Keywords: market intensity, 5 Forces diagram, industry, competition, Physical Sciences

This research considers Michael Porter's "Five Forces" diagram for what forces firstly mean, a strength or an energy as an attribute of physical action or movement. Here, our objective is to build a competitive rivalry assessment tool to assist decision-making processes. Since the literature review has proven that such improbable couples make sense, we have invited Physics to collaborate with Management Sciences. After screening both Structured and Expert diagrams, we came up with a quantitative solution, which might be a new asset to diagnose the intensity of a market's rivalry through the calculation of every force's real weight. A theoretical application to art and salt sectors anchors the roots of further research.

Résumé : Dans cette recherche, les 5 forces du diagramme de Porter sont prises dans leur acception première, celle d'une énergie liée à une action physique. Notre objectif est ici de construire un outil diagnostic de l'intensité concurrentielle permettant d'accompagner les processus décisionnaires. Pour cela, nous avons invité les Sciences Physiques à collaborer avec les Sciences de Gestion, un couple improbable comme la littérature en révèle pourtant beaucoup. Après étude des diagrammes Expert et Structuré, nous avons bâti une solution quantitative qui pourrait être un nouvel atout dans le diagnostic d'une intensité concurrentielle par le calcul du poids réel de chaque force. Une application théorique aux secteurs de l'art et du sel pose ici les bases de futures recherches. Mots-clés : intensité concurrentielle ; modèle des 5 Forces ; industrie ; concurrence ; Sciences Physiques.

Resumen: En este estudio, las 5 fuerzas del diagrama de Michael Porter son consideradas por su primera explicación, aquella que la energía es vinculada a una acción física. Nuestro objetivó es construir una herramienta diagnóstica de la intensidad competitiva para acompañar los procesos decisorios. Para eso, invitamos a las Ciencias Físicas à colaborar con las Ciencias de Gestión, al final como una pareja improbable desde que la literatura lo revela mucho. Después de estudiar los diagramas Experto y Estructurado, hemos construido una solución cuantitativa que podría ser una nueva ventaja en el diagnóstico de la intensidad competitiva por el cálculo del peso real de cada fuerza. Una aplicación teórica en los sectores del arte y de la sal establece aquí las bases de futuros estudios. Palabras claves: intensidad competitiva; 5 fuerzas; industria; competencia; Ciencias Físicas.

After all, Science remains Science

Management sciences have experienced significant development during the last sixty years [START_REF] Bain | Barriers to New Competition[END_REF], with an intensification of business activities, alongside the theorization of entrepreneurial practices on a global academic and professional scale [START_REF] Camerer | Does strategy research need game theory?[END_REF][START_REF] Brandenberger | The right game: Use game theory to shape strategy[END_REF][START_REF] Whittington R | Strategy as a practice[END_REF][START_REF] Stam | Social capital of entrepreneurs and small firm performance: A meta-analysis of contextual and methodological moderators[END_REF]. By including formal sciences in their construction, management sciences proved not to be just a set of independent disciplines [START_REF] Dorn | Strategic Management Einstein: Still No Equal[END_REF][START_REF] Suddaby R | Rhetorical history as a source of competitive advantage[END_REF]. Although Isaac Newton was mentioned only once in all of Adam Smith's works, Newton had a considerable influence on the father of the modern economy [START_REF] Montes | Newton's real influence on Adam Smith and its context[END_REF], who, in return, also had a great affinity with natural philosophy and a strong willingness to link both the social and the physical worlds. Like in Adam Smith's perception, this mind independence and freedom of thinking, combining formal and soft analysis anchors our paradigm. Here, after a broad review of different strategic tools emphasizing the fact that companies are constantly looking for tangible tools to analyse their current situations and predict the future [START_REF] Hatchuel | Strategy as innovative design: An emerging perspective[END_REF][START_REF] Pech | Prophets and losses: the predictive impulse[END_REF][START_REF] Hyytinen | Does innovativeness reduce start-up survival rates?[END_REF], we present technics to assess industries since environment affect companies' profitability. Then, we explain why Porter's Five Forces diagram remains one of the most suitable tools to assess markets' rival intensity. Finally, we use formal, scientific and rigorous methods to quantify these forces.

Diving from Macro to Micro Relevance

Michael Porter, a foregone conclusion

It has been decades since academics nourish the already abundant strategic analysis tools for business leaders trying to link theory building and testing [START_REF] Colquitt | Trends in theory building and theory testing: A five-decade study of the Academy of Management Journal[END_REF]. Indeed, this confirms that strategy's boundaries expand, merely through the assessment of its performativity components [START_REF] Cabantous | The performativity of strategy: Taking stock and moving ahead[END_REF].

Consequently, from macro to micro analysis, strategic analytical tools should be considered as the foundation stone of all strategic decisions, either from local, international, corporate, or entrepreneurial business perspective [START_REF] Degeorge | L'opportunité entrepreneuriale au coeur d'un processus de réflexion et d'action[END_REF]. This is how our thought took us to consider Porter's most famous diagram as a serious candidate to welcome our theory (see figure 1).

Table 1 highlights our thinking's evolution from our original statement. Just like managers do, organisations also develop natural strategic decisional tendencies, among which the approach through competitive environments investigation clearly roots our motivation to sharpen Porter's Five Forces diagram's relevance. Indeed, we here predicate that organisations that make strategic decisions upon diagnosing their external and internal surroundings naturally seek related levers able to tactically make decision making more flexible as well as more relevant. The latter results in methodologically consider past academic contributions in order to enrich them with further perspectives. In table 1, this intellectual process is made clear.

Corporate Strategic Personalities

Game Theory

Uncertainty Environment Resource

Corporate Strategic Behaviours

Organisations' interactions are serial because they are anchored in fixed variables [START_REF] Camerer | Does strategy research need game theory?[END_REF][START_REF] Brandenberger | The right game: Use game theory to shape strategy[END_REF] Organisations are either risk-averse or pioneers [START_REF] Wernerfelt | Competitive Strategy under Uncertainty[END_REF] Organisations consider external elements such as market atomicity, environmental rigidity, and consumer behaviour the foundation of all relevant strategic decisions [START_REF] Porter | The contribution of industrial organization to strategic management[END_REF] Organisations resort to both tangible and intangible resources according to the latter's availability [START_REF] Wernerfelt | A resource-based view of the firm[END_REF])

Corporate Tactical Behaviours

Mathematical & econometric approach [START_REF] Dorn | Strategic Management Einstein: Still No Equal[END_REF] Historical & cultural approach (Prasad, 2011)

