed by a Fisher Protected Least Significance (PLSD) or by a Student's unpaired t-test (peptide versus vehicle) were used to assess the significance of the results.

RESULTS

Identification of Apelin Receptor Ligands by a Screening Approach Based on FRET

When the project started, no convenient radioactive binding assay was available for the apelin receptor. An alternative approach using FRET between fluorescently tagged proteins and collections of fluorescent molecules just emerged [START_REF] Hibert | Collection of traceable compounds and uses thereof[END_REF](29)[START_REF] Ilien | Fluorescence resonance energy transfer to probe human M1 muscarinic receptor structure and drug binding properties[END_REF][START_REF] Vollmer | Subcellular compartmentalization of activation and desensitization of responses mediated by NK2 neurokinin receptors[END_REF]. We thus decided to adapt this technique to set up a screening assay for apelin receptor. We used the human truncated apelin receptor EGFP-tagged (EGFP-∆ 16 human apelin receptor) at the N-terminal part as an energy donor and ligands carrying a lissamine fluorophore as energy acceptors. The principle of the assay consisted in identifying molecules from the library that reduced EGFP fluorescence emission as a result of FRET. We screened a library of 800 nonpeptidic compounds [START_REF] Hibert | Collection of traceable compounds and uses thereof[END_REF]29) on cells expressing the EGFP-∆ 16 human apelin receptor. We identified three hit compounds inducing FRET including the compound E339-3D6. It displayed the highest affinity for the EGFP-∆ 16 human apelin receptor (K d = 9 ± 1 x 10 -8 mol/L), as assessed by determining the amplitude of EGFP extinction at different concentrations of E339-3D6 (Figure 1A,B, Table S1). Finally, FRET was also detected with the non truncated human apelin receptor, although less intensely (data not shown).

Binding Affinity of E339-3D6

The affinity of E339-3D6 for the apelin receptor was confirmed by determining its ability to displace [ 125 I]-pE13F (2 x 10 -10 mol/L) binding on membrane preparations from HEK 293 cells or CHO cells stably expressing respectively the EGFP-∆ 16 human apelin receptor and the rat apelin receptor tagged at its C-terminal part with EGFP previously characterized [START_REF] De Mota | Apelin, a potent diuretic neuropeptide counteracting vasopressin actions through inhibition of vasopressin neuron activity and vasopressin release[END_REF].

The total binding for a concentration of [ 125 I]-pE13F of 2 x 10 -10 mol/L was 302 ± 27 fmol/mg protein, and the specific binding was 290 ± 23 fmol/mg protein. We first determined the apparent dissociation constant (K d ) of pE13F (K d = 1 x 10 -10 mol/L) by Scatchard analysis of the saturation curves (data not shown). E339-3D6, K17F and pE13F dose-dependently inhibited specific binding to the EGFP-∆ 16 human apelin receptor with Ki values of 3.9 ± 0.7

x 10 -7 mol/L, 5.9 ± 1.6 x 10 -11 mol/L and 2.9 ± 0.4 x 10 -10 mol/L, respectively and to the rat apelin receptor with Ki values of 3.8 ± 0.4 x 10 -7 mol/L, 5.0 ± 0.8 x 10 -11 mol/L and 2.1 ± 0.9

x 10 -10 mol/L, respectively (Table S1). Corresponding Hill coefficient values were close to unity, compatible with a single-site competitive model.

Effects of E339-3D6 on Forskolin-Induced cAMP Production

Incubation of CHO cells stably expressing the rat apelin receptor-EGFP with increasing concentrations of E339-3D6 (10 -8 to 10 -4 mol/L), K17F (10 -12 to 10 -7 mol/L) and pE13F (10 -12 to 10 -7 mol/L) resulted in a concentration-dependent inhibition of forskolin-induced cAMP production (Figure 1C) with IC 50 of 6.4 ± 1.3 x 10 -7 mol/L, 8.3 ± 1.2 x 10 -11 mol/L and 2.3 ± 1.2 x 10 -10 mol/L, respectively. The maximal inhibitory effect of E339-3D6 (8135 ± 685 fmoles/100,000 cells, n=5) corresponding to 60% of inhibition of forskolin-induced cAMP production (20136 ± 1361 fmoles/100,000 cells, n=10) occured for a concentration equal or superior to 10 -5 mol/L and was significantly different from those induced by K17F (1410 ± 260 fmoles/100,000 cells, n=10, p<0.001) and pE13F (2477 ± 602 fmoles/100,000 cells, n=5, p<0.005). The maximal inhibitory effects of K17F (93 %) and pE13F (87 %) on forskolininduced cAMP production occured for concentrations equal or superior to 10 -8 mol/L. Under basal conditions, E339-3D6, K17F and pE13F applied separately at the concentrations of 10 -4 mol/L or 10 -7 mol/L, did not significantly decrease basal cAMP levels (data not shown).

Capacity of E339-3D6 to Trigger Apelin Receptor Internalization

Confocal microscope analysis of CHO cells stably expressing the rat apelin receptor-EGFP in resting conditions displayed intense apelin receptor-EGFP fluorescence at the plasma membrane as previously described (9) (Figure 2A). Incubation with 10 -7 mol/L K17F or 10 -7 mol/L fluorescent apelin 13 (i.e. lissamine-apelin 13: Lissamine-SO 2 -NH-(CH 2 ) 10 -CO-Gln-Arg-Pro-Arg-Leu-Ser-His-Lys-Gly-Pro-Met-Pro-Phe-OH) (Figure 2A, S1) for 20 min resulted in the internalization of the apelin receptor-ligand complexes (Figure S1A arrowheads). We then tested the capacity of E339-3D6 to trigger internalization of apelin receptor-EGFP. Incubation of CHO cells with increasing concentrations of E339-3D6 (10 -7 to 10 -4 mol/L) for 20 min resulted in the dose-dependent internalization of the rat apelin receptor-EGFP (Figure 2B). The maximal internalization occurred for E339-3D6 concentrations comprised between 3 x 10 -6 mol/L and 10 -4 mol/L. Moreover, since E339-3D6 was fluorescent due to the presence of lissamine, its cellular localization was followed during the internalization process. After 20 min of incubation with 3 x 10 -6 mol/L and 10 -4 mol/L E339-3D6, an extensive overlap was observed in labeling for the apelin receptor-EGFP and E339-3D6 within cytoplasmic vesicles, indicating that the internalization of the apelin receptor implies apelin receptor/E339-3D6 complex formation (Figure 2B). In contrast, incubation of CHO cells stably expressing the rat angiotensin II receptor type 1A fused at its C-terminus with EGFP (AT1a-EGFP) with 10 -4 mol/L E339-3D6 neither induced internalization of the AT1a-EGFP receptor nor the internalization of E339-3D6 itself (Figure 2C). The time-course of the rat apelin receptor-EGFP internalization induced by 3 x 10 -5 mol/L E339-3D6 (Figure S1B) showed the appearance of fluorescent vesicles located under the plasma membrane after 5 min of incubation. E339-3D6-induced endocytosis was more pronounced after 10 min of incubation and maximal after 20 min. After 45 min of incubation, vesicles fused and moved in clusters to the perinuclear region of cells with a slight sorting to the plasma membrane (Figure S1B). To evaluate if E339-3D6 and K17F induce the same maximal extent of internalization, we quantified at a maximal dose of K17F (10 -7 mol/L) and E339-3D6 (10 -4 mol/L), the extent of ligand-induced rat apelin receptor-EGFP internalization by confocal microscopy coupled to digital image analysis (Figure 3) as previously described [START_REF] Lenkei | A highly sensitive quantitative cytosensor technique for the identification of receptor ligands in tissue extracts[END_REF]. First, the confocal analysis showed that as well for K17F as for E339-3D6, 100% of analyzed cells (100 cells for each condition) exhibited a profile of internalization (Figure 3A). Secondly, quantification of membrane/intracellular fluorescent ratios (M/I) induced by 10 -7 mol/L K17F (M/I = 1.07 ± 0.15, number of cells analyzed=17) or 10 -4 mol/L E339-3D6 (M/I = 1.03 ± 0.11, number of cells analyzed=17) indicated that E339-3D6 and K17F induced the same maximal extent of internalization (not statistically different, p=0.95) (Figure 3B,C).

Comparison of the Binding Properties of E339-3D6 versus the Apelin Peptide, pE13F, by Molecular Modeling

The in silico models of the human apelin receptor/pE13F and human apelin receptor/E339-3D6 complexes obtained after molecular simulation, revealed that both ligands bind in the same way within the receptor pocket. Both, pE13F and E339-3D6, exhibited a preferred elongated conformation when docked within the receptor cavity. In the complexes, the Cterminal phenylalanine of pE13F and the lissamine chromophore of E339-3D6 were surrounded by aromatic side-chains of the receptor cavity forming a stable network of π-π interactions (Figure 4). In fact, E339-3D6 missing lissamine has much lower affinity for the receptor (data not shown). However, lissamine itself even at a high concentration of 10 -4 mol/L did not inhibit specific binding to the rat apelin receptor-EGFP and was not able to induce apelin receptor internalization (data not shown).

Vasorelaxing Effects of E339-3D6 on Isolated Rat Aorta

In aortic rings precontracted with 3 x 10 -6 mol/L NA, a concentration-dependent relaxation was induced by Ach, K17F and E339-3D6. All these effects were almost abolished in the absence of endothelium (Figure 5). The sensitivity (pD 2 ) to K17F and E339-3D6 (8.1 ± 0.3 and 8.6 ± 0.4, respectively) were significantly different (P < 0.05 and P < 0.01 respectively) from the corresponding value for Ach (6.8 ± 0.4) whereas the maximal relaxations (% of induced tone) were similar (E max of 92 ± 2, and 102 ± 1 and 94 ± 5, respectively).

