Shivendra Kumar

Fabrice Mathieux
email: fabrice.mathieux@g-scop.inpg.fr

Godfrey Onwubolu
email: onwubolu_g@usp.ac.fj

Vineet Chandra
email: v_v_chandra@yahoo.com

A novel powder metallurgy-based method for the recycling of aluminum adapted to a small island developing state in the Pacific

Keywords: End-of-life products management, Aluminum, powder metallurgy, recycling technology, solid waste, SIDS (Small Island Developing States)

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A novel powder metallurgy-based method for the recycling of aluminum adapted to a small island developing state in the Pacific.

Introduction: Sustainable Development and SIDS

Sustainable Development requires that "special attention [is paid] to the developmental needs of Small Island Developing States (SIDS) and the Least Developed Countries" [START_REF]The Johannesburg Declaration on Sustainable Development[END_REF].

Indeed, SIDS from the Caribbean, the Indian Ocean and the Pacific are small, remote, in-development, rich with traditional knowledge and made up of unique and fragile ecosystems. Therefore, SIDS are especially vulnerable to current development trends, from economical, social and ecological perspectives [START_REF]Ministerial SIDS Inter-regional Preparatory Meeting[END_REF][START_REF] Briguglio | Small island developing states and their economic vulnerabilities[END_REF].

For Pacific SIDS, many issues are extremely important for their sustainable development, like e.g. global warming, capacity building, water and oceans protection, but also waste management [START_REF] Usp | USP Team report on the World Summit on Sustainable Development[END_REF]. Indeed, SIDS face new challenges with solid waste, especially in urban centers, as both space for landfills and the recycling activity are extremely limited.

Another important issue for Pacific SIDS concern Science and Technology [START_REF] Usp | USP Team report on the World Summit on Sustainable Development[END_REF]. Science and Technology (S&T) has enormous potential to contribute to sustainable development.

In particular, governments should give higher priority to investments in S&T to find solutions for pressing environmental and developmental challenges, and should build up a threshold S&T capacity. Also, integration of indigenous knowledge, emphasis on local, culturally appropriate and low-cost technologies must be supported.

This paper focuses on solid waste management and development of science and technology in Pacific SIDS. It reports recent research aiming at developing adapted and low-cost technologies for recycling and/or adding value to scrap aluminium. First, some background of solid waste management in Fiji is given. Then, current local casting process for recycling aluminium is described and discussed. The third part report potentialities of the alternative powder technology and experimental procedures to test its benefits if introduced in the local industry. Initial results of the experiments are presented and discussed in the fourth part and the paper sums up with conclusions and perspectives.

Background of solid waste management in SIDS and in Fiji

Current situation

The management of solid waste is recognized as "a critical issue for Small Island Developing States" in the Pacific [START_REF]UN, Programme of Actions for Small Island States[END_REF]. This is in particular true in the Fiji Islands, where the quantity of waste produced by individuals has been continuously increasing during the last 50 years [START_REF]Naboro Landfill -EIS and engineering studies[END_REF]. With almost 1 kg of domestic solid waste produced per day and per capita in the capital Suva, the waste production in Fiji will soon become as large as in developed countries.

With economic development and modernization, the composition of solid waste has been also strongly evolving in SIDS: a few decades ago, solid waste was almost exclusively organic; today, it contains more and more industrial products and materials, i.e. glass, metals, plastics and hazardous substances [START_REF]Guidelines for municipal solid waste management planning in Small Islands Developing States in the Pacific Region[END_REF]. Scrap metals represent as much as 3% in mass of the solid waste produced in Suva and disposed of in the landfill [START_REF]Naboro Landfill -EIS and engineering studies[END_REF].

In front of this problem, many donor agencies and the government have been favoring

final disposal facilities, when building brand new sanitary landfills, or upgrading former rubbish dumps. However, these end-of-pipe solutions have limited lifetimes, and due to limited land capacity in the Pacific, SIDS will again face problems in the future to find lands for landfills.

The environmental dilemmas experienced by SIDS from the Caribbean, the Indian Ocean and the Pacific associated with their economic development are starting to be reported in scientific publications. A recent paper deals for example with environmental impacts of intensive agriculture in the Kingdom of Tonga in the Pacific: [START_REF] Van Der Veldea | Sustainable development in small island developing states: next term Agricultural intensification, economic development, and freshwater resources management on the coral atoll of Tongatapu[END_REF]: benefits of the increase of export of squash are unfortunately balanced by higher costs for importing chemical fertilizers and by large impacts on ecosystems. This example illustrates well the reliance of SIDS upon developed countries as well as the fragility of their ecosystems.

Another publication reviews the potential of renewable energy systems for SIDS, in particular for Mauritius, in terms of reduction of transportation and distribution costs, of security of energy supply [START_REF] Weisser | Costing electricity supply scenarios: A case study of promoting renewable energy technologies on Rodriguez, Mauritius[END_REF]. However some typical SIDS barriers such as limited institutional capacity restrain the development of these alternative energies. Another recent paper reports the impact of the economic development, in particular with tourism, on fisheries biodiversity in Mauritius [START_REF] Sobhee | Fisheries biodiversity conservation and sustainable tourism in Mauritius[END_REF]. Unfortunately, very few publications report researches on integrated solid waste and industrial resources management in SIDS, in relation to its economic development: it was only possible to find a paper on the application of the industrial ecology concept to a closed and bounded system like one

Caribbean SIDS [START_REF] Deschenes | An Island approach to Industrial Ecology: Towards sustainability in the island context[END_REF]. The most comprehensive appraisal of solid waste management in SIDS are nevertheless covered by United Nations publications, such as [START_REF]Proceedings of Waste Management Expert Meeting in SIDS[END_REF], or by reports written by local stakeholders.

Towards improved system in Fiji

To deal with the waste problem in Fiji efficiently, more preventative approaches should actually be developed. This is actually the main objective of the Integrated Solid Waste Management Project that was initiated at the University of the South Pacific in 2003 [START_REF]PACE-SD/USP Homepage. Integrated solid waste management project[END_REF].

In particular, recoverable solid waste should be diverted from the flow that is currently buried. A few experiences in Fiji concentrate on the collection and sometimes on the recovery of solid waste. The following experiences have been identified:

• One soft drinks producer maintains a take-back system for PET bottles: individuals bring back and sell post-consumer bottles in three collection points in the country;

unfortunately, no information on the collection rate achieved is currently available [START_REF] Mathieux | The value-added of engineering students in the search of adapted solutions to solid waste management in Small Islands Developing States: the example of the Fiji Islands[END_REF],

• Waste oil from automotive and process industries are collected to serve as feed in a steel rolling process [START_REF] Aalbersberg | Assessment of Scrap metals, derelict Vehicles, Batteries and Tires (SVBT) in Fiji[END_REF],

• The University of the South Pacific and a few hotels are currently setting-up composting units for their own organic waste made of food scrap and green waste [START_REF] Mathieux | The value-added of engineering students in the search of adapted solutions to solid waste management in Small Islands Developing States: the example of the Fiji Islands[END_REF]: however, the implementation is long as no adapted process and know-how is already available: as was recently done in an African country such as Tanzania (cf. [START_REF] Mbuligwe | Potential and constraints of composting domestic solid waste in developing countries: findings from a pilot study in Dar es Salaam, Tanzania[END_REF]), key parameters of the composting unit such as carbon-nitrogen (C/N) ratio of raw organic waste, the variation of temperature, pH and moisture content will have to be determined for optimal processing, • Four scrap metals dealers collect scrap metals such as aluminum, copper, lead and ferrous metals from companies and mainly from individuals, prepare them and ship them abroad (mainly Australia, New Zealand and Asia) for recycling [START_REF] Mathieux | Solid waste management and recycling in The Fiji Islands. Preliminary analysis of the situation in the Western region of Fiji and the handling of some difficult solid waste[END_REF],

• One company collects scrap metals (steel, copper and aluminium) from individuals and garages and recycle them through casting to produce parts to be used by the local market [START_REF] Mathieux | Solid waste management and recycling in The Fiji Islands. Preliminary analysis of the situation in the Western region of Fiji and the handling of some difficult solid waste[END_REF].

Although very valuable, theses initiatives are only economically-driven and the scrap without sufficient value is still landfilled. Therefore, the collected flows remain very limited. Also, this has got impact on the Fiji economy as Fiji imports high value of finished products (i.e. end of life machinery such as tractors and cars, construction materials and consumer goods) and exports low value scraps contained in these products.

As shown in [START_REF] Van Der Veldea | Sustainable development in small island developing states: next term Agricultural intensification, economic development, and freshwater resources management on the coral atoll of Tongatapu[END_REF] for the Kingdom of Tonga, Fiji is a typical SIDS for which imports are usually larger than exports, which result in a strongly negative trade balance. The country could therefore benefit further if processes aiming at recycling or adding value to scrap metal are developed and adopted locally. In addition to more jobs being created, the low cost of recycled metal could assist the cost of some metallic items manufactured locally to drop, or the price of exported scrap materials to increase.

