Romain De Nijs
email: romain.de-nijs@polytechnique.edu

Bernard Caillaud

Bruno Jullien

Philippe Février

Laurent Linnemer

Régis Renault

Jean Tirole

Cédric Argenton

Volker Nocke

Jérôme Pouyet

Patrick Rey

Idrissa Sibailly

Lars Stole

Luis Vasconcelos

J Miguel

Behavior-based price discrimination and customer information sharing *

Keywords: Price discrimination, Dynamic pricing, Privacy, Information sharing. JEL: L1, D4

This article investigates the incentives and the effects of information sharing among rival firms about the identities of their past customers in a two-period model with behaviorbased price discrimination (BBPD). An unilateral information exchange between the two periods takes place in a subgame-perfect equilibrium. This exchange increases the ability of the industry to price discriminate consumers according to their profiles and boosts the profitability of BBPD at the expense of consumers.

Introduction

In many markets, firms share individual-level customer information with their competitors.

For example, in the grocery and drugstore markets, Catalina Marketing organizes information sharing of purchase history data among retailers to help them in designing their promotion campaigns [START_REF] Pancras | Optimal Marketing Strategies for a Customer Data Intermediary[END_REF]). Airline companies, through code-sharing agreements, exchange data on passengers to customize their services and prices [START_REF] Czerny | Code-sharing, price discrimination and welfare losses[END_REF]). One feature common to these examples 1 is that information sharing tends to facilitate price discrimination.

Inspired by these examples, this article investigates a model in which firms can share their proprietary data on consumers when they compete with price discrimination and then evaluates the effects of such exchanges on market functioning. More specifically, I consider a two-period model in which firms that engage in behavior-based price discrimination (BBPD) in the second period may share their private information regarding the identities of their previous customers.

BBPD is a very simple and common form of price discrimination that consists in offering different prices to different customers according to their past purchase history.

Information exchanges on consumer identities have two potential effects. A first static effect occurs in the mature phase of a market: information exchanges enable firms to more finely price discriminate their customers and hence have a priori ambiguous effects on firms profits. The second dynamic effect is fostered by the prospect of future information exchange that modifies the incentives of firms to initially acquire information on their consumers in a new market. I am interested in understanding these two effects of information exchanges and their consequences on firms profits and consumers surplus.

To investigate these issues, I study a two-period model with repeated purchases and three rival firms that compete on price to sell horizontally differentiated goods. Each firm offers a product that matches a consumer's preferences with a certain probability. Consumers fall into four segments, depending on the number of products they value: consumers who value no product, captive consumers who value only one product, local shoppers who value two products and global shoppers who value all three of the products. In the first period, there is no purchase history; therefore, firms use uniform prices. In the first-period, firms set their prices according to an absolutely continuous price distribution so that they can be ranked ex post according to said prices. The highest-price firm (hereafter the small firm) serves only its captive consumers.

The intermediate-price firm (hereafter the medium firm) serves its captive consumers and the local shoppers between its product and that of the small firm. The lowest-price firm (hereafter the large firm) serves all the other consumers. In the second period, firms can recognize their past customers and charge them a different price than they charge their new customers. When information sharing is allowed, each firm decides between the two periods whether to sell its list of past customers and its price, as well which databases sold by its rivals it purchases.

Depending on which additional customer lists a firm has acquired, it can eventually more finely price discriminate among consumers in the second period. I show that customer information sharing between rival firms takes place in a subgame perfect equilibrium and that it increases the profitability of BBPD at the expense of consumers in both periods. In the terminology of [START_REF] Liu | Customer information sharing among rival firms[END_REF], there is a "one-way information sharing" from the medium or the large firm to the small one. The acquisition of this new information increases the surplus extraction power of the small firm, which is then able to offer three different prices -one price for each firm's previous customers -as opposed to just two prices in the absence of information sharing. Information sharing clearly benefits the small firm but does not hurt the profits of the medium firm or the large firm. This corresponds to the static effect of information sharing. Because they do not suffer from information sharing in the second period, both the medium and the large firms are then willing to sell the identity of their previous customers to the small firm between the two periods. The small firm only needs one additional customer list from either the medium or the large firm to more finely price discriminate consumers, so that direct competition between the medium and the large firms leads them to sell their databases at a price equal to zero. However, all firms benefit from information sharing across the entire game because information sharing further increases the profit of the small firm in the second period. This phenomenon reinforces the incentives of firms to charge a high price in the first period to secure the small firm position. The first period competition is therefore softer with the prospect of BBPD and information sharing than with BBPD alone: firms' profits increase and consumer surplus decreases. This corresponds to the dynamic effect of information sharing.

