Ducruet César

Notteboom Theo
email: theo.notteboom@ua.ac.

Developing Liner Service Networks in Container Shipping

After a brief background about the development of containerization in recent decades, this chapter reviews the current characteristics of liner shipping networks under three main themes. First, it provides an overview of the different service types of shipping lines and dynamics in liner service configuration and design. Second, a global snapshot of the worldwide liner shipping network is proposed by means of vessel movement data. The changing geographic distribution of main inter-port links is explored in the light of recent reconfigurations of liner shipping networks (e.g. multiplication versus rationalisation of port calls). We also discuss the position of seaports in liner shipping networks referring to concepts of centrality, hierarchy, and selection factors. The chapter concludes by elaborating on the interactions and interdependencies between seaport development and liner shipping network development notably under current economic changes.

INTRODUCTION: BACKGROUND ON LINER SHIPPING

Container liner shipping has a relatively short history. In 1956 Malcolm McLean launched the first containership Ideal X. Ten years later the first transatlantic container service between the US East Coast and North Europe marked the real start of long distance scheduled container liner services. The first specialized cellular containerships were delivered in 1968. In the 1970s the containerization process expanded rapidly due to the adoption of standard container sizes and the awareness of industry players about the advantages and cost savings containerization brought [START_REF] Rodrigue | The future of containerization: perspectives from maritime and inland freight distribution[END_REF][START_REF] Levinson | The Box: How the Shipping Container Made the World Smaller and the World Economy Bigger[END_REF]. Although container shipping occupies a relatively minor share of the whole maritime fleet (about 12 per cent), it is the fastest growing sector and currently concentrates more than half of world trade value, regularly expanding to other commodities (e.g. neo bulks).

The world container traffic, the absolute number of containers being carried by sea, increased from 28.7 million TEU in 1990 to 152 million TEU in 2008 or an average annual increase of 9.5 per cent. Worldwide container port throughput increased from 36 million TEU in 1980and 88 million TEU in 1990to about 535 million TEU in 2008. A comparison between world container traffic and world container port throughput reveals a container on average was handled (loaded or discharged) 3.5 times between the first port of loading and the last port of discharge in 2008. This figure amounted to 3 in 1990. The rise in the average number of port handlings per box is the result of more complex configurations in liner service networks as will be explained later in this chapter. Furthermore, the centre of gravity of these liner service networks has shifted to Asia. The dominance of Asia is reflected in world container port rankings. In 2009 fourteen of the twenty busiest container ports came from Asia, mainly from China. In the mid 1980s there were only six Asian ports in the top 20, mainly Japanese load centres. The emerging worldwide container shipping networks helped to reshape global supply chain practices and supported the globalization in production and consumption. New supply chain practices in turn increased the requirements on container shipping service networks in terms of frequency, schedule reliability/integrity, global coverage of services and rate setting. This chapter analyses liner service networks as configured by container shipping lines. In a first section we discuss the drivers of and decision variables in liner service design as well as the different liner service types. Next, the chapter provides a global snapshot of the worldwide liner shipping network based on vessel movement data. The changing geographic distribution of main inter-port links is explored in the light of recent reconfigurations of liner shipping networks. Third, we zoom in on the position of seaports in liner shipping networks referring to concepts of centrality, hierarchy, and selection factors. The chapter concludes by elaborating on the interactions and interdependencies between seaport development and liner shipping network development notably under current economic changes.

CONFIGURATION AND DESIGN OF LINER SHIPPING SERVICES

2.1. The configuration of liner shipping services and networks Liner shipping networks are developed to meet the growing demand in global supply chains in terms of frequency, direct accessibility and transit times. Expansion of traffic has to be covered either by increasing the number of strings operated, or by vessel upsizing, or both. As such, increased cargo availability has triggered changes in vessel size, liner service schedules and in the structure of liner shipping.

When designing their networks, shipping lines implicitly have to make a trade-off between the requirements of the customers and operational cost considerations. A higher demand for service segmentation adds to the growing complexity of the networks. Shippers demand direct services between their preferred ports of loading and discharge. The demand side thus exerts a strong pressure on the service schedules, port rotations and feeder linkages. Shipping lines, however, have to design their liner services and networks in order to optimize ship utilization and benefit the most from scale economies in vessel size. Their objective is to optimize their shipping networks by rationalizing coverage of ports, shipping routes and transit time [START_REF] Zohil | The MED rule: the interdependence of container throughput and transshipment volumes in the Mediterranean ports[END_REF][START_REF] Lirn | An application of AHP on transshipment port selection: a global perspective[END_REF]. Shipping lines may direct flows along paths that are optimal for the system, with the lowest cost for the entire network being achieved by indirect routing via hubs and the amalgamation of flows. However, the more efficient the network from the carrier's point of view, the less convenient that network could be for shippers' needs [START_REF] Notteboom | The time factor in liner shipping services[END_REF].

Bundling is one of the key drivers of container service network dynamics. The bundling of container cargo can take place at two levels: (1) bundling within an individual liner service and (2) bundling by combining/linking two or more liner services.

[ Insert Figure 6.1 about here ] The objective of bundling within an individual liner service is to collect container cargo by calling at various ports along the route instead of focusing on an end-to-end service. Such a line bundling service is conceived as a set of x roundtrips of y vessels each with a similar calling pattern in terms of the order of port calls and time intervals (i.e. frequency) between two consecutive port calls. By the overlay of these x roundtrips, shipping lines can offer a desired calling frequency in each of the ports of call of the loop [START_REF] Notteboom | The time factor in liner shipping services[END_REF]. Line bundling operations can be symmetric (i.e. same ports of call for both sailing directions) or asymmetric (i.e. different ports of call on the way back), see Figure 1. Most liner services are line bundling itineraries connecting between two and five ports of call scheduled in each of the main markets. The Europe-Far East trade provides a good example. Most mainline operators and alliances running services from the Far East to North Europe stick to line bundling itineraries with direct calls scheduled in each of the main markets. Notwithstanding diversity in calling patterns on the observed routes, carriers select up to five regional ports of call per loop. Shipping lines have significantly increased average vessel sizes deployed on the route from around 4500 TEU in 2000 to over 8000 TEU in early 2011. These scale increases in vessel size have put a downward pressure on the average number of European port calls per loop on the Far East-North Europe trade: 4.9 ports of call in 1989, 3.84 in 1998, 3.77 in October 2000, 3.68 in February 2006, and 3.35 in December 2009. Two extreme forms of line bundling are round-the-world services and pendulum services.

[ Insert Figure 6.2 about here ] The second possibility is to bundle container cargo by combining/linking two or more liner services. The three main bundling options in this category include a hub-and-spoke network (hub/feeder), interlining and relay (Figure 2). The establishment of global networks has given rise to hub port development at the crossing points of trade lanes. Intermediate hubs emerged since the mid-1990s within many global port systems: Freeport (Bahamas), Salalah (Oman), Tanjung Pelepas (Malaysia), Gioia Tauro, Algeciras, Taranto, Cagliari, Damietta and Malta in the Mediterranean, to name but a few. The role of intermediate hubs in maritime hub-andspoke systems has been discussed extensively in recent literature (see for instance [START_REF] Baird | Optimising the container transshipment hub location in northern Europe[END_REF][START_REF] Fagerholt | Designing optimal routes in a liner shipping problem[END_REF][START_REF] Guy | Shipping line networks and the integration of South America trades[END_REF][START_REF] Mccalla | The Caribbean basin: adjusting to global trends in containerization[END_REF]. The hubs have a range of common characteristics in terms of nautical accessibility, proximity to main shipping lanes and ownership, in whole or in part, by carriers or multinational terminal operators. Most of these intermediate hubs are located along the global beltway or equatorial round-the-world route (i.e. the Caribbean, Southeast and East Asia, the Middle East and the Mediterranean). These nodes multiply shipping options and improve connectivity within the network through their pivotal role in regional hub-and-spoke networks and in cargo relay and interlining operations between the carriers' east-west services and other inter-and intra-regional services. Container ports in Northern Europe, North America and mainland China mainly act as gateways to the respective hinterlands.

Two developments undermine the position of pure transhipment/interlining hubs [START_REF] Rodrigue | Foreland-based regionalization: integrating intermediate hubs with port hinterlands[END_REF]. First of all, the insertion of hubs often represents a temporary phase in connecting a region to global shipping networks. Hub-and-spoke networks would allow considerable economies of scale of equipment, but the cost efficiency of larger ships might be not sufficient to offset the extra feeder costs and container lift charges involved. Once traffic volumes for the gateway ports are sufficient, hubs are bypassed and become redundant (see also [START_REF] Wilmsmeier | Determinants of liner shipping network configuration: a two region comparison[END_REF]. Secondly, transhipment cargo can easily be moved to new hub terminals that emerge along the long distance shipping lanes. The combination of these factors makes that seaports which are able to combine a transhipment function with gateway cargo obtain a less vulnerable and thus more sustainable position in shipping networks.

In channelling gateway and transhipment flows through their shipping networks, container carriers aim for control over key terminals in the network. Decisions on the desired port hierarchy are guided by strategic, commercial and operational considerations. Shipping lines rarely opt for the same port hierarchy in the sense that a terminal can be a regional hub for one shipping line and a secondary feeder port for another operator. For example, Antwerp in Belgium and Valencia in Spain are some of the main European hubs for Mediterranean Shipping Company (MSC) while they receive only few vessels from Maersk Line. Zeebrugge and Algeciras are among the primary European ports of call in the service network of Maersk Line while these container ports are rather insignificant in the network of MSC.

