Luigi Sportiello

Stefano Zanero

Context

Keywords:

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Introduction

An important task in digital forensics is the retrieval of deleted files from storage media. Since files are typically stored as sequences of data blocks, the retrieval process involves the identification and collocation of the original blocks of each file. This is often performed using file system structures, which due to the persistence of file system metadata, still point to deleted data. When old data has to be retrieved or when the file system has extensive damage, it is necessary to use a "file carving" technique that reconstructs files based on their content. This is usually performed by relying on the signatures of known file headers and footers to detect the beginning and the end of each file on the storage media [START_REF] Richard | Scalpel: A frugal, high performance file carver[END_REF]. Of course, this creates problems when reassembling fragmented files in which blocks belonging to different files are interleaved [START_REF] Garfinkel | Carving contiguous and fragmented files with fast object validation[END_REF].

To perform file carving without relying on headers and contiguity, it is useful to classify blocks according to their file types based entirely on their content. Another application of file block classification is the detection of data hidden in locations that are not pointed to by the file system or residual data (e.g., in memory dumps or swap files and temporary files). A review of file carving techniques [START_REF] Pal | The evolution of file carving[END_REF] underscores the importance of block classification in creating novel file carving solutions.

When using a classifier to perform file block classification, it is important to have a high detection rate because missing blocks may compromise file reconstruction. Also, it is necessary to have a low false positive error rate to reduce the computational complexity of file recovery.

However, existing classifiers [START_REF] Calhoun | Predicting the types of file fragments[END_REF][START_REF] Sportiello | File block classification by support vector machines[END_REF][START_REF] Veenman | Statistical disk cluster classification for file carving[END_REF] exhibit two problems. First, classification performance is far from perfect: false positives and false negatives are present, and they may hinder the reconstruction process. Thus, an improvement of the block-by-block classification approach is required. Second, some file types (e.g., doc and pdf) are inherently "compound" in nature, meaning that they may contain data encoded in other file type formats (e.g., an image embedded in a pdf file); this must be taken into consideration when creating the classifier. This paper first demonstrates the impact that compound files have on a statistical block classification approach; this underscores the importance of considering compound files when designing and testing a statistical block classification approach. Next, a context-based classification approach is proposed that improves on block-by-block classification schemes of compound files by exploiting the contiguity of file blocks belonging to the same file on the storage media.

Related Work

Two primary approaches for classifying file blocks into their original file types are: (i) using a distance measure between a given input block and each reference model/sample; and (ii) applying machine learning techniques to create an appropriate classifier.

A distance-based approach [START_REF] Karresand | File type identification of data fragments by their binary structure[END_REF][START_REF] Karresand | Oscar -File type identification of binary data in disk clusters and RAM pages[END_REF] performs the classification based on the frequencies of byte values and the differences between the values of consecutive bytes in a block. A set of files of each file type is used to compute the frequency model. If the distance between the frequencies of an unclassified block and one of the models is below a threshold, then the block is associated with the corresponding file type. Some solutions (e.g., [START_REF] Axelsson | The normalized compression distance as a file fragment classifier[END_REF][START_REF] Axelsson | Using normalized compression distance for classifying file fragments[END_REF]) measure the distance between a pair of blocks by comparing the compression of the two individual blocks with the compression of their concatenation. In this case, a block is classified by computing its distance from sample blocks representing different file types and associating the block with the file type of the closest sample block.

A machine learning approach uses statistical values as a set of input features in a classification algorithm. The algorithm is then used to classify new samples based on the learned model. A learning algorithm typically employs Fisher's linear discriminant [START_REF] Calhoun | Predicting the types of file fragments[END_REF][START_REF] Veenman | Statistical disk cluster classification for file carving[END_REF] or a support vector machine (SVM) [START_REF] Sportiello | File block classification by support vector machines[END_REF]. Since multiple file types exist, the problem can be categorized as multi-class classification. Binary classifiers can be used to distinguish between the multiple file types by generating several one-toone classifiers [START_REF] Calhoun | Predicting the types of file fragments[END_REF] that distinguish individual file types from each other. Alternatively, one-to-many classifiers [START_REF] Sportiello | File block classification by support vector machines[END_REF] can be created to separate a single file type from other file types. Two variations of multiple classifiers for identifying blocks have been proposed: one directly assigns a file type to a given input block; the other discerns specific file type blocks in a block set [START_REF] Veenman | Statistical disk cluster classification for file carving[END_REF].

File Block Classifiers

The purpose of file block classification is to assign a file type to a file block based only on its content. This work addresses the problem of detecting all the blocks belonging to a specified target file type in a block set (e.g., disk image). This is structured as a one-to-many classification problem in which a binary classifier for each file type is trained to discern blocks of the target type from among all the other file types.

A support vector machine (SVM) [START_REF] Burges | A tutorial on support vector machines for pattern recognition[END_REF] is used as the classifier. An SVM performs binary classification based on a training set (x i , y i ), i = 1, . . . , l, with each sample x i represented by n attributes (features) in the space R n and labeled with a class y i ∈ {1, -1}. The SVM obtains a maximally separating hyperplane of the form w • x + b = 0 by solving the optimization problem: min w,b,ξ

1 2 w T w + C l i=1 ξ i subject to y i (w T φ(x i ) + b) ≥ 1 -ξ i ξ i ≥ 0
Such a hyperplane linearly separates the space R n into two regions representing the two classes {1, -1}. A new data sample is assigned to a class according to the side of the hyperplane where the sample lies. Since the training samples x i may not be linearly separable in R n , to improve the classification, they are mapped to a higher dimensional space by the function φ. The linear separation is achieved in the higher dimensional space, resulting in a non-linear separation in the original space R n .

The mapping uses a suitable kernel function K(x i , x j ) ≡ φ(x i ) T φ(x j ). In our approach, we use the RBF kernel function K(x, y) = e γ∥x-y∥ 2 , γ > 0, a popular choice that is suitable for many problems [START_REF] Hsu | A Practical Guide to Support Vector Classification[END_REF]. Thus, the classifier parameters are: γ (kernel function parameter) and C (misclassification penalty parameter).

In a classification process, it is important to represent the samples using a set of features that highlight the differences between items from different classes. Our classification technique uses the features defined by Sportiello and Zanero [START_REF] Sportiello | File block classification by support vector machines[END_REF]. These features include:

Byte Frequency Distribution (BFD): This feature is based on the frequencies f v of each possible byte value v ∈ {0, . . . , 255} in the block.

Rate of Change (RoC):

This feature is based on the frequencies of the differences between two consecutive bytes b i and b i+1 in a block (i.e., distribution of b ib i+1 ).

Word Frequency Distribution (WFD): This feature is similar to BFD, but it considers a block as a sequence of 16-bit words and computes their frequency values.

Mean Byte Value: This feature interprets a block as a sequence of bytes and computes the average value.

Entropy: This feature interprets a block as a sequence of bytes and computes the entropy value.

Lempel-Ziv Complexity: This feature interprets a block as a binary stream and computes the Lempel-Ziv complexity.

Various combinations of these features were tested. The feature combinations that provided the most accurate classification results were used for file block classification.

Compound File Problem

Storage media may contain a variety of file types. Many of these file types encode data in a similar pattern throughout the length of the file. Good examples are image and video files. They are called "primitive" files because the blocks corresponding to these file types tend to present common features that can be exploited in identifying the blocks.

On the other hand, "compound" files have distinctive file structures in which data encoded in the form of primitive types may be embedded.

Such files are created, for example, when an image is embedded in a document file by a word processor or when a video is included in a presentation. In these cases, the file blocks corresponding to the file do not present uniform properties because the blocks related to the embedded data are statistically different from the other file blocks.

To perform file block classification in the presence of compound files, Roussev and Garfinkel [START_REF] Roussev | File fragment classification -The case for specialized approaches[END_REF] recommend classifying each block as a primitive type and then, during compound file recovery, use their internal structures to recover the complete file. Conducting this type of recovery requires the ability to distinguish blocks that constitute the basic structure of a compound file from the data blocks of other formats. This is, in fact, the focus of this paper.

Certain problems posed by compound files must be taken into consideration when creating models for block classification. In particular, with regard to a supervised learning approach, the training set should be prepared in a proper manner. To understand this, consider the training set for a given primitive file type. The training set would comprise examples and counterexamples. However, the training set would mislead the classifier if blocks from a compound file with embedded data that is encoded according to the target type format were to be mistakenly included among the counterexamples. In fact, the training set would comprise blocks of the target file type data that are labeled as both target and non-target, inducing poor classification performance. For this reason, when compound files are included in a training set, it is prudent to use compound file types that do not contain embedded data.

Experimental Setup

Classifiers were constructed using the methodology described in [START_REF] Sportiello | File block classification by support vector machines[END_REF], except for differences in data set preparation to account for the compound file problem. The experiments used the same data set of randomly downloaded files as in [START_REF] Sportiello | File block classification by support vector machines[END_REF], which includes bmp, doc, exe, gif, jpg, mp3, odt and pdf files. Because doc, exe, odt and pdf are compound file types, the corresponding files were inspected for embedded data. In constructing the file block classifiers, all files containing embedded data were replaced with files without embedded data that were randomly downloaded from the Internet using the same approach as in [START_REF] Sportiello | File block classification by support vector machines[END_REF].

The collected files were decomposed into 512-byte blocks, yielding an average of 28,000 blocks per file type. The length of 512 bytes was selected because it is the smallest block size that is commonly used to manage storage media. Also, as described in [START_REF] Sportiello | File block classification by support vector machines[END_REF], a smaller block size renders the classification task more difficult; therefore, using the smallest value yields conservative performance results.

