Christian Riebeling

Jean-Pascal Piret

Bénédicte Trouiller

Inge Nelissen

Christelle Saout

Olivier Toussaint

Dr Andrea Haase
email: andrea.haase@bfr.bund.de

A guide to nanosafety testing: considerations on cytotoxicity testing in different cell models

Keywords: Nanoparticles, cytotoxicity, cell viability, reproducibility ACCEPTED

Safety testing of nanoparticles (NPs) relies on robust, predictive, and reproducible methods.

Integrated test strategies usually start with the assessment of in vitro cytotoxicity. Methods for this purpose are only partially established for NPs. The aim of this study was to evaluate three cytotoxicity assays for their applicability to NPs in several human cell models representing the most important NP target organs and to compare cell responses.

The assays covered different biological principles, i.e. ATP content, redox metabolism, and membrane permeability, as well as different read-out principles, i.e. luminescence, colorimetry, and fluorescence. For all assays standard operating procedures were developed. A549 and NCI-H441 lung epithelial cells in single and in co-culture with THP-1 derived macrophages served as lung epithelial models. Normal human epidermal keratinocytes and in vitro reconstituted human epidermis were employed as skin models.

Naïve and differentiated Caco-2 gut epithelial cells were applied as gut epithelial models. Primary CD34-derived dendritic cells (CD34-DCs) served as an immune model. We used well-characterized model NPs, i.e. 50 nm amine-modified and 40 nm carboxyl-modified polystyrene NPs (PS-NH 2 and PS-COOH, respectively). PS-COOH NPs showed no effect in any test while PS-NH 2 displayed cytotoxicity in most cell models. CD34-DCs was the most sensitive cell model tested. The tetrazolium-based MTS assay was the most robust assay in our study, applicable to all cell models investigated.

Therefore, this assay could become an integral part of a NP testing strategy. Other assays might be also useful, depending on the cell model or the type of investigation.

Introduction

The development of in vitro methods for toxicological testing of nanoparticles (NPs) is a major research endeavor. NPs can be synthesized in many different variants by varying size, shape or by applying chemical surface modifications. Therefore in order to save costs and time and to reduce animal testing reliable in vitro methods are urgently needed to screen for hazardous NPs, to prioritize NPs for further testing and also to obtain initial data for risk assessment. Most toxicological endpoints require prior knowledge of the acute toxicity of the test compound. Many pathways of (regulated) cell death have evolved in multicellular organisms. Some can be specific for certain cell types and/or stimuli, including several types of regulated necrosis and apoptosis [START_REF] Galluzzi | Molecular mechanisms of regulated necrosis[END_REF][START_REF] Green | Cell biology. Metabolic control of cell death[END_REF]. While mechanistic insight is important in some cases, a mechanism-independent measurement often gives an initial starting point for more elaborate testing. In vivo, many forms of cell death (e.g. apoptosis) lead to early removal of the dying cells by macrophages. However, in vitro, in the absence of macrophages, all forms of cell death will eventually cause loss of integrity of the cellular membrane. Certain dyes such as trypan blue and propidium iodide will cross only compromised membranes and can be used to specifically stain dead cells. In addition, the concomitant release of cytosolic enzymes into the cell culture medium can be used as a read-out since the released enzyme activity correlates with the extent of membrane rupture and therefore is a measure for cell death. Most often lactate dehydrogenase (LDH) activity is measured in cell culture medium using the NADHdependent conversion of a dye [START_REF] Korzeniewski | An enzyme-release assay for natural cytotoxicity[END_REF]. Another approach is the measurement of the number of viable cells after treatment. Arguably the most popular method is based on the reduction of a tetrazolium salt to a colored formazan salt derivative by cellular NAD(P)H-dependent oxidoreductase enzymes, introduced by [START_REF] Mosmann | Rapid colorimetric assay for cellular growth and survival: application to proliferation and cytotoxicity assays[END_REF].

The reduction of 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) represents a measure for the cellular metabolic capacity and correlates with the number of cells [START_REF] Mosmann | Rapid colorimetric assay for cellular growth and survival: application to proliferation and cytotoxicity assays[END_REF]. Oxidoreductases responsible for the dye reduction are mainly located in the cytoplasm, i.e. in the endosomal/ lysosomal compartment and at the plasma

A C C E P T E D M A N U S C R I P T

-4 -membrane, and are also partly associated with oxidative phosphorylation in mitochondria [START_REF] Stepanenko | Pitfalls of the MTT assay: Direct and off-target effects of inhibitors can result in over/underestimation of cell viability[END_REF]. Meanwhile a couple of assays using similar dyes such as MTS, WST or XTT are available as well. In some forms of cell death such as apoptosis mitochondria play an important role. During apoptosis cellular proteins involving the proapoptotic bcl-2 family disrupt mitochondrial function [START_REF] Galluzzi | Molecular mechanisms of regulated necrosis[END_REF][START_REF] Green | Cell biology. Metabolic control of cell death[END_REF].

In other forms of cell death such as necrosis, loss of mitochondrial function and of oxidoreductases is a later event coinciding with plasma membrane rupture [START_REF] Aki | Regulated necrosis and its implications in toxicology[END_REF][START_REF] Green | Cell biology. Metabolic control of cell death[END_REF]). Thus, dyes, that accumulate in mitochondria to test their function can be used as well to assess cell viability [START_REF] Petronilli | The mitochondrial permeability transition, release of cytochrome c and cell death. Correlation with the duration of pore openings in situ[END_REF]. Finally, metabolic capacity can also be measured by determination of the intracellular ATP content using luciferase or other ATP consuming reactions [START_REF] Kangas | Bioluminescence of cellular ATP: a new method for evaluating cytotoxic agents in vitro[END_REF]. Viable cells under normal culture conditions maintain a relatively constant amount of ATP. It is assumed that the capacity for ATP synthesis is lost early during cell death and remaining ATP is rapidly used up or eliminated by ATPases.

For cytotoxicity measurements it would be optimal to use non-dividing cells as otherwise it may become difficult to distinguish effects on cell viability from effects on cell proliferation.

However, often this is not possible. Thus, it is very important to understand and to precisely describe how a particular assay is performed, as this affects the interpretation of the results.

Furthermore, cytotoxicity should be assessed under similar conditions (i.e. concentrations, time points) as other toxicological endpoints of interest, which might be exploited later on in the test strategy.

It should be noted that specific issues arise when testing cytotoxic effects of NPs in vitro. In particular, enzyme-and dye-based assays need to be carefully checked for possible interferences from NPs. Assay reagents might react or be immobilized on the NP's surface.

Especially assays using spectrophotometric read-out might be impaired as NPs can scatter or absorb light or exhibit interfering plasmon resonance [START_REF] Bonvin | Assessment of nanoparticles' safety: corrected absorbance-based toxicity test[END_REF][START_REF] Kroll | Interference of engineered nanoparticles with in vitro toxicity assays[END_REF][START_REF] Monteiro-Riviere | Limitations and relative utility of screening assays to assess engineered nanoparticle toxicity in a human cell line[END_REF][START_REF] Stone | Development of in vitro systems for nanotoxicology: methodological considerations[END_REF][START_REF] Tournebize | Pitfalls of assays devoted to evaluation of oxidative stress induced by inorganic nanoparticles[END_REF]. Several consortia have already worked on evaluation and harmonization of cytotoxicity assays for NPs (Elliott et al. 2017;[START_REF] Piret | Pan-European inter-laboratory studies on a panel of in vitro cytotoxicity and pro-inflammation assays for nanoparticles[END_REF][START_REF] Rösslein | Use of Cause-and-Effect Analysis to Design a High-Quality Nanocytotoxicology Assay[END_REF][START_REF] Xia | Interlaboratory evaluation of in vitro cytotoxicity and inflammatory responses to engineered nanomaterials: the NIEHS Nano GO Consortium[END_REF]. Moreover, a standard assay specifically dedicated to NPs is currently in preparation using the MTT variant 3-(4,5-

A C C E P T E D M A N U S C R I P T -5 -
dimethylthiazol-2-yl)-5-(3-carboxymethoxyphenyl)-2-(4-sulfophenyl)-2H-tetrazolium (MTS)
assay as an in vitro cytotoxicity assay (ISO/AWI 19007 Modified MTS assay for measuring the effect of NPs on cell viability, ISO, Geneva, Switzerland).

In this study we decided to further explore three assay types. As a late response assay the LDH assay was chosen. For an earlier response assay the MTS assay was chosen. Finally, ATP level measurements were selected as an early marker of cellular stress.

In toxicology, automated testing based on high-throughput screening (HTS) or high-content screening (HCS) is becoming increasingly important [START_REF] Hartung | Towards a new toxicologyevolution or revolution?[END_REF][START_REF] Judson | Perspectives on Validation of High-Throughput Assays Supporting 21st Century Toxicity Testing[END_REF]). It may be considered that in the future HTS/ HCS might be an integral part of the decisionmaking tools used routinely for the toxicity screening of chemicals, as well as NPs [START_REF] Judson | Perspectives on Validation of High-Throughput Assays Supporting 21st Century Toxicity Testing[END_REF]). Thus, another goal of this study was to identify assays which can be used in HCS approaches to investigate the cytotoxicity of NPs. The cell membrane impermeable TOPRO nucleic acid stain may be considered as an LDH equivalent assay for an HCS approach [START_REF] Anguissola | High content analysis provides mechanistic insights on the pathways of toxicity induced by amine-modified polystyrene nanoparticles[END_REF]. As an HCS equivalent for the MTS assay cyanine monomers as nucleic acid counterstains such as TO-PRO-1/3/5 and similar dyes may be considered [START_REF] Anguissola | High content analysis provides mechanistic insights on the pathways of toxicity induced by amine-modified polystyrene nanoparticles[END_REF]. For the ATP assay we have not identified an HCS equivalent.

To generate data relevant to toxicological problems, the route of exposure should be taken into account. In terms of potential NP hazards, inhalation has been identified as the most important exposure route. Experimental studies suggest that NPs can cross from the lung into the bloodstream [START_REF] Balasubramanian | The effect of primary particle size on biodistribution of inhaled gold nano-agglomerates[END_REF][START_REF] Kendall | Health impact and toxicological effects of nanomaterials in the lung[END_REF][START_REF] Kreyling | Air-blood barrier translocation of tracheally instilled gold nanoparticles inversely depends on particle size[END_REF]. Cutaneous and oral exposure to NPs, which may be a fraction of bulk materials and not necessarily specifically engineered NPs, are already commonplace for some materials such as silica and titanium dioxide [START_REF] Peters | Presence of nano-sized silica during in vitro digestion of foods containing silica as a food additive[END_REF][START_REF] Weir | Titanium dioxide nanoparticles in food and personal care products[END_REF]. Historical data for oral intake of these materials show no adverse effects [START_REF] Lewinson | Characterization and toxicological behavior of synthetic amorphous hydrophobic silica[END_REF][START_REF] Martin | The chemistry of silica and its potential health benefits[END_REF][START_REF] Skocaj | Titanium dioxide in our everyday life; is it safe?[END_REF]. Dermal penetration is considered negligible at least for healthy skin and for the most commonly used metal and metal oxide NPs (Labouta and Schneider 2013; [START_REF] Nohynek | Nano-sized cosmetic formulations or solid nanoparticles in sunscreens: a risk to human health?[END_REF][START_REF] Piret | Sonicated and stirred copper oxide nanoparticles induce similar toxicity and pro-inflammatory response in N-hTERT keratinocytes and SZ95 sebocytes[END_REF]). In the case that NPs breach the barriers of the first-line exposure organs, blood and lymphatic system are usually the first compartments reached, prior to secondary organs such as liver and kidney. In order to take into account different possible routes of exposure we tested several widely used cell lines representing first-line exposed organs, skin, intestinal tract and respiratory tract, and also the immune system. A549 and NCI-H411 are epithelial cells derived from human, male, lung carcinomas, and were used to represent lung epithelial models. A549 cells have an alveolar type II-like phenotype [START_REF] Foster | Characterization of the A549 cell line as a type II pulmonary epithelial cell model for drug metabolism[END_REF]) and NCI-H411 exhibit both alveolar (i.e. type II cell) and bronchiolar (i.e. club cell) epithelial phenotype [START_REF] Hermanns | Lung epithelial cell lines in coculture with human pulmonary microvascular endothelial cells: development of an alveolo-capillary barrier in vitro[END_REF][START_REF] Salomon | The cell line NCl-H441 is a useful in vitro model for transport studies of human distal lung epithelial barrier[END_REF].

A C C E P T E D M A N U S C R I P T -6 -
Caco-2 cells are epithelial cells derived from human, male, colorectal adenocarcinoma.

These cells can undergo enterocytic differentiation [START_REF] Jumarie | Caco-2 cells cultured in serum-free medium as a model for the study of enterocytic differentiation in vitro[END_REF][START_REF] Piret | Differential toxicity of copper (II) oxide nanoparticles of similar hydrodynamic diameter on human differentiated intestinal Caco-2 cell monolayers is correlated in part to copper release and shape[END_REF]) and represent the gastro-intestinal tract in this study. THP-1 cells are monocytes derived from human, male, acute monocytic leukaemia, and can be differentiated into macrophage-like cells [START_REF] Chanput | THP-1 cell line: an in vitro cell model for immune modulation approach[END_REF].

Generally, primary cells are considered to closer resemble the in vivo phenotype while cell lines are mostly of tumor origin and/or virus-transformed. Therefore, in this study we also included two primary cell types. Normal human epidermal keratinocytes (NHEK) from a pool of three donors were used as skin in vitro model. CD34 dendritic cells (CD34-DCs) were derived through differentiation of progenitor cells from different human cord blood donors.

Dendritic cells are part of the immune system; they process antigen material and present it on the cell surface to the T cells of the immune system. However, when using primary cells in toxicology one needs to consider that access to them is often limited and variability in test results will be higher due to donor variability.

In addition, we included a few more complex models of the target organs such as 3D models and co-cultures. The use of complex cellular models has been discussed for some time as superior to the single layer dish culture [START_REF] Guguen-Guillouzo | Maintenance and reversibility of active albumin secretion by adult rat hepatocytes co-cultured with another liver epithelial cell type[END_REF]. For their use in toxicology it is expected that such models could yield more relevant data requiring less modelling or extrapolation [START_REF] Nam | Biomimetic three-dimensional tissue models for advanced high-throughput drug screening[END_REF]. Thereby, they are assumed to more reliably

A C C E P T E D M A N U S C R I P T -7 -
identifying a human hazard [START_REF] Grainger | Cell-based drug testing; this world is not flat[END_REF]. As a more complex model we employed a commercially available reconstituted human epidermis (RHE) produced from a pool of human keratinocytes from different donors. These RHE are characterized by the presence of the 4 typical epidermis layers (basal, spinous, granular, and topical cornified layer) observed in in vivo epidermis and express specific differentiation markers such as keratin 14, keratin 10 or involucrin [START_REF] Poumay | A simple reconstructed human epidermis: preparation of the culture model and utilization in in vitro studies[END_REF]. RHE were treated at the air-liquid interface.

Furthermore, Caco-2 cells were differentiated on filters forming continuous polarized monolayers, and acquire several of the characteristics of normal enterocytes [START_REF] Hillgren | In vitro systems for studying intestinal drug absorption[END_REF][START_REF] Volpe | Variability in Caco-2 and MDCK cell-based intestinal permeability assays[END_REF]. Co-cultures of aforementioned A549 and NCI-H411 lung epithelial cells with THP-1 cells differentiated into macrophage-like cells were used to represent a situation more close to the lung surface [START_REF] Hjort | Alveolar epithelial cell-macrophage interactions affect oxygen-stimulated interleukin-8 release[END_REF][START_REF] Wottrich | Biological effects of ultrafine model particles in human macrophages and epithelial cells in mono-and co-culture[END_REF]).

All cell models employed in our study are listed in Supplemental Table 1. The selection of cell models was based on a literature survey, initial experiments, as well as on availability and experience in the participating laboratories. The aim of our study was to investigate the applicability of three cytotoxicity assays in a variety of different cell models representing the most important target organs for NPs. Amine-modified polystyrene NPs (PS-NH 2 ) and carboxyl-modified polystyrene NPs (PS-COOH) were used as 'model' NPs as they exhibit narrow size ranges and low agglomeration [START_REF] Bexiga | RNAi-mediated inhibition of apoptosis fails to prevent cationic nanoparticle-induced cell death in cultured cells[END_REF][START_REF] Kim | Low dose of amino-modified nanoparticles induces cell cycle arrest[END_REF][START_REF] Wang | The biomolecular corona is retained during nanoparticle uptake and protects the cells from the damage induced by cationic nanoparticles until degraded in the lysosomes[END_REF]. Moreover, the cationic PS-NH 2 have been demonstrated to induce cytotoxicity at doses below 50 g/ml [START_REF] Anguissola | High content analysis provides mechanistic insights on the pathways of toxicity induced by amine-modified polystyrene nanoparticles[END_REF][START_REF] Bexiga | Cationic nanoparticles induce caspase 3-, 7and 9-mediated cytotoxicity in a human astrocytoma cell line[END_REF][START_REF] Ruenraroengsak | Respiratory epithelial cytotoxicity and membrane damage (holes) caused by amine-modified nanoparticles[END_REF][START_REF] Wang | The biomolecular corona is retained during nanoparticle uptake and protects the cells from the damage induced by cationic nanoparticles until degraded in the lysosomes[END_REF][START_REF] Xia | Cationic polystyrene nanosphere toxicity depends on cell-specific endocytic and mitochondrial injury pathways[END_REF]. We have developed standard operation procedures (SOPs) and discussed some limitations and pitfalls. All SOPs are included in the Supplementary Information.

A C C E P T E D M A N U S C R I P T

-8 -

Material and methods

Chemicals and nanoparticles

Benzalkonium chloride (BC, Sigma-Aldrich), known to induce cell death in multi-layered engineered tissues like RHE and in differentiated Caco-2 monolayers was used as positive control (Supplemental Table 2) [START_REF] Coquette | Analysis of interleukin-1alpha (IL-1alpha) and interleukin-8 (IL-8) expression and release in in vitro reconstructed human epidermis for the prediction of in vivo skin irritation and/or sensitization[END_REF][START_REF] Piret | Differential toxicity of copper (II) oxide nanoparticles of similar hydrodynamic diameter on human differentiated intestinal Caco-2 cell monolayers is correlated in part to copper release and shape[END_REF][START_REF] Vankoningsloo | Cytotoxicity of multi-walled carbon nanotubes in three skin cellular models: effects of sonication, dispersive agents and corneous layer of reconstructed epidermis[END_REF]. Zinc sulphate (Sigma-Aldrich) and Triton X-100 were used as positive chemical control for CD34-DCs or A549/THP-1 cells respectively (Supplemental Table 2).

Monodisperse

Cell culture

A549 cell line

Human lung epithelial A549 cells (obtained from LGC Standards, catalog number: ATCC ® CRM-CCL-185 TM ) were cultured in RPMI 1640 medium (Gibco) supplemented with 10% (v/v) heat-inactivated (20 min, 56°C) fetal calf serum (FCS) (Sigma-Aldrich), 1% (v/v) penicillinstreptomycin (Invitrogen). Cells were seeded at 10 4 cells/well in 96-well plates 42 h prior to exposure.

NCI-H411 cell line

Human lung epithelial NCI-H411 (catalog number: ATCC ® HTB-174 TM ) cells obtained from

LGC Standards were cultured in RPMI 1640 medium (Gibco) supplemented with 10% (v/v)

A C C E P T E D M A N U S C R I P T

-9 -heat-inactivated FCS (Sigma-Aldrich), 1% (v/v) penicillin-streptomycin (Invitrogen). Cells were seeded at 10 4 cells/well in 96-well plates 42 h prior to exposure.

Naïve Caco-2 cell line and differentiated monolayers

Human colon carcinoma Caco-2 cells were purchased from European Collection of Cell Cultures (ECACC, Salisbury, UK, catalog number: ECACC 86010202). Caco-2 cells were maintained in culture in 75-cm 2 polystyrene flasks (Costar) with DMEM High Glucose (Gibco) containing 100 U/ml penicillin, 100 µg/ml streptomycin (BioWhittaker), 1% (v/v) non-essential amino-acids (Gibco) and 10% FCS (Gibco).

To obtain a tight intestinal epithelium, Caco-2 cells were differentiated as described previously [START_REF] Piret | Differential toxicity of copper (II) oxide nanoparticles of similar hydrodynamic diameter on human differentiated intestinal Caco-2 cell monolayers is correlated in part to copper release and shape[END_REF]. Briefly, 50,000 Caco-2 cells were grown on transwell polycarbonate filters (12 wells, pore diameter of 3 µm, 1.12 cm 2 growth area, Corning Costar) for 16 days. Medium (500 µL in the insert -apical side-, and 1.5 mL in the well -basolateral side) was renewed every 48 h. The tightness of the intestinal monolayers was controlled by transepithelial electrical resistance (TEER) measurement, performed with probes connected to a Millicell-ERS Volt-Ohm meter (Millipore). TEER data were calculated by subtracting the value of blank inserts and normalized for growth area (.cm 2 ). For toxicity assays, intestinal epithelia were incubated for 24 h with 500 µL of complete DMEM containing increasing concentrations of PS-NH 2 .

THP-1 cell line-derived macrophages and co-culture with lung epithelial cells

Human acute monocytic leukaemia THP-1 cells (obtained from LGC Standards, catalog number: ATCC ® TIB-202 TM ) were cultured in RPMI 1640 medium (Gibco) supplemented with 10% FCS (Sigma-Aldrich), 1% L-glutamine, 1% penicillin/streptomycin (Invitrogen), 1%

Hepes. THP-1 cells (non-adherent monocytic cells) were seeded at 8×10 4 cells/well in a 96well plates and differentiated in macrophage-like adherent cells according to the method previously described [START_REF] Lanone | Comparative toxicity of 24 manufactured nanoparticles in human alveolar epithelial and macrophage cell lines[END_REF]) with 30 ng/mL of phorbol-12-myristate 13-acetate (Sigma-Aldrich) in RPMI medium 40 h prior to exposure. For co-culture, differentiated THP-1 cells were detached with 0.25% trypsin and seeded for 2 h on top of epithelial cells at a ratio of 1 macrophage for 2 epithelial cells on the day of exposure.

A C C E P T E D M A N U S C R I P T

-10 -

Normal human epidermal keratinocytes (NHEK)

Neonatal NHEK (pool of three donors) were purchased from Lonza (catalog number:

192906). NHEK were grown in 75 cm 2 flasks (Corning) to 80% confluence in presence of keratinocytes SFM medium supplemented with bovine pituitary extract and recombinant human epidermal growth factor (Gibco). Importantly, all steps were performed at 4°C when splitting NHEK in order to prevent differentiation.

Reconstituted human epidermis (RHE)

RHE (EPI/001) differentiated for 14 days and the related culture media were purchased from StratiCELL s.a. (Les Isnes, Belgium). RHE were produced from a pool of human keratinocytes from different donors seeded on a 0.63 cm² polycarbonate filters and grown at the air-liquid interface. RHE were maintained in culture according to the manufacturer's recommendations.

Primary CD34-derived dendritic cells (CD34-DCs)

CD34 + -progenitor-cell isolation and culture procedures have been described previously [START_REF] Nelissen | MUTZ-3-derived dendritic cells as an in vitro alternative model to CD34+ progenitor-derived dendritic cells for testing of chemical sensitizers[END_REF]). Briefly, human cord blood samples were collected from umbilical blood All cell lines and complex in vitro models were maintained at 37°C in a 5% CO 2 incubator with humidified air.

Cytotoxicity testing

The detailed SOPs for MTS, LDH and ATP assays are included in supplemental files.. Cells were seeded as described above and treated with NPs in concentrations ranging from 1-100 µg/ml, corresponding solvents or control chemicals, or left untreated for a period of 24 h.

Afterwards the different assays were applied. We have carefully checked for possible interferences for each of the assays, instructions are included in the SOPs.

MTS assay

For the MTS assay CellTiter 96 ® AQueous One Solution Reagent from Promega was used.

After 24 h treatment with NPs, the medium was removed from each well and replaced by MTS solution, which has been prepared according to manufacturer instructions. The plate was then incubated at 37°C in a cell incubator for another 1 h. The absorbance was read at 490 nm. This SOP has been tested in an inter-laboratory comparison study (Nelissen et al, under review).

LDH assay

For the LDH assay, CytoTox-ONE TM Homogeneous Membrane Integrity Assay from Promega was used. After 24 h treatment, the medium was collected from each well and centrifuged at 15,700×g for 30 min to remove potentially interfering NPs and also cell debris.

Afterwards 100 l cell supernatant was mixed with 100 l CytoTox-ONE TM Reagent, incubated for 10 min at room temperature. Then, 50 μl stop solution was added and fluorescence was recorded within one hour with an excitation wavelength between 530-570 nm and an emission wavelength between 580-620 nm.

A C C E P T E D M A N U S C R I P T

-12 -

ATP assay

For this purpose, ATPlite TM kit (PerkinElmer) was used according to the manufacturer's instructions. Additionally, total cellular protein content was assessed according to the Bradford method (BioRad) [START_REF] Bradford | A rapid and sensitive method for the quantitation of microgram quantities of protein utilizing the principle of protein-dye binding[END_REF], and the ATP content normalized to it.

Data analysis

A C C E P T E D M A N U S C R I P T

-13 -

Results

In this study we evaluated the suitability of three different cytotoxicity assays for NP testing.

We included six different single cell models and four complex cell models (Supplemental Table 1).

Skin cell models

For the skin models we used NHEK and RHE and applied three different assays (i.e. MTS, LDH and ATP assay). The quaternary ammonium compound and surfactant BC was used at 0.005 mg/mL (NHEK, MTS and LDH assays; 0.0025°mg/mL ATP assay) or 1 mg/mL (RHE) as a positive control and quality control to verify the performance of the assays (Supplemental Table 2).

In NHEK cells, we detected a clear dose dependent effect of PS-NH 2 on cell viability using the ATP and MTS assays (Figure 1 A and B), with ED 50 values of 37 µg/mL and 25 µg/mL, respectively (Table 1). For PS-COOH we could not observe any effect for all the tested concentrations. Please note, that although the manufacturer of the ATP assay recommended the normalization of the results using the sample's protein content, in our experience normalization introduced high variation in the results as seen for NHEK (Supplemental Figure 1A). Therefore, and because plated cell numbers were tightly controlled in the experiments, the ATP assay results are reported without such normalization. However, reduced ATP levels and reduced cell numbers cannot be distinguished in our results. The LDH assay proved to be less sensitive and a cytotoxic effect was only detected for PS-NH 2 at a dose of 100 µg/mL (Figure 1C).

In parallel, in vitro RHE were treated with PS-NH 2 or BC at the air-liquid interface. As the use of RHE was cost intense and all previous tests using PS-COOH on NHEK did not show an effect on viability, only PS-NH 2 at a dose of 31.25 µg/cm 2 (equivalent to 100 µg/mL inducing the highest toxic effect on NHEK, Figure 1) was tested. NPs were added topically as a liquid dispersion on the cornified layer and incubated for 24 h. Phosphate Buffered Saline (PBS) solution was used to evaluate the potential effect of a buffered liquid on RHE viability. No toxic effect of PS-NH 2 on epidermises was detected using either the MTS or LDH assay

A C C E P T E D M A N U S C R I P T

-14 -(Figure 2), while the positive control BC caused a dramatic loss of epidermis viability measured with both assays (Supplemental Table 2).

Lung cell models

As a model for lung exposure we used two lung related cells lines (A549 cells or NCI-H411 cells) in monoculture as well as in co-culture with THP-1-differentiated macrophages. Triton X-100 was used as a positive control in the lung cell experiments (Supplemental Table 2).

The lung cells models alone (i.e. A549 or NCI-H441 cells) reached ED 50 values of 117 µg/mL and 77 µg/mL in the MTS assay, respectively (Figure 3, Supplemental Figure 2, and Table 1). In contrast, monocultures of THP-1 cells were observed to be more sensitive with an ED 50 of 54 µg/mL in the MTS assay (Table 1). Co-cultures showed similar effects compared to the lung cell models (i.e. A549 and NCI-H441) alone with ED 50 values of 112 µg/mL and 64 µg/mL in the MTS assay, respectively (Table 1). Similar results were derived in the ATP and the LDH assays (Figure 3, Supplemental Figure 2, and Table 1). In all assays THP-1 cells alone were the most sensitive cells. In addition, PS-COOH were tested in the three assays and the five airway cell models, and no cytotoxicity was detected (Figure 3, Supplemental Figure 2).

Intestinal cell models

The response of Caco-2 cells to NPs was tested using undifferentiated and differentiated cells as intestinal model, representing oral intake. BC was used at as positive control [START_REF] Piret | Differential toxicity of copper (II) oxide nanoparticles of similar hydrodynamic diameter on human differentiated intestinal Caco-2 cell monolayers is correlated in part to copper release and shape[END_REF] to verify the performance of the assays (Supplemental Table 2). In addition, TEER was measured on differentiated Caco-2 cells to evaluate the effect of NPs on the intestinal epithelium integrity.

Undifferentiated Caco-2 cells exhibited a dose dependent loss of viability after PS-NH 2 exposure with an ED 50 of 73 µg/mL in the MTS assay and a somewhat higher ED 50 of 103 µg/mL in the ATP assay (Figure 4A andB, Table 1). PS-COOH did not show any cytotoxicity in both assays (Figure 4A andB). Using the LDH assay, low levels of fluorescence signals and high background were encountered with this cell line.

A C C E P T E D M A N U S C R I P T

-15 -Recommended changes to the protocol to overcome these issues such as reduction of serum (that may contain LDH) to reduce high background and the use of pyruvate-free medium together with an increase of incubation time and temperature to reduce low overall fluorescence signals were unsuccessful. Therefore, no LDH assay results were obtained for this cell line.

The impact of PS-NH 2 on viability and integrity of intestinal Caco-2 monolayers obtained after differentiation was evaluated by the MTS assay and measurement of the TEER, respectively.

In order to respect the same NP doses/cm² used with undifferentiated Caco-2 cells during viability assays (using 96-well plates), intestinal monolayers (using transwells) were exposed to PS-NH 2 concentrations between 70 µg/mL (i.e. 100 µg/mL on undifferentiated Caco-2 cells, corresponding to 31.25 µg/cm² in both the cases) and 0.7 µg/mL (i.e. 1 µg/mL on undifferentiated Caco-2 cells, corresponding to 3.125 µg/cm² in both the cases). No effect on intestinal epithelium viability and integrity was observed with PS-NH 2 independently of the concentration (Figure 4B and C, respectively), while BC reduced viability to 20% and caused complete breakdown of the electrical resistance (Supplemental Table 2). PS-COOH were not tested since no effect was seen on naïve Caco-2 cells (Figure 4A andB).

Immune cell models

In addition to the THP-1 cells discussed earlier (Figure 3, Supplemental Figure 2), CD34-DCs prepared from different human cord blood donors were investigated as models of the immune system. As expected, donor variability had a strong impact on the quality of the assay results. However, similar trends to THP-1 cells were observed as PS-NH 2 induced cytotoxicity at low doses for some donors (Figure 5). Zinc sulphate (144 nM) was used as a positive control showing less than 20% cell viability or of ATP content compared to untreated controls cells in all donors (Supplemental Table 2).

The average ED 50 for PS-NH 2 in the ATP assay was 26 µg/mL (Figure 5A and Table 1), and for the MTS assay the ED 50 was 82 µg/ml (Figure 5B and Table 1). LDH release was observed only at doses of 100 µg/mL (Figure 5C and Table 1), which was not significant. In all cases, PS-COOH did not display any cytotoxic effect to the cells (Figure 5).

A C C E P T E D M A N U S C R I P T

-16 -

Discussion

We compared three cytotoxicity assays (MTS, ATP, and LDH assays) that cover different biological principles (redox metabolism, ATP content, membrane permeability) and different read-out principles (absorption, fluorescence and luminescence) on different cell models (Supplemental Table 1). The cell models covered the main routes of exposure to NPs (cutaneous exposure, ingestion, inhalation, and i.v. injection) and therefore represented the most important target organs. PS-NH 2 and PS-COOH were used as an example of ideal NPs as they were available with a perfect spherical shape with a very narrow size distribution and were easy to disperse by vortexing yielding stable dispersions. Moreover, previous studies already suggested that PS-NH 2 induce cytotoxicity and PS-COOH are of low or no cytotoxicity [START_REF] Anguissola | High content analysis provides mechanistic insights on the pathways of toxicity induced by amine-modified polystyrene nanoparticles[END_REF][START_REF] Bexiga | Cationic nanoparticles induce caspase 3-, 7and 9-mediated cytotoxicity in a human astrocytoma cell line[END_REF][START_REF] Ruenraroengsak | Respiratory epithelial cytotoxicity and membrane damage (holes) caused by amine-modified nanoparticles[END_REF][START_REF] Wang | The biomolecular corona is retained during nanoparticle uptake and protects the cells from the damage induced by cationic nanoparticles until degraded in the lysosomes[END_REF][START_REF] Xia | Cationic polystyrene nanosphere toxicity depends on cell-specific endocytic and mitochondrial injury pathways[END_REF]. Indeed, in this study all assays showed that PS-COOH had no effect on the cell viability within the tested dose range up to 100 µg/mL in none of the cell models. In contrast, PS-NH 2 induced a dose-dependent cytotoxicity in most cell models tested in all assays. We thoroughly tested for possible interference of the NPs in all the test assays and we could confirm absence of interferences. Therefore, our data confirm that PS-COOH and PS-NH 2 are very good candidates for negative and positive control nanomaterials, respectively, for in vitro cytotoxicity measurements.

The ATP assay was the most sensitive assay, yielding the lowest ED 50 values in most cell models. However, ED 50 values of the MTS assay were often in the same range as those of the ATP assay. This might be related to the fact that ATP results were not normalized by the protein content. Importantly, for a conclusion on general cytotoxicity such a conflation is unproblematic provided that cells were not significantly proliferating during the treatment period. Therefore, the MTS and the ATP assays both appear suitable for cytotoxicity testing of NPs as previously described [START_REF] Piret | Pan-European inter-laboratory studies on a panel of in vitro cytotoxicity and pro-inflammation assays for nanoparticles[END_REF]. While the LDH assay also indicated cytotoxicity in most cell models it proved to be the least sensitive assay. Often ED 50 values were not reached within the tested dose range. These results are in line with the biological basis of the methods which suggests ATP levels to be affected earliest, NAD(P)H

A C C E P T E D M A N U S C R I P T

-17 -oxidoreductases somewhat later and loss of membrane integrity as a late effect. However, some issues of high background and low overall levels of fluorescence signals in the LDH assay were encountered with certain cell models, i.e. Caco-2 cells and CD34-derived dendritic cells. These render this assay not recommendable for those cell models.

The two tissue models involving monocultures of differentiated cells, i.e. reconstituted epidermis and differentiated (intestinal) Caco-2 cell monolayer were unaffected by PS-NH 2 in the tested dose range regardless of the type of viability assay. This was in contrast to the results obtained in the respective cell models that were used to grow these more complex models, i.e. NHEK and Caco-2 cells. In vitro reconstituted human epidermises exhibit the barrier function of cornified cells, emphasizing the excluding/protective role of the upper layer of the epidermis in case of acute exposure to PS-NH 2 The susceptibility of NHEK cells indicated a potential toxic effect in case of a breach of the barrier function. In the case of differentiated Caco-2 cells [START_REF] Jumarie | Caco-2 cells cultured in serum-free medium as a model for the study of enterocytic differentiation in vitro[END_REF], the change of cellular features such as cell polarity, microvilli formation, tight junctions, domes and increased brush-border membrane enzymatic activities, and the secretion and assembly of an extracellular matrix because of the long undisturbed culture period might be responsible for protecting them from the cytotoxic effect of PS-NH 2 that was observed in the undifferentiated counterparts. In addition, the integrity of the intestinal epithelia generated by the Caco-2 differentiation was not affected by the NPs as confirmed by TEER measurements. All together, these data underline the importance of considering more complex cell models in the evaluation of NP toxicity as differences were observed depending on the differentiation status and complexity of cell models.

Cells of the immune system are expected to be among the first cells to encounter NPs after they breach barriers and become systemically available. Here we tested two types of models for immune cells, phorbol ester-primed THP-1 monocytes and CD34-DCs. Both cell types showed a higher sensitivity towards PS-NH 2 compared to the epithelial cell types. However, in a co-culture of THP-1 cells with lung cells the co-cultures exhibited ED 50 values more comparable to the lung epithelial cells alone, arguing that within the incubation time the effect

A C C E P T E D M A N U S C R I P T -18 -
of the immune cells is rather low [START_REF] Loret | Air-liquid interface exposure to aerosols of poorly soluble nanomaterials induces different biological activation levels compared to exposure to suspensions[END_REF]. However, immune cells, especially macrophages, can elicit a so-called respiratory burst during which they release reactive oxygen species and cytokines into the surrounding tissue [START_REF] Forman | Reactive oxygen species and cell signaling: respiratory burst in macrophage signaling[END_REF]. The respiratory burst has been observed in response to NP treatment [START_REF] Cruz | Interaction between polyalkylcyanoacrylate nanoparticles and peritoneal macrophages: MTT metabolism, NBT reduction, and NO production[END_REF][START_REF] Prietl | Nano-sized and micro-sized polystyrene particles affect phagocyte function[END_REF]. Prolonged exposure and hence release of reactive oxygen species and cytokines can lead to inflammation. Upon chronic stimulation this might lead to tissue scarring and reduced organ function and possibly tumorigenesis [START_REF] Babior | Phagocytes and oxidative stress[END_REF]. This argues, that macrophages should always be included when testing NPs because of the necessity to record possible respiratory burst reactions at least for lung epithelial models.

Table 2 summarizes the findings of this study with respect to the different cell models.

Overall, we achieved to develop robust SOPs for all three assays, i.e. MTS, ATP and LDH assays, which have been tested for a variety of important cell models. However, several possible improvements of these SOPs can be considered in future. Improvements could concern the exposure of the NPs, such as pre-incubation of the NPs with a more realistic biofluid prior to cell treatments. For instance, surfactant can be used in the case of lung models [START_REF] Mckenzie | Surfactant protein A (SP-A) inhibits agglomeration and macrophage uptake of toxic amine modified nanoparticles[END_REF][START_REF] Raesch | Proteomic and Lipidomic Analysis of Nanoparticle Corona upon Contact with Lung Surfactant Reveals Differences in Protein, but Not Lipid Composition[END_REF] and for oral uptake the different consecutive pH values encountered along the stages of the gastro-intestinal tract can be simulated [START_REF] Peters | Presence of nano-sized silica during in vitro digestion of foods containing silica as a food additive[END_REF]. Furthermore, skin and lung models may be exposed at the airliquid interface [START_REF] Stoehr | Assessment of a panel of interleukin-8 reporter lung epithelial cell lines to monitor the pro-inflammatory response following zinc oxide nanoparticle exposure under different cell culture conditions[END_REF] while ingestion models could include a more realistic digestion fluid on one side. Finally, cell models are being refined, and new developments such as microfluidic systems and organoids promise to allow more realistic models and to eliminate the species differences in toxicological data by using human cells [START_REF] Esch | Body-on-a-chip simulation with gastrointestinal tract and liver tissues suggests that ingested nanoparticles have the potential to cause liver injury[END_REF][START_REF] Tralau | Regulatory toxicology in the twenty-first century: challenges, perspectives and possible solutions[END_REF].

Conclusions

All assays investigated here should be applicable for all NPs when known issues of interference and artefacts have been controlled for [START_REF] Bonvin | Assessment of nanoparticles' safety: corrected absorbance-based toxicity test[END_REF][START_REF] Kroll | Interference of engineered nanoparticles with in vitro toxicity assays[END_REF][START_REF] Monteiro-Riviere | Limitations and relative utility of screening assays to assess engineered nanoparticle toxicity in a human cell line[END_REF][START_REF] Stone | Development of in vitro systems for nanotoxicology: methodological considerations[END_REF][START_REF] Tournebize | Pitfalls of assays devoted to evaluation of oxidative stress induced by inorganic nanoparticles[END_REF]. In summary, the MTS assay was applicable for all tested cell models, including primary cells and more

A C C E P T E D M A N U S C R I P T

-19 -complex models and therefore represents the most robust and reproducible assay out of the three tested assays. Thus, our data confirm previous findings that the MTS assay should be integrated in an in vitro NP test strategy [START_REF] Piret | Pan-European inter-laboratory studies on a panel of in vitro cytotoxicity and pro-inflammation assays for nanoparticles[END_REF]. However, other assays can be useful as well, depending on the NP type and the cell model. The ATP assay proved to be more sensitive in several cell models investigated here and thus could be useful in a test strategy too, provided that the issues with normalization can be overcome. Our data also

show that combining different assays, which represent different biological read-out gives a more complete picture especially on the chronological sequence of different events and allows already first conclusions on possible underlying mechanisms of cytotoxicity.

Furthermore, our data suggest that PS-COOH and PS-NH 2 may be used as negative and positive control NPs, respectively, for NP cytotoxicity testing.

6.

A

  aqueous dispersions of polystyrene (PS) NPs with amine (PS-NH 2 ) and carboxyl (PS-COOH) surface functionalization (nominal size of 50 nm diameter) were obtained from Bangs Laboratories (catalog number PA02N) and Polysciences, Inc. (catalog number 15913-10), respectively. These NPs can be easily dispersed by simple vortexing as has been assured in an accompanying round robin using Dynamic Light Scattering (DLS) and Differential Centrifugal Sedimentation (DCS) (Langevin et al., under review). NPs were dispersed in cell culture media by vortexing for 30 sec immediately prior to experiments. Particle size distributions in water and in cell culture medium with FCS were determined using Dynamic Light Scattering (Langevin et al., under review).

  vessels of placentas of full-term infants, born at the Heilig Hart hospital in Mol and the St.Dimpna hospital in Geel, Belgium. Informed consent was given by the mothers and the study was approved by the ethical commission of both hospitals. Mononuclear cells were separated from the cord blood by density gradient centrifugation (Ficoll-Paque™ plus, GE Healthcare). Subsequently, CD34 + progenitor cells were extracted using positive immunomagnetic selection (EasySep® human CD34 positive selection kit, Stemcell Technologies, Grenoble, France) according to the manufacturer's guidelines. The progenitor cells were cultured for 12 days in Iscove's Modified Dulbecco's Medium (IMDM, Gibco) supplemented with 10% fetal bovine serum (PAA laboratories, Pasching, Austria), 2% penicillin/streptomycin (Gibco) and 1% bovine serum albumin (Sigma-Aldrich) in the presence of TNF-α (250 U/ml, Roche applied science), granulocyte macrophage colonystimulating factor (5000 U/ml; Gentaur, Brussels, Belgium), stem cell factor (5 U/ml, , Belgium) and interleukin 4 (1000 U/ml, Biosource) to induce proliferation and differentiation towards immature CD34-DCs.

  Half maximal effective doses (ED 50 values) were determined by curve fitting performed in the R statistical computing environment (R Development Core Team, 2009) in combination with the drc package (Ritz and Streibig, 2005) using the five parametric logistic function. Mean ED 50 values were calculated from the averages of the logarithm of the ED 50 of independent biological experiments. Upper and lower bounds correspond to the standard deviation of the logarithmic mean. Student's t-test with a significance cut-off level p of 0.05 was used to compare mean values of NP-treated with those of untreated cells.

  Figure 1

Figure 2

 2 Figure 2

Figure 3

 3 Figure3

Figure 4

 4 Figure 4

Figure 5

 5 Figure 5

Fig. 4 A

 4 Fig. 4

Acknowledgements

We are grateful to the excellent technical assistance of Kelly Blazy and Christelle Gamez (INERIS, France) as well as of An Jacobs (VITO, Belgium).

Funding

We would like to acknowledge funding from the EC FP7 project QualityNano (INFRA-2010-1.1.31-262163)

Declaration of Interest

The authors declare no competing interests.

A C C E P T E D M A N U S C R I P T

-21 -10.

A C C E P T E D M A N U S C R I P T

-25 - 2) Comparison of ten cell models, i.e. cell lines, primary and complex cell models 3) Cell models are representative of the main routes of exposure to NPs 4) MTS assay was applicable to all, and was the most robust and reproducible assay 5) PS-COOH and PS-NH2 are useful negative and positive controls for cytotoxicity testing "include 3 to 5 bullet points (maximum 85 characters, including spaces, per bullet point)"

Tables


E D Coarita-Tintaya

M Souley

M N Vu

F Golfier

NUMERICAL ANISOTROPIC MODELLING OF A DEEP DRIFT AT THE MEUSE/HAUTE-MARNE URL

The development of nuclear energy raised the issue of radioactive waste disposal in deep geological formations. Clay formations in their natural state show very favorable confining conditions as repositories for radioactive waste due to their generally low values of hydraulic conductivity and significant retention capacity for radionuclides. Three clay formations in Europe

are currently being studied in details as potential host rocks for high level radioactive waste disposal, i.e. the Boom Clay in Belgium, the Opalinus Clay in Switzerland and the Callovo-Oxfordian Claystone (COx) in France. Lastly, the French national radioactive waste management agency (Andra) started building the underground research laboratory (URL) at Bure in the Meuse district. Several specific in situ experiments were carried out with the main objectives in term of underground civil engineering to: (a) evaluate the geological environment of the future Deep Geological Repository (DGR), (b) collect data from the clay formation in order to assess the performances of the future DGR, (c) test and demonstrate the feasibility of the future DGR. The feedback of these experiments and in situ measurements as well as the numerous theoretical analyzes and numerical modeling works undertaken on the COx allowed to identify the key mechanisms (mainly in the framework of THM processes) governing the behavior of the COx [START_REF] Armand | Fundamental aspects of the hydromechanical behavior of Callovo-Oxfordian claystone: From experimental studies to model calibration and validation[END_REF]. In addition to the THM processes and the associated parameters, it appears necessary to consider the anisotropies (mechanical, hydraulic and thermal) of such rock, but also its time-dependent behavior. This paper deals with these issues.

Saturated hydromechanical coupling was investigated on a circular experimental gallery of radius 2.6 m (GCS drift) excavated following the principal major stress and without rigid lining at the URL. The plane strain assumption is considered. Then a cross section of the GCS drift with a drainage condition along its wall, was modeled with COMSOL Multiphysics software. The initial pore pressure and in situ stress fields at the main level (-490 m) are given in figure 1. The model geometry and the poromechanical properties are presented in figure 1 and table 1, respectively. For simplicity, the creep of COx is described by a power law (A=1.26x10 -26 Pa 1/n s -1 and n=1.98).

The instantaneous (t=0+) response induced by excavation leads to overpressures with a maximum value of 1.3 MPa in the horizontal direction (i.e. direction of initial minor stress) and underpressures in the vertical direction (i.e. direction of initial major stress) with respect to the GCS section (figure 2). These pore pressure distributions are in good agreement with a poroelastic analysis presented by [START_REF] Guayacán-Carrillo | Effect of anisotropy and hydro-mechanical couplings on pore pressure evolution during tunnel excavation in low-permeability ground[END_REF]. Moreover, with time and in the horizontal direction, the pressure peak, shift towards the interior of the rock mass and decreasing with time at a very small rate due to the low permeability of the COx. This remains in agreement with the in-situ observations [START_REF] Seyedi | Transverse Action" -A model benchmark exercise for numerical analysis of the Callovo-Oxfordian claystone hydromechanical response to excavation operations[END_REF].

It is observed that the evolution of the pore pressure around drift, is not influenced by creeping (figure 2), in accordance with the creep model used (not dependent on the mean stress). Conversely, the creep relaxes the von Mises stress until a distance of 2 diameters (depending on the used creep model and associated parameters) from the wall (figure 3 b) at 100 years. The radial displacement (figure 3 a) highlights a significant increase in magnitude from 10 years. These results will be compared with the in-situ measurements.

Figure 2 .

 2 Figure 2. Pore pressure profiles (a) horizontal direction (b) vertical direction (continue = poroelastic, dashed = poroviscoplastic).

Figure 3 .

 3 Figure 3. (a) Radial displacement and (b) von Mises stress in the horizontal direction (continue = poroelastic, dashed = poroviscoplastic).

Table 1 .

 1 Poromechanical properties. Figure1. Geometry and boundary conditions.

	Parameter Unit	Value
	Ex / Ey	(GPa)	5.2 / 4
	vxz / vyx	(-)	0.2 / 0.3
	Gxy	(MPa)	1740
	b	(-)	0.6
	kx / ky	(m 2 )	4/1.33 (x10 -20 )
	ϕ0	(-)	0.18


Marti ´n

I Idiart

Pedro Ponte
email: ponte@seas.upenn.edu

Castan ˜eda

Variational linear comparison bounds for nonlinear composites with anisotropic phases. I. General results

Keywords: nonlinear homogenization, variational methods, anisotropy

This work is concerned with the development of bounds for nonlinear composites with anisotropic phases by means of an appropriate generalization of the 'linear comparison' variational method, introduced by Ponte Castan ˜eda for composites with isotropic phases. The bounds can be expressed in terms of a convex (concave) optimization problem, requiring the computation of certain 'error' functions that, in turn, depend on the solution of a non-concave/non-convex optimization problem. A simple formula is derived for the overall stress-strain relation of the composite associated with the bound, and special, simpler forms are provided for power-law materials, as well as for ideally plastic materials, where the computation of the error functions simplifies dramatically. As will be seen in part II of this work in the specific context of composites with crystalline phases (e.g. polycrystals), the new bounds have the capability of improving on earlier bounds, such as the ones proposed by deBotton and Ponte Castan ˜eda for these specific material systems.

Introduction

For a linear-elastic constitutive response, there are well-established methods to estimate the effective or overall behaviour of composite materials. These so-called homogenization methods include the variational principles of [START_REF] Hashin | A variational approach to the theory of the elastic behavior of multiphase materials[END_REF], which can be used to bound the effective modulus tensor of linear-elastic composites. A comprehensive review of this and other works on linear composites is given, for example, in the recent monograph by [START_REF] Milton | The theory of composites[END_REF].

On the other hand, for nonlinear (e.g. plastic, viscoplastic, etc.) composites, rigorous methods have not been available until more recently, even though efforts along these lines have been going on for some time, particularly in the context of ductile polycrystals (e.g. [START_REF] Hill | Continuum micro-mechanics of elastoplastic polycrystals[END_REF][START_REF] Hutchinson | Bounds and self-consistent estimates for creep of polycrystalline materials[END_REF]. Making use of a nonlinear extension of the Hashin-Shtrikman (HS) variational principles, due to [START_REF] Willis | The overall response of composite materials[END_REF], the first bounds of the HS type for nonlinear composites were derived by [START_REF] Talbot | Variational principles for inhomogeneous nonlinear media[END_REF]. Further applications of this methodology to composites with nonlinear anisotropic phases (e.g. polycrystals) were given by [START_REF] Willis | Upper and lower bounds for nonlinear composite behavior[END_REF]. A more general variational approach making use of the notion of optimally chosen 'linear comparison composites' was proposed by Ponte Castan ˜eda (1991). This approach is not only capable of delivering bounds of the HS type for nonlinear composites, but also, in addition, can be used to generate bounds and estimates of other types, such as self-consistent estimates and three-point bounds [START_REF] Ponte Castan ˜eda | New variational principles in plasticity and their application to composite materials[END_REF]. The connections between these two different approaches were explored by [START_REF] Willis | On methods for bounding the overall properties of nonlinear composites: correction and addition[END_REF] and [START_REF] Talbot | Some explicit bounds for the overall behavior of nonlinear composites[END_REF]. [START_REF] Suquet | Overall potentials and extremal surfaces of power law or ideally plastic materials[END_REF] and [START_REF] Olson | Improvements on Taylor upper bound for rigid-plastic composites[END_REF] proposed alternative, but equivalent methods for the special classes of power-law and ideally plastic materials, respectively. [START_REF] Suquet | Overall properties of nonlinear composites: a modified secant moduli theory and its link with Ponte Castan ˜eda's nonlinear variational procedure[END_REF] also gave a novel physical interpretation of the variational estimates of Ponte Castan ˜eda (1991) in terms of 'secant' moduli and the second moments of the local fields in the linear comparison composite. In addition, deBotton & Ponte Castan ˜eda (1995) proposed an extension of the variational principle of Ponte Castan ˜eda (1991) to generate bounds for composite materials with (anisotropic) viscoplastic crystalline phases. This variational approach was, in turn, generalized further by Suquet (see Ponte Castan ˜eda [START_REF] Ponte Castan ˜eda | Nonlinear composites[END_REF]) to include more general types of anisotropies in the context of nonlinear composites. Finally, it should be mentioned that there is another approach, called the 'translation' method, which has been applied successfully to polycrystals, but thus far only for scalar potential problems [START_REF] Kohn | Some model problems of polycrystal plasticity with deficient basic crystals[END_REF][START_REF] Nesi | Improved bounds for the yield stress of a model polycrystalline material[END_REF][START_REF] Goldsztein | Rigid perfectly plastic two-dimensional polycrystals[END_REF][START_REF] Garroni | Some three-dimensional problems related to dielectric breakdown and polycrystal plasticity[END_REF].

The present work is concerned with an extension of the variational method of Ponte Castan ˜eda (1991,1992) for composites with nonlinear (viscoplastic) anisotropic phases. It will be shown that this generalization, which is more closely tied to the original formulation, has the capability to give improved bounds relative to the generalizations provided by deBotton & Ponte Castan ˜eda (1995) in the specific context of viscoplastic polycrystals, and the further generalization provided by Ponte Castan ˜eda & [START_REF] Ponte Castan ˜eda | Nonlinear composites[END_REF] for more general anisotropies.

Thus, attention will be focused on composite materials made of N homogeneous constituents, or phases, that are randomly distributed in a specimen occupying a volume U at a length-scale which is much smaller than the size of the specimen and the scale of variation of the loading conditions. The constitutive behaviour of the nonlinear (viscoplastic) anisotropic phases is characterized by convex strain (strain-rate) potentials w (r) (rZ1, ., N ), satisfying the conditions w (r) (0)Z0 and w (r) (3)/N as j3j/N. The microstructure is defined by characteristic functions c (r) that are equal to 1 if the position vector x is in phase r (i.e. x2U (r) ), and 0 otherwise. In this case, the relation between the stress s and the strain (strain-rate) 3 is defined by a local strain potential w, such that s Z vw v3 ðx; 3Þ; wðx; 3Þ Z X N rZ1 c ðrÞ ðxÞw ðrÞ ð3Þ: ð1:1Þ

The problem is then that of determining the effective behaviour of the composite, which is defined as the relation between the average stress sZ hsi and the average strain 3Z h3i. Here, the symbols h$i and h$i (r) will be used to denote volume averages over the composite (U) and phase r (U (r) ), respectively. For the class of materials characterized by relations (1.1), the effective behaviour is determined by the effective strain potential, defined by

wð 3Þ Z inf 32Kð 3Þ hwðx; 3Þi Z inf 32Kð 3Þ
X N rZ1 c ðrÞ hw ðrÞ ð3Þi ðrÞ ; ð1:2Þ

where c (r) Zhc (r) i denotes the volume fraction of phase r; and Kð 3Þ is the set of 'kinematically admissible' strain fields, such that there is a continuous displacement field u satisfying 3Z ð1=2Þ½VuC ðVuÞ T in U, and uZ 3x in vU. It is important to recall that (strict) convexity of the local strain potential w on the local strain 3 implies (strict) convexity of the effective potential w in the applied strain 3. The effective stress-strain relation is then known to be given by s Z v w v 3 ð 3Þ; ð1:3Þ

where it has been assumed that w is differentiable, as has been done for the local potential w in expression (1.1). In practice, the local and effective potentials can be assumed to be differentiable for most material models of interest, except in some special cases, including the ideally plastic materials to be considered further in §4b, where the potential is not differentiable (everywhere), but is still convex (although not strictly so). In this case, it is most natural to work with the subdifferential of convex analysis [START_REF] Ekeland | Convex analysis and variational problems[END_REF]. Here, for simplicity, the distinction will not be made between standard (Gateaux) derivatives and subdifferentials, except when it becomes necessary or convenient to do so. Alternatively, the behaviour of phase r may be characterized by a convex stress potential u (r) , defined by the Legendre-Fenchel transform, or convex dual [START_REF] Ekeland | Convex analysis and variational problems[END_REF], of w (r) where Sð sÞ is the set of 'statically admissible' stress fields s that are divergencefree in U, satisfying the traction continuity conditions across phase boundaries, and are such that hsiZ s. Note that ũ is convex, and it has also been assumed to be differentiable, for simplicity.

Given the assumed convexity of the phase potentials, the variational formulations (1.2) and (1.6) 2 for the effective behaviour of the nonlinear composites can be shown to be completely equivalent, in the sense that the functions w and ũ are Legendre duals of each other, ũZ wà . Thus, the problem of finding the constitutive relation for the composite reduces to that of determining the effective potentials w or ũ. It is well known (e.g. [START_REF] Hutchinson | Bounds and self-consistent estimates for creep of polycrystalline materials[END_REF]) that upper and lower bounds for w and ũ may be obtained by making use of uniform trial fields in the context of definitions (1.2) and (1.6) 2 . In the linear case, they are known as the bounds of Voigt and Reuss.

Linear comparison materials

Consider a linear-elastic comparison material with positive-definite, fully symmetric stiffness tensor C 0 , such that its strain potential is given by w 0 ðx; 3Þ Z 1 2 3$C 0 ðxÞ3: ð2:1Þ

Assuming that w (r) has 'weaker-than-quadratic' growth at infinity, implying that (w (r) Kw 0 )/KN as j3j/N (for fixed x), suggests the following definition. Definition 2.1. Let v (r) be the phase 'error' function, serving as a measure of the nonlinearity in the original material with the strain potential w (r) , given by

v ðrÞ ðC 0 Þ Z sup 3 w ðrÞ ð3ÞK 1 2 3$C 0 3 : ð2:2Þ Remark 2.1. Definition (2.
2) is still valid when the stiffness tensor C 0 is not positive definite, but in this case v (r) takes on infinite values. This is easy to see, because choosing 3 to be proportional to the eigentensor corresponding to any zero, or negative eigenvalue of C 0 , leads to an infinite value for the supremum in expression (2.2). In fact, v (r) tends to infinity, when any eigenvalue of (a positive definite) C 0 tends to zero. Also, note that, by definition, v ðrÞ ðC 0 ÞR w ðrÞ ð0ÞZ 0, and hence v (r) is a non-negative function. In addition, being the pointwise supremum of a collection of affine functions of C 0 , v (r) is a convex function of C 0 . Furthermore, assuming smoothness of the functions w (r) (r) , denote all the strain tensors for which the maximum is attained in (2.2) for a given C 0 . (Note that M (r) can be infinite in some instances, such as when the potential w (r) and C 0 are both isotropic.) It follows from definition (2.2) that

w ðrÞ ð3Þ% 1 2 3$C 0 3 C v ðrÞ ðC 0 Þ; ð2:4Þ
for any C 0 , and hence that

w ðrÞ ð3Þ% inf C 0 1 2 3$C 0 3 C v ðrÞ ðC 0 Þ : ð2: 5Þ
But, as has been pointed out earlier, the function v (r) is infinite if C 0 is not positive definite, and therefore expression (2.5) can be equivalently written as

w ðrÞ ð3Þ% inf C 0 O0 1 2 3$C 0 3 C v ðrÞ ðC 0 Þ ; ð2:6Þ
where the notation C 0 O 0 has been used to signify that C 0 is positive definite.

In general, the inequality in expression (2.6) will hold. However, there may be classes of nonlinear materials, for which the equality would hold. Noting that the right-hand side of expression (2.6) is concave in the variable 353 (because it is the pointwise infimum of a set of affine functions in this variable) suggests that in order for equality to hold in expression (2.6), the potential w must be concave in the variable 353. Accordingly, it will be assumed that the potentials w of interest here satisfy the following 'square concavity' hypothesis.

Hypothesis 2.1. It is assumed that there exist functions f (r) that are concave in the space of fourth-order tensors, e j j , possessing the usual symmetry properties of elasticity tensors, and satisfying the following properties: (i) f ðrÞ ðOÞZ 0 and (ii) f (r) /N as je j j j/N, such that the phase potentials w (r) may be expressed as w ðrÞ ð3Þ Z f ðrÞ 1 2 3 5 3 : ð2:7Þ

This hypothesis was introduced by Ponte Castan ˜eda (1992) for isotropic materials and generalized as described previously for anisotropic materials by Suquet (see Ponte Castan ˜eda [START_REF] Ponte Castan ˜eda | Nonlinear composites[END_REF]. It should be emphasized that the functions f (r) are not uniquely determined. Also, a consequence of hypothesis 2.1 is that the potentials w (r) , as defined by relation (2.7), should satisfy the relation w ðrÞ ðK3ÞZ w ðrÞ ð3Þ. In addition, it should be noted that this hypothesis is consistent with the earlier assumption that the potentials w (r) exhibit weaker-than-quadratic growth on the strain 3. It is shown next that the equality holds strictly in expression (2.6) when the stronger square concavity hypothesis is made.

Lemma 2.1. Consider a composite with 'square concave' phase potentials w (r) , as defined in hypothesis 2.1. It then follows that w ðrÞ ð3Þ Z inf

C 0 O0 1 2 3$C 0 3 C v ðrÞ ðC 0 Þ ; ð2:8Þ
where the error functions v (r) have been defined by relations (2.2). It also follows that wðx; 3Þ Z inf

C 0 O0 fw 0 ðx; 3Þ C vðx; C 0 Þg; ð2:9Þ
where w 0 is given by (2.1), and v is the function defined by vðx; C 0 Þ Z sup 3 fwðx; 3ÞKw 0 ðx; 3Þg: ð2:10Þ

Proof. The concave Legendre-Fenchel transform of f (r) is defined as

f ðrÞ Ã ðC 0 Þ Z inf e j j
fC 0 $e j j Kf ðrÞ ðe j j Þg: ð2:11Þ

Note that f ðrÞ Ã is a non-positive, concave function of C 0 , such that f ðrÞ Ã ðC 0 ÞZKNwhen C 0 is not positive definite. It then follows from the assumed concavity of f (r) that f ðrÞ ðe j j Þ Z inf C 0 O0
fC 0 $e j j Kf ðrÞ Ã ðC 0 Þg; ð2:12Þ

where the restriction to positive definite C 0 has been made since, otherwise, the righthand side would be infinite.

Using (2.7) in relation (2.12), it is concluded that

w ðrÞ ð3Þ Z inf C 0 O0 1 2 3$C 0 3Kf ðrÞ Ã ðC 0 Þ : ð2:13Þ
It should be emphasized that this result, first given by Ponte Castan ˜eda & Suquet (1998), is valid for any 'concave extension' f (r) of the phase potential w (r) , as provided by hypothesis 2.1. Now, it follows from (2.11) that

f ðrÞ Ã ðC 0 Þ% inf e j j Zð1=2Þ353
fC 0 $e j j Kf ðrÞ ðe j j Þg Z inf 3 1 2 3$C 0 3Kw ðrÞ ð3Þ ; ð2:14Þ and hence that Kf ðrÞ Ã ðC 0 ÞR v ðrÞ ðC 0 Þ; ð2:15Þ where v (r) is the phase error function, as defined by (2.2). (It is important to emphasize that, in general, the functions v (r) cannot be identified with the functions Kf ðrÞ Ã .) Therefore, from expression (2.13), it is concluded that

w ðrÞ ð3ÞR inf C 0 O0 1 2 3$C 0 3 C v ðrÞ ðC 0 Þ : ð2:16Þ
But this expression, together with (2.6), which is itself independent of any concavity hypothesis, indeed implies that the identity (2.8) holds under the square concavity hypothesis 2.1. Next, recalling that w is defined in terms of the phase potentials w (r) by relation (1.1), and defining v similarly in terms of the functions v (r) by

vðx; C 0 Þ Z X N rZ1 c ðrÞ ðxÞv ðrÞ ðC 0 Þ; ð2:17Þ
it is concluded that relations (2.9) and (2.10) also hold true, where use has been made of relation (2.1). & Remark 2.2. On account of the convexity of v (r) on C 0 , and of the fact that the restriction to positive definite C 0 is artificial, the optimality condition in expression (2.8) for w (r) is given by the condition that zero should be included in the subdifferential of the terms in curly brackets, which may also be written as

K 1 2 3 5 3 2v C 0 v ðrÞ ð Ĉ0 Þ; ð2:18Þ
where Ĉ0 is the optimal value of C 0 , and v C 0 v ðrÞ is the subdifferential of the function v (r) . In this connection, it should be emphasized that the functions v (r) are not smooth, even when the functions w (r) are. (This is essentially because the optimal solutions (2.3) in expressions (2.2) for the functions v (r) can change abruptly with changes in C 0 , leading to 'corners' and 'edges' in v (r) .) As a consequence, the eigendirections of Ĉ0 may be determined by these sharp edges, leaving the corresponding eigenvalues to be determined by appropriate stationarity conditions. In this connection, it is useful to note that the subdifferential of v (r) , which is, by definition, a 'max-function', may be expressed as (e.g. Exercise 10 in section 6.1 of Borwein & Lewis 2000) (r) , denote all the strain vectors for which the maximum in (2.2) is attained for a given C 0 (see also the optimality relations (2.3)). In other words, the subdifferential (2.19) of the function v (r) is the set of all convex combinations of the rank-one tensors ðK1=2Þ3 ðrÞ ðmÞ 53 ðrÞ ðmÞ , formed by the optimal strain vectors 3ðrÞ ðmÞ in expression (2.2) for the functions v (r) .

v C 0 v ðrÞ ðC 0 Þ Z e j j je j j ZK 1 2 X M ðrÞ mZ1 a ðrÞ
Result 2.1. The stress-strain relation associated with the strain potential w (r) , as determined by expression (2.8), is given by s Z vw ðrÞ v3 ð3Þ Z Ĉ0 ð3Þ3; ð2:20Þ

where Ĉ0 ð3Þ is the optimal value of C 0 in expression (2.8), evaluated at 3.

Proof. Given the convexity of w (r) and the fact that the optimality condition in expression (2.8) is not a simple stationarity condition, it is best to work here with the subdifferential. Now, if s2vw (r) (3), then s$ð3 0 K3Þ% w ðrÞ ð3 0 ÞKw ðrÞ ð3Þ; for all 3 0 ; ð2:21Þ

or, using expression (2.8) for w (r) ,

s$ð3 0 K3Þ% inf C 0 O0 1 2 3 0 $C 0 3 0 C v ðrÞ ðC 0 Þ K inf C 0 O0 1 2 3$C 0 3 C v ðrÞ ðC 0 Þ % 1 2 3 0 $C 0 0 3 0 Kv ðrÞ ðC 0 0 ÞK 1 2 3$ Ĉ0 3 C v ðrÞ ð Ĉ0 Þ; for all 3 0 ; C 0 0 ;
where Ĉ0 is the optimal value of C 0 evaluated at 3. In particular, for C 0 0 Z Ĉ0 ð3Þ, it follows that

s$ð3 0 K3Þ% 1 2 3 0 $ Ĉ0 ð3Þ3 0 K 1 2 3$ Ĉ0 ð3Þ3; for all 3 0 ; ð2:22Þ
which means that s2vw 0 , where w 0 Z ð1=2Þ3$ Ĉ0 3, with Ĉ0 fixed. But w 0 is differentiable, with derivative Ĉ0 3, and hence vw 0 ð3ÞZ f Ĉ0 ð3Þ3g. In conclusion, it has been shown that if s2vw (r) (3), then sZ Ĉ0 ð3Þ3, which is the desired result. & Remark 2.3. Result 2.1 should not be confused with the optimality condition (2.3), which is only a prescription for the optimal 3ðrÞ ðmÞ , for some given C 0 , in the definition of the error function v (r) . Interestingly, although expression (2.3) is commonly used to define (in a non-unique fashion) the secant modulus tensor (e.g. Ponte Castan ˜eda & Suquet 1998), it follows from result 2.1 that, in fact, a better definition of the secant modulus, in the general anisotropic case, is provided by the optimal Ĉ0 in expression (2.8). This leads to a well-defined prescription that can be shown to be consistent with the standard prescription in the isotropic case.

It is also possible to obtain an alternative representation starting from the local stress potentials (2.2). To accomplish this, note that (1.4) can be re-expressed as

v ðrÞ ðC 0 Þ Z sup 3 sup s n 3$sKu ðrÞ ðsÞ o K 1 2 3$C 0 3 Z sup s sup 3 3$sK 1 2 3$C 0 3 Ku ðrÞ ðsÞ Z sup s 1 2 s$ðC 0 Þ K1 sKu ðrÞ ðsÞ ; ð2:23Þ
where use has been made of the fact that w ðrÞ Z u ðrÞ* and of the positive definiteness of C 0 , as well as of the fact that the order of suprema can be interchanged. This allows the definition of a new function v ðrÞ ðS 0 ÞZ v ðrÞ ðC 0 Z S K1 0 Þ, as follows. Definition 2.2. The phase 'error' functions can alternatively be expressed in terms of the phase compliance tensors S 0 via the relations v ðrÞ ðS 0 Þ Z sup s 1 2 s$S 0 sKu ðrÞ ðsÞ : ð2:24Þ

Remark 2.4. v ðrÞ is a convex, non-negative function of S 0 ðZC K1 0 Þ, such that v ðrÞ Z 0 when S 0 % 0. Note that in the following there will be no risk of confusion, and, for simplicity, no attempt will be made to distinguish between v ðrÞ and v (r) , henceforth writing simply as v ðrÞ ðS 0 Þ. Also, assuming smoothness of the functions u (r) , the solutions of the non-concave optimization problem defined by ( 2 (r) , denote all the stress tensors for which the maximum is attained in (2.24) for a given S 0 .

Then, introducing the stress potential of the linear comparison material via

u 0 ðx; sÞ Z 1 2 s$S 0 ðxÞs; ð2:26Þ
it is also possible to define the local 'error' function vðx; S 0 Þ Z sup s fu 0 ðx; sÞKuðx; sÞg: ð2:27Þ

From these definitions, the next result follows.

Lemma 2.2. Consider a composite with the phase stress potential u ðrÞ Z w ðrÞ* , as defined by relation (1.4), where w (r) are square concave (see hypothesis 2.1). Then, u (r) are 'square convex' and u ðrÞ ðsÞ Z sup

S 0 R0 1 2 s$S 0 sKv ðrÞ ðS 0 Þ ; ð2:28Þ
where the functions v ðrÞ Z v ðrÞ are defined by (2.24). Similarly, the local stress potential uZw à , as defined by relation (1.5) 2 , is given by uðx; sÞ Z sup

S 0 R0
fu 0 ðx; sÞKvðx; S 0 Þg; ð2:29Þ

where u 0 and v are the functions defined by (2.26) and (2.27), respectively.

Proof. On account of (2.8), it follows that

u ðrÞ ðsÞ Z sup 3 3$sK inf C 0 O0 1 2 3$C 0 3 C v ðrÞ ðC 0 Þ Z sup C 0 O0 sup 3 3$sK 1 2 3$C 0 3 Kv ðrÞ ðC 0 Þ Z sup C 0 O0 1 2 s$ðC 0 Þ K1 sKv ðrÞ ðC 0 Þ ; ð2:30Þ
which leads to result (2.28) upon setting S 0 Z C K1 0 (recall that C 0 O 0). This result also demonstrates the existence of a convex function g (r) (in the space of fully symmetric, fourth-order tensors), such that u ðrÞ ðsÞ Z g ðrÞ 1 2 s5s : ð2:31Þ

Thus, square concavity for w (r) implies square convexity for u (r) . Note also that square convexity is consistent with 'super-quadratic' growth for u (r) . The derivation of (2.29) is now straightforward in view of (1.5) 2 , (2.17) and (2.26). &

Variational principles for anisotropic materials

In this section, use is made of expressions (2.8) and (2.28) for the local potentials w and u in terms of linear comparison materials to derive alternative variational representations for w and ũ that are equivalent to (1.2) and (1.6) 2 . These representations have the advantage that they can be used to generate improved bounds-relative to the classical bounds of Voigt and Reuss-for the effective potentials.

Proposition 3.1. Under the 'square concavity' hypothesis 2.1, the effective potential w defined by (1.2) admits the alternative representation wð 3Þ Z inf Noticing that the inner infimum over 3 affects only the first term inside the curly brackets, and that VðC 0 ÞZ hvðx; C 0 ðxÞi, one arrives at the desired result. & Remark 3.1. The variational representation (3.1) generalizes a corresponding variational representation for nonlinear composites with isotropic phases first given by Ponte Castan ˜eda (1992). It expresses the effective properties of the nonlinear composite (through its potential w) in terms of two functionals. The functional w0 is the elastic energy of a fictitious linear heterogeneous solid, called the linear comparison material, made up of phases with stiffness C 0 ðxÞ at a point x, whose properties are determined by the solution of the variational representation (3.1) itself. The functional V depends on the phase error functions v (r) , and thus provides a measure of the nonlinearity of the actual material. The representation (3.1) is exact and strictly equivalent to the variational characterization of w given in (1.2). But it requires the exact solution of the homogenization problem (3.2), which is a difficult task in view of the fact that it corresponds to a linear composite material with infinitely many phases. However, as will be seen in §4, this variational representation can be used to generate estimates for w, by making use of suitable trial fields for the moduli tensor C 0 of the linear comparison material.

Remark 3.2. Representation (3.1) also provides an interpretation of the strain field in the actual nonlinear composite as the strain field in the optimal linear comparison solid. Indeed, although the strain field 3 0 associated with the minimizer in the linear comparison problem (3.2) depends on C 0 , and is therefore different, in general, from the strain field 3 associated with the original nonlinear problem (1.2), under the hypothesis of square concavity of w, it follows from (3.1) that the 'linear' minimizer 3 0 arising from the optimal choice of C 0 is precisely the nonlinear minimizer 3.

It is also possible to start from the identity (2.29) for the local stress potential u to generate a corresponding representation for the effective stress potential ũ.

Proposition 3.2. Assuming 'square convexity' of the local stress potential u (or 'square concavity' of the corresponding strain potential w), the effective stress potential ũ defined by ( 1 where the supremum over S 0 has been taken out of the triangular brackets. But u 0 is convex in s and Kv is concave in S 0 , and, therefore, by the saddle point theorem (Ekeland & Teman 1999), the order of the infimum and supremum can be interchanged, and hence ũð sÞ Z sup 3.5), are exactly equivalent, under the given hypothesis on the potentials w and u, respectively. (In other words, result (3.5) for ũ can be shown to follow directly from result (3.1) for w by using the fact that ũZ wà .) It should also be emphasized that the variational principles (3.1) and (3.5) for composites with anisotropic phases are entirely consistent with the earlier variational representations (Ponte Castan ˜eda 1992) for composites with isotropic phases. Indeed, it can be shown that, for isotropic nonlinear composites, the optimal choice for the comparison elasticity and the compliance tensors in the context of the general anisotropic forms (3.1) and (3.5) is provided by isotropic fourth-order tensors.

Remark 3.4. Finally, it is noted that an alternative version of the variational representations (3.1) and (3.5) for composites with anisotropic phases was presented by Ponte Castan ˜eda & Suquet (1998), generalizing earlier estimates for the special case of crystalline plasticity by deBotton & Ponte Castan ˜eda (1995). These variational principles have a form identical to the representations (3.1) and (3.5), except that the functions v (r) are defined differently. Essentially, v (r) are identified with the functions Kf ðrÞ Ã introduced in expressions (2.11), which, as has already been observed (see equations (2.15)), are different, in general, from the functions v (r) , as defined by relations (2.2).

Bounds and estimates via piecewise constant moduli

Unlike the stress and strain trial fields in the context of the classical variational representations (1.2) and (1.6) 2 , the trial fields of stiffnesses and compliances in the variational representations (3.1) and (3.5) can be chosen to be constant in each phase (not necessarily the same constant). Thus, the optimization over the elasticity moduli C 0 ðxÞ can be restricted to the set of piecewise constant moduli where the functions v (r) are defined by relations (2.2), and

w0 ð 3Þ Z 1 2 3$ C0 C ðsÞ 0 3; ð4:3Þ
is now the effective strain potential (3.2) of a linear comparison composite (LCC) with the uniform stiffness tensors C ðrÞ 0 in each of the phases (rZ1, ., N)-and, hence, the same microstructure as the nonlinear composite-and with the effective stiffness tensor C0 .

Remark 4.1. The form (4.2) for the bound wC ð 3Þ involves a convex optimization problem for the stiffness tensors C ðrÞ 0 . This follows, for example, from lemma 2.1 of [START_REF] Ekeland | Convex analysis and variational problems[END_REF], because the terms inside the curly brackets may be written as the infimum (over the variables 3 2Kð 3Þ) of a convex function in the arguments 3 and C ðrÞ 0 (refer to equation (3.3)). In this connection, it is important to recall that the tensors C ðrÞ 0 need not be restricted to be positive definite (cf. the passage from relations (2.5) to (2.6)); it is simply that the infimum cannot occur for C ðrÞ 0 that are not positive definite, since the value of the functions v (r) would be infinite in that case. Thus, given that the optimization problem for the bound wC is convex for the whole space of stiffness tensors C ðrÞ 0 -and the fact that the functions v (r) need not be smooth-a necessary and sufficient condition for the minimum to be attained is that zero be included in the subdifferential of the terms inside the curly brackets in equation (4.2), for the optimal stiffness tensors ĈðrÞ 0 . Now, since w0 is differentiable (with respect to C ðrÞ 0 ), this optimality condition can also be written as is satisfied? Thus, it is seen here too that the optimal values ĈðrÞ 0 may be related to the second moments h3 L 53 L i ðrÞ of the strain field in the LCC, as has been shown by [START_REF] Suquet | Overall properties of nonlinear composites: a modified secant moduli theory and its link with Ponte Castan ˜eda's nonlinear variational procedure[END_REF] for the special case of composites with isotropic phases. In general, the tensor of the second moments of the strain is of full rank, and, consequently, the optimality conditions (4.6) require that ĈðrÞ 0 be such that the maximum in the function v (r) be attained simultaneously at several strains 3ðrÞ ðmÞ (i.e. M (r) should be sufficiently large), so that the fourth-order tensors on the two sides of (4.6) have the same rank. In turn, this implies that the optimal ĈðrÞ 0 should be precisely in regions where the functions v (r) are not differentiable (M (r) O1). Thus, the fact that the functions v (r) have sharp edges implies that the optimal tensors ĈðrÞ 0 have certain preferred orientations, depending on the anisotropy of the functions w (r) , and correspondingly depend only on certain traces of the second moment strain tensor h3 L 53 L i ðrÞ .

Remark 4.2. It is also possible to make use of expressions (2.2) for the functions v (r) in expression (4.2) for wC , to rewrite the problem as an inf-sup optimization. (Note that this problem is not concave in the variables 3, and therefore the order of the inf and the sup cannot be interchanged in general.) Then, using the optimality conditions (4.6), it can be shown that the bound wC can be formally written in the form

wC ð 3Þ Z X N rZ1 X M ðrÞ mZ1
c ðrÞ a ðrÞ m w ðrÞ 3ðrÞ ðmÞ ; ð4:7Þ where 3ðrÞ ðmÞ are the optimal solutions of problem (2.2) for the functions v (r) , satisfying the secant condition (2.3), and evaluated at the optimal values of ĈðrÞ 0 , as determined by relation (4.2). This form generalizes the form proposed by [START_REF] Suquet | Overall properties of nonlinear composites: a modified secant moduli theory and its link with Ponte Castan ˜eda's nonlinear variational procedure[END_REF] for composites with isotropic phases, but it should be emphasized that in the more general case of anisotropic phases the (difficult) inf-sup problem must be solved anyway to determine the correct choice for 3ðrÞ ðmÞ . In general, the estimate (4.2) for w requires numerical treatment, and therefore to determine the associated stress directly from this expression, the relevant derivative with respect to 3 would have to be computed numerically. However, the following result avoids this complication.

Result 4.3. The effective stress-strain relation of the nonlinear composite is approximated by

s Z v wC v 3 ð 3Þ Z C0 ĈðsÞ 0 3; ð4:8Þ
where ĈðsÞ 0 are the stiffness tensors satisfying the optimality conditions in (4.2). Remark 4.4. The demonstration of this result is analogous to the derivation of result 2.1, and will not be detailed further here for conciseness. It should be emphasized, however, that relation (4.8) is an approximation to the exact stress-strain relation of the nonlinear composite. This relation is a generalization of a result first given in the context of composites with isotropic phases by deBotton & Ponte Castan ˜eda (1993). In spite of its appearance, this effective stress-strain relation is nonlinear, due to the nonlinear dependence of the optimal ĈðrÞ 0 on the average strain 3. In fact, in parallel with remark 2.3 concerning the interpretation of ĈðsÞ 0 as the secant moduli tensors of the phases, relation (4.8) suggests that C0 can be interpreted as the secant modulus tensor of the nonlinear composite, generalizing earlier interpretations for the isotropic case by [START_REF] Suquet | Overall properties of nonlinear composites: a modified secant moduli theory and its link with Ponte Castan ˜eda's nonlinear variational procedure[END_REF].

A bound that is equivalent to (4.2) can be obtained by considering the stress potential ũ and its variational representation (3.5). Thus, restricting the optimization in (3.5) to piecewise constant compliances S ðrÞ 0 in the variational representation (3.5) leads to a lower bound for ũ. In addition, an estimate for the stress-strain relation may be obtained by differentiation, which is completely analogous to the estimate (4.8) in result 4.3. where the functions v (r) have been defined by relations (2.24), and where ŜðsÞ 0 are the optimal values of S ðsÞ 0 in expression (4.9). Remark 4.5. Expression (4.9) for the bound ũK involves a concave optimization problem for the compliance tensors S ðrÞ 0 . This follows from the facts that the functions v (r) are convex in the tensors S ðrÞ 0 , while the term arising from ũ0 is, on the other hand, concave in the tensors S ðrÞ 0 . (This last observation is easy to show making use of definition (3.6) and of the fact that the infimum of the sum is greater than the sum of the infima.) Then, the analysis carried out in the context of remarks 4.1 and 4.2 can be repeated step by step, with results completely analogous to expressions (4.4), (4.6) and (4.7).

ũ 0 ð sÞ Z 1 2 s$ S0 S ðsÞ
(a ) Special results for power-law materials

The above bounds for the effective potential may be given simpler, alternative forms for power-law composites by exploiting the homogeneity of the relevant potentials and error functions. Such results generalize earlier results for composites with isotropic phases by [START_REF] Suquet | Overall potentials and extremal surfaces of power law or ideally plastic materials[END_REF], who made use of Ho ¨lder's inequality for their derivation. Thus, when phase r is of the (incompressible) power-law type with exponent m, such that 0%m%1, the strain potential w (r) is positively homogeneous of degree mC1, w ðrÞ ðl3Þ Z l mC1 w ðrÞ ð3Þ; clR 0: ð4:12Þ Replacing 3 by l3 in expression (2.2), using property (4.12), and optimizing with respect to l, the function v (r) can be written in the form

v ðrÞ ðC 0 Þ Z 1Km 2ð1 C mÞ sup 3 ½ð1 C mÞw ðrÞ ð3Þ 2=ð1KmÞ ½3$C 0 3 ð1CmÞ=ð1KmÞ
( )

; ð4: 13Þ
where the optimization variables 3 can now be chosen to have magnitude 1 (and therefore belong to a bounded space). Note that it follows from this expression that v (r) is a homogeneous function of degree (mC1)/(mK1) in the variable C 0 . When all the phases in a composite are made of (incompressible) power-law materials with the same exponent m, the composite is itself an (incompressible) power-law material (Ponte Castan ˜eda & Suquet 1998). In other words, the effective potential w is also homogeneous of degree mC1. Then, letting C 0 ðxÞZ tC 0 ðxÞ, for arbitrary positive t, noting that w0 and v (r) , as defined by relations (4.3) and (4.13), are homogeneous functions of degrees 1 and (mC1)/ (mK1) in C 0 , respectively, and optimizing with respect to t, the following alternative representation for the bound wC is obtained. ð4:14Þ where v (r) are given by (4.13).

A corresponding representation can be given for the lower bound ũK when all the individual phases are power-law materials with the same exponent nZ1/m. The details are omitted here for brevity, but the result is as follows.

Result 4.6. The effective stress potential ũ for power-law composites is bounded below by Even though simpler than the general form of the bounds, the alternative forms of the bounds (4.14) and (4.15) for power-law composites still require the solution of difficult non-concave problems (4.13) and (4.16) for the functions v (r) . In the limiting case of ideal plasticity (n/N), however, this problem simplifies considerably. This is because, in that case, the functions v (r) , defined by (2.24), can be written in the form

ũK ð sÞ Z 2 n C 1 sup S ðrÞ 0 O0 rZ1;.;N ½ũ 0 ð sÞ ðnC1Þ=2 n C 1 nK1 X N rZ1 c ðrÞ v ðrÞ S ðrÞ
v ðrÞ ðS 0 Þ Z sup s2P ðrÞ 1 2 s$S 0 s ; ð4: 17Þ
where P (r) is the strength domain of phase r, defined by the conditions u (r) (s)Z0 if s2P (r) , and N otherwise. Note that P (r) is the closed, convex set of stress tensors bounded by the yield surface of phase r. The advantage of rewriting the functions v (r) in this way is that its computation reduces to the well-studied problem of finding the maximum of a convex function relative to a convex set. Indeed, it can be shown (see corollary 32.3.2 in Rockafellar 1970) that the maximum in (4.17) is attained at one or more of the extreme points of the set P (r) , which means that the optimal ŝðrÞ ðmÞ are necessarily on the yield surface of phase r. Furthermore, in many cases, such as that of crystalline phases considered in part II of this work, the number of extreme points of P (r) is finite, and so the function v (r) is the total number of extreme points of P (r) , and s ðrÞ ðkÞ denotes the stress vector associated with the kth extreme point. For a given S 0 , this expression is very simple to evaluate. Finally, considering the limit as n/N in expression (4.15) for ũK , the following result is obtained for the effective strength domain P of the ideally plastic composite, which is defined by the conditions ũð sÞZ 0 if s 2 P, and N otherwise.

Concluding remarks

In §4, bounds have been derived for nonlinear composites with anisotropic phases, including special forms for power-law and ideally plastic composites, making use of the variational representations derived in §3. As already noted (remark 3.4), alternative forms for the variational representations have been proposed by Ponte Castan ˜eda & Suquet (1998). These alternative representations for w and ũ can be used to generate bounds that exhibit precisely the same forms as bounds (4.2) and (4.9) for general square concave and square convex phase potentials w (r) and u (r) , respectively, except that the functions v (r) , as defined by relations (2.2) (or (2.24)), must be replaced by the functions Kf ðrÞ Ã , as defined in expression (2.11). Similarly, special forms of the bounds may be derived for power-law and ideally plastic composites that correspond exactly to the forms (4.14), (4.15) and (4.19), but with the same caveat for the functions v (r) . Now, it follows trivially from inequalities (2.15), v ðrÞ %Kf ðrÞ Ã , that the bounds proposed in this work are at least as good as the corresponding bounds given by Ponte Castan ˜eda & [START_REF] Ponte Castan ˜eda | Nonlinear composites[END_REF]. The question of interest, however, is whether the bounds given in this work are actually better, in general, as suggested by the above-mentioned inequality. This question will be partially answered in part II of this work in the context of composites with crystalline phases. In fact, for the specific class of crystalline materials, the bounds of Ponte Castan ˜eda & Suquet (1998) have already been shown to recover exactly (for some appropriate choice of the 'extension' functions f (r) ) the bounds of deBotton & Ponte Castan ˜eda (1995). In part II, it will be shown that the new bounds derived in this work are sharper, in general, than the bounds of deBotton & Ponte Castan ˜eda (1995) for crystalline materials. This suggests that the new bounds may also be sharper, in general, than the bounds of Ponte Castan ˜eda & [START_REF] Ponte Castan ˜eda | Nonlinear composites[END_REF].

It should be emphasized, however, that the earlier bounds have a significant advantage over the new bounds proposed in this work in terms of computational efficiency. Indeed, the error functions v (r) needed in the computation of the new bounds require solving a non-concave (non-convex) problem, while the corresponding functions Kf ðrÞ Ã in the earlier bounds require the solution of a concave problem (in an enlarged space). This additional structure, which makes the problem for the bounds concave in one set of variables and convex in the other, allows the use of the saddle point theorem to simplify the computation of the earlier bounds. In particular, the form of the bounds corresponding to expression (4.7) depends directly on the second moments of the strain in the linear comparison composite. (On the other hand, the earlier bounds could depend on the concave extension f (r) of the phase potentials w (r) , while the new ones, given in this work, depend only on the actual potentials w (r) .) Thus, in conclusion, the new bounds developed in this work can give improved results, relative to the earlier bounds, but any improvement will come at an increased computational cost.

where

  VðC 0 ÞZ hvðx; C 0 ðxÞi, with v given by (2.10) and w0 denotes the effective potential of the linear comparison material with local potential (2.1), i.e. w0 ð 3Þ Z inf 32Kð 3Þ hw 0 ðx; 3Þi: ð3:2Þ Proof. Lemma 2.1 is used together with expression (1.2) to deduce the following result for the effective strain-energy density: wð 3Þ Z inf 32Kð 3Þ inf C 0 ðxÞO0 fhw 0 ðx; 3Þi C hvðx; C 0 ðxÞig; ð3:3Þ where the infimum over C 0 has been taken out of the volume integral implied by the triangular brackets. (Note that C 0 is now a function of position x.) But the order of infima can be interchanged, and hence wð 3Þ Z inf C 0 ðxÞO0 inf 32Kð 3Þ fhw 0 ðx; 3Þi C hvðx; C 0 ðxÞig: ð3:4Þ

  It can be shown directly that the two versions of the variational representations, (3.1) and (

  to be constant. Making use of this trial field in the variational statement (3.1) for w then leads to the following bound, which is a generalization for composites with anisotropic phases of bounds that were first given for composites with isotropic phases by Ponte Castan ˜eda(1991).

  Result 4.1. The effective strain potential w of the nonlinear composite is bounded above by wC ð 3Þ Z inf

  use of the following identity for linear composites (e.g.[START_REF] Parton | Stress fluctuations in elastic composites[END_REF] h3 L 53 L i ðrÞ Z 1 is the local strain field in the LCC, as well as of expression (2.19) for the subdifferential of v (r) , the optimality conditions (4.4) for C ðrÞ 0 may also be expressed as follows: do there exist an integer M (r) and constants a ðrÞ m (mZ1, ., M(r)

Result 4. 4 .

 4 The effective stress potential ũ of the nonlinear composite is bounded below by ũK ð sÞ Z sup S

  effective stress potential of a linear comparison composite (LCC) with uniform compliance tensors S ðrÞ 0 in each of the phases (rZ1, ., N ) and the effective compliance tensor S0 . Moreover, the following estimate is obtained for the effective stress-strain relation of the nonlinear composite 3

Result 4. 5 .

 5 The effective strain potential w for power-law composites is bounded

  Special results for rigid-ideally plastic materials

Result 4. 7 .

 7 The effective strength domain P of the ideally plastic composite is bounded from the outside by Note that this set corresponds to the intersection of all quadratic functions of the average stress s, which are defined by the conditions ũ0 ð sÞ% P N rZ1 c ðrÞ v ðrÞ ðS ðrÞ 0 Þ, for each possible choice of the compliance tensors S ðrÞ 0 R 0 of the LCC.

  , u ðrÞ ðsÞ Z ðw ðrÞ Þ Ã ðsÞ Z sup

									h	s$3Kw ðrÞ ð3Þ	i :	ð1:4Þ
								3
	Then, the local stress and strain tensors are related by
				3 Z	vu vs	ðx; sÞ;	uðx; sÞ Z	rZ1 X N	c ðrÞ ðxÞu ðrÞ ðsÞ;	ð1:5Þ
	and the effective behaviour can be described in terms of the effective stress
	potential ũ, such that
	3 Z	vũ v s	ð	sÞ;	ũð	sÞ Z inf s2Sð sÞ huðx; sÞi Z inf s2Sð sÞ	rZ1 X N	c ðrÞ hu ðrÞ ðsÞi ðrÞ ;	ð1:6Þ

  .6) 2 admits the alternative representation

	ũð	sÞ Z sup	f ũ 0 ð	sÞKVðS 0 Þg;	ð3:5Þ
			S 0 ðxÞR0
		ũ0 ð sÞ Z inf s2Sð sÞ hu 0 ðx; sÞi:	ð3:6Þ
	Proof. Lemma 2.2 is used together with expression (1.6) 2 to deduce the
	following result for the effective stress potential:
	ũð sÞ Z inf s2Sð sÞ	S 0 ðxÞR0 sup fhu 0 ðx; sÞiKhvðx; S 0 ðxÞig;	ð3:7Þ

where VðS 0 ÞZ hvðx; S 0 ðxÞi, with v given by (2.27) and ũ0 denotes the effective potential of the linear comparison material with local potential (2.26), i.e.

  can be written asv ðrÞ ðS 0 Þ Z max

	kZ1;.;K e ðrÞ	1 2	s	ðrÞ ðkÞ $S 0 s	ðrÞ ðkÞ	;	ð4:18Þ
	where K e ðrÞ

This paper is based upon the work supported by the National Science Foundation under grants CMS-02-01454 and OISE-02-31867.