Dr Renaud Bellais
email: renaud-bellais@voila.fr

TECHNOLOGY AND THE DEFENSE INDUSTRY: REAL THREATS, BAD HABITS, OR NEW (MARKET) OPPORTUNITIES?

Keywords: Defense, R&D, technology, security, procurement JEL H56, H57, O32

Technology has been playing a central role in defense spending or arms-producing countries since World War II. Although there has been no major threat or conflict since the 1990s, defense R&D absorbs a large share of military expenditures, as well as public R&D. This technology-centric paradigm results from uncertainties surrounding defense matters and the need to avoid strategic surprises. However, one can wonder whether such a paradigm is still adapted to today's defense needs. This is a trend strongly driven by the supply side: defense firms have developed a business model that cannot survive without launching new programs, hence a high level of defense R&D. This explains both an overinvestment in technology, resulting in the development of unaffordable technologies or unsustainable performance targets, and the technology-centric model that defense firms favor in side markets like security.

INTRODUCTION

Despite the end of Cold War, armed forces continue to allocate a large share of their investment effort to defense R&D, particularly in arms-producing countries. Emerging powers like China, India and Brazil are also dedicating an increasing share of their defense budgets to R&D while these countries were not use to invest so heavily in this domain.

One may wonder whether such investment results from the need for new capabilities, linked to existing or emerging threats, and whether the outputs of defense R&D match the expectations. Defense R&D has been widely criticized since the 1970s in the economic literature because of its opportunity cost and its relative ineffectiveness compared to civilian R&D [START_REF] Lichtenberg | Economics of Defense R&D[END_REF]. It is therefore quite surprising that this part of defense spending was sheltered or at least preserved from budget cuts after the end of Cold War although there were mainly low-tech conflicts since the beginning of 1990s.

Such a situation reflects a paradigm shift in the twentieth century. The defense industry and armed forces' procurement have been driven by technology since World War II and this industrial model has survived after the collapse of the Soviet Union. Nevertheless one can ask if a technology-driven defense market is still able to deliver capabilities that armed forces require for today's conflicts.

Moreover does this technology-centric approach correspond to today's business model of the defense industry and does it respond to a true need to face potential threats? This focus on technology has characterized the so-called "military-industrial complex" for four decades. However, is this approach still relevant in the post-Cold War era? Or are defense firms mainly trying to preserve their technologydriven industrial model? This paper aims to address these questions. Part one underlines how the technology-centric paradigm has been dominating defense capabilities since the end of World War II and why it is still dominant in arms-producing countries despite the end of Cold War. Part two looks at the limit reached by such a paradigm when dealing with current conflicts or ongoing programs, even though advanced technology seems to remain a core parameter in current and future defense efforts. This leads to look at the predominance of technology in defense firms' strategy, which explains both the limits reached through a technology-centric approach and the repositioning of these firms on the emerging market of security.

THE PERSISTENCE OF A TECHNOLOGY-CENTRIC DEFENSE

While military capacities have always played an important role in defense matters, a new paradigm emerged in the 1940s centered on technological issues. Such a paradigm was steadily reinforced by the Cold War and it survived after 1990, since a technology-centric defense is still considered the best approach to solve security challengesat least for the largest military spenders.

Technology disruption as the driver of military capabilities

Since the 1940s, armed forces have fielded advanced technology through an aggressive pursuit of R&D and the development of a high-technology defense industrial base of unprecedented scale.

Technological superiority is considered to be a key element to achieve defense effectiveness, and R&D plays a major role in accessing relevant advanced technologies. This explains why the United States and Western European countries have been pouring a large share of their R&D expenditures into applications for defense since World War II.

Technology played a minor role before the middle of the twentieth century, but World War I and even more World War II marked a radical change. This latter demonstrated the potential that scientific and technical progress can provide to armed forces, thus legitimizing large R&D budgets in most industrial countries. This was a major change, which has given birth to what [START_REF] Freeman | The Economics of Industrial Innovation[END_REF] called "Big Science." For instance R&D spending in the United States represented only 0.8% of federal budget in 1940, but more than 10% by 1960and defense R&D received most of these additional resources.

Such a dedication proceeded from major successes in defense R&D like the "Manhattan" project. As Väyrynen (1983, p. 63) states, "the invention of the atomic bomb modified entirely the importance of technology towards national military policies." The greatest advocate of defense R&D was unquestionably Vannevar Bush, who supervised technological projects for the Pentagon during World War II. In Science, The Endless Frontier, he popularized the idea that government-funded R&D produces large technological spin-offs. His book revealed how much the war effort generated technological opportunities and argued that it is fruitful to maintain it in peacetime. Vannevar Bush created a founding myth, which soon became the justification of government-funded R&D (both military and civilian).

The Cold War reinforced this technology-centric paradigm, as the USSR and the United States were looking for means that would provide strategic superiority at whatever costs. In a seminal paper on industrial policy, Reich (1982, p. 864) states: "Large scale defense and aerospace contracts have provided emerging industries in the United States with a ready market for which they have expanded production and thereby gained valuable experience, know-how and scale economies. The Pentagon's willingness to pay a high premium for quality and reliability, moreover, has helped emerging industries bear the costs of refining and 'debugging' their products." Indeed armed forces have been able to spend large amounts of R&D to sustain disruptive technology as well as achieve incremental innovations, because a "small edge in performance can mean survival" (Alic et al., 1992, p. 114). This paradigm is not only a result of the quest for technological superiority, but also from the need to avoid strategic surprises that can result from scientific and technological path-breaking innovations. As Gansler (1989, p. 217-218) puts it, "we have what refer to as the technological imperative: because we can have it, we must have it (...) Technological opportunities resulting in incremental changes are rapidly accepted and enthusiastically pushed forward-perhaps too unquestioningly." Such a quest for technological superiority and innovation makes sense when armed forces are trying to preserve and reinforce their strategic superiority. However, by nature, scientific and technological opportunities are unpredictable. In order to face such uncertainty, armed forces try to avoid strategic surprises by anticipating any possible technological disruption. [START_REF] Saunders | Priority-Setting and Strategic Sourcing[END_REF] explain that it is imperative to prevent "technological surprises," which threaten or jeopardize national security. Services are extremely risk-averse, in the sense that the cost of failure to find the best technology (technically speaking, of course) can appear very high. Research by definition is risky, and to maintain technological superiority armed forces must allocate a significant share of their resources to basic and applied research in areas of high potential technologies.

Regardless of whether such investments succeed, failure can actually be one of the desirable features of advanced R&D, because it demonstrates that a given technology cannot be used by an enemy. If there are not some failures, technologies might not be pushed far enough nor all its military uses explored.

By exploring alternative technologies, defense R&D helps maintain technology scouting by continuing "experimenting long after a market would have standardized on one technology" (Cowan and Foray, 1995, p. 865). Defense R&D in new technologies eventually makes these technologies affordable by moving them along learning curves, lowering costs for later civilian adopters. Defense R&D can then be considered as a complementary action vis-à-vis commercial firms, as it can provide what market mechanisms fail to supply. When the military supports more than one technology in the development phase, it increases the choice of civilian users, especially in the case of new, complex and/or expensive technologies.

Once driven by the East-West arms race, technological superiority continues to be the core of defense procurement. Although Western countries no longer face the threat posed by a global peer competitor like the former Soviet Union, they still live in a very dangerous world. Western armed forces feel that strategic surprises can be avoided by mastering innovation and disruptive technology. This explains why defense R&D remains a key dimension of defense spending.

Still a large effort of R&D in defense

Trends in military spending show that a defense technology race remains at the heart of defense procurement in the Western world. Despite the end of Cold War, defense R&D absorbs a large share of defense spending although no country is able to challenge the United States or NATO countries in the field of defense technology, at least in the medium term.

Conflicts since the 1990s can be identified as low-tech wars. Gulf Wars I and II demonstrated this by proving that the most dangerous enemies or troublemakers did not possess high-tech capabilities; and that there is a major technological gap between NATO countries and most other countries. There is indeed a technological barrier to reach the most advanced military capabilities [START_REF] Middleton | The effect of defense R&D on military equipment quality[END_REF], which naturally protect NATO countries for several years from any technologically-sound military competitorincluding China, India and Russia.

Nevertheless, defense R&D remains a central component of scientific and technology investment in large arms-producing countries, as illustrated in the graph below. Through all transformations of 1990s and 2000s, armed forces have preserved their ability to invest in R&D. One can note that the Pentagon was very careful to "sanctuarize" such budget in the 1990s, when "peace dividends" led to a one-third reduction of its budget. Even though defense R&D was less protected in other large armsproducing countries, they remained larger than expected, especially when the economic impacts of defense R&D were highly challenged [START_REF] Lichtenberg | Economics of Defense R&D[END_REF][START_REF] Serfati | Production d'armes, croissance et innovation[END_REF].

Graph 1 -Defense R&D as percentage of total GBAORD GBAORD: Government Budget Appropriations or Outlays for R&D Source: OECD, Main Science and Technology Indicators, various issues.

Therefore defense R&D has kept a key role in the national innovation systems of arms-producing countries. One may wonder why armed forces have not changed their strategy, especially since their budgets are under pressure (except for the United States since 2001) and recent wars have not required advanced capabilities. In fact, the post-Cold War era demonstrates that the technologycentric paradigm was not only related to this specific period but translates a radical change in the way armed forces conceive military matters. This is particularly relevant in the United States, where technology has become and will remain the essential dimension to win wars.