Effects of Intracerebroventricular Injection of E339-3D6 on Systemic Vasopressin

Release in Conscious Mice Deprived of Water for 24 h

Water deprivation of mice for 24 h significantly increases plasma AVP levels (87.6 ± 14.2 pg/mL , n=11 versus 33.2 ± 3.3 pg/mL , n=11; P< 0.005) (Figure 6). As previously described [START_REF] Reaux | Physiological role of a novel neuropeptide, apelin, and its receptor in the rat brain[END_REF], i.c.v. injection of K17F in water-deprived mice at the dose of 1 µg (468 pmol) (significantly decreased plasma AVP levels (48.5 ± 7.3 pg/mL ) compared with waterdeprived mice injected with saline (87.6 ± 14.2 pg/mL ) (P< 0.05) (Figure 6). I.c.v. injection of E339-3D6 in increasing doses (from 0.03 to 2 µg corresponding to 21 to 1422 pmol) to water-deprived mice induced a dose-dependent decrease in plasma AVP levels with an ED 50 of 0.09 μg (64 pmol) (Figure 6, inset). The maximal decrease in AVP release induced by 1 µg (711 pmol) E339-3D6 (-67 %) was similar to that observed with 1 µg (468 pmol) K17F (-72 %) (Figure 6).

DISCUSSION

The endogenous peptide apelin improves cardiac contractility and decreases cardiac loading in vivo [START_REF] Ashley | The endogenous peptide apelin potently improves cardiac contractility and reduces cardiac loading in vivo[END_REF]20). Together with its ability to decrease arterial BP and systemic AVP release inducing an increase in aqueous diuresis [START_REF] Lee | Characterization of apelin, the ligand for the APJ receptor[END_REF][START_REF] De Mota | Apelin, a potent diuretic neuropeptide counteracting vasopressin actions through inhibition of vasopressin neuron activity and vasopressin release[END_REF][START_REF] Reaux | Physiological role of a novel neuropeptide, apelin, and its receptor in the rat brain[END_REF], this suggests a prominent role for the apelin system in body fluid and cardiovascular homeostasis. Therefore the development of a specific and selective apelin receptor agonist offers the possibility of exploring the role played by this peptide further upstream in cardiovascular diseases and water retention and/or hyponatremic disorders and might be useful for the treatment of heart failure. The present study identifies the first nonpeptidic agonist for the apelin receptor and describes its pharmacological properties in vitro as well as its in vivo biological effects.

At the beginning of our search for APJ receptor ligands, there was no available binding assay suitable for medium or high throughput screening. As an alternative, we decided to use a novel emerging strategy based on FRET [START_REF] Vollmer | Subcellular compartmentalization of activation and desensitization of responses mediated by NK2 neurokinin receptors[END_REF][START_REF] Galzi | Use of a fluorescent protein for detecting interaction between a target protein and its ligand[END_REF]. The human apelin receptor was expressed fused to EGFP on its N-terminal. In parallel, libraries of fluorescent, non-peptide compounds were synthesized [START_REF] Hibert | Collection of traceable compounds and uses thereof[END_REF]29). The fluorophore, lissamine, was chosen for its ability to absorb at EGFP emission wavelength via FRET. About 800 tagged molecules were thus prepared and screened on the EGFP-apelin receptor. As previously validated, such an assay allows the detection of molecules binding to the tagged receptor on orthosteric or allosteric sites [START_REF] Ilien | Fluorescence resonance energy transfer to probe human M1 muscarinic receptor structure and drug binding properties[END_REF][START_REF] Bonnet | A rapid and versatile method to label receptor ligands using "click" chemistry: Validation with the muscarinic M1 antagonist pirenzepine[END_REF][START_REF] Galzi | Method for isolating an allosteric effector of a receptor[END_REF] without need of a radioligand. This method is of particular interest for orphan receptors and when no classical binding assay is easily accessible. In the particular case of the human apelin receptor, by using this new screening approach, we have isolated three hit compounds including the compound E339-3D6 [START_REF] Cortes | APJ receptor ligands and uses thereof[END_REF]. E339-3D6 displayed an affinity of 9 x 10 -8 mol/L for the human truncated apelin receptor determined by FRET and a Ki of 3.9 x 10 -7 mol/L for the human truncated apelin receptor and a Ki of 3.8 x 10 -7 mol/L for the rat apelin receptor determined by radioligand binding assay. E339-3D6 exhibits a high affinity and selectivity for both rat and human apelin receptor. For instance, the molecule does not lead to FRET detection on the following human GPCRs, vasopressin V1a, oxytocin OTR, muscarinic m1R, chemokine CXCR4 and CCR5, melanocortin MC3 nor on rat takykinin NK2 receptors (data not shown). E339-3D6 provides an original research probe that validate the FRET-based strategy that could be applied to other GPCRs, including orphan receptors.

E339-3D6 inhibits forskolin-stimulated cAMP accumulation but the maximal response was only 60% of the maximal responses elicited by the natural full agonists K17F and pE13F, showing that E339-3D6 displays partial agonist activity with regard to cAMP production. In agreement with the partial agonist behavior, the binding sites for E339-3D6 and pE13F are overlapping as shown by the docking of these compounds in the 3D model of the apelin receptor. In the α2-adrenergic receptor, full and partial agonists having different intrinsic efficacy were shown to induce and/or stabilize distinct conformational states of the receptor [START_REF] Ghanouni | Functionally different agonists induce distinct conformations in the G protein coupling domain of the beta 2 adrenergic receptor[END_REF][START_REF] Swaminath | Sequential binding of agonists to the beta2 adrenoceptor. Kinetic evidence for intermediate conformational states[END_REF]. This suggests that pE13F and E339-3D6 could stabilize different conformations of the apelin receptor, pE13F would exclusively bind the active conformation of the receptor whereas E339-3D6 would bind both active and inactive conformations of the receptor, thus generating quantitatively different amounts of active receptors.

However, confocal analysis of CHO cells stably expressing the rat apelin receptor-EGFP stimulated by K17F or E339-3D6 at maximal concentrations, showed that E339-3D6 and K17F induced the same maximal extent of internalization. This shows that E339-3D6 is a full agonist with regard to internalization. Since the apelin receptor is internalized by the early endosomes via a clathrin-dependent mechanism, probably involving the perinuclear recycling compartment [START_REF] El Messari | Functional dissociation of apelin receptor signaling and endocytosis: implications for the effects of apelin on arterial blood pressure[END_REF], it could be concluded that E339-3D6 would be a biased agonist and even an imperfect bias where selectivity for different signaling pathways is a matter of degree as reviewed by Violin and Lefkowitz (37). This is in line with the concept that different ligands can stabilize distinct receptor conformations that may differ in their signaling partner preference [START_REF] Drake | beta-arrestin-biased agonism at the beta2-adrenergic receptor[END_REF][START_REF] Galandrin | Distinct signaling profiles of beta1 and beta2 adrenergic receptor ligands toward adenylyl cyclase and mitogen-activated protein kinase reveals the pluridimensionality of efficacy[END_REF][START_REF] Shukla | Distinct conformational changes in beta-arrestin report biased agonism at seven-transmembrane receptors[END_REF], thus inducing different biological responses. Indeed, we previously showed that K16P (the C-terminal phenylalanine-deleted fragment of K17F) and K17F bind the apelin receptor with similar affinities and equally inhibit forskolin-induced cAMP production, whereas in contrast to K17F (9), K16P does not induce apelin receptor internalization and does not decrease arterial BP [START_REF] El Messari | Functional dissociation of apelin receptor signaling and endocytosis: implications for the effects of apelin on arterial blood pressure[END_REF]. This indicates that a specific receptor conformation inducing internalization is required to decrease arterial BP. In the 3D model of the apelin receptor, the C-terminal phenylalanine in pE13F is positioned in an aromatic pocket within the receptor binding site.

Interestingly, we found that the lissamine chromophore in E339-3D6 is similarly embedded, which could account in part for the full agonist activity of E339-3D6 with regard to apelin receptor internalization.

Since the apelin induced-decrease in BP was proposed to occur through a NO-mediated arterial vasodilatation [START_REF] Tatemoto | The novel peptide apelin lowers blood pressure via a nitric oxidedependent mechanism[END_REF][START_REF] Ishida | Regulatory roles for APJ, a seven-transmembrane receptor related to angiotensin-type 1 receptor in blood pressure in vivo[END_REF], we first evaluated ex vivo, the effect of E339-3D6 on rat aortic vascular tone. To our best knowledge, the effects of K17F on vascular responsiveness of nonobese normotensive rats were poorly investigated. In db/db mice, apelin 36 restores the altered aortic vascular responsiveness to Ach and AngII by potentiating phosphorylation of Akt and eNOS [START_REF] Zhong | Apelin modulates aortic vascular tone via endothelial nitric oxide synthase phosphorylation pathway in diabetic mice[END_REF]. Furthermore, apelin 13 produced relaxation in normal human splanchnic arteries via NO release after activation of apelin receptors located in the endothelium [START_REF] Salcedo | Apelin effects in human splanchnic arteries. Role of nitric oxide and prostanoids[END_REF].

More recently, apelin 36 and pE13F administration in man was shown to cause NO-dependent arterial vasodilation [START_REF] Japp | The apelin-APJ system in heart failure: pathophysiologic relevance and therapeutic potential[END_REF]. Moreover, we recently reported that K17F caused NO-dependent vasorelaxation of rat renal glomerular arterioles precontracted with AngII [START_REF] Hus-Citharel | Effect of apelin on glomerular hemodynamic function in the rat kidney[END_REF]. In the present work, K17F and E339-3D6 induced a similar concentration-and endothelium-dependent vasorelaxation of NA-precontracted aortic rings from normotensive rats. Since the apelin induced-decrease in BP was proposed to occur through a NO-mediated arterial vasodilatation [START_REF] Tatemoto | The novel peptide apelin lowers blood pressure via a nitric oxidedependent mechanism[END_REF][START_REF] Ishida | Regulatory roles for APJ, a seven-transmembrane receptor related to angiotensin-type 1 receptor in blood pressure in vivo[END_REF], the apelin-and E339-3D6-induced aorta vasodilatation could be also mediated by endocytosis of the apelin receptor. Since E339-3D6 is full agonist with regard to internalization, this could explain in part why E339-3D6 has a maximal vasorelaxant effect equal to K17F. Nevertheless, the vasorelaxant action of E339-3D6, even if its affinity for the apelin receptor is weaker than that of the endogenous peptide K17F, could be also related to higher metabolic stability and biological activity of E339-3D6 as compared to K17F, as expected from its non-peptidic structure.