Recycling and adding value to solid waste is indeed an important strategy for Fiji to avoid environmental as well as economical impacts of solid waste. However, due to the small size of the economy, the quantity of waste produced in Fiji, although growing, remains limited and does not justify the implementation of large scale conventional recovery plants. There is therefore a need to develop adapted technologies and systems to recycle and add value to solid waste.

Current research on scrap metals

This paper reports research aiming at encouraging and facilitating the recycling in Fiji of a valuable scrap metal like aluminium. An integrated approach has been developed and the research focused on various issues such as: evaluation of quantities and of sources of scrap aluminium; improvement of the scrap collection and preparation; development of a new recycling technology; marketing of the produced material. The results presented in the paper concern all these issues but will mainly focus on the development of a novel recycling technology. In particular, the methods of casting and powder metallurgy for recycling aluminum will be discussed. The method of casting is currently used by a local company while the proposed alternative, the powder metallurgy process is still in the process of being introduced to local industry. The quantity of scrap metal discarded by industries and households in the Fiji Islands are ever increasing but the discarded quantities that are forwarded for recycling is significantly less than that of developed countries. This is firstly because of the relatively smaller population and economic development, and secondly, due to the lack of knowledge of the general population on such issues. This puts pressure on local industries to adopt a process that works best with smaller quantities of scrap aluminum.

3.

Current local process of aluminum recycling and their limitations

Presentation of the process

Casting is a conventional manufacturing process that can utilize scrap aluminum. Casting of scrap materials to new usable products is done in Fiji by a local industry. It is a relatively simple process comprising five generic processes;

1. Pattern making -the pattern is representative of the product to be realized and is made of wood for reusability and dimensional stability. The pattern is divided into two parts so that the shape can be easily removed without damaging the mould.

2. Moulding -the mould is made of quarry sand and this is reused numerous times.

Sand is shoveled around the pattern and rammed using a pneumatic hammer.

3. Melting-the materials to be melt are put in a cupola and heated. These materials are pre-sorted. Since the volume drops with melting, raw material is continually added. To accelerate the process, the molt is regularly stirred. This process lasts for approximately 3 hours.

4. Pouring -the molten liquid is poured in the mold using the two man ladle method. The pouring rate is kept in control so that air inside the mold can easily escape.

5. Machining -once solidified and cool, produced parts that do not have the desired surface and shape are then machined.

A detailed illustration of the energy and material flows in the current casting process is given in Figure 1. Further details are given in [START_REF] Kalpakjian | Manufacturing Processes for Engineering Materials[END_REF]. It was not possible to measure accurate flows for the process used at the local company, but this will be done in the future.

Limits of the process

Output quality

While casting provides parts of various sizes and shapes, stringent quality control procedures restrict its usage by industrial consumers. This is because such cast products are a mixture of various grades of aluminum : therefore their mechanical properties are non-standard and possibly inferior when compared to pure aluminum products or its alloys. Hence, the market for such items is limited. Since the final product of casting is sensitive to the process parameters, there is limited flexibility even in the end shapes that can be produced.

Cast products require sophisticated equipment for quality control particularly to check the presence of trapped air inside the cast and to check for any micro-structural deformities. The local industry is not profitable enough to invest into such test equipment. Nevertheless, when needed by the customers, the product samples are sent abroad for internationally accredited testing.

Production planning

Moreover, to ensure profit, a definite quantity of aluminum must be melted in each run.

It also means that there should be enough products ready to be cast. This instantaneous supply and demand balance can sometimes be difficult and expensive to achieve, when considering that the main customer activity (i.e. sugar cane mills) are highly seasonal. In addition, molten aluminum which is surplus and not used in casting, will again have to be molten, resulting in extra consumption of energy.

Skilled employees are needed to carry out the operations and since the operations are sequential, the employees cannot perform more than one task. Casting therefore has a heavy human resource demand and it is something that makes this process weaker than some new alternatives.

Overall impact on resources management

Despite the local firms success, the positive effects of it on overall solid waste management are not felt in the country. It only recycles a fraction of the total aluminum scrap produced in Fiji. This is summarized in Table 1 below. These figures should be compared to the quantity of aluminium that is supposed to be sent to the various sanitary landfills and rubbish dumps of the country. Considering only the capital Suva and six municipal councils of the main island Viti Levu, we estimate that at least 300 tons of scrap aluminum were sent to the landfill in Fiji in 2003 (after [START_REF] Mathieux | Solid waste management and recycling in The Fiji Islands. Preliminary analysis of the situation in the Western region of Fiji and the handling of some difficult solid waste[END_REF]).

End of life products with significantly lower aluminum content, but which are more in quantity, are left out. Some examples of products left out include aluminum cans, appliance covers and scraps from the building and construction industry. Depending on their value and availability, theses products are either:

• Exported for recycling overseas, implying resource loss and limited economic gains for Fiji, or • Landfilled locally.

It was shown in this paragraph that there is indeed a need to develop other processes for recycling aluminium scrap more adapted to the Fiji context. In particular, the process should lead to:

• Higher quality materials produced, that should better answer user's requirements and therefore open up new markets both in Fiji and overseas; • More efficient production planning that should lead to production cost reductions;

• Local recycling of a larger quantity of scrap aluminium, that should eventually reduce the quantity of waste landfilled.

A new recycling process should lead to an improved system that is represented on Fig. 2 with Sankey diagram notations, where in particular scrap would be less accumulated in landfills. proposed. The powder metallurgy process is a combination of sub-processes that include powder formation, blending, compaction, sintering and finishing [START_REF] Kalpakjian | Manufacturing Processes for Engineering Materials[END_REF]. The common powder formation processes use molten material to produce powdered particles.

Amongst various methods of powder production are atomization, reduction, electrolytic deposition, comminution and mechanical alloying. Production of uniformly sized powdered particles are important. The blending stage involves the mixing of various sizes and shapes of the produced powder particles. The ratio of added particles in the mixture depends on the mechanical and physical requirements of the end product. Mechanical and physical properties such as hardness and toughness will determine how the product behaves under load. Lubricants and binders are also added at this stage to improve the flow of particles in the die and hold the particles together after sintering respectively. In the third step, compaction, blended powders are pressed into desired shapes in dies using either mechanical or hydraulic presses. The produced part is referred to as the green compact. Sintering is the process where the green compact is heated in a controlled atmosphere furnace to a temperature below its melting point, but high enough to allow the fusion of individual particles. They are several finishing processes that are dependent on the end use of the produced part. Materials commonly used for this process include Aluminum, Titanium, Copper and some Nickel based alloys [START_REF] Kalpakjian | Manufacturing Processes for Engineering Materials[END_REF][START_REF] Lee | Defect enhanced solid-state reaction behavior of shock modified Ti+C powder mixture compacts[END_REF][START_REF] Li | Liquid Phase Sintering of Metal Matrix Composites[END_REF][START_REF] Matsura | Microstructure and Mechanical properties of NiAl intermetallic compound synthesized by reactive sintering under pressure[END_REF][START_REF] Rossi | A model for the simulation of powder compaction processes[END_REF][START_REF] Satoh | High temperature deformation behavior of aluminum alloys produced form centrifugally-atomized powders[END_REF][START_REF] Smith | Simulation of metal powder compaction, for the development of a knowledge based metallurgy process advisor[END_REF].

Figure 3 illustrates the powder metallurgy process in detail. On a general view, there is twice the number of major processes as in the casting method. Of these, the first two are the same, except that the chemicals are not used in melting. For the current experimental process, the energy consumption is approximately the same as with the casting method.

However, the energy consumption should be lower if economies of scale are achieved.

Exact figures on flows crossing the powder metallurgy process are today not available but will be explored in the future. These figures might allow in the future a comparison of the environmental performances of the two processes according to various criteria, not only on the quantity of waste treated locally.

Figure 3 The proposed powder metallurgy process

Previous Research in Powder Metallurgy

Various researchers have in the recent past looked at distinct aspects of the process.

Satoh et al researched on the production of powder of an alloy of aluminum and silicon using the centrifugal atomization process which is a process in which the molten metal is poured over a rotating plate in a vacuum chamber [START_REF] Satoh | High temperature deformation behavior of aluminum alloys produced form centrifugally-atomized powders[END_REF]. The rotating plate breaks the flow stream sending powdery particles around its periphery. While traditionally presses have been used for compaction, Lee and Thadhani have experimented with shock compression techniques [START_REF] Lee | Defect enhanced solid-state reaction behavior of shock modified Ti+C powder mixture compacts[END_REF]. Results indicate superior packing densities. Li et al proposed the liquid phase sintering method to sinter metal matrix composites [START_REF] Li | Liquid Phase Sintering of Metal Matrix Composites[END_REF]. Their method deviates from the traditional practice of leaving the material in solid phase during sintering by allowing the green compact to transform to partial liquid form. Traditional methods have nevertheless been in the spotlight. Researchers in Japan have looked at reactive sintering under pressure [START_REF] Matsura | Microstructure and Mechanical properties of NiAl intermetallic compound synthesized by reactive sintering under pressure[END_REF]. All the methods have results suitable for specific applications but need tailor made equipment and are capitally intensive.