Even though technology is neither the unique answer nor effective by its own right, it is perceived as crucial to analyze threats and to maintain military dominance. This is the reason why merely sustaining the technological effort is not considered an option but a necessity to counter unexpectedand sometimes unpredictablefoes. This view also explains why the United States and Europe have kept spending over €65 billion a year in defense R&D over the last decade, according to the European Defense Agency.

Maintaining technological superiority requires an effective technology policy both in terms of operational and budgetary impacts. This is the reason why R&D represents between a quarter to half of equipment spending of large NATO countries, as shown in the table below. Such a policy can appear as expensive, and it can be questioned whether the technology-centric paradigm is still relevant to defense missions. Similarly, the United Kingdom has preserved its defense R&D effort in spite of budgetary tensions.

Although the above graph and table show that R&D expenditures have been decreasing over the last few years, the British ministry of defense still considers technology as a vital dimension of its strategy.

This position was reaffirmed in February 2012 when the British government published the policy paper entitled National Security Through Technology -Technology, Equipment, and Support for UK Defence and Security. It is noticeable that, in this specific approach, the preservation of defense-related R&T goes through a more global perimeter of activities, because this policy paper explicitly takes into account civilian security.

Beyond the "historical" leaders of defense R&D, emerging (or re-emerging) powers seem to follow the same technology-centric paradigm. Non-western countries such as Russia, India, or China have been increasing their defense R&D since the 1990s, while they have yet to reach a maturity level similar to that of the biggest NATO spenders. Finding reliable statistics about defense R&D spending in these countries proves very difficult and the reliability of data is questionable. Even the Pentagon is very cautious when it estimates China's military spending.

3

Nevertheless, emerging countries are focusing more and more on advanced technologies for their defense capabilities, and the situation is very similar in Russia, India, Brazil and Turkey. It appears that any country, which has military ambitions and enough resources, is dedicating a large share of its defense budget and economic resources to defense technology.

The predominance of a technology-centric paradigm can be explainedat least partiallyby the evolutions of international relations since the end of 1980s, as well as the transformation of threats. In other words, armed forces can perceive technology as a means to anticipate threats (in a technical and operational perspective) and increase their ability to manage more and more heterogeneous configurations.

Managing uncertainty through technology

Since the 1990s threats have become dramatically more diverse than ever. Most armed forces consider advanced technology as an appropriate and adequate answer to both traditional enemies 2 Livre blanc sur la défense et la sécurité nationale, particularly from p. 269. when uncertainty is high about potential threats and risks, as is the case in the post-post-Cold War era. As Donald Rumsfeld (then U.S. Secretary of Defense) put it, "our challenge in this new century is a difficult one: to defend our nation against the unknown, the uncertain, the unseen, and the unexpected" (Rumsfeld, 2002b: 23). This analysis of an uncertain as well as unpredictable world is at the core of the defense policy of major countries, particularly the United States 4 , the United Kingdom 5 and France.

6 [START_REF] Sapolsky | Buying Weapons Without an Enemy[END_REF] underlines that this is the main reason why the so-called "Revolution in Military Affairs," launched by the Pentagon in the 1990s, appears so jumbled. A true revolution in military affairs is more than building new high-tech weapons; this process is also about new ways of thinking defense and fighting. Armed forces do not only require improved systems but also alternative ones and new doctrines to employ both new and old systems.

Such uncertainty seems even more important because of asymmetric threats and the lack of a clearly identifiable enemy. Defense planners cannot know precisely what kind of risks they have to avoid or fight, so that they must assure that forces can rapidly counter them by increasing their conceptual and organizational adaptability and flexibility. Such an evolution has strong implications at the procurement level, for acquired systems must be reoriented towards new missions, and new needs must be satisfied quickly.

This implies that armed forces allocate a substantial amount to defense R&D for two reasons. How can armed forces manage their procurement policy when the rules of war are changing, at least partially? Current defense procurement principles become obsolete, but are not replaced by new ones.

It is possible to draw a list of asymmetric threats, and many analysts do so. For instance, [START_REF] Mckenzie | The rise of asymmetric threats: Priorities for defense planning[END_REF] identifies a typology of six potential asymmetric threats: nuclear, chemical, biological, information operations, operational concepts, and terrorism. However this kind of list is not necessarily useful here. It is more relevant to understand how armed forces can prepare themselves to emerging, unexpected threats rather than targeting only those that have been identified. In a way military powers are still trapped in a "Maginot line" system, relevant during the Cold War but less and less effective today. In fact, "as in judo, notes Saint Germain ( 2001), [our potential adversaries] are going to challenge us not with the same weapons but in targeting our weaknesses".

NATO countries focus their reply mainly on technological innovations and weapons superiority, notably developing multi-purpose systems. While multipurpose systems are seldom as efficient as singlepurpose ones, they may make the most sense because of the great uncertainty about foreseeable threats. Nonetheless single-purpose systems remain relevant when the strategic environment becomes clearer and specific threats are identified. Procurement strategies sould be revisited and modified to integrate these new constraints, leaving more room for new or unconventional programs.

Interpreting the impacts of 9/11 events Donald Rumsfeld used an eloquent image to describe the situation Western armed forces have to face: "It's like dealing with burglars. You can't possibly know who wants to break into your home, or when they might try it-but you do know how they might try itand that you need good, solid deal bolt on your front door. You know that they might try breaking through a window-and that you need a good alarm system. You know it's better to stop them before they get in-and that you need a police force to patrol the neighborhood and keep the bad guys off the streets." (Rumsfeld, 2002a) Weapons acquisition takes place today at two levels: non-symmetric and asymmetric. Classical Preparing for the future in an uncertain world requires innovative visions, as well as forces and capabilities that can adapt quickly to new challenges and unexpected circumstances. As Metz and Johnston (2001, p. 15) explain: "While iconoclasts and nonconformists should not rule the military, they should be valued, preserved and heard." However such an evolution does not appear as selfevident. It requires not only a complete revolution in the way military procurement is handled, but also a great awareness on more and more diversified scientific knowledge and technology.

The loop "identified threat/corresponding technology" appears necessary but not sufficient, and armed forces can no longer rely on pure defense technology. More than ever armed forces must prevent military "surprises". The latter were essentially localized in technological fields, but nowadays emerge from various sources… This can lead to an increase rather than a reduction in defense R&D as a means to increase technology scouting and enhance the awareness about possible strategic disruptions.

If such a strategy is necessary, is it enough and truly appropriate? For instance the overemphasis on technology in Western countries could lead to underestimating the many low-tech means and ways by which adversaries could asymmetrically respond to one country's technological supremacy. Such supremacy did not prevent the Netherlands' defeat in Indonesia, France's defeats in Indochina and Algeria, or more recently Russia's difficulties in Chechnya. Even the United States cannot rely solely on its technological superiority, since it does not automatically guarantee victory on the battlefield (as the Vietnam War demonstrated). Actually it is not possible to find a technological silver bullet for each asymmetrical threat. Although technology remains a crucial element in military battles, an alternative approach might be found.

DEFENSE FIRMS' STRATEGY IN A STALEMATE?

Today's defense R&D model, set up during the Cold War, appears less and less adapted to deliver defense innovation at a sustainable cost. Not only budgets seem out of control, but one could also question the choices made in terms of R&D targets. This leads to wonder whether defense R&D provides military capabilities or rather helps sustain the business model of the defense industry. This second hypothesis can be validated when analyzing how these firms are repositioning on the security market and how they are molding such a market through a technology-centric paradigm.

The drifting costs of defense R&D

Even though advanced technologies can support some missions and maintain strategic dominance for arms-producing countries, defense R&D has become increasingly expensive while being less and less able to generate disruptive technologies. [START_REF] Serfati | Production d'armes, croissance et innovation[END_REF] had already underlined the fact that defense R&D was focusing on incremental innovations, which are by definition much more difficult to achieve and therefore increasingly costly. Almost two decades later, one can note that such an approach has not changed when reading official reports on the management of defense programs. Almost all programs of complex systems are encountering major difficulties concerning either technological developments or budgetary targets.

As If the Pentagon provides the most obvious example of such drift, major arms-producing countries are in a quite similar situation. [START_REF] Kirkpatrick | Trends in the costs of weapon systems and the consequences[END_REF] or [START_REF] Pugh | Retrospect and prospect: Trends in cost and their implications for UK aerospace[END_REF] underline that fixed costs represent a larger and larger share of defense programs, and R&D constitutes a key element in these costs. In other words, such programs appear overambitious in terms of technology and performance. One can wonder whether these outcomes result from the management of defense R&D or from inadequate objectives.

Moreover, one can wonder whether defense R&D is truly useful, since economic literature shows that its effectiveness results from both the spending level in a given technology and how this spending is managed. In fact, threshold affects results from the evolution of defense-related technologies. In particular, one cannot expect a linear relationship between R&D spending and its output: below a certain level of investment in a given technology, the output of R&D falls rapidly. Exploring investments in integrative technologies in a dynamic optimization framework, Setter andTishler (2006, 2007) find that under nonlinear, convex development costs it is not optimal to build military forces using a myopic, short-term approach. It is difficult to transform an armed force within a few years. Consequently, early investment in technological infrastructure is required because entry cost in technology is high and transformation period extends over more than a decade.

If a country's investment in a given technology is too limited, it cannot expect to keep pace with the state of the art, and it is not worth investing in that technology. This explanation can cover part of technology issues, especially for countries with limited budgetary resources. When looking at EDA statistics, it becomes obvious that defense R&D is not optimally used. Nevertheless it seems insufficient to understand the limits or stalemates of current defense R&D.