Interestingly, both K17F and E339-3D6 exhibited a significantly more potent vasorelaxant effect than Ach in our experimental conditions, as their pD 2 were significantly higher than that of Ach despite similar maximal relaxant effects. The endothelium-dependent vasorelaxant effect of E339-3D6, similar to that of K17F, is in agreement with the presence of apelin receptor binding sites in human aorta [START_REF] Katugampola | Discovery of recently adopted orphan receptors for apelin, urotensin II, and ghrelin identified using novel radioligands and functional role in the human cardiovascular system[END_REF] and mRNA apelin receptor expression in endothelial cells lining large conduit vessels of various organs [START_REF] Japp | The apelin-APJ system in heart failure: pathophysiologic relevance and therapeutic potential[END_REF][START_REF] Hus-Citharel | Effect of apelin on glomerular hemodynamic function in the rat kidney[END_REF]. E339-3D6 is the first apelin receptor agonist decreasing arterial vascular tone in normotensive rats, suggesting that E339-3D6 could constitute a lead compound for the future development of a new class of vasodilator agents.

Another important part of our study addresses the question whether central administration of E339-3D6, like apelin, is able to decrease systemic AVP in 24h-water deprived mice. E339-3D6 significantly decreased dehydration-induced AVP release and its potency did not differ from that of the natural ligand K17F [START_REF] Reaux | Physiological role of a novel neuropeptide, apelin, and its receptor in the rat brain[END_REF]. These data suggest that E339-3D6 similarly to K17F, when i.c.v injected, is able to rapidly reach the hypothalamic structures involved in AVP release and, by acting on apelin receptors expressed by magnocellular vasopressinergic neurons, inhibits the phasic electrical activity of these neurons. This subsequently induces a decrease in AVP release in the blood circulation. As previously shown for the central action of apelin [START_REF] De Mota | Apelin, a potent diuretic neuropeptide counteracting vasopressin actions through inhibition of vasopressin neuron activity and vasopressin release[END_REF], we can hypothesize that an apelin receptor agonist such as E339-3D6, by counteracting AVP actions would have aquaretic effects. In this context, apelin receptor agonists would be particularly interesting for the treatment of water retention and/or hyponatremia, avoiding the excessive loss of sodium and potassium commonly found with the use of diuretics.

In conclusion, we described the discovery of the first nonpeptidic specific agonist of the apelin receptor, E339-3D6, by an original and efficient screening approach. We demonstrated ex vivo its ability to produce vasorelaxation of rat aorta as well as its in vivo potential to inhibit water deprivation-induced AVP release in the blood circulation. Such compound would be useful for a better understanding of the physiopathological roles of apelin and its receptor and to evaluate the therapeutic potential of apelin receptor agonists in different animal models of pathologies. This ligand represents a unique and very interesting lead compound for which a medicinal chemistry program has been undertaken in order to optimize its affinity and its bioavailability. Optimization of the structure towards a compound meeting the clinical candidate status requirements will also be conducted in order to develop new agonists of the apelin receptor which could represent a new generation of vasodilator and aquaretic agents.

Figure 1 .

 1 Figure 1. In vitro pharmacological characterization of E339-3D6 (A) structure of compound

Figure 2 .

 2 Figure 2. Effects of K17F and E339-3D6 on rat apelin receptor-EGFP internalization in CHO

Figure 3 .

 3 Figure 3. Quantification of rat apelin receptor-EGFP internalization. (A) Confocal images of

Figure 4 .

 4 Figure 4. Docking of pE13F or E339-3D6 into the human apelin receptor 3D model. The

Figure 5 .

 5 Figure 5. Effects of E339-3D6 on rat aorta vascular tone. Cumulative concentration-response

Figure 6 .

 6 Figure 6. Effects of i.c.v. injection of E339-3D6 in mice on water deprivation-induced

  Figure 1

  Figure 2

Figure 5 LigandFigure 6

 56 Figure 5

ACKNOWLEDGMENTS

This work was supported by the Agence Nationale pour la Recherche (ANR) "Cardiovasculaire Obésité et Diabète 2005". Xavier Iturrioz was supported by a grant from the Société Française d'Hypertension Artérielle. Romain Gerbier was supported by a grant from the Ile de France "Domaine d'Intérêt Majeur-Maladies Cardiovasculaires, diabète et obésité".

COMPETING INTERESTS STATEMENT

The authors declare that they have no competing financial interests.


Multiaxial fatigue criteria based on a multiscale approach

K.

Multiaxial Fatigue

Criteria Based on a Multiscale Approach

K. DANG VAN

Laboratoire de Mechanique des Solid, Ecole Polytechnique, 91128 Palaiseau, France 6.9.1 VALIDITY Prediction of high-cycle fatigue resistance is of great importance for structural design. In spite of this clear industrial need, until now modeling of metal behavior in a high-cycle fatigue regime was often based on empirical approaches: the Wohler curve and the Goodman-Haigh or Gerber diagrams are still the very popular tools for engineers. However, these concepts are not appropriate when studying the multiaxial stress cycles that are frequently encountered on modem mechanical components. These multiaxial stresses arise from factors such as external loadings, the geometry of the structure, which can induce multiaxiality even if the loading is uniaxial, and finally residual stresses.

BACKGROUND

In order to derive a multiaxial endurance fatigue criterion, an original method of computing based on a multiscale approach was proposed by Dang Van.

1

This method is quite different from existing fatigue approaches. It arises from the observation that generally the first fatigue damage processes begin in grains which have undergone plastic deformation, with the appearance of slip bands in some grains which broaden progressively with the applied cycles; this stage is then followed by localized damage corresponding to formation of intragranular microcracks; these microcracks can be arrested by grain boundaries, but they may also propagate. After a certain number of cycles, a main crack initiates, grows in size, and shields the other defects and consequently leads to the final rupture of mechanical structure. In a high cycle fatigue regime, even if it is necessary to have plastic deformation at the micro-or the mesoscale (corresponding to the grain size), most of the time no visible irreversible deformation at the macroscopic level can be detected. It is thus characterized by a large heterogeneity of plastic deformation from grain to grain: only certain misoriented crystals undergo plastic slips, and in this way a very heterogeneous distribution of microcracks can be observed. In this fatigue regime, the initiation of the first visible cracks takes a large part of the fatigue life of the structure. It demonstrates the importance of studying conditions governing the crack initiation process.

In most of the existing fatigue models, these conditions are described with the help of macroscopic parameters which are evaluated according to different assumptions of homogeneity and isotropy. For example, engineers evaluate the stress I: not at a point of a structure but over a finite volume V that defines the macroscopic scale which is used (see Fig. 6.9.1). Ty pically V is of the order of the dimension of the strain gauges. However, because the phenomena which cause fatigue initiation are microscopic, the local parameters (for example, local stress a) differ from the macroscopic ones. Thus the use of classical macroscopic engineering parameters does not seem pertinent, since, FIGURE 6.9.1 Different scales of material description: the macro-and mesoscopic scales.

at that local scale, the material cannot be considered anymore as homogeneous. In particular, the local redistributions induced by the inhomogeneous incompatible strains llp, and as a consequence local residual stress p, are not accounted for.

FORMULATION

The originality in Dang Van's proposal is precisely the use of local mesoscopic mechanical parameters 0' to derive fatigue resistance criteria. These parameters are evaluated from the macroscopic parameters thanks to a hypothesis of elastic shakedown. More precisely, it is postulated that, near the fatigue limit threshold, the mechanical structure shakes down elastically at all scales of material description. Under this assumption, the precise knowledge of the local constitutive equations, which is not possible to evaluate, is not necessary.

The physical interpretation of this hypothesis is that after a certain number of loading cycles the response is purely elastic (or at least the plastic dissipation rate becomes negligible). Then, using shakedown theorems (Melan's theorem and its generalization by different authors; (see, for instance, Reference [ l]), it is possible to derive a method for estimating the apparent stabilized stress (tensorial) cycle at the macroscopic and mesoscopic level which intervenes in the proposed fatigue criteria, provided that the material is considered a structure made of grains of different crystallographic orientations. Theoretical developments of this theory are presented in Reference [START_REF] Van | Introduction to fatigue analysis in mechanical design by the multiscale approach[END_REF]. For practical applications, it is only necessary to remember how to derive local parameters from macroscopic stress cycles near the fatigue limit, which is presented in following text.

The general relation between macroscopic and local stress tensor is O'(m, t) = A.l:(M, t) + p(m, t)

This relation is well known in the theory of polycrystalline aggregates. In this equation A is an elastic localization tensor which depends on the microstructure; l:(M, t) is the macroscopic stress tensor at time t in the representative volume element V(M) surrounding M, and O"(m, t) and p(m, t)

are, respectively, the local stress tensor and the residual stress tensor at any point m of V(M) . For the sake of simplicity, let us assume that A= identity (elastic homogeneity); then p characterizes the local stress fluctuation in V(M) .