For each file type, we constructed an SVM classifier to detect the corresponding blocks. For each classifier, it was necessary to set the relative values of the parameters γ and C, and to select the features for block representation. To identify the best feature-parameter combination for each file type, a seven-fold cross validation was conducted, which split the data set into a training set and test set containing 2,000 and 8,000 blocks, respectively. In each training set, half of the blocks were of the target file type, while the remaining blocks uniformly represented the other file types. For each file type we trained and tested a series of classifiers by varying the parameters γ ∈ {2 -15 , 2 -13 , 2 -11 , . . . , 2 5 , 2 7 } and C ∈ {2 -5 , 2 -3 , 2 -1 , . . . , 2 13 , 2 15 }, and attempting combinations of all the features as well as reduced versions (e.g., BFD related only to ASCII byte values) and concatenations (e.g., Entropy-Complexity-BFD) [START_REF] Sportiello | File block classification by support vector machines[END_REF]. The feature set selected was the combination that maximized the function 0.5 • TP + 0.5 • (1 -FP), where TP and FP denote the true positive and false positive error rates, respectively. After the best combinations for each file type were identified, they were used to create a final set of classifiers that relied on the entire block collection, with training sets of 28,000 blocks and test sets of 112,000 blocks.

Experimental Results

Table 1 lists the features used to test file type classification. Table 2 presents the SVM parameters used for each file type and the final classification results. The concatenation Entropy-Complexity-BFD proved to be the most effective feature representation for all the file types, except for bmp, for which RoC is marginally better. Thus, this file block representation can work well for most different file types.

Because the compound files were removed, the classification results are better than those presented in [START_REF] Sportiello | File block classification by support vector machines[END_REF]. This is particularly evident for the FP rate of the doc classifier, which decreased from 19.8% in [START_REF] Sportiello | File block classification by support vector machines[END_REF] to the current value to 2.4%. Also, the FP rates against gif and jpg (two common types of embedded data in doc files) reduced by 6% and 28%, respectively. Likewise, the elimination of compound files improved the gif and jpg classifiers; their specific FP rates against doc files decreased by 3% and 15%, respectively. The classifiers support data recovery of primitive file types (e.g., bmp and jpg). However, because the recovery of compound files (e.g., doc and pdf) requires the handling of embedded data, the classifiers provide limited support for compound file types (techniques such as those described in [START_REF] Roussev | File fragment classification -The case for specialized approaches[END_REF] should be used for these file types).

Context-Based Block Classification

Most of the files of interest in a forensic recovery process (e.g., documents, images and videos) are typically not small in size and, thus, span multiple blocks [START_REF] Garfinkel | Carving contiguous and fragmented files with fast object validation[END_REF]. Modern file systems tend to reduce file fragmentation, meaning that blocks belonging to the same file are stored in contiguous locations to the extent possible. When fragmentation occurs, the most common scenario is bi-fragmentation, where a file is stored as two series of contiguous blocks far from each other on the media [START_REF] Garfinkel | Carving contiguous and fragmented files with fast object validation[END_REF]. Thus, it follows that blocks are typically surrounded by other blocks of the same file (and the same file type), except for those at the beginning and at the end of a file or its fragments.

Even if the error rates of a block classifier are low, the misclassifications that occur are multiplied when dealing with many terabytes of data; this can significantly impact the final results. Therefore, it is necessary to improve the precision of block classification. Also, when there is insufficient information to make a correct decision, it is preferable to mark a block as "not-classified" instead of giving it an incorrect classification. This helps reduce errors in the processes that use the classification information [START_REF] Roussev | File fragment classification -The case for specialized approaches[END_REF].

Our approach exploits file block contiguity. When classifying a block B i , we rely on the classifications of its neighboring blocks because they tend to belong to the same file. We call such blocks the "context" of B i (Figure 1). We use the ContextSize parameter to denote the number of neighbor blocks to be considered on each side of B i , and combine the classifications of the blocks with the classification C i of the block itself. This yields a "context-based" classification

CC i = f (C i-ContextSize , . . . , C i-1 , [C i ], C i+1 , . . . , C i+ContextSize ).
The idea is that if a good, but not perfect, classifier is available, a more robust evaluation may be achieved by relying on a series of classifications (context).

Block Context Evaluation

The block context evaluation classification model is binary: for a given target file type, for each generic block B i , the classifier outputs classification C i ∈ {-1, 1}, where 1 means the block is of the target type and -1 means it is not.

The context-based classification CC i for a block B i is performed according to Algorithm 1. The classification of the context blocks preced-

Linear P n i=1 C i (1-1 n (i-1 2 )) weight with weight = P n i=1 1 -1 n (i -1 2 )
or negative value, respectively). The context blocks following B i ("right" context) are evaluated in a similar manner.

Our experiments compared two variants of the algorithm. The first variant ignores the classification C i of B i and relies only on the context. The second variant includes C i with the context during classification. The final classification CC i is deemed a "target" if both the evaluations of the context have positive values (this is interpreted as B i being in the middle of the target file). Similarly, if the two evaluations present negative values, then the block is deemed a "non-target." In the second variant, where C i is taken into account, if its value agrees with the two context evaluations, then the final classification is the same as that obtained using the first variant; otherwise, the output is "not-classified" because of the mismatch in the available information. Some special cases are considered. If one of the two context evaluations returns zero (perhaps because the context is placed over the blocks of two different contiguous files), then the block classification C i is compared with the non-null evaluation; if they agree, then the block is classified accordingly, otherwise, "not-classified" is returned. Also, if the two evaluations disagree or are both equal to zero, then the block is labeled as not-classified. This is because the available information is inconsistent.

Note that the left context is null for the first block of a disk image. It grows in size for each following block until it reaches the defined ContextSize. Symmetric behavior occurs at the end of the disk image, where RightEvaluation = 0 for the last block.

Table 3 shows the three ContextEvaluation functions that were tested. The Uniform function computes the average of the classifications of all the blocks in a context without any weights. The Exponential and Linear functions give higher weights to classifications of blocks that are closer to B i . The idea is that these blocks are more likely to be part of the same file as B i , so their support for the final decision is higher.

Experimental Setup

The second set of experiments focused on improving the detection of gif files using context-based classification. The gif files were chosen because they are a primitive file type whose classifier, as shown in Table 2, performs well. In fact, we show that a good classifier can be used as a basis in our context-based algorithm to achieve better classification. Note, however, that the concept is general although our experiment uses SVM classifiers as described above.

The experiments used a new data set comprising 1.5 MB of files downloaded from the Internet, each file was roughly 100 KB in size. All the collected files were divided into 512-byte block sequences. Four "disk image" scenarios were created: Scenario 1: The files were randomly concatenated to create a disk image without fragmentation.

Scenario 2:

The files were split into two equal fragments that were randomly concatenated to create a bi-fragmented disk image.

Scenario 3:

The files were split into three equal fragments that were randomly concatenated to create a tri-fragmented disk image.

Scenario 4:

The files were split into ten equal fragments that were randomly concatenated to create a ten-fragmented disk image. These scenarios were designed to test the context-based classification technique on common situations (corresponding to Scenarios 1-3) [START_REF] Garfinkel | Carving contiguous and fragmented files with fast object validation[END_REF]. Scenario 4 corresponds to a possible, albeit unlikely, high fragmentation situation.

The experiments involved two stages. First, the gif classifier was used on all the blocks of a generated disk image to yield their basic classifications. Next, the context-based classification for each block was computed using the output of the gif classifier. The experiments compared the two variants of the algorithm that include and ignore the classification C i of the block itself. Also, they explored ContextSize parameter values of 3, 5, 8 and 10.

Experimental Results

Figures 2 and 3 present the true positive (TP), true negative (TN), false positive (FP) and false negative (FN) rates, along with the notclassified (NC) rates, since TP + FN < 100% and TN + FP < 100% due to the presence of NC blocks. Figure 2 file scenarios. Figure 3 shows the context-based file block classification results for the tri-fragmented and ten-fragmented file scenarios. The graphs also show the basic classification results, achieved using the gif classifier without a context as a baseline.

The first key result concerns the FP and FN rates. Both the error rates are strongly improved (i.e., decreased) compared with the basic classification for all the fragmentation scenarios and parameter values. For the first three scenarios, the TN rates are improved when the classification of the block itself is ignored. On the other hand, the TP rate decreases with increasing fragmentation, but increases in Scenario 1 (note that that non-fragmentation is the most likely condition for a file). These results can be explained by the presence of a long series of contiguous non-gif blocks that provide a wide base for the context-based classification of non-target blocks, while smaller block series related to gif files and their fragments are present to be exploited in target block identification. Also, roughly 3% of all blocks in Scenarios 1-3 are not-classified due to the gif blocks being incorrectly classified as target blocks. This is clearer in the graphs for Scenario 4, where a drop in the TP rate is correlated with an increase in the NC rate.

As expected, the classification performance decreases when the fragmentation rate increases. However, in context-block classification, the TP and TN rates tend to be converted to NC rather than to FP and FN, keeping the rates low and maintaining a lower classification error.

Increasing the ContextSize parameter helps reduce the FP and FN rates; the TN rate is also slightly increased, except that for the last high fragmented scenario. However, the TP rate does not seem to be positively affected, probably due to the absence of target block series long enough to be exploited by the enlarged context. A ContextSize value of five appears to represent a good trade-off in our experiments.

The results suggest that it is better to ignore the classification of the block itself when classifying a block. In fact, taking C i into account negatively affects the TP and TN rates, leaving the FP and FN rates substantially similar, due to an increased NC rate.

The three ContextEvaluation functions exhibit comparable performance. The Linear function performs worse than the other two functions for a ContextSize value of three, probably because the short context causes the function to focus mainly on the two contiguous blocks of B i . On the other hand, the Linear function outperforms the others in the high fragmentation scenario. In this scenario, it effectively reduces the effect of the non-target blocks in the context (a typical situation close to fragment boundaries), relying on the context blocks closer to the block under classification.

The classification approach tends to concentrate NC blocks mostly at the boundaries of files or fragments. A file carver could exploit this fact to identify block regions belonging to a specific file type and related to a file or fragment on the storage media, but with "faded" boundariescorrect classification in the middle of fragments with NC values near the ends. Then, the file carver could attempt to collate these block regions, varying their length in the range identified by the relative NC areas, until a complete file is recovered (the various combinations could be checked using a file validator [START_REF] Garfinkel | Carving contiguous and fragmented files with fast object validation[END_REF]).

Conclusions

This paper makes two contributions to file block classification for forensic data carving applications. The first is a technique that improves the performance of single block file block classifiers -false positive and false negative error rates are reduced using training sets restricted to primitive file types. The second is a context-based classification methodology that exploits the spatial coherence of data, i.e., the contiguity of blocks related to a given file. This methodology improves block classification performance by covering misclassifications at the cost of introducing a limited number of non-classified blocks. The methodology is general and can be applied in conjunction with other content-based file block classification algorithms. Our future research will focus on this aspect and will also assess the impact of the methodology on other block classification strategies.