Reports from public audit organizations in France 7 and the United Kingdom 8 put into relief that armed forces choose immature or unproven technologies that could jeopardize the delivery of capacities, while such specifications are not essential to achieve armed forces' missions. As [START_REF] Rogerson | Quality vs. Quantity in Military Procurement[END_REF][START_REF] Rogerson | Economic Incentives and the Defense Procurement Process[END_REF] demonstrates, there is a systematic bias in the defense procurement in favor of quality even though this choice results in a reduced quantity of systems.

Such a context has nurtured an industrial environment in which firms are encouraged to promote advanced technology rather than look to minimize unit costs [START_REF] Serfati | Production d'armes, croissance et innovation[END_REF][START_REF] Bellais | Production d'Armes et Puissance des Nations[END_REF]. Therefore today's drift in R&D costs reflects, for a large part, the business model which has characterized the defense industry since World War II. The blame should be put on the technology-centric paradigm that defines the essence of such an industry.

Missed targets or wrong strategy?

One may therefore wonder whether current R&D policies can achieve their goals or whether they should be reviewed, especially to manage tensions between capabilities required immediately and the development of advanced systems that can take years or even decades. When U.S. armed forces were deployed to Iraq and Afghanistan, the Pentagon was accused of neglecting the protection of ground forces while allocating fantastic budgets to programs like the F-22 fighter aircraft whose relevance to these operations was inexistent.

Two dimensions can be criticized about the way defense programs are defined: armed forces focus on incremental innovations that appear more and more costly; and both armed forces and the industry tend to favor technology-intensive solutions, which is not always the best way of solving issues.

First, armed forces seem to have too many weapon systems they do not necessarily need and too short of those they do need. "But in spite of these shortages, as underlined by Rumsfeld (2002b: 28), the department postponed the needed investments, while continuing to fund what were, in retrospect, less valuable programs." How can we understand such a situation?

The relative stability in arms production results from armed forces' wish to maintain production lines that are considered as crucial for national security. Since they are responsible for national security, armed services may have an industrial and technological base at their disposal to respond to an emergency. As a result, some production units represent a strategic resource and must be preserved.

"The Defense Department would find it risky and even reckless, notes Kurth (1993, p. 308), to allow a large production line to wither and die for lack of a large production contract". This is the reason why armed forces have sometimes launched new productions when a major program ended to avoid closing a unique and essential capability, even though such systems do neither provide path-breaking innovations nor fill a crucial lack of equipment.

This "follow-on principle" [START_REF] Kurth | The Political Economy of Weapons Procurement[END_REF] has two main consequences: a relative stability of the main arms-producing firms, and a technological continuity-resulting from both firms' lobbying and armed forces' conservatism. This explains why the military-industrial complex "has endured for several decades, in some cases dating back to the Second World War, despite the ebbs and flows, the booms 7 Cour des Comptes.

8 National Audit Organization. and busts in defense spending" (Kurth, 1993, p. 307). Such a system constitutes a truly high barrier against technological disruptions and the introduction of new, alternatives defense systems… as long as the tension between available weapon systems and operational needs is not too strong. "The striking fact about the post-Cold War period, mergers and acquisitions notwithstanding, is that essentially all the military aircraft, armor vehicle, and warship building lines opened at the end of Cold War stayed open and productive", explains Sapolsky (2002, p. 35). Similar conclusions could be drawn in most European countries.

Such a bias induces a tacit agreement with defense firms that promote the renewal of existing systems-relying on assets, technology and know-how they master. This strategy can be useful to maintain the defense industrial and technological base, but it has strong negative effects especially in an age of uncertainty and geopolitical evolution. Indeed the follow-on principle favors existing systems against new ones, which do not have the same backing from firms, armed forces and decision-makers for economic, operational and political reasons respectively… Moreover such "follow-on" programs require huge credits and then reduce the available budget for other programs. Here the eviction effect might be strong (and often it is), especially when a less clear geopolitical context undermines the justifications of defense expenditure in budget debates. 9

Second, the technology-centric paradigm leads armed forces and the industry to look for solutions which appear quite expensive and not optimal with regard to operational, technical and budgetary dimensions. Iraq and Afghanistan operations give a good illustration of such a trend.

As mentioned before, armed forces have faced new threats for which existing capabilities were not fully adapted. Thanks to the massive increase of its budget, The Pentagon has been able to launch new, disruptive programs. Nevertheless, a closer look reveals that choices favor the reproduction of Cold-War program targeting.

In fact, the Pentagon picked up solutions that are technology-intensive, relying mostly on unproven technology and architecture. Two examples that illustrate this trend: first, the Pentagon spent billions of dollars to develop counter-IED devices 10 and yet most of these programs did not deliver expected outcomes (even though some R&D investments were able to reduce risks, at least in Iraq). Similarly, in order to reduce the sniper threat in Afghanistan, the Pentagon promoted the XM25 Counter-Defilade Target Engagement Systems, which appears as a "technological tool" not fully adapted for operational deployment and not fully reliable.

There are many examples of technological over-specifications in defense programs in the United States (For instance, does the U.S. Air Force needs the F-22?) and in major arms-producing countries. The failure of NIMROD in the United Kingdom illustrates a similar situation, as well as FELIN electronic systems for infantry in France.

9 "Democracies, comments Sapolsky (2001, p. 34), are motivated to buy weapons for what can be best expressed as combinations of survival panics and desire for pork […] Without an acknowledged and believable enemy nearly all weapon purchases tend to look like the distribution of pork to most outside and even some inside observers." 10 Improvised Explosive Devices.

Even though technology-driven solutions can deliver capabilities for certain threats, it appears that they mainly follow an industrial model where armed forces look for advanced technologies and where the industry is organized to deliver such high-tech systems. This interactive model leads to today's troubles (technological underperformance, unmanageable costs), and it explains why arms-producing countries allocate such a large share of their defense spending to R&D. This does not mean that defense R&D is unnecessary, but the way it is used could be significantly improved to increase its output and reduce its costs [START_REF] Oudot | Choix contractuels et innovation : le cas de l'approvisionnement de défense[END_REF].

The security market as a substitute for defense firms

As defense budgets are bound to decrease in forthcoming years, the ability of the defense industry to preserve its business model will be challenged. It appears that this industry has already looked for substitution markets, and it has deeply invested in the security market. This trend was obvious in the United States, when the Bush Administration created the Department of Homeland Security and gave it billions of dollars, mainly to procure systems in the same way the Pentagon does.

It is also interesting to analyze how the defense industry has influenced the structuring of the security market in Europe (Boulanin and [START_REF] Bellais | Production d'Armes et Puissance des Nations[END_REF]. Indeed, even though this market is smaller than the American one, its emergence demonstrates how defense firms try to sustain their technology-centered economic model by duplicating it in alternative markets.

In 2003, following the release of its security strategy "For a Secure Europe, in a better world," the European Commission decided to start a research program on security. As part of the program, the Commission gathered a "Group of Personalities," whose role would be to identify needs and research priorities and provide concrete recommendations for the design of an ad hoc research program. This group was mainly composed of defense actors from both public institutions and the industry, including four main European defense firms (EADS, BAE Systems, Thales and Finmeccanica).

Arms-producing firms have been ostensibly associated with the agenda setting process and their influence has been remarkable if one considers recommendations formulated by the Group of Personalities. First, the Group of Personalities clearly promotes technology as a central component to the EU's security strategy, "as a force enabler for security in Europe." The key argument being the following: "Technology itself cannot guarantee security, but security without the support of technology is impossible" (Group of Personalities, 2004: 6). Such promotion of technology is typical of the intellectual paradigm that dominates the defense sector. For example, in the American context in particular, the defense industry offers to respond to each kind of threat with a specific and adapted technology.

Secondly, the report of Group of Personalities underlines that many possible synergies could be found between civil and defense industries (Group of Personalities, 2004, p. 12). In terms of industrial knowhow, some military technologies have found some civilian commercial applications and on the other hand, military network-centric systems are largely developed on civilian commercial technologies. The technological needs of police and military forces are also said to be increasingly overlapping (Group of Personalities, 2004, p. 12).

This report takes, for instance, the case of UAVs, suggesting that they can be used both for border surveillance by customs authorities and crisis management missions or surveillance missions carried out abroad by military forces. Articulating that there is more and more of a "continuum" between the technological base of defense and civil security applications, the Group of Personalities recommended to the EU to support a greater integration of defense and security technologies, in particular by favoring the development of sophisticated security goods and services and by taking specific measure to support the European defense industrial base (Group of Personalities, 2004: 20). PASR 2004PASR -2006) ) and FP7, arms-producing firms seem to have benefited to a great extent from allocated research credits.

Out of 24 projects funded under PASR, 17 were led by firms operating mainly in the defense sector.

EADS, BAE Systems, Finmeccanica and Thales, the four firms sitting on the Group of Personalities, were here highly represented (Hayes, 2004).

One can make a similar appraisal by looking at projects funded under FP7. Even if the ratio of projects led by arms-producing firms is lower than under PASR, the defense industry remains a pampered beneficiary of allocated credits. For instance, out of the 45 projects funded in 2009, 17 were led by firms operating mainly in defense, and 5 were led by firms operating mainly in security sector. One can also note the participation of other non-industrial defense related agents, such as state defense research agencies from the Netherlands (TNO) and Sweden (FOI).