The liner service configurations in Figures 1 and 2 are often combined to form complex multilayer networks. The advantages of complex bundling are higher load factors and/or the use of larger vessels in terms of TEU capacity and/or higher frequencies and/or more destinations served. Container service operators have to make a trade-off between frequency and volume on the trunk lines: smaller vessels allow meeting the shippers' demand for high frequencies and lower transit times, while larger units will allow operators to benefit from economies of vessel scale. The main disadvantages of complex bundling networks are the need for extra container handling at intermediate terminals and longer transport times and distances. Both elements incur additional costs and as such could counterbalance the cost advantages linked to higher load factors or the use of larger unit capacities. Some have suggested that the most efficient east/west pattern is the equatorial round-the-world, following the beltway of the world (e.g. [START_REF] Ashar | Revolution Now, Containerisation International[END_REF][START_REF] Monie | The global economy, very large containerships and the funding of mega-hubs, Port Finance Conference[END_REF]. This service pattern focuses on a hub-andspoke system of ports that allows shipping lines to provide a global grid of east/west, north/south and regional services. The large ships on the east/west routes will call mainly at transhipment hubs where containers will be shifted to multi-layered feeder subsystems serving north/south, diagonal and regional routes. Some boxes in such a system would undergo as many as four transhipments before reaching the final port of discharge. The global grid would allow shipping lines to cope with the changes of trade flows as it combines all different routes in a network.

Existing liner shipping networks feature a great diversity in types of liner services and a great complexity in the way end-to-end services, line bundling services and transhipment/relay/interlining operations are connected to form extensive shipping networks.

Maersk Line, MSC and CMA-CGM operate truly global liner service networks, with a strong presence also on secondary routes. Especially Maersk Line has created a balanced global coverage of liner services. The networks of CMA-CGM and MSC differ from the general scheme of traffic circulation through a network of specific hubs (many of these hubs are not among the world's biggest container ports) and a more selective serving of secondary markets such as Africa (strong presence by MSC), the Caribbean and the East Mediterranean.

Notwithstanding the demand pull for global services, a large number of individual carriers remains regionally based. Asian carriers such as APL, Hanjin, NYK, China Shipping and HMM mainly focus on intra-Asian trade, transpacific trade and the Europe -Far East route, partly because of their huge dependence on export flows generated by the respective Asian home bases. MOL and Evergreen are among the few exceptions frequenting secondary routes such as Africa and South America. Profound differences exist in service network design among shipping lines. Some carriers have clearly opted for a true global coverage, others are somewhat stuck in a triad-based service network forcing them to develop a strong focus on cost bases. Alliance structures (cf. Grand Alliance, New World Alliance, and CYKH) provide its members easy access to more loops or services with relatively low-cost implications and allow them to share terminals.

The process of designing a liner service

Figure 3 summarizes the liner service design process. Before an operator can start with the actual design of a regular container service, he will have to analyse the targeted trade route(s).

The analysis should include elements related to the supply, demand and market profile of the trade route. Key considerations on the supply side include vessel capacity deployment and ulitzation, vessel size distribution, the configuration of existing liner services, the existing market structure and the port call patterns of existing operators. At the demand side, container lines focus on the characteristics of the market to be served, the geographical cargo distribution, seasonality and cargo imbalances. The interaction between demand and supply on the trade route considered results in specific freight rate fluctuations and the overall earning potential on the trade.

[ Insert Figure 6.3 about here ] The ultimate goal of the market analysis is not only to estimate the potential cargo demand for a new liner service, but also to estimate the volatility, geographical dispersion and seasonality of such demand. These factors will eventually affect the earning potential of the new service.

Once the market potential for a new service has been determined, the service planners need to take decisions on several inter-related core design variables. These design variables are indicated in dark gray/shaded boxes in Figure 3 and mainly concern (1) the liner service type,

(2) the number and order of port calls in combination with the actual port selection process,

(3) vessel speed, (4) frequency and (5) vessel size and fleet mix.

The array of liner service types and bundling options available to shipping lines was discussed in the previous section.

Limiting the number of port calls shortens round voyage time and increases the number of round trips per year, thereby minimizing the number of vessels required for that specific liner service. However, fewer ports of call mean poorer access to more cargo catchment areas.

Adding port calls can generate additional revenue if the additional costs from added calls are offset by revenue growth. The actual port selection is a complex issue. Traffic flows through ports are a physical outcome of route and port selection by the relevant actors in the chain.

The most relevant service-related and cost factors explaining port selection by the main players of the transport chain (e.g. shippers, ocean carriers, and forwarders) are identified in the scientific literature on port choice, see e.g. [START_REF] Murphy | Port selection criteria: An application of a transportation research framework[END_REF], [START_REF] Murphy | A comparative analysis of port selection factors[END_REF], [START_REF] Malchow | A disaggregate analysis of factors influencing port selection[END_REF], [START_REF] Tiwari | Shippers' port and carrier selection behavior in China: A discrete choice analysis[END_REF], [START_REF] Nir | Port choice behaviour -from the perspective of the shipper[END_REF], [START_REF] Chou | Comparison of two models for port choice[END_REF], [START_REF] Song | A competitive analysis of Chinese container ports using the analytic hierarchy process[END_REF], [START_REF] Guy | Port selection and multicriteria analysis: An application to the Montreal-New York alternative[END_REF] and [START_REF] Wiegmans | Port and terminal selection by deep-sea container operators[END_REF]. Port choice has increasingly become a function of the overall network cost and performance. Figure 3 incorporates the approach of [START_REF] Notteboom | Complementarity and substitutability among adjacent gateway ports[END_REF] to group port selection factors together in the demand profile of the port, the supply profile of the port, and the market profile of the port.

Human behavioural aspects might impede carriers from achieving an optimal network configuration. Incorrect or incomplete information results in bounded rationality in carriers' network design, leading to sub-optimal decisions. Shippers sometimes impose bounded rational behaviour on shipping lines, e.g. in case the shipper asks to call at a specific port. [START_REF] Wiegmans | Port and terminal selection by deep-sea container operators[END_REF] argue that port selection by shipping lines can also be heavily influenced by the balance of power among the shipping lines of the same strategic alliance, or the carrier's objective to make efficient use of its dedicated terminal capacity in specific ports.

The choice of vessel speed is mainly affected by the technical specifications of the vessel deployed (i.e. the design speed), the bunker price (see [START_REF] Notteboom | The effect of high fuel costs on liner service configuration in container shipping[END_REF], environmental considerations (e.g. reduction of CO 2 through slow steaming) and the capacity situation in the market (i.e. slow steaming can absorb some of the vessel overcapacity in the market, see e.g. [START_REF] Cariou | Bunker costs in container liner shipping: are slow steaming practices reflected in maritime fuel surcharges?[END_REF]Notteboom, 2011 and[START_REF] Notteboom | The organizational and geographical ramifications of the 2008-09 financial crisis on the maritime shipping and port industries[END_REF].

The number and order of port calls, the total two-way sailing distance and the vessel speed are the main determinants of the total vessel roundtrip time. The theoretical/optimal roundtrip time will seldomly be achieved in practice due to delays along the route and in ports giving rise to schedule reliability problems. Low schedule integrities can have many causes ranging from weather conditions, delays in the access to ports (pilotage, towage, locks, tides) to port terminal congestion or even security considerations [START_REF] Notteboom | The time factor in liner shipping services[END_REF]. A shipping line can insert time buffers in the liner service to cope with the chance of delays. Time buffers reduce schedule unreliability, but increase the vessel roundtrip time.

When it comes to the service frequency, carriers typically aim for a weekly service. The service frequency and the total vessel roundtrip time determine the number of vessels required for the liner service. Carriers have to secure enough vessels to guarantee the desired frequency.

Given the number of vessels needed and the anticipated cargo volume for the liner service, the shipping line can then make a decision on the optimal vessel size and fleet mix. As economies of vessel size are more significant on longer distances, the biggest vessels are typically deployed on long and cargo-rich routes.

Decisions on all of the above key design variables will lead to a specific slot capacity offered by the new liner service. The resulting slot capacity should be in line with the actual demand as to maximize average vessel utilization (given expected traffic imbalances, cargo dispersion patterns and cargo seasonality and volatility).

SHIPPING ROUTES, NETWORK PATTERNS, AND PORT CENTRALITY

The aforementioned services altogether form a global maritime network within which local, regional and global links among ports become interconnected through the establishment of hub, interlining and relay ports.

The distribution of container flows

The weight and growth of major trade routes measured in TEUs provides evidence about the imbalanced structure of the global liner shipping network based on the offer of services (Table 1). Their distribution confirms the predominance of the Europe-Asia link both in terms of weight and growth, closely followed by Asia-USA but with lower growth, while other links lag far behind in terms of the capacity deployed. This confirms the study by [START_REF] Frémont | Transport maritime conteneurisé et mondialisation[END_REF] of the global container shipping network through the mapping of the top shipping lines' service offers among world regions. They explain the dominance of Asia by the role of the Newly Industrialised Countries (NICs) that provide consumers goods to industrialized countries, thus intensifying trans-Pacific flows at the expense of transatlantic flows. They also calculated that in 2002, such relations among the main economic poles of the "Triade" concentrated about 67 per cent of total service capacity, 22 per cent only remaining for North-South relations with these poles, and South-South relations being negligible in size.

[ Insert Table 6.1 about here ] A more precise method for measuring the weight of links is to trace the worldwide circulation of container vessels (Table 2). Each time a vessel calls at one port, its capacity (in deadweight tonnage, DWT) is added to the port and to the inter-port link. The yearly total is thus an expression of the frequency and capacity of the links formed on various levels (i.e. ports, regions, continents) in an origin-destination matrix. One important aspect of the methodology is to have considered all ports of the same vessel voyage being interconnected, should they be or not adjacent calls in the sequence. This allows for a better view of the distribution of links and traffics.