If elastic shakedown happens, then p must become independent of time after a certain number of cycles, so that the local plastic yield criterionf(m) is no longer violated. Assuming that an approximate elastic shakedown occurs if the loading cycles are near the fatigue limit, then there must exist a local fixed If the Mises criterion is chosen, then G( m, t) belongs to the hypershere in five dimensional space, representing the limiting value of the Mises norm for which elastic shakedown is possible. Mandel et al. [START_REF] Mandel | Adaptation d'une structure elastoplastique a ecrouissage cinematique[END_REF] showed that p* ( m) can be approximately taken to be the center of the smallest hypersphere surrounding the loading path in the deviatoric macroscopic stress space represented in Figure 6.9.2. In this figure, S (resp. s) represents the deviatoric stress corresponding to I: (resp. G). Because of the shakedown hypothesis at all scales of material description, S at stabilization is also elastic. Finally, by that construction, the local stress state is known at any time t of the apparent stabilized state. One must still choose a fatigue criterion. Since the local stress is approximately known at any time t, it is natural to try to take account of the characteristic of the loading path (as in plasticity). Thus a reasonable fatigue criterion could be stated as follows.

Crack initiation will occur in a critically oriented locus (usually corresponding to a grain) within V(M) that has undergone plastic deformation, if, for at least one time instant t of the stabilized cycle, one has F[( G(m, t)] ;::: 0 for m E V(M)

In such a criterion, the current stress is considered, in contrary to most (or all) existing fatigue criteria. Damage arises over a precise portion of the loading path (or equivalently, over a precise time interval of the loading period). Since cracks usually occur in transgranular slip bands, the local shear acting on these planes is an important parameter. Moreover, the normal stress acting on these planes accelerates damage formation. However, this quantity is rather difficult to compute generally because it depends of the considered plane. For this reason, hydrostatic stress is preferred because it is much easier to use, being an invariant scalar. Furthermore, it can be interpreted as the mean value of the normal stresses acting on all the planes that pass through the considered point of the structure. Based on these remarks, F( cr) is chosen to be a function of the local shear r and the local hydrostatic stress p.

IDENTIFICATION OF THE PARAMETERS

The simplest criterion that can be conceived is a linear relation between these quantities,

F(cr) = r + ap-b
where a and b are material parameters that can be determined by two simple types of fatigue experiments: uniaxial tests of tension-compression and torsion on classical fatigue test machines. Iff is the fatigue limit strength in alternate tension-compression, and t is the fatigue limit in alternate twisting, then t-f /2 a= ---rf3 ' b=t

The safety domain (no fatigue crack initiation) is delimited by the two straight lines represented in Figure 6.9.3. The ordinate is the algebraic shear stress acting in an oriented direction, and the abscissa is the hydrostatic stress p. On the same figure the loading paths for fatigue limit in tension compression and in twisting are represented. Two more loading paths are shown. Path r 1 is nondamaging because it lies entirely within the straight lines that delimit the safe domain, whereas a small portion of path r 2 induces damage.

HOW TO USE THE MODEL

To check automatically the fatigue resistance of a structure is a rather difficult task, because at each point one has to consider the plane on which the loading path (r[t], p[t]) is a "maximum" relative to the criterion. This computation can t tension-compression p FIGURE 6.9.3 Fatigue limit domain and loading paths.

be simplified as follows. The maximum shear stress according to Tresca's measure is calculated over the cycle period:

r(t) = Tresca[o-(t)]
It is useful to notice that Another possibility is to use the octahedral shear J2[o-(t)] instead of r(t). However, this method does not give the critical facets.

Another interesting proposal derived from the multiscale approach was given by l.V.Papadopoulos: the fatigue limit for a given periodic loading J.Q(t) corresponds to the limit of the intensity A such that elastic shakedown is possible. Beyond this limit, plastic shakedown or ratchet phenomena will induce damage and fracture because of subsequent softening. The limit size k* of the hypersphere surrounding the loading path (as explained previously) is one possible and natural way to characterize this state. If k* is greater than some limit value which depends on the local maximum hydrostatic stress in the cycle, fatigue will occur. This corresponding fatigue criterion is

k* + LXPmax -[3 > 0
As previously, the parameters ex and f3 can be identified by two different tests. By this method it is no longer necessary to describe the whole loading path once k* is determined. In many cases, the predictions are very similar to the current state methodology as presented previously.

FIGURE 6.9. 2

 2 FIGURE 6.9.2 Determination of the local residual stress P* at the shakedown state.

  Tresca[o-(t)] = Tresca[s(t)] = Maxulo-r(t)-O"J(t)l/2 The stresses o-1(t) and o-1(t) are principal local stresses at timet. The quantity d that quantifies the danger of fatigue failure defined byd = Max r(t) t b-ap(t)is calculated over the loading period. The maximum is to be taken over the cycle. If d > 1, the fatigue failure will occur.Working this way, all couples (r,p) are situated in the positive part of r. All fa cets which could be involved by the crack initiation are automatically reviewed. Couples (r, p) verify ing the condition d > 1 are associated with specific fa cets. Therefo re, the criterion also provides the direction of crack initiation.


Andre ´zaoui

Continuum Micromechanics: Survey

Keywords: Micromechanics, Surveys, Heterogeneity, Material properties

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

Introduction

The problem of the transition from the microscale to the macroscale in view of the prediction of the macroscopic mechanical constitutive behavior of materials has been the matter of intensive research for several decades. The microscale considered here refers to the inhomogeneous constitution of the considered materials and to the associated inhomogeneity of their mechanical characteristics. Whereas preliminary results in this field were obtained as early as in 1887 for the first ''law of mixtures'' ͑the wellknown Voigt estimate͒ and in 1929 for another one ͑the Reuss estimate͒, the actual foundations of homogenization techniques are only about forty years old. Two main underlying motivations have been active. The first one originated from metal forming and from the will of a few pioneer material scientists to predict the plastic flow of polycrystals from plastic glide in single crystals: the Sachs model ͑Sachs 1928͒ as well as the popular Taylor model ͑1938͒ actually addressed very difficult problems of nonlinear micromechanics, though from what can be appreciated today as a rather empirical point of view. The preliminary development of composite materials progressively motivated a new interest in this field, mainly for linear elasticity, on the firm basis of continuum mechanics: the basic concepts of ''representative volume element'' with adequate boundary conditions and of ''homogeneous equivalent medium'' were defined and powerful variational approaches were achieved from which rigorous bounds, instead of approximate estimates, were derived. At the same time, the early Voigt, Reuss, and Taylor estimates were re-assessed as bounds whereas the basic inclusion problem was given a general solution, opening the way to improved estimates. In the early 1960s the main basis of the so-called ''continuum micromechanics,'' according to Hill ͑1965͒, can be considered as founded.

New results in the field of linear elasticity have been continuously derived from that period, with special mention of refined bounding associated with an improved morphological description and of thorough theories for the special case of media with a periodic microstructure ͑see, e.g., [START_REF] Bensoussan | Asymptotic analysis for periodic structure[END_REF][START_REF] Sanchez-Palencia | Non-homogeneous media and vibration theory[END_REF][START_REF] Suquet | Elements of homogenization for inelastic solid mechanics, Homogenization techniques for composite media[END_REF]. Nevertheless, it can be considered that the main novel developments during the last 30 years have been concerned with nonlinear continuum micromechanics. A striking feature of this period lies in the fruitful interaction which has developed between two different points of view, attached to the derivation of bounds and estimates, respectively: while they have been going forward independently for more than two decades, it has been possible at some stage to make comparisons between them in specific situations. Consequently, some unanimously adopted estimates were shown to violate rigorous bounds whereas variational procedures were used to obtain new estimates. Despite significant advances in this field, many problems still remain open today, especially for constitutive behavior which does not derive from one single potential. Another important new ͑or renewed͒ current field of research, especially promoted by the increasing facilities of numerical treatment and the associated ''multiscale'' approaches, is concerned with situations where the basic concept of ''representative volume element'' is failing and where internal length scales must be taken into account.

In this brief survey of continuum micromechanics, the stress is deliberately laid on the basic foundations and on the more classical and simpler case of linear elasticity, including the various situations for which eigenstrains or eigenstresses are present. The main questions and approaches concerning nonlinear micromechanics are only reported shortly and qualitatively while the limits of the considered theories are briefly discussed in conclusion.

Foundations of Continuum Micromechanics

Scale Separation

Classical homogenization techniques aim at replacing an actual inhomogeneous complex body by a fictitious homogeneous one which behaves globally in the same way. Continuum micromechanics is mainly concerned with statistically homogeneous materials for which it is possible to define a ''representative volume element'' ͑RVE͒ and an ''equivalent homogeneous medium'' ͑EHM͒ which are equivalent to each other from a mechanical point of view. This means that their overall responses to any mechanical loading have to be the same. In other words, the EHM must be such that the stress and strain fields ⌺(X គ ) and E(X គ ) derived at the macroscale by solving the boundary value problem of a homogeneous body constituted by this fictitious homogeneous material are the average values, over any RVE centered at X គ , of the local stress and strain fields (x គ ), and (x គ ) which would have been derived at the microscale if the considered microstructural inhomogeneities could have been accounted for in such a calculation.

This requires at least that, for the mechanical behavior under investigation, the characteristic length, say d, of the considered inhomogeneities and deformation mechanisms to be much smaller than the size, say l, of the studied volume element, so that this element could be considered as representative of the studied material whatever its location in the macroscopically ͑or statistically͒ homogeneous body. Moreover, l must be sufficiently smaller than the characteristic dimensions L of the whole body and than the fluctuation length of its prescribed mechanical loading so as to make possible the use of the classical integral and differential tools of structural analysis. In addition, the smallest characteristic length d has to be compatible with the use of the basic concepts ͑strain and stress tensors, etc.͒ of continuum mechanics. To sum up, we must have

d 0 ӶdӶlӶL, lӶ (1)
where d 0 ϭlower length bound under which continuum mechanics is no more valid. Formula ͑1͒ gives only necessary scale separation conditions for such a problem not to be meaningless: these conditions may be not sufficient when percolation phenomena or long-range correlation effects are prominent.