11 The European Security Research Advisory Board.

The defense industry has played a central role in shaping and monitoring the EU's security research program. There is also little doubt that security-related R&D can appear as an alternative funding channel for European arms-producing countries that need to support their defense industrial bases.

On the other hand, security markets related to border security or counter terrorism represent interesting opportunities in the context of a declining defense market. Nevertheless this fact does not determine whether defense R&D is effectively employed in the most effective manner. Indeed this technology-centric paradigm favored the development of a high-tech defense industry, and these firms appear to try to preserve their "business model" rather than reshuffle this model to adjust to armed forces' current needs. Such hysteresis effect is perceivable when analyzing the emergence of a security market, which has been structured by some defense firms to promote a technology-centric approach.

CONCLUSION

Trends in defense and security markets highlight the bias possibly introduced by firms when they are able to structure such markets. This leads to question how such state-managed and state-centered markets can be regulated, where competitive pressures cannot be the only driver to avoid manipulation. Indeed, defense and security constitute specific markets as they are not competitiondriven markets. Therefore such markets require an ad hoc regulation, preserving an industrial base whilst keeping firms under control.

Beyond administrative regulation, the need for reintroducing democratic processes into the decision loop should be raised. Is it acceptable that critical decisions such as defense spending and military capabilities be delegated to experts and de facto exclude citizens from decision-making because of the technicality of such choices? Whilst technology remains critical to provide adequate capabilities to armed forces, the effectiveness of public spending and the ability to deliver the right capabilities on time require giving citizens and political decision-makers adequate tools to oversee proposed solutions and investment choices.

  First, as potential foes can also have access to much state-of-the-art technology and capabilities, accessible on the open global market, it is necessary to "run faster" than others. Second, potential foes might adopt non-conventional means to overcome the technological superiority of today's military powers. It is therefore necessary to explore a larger scope of technology that could be used for military purposes and evaluate their possible impacts on the military balance.In fact current armed forces are structured in battle systems prepared to face classical conflicts, where professional units try to acquire the means of tactical and strategic superiority thanks to training and equipment. The East-West rivalry was completely based on such a philosophy, with doctrines aimed at guaranteeing a victory through a major conflagration: the decisive battle. The arms race resulted from the attempt to gain technological and quantitative superiority, but both sides shared a similar vision of war. "Daughter of World War II's confrontations, notes LaMaisonneuve (1997, p. 196), the Cold War favored the accumulation of armaments and a race to their modernization. Modern armed forces are those of quantity." Indeed, for at least a century, armament policies have been a quest for strategic balance, characterized sometimes by great oscillations but always corrected quite rapidly. However today's geopolitical situation is rather characterized by an out-of-balance situation. 4 U.S. Department of Defense. 2012, Sustaining U.S. Global Leadership: Priorities for the 21st Century Defense, Washington D.C., Office of the Secretary of Defense, January. 5 British Defense Secretary: Hammond P. 2012, Shared Security: Transforming Defence to Face the Future, Berlin, speech given to the German Council on Foreign Relations, 2 May. 6 Délégation aux affaires stratégiques. 2012, Horizons stratégiques, Paris, French Ministry of Defense, April.

  armament remains important in a large field of international relations. Even though no classical threat may rise in foreseeable years and the American dominance appears uncontested, it seems unwise to get rid of major weapon systems since they remain the basis of the military balance between States in the long term. The United States has then developed a preemptive strategy to prevent potential rivals from investing in defense technology by raising entry barriers thanks to their own defense R&D effort… Nevertheless this dimension is not sufficient because of asymmetric threats. It is also necessary to develop a new way of thinking to prevent and anticipate "flaws in the wall." Uncertainty concerning intentions of asymmetric actors and the possible evolution of threats implies a divide of the strategic field in numerous scenarios.

  Technology has been essential to military capabilities since World War II. If this trend characterized particularly the Cold War, defense R&D has maintained a central role after the fall of Berlin Wall. This demonstrates that the technology-centric paradigm in defense does not translate only to the specific era of East-West confrontation alone, but represents an essential feature of defense in the twentieth and twenty-first centuries due to uncertainties surrounding international relations.

Table 1 -R&D share in equipment spending

 1 European Defense Agency, statistical surveys, various years. The predominance of this technology-centric paradigm can be easily identified through the official publications of large arms-producing countries. For instance, in June 2008, France released a defense review 2 that called for an increased effort of defense R&T and for the launch of technological demonstrators. The subsequent Defense Program Law emphasizes such a trend, with a significant increase of effective spending since 2008.

		2005	2006	2007	2008	2009	2010
	France	38.1%	37.4%	33.4%	34.4%	35.0%	30.2%
	Germany	23.2%	21.9%	25.2%	18.2%	17.3%	20.5%
	Spain	8.1%	7.9%	10.5%	11.0%	10.4%	n.a.
	Sweden	n.a.	18.7%	18.8%	20.7%	16.0%	9.4%
	United Kingdom	35.3%	34.8%	31.4%	29.4%	26.5%	25.5%

  disruptive ones. Peacetime, or a close equivalent, is a difficult time for a nation's arms buyers, since there is no readily available guide to what should be acquired. Such a difficulty is increased

3 See for instance U.S. Department of Defense. 2012, Military and Security Developments Involving the People's Republic of China 2012, Washington D.C., Office of the Secretary of Defense, May.

and

Table 2 -Changes in DOD's 2011 Portfolio of Major Defense Acquisition Programs

 2 the biggest military spender, the United States provides the most relevant experience of the limits of the development cost of selected technologies and if the industrial and technological base is unable to deliver expected outcomes.

	in the dominant model of defense R&D. In spite of doubling its global budget from 2000 to 2008, with
	outlays growing from $386 billion to $736 billion (constant FY 2012), the Pentagon is experiencing
	technology shortcomings and failures. One can be surprised, since over the same period, the RDT&E

line of Department of Defense (DoD) budget increased from $47 to $82 billion. As underlined in the table below, defense technology appears more and more expensive. One can ask if there is a systematic underestimation Source: U.S. Government Accountability Office, Assessments of Selected Weapon Programs, GAO-12-400SP, Washington D.C., March 2012, page 171.

  These recommendations, which are largely favorable to the defense sector, were by and large retained by the Commission and integrated into a policy paper setting the agenda of the security research program entitled Security Research: The Next Steps(European Commission, 2004). The participation of the defense industry in shaping the European program did not come to an end with the agenda-setting phase. Arms-producing firms took part in developing the research program itself.One of the final recommendations presented by the Group of Personalities, in its report was indeed to set up an advisory board. As for the Group of Personalities, the private-sector side of ESRAB 11 was largely composed of arms-producing firms, mostly the ones sitting at the Group of Personalities.The defense industry's footprint on the development of this research program is clearly visible. The capacity-based approach that structures the research agenda recalls to a great extent the logic that dominates arms procurement in the United States and European Union. Consequently, it is not fundamentally surprising to note that the actual implementation of research program largely benefits to arms producers. Looking at the various projects funded under the framework of the Preparatory Action for Security Research (

	Recommendations addressed by ESRAB regarding the 7th European R&D Framework Program (FP7)
	had undertones of a remarkable "defense approach."

For each of the four identified key security missions (protection against terrorism and organized crime; border security; protection of critical infrastructure; and restoring security in crisis situation), ESRAB recommended to tune the research agenda according to a capacity-based approach

(ESRAB, 2006, p. 6)

. This concept was directly borrowed from defense. ESRAB pointed out two generic objectives: respond to security needs and develop the global competitiveness of European technological supply

(ESRAB, 2006, p. 6

).

The author is grateful to Susan Dewavrin, Josselin Droff, Stanislas de Lauriston and Enno Schumacher as well anonymous referees for their fruitful comments. All usual disclaimers apply.


Kamel Madi
email: kamel.madi@port.ac.uk

Gianluca Tozzi

Qinghang H Zhang

Jie Jong

Andrew Cossey

Andrew Au

David Hollis

François Hild

J Tong

F Hild ++

Computation of full-field displacements in a scaffold implant using Digital Volume Correlation and Finite Element Analysis

Keywords: micro-computed tomography, cellular material, digital volume correlation, local approach, global approach, finite element analyses, measurement uncertainties

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

Introduction

Studies on the relationships between the morphology and the mechanical behaviour of biological tissues and biomaterials are critical to biomedical engineering applications of joint repair and replacement materials. With the recent and rapid progress of micro-focus computed tomography (µCT) combined with the development of in situ experiments [START_REF] Nazarian | Time-lapsed microstructural imaging of bone failure behavior[END_REF][START_REF] Youssef | Finite element modeling of the actual structure of cellular materials determined by X-ray tomography[END_REF][START_REF] Buffière | In Situ Experiments with X ray Tomography: an Attractive Tool for Experimental Mechanics[END_REF], Digital Volume Correlation (DVC) has become a powerful tool to capture 3D strain distributions in solid and cellular materials [START_REF] Bay | Digital Volume Correlation: three-dimensional strain mapping using X-ray tomography[END_REF][START_REF] Benoit | 3D analysis from micro-MRI during in situ compression on cancellous bone[END_REF][START_REF] Bornert | Mesure tridimensionnelle de champs cinématiques par imagerie volumique pour l'analyse des matériaux et des structures[END_REF][START_REF] Germaneau | 3D strain field measurement by correlation of volume images using scattered light: recording of images and choice of marks[END_REF][START_REF] Hild | Three dimensional analysis of a compression test on stone wool[END_REF][START_REF] Liu | Accuracy and precision of digital volume correlation in quantifying displacements and strains in trabecular bone[END_REF][START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF][START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF][START_REF] Verhulp | A three-dimensional Digital Image Correlation technique for strain measurements in microstructures[END_REF].