This article is firstly related to a recent vein of research that studies information sharing among rival firms in dynamic frameworks for price discrimination purposes2 . [START_REF] Liu | Customer information sharing among rival firms[END_REF] 3 investigate a two-period duopoly model, with perfect price discrimination in the second period based on first-period information. They show that information sharing occurs if firms are sufficiently asymmetric in their customer bases. With sufficient asymmetry, the smaller firm has an incentive to share its customer information with the larger one. My model gives a different prediction in that the shared information goes from one of the two largest firms to the smallest firm. In addition, I consider that firms can only condition their prices on past purchase histories and, hence, cannot perfectly price discriminate as in [START_REF] Liu | Customer information sharing among rival firms[END_REF]. [START_REF] Kim | Customer information sharing: strategic incentives and new implications[END_REF] study a two-period model with BBPD and information sharing that reveals whether the rival products are complement or are substitute for each other. In the current analysis, firms know that their products are substitutes, and information sharing enables them to refine their knowledge regarding each consumer's preferences. [START_REF] Shy | Investment in customer recognition and information exchange[END_REF] investigate how information exchange affects the incentives of firms to invest in the costly acquisition of information on customers tastes. They show that the exchange of acquired information is bad for firms' profits and hurts consumers, but that such exchanges do not occur in equilibrium.

The current analysis is also related to articles on information sharing for price discrimination purposes in static frameworks4 . (See [START_REF] Chen | Individual marketing with imperfect targetability[END_REF], Shy and[START_REF] Shy | Customer Privacy and Competition[END_REF][START_REF] Jentzsch | Targeted pricing and customer data sharing among rivals[END_REF]).

This article is also related to the literature on competitive BBPD pioneered by [START_REF] Chen | Paying Customers to Switch[END_REF], Villas-Boas (1999) and [START_REF] Fudenberg | Customer Poaching and Brand Switching[END_REF] (See [START_REF] Chen | Oligopoly Price Discrimination by Purchase History[END_REF], [START_REF] Fudenberg | Behavior-Based Price Discrimination and Customer Recognition[END_REF], and Esteves (2009b) for surveys). Technically, the current model builds on those of Esteves (2009a) and [START_REF] Chen | Dynamic targeted pricing with strategic consumers[END_REF] 5 , by including one additional firm and introducing the possibility for firms to exchange their databases. As further discussed in the analysis, the current article is also importantly related to Esteves and Vasconcelos (2014) who investigate BBPD and mergers in a triopoly model. My model differs in that I consider a different segmentation of consumers and study information sharing rather than merger. In terms of results, I

show that without merger or information sharing, BBPD is still profitable thanks to the presence of local shoppers. I also show that information sharing is an equilibrium outcome with anticompetitive effects which is not the case in the environment of [START_REF] Esteves | Price discrimination under consumer recognition and mergers[END_REF].

The rest of this article is organized as follows. Section 2 presents the model. Section 3 investigates the case of competition with BBPD but without information sharing. Section 4 considers competition with BBPD and information sharing. Sections 5 concludes. All missing proofs are relegated in an appendix.

The model

Consider a model with two periods of consumption and repeated purchases. There is a mass one of consumers willing to buy at most one product each period. A consumer derives a utility v > 0 if the product matches her preferences and 0 otherwise. There are 3 firms i =1, 2 and 3, that compete on price. Each firm provides a good that matches a consumer's preferences with a probability 6 θ ∈ [0, 1]. Consequently, each firm has a base θ(1 -θ) 2 of captive consumers who only consider its product. Each pair of firms has a fraction θ 2 (1 -θ) of consumers who consider only their two products (hereafter local shoppers). Last, a fraction θ 3 of consumers consider the three products as suitable (hereafter global shoppers). Consumers are myopic: they only care about the price to be paid in the current period. Hence, they buy in each period the lowest-price product that match their preferences.

Forward-looking firms have zero marginal costs and a discount factor equal to one7 . In the first period, firms use uniform prices. In the second period, when information sharing is not allowed, firms can engage in BBPD by charging a different price to their new and past customers. When information sharing is allowed, each firm decides simultaneously between the two periods whether and at which prices to sell its entire list of past customers, as well as, which databases sold by its rivals it purchases. Depending on which additional customer lists a firm has acquired, a firm can eventually more finely price discriminate among consumers in the second period.

Direct application of [START_REF] Stahl | Oligopolistic Pricing and Advertising[END_REF] with exogenous consumer segmentation shows that when BBPD is not allowed, there exists a symmetric equilibrium in which in each period, firms randomize their price according to the cumulative distribution function (cdf)

F up (p) = 1 - 1-θ θ (( v p ) 1/2 -1) with support [p up = (1 -θ) 2 v, v].
Each firm earns an expected profit θ(1 -θ) 2 v per period. As a consequence, in the benchmark case of uniform price competition, each firm earns a total expected profit Π tot up = 2θ(1 -θ) 2 v.

3 BBPD without information sharing

Second period

Define p i,t as firm i's price at period t. Without loss of generality, assume that p 1,1 > p 2,1 > p 3,18 .

In this case, firm 1 (hereafter the small firm) has served only its captive consumers, and can perfectly recognize them in the second period. Firm 2 (hereafter the medium firm) has served both its captive consumers and local shoppers between products 1 and 2. Firm 2 has therefore only partially told apart its segment of captive customers. Firm 3 (hereafter the large firm) has served all other consumers and hence has learned nothing about the type of its customers.