The views expressed in this paper are those of the authors and do not reflect the official policy or position of the European Commission.

Figure 1 .

 1 Figure 1. Context-based file block classification.

Figure 2 .Figure 3 .

 23 Figures2 and 3present the true positive (TP), true negative (TN), false positive (FP) and false negative (FN) rates, along with the notclassified (NC) rates, since TP + FN < 100% and TN + FP < 100% due to the presence of NC blocks. Figure2shows the context-based file block classification results for the non-fragmented and bi-fragmented

Table 1 .

 1 Feature set used to represent file blocks.

	Feature	Description
	Entropy	File block entropy
	Complexity	File block Lempel-Ziv complexity
	BFD	Frequency of byte values in the file block
	Entropy-Complexity-BFD Concatenation of entropy, complexity and BFD
	RoC	Frequency of differences between two consecutive
		byte values in the file block

Table 2 .

 2 File block classification by SVMs (no embedded data in compound files).

			Feature				γ, C	TP	FP
	bmp		RoC				2 1 , 2 9 99.6	1.7
	doc		Entropy-Complexity-BFD 2 3 , 2 3 91.0	2.4
	exe		Entropy-Complexity-BFD 2 1 , 2 5 87.1	0.1
	gif		Entropy-Complexity-BFD 2 5 , 2 1 95.5	3.9
	jpg		Entropy-Complexity-BFD 2 5 , 2 1 96.4	3.9
	mp3		Entropy-Complexity-BFD 2 5 , 2 1 96.9	2.8
	odt		Entropy-Complexity-BFD 2 5 , 2 1 96.8 16.7
	pdf		Entropy-Complexity-BFD 2 3 , 2 1 94.4 19.8
	Average						94.7	6.4
				FP per File Type
		bmp doc exe	gif	jpg mp3 odt pdf
	bmp	-	8.0	3.0	0.1	0.2	0.2	0.1	0.3
	doc	10.4	-	5.2	0.2	0.2	0.7	0.1	0.3
	exe	1.3	3.6	-	0.3	0.2	0.1	0.0	0.3
	gif	0.7	5.3	1.7	-	3.2	2.7	6.8	6.9
	jpg	0.4	0.5	1.2	2.1	-	6.0	10.7	6.2
	mp3	0.9	0.6	2.9	2.6	5.3	-	4.8	2.5
	odt	0.4	6.6	8.5	12.3 20.4	10.6	-	57.7
	pdf	0.5	6.2	7.7	16.1 16.9	7.2	84.0	-

Table 3 .

 3 Context evaluation functions.

		ContextEvaluation(C1, C2, ..., Cn )
	U nif orm Exponential	1 n P n P n i=1 Ci i=1 C i e -(i-1 2 weight	)/n	with weight =	P n i=1 e -(i-1 2 )/n

Acknowledgements

This research was partially supported by the Prevention, Preparedness and Consequence Management of Terrorism and Other Security-Related Risks Program of the European Commission under Project i-Code: Real-Time Malicious Code Identification; and by the EU Seventh Framework Program (FP7/2007-2013) under Grant No. 257007 -SysSec.

ing B i (i.e., "left" context) is performed using the ContextEvaluation function, which returns a value from {-1, 1} expressing the number of considered blocks that correspond to the target or non-target (positive


F Hannard

T Pardoen

Éric Maire

Christophe Le Bourlot

R Mokso

E Maire

A Simar

Characterization and micromechanical modelling of microstructural heterogeneity effects on ductile fracture of 6xxx aluminium alloys

Keywords: damage initiation, aluminium alloys, X-ray tomography, Intermetallics, Cellular automaton model

come   L'archive ouverte pluridisciplinaire

Introduction

Ductile fracture results from the nucleation, growth and coalescence of small internal cavities. Microvoid nucleation is usually associated with the fracture of brittle particles or interface decohesion [START_REF] Tanaka | Cavity formation at the interface of a spherical inclusion in a plastically deformed matrix[END_REF][START_REF] Argon | Cavity formation from inclusions in ductile fracture[END_REF]. After nucleation, voids grow by plastic deformation of the surrounding matrix [START_REF] Benzerga | Ductile fracture by void growth to coalescence[END_REF]. During the last four decades, ductile fracture has been a subject of intense investigation in the solid mechanics and material science communities [START_REF] Benzerga | Ductile fracture by void growth to coalescence[END_REF]. The first micromechanical treatments due to McClintock [START_REF] Mcclintock | A criterion for ductile fracture by the growth of holes[END_REF] and Rice and Tracey [START_REF] Rice | On the ductile enlargement of voids in triaxial stress fields[END_REF] considered isolated voids. Stable growth proceeds until plasticity localizes within the ligament between closely spaced cavities setting the onset of the coalescence mechanism [START_REF] Thomason | Ductile Fracture of Metals[END_REF]. The accumulation of void linkages leads to the formation of a dominant macroscopic crack that finally propagates into the entire material.

The understanding and modelling of the different steps of the process of nucleation, growth, and coalescence of voids have been enhanced over the years, as covered in several recent review papers [START_REF] Benzerga | Ductile fracture by void growth to coalescence[END_REF][START_REF] Pineau | 2.06 -failure of metals[END_REF][START_REF] Pardoen | Multiscale modeling of ductile failure in metallic alloys[END_REF]. However, major questions remain about the understanding and proper modelling of ductile fracture in metals involving significant microstructure heterogeneities. This introduction proceeds with a review of the main issues related to heterogeneities and how they affect damage evolution and complexify the application of existing models.

Void nucleation is inherently a discontinuous process made of a succession of discrete nucleation events [START_REF] Pineau | 2.06 -failure of metals[END_REF]. The effect of particle size, associated with the distribution of internal or interfacial defects, can be a dominant feature of the inhomogeneity of the damage process. Void nucleation becomes increasingly difficult with decreasing particle size since the largest defects are more likely to be found in the largest inclusions. The particle shape combined to the loading orientation can also contribute to the heterogeneity of the nucleation process [START_REF] Beremin | Cavity formation from inclusions in ductile fracture of A508 steel[END_REF][START_REF] Lassance | Micromechanics of room and high temperature fracture in 6xxx Al alloys[END_REF]. Furthermore, the interfacial resistance between second phase particles and surrounding matrix is dependent on the local chemical composition [START_REF] Pineau | 2.06 -failure of metals[END_REF]. Nucleation can be inhomogeneous also because the load transfer in the particles depends on local microplasticity effects related to microstructure details and local crystallography, see e.g. [START_REF] Bugat | Micromechanical modeling of the behavior of duplex stainless steels[END_REF]. Heterogeneities in the spatial distribution of the second phases which can be spatially clustered lead to local stress concentrations [START_REF]Proceedings of 6th International Symposium on Plasticity and its Current Applications[END_REF]. All these effects could in principle be rationalized using micromechanics based nucleation criteria. The Beremin group proposed a void nucleation model valid for a range of particle sizes formulated within a continuum approach [START_REF] Beremin | Cavity formation from inclusions in ductile fracture of A508 steel[END_REF], based on a critical value of maximum principal stress in the particle or at the interface. The model accounts for plastic strain incompatibility between the matrix and inclusion and for particle shape. The effects of inclusion orientation and of complex loading path on nucleation have been addressed by finite element(FE) simulations on complex multiparticle cells [START_REF] Roux | A new finite element approach for modelling ductile damage void nucleation and growthanalysis of loading path effect on damage mechanisms[END_REF].

Voids nucleated by particle fracture are initially flat and become more rounded with plastic deformation.

Voids nucleated by interface decohesion are initially rounded and tend to elongate in the principal loading direction under low stress triaxiality. Heterogeneities in the void distribution do not significantly affect the void growth rates [START_REF] Needleman | An analysis of void distribution effects on plastic flow in porous solids[END_REF][START_REF] Huang | The role of nonuniform particle distribution in plastic flow localization[END_REF][START_REF] Thomson | Void coalescence within periodic clusters of particles[END_REF][START_REF] Pardoen | An extended model for void growth and coalescence[END_REF] and the interaction between voids during the growth phase is usually weak in industrial alloys (i.e. with low volume fraction of constituent particles). However, these interactions become significant in void clusters of highly heterogenous materials (e.g., [START_REF] Achon | Comportement et tenacite d'alliages d'aluminium a haute resistance[END_REF]). For instance, small voids in the neighborhood of large cavities grow much faster than in a single size population [START_REF] Tvergaard | Interaction of very small voids with larger voids[END_REF][START_REF] Fabrègue | A constitutive model for elastoplastic solids containing primary and secondary voids[END_REF]. The void growth rate depends also on the local crystallographic orientation [START_REF] Yerra | Void growth and coalescence in single crystals[END_REF]. Aside from the simple analytical models for isolated voids, constitutive models for porous plastic materials were developed based on homogenization theory but without interaction effects, see in particular the Gurson model [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth,1. Yield criteria and flow rules for porous ductile media[END_REF] and a realm of its extensions [START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals[END_REF][START_REF] Pardoen | An extended model for void growth and coalescence[END_REF][START_REF] Besson | An extension of the Green and Gurson models to kinematic hardening[END_REF][START_REF] Liao | Approximate yield criteria for anisotropic porous ductile sheet metals[END_REF][START_REF] Benzerga | Plastic potentials for anisotropic porous solids[END_REF][START_REF] Keralavarma | An approximate yield criterion for anisotropic porous media[END_REF][START_REF] Leblond | Exact results and approximate models for porous viscoplastic solids[END_REF][START_REF] Klöcker | Growth and coalescence of non-spherical voids in metals deformed at elevated temperature[END_REF][START_REF] Lassance | Micromechanics of room and high temperature fracture in 6xxx Al alloys[END_REF][START_REF] Scheyvaerts | The growth and coalescence of ellipsoidal voids in plane strain under combined shear and tension[END_REF][START_REF] Nahshon | Modification of the Gurson model for shear failure[END_REF].