Two approaches to image correlation have been mainly used for quantifying continuum-level strain measurements. The first approach, the most commonly used, is a local analysis, which consists of dividing the reference and the deformed images into smaller interrogation windows that are then individually correlated. It was developed and first applied in solid mechanics at the beginning of the 1980's, in 2D [START_REF] Sutton | Determination of displacements using an improved digital correlation method[END_REF][START_REF] Peters | Digital Imaging Techniques in Experimental Stress Analysis[END_REF], then applied to a variety of methods [START_REF] Bay | Texture correlation -a method for the measurement of detailed strain distributions within trabecular bone[END_REF][START_REF] Sjödahl | Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy[END_REF] and extended to 3D surface [START_REF] Luo | Accurate measurement of three-dimensional displacement in deformable bodies using computer vision[END_REF][START_REF] Helm | Improved three-dimensional image correlation for surface displacement measurement[END_REF] and volume measurements [START_REF] Bay | Digital Volume Correlation: three-dimensional strain mapping using X-ray tomography[END_REF][START_REF] Bay | Experimental measurement of three dimensional continuum-level strain fields in trabecular bone[END_REF][START_REF] Smith | Digital Volume Correlation including rotational degrees of freedom during minimization[END_REF]. Since then, several robust and accurate DVC algorithms have been developed, based mainly on cross-correlation and sum of squared differences criteria, with displacement uncertainties ranging between 2 10 -voxel [6,21-23] and 1 10 -voxel [START_REF] Bay | Digital Volume Correlation: three-dimensional strain mapping using X-ray tomography[END_REF][START_REF] Germaneau | 3D strain field measurement by correlation of volume images using scattered light: recording of images and choice of marks[END_REF][START_REF] Liu | Accuracy and precision of digital volume correlation in quantifying displacements and strains in trabecular bone[END_REF][START_REF] Verhulp | A three-dimensional Digital Image Correlation technique for strain measurements in microstructures[END_REF][START_REF] Bay | Experimental measurement of three dimensional continuum-level strain fields in trabecular bone[END_REF][START_REF] Germaneau | Comparison between X-ray micro-computed tomography and optical scanning tomography for full 3D strain measurement by digital volume correlation[END_REF][START_REF] Scarano | Advances in iterative multigrid PIV image processing[END_REF]. The second approach, referred to as global, estimates the displacement fields from pairs of images based on a continuous and global field, as commonly used in finite element simulations. Developed first in 2D [START_REF] Cheng | Full-field speckle pattern image correlation with B-spline deformation function[END_REF][START_REF] Sun | Finite element formulation for a digital image correlation method[END_REF][START_REF] Hild | Digital Image Correlation: from displacement measurement to identification of elastic properties -a Review[END_REF], this approach has been extended to 3D [START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF] and applied with success to solid and cellular materials [START_REF] Benoit | 3D analysis from micro-MRI during in situ compression on cancellous bone[END_REF][START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF][START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF]. Although improvements have been made for both local and global approaches to DVC, a systematic comparison of the two methods in terms of performance and local strain fields has yet to emerge, especially for materials with foam-like morphologies [START_REF] Benoit | 3D analysis from micro-MRI during in situ compression on cancellous bone[END_REF][START_REF] Liu | Accuracy and precision of digital volume correlation in quantifying displacements and strains in trabecular bone[END_REF][START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF]. In 2D, the global approach has been shown to be more accurate in solid materials when compared to FFT-based or other local algorithms [START_REF] Hild | Digital Image Correlation: from displacement measurement to identification of elastic properties -a Review[END_REF][START_REF] Hild | Comparison of Local and Global Approaches to Digital Image Correlation[END_REF], but such comparison in 3D has never been tested in either solid or cellular materials. Since the local and global approaches to DVC are close in spirit to the two-dimensional case, it is hypothesized that the global approach may also outperform the FFT-based local approach.

Micro-finite element (micro-FE) models generated from high resolution micro-CT images (up to ~10 μm resolution) have proven to be a powerful tool to simulate the mechanical behaviour of cellular materials such as polymeric foams [START_REF] Saadatfar | Mechanical and Transport Properties of Polymeric Foams Derived from 3D Images[END_REF][START_REF] Fischer | Numerical Simulation of Mechanical Properties of Cellular Materials Using Computed Tomography Analysis[END_REF], metallic foams [START_REF] Youssef | Finite element modeling of the actual structure of cellular materials determined by X-ray tomography[END_REF][START_REF] Singh | Characterization of the deformation behavior of intermediate porosity interconnected Ti foams using micro-computed tomography and direct finite element modeling[END_REF], scaffolds [START_REF] Jones | Hierarchical porous materials for tissue engineering[END_REF][START_REF] Lacroix | Computer-aided design and finite-element modelling of biomaterial scaffolds for bone tissue engineering[END_REF] and more extensively trabecular bone [START_REF] Hollister | A homogenization sampling procedure for calculating trabecular bone effective stiffness and tissue level stress[END_REF][START_REF] Müller | Three-dimensional finite element modelling of non-invasively assessed trabecular bone structures[END_REF][START_REF] Van Rietbergen | Computational strategies for iterative solutions of large FEM applications employing voxel data[END_REF][START_REF] Yeni | Finite element calculated uniaxial apparent stiffness is a consistent predictor of uniaxial apparent strength in human vertebral cancellous bone tested with different boundary conditions[END_REF][START_REF] Niebur | Convergence behavior of high-resolution finite element models of trabecular bone[END_REF][START_REF] Verhulp | Micro-finite element simulation of trabecular-bone post-yield behaviour--effects of material model, element size and type[END_REF]. Despite the significant modelling progress, the accuracy of this approach in predicting the material constitutive behaviour could still be improved. Amongst the limitations that have been extensively discussed in the previous studies [START_REF] Müller | Three-dimensional finite element modelling of non-invasively assessed trabecular bone structures[END_REF][START_REF] Van Rietbergen | Computational strategies for iterative solutions of large FEM applications employing voxel data[END_REF][START_REF] Yeni | Finite element calculated uniaxial apparent stiffness is a consistent predictor of uniaxial apparent strength in human vertebral cancellous bone tested with different boundary conditions[END_REF][START_REF] Niebur | Convergence behavior of high-resolution finite element models of trabecular bone[END_REF][START_REF] Verhulp | Micro-finite element simulation of trabecular-bone post-yield behaviour--effects of material model, element size and type[END_REF], the boundary conditions are cited as a possible source of errors. Comparing finite element predictions with DVC measurements was attempted for cancellous bone to validate the strain predictions of the FE model [START_REF] Zauel | Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D Digital Volume Correlation measurements[END_REF]. A good agreement was obtained in the loading direction, yet less accuracy was achieved in the transverse directions. It is hypothesised that parameters affecting the accuracy may include lateral and vertical boundary conditions.

The above hypotheses are tested by first measuring the uncertainty levels and analysing the local strain fields of a polymer foam implant in a FFT-based local approach and a global approach to DVC.

Secondly, the results from the two DVC approaches are compared with those obtained from the FE model, the latter being developed from the same µCT data sets as used in DVC. The experimental displacements obtained by the two DVC methods are applied as boundary conditions on the external faces of the FE model.

Methods

Experimental methods

Material and specimen

A scaffold implant material (Smith & Nephew) was used for this study. The scaffold material was taken from a dual-layer cylindrical implant that mimics cartilage and bone for knee repair purposes [START_REF] Slivka | Porous, Resorbable, Fiber-Reinforced Scaffolds Tailored for Articular Cartilage Repair[END_REF] and made of 85:15 Poly(D,L-lactide-co-glycolide). Only the bone layer was retained, with a diameter of 8.57 mm and a height of 14.7 mm.

Micro-compression device

A novel customised micro-mechanical loading device (Deben Ltd, UK), equipped with a 3kN miniature load cell, was used to load the sample in the micro-CT chamber (Nikon XTH-225 X-Ray & CT Inspection System, see Fig. 1a). This loading device, as shown in Fig. 1b, has been used in previous studies [START_REF] Tozzi | 3D real-time micromechanical compressive behaviour of bonecement interface: experimental and finite element studies[END_REF]. It has a torque accurately applied by a DC motor, which is then transmitted to a ballscrew system via a set of gears, resulting in the application of precise strain steps without introducing fluctuation errors. The nominal torque is 23 mNm and calculated to provide a required nominal axial force of about 70 N. The system drives the jaws symmetrically in opposite directions, with screws S1 and S3 driving the complex carbon fibre tube-actuator from the top and S2 driving the sample from the bottom simultaneously. The loading stage employs an analogue resistive extensometer with a 16-bit AD converter for accurate elongation measurement, achieving a nominal strain resolution of around 15 με. Using the same converter, a typical resolution range of 1000:1 (dynamic) to 2000:1 (static) may be achieved for the load cell (i.e. 3N resolution for a 3kN load cell), with an overall uncertainty of +/-1% of full scale. The motor is linked to a board card and to the extensometer using a closed-loop system for accurate position and speed control via the Deben Microtest software (Microtest, Deben Ltd, East Grinstead, UK), where data acquisition in the form of force-extension curves are displayed and recorded.