Each firm is potentially able to charge one price to its past customers and one price to its new consumers. However, the large firm has served all its potential consumers in the first period.

Consequently, the large firm cannot attract new customers and hence, charges only one price to its past customers.

Proposition 1 In the second period when BBPD is allowed, there exists an equilibrium in which:

• The small firm charges its past customers the price v and charges its new customers a price randomized according to the cumulative distribution function (cdf )

F n s (p) = 1 -1-θ θ ( v p -1) on [p = (1 -θ)v, v].
• The medium firm charges its past customers a price randomized according to the cdf

F o m (p) = 1-(1-θ)v p on [(1-θ)v, v[ with a mass (1-θ) on v;
and charges its new customers a price randomized according to the cdf

F n m (p) = 1-1-θ θ ( (1-θ)v p -1) on [p = (1-θ) 2 v, (1-θ)v].
• The large firm randomizes its price according to the cdf

F o,n l (p) = 1-(1-θ) 2 v p on [(1-θ) 2 v, v]
with a mass (1 -θ) 2 on v.

Firms' second-period expected profits are:

Π s = θ(1 -θ) 2 v + θ(1 -θ) 2 (1 -(1 -θ) 2
)v for the small firm.

Π m = θ(1 -θ) 2 v + θ(1 -θ) 2 (1 -(1 -θ))v for the medium firm. Π l = θ(1 -θ) 2 v
for the large firm.

The first term in expected profits θ(1 -θ) 2 v is the profit each firm can guarantee itself by charging the monopoly price on its base of captive consumers. It also corresponds to the one-period profit under uniform price competition [START_REF] Stahl | Oligopolistic Pricing and Advertising[END_REF]).

The second term is the extra profit a firm derives from its ability to price discriminate. It is immediate to check that Π s > Π m > Π l so that a firm derives a higher profit when it has more accurately recognized its segment of captive consumers. A firm is said to have more accurately recognized its captive consumers when it can build a smaller list of consumers that contains its captive consumers.

The small firm has perfectly recognized its captive consumers, as its list of past customers contains only its captive consumers. The medium firm has only imperfectly recognized its captive consumers because its list of past customers contains both its captive consumers and local shoppers with the small firm. However, the medium firm has more accurate information about its captive consumers than the large firm, whose list of past customers contains all of its potential customers.

Proposition 1 shows that the profitability of BBPD studied by Esteves (2009a) and [START_REF] Chen | Dynamic targeted pricing with strategic consumers[END_REF] can survive in a model with more than 2 firms. However, it contrasts with some results of [START_REF] Esteves | Price discrimination under consumer recognition and mergers[END_REF] who study BBPD and mergers in a triopoly model with captive consumers and global shoppers. Among other things, they show that (without merger) firms earn the same profits with or without BBPD. This result occurs because, in the second period, discriminating firms compete à la Bertrand for price-sensitive consumers. As a consequence, they make no extra profits from their ability to price discriminate. In the current article, there is no full-fledged competition effect because there are always consumers with only two firms offering a good they value, namely local shoppers. Consequently, in the second period, discriminating firms have market power on a fraction of price-sensitive consumers from rivals,

and hence, can derive extra profits from their ability to price discriminate 9 .

As usually found in the literature on BBPD without long-term contract, discriminating firms offer higher prices to their past customers who have revealed a stronger preference for the firm's product 10 . Here, a novelty arises for the small firm: it randomly draws the price it charges to its new customers on the interval [(1 -θ)v, v]. Doing so, it specifically targets local shoppers with the medium firm's product and local shoppers with the large firm's product. The small firm indeed has no chance to attract global shoppers because the medium firm randomly chooses the price it charges to its new customers (among which are the global shoppers) on the

interval [(1 -θ) 2 v, (1 -θ)v] which is below the interval [(1 -θ)v, v].
The equilibrium described in Proposition 1 still exists with forward-looking consumers as long as their discount factor is low enough. The proof is available upon request.

First period

In the first period, a firm makes its pricing decision rationally anticipating how this decision affects its current and future profits. One can show that there exists no pure-strategy equilibrium, but a symmetric mixed-strategy equilibrium exists in which firms mix their price according to a cdf F pd (.) on [p pd , v]. The total expected profit of a firm that charges a price p in the first period then writes:

Π tot pd (p) = pθ(1 -θF pd (p)) 2 + (1 -F pd (p)) 2 Π l + 2(1 -F pd (p))F pd Π m + F 2 pd (p)Π s (1)
The first term is identical to that of the benchmark case with uniform pricing [START_REF] Stahl | Oligopolistic Pricing and Advertising[END_REF]).