Structure Conduct

Performance (SCP): profitability comes from organisations' diagnosing agility and resilience capabilities [START_REF]Price and production policies of large-scale enterprise[END_REF][START_REF] Bain | Barriers to New Competition[END_REF] Financial approach [START_REF] Gallo | A refresher on return on assets and return on equity[END_REF]

Academic contributions

Abduction & stochastic paradigms [START_REF] Peirce | Collected Papers of Charles Sanders Peirce[END_REF] Resilience [START_REF] Werner | Vulnerable but invincible: A study of resilient children and youth[END_REF])

Porter and the Expert Diagram: offsetting personal interpretation issues by involving experts, and introducing a mathematical intensity weighting system

Economic & business intelligence [START_REF] Harbulot | Manuel d'intelligence économique, Major[END_REF])

𝑭 𝒋 = $ 𝐄 𝐢 𝐥 ⁄ 𝐄 𝐦𝐚𝐱 𝐧 𝐢-𝟏 × 𝐰 𝐢
where the total forces' intensity is

𝑭 = $ 𝑭 𝒋 𝟓 𝒋-𝟏
and the industry's attractiveness is Aind = 5-F [START_REF] Porter | Competitive Strategy: Techniques for Analyzing Industries and Competitors[END_REF][START_REF] Porter | The contribution of industrial organization to strategic management[END_REF][START_REF] Porter | Towards a dynamic theory of strategy[END_REF][START_REF] Rosenthal R | Experimenter Effects in Behavioral Research[END_REF][START_REF] Shkurko | Shvindina // Економічні проблеми сталого розвитку : матеріали Міжнародної науковопрактичної конференції[END_REF]) From this synthesis, among all approaches developed to study firms' interactions, we knowingly consider the organization as part of a more complex ecosystem since fixed conditions and variables become less relevant in changing environments when technological changes, modified buyer needs, and industry structural mutations [START_REF] Porter | Towards a dynamic theory of strategy[END_REF][START_REF] Kibler | Attributes of legitimate venture failure impressions[END_REF] occur. Michael Porter's original study field was the Industrial Organisation. Indeed, he then quickly focused his research on investigating changing business environments. At the beginning of the 80's, Porter then came up with a Five Forces diagram with the intent to timelessly summarise this complexity through the action of five distinct actors. It is from this perspective that we have considered Michael Porter's Five Forces Diagram as the most relevant tool to welcome our tentative to make transdisciplinary theories meet (see Figure 1). In this diagram, Porter explains that every organisation should raise five specific questions before breaking into a market or developing a competitive position:

Our contribution N/A N/
• (1) Is the organisation a pioneer or is competition already a reality?

• (2) Is there any organisation aiming at breaking into the same market?

Following the two previous questions, is the organisation able to cope with existing key success factors -i.e. elements that are common to all competitors, here to be distinguished from competitive and comparative advantages -or able to create sufficiently demanding key success factors if it comes to be a pioneer?

• (3) What is the level of bargaining power of clients?

• (4) What is the level of bargaining power of suppliers?

• And (5) is the market elastic to price so that there is a competitive threat from substitutes? Those questions shall, of course, be raised considering the business localisation's governmental rules, regulations and influence.

Notwithstanding the latter, they are essential to determine the degree of rigidity of a market. Consequently, we here outlined a series of mechanisms leading to a new quantified Porter's Five Forces diagram to better implement its related strategies [START_REF] Akan | Critical tactics for implementing Porter's generic strategies[END_REF] keeping in mind that if quantification cannot fully replace critical analysis, it can still bring extra information leading to consider competitive environments under new angles.

Towards a new diagram?

Here, the importance to offer a fully customizable tool is at stake. Therefore, unlike others quantitative diagrams aiming at managing complex decision making [START_REF] Quintus R | Emergent Strategies and Their Consequences: A Process Study of Competition and Complex Decision Making[END_REF], ours is data-based and relies on limited qualitative expertise with the possibility to freely set industry boundaries, products/services and their substitutes in the forces' intensity calculation.

Methodological Premises

To make a first draft of our diagram, we chose the induction method which refers to the process of inferring a general law or a principle from the observation of artefacts [START_REF] Rothchild | Induction, deduction and the scientific method: An eclectic overview of the practice of science[END_REF].

Since Newtonian mechanical science is the study of forcegoverned bodies' physical movement, we see Porter's Five Forces diagram as a system where the industry is the studied object that five distinctive forces influence the direction and the structure. In classical mechanics, the force always equals the dimension of a length and of a mass divided by time squared. In view of the different natures of Porter's forces, it seems difficult to fully transpose the notion of forces' homogeneity. Consequently, to respect this notion, the quantified expression of each one of Porter's forces is neither assigned to a dimension nor to a measurement unit.

Non-structural forces

We followed Porter's definition of industry as a group of similar companies competing against each other [START_REF] Porter | Competitive Strategy: Techniques for Analyzing Industries and Competitors[END_REF][START_REF] Porter | The contribution of industrial organization to strategic management[END_REF]. Some non-structural forces have been introduced since then [START_REF] Kibler | Attributes of legitimate venture failure impressions[END_REF] but not taken into consideration here because their very action is specific to specific industries.

Forces' variables

Each Porter's force is commonly associated with variables represented in table 2. We noticed that for each force, several non-quantifiable variables could be compiled through a function that equals key variables. Since profitability is a function of price and cost, we sought to relate forces to the latter. Another task was to distinguish tangible and intangible variables. Consequently, to determine each force's formula, we applied four conditions on key variables: 1) Tangibility: Key variables are numerically tangible and accessible.

2) Importance: Like a physics formula, their role is significant in both the direction and the intensity of the considered force.

3) Relevance: They receive strong influence from the force's nonquantifiable variables.

4) Practical Use: The formula of the force deriving from them has to be functional.

Quantitative interpretation

Industry rivalry. Four variables respecting the three conditions, were considered.

"Growth index c", divided into four stages -emergence, growth, maturity and decline -and referring to the industry life cycle, is a whole number from one to four.

We also considered the average differentiation cost either spent to improve and maintain the existing key success factors or to produce a unique selling proposal. If this cost is important, competitors shall attempt to adopt a cost domination strategy instead; hence it is in the numerator of the formula.