In situ testing and time-lapsed imaging

Before testing, the sample was glued on the lower compressive platen of the stage and a preload of 5N was applied at the top platen. A complete scan was performed at the preloaded state prior to testing (Fig. 2). The scanner settings were set to a voltage of 60 kV, current of 51 μA and an isotropic voxel size of 20 μm. The specimen was then step-wise compressed to -0.441 and -0.879 mm displacements (i.e. -3% and -6% strains) at a constant displacement rate of 0.01 mm/s. Following the application of each displacement step, the specimen was allowed to relax for 15 min, due to its viscoelastic properties, before the micro-CT acquisition procedure was repeated. Approximately one hour scan time was required for scanning and additional 10-min time for reconstruction from 1500 angular projections using CT-Pro XT software (XT Software suite, 2.2, Nikon Metrology Ltd, Derby, UK).

Displacement measurement by Digital Volume Correlation (DVC)

Two DVC methods were used to compute the 3D displacement and strain fields from the same pair of images, namely, the reference reconstructed image obtained in the preloaded stage and the deformed image obtained after compressing the specimen to -3 % strain along (Oz) direction (Fig. 2).

Figure 3 shows the differences between the local and the global approaches used herein. The spatial resolution is defined as the length scale of the displacement and strain measurements, i.e. the size of the correlation window (local approach) or twice the element size (global approach).

Local correlation algorithm

The local correlation algorithm (LA-DVC), implemented into the LaVision DVC software (Davis, 8.0.1, LaVision GmbH, Goettingen, Germany), uses fast Fourier transform (FFT) cross-correlation to compare sub-volumes (Fig. 3a), and a multi-pass approach that uses the displacement gradient information from the previous passes to deform the sub-volumes on the subsequent passes [START_REF] Fonseca | Full-field strain mapping by optical correlation of micrographs acquired during deformation[END_REF]. In the present case, final sub-volumes of 32 × 32 × 32 voxels 3 overlapped by 50% were reached after successive passes using sub-volumes of 96 × 96 × 96 voxels 3 and 64 × 64 × 64 voxels 3 . From the displacements at the centre of the sub-volumes, all the components of the Green-Lagrange strain tensor were calculated using a centred finite differences scheme [START_REF] Germaneau | Comparison between X-ray micro-computed tomography and optical scanning tomography for full 3D strain measurement by digital volume correlation[END_REF].

Global correlation algorithm

The second correlation code (Correli-C8, 1.2, LMT-Cachan, Cachan, France) is based on a continuous and global approach to DVC [START_REF] Benoit | 3D analysis from micro-MRI during in situ compression on cancellous bone[END_REF][START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF] and will be designated by GA-DVC. The whole volume of interest is analysed and the sum of squared differences is minimized with respect to the unknown kinematic degrees of freedom by resorting to a multiscale procedure (Fig. 3b). In the present case, a Galerkin approach based upon 8-nodebrick (C8) elements is chosen. A trilinear displacement interpolation is used, and a spline grey level interpolation. When the correlation procedure has converged, its quality is evaluated using the dimensionless correlation residuals (Equation (2)), DVC GA r -, that are computed as the absolute difference between the reference volume, and the deformed one corrected by the measured displacement field. The only parameter to be chosen by the user is the element edge size. For comparison purposes, an element size of 32 voxels was selected and the mean Green-Lagrange strains per element were computed.

Performance evaluation

Successive sub-voxel displacements over a range of one voxel were artificially prescribed in each direction by using the shift / modulation property of FFTs, thereby creating new 'deformed' volumes. These artificial volumes were then correlated with the original one using final sub-volumes of 32 × 32 × 32 voxels 3 and 64 × 64 × 64 voxels 3 for the local approach. Final sub-volumes of 32 × 32 × 32 voxels 3 overlapped by 50% were reached following the procedure described in Section 2.2.1. Final sub-volumes of 64 × 64 × 64 voxels 3 overlapped by 50% were reached after passes using sub-volumes of 96 × 96 × 96 voxels 3 .

Since the global approach is based on finite elements, the evaluation of any degree of freedom of an interior node uses the 8 elements with which it is connected. Therefore the spatial resolution of the global approach is twice the element size. For comparison purposes, volumes with element sizes of 16 × 16 × 16 voxels 3 and 32 × 32 × 32 voxels 3 are analysed.

The errors in the vertical and lateral displacements (u x , u y and u z ) (resp. normal strains, xx ε , yy ε and zz ε ) were estimated from the systematic error (or bias) and the corresponding standard deviation (or standard uncertainty). The systematic error was calculated only from the difference between the spatial average of the measured displacement and the prescribed translation. The systematic error of the strain was not considered due to exact zero strain predicted by the two correlation methods.

The standard displacement and strain uncertainties ( u σ , ε σ ) were measured from the standard deviation of the displacement and strain fields respectively, relative to their mean values.

Reliability and accuracy of the displacement measurements

The reliability of the displacement measurements was assessed in terms of the quality of the correlation scheme for both DVC methods. For LA-DVC, a normalised cross-correlation coefficient, DVC LA r -, based on grey level gaps is used:

) 1 ( ) ( ) ( ) ( ) ( * 2 * 2 * ∑ ∑ ∑ ∈ ∈ ∈ = - VOI X VOI X VOI X X g X f X g X f r DVC LA
where, X and X* refer to coordinates (in voxels) of the same material point in the reference and the deformed state; f and g are, respectively, the grey levels of the initial and deformed volumes. A correlation coefficient value, DVC LA r -, close to 1 indicates a perfect match. For GA-DVC, a dimensionless correlation residual, DVC GA r -, between the two volumes f and g is used:

) 2 ( )) ( min( )) ( max( | ) ( ) ( | * X f X f X f X g r DVC GA - > - < = -
where < . > is the average value over the volume of interest (VOI), max( . ) and min( . ) are the maximum and minimum values of the grey levels present in the VOI. This indicator is equal to 0 when the two images are identical or when no correlation error occurs.

Prior to the analysis, the accuracy of the reported strain measurements was validated by measuring the Young's modulus of nodular cast iron material using both DVC methods. Five tomographic scans were acquired from a tensile test along the longitudinal axis [START_REF] Réthoré | Digital volume correlation of synchrotron tomographic images[END_REF], where the applied load was equal to 22 (reference), 150, 200, 250 and 273 N. The mean longitudinal strains were measured using the GA-DVC algorithm with an element size of 16 voxels from which a stress-strain curve of the specimen (cross section: 1.6 × 0.8 mm 2 ) was obtained [START_REF] Réthoré | Digital volume correlation of synchrotron tomographic images[END_REF]. The same micro-CT scans were used for an analysis using the local approach, LA-DVC, following the procedure described in Section 2.2.1.

These values are finally compared with independent studies on the same material [START_REF] Dierickx | Etude de la microstructure et des mécanismes d'endommagement des fontes G.S. ductiles : influence des traitements thermiques de ferritisation[END_REF].

Finite element modelling

A volume of interest (VOI) was selected for this study, and resulted in a cube of size 5.12 mm (256 voxels), extracted from the core of the upper part of the cylinder (Fig. 4). To be comparable to the histograms and the strain maps measured using the DVC methods, the solid phase and the void phase were meshed together. The average strain of the mixture is the sum of the average strains of the solid and void phases.

Finite element mesh

The micro-CT scan images of the implant were imported into the Avizo software (Avizo Standard Edition, 6.3.0, VSG, an FEI company, Burlington, USA) to generate unstructured tetrahedral meshes using the methodology described in [START_REF] Madi | Finite element simulations of the deformation of fused-cast refractories based on X-ray computed tomography[END_REF]. The number of elements was fixed to 21,138 so that the average element size, 21 voxels (0.42 mm, Fig. 4b), matches as best as possible the DVC cell size, 32 voxels (0.64 mm), to allow the displacements obtained from FE to be compared with the measured displacements using the DVC methods. Increasing further the element size would reduce the quality of the mesh. A sensitivity analysis, where the number of elements was varied from 21,138 to 502,726 elements (0.127 mm element size, Fig. 4c), revealed that the average strains are independent of the mesh density (Fig. 5a), which is essentially attributed to the boundary conditions. The experimental displacements are applied here as boundary conditions on all the external faces of the meshes (Section 2.3.3). According to the theorem of spatial averaging in the homogenisation theory [START_REF] Hori | On two micromechanics theories for determining micro-macro relations in heterogeneous solids[END_REF], as the volume fraction is preserved in all the meshes, the average strains are unique and similar. The local convergence was reached for an element size of 0.168 mm (Fig. 5b). Thus, the strain maps obtained with an element size of 0.168 mm or 0.127 mm are expected to be more heterogeneous than those obtained with 0.42 mm (Section 3.3).

Mechanical properties of the constituents

The mechanical properties of the solid phase were identified separately after calibration using an inverse method [START_REF] Verhulp | Micro-finite element simulation of trabecular-bone post-yield behaviour--effects of material model, element size and type[END_REF]. A linear elastic analysis was first performed up to a nominal apparent strain of 0.1% and the tissue elastic modulus was determined by matching the predicted apparent modulus of the model to the one deduced from the experimental force versus displacement curve (Fig. 2a).