The dominant parameter controlling the transition from stable growth to coalescence is the relative void spacing which can be highly heterogeneous if particles are clustered. If coalescence proceeds after significant amount of void growth, the failure is said "void controlled". When nucleation is rapidly followed by void coalescence with limited amount of void growth, the fracture mechanism is said to be "nucleation controlled" such as observed for instance in high strength Al alloys [START_REF] Worswick | Damage characterization and damage percolation modelling in aluminum alloy sheet[END_REF]. Significant progress has been made over the past two decades in the micromechanical modeling of coalescence, mainly along the lines of the approach developed by Thomason [START_REF] Thomason | Ductile Fracture of Metals[END_REF] with several generalizations [START_REF] Pardoen | An extended model for void growth and coalescence[END_REF][START_REF] Fabrègue | A constitutive model for elastoplastic solids containing primary and secondary voids[END_REF][START_REF] Zhang | A new failure criterion for the Gurson-Tvergaard dilational constitutive model[END_REF][START_REF] Benzerga | Micromechanics of coalescence in ductile fracture[END_REF][START_REF] Lecarme | Void growth and coalescence in ductile solids with stage iii and stage iv strain hardening[END_REF][START_REF] Tekoglu | A criterion for the onset of void coalescence under combined tension and shear[END_REF][START_REF] Scheyvaerts | A new model for void coalescence by internal necking[END_REF][START_REF] Morin | Coalescence of voids by internal necking: Theoretical estimates and numerical results[END_REF]. The proximity within clusters of voids, see e.g. [START_REF] Thomson | Void coalescence within periodic clusters of particles[END_REF], has a huge effect on coalescence because the dominant parameter is the relative void spacing.

The impact of heterogeneities on damage accumulation until cracking can be modelled in many different ways, see review [START_REF] Pardoen | Multiscale modeling of ductile failure in metallic alloys[END_REF]. One approach consists in averaging the response on large RVE through statistical treatment of void nucleation. This statistical treatment can be connected to the Beremin model [START_REF] Beremin | Cavity formation from inclusions in ductile fracture of A508 steel[END_REF] based on a weakest link approach through a Weibull distribution for particle fracture stress [START_REF] Babout | Damage initiation in model metallic materials: X-ray tomography and modelling[END_REF] or based on more simple distributions, see [START_REF] Tekoglu | A micromechanics based damage model for composite materials[END_REF][START_REF] Huber | Predictive model for void nucleation and void growth controlled ductility in quasi-eutectic cast aluminium alloys[END_REF]. A mean average porosity is assumed for the void growth phase and coalescence is treated by accounting for a nearest neighbours void spacing [START_REF] Wilkinson | A model for damage in a clustered particulate composite[END_REF]. More advanced approaches consist in representing the microstructure and property heterogeneities in a systematic way. For instance, damage percolation models have been developed to capture cluster effects on coalescence and cracking [START_REF] Worswick | Damage characterization and damage percolation modelling in aluminum alloy sheet[END_REF][START_REF] Chen | Application of the complete percolation model[END_REF].

Heterogeneous damage nucleation can also be taken into account in FE calculations with different void nucleation rates in each element [START_REF] Devillers-Guerville | Notch fracture toughness of a cast duplex stainless steel: modelling of experimental scatter and size effect[END_REF]. Probably the most sophisticated approach is to generate a large ensemble of particles, following physical distribution, and to simulate all steps of the damage process without the need for approximate nucleation, growth and coalescence laws [START_REF] Moorthy | A voronoi cell finite element model for particle cracking in elastic-plastic composite materials[END_REF].

From an experimental view point, 3D X-ray tomography is perfectly suited to study the effect of microstructure heterogeneity on all steps of ductile failure. It allows non-destructive local characterization of the microstructure and quantitative analysis of the morphology of the particles associated to void nucleation [START_REF] Babout | Damage initiation in model metallic materials: X-ray tomography and modelling[END_REF]. The results of in situ tensile tests in X-ray tomography have lead to a quantitative characterization of the initiation [START_REF] Landron | Characterization and modeling of void nucleation by interface decohesion in dual phase steels[END_REF], growth [START_REF] Landron | Validation of void growth models using x-ray microtomography characterization of damage in dual phase steels[END_REF] and coalescence [START_REF] Landron | Experimental investigation of void coalescence in a dual phase steel using x-ray tomography[END_REF] in high strength steels, Al and Ti alloys.

The goal of the present study is to understand and model the effect of microstructure heterogeneities on damage accumulation in three 6xxx series Al alloys, but with a methodology that can be directly extended to any other metallic alloy. These alloys, i.e. Al 6005A, Al 6061 and Al 6056, exhibit a volume fraction of iron rich particles close to 1 %. However, the variation of fracture strain as a function of yield stress shown in Fig. 1.a (and explained further in this paper) is very different, involving a factor 10 in the reduction of area between the extreme cases (i.e. factor 3 in true "log" fracture strain). In a preliminary study [START_REF] Simar | Micro-mechanical modelling of ductile failure in 6005a aluminium using a physics based strain hardening law including stage {IV }[END_REF], a recent advanced extension of Gurson's model for porous ductile metals has been applied in order to reproduce this effect. However, because of the very small difference in volume fraction of Fe based particles, the model was not rich enough to allow capturing the differences between the three alloys. This negative outcome motivated the formulation of a physical model which does not rely on an average description of the microstructure. Furthermore, damage models are rarely challenged in applications involving wide datasets encompassing several materials and heat treatments. This is the purpose of the paper to describe the key experimental results regarding damage evolution motivating the need for a model that accounts for microstructural heterogeneities.

The outline of the paper is the following. Section 2 introduces the experimental methods used for the microstructural and mechanical characterization of the three alloys. The damage model is described in section 3. Subsequently, section 4 starts with the experimental results which are used for the identification of the damage model. The model is then validated by comparison with additional experimental data.

Experimental methods

Materials

The fracture of 6005A, 6061 and 6056 Al alloy, is investigated under a wide range of heat treatment conditions in order to allow cross-comparisons at identical yield strength. Table 1 presents the chemical composition as measured by inductively coupled plasma optical emission spectroscopy. The alloys are provided in extruded plates (6005A alloy) or cold rolled plates (6061 and 6056 alloys), the rolling or extrusion directions are refered as the "primary" processing direction. The thickness of the plates is equal to 6 mm for the 6005A and 6056 alloys and 4.7 mm for the 6061 alloy.

The alloys are initially in T4 (6056 alloy) or T6 conditions (6005A and 6061 alloys). Some plates provided in T6 condition are transformed into T4 condition by a solution heat treatment of 1 hour at 535C for the 6061 alloy and at 530 C for the 6005A alloy. Heat treatments at 180C are performed on samples in the T4 state. Some 6056 plates in T4 state are transformed into T6 by a heat treatment at 180C for 8 hours. Heat treatments at 300C and 350C for various times ranging from 10 seconds to 20 days are performed on samples initially in the T6 state. Heat treatments are systematically followed by water-quenching and natural aging for at least one week before further artificial aging.

Microstructure characterization of base alloys

After standard polishing, the intermetallics and dispersoids particles are characterized using a Field Emission Gun Scanning Electron Microscope (FEG-SEM) operated at 15 kV under electron backscattered mode. X-ray microtomography are performed at the TOMCAT beamline of the Swiss Light Source [START_REF] Stampanoni | Tomographic hard xray phase contrast micro and nano imaging at tomcat[END_REF]. The sample is probed by X-ray beam tuned to 20 keV using a double crystal multilayer monochromator. 1800 projections are recorded in absorption mode using the 16 bit PCO.edges CMOS detector coupled with a 20X optical magnifying lens to a 20 micrometer thick LuAG:Ce scintillator. The observed region is a cube of 260 µm side length with an isotropic voxel size of 325 x 325 x 325 nm 3 .

Characterization of mechanical properties and damage

Macroscopic tensile tests

Uniaxial tensile tests are performed on an universal Zwick 50 kN testing machine under displacement control at constant velocity (1 mm/min). The cylindrical samples, 4 mm in diameter, are extracted perpendicular to the primary direction. The initial gauge length is equal to 30 mm. The initial yield stress σ 0 and the strain hardening parameters are extracted from the stress-strain curves based on an extended Voce law, i.e. ∂σ/∂ǫ p = Θ -βσ + Θ IV , where σ is the true stress and ǫ p is the true plastic strain. The strain hardening parameters are the dislocation storage rate Θ, the dynamic recovery rate β and the stage IV hardening Θ IV following the extended Voce law, see [START_REF] Simar | Micro-mechanical modelling of ductile failure in 6005a aluminium using a physics based strain hardening law including stage {IV }[END_REF]. The strain hardening exponent at necking is defined as n necking = ǫ σ ∂σ ∂ǫ | ǫ=ǫu , where ǫ u is the true strain at necking. The local true strain is defined as ε loc = ln A0 A , where A 0 and A are the initial and current cross-section area, respectively. The true fracture strain is thus defined as ε f = ln A0 A f , where A f is the final cross-section area.

Metallographic observations on broken samples

Three broken tensile samples per alloy are polished down to mid-thickness along transverse section near the fracture surface: one in T4 state, one in T6 state and one in overaged state. FEG-SEM images at a magnification of 1500x are taken along a line perpendicular to the fracture surface every 0.2 mm, see Fig. 2.

Particle fracture is the dominant nucleation mechanism with almost no interface decohesion. The surface and best fitting ellipse is recorded for every unbroken particle. If the particle is broken, it is reconstructed by merging all fragments and the best fitting ellipse and surface is recorded. About 10,000 particles have been analysed.

In-situ tensile testing in X-ray microtomography

A dedicated in-situ tensile testing machine described in [START_REF] Buffière | Characterization of internal damage in a MMCp using X-ray synchrotron phase contrast microtomography[END_REF] has been mounted on the rotation stage of the 3D tomography setup [START_REF] Maire | Initiation and growth of damage in a dual-phase steel observed by X-ray microtomography[END_REF] of the ID15A beam line at the European Synchrotron Radiation Facility (ESRF)

in Grenoble, France. The stage rotates at a constant speed while the tensile test is continuously performed at a displacement rate equal to 2 • 10 -7 m.s -1 for the 6061 and 6056 alloys and at 5 • 10 -4 m.s -1 for the 6005A alloy. The tensile force F is recorded. Three mini notched tensile specimens, with yield stress close to 120

MPa, outer diameter of 3 mm, reduced diameter of 1 mm and notch radius of 1 mm are deformed up to fracture.

