Willy Picard
email: picard@kti.ue.poznan.pl

Resilient and Robust Human-Agent Collectives: A Network Perspective

Keywords: Human-Agent Collectives, HAC, resilience, robustness, interdependent networks, small-world networks

Human and software agents are more and more often interacting in groups in which directives are originated as well as addressed by humans or software agents. Among challenges raised by the rise of such heterogeneous groups, referred to as Human-Agent Collectives (HACs), resilience and robustness remain an open question. In this paper, the factors that influence the resilience and robustness of HACs are identified based on former research concerning interdependent and complex networks. The two main factors identified in the paper are 1) interactions between HACs with their opennetworked environment, 2) the structure of the interactions among members of the HACs.

Introduction

With the rise of the internet of think and the ubiquity of ICT, humans are interacting more and more with various electronic devices and information systems. These devices and systems provide not only means to sense and act on the surrounding world, they also allow for distant, often complex, interactions between humans. As such, humans and devices often collaborate within limited temporal frames to achieve a common goal. An example of such interactions is the process of online booking at airport during which many actions are performed by humans with the help of various systems, depending if the booking process is taken from the perspective of the traveler, of the airline company officer that checks the travelers, or the immigration officer.

The collaboration with devices and information systems may take different forms: in some cases, the human being is controlling the device, by asking the device to perform in a given manner. However, in more and more frequent cases, the device is asking the human to behave in a given manner: in the case of online booking at the airport, the information system instructs the traveler about the information to be provided to the system. Another example of a system that instruct human being are most hotlines, especially in the e-banking sector: the system directly asks the customer to press some keys to perform certain tasks.

Therefore collaboration may take place within groups whose members can be humans or agents. In the remaining of this paper, the word "agents" refers to electronic devices and information systems. Importantly, the social relations between humans and agents are not based on a fixed hierarchy: humans may instruct other humans or agents as well as agents may instruct other humans or agents.

Human-Agent Collectives (HACs) have been proposed by Jennings et al. [START_REF] Jennings | Human-Agent Collectives[END_REF] as a new concept to capture these type of collaborative systems. Jennings et al. define HAC as "a new class of socio-technical systems in which humans and smart software (agents) engage in flexible relationships in order to achieve both their individual and collective goals. Sometimes the humans take the lead, sometimes the computer does and this relationship can vary dynamically" [START_REF] Jennings | Human-Agent Collectives[END_REF]. Among key research challenges related to HACs, they have identified, among others, "achieving flexible autonomy between humans and the software, and constructing agile teams that conform and coordinate their activities".

Underpinning the question of flexible autonomy, the question of resilience and robustness for HACs is not mentioned in [START_REF] Jennings | Human-Agent Collectives[END_REF] although it has to be addressed, especially in the concept of open environments. There is a need to identify factors that influence the resilience and robustness of human-agent collectives. Although other aspects of HACs, such as flexibility, agility, efficiency, effectiveness, responsiveness, and stability, still need to be addressed, we choose to focus in this paper on the aspects of resilience and robustness.

In this paper, we will present an analysis of the problem of resilience and robustness of HACs from the perspective of networks and based on the results of former works on complex and interdependent networks. The basic concepts underlying this paper are presented in Section 2, starting from robustness and resilience, followed by human-agent collectives. In Section 3, the interactions between HACs with their open-networked environment are presented as an important factor influencing the resilience and robustness of HACs. In Section 4, the structure of the interactions among members of the HACs, especially when structured as smallworld networks, are discussed as a second important factor influencing the resilience and robustness of HACs. Finally, Section 5 concludes this paper.

Basic Concepts

Robustness and Resilience

The concepts of robustness and resilience have been studied in various scientific fields such as ecology [START_REF] Meerow | Resilience and Complexity: A Bibliometric Review and Prospects for Industrial Ecology[END_REF], transportation [START_REF] Gluchshenko | Definitions of Disturbance, Resilience and Robustness in ATM Context[END_REF], digital forensics [START_REF] Kim | Supply network disruption and resilience: A network structural perspective[END_REF], or supply chains [START_REF] Amann | Designing robustness and resilience in digital investigation laboratories[END_REF]. Currently, there is no consensus on a broadly accepted definition of these concepts.

Although these concepts are related, they are different and refer to two distinct characteristics of systems that should not be considered as synonyms. The concept of robustness refers to "the ability to withstand or survive external shocks" [START_REF] Bankes | Robustness, Adaptivity, and Resiliency Analysis[END_REF]. As an illustration, most nuclear plants are robust, as they may continue to operate even in the case of external shocks.

In this paper, we adopt the definition of resilience given by Haimes as "the ability of the system to withstand a major disruption within acceptable degradation parameters and to recover within an acceptable time and composite costs and risks" [START_REF] Haimes | On the Definition of Resilience in Systems[END_REF]. The concept of resilience is related with the idea that a resilient system may absorb shocks and adapt to the damages caused by these shocks. While a robust system withstands shocks by construction, thanks to its internal constitution, a resilient system withstands shocks by behavior, thanks to its adaptation capabilities. As an example, urban road transportation systems are resilient as, although some traffic jam may congest a large part of a city, the traffic will be redirected to other streets and arterial roads to keep the transportation system working.

Human-Agent Collectives

The concept of Human-Agent Collectives (HACs) has been forged by Jennings et al. in [START_REF] Jennings | Human-Agent Collectives[END_REF]. The term HAC aims at reflecting "the close partnership and the flexible social interactions between the humans and the computers. As well as exhibiting increased autonomy, such systems are inherently open and social. This openness means participants need to continually and flexibly establish and manage a range of social relationships".

One may recognize in HACs characteristics of Collaborative Networks defined as "a network consisting of a variety of entities-organizations and individuals-that are largely autonomous, geographically distributed, and heterogeneous in terms of their operating environment, culture, social capital and goals, which collaborate to better achieve common or compatible goals, and whose interactions are supported by computer networks" [START_REF] Camarinha-Matos | Ecolead And CNO Base Concepts[END_REF]. Therefore HACs are CNs in which entities may be either humans or agents and the interactions are not only supported by computer networks, but they can also be initiated by entities existing only on computer networks (i.e., agents). In HACs, the control usually shifts between humans and agents in a flexible manner.

HACs exhibit the following characteristics as collaborative networks:

-Socially heterogeneous -humans and agents are related by very different social bounds. Power is exerted in a different manner between humans, between humans and agents, and between agents; -Embedded in an open environment -when both humans and agents are always connected to the Internet, HACs have to face the possibility of a broad variety of competitive entities (humans or agents) that may potentially perform the tasks of some entities of the HAC in a competitive manner; -Adaptive -Not only humans have to react to changes in the HAC and its environment, but agents should also be able to adapt to highly adaptive human decision making processes. As a consequence, the whole HAC itself often has to adapt by changing its structure, shifting control from members to members.

These characteristics of HACs lead to a set of major research challenges, among which Jennings et al. pointed out:

1. Flexible autonomy -"flexible autonomy […] allows agents to sometimes take actions in a completely autonomous way without reference to humans, while at other times being guided by much closer human involvement". 2. Agile teaming -agile teams should be able to "come together on an ad hoc basis to achieve joint goals and then disband once the cooperative action has been successful". 3. Incentive engineering -as a result of incentive engineering, "the actors' rewards are designed so the actions the participants are encouraged to take generate socially desirable outcomes" 4. Accountable information infrastructure -an "accountable information infrastructure […] allows the veracity and accuracy of seamlessly blended human and agent decisions, sensor data, and crowd-generated content to be confirmed and audited".

One may notice that the second and third points are objects of research for the community focusing on CNs, and that HACs could probably benefit from the results concerning CNs in these areas.

Robustness of HACs in Interdependent Networked Environments

Interdependent Networks

HACs operate in open, networked environments. In these networks, it is frequent to observe various networks, interacting one with another. A well-known example of this fact is the structure of IT networks according to the Open Systems Interconnection model, known as the OSI model [START_REF] Zimmermann | OSI Reference Model -The ISO Model of Architecture for Open Systems Interconnection[END_REF]. In this model, 7 layers, corresponding to various networks, are defined and their potential interactions are defined. In HACs, agents are usually related to various networks, such as telecommunication networks, application networks, social networks, corporate networks… Interdependent networks are defined as a set of networks whose nodes and links are connected by epilinks. We define an epilink as a link connecting two nodes, or two links, or a link and a node from two networks. The concept of an interdependent network captures the idea that various networks may influence each other. For, example, in metropolises, the exchange of ideas and information via social media relies on infrastructural networks, such as the wired and wireless Internet or cable television. Similarly, attendance of a child at a given school is often related with the existence of appropriate transportation means, which illustrates the interdependence between social networks and infrastructure networks. Interdependence between networks is usually reciprocal, i.e., if a network A influences a network B, then the network B often influences the network A as well. As an example, not only is the school attendance of children correlated with available transportation means, but also transportation means, especially the capacity and timetable of public transportation, are usually evolving to appropriately support the population of children attending schools.

Failure Propagation in HACs and Surrounding Interdependent Networks

Buldyrev et al. [START_REF] Buldyrev | Catastrophic cascade of failures in interdependent networks[END_REF] have shown that the dynamics of interdependent networks is a novel surprising field of study. They have studied the robustness of a set of interdependent networks, i.e., the vulnerability of interdependent networks to the removal of nodes and links. The results of this study are that interdependent networks are less robust than each network is isolation due to cascades of failure from one network to another. And "surprisingly, analysing complex systems as a set of interdependent networks may destabilize the most basic assumptions that network theory has relied on for single networks" [START_REF] Buldyrev | Catastrophic cascade of failures in interdependent networks[END_REF].