Moreover, Smith et al simulated the powder compaction process to allow researchers to carry out analysis on the combination of input parameters to achieve the desired product characteristics [START_REF] Smith | Simulation of metal powder compaction, for the development of a knowledge based metallurgy process advisor[END_REF].

Apart from Hywon et al [START_REF] Hywon | Manufacturing of aluminum flake powder from foil scrap by dry ball milling process[END_REF] who reported on a feasibility study for producing aluminum powder from aluminum foil scrap by dry ball milling using argon and oxygen, there is no literature available on the use of the powder metallurgy process for the recycling of aluminum in both developed and developing countries. This paper therefore communicates a new and innovative process for the recycling of scrap aluminum.

Merits of the Powder Metallurgy for applications in SIDS

The powder metallurgy process is cheaper than casting in terms of the resources needed.

Other researchers like [START_REF] Rossi | A model for the simulation of powder compaction processes[END_REF] have earlier mentioned that the powder metallurgy process is economical. [START_REF] Rossi | A model for the simulation of powder compaction processes[END_REF] mentions that the powder metallurgy process allows the manufacture of near net shape parts which have complex geometries. Moreover, the powder metallurgy process can recycle lightweight aluminum and can even be designed to look at low volumes of scraps. Since the sub-processes involved are independent, the process is highly flexible and can be decentralized using different facilities at different locations.

Many other advantages of the powder technology are foreseen in Fiji. Two of them are listed below:

Reduction at source and internal recycling

Instead of machine dust (as in casting), this process produces very fine dust during the atomization process and this cannot be contained. The quantity of this is however, negligible. Moreover, over-flown aluminum powder in the compaction process can be used directly in the next compaction process without incurring any additional energy expense. In the case of over-flown aluminum in casting, it can re-enter the process only if it is remelted.

Higher flexibility for the production

Another advantage of this method, is that the produced powder can be stored for later use. Comparatively, the material in casting has to be used right away. This usually translates to sufficient moulds being ready in time to utilize the entire produced molt. In the powder metallurgy case, the powder production phase can happen without the mold being ready. This also allows flexibility in the collection of scrap as the frequency of collection can be lower than casting and can also open up local or international market for aluminum powder.

Experimental procedure

In order to validate its adaptability in Fiji, some experiments were carried out. These experiments aimed at determining the properties of the produced parts and also providing ideas on how to improve the process. Furthermore, experiments were needed to verify if scrap aluminum from various sources could be useful as input of the recycling process.

In the reported experiment, the raw material was scrap aluminum from the construction industry and was classified into various categories before melting in the 1032 o C furnace at 693 o C. The composition and behavior of construction aluminum is standardized and hence is the most appropriate for comparison. The average volume of the scrap pieces were 0.001m 3 and were collected from the waste disposal site of a construction firm. The pieces were intended for use as aluminum window frames. The temperature during melting was constantly monitored using a Lutron digital thermometer. The temperature of the aluminum was kept higher than the nominal melting temperature of aluminum (660 o C) to prevent it from solidifying when poured from the heating cupola.

A two men ladle method was used to pour the liquid aluminum from the cupola to a sprue. The steel sprue was preheated to prevent it from causing the aluminum to solidify.

A 10mm hole in sprue provided an exit for the melt. Compressed air at over 400kPa was then applied perpendicular to the direction of flow of the melt. It caused the melt to be blown away from the sprue in forming fine particles.

Formed powder was collected in a 3 m long collection chamber. Various sizes of powder were seen and these were then sieved using sieves of 9.5 mm, 2.5 mm, 500 μm and 250 μm aperture. The 9.5 mm sieve was used first and the 250 μm the last. To bind and lubricate the powder, 3% volume ethanol and 1% mass zinc stearate were used. These were then mixed in a glass container.

To make analysis easier, a cylindrical die and punch were prepared. It had a diameter of 21 mm. The powder, mixed with the binder and lubricant was then compressed using the die and punch fitted to a hydraulic press at 3.5MPa. After removal from the die, the shaped compacts were sintered at 585 o C in atmospheric pressure.

Before the surface analysis, the shaped compacts were polished and then etched with a solution containing 2% potassium dichromate, 5% nitric acid and 1% hydrochloric acid.

In addition, the hardness tests were carried out using a 11 kg load on a 0.59 mm indentor. The hardness number was verified using a Hardmatic digital hardness tester.

Table 2 summarizes the process parameters used.

Results of the experiments

Basic experiments to characterize the produced parts were carried out at the university metallurgy laboratory. Although not exhaustive, these tests give a fair idea of the products mechanical and physical properties. Moreover, most of these tests are simple and use basic equipment and can therefore be easily re-produced by local industries.

Density vs pressure

The first experiment focused on compaction with respect to applied pressure on the powder during the compaction process. Compaction was measured via the density of the produced compact before sintering. Figure 4 illustrates the logarithmic increase in compaction as the pressure is increased. The high pressure used to achieve required densities is available through hydraulic presses, however, it can also be achieved easily through impact drivers. Densities of the products will be dependent on the application it is destined for.

Compression vs grain size

The grain size can also affect the compression. As Figure 5 illustrates, the compaction percentage increases with grain size. This is largely attributed to the intra-grain gaps mainly found in compacts of large grain size which reduce significantly during compression as the size of the gaps reduce. Smaller grain sizes have smaller gaps which reduce only by a smaller margin. the grain size is larger, higher compression will be required.

Hardness

The Brinell hardness number (BHN) of the new product in different grain sizes was compared to that of the raw scrap aluminum which has a hardness of HB 51. The results are summarized by Table 3. The BHN of scrap aluminum is higher than pure aluminum as the scrap aluminum is not pure aluminum but aluminum alloys. From the above results, the hardness number decreases with decreasing grain size. Larger grains have high internal hardness and thus are much more resistant to wear. However, the produced part compares very well with the original hardness number of the material.

A cast sample has a BHN of 76. This however can vary depending on the cooling method used on the fresh cast. Quenched cast samples, as in this one, have high hardness. The BHN demand from customers will vary on application.

Conclusions from the experiments

The results of the experiments give an overview of the characteristics of the part produced. The density and hardness numbers compare well with the raw scrap material and suggest that it could be used for similar applications. However, due to the different nature of the two processes, the grain sizes cannot be compared. The density indicates that the part maintains the primary characteristic of aluminum being a relatively light material and promotes its use in such applications. Moreover, the hardness numbers are indicative of its mediocre resistance to wear compared to the quenched cast products.

Despite slight inferior mechanical characteristics of produced parts, the novel process seems worth being developed considering its numerous advantages at the production plant.

Discussion

Possible industrial implementation of the process

As mentioned in Section 1, high capital costs have prevented the establishment of a aluminum recycling plant in Fiji. Moreover, despite having large quantities of scrap aluminum, the quantity is not sufficient for the long term sustainability of a conventional aluminum recycling plant. However, we showed earlier that a powder metallurgy-based process could be set-up at an industrial scale. The key question is now to define which of the two following models should be adopted.

1. Only half of the entire process is invested. This would include the sorting, melting and atomization processes. The powder could then be sold or exported to firms for making numerous end products. This would add value to the scrap and, with higher density of the shipment, this could decrease the cost of trucks and boat transport of the processed scrap, and therefore bring obvious benefits to the Fiji economy. The savings could be used to implement more efficient scrap aluminum collection methods. Such a process could be implemented in several islands in Fiji.

2. The entire process plant is set-up. Since there will not be enough aluminum to keep the plant running daily, days could be dedicated to individual processes. For example, sorting could happen on day 1, and the rest of the processes could happen on consecutive days. A day also could also be dedicated to the collection of scrap aluminum. The plant could be run daily if Fiji became a transfer platform for all scrap coming from other SIDS in the Pacific where value would be added to scrap.

The choice between the two models should be done in the future by investors and the government according to several criteria such as: investment and maintenance costs, annual economic profitability, number of job created, recycling rate of scrap aluminium, and overall environmental performances.

6.2. Foreseen benefits for the overall solid waste management in Fiji Both models cannot be used for the conventional process of casting as the processes in casting are strongly bounded by the material flow time. The new method can easily be adopted by local industries. Even firms already involved in casting can diversify into this new process. Approximately 10 kg aluminum is needed for a single process. This equates to 0.004 cubic meters of recycled aluminum that can be produced from it (given that maximum density is equal to 2.5g/cm 3 ). With only one production line, running the process twice daily, approximately 6000 kg per year of useless aluminum could be diverted from the landfill and could find its place in the market again. The setting-up of several dozen of lines in the country would bring not only space economy in landfills, but also added value to the Fiji economy and create job.