The polarizing role of Asia appears even more explicitly, since most regions have their largest flow link directed to it at both years (Middle East, Oceania, North Europe, North America), or only in 2006 (Africa, South Europe, Latin America). In fact the latter regions have shifted their main traffic flow from North Europe, North America and South Europe respectively [ Insert Table 6.2 about here ] The importance of intra-regional traffic is estimated based on the sequences of calls that are internal or external to LMIU regions. Such distinction provides a rough estimate on the extent to which different regions have different shipping dynamics. The intensity of intra-regional traffic in total traffic (Table 3) can be explained by various factors such as coastal morphology, the presence of hub ports, and the level of trade integration within the region.

For instance, the low share of Africa and the Middle East in 1996 clearly reflects the lack of internal cohesion and integration, but the figure has changed dramatically in 2006, due to greater interdependency among regional ports. Shipping networks are thus a good revelatory of trade and regionalization dynamics [START_REF] Lemarchand | Regional integration and maritime range, in Ports in Proximity: Competition and Coordination Among Adjacent Seaports[END_REF]. Regions with high internal connectivity through the extensive use of hub-and-feeder systems often have a high share of intra-regional traffic, such as Asia and North Europe, but also Latin America, which includes the Caribbean port system, whereas for North America, it is more the increase of multiple calls along East and West coasts, notably with the shift of major container traffic and intermodal facilities to the Southeast (e.g. Hampton Roads, Jacksonville, Miami).

[ Insert Table 6.3 about here ]

Topology and the role of distance

Although maritime transport does not use an infrastructure of tracks like in road or rail transport, [START_REF] Ducruet | The worldwide maritime network of container shipping: Spatial structure and regional dynamics[END_REF] calculated that the overall length of the network using orthodromic distance doubled between 1996 and 2006, from five to ten million kilometres. The length of the longest inter-port link has remained constant (10,000 km) but the average length has slightly increased from 1,000 to 1,200 km, as well as the traffic density from 331 to 407 TEU per kilometre. Such evidences validate the fact that shipping networks have constantly expanded geographically during this period.

In addition to these results, [START_REF] Ducruet | The worldwide maritime network of container shipping: Spatial structure and regional dynamics[END_REF] also underline the influence of distance on traffic concentration. They show that most traffic occurs across relatively short distances: about 80 per cent of total worldwide traffic concentrates over direct links of 500 km or less, while links of 100 km or less support more than half. Besides the influence of coastal morphology and the necessity following successive calls in relative proximity, such figures can be explained by some local service configurations, as in the case of adjacent seaports serving shared hinterlands (e.g. Le Havre-Hamburg range) or acting as dual hubs (e.g. Busan and Gwangyang), which often receive multiple calls for the same vessels or liner services.

The noticeable increase of the longest links can be explained by stronger trans-Pacific ties and also by rapid technological progress in the shipping industry, allowing longer sailing distances between two ports: links of over 5,000 km concentrate 7 per cent and 10 per cent of worldwide traffic in 1996 and 2006 respectively. Overall, it could be calculated that the top 100 direct inter-port links in terms of traffic volume represent no less than 52 per cent and 39 per cent of worldwide container traffic in 1996 and 2006, respectively, thus confirming a trend of de-concentration due to the multiplication of links. The spatial distribution of these top links also shows the dominance of intra-regional relations, with the exception of trans-Pacific links. The maps in Figure 6.4 retain only interregional inter-port (direct) links based on the definition of large world regions by the United Nations (i.e. Europe, Americas, Asia, Oceania, and Africa). We clearly observe a reduction and simplification of transatlantic and trans-Mediterranean links together with the appearance of new links in the top 100 such as Europe-Brazil links and Asia-Mexico links. There is, however, also some continuity, since Le Havre -New York is the heaviest direct link connecting Europe with the world in both years, and Trans-Pacific links remain at centre stage, but with a shift of main links from Japanese to Chinese ports.

[Insert Figure 6.4 about here]

The extent to which the strategies of shipping lines are reflected in the topological structure of the network can also be verified by applying some measures from graph theory and complex networks. On a world level, [START_REF] Hu | Empirical analysis of the worldwide maritime transportation network[END_REF] were the first to confirm that container shipping networks belong to the category of so-called "scale-free" and "small-world" networks, i.e. where a limited number of nodes have the majority of links, the latter's frequency being distributed along a power-law, and with high cluster densities among smaller nodes outside hubs. Although [START_REF] Kaluza | The complex network of global cargo ship movements[END_REF] contradict [START_REF] Deng | Worldwide marine transportation network: Efficiency and container throughput[END_REF] about the extent to which the global maritime network is more or less "efficient" (i.e. low average number of stops between two nodes) than other transport networks such as airlines, Ducruet and Notteboom (2010) underlined an increase in efficiency between 1996 and 2006, which is attributed to the expansion of the network as well as to the emergence of new hub ports.

Another important trend topologically speaking is the decreasing hierarchical structure of the network, as observed by Ducruet and Notteboom (2010) on a world level and by Ducruet et al. (2010aDucruet et al. ( , 2010b) ) in Northeast Asia and the Atlantic regions. Such trend results from the combination of various factors such as regional integration processes (multiplication of intraregional links, opening of new direct call and multi-port services), diseconomies of scale in large gateway and hub ports, and competition between existing and emerging hub ports.

The centrality of container ports

The impact of liner shipping network's operation on container ports is often analysed in terms of throughput, the most widely available indicator of port performance in official statistics.

Table 4 shows the classic port hierarchy with regard to the number of containers (TEUs) handled by top ports since the 1970s, regardless of the function of ports in the network.

However, the network perspective allows for calculating the connectivity of ports, which is critically lacking in the related literature [START_REF] De Langen | New indicators to measure port performance[END_REF]. Two main measures of centrality in networks can be obtained based on the configuration of inter-port links in a binary port-to-port matrix (i.e. presence or absence of links between two given ports). First, betweenness centrality counts the number of positions of a node on possible shortest paths among all nodes in the entire network [START_REF] Ducruet | Graph theory: Measures and indices[END_REF]. It is a measure of accessibility or reachability. Second, degree centrality is the number of adjacent neighbours, which simply counts the number of ports connected to a given port. These are two very classic measures in network analysis across all fields of investigation from physics to sociology [START_REF] Wasserman | Social Network Analysis. Methods and Applications[END_REF], which can provide answers to theoretical configurations notably provided by [START_REF] Fleming | Spatial characteristics of transportation hubs: Centrality and intermediacy[END_REF] on the centrality and intermediacy of transportation hubs. When it comes to ports, these measures can reveal other dimensions than sole throughput, with which they can be highly correlated.

[ Insert Table 6.4 about here ]

A first look at the top 25 central ports in the worldwide network provides some evidence about the usefulness of the measures and how they characterize the position of ports in the network. Unlike airline networks where anomalous centralities depict the peculiar position of very central airports (betweenness) with few direct connections (degree) (Guimera et al., 2005), liner shipping shows a good fit between betweenness and degree [START_REF] Deng | Worldwide marine transportation network: Efficiency and container throughput[END_REF].

Thus, very central ports in the entire liner shipping network are also those multiplying their connections towards other ports. This would mean that hub ports have many connections while being very central, unlike relay hubs in airline networks (e.g. Anchorage). Some exceptions, however, are visible in the results about ports, in light of the overall drop in the linear correlation among betweenness and degree from 0.84 in 1996 to 0.72 in 2006. This change suggests a more complex relationship between the two variables. Indeed in 2006, the peculiar position of some ports having less degree than betweenness appears with Surabaya and Miami. Those ports thus tend to have a role as regional hubs, with fewer connections to local ports that are not well connected to the rest of the network, and have no option but to go through Surabaya and Miami, such as several Indonesian and Caribbean ports. Surabaya and Miami thus benefit from their bridge position towards such smaller ports to raise their centrality in the global network. Such trend is also visible in the work of [START_REF] Ducruet | Local strength and global weakness: A maritime network perspective on South Korea as Northeast Asia's logistics hub[END_REF] showing how Busan has increased its centrality within Northeast Asia but has simultaneously seen its centrality lowering in the worldwide network.

[ Insert Table 6.5 about here ] The extent to which network position relates with the hierarchy of container throughput is a crucial question that can be tested in Figure 5. Interestingly, the correlation with betweenness and with degree has increased between 1996 and 2006, showing a better fit with container throughput. In terms of variance, betweenness centrality explains 40 per cent and 47 per cent of total throughput, while degree centrality explains 57 per cent and 66 per cent at respective years. This would suggest that network indicators are very good tools for understanding overall port performance, although they do not include land-based dimensions of hinterland connectivity, coverage, and other aspects of performance such as technical standards and the availability, quality, size, and cost of terminal handling facilities and services. Overall, betweenness is less related with throughput than is degree, with regard to correlation levels and to the slope of the power-law line. Degree centrality scales superlinearly with throughput, which means that the number of connections is highly concentrated at large throughput ports.

At the top of the hierarchy, large gateway ports such as Shenzhen and Yokohama may have less betweenness centrality than transhipment hubs, while ports combining both functions (cf. section 2.1) may rank high in the three indicators. Further analyses may better explain the role of network position on throughput performance as empirically tested by Ducruet et al. (2011).

Overall, the position of ports in shipping networks seems to explain a large part of their overall activity.

[Insert Figure 6.5 about here]

CONCLUSIONS

The extensive worldwide container shipping networks are key to globalization and global supply chains. The requirements on container shipping service networks have tightened in terms of frequency, schedule reliability/integrity, global coverage of services and rate setting.