On the other hand, market size counters the effect of high exit cost. The bigger this size, the more firms shall preferably focus on their market / segment position through differentiation. Consequently, the formula is:

Rivalry Weight = c × @exıt cost EEEEEEEEEEE + dıfferentıatıon cost EEEEEEEEEEEEEEEEEEEEEEEEE J market size
Threat of new entrants. New entrants' strength mainly depends on their capability to acquire the industry's key success factors, and maintain this know-how at a competitive level through research and development. These barriers are divided into two types of cost: the average entry cost, and the sum of all average costs associated with i different attributes of the products/services, which make the key success factors of the latter. The strength of new entrants will also depend on the size of the market. The larger this size, the higher and diversified the demand, thus increasing the possibility for new entrants to break into new markets. The new entrants' strength formula, symmetrical to Rivalry weight, is translated as:

New entrants P strength = market size entry cost EEEEEEEEEEEEE + ∑ attrıbute ı cost EEEEEEEEEEEEEEEEEEE T U
Bargaining power of buyers. A buyers' strength formula implying "buyers' switching cost" and "price sensitivity" could hardly be practical due to a very in-depth industry study to access data. Consequently, it does not respect the Practical Use condition. Therefore, we found an alternative: we first defined a high average price p WXYW EEEEEEE applied by the biggest competitors in the industry, which shapes the latter.

Knowing that the products and services of the biggest competitors have a better performance and a greater level differentiation than those who hope to dominate the market solely through low average price (p Z[\ EEEEEE), the relationship from the companies' perspective is implied as:

Buyers P strength = p Z[\ EEEEEE p ^_`Ê EEEEE
In table 3, we vary the highest and the lowest prices to test correlations between mathematical and analytical results. Bargaining power of suppliers. To determine our first formula, "suppliers' strength", we focused on the "suppliers' cost" variable. Hypothetically, "suppliers' cost" is translated into a function of three variables: their concentration level, the differentiation level of the assets they sell [START_REF] Gallo | A refresher on return on assets and return on equity[END_REF] and the risk of integrating the activities of their clients: Suppliers cost= f (concentration level, differentiation level, integration risk).

When suppliers cost is high, either suppliers' resources are differentiated (e.g. unique employees skills, rare resources) or their bargaining power became important due to their strong concentration level [START_REF] Wernerfelt | A resource-based view of the firm[END_REF]. Suppliers cost can also be high when suppliers make economies of scale, the integration risk still being valid.

When suppliers cost is low, either suppliers' resources are undifferentiated, or concentration level is low; consequently, their bargaining power is limited. Thus, suppliers' strength moves in the same direction than suppliers' cost, and respects Tangibility, Importance, and Relevance conditions. The suppliers' strength has a general formula which is the average industry profit and the average suppliers' costs ratio so that the user has a comparative basis between different industries.

Suppliers P strength = external supplıers cost + ınternal supplıers cost EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE ındustry net ıncome EEEEEEEEEEEEEEEEEEEEEEEEEE Threat of substitutes. Substitutes' strength comes from two factors: The practical and qualitative benefits of using low-cost alternative products/services and the low buyers' switching cost associated with substitutes. The substitute's strength formula is related to Buyers' strength and can be expressed by the ratio of the average price of products/services of the industry and the average price of substitutes, such as:

Substitutes′ strength = prıce the ındustry charges EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE substıtutes′ prıce EEEEEEEEEEEEEEEEEEEEEE

Formulas

To assess the level of an industry's profitability, the Five Forces formulas developed above are multiplied:

Total strength = supplıers cost EEEEEEEEEEEEEEEEEEE ındustry net ıncome EEEEEEEEEEEEEEEEEEEEEEEEEE × p Z[\ EEEEEE p ^_`Ê EEEEE × c × @exıt cost EEEEEEEEEEE + dıfferentıatıon cost EEEEEEEEEEEEEEEEEEEEEEEEE J entry cost EEEEEEEEEEEEE + ∑ attrıbute ı cost EEEEEEEEEEEEEEEEEEE T U

× prıce the ındustry charges EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE substıtutes′ prıce EEEEEEEEEEEEEEEEEEEEEE

Even if this formula is functioning for companies, we simplified it as much as possible, under reasonable approximations, for those who meet obstacles in obtaining confidential data. Firstly, we expressed the net profit margin:

Net profit margin = net income sales revenue = sales revenue -suppliers cost sales revenue

By deduction and averaging on industry level:

sales revenue EEEEEEEEEEEEEEEEEE -industry net profit margin × sales revenue EEEEEEEEEEEEEEEEEE net ıncome EEEEEEEEEEEEEE

= supplıers costs EEEEEEEEEEEEEEEEEEE net ıncome EEEEEEEEEEEEEE

Therefore, Suppliers' strength is also equal to:

Suppliers′ strength = 1 industry net profit margin -1
The real difficulty lies in the factor:

Rivalry weight × New entrants′ strength

At the embryonic stage, investments are low due to uncertainty; entry and exit costs are similar, and essentially depend on the demand. Costs are not associated with key success factors or unique selling proposals. These suppositions implied that the factor between rivalry weight and new entrants' strength could be equal to "growth index c" at this stage.

At the growth stage, companies begin to position themselves and standardize their products/services around a dominant design and technology. Even if entry and exit costs incrementally increase, they remain low, key success factors are still not well defined, and unique selling proposals are not identified. Thus, their costs are bot low and on the same order of magnitude. At this stage, the factor is also equal to "growth index c".

For a mature industry, entry and exit costs are high but due to important investments made to obtain key success factors and unique selling proposals, exit costs could be superior to entry costs. Key success factors' costs increase quicker than costs that are linked to unique selling proposals, which are more episodic.

Therefore, we can issue the hypothesis that the factor could be equal to "growth index c" at this moment. We ten assume that this factor is equal to "growth index c" regardless of the industry stage. Thus, the simplified formula is:

Simplified total strength = f 1 industry net profit margin -1g × p Z[\ EEEEEE p ^_`Ê EEEEE × c

× prıce the ındustry charges EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE substıtutes′ prıce EEEEEEEEEEEEEEEEEEEEEE

As "growth index c" is the only variable which appeals to a qualitative expertise, we use a logarithmic scale, firstly to minimize misjudgement and, secondly, to obtain ranges of industry profitability averaging values: 𝑆𝑐𝑎𝑙𝑒𝑑 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑑 𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ = log yz (𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑑 𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ)

Application

We have applied the simplified formula on two industries, the salt and art industries, to demonstrate our arguments and we compared qualitative interpretation and numerical results, as shown in table 4.

We have chosen those sectors for two precise reasons: Firstly, to define limits, indeed a maximum total force and a minimum total force possibly reachable by all industries, and secondly, to show the underlying logic behind the formula. As far as the art industry is concerned, logarithmic force tends towards plus infinite. In this case, buyers are so powerful that trade-offs are non-existent: they buy or they do not buy regardless of the masterpiece's price. Buyers establish price lists and artists are completely subjected to the market's requirements.