7.

Conclusion and perspectives

A novel method for aluminum recycling in the Fiji Islands has been proposed after studying a conventional recycling method, casting, which cannot easily recycle light and low volume end of life aluminum products. The preliminary results indicate the viability of this process in Fiji and characteristics of the products further strengthen the point.

Higher flexibility and waste reduction at source makes the process competitive as other processes.

Future work includes:

• further experiments on the powder metallurgy process, in particular testing the influence of several types of scrap on the process performances and on the output quality, • a comprehensive materials flow analysis of aluminum in the country to identify future sources of inputs for the process,

• a material, energy and labor balance of the two concurrent processes, to identify environmental and social advantages and drawbacks and

• the extension of the process to cater for other scrap metals such as steel.

8.

Figure 1

 1 Figure 1 The sand casting process used at a local industry

Figure 2 . 4 .of a novel process to recycle aluminum based on powder metallurgy 4 . 1 .

 2441 Figure 2. (a) Current and (b) future improved system for the management of the aluminium resources within the Fiji Islands system.

Figure 4

 4 Figure 4 Graph of Density vs Pressure for green compacts

Figure 5

 5 Figure 5 Relationship of percentage compression and grain size

Table 1 .

 1 Estimated quantity of scrap aluminium collected and exported for recycling compared to the quantity recycled locally in 2003 (after[START_REF]Guidelines for municipal solid waste management planning in Small Islands Developing States in the Pacific Region[END_REF]).

	Estimated quantity of aluminium	Estimated quantity of aluminium
	collected and shipped abroad for	collected and recycled by casting

Table 2

 2 Process parameters and characteristics

	Parameter			Characteristics
	Furnace Temperature		1032 o C
	Melting Period			3-4 hours
	Heating	Temperature	(Melting	700 o C (660 o C)
	Temperature of Aluminum)
	Sprue			Preheated to 460 o C
	Average Air Pressure		400kPA
	Binders and Lubricants		1% mass Zinc Stearate; 3% volume
				Ethanol
	Die Diameter			21 mm
	Compacting Pressure		35MPa
	Sintering Temperature (Duration)		585 o C (30 mins)
	Polishing Solution (Surface)		Aluminum Oxide (Emery Paper P400)
	Etchant			2% Potassium Dichromate, 5% Nitric
				Acid, 1% Hydrochloric Acid

Table 3

 3 Comparison of Brinell hardness number

	Grain Size	BHN
	9.5 mm	45

Acknowledgements

The authors acknowledge the financial assistance provided towards this project by the University of the South Pacific Research Committee (South Pacific), Fonds Pacifique and Champagne Ardennes Regional Government (France). We also thank all the students who have been involved so far in the Integrated Solid Waste Management project.


Marti ´n

I Idiart

Pedro Ponte
email: ponte@seas.upenn.edu

Castan ˜eda

Field statistics in nonlinear composites. II. Applications

Keywords: nonlinear homogenization, variational methods, field fluctuations

Part I of this work provided a methodology for extracting the statistics of the local fields in nonlinear composites, from the effective potential of suitably perturbed composites. In particular, exact relations were given for the first and even moments of the fields in each constituent phase. In this part, use is made of these exact relations in the context of the 'variational', 'tangent second-order' and 'second-order' nonlinear homogenization methods to generate estimates for the phase averages and second moments of the fields for two-phase, power-law composites with isotropic and transversely isotropic microstructures. The accuracy of these estimates is assessed by confronting them against corresponding exact results for sequentially laminated composites. Among the nonlinear homogenization estimates considered in this work, the second-order estimates are found to be, in general, the most accurate, especially for large heterogeneity contrast and nonlinearity. Thus, these estimates are able to capture, for example, the strong anisotropy in the strain fluctuations that can develop inside nonlinear porous and rigidly reinforced composites.

Introduction

In part I of this work (Idiart & Ponte Castan ˜eda in press), we presented a means for extracting the statistics of the local fields in nonlinear composites, from the effective potential of suitably perturbed composites. More precisely, the idea is to perturb the local potentials by adding a term that contains a parameter, generally a tensor, such that the derivative of the perturbed effective potential with respect to that parameter, evaluated at the parameter equal to zero, yields the average of the desired quantity in the unperturbed problem. The usefulness of the resulting relations is that they allow the determination of rigorous homogenization estimates for the field statistics, from corresponding estimates for the effective potentials. Such relations were used in part I in the context of the so-called variational, tangent second-order and second-order nonlinear homogenization methods [START_REF] Ponte Castan ˜eda | The effective mechanical properties of nonlinear isotropic composites[END_REF][START_REF] Ponte Castan ˜eda | Exact second-order estimates for the effective mechanical properties of nonlinear composite materials[END_REF], 2002a) to generate estimates for the phase averages and second moments of the local fields. It is recalled that these methods are based on suitably designed variational principles making use of 'optimally' chosen, fictitious linear comparison composites (LCCs), which can be homogenized using any appropriate linear homogenization estimate, so that corresponding estimates are then generated for the effective behaviour of the nonlinear composites.

Recently, several works have been concerned with the use of these nonlinear homogenization methods to estimate the field fluctuations, i.e. higher-order field statistics, in nonlinear random composites. Following Ponte Castan ˜eda (2002b), [START_REF] Idiart | Field fluctuations and macroscopic properties in nonlinear composites[END_REF] first made use of the second-order homogenization method to generate estimates for the field fluctuations in twophase composites with 'particulate' microstructures. Field fluctuations in this class of composites were also studied by [START_REF] Moulinec | Intraphase strain heterogeneity in nonlinear composites: a computational approach[END_REF], 2004), by means of the earlier variational homogenization method and full-field numerical simulations. More recently, Idiart et al. (2006) have provided comparisons between these numerical simulations and the more sophisticated tangent second-order and second-order methods. Similar studies have also been carried out by Lebensohn et al. (2004a,b) and Liu & Ponte Castan ˜eda (2004a,b) for two-and three-dimensional viscoplastic polycrystals. [START_REF] Brenner | Mechanical field fluctuations in polycrystals estimated by homogenization techniques[END_REF] also studied field statistics in viscoplastic polycrystals by means of the 'affine' method, which is a simplified, but less accurate version of the tangent second-order method, as well as the 'classical secant' approach, which is also known to become rather inaccurate at high nonlinearity. In all these works, the homogenization estimates for the field statistics were obtained by making use of various ad hoc assumptions based on the conjecture that the first and second moments of the local fields in the relevant LCC constitute reasonable approximations for the corresponding nonlinear quantities. However, it has been shown in part I, by means of the rigorous procedure described in the previous paragraph, that this conjecture is valid only in the context of the variational method, but not in the context of the tangent second order and second-order methods, where 'correction' terms arise due to the lack of full stationarity of the relevant functionals with respect to the properties of the LCC. Here, the more consistent homogenization estimates proposed in part I for the field statistics are determined for the specific case of two-phase, power-law random composites with isotropic and transversely isotropic microstructures, and their accuracy is assessed by comparing them with exact results available from the literature [START_REF] Debotton | High-rank nonlinear sequentially laminated composites and their possible tendency towards isotropic behavior[END_REF].

Application to two-phase, power-law composites

In this work, the focus will be on two-phase composites with random particulate microstructures, with clearly defined (1) 'matrix' and (2) 'inclusion' phases. Both phases are assumed to be isotropic, incompressible materials characterized by power-law potentials of the form where mZ1/n is the strain-rate sensitivity, such that 0%m%1, s ðrÞ 0 is the flow stress of phase r, and 3 0 is a reference strain. Note that the limiting values, mZ1 and mZ0, correspond to linear and rigid-ideally plastic behaviours, respectively. For simplicity, both phases are assumed to have the same exponent m and reference strain 3 0 . Then, from the homogeneity of the local potentials (2.1), it follows that the effective potentials can be written as

f ðrÞ ð3 e Þ Z 3 0 s ðrÞ
W ð 3Þ Z 3 0 s0 1 C m 3 e 3 0 1Cm ; Ũ ð sÞ Z 3 0 s0 1 C n s e s0 1Cn ; ð2:2Þ
where s0 is the effective flow stress of the composite, and 3 e and s e are the equivalent macroscopic strain and stress.

Two different classes of composites are considered in this work. The first one corresponds to fibrous composites with transversely isotropic microstructures, subjected to isochoric in-plane loadings. In this case, the effective flow stress s0 is a function of the strain-rate sensitivity, the heterogeneity contrast and the inclusion concentration. The second one corresponds to isotropic composites, in which case the flow stress s0 exhibits additional dependence on the macroscopic strain invariant q, defined by cosð3qÞZ 4 detð [START_REF] Ponte Castan ˜eda | Exact second-order estimates for the effective mechanical properties of nonlinear composite materials[END_REF].