Homogenization Methodology

Once the scales have been defined, the RVE's description, for both its geometrical and mechanical characteristics, has first to be achieved. A basic difference must be emphasized at this point between two kinds of situations. The first one is concerned with materials whose microstructure can be completely described; this is especially the case for media with a periodic microstructure.

For such materials, deterministic approaches, which are not basically different from those used to solve classical boundary-value problems of continuum mechanics, have been developed within a rigorous mathematical framework; approximate numerical treatments, resorting to the popular ''unit-cell'' method, have also been finalized. This case is not considered in the sequel and attention is focused on heterogeneous materials with a random microstructure. For such materials, a statistical ͑or stochastic͒ approach is necessary. This needs first the specification of the considered mechanical ''phases'' and the determination of their geometric and mechanical characteristics, and then the statistical description of the spa-tial distribution ͑or the ''morphology''͒ of the phases: the texture function and the correlation functions of the lattice orientations for a polycrystal, the volume fraction, and the shape and orientation distribution of the particles or fibers for a composite material, etc. The crucial point is that, from a practical point of view, the spatial distribution of the phases cannot be completely determined, even in a statistical sense. Consequently, without appropriate assumptions or approximations, the constitutive behavior of the EHM cannot be determined unambiguously.

In other words, for a given description of the RVE, the concentration ͑or ''localization''͒ problem, which is concerned with the mechanical modeling of the interactions between the phases and the associated derivation of the local fields (x គ ), (x គ ),..., within the RVE from the knowledge of the macroscopic quantities ⌺, E,..., cannot be solved in general. One can then proceed along two routes: either one makes additional assumptions or approximations in order to derive some estimates for the overall behavior or one looks for bounds for the overall mechanical properties. Such estimates are more or less pertinent according to the adequacy of the underlying assumptions; as for the bounds, they are derived by considering all the EHMs which are consistent with the limited available information on the phase distribution and by computing the extremal properties which could be reached in this way. This can be done by using variational approaches which deal with admissible strain or stress fields, i.e., compatible strain or equilibrated stress fields obeying the displacement or tension boundary conditions on the RVE, respectively: this needs an adequate definition of these conditions.

As a matter of fact, the original concentration problem is not a ''well-posed'' boundary-value problem since the detailed conditions on the boundary are not known. So, this problem is first transformed into a simpler one by assuming homogeneous boundary conditions on the RVE ͑Hill 1963;1967;[START_REF] Hashin | Analysis of composite materials: a survey[END_REF]. According to the homogeneous stress boundary conditions, surface tractions T គ g are supposed to be prescribed at the boundary and given by

T គ g ϭ⌺"n គ (2)
where the constant tensor ⌺ϭknown macroscopic stress tensor and n គ ϭunit outward normal at the boundary. This approximation is valid far enough from the boundary, i.e., almost everywhere in the RVE as long as conditions ͑1͒ are fulfilled. From Eq. ͑2͒, one proves that ⌺ equals the volume average stress ͗͘ in the RVE, denoted by V; in fact, for any equilibrated ͑i.e., divergence free͒ stress field *(x គ ) which obeys Eq. ͑2͒, one has

͗*͘ϭ͑1/V͒ ͵ V *͑x គ ͒dVϭ⌺ (3)
Similarly, homogeneous strain boundary conditions are associated to prescribed displacements u គ g at the boundary given by

u គ g ϭE•x គ (4)
where Eϭmacroscopic ͑constant͒ strain tensor and x គ belongs to the boundary. This approximation is satisfactory as long as d Ӷl. One gets Eϭ͗͘ since, for any compatible ͑i.e., derived from a displacement field͒ strain field Ј(x គ ) obeying Eq. ͑4͒, one gets While in principle, Eqs. ͑8͒ and ͑9͒ are not equivalent, they tend to be so when d/l tends to zero ͑Hill 1967; Mandel 1972͒, as assumed in the sequel, according to Eq. ͑1͒. On the contrary, for periodic media and d of the order of l, Eqs. ͑8͒ and ͑9͒ would lead to substantial differences ͑Suquet 1987͒.

Linear Elasticity

In this section, a natural initial state is considered, i.e., no eigenstrains or eigenstresses are present. Due to the linearity of the constitutive equations and to the unicity of the solution, Eqs. ͑7͒ must be linear and homogeneous with respect to E or ⌺. This means that the strain or stress concentration operators AЈ and B* reduce to simple fourth-order tensors fields AЈ(x គ ) and B*(x គ ), respectively

Ј͑x គ ͒ϭAЈ͑ x គ ͒:E or *͑x គ ͒ϭB*͑ x គ ͒:⌺ (
where S* and CЈ can be viewed either as estimates or as bounds ͑through the use of the minimum principles of linear elasticity, i.e., the theorems of potential and complementary energy͒ for the overall effective compliances and moduli S eff and C eff , respectively. Thanks to Hill's lemma, the energetic definition of these tensors, namely

P Hi jkl 0 ϭϪ ͩ ͵ H G ik 0 ͑ x គ Ϫx គ Ј͒dVЈ ͪ l j ͑ i j͒͑kl͒ x គ H
where (i j)(kl)ϭsymmetrization with respect to (i j) and (kl) and G 0 (x គ Ϫx គ Ј)ϭGreen tensor for the infinite medium with moduli C 0 ͑i.e., the tensor which correlates the displacement at x គ to the unit point force applied at x គ Ј͒. The tensor P 0 is closely related to the Eshelby tensor S 0 Esh ͑namely, S H 0 Esh ϭP H 0 :C 0 ͒. For a sphere and isotropic elasticity, one has

S Sph 0 Esh ϭ␣ 0 Jϩ␤ 0 K IϭJϩK J i jkl ϭ͑1/3͒␦ i j ␦ kl (19) ␣ 0 ϭ 3k 0 3k 0 ϩ4 0 ␤ 0 ϭ 6͑k 0 ϩ2 0 ͒ 5͑3k 0 ϩ4 0 ͒
with 0 and k 0 ϭshear and bulk moduli of the reference medium, respectively.

Inclusion-based estimates for the overall moduli can be defined from this solution by estimating the average mechanical state in each phase ͑r͒ of the RVE subjected to the macroscopic strain E as that of an ellipsoidal inhomogeneity (H r ) with the same moduli c r , embedded in an infinite matrix with arbitrary moduli C 0 subjected to some adequate uniform strain at infinity E 0 . The shape and orientation of (H r ) can be specified from what is known about the geometry of phase ͑r͒ whereas E 0 is determined by the average equation ͗͘ϭE. This results in the following set of equations:

r ϭ͓IϩP r 0 :͑c r ϪC 0 ͔͒ Ϫ1 :E 0 Eϭ͗͘ϭ͓͗IϩP 0 :͑cϪC 0 ͔͒ Ϫ1 ͘:E 0 (20) C 0 est ϭ͗c:͓IϩP 0 :͑cϪC 0 ͔͒ Ϫ1 ͘:͓͗IϩP 0 :͑cϪC 0 ͔͒ Ϫ1 ͘ Ϫ1
As for the moduli C 0 , they can be chosen at will in order to express at best the specific morphology of the considered material. Several classical estimates correspond to special choices for C 0 or S 0 : the law of mixtures is recovered with vanishing values for C 0 or S 0 ; the Mori-Tanaka model ͑Mori and Tanaka 1973; Benveniste 1987͒, which is devoted to particle reinforced composites for small volume fractions of particles, is associated to the choice C 0 ϭC mat , i.e., the elastic moduli of the matrix phase, expressing the fact that the continuous matrix plays a prominent morphological role; the self-consistent estimate ͑Hershey 1954; Kro ¨ner 1958͒ corresponds to the choice C 0 ϭC SC , with C SC the searched overall moduli, which is adequate for materials, such as polycrystals, whose phases are dispersed in the RVE so that none of them plays any specific morphological role. In the latter case, Eqs. ͑20͒ for C SC become integral equations, since P 0 ϭP SC and C 0 ϭC SC are not known in advance.

Bounds for Effective Moduli and Compliances

Bounds for C eff or S eff can be derived with use of the minimum principle for elastic media. For homogeneous boundary conditions, they read ͗*:s:*͘у͗:s:͘ where the tensor T A is transposed from A ( T A i jkl ϭA kli j ).

Bounds can be derived from Eqs. ͑23͒ and ͑24͒ whenever admissible strain or stress fields, Ј(x គ ) or *(x គ ), are found which make this derivation possible from the known statistical description only. When the phase moduli and volume fractions are known, uniform fields can be chosen, namely, *ϭ⌺ ͑so that B*ϭI͒ or ЈϭE, ͑AЈϭI͒, which leads to the well-known Reuss and Voigt bounds

⌺:͑͗s͘ϪS eff ͒:⌺у0 ᭙⌺ ͑ Reuss͒ (25) E:͑͗c͘ϪC eff ͒:Eу0 ᭙E ͑ Voigt͒ (26)
If nothing more is known about the space distribution of the phases, S eff and C eff are arbitrarily anisotropic, so that Eqs. ͑25͒ and ͑26͒ lead to a rather loose bounding of the 21 corresponding elastic constants. On the other hand, if the symmetry of the global anisotropy is known in advance, the Reuss and Voigt bounds are more efficient, but they are certainly not optimal. If the inhomogeneous material is known to be isotropic at the micro and the macro scales, the overall shear and bulk moduli eff and k eff are bounded by

͗ Ϫ1 ͘ Ϫ1 р eff р͗͘ and ͗k Ϫ1 ͘ Ϫ1 рk eff р͗k͘ (27)
Note that the laws of mixtures, which estimate any overall parameter as the mean value of the corresponding local ones, are only valid here as bounding properties. This is no more the case for other elastic constants, such as the Young modulus or the Poisson ratio, and it is completely wrong for anisotropic elasticity.