Consequently, since those two sectors constitute the lowest and the highest reachable total force, we here draw a parallel between numerical and analytical results: On the one industries manufacturing necessity goods are infinitely profitable because they both do not have a typical product life cycle and are needed regardless of the economic cycle. On the other hand, industries manufacturing ultra-luxury goods are subjected to powerful buyers. Hence, the forces that are at stake are particularly important here. Consequently, analytical and numerical results converge towards the same conclusion.

Conclusion

Through Adam Smith's inspiration from Isaac Newton's mechanical science, we saw that Management Sciences are arguably at the crossroads of all sciences, becoming the privileged bridge that links these to business affairs. Moreover, in a context where decisions concerning business administration must be based on physical data and concrete tools, the development of a quantified diagram from Porter's Five Forces proved to be a solution achieving two goals at once. On the one hand, we showed that this series of tools and theories could be an adequate way to contribute to management sciences, and, on the other hand, we built a strategic analysis tool to make it fully functional to assist managerial decisions more effectively.

Through a series of assumptions and methodical reasoning, we determined a total strength formula to help assessing industry's level of competitive intensity. Obtaining cost data is a difficult task but we circumvented the problem by determining a simplified formula. However, future research shall deeper explain our theoretical contribution, as well as explore new sectors, ideally with the benefit of stochastic data.

Active cooperation between formal and management sciences, that we are trying to resuscitate here, gives some perspectives that should be considered in the future.

Paradoxically, although Smith's division of labour has led to an increasing specialization of skills, thus putting barriers between sciences, we believe that this choice to vertically improve a science can be accompanied by horizontal enrichment from other of study.

The considerable development of management sciences supported by the adoption of a distinctive language and representative methods shows that we are still at the dawn of the ambition to raise further epistemological questions.

Figure 1 .

 1 Figure1.Michael Porter's Five Forces Diagram (source: Mouillot, 2007)

Table 1 .

 1 A The foregone 5 Forces compromise.

	Newtonian method
	associated with
	Induction method here
	leads to a mathematical	N/A
	weighting system of
	Michael Porter's Five
	Forces Diagram

Table 2 .

 2 Common Porter's Five Forces components.

Table 3 .

 3 Testing "low & high price" as key variables in buyers' strength formula.

Table 4 .

 4 Numerical applications on salt and art industries of our simplified total formula. Salt industry's total force is zero because salt is a nonsubstitutable basic food, which makes it infinitely profitable.

			Salt	Art
		Numerical value	Comments	Numerical value	Comments
	Net margin	-		-	Same as for
	Suppliers' strength	-	Regardless the data on	-	the salt
			buyers, suppliers,		industry,
	Max price (US$)	-	rivalries and new	-	investigation
	Min price (US$)	-	entrants, their	-	is not
	Buyers' strength	-	investigation is not	-	necessary
	Growth index	-	necessary regarding the	-	regarding the
			substitutes' situation		substitutes'
	Price (US$)	-		-	situation.
					Here, if the
					price of the
					substitutes is
			A substitute that doesn't exist cannot be		zero, it is merely
	Substitutes price (US$)	+¥	purchased; consequently, the price is infinite.	0	because art industry's clients buy something
					with high
					emotional
					value.
					A ratio of
			A ratio of a positive certain value and		positive certain value
	Substitutes' strength	0	positively infinite value is zero	+¥	and zero is then positively
					infinite
	Total force	0		+¥
	Logarithmic force	-¥		+¥


N Dahdah

N Limodin

A El Bartali

J F Witz

R Seghir

E Charkaluk

J Y Buffiere

Damage investigation in a319 aluminium alloy by X-ray tomography and digital volume correlation during in situ high temperature fatigue tests

Keywords: Fatigue, A319, Digital Volume Correlation, in situ synchrotron X-ray tomography, High temperature

The aim of this work is to analyse the strain fields heterogeneity in an aluminium alloy subjected to low cycle fatigue at high temperature. In the cylinder heads produced by a lost foam casting process, the microstructure of the studied alloy consists in hard intermetallic phases and large gas and microshrinkage pores. In order to study the influence of this complex 3D microstructure on fatigue crack initiation and propagation at 250°C, an experimental protocol using laboratory and synchrotron tomography, Finite Element simulation and a new Digital Volume Correlation platform has been used. The results showed the role of pores in the crack nucleation and highlighted the importance of hard phases in the crack propagation thanks to the resolution on the DVC measurement.

INTRODUCTION

Aluminium alloys are extensively used in the automotive industry, especially the Al-Si alloy for different engine parts. Due to its good castability and mechanical properties, the A319 alloy is one of the preferred choice in automotive applications notably for the cylinder head. This part and in particular the inter-valve zone in fire deck, is subjected to severe thermomechanical stresses with temperature gradient from 20 to 250°C and high cyclic loadings that can lead to component failure in service [START_REF] Tab B An | TMF a fo Los Foa Cas ng alu nu alloys: TMF OF LOST FOAM CASTING ALUMINUM ALLOYS[END_REF]. In the automotive industry, the conventional die casting process is progressively being replaced by the Lost Foam Casting (LFC) process for geometry optimization, cost reduction and consumption control purposes. The main benefits of LFC over conventional casting are its ability to produce complex shapes with internal sections at reduced grinding and finishing costs [START_REF] Albon | Po os y and In all Fo a on n Los Foa Cas ng of 356 Alloy[END_REF]. However, aluminium alloy automotive parts produced by the LFC process have a coarser microstructure and more porosity defects than parts produced with conventional casting processes at faster cooling rates. As a result, the A319 cast Aluminium-Silicon alloy produced by LFC process and used for automotive cylinder heads has a multiscale microstructure: porosity with a size that ranges from small gas pores to large microshrinkage cavities, iron intermetallics, eutectic Al2Cu and eutectic silicon (Fig. 1.a). Above a critical size, pores [START_REF] Wang | Fa gu b av o of A356-T6 aluminum cast alloys[END_REF] play a decisive role on the fatigue properties by providing preferential crack initiation sites. However, an influence of oxides [START_REF] Cao | T nu l a on of F -rich phases on oxide films in Al-11.5 Si-0.4 Mg as alloys[END_REF], iron-based intermetallics [START_REF] Tab B An | TMF-LCF life assessment of a Lost Foa Cas ng A319 alu nu alloy[END_REF] and Si particles [START_REF] Buffière | Exp n al s udy of po os y and s la on to fatigue mechanisms of model Al-Si7-Mg0.3 as Al alloys[END_REF] are also observed. As a matter of fact, cracks can initiate at any large pores or hard phases present in bulk. The coarser microstructure produced by the LFC process has an effect on the mechanical properties and can drastically reduce the fatigue lifetime of the alloy. 2D studies and post-mortem observations (Fig. 1.b) reveal that the damage mechanisms are strongly limited to the presence of hard phases [START_REF] Tab B An | B av o , da ag and fa gu l f assessment of lost foam casting aluminum alloys under thermo-an al fa gu ond ons[END_REF]. However in this kind of material, the crack often nucleates inside the material; therefore, it is essential to characterize the mechanisms in 3D. In order to study the influence of the LFC microstructure upon the low cycle fatigue behaviour, an experimental protocol has been set up using X-ray tomography and 3D Digital Volume Correlation (DVC).