The other terms correspond to the profit a firm will earn in the second period according to 9 See [START_REF] Taylor | Supplier surfing: competition and consumer behavior in subscription markets[END_REF] and [START_REF] Chen | Oligopoly Price Discrimination by Purchase History[END_REF] for a similar argument in markets with switching costs. 10 For model with rewards for past customers, see [START_REF] Shaffer | Preference-Based Price Discrimination in Markets with Switching Costs[END_REF], [START_REF] Shin | A Customer Management Dilemma: When Is It Profitable to Reward One's Own Customers?[END_REF], [START_REF] Caillaud | Strategic loyalty reward in dynamic price discrimination[END_REF], and De Nijs and Rhodes (2013) when firms cannot commit to future prices, and Chen and Pearcy (2010) when firms can use long-term contracts.

its first-period price ranking. For instance, with a probability (1 -F pd (p)) 2 a firm charges the lowest first-period price and hence, earns Π l in the second period. Because a firm must be indifferent between all prices in the support of F pd (.), one has v]. This equality gives F pd (.). The regularity condition

Π tot pd (p) = Π tot pd (v) = 2θ(1 -θ) 2 v + θ(1 -θ) 2 (1 -(1 -θ) 2 )v for all p ∈ [p pd ,
F (p pd ) = 0 yields: p pd = (1 -θ) 2 (2 -(1 -θ) 2 )v.
The formal proof for Proposition 3 is a direct application of Varian (1980) and [START_REF] Narasimhan | Competitive promotional strategies[END_REF] and is therefore omitted.

Proposition 2 When BBPD is allowed in the second period, there is a symmetric price equilibrium in the first period in which each firm randomizes its price according to the cdf:

F pd (p) = 1 -1-θ θ (( (1-(1-θ) 2 )v p-(1-θ) 2 v ) 1/2 -1) on [p pd = (1 -θ) 2 (2 -(1 -θ) 2 )v, v]. Each firm earns an total expected profit Π tot pd = 2θ(1 -θ) 2 v + θ(1 -θ) 2 (1 -(1 -θ) 2 )v.
The first term 2θ(1-θ) 2 v in the expected profit Π tot pd is the guaranteed profit each firm would derive in a two-period game with uniform pricing. The second term θ

(1 -θ) 2 (1 -(1 -θ) 2 )v is
the extra profit each firm expects to earn thanks to BBPD, viewed from the beginning of the game. It is simple to check that F pd (.) has first-order stochastic dominance over F up (.). This property clearly delineates the incentive firms have to price high in the first period to more accurately recognize their captive consumers. This "race for discrimination effect" has first been identified and investigated by Esteves (2009a) and [START_REF] Chen | Dynamic targeted pricing with strategic consumers[END_REF] in duopoly markets. Because BBPD is profitable in the second period of the current model, the "race for discrimination effect" is also present, whereas it is not the case in Esteves and Vasconcelos (2009).

BBPD with information sharing 4.1 Equilibrium in the second period

Assume that p 1,1 > p 2,1 > p 3,1 . The only firm that is likely to be interested in acquiring customers lists from its rivals is the small firm (firm 1). Indeed, if the small firm learns the identities of its rivals' past customers, it is then able to charge three different prices (instead of two): one price for its own past customers and one price for each of the other two firms' past customers. The medium firm (firm 2) is not interested in acquiring information regarding the small firm's past customers' identities because these consumers do not value its product.

In addition, firm 2 does not value the customers list of the large firm because this list would not allow it to more finely price discriminate consumers. Last, the large firm does not value the small and the medium firms' past customer lists because these customers do not value its product. Therefore, I consider the scenario in which the small firm has acquired the list of past customers of either the medium or the large firm.

Proposition 3 In the second period, when BBPD is allowed and the small firm has acquired the list of past customers of the medium or the large firm, there exists an equilibrium in which:

• The small firm charges its past customers a price v, randomizes its prices to the medium firm's past customers according to the cdf

F m s (p) = 1 -1-θ θ ( v p -1) on [(1 -θ)v, v]
, and randomizes its price to the large firm' s past customers according to the cdf

F l s (p) = 1 -1-θ θ (( v p ) 1/2 -1) on [(1 -θ) 2 v, v].
• The medium firm randomizes its price to its past customers according to the cdf

F m m (p) = 1 -(1-θ)v p on [(1 -θ)v, v]
, with a mass (1 -θ) on v, and randomizes its price to the large firm's past customers according to the cdf

F l m (p) = 1 -1-θ θ (( v p ) 1/2 -1) on [(1 -θ) 2 v, v].
• The large firm randomizes its price to its past customers according to the cdf

F l l (p) = 1 -(1 -θ)( v p ) 1/2 on [(1 -θ) 2 v, v] with a mass (1 -θ) on v.
Firms' second-period expected profits are:

Πs = θ(1 -θ) 2 v + 2θ 2 (1 -θ) 2 v
for the small firm.

Πm = θ(1 -θ) 2 v + θ 2 (1 -θ) 2 v
for the medium firm.

Πl = θ(1 -θ) 2 v for the large firm.