In addition, when the spatial distribution of the phases is known to be isotropic, the Voigt-Reuss bounding can be improved, as shown by Hashin and Shtrikman ͑1963͒. The corresponding derivation is based on an optimization procedure with respect to an infinite set of admissible trial strain fields obtained as the solution of the following problem: an infinite elastic body with uniform moduli C 0 is subjected to an arbitrary distribution of fictitious body forces f គ (x គ ) derived from a polarization stress field p(x គ ) by f គ ϭdiv p. The associated strain field can be found by using the same Green techniques as those mentioned above for the solution of the inhomogeneity problem; for polarization stress fields which are piecewise constant per phase, p r , the property of an isotropic distribution of the phases can be used to derive explicit bounds which have still to be optimized through an adequate choice of p(x គ ) and C 0 . The corresponding bounding for local and global isotropy reads as follows:

͗͑ Ϫ ϩ␤ Ϫ ␦ Ϫ ͒ Ϫ1 ͘•͗͑ Ϫ ϩ␤ Ϫ ␦ Ϫ ͒ Ϫ1 ͘ Ϫ1 р eff р͗͑ ϩ ϩ␤ ϩ ␦ ϩ ͒ Ϫ1 ͘•͗͑ ϩ ϩ␤ ϩ ␦ ϩ ͒ Ϫ1 ͘ Ϫ1 (28) ͗k͑k Ϫ ϩ␣ Ϫ ␦k Ϫ ͒ Ϫ1 ͘•͗͑k Ϫ ϩ␣ Ϫ ␦k Ϫ ͒ Ϫ1 ͘ Ϫ1 рk eff р͗k͑k ϩ ϩ␣ ϩ ␦k ϩ ͒ Ϫ1 ͘•͗͑k ϩ ϩ␣ ϩ ␦k ϩ ͒ Ϫ1 ͘ Ϫ1 (29)
where Ϫ ϭinf(); k Ϫ ϭinf(k); ϩ ϭsup(); k ϩ ϭsup(k); ␦ Ϯ ϭϪ Ϯ ; ␦k Ϯ ϭkϪk Ϯ ; and ␤ Ϯ and ␣ Ϯ are defined from Eq. ͑19͒. When k and are ''well ordered'' i.e., Ϫ and k Ϫ refer to the same ''softest'' phase and ϩ and k ϩ to the same ''stiffest'' phase͒, the Hashin-Shtrikman bounds belong to the set of estimates defined by Eq. ͑20͒. This result is an illustration of the strong connection which exists between heterogeneity and polarization stresses p(x គ ), which can be viewed as eigenstresses ͑associated with eigenstrains T through p(x ᠪ)ϭϪC 0 : T (x គ )), as well as between Eshelby's problems of the heterogeneity and of the inclusion ͑see hereafter͒.

''Point'' Versus ''Pattern'' Approaches

The Hashin-Shtrikman bounds have been derived for more general cases, especially when the phase distribution is no more isotropic but has an ellipsoidal symmetry ͑Willis 1977͒. They can also be integrated in the so-called ''systematic theory'' developed by Kro ¨ner ͑1977͒ for elastic random materials. This theory is based on the solution of the general equation for inhomogeneous elasticity by use of Green techniques and on the description of random media by correlation functions of their elastic moduli. The effective moduli ͑or compliances͒ can be written as formal infinite series whose general term of order n is associated with the n-point correlation functions of the elastic moduli. By truncation of the series at any rank, bounds for the effective moduli ͑or compliances͒ are derived: they are closer and closer when the rank of truncation increases. In the case of ''graded disordered'' materials, which satisfy some property of the correlation functions up to a given rank n, the Voigt-Reuss bounds are recovered for nϭ1, the Hashin-Shtrikman bounds for nϭ2, and the selfconsistent estimate for n→ϱ, which corresponds to ''perfectly disordered'' materials ͑Kro ¨ner 1978͒.

This theory is essential for a thorough understanding of the elasticity of random media, but it is of limited practical help. Besides some mathematical difficulties in the effective computation of bounds, the use of ͑point͒ correlation functions happens not to be very convenient in many cases: their experimental identification can be performed only up to limited orders while some major primary morphological characteristics, such as the phase connectedness or disconnectedness, need quite high orders to be accounted for efficiently. Fifteen years before Kro ¨ner's theory, this crucial point of the geometrical description of ''matrix/ inclusion''-type morphologies had been given a quite attractive tentative solution by Hashin ͑1962͒ according to a completely different point of view: the ''composite spheres ͑or cylinders͒ assemblage'' ͑CSA, say, or CCA͒ is made of spherical particles ͑or cylindrical parallel fibers͒ surrounded by a concentric matrix layer whose thickness is such that the phase volume fraction of each composite element equals the prescribed volume fractions of the composite; these similar composite elements have a variable size going to zero so as to fill up the whole space and to maintain the connectedness of the matrix; they are distributed in an isotropic ͑or transversely isotropic͒ manner according to the case ͑particles or fibers͒.

Unfortunately, the mechanical treatment of such assemblages proved to be disappointing: Hashin's bounds for the CSA ͓or Hashin-Rosen's bounds ͑1964͒ for the CCA͔ for the shear ͑or transverse shear͒ modulus could be very distant from each other, even more than the Hashin-Shtrikman's ones which consider any isotropic ͑or transversely isotropic͒ distribution instead of the very specific ones expressed by the CSA ͑or the CCA͒. Later, this approach was resumed by Christensen and Lo ͑1979͒ in view of estimating the overall shear modulus: by transferring to this case the basic idea of the self-consistent procedure, they proposed, after Kerner ͑1956͒, to consider one composite sphere ͑or cylin-der͒ embedded in an infinite matrix with the effective ͑unknown͒ moduli and to determine these effective moduli through a selfconsistent energy condition. The results, as compared with those derived from several micromechanics models of the same class ͑Christensen 1990͒, were more satisfying; but the morphological meaning of this ''three-phase model'' ͑or ''generalized selfconsistent scheme''͒ was less clear than the initial one.

Going back to Hashin's initial approach, one can save the basic idea of finite composite elements and improve the geometrical description of their spatial distribution. This leads to combine a deterministic description of small, but finite, well-chosen ''composite patterns'' and a statistical representation of their distribution: in many cases, essential morphological features can be expressed in this way much more directly and easily than by using point correlation functions. According to this ''pattern approach'' ͑Stolz and Zaoui 1991͒, a given multiphase material is decomposed, after some morphological analysis, into N patterns or ''morphological phases'' ͑͒ consisting of identical composite representative domains D l . The general Hashin-Shtrikman procedure is then applied with a polarization stress field p (y គ ) which is no more uniform within the morphological phases ͑͒ but has identical values at homologous points y គ of the domains of the same pattern. With help of adequate definitions of strain and stress averages over homologous points in each pattern and use of the Green techniques, the polarization stress field can be optimized and new Hashin-Shtrikman-type bounds can be obtained if the distribution of the pattern centers is isotropic or ellipsoidal. Like for the classical ''point'' Hashin-Shtrikman bounds, these bounds can be obtained from the solution of N elementary inclusion-matrix problems where each domain D i is embedded in an infinite homogeneous matrix with extremal moduli. These problems can be solved numerically ͑Bornert et al. 1996͒ in the general case. For the CSA and isotropy or for the CCA and transverse isotropy, an analytical solution can be derived ͑Herve ´et al. 1991͒ and extended to n-layered spheres or cylinders according to Herve ´and Zaoui ͑1993, 1995͒. They improve significantly on classical Hashin or Hashin-Shtrikman bounds. Similarly, new generalized ''pattern-based'' self-consistent estimates can be obtained ͑Bornert 1996͒.

Linear Elasticity with Eigenstrains

A number of interesting physical situations can be analyzed from an extension of the foregoing developments to the case where eigenstrains T (x គ ) or eigenstresses p(x គ ) are present in the initial state. A typical situation of this kind is concerned with thermal strains th (x គ ), which can derive, for instance, from a uniform temperature variation ⌬T in a heterogeneous material whose constituents ͑r͒ have different elastic moduli c r and different tensors of thermal expansion ␣ r ͑or coefficients ␣ r in case of local isot-ropy͒. It may be useful to predict the overall thermal strain E th as well as the effective tensors of thermal expansion ␣ eff ͑or coeffi-cients ␣ eff in case of global isotropy͒. This cannot be derived by direct averaging of the local quantities since, while the total strain field (x គ )ϭ el (x គ )ϩ th (x គ ) is still compatible, so that the overall strain tensor E is the volume average of (x គ ), neither the elastic nor the thermal strain fields are generally compatible; consequently, E th and ␣ eff do not reduce to ͗ th ͘ and ͗␣͘, respectively. Similar questions arise when electric or magnetic effects as well as various environmental phenomena ͑moisture, oxidation, corrosion, etc.͒ are considered in heterogeneous elastic media, i.e., whenever stress-independent eigenstrains are superimposed on the elastic strains; prestresses or eigenstresses can be dealt with in the same manner, e.g., when gas ͑Fen-Chong et al. 1999͒ or liquid ͑Dormieux and Maghous 2000͒ pressure is present in porous elastic materials. Though stress dependent, even plastic strains can also be considered as long as they are fixed.

The main result which is easily derived from the analysis of such problems by repeated use of Hill's lemma ͑6͒, is the following:

E T ϭ͗ T :B͘, E el ϭ͗ el :B͘ (30) Pϭ͗p:A͘, C eff :Eϭ͗:c:A͘ which leads directly to the effective tensors of thermal expansion ␣ eff or to the effective ''tensors of thermal coefficients'' eff ͑defined from the local tensors of thermal coefficients ϭc:␣͒.