An exhaustive list of studies of interdependent networks is provided in [START_REF] Zio | Modeling Interdependent Network Systems for Identifying Cascade-Safe Operating Margins[END_REF] and [START_REF] Gao | Networks formed from interdependent networks[END_REF].

As HACs are embedded in interdependent open networks, they are also exposed to the weaknesses pointed out by Buldyrev et al.: as an HAC is connected to an interdependent network, the robustness of the HAC may be deeply affected by potential cascades and propagations in the surrounding interdependent network. The case studied by Buldyrev et al. was related with a failure in the Italian electric grid, which has propagated to the Italian Internet (Internet routers needed energy to work), which has propagated back to the electric grid (as power stations are controlled remotely via the Web). Other examples of cascades and propagations in interdependent networks may be found in the area of infrastructure networks, transportation networks, and the financial sector and its recent crisis.

In the scenario of a failure in the HAC, a similar scenario may happen: not only the other members of the HAC may be directly affected by the failure, but they may also be affected in an indirect manner: the member that fails may by connected by one network to the surrounding interdependent open environment. In the surrounding environment, the failure may propagate to another network via epilinks, which may further affect another member of the HAC. Figure 1 illustrates this scenario.

The risk that the surrounding open interdependent environment introduces for HACs may be mitigated:

1) by reducing the number of interactions with the environment, which reduces the probability of a failure directly cause by the surrounding network. An entity may play the role of a façade to the HAC with regards to the surrounding environment, hiding the remaining member of the HAC from the environment, and therefore protecting them;

2) by limiting the number of networks to which the various members of the HACs are connected to in the surrounding environment. As a result, failures propagated via epilinks will less probably cause failures to HAC members connected to the surrounding interdependent network. While the interdependent structure of the surrounding environment has an influence on the robustness of HACs, the key factor of resilience for HACs is related to their internal structure, which is defined, to a large extent, by its network structure.

Popular Network Structures

Networks have been the object of intense research works since the late 1990's.

Among key characteristics of networks, the clustering coefficient, the scale-free and the small-world properties of some networks have been intensively scrutinized. The clustering coefficient is a measure of the tendency of nodes to connect to other nodes within groups of nodes. The highest the clustering coefficient, the highest the number of connected triplets of nodes. Scale-free networks are networks in which the probability that a randomly selected node has k links, i.e., degree k follows P(k)~k -γ , where γ is the degree exponent. In scale-free networks, a limited number of nodes have a large number of links (a large degree), while a large number of nodes have a small number of links (a small degree). Small-world networks are graphs in which the clustering coefficient is high and the average path length is small. In small-world networks, a relatively small number of nodes separate any two of them, even if most nodes are not connected to each other.

Popular models of structure of networks are:

• the Erdős-Rényi model -this model is a random model in which the links between nodes are added in a random manner. Clustering of Erdős-Rényi networks is low;

• the Barabàsi-Albert model -in this model, a newly added node is connected to other nodes of the network, such that the probability to connect to a given node is proportional to the degree of this node. Barabàsi-Albert networks are scale-free and their clustering coefficient is usually higher than in Erdős-Rényi networks;

• the Watts-Strogatz model: -in this model, a regular network, in which each node is connected to its neighbors, is modified by replacing with a given probability β the end destination node of each link by a randomly chosen different node. The Watts-Strogatz model may generate random networks, similar to Erdős-Rényi networks, as well as regular networks, depending on the value of β. Even small values of β lead the generation of small-world networks, although not scale-free ones;

• the hierarchical model-in this model, a network pattern is iteratively replicated, leading to a hierarchical organization. Hierarchical networks may be both small-world networks and have a high clustering coefficient.

Resilience of Networks

The resilience of networks has been largely studied in the literature, mostly from a topological perspective. It has been shown that the resilience of a networks depends not only of its structure but also depends on the type of attack. Albert et al. have demonstrated that scale-free networks are highly resistant to attacks focused on random nodes, but they are very vulnerable to attacks targeting the hubs, i.e., highly interconnected nodes.

In [START_REF] Watts | Six Degrees: The Science of a Connected Age[END_REF], Watts provides an analysis of the 1997 Toyota-Aisin crisis with regards to resilience and network topology. On February 1, 1997, a fire started in the Toyota Aisin factory. This factory was producing almost all p-valves, a valve used to control the fluid pressure in brakes. Without almost any stock, the destruction of the Aisin factory may cause the interruption of the whole Toyota production. However, within 4 days, Toyota was producing its first p-valves with the help of its other factories and business partners.

According to Watts, the resilience of Toyota in the Aisin incident was due to the structure of the social and organizational network. On the one hand, the Japanese culture has imposed a strongly hierarchical organization within Toyota. On the second hand, due to internal policies providing strong incentives for employee mobility, many horizontal relations between various factories have been created by employees visiting other factories for a few months.

This particular network structure, consisting of a highly regular structure and many shortcuts, allows to control the exchange of information (on the regular structure) but also to speed up and to recover from failure (with shortcuts) if the regular structure is partially destroyed. One may notice that the idea of a regular structure rewired with shortcuts is at the heart of small-world networks [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF]. Figure 2 illustrates this type of network structure. Following the Aisin-Toyota case, HACs could rely on their internal structure, especially shortcuts between their members to increase resilience. In case of a failure of an HAC member, shortcuts can be used to recover efficiently by increasing the speed of information exchange among HAC members.

Conclusions

HACs are a novel type of collaborative networks in which both humans and agents can ask their counterparts to take some actions. HACs requires additional works in many areas, including technical and business ones.

As far as resilience and robustness are concerned, HACs are facing various important challenges. For the robustness of HACs, the open interdependent networked environment in which they exist is an important challenge as the interdependency of the networks in the surrounding environment is potentially strongly degrading the robustness of HACs.

From a broader perspective, although interdependent networks have been studied in various contexts, e.g., critical infrastructure or population mobility, there is still a need for a unifying theory of interdependent networks that would encompass and largely extend former studies in a structured and organized manner.

For the resilience of HACs, their internal structure plays an important role: an appropriate structure, with shortcuts, may allow an HAC to rapidly recover from a failure by rapidly exchanging information among the remaining HAC members, which speed up the decision making process related with recovery.

The case of Toyota is a very interesting example of resilience within networks of networks. Or to be precise, Toyota may be considered as an organization consisting of various organizational entities. These organizational entities are collaborating in a similar way to collaborative networks. The case of the fire in the break factory of Toyota illustrates the idea that collaborative networks, if appropriately structured can be resilient.

Finally, this paper is in not proposing an exhaustive treatment of the hard question of robustness and resilience of HAC, leaving many questions not only unanswered but even unformulated: what is the role of trust for the resilience and robustness of HACs and how to measure/improve it? Is it possible to adopt preventive measures to improve the robustness and resilience of HACs? These questions remain open and are waiting for further attention.

Fig. 1 .

 1 Fig. 1. Failure propagation in HACs and surrounding interdependent open networks

Fig. 2 .

 2 Fig. 2. A resilient network with a regular hierarchical network and shortcuts.


Janaina Macke
email: janaina.macke@imed.edu.br

João Alberto

Rubim Sarate

Collaborative Networks Dynamics and Social Capital: A Brazilian Case

Keywords: collaborative networks, social capital, survey, Brazil

The ability to mobilize resources through collaborative action is understood as the social capital present in a territory. The presence of social capital can indicate how a collaborative network can be managed. Using a quantitative method, this article aims to analyze the social capital present in collaborative networks, by accessing the case of three collaborative networks in southern Brazil. The results show that social capital can be analyzed according to four factors: proximity, territorial anchorage, reciprocity and collective memory. The elements found can indicate barriers or motivators for the operation of collaborative networks.

Introduction

Potential partners in a collaborative network (CN) are expected to collaborate with each other in problem solving processes, in the compliance with operational standards, in the establishment of forms of cooperation, in the acceptance of agreements, building trust among its members. This collaboration process requires effort, time, dedication, and disagreements might occur. Transparent governance benefits from an incentive system that encourages proactive participation and recognition of individual contributions [START_REF] Camarinha-Matos | Collaborative networked organizations: concepts and practice in manufacturing enterprises[END_REF].

New researches demonstrate that elements, such as values, trust and norms can influence collaboration processes, shape networks and members' behaviors in a significant way and, consequently, indicate the chances of success (or failure) of collaborative networks. On the other hand, competence or technologic affinities can be important during the preliminary stage of the development of a network [START_REF] Rosas | An approach to assess collaboration readiness[END_REF].

One element of social theory that highlights the potential of a collaborative network is the social capital [START_REF] Macke | Social Capital in Collaborative Networks Competitiveness: the case of the Brazilian Wine Industry Cluster[END_REF], because it represents the "sum of the actual and potential resources embedded within, available through and derived from the network of relationship possessed by an individual or social unit" [START_REF] Nahapiet | Social capital, intellectual capital and the organizational advantage[END_REF]. In this sense, social capital is a strongly competitive resource, enhancing the individual and collective capacities through collaborative practices.

Social capital favors collective actions due to the dynamics and density of interactions, changes and learning processes. These learning processes, on its turn, define the ability to intervene and manage change processes, allowing people to act as active agents rather than as merely passive recipients of outside demands [START_REF] Barquero | Desenvolvimento endógeno em tempos de globalização[END_REF] [START_REF] Albagli | Capital social e desenvolvimento local[END_REF].