The evolutionary path of liner shipping networks and port operations is characterised by drastic changes as well as permanencies. Shipping lines have embraced a wide range of bundling concepts and liner service configurations to drive container service network dynamics. As global trade expands in economical and geographic terms, despite difficult conjunctures such as the global financial crisis, new ports and new shipping networks are regularly created to cope with demand. Shipping lines logically adapt to such trends as well as influence them, sometimes by refining their services through rationalization or by creating new service configurations through a combination of line bundling itineraries and transhipment/relay/interlining operations at pivotal ports of the network. This chapter provided evidence about the increasing complexity and number of cargo movements that occurs in parallel with increased concentration and polarisation, depending on the measures and methodologies applied for revealing such trends. It discussed some fundamental aspects, such as the economic and geographic dimension of the variety of services offered by the industry, as well as the strong and growing interdependency between maritime centrality and port throughput for container ports, although in this simple equation, hinterland connectivity and port efficiency are not included. Looking at the distribution of main trading routes as well as disaggregated interregional and inter-port shipping links, the latter being compared with kilometric distance, we observed that the overall network is growing in size and length notably thanks to a catching-up of South-South linkages versus North-North and North-South linkages. However, most worldwide traffic still concentrates over very short distances, that is more specific to maritime transport than to air transport due to adjacent calls between ports.

In light of our results, further research on container shipping networks should go deeper in the analysis of the causal relationship between throughput and centrality for container ports, while better identifying specific cases and outliers. Another avenue of future research would be to test the impact of the global financial crisis on the overall structure of regional and global liner shipping networks, as well as on the position of individual container ports, which would complement the classic view of shipping based on aggregated cargo flows among major trade routes. The global database on vessel movements is being expanded to other years and other types of vessels so as to better appreciate the linkages between port hierarchy, global/regional trade patterns, and the evolution of network structure. Last but not least, the analysis of the situation of ports and cities within combined maritime and land-based networks would prove helpful for the study of logistics chains, the hinterland-foreland continuum, intermodal transport systems, and port competitiveness. 1996 2006 1996 2006 1996 2006 1996 2006 1996 2006 1996 2006 1996

  (in 1996) toAsia (in 2006), thereby illustrating the continuous influence of Asia on world trade patterns. Links can also be differentiated by their traffic growth rate in a descending order, confirming the faster growth of South-South linkages versus North-North and North-South linkages (albeit in smaller volumes than main routes): Very fast growth (over 500 per cent): Latin America-Oceania, Latin America-Middle East, and Middle East-Africa;  Fast growth (over 250 per cent): Latin America-South Europe, Latin America-Africa, Latin America-South & East Asia, South Europe-South & East Asia, South & East Asia-Middle East;  Significant growth (over 100 per cent): South Europe-Middle East, South Europe-Oceania, North Europeall regions, South & East Asia-Oceania, South & East Asia-North Europe, North America-all regions; Moderate growth (100 per cent or less): Africa-Oceania, Oceania-Middle East, Africa-South Europe, North America-South & East Asia, South & East Asia-North Europe, North America-North Europe.

Figure

  Figure 6.1: Bundling within an individual liner service Line bundling service (symmetric and asymmetric)

Figure 6 . 5 :

 65 Figure 6.5: Centrality in liner shipping networks and container throughput

Table 6 .1: World's major trade routes in 2007

 6

	Main route	Transpacific Asia-USA USA-Asia	Europe-Asia Asia-Europe Europe-Asia USA-Europe Europe-USA Transatlantic
	Cargo flows (million TEUs)	15.4	4.9	17.7	10.0	2.7	4.5
	Growth 2006-2007 (per cent)	2.8	3.0	15.5	9.0	7.3	1.6
			Source: Containerisation International

Table 6 .2: Distribution of interregional flows in 1996 and 2006 (million DWT)

 6

	Region	OCEANIA	SOUTH EUROPE	LATIN AMERICA	AFRICA	NORTH EUROPE	SOUTH & EAST ASIA	NORTH AMERICA
	Year

Table 6 .3: Share of intraregional traffic in total regional traffic (per cent DWT)

 6

	Region	1996 2006
	SOUTH & EAST ASIA	69.8	70.6
	OCEANIA	49.8	53.9
	LATIN AMERICA	59.1	57.1
	NORTH EUROPE	48.4	52.2
	World average	46.7	48.6
	AFRICA	34.7	46.5
	SOUTH EUROPE	47.1	43.2
	MIDDLE EAST	32.4	33.3
	NORTH AMERICA	32.2	32.1
	Source: own elaboration based on LMIU data


C Guerra

J Scheibert

D Bonamy

D Dalmas

Understanding fast macroscale fracture from microcrack post mortem patterns

Dynamic crack propagation drives catastrophic solid failures. In many amorphous brittle materials, sufficiently fast crack growth involves small-scale, high-frequency microcracking damage localized near the crack tip. The ultra-fast dynamics of microcrack nucleation, growth and coalescence is inaccessible experimentally and fast crack propagation was therefore studied only as a macroscale average. Here, we overcome this limitation in polymethylmethacrylate, the archetype of brittle amorphous materials: We reconstruct the complete spatio-temporal microcracking dynamics, with micrometer / nanosecond resolution, through post mortem analysis of the fracture surfaces. We find that all individual microcracks propagate at the same low, load-independent, velocity. Collectively, the main effect of microcracks is not to slow down fracture by increasing the energy required for crack propagation, as commonly believed, but on the contrary to boost the macroscale velocity through an acceleration factor selected on geometric grounds. Our results emphasize the key role of damage-related internal variables in the selection of macroscale fracture dynamics.

The fracture of brittle amorphous materials is usually described using the linear elastic fracture mechanics (LEFM) framework [START_REF] Freund | Dynamic Fracture Mechanics[END_REF][START_REF] Fineberg | Instability in dynamic dracture[END_REF][START_REF] Ravi-Chandar | Dynamic Fracture[END_REF][START_REF] Cox | Modern topics and challenges in dynamic fracture[END_REF], which considers the straight propagation of a single smooth crack. All dissipative processes (e.g. plastic deformation or bond breaking) are assumed to be localized in a small zone around the crack tip (fracture process zone, FPZ). Crack velocity, v, is then predicted from the balance between the flux of mechanical energy relased from the surrounding elastic material into the FPZ [START_REF] Livne | The near-tip fields of fast cracks[END_REF], and the dissipation rate within this zone. The former is computable within continuum theory and connects to the stress intensity factor, K, which describes the macroscopic forcing applying on the crack tip and depends on the external loading and specimen geometry only. The dissipation rate is quantified by the fracture energy, Γ, required to expose a new unit area of cracked surfaces, to be measured experimentally. The resulting equation of motion reads [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] where E and c R denote the material's Young's modulus and Rayleigh wave speed, respectively. Polymethylmethacrylate (PMMA) is often considered as the archetype of nominally brittle materials and, therefore, has been one of the most widely used materials against which theories have been confronted from the early stages of fracture mechanics. Yet, in PMMA, single smooth cracks are actually observed for slow propagation only. Fast enough cracks (v > v a 0.2 c R [START_REF] Scheibert | Brittle-quasibrittle transition in dynamic fracture: An energetic signature[END_REF]) propagate through the nucleation, growth and coalescence, in the fracture plane, of individual microcracks [START_REF] Scheibert | Brittle-quasibrittle transition in dynamic fracture: An energetic signature[END_REF][START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF][START_REF] Ravi-Chandar | Dynamic fracture of nominally brittle materials[END_REF][START_REF] Hull | Fractography: Observing, measuring and interpreting fracture surface topography[END_REF]. Cracks faster than v b 0.4 c R also involve aborted out-of-plane secondary cracks known as microbranches [START_REF] Fineberg | Instability in dynamic dracture[END_REF][START_REF] Sharon | Local crack branching as a mechanism for instability in dynamic fracture[END_REF][START_REF] Sharon | Confirming the continuum theory of dynamic brittle fracture for fast cracks[END_REF], which prevent LEFM from being applicable [START_REF] Sharon | Confirming the continuum theory of dynamic brittle fracture for fast cracks[END_REF]. LEFM has been shown to agree with experiments as long as no microbranch is involved [START_REF] Scheibert | Brittle-quasibrittle transition in dynamic fracture: An energetic signature[END_REF][START_REF] Sharon | Confirming the continuum theory of dynamic brittle fracture for fast cracks[END_REF][START_REF] Bergkvist | Some experiments on crack motion and arrest in polymethylmethacrylate[END_REF][START_REF] Goldman | Acquisition of inertia by a moving crack[END_REF], i.e. even in the presence of microcracks, provided a suitable velocity dependence of the fracture energy, Γ(v), is prescribed [START_REF] Scheibert | Brittle-quasibrittle transition in dynamic fracture: An energetic signature[END_REF][START_REF] Sharon | Confirming the continuum theory of dynamic brittle fracture for fast cracks[END_REF].

FIG. 1. Fracture energy Γ as a function of macroscale crack velocity v (adapted from [START_REF] Scheibert | Brittle-quasibrittle transition in dynamic fracture: An energetic signature[END_REF]). Different symbols correspond to different experiments. The two vertical dashed lines correspond to v a (microcracking onset) and v b (microbranching onset). Below v b , all the experimental points collapse onto a a single Γ(v) curve. The slope of this curve exhibits a drop at v a .