EXPERIMENTAL PROCEDURE

X-RAY TOMOGRAPHY

Small dog bone specimens were cut out from automotive cylinder heads in the inter-valve area (Fig. 2.a). A preliminary characterization by laboratory X-ray microtomography was performed at the MATEIS laboratory (Lyon, France) with a 80 kV acceleration voltage to ensure a 10% transmission of the X-ray beam through the cross-section of the sample. The scan was made at a medium resolution with a voxel size of 4.5μm and an acquisition time per image of 500 ms; these medium resolution images allow to: (1) select the most suitable specimens, (2) characterize in 3D the population of defects in the specimen gauge length, (3) obtain a 3D mesh of the porous aluminium matrix in order to compute the associated strain fields using a finite element software (Abaqus). These preliminary steps help focusing the scanned volumes to a region of interest where cracks are more likely to initiate. Low Cycle Fatigue tests were then performed under synchrotron X-ray tomography at the European Synchrotron Radiation Facility (ID19 beamline) using an in-situ test rig [START_REF] Dezecot | S p a Ma al a In s u 3D a a za on of g p a u fa gu da ag an s s n a as alu nu alloy using synchrotron X-ray o og ap[END_REF] developed at MATEIS laboratory (see Fig. 3). were performed with a load ratio of 0.1 and a maximum stress of the order of 150% of the yield stress at the corresponding temperature. Tests were conducted until failure with periodic interruptions in order to detect and follow fatigue crack

Unit in mm initiation and growth through tomographic scans which were recorded under minimum and maximum load of the considered cycle. All the 3D images were reconstructed by using both the classical filtered backprojection algorithm [START_REF] Kak | P n pl s of Co pu z d To og ap I ag ng[END_REF] and the Paganin reconstruction algorithm [START_REF] Pagan N | S ul an ous p as and a pl ud x a on f o a s ngl d fo us d ag of a o og n ous obj[END_REF] to have a better contrast between the different phases including eutectic Silicon. For the analysis, only the images reconstructed in Paganin were used. The grey level images were cropped to reduce the amount of data and the time needed to process them. Synchrotron tomography allowed visualizing the eutectic Al-Al2Cu, iron based intermetallic phases and above all eutectic Si (see Fig. 1.a), which could not be distinguished with laboratory tomography (see Fig. 4). Thus, the total volume fraction of hard inclusions that can serve as natural markers for DVC is about 17% including about 9% of eutectic Si [START_REF] Wang | Influ n of as ng os u u on da ag an s s n Al-Si alloys by using 2D and 3D in-situ analys s[END_REF]. In comparison, in a nodular cast iron where the spherical graphite nodules are the only natural markers, the measurement accuracy of DVC is virtually the same either by using laboratory [START_REF] Ré O É | Analys s and A fa Co on fo Volu Correlation Measurements Using Tomographic Images from a Laboratory X-ay Sou[END_REF] or Synchrotron tomography [START_REF] Ré O É | Influ n of closure on the 3D propagation of fatigue cracks in a nodular cast iron investigated by X-ray tomography and 3D volu o la on[END_REF]. In A319 alloy, Synchrotron tomography allows a good contrast to be achieved in the interdendritic space where strain heterogeneity is expected to occur during the fatigue test. As more natural tracers are observed with SR-CT than with lab-CT, a higher measurement accuracy could be obtained for the former with a smaller uncertainty and a higher spatial resolution (ref 3DMS Long). The microstructure of A319 alloy provides a suitable natural speckle for DVC. The 3D displacement field was measured with the DVC technique [START_REF] Rethore | D g al volu o la on analys s of syn o on o og ap ag s[END_REF]. In the present study, a 3D image correlation platform called YADICS [16] recently developed at LML laboratory (Lille, France) has been used. This platform is based on C++ and it has been optimized to process large 3D volumes in a limited time. In this section, the finite element simulation is first detailed then the DVC method is explained.

FINITE ELEMENT SIMULATION

FEM simulation was performed before the in-situ fatigue test in order to predict the most strained area where crack initiation is expected. 3D images obtained by laboratory tomography were used to generate microstructurally realistic meshes of the microstructure within the pristine samples (pores + matrix, no intermetallic second phase) thanks to Avizo software (Fig. 5.a). A triangular surface was reconstructed, the minimum and maximum sizes of the triangle element were set as 0.2 and 10 pixels respectively. The mesh is built in such a way that elements of the mesh generated around the pores have a smaller size than in other areas in order to decrease the number of elements. Then, a volumetric tetrahedral grid was build from the triangular surface mesh (Fig. 5.b). Those meshes were used to perform Finite Element (FE) simulation of the strain/stress distribution during monotonic loading (elasto-plastic calculation), i.e. at the maximum load of the first half-cycle (Fig. 5.

c). In this way, a von

Mises type elasto-plastic constitutive law has been used, with an isotropic hardening. The entire stress-strain curve coming from a tensile test has been introduced in the Abaqus software. A displacement of l=0.018mm is applied on upper and lower side of the FE model to obtain a maximum strain ε=0.4% at a length of the specimen L=9mm. In order to localize potential critical area, the cumulated plastic strain distribution is computed and plotted, as shown on Figure 5.c. This FE simulation are therefore of great help to define a consistent zone of interest for the tests.