The present paper aims at identifying and analyzing elements of social capital present in collaborative networks, focusing on local aspects and on the links among collaborative networks and the territories in which they are located. By analyzing these elements, we expect to contribute to the debate on inter-organizational issues, such as the elimination of obstacles generated by the environment and/or other organizations which might hamper collaboration processes.

Theoretical Background

Social capital is a set of informal norms and values common to the members of a specific group which allow cooperation among them.

Since its creation, the concept of social capital is being used to explain specific social phenomena. Most researches have focused on its role in the human capital development [START_REF] Coleman | Capital in the creation of Human Capital[END_REF] [START_REF] Loury | Why would we care about group inequality? Social Philosophy & Policy[END_REF], in economic performance [START_REF] Baker | Market networks and corporate behavior[END_REF], in regional [START_REF] Putnam | Making Democracy Work: civic traditions in modern Italy[END_REF] and national development [START_REF] Fukuyama | Trust: the social creation virtues and the creation of prosperity[END_REF].

In a study that relates social capital and intellectual capital, Nahapiet and Ghoshal [START_REF] Nahapiet | Social capital, intellectual capital and the organizational advantage[END_REF] propose three social capital macro-dimensions: structural, relational and cognitive. However, such analytical division does not exclude the existence of a close association between their main features.

The structural dimension of social capital is related to the presence or absence of interactions between the members and the configuration or morphology of the network by describing the standards of connections through variables, such as density, connectivity, network configuration, stability, and ties. The relational dimension describes the kind of personal relationship developed through a history of interactions. This concept focuses on aspects that influence behaviors, such as trust and distrust, norms, obligations, expectations and identity. Finally, the cognitive dimension refers to resources that originate shared visions, interpretations and systems of meaning, mainly codes, shared narratives, values, and other cultural elements. Some authors believe that the cognitive dimension is not sufficiently explored in the literature [START_REF] Nahapiet | Social capital, intellectual capital and the organizational advantage[END_REF].

The promotion and strengthening of solid networks of relationships based on trust, reciprocity and values is more easily done in a micro rather than in macro society where relationships tend to be more formalized and impersonal. In this sense, it is necessary to analyze the three main characteristics of the collaborative processes that contribute to the creation of a territory [START_REF] Pecqueur | A guinada territorial da economia global[END_REF]: a) the society and the community are in balance. The main characteristic is an economy that is autonomous in relation to politics and the functioning of society itself, which brings out the concept of territorial anchorage; b) historicity (collective memory), i.e. the social construction of collective cognitive reserves and the learning ability of the agents involved; c) reciprocity which determines the relationship between the agents recognized by their life beyond purely commercial transactions.

Pecqueur's elements [START_REF] Pecqueur | A guinada territorial da economia global[END_REF] [13] can be associated with Putnam's [14] and Coleman's ideas [START_REF] Coleman | Capital in the creation of Human Capital[END_REF]. The characteristics of the regionalized production model lead to the establishment of a new local/global relationship around which territorial anchorage and non-spatial production are complexly articulated. In other words, the territory becomes central to the coordination of collaborative actions among players interested in solving unprecedented problems [START_REF] Pecqueur | A guinada territorial da economia global[END_REF]. This can be achieved by the mobilization of the existing social capital and the promotion of collaborative networks.

Local agents able to activate and evaluate the area's social capital, as well as, to change generic into specific resources are increasingly important. Specific resources are unique and differentiated and, consequently, difficult to transpose or translate, being one of the keys to territorial competitiveness and development.

Method and Context

Data was collected through survey in order to identify and analyze the social capital elements present in collaborative networks and its relation with the territory.

The authors conducted an extensive literature review in order to identify the most significant studies on social capital [START_REF] Nahapiet | Social capital, intellectual capital and the organizational advantage[END_REF] [15] [START_REF] Woolcock | Social capital for social policy: lessons from international research and policy[END_REF], collaborative networks [START_REF] Camarinha-Matos | Collaborative networked organizations: concepts and practice in manufacturing enterprises[END_REF] and territories [START_REF] Pecqueur | A guinada territorial da economia global[END_REF] [START_REF] Wu | Dimensions of Social Capital and Firm Competitiveness Improvement: The Mediating Role of Information Sharing[END_REF]. The variables to be used in the survey were decided upon after meta-analysis.

All items were measured using a five-point Likert scale (1=strongly disagree; 2= somewhat disagree; 3= neither agree; nor disagree; 4 = somewhat agree; 5=strongly agree). Before formal survey, we conducted semi-structured interviews with three experts in collaborative networks and social capital theory to validate our scale items (these experts are researchers and professors of graduate programs and members of the Social Theory Research Group). We also ran a pre-test with ten respondents.

The survey was conducted in three Brazilian collaborative networks (Caminhos de Pedra, APROBELO and APROVALE) immersed in the same regional culture. These collaborative networks are located in three areas of the Serra Gaúcha region, state of Rio Grande do Sul (Southern Brazil). The APROVALE (Association of Producers of Fine Wines of the Valley of Vinhedos) consists of 31 wineries, and 43 members to support tourism, including hotels, hostels, restaurants, handcraft and antique shops.

The APROBELO (Association of Producers of Fine Wines of Monte Belo) comprises 11 wineries; the Caminhos de Pedra Association has 23 members, including restaurants, hostels and small family businesses.

The networks have a diversified economy, and are located in the largest wineproducing region of Brazil (with approximately 40,000 hectares of vineyards). The region is also characterized by family farms and lower mechanization level, because of the mountainous terrain. Nowadays, rural wine tourism is also an economic resource being explored.

The survey was conducted with employees and company owners. The sample of 206 respondents was chosen by convenience [START_REF] Hair | Essentials of Business Research Methods[END_REF]. We used the software PASW statistic 18, to analyze data descriptive and factorial analysis.

Results and Discussion

The responses were submitted to PCA (Principal Component Analysis) factor analysis with varimax rotation and pairwise treatment (considering all valid observations of each variable) for missing data. The Kaiser-Meyer-Olkin index (KMO) of sampling adequacy was 0.837 and Bartlett's test of sphericity (significant 0.001) indicated the factorability of the data. The answers submitted to descriptive analysis revealed averages ranging between 2,158 and 4,356, with standard deviations from 0.906 to 1.836. The Cronbach's alpha measured for the instrument with the 23 social capital variables resulted in 0.829, demonstrating excellent internal consistency of analyzed variables [START_REF] Pestana | Análise de Dados para Ciências Sociais: A Complementaridade do SPSS[END_REF]. Moreover, few cases of missing values were observed.

The final factor analysis resulted in four elements; the percentage of explained variance was 54.52%, which means that the variables chosen and the resulting factors can explain 54% of the area's social capital of the collaborative networks studied. Applied social researches consider that a good result: the classic study of social capital in Australian communities conducted by Onyx and Bullen [START_REF] Onyx | Measuring Social Capital in Five Communities[END_REF], whose factor analysis explained 49.3% of the variance, is a reference in social capital measurement.

In order to check the consistency of the variables in each factor, values for Cronbanch's alpha were calculated. The ideal value in social sciences exploratory studies should be higher than 0.6 [START_REF] Malhotra | Pesquisa de marketing: uma orientação aplicada[END_REF]. Considering the whole instrument, Cronbach's alpha was 0.833.

The fourth factor presented an Alpha value considered low (0.567). Its removal was not recommended given the significant reduction it would cause in the explained variance and even in the KMO index.

The first factor identified was proximity with alpha of 0.813, which is considered a very good result [START_REF] Pestana | Análise de Dados para Ciências Sociais: A Complementaridade do SPSS[END_REF]. The variables in this factor concern the participation in associations and organizations, volunteering, participation in the community's festivals and celebrations. These variables are related with situations that promote the coming together of people and groups, either by shared goals (associations/organizations) through volunteer work or at parties and celebrations. The literature describes an oscillatory movement because the coming together is not a "state" but a "tension". It does not abolish the distance and it is defined by oppositions [START_REF] Lefebvre | La 'proximité': nouveau capital social au chevet de la démocratie?[END_REF].

The question of proximity can also be understood through the analysis of the three mechanics of social capital (bonding, bridging and linking). A significant difference between bridging and bonding, for example, indicates the distance between heterogeneous groups in the same way that low linking levels indicate the presence of asymmetric power relations [START_REF] Onyx | Measuring Social Capital in Five Communities[END_REF]. "Taking part in a network" allows discussion of common problems, exchange of information and practical experience, facilitated by territorial proximity. This aspect is decisive for the development of innovations and the building of a sense of inhabiting.

The second factor, territorial anchorage, displays a set of variables in which the collaborative network illustrates the sense of experiencing the territory. The social approval of the territory and its resources and the collective investment and believe in its development characterize territorial anchorage.

The variable "people treat well those coming from outside" shows willingness to accept the "different" and indicates a certain degree of openness (51% of the respondents worked for the network and also reside in the territory). The coexistence between rich and poor suggests that, despite economic differences, there is a movement towards the horizontalization of the social relationships.

The variable "feel proud to work to the network" obtained the best performance. Even those who do not live in the area, feels connected through the work. To consider people trustworthy is a key indicator of the presence of social capital; trust allows the establishment of informal standards systems, which facilitates the coordination of existing regional resources, strengthening territorial anchorage. Finally, to "believe that the territory has a future" summarizes the belief that it is worth investing in collaborative network, i.e. there is adherence of individuals and groups to future projects.

The third factor, reciprocity, relates to the feeling of obligation one feels to return a favour and to the social embarrassment of anyone who does not cooperate or violates agreed norms [START_REF] Putnam | Bowling Alone: the collapse and revival of American community[END_REF]. To count on neighbours to take important decisions, to recognize other people's help (and be socially compelled to repay it) and to consider that the organization is a place of exchange (mutual assistance) are reciprocity indicators. In the literature, norms of reciprocity and participation systems are the main evidence of the presence of social capital.