Secondly, a nonlinear analysis was carried out to adjust the tissue yield stress by fitting the predicted apparent yield point to the experimental yield stress value using the 0.2% offset method. The mesh of the solid phase resulted in 437,159 elements and 572,178 nodes. The implant tissue was assumed to be elastic perfectly plastic due to the lack of mechanical data available. Poisson's ratio was assumed to be 0.3 [START_REF] Terrier | Biomechanical evaluation of porous biodegradable scaffolds for revision knee arthroplasty[END_REF], while Young's modulus and yield stress were found to be 600 MPa and 12.5 MPa, respectively. A very soft linear elastic behaviour was chosen for the void phase. The contrast defined as the ratio between Young's moduli of the solid phase and the void phase was fixed to 10,000.

Boundary conditions

The experimental displacements measured by the two DVC methods were separately applied as boundary conditions on the mesh external faces. All the FE analyses were performed on the FE solver COMSOL Multiphysics (COMSOL Multiphysics, 4.2.a, COMSOL Group, Boston, USA), using large deformation to account for geometrical nonlinearity and large rotations.

Results

Performance study

The average displacements computed over the VOI (Fig. 4) and the standard displacement/strain uncertainties (Table 1 and Fig. 6) are shown as functions of the prescribed displacement. The correlation coefficient is extremely good ( DVC LA r -= 0.99 and DVC GA r -≈ 0) for all the computations as a simple rigid body motion is prescribed with no additional noise. With the increase of the prescribed displacement, the systematic error of the local correlation method varies between 0.005 voxel (0.1 μm) to 0.016 voxel (0.32 μm), while that of the global correlation method is less than 0.00012 voxel (0.0024 μm). Displacement uncertainties ranging between 0.006 voxel (0.12 μm) and 0.02 voxel (0.4 μm) are observed for both approaches using a sub-volume of size 32 voxels or an element size of 16 voxels. The highest displacement uncertainty is reached for the prescribed displacement of 0.5 voxel. Before correlation, the artificial images were shifted and interpolated in the sub-voxel region using FFT. This kind of interpolation is known to introduce small errors that increase with the shift level, with a maximum reached when the shift comes close to half a voxel [START_REF] Germaneau | 3D strain field measurement by correlation of volume images using scattered light: recording of images and choice of marks[END_REF][START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF][START_REF] Inglada | Analysis of artifacts in subpixel remote sensing image registration[END_REF].

When increasing the element size, from 16 voxels to 32 voxels, the impact on the systematic error is negligible and the displacement uncertainty is reduced by a factor of 4. The strain uncertainty is decreased by a factor of 9 and is two times lower than that of the local approach, which ranges between 60 με and 200 με. Since the displacement estimates are biased for the local approach, the strain uncertainty estimates are lower bounds [START_REF] Bergonnier | Digital image correlation used for mechanical tests on crimped glass wool samples[END_REF]. Increasing the correlation window of the local approach reduces slightly the displacement and strain uncertainties. Interestingly, increasing the number of intermediary sub-volumes during the multi-pass approach allows us to reduce the displacement uncertainty by a factor of about 4, and the strain uncertainty by a factor of about 3 (Fig. 7), even though the systematic error does not vary.

To validate the reported strain measurements accuracy, the stress-strain curve of a nodular graphite cast iron was deduced from the DVC analysis of five micro-CT scans (Fig. 8). A very good agreement is observed for both DVC methods. The reported displacement uncertainty can reach up to 0.02 voxel using a sub-volume size of 32 voxels or an element size of 16 voxels. The corresponding standard uncertainty of the mean longitudinal strain, derived from [START_REF] Réthoré | Digital volume correlation of synchrotron tomographic images[END_REF], is less than 5 10 -for the local and global approaches, which is much lower than the measured levels. The identified value of Young's modulus is 156 GPa using the global approach and 155 GPa using the local approach, which is very close to published values of the same material (160 ± 3 GPa [START_REF] Dierickx | Etude de la microstructure et des mécanismes d'endommagement des fontes G.S. ductiles : influence des traitements thermiques de ferritisation[END_REF]).

Displacement measurements

Correlation quality

The histograms of the correlation coefficient, DVC LA r -, and the correlation residual, DVC GA r -, are represented in Figs. 9a and10a, respectively. All the curves are bell-shaped with a pronounced peak.

The mean value and standard deviation of each distribution are reported in Table 2. The overall quality of the correlation is very satisfactory with a sub-volume size of 32 voxels ( 2) and illustrates the benefit of using the multipass approach. For GA-DVC, as the element decreases in size, more degrees of freedom are available to reduce the correlation residual. This effect is clearly seen when comparing Figs. 10b and10c. The mean residual reached for 16 voxel elements is equal to 2 % of the dynamic range versus 2.8 % for 32 voxel elements, which indicates that the matching is better for small size elements. This effect is typical of localised phenomena [START_REF] Hild | Three dimensional analysis of a compression test on stone wool[END_REF].

The maps of the correlation coefficient, DVC LA r -, and the correlation residual, DVC GA r -, are represented in Figs. 9b,c and Figs. 10b,c, respectively. For the local approach, the correlation coefficient is smaller in regions at the top of the sample where the deformation is the highest. Elsewhere, the correlation coefficient is very high and continuously distributed, especially with a sub-volume size of 32 voxels.

For the global approach, the correlation residuals are not only small on average but also locally. The highest correlation residuals seem to be homogenously distributed over the region of interest, which is likely to be attributed to the noise level [START_REF] Benoit | 3D analysis from micro-MRI during in situ compression on cancellous bone[END_REF].

Displacement maps

The effect of cell size on the vertical displacement distribution was assessed for the two DVC methods before the analysis (Fig. 11). When decreasing the cell size, the convergence of LA-DVC was reached after 48 voxels while faster convergence was observed for GA-DVC. As a precaution, a subvolume or element size of 32 voxels was used. This size is the most suitable for a detailed comparison of the strain fields (Section 3.3) as similar strain uncertainties are obtained with this size, using both approaches (Fig. 6b), and with a good correlation quality (Figs. 9 and 10). The displacement maps obtained on the VOI are represented in Figure 12. Similar results are obtained between the two correlation techniques. The vertical displacement map (Fig. 12c) reveals that the compression is not symmetric with respect to the loading axis (Oz). The sample has probably slightly rotated along (Oy) during the experiment. The lateral displacement maps (Figs. 12a and12b) also indicate that twisting of the sample has occurred in the (xy) plane during the experiment. This asymmetry is probably a consequence of the misalignment of the sample as illustrated by the displacement maps. Each distribution was characterised by its mean value and standard deviation, as summarised in Table 3, and the results compare well with each other. The maximum error, calculated as the absolute value of the difference between the mean strains obtained by the two DVC methods, is 0.08%. The mean strains obtained with the FE model are also independent of the mesh size, although slightly more scatter is observed when decreasing the mesh size (Fig. 13d and Table 3). For each DVC method, the mean strains and standard deviations compare well with the corresponding FE model (maximum error: LA-DVC ~ 0.04%; GA-DVC ~ 0.01%). The vertical strain maps obtained by the two DVC methods are illustrated in Figure 14, and similar results are obtained.

Comparison of the strain fields

The levels of the lateral strains are similar and omitted here for clarity. Highly localised vertical strain is evident near the top (Fig. 14c), similar to the lateral strains (Fig. 15c). Elsewhere, the strains obtained by DVC seem to be lower and more continuously distributed. The strain maps predicted by the FE analysis with an average element size of 0.42 mm are similar to the DVC measured strains (Figs. 14, 15c vs. 15e and 16c vs. 16e). Decreasing the average mesh size to 0.127 mm would lead to more local heterogeneity of the strain distributions (Figs. 15c vs. 15d and 16c vs. 16d).

Discussion

The use of FFT, combined with the multi-pass approach in the local correlation algorithm (LA-DVC), allowed fast calculations with systematic errors or displacement bias ranging between 0.0052 voxel (0.104 μm) and 0.0129 voxel (0.258 μm), standard displacement uncertainties ranging between 0.006 voxel (0.12μm) and 0.02 voxel (0.4 μm) and strain uncertainties ranging between 60 and 200 με, with a correlation window of 32 voxels (640 μm). The performance compares well with the data reported in the literature [START_REF] Bay | Digital Volume Correlation: three-dimensional strain mapping using X-ray tomography[END_REF][START_REF] Bornert | Mesure tridimensionnelle de champs cinématiques par imagerie volumique pour l'analyse des matériaux et des structures[END_REF][START_REF] Liu | Accuracy and precision of digital volume correlation in quantifying displacements and strains in trabecular bone[END_REF][START_REF] Bay | Experimental measurement of three dimensional continuum-level strain fields in trabecular bone[END_REF][START_REF] Zauel | Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D Digital Volume Correlation measurements[END_REF], where the displacement uncertainty ranged between 0.009 and 0.094 voxel (0.315 and 3.39μm) and the strain uncertainty ranged between 100 and 800 με, using local DVC algorithms. The global correlation method, GA-DVC, gives similar displacement uncertainties but lower systematic errors are obtained (below 0.00012 voxel. i.e. 0.0024 μm), independent of the element size. Increasing the number of intermediary sub-volumes using the local approach reduces the displacement and the strain uncertainties by a factor of 4 and 3, respectively. The same levels of displacement and strain accuracies can also be achieved using the global approach by simply increasing the size of the element. Solutions may be developed to reduce the displacement uncertainty of the local correlation algorithm LA-DVC, such as taking into account higher-order displacement gradients during the multi-pass approach, improving further the peakfinding algorithm [START_REF] Pan | Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review[END_REF] or reducing the numerical artefacts generated by FFTs [START_REF] Okamoto | Comparison of sub-pixel estimation of PIV using frequency response analysis[END_REF]. On the other hand, this study reveals that the requirement of global displacement continuity, provided by the GA-DVC algorithm, appears to be one of the keys in significantly reducing both displacement uncertainty and systematic error [START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF][START_REF] Hild | Digital Image Correlation: from displacement measurement to identification of elastic properties -a Review[END_REF][START_REF] Hild | Comparison of Local and Global Approaches to Digital Image Correlation[END_REF][START_REF] Leclerc | Digital Volume Correlation: What are the limits to the spatial resolution? Mech Indus[END_REF] when compared to FFT-based local approaches.