3 d = 3 e Þ (Ponte
In order to analyse the statistics of the local fields in these composites, it is convenient to identify two 'components' of the deviatoric strain (resp. stress) tensor which represents its projections 'parallel', 3 s (resp. s s ) and 'perpendicular', 3 t (resp. s t ), to the macroscopic strain (resp. stress). These components can be determined (up to a sign) by the two orthogonal fourthorder projection tensors E and F as given by expressions (4.52) of part I with 3 ðrÞ Z 3, through the following relations: 3 2 s Z ð2=3Þ ð3$E3Þ, 3 2 t Z ð2=3Þ ð3$F3Þ, s 2 s Z ð3=2Þ ðs$EsÞ and s 2 t Z ð3=2Þ ðs$FsÞ. They are such that 3 2 e Z 3 2 s C 3 2 t and s 2 e Z s 2 s C s 2 t . In addition, it can be shown that the strain and stress fields are homogeneous of degree one in 3 e and s e , respectively, so that normalized quantities such as ð 3 ðrÞ e = 3 e Þ and ð s ðrÞ e = s e Þ depend only on material and microstructural parameters, and on q.

The nonlinear homogenization methods described in part I require the use of estimates for the effective behaviour of linear elastic and linear thermoelastic composites with phases characterized by a modulus tensor L ðrÞ 0 , a 'thermal stress' t ðrÞ 0 , and a 'specific heat' f ðrÞ 0 (see expression (4.31) in part I). In this work, use is made of the generalized Hashin-Shtrikman (HS) estimates of [START_REF] Willis | Variational and related methods for the overall properties of composites[END_REF]. For two-phase, linear elastic composites, the effective modulus tensor can be written as

L0 Z L ð1Þ 0 A ð1Þ C L ð2Þ
0 A ð2Þ , where A (r) are strain concentration tensors such that c ð1Þ A ð1Þ C c ð2Þ A ð2Þ Z I.

Defining DL 0 Z L ð2Þ
0 KL ð1Þ
0 , the HS estimates for the strain concentration tensor in the inclusion phase A (2) can be written as

A ð2Þ Z I C c ð1Þ P ð1Þ 0 DL 0 h i K1 : ð2:3Þ
In this expression, the microstructural tensor P ð1Þ 0 is given by

P ð1Þ 0 Z 1 4p det Z ð jxjZ1 H ð1Þ ðxÞjZ K1 xj K3 dSðxÞ;
where H ð1Þ ijkl ðxÞZ N ð1Þ ik x j x h j ðijÞðkhÞ , N ð1Þ Z K ð1Þ K1 , K ð1Þ ik Z L ð1Þ 0 ijkh x j x h , and the second-order tensor Z serves to characterize the 'shape' of the assumed 'ellipsoidal' two-point correlation functions, such that ZZdiag(1, 1, 1) and ZZdiag(1, 1, 0) correspond to isotropic and transversely isotropic microstructures, respectively. For two-phase, linear thermoelastic composites, use can be made of the Levin relations [START_REF] Levin | Thermal expansion coefficients of heterogeneous materials[END_REF] to relate the 'thermo-elastic' quantities to the purely 'elastic' ones by means of standard relations, which will not be included here for conciseness.

These linear HS estimates are known to be quite accurate for particulate random systems like the ones of interest in this work, up to moderate concentrations of inclusions.

In order to assess the accuracy of the nonlinear homogenization estimates, exact results have been generated for power-law composites with a special type of particulate microstructures called multiple-rank sequential laminates, following the procedure described in deBotton & Hariton (2002). These authors showed that there are two-dimensional lamination sequences for which the macroscopic behaviour tends to be more transversely isotropic as the rank increases. In fact, making use of a differential scheme as described by deBotton (2005), it can be shown analytically that the macroscopic behaviour of these composites does become transversely isotropic in the limit of infinite rank. Similarly, threedimensional lamination sequences can also be found so that in the limit of infinite rank, the macroscopic behaviour becomes isotropic (Idiart in preparation). The interest in composites with this class of (transversely) isotropic microstructures is that, in the linear case, these reproduce exactly the effective behaviour of the above-mentioned HS estimates, for any values of the modulus tensors of the phases [START_REF] Milton | The theory of composites[END_REF]. For this reason, homogenization estimates of the HS type, like the ones considered here for the LCC, are particularly appropriate for nonlinear composites with this class of microstructures, since the effective behaviour of the LCC is being computed exactly in that case, and therefore there is only one level of approximation involved, namely, at the linearization stage. In addition, since this property holds for any type of linearization scheme, this class of nonlinear composites provide an ideal test case to compare LCC-based homogenization methods by making use of different linearization schemes. Furthermore, a peculiarity of this class of nonlinear composites is that, by construction, the local fields in the inclusion phase are uniform, independently of the behaviour of the phases. In this connection, it is relevant to emphasize that the sequentially laminated microstructures are intrinsically different from the 'composite cylinder assemblage' (CCA) microstructures considered by [START_REF] Moulinec | Intraphase strain heterogeneity in nonlinear composites: a computational approach[END_REF], 2004) also for two-phase, power-law composites with transversely isotropic symmetry. While these two very different types of microstructures are found to exhibit very similar in-plane behaviours when the phases are linear, their behaviours become progressively more different as the nonlinearity increases, the difference being most notable for the case of an ideally plastic matrix containing weaker inclusions. In this case, the most striking difference is the fact that the strain fluctuations in the inclusion phase are infinite in CCA composites, but identically zero in sequential laminates. As the HS-type nonlinear homogenization estimates are consistent with zero fluctuations in the inclusion phase, in this work, we have chosen to compare these nonlinear estimates with the corresponding (infinite-rank) laminate results, instead of the CCA results, as in Idiart et al. (2006).

In the sections to follow, comparisons are provided among the exact results for power-law laminates (LAM ), and the variational (VAR), tangent second-order (TSO) and second-order (SO) estimates described in part I, for fibre-reinforced ðs ð2Þ 0 =s ð1Þ 0 Z 5Þ and fibre-weakened ðs ð2Þ 0 =s ð1Þ 0 Z 0:2Þ composites, as well as for isotropic, rigidly reinforced and porous composites. It is recalled that the SO and TSO estimates exhibit a duality gap, and therefore two sets of these estimates corresponding to the strain (W ) and stress (U ) formulations are shown. It should also be mentioned that the SO estimates provided in this work make use of 3 and s as the 'reference' strain and stress tensors (see part I). Finally, the classical bounds of Voigt and Reuss for the effective behaviour are also included for comparison purposes.

Transversely isotropic, fibre-reinforced composites (a ) Effective behaviour

In figure 1, plots are provided for the effective flow stress s0 of a fibrereinforced composite, normalized by the flow stress of the matrix phase s ð1Þ 0 . Figure 1a shows plots as a function of the strain-rate sensitivity m. Several observations are relevant in the context of this figure. First, all homogenization estimates of the HS type coincide for mZ1 with the linear HS estimates and the LAM results, but give different predictions for other values of m. The main observation, however, is that both versions (W and U ) of the SO and TSO estimates are in very good agreement with the exact LAM results, for all values of the strain-rate sensitivity m. In particular, the agreement is excellent for the 2) , for a given strain-rate sensitivity (mZ0.2). Comparisons between the 'second-order' (SO), 'tangent second-order' (TSO) and 'variational' (VAR) estimates of the Hashin-Shtrikman type, and exact results for power-law laminates (LAM ). SO(W ) estimates. In addition, all these estimates are found to tend to the Reuss lower bound as m/0, which seems to be the case for the LAM results as well (for numerical reasons, it was not possible to reach mZ0). The VAR estimates, on the other hand, are seen to overestimate the LAM results for all values of m different from 1, which is not suprising given the fact that the former are known to provide rigorous bounds for the latter. It is also observed that the strain (W ) and stress (U ) versions of the SO and TSO estimates give slightly different predictions, i.e. they exhibit a duality gap, as anticipated. However, this gap can be shown to vanish not only in the linear case, but also in the ideally plastic limit (mZ0). Figure 1b shows plots as a function of the fibre concentration for a moderate nonlinearity (mZ0.2). The main observation is again the good accord of the SO and TSO estimates with the LAM results, even at high-fibre concentrations. It is also interesting to note that the SO(W ) estimates are more accurate than the SO(U ) estimates for low-to-moderate values of c (2) , while the converse is true for high values of c (2) .