Recent experiments using PMMA showed that, above v a , the slope of Γ(v) drops [START_REF] Scheibert | Brittle-quasibrittle transition in dynamic fracture: An energetic signature[END_REF] (see Fig. 1), suggesting that microcracks make macroscale cracks dissipate less or/and propagate faster than a single crack would. This is at odds with the common view that damage through opening mode microcracks slows down crack propagation by increasing energy dissipation [START_REF] Ravi-Chandar | Dynamic Fracture[END_REF][START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF][START_REF] Washabaugh | A Reconciliation of dynamic crack velocity and Rayleigh-wave speed in isotropic brittle solids[END_REF]. Understanding this counter-intuitive behaviour requires unravelling the coupling between (i) the space-time dynamics of damage at the FPZ scale and (ii) the crack dynamics at the macroscale. The time interval between two successive microcrack nucleation events is typically a few tens of nanoseconds. This makes realtime local measurements of microcracking dynamics beyond current researchers' reach. Hence, fast crack propagation has been studied only through measurements of the average dynamics of the macroscopic crack front [START_REF] Livne | The near-tip fields of fast cracks[END_REF][START_REF] Scheibert | Brittle-quasibrittle transition in dynamic fracture: An energetic signature[END_REF][START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF][START_REF] Ravi-Chandar | Dynamic fracture of nominally brittle materials[END_REF][START_REF] Sharon | Local crack branching as a mechanism for instability in dynamic fracture[END_REF][START_REF] Sharon | Confirming the continuum theory of dynamic brittle fracture for fast cracks[END_REF][START_REF] Bergkvist | Some experiments on crack motion and arrest in polymethylmethacrylate[END_REF][START_REF] Goldman | Acquisition of inertia by a moving crack[END_REF][START_REF] Washabaugh | A Reconciliation of dynamic crack velocity and Rayleigh-wave speed in isotropic brittle solids[END_REF][START_REF] Boudet | Dynamics of crack propagation in brittle materials[END_REF][START_REF] Rosakis | Cracks faster than the shear wave speed[END_REF]. Quantitative fractography is an appealing tool to probe microscale damage mechanisms. Fracture surfaces are indeed known to record fracture processes down to the nanoscale [START_REF] Hull | Fractography: Observing, measuring and interpreting fracture surface topography[END_REF][START_REF] Kermode | Low-speed fracture instabilities in a brittle crystal[END_REF]. In particular, in many materials including PMMA (Fig. 2A), microcracks leave characteristic conic-like markings on fracture surfaces [START_REF] Ravi-Chandar | Dynamic Fracture[END_REF][START_REF] Scheibert | Brittle-quasibrittle transition in dynamic fracture: An energetic signature[END_REF][START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF][START_REF] Ravi-Chandar | Dynamic fracture of nominally brittle materials[END_REF][START_REF] Hull | Fractography: Observing, measuring and interpreting fracture surface topography[END_REF]. These patterns are commonly understood through a geometrical model first

A 1 mm A 1 mm B c 1 t c2 (t-τ) C O 1 O 2 × × P 12 × d n d n 100 µm case c 1 =c 2 =c m (macroscale) crack propagaƟon
developed in [START_REF] Smekal | Zum Bruchvorgang bei sprodem Stoffverhalten unter ein-und mehrachsigen Beanspruchungen[END_REF] and improved in e.g. [START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF][START_REF] Rabinovitch | Origin of mist and hackle patterns in brittle fracture[END_REF]. In this model, each conic-like marking corresponds to the intersection of two penny-shaped microcracks, nucleated at point-like nucleation centers and growing at speeds c 1 and c 2 along two slightly different planes (Fig. 2B). The numerical implementation of this model demonstrated that microcracking is responsible for some of the complexity of macroscopic crack growth [START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF], e.g. mist fracture surfaces decorated by conic-like markings and strong fluctuations in the velocity signal, v(t). However, the agreement remained only qualitative because simplifying prescriptions were used for the microcracking dynamics [START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF], namely (i) the location of nucleation centers, (ii) c 2 /c 1 and (iii) the nucleation criterion. with parameter ρa 2 . The fitting parameter ρ is a-independent, indicating homogeneous uncorrelated random distribution with mean surface density ρ. Similar results hold for all K (Fig. S3). Bottom: ρ(K) curve (black disks) superimposed to that obtained from the data reported in [START_REF] Scheibert | Brittle-quasibrittle transition in dynamic fracture: An energetic signature[END_REF] (gray triangles, vertical lines indicate standard deviation). Red line: fit using Eq. (S3) in the range K a =2.1 MPa.m 1/2 < K < K b =3.1MPa.m 1/2 (see SI text). ρ sat 45.5mm-2. (B) Cumulative distribution, for various ρ, of the velocity ratio c 2 /c 1 obtained via fitting fractographic branches using the geometrical model (see Eq. (S1) and SI text). Red line: Fitted normal distribution (average 0.98 and standard deviation 0.03). Similar results hold for each ρ (Fig. S4). (C) Top: cumulative distribution of d n for ρ =27.5mm -2 . Red line: best two-parameters fit

P(d n ) = ((d max -d n )/(d max - d min ))2.
Here d min =4 m, d max =77 m. Similar fits hold for all ρ (Fig. S5). Bottom: mean distance at nucleation as a function of ρ. Error bars: one standard deviation. Red line: fit using Eq. (S4) up to a saturating value 50 m reached at ρ sat (see SI text).

EXPERIMENTAL DETERMINATION OF THE DYNAMICS OF INDIVIDUAL MICROCRACKS

Here, we determine experimentally the microscopic rules for the nucleation and growth of microcracks, by analyzing the morphology of each individual conic-like marking on different millimeter-sized fracture surfaces (see e. g. Fig. 2A) corresponding to different K (i.e. to different v in the range 0.23-0.49 c R ) (Materials and Methods). We first find that, irrespective of K, the spatial distribution of nucleation centers is Poissonian (see Fig. 3A, top and Fig. S3), i.e. the centers are homogeneously and randomly distributed in space, without correlation. This is consistent with the usual view that microcracks nucleate at some preexisting weak defects randomly distributed within the material's volume, when a crack tip running in their vicinity sufficiently enhances the stress field [START_REF] Ravi-Chandar | Dynamic Fracture[END_REF][START_REF] Hull | Fractography: Observing, measuring and interpreting fracture surface topography[END_REF]. The increase in mean surface density of nucleation centers, ρ, with K (Fig. 3A, bottom) is attributed to the increase in FPZ size with K, which yields more volume defects turning into microcracks (see [START_REF] Scheibert | Brittle-quasibrittle transition in dynamic fracture: An energetic signature[END_REF] and SI text). Because ρ completely characterizes Poisson distributions, it will be used hereafter as the parameter as a function of which the various quantities will be plotted.

Stationarity of macroscopic crack propagation at the scale of each millimetric-sized image requires the ratio c 2 /c 1 of the velocities of two successive microcracks to be 1, on average. A smaller (larger) value would indeed produce a decelerating (accelerating) macroscale crack. This requirement has consequences on the geometry of conic-like markings (see SI text and Fig. S2), which were checked:

We fitted all individual markings with the shape predicted using the geometrical model (see Fig. 2B and SI text), with c 2 /c 1 being the only adjustable parameter. Irrespective of ρ, c 2 /c 1 is found equal to 1 within 4% standard deviation (see Fig. 3B and Fig. S4). In the following, we will neglect the slight dispersion of c 2 /c 1 and consider that, for any given ρ, all microcracks propagate at the same velocity:

c 2 = c 1 = c m
, where c m denotes the speed of individual microcracks and a priori depends on the macroscopic external loading K (or equivalently on ρ).

In these conditions, the intersection between two microcracks is a true conic. Its focus coincides with the nucleation center of the triggered microcrack, and the apex-to-focus distance is half the distance d n between the triggering front and the triggered center at the instant of nucleation (see Fig. 2C and SI text). Hence, d n defines the nucleation criterion. Its cumulative distribution is well fitted by a twoparameters parabolic function, irrespective of ρ (Fig. 3C, top and Fig. S5). Variations of the mean value with ρ exhibit two regimes: an initial linear increase followed by a saturating plateau, when ρ exceeds a value ρ sat (Fig. 3C,bottom). The linear behaviour comes from the fact that and ρ both scale linearly with the FPZ size (see SI text). The transition is understood as the point where becomes comparable with the mean distance between nucleation centers (see SI text and Fig. S6).

DETERMINISTIC RECONSTRUCTION OF THE MICROCRACKING DAMAGE HISTORY

The analyses performed up to now permit a full characterization of the statistics of microcrack nucleation, growth and coalescence within the FPZ. To unravel how this FPZ quantitatively operates to relate the macroscale crack velocity, v, to the microscale velocity, c m , of individual microcracks, we feed the geometrical model with the observed locations of all individual nucleation centers and the corresponding distances at nucleation, d n . We then simulate the space-time evolution of the fracturing process with the constraint that all microcracks propagate at the same velocity c 1 = c 2 = c m (Materials and Methods). Note that, at this point, c m is constant within the FPZ but can depend on K (or equivalently on ρ). Figures 4A-B show typical snapshots of the simulated crack dynamics (see supporting movies). Apart from edge effects (see SI text and Fig. S7), the matching between the experimental conics and the simulated ones is quite satisfactory (Fig. 4C-D) for all values of ρ. As expected, the simulated dynamics thus provide a deterministic reconstruction of the ultra-fast microcracking dynamics. The spatial resolution of 2 m (pixel size) and the time resolution of 10 ns (pixel size divided by c m , demonstrated hereafter to be a load-independent constant close to a value c m 200 m/s) are much beyond standard experimental mechanics methods like acoustic emission or fast imaging. Similar deterministic nucleation and geometrical growth models are used in a broad range of fields including metallurgy [START_REF] Mukhopadhyay | A cellular operator model for the simulation of static recrystallization[END_REF], biology [START_REF] Jettestuen | Competition on the rocks: Community growth and tessellation[END_REF] and superconductivity [START_REF] Vestgarden | Flux distribution in superconducting films with holes[END_REF]. To shed light on the macroscale effect of microcracking damage, we now focus on the time evolution of the average location of the simulated crack front (Fig. 5A). For each ρ, this evolution is linear, meaning that the average front has a constant velocity, A × c m , the value of which was found insensitive to edge effects (see SI text and Fig. S7). Figure 5B shows that the acceleration factor A equals 1 only for ρ =0, and then increases with ρ. The time evolution of the position of a single point of the front (Fig. 5A) sheds light on the origin of this effective acceleration. The point motion is jerky, with sudden jumps corresponding to microcrack coalescence events, and the velocity between jumps is close to c m . Hence, as the rate of coalescence events increases with ρ, A also increases with ρ. A simple mean-field lattice model, which evaluates the rate of coalescence events, yields (see SI text and Fig. S8):

MACROSCALE CRACK DYNAMICS

(1) where b is a numerical factor 1. This equation, when combined with the observed evolution of with ρ (Fig. 3C, bottom, red line), gives the red line in Fig. 5B, which is in very good agreement with reconstructed velocities.