The fourth factor is designated as collective memory, due to variables that show the families' efforts so their children and grandchildren follow customs and are aware of the region's history. The other two variables -consider co-workers friends and share a family meal at least once a day -seem, at first, unrelated to the others. However, when we analyzed the difference between the responses of resident and non-resident workers we realized that the residents perform better, i.e. "work" seems to be the link between the variables. Working and living in the same territory offer more opportunities to share family meals and to work with the family, which encourage the teaching of values and customs and contribute to the preservation of the territory's collective memory.

Among the factors found, three had already been reported in the literature [START_REF] Pecqueur | A guinada territorial da economia global[END_REF] as essential to the process of collective construction. The authors identified these elements through the analysis of groups and of specific literature. Pecquer's typology [START_REF] Pecqueur | A guinada territorial da economia global[END_REF] failed to include an element (factor 1 in this study), which the authors linked to the concept of proximity.

Final Remarks

This study allowed the identification of the members' perception about their interactions and purposes, in other words, the social dynamic. The interaction between members and organizations appears to broaden options to reach common interests and projects, and to break through bureaucratic barriers. The idea is to preserve the group's heterogeneity and seek flexibility, focusing on cooperation without eliminating constructive conflict and competition [START_REF] Loyola | Análise de Redes: uma Contribuição aos Estudos Organizacionais[END_REF].

Social capital is not disconnected from historical or geographical influences. Therefore, the results were strongly influenced by the context in which they were inserted. Analyzing the main evidences according to that theoretical framework, the authors could collect evidences about the dynamics of collaborative networks and their links to social capital.

The present study identified and analyzed the elements of social capital in territories with collaborative networks based on four main factors: proximity, territorial anchoring, reciprocity and collective memory. In an initial stage, usually technical skills were essential to the network's functioning. Nevertheless, research on the role of social players in territorial building can contribute to broaden the debate on the role of collaborative networks in local development.

The theoretical contribution of the paper is the identification of a new element necessary for the maintenance of collaborative networks in communities: the concept of proximity. Moreover, from the practical perspective, the study provides a social capital assessment tool to access social capital levels in collaborative networks.

Many questions remain to be answered, such as: how to develop new ways of network management capable of dealing with the territorial multiplicity in which we are inserted? How to reinstate a territorialization concept that means not only "controlling" the space, but also its production and experience taking into consideration the collaborative networks?

In conclusion, social capital is not an instrument to be used in isolation, nor it claims to be the single tool to understand the role of collaborative networks embedded in a territory. The authors expect that this study can contribute to the reflection on the obstacles to the establishment of collaborative networks.

Table 1

 1 Social capital elements in collaborative networks

	Variables	1	2	3	4
	11 -I participate in the network	0.797
	12 -I do volunteer work	0.794
	17 -I participate in parties and	0.790
	celebrations
	13 -I believe that people treat well
	those arriving from outside to work		0.708
	in the network
	16 -The rich and poor live well
	together		0.627
	15 -I feel proud to work to the network		0.599
	19 -The local government supports
	local development		0.584
	20 -I consider locals trustworthy
			0.505
	22 -I believe in the future of the territory		0.480
	2 -I can count on other members to			0.773
	take important decisions
	8 -I would not have made it without			0.677
	the help of others
	7 -People help each other in the			0.561
	territory
	1 -Co-workers are friends				0.712
	14 -I can share a family meal at				0.706
	least once a day
	4 -The children and grandchildren				0.499
	are encouraged to follow traditions
	6 -I know local history				0.452
	Source: present study

Acknowledgments. This work is being supported by the Faculty Meridional (Faculdade IMED), University of Caxias do Sul (Universidade de Caxias do Sul -UCS), the Coordination for the Improvement of Higher Education Personnel (Coordenação de Aperfeiçoamento de Pessoal de Nível Supeior -CAPES), and the National Counsel for Scientific and Technological Development (Conselho Nacional de Desenvolvimento Científico e Tecnológico -CNPq).


Jean Sulem
email: sulem@cermes.enpc.fr

Hichem Ouffroukh

Cermes

Keywords: Sandstone, shear-band, porosity, permeability, image analysis

Shear-banding in drained and undrained triaxial tests on

a saturated sandstone ; porosity and permeability evolution.

Introduction

Transport properties of porous rocks such as permeability are strongly influenced by the deformation process and controlled by the evolution of the structure of the grain geometry and the pore space (volume, connectivity, tortuosity, shape, etc.). Shear bands which in brittle rocks are zones of localised intense damage appear as weaker than the surrounding rock.

Depending on the initial porosity of the rock and on the stress level, compaction and strainhardening or, on the contrary, dilation and strain softening may occur. In the last decade important experimental and theoretical studies have been made in this area for understanding deformation mechanisms and permeability evolution in granular rocks [START_REF] Aydin | Analysis of faulting in porous sandstones[END_REF][START_REF] Antonellini | Microstructure of deformation bands in porous sandstones at Arches National Park,Utah[END_REF][START_REF] Menendez | Micromechanics of brittle faulting and cataclastic flow in Berea sandstone[END_REF][START_REF] Wong | The transition from brittle faulting to cataclastic flow: Mechanical deformation[END_REF][START_REF] Zhu | Permeability reduction in a dilatant rock: network modeling of damage and tortuosity[END_REF][START_REF] Zhu | The transition from brittle faulting to cataclastic flow: Permeability evolution[END_REF][START_REF] Zhu | Network modeling of the evolution of permeability and dilatancy in compact rock[END_REF][START_REF] David | Mechanical compaction, microstructures and permeability evolution in sandstones[END_REF][START_REF] Mair | Sequential growth of deformation bands in the laboratory[END_REF][START_REF] Main | Experimental constraints on the mechanical and hydraulic properties of deformation bands in porous sandstones: a review[END_REF][START_REF] Bernabé | Permeability-porosity relationships in rocks subjected to various evolution process[END_REF][START_REF] Bésuelle | Experimental characterisation of the localisation phenomenon inside a Vosges sandstone in a triaxial cell[END_REF][START_REF] Ord | Shear band formation in Gosford sandstone[END_REF][START_REF] Ngwenya | Permeability evolution during progressive development of deformation bands in porous sandstones[END_REF][START_REF] Fortin | Elastic wave velocities and permeability evolution during compaction of Bleurswiller sandstone[END_REF]. Shear localisation results from the coalescence near the peak stress of clusters of oriented microcracks due to Hertzian fractures initiated at grain contacts.

For rocks with a relatively large porosity (> 10%) under hydrostatic loading, it has been identified by many authors (e.g. Brace, [START_REF] Brace | Volumes changes during fracture and frictional sliding: a review[END_REF], Zhang et al. [START_REF] Zhang | Micromechanics of pressure-induced grain crushing in porous rocks[END_REF]) that the volumetric strain curve exhibits an inflection point for a stress of several hundreds of MPa beyond which an important inelastic volumetric compaction occurs. Microstructural observations show that significant grain crushing and pore collapse is associated with the inelastic compaction. The high density of cracks in the shear band is comparable with that of samples 'homogeneously' deformed in the cataclastic flow regime. Detailed observation of shear zones may thus give clues to understand the behaviour of rocks at large stress when pore collapse and grain crushing are important.

A challenging issue is the modelling of grain crushing phenomenon and its influence on the hydro-mechanical properties. Grain crushability can be included in the modelling procedure using discrete element analysis [18,[START_REF] Tsungui | Numerical model of crushing of grains inside two-dimensional granualar materials[END_REF]. Discrete element models can mimic the macroscopic behaviour of real including elasticity, anisotropic damage, post-peak softening [START_REF] Potyondy | A bonded-particle model for rock[END_REF][START_REF] Tang | Numerical tests on micro-macro relation ship of rock failure under uniaxial compression -Part I: effect of heterogeneity[END_REF][START_REF] Tang | Numerical tests on micro-macro relation ship of rock failure under uniaxial compression -Part II: constraint, slenderness and size effect[END_REF].

However up to now these models do not consider the influence of the evolution of the microstructure in due course of the deformation process on the hydro-mechanical properties of the rock. More recently a continuous approach including a degradation law for the rock skeleton was proposed by Yuan and Harrison [START_REF] Yuan | Modelling, enhanced fluid flow and fault sealing due to fracturing in heterogeneous rock[END_REF] in order to account for the effect of porosity evolution on the permeability.

In a previous paper [START_REF] Bied | Microstructure of shear zones in Fontainebleau sandstone[END_REF] the porosity change of shear bands formed in dry samples of Fontainebleau sandstone of initial porosity of 21% tested in triaxial compression has been analysed using a simple evaluation technique. This technique was based on the concept of linear porosity applied to microphotographs of the shear zone. It was shown that for specimens tested under relatively low confining pressure (7 MPa), the porosity increases up to about 30% inside the band and decreases rapidly towards the initial value outside the band. This was interpreted as dilating shear banding at low confining pressure. For specimens tested under relatively high confining pressure (28 MPa), a different patterning was observed. A compacting zone with high grain crushing and low porosity (values between 2 and 19 %) is observed in the centre of the band. This zone is surrounded by a dilating one with grain cracking and high porosity (up to 36 %). However as mentioned in this paper, this simple technique is time consuming and tedious, and cannot be used for systematic analysis of a large data base.