␣ eff ϭ͗␣:B͘, eff ϭ͗:A͘ (31)

It is noteworthy that this result depends, through the strain or stress concentration tensors A or B, only on the solution of the purely elastic problem ͑i.e., without eigenstrains͒-which, by the way, proves that it is inconsistent to use the direct law of mixtures ͑i.e., AЈϭI͒ for estimating C eff as ͗c͘ and ␣ eff as ͗␣͘ simultaneously. For a two-phase material, these concentration tensors are not even needed if the overall elastic moduli or compliances are known ͑possibly from direct measurement instead of modelling͒, according to Levin's theorem ͑Levin 1967͒

␣ eff ϭ͗␣͘ϩ͑S eff Ϫ͗s͒͘:͑s 2 Ϫs 1 ͒ Ϫ1 :͑␣ 2 Ϫ␣ 1 ͒ (32) eff ϭ͗͘ϩ͑C eff Ϫ͗c͒͘:͑c 2 Ϫc 1 ͒ Ϫ1 :͑ 2 Ϫ 1 ͒ Associated results can be derived for the average thermal stresses in both phases without additional knowledge of the concentration tensors.

In the general case, the local mechanical state within the phases can be estimated according to adequate models. Inclusionbased models have been developed from the solution of Eshelby's inclusion problem which refers to an infinite unloaded homogeneous elastic body with the moduli C 0 an ellipsoidal part ͑I͒ of which undergoes the uniform ͑''stress-free''͒ eigenstrain T ͑or the eigenstress p I ϭϪC 0 : T ͒. The resulting strain and stress tensors in the inclusion are found to be uniform and given by

I ϭS I 0Esh : T ϭϪP I 0 :p I , I ϭC 0 :͑ I Ϫ T ͒ (33)
where the tensors S I 0Esh and P 0 have already been defined in Eqs. ͑19͒ and ͑18͒ for the solution of Eshelby's inhomogeneity problem. If the inclusion is inhomogeneous, with the moduli c, and if the matrix undergoes the uniform stress-free strain E T and is subjected to the uniform strain E at infinity, the strain and stress tensors in (IH) are still uniform and given by IH ϭ͓IϩP IH 0 :͑cϪC 0 ͔͒ Ϫ1 :͓EϩP IH 0 :͑c: T ϪC 0 :E T ͔͒ (34)

IH ϭc:͑ IH Ϫ T ͒
These results can be used to derive sets of estimates in a way similar to the one already reported when no eigenstrains are present. For example, for a polycrystal with isotropic ͑and then uniform͒ elasticity and an isotropic distribution of the constituents ͑e.g., the grain families ͑r͒ with the same lattice orientation͒ undergoing uniform isochoric eigenstrains r T , the concentration equation associated with a self-consistent scheme will read, according to Eqs. ͑19͒ and ͑34͒

r ϭEϩP Sph 0 :C 0 :͑ r T ϪE T ͒ϭEϩS Sph 0 :͑ r T ϪE T ͒ ϭEϩ␤ 0 ͑ r T ϪE T ͒ (35) or r ϭ⌺ϩ2 0 ͑ 1Ϫ␤ 0 ͒͑ E T Ϫ r T ͒ (36)
with E T ϭ͗ T ͘ ͑because of uniform isotropic elasticity, BϭI͒ and ⌺ϭ͗͘.

Outlines of Nonlinear Micromechanics

Rate-Independent Elastoplasticity

Except for pioneering contributions based on simple assumptions of uniform strain or stress and their variants, such as the Taylor model and its variants, nonlinear micromechanics was first addressed significantly through the self-consistent prediction of the overall elastoplastic response of polycrystals by Kro ¨ner ͑1961͒ on the basis of Eshelby's solution of the inclusion problem. He proposed to consider the plastic strain p as a ''stress-free strain'' in the sense of Eshelby and to model the mechanical interaction between one phase r ͑i.e., one set of grains with the same lattice orientation͒ and all the other phases as the one between an ellipsoidal inclusion with the uniform plastic strain r P and the surrounding infinite matrix with the macroscopic ͑uniform͒ plastic strain E P . For isotropic elasticity with the moduli (,k), spherical inclusions and isochoric plastic strains, this corresponds to the problem solved by Eqs. ͑35͒ and ͑36͒, namely, to the concentration equation

r ϭ⌺ϩ2͑1Ϫ␤ ͒͑ E P Ϫ r P ͒ (37)
which is the so-called ''Kro ¨ner interaction law.''

This approach is open to the criticism that elastoplastic interactions cannot reduce to the elastic ones which are concerned with Eshelby's inclusion problem; actually, plastic flow is stress dependent and the plastic strain is not a ''stress-free strain'' when plastic flow is considered. As a matter of fact, predictions derived from Kro ¨ner's model proved to be very close to those derived from an assumption of uniform strain, which can be guessed from Eq. ͑37͒ where the plastic strain deviation is forced to be very small because of the high value of the elastic shear modulus . This means that the mechanical interactions between the phases are overestimated by this model which yields too ''stiff'' estimates. Four years later, Hill ͑1965͒ proposed an alternative use of Eshelby's solution: by linearization of the constitutive equations of elastoplasticity and use of the tangent ͑multibranch͒ local and global instantaneous moduli, denoted as l r and L SC , respectively, the solution of the inhomogeneity ͑instead of the inclusion͒ problem can be integrated in the linear self-consistent concentration equation, as derived from Eq. ͑20͒, namely ˙rϭ͓IϩP r SC :͑l r ϪL SC ͔͒ Ϫ1 :E ˙0 E ˙ϭ͗ ˙͘ϭ͓͗IϩP SC :͑lϪL SC ͔͒ Ϫ1 ͘:E ˙0

L SC ϭ͗l:͓IϩP SC :͑lϪL SC ͔͒ Ϫ1 ͘:͓͗IϩP SC :͑lϪL SC ͔͒ Ϫ1 ͘ Ϫ1

For similar and aligned ellipsoids, the concentration equation can be put in the more convenient form ˙ϭ⌺ ˙ϪL*:͑ ˙ϪE ˙͒ (39) where Hill's ''constraint tensor'' L* is given by

L*ϭ͑P SC ͒ Ϫ1 ϪL SC ϭL SC :͓͑S EshSC ͒ Ϫ1 ϪI͔ (40)
The fundamental interest of Hill's approach lies in the fact that it defines and applies the basic concept of linearization of the constitutive equations in view of nonlinear homogenization; consequently, the elastoplastic nature of the intergranular accommodation, which was reduced to an elastic one by Kro ¨ner's treatment, is restored and expressed through an incremental formulation of the concentration equation, in adequation with the flow theory of plasticity. Hill's incremental formulation makes the recourse to Eshelby's elastic problem of the inhomogeneity possible, whereas Kro ¨ner's assimilation of the plastic strains to eigenstrains of the inclusion problem constrains the solution of the concentration problem within an elastic framework. A number of applications of Hill's model to metal forming, initiated by Hutchinson ͑1970͒ and extended at finite strain by Iwakuma and Nemat-Nasser ͑1984͒ or Lipinski et al. ͑1990͒ have shown this model to be a significant improvement with respect to Kro ¨ner's model. The reason for that can be illustrated by the simplified isotropic version of Hill's model proposed by Berveiller and Zaoui ͑1979͒ within the deformation theory of plasticity, under the assumption of local and global proportional loading: for isotropic elasticity and phase distribution and isochoric plastic strain, the corresponding ''secant'' approach leads to the following concentration equation:

r ϭ⌺ϩ2͑1Ϫ␤ ͒␣ ep ͑ E P Ϫ r P ͒ (41)
which only differs from Kro ¨ner's law by the scalar ''elastoplastic accommodation factor'' ␣ ep . This factor was shown to be equal to one in the elastic domain and to rapidly decrease by one or two orders of magnitude in the plastic regime, which allows the plastic strain deviation to increase significantly and the corresponding overall response to be far ''softer'' than the Taylor or Kro ¨ner one. That is the reason why Hill's approach rapidly gained acceptance so as to have been considered for a long time the standard for deriving nonlinear estimates. This is unfortunately no longer true today, because of the development of new variational approaches for nonlinear elasticity or viscoplasticity.

Nonlinear Elasticity and Viscoplasticity

The case of nonlinear elasticity or of ''nonhereditary'' viscoplasticity is considered now. Though the current response does not depend on the loading path, Hill's incremental formulation was adopted by Hutchinson ͑1976͒ for predicting the overall creep response of viscoplastic polycrystals: Hill's method was applied now to linearized relations between the stress rate tensor and the second time derivative of the strain tensor ͑instead of relations between the strain rate and stress rate tensors͒. Hutchinson proved in addition that the self-consistent prediction of power-law creep with the same exponent for all the slip systems can be reduced to a ''total'' treatment dealing directly with stresses and strain rates through the use of adequate secant creep compliances. About ten years later, a different treatment of the same problem was developed at finite strain by Molinari et al. ͑1987͒: since an incremental formulation was not necessary for such a nonhereditary behavior, they proposed to approximate the nonlinear local behavior at each stage by a linear relation between the stress and the strain rate tensors by making use of the tangent creep compliances and of some adequate prestress or prestrain rate tensor. Initially given in some approximate isotropic form, this ''tangent model'' was fully implemented by Lebensohn and Tome ´͑1993͒ who observed this scheme to tend to the Reuss-type lower bound for high nonlinearities; it also suffers from several limitations, primary among which is the fact that it is restricted to ͑a specific͒ self-consistent type of model and to power-law creep. More recently, a more general approach based on the same tangent approximation of the nonlinear behavior of the constitutive phases has been proposed by Zaoui and Masson ͑2000͒ and extensively discussed by Masson et al. ͑2000͒. This ''affine'' formulation, which deals with the linearized constitutive equations as thermoelastic ones ͑see the section ''Linear Elasticity with Eigenstrains''͒, can be applied to any type of microstructure and to any form of stress/strain rate relation; it leads to predictions intermediate between the secant and the tangent approaches, without recourse to any appropriately fitted parameter.