The data acquisition is made at regular discrete time steps, with a voxel size of 1.06µm 3 . The procedure used to separate the voids from the Al matrix is described in [START_REF] Landron | Validation of void growth models using x-ray microtomography characterization of damage in dual phase steels[END_REF]. Only the central zone of the specimen is selected for the quantitative analysis of damage because it undergoes the largest stress triaxiality and thus the fastest damage rates [START_REF] Maire | Initiation and growth of damage in a dual-phase steel observed by X-ray microtomography[END_REF]. A cube of 200µm x 200µm x 200µm is analyzed which is large enough to involve a sufficient number of voids and to be representative of the damage process, see further.

For each scanning step, the voids are distinguished from the Al matrix and the geometrical characteristics are recorded. A tracking procedure follows the cavities from one step to the next. The results of the tracking carried out on the Al 6056 specimen provides the experimental void growth rate. Experimental data were analysed by automatic tracking following the algorithm in [START_REF] Lecarme | Heterogenous void growth revealed by in situ 3-d x-ray microtomography using automatic cavity tracking[END_REF], and double-checked by manual tracking. Only the cavities tracked for a minimum of 8 consecutive steps are used. The cavities leaving the analyzed box during straining are not considered. About one hundred tracks have been analyzed.

Cellular automaton micromechanics based damage model

Synopsis of the model

A cellular automaton model has been developed based on an initial distribution of n particles described by their position, dimensions (and thus shape) and orientation in a volume of material. A nucleation stress is associated to each particle following a distribution law which can be identified experimentally or predicted theoretically. The material is deformed under a prescribed stress and strain history which can be homogeneous or not at the scale of the volume. Nucleation, growth and coalescence of voids are described using analytical micromechanical models presented in sections 3.2 to 3.4. The model assumes no interactions between particles and/or voids during the nucleation and growth stages. The presence of the voids do not affect the applied stress and strain. At each deformation step, the nucleation, growth and coalescence of cavities are updated until a final failure criterion is verified (see section 3.5). The material is modeled using isotropic J 2 flow theory. Since the volume fraction of second phase particles f p is close to 1 %, the reinforcement effect of these particles is weak as well as the loss of strength resulting from particle fracture. The matrix stress state is thus accurately described with the strain hardening parameters identified on the overall experimental flow stress (see section 2.3.1).

Void nucleation model

Stress within the particle

Particle fracture depends on the stress transfer in the particle and on the interface which itself depends on microstructural parameters such as particle size, shape and relative orientation to the loading direction [START_REF] Beremin | Cavity formation from inclusions in ductile fracture of A508 steel[END_REF]. Void nucleation via interface decohesion is not taken into account as particle fracture is the dominant failure mechanism [START_REF] Simar | Micro-mechanical modelling of ductile failure in 6005a aluminium using a physics based strain hardening law including stage {IV }[END_REF], see also later.

The stress in the particle is decomposed as the sum of the stress in the matrix and of an additional stress transfer arising from the deformation mismatch between particle and matrix, which is calculated using Eshelby's theory with a plastic equivalent modulus for the matrix [START_REF] Berveiller | An extension of the self-consistent scheme to plastically-flowing polycrystals[END_REF] as in the Beremin approach [START_REF] Beremin | Cavity formation from inclusions in ductile fracture of A508 steel[END_REF]. Hill [START_REF] Hill | A self-consistent mechanics of composite materials[END_REF] developed a self-consistent scheme for the problem of an ellipsoidal elastoplastic inclusion embedded in an infinite elastoplastic matrix. The elastoplastic behavior of the matrix is described by the effective moduli L ef f which links the macroscopic stress rates Σ and the macroscopic strain rates εmacro as

Σ = L ef f : εmacro . ( 1
)
The homogeneous stress rate in the inclusion σ is then written as

σ = Σ + L ef f : (S -1 -I) : ( εmacro -εinc ) ( 2
)
where S is Eshelby's tensor and εinc is the strain rate in the inclusion. The isotropic approximation of the effective moduli proposed by Berveiller and Zaoui [START_REF] Berveiller | An extension of the self-consistent scheme to plastically-flowing polycrystals[END_REF] with the isotropic approximation of the effective moduli called L ′ writes

L ′ ijkl = µ * 2ν * 1 -2ν * δ ij δ kl + δ ik δ jl -δ il δ jk (3)
where δ ij is the Dirac delta function. The expression for the secant "elasto-plastic" shear modulus µ * and Poisson ratio ν * are given in [START_REF] Berveiller | An extension of the self-consistent scheme to plastically-flowing polycrystals[END_REF].

Introducing these secant elasto-plastic moduli into equation ( 2) gives the resulting interaction law relating finite quantities instead of their rates :

σ = Σ + L ′ : (S -1 -I) : (ε macro -ε inc ) . ( 4
)

Void nucleation model

Particles are treated as brittle solids and fracture is assumed to occur when the maximum principal stress σ max P art (eq.( 4)) reaches the critical stress σ i c associated to this particle. The particle fracture stress distribution is assumed to follow a Weibull statistics given by

P (σ max part , V part ) = 1 -exp - V part V 0 ( σ max part σ w ) m ( 5
)
where P is the probability of particle fracture for a given σ max part and volume of particle V part , see Babout et al. [START_REF] Babout | Damage initiation in model metallic materials: X-ray tomography and modelling[END_REF] for example; V 0 is the reference particle volume and σ w and m are the parameters of the distribution.

The nucleated void is assumed to be penny-shaped and its location is given by the intersection of the particle with a plane perpendicular to the maximum principal stress direction, see Fig. 3(a). The newly nucleated void is assumed to be ellipsoidal with main axis a, b and c. The crack is assumed to appear at the middle of the particle. The initial void aspect ratio W 0 is set equal to 0.01 (with c = W 0 √ ab, see Fig. 3(a)).

Lassance et al. [START_REF] Lassance | Growth and coalescence of penny-shaped voids in metallic alloys[END_REF] have shown that penny-shaped voids rapidly open and that the predicted void growth rate is independent of the choice of the initial porosity when imposing a value for W 0 sufficiently small.

Void growth model

The Rice and Tracey (R&T) void growth model [START_REF] Rice | On the ductile enlargement of voids in triaxial stress fields[END_REF] has been selected. The original version was developed for a spherical void in an infinite, rigid perfectly plastic material under uniform remote strain ǫ ij and stress σ ij . The volume change is a function of the hydrostatic stress only while the change of shape is dictated by the deviatoric stress component. The strain rate field is expressed in terms of the principal components ε1

≥ ε2 ≥ ε3 .
A version of the model taking into account the aspect ratio evolution elaborated by Thomason [START_REF] Thomason | Ductile Fracture of Metals[END_REF] links the rate of change of the radii in the three directions of space (R i , i=1,2,3) to the magnitude of

R mean = R1+R2+R3 3 : Ṙk = (1 + E) εk + 2 3 εl εl D R mean ( 6
)
where D is given by D = α T 0.25 exp( 3 2 T ) for T ≤ 1 , and D = αexp( 3 2 T ) for T ≥ 1, with the stress triaxiality T and α equal to 0.427 as corrected by Huang [START_REF] Huang | Accurate dilatation rates for spherical voids in triaxial stress fields[END_REF]. Here α can be taken as a free parameter that can be adjusted to account for the approximations in the model such as the absence of strain hardening. The factor D characterizes the dilatational contribution and (1 + E) characterizes the change of aspect ratio. The factor (1 + E) is fixed and equal to 5 3 as initially predicted by Rice and Tracey [START_REF] Rice | On the ductile enlargement of voids in triaxial stress fields[END_REF], see [START_REF] Worswick | Void growth and constitutive softening in a periodically voided solid[END_REF] for a more elaborated expression not used in this study. This void growth model has been initially developed for voids with the main axes aligned with the principal loading directions (Σ 1 ,Σ 2 ,Σ 3 ). An heuristic procedure is proposed in order to estimate the void growth rates in the case of an ellipsoidal void under general loading conditions. The principal radii of the ellipsoidal void are replaced by the three intersections of the void with the principal stress directions. This equivalent void is then assumed to follow the growth model given by equation ( 6), see Fig. 3(b). The void growth procedure is detailed in the Appendix. Since voids nucleate by particle cracking rather than by decohesion, a freshly nucleated void is initially penny-shaped and the mode of deformation is dominated by opening and blunting of the void/crack tip [START_REF] Tvergaard | Void shape effects and voids starting from cracked inclusion[END_REF].

At low T, the growth model given by equation ( 6) predicts a decrease of the void size in the transverse directions Σ 2 and Σ 3 while, in reality, the particle constraints the transverse contraction same as for the case where the particle is inside the cavity as analysed by [START_REF] Lassance | Growth and coalescence of penny-shaped voids in metallic alloys[END_REF]. A slight modification of the model has thus been made. The term controlling the transverse contraction due to the shape changing component of the remotely applied deformation field is deleted for the transverse directions (1 + E = 0 for Ṙ2 , Ṙ3 ). This version of the model will be referred next as the "constraint" growth model.

Void coalescence model

The void coalescence condition developed by Thomason is used in its extended form [START_REF] Thomason | Ductile Fracture of Metals[END_REF][START_REF] Pardoen | An extended model for void growth and coalescence[END_REF][START_REF] Scheyvaerts | The growth and coalescence of ellipsoidal voids in plane strain under combined shear and tension[END_REF]:

Σ n σ y = (1 -χ 2 ef f ) α T h 1-χ ef f W ef f χ ef f 2 + 1.24 1 √ χ ef f (7)
where Σ n is the overall stress component normal to the localization plane; χ ef f and W ef f are the effective void aspect ratio and effective void spacing respectively defined in eq. ( 8) and ( 9), see later. The parameter α T h depends on the strain hardening exponent n as proposed by Pardoen and Hutchinson [START_REF] Pardoen | An extended model for void growth and coalescence[END_REF] :

α T h = 0.1 + 0.217n + 4.83n 2 .
The coalescence model has been developed for identical neighboring voids having the ligament joining their centers normal to the principal loading direction Σ 1 . This is not representative of an actual microstructure.