In the present study the same rock (Fontainebleau sandstone) is used and geometrical characteristics of the shear band (porosity, specific surface, grain size) are evaluated by using image processing software. The previous study is also extended to fluid saturated samples tested in drained and undrained conditions. The influence of the fluid and of the drainage conditions on shear band formation and shear band microstructure is carefully examined.

Several authors have used image analysis of the pore structure of sandstone for predicting the permeability (e.g. Lock et. al [START_REF] Lock | Predicting the permeability of sandstone from image analysis of pore structure[END_REF]). Usually these models are based on a reconstruction of a network of connected conductors which requires the knowledge of the geometrical properties of the pores. The approach that we use here is more empirical. The permeability model of Walsh and Brace [START_REF] Walsh | The effect of pressure on porosity and the transport properties of rock[END_REF] is used to correlate porosity and permeability.

Experimental setting

A triaxial cell with internal diameter of 140 mm is used (Fig. 1). This cell can sustain a confining pressure up to 60 MPa. It contents a system of hydraulic self-compensated piston.

The loading piston is then equilibrated during the confining pressure builds up and directly applies the deviatoric stress. The axial and radial strains are measured directly on the sample inside the cell with two axial transducers and four radial ones of LVDT type (see section AA on Fig. 1 and Fig. 2). These internal devices allow to avoid the main errors of strain measurements of devices external to the cell such as the compliance of the loading device, the tilting of the specimen, the bedding errors at the ends of the specimen. This device is also more adequate that strain gages which are currently used for strain measurements on rocks.

Strain gages can perturb the porous specimen because of glue penetration inside the sample and are useless beyond the onset of strain localisation. A correction for the radial deformation of the membrane during the test has been calibrated. The lower base of the cell is equipped with a pore pressure transducer. Pore pressure measurements during test are then performed very close to the sample base. The confining pressure is applied by a servo controlled high pressure generator which consists of a piston moving in a pressure chamber. The pressure generator is guided by an electronic regulator receiving an analogical signal from an external transducer. The regulators are programmed by a PC computer.

Axisymmetric strained-controlled compression tests in drained and undrained conditions have been performed on Fontainebleau sandstone at various confining pressures [START_REF] Zhu | Network modeling of the evolution of permeability and dilatancy in compact rock[END_REF][START_REF] Ngwenya | Permeability evolution during progressive development of deformation bands in porous sandstones[END_REF][START_REF] Labuz | Reducing frictional constraint in compression testing through lubrication[END_REF]40 and 50 MPa). The axial displacement is servo-controlled via the internal radial transducers at a rate corresponding to a constant radial strain rate of 10 -5 sec -1 which is slow enough to insure fully drained deformation. The servo-control system allows to control failure and post-peak behaviour when the control parameters have been optimised for a fast response of the system [START_REF] Fairhurst | Draft ISRM suggested method for the complete stress-strain curve for intact rock in uniaxial compression[END_REF]. The choice of the radial displacement as the control variable is made because this quantity is increasing rapidly with the axial stress close to the peak. The loading rate is thus actually very slow close to the peak strength. The tests have been stopped close after the maximum axial load has been reached in order to keep the samples in a state as close as possible to the shear band initiation and observe the bands after the tests with an electron scanning microscope as described in section 4. In order to make good observation it is important to obtain at the end of the test a specimen with one or several bands clearly formed but without any material discontinuity. Failure along a unique shear band is actually obtained for confined tests at low confining pressure and eventually along several bands for tests at high confining pressure. The interface between the rock specimen and the loading platens of the testing machine is lubricated to avoid friction [START_REF] Labuz | Reducing frictional constraint in compression testing through lubrication[END_REF][START_REF] Papamichos | Constitutive testing of a weak sandstone[END_REF][START_REF] Bied | Adaptation du mode opératoire pour la réalisation d'essais en compression « vraie » sur les roches[END_REF]. Cylindrical specimens with a diameter of 37.5 mm and a height of 77 mm are used.

The rock chosen for this study is the Fontainebleau sandstone with a porosity of 21%. It is a uniformly graded rock with a grain size of 0.25 mm (Fig. 3). Its composition is 98% quartz.

3 Experimental results

Stress-Strain response

In the range of confining pressures used in our tests the behaviour of Fontainebleau sandstone is rather brittle. The tests have been stopped just after the peak of axial load. The corresponding axial strain is around 0.5% for low confining pressure and 1% for higher ones.

The overall stress-strain response is presented on Fig. 4 for drained tests and on Fig. 5 for undrained tests. We observe that for the undrained test at 7 MPa of confining pressure, the axial strain is decreasing after the peak whereas the radial strain is always increasing (Fig. 5a).

As explained above, the radial displacement is actually the control variable of the system and it is increased at constant rate. The decrease of the axial displacement after the peak stress as measured by the transducers reflects the strain localisation process. When shear banding occurs the strain field is not homogeneous inside the sample and rigid block sliding along the shear surface can occur.

Use is made here of the Terzaghi effective stress concept. Note that due to the fact that the fluid compressibility (4.0*10 -4 MPa -1 at 20°C) and the quartz grain compressibility (2.75*10 -5 MPa -1 ) are comparable to the saturated rock compressibility (10 -4 MPa -1 ), undrained tests are not isochoric (i.e. at constant volume) as in classical Soils Mechanics (Fig. 5b). The maximum effective stress is increasing with effective confining pressure. On these graphs, negative (respectively positive) volumetric strains correspond to contractant (respectively dilatant) behaviour. The samples undergo initial compaction and then dilatant strain hardening. The contractant phase is less pronounced for undrained tests due to pore pressure generation.

The experimental results for drained and undrained tests on saturated samples are consistent with those presented in [START_REF] Bied | Microstructure of shear zones in Fontainebleau sandstone[END_REF] for dry samples (Fig. 6). The effective yield stress can be fitted with a linear Mohr-Coulomb criterion which in case of axisymmetric compression is written as

( ) ( ) 2 / 4 / tan ' 2 / 4 / tan C 2 ' 2 radial axial φ + π σ + φ + π = σ (1)
with C=15.5 MPa and φ=41.3°.

(2)

Pore pressure generation in undrained tests

The generated pore-pressure in undrained tests is plotted versus the axial strain for the various confining pressures on Fig. 5c. The magnitude of the induced pore pressure in the contractant phase depends upon the porosity of the rock, the drained compressibility of rock, and the compressibilities of the pore fluid and rock grains as described in [START_REF] Zimmerman | Compressibility of sandstones[END_REF]. Theses curves are consistent with the volumetric response of the samples (Fig. 5b). They exhibit a maximum at the contractance-dilatance transition. Dilatancy is caused by the formation and extension of open microcracks which results in pore pressure dissipation. The maximum of the generated pore pressure increases with increasing confining pressure. At low confining pressures (7 and 14 MPa) the pore pressure at yield effective stress is lower than the initially applied back pressure. Such cavitation phenomenon has been observed for dense sand [START_REF] Roger | Experiments on strain localisation in dense sand under isochoric conditions[END_REF] and for sandstone under low confining pressure [START_REF] Oka F Eds | A study on the strain and pore water pressure characteristics of sandstone during repeated loading[END_REF].

Shear band formation

For all the tests in drained and undrained conditions, the specimens failed with the formation of one or several shear bands. In most experiments it was possible to control the post-peak behaviour and the unloading phase of the sample in order to avoid the formation of a discontinuity surface. These shear bands are clearly visible on the surface of the sample. The zone of localised failure is clearly identified due to the white coloration of the band visible to the naked eye (Fig. 7). Similar coloration of the shear zone in sandstone was observed by Ord et al. [START_REF] Ord | Shear band formation in Gosford sandstone[END_REF], Bésuelle et al. [START_REF] Bésuelle | Experimental characterisation of the localisation phenomenon inside a Vosges sandstone in a triaxial cell[END_REF], Mair et al. [START_REF] Mair | Sequential growth of deformation bands in the laboratory[END_REF], El Bied et al [START_REF] Bied | Microstructure of shear zones in Fontainebleau sandstone[END_REF]. This colour change is due to the crushing of quartz grains and allows for a precise evaluation of the thickness and the orientation of the shear band.

Extensive presentation of shear band analysis in geomaterials can be found in Vardoulakis and Sulem [START_REF] Vardoulakis | Bifurcation Analysis in Geomechanics[END_REF]. In globally undrained conditions, the heterogeneity of the damage pattern at shear banding as it will be presented in the next section, allows for local fluid exchanges. This means in turn that the heterogeneous development of micro-fractures in a granular rock during deviatoric loading leads to local drainage of the pore fluid. As shown in Vardoulakis [START_REF] Vardoulakis | Deformation of water saturated sand: I. Uniform undrained deformation and shear banding[END_REF][START_REF] Vardoulakis | Deformation of water saturated sand: II. The effect of pore-water flow and shear banding Géotechnique[END_REF] (see also Vardoulakis and Sulem [START_REF] Vardoulakis | Bifurcation Analysis in Geomechanics[END_REF]), the possibility of locally undrained shear banding is excluded as this situation would induce pore-water pressure shocks across the shear band boundaries.

The onset of shear banding has been studied experimentally using the independent axial and lateral displacement measurements on the specimen. Detection is performed by comparing the two axial strain measurements and/or the four radial strain measurements. The difference between the measurements of two opposite points is plotted versus the applied axial strain as

shown for example on Fig. 8. This difference is very small and almost constant as long as the deformation is reasonably homogeneous inside the sample but diverges rapidly at some critical point corresponding to the threshold of loss of homogeneous deformation. This point indicate the limit after which the response of the specimen cannot be considered as representative of the material behaviour but as a structural response of a system. Without precise monitoring of the displacement field inside the sample during the test using visualisation techniques it is not possible to know if this point is the onset of the coalescence of localised damage zones or the onset of movement along a through-going shear band.