DIGITAL VOLUME CORRELATION (DVC)

The DVC technique is a 3D extension of the well-developed DIC method and their principles are similar. DVC was developed from the end of 1990s [START_REF] Bay | D g al volu o la on: T -dimensional strain mapping using X-ay o og ap[END_REF], and is now widely used due to the development of 3D advanced imaging devices [START_REF] Su On | I ag o la on fo s ap , o on and d fo a on asu n s[END_REF] [19] [START_REF] Pan | In nal d spla n and s a n asu n us ng d g al volu o la on: a least-squa s f a wo k[END_REF]. The principle of DVC is to find the transformation between a fixed image and a moving one for a given metric using an optimization scheme (Fig. 6). To identify the transformation, several steps are necessary: define a metric, choose a sampling and an interpolator, then an optimizer and finally a regularization method. The first one is to define a metric with the type of the transformation as a variable, the fixed image and the moving image. The DVC formulation used here is based on texture conservation recalled in Equation 1, where x is the vector denoting the position of any voxel and u is the unknown displacement vector field. Each image is identified by its grey level function representing the volume of the sample: f for the reference 3D image taken before deformation, g for the deformed 3D image taken after deformation [START_REF] Roux | D g al I ag Co la on : F o D spla nt Measurement to identification of Elastic P op s-a R v w[END_REF] [22]. In this case, we used the optical flow equation which assumes that the grey level variations are due to the displacement of the material points only:

f(x) = g(x + u(x)) (Eq.1)
In this study, global (rigid and homogeneous) and elastic transformations were chosen in order to compute the metric defined by the Sum of Squared Difference (SSD) (Eq.2) which corresponds to a least squares problem. It consists in finding the best displacement field by minimizing:

- (Eq.2)
With YADICS software, the metric is evaluated on the whole image: it is a total sampling. Then, an optimizer method has to be chosen and in the present case a gradient descent method is used for optimization. The gradient descent method allows to converge quickly to a minima. The first step is to compute the gradient of the metric regarding the transform parameters (Eq.3):

The optimization can be performed with any iterative numerical algorithm. Once the metric gradient has been computed, an optimization strategy is chosen. The gradient method principle is explained in (Eq.4):

The gradient step can be constant or updated at every iteration and defined by: where γk=1 and .

Figure 6 : The basic registration components of YADICS [START_REF] Su On | I ag o la on fo s ap , o on and d fo a on asu n s[END_REF] image correlation platform at LML (see the text for details)

One notices that optimization rules used herein is not the classical Gauss-Newton used in [START_REF] Kak | P n pl s of Co pu z d To og ap I ag ng[END_REF], [START_REF] Bay | D g al volu o la on: T -dimensional strain mapping using X-ay o og ap[END_REF]. In order to treat such large volumes with a high Degree of Freedom (superior to 1 million), it is easier to use a steepest gradient algorithm [START_REF] Cau | Mé od géné al pou la ésolu on d s sys è s d'équa ons s ul ané s [T ansla d[END_REF] as there is no matrix problem to solve [START_REF]Nonl n a p og a ng[END_REF]. Finally, a bi-cubic interpolator is employed to apply the transformation to the moving image at each iteration. Cachier et al. [START_REF] Cac | I on f a u bas d non g d g s a on: PASHA algo[END_REF] demonstrated that the problem of minimizing image and mechanical energy can be reformulated in solving the energy image then applying a Gaussian filter at each iteration. Yadics software uses this strategy and adds a median filter (see the arrow in Fig. 6 from "Regularization" to "Transformation"), here, with a window of 3x3x3 elements, allowing to avoid local minima while preserving discontinuities. A median filter removes the noisy pixels by replacing the tested pixel with the median value of the pixels in the window [START_REF] Mandal | I ag F l ng Us ng All N g bo D onal W g d P x ls : Op za on Us ng[END_REF]. The median filtering is a nonlinear filtering technique that is known for preserving sharp changes in signal and for being particularly effective in removing impulsive noise [START_REF]An p ov d u s v d an f l ng s fo ag p o ss ng[END_REF]. One effective use of median filters has been the reduction of high-frequency and impulsive noise in digital images without the edge destruction associated with linear filters. Thus, the discontinuities, i.e. the sharp change of the displacement field, are preserved because it is a no diffusive method [START_REF] Arias-Cas O | Do s d an f l ng uly p s v dg s b an l n a f l ng?[END_REF].

The correlation is based on a multiscale resolution strategy like a pyramid scheme (see Fig. 7) allowing to save time as the image contains much fewer voxels in the first resolution levels. In the present case, six resolutions or scales were used; the coarsest one is "scale 5" where one "macro" voxel is averaged over 2 Before displacement field measurements, all the tomographic images were first rigidly registered with the reference image, i.e., the image at the minimum load of the first cycle. Therefore, a calculation with several sequences of OFI searching all the translations and rotations between two images was computed before DVC measurement.

INITIAL OPTIMIZATION OF DVC-ROUTINE

General statistical methods for experimental data processing were computed to show the efficiency of the regularization with a median filter. A cube of 250x250x250 voxels was chosen in a zone with a high strain concentration, and a comparison of standard deviation and mean values measured with different element sizes for DVC was realized. The results showed that the standard deviation of the strain is quite similar with an element size of 8x8x8 voxels (std(ɛ)=0.0056)

and with an element size of 4x4x4 voxels (std(ɛ)=0.0047). The calculations of the mean strain confirm the preservation of the strain amplitude between an element size of 8x8x8 voxels and 4x4x4 voxels, since the ɛ(mean) is equal to 0.0179 and 0.0178 for an element size of 8x8x8 voxels and 4x4x4 voxels, respectively. The originalities of this platform are this regularization technique, which is quite new in the correlation methods, and also the multi-resolution scheme where different strategies can be adopted for different scales.

The spatial resolution is set by the size of element selected for DVC computation. In this work different sizes of isotropic element were tested with 16, 8 and 4 voxels edge. The results of these calculations are shown in the Fig. 8. A phenomenon of diffusion appears due to the size of element. Indeed, when the size increases artefacts become visible around the crack.

This phenomenon is essentially caused by the discontinuity of the crack. In fact, the median filter used allows conserving the discontinuities and the strain amplitude. By using the above parameters of correlation, one DVC measurement takes about 90 minutes with a size of element of 16 voxels edge, 150 minutes with 8 voxels edge and 480 minutes with 4 voxels edge. Although, the duration of the calculation is longer than the others, an element with a 4 voxels edge appears to be the best for this study, as the strain fields are localized in the same area than with larger elements but show less diffusion. With sample. Then, the standard deviation of the displacement is calculated for each component in the three different directions permitting to verify if there is one favoured direction in the noise. Figure 9 shows an uncertainty range from about 0.021 voxel (0.058µm) to 0.058 voxel (0.16µm) for element sizes ranging from 32 to 4 voxels (88-11µm). The spatial resolution of the measured displacement in 3D shows encouraging results. Although, the uncertainty for high spatial resolution is about three times higher than at low resolution, the strain localization around the crack at an element size of 4 voxel is better assessed than at coarser resolution. In what follows, the results are shown with a 4 voxel element size. Figure 10 represents the different scales used in this work: the sample analysed is about 2x2 mm 2 after mechanical polishing, the size of the pores from which the cracks are initiated varies between 450 and 1100µm, and the smaller element used is 11µm.