The question remains of the possible dependence of c m with ρ. Figure 5B shows that, if we chose a ρindependent c m =217±3 m/s = 0.24±0.01 c R , the ρ-dependence of the reconstructed acceleration factor

A is identical to the ratio of the measured macroscopic crack speed, v, over c m . This means that the propagation speed of microcracks is not only identical for two successive microcracks, but also all along the crack path, irrespective of ρ -and hence of K. Note that c m is found very close to the maximum speed, 204 m/s or 0.23 c R , of individual crack fronts in PMMA originating from the fracture energy increase with FPZ size [START_REF] Scheibert | Brittle-quasibrittle transition in dynamic fracture: An energetic signature[END_REF]. The change in the ρ-dependence of A observed at ρ sat in Fig. 5B corresponds to a macroscopic crack speed 1.67 c m 0.41 c R . This velocity is very close to the onset of the microbranching instability (0.36 c R in PMMA [START_REF] Sharon | Confirming the continuum theory of dynamic brittle fracture for fast cracks[END_REF]), which suggests that this instability could be related to the steric effect responsible for the saturation of (ρ) above ρ sat .

DISCUSSION

In dynamic fracture, the relationship between the opening force and the speed at which a macroscopic crack moves forward is controlled by dissipative and nonlinear processes that develop at the microscale within the FPZ. The space and time scales associated with the FPZ dynamics are usually too small to enable a real time and space monitoring of these processes. Here, we demonstrate that such a detailed monitoring is actually possible in PMMA, the archetype of nominally brittle materials, by analysing post mortem the patterns left on fracture surfaces by microcracking damage.

Our results show that, in PMMA, the true local propagation speed of single cracks is limited to a fairly low value c m , about 0.23c R , while the apparent speed, v, measured at the continuum-level scale can be much higher. When v c m , the macroscopic crack is actually found to progress through the coalescence of microcracks, all growing at the same constant velocity c m . The main effect of microcracking damage, therefore, is not, as commonly believed [START_REF] Ravi-Chandar | Dynamic Fracture[END_REF][START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF][START_REF] Washabaugh | A Reconciliation of dynamic crack velocity and Rayleigh-wave speed in isotropic brittle solids[END_REF], to slow down fracture by increasing the energy required to further propagate a crack, but on the contrary to boost the macroscopic (group) crack velocity to a value larger than what would have been obtained in their absence.

We conjecture that the limiting value c m of the local crack speed is set by the material-dependent dissipative and non-linear processes that develop in the highly stressed/strained zones in the very vicinity of the (micro)crack tips, like e.g. thermal [START_REF] Estevez | Modeling of the competition between shear yielding and crazing in glassy polymers[END_REF], viscoelastic [START_REF] Boudet | Dynamics of crack propagation in brittle materials[END_REF][START_REF] Persson | Crack propagation in viscoelastic solids[END_REF] or hyperelastic [START_REF] Livne | The near-tip fields of fast cracks[END_REF][START_REF] Buehler | Hyperelasticity governs dynamic fracture at a critical length scale[END_REF] processes. As for the subsequent boost from c m to the continuum-level scale velocity v, it is shown here to take the form of a purely geometric factor controlled by two microscopic quantities: (i) The density of nucleation centers ρ and (ii) the mean distance at nucleation d n . These two internal variables characterize the damaging state, and evolve with the amount of mechanical energy flowing into the FPZ. As such, they are material-dependent functions of the external loading K, the knowledge of which permits to fully relate v and c m . This enhanced description of dynamic brittle fracture, demonstrated on PMMA, can likely be extended qualitatively to all materials involving propagation-triggered microcracks, e.g. oxyde glass [START_REF] Ravi-Chandar | Dynamic Fracture[END_REF][START_REF] Rountree | Atomistic aspects of crack propagation in brittle materials: Multimillion atom molecular dynamics simulations[END_REF], polymeric glasses [START_REF] Ravi-Chandar | Dynamic Fracture[END_REF][START_REF] Hull | Fractography: Observing, measuring and interpreting fracture surface topography[END_REF], polycrystals [START_REF] Ravi-Chandar | Dynamic Fracture[END_REF], rocks [START_REF] Moore | The role of microcracking in shear-fracture propagation in granite[END_REF][START_REF] Kobchenko | 4D imaging of fracturing in organic-rich shales during heating[END_REF] and bones [START_REF] Nalla | Mechanistic fracture criteria for the failure of human cortical bone[END_REF]. Further work is required to check this conjecture, and subsequently to quantitatively determine how c m , ρ and are selected in these materials. From the geometric nature of the acceleration factor, we also anticipate that fast macroscopic cracks in other fracture modes could similarly originate from the collective motion of many slow microcracks.

MATERIALS AND METHODS

Experiments

Fracture surfaces were obtained from the experiments described in reference [START_REF] Scheibert | Brittle-quasibrittle transition in dynamic fracture: An energetic signature[END_REF]. Dynamic cracks were driven in PMMA (Young's modulus E = 2.8 GPa and Poisson's ratio ν= 0.36, yielding a Rayleigh wave speed c R =880 m/s) using the Wedge-Splitting geometry sketched in Fig. S1. Specimens were prepared from parallelepipeds of size 140×125×15 mm 3 in the propagation, loading, and thickness directions, respectively. Subsequently, a notch was formed (i) by cutting a 25×25 mm 2 rectangle from the middle of one of the 125×15 mm 2 edges; and (ii) by subsequently adding a 10 mm groove deeper into the specimen. A circular hole with a radius ranging between 2 and 8 mm was eventually drilled at the tip of the groove. Two steel jaws equipped with rollers were placed on both sides of the rectangular cut-out and a steel wedge of semi-angle 15 degrees was pushed between them at constant velocity 38 m/s up to crack initiation. Crack speed was measured using a modified version of the potential drop technique: A series of 90 parallel conductive lines (2.4 nm-thick Cr layer covered with 23 nm-thick Au layer), 0.5 mm wide with a period of 1 mm (space accuracy 40 m) were deposited on one of the two 140×125 mm 2 sides of the specimen, connected in parallel and alimented with a voltage source. As the crack propagated, the lines were cut at successive times detected with an oscilloscope (time accuracy 0.1 s) and allowed to record the instantaneous macroscopic crack velocity v, with better than 10% accuracy. The variations of the quasi-static stress intensity factor K were computed using 2D finite element calculations (software CASTEM 2007) on the exact experimental geometry, assuming plane stress conditions, and a constant wedge position throughout failure of the specimen. Values for the fracture energy Γ were directly obtained from the equation of motion by combining the v measurements and the K calculations.

Post mortem topography images were obtained with an optical profilometer (M3D, Fogale Nanotech, ×5 objective yielding square pixels of size 1.86 m) at various locations along the fracture surfaces in different broken specimens -each zone of observation is characterized by a given value of K. For each location, nine neighbouring images were gathered to provide an observation field of at least 2×2 mm 2 , large enough to carry out statistical analyses. The presence of a highly reflective area at the focus of each conic-like marking results from plastic deformations at microcrack nucleation and allows locating unambiguously all nucleation centers (see Fig. 2A). For many microcracks, fragmentation lines focusing on the nucleation center were also observed, and helped increasing the accuracy of the location. For each marking, we made an initial guess about which microcrack triggered its nucleation. The apex of the marking was defined as the intersection between the segment linking the triggering and triggered centers and the conic-like marking. A new guess was made if the simulated marking did not resemble the observed one.

Simulation

The macroscopic crack front was initially straight, vertical and located on the left of the image. It started propagating towards the right at constant velocity (1 pixel/time step). When the macroscopic crack front reached a distance d n from the closest nucleation center, a microcrack was nucleated and made grow radially at the same velocity. The total front was then made of both the initially straight translating front and the newly created radially growing circular front. When these two coincided, propagation was continued in the unbroken part of the specimen only. Intersection points defined the conic-like marking. The same procedure was applied each time the shortest distance between the total front and another nucleation center was found to have decreased down to the distance at nucleation d n associated with this center. Edge effects were minimized in the evaluation of A by considering only the times after all points of the initial front coalesced with a nucleated microcrack, and before the first point of the total front reached the right edge of the image.

Supporting Information

SI Text

Relative speed of interacting microcracks as deduced from the geometry of conic-like markings.

Figure S2(A) depicts the interaction between two microcracks growing radially at velocities c 1 and c 2 , respectively. Calling ∆ the distance between the two nucleation centers and τ the time interval between the nucleations of the first and second microcracks, the equation describing the successive locations of intersection points (and hence the conic-like marking) is given by:

( ) ( ) ( ) ( ) 2 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 1 1 2 1 1 2 1 1 ∆ + - ∆ - ∆ - + - + ∆ - ∆ - + + ∆ - ± = ∆ τ τ τ c c c x c c c c x c c c c x y , (S1)
where c=c 2 /c 1 and (x,y) are expressed in a frame the origin of which is the first nucleation center and the x-axis goes through both centers (see Fig. S2(A)). As can be seen in Figs. S2(B) and (C), qualitatively different shapes are predicted by this equation depending on the velocity ratio c 2 /c 1 : True (mathematical) conics when c 2 /c 1 =1, egg shapes when c 2 /c 1 <1 and flared shapes when c 2 /c 1 >1. The fact that the markings observed on the post-mortem fracture surfaces look like conics hence suggests a velocity ratio close to unity.