0 0.2 0.4 0.6 0.8 1.0 TSO(U) m 0 0.2 0.4 0.6 0.8 1.0 c (2) VAR LAM Reuss TSO(W) SO(W) SO(U ) s 0 (1) s 0 1 2 3 4 5 LAM VAR m = 0.2 Voigt Reuss TSO(U) TSO(W) SO (U) SO(W) s 0 (1) s 0 s 0 (1) (2) = 5 s 0 s 0 (1) (2) c (2) = 0.21 = 5 s 0

(b ) First moments of the local fields

In figure 2, results are given for the corresponding first moments (phase averages) of the local fields. Figure 2a,b shows plots for the equivalent part of the average strains in each phase 3 ðrÞ e , normalized by the equivalent macroscopic strain 3 e . It can be seen that all homogenization estimates are in excellent agreement with the LAM results, for all values of m and c (2) . Thus, it is found in figure 2a that the strain in the fibres decreases with increasing nonlinearity (decreasing m) until it vanishes in the ideally plastic limit, meaning that the fibres behave like rigid inclusions in this strongly nonlinear limit, even though they are not rigid, and all the macroscopic deformation is carried in the matrix phase. Figure 2b shows that, for a moderate nonlinearity (mZ0.2), the strain in the fibres remains very small for most values of c (2) , and hence the average strain in the matrix is 3 ð1Þ e z 3 e =c ð1Þ . The corresponding results for the average stresses in each phase s ðrÞ e are shown in figure 2c,d, normalized by the equivalent macroscopic stress s e . Figure 2c shows plots as a function of m. Once again, it is seen that the SO and TSO estimates are in good agreement with the LAM results for all values of m, while the VAR estimates are rather inaccurate. In particular, the agreement is found to be excellent for the SO(W ) estimates. Thus, these estimates predict a higher average stress in the (stronger) fibres than in the matrix phase, as expected, but as the nonlinearity increases, this difference becomes smaller and finally vanishes as m/0, so that s

ð1Þ e Z s ð2Þ
e Z s e in this limit. This is consistent with the stress field becoming uniform throughout the composite in the ideally plastic limit, which in turn is consistent with the effective behaviour being given by the Reuss lower bound (see figure 1a). Figure 2d shows that the relative merits of the different homogenization estimates for the average stresses in fact change with fibre concentration. Thus, while the SO(W ) estimates are seen to be the most accurate ones for fibre concentrations below 0.5, they deteriorate significantly at higher concentrations. This deterioration can be traced to the use of 3 as the reference strain, which suggests that this choice is not good for very large concentrations. In contrast, the use of s as the reference stress in the context of the SO(U ) estimates is seen to lead to better behaved, albeit sometimes less accurate, estimates for all values of c (2) .

(c ) Second moments of the local fields

Plots for the corresponding second moments of the strain field are given in figure 3,normalized by 3 2 e . Figure 3a shows plots for the second moments of the parallel component of the strain in each phase, as a function of m. It can be seen in this figure that, in the matrix phase, the LAM results increase significantly with increasing nonlinearity, and tend to become unbounded as m/0, unlike the first moments of the strain, which remain finite in this limit (see figure 2a). This implies that the spatial distribution of the strain in the matrix phase becomes significantly more heterogeneous with increasing nonlinearity. On the other hand, figure 3c shows that the LAM results for the second moments of 3 t in the matrix drop to zero as m/0, meaning that in this limit the strain field becomes 'aligned' with the macroscopic strain throughout the composite. In addition, for a finite value of m, the second moments of both components in the matrix are seen in figure 3b,d unbounded as c (2) /1. Both versions of the SO and TSO estimates are found to be consistent with these observations, being in good qualitative agreement with the LAM results for all values of m and c (2) . On the other hand, the VAR estimates are found, once again, to be the least accurate among the nonlinear homogenization estimates. Finally, it is recalled that the local fields in the inclusion phase of the power-law laminates are uniform for any value of the material parameters. Thus, the LAM results in the inclusion phase are such that s

e =s ð2Þ
0 Z ð 3 ð2Þ e =3 0 Þ m , h3 2 s i ð2Þ Z ð 3 ð2Þ
e Þ 2 and h3 2 t i ð2Þ Z 0. In this connection, it is noted that, while the SO and VAR estimates satistfy these relations, thus being consistent with uniform fields in the fibres, the TSO estimates do not satisfy the first two. In fact, the TSO estimates for h3 2 e i ð2Þ are found to be slightly less than those for ð 3 ð2Þ e Þ 2 , and therefore violate the rigorous inequality h3 2 e i ð2Þ R ð 3 ð2Þ e Þ 2 . This inconsistency demonstrates that the TSO estimates are less good than the SO estimates. Plots for the corresponding second moments of the parallel s s and perpendicular s t components of the stress, normalized by s 2 e , are given in figure 4. Figure 4a,c shows plots as a function of m. The LAM results show that the second moments of the stress remain bounded as m/0, unlike those of the strain and seem to be consistent with vanishing fluctuations in the ideally plastic limit. It can be seen that SO and TSO estimates are in good qualitative agreement with the LAM results for all values of m, and predict no stress fluctuations in the ideally plastic limit, which is consistent with the fact that the corresponding estimates for s0 attain the Reuss lower bound in this limit. In particular, the agreement is seen to be excellent for the SO(W ) estimates, at least for this moderate value of fibre concentration (c (2) Z0.21). Figure 4b,d shows that the SO(W ) estimates remain the most accurate among the homogenization estimates up to fairly large fibre concentrations. However, as c (2) becomes larger, the SO(W ) estimates for the second moments of s s in the matrix deteriorate •s 2 (1)

^ • •s 2 (1) ^ • •s 2 (1) || • •s 2 (1) || • •s 2 (2) || • •s 2 (2) || • Figure 4
. Second moments of the strain field in fibre-reinforced composites. 'Parallel' s s and 'perpendicular' s t components, normalized by s 2 e .

significantly, while the SO(U ) estimates remain well behaved, which is consistent with the observations made in the context of figure 2d. Finally, as already noted in the context of figure 3, it should be mentioned that, while the SO and VAR estimates are consistent with uniform fields in the inclusion phase, in agreement with the LAM results, both versions of the TSO estimates for the average and second moments of the stress in the inclusion phase are not so and violate the inequality hs 2 e i ð2Þ R ð s ðrÞ e Þ ð2Þ . Again, this suggests that the TSO estimates should be less reliable, in general, than the SO estimates, as expected. Figure 5a shows plots as a function of the strain-rate sensitivity m. The main observation in the context of this figure is that both versions of the SO estimates are found to be in very good agreement with the LAM results, not only for weak nonlinearities, but more importantly, also for strong nonlinearities. In contrast, the TSO estimates are seen to be in very good agreement with the exact LAM results for weak-to-moderate nonlinearities, but, unlike in the case of stronger fibres, they are seen to deteriorate and deviate significantly from each other (large duality gap) for strong nonlinearities. In this connection, it is observed that, as m/0 the TSO(W ) estimates rapidly decrease to a finite value well 2) , for a given strain-rate sensitivity (mZ0.1). Comparisons between the 'secondorder' (SO), 'tangent second-order' (TSO) and 'variational' (VAR) estimates of the Hashin-Shtrikman type, and exact results for power-law laminates (LAM ). below the LAM results, while the TSO(U ) estimates tend to go to the Voigt upper bound, violating the sharper bound provided by the VAR estimates. Finally, it can be seen in part (b) that, for a strong nonlinearity (mZ0.1), the SO estimates remain the most accurate for all values of the fibre concentration.

(b ) First moments of the local fields

Plots for the corresponding phase averages of the local fields are provided in figure 6. Figure 6a shows the equivalent average strains 3 ðrÞ e in each phase, normalized by 3 e , as a function of m. It can be seen in this figure that the LAM results show an average deformation that is higher in the (weaker) fibres than in the matrix, as expected, and that increases with increasing nonlinearity. The SO(W ) estimates are found to be in excellent agreement with the LAM results for all values of m, while the agreement is less good for the SO(U ) estimates, which exhibit a peculiar, non-monotonic behaviour close to the ideally plastic limit. On the other hand, the TSO estimates are in very good agreement with the LAM results up to moderate nonlinearities, but deteriorate significantly as m/0. In particular, the TSO(U ) estimates give 3 ð1Þ e Z 3 ð2Þ e Z 3 e , which is consistent with the TSO(U ) estimates for s0 being given by the Voigt bound (see figure 5a). Finally, the VAR estimates are seen to capture the right trends, even if they underestimate significantly the deformation of the inclusion phase for strong nonlinearities. Figure 6b shows that the SO(W ) estimates remain the most accurate for all values of the fibre concentration. These estimates agree with the LAM results in that the deformation in the inclusion phase increases with decreasing fibre concentrations, and that this trend is significantly enhanced by nonlinearity.

Figure 6c,d shows plots for the equivalent average stresses s ðrÞ e in each phase. We begin by noting that the LAM results show an average stress that is lower in the (weaker) fibres than in the matrix, as expected, and that decreases with increasing nonlinearity. The main observation in the context of these figures, however, is that all homogenization estimates are in very good agreement with the LAM results, for all values of m and c (2) . This is not surprising, for in the limit of void fibres ðs ð2Þ 0 Z 0Þ all the estimates predict the correct ratios s ðrÞ e = s e . However, it is observed in part (c) that, as m/0, the TSO(W ) estimates exhibit a peculiar behaviour, deviating from the exact LAM results and thus becoming less accurate than the other estimates in this strongly nonlinear limit.