RESULTS AND DISCUSSION

CRACK INITIATION

For fatigue tests, a maximum stress of 75 MPa was applied at 250°C, with an expected average strain εzz of 0.4% obtained from FE computation which is in accordance with an average strain of 0.5% measured by DVC between the minimum and the maximum loads of the first cycle. The in-situ tests highlighted the porosity as the main influent parameter on crack initiation as crack always initiated close to a pore, and often in the bulk independently of the temperature (Fig. 11a& b).

Results obtained at room temperature [START_REF] Dahdah | STUDY OF DAMAGE MECHANISMS IN A319 ALUMINIUM ALLOY BY X-RAY TOMOGRAPHY AND DIGITAL VOLUME CORRELATION[END_REF] [30] at different numbers of fatigue cycles showed crack initiation close to the largest pores and crack propagation along the hard inclusions towards the free surface. At temperatures above 200°C (Fig. 12), cracks were detected in silicon particles (Fig. 12b) around the main pore that drove to failure but also in other areas of the specimen gauge length (Fig. 12c). For all the specimens cracks initiated around large internal pores most of the time, during the first loading step. The FE simulations are shown in Fig. 13.a: a large strain concentration can be seen close to the pore cluster where crack initiation was indeed observed.

CRACK PROPAGATION

A correlation between the localization of damage and the strain localization can be performed based on the reconstructed 3D images. The internal microstructure of the material as well as the displacement and strain fields at 250°C between the minimum and maximum of each cycle are shown on Fig. 14 for different numbers of fatigue cycles. After the first cycle, a strain localization is noticed around a subsurface pore where the crack nucleated (see the arrow in Fig. 14.c). After 10 cycles, the strain localization extends to be visible, especially in the area between the pore initiating the crack and a neighbouring pore (see the red circle in Fig. 14.f). Crack propagation is observed in Fig. 14.h&i, after 30 cycles. The displacement discontinuities (Fig. 14.b,e,h) visible in the crack zone are in good agreement with the strain localization (Fig. 14.c,f,i) and the crack location (Fig. 14.a,d,g). In this zone, cracks are observed in numerous phases. In order to understand the propagation, a zoom was made between the two pores. Figure 12 shows the strain localization at different cycles. After the first cycle (Fig. 15.a), a strain localization is observable around an Al2Cu particle between the two pores, the local strain around the Al2Cu particle is about ε zz (Al2Cu) = 2.6% while the mean strain is εz z (mean) =0.4 %. The strain localization is also observed after 5 cycles (Fig. 15.b) on hard inclusions between the pores while no crack could be detected in this area at the same number of cycles, εzz (Al2Cu)= 4.9% and εzz (Si)= 3.3% w l εzz (mean)= 0.43% . Indeed, after 10 cycles (Fig. 15.c), micro-cracks are observed in the Al2Cu particle and in the silicon particles, matching with the previous strain localization. After 20 cycles (Fig. 15.d), the cracks propagate along the hard inclusions including iron intermetallics, copper containing phases and above all silicon phase leading to the final failure. From these observations, it appears that the presence of Si particles is an essential feature for the propagation of damage for the experimental conditions investigated here. Several explanations can be suggested to understand the role of Si particles on the propagation. The first one is that the silicon particles form an extended network [31] [32] in the whole volume of the specimen contrary to the copper and iron intermetallics phases [START_REF] Wang | Influ n of as ng os u u on da ag an s s n Al-Si alloys by using 2D and 3D in-situ analys s[END_REF]. The second explanation is the coarser morphology of the silicon particles which generate large stress concentration [START_REF] Ba O | A o P ob Analys s of S D s bu on n AlS Found y Alloys[END_REF] in this material that has been Sr modified but has not undergone a solution heat treatment. Lastly, the eutectic silicon is more brittle than other hard inclusions at high temperatures [START_REF] Sadler | C a a sa on of In all P as s n Mul o pon n Al-Si Casting Alloys for Engineering Appl a ons[END_REF]. Strain localizations were observed in zones where cracks propagations were observed. Moreover, the localization is noticeable before the crack observation and the strain level at cracked particles can now be quantitatively assessed thanks to DVC. The local strain level that leads to the cracking of particles ranges between 4 to 8 % for the Al2Cu particles and between 3 to 6 % for the Si particles.

This study highlights the possibility to understand the roles played by the pores and the various hard phases of the microstructure through the local measurement of strain. The DVC measurement is in accordance with the FE simulation based on a realistic 3D microstructure since the localization zones and the strain levels are quite equivalent. Some studies [START_REF] Gall | F n l n analys s of s ss d s bu ons n a da ag d S pa l lus s n as Al±S alloys[END_REF], [START_REF] Fan | Cy l plas y a po s and n lus ons n as Al-Si alloys[END_REF] used FE computation to show the influence of pores and Si particles size, morphology and configuration on particle fracture in Al-Si alloy, but these analyses are realized for simplified microstructure that does not take in consideration the complex structure of the alloy. Therefore, DVC technique gives more realistic results. Thanks to the high spatial resolution, a localization around the hard phases (Al2Cu and Si) in the neighbourhood of pores is pointed out.

This level of representativeness could not be accessible by FE computation as it meets several difficulties such as the mesh size (expensive calculation) and also the mechanical properties of the hard phases which are poorly understood at 250°C. Thus, the FE simulation allows only to have an idea of the crack location since it is around the pores that the crack nucleated. Then thanks to DVC measurement, the observation of the crack path through the cracked particles is possible.

The 3D images allow access to the 3D arrangement of the hard phases to understand the complex crack path and extract the 3D crack.

CONCLUSIONS

The efficiency of the experimental protocol using laboratory and synchrotron tomography, FE simulation and 3D DVC to study the influence of the casting microstructure upon the mechanical properties of an Al-Si alloy has been proved. To the best of the authors' knowledge, it is the first time that DVC images are obtained at high temperature in an optically opaque material and with such a resolution, i.e. at 4 voxel element size which results in a high Degree of Freedom (40 million).