On Fig. 8 corresponding to a drained test at 28MPa of confinement, the points of divergence for axial and radial strain measurements (points C 1 and C 2 respectively) are almost simultaneous and are compared with the threshold of contractant/dilatant behaviour (point B)

and the point of loss of linearity of the axial strain-axial stress response (point A). Similar technique for the detection of strain localisation has been used by Santarelli [START_REF] Santarelli | Etude expérimentale de la bifurcation dans les roches[END_REF], Tatsuoka

and Kim [START_REF] Tatsuoka | Deformation of shear zone in sedimentary soft rock observed in triaxial compression[END_REF], Bésuelle et al [START_REF] Bésuelle | Experimental characterisation of the localisation phenomenon inside a Vosges sandstone in a triaxial cell[END_REF]. The results for all the tests are shown on Fig. 9. The axial and the radial measurements give in general similar results. The loss of homogeneous deformation occurs before the peak and is delayed at higher confining pressure in globally undrained conditions. On Fig. 10, the threshold from contractant to dilatant behaviour is compared with the onset of localisation for drained and undrained tests. This graphs shows that for drained tests the transition from contractant to dilatant behaviour which corresponds to the initiation of microcracking occurs close before the onset of localisation. For undrained tests, due to the effect of pore pressure generation, the transition from contractant to dilatant behaviour occurs much earlier than for drained tests and is increasing for higher confining pressure. These two thresholds are observed not only for sandstones but also for crystalline rocks with the crack initiation and crack damage thresholds as defined for example for granite [START_REF] Martin | The progressive fracture of Lac du Bonnet granite[END_REF][START_REF] Diederichs | Damage initiation and propagation in hard rock during tunnelling and the influence of near-face stress rotation[END_REF].

The orientation of the shear band is measured on the central part of the specimen, as the shear band is often deviated close to the extremities of it. The measured orientation of the shear band with respect to the minor (in absolute value) principal stress axis is presented on Table 1.

These results show that the orientation of the shear band decreases when the effective confinement pressure increases as commonly observed (e.g. [START_REF] Bésuelle | Experimental characterisation of the localisation phenomenon inside a Vosges sandstone in a triaxial cell[END_REF][START_REF] Bied | Microstructure of shear zones in Fontainebleau sandstone[END_REF]). This can be explained by the pressure sensitivity of granular rocks for which the mobilised friction angle and dilatancy angle decrease with increasing effective mean stress. The difference observed in the shear band orientation between drained and undrained tests is due to the fact that in Table 1, the total confining pressure is referenced. As shown in the following, in terms of effective confining pressure, the results for undrained tests are consistent with those of drained ones (Fig. 12).

Strain localisation in the form of a shear band corresponds to weak discontinuities for the incremental displacement and occurs first when the acoustic tensor Γ Γ Γ Γ becomes singular [START_REF] Hill | Acceleration waves in solids[END_REF]:

l k ijkl ij ij n n L with , 0 ) det( = Γ = Γ ( 3
)
where L is the incremental constitutive tensor of the material and n i are the cosines of the normal to the shear band.

As shown in Vardoulakis and Sulem [START_REF] Vardoulakis | Bifurcation Analysis in Geomechanics[END_REF], the shear band orientation θ B with respect to the major (in absolute value) principal stress axis can be approximated by a simple formula which was first proposed by Arthur et al [START_REF] Arthur | Plastic deformation and failure in granular media[END_REF] on the basis of experimental observations and 10 subsequently proven theoretically and supported experimentally by Vardoulakis [START_REF] Vardoulakis | Shear band inclination and shear modulus of sand in biaxial tests[END_REF] for biaxial tests

4 / 4 / 45 B ψ + φ + °= θ (4)
where φ is the friction angle and ψ is the dilatancy angle at the onset of strain localisation. In a simple analysis it is assumed that shear banding occurs at maximum deviatoric stress corresponding to a mobilised friction angle of 41.3°. The dilatancy angle is evaluated on experimental data at various effective stress. For drained and undrained tests, it is found that the dilatancy angle at peak stress evolves in a consistent manner with the effective confining pressure. According the data are fitted with the following linear relation (Fig. 11):

3 . 48 ) MPa in ( ' -0.76 degrees) in ( rad + σ = ψ (5)
where σ' rad is the effective radial stress expressed in MPa.

As shown on Fig. 12a the Arthur's formula gives an acceptable evolution of the shear band inclination with respect to the effective confining pressure.

Remark: For our tests in the range of confining pressures used, the maximum mobilised friction angle was found to be independent of the confining pressure and equal to 41.3. The classical Mohr-Coulomb solution for the orientation of the failure plane is 45°+ φ/2 = 65.65°.

The Coulomb solution cannot reflect the decrease of the shear band orientation with confining pressure. We show here the important effect of the dilatancy angle on the shear band orientation.

A more precise evaluation of the shear band orientation can be obtained by performing a complete a complete strain localisation analysis which gives good predictions of the shear band inclination (Fig. 12b). The analysis is presented in Appendix.

4 Microstructural observations

Sample preparation

As explained in section 2, the test is stopped close after the peak strength. A good control of the post-peak phase allows to avoid a violent failure of the sample. Therefore the sample shows a clear shear band but no material discontinuity.

At the end of the compression test, each sample is carefully removed from the loading cell while still encased in its rubber sleeve and put into a vacuum bell-jar where it is impregnated with epoxy resin. A plane cutting of the specimen allows a clear observation of the shear band inside the rock to evaluate its thickness and orientation. A metallization is performed by pulverising gold. This operation allows better observations with the electron scanning microscope. Examples of microphotographs of the shear band zone are presented for an undrained test under 7 MPa of confining pressure (Fig. 13) and a drained test under 50 MPa of confining pressure (Fig. 14).

Image analysis software

Image analysis is nowadays currently used to study rock microstructure (e.g. [START_REF] Antonellini | Microstructure of deformation bands in porous sandstones at Arches National Park,Utah[END_REF][START_REF] Menendez | Micromechanics of brittle faulting and cataclastic flow in Berea sandstone[END_REF][START_REF] Lock | Predicting the permeability of sandstone from image analysis of pore structure[END_REF][START_REF] Ioannidis | Statistical analysis of the porous microstructure as a method for estimating reservoir permeability[END_REF][START_REF] Liang | Permeability and electrical conductivity of porous media from 3D stochastic replicas of the microstructure[END_REF][START_REF] Cerepi | Pore microgeometry analysis in low-resistivity sabdstone reservoirs[END_REF]). It is of major importance to have a good resolution of the input image and a good tuning of the contrast and of the threshold of grey level to discriminate between solids and voids and obtain reliable image analysis results. In this study the image analysis software 'IMAQ-VISION Builder 6' of National Instruments is used. Each image taken from the electron scanning microscope is represented by a grey level matrix with values between 0 and 255. A binary image is built by the choice of a threshold for the grey level which attributes the value 0 for voids and 1 for the solid phase. The proper choice of this threshold as well as the good resolution of the original image are key points to obtain good results in computer-aided image analysis. The ratio of magnification is chosen for each micrograph and the threshold of the grey level is first calibrated for each image by analysing the grey level histogram and by comparing the binary image and the original one (Fig 15). In particular one must insure that the small particles are not lost or clustered which are the main sources of inaccuracies for porosity evaluation. The binary image is then analysed for the determination of the porosity, the specific surface, the orientation and the size of the particles.

This technique was first applied to intact Fontainebleau sandstone specimens with different porosities between 7 and 20%. The results of the image analysis have been compared to the ones obtained from classical experimental porosity determination such as water imbibation and mercury intrusion (Table 2). They show a good agreement between the experimental technique and the image analysis determination. The specific surface is the pore surface area per unit volume. Image analysis of the intact Fontainebleau sandstone specimens is used to evaluate this quantity. The results have been compared to those published by Fredrich et al [START_REF] Ioannidis | Statistical analysis of the porous microstructure as a method for estimating reservoir permeability[END_REF] for Fontainebleau sandstone specimens with similar porosity and have also shown good agreement (Table 3).

Image analysis of shear zones

The structure of the shear zones for the specimens tested at relatively low confining pressure (7 and 14 MPa) are similar for drained and undrained tests (Figs. 13 and16). The shear band thickness is comprised between 200 and 500 µm i.e. one to two intact grain size. Inside the band the grains are broken with little localised crushing.

For the specimens tested in drained conditions under higher confining pressure (28, 40 and 50

MPa) the shear band is characterised by intense grain crushing and pulverisation (Fig. 17).

Major changes in the grain size distribution is clearly visible. Very small particles with size of the order of micrometer are produced. The fragments are densely packed, which gives an aspect of flour or white powder visible to the naked eye. Similar observations have been made

for dry specimens of Fontainebleau sandstone [START_REF] Bied | Microstructure of shear zones in Fontainebleau sandstone[END_REF].

For specimens tested in undrained conditions under high confining pressure [START_REF] Labuz | Reducing frictional constraint in compression testing through lubrication[END_REF][START_REF] Diederichs | Damage initiation and propagation in hard rock during tunnelling and the influence of near-face stress rotation[END_REF]50 MPa) the structure of the shear zone is quite heterogeneous with zones of compaction (as in the drained tests) and zones of dilation (Fig. 18). This heterogeneity can be related to the Outside the shear band, near the limits of the shear band, the grains are cracked but not crushed. The cracking phenomenon decreases rapidly outside the band.