The quality of the correlation results was very satisfactory for both DVC methods and the accuracy of the reported strain measurements was validated against the experimental Young's modulus of a model material. The strain uncertainties are also much lower than the mean strain values (between 0.9 and 1.8%). In addition, the strain maps and histograms obtained with the local correlation approach are in a good agreement with those obtained with the global approach. This indicates that the image registration performed by the two DVC methods was successful and that the measured displacements and strains are reliable. Both correlation techniques predicted highly localised strains at exactly the same region (top end of the sample). Collapse of the foam cells was observed precisely in this zone after compressing the sample to -6% strain (see Fig. 2). The occurrence of localised strains at the top might be due to end artefacts, which may be removed by fixing both ends of the sample with pre-aligned end-caps [START_REF] Nazarian | Time-lapsed microstructural imaging of bone failure behavior[END_REF] to prevent the sample from bending or twisting and to allow deformation to occur preferentially at the midsection.

The FE method predicts mean strains and standard deviation in the axial and lateral directions that are similar to those measured using both DVC methods. Note that the length scale of the local displacement measurements in the DVC methods (0.64 mm) is five times larger than the mean size of an individual strut (0.15mm+/-0.04mm) and three times larger than the mean size of the spacing between the struts [START_REF] Madi | Viscoelastic modelling and full-field strain computation of biphasic scaffold for osteochondral defect repair[END_REF]. Therefore, the DVC methods assume the struts to be a continuum medium.

The FE model with an average mesh size of 0.127 mm is able to operate at the strut level. Although the mean strain values compare well with the DVC methods, the FE strain maps reveal, logically, more local heterogeneity. By increasing the average mesh size of the FE model to 1.5 times the size of the DVC correlation window (0.42 mm vs 0.64 mm), to reduce the resolution of the FE displacements at a level comparable to DVC methods, similar strain maps may be obtained. In [START_REF] Zauel | Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D Digital Volume Correlation measurements[END_REF], smaller lateral deformations were predicted than measured using DVC. In addition to the limitations addressed by the authors, another assumption might be added to explain such a difference. The presence of apparent localisation observed from the DVC strain map at the top of the bone sample (along the lateral direction), seems not to be entirely captured by the FE model. This might reflect the need for using DVC displacements to constrain also the lateral faces in order to catch properly this localisation.

The present analysis offers a tool for accurately determining the mechanical properties of porous scaffold materials in clinical applications of musculoskeletal tissue engineering [START_REF] Henson | The use of scaffolds in musculoskeletal tissue engineering[END_REF]. For example, in vivo bone compressive strains in the knee can range from 414 με during walking up to 2000 με during vigorous activities [START_REF] Burr | In vivo measurement of human tibial strains during vigorous activity[END_REF]. These values are much higher than the DVC strain uncertainties and thus both DVC techniques are viable to characterise the mechanical properties of scaffolds within the physiological strain range. The proposed method can also be used for trabecular bones and might have some clinical significance in better predicting the risk for osteoporotic fracture [START_REF] Yeni | Finite element calculated uniaxial apparent stiffness is a consistent predictor of uniaxial apparent strength in human vertebral cancellous bone tested with different boundary conditions[END_REF]. The use of realistic boundary conditions in the models might produce more accurate description of the internal deformation hence improve the predictions of the stiffness and strength currently available [START_REF] Zauel | Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D Digital Volume Correlation measurements[END_REF][START_REF] Van Rietbergen | Computational strategies for iterative solutions of large FEM applications employing voxel data[END_REF][START_REF] Yeni | Finite element calculated uniaxial apparent stiffness is a consistent predictor of uniaxial apparent strength in human vertebral cancellous bone tested with different boundary conditions[END_REF].

Conclusions

This paper compares 3D full-field displacements in a porous scaffold material measured using two digital volume correlation methods and predicted with finite element simulations. The results demonstrate that both DVC methods are applicable to the porous material and the measured displacements and strains compare well between the two DVC methods, and with those obtained from the FE model. One key ingredient for this agreement is related to the fact that measured boundary conditions were applied to the numerical models. For applications where displacement uncertainties of 0.02 voxel (0.4 μm) and strain uncertainties of the order of 200 με are acceptable, the local approach is appropriate and gives a good compromise between accuracy and computational cost. For applications where small strain levels and both low systematic error and low displacement uncertainty are required (below 0.02 voxel) with the same level of strain uncertainty, the global approach is a good alternative. Lastly, these displacement (and strain) measurement techniques seem to be promising for a range of applications in biomedical engineering. prescribed displacement of 0.5 voxel was applied. Notations x 1 >x 2 ...>x n refer to successive passes using sub-volumes of x 1 × x 1 × x 1 voxels 3 , x 2 × x 2 × x 2 voxels 3 ....to reach the final sub-volumes of x n × x n × x n voxels 3 . Results are compared with those obtained with the global approach, using an element size of 32 voxels. ε : DVC (sub-volume size: 0.64 mm), (d) yy ε : FE model (average mesh size: 0.127 mm), (e) yy ε : FE model (average mesh size: 0.42 mm). The macroscopic strain was applied along (Oz) direction. Similar maps were obtained for LA-DVC and GA-DVC and, for the sake of clarity, only LA-DVC is represented. The DVC maps were obtained with 50% overlap and a bicubic interpolation.

O

Table 1. Average displacements over the domain as functions of the prescribed value (correlation window: 32 and 64 voxels, i.e. 0.64 and 1.28 mm respectively). The spatial resolution of the global approach is twice the element size. Consequently, additional results with element sizes of 16 × 16 × 16 voxels 3 (320 × 320 × 320 μm 3 ) are also analysed. 3. Mean, μ and standard deviation, µ σ , of the axial and lateral strain distributions obtained using the FE model and both DVC methods (see Fig. 13). FE LA-DVC (resp. FE GA-DVC ) refers to the FE model response calculated with experimental displacements obtained with the local (resp. global) approach, LA-DVC (resp. GA-DVC).

  018). The results with a final sub-volume of size 32 voxels are obtained after a previous pass with a 64 voxels sub-volume size, which improves the correlation quality (Table

Figure 13

 13 Figure 13 depicts the histograms of the axial strain, zz ε , and the lateral strains, xx ε and yy ε . For each

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. (a) The micromechanical testing device positioned in the micro-CT chamber, (b) the linear compression applied to the sample is generated through a ballscrew system (S1,S2,S3) that drives the jaws (and consequently the platens) symmetrically in opposite directions.

Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .

 4567 Fig. 4. (a) 3D reconstruction of the cylindrical specimen (diameter of 8.57 mm and height of 14.7 mm); FE mesh generated from the volume of interest (marked in red in Fig. 4a) with an average element size of 0.42 mm (21,138 elements), (b) and 0.127 mm (502,726 elements), (c). The solid phase is yellow and the porous phase is red. The volume of interest is a cube of size 5.12 mm.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Stress-strain curve of a nodular graphite cast iron deduced from the DVC analysis of five micro-CT scans, for the two correlation methods: (a) LA-DVC (sub-volume size: 32 voxels), (b) GA-DVC (element size: 16 voxels). The spatial resolution of the global approach is twice the element size.

Fig. 10 .

 10 Fig. 10. (a) Histogram of the correlation residual, DVC GA r -, over the VOI (Fig. 4); 3D visualisation of the

OFig. 11 .Fig. 12 .Fig. 13 .

 111213 Fig. 11. The effect of the sub-volume or element size on the vertical displacement distribution (u z )for the two correlation techniques. The distributions were obtained from the displacements computed over the entire cylindrical specimen (diameter of 8.57 mm and height of 14.7 mm). Note that the volume of interest (Fig.4) was cropped at a distance of ~1 mm from the top (maximum vertical displacement ~-254 μm, as illustrated in Fig12c). The spatial resolution of the global approach is twice the element size.

Fig. 14 .OFig. 15 .OFig. 16 .

 141516 Fig.14. 3D visualization of the vertical strain, zz ε , of the VOI (Fig. 4): (a) LA-DVC (sub-volume size: 0.64 mm with 50% overlap), (b) GA-DVC (element size: 0.64 mm), (c) FE model with boundary conditions applied from LA-DVC (average mesh size: 0.42 mm), (d) FE model with boundary conditions applied from GA-DVC (average mesh size: 0.42 mm). The macroscopic strain was applied along (Oz) direction.

Table 2. Mean, μ and standard deviation, µ

Acknowledgements

The authors would like to acknowledge Smith & Nephew for providing the samples. This work is partially funded by the Higher Education Innovation Fund (HEIF4).

Conflict of interest statement

The authors confirm that no conflict of interest is related to the enclosed manuscript.

Ethical approval

Not required. σ , of the correlation coefficient, DVC LA r -, and correlation residual, DVC GA r -, obtained on the considered VOI (Fig. 4) using both DVC methods (see Figs. 9a and10a).