In comparison to competition with uniform pricing, the small and the medium firms earn an additional profit θ 2 (1 -θ) 2 v from each sub market of past customers of their rivals wherein they compete. In comparison to competition with BBPD but without information sharing, the small firm earns an additional profit θ 3 (1 -θ) 2 v and the medium and the large firms earn the same profits.

To increase its profit, the small firm only needs to acquire the list of past customers of the medium or the large firm. Consequently, the medium and the large firm compete à la Bertrand to sell their customer lists to the small firm so that they offer their lists of past customers at a price equal to zero. Therefore in equilibrium, information sharing occurs from the medium or the large firm to the small firm, and total industry profits increase with respect to the situation with BBPD and no information sharing. Last, as the total welfare is fixed to (1

-(1 -θ) 3 )v
by period in this model, consumers are hurt by information sharing in the second period.

Proposition 4 There exits an equilibrium in which the small firm acquired the list of past customers of the medium or the large firm at a price equal to zero. This information sharing increases the second-period industry profits at the expense of consumers.

It is interesting to compare the effects of information sharing with those of mergers as in [START_REF] Esteves | Price discrimination under consumer recognition and mergers[END_REF]. In the current model it is possible to find a two-firm merger that is more profitable that information sharing. Indeed, consider the situation in which the small and the medium firms merge at the beginning of the second period. In the second period of the game, the merging entity uses three prices: one price v charged to their 2θ(1 -θ) 2 + θ 2 (1 -θ) common past customers, one price charged by the small firm to the past customers of the large firm, and one price charged by the medium firm to the past customers of the large firm. Application of Proposition 3 shows that the merging entity makes a profit 2θ 2 (1 -θ) 2 v from past customers of the large firm. The bottom line, is that the merging entity makes a total second-period profit equal to 2θ(1

-θ) 2 + θ 2 (1 -θ) + 2θ 2 (1 -θ) 2 v which is
greater that the joint profit of the small and the medium firms with information sharing which

is equal to 2θ(1 -θ) 2 + θ 2 (1 -θ) 2 + 2θ 2 (1 -θ) 2 v.
This implies that information sharing is less detrimental than a merger for consumers. However, in contrast to [START_REF] Esteves | Price discrimination under consumer recognition and mergers[END_REF], information sharing has still substantial anticompetitive effects.

First period

In the first period, firms rationally anticipate future information sharing and BBPD. A similar reasoning than in the case of BBPD without information sharing leads to Proposition 5. The formal proof for Proposition 5 is a direct application of Varian (1980) and [START_REF] Narasimhan | Competitive promotional strategies[END_REF] and is therefore omitted.

Proposition 5 When BBPD and information sharing are allowed in the second period, there is a symmetric price equilibrium in the first period in which each firm randomizes its price according to the cdf:

F pd,is (p) = p-(1-θ)v-(1-θ)(((1-θ) 2 v+(2θ-1)p)v) 1/2 θp on [p pd,is = (1+2θ)(1-θ) 2 v, v].
Each firm earns an expected total profit

Π tot pd,is = 2θ(1 -θ) 2 v + 2θ 2 (1 -θ) 2 v.
Viewed from the beginning of the game, all firms benefit in expectation from future information with BBPD, even if, only one of them will eventually be the second-period small firm that actually increases its profits thanks to information exchange. One can easily show that F pd,is (.) has first-order stochastic dominance over F pd (.). This property shows that the "race for discrimination" is exacerbated when information sharing occurs on top of BBPD in the second period. This is because the small firm position becomes even more profitable with information sharing. This dynamic effect of information sharing is similar to the first-period effect of merger studied by [START_REF] Esteves | Price discrimination under consumer recognition and mergers[END_REF]. If consumers were forward-looking, the equilibrium described in Proposition 5 might be disrupted by strategic behaviors of consumers. Indeed, forward-looking consumers might want to make different purchasing decisions than myopic consumers to manipulate the information collected by firms and eventually benefit from better deals in the second period. The analysis of this situation is technically very difficult and is beyond the scope on this article.

One has Π tot pd,is ≥ Π tot pd ≥ Π tot up . This result means that the profitability of BBPD is magnified with information sharing. The total welfare being fixed and equal to 2(1 -(1 -θ) 3 )v, an important consequence is that consumers are hurt by information sharing in both periods.

Conclusion

This article has shown that information sharing between rival firms that compete with BBPD can occur in equilibrium and that such an exchange magnifies the profitability of BBPD at the expense of consumers. This article has implications for privacy regulation. Several qualitative analysis have shown that consumers dislike the practice of targeted pricing. For instance, the Office of Fair Trading argues in its report on targeted advertising and pricing (OFT (2010)) that "a 2005 survey conducted in the US found that 87 percent of respondents objected the practice of online stores charging people different prices, for the same products based on information collected about their shopping habits". The Office of Fair Trading is also concern by the "deterioration in trust in online markets" that behavioral pricing could foster (OFT (2010) § § 5.26 and 5.27). The current analysis confirms previous results (Esteves (2009a) and [START_REF] Chen | Dynamic targeted pricing with strategic consumers[END_REF]) that BBPD with proprietary data can increase firm profits at the expense of consumers but also reaches the novel conclusion that consumers can be even more hurt by behavioral pricing under weak protection of their personal data that could be exchanged between rival firms for price discrimination purposes.