From image analysis processing, the porosity of the material is evaluated at various places inside and outside the shear band. Inside the shear zones, the grain size distribution of the rock and the pore space are heterogeneous. Therefore the analysis is performed on ten different images taken at various locations along a parallel to the shear band axis. The porosity is evaluated for each image and the mean value is calculated. The size of each image on which the porosity is evaluated is about 20 intact grain size which can be considered as a representative element. An example of an image on which the porosity is calculated is shown on Fig. 20. This micrograph is a magnification of a zone of Fig. 17.a. In a previous paper [START_REF] Bied | Microstructure of shear zones in Fontainebleau sandstone[END_REF],

the evolution of the grain size distribution inside the band was discussed. For example, it was shown that, for the specimen tested under 28 MPa, 74% of the particles have a size smaller than D 50 /10 and 26% have a size comprised between D 50 /10 and D 50 . These numbers show that the dominant effect is here pulverisation of the grains. The smallest grain size observed is 1µm. Only few grains are simply fractured. All grain size measurements are smaller that D 50 .

For the permeability evaluation proposed in section 4, the permeability model of Walsh and

Brace [START_REF] Walsh | The effect of pressure on porosity and the transport properties of rock[END_REF] used here does not include explicitly the grain size of the rock. The only geometric quantities involved are the porosity and the specific surface.

The results of the evaluation of the porosity inside the bands are shown on Fig. 21a for the drained tests and on Fig. 21b for the undrained tests. As in the undrained tests at high confining pressure it was observed that the structure of the band is heterogeneous with zones of dilation and zones of contraction, porosity evaluation on these specimens has been preferably performed on the compacting zones. The average porosity inside the band is plotted versus the confining pressure for drained and undrained tests on Fig. 22. For each confining pressure the two values are very close. A sharp transition occurs for a critical confining pressure of about 20 MPa. At lower confining pressure that this critical value, the porosity inside the band is higher that the one of the original sample (here 21%). For higher ones it is smaller and reaches a value of about 10% for tests at 50 MPa of confinement.

Permeability model

It is known that there is no unique relationship between porosity and permeability applicable to all porous media and that the geological evolution process of the pore space influence the permeability-porosity relationship (e.g. [START_REF] Bernabé | Permeability-porosity relationships in rocks subjected to various evolution process[END_REF]). In classical permeability models such as the Carman-Kozeny model, the permeability is related to pore geometry parameters, i.e. porosity, hydraulic radius, tortuosity and specific surface. For the Fontainebleau sandstone, it was shown by Fredrich et al [START_REF] Fredrich | Pore geometry and transport properties of Fontainebleau sandstone[END_REF] that the observed bench permeabilities for samples with porosities between 10 and 20% agree well with those using the equivalent channel model of Walsh and Brace [START_REF] Walsh | The effect of pressure on porosity and the transport properties of rock[END_REF] with the following expression for the permeability k Fb

1 S n k 2 v         = (6)
where S v is the specific surface, n the porosity, b is a shape factor and F is a formation factor given by

f 2 n F ρ ρ = τ = (7)
In the above expression for the formation factor F, τ 2 is the tortuosity and ρ and ρ f are the resistivities of the saturated rock and of the fluid respectively. The shape factor b is equal to 2

for circular tubes and equal to 3 for cracks. The Archie [START_REF] Archie | The electrical resistivity log as aid in determining some reservoir characteristics[END_REF] law is used to approximate the formation factor

m f n a F = ρ ρ = ( 8
)
where m is the cementation factor and a is a pore geometry coefficient.

Data published in the literature on measurements of pore geometry characteristics and transport properties of Fontainebleau sandstone samples of various porosities by Bourbié and Zinsner [START_REF] Bourbie | Hydraulic and acoustic properties as a function of porosity in Fontainebleau sandstone[END_REF], David and Darot [START_REF] David | Permeability and conductivity of sandstones[END_REF], Fredrich et al [START_REF] Fredrich | Pore geometry and transport properties of Fontainebleau sandstone[END_REF] have been used to calibrate the various parameters of the Walsh and Brace [START_REF] Walsh | The effect of pressure on porosity and the transport properties of rock[END_REF] model.

The two parameters a and m are calibrated on data from [START_REF] Fredrich | Pore geometry and transport properties of Fontainebleau sandstone[END_REF] and [START_REF] David | Permeability and conductivity of sandstones[END_REF]. The results are shown on Fig. 23. The following values have been found: a=1.844, m=1.427.

On the basis of a large data base of 240 samples of Fontainebleau sandstone with various porosities, Bourbié and Zinsner [START_REF] Bourbie | Hydraulic and acoustic properties as a function of porosity in Fontainebleau sandstone[END_REF]

- < = - > = - (9)
The evaluation of permeability on image analysis of intact samples has been performed using the equation (6-8) with b=2 and b=3 and compared to the power law relationship [START_REF] Mair | Sequential growth of deformation bands in the laboratory[END_REF]. The results are shown on Fig. 24. The evaluated permeability reproduce correctly the data of Bourbier and Zinsner [START_REF] Bourbie | Hydraulic and acoustic properties as a function of porosity in Fontainebleau sandstone[END_REF]. These results validate the calibration procedure.

From the evaluation of porosity and specific surface inside the bands of our samples, the permeability has been estimated using the Walsh and Brace model with the above values of the parameters. The results are shown on Fig. 25 for drained tests and on Fig. 26 for undrained tests. In both case the value b=2 is assumed. They are similar for drained and undrained tests. For high confining pressure with a compacted shear band, the porosity reduction is accompanied by an important permeability reduction of more than 3 orders of magnitude. For low confining pressure with a dilatant shear band, the porosity increase is accompanied also with a decrease of the permeability of 1 to 2 order of magnitude. This counterintuitive result is due to the fact that despite the increase of porosity, the increase of the specific surface inside the band induces a permeability reduction. Similar observations have been made for sandstones with a porosity higher than 15% [START_REF] Zhu | Permeability reduction in a dilatant rock: network modeling of damage and tortuosity[END_REF][START_REF] Zhu | The transition from brittle faulting to cataclastic flow: Permeability evolution[END_REF].

Conclusion

Transport properties of rocks such as permeability are strongly influenced by the deformation process. On the basis of image analysis, porosity, specific surface and grain-size changes have been studied inside the shear bands formed in triaxial testing on a Fontainebleau sandstone with initial porosity of 21% under drained and undrained conditions at various confining pressures.

It is obtained that for specimens tested under relatively low confining pressure (7 MPa and 14 MPa), the porosity reaches a maximal value of about 30% inside the band and decreases rapidly towards the initial value of 21% outside the band. This is interpreted as dilating shear banding at low confining pressure. For specimens tested under high confining pressure (>25 MPa), a different patterning is observed. A compacting zone with high grain crushing and low porosity is observed in the centre of the band. The porosity reduction is more pronounced at higher confining pressures. From the porosity and specific surface data evaluated from image processing inside the shear zone, the changes in permeability are quantified by using the permeability model of Walsh and Brace [START_REF] Walsh | The effect of pressure on porosity and the transport properties of rock[END_REF]. It is shown that, porosity increase as observed in the band at low confining pressure and porosity decrease as observed at high confining pressure are both accompanied by a reduction of permeability inside the shear band due to the increase of tortuosity and specific surface. However, this permeability reduction is much more important at high confining pressure and can reach values three orders of magnitude smaller than the permeability of the intact material.

In globally undrained conditions, similar changes of microstructure of the rock are observed depending on the confining pressure. This results into local fluid exchanges between compacting zones that expel fluid and dilating ones that absorb it. The observations demonstrate also that, although no fluid exchanges is allowed between the sample and the exterior of the system, local fluid exchanges inside the sample occur at shear banding resulting into an heterogeneous damage pattern along the shear band. At high confinement, pore pressure generation inside the band leads locally to fluidisation of the crushed material which results into the formation of connected channels in the heart of the band.

At large scale, similar phenomena are observed in faulted zones when sheared. Usually in fault zones, two main domain can be identified: a fault core of small thickness constituted of highly comminuted ultra-cataclasites is surrounded with a damage zone which consists of fractured host rock (e.g [START_REF] Chester | Ultracataclasite structure and friction processes of the Punchbowl fault, San Andreas system, California[END_REF][START_REF] Micarelli | Structural properties of rift-related normal faults: the case study of the Gulf of Corinth, Greece[END_REF][START_REF] Wibberley | Internal structure and permeability of major strike-slip fault zones: the Median Tectonic Line in Mie Prefecture, Southwest Japan[END_REF]). The ultracataclastic structure is the results of numerous earthquake ruptures. As shown for example in a recent study of Sulem et al [START_REF] Sulem | Thermo-poro-mechanical properties of the Aigion Fault clayey gouge -Application to the analysis of shear heating and fluid pressurization[END_REF] dedicated to the characterisation of Aigion fault material in the Gulf of Corinth (Greece), the permeability of the fault core is very low so that this zone acts as an impervious barrier to transverse fluid flow, whereas the highly fractured damage zone around acts mostly as conduits for nearly along-strike flow. Similar observations can be found in the work of Wibberley and Shimamoto [START_REF] Wibberley | Internal structure and permeability of major strike-slip fault zones: the Median Tectonic Line in Mie Prefecture, Southwest Japan[END_REF].

Appendix: Shear band analysis of axisymmetric triaxial test

Extensive presentation of shear band analysis in geomaterials can be found in Vardoulakis and Sulem [START_REF] Vardoulakis | Bifurcation Analysis in Geomechanics[END_REF].