Thanks to the use of synchrotron X-rays, a detailed description of the material microstructure and, hence a high density of markers that can be used for the correlation can be obtained (detection of Si particles through phase contrast) and the scan duration (45s) is small enough to avoid any creep effect in the mechanically loaded sample with a voxel size in the micrometre range. DVC measurement allows to access local strain levels which help in understanding of the influence of the pores and hard phases on the fatigue mechanisms. Crack initiation is porosity driven while propagation is correlated with the presence of hard intermetallic phases. The crack nucleates during the first loading cycle and propagates along the hard phases until failure. Micro-cracks are mainly observed in the silicon particles due to the stress concentration around them although some cracks are also visible in the hard second phases such as Al2Cu. These experiments at high temperature highlighted the role of pores on strain localization: the pores generate enough stress concentration to facilitate the propagation in this area. Hard phases and strain localization between pores seem to have a high influence in the propagation path. The feasibility of DVC measurement at this finer scale resolution was assessed by the uncertainty measured. DVC measurement with an isotropic element of 4 voxel edge is proved feasible with an uncertainty that remains at the sub voxel scale. This protocol also gives some very interesting information on the degree of complexity which has to be taken into account in so-called "realistic" microstructures to reproduce actual strain development within the samples.

Figure 1 :

 1 Figure 1: (a) Reconstructed tomographic slice (synchrotron tomography) showing the various microstructural features of alloy A319 produced by LFC process, (b) SEM optical microscopy showing crack propagation from an internal pore along hard phases This paper presents first the experimental protocol used for in-situ fatigue tests at high temperature thanks to Synchrotron tomography and DVC. In a second part, the strain and displacement fields are analysed by comparing the experimental and computed fields under monotonous loading, i.e. under the first half cycle of fatigue, and also by following the evolution of the fatigue crack in the specimen.

Figure 2 :Figure 3 :

 23 Figure 2: (a) Optical image of the intervalve region of a cylinder head where 36 fatigue samples have been extracted by electro discharge machining; (b) Geometry of the specimen (dimensions in mm)

Figure 4 :

 4 Figure 4: Slice of an A319 alloy from: (a) Laboratory tomography and (b) Synchrotron tomography allowing the characterization of the different phases. The eutectic Si can be distinguished only with Synchrotron tomography.

Figure 5 :

 5 Figure 5 : (a) 3D rendering of pores with Avizo software, (b) 3D volume mesh (5M tetrahedron), (c) FE calculation with Abaqus software showing Ɛzz

  the full resolution image corresponds to "scale 0". The correlation calculation is a sequence of Optical Flow Integrated (OFI) and Finite Element methods (OFFEM). For easy and fast convergence at the coarsest scales, only global transformations are searched for, using OFI algorithms while at the finest scales, the more computing intensive OFFEM algorithm was used to compute local displacements. The correlation sequence per image couple was defined with the following analysis sequence OFI_H, OFI_H, OFI_H, OFFEM4, OFFEM4, OFFEM4 where:OFI_H is defined by the OFI method, and assumes a homogeneous deformation such as rigid body motions, i.e, translations (Tx, Ty, Tz) and rotations (Rx,Ry,Rz), and global strains (Exx, Eyy, Ezz, Exy, Exz, Eyz). It measures the similarity by using Sum of Squared Differences (SSD). OFI was used for global transformation for scales 5 to 3, as it can reduce the dimensionality of the problem and it has a low sensitivity to noise. OFFEM is the Optical Flow elastic transformation based on Finite Element Methods, it is used for the elastic deformation, as it has a good spatial coherence. OFFEM4 mean the used element size is 4x4x4 voxels, respectively.

Figure 7 :

 7 Figure 7: Pyramidal process used in YaDICs DVC was performed between the reference image at minimum load and the deformed image at maximum load of each considered cycle. Displacement field measurements were performed for a region of interest of 1200x1200x601 voxels 3 .

  a1200x1200x601 voxels tomographic image and a 4 voxel element size, the number of Degrees of Freedom reaches 40 million.

Figure 7 :

 7 Figure 7: Strain field Ɛzz showing the effect of the element size on the localization of strain: (a) 16 voxels, (b) 8voxels, (c) 4 voxelsThe uncertainty of the measured displacement field was calculated to assess the feasibility of DVC. This uncertainty is determined from the standard deviation of the displacement computed between two images, one at the reference position and another after a 10µm translation in a direction at 90° with respect to the X-ray beam and to the rotation axis of the

Figure 9 :Figure 10 :

 910 Figure 9: Uncertainty in displacement fields in the 3 directions measured with DVC

Figure 8 (

 8 Figure 8

Figure 13 .

 13 b also shows the strain field calculated by DVC along the loading direction at the first cycle between the minimum and maximum load; the image of the microstructure was superimposed to this field to allow comparison of the crack path with local deformation. The displacement field measurements allowed identifying and tracking the development and localization of deformation, and allowed identifying the initiation sites of microcracks during the low cycle fatigue test in relation with the observed microstructure. The DVC results are in good agreement with the FE simulation regarding the strain concentration around the pores; however a large strain accumulation close to clusters of hard second phase particles is also detected by DVC but not by the FE simulation which is based on a simplified mesh where only pores are considered (see red circles). These results underline the importance of the hard phases on the crack path. More efforts could be necessary to introduce hard particles in the FE simulation.

  Figure 10 simulation during monotonic loading (elasto plastic calculation), i.e. at the maximum load of the first half-cycle of fatigue and (b) Strain field measured with DVC between the minimum and maximum of the first cycle; strain localization at hard second phase particles in the DVC results are circled in red.

Figure 11 :Figure 12 :

 1112 Figure 11: Slice view at 477 µm below the surface, (250°C, max= 75MPa), at (a,b,c) Ncycles=0, (d,e,f) Ncycles=10 & (g,h,i)Ncycles=30, showing (a,d,g) the crack location, (b,e,h) 3D displacement field (in voxels; 1 voxel=2.75µm) along the loading direction z and (c,f,i) εzz strain

ACKNOWLEDGMENTS

The authors wish to thank the ANR (Agence Nationale de la Recherche) MatetPro project INDiANA (ANR-12RMNP-0011) for funding the study on Al-Si aluminum alloys. The ESRF (European Synchrotron Radiation Facility) is also acknowledged for providing beamtime on the ID19 beamline; Elodie Boller and Vincent Fernandez from ESRF are particularly acknowledged for their help and advices with the Synchrotron tomography experiment. The ISIS4D X-Ray CT platform has been funded by International Campus on Safety and Intermodality in Transportation (CISIT), the Nord-Pas-de-Calais Region, the European Community and the National Center for Scientific Research. The authors gratefully acknowledge the support of these institutions.