The analysis has also implications for competition policy. It shows that information sharing with BBPD is an equilibrium outcome that can restrict competition without collusion. This means that information sharing among firms that use BBPD may infringe upon article 101(1) of the Treaty on the Functioning of the European Union (TFEU), which prohibits practices likely to restrict competition. According to the guideline on the applicability of the Article 101 to horizontal co-operation agreements (European Commission (2010)), customer lists fall into the category "of commercially sensitive, i.e., strategically useful data" that "can give rise to restrictive effects on competition if its reduces the parties' decision making independence by decreasing their incentives to compete". In addition, according to Fine (2010) "it is not necessary for the application of Article 101(1) that the parties engage in bilateral or multilateral exchanges of sensitive information; even a one-way communication may establish liability" 11 , which is exactly the situation that happens in the model developed in this article. Another implication of the analysis is that the anti-competitive effects of an information exchange can occur even before the exchange itself occurs. This is because forward-looking firms that anticipate the extra profits generated by information exchanges will modify their current pricing strategies through higher prices to acquire more valuable information. A practical consequence is that the threat of punishment by competition authorities has to be strong enough to deter firms from even considering the possibility of such consumer-detrimental exchanges, thus preventing them from modifying their current behaviors.

Villas-Boas, J.M., 1999, "Dynamic Competition with Customer Recognition", Rand Journal of Economics, 30, 604-631.

A Proof of Proposition 1

As in [START_REF] Stahl | Oligopolistic Pricing and Advertising[END_REF], there does not exist an equilibrium in which all firms use pure strategies.

There exists an equilibrium with mixed strategies wherein firms draw their prices according to continuous distribution functions. Let's introduce some notations :

• F m i (.
) is the price distribution of firm i = s, m, l (for small, medium and large firm) in its market k ∈ {o, n} with o that stands for own previous customers and n for new customers.

• Π i is the total expected second-period profits of firm i = s, m, l.

• Π m i is the expected second-period profits of firm i = s, m, l in its market k ∈ {o, n}.

The small firm charges two different prices :

• One price to its previous consumers from who it earns a profit Π o s (p) = pθ(1 -θ) 2 . Optimally, the small firm charges its previous customers the monopoly price v.

• One price to its new customers from who it earns an expected profit Π

n s (p) = p(θ 2 (1 - θ)(1 -F o m (p)) + θ 2 (1 -θ)(1 -F o,n l (p)) + θ 3 (1 -F n m (p))(1 -F o,n l (p)))
. Indeed, there is a fraction θ 2 (1 -θ) of consumers who consider both the small and the medium firms' products and they choose to buy to the small firm only when this firm charges a lower price p than the medium one which occurs with probability (1 -F o m (p)). By the same reasoning with the large firm, one can obtain the second term of Π n s (p). Last, there is fraction θ 3 of consumers who consider all the thres firms' products and they choose to buy to the small firm only when this firm charges a lower price p than the medium and the large ones which occurs with probability (1 -

F n m (p))(1 -F o,n l (p))).
The medium firm charges two prices :

• One price to its previous customers from who it earns an expected profit:

Π o m (p) = p(θ(1 -θ) 2 + θ 2 (1 -θ)(1 -F n s (p))).
• One price to its new customers from who it earns an expected profit Π

n m (p) = p(θ 2 (1 - θ)(1 -F o,n l (p)) + θ 3 (1 -F n s (p))(1 -F o,n l (p))).
The large firm charges only one price (it cannot attract new customers) :

• One price to its previous customers from who it earns an expected profit Π

o,n l (p) = p(θ(1 -θ) 2 + θ 2 (1 -θ)(1 -F n m (p)) + θ 2 (1 -θ)(1 -F n s (p)) + θ 3 (1 -F n m (p))(1 -F n s (p))).
The construction of the equilibrium runs as follows :

• First step: The medium firm never charges its previous customers a price below p =

(1 -θ)v, as it would be better-off charging v rather than any price below p. Consequently:

Π o m (p) = Π o m (v) = θ(1 -θ) 2 v for all p ∈ [p, v]. This yields F n s (p) on [p, v].
• Second step: The large firm never charges a price below p = (1-θ) 2 v, as it would be betteroff charging v rather than any price below p. • Fourth step: Let's assume that the expected profit the small firm earns from its new

Consequently Π o,n l (p) = Π o,n l (v) = θ(1 -θ) 2 v for all p ∈ [p, v].
customers is Π n s (p) = θ 2 (1 -θ)(p + p) for p ∈ [p, v]. This yields F o m (p) on [p, v].
It remains to check that no firm has a profitable deviation from the putative equilibrium constructed above and described in Propostion 1.