(c ) Second moments of the local fields

Figure 7 shows plots for the second moments of the parallel 3 s and perpendicular 3 t components of the strain in each phase, as a function of m (figure 7a,c) and fibre concentration (figure 7b,d). We begin by noting that the general agreement between the exact LAM results and the different homogenization estimates is worse than that found for the first moments of the strain (see figure 6a,b). Of all the homogenization estimates, the SO(U ) estimates seem to be the most consistent with the LAM results, in general, even though they exhibit the peculiar behaviour for strong nonlinearities already observed in the context of figure 6a. On the other hand, while better for the second moments of the strain in the inclusion phase, in the matrix phase the SO(W ) estimates underestimate considerably the second moments of 3 t . However, both versions of the SO estimates are consistent with uniform fields in the inclusion phase, in agreement with the exact LAM results. In contrast, both versions of the TSO estimates are relatively good for weak-to-moderate nonlinearities, but become meaningless for sufficiently strong nonlinearities, violating once again, the rigorous inequality h3 2 e i ðrÞ R ð 3 ðrÞ e Þ 2 in the inclusion phase (rZ2) and sometimes even in the matrix phase (rZ1). It should be noted that, in the limit m/0, the TSO(U ) estimates for Ũ deviate from the Voigt bound when the perturbation parameters l (r) (see part I) are non-zero, and therefore the TSO(U ) estimates for the second moments of the strain do not agree with uniform strain fields, unlike those for the phase averages.

Figure 8 shows plots for the second moments of the parallel s s and perpendicular s t components of the stress in each phase, as a function of m (figure 8a,c) and fibre concentration (figure 8b,d). Once again, it is noted that the general agreement between the exact LAM results and the different homogenization estimates is worse than that found for the first moments of the stress (see figure 6c andd), even though the stress statistics are less sensitive than the corresponding strain statistics in the case of weaker fibres. Of all homogenization estimates, the SO(W ) seem to do best for the second moments of the stress, as opposed to the SO(U ) estimates for the corresponding strain quantities, the agreement with the LAM results being good for all values of m and c (2) . In contrast, both versions of the TSO estimates are found to be in good agreement with the LAM results for weak-to-moderate nonlinearities, but deteriorate significantly as m/0, where they violate the inequality hs 2 e i ð2Þ R ð s ð2Þ e Þ 2 . In particular, the TSO estimates for the second moments of s t in the matrix phase are seen to blow up in this limit, which is at odds with the exact LAM results.

m = 0.1 m = 0.1 0.2 0.4 0.6 0.8 1.0 m 0.2 0.4 0.6 0.8 1.0 c (2) •e 2 (1) || • •e 2 (1) || • •e 2 (2) || • •e 2 (2) || • •e 2 (1) ^ • •e 2 (1) ^ • Figure 7
. Second moments of the strain field in fibre-weakened composites. 'Parallel' 3 s and 'perpendicular' 3 t components, normalized by

3 2 e .
Noting that the plots for the stress statistics provided in figures 6 and 8 are appropriately normalized by s e , which should be set equal to s0 in the ideally plastic limit for the composite to flow, it is inferred that, as m/0, the exact LAM results for the second moments of the equivalent stress in the matrix phase ffiffiffiffiffiffiffiffiffiffi ffi

hs 2 e i ð1Þ p tend to the flow stress s ð1Þ 0 , while those for s ð1Þ e remain below s ð1Þ 0 . In addition, the LAM results for these quantities in the (weaker) inclusion phase tend to the flow stress s ð2Þ 0 in this limit. This means that in ideally plastic, sequential laminates like those considered here, every point is at yield in both phases. However, in the matrix phase, the 'direction' of the stress tensor varies with position, thus giving rise to intraphase stress fluctuations even though s e Z s ð1Þ 0 everywhere. This is a very special behaviour that would not be expected to happen in more realistic microstructures, as, for example, in those considered by Idiart et al. (2006). It should be emphasized, nevertheless, that, independently of the microstructure, the stress statistics in ideally plastic composites are such that s ðrÞ e % ffiffiffiffiffiffiffiffiffiffi ffi

U W •s 2 (2) || • •s 2 (2) || • •s 2 (1) || • •s 2 (1) || • •s 2 (1) ^ • •s 2 (1) ^ • 0.2 0.4 0.6 0.8 1.0 m 0.2 0.4 0.6 0.8 1.0 c (2)
hs 2 e i ðrÞ p % s ðrÞ 0 , and therefore accurate estimates should satisfy these inequalities. Even though not shown here (see Idiart (in preparation) for details), it is noted that the SO and VAR estimates do satisfy these inequalities in the ideally plastic limit, but the TSO estimates sometimes violate them. In this connection, it is recalled that the so-called affine method, which amounts to obtaining the estimates directly from the LCC of the TSO method and therefore delivers estimates even less accurate than the TSO estimates, is already known to predict second moments of the 'resolved' shear stresses that are larger than the flow stress of the slip system in the context of viscoplastic polycrystals (see [START_REF] Brenner | Mechanical field fluctuations in polycrystals estimated by homogenization techniques[END_REF]).

Isotropic, rigidly reinforced composites

In this section, results are provided for the effective behaviour and associated average stresses in isotropic power-law composites with rigid inclusions, subjected to axisymmetric shear (qZ0) and simple shear (qZp/6) loadings. For the sake of brevity, only one value of the strain-rate sensitivity (mZ0.1) is considered. However, it is worth emphasizing that, while the TSO and SO estimates for simple shear are found to exhibit a dependence on m that is similar to that observed in the context of fibre-reinforced composites under in-plane shear, the corresponding estimates for axisymmetric shear predict a reinforcement effect, as well as inhomogeneous stress fields, even in the ideally plastic limit, which seems to be consistent with the exact LAM results (LAM results could not be obtained for mZ0).

In figure 9, plots are given for the effective flow stress s0 , normalized by the flow stress of the matrix s ð1Þ 0 , and the corresponding equivalent average stresses s ðrÞ e , normalized by the equivalent macroscopic stress s e , as a function of the reinforcement concentration c (2) . The main observation in the context of this figure is that, among the nonlinear homogenization estimates for s0 , the SO estimates are found to be the most accurate for all values of the reinforcement concentration under both types of loading conditions considered, as can be seen in figure 9a,b. Thus, these estimates predict a lower s0 in simple shear than in axisymmetric shear loading, in agreement with the trend found for the exact LAM results. As for the case of fibre-reinforced composites (see figure 1b), it is seen that for low-to-moderate values of c (2) , the SO(W ) estimates are closer to the exact LAM results than the SO(U ) estimates, while for large values of c (2) the converse is true. It can be seen that, while the TSO estimates also capture the right dependence on q, they are found to overestimate significantly the exact LAM results. The TSO(U ) estimates for qZ0 even violate the rigorous bound provided by the VAR estimates. Finally, the VAR estimates not only overestimate the exact LAM results, but also they are unable to capture any dependence on q.

It is recalled that, since the inclusions are rigid, the average strains are trivial in this case, namely 3 ð2Þ Z 0 and 3 ð1Þ Z 3=c ð1Þ . The corresponding average stresses are shown in figure 9c,d. It is seen that the LAM results exhibit an average stress in the inclusion phase s ð2Þ e that is larger in axisymmetric shear (qZ0) than in simple shear (qZp/6), even though the difference is relatively small. The SO and TSO estimates are able to capture this dependence on q, unlike the VAR estimates, but they are seen to overestimate s ð2Þ e for dilute reinforcement concentrations. Again, it is observed that, for low-to-moderate values of c (2) , the SO(W ) estimates are in better agreement with the LAM results than the SO(U ) estimates, while the opposite is true at large c (2) . In this connection, the SO(W ) estimates for s ð1Þ e deteriorate significantly as c (2) /1, as already noted in the case of fibre-reinforced composites. Results for the corresponding second moments of the fields are not given here for brevity, and the reader is referred to Idiart (in preparation). Finally, it should be mentioned that the 'optimal' LCC associated with the SO(W ) estimates for qZ0 was found to involve an L ð1Þ 0 that is not strongly elliptic at some moderate values of c (2) , for small values of m (smaller than mZ0.1). The fact that this negative feature is manifested in this 'extreme' case is surely related to the non-optimal choice of the reference tensor 3 (1) , which has been set equal to 3 in this work (see part I). However, in practice, it does not seem to have a big effect on the predictions for the effective behaviour and phase averages.

Isotropic porous materials

In this section, results are provided for the effective behaviour and associated average strains in isotropic, ideally plastic (mZ0) composites with incompressible pores, subjected to axisymmetric shear (qZ0) and simple shear (qZp/6) loadings. In figure 10, plots are given for the effective flow stress s0 , normalized by the flow stress of the matrix s ð1Þ 0 , as well as for the corresponding equivalent average strains 3 ðrÞ e , normalized by the equivalent macroscopic strain 3 e , as a function of the porosity c (2) . The main observation in the context of this figure is that both versions of the SO estimates for the effective flow stress s0 are in excellent agreement with the exact LAM results, for all values of the porosity and both types of loading conditions (see figure 10a,b), thus capturing the right dependence on q, which happens to be subtle in this case. In contrast, the TSO(W ) estimates for s0 are found to be accurate for axisymmetric shear, but highly inaccurate for simple shear. In particular, the TSO(U ) estimates are found to coincide with the Voigt upper bound, thus violating the sharper bound provided by the VAR estimates. On the other hand, the VAR estimates, even though insensitive to q, are in relatively good agreement with the LAM results in this case.

It is recalled that, since the inclusions are incompressible pores, the deviatoric parts of the average stresses are trivial in this case, namely s ð2Þ d Z 0 and s ð1Þ d Z s d =c ð1Þ . The corresponding average strains are shown in figure 10c,d. It is seen that, among the various homogenization estimates, the SO estimates are in general the most accurate, their agreement with the LAM results being good for all values of the porosity, under both loading conditions. It is interesting to note that, as c (2) /0, these estimates predict that the average strain in the pores remains finite in axisymmetric shear (see figure 10c), but blow up in simple shear (see figure 10d). Even though the available LAM results are not conclusive in this respect, they strongly suggest that these predictions are indeed correct, which, if true, would constitute a remarkable result. In contrast, the TSO(W ) estimates, while relatively good in axisymmetric shear, become meaningless in simple shear where they predict a zero average strain in the matrix phase. In addition, the TSO(U ) estimates predict a uniform strain field in the composite, i.e.

3 ð1Þ Z 3 ð2Þ Z 3, completely at odds with the exact LAM results, but consistent with the fact that the corresponding estimates for s0 coincide with the Voigt bound. Finally, the VAR estimates are found to underestimate the average strain in the pores for low values of the porosity, and are insensitive to q.

Concluding remarks

The nonlinear homogenization estimates for the effective behaviour and field statistics proposed in part I of this work have been applied to the specific cases of two-phase, power-law composites with isotropic and transversely isotropic microstructures, and have been compared with available exact results for composites with sequentially laminated microstructures. The main findings are as follows. Globally, the second-order estimates are found to be superior to the tangent second-order and variational estimates, which can lead to inconsistent predictions in some cases. In particular, the TSO estimates have been found to give predictions for the second moments of the stress that are inconsistent with inequalities of the type hs 2 e i ðrÞ R ð s ðrÞ e Þ 2 , for sufficiently strong nonlinearities. More specifically, in the case of reinforced composites, both versions of the second-order and tangent second-order estimates for the effective behaviour, as well as for the field statistics, were found to be in fairly good agreement with the exact results for all values of the nonlinearity. All these estimates (SO and TSO) are able to capture the anisotropic character of the field fluctuations and the fact that certain components of the strain fluctuations in the matrix become unbounded in the strongly nonlinear ideally plastic limit. In contrast, the variational estimates were found to significantly overestimate the effective behaviour, in agreement with their upper bound character, and to give qualitatively incorrect predictions for the field statistics, failing to capture the strong anisotropy of the strain fluctuations at high nonlinearities.

For the cases of fibre-weakened and porous composites, which are more 'demanding' than those of reinforced composites, the second-order estimates for the effective behaviour and first moments of the local fields were found to be in good agreement with the exact results even for strong nonlinearities. In general, the accuracy of the corresponding estimates for the second moments of the fields was found to be worse, which is not surprising in view of the fact that they correspond to more sensitive information. However, these estimates agree with the exact results in that, unlike in the case of reinforced composites, the second moments of the strain remain bounded in the ideally plastic limit, thus capturing the relative differences between the deformation patterns in the weaker-and stronger-particle cases. On the other hand, the tangent second-order estimates were found to be fairly accurate for weak-to-moderate nonlinearities, but deteriorate significantly for strong nonlinearites. In turn, the variational estimates were found to be relatively good for the effective behaviour, but qualitatively incorrect for the field statistics.

It is worth mentioning (see Idiart (in preparation) for more details) that accounting for the correction terms derived in part I in the context of the secondorder and tangent second-order, estimates for the field statistics always had a beneficial effect, in that they improve the predictions arising from the sole use of the LCC. This improvement can be quite significant at strong nonlinearities, especially for the statistics of the dual field (i.e. the strain statistics arising from the stress formulation of the methods, and vice versa). Finally, it is emphasized that even though the 'optimal' choice of the 'reference tensors' in the context of the second-order estimates remains an open question, the results provided in this work show that accurate estimates can be obtained by making use of the simplest possible prescriptions for these tensors, namely, the macroscopic fields themselves. In addition, these choices have been shown (see part I) to lead to relatively simple analytical estimates for the first and second moments of the strain and stress fields.

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

Figure 1 .

 1 Figure 1. Effective flow stress s0 , normalized by the flow stress of the matrix s ð1Þ 0 , for power-law, fibre-reinforced ðs ð2Þ 0 =s ð1Þ 0 Z 5Þ composites subjected to in-plane shear: (a) as a function of the strain-rate sensitivity m, for a given concentration of fibres (c (2) Z0.21), (b) as a function of fibre concentration c (2) , for a given strain-rate sensitivity (mZ0.2). Comparisons between the 'second-order' (SO), 'tangent second-order' (TSO) and 'variational' (VAR) estimates of the Hashin-Shtrikman type, and exact results for power-law laminates (LAM ).

Figure 2 .

 2 Figure 2. First moments (phase averages) of the local fields in fibre-reinforced composites. Equivalent average strain 3 ðrÞ e and stress s ðrÞ ein each phase, normalized by the macroscopic equivalent strain 3 e and stress s e , respectively.

Figure 3 .

 3 Figure 3. Second moments of the strain field in fibre-reinforced composites. 'Parallel' 3 s and 'perpendicular' 3 t components, normalized by 3 2 e .

4.

  Transversely isotropic, fibre-weakened composites (a ) Effective behaviour In figure 5, plots are provided for the effective flow stress s0 of a fibreweakened composite, normalized by the flow stress of the matrix phase s ð1Þ 0 .

Figure 5 .

 5 Figure 5. Effective flow stress s0 , normalized by the flow stress of the matrix s ð1Þ 0 , for power-law, fibre-weakened ðs ð2Þ 0 =s ð1Þ 0 Z 0:2Þ composites subjected to in-plane shear: (a) as a function of the strain-rate sensitivity m, for a given concentration of fibres (c (2) Z0.21), (b) as a function of fibre concentration c (2) , for a given strain-rate sensitivity (mZ0.1). Comparisons between the 'secondorder' (SO), 'tangent second-order' (TSO) and 'variational' (VAR) estimates of the Hashin-Shtrikman type, and exact results for power-law laminates (LAM ).

Figure 6 .

 6 Figure 6. First moments (phase averages) of the local fields in fibre-weakened composites. Equivalent average strain 3 ðrÞ e and stress s ðrÞ ein each phase, normalized by the macroscopic equivalent strain 3 e and stress s e , respectively.

Figure 8 .

 8 Figure 8. Second moments of the strain field in fibre-weakened composites. 'Parallel' s s and 'perpendicular' s t components, normalized by s 2 e .

Figure 9 .

 9 Figure 9. Estimates and exact results for power-law rigidly reinforced composites, as a function of the reinforcement concentration c (2) , for a given strain-rate sensitivity (mZ0.1). Effective flow stress s0 , normalized by the flow stress of the matrix s ð1Þ 0 , for (a) axisymmetric shear (qZ0) and (b) simple shear (qZp/6). Parts (c) and (d) show corresponding equivalent average stresses in each phase s ðrÞ e , normalized by the equivalent macroscopic stress s e . Comparisons between the 'secondorder' (SO), 'tangent second-order' (TSO) and 'variational' (VAR) estimates of the Hashin-Shtrikman type, and exact results for power-law laminates (LAM).

Figure 10 .

 10 Figure 10. Estimates and exact results for ideally plastic (mZ0) porous composites, as a function of the porosity c (2) . Effective flow stress s0 , normalized by the flow stress of the matrix s ð1Þ 0 , for (a) axisymmetric shear (qZ0) and (b) simple shear (qZp/6). Parts (c) and (d) show corresponding equivalent average strains in each phase 3 ðrÞ e , normalized by the equivalent macroscopic strain 3 e . Comparisons between the 'second-order' (SO), 'tangent second-order' (TSO) and 'variational' (VAR) estimates of the Hashin-Shtrikman type, and exact results for power-law laminates (LAM ).