The constitutive equations of Mohr-Coulomb flow elastoplasticity theory are used here. In the coordinate system of principal axes of initial stress, the incremental equations are:

G 2 L L L L ε ∆ = σ ∆ ε ∆ + ε ∆ = σ ∆ ε ∆ + ε ∆ = σ ∆ (A1)
where for axisymmetric compression

( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )  
    κ + - κ + - - + κ =       κ + - κ + - - κ =       κ + κ + - - - κ =       κ + κ + - + κ = d 3 / 2 f 3 / 2 H 1 3 / 4 G L d 3 / 2 f 3 / 1 H 1 3 / 2 G L d 3 / 1 f 3 / 2 H 1 3 / 2 G L d 3 / 1 f 3 / 1 H 1 3 / 4 G L
= + θ + θ (A3)
where θ is measured with respect to the minor principal axis direction and

( )( ) ( ) 22 21 12 2 22 11 11 L c G L G L G L L b L a = - + - + = = (A4)
The condition for shear band bifurcation is derived from the requirement that the above data from [START_REF] Fredrich | Pore geometry and transport properties of Fontainebleau sandstone[END_REF] and [START_REF] David | Permeability and conductivity of sandstones[END_REF] Fit : F=a*n -m ; a=1.844, m=1.427

CAPTIONS TO TABLES AND FIGURES

  heterogeneity of the pore pressure generated during the loading of the sample. This results into local fluid exchanges between compacting zones that expel fluid and dilating ones that absorb it. The observations demonstrate also that, although no fluid exchanges is allowed between the sample and the exterior of the system during globally undrained tests, local fluid exchanges inside the sample occur at shear banding resulting into an heterogeneous damage pattern along the shear band. At high confinement, pore pressure generation inside the band leads locally to fluidisation of the crushed material which results into the formation of connected channels in the heart of the band (Fig.19). On Fig.19bone can clearly observe that the fines are accumulating along the boundary of the shear band and form a clogged zone of clustered fines. In our understanding the observed channels in undrained tests cannot be attributed to the formation of open fractures during unloading or resin impregnation process as such features are not observed in samples tested in drained conditions.

  equations (A1), G is the elastic shear modulus, κ = 2(1+ν)/3(1-2ν) (ν is the Poisson'modulus (h is the hardening modulus), f is the friction coefficient and d is the dilatancy coefficient. These parameters are deduced from the calibration of the elasto-plastic parameters on the triaxial test data.The characteristic equation for the shear band inclination θ

Fig. 1 Fig. 2 :Fig. 3 :

 123 Fig.1 Testing device Fig. 2: Instrumentation of the sample inside the cell Fig. 3: Microphotograph of an intact Fontainebleau sandstone.

Fig. 4 :Fig. 5 :Fig. 6 :Fig. 7 :Fig. 8 :Fig. 9 :Fig. 10 :Fig. 11 :Fig. 12 :Fig. 13 :Fig. 14 :Fig. 15 :Fig. 16 :Fig. 17 :Fig. 19 :Fig. 20 :Fig. 21 :Fig. 22 :Fig. 23 :Fig. 24 :Fig. 25 :Fig. 26 :Fig. 27 :Fig. 28 :

 456789101112131415161719202122232425262728 Fig. 4: Drained tests: Deviatoric stress and volumetric strain versus axial stress Fig. 5: Undrained tests: Deviatoric stress. volumetric strain and pore pressure versus axial stress Fig. 6: Failure criterion for drained and undrained tests Fig. 7: Lateral and radial surface of samples with a shear-band: (a) undrained test under 7MPa of confinement, (b) drained test under 28 MPa of confinement Fig. 8: Example of determination of the threshold of loss of homogeneous deformation (drained test at 28 MPa of confining pressure) Fig. 9: Onset of strain localisation: (a) drained tests, (b) undrained tests Fig. 10: Onset of dilatancy and onset of strain localisation Fig. 11: Dilatancy angle versus effective confining pressure Fig. 12: Shear band orientation versus the minor principal stress for drained and undrained tests Fig. 13: Microphotographs of a shear band formed in undrained test under 7MPa of confinement: General view and details Fig. 14: Microphotographs of a shear band formed in a drained test under 50MPa of confinement: General view and detail Fig. 15: Example of binary image (b) obtained from the original grey level image (a) Fig. 16: Microphotographs of a shear band formed under 14 MPa of total confinement: (a) drained tests; (b) undrained test Fig. 17: Microphotographs of a shear band formed in drained tests: (a) 28 MPa; (b) 40 MPa; (c) 50 MPa Fig. 18: Microphotographs of a shear band formed in undrained tests: (a) 28 MPa; (b) 40 MPa; (c) 50 MPa Fig. 19: Channels formed in undrained tests: (a) 28 MPa; (b) 40 MPa; (c) 50 MPa Fig. 20: Example of a image on which the porosity is evaluated Fig. 21: Porosity evolution in the direction perpendicular to the major (in absolute value) principal stress axis: (a) drained tests; (b) undrained tests Fig. 22: Average porosity evaluated inside the shear band for drained and undrained Fig. 23: Evaluation of the formation factor Fig. 24: Permeability evaluation on intact samples and comparison with Bourbier and Zinsner [49] power law Fig. 25: Evaluation of the permeability inside the shear bands for drained tests Fig. 26: Evaluation of the permeability inside the shear bands for undrained tests Fig. 27: Critical hardening modulus at shear banding Fig. 28: Computed shear band orientation

Fig. 1 Fig. 2 :Fig. 3 :Fig. 4 :Fig. 5 :Fig. 6 :Fig. 7 :Fig. 8 :Fig. 9 :Fig. 10 :Fig. 11 :Fig. 13 :Fig. 14 :Fig. 15 :Fig. 16 :Fig. 17 :Fig. 18 :Fig. 19 :evolution 48 -Fig. 21 :Fig. 22 :

 123456789101113141516171819482122 Fig.1 Testing device

Fig. 23 :

 23 Fig. 23: Evaluation of the formation factor

Fig. 24 :Fig. 25 :Fig. 26 :Fig. 27 :

 24252627 Fig.24: Permeability evaluation on intact samples and comparison with Bourbier and Zinsner[START_REF] Bourbie | Hydraulic and acoustic properties as a function of porosity in Fontainebleau sandstone[END_REF] power law

Table 1 :

 1 Shear band orientation for drained and undrained tests

Table 2 :

 2 Measured and estimated porosity

Table 3 :

 3 Specific surface for Fontainebleau sandstone of various porosities (after[START_REF] Fredrich | Pore geometry and transport properties of Fontainebleau sandstone[END_REF])

Table 4 :

 4 Measured and estimated permeability

Table 1 :

 1 Shear band orientation for drained and undrained tests

	Total confining	Shear band	Shear band
	pressure	orientation	orientation
	(MPa)	Drained tests	Undrained tests
		(°)	(°)
	7	69.7	70
	14	63.4	69
	28	61.9	66-68
	40	57-60	63.5-66.5
	50	54	60-62

Table 2 :

 2 Measured and estimated porosity

	Sample	Porosity	Specific surface
		(%)	(mm -1 )
	F1	4.6	9.36±2.76
	F4	10.7	20.32±4.76
	F7	15.9	22.44±3.64
	F8	20.5	19.76±2.28

Table 3 :

 3 Specific surface for Fontainebleau sandstone of various porosities (after[START_REF] Fredrich | Pore geometry and transport properties of Fontainebleau sandstone[END_REF])

	Test	Evaluated	Evaluated	Estimated	Estimated	Measured
	reference	porosity	specific surface	permeability	permeability	permeability
		(%)	(mm -1 )	(mDarcy)	(mDarcy)	(mDarcy)
				Walsh and	Walsh and	(eq. 10, after
				Brace model	Brace model	[49])
				(b=2)	(b=3)
	877E	7.08	15.55	128.5	85.6	118.6
	878B	9.6	15.06	389.1	259.4	300.2
	878A	14.05	16.08	1258.9	839.3	959.1
	878E	17.63	19.29	1902.7	1268.5	1916.5
	878D	21.13	22.03	2713.3	1808.9	3329.5

Table 4 :

 4 Measured and estimated permeability

µm 200 µm

Acknowledgements

The authors wish to acknowledge the EU projects DG-Lab Corinth (EVR1-CT-2000-40005) and Fault, Fractures and Fluids: Golf of Corinth, in the framework of program Energy (ENK6-2000-0056) and the 'Groupement de Recherche Corinthe' (GDR 2343), INSU-CNRS for supporting this research. They wish also to thank François Martineau (Laboratoire Central des Ponts et Chaussées) for his assistance in the testing program.

For axisymmetric compression, the friction coefficient f and the dilatancy coefficient d are expressed in terms of the mobilized friction angle φ and dilatancy angle ψ

In a simplified analysis, we take here the values of the friction and dilatancy coefficients at peak stress. For the Fontainebleau sandstone Poisson's ratio is ν=0.2, the friction angle is φ=41.3°, and the dilatancy coefficient is fitted on the experimental data with a linear fit as a function of the confining pressure σ 3 (Fig. 11)

with d 1 = -0.02 and d 2 =1.32

When the shear band bifurcation condition (A5) is met two symmetric shear band directions exist given by ( ) ( )

The critical hardening modulus h B is plotted on Fig. 27 versus the confining pressure. In flow theory of elastoplasticity, it is obtained that for axisymmetric compression conditions, shear banding occurs in the softening regime (h < 0) [START_REF] Vardoulakis | Bifurcation Analysis in Geomechanics[END_REF]. The corresponding shear band orientation is presented on Fig. 28.