A generalization has been proposed by Scheyvaerts et al [START_REF] Scheyvaerts | The growth and coalescence of ellipsoidal voids in plane strain under combined shear and tension[END_REF] in order to account for the orientation of the inter-void ligament. The equivalent maximum in-plane tensile stress Σ 1 P lane is found using a simple procedure detailed in [START_REF] Butcher | A multi-scale damage percolation model of ductile fracture[END_REF]. To apply this procedure, the two voids are transformed into an effective configuration made of two identical voids (Fig. 3 (c)) in order to define W ef f and χ ef f . The later is determined from the center-to-center distance L center and R center,1 and R center,2 the radii measured along the center-to-center direction (Fig. 3 (c)) as

χ ef f = R center,1 + R center,2 L center . ( 8
)
The effective void aspect ratio W ef f is taken as a mean of the aspect ratio of the two neighboring voids and is defined as :

W ef f = 1 2 R perp,1 √ a 1 b 1 + R perp,2 √ a 2 b 2 (9)
where (a 1 ,b 1 ) and (a 2 ,b 2 ) are the half-axes of the two ellipses resulting from the cross section of each void taken in the transverse direction to the in-plane stress direction. The equivalent voids resulting from this procedure are shown in Fig. 3(d), where R center = χ ef f Lcenter 2

and R perp = W ef f R center . Furthermore, the minimum distance between each void is checked to be larger than zero in order to ensure no overlapping.

This heuristic extension has been partly assessed by [START_REF] Scheyvaerts | The growth and coalescence of ellipsoidal voids in plane strain under combined shear and tension[END_REF], although only for spheroids of identical dimension.

This extension is certainly an open area for improvement. The local displacements of the centroid of the particles and cavities during a strain increment are calculated by interpolation of the FE displacement field.

Once the coalescence criterion is satisfied for two voids, failure of the inter-void ligament is modelled by merging the two voids and replacing them by the minimum ellipsoid that envelops them both (Fig. 3 (e)).

In this case, the axis of the cavity can be significantly out of the principal loading direction justifying the correction for non aligned ellipsoid (see Appendix). The coalescence criterion is then checked for this new void and each neighboring void, allowing the coalescence process to locally flourish. This procedure neglects the small additional strain needed to bring the two voids from the onset of coalescence to final coalescence (see [START_REF] Scheyvaerts | A new model for void coalescence by internal necking[END_REF] for discussion of details). More advanced versions of a coalescence model accounting for the shear component along the localization plane [START_REF] Tekoglu | A criterion for the onset of void coalescence under combined tension and shear[END_REF] will be considered in future extensions.

Final failure criterion

Several authors [START_REF] Marini | Ductile rupture of a508 steel under nonradial loading[END_REF][START_REF] Pardoen | Assessment of void growth models from porosity measurements in cold-drawn copper bars[END_REF][START_REF] Pardoen | An extended model for void growth and coalescence[END_REF] have shown the inadequacy of models based on a critical porosity value to predict the strain to failure. Indeed, the porosity at coalescence is found to depend on the initial void volume fraction f 0 , the flow properties and stress state and is thus not a "universal" value. The onset of coalescence in a small material volume as predicted by eq.( 7) is not a sufficient condition for fracture in the case of heterogeneous materials. Thomson et al. [START_REF] Thomson | Void coalescence within periodic clusters of particles[END_REF] have shown that coalescence of voids within a cluster consisting of a few particles is a stable event while the collapse of inter-cluster ligaments leads to imminent material failure. This inter-cluster ligaments collapse is associated to a significant and sudden increase of the porosity (i.e. a large rate of porosity increase). An empirical failure criterion has thus been chosen here to be a critical rate of porosity increase ∂f ∂ǫ p c

. The physical meaning of this critical rate of porosity increase is associated to voids linkage over a macroscopic scale that triggers catastrophic ductile failure. In a more sophisticated model, crack propagation from the most critical coalesced patch of voids would be the most correct approach but this goes out of the scope of the current state of this work.

Summary of the parameters of the model

The following set of parameters is needed to run a simulation with the cellular automaton model (see Fig. 4). The microstructural parameters, i.e. particles and initial cavities parameters 1 :

• the positions x i , y i , z i ;

• the dimensions R x,i , R y,i , R z,i ;

• the three angles which define the orientation in space θ 1,i , θ 2,i , θ 3,i .

The flow properties of the matrix :

• the initial yield stress σ 0 ;

• the extended Voce law parameters : Θ, β, Θ IV .

The damage model parameters :

• the Weibull distribution parameters 2 : V 0 , σ w , m ;

• the α parameter of the R&T model which is experimentally measured to correct for the limitations of the model : α growth ; • the critical rate of porosity increase ∂f ∂ǫ p c .

Numerical procedures

A multi-scale approach combines FE simulations and the use of the cellular automaton model at a local scale, see Fig.

Results and discussion

Microstructure and mechanical properties

The reconstructed X-ray microtomography volume of the three alloys in their initial states (as observed with 0.3 µm voxel size microtomography at the TOMCAT beamline in SLS) are illustrated in Fig. 5. Particles and initial porosities are respectively highlighted in blue and red. Fig. 5(b,c) presents the size distribution for the intermetallic particles based on SEM micrographs and on 3D X-ray microtomography images. The equivalent diameter D eq corresponds to the diameter of a sphere with the same volume as the particle. Table 2 gives the corresponding averaged parameters. The differences between the two characterization techniques are probably due to the difference of resolution. The SEM micrographs were taken at a magnification of 2000X allowing a statistical analysis of particles larger than about 500 nm. The X-ray microtomography images have a voxel size of 325x325x325 nm 3 (section 2.2) allowing to reliably identify particles larger than about 700 nm (i.e. 2 voxels). Indeed, it will be shown subsequently (section 4.2.1, Fig. 7) that the particles 1 Identified from X-ray microtomography data of the base alloys (section 2.2) 2 Identified in order to satisfy the Weibull distribution (section 2.3.2)

that will actually lead to damage are essentially the largest one (D eq > 1 µm). Thus, the exact identification (size, position, number density) of the smallest particles is of secondary importance for the prediction of the fracture strain by the cellular automaton model. Furthermore, the SEM micrographs allow the observation of the near surface of the polished sample only. As the particles are not necessarily intersected close to the equator, the value of the equivalent diameter measured in the plane section is smaller than the actual 3D value and the 2D equivalent diameter has thus been multiplied by a stereological factor 4 π . Globally, Fig. 5(b,c) shows that the trends remain in good agreement, irrelevant of the characterization technique. In particular Table 2 indicates that the particle volume fraction of all three alloys is close to 1 % (the value for the 6005A is underestimated due to the small size of the particles in that alloy). All alloys have a particle size shape factor close to 2 that is measured as slightly lower by SEM, but 2D imaging is not ideal for the identification of this parameter. The mean equivalent diameter is ˜1 µm for the 6005A alloy, ˜2 µm for the 6061 alloy and slightly above 2 µm for the 6056 alloy which includes a high fraction of very large particles (D eq >5 µm) as demonstrated in the inset of Fig. 5(b). Initial porosities are observed in the Al 6056 and Al 6061 both in the X-ray microtomography images and SEM micrographs. None of these porosities are observed in the Al 6005A at the resolution of the Xray microtomography images. The initial porosity f 0 , equivalent diameter and shape factor are provided in Table 3. Some of these cavities originate from the gathering during Al solidification of the hydrogen easily dissolved in liquid Al [START_REF] Toda | The true origin of ductile fracture in aluminum alloys[END_REF] while others originate from the fracture of intermetallic particles during the forming process. Fig. 6 presents the yield stress σ 0 of the three alloys as a function of the various heat treatments durations.

In each alloy, the yield stress peak corresponds to the T6 state. Fig. 1 summarizes the results for the three alloys in terms of true fracture strain (Fig. 1.a) and of true strain at onset of necking (Fig. 1.b) as a function of the yield stress. While the fracture strains are very different for the three alloys, the necking strains follow the same "banana" shaped trend with limited exceptions. This clearly indicates that uniform elongation is absolutely not representative of the true fracture strain. Fig. 1(c,d) provide the dislocation storage rate θ 0 and parameter β. These parameters are needed to account for strain hardening in the model (Fig. 4, input 3).

Damage evolution: experimental results and model parameter identification 4.2.1. Void nucleation

The magnitude and distribution of particle fracture stress can be identified in the following way. Metallographic observations of the polished sections of broken specimens (cf. section 2.3.2) are used instead of in-situ microtomography tests because the void nucleation kinetics is not accurately captured with the resolution of about 1 µm 3 (cf. section 2.3.3) when compared to the particle size (1 to 4 µm). The maximum principal stress σ max P art in each particle can be estimated using equation ( 4) based on the overall stress and strains calculated by standard FE simulations at the location of the particles. If a given observed particle is broken, the critical stress at which void nucleation has occurred lies between 0 and σ max P art . The nucleation stress of the broken particles is said to be interval-censored and falls in the interval (0, σ max P art ]. On the other hand, if a given particle is not broken, it can only be concluded that the critical stress at which void nucleation would occur is larger than σ max P art . The nucleation stress of the unbroken particles is said to be right-censored and falls in the interval (σ max P art , ∞]. The exact nucleation stress of the i th particle is thus not exactly known but is associated to an interval (0, σ max P art ] or (σ max P art , ∞]. A mathematical iterative procedure is needed in order to retrieve from these data, the nucleation stress distribution. A set of Weibull parameters is selected and for all the particles used to generate the curve of Fig. 7, a critical stress σ i c is assigned to each particles. σ max P art is calculated for each particle and if the critical stress σ i c is larger than σ max P art , the particle is considered as broken. The criterion is checked for all particles and the prediction of "broken" versus "unbroken" is compared to the experimental record. This procedure is repeated for several sets of Weibull parameters and the best parameters are found to be σ w = 1200 MPa, m=7, and V 0 corresponding to an equivalent diameter

d 0 = 2.2 µm.
A very strong assumption is made regarding the void nucleation stress: the magnitude and distribution of void nucleation stress is taken identical for the three alloys, depending only on the particle size. Alternatively, different critical stress distribution could have been identified but, for the sake of simplicity, and mainly to focus on the primary issue of particle size and spatial distribution, it was decided to aggregate all data together. This implicitly assumes that the intermetallic particles in the three alloys have a similar structure and mechanical behaviour. Fig. 7 shows the evolution of the fraction of broken particles as a function of particles size as observed experimentally and as obtained with the best Weibull parameters (σ w = 1200 MPa, m=7, and V 0 corresponding to an equivalent diameter d 0 = 2.2 µm) found in section 3.2. The predictions agree with the experimental evolution. Consequently, a Weibull distribution using the same parameters for the 3 alloys seems to be a reasonable approximation partly validating our initial choice. The fraction of broken particles increases with particle size. Fig. 7 also shows that smaller particles give rise to void nucleation in the 6005A alloy (Fig. 5(b)), but at relatively large equivalent strains (as it will be shown in section 4.3) corresponding to sufficiently large stress needed to break small particles. The alloy 6061 presents an intermediate behavior consistent with its reduced proportion of large particles (Fig. 5(b)).