To assess quantitatively the value of the velocity ratio, we directly extracted c 2 /c 1 for each pair of interacting microcracks from the marking's geometry on the fractographic images (see Fig. 1(B) and Fig. S4(top)). Note that seemingly continuous conic-like markings often result from the successive interactions between more than two microcracks, and are hence actually made of several branches, each of them associated with a single pair of interacting microcracks. The analysis procedure we developed is the following: (i) the nucleation centers of two interacting microcracks are selected on the image; (ii) the apex of the associated marking is determined as the intersection of the segment relating the two centers and the marking's branch lying in between; (iii) the marking's shape predicted by equation (S1) is plotted for various ratio c 2 /c 1 while adjusting c 1 τ/∆ so that the apex position remains fixed; and (iv) the value c 2 /c 1 that best fits the experimental marking is selected. Typical examples of the cumulative distributions obtained for c 2 /c 1 are presented in Fig. 2(B) and Fig. S4(bottom). For all the fractographic images analysed, the distributions were found to be roughly Gaussian, with mean values around 0.98-0.99 and standard deviation of 0.02-0.03, irrespective of the density of conic-like markings. Note that local stationarity of the average crack front over the millimeter length-scales of the analyzed fractographic images should imply a geometric mean value of c 2 /c 1 strictly equal to 1. The dispersion smallness around 1 allows identification of this geometric mean with the standard arithmetic one. The observation of a mean value systematically slightly smaller than unity is attributed to an initial accelerating transient in microfracturing events that is not taken into account to derive Eq. S1.

It is then justified to assume that all microcracks propagate at the same constant velocity c 1 =c 2 =c m on a given fractographic image. Equation (S1) can then be simplified to:

2 2 2 2 2 4 4 ) 2 ( ) 2 ( ) 2 ( ) 2 ( 4 ) 2 ( ) 2 ( 4 ∆ + ∆ - - ∆ - ∆ + ∆ - ∆ - ∆ - ∆ - ∆ - ∆ ± = ∆ n n n n n n n n n n n d d d d d x d d d x d d d y , (S2)
where d n =∆-c m τ is the distance between the triggering front and the nucleation point at the instant of microcrack nucleation and is equal to twice the distance between the marking's apex and focus. Note that, contrary to Eq. (S1), Eq. (S2) describes a true mathematical conic (see Fig. 1(C)), the eccentricity and focal parameter of which depend on both d n and ∆.

Variation of microcrack density with stress intensity factor

The K dependency of (Fig. 2(A)bottom) has been understood in Ref. (S1) by assuming the material to contain a population of local weak zones, so-called "source-sinks" (SS). Each one is able to turn into a microcrack provided two conditions are met: (i) the local stress at the considered SS reaches a threshold value * σ (smaller than the local yield stress Y σ ) and (ii) the SS is located at a distance from the crack front larger than d a . Calling v ρ the SS volume density, the surface density is then equal to the number of activated SS beyond d a per unit of fracture area, i.e.

{ }

V d h a v ρ ρ - - ⊥ 2
, where ⊥ h is the thickness (size orthogonal to the fracture plane) of the FPZ, i.e. the layer in which the stress is larger than * σ . V is the excluded volume around nucleated microcracks. The universal square root singular form taken by the elastic stress field around the tip of the growing crack gives α is a dimensionless constant close to unity, and d K relates to K via

K c k K m d ) ( = with 81 . 0 / 1 / ) / 1 ( ) ( ≈ - - = D m R m m c c c c c k (Ref. (S3)
). Finally, one gets:

( )

2 2 a K K C - = ρ with 2 * 1 2 ) ( 1 σ α ρ ρ m v v c k V C + = and ) ( 2 2 2 * 1 2 m a a c k d K σ α = (S3)
The parameter K a represents then the value of K at the onset of microcracking. Below this value, microcracks cannot nucleate and =0. Equation (S3) is found to reproduce the experimental data fairly well (see Fig. 1(A α is a dimensionless constant close to unity. This relation together with Eq. (S3) yields :

a v v n d V d 2 1 2 1 2 1 α α ρ ρ ρ α α + + = (S4)
This equation is found to reproduce the experimental data fairly well (see Fig. (presented in Fig. 2(C)bottom) to the variation of the mean nearest-neighbour distance <∆r> with ρ. For a Poissonian distribution, <∆r>= ) 2 ( 1 ρ while the variance is given by σ 2 ∆r = (4-π) / (4πρ) 0.068/ρ. When ρ becomes of the order of ρ sat , the distance between some nucleation centers and the centers of their triggering microcrack becomes smaller than the distance necessary for them to nucleate. Hence, both centers will open almost simultaneously (within the same time step), and an avalanche will occur. This steric effect yields an effective mean distance at nucleation n d (as measured from fracture surfaces) that saturates above ρ sat at a value m d n µ 50 ≈

. The proportion of micro-cracks involved in nucleation avalanches increases significantly in the vicinity of ρ sat (solid squares in Fig. S6). However, this proportion remains small, less than 7% over the explored range of densities. This explains why the mean-field model yielding Eq. ( 1) captures so well the increase of the acceleration factor with ρ, even above ρ sat (see Fig. 4(B)bottom).

Edge effects in the deterministic reconstruction

Once the growth rule c 2 =c 1 has been ascertained experimentally, and once the nucleation center location and distance at nucleation of all individual microcracks have been determined from image analysis, the main source of mismatch between the reconstruction and the actual fracture surfaces are errors in the time succession of nucleation events. Such errors are unavoidable when using partial images (a few mm 2 ) of the complete fracture surface (a few cm 2 ). Along the top and bottom sides of the image, we lack for the possibility of being triggered by microcracks outside the field of view. From the left side of the image, we lack for a realistic initial front shape which would contain all the information about the precise instants at which the leftmost centers have to be nucleated during the reconstruction. Since no information could be obtained about this initial front shape, we arbitrarily chose a straight vertical front as an initial condition.

In Fig. S7, we illustrate the degree of sensitivity of the reconstruction results to changes in the initial front shape, by running the simulation with a sinusoidal initial front. Its period is chosen to roughly match the average vertical distance between simultaneously propagating microcracks. Its amplitude is set to the mean value of the standard deviation of the horizontal location of the total front as obtained using an initial straight front. The same sinusoidal shape, but translated vertically by half a period, was also tested. As the front involves more generations of microcracks, the patterns become more alike, and hence less affected by the initial conditions. However, some differences can propagate over the whole image. We emphasize that nearly perfect reconstructions could have been obtained through the analysis of the images which include the region where the first microcracks nucleate, because it would have allowed for the determination of the actual initial front shape.

The crucial point for our study is to check that the reconstruction errors induced by edge effects do not quantitatively affect the value A of the acceleration factor. Fig. S7 shows that, for the three very different initial front shapes tested, A is found constant within less than 0.6%. This means that the average velocity measurement hardly depends on edge effects. These results were found to be robust to changes in period and amplitude of the sinusoidal initial front.

Lattice model

The variations of the ratio between the macro and micro-scale velocities, i.e. the acceleration factor A , as a function of microcrack density ρ can be captured by a simple meanfield model. In the model, the nucleation centers are placed at the nodes of a square lattice, so that the distance between two neighbouring centers is kept constant and equal to ρ 1 = . This model arrangement is depicted in Fig. S8, where x and z axes are chosen parallel to mean crack propagation direction and to mean crack front, respectively. At t = 0, the leftmost nucleation centers are turned into microcracks and start to grow radially at velocity c m . The fronts then trigger the nucleation of the next centers when the shortest distance between the nucleation centers and the fronts reaches a distance of n d . The nucleated microcracks then grow radially, coalescing with each other and with the primary crack. The new microcracks trigger the nucleation of the next microcracks, and so on. Because of the invariance to translation along the z-axis the crack can be considered to propagate in the x direction only (see Fig. S8). When the main front has travelled over a distance L = Ac m t along this line, it has triggered ρ L L = micro-cracks. And because each coalescence with a microcrack makes the rightmost point jump over a distance n d , while the crack velocity is c m between these coalescence events, one also gets

ρ n m d L t c L + =
. From the two expressions for L, it can be deduced that ) 1 ( 1

ρ n d A - =
. In real materials, the centers are not aligned along lines parallel to the direction of mean crack propagation but are distributed randomly. We thus propose to modify this equation into:

ρ n d b A - = 1
1 (S5)
where the geometrical constant b (expected to be close to 1) accounts for the projection onto the x axis of (i) the real distance between successive nucleation centers and (ii) the distance jumped during a coalescence event with a non-aligned micro-crack. This equation is the one proposed in the main text (Eq. ( 1)). The two grey circles correspond to both microcrack fronts at time t, if microcrack interaction was ignored. In reality, the intersection of the two fronts leaves on the fracture surface a marking that develops as the blue curve as time t increases. In dimensionless coordinates x/∆ and y/∆, the marking aspect is set by the ratios c 2 /c 1 and c 1 τ/∆. The forms obtained for c 2 /c 1 =0.9 (blue), c 2 /c 1 =1 (red) and c 2 /c 1 =1.1 (green) are plotted in (B) and (C), for c 1 τ/∆=0.05 and c 1 τ/∆=0.2 respectively. where the fitting parameter ρ is the same in all four curves and hence defines the surface density of centers in each image. Note that the same Poissonnian distribution was assumed in Ref. (S4). The stress intensity factor K applying on the macroscopic crack front at these points was computed using finite element calculations (Materials and Methods). Its value together with the fitted value ρ is reported in each graph inset. Typical examples of investigated zones (985×745µm 2 in size) at three different microcrack densities. Each conic branch has been attributed a given color and the nuclei of the two corresponding interacting microcracks has been joined by a dotted segment of the same color. Note that a conic mark is often made of several of these branches. The ratio c 2 /c 1 is the only adjustable parameter in equation (S1) to determine the branch geometry once the nuclei position and the branch apex are set. Bottom: Corresponding distributions for c 2 /c 1 . In the three cases, the distributions are found to fit normal distributions of mean value ~0.98-0.99 and standard deviation ~0.03-0.04, irrespective of . ). Three different initial conditions were used. A: Straight vertical line. B: Vertical sinusoidal shape with a period of 186µm and a peak-to-peak amplitude of 242µm. C: Same sinusoidal shape, but translated vertically over half a period. In all cases, red dots indicate nucleation centers. Bottom: Time evolution (scaled by c m ) of the average location of the simulated crack front for the three different initial front shapes. For each curve, the dashed line is a linear fit of the data between the two black dots. The fitted slopes between the black dots, which directly give the value of the acceleration factor A, are 1.679±0.001 (straight), 1.666±0.001 (sinus 1) and 1.687±0.001 (sinus 2). This shows that the value of A is hardly sensitive to edge effects in the reconstruction.