The large firm has no profitable deviation. Pricing above v would lead to zero demand and the large firm is always better-off charging v than any price below p. If the large firm charges a price p ∈ [p, p] then its profit writes Π l = p(θ(1 -θ) 2 + θ(1 -θ) 2 + θ 2 (1 -θ) 1-θ θ ( (1-θ)v-p p ) + θ 3 1-θ θ ( (1-θ)v-p p )) = θ(1 -θ) 2 v. If the large firm charges a price p ∈ [p, v], then its profit writes:

Π l (p) = p(θ(1 -θ) 2 + θ 2 (1 -θ) 1-θ θ ( v-p p )) = θ(1 -θ) 2 v
. By the same reasoning the medium firm has also no incentive to charge a price outside [p, v] on its market of previous customers. Let's consider a deviation by the medium firm on its market of new customers. Charging a price strictly below p would not be profitable as this does not lead to higher demand but decreases its margin. If the medium firm charges a price p ∈ [p, v] to its new customers, then its profit writes: Π n m (p) = θ 2 (1 -θ) 3 v + θ 2 (1 -θ) 3 ( v p -1) which is maximized in p where it takes the value θ 2 (1 -θ) 2 v, so that the medium firm has no strictly profitable deviation.

Last, let's consider a deviation by the small firm. It is clear that the small firm has no incentive to charge its new customers above v or below p. If the small firm charges its new customers a price p ∈ [p, p], then its profit writes: Π n s (p) = θ 2 (1 -θ)

3 v + θ 2 (1 -θ)p + (1 -θ) 2 θ 2 v( (1-θ)v p -1)
which is U shape convex function in p on [p, p] and is therefore maximized either at p or at p.

One has Π

n s (p) = Π n s (p) = θ 2 (1 -θ)(p + p) = θ 2 (1 -θ) 2 (2 -θ)v = θ(1 -θ) 2 (1 -(1 -θ) 2
)v so that the small firm has no strictly profitable deviation.

B Proof of Proposition 3

I look for an equilibrium in mixed strategies with continuous distribution functions. Let's introduce some notations :

• p j j is the price charges by firm i = s, m, l (for small, medium and large firm) on the submarket of firm j' previous customers with j = s, m, l.

• F j j (.) is the distibution used by firm i = s, m, l to randomize its price on the submarket of firm j' previous customers with j = s, m, l.

• Πi is the total expected second-period profits of firm i = s, m, l.

• Πj j is the expected second-period profits of firm i = s, m, l on the submarket of firm j' previous customers with j = s, m, l.

The small firm charges three prices :

• One price to its previous customers p s s = v.

• One price to the medium firm's previous customers p m s drawn from F m s (.).

• One price to the large firm's previous customers p l s drawn from F l s (.).

The medium firm charges two prices :

• One price to its previous customers p m m drawn from F m m (.).

• One price to the large firm's previous customers p l m drawn from F l m (.).

The large firm charges one price:

• One price to its previous customers p l l drawn from F l l (.).

On the submarket of the medium firm's previous customers, expected profits write as follows:

  This is true in particular for p ∈ [p, p] where F n

	s (p) = 0. This yields
	F n m (p) on [p, p].

• Third step: Let's assume that the expected profit the medium firm earns from its new customers is Π n m (p) = θ 2 (1 -θ)p for p ∈ [p, p]. This yields F o,n l (p) on [p, p] and by continuation on [p, v].

See Taylor (2004) andCalzoni and Pavan (2006) for information sharing among non-rival firms.

See Liu and Serfes (2004) who also endogenize the information used by firms to engage in price discrimination in an initial investment stage.

There also exists a branch of the literature on evaluations of information exchanges in credit markets (See for instance[START_REF] Pagano | Information Sharing in Credit Markets[END_REF],Padilla and Pagano (1997, 2000), and[START_REF] Gehrig | Information Sharing and Lending Market Competition with Switching Costs and Poaching[END_REF]

See also[START_REF] Caillaud | Strategic loyalty reward in dynamic price discrimination[END_REF] and De Nijs (2013) for related models, and[START_REF] Esteves | Pricing with Customer Recognition[END_REF] for another model of BBPD with discrete consumer preferences.

[START_REF] Chen | Paid-Placement: Advertising and Search on the Internet[END_REF] make similar assumptions.

Considering a discount factor smaller than one adds notation without any additional result.

The probability of a tie in prices is zero in any first-period equilibrium (See Section 3.2), and one can always renumber firms.