Void growth

The results of the tracking carried out on the Al 6056 specimen provides the experimental void growth rate for each cavity (Fig. 8(a)), with clear dispersion in the growth kinetics. Fig. 8(b) shows a comparison of the experimental and predicted evolution of the relative rates of change R eq /R eq,0 of the equivalent radius of the voids. These rates have been averaged over all voids. The predictions of the R&T (cf. eq 6, section 3.3) in the original and "constraint" versions are provided in Fig. 8(b) for the void growth, and in Fig. 8(c) for the void aspect ratio evolution. The void aspect ratio W void is defined as W void = R1 √ R2R3 , see section 3.3. Knowing the values of T and of the strain around the cavity for each microtomography measurement, an inverse procedure is carried out in order to calculate the value of α (eq. 6) which best corresponds to the experimental relative growth rate for each version of the model. The best value associated to the void growth model is α ≈ 0.84. With α ≈ 0.84, one can observe that the rate of change of the void aspect ratio is then significantly over predicted (Fig. 8(c)). The same procedure has been carried out for the "constraint" growth model. In this case, the best value for the parameter α is found to be 0.57, much closer to the theoretical value 0.427 [START_REF] Huang | Accurate dilatation rates for spherical voids in triaxial stress fields[END_REF], and the rate of change of the void aspect ratio is more consistent with the experimental data. Landron [START_REF] Landron | Validation of void growth models using x-ray microtomography characterization of damage in dual phase steels[END_REF] and Lecarme et al. [START_REF] Lecarme | Heterogenous void growth revealed by in situ 3-d x-ray microtomography using automatic cavity tracking[END_REF] have largely discussed several possible reasons that justify why α can be different from the theoretical value. Now, this value α = 0.57 and constraint growth model are used for all other materials and all thermal treatments.

Exploitation and assessment of the model

The model is assessed in two steps. First, by direct comparison of the full model with microtomography results obtained on the three alloys with a similar yield strength. Second, all heat treatment conditions are modelled and compared to the experimental ǫ f , as provided in Fig. 1.a.

The growth model is now coupled to the nucleation and coalescence model, which was not the case in Fig. 8. Fig. 9 shows the evolution of the void volume fraction f . The predicted evolution of f agrees with the experimental evolution. The microstructural parameters used to run these simulations (see section 3.6) are identified from the X-ray microtomography data of section 2.2, with a resolution of about 0.034 µm 3 . Indeed, the resolution of the first scan of the material (i.e. in its undeformed state) of in-situ microtomography tests is about 1 µm 3 (cf. section 2.3.3) and is not sufficient to accurately capture the particle size distribution ranging mainly between 1 to 4 µm. This explains why the initial void volume fraction predicted by the model is different from the initial void volume fraction measured during in-situ microtomography (see Fig. 9.b).

However, the fact that the predicted f agrees with the experimental evolution is an indication that the RVE has a sufficient size from a statistical viewpoint to reproduce the experiments.

For the alloy 6005A, the rapid increase of f is predicted later than in the experiments. This could have been corrected by selecting different void nucleation Weibull parameters for each alloy. However, using the same nucleation stress distribution versus particle size, i.e. neglecting possible differences of particles properties between the three alloys, highlights the major effect of particle size distribution on the fracture strain. Fig. 10 shows in more details results from the model applied to the Al 6056 alloy for three heat treatments (namely T6, T4 and overaged with σ 0 ≃ 60 MPa) and results for the three alloys in the T6 state. The point on each curve corresponds to the predicted fracture strain, i.e. the strain at which the critical rate of porosity ḟc = ∂f ∂ǫ loc c is satisfied.

Results obtained with the model for the three alloys in the T6 state explain the large differences in ε f . Fig. 10(b) shows a fast increase of the fraction of broken particles, i.e. a large nucleation rate, for the Al 6056 and Al 6061. This fast increase corresponds to very large particles which break at very low strain. Very large particles are present in larger proportion in the alloy 6056, see inset of Fig. 5(b), and it exhibits the highest fraction of broken particles. On the opposite, this initial fast increase of the fraction of broken particles is much lower for the Al 6005A due to the absence of large particles in this alloy, see inset of Fig. 5(b). After a few percents of deformation, the increase of the fraction of broken particles reaches a linear evolution.

The rate of increase in this linear regime is the highest for Al 6056, and higher for Al 6061 compared to Al 6005A. This is consistent with the particle size distribution (see Fig. 5(c)), similarly to what was concluded from Fig. 7. Fig. 10(c) shows that the differences in the fractions of coalesced voids between the three alloys correlate with the trends in void nucleation. This is due to the fact that more void nucleation increases the void density, and this locally increases probability that two voids satisfy the coalescence criterion. Fig. 10(d) shows the variation of the total number of voids, i.e. the sum of the initial cavities and nucleated voids minus the number of coalescence events. In each case, the total number of voids exhibits first an increase, as it is dominated by void nucleation, up to some point where voids linkage spread over. At this point, the number of voids starts to decrease and the void volume fraction increases very fast (Fig. 10(a)).

Predictions of the model applied to the Al 6056 alloy for three heat treatments (T6,T4,OA) explain why, for a given alloy, ε f decreases with increasing σ 0 . This well known evolution [START_REF] Simar | Micro-mechanical modelling of ductile failure in 6005a aluminium using a physics based strain hardening law including stage {IV }[END_REF] is associated to the larger strains needed to reach the same stress level in particles surrounded by a soft matrix (Fig. 10(b)) and is consistent with a stress criterion for the fracture of the particles. The density of voids and the number of coalescence events, at a given strain, is thus lower if σ 0 decreases (see Fig. 10(c,d)).

If the final failure criterion is chosen as a given fraction of coalesced voids (10% for exemple, see Fig. 10(c)) or as the beginning of the fast decrease in the total number of voids (Fig. 10(d)), the predicted ε f is different, but the differences between the three materials or between the heat treatments exhibit the same trend. This is due to the fact that the fast increase of void volume fraction is associated to voids linkage over a macroscopic scale that triggers catastrophic ductile failure.

The model is now applied to predict the evolution of ε f with σ 0 for all heat treatments conditions. The critical ḟ , ḟc , was identified as equal to 24 ( ḟ = ∆f ∆ε Loc , with ∆ε Loc =0.01) to capture one data point of one of the three alloys and then kept constant for all other conditions. The results are shown in Fig. 11. The model nicely captures the differences in ε f among the three alloys as well as the decrease of ε f with the increase of σ 0 . The difference between the alloys is thus primarily related to the particle size distribution. Once again, the cellular automaton model parameters have been identified directly from experimental data with a clear physical meaning with no additional fitting or adjustment of tuning parameters except for ḟc and α (with for the last one a value that remains close to the theoretical one).

Sensitivity analysis

Fig. 12 presents the sensitivity analysis of the CA to the inclusion orientation and spatial distribution.

The analysis also shows the effect of using the two versions of the void growth model and the effect of the void aspect ratio definition used in the Thomason criterion. The sensitivity analysis has been performed for the Al 6056 alloy under the three different heat treatments (namely T6, T4 and overaged with σ 0 ≃ 60MPa) which have already been studied in Fig. 10. The version of the CA developed in section 3, and applied to the microstructure extracted form X-ray microtomography volume, will be referred as the "full model". Each effect has been tested independently.

Sensitivity to inclusion orientation

The sensitivity of the model to inclusion orientation has been analysed by testing the extreme case for which all the particles have their major axis aligned with the main loading direction, keeping their volume and spatial distribution unchanged. Fig. 12 shows that for this configuration, the predicted ε f is nearly unchanged in the T6 state, slightly modified in the T4 state and lower in the overaged case by 15 %. The orientation of the particles mainly affects the void nucleation stage. When the major axis of the particles is aligned with the main loading direction, the load transfer arising from the deformation mismatch between particle and matrix increases (see section 3.2), as already shown in Ref. [START_REF] Beremin | Cavity formation from inclusions in ductile fracture of A508 steel[END_REF]. In the T6 and T4 cases, void nucleation is already quite fast (see Fig. 10(b)) in any cases but in the overaged case the impact is more significant. The increased void nucleation rate due to the reorientation of the particles has thus a small to moderate impact on ε f .

Sensitivity to spatial distribution of inclusions

The spatial distribution of the particles used in the full model has been directly extracted from microtomography. A periodic cubic arrangement of particles has been chosen to study the effect of particle distribution on the damage process as a sort of extreme ideal case. The particles shape and orientation and the overall particles volume fraction remain unmodified compared to the full model. Fig. 12 shows that there is a very large effect of the particle distribution on the predicted ε f . Spatial distribution of particles influences the void coalescence process due to its effect on the local void spacing. On the opposite, the spatial distribution doesn't impact void nucleation and growth since interactions are neglected. The increased ductility with the periodic distribution can be explained by the fact that the voids linkage percolating over a sufficient length scale to trigger catastrophic ductile fracture is significantly postponed. Indeed, as shown in Fig. 5(a), the actual spatial distribution of particles is not homogeneous, with some local clustering. The full analysis of the effect of particles spatial distribution, for instance using percolation indicator, goes beyond the scope of the present analysis but this result indicates the need for further investigation on this topic in order to define the right topological indicators to characterize the microstructure heterogeneity with respect to this specific mechanism.

Sensitivity to initial porosity

Fig. 12 shows that the initial porosity has a large effect on the predicted ε f . Indeed, even though f 0 is very low (see Table 3), the cavities which are initially present in the material are quite large compared to the penny-shaped voids nucleated by particle fracture. The presence of an initial porosity facilitates voids linkage over a sufficient length scale to trigger earlier catastrophic ductile failure. The effect is more important in the overaged case since void nucleation is a dominant phase in the T6 and T4 cases (see Fig. 10(b)), attenuating the impact of the presence of an initial porosity. Now, looking back at the comparison between the 3 alloys tested experimentally, the initial porosity is also a factor that can justify the large difference in fracture strain among the three alloys as ε f clearly decrease with increasing f 0 , see Table 3.

Sensitivity to the void growth model

As explained in section 3.3, the R&T void growth model has been slightly modified to take into account the effect of the presence of the particle on void growth. Furthermore, the best value for the parameter α growth has been shown to be 0.57 instead of the theoretical value 0.427. This modified void growth model has been shown to give a better agreement with the experimentally observed void growth (section 4.2.2). In order to assess the significance of this correction, the model has been applied with the original version of the R&T void growth model (i.e. α growth = 0.427 and no effect of the particle on the void growth). Fig. 12 shows that there is a strong increase of the predicted ε f without the correction. Indeed, reducing the parameter α growth from 0.57 to 0.427 induces a slower void growth rate retarding fracture. Furthermore, at low stress triaxiality, the original growth model predicts a decrease of the void size in the transverse directions while, in reality, the particle constraints the transverse contraction (see section 3.3). This delays the coalescence process as the relative void spacing is reduced and this shows that fracture in these alloys is also partly dominated by the void growth phase.

Sensitivity to coalescence model

A heuristic extension of the Thomason criterion has been used in order to define an effective void aspect ratio W ef f (see section 3.4). As mentioned previously, this extension has been partly assessed in [START_REF] Scheyvaerts | The growth and coalescence of ellipsoidal voids in plane strain under combined shear and tension[END_REF], although only for spheroids of same dimension. To test the impact of the assumptions made to estimate W ef f , the effective void aspect ratio W ef f used in eq.( 7) has been fixed equal to 1 as an extreme case. In other words, the coalescence criterion is evaluated as if the two voids were spherical. This extreme assumption is used to evaluate the void coalescence criterion only; the shape of the actual void is used in the rest of the model. Fig. 12 shows that the predicted ε f remains nearly unchanged in the overaged case. In this ductile condition, a significant void growth occurs before the coalescence criterion is met with a shape that gets close to equiaxed, see Lassance et al. [START_REF] Lassance | Growth and coalescence of penny-shaped voids in metallic alloys[END_REF] for more detailed analysis of the void growth and coalescence of initially penny-shaped voids. In the T6 state, void nucleation is the dominant stage (see Fig. 10(b)) and coalescence events occurs at smaller strains (see Fig. 10(c)). Since the void growth stage is reduced, the opening of the penny-shaped voids is reduced and the aspect ratio is significantly smaller than one when coalescence starts.

As a conclusion to the sensitivity analysis, it shows first that the particle distribution is the most significant factor. It also shows that it is important to test the validity of the three elements of the model; void nucleation, void growth and void coalescence as they can play a first order role depending on the hardening of the matrix.

Conclusion

The hardening and fracture properties of three 6xxx series aluminum alloys have been measured for a wide variety of heat treatment conditions. The tensile tests revealed major differences in the fracture strain for the same yield strength, even though the volume fractions of second phase particles are very similar (˜1%). The fracture strain decreases with increasing yield strength with about the same slope for each alloy. A cellular automaton model, based on simple micromechanical models for void nucleation, growth and coalescence taking into account microstructural heterogeneities is presented. It provides very good predictions regarding the different fracture strain variations with only two fitting parameters, out of which one is close to the theoretical estimate.

The main conclusions of the study are the following :

-the fracture strain is primarily dictated by the effect of particle size distribution with the alloy exhibiting the smallest particles being the most ductile; -the use of the same magnitude of nucleation stress and distribution with particles sizes for the three alloys is sufficient to deliver adequate predictions indicating that the intermetallic particles properties must be relatively similar; -particle spatial distribution when compared to periodic distribution plays also a first-order effect on controlling the ductility; -the presence of pre-existing voids has also a first order effect on the predicted fracture strains; -the experimental scattering in the void growth rate in aluminum is significant, such as recently concluded for a titanium alloy (see Lecarme et al [START_REF] Lecarme | Heterogenous void growth revealed by in situ 3-d x-ray microtomography using automatic cavity tracking[END_REF]); -a cellular automaton model involving simple damage criteria is a relevant approach to address the fracture strain prediction in the case of a metallic alloy exhibiting significant heterogeneity effects. eigenvectors. This procedure for the arbitrary stretching of an ellipsoid is detailed in [START_REF] Butcher | A multi-scale damage percolation model of ductile fracture[END_REF]. Using this void growth model, the void will grow and reorient itself to be aligned with the preferential loading direction. If the directions of the semi-axes and principal loading directions are aligned, the void will not rotate. The center of the ellipsoid remains unchanged.

  [START_REF] Mcclintock | A criterion for ductile fracture by the growth of holes[END_REF]. FE simulations of the tensile specimens are conducted with the general purpose software Abaqus[63] using J 2 flow theory and hardening law prescribed in section 2.3.1. The aim is to transfer to the cellular automaton the local stress and strain variations accounting here for specimen necking. The FE procedures are fully standard and convergence of the results was carefully checked. A Matlab[64] program has been written for the cellular automaton (CA). For each deformation step, one iteration of the CA is performed, see Fig. 4. Fig. 3(f) shows the full representative volume element (RVE) used in the cellular automaton for a local strain ε loc = 0.25. It appears from this figure that the size of the RVE must be large enough to capture the local variation of spatial distribution of voids and particles inducing local coalescence processes (voids in grey in Fig. 3(f)) and to enable damage percolation over a macroscopic scale that triggers catastrophic ductile failure.

Figure 1 :

 1 Figure 1: Variation of the different parameters extracted from the uniaxial tensile tests on the three alloys as a function of the yield stress corresponding to the different heat treatments. (a) True fracture strain of three Al alloys; (b) true strain at the onset of necking; (c) dislocation storage rate θ 0 of the Voce law; (d) dynamic recovery parameter β.

Figure 2 :Figure 3 :

 23 Figure 2: Schematic representation of the method used to analyse the probability of fracture of intermetallic iron-rich particles at various distances from the fracture surface of broken tensile samples polished down to mid-thickness; for each particle, record position, dimensions and broken or not broken (here a 6056 alloy specimen), micrograph obtained by FEG-SEM. The color map shows the finite element simulation predictions of the tensile test.

Figure 4 :Figure 5 :

 45 Figure 4: Overview of the modelling strategy

Figure 6 :Figure 7 :Figure 8 :

 678 Figure 6: Yield stress σ 0 corresponding to the various heat treatments performed on the three alloys.

Figure 10 :

 10 Figure 10: Results of the model applied to the Al 6056 alloy for three heat treatments (namely T6, T4 and overaged with σ 0 ≃ 60 MPa). Comparaison between : (a) the variations of the void volume fractions; (b) fraction of broken particles extracted from the model; (c) number of coalescence events; (d) total number of voids. All the data are extracted from the model and their evolution are presented as a function of the local strain in the necking zone. The point on each curve corresponds to the predicted fracture strain.

Figure 11 :

 11 Figure 11: Comparaison between the experimental fracture strain and the fracture strain of the model as a function of yield stress. The plain line corresponds to the linear regression of experimental values. The dotted line corresponds to the linear regression of the values from the model.

Figure 12 :

 12 Figure 12: Sensitivity analysis on different contributions of the cellular automaton model. The "full model" corresponds to the predictions of the model derived in the present paper and used in Fig. 10 and Fig. 11 for comparison to experimental data. The "modified models" are degraded versions of the model.

Figure 13 :

 13 Figure 13: Definition of the effective void dimensions used in the growth model.

Table 1 :

 1 .47 0.60 0.09 0.23 0.11 < 0.01 0.07 0.02 98.73 Al 6061 0.88 0.58 0.23 0.39 0.05 0.18 0.03 0.02 98.44 Al 6056 0.75 0.95 0.64 0.11 0.61 < 0.01 0.18 0.01 97.5 Chemical composition measured by ICP-OES of the 6005A, 6061 and 6056 Al alloys [wt.%] Al content is the balance. ± 0.7 µm 2.1 ± 1.3 µm 3.1 ± 2.1 µm M icrotomography 0.9 ± 0.7 µm 1.8 ± 1.3 µm 2.1 ± 1.8 µm

	Alloy	M g	Si	Cu F e M n	Cr	Zn	T i	Al
	Al 6005A 0

Table 2 :

 2 Mean volume fraction, shape factor, diameter (based on the surface area) of the intermetallic particles in the three aluminium alloys; results based on FEG-SEM image analysis; observations performed at mid-thickness viewed from the top of the plates.

Acknowledgements

The authors would like to thank the ESRF for the provision of synchrotron radiation at the ID15A beamline. The authors acknowledge the financial support of the Interuniversity Attraction Poles Program from the Belgian State through the Belgian Policy agency, contract IAP7/21 INTEMATE. F. Hannard acknowledges the financial support of the FRIA, Belgium. The collaboration between UCL and INSA-Lyon is financed by a Tournesol 2013 project.

Appendix

The void growth model has been developed for voids with the main axes aligned with the principal loading directions (Σ 1 ,Σ 2 ,Σ 3 ). An heuristic procedure is proposed in order to estimate the void growth rates in the case of an ellipsoidal void under general loading conditions. The principal radii of the ellipsoidal void are replaced by the three intersections of the void with the directions of the principal stress (R 1 ,R 2 ,R 3 , see Fig. 13). This equivalent void is then assumed to follow the growth model given by equation [START_REF] Thomason | Ductile Fracture of Metals[END_REF]. The problem can now be treated as the stretching of an ellipsoid along three arbitrary directions (see Fig. 13) which requires an eigenvalue solution where the new void semi-axes and orientation is defined by the eigenvalues and the

List of Tables