Fig. S4: Distribution of microcrack relative velocity

FIG. 2 .

 2 FIG. 2. Fractographic signature of microcracking in the dynamic fracture of PMMA. (A) Typical fractographic microscope image (K 3.10 0.05 MPa.m 1/2 ). Bright regions correspond to microcrack nucleation centers (Materials and Methods). (B) Red dashed circle arcs sketch successive front locations of two interacting microcracks (nucleated at t=0 and t=τ) growing radially at speeds c 1 and c 2 . Fitting of fractographic branches (color lines) with a geometrical model (Eq. (S1)) allows measuring c 2 /c 1 . (C) When c 2 = c 1 , markings (green line) are conic branches (Eq. (S2)) and the distance d n between the triggered microcrack center and the triggering front at the nucleation time t = τ (highlighted in red) is twice the apex-to-focus distance, O 2 P 12 .

FIG. 3 .

 3 FIG. 3. Microcrack dynamics deduced from fracture surfaces. (A) Top: cumulative distributions for the number of nucleation centers in square regions of size a, for K 4.15 0.07 MPa.m 1/2 . Solid lines: Poisson function

FIG. 4 .

 4 FIG. 4. Deterministic reconstruction of microscale damage and fracture processes. (A-B): successive snapshots of the reconstructed crack propagation and associated conic markings for ρ =64.6 mm -2 . Crack propagates from left to right. (C-D): Fracture surface images (grey level) for (C) ρ =27.5mm -2 (K 2.77 MPa.m 1/2 ) and (D) ρ =64.6 mm -2 (K 4.18 MPa.m 1/2 ) compared to the reconstructed conic markings (red lines). Red dots indicate nucleation centers.

FIG. 5 .

 5 FIG. 5. From slow microcracks to fast collective macroscopic crack motion. (A) Time evolution (scaled by c m ) of the average location of the simulated crack front for various ρ. The fitted slopes of these curves define the acceleration factor A. A=1 for ρ =0. Inset: Evolution of the location of a single point of the simulated front, for ρ=64.6mm 2 , together with that expected for ρ=0 (slope 1). Jumps correspond to coalescence events with microcracks. Between jumps, the slope is close to 1. (B) Black dots: Evolution of the reconstructed acceleration factor A as a function of ρ. Triangles: Ratio of the measured macroscopic crack speed, v, over the microscopic velocity fitted to be c m = 217 3 m/s. Error bars indicate the minimum and maximum measured velocities within the considered fractographic image. Thick red line: Eq. 1 with b=1.19 0.02. A change in regime occurs for ρ = ρ sat (vertical dashed line), which corresponds to a velocity of 1.67 c m (horizontal dashed line). stands for 95% confidence interval.

K

  )bottom in the main text) up to a value K b 3.1MPa.m 1/2 which is associated with the microbranching onset (see Ref. (S1)). The loss of agreement above K b is actually expected since LEFM is invalid in the presence of microbranches (see Ref. (S2)). The fitted parameters C and a of mean distance at nucleation with microcrack density In the scenario invoked above, the relation between the mean distance at nucleation n d and d K is deduced from the universal square root singular form taken by the elastic stress field around the tip of the growing crack (see e.g. Ref. (S3)). The mean distance n d at which the stress level reaches *

Figure

  Figure S6 superimposes the variation of the mean distance at nucleation n d with ρ

Fig. S1 :

 S1 Fig. S1: Sketch of the experimental setup.

Figure S1 :

 S1 Figure S1: Sketch of the so-called wedge splitting geometry used to grow dynamic cracks in PMMA (see Materials and Methods and Ref. (S1)).

Fig. S2 :

 S2 Fig. S2: Geometry of conic markings.

Figure

  Figure S2: (A) Sketch underlying equation (S1). A first microcrack nucleates at time t=0 at frame origin O 1 , and subsequently grows at velocity c 1 . A second microcrack nucleates at time τ at point O 2 of coordinates (x=∆, y=0), and subsequently grows at velocity c 2 . The two

Fig. S3 :

 S3 Fig. S3: Spatial distribution of nucleation centers.

Figure S3 :

 S3 Figure S3: Cumulative distribution of the number of nucleation centers contained in square regions of lateral size a, for each of the eight fractographic images analyzed. The cumulative distribution is defined as the proportion of square regions containing a number of nucleation centers which is strictly superior to the value in abscissa. Four values of a were chosen, namely a = 100, 200, 300 and 400 µm. In each graph, solid lines represent Poisson fits =

Figure S4 :

 S4 Figure S4: Direct extraction of the relative speed between two interacting microcracks. Top:Typical examples of investigated zones (985×745µm 2 in size) at three different microcrack densities. Each conic branch has been attributed a given color and the nuclei of the two corresponding interacting microcracks has been joined by a dotted segment of the same color. Note that a conic mark is often made of several of these branches. The ratio c 2 /c 1 is the only adjustable parameter in equation (S1) to determine the branch geometry once the nuclei position and the branch apex are set. Bottom: Corresponding distributions for c 2 /c 1 . In the three cases, the distributions are found to fit normal distributions of mean value ~0.98-0.99 and standard deviation ~0.03-0.04, irrespective of .

Fig. S5 :

 S5 Fig. S5: Distribution of distances at nucleation.

Figure S5 :

 S5 Figure S5: Cumulative distribution, P, of the distance d n between the triggering crack and the nucleation center at the time of nucleation, determined from each of the analyzed fractographic images. In each graph, the solid line shows the fit of the form P(d n ) = ((d max d n ) / ( d max d min )) 2 , where d min and d max are positive quantities. The fitted values d min and d max together with the surface density of nucleation centers ρ are reported in inset in each graph. d min decreases with ρ and becomes equal to zero when ρ is larger than 36 mm 2 . d max increases with ρ over the whole explored range. Neglecting the small value of d min allows us to use a single parameter (e.g. n d (ρ)) to define the whole distribution and its variations.

Fig. S6 :

 S6 Fig. S6: Saturation of n d and avalanches.

Figure S6 :

 S6 Figure S6: Evolution of the fitted mean distance at nucleation, n d , (thick red line, see Fig.

ρ

  2(C)bottom), as a function of ρ. It is compared to the mean nearest-neighbor distance in a (black solid line, error bars correspond to ± one standard deviation ρ squares indicate the proportion of micro-cracks involved in avalanches, as computed from the reconstruction.

Fig. S7 :

 S7 Fig. S7: Influence of edge effects on the reconstruction and average velocity

Fig. S8 :

 S8 Fig. S8: Sketch of the lattice model.

Figure S8 :

 S8 FigureS8: Sketch of the mean field model used to establish the relationship between the acceleration factor A, the microcrack density ρ, the mean distance at nucleation n d and the microscopic velocity c m of micro-crack growth (Eq. (1) in the main text). The nucleation centers are placed on the nodes of a square lattice. The crack front is plotted at 10 successive times separated by a constant interval n d /c m (from blue to red). At each time, the rightmost point of the front is projected along the x axis (thick black ticks on the x axis). It propagates at a constant velocity c m between coalescence events, and jumps over a distance n d at coalescence.

We thank A. Prevost for his help with the profilometry measurements. We thank K. Ravi-Chandar for helpful discussions. We thank L. Barbier, G. Debrégeas, A. Malthe-Sorenssen, J. Mathiesen, K.J. Måloy, P. Meakin, and C. Rountree for careful reading of the manuscript. We acknowledge funding from French ANR through Grant No. ANR-05-JCJC-0088, from Triangle de la Physique through Grant No. 2007-46, from Mexican CONACYT through Grant No. 190091 and from the European Union through Marie Curie grant No. PIEF-GA-2009-237089. This Letter was supported by a Center of Excellence grant to PGP from the Norwegian Research Council.

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Scheibert J, Guerra C, Célarié F, Dalmas D, Bonamy D (2010) Brittle-quasibrittle transition in dynamic fracture: An energetic signature. Phys. Rev. Lett. 104:045501. S2.

Sharon E, Fineberg J (1999) Confirming the continuum theory of dynamic brittle fracture for fast cracks. Nature 397:333-335. S3.

Freund LB (1990) Dynamic Fracture Mechanics (Cambridge University Press, Cambridge, England). S4.

Ravi-Chandar K, Yang B (1997) On the role of microcracks in the dynamic fracture of brittle materials. J. Mech. Phys. Solids 45:535-563.

NOTE:

The following pages correspond to figures S1 to S8, in the order. ρ=24.3mm -2 ρ=36.1mm -2 ρ=64.6mm -2