In the same field of viscoplasticity or nonlinear elasticity, a very different approach was developed in parallel according to a variational point of view; since the pioneering work of Talbot and Willis ͑1985͒, several contributions ͓Ponte Castan ˜eda 1991; [START_REF] Suquet | Overall potentials and extremal surfaces of power law or ideally plastic materials[END_REF][START_REF] Willis | Upper and lower bounds for nonlinear composite behavior[END_REF]; see also Ponte Castan ˜eda and Suquet ͑1998͒, for a general review͔ have provided rigorous Hashin-Shtrikman-type bounds for the effective dissipation potential, which, for the kind of nonlinearities exhibited by viscous materials, are upper bounds. It became then possible to compare these new bounds to the above estimates and it has been shown ͑Gilormini 1996͒ that all these approximate schemes, and especially the Hill-Hutchinson model which is the stiffest, may violate the bounds, at least for some particular combination of the parameters. This has motivated a novel interest in the development of ''variational estimates'', which would a priori conform with them ͑deBotton and Ponte Castan ˜eda 1996; [START_REF] Nebozhyn | Variational self-consistent estimates for cubic viscoplastic polycrystals: the effects of grain anisotropy and shape[END_REF] and of new formulations and procedures which would behave better with respect to these bounds. Let us quote especially the secondorder procedure ͑Ponte Castan ˜eda 1996͒ which is based on second-order Taylor expansions of the relevant potentials for the constituent phases; while similar to the affine formulation at the local level, it differs from it for the derivation of the overall response. The method has the distinctive advantage that it leads to estimates that are exact to second order in the heterogeneity contrast, but has the disadvantage that it exhibits a duality gap. It has recently been applied in the context of two-phase systems as well as polycrystals ͑Bornert et al. 2001͒ and the resulting estimates appear to be more accurate, particularly when compared to rigorous bounds, than earlier estimates such as those that are based on the Hill incremental or the Hutchinson total formulations.

The reason for that can be understood more easily by referring to the notion of the ''linear comparison composite'' ͑Ponte Castan ˜eda 1991͒. According to this view, the actual nonlinear heterogeneous RVE is associated at any stage with a fictitious composite which, at each point, obeys the linear behavior derived from the chosen linearization procedure ͑let it be ''tangent,'' ''secant,'' ''total,'' ''affine,'' or of ''second-order,'' etc.͒ of the local constitutive equations; this ''comparison composite'' is continuously heterogeneous, so that the homogenization procedure cannot be achieved on it without adequate simplification. According to Hill's formulation, as well as to most of the current ones, this simplification consists in attributing to each phase, at any given stage, a homogeneous behavior referred to the average stress or strain ͑or strain rate͒. Due to the actual nonlinear behavior, which causes any inhomogeneity of the strain or stress field to provoke an equivalent inhomogeneity of the local moduli or compliances, this approximation amounts to neglecting the intraphase heterogeneity and, for the kind of nonlinearity exhibited by usual elastic or viscous materials, to overestimating the phases stiffness with more or less strong consequences on the overestimation of the overall stiffness, according to the chosen formulation. All current and future advances in this field are likely to depend on to what extent this intraphase heterogeneity of the linear comparison composite is accounted for in a better way.

Nonlinear Viscoelasticity and Elastoviscoplasticity

Viscoelasticity, whether it is linear or not, reveals the specific difficulty of the coupling between elasticity and viscosity which is responsible for a complex ''long-range memory effect'' which has to be captured by the micromechanical treatment: delayed phenomena are associated to the viscoelastic nature of the mechanical interactions between the constituents so that the overall behavior of heterogeneous viscoelastic media is much more complicated than the one of each phase. An illustration of that is given by the fact that a mixture of Maxwellian constituents is no more Maxwellian: while each phase has only one relaxation time, the simplest two-phase material exhibits, according to the classical selfconsistent scheme, a complex continuous relaxation spectrum which is especially intricate if more general morphological situations or the existence of an interphase are taken into account ͑Rougier et al. 1993; Beurthey and Zaoui 2000͒. Additionally, the viscoelastic coupling is responsible for the fact that the concerned behavior cannot be described by one single potential, so that no variational approach from which bounds could be derived for an arbitrary loading path is available at the time being.

Another specific difficulty attached to nonlinear viscoelasticity or to rate-dependent elastoplasticity is associated with the fact that derivatives of different orders occur simultaneously in the constitutive equations, which makes the definition of tangent moduli or compliances problematical: actually, such a definition, which is straightforward for elasticity, viscosity, viscoplasticity or elastoplasticity, was the key to linearized treatments in what precedes. Fortunately, in the case of linear nonaging viscoelasticity, the Laplace technique can be used to transform, via the correspondence principle, a viscoelastic problem into a ͑symbolic͒ elastic one: this trick was already used more than two decades ago by Laws and McLaughlin ͑1978͒ for the self-consistent scheme but it is of no evident use as soon as nonlinearity is present.

This may be the reason why the development of the micromechanics of nonlinear viscoelastic ͑or rate-dependent elastoplastic͒ heterogeneous materials has been rather late and chaotic ͑Zaoui 1997͒. The first significant attempt in this field was made by Weng ͑1981͒ who proposed to adopt the Kro ¨ner model by arguing that, unlike the plastic strain for rate-independent elastoplastic materials, the viscoplastic strain would be a true stress-free strain in the sense of Eshelby. Using different approaches, Nemat-Nasser and Obata ͑1986͒ as well as Harren ͑1991͒ adopted the same Kro ¨ner-type framework in the context of finite strain. On the contrary, Zaoui and Raphanel ͑1991, 1993͒ proved that Hill's criticism of Kro ¨ner's treatment of rate-independent elastoplasticity was still valid for this case and a new model Zaoui and Masson ͑1998͒ was proposed for rate-dependent elastoplasticity, associated first to the linear self-consistent scheme ͑Rougier et al. 1994͒ and then, within the ''affine'' framework, to any linear transition model ͑Masson and Zaoui 1999; [START_REF] Pouya | Line ´arisation et homoge ´ne ´isation en viscoe ´lasticite ´[END_REF][START_REF] Masson | An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals[END_REF]. Unfortunately, no bounds are still available for this kind of behavior: any progress in this field would surely stimulate new advances.

Conclusion

As suggested by this paper, continuum micromechanics is now a well-developed scientific area and a number of problems and materials can be analyzed according to this approach. It is frequently claimed that homogenization techniques are restricted to the derivation of overall properties and are of no use for local analyses. This is a reducing statement: as a matter of fact, several modern micromechanical approaches such as the above-mentioned ''pattern approach,'' especially when they are coupled with developing micromechanical experimental investigations and numerical simulations, allow to tackle better and better local states and local effects. Nevertheless, many questions are still open and many fields have not yet been investigated significantly: this is especially the case for damage modeling for which, despite valuable advances already achieved ͑e.g., Kachanov 1992͒, important progress is likely to occur in the next future.

Note in conclusion that all the above-reported developments were concerned with problems and materials obeying the basic conditions ͑1͒ of ''macro-homogeneity.'' A number of important situations conflict with such conditions, because either d is of the order of d 0 ͑elasticity of metallic alloys, influence of vacancies, shearing of precipitates, precipitation hardening, etc.͒ or l is of the order of L ͑thin sheets, long range correlations, percolation effects, etc.͒ or of ͑short waves, stress gradients, etc.͒. In most such cases, one additional length scale, at least, would be needed in order to allow comparison of the standard size d either to a finite macroscopic length ͑distance to a free surface, fluctuation length of the macroscopic stress field, etc.͒ or to nonvanishing microscopic quantities ͑dislocation dissociation width, adequate mean free paths, precipitate size, etc.͒. This is a matter of current intensive research along several directions: multiscale approaches, second-gradient theories, generalized continua, etc. Answers to many open questions which are out of reach of classical micromechanical treatments are expected from such developments ͑influence of grain size on the yield stress, width of shear or persistent slip bands, instability and fracture analysis, etc.͒.

  10) with, because of Eqs. ͑3͒ and ͑5͒, ͗AЈ͘ϭ͗B*͘ϭI ͑I is the fourthorder symmetric unit tensor͒. With the local constitutive equations ϭc r : or ϭs r : (11)where the elastic moduli c r and compliances s r are known and uniform in every phase ͑r͒, Eqs. ͑8͒ and ͑9͒ become

  by combining Eqs. ͑10͒, ͑14͒ and ͑15͒, ͑21͒ and ͑22͒, we get ⌺:͑͗ T B*:s:B*͘ϪS eff ͒:⌺у0 ᭙⌺ (23) E:͑͗ T AЈ:c:AЈ͘ϪC eff ͒:Eу0 ᭙E (24)

  LM ϭ͗c͘ or to S LM ϭ͗s͘. Various other estimates have been derived by use of Eshelby's solution of the inhomogeneity problem ͑Eshelby 1957͒. Consider an ellipsoidal inhomogeneity ͑H͒ with the moduli c H , perfectly bonded to an infinite homogeneous elastic matrix ͑the ''reference medium''͒ with the moduli C 0 ͑or compliances S 0 ͒ subjected to the homogeneous strain E or stress ⌺ at infinity. The stress and strain fields H and H in ͑H͒ are found to be uniform and given by H ϭ͓IϩP H 0 :͑c H ϪC 0 ͔͒ Ϫ1 :E, H ϭc H :

͗:s:͘ϭ⌺:S eff :⌺ (14) ͗:c:͘ϭE:C eff :E (15) can be proved to be equivalent to the direct one Eϭ͗͘ϭ͗s:͘ϭS eff :⌺ (16) ⌺ϭ͗͘ϭ͗c:͘ϭC eff :E (17) Estimates for Effective Moduli and Compliances Estimates for C eff or S eff can be obtained whenever AЈ or B* are estimated. The simple choice AЈϭI or B*ϭI leads to the popular direct or inverse ''laws of mixtures,'' i.e., either to C H (18)