Sylvain Patinet
email: sylvain.patinet@espci.fr

Damien Vandembroucq
email: damien.vandembroucq@espci.fr

Alex Hansen
email: alex.hansen@ntnu.no

Stéphane Roux
email: stephane.roux@lmt.ens-cachan.fr

Cracks in random brittle solids: From fiber bundles to continuum mechanics

come

Introduction

It is needless to emphasize the importance of brittle fracture for safety and reliability issues. In fact, the condition for crack propagation is well understood since Griffith pioneering work [1]. Critical energy release rates or equivalently stress intensity factors are known to be material characteristics defining the onset of crack propagation for an existing crack. However, a resisting issue concerns crack initiation. Indeed, for a long period, the common view was that cracks would initiate from preexisting defects. Brittleness was invoked to consider that as soon as initiated a crack would propagate in an unstable fashion down to complete failure. Thus, amazingly, solid mechanics for the most part disappeared from strength predictions. Because the initiation defect was local, the failure load remained a statistical issue, with probability distribution which simply resulted from that of the defect distribution after an elementary elastic computation (together with some assumptions for the defect geometry). This was formalized within "weakest links" type of approaches, that can be traced back to the pioneering work of Weibull. [2] These approaches had also the merit of shedding some light on salient features unexpected in the mechanics of homogeneous materials, such as the occurrence of systematic size effects. A recent review on this topic and extensions thereof can be found in Ref. [3].

Alternative approaches based on a purely deterministic picture was proposed by Francfort and Marigo [4], as an extension of Griffith's theory. This theory, although appealing, raises subtle mathematical issues -which may in fine resume to physical questions -related to how small scales may or may not disappear at the continuum limit for such singular problems. Another direction of attack consists of introducing a more complicated picture for the crack tip, through a "fracture process zone" whereby, at microscopic scales, a displacement discontinuity can still allow for the transmission of stress. Such models following the initial work of Barenblatt [5] illustrate the fact that initiation and propagation can be incorporated in the same picture if in addition to toughness, a critical stress is also considered. This is yet another way to stress that some microscopic length scale will survive in this macroscopic picture. Leguillon [6] for instance proposes such a dual criterion for accounting for crack initiation in the spirit of these fracture process zone models. Although this latter class of theories emphasizes the importance of small scales, materials are always treated as if they were homogeneous.

Thus, from a continuum mechanics point of view, the choice is left between approaches that either emphasize the geometry of defects (without mechanical manifestation other than linear elasticity) or focus on fine details of local variations of mechanical free energy (not to say damage) but ignores material variability.

It is therefore interesting to consider discrete models (where no ambiguity is left about microscopic limits) and where disorder is explicitly considered. On the one hand, such models may be considered as too simplistic for portraying any specific materials, however, on the other hand, they may shed some light on statistical size effects that may be expected. This is very important as scaling is not neutral. If interpolation based on Weibull distribution is fair when the same sizes and level of probabilities as determined in identification, it becomes very fragile when extrapolation is needed.

Hans Herrmann [7][8][9] has been one of the pioneers in this matter with first papers on electrical analog, "fuse networks", to brittle material published in 1985. However, a direct numerical simulation reveals to be a difficult pathway to reaching definitive conclusions, as results are prone to slow corrections to scaling, and become expensive in computational power as one moves to two or three dimensions. [10] Nevertheless, those simulations could demonstrate quite clearly that the weakest link approach could not be applied at the smallest scales. Therefore, it was already clear from the start that neither approach based on prior defect statistics only nor on a more sophisticated interaction model but within a homogeneous medium description were applicable, and hence the merit of the statistical laws and the attached corresponding size effects could be questioned.

To progress along these lines, fiber bundle models [11,12] play a key role. Indeed despite their simple definition, and their introduction about 90 years ago, an exhaustive characterization of their behavior has only been achieved in some particular cases [13][14][START_REF] Hansen | The fiber bundle[END_REF].

These models consist of a collection of elastic brittle fibers with random strength loaded in parallel. One very important feature of these models leading to different scaling regimes is the way load is distributed amongst the unbroken fibers. In the "Equal Load Sharing" (or ELS), [12] load is equally distributed over all surviving fibers. This corresponds to clamping all fibers onto a rigid substrate onto which the external load is applied. Because of this even distribution, the dimensionality of the substrate plays no role. This limit can thus be seen as a mean-field model which is analytically tractable.

The opposite limit of Local Load Sharing, (or LLS), is based on a local redistribution of the load carried by a fiber onto its unbroken neighboring fibers [START_REF] Hansen | The fiber bundle[END_REF][START_REF] Harlow | [END_REF].

Here the breakdown scenario is very different from ELS, in the sense that after a first diffuse damage regime, (where fibers breakage is spread over the entire system), a localized region hosts a crack which -after its nucleation -grows without limits. Macroscopically, this failure appears as brittle but after a significant number of fibers have been broken. However, the LLS model suffers from some ambiguity on the way the load is redistributed after fiber breakage. This ambiguity becomes even more pronounced in two dimensions and more. In one dimension, some aspects of this model are analytically tractable [START_REF] Hansen | The fiber bundle[END_REF]17]. However, mostly, numerical methods must be used.

There are also other more realistic variants of the fiber bundle model. The Soft Clamp Model [18][19][20][21] has the fibers attached to clamps that respond elastically. The clamps are infinite half spaces and the force distribution among the fibers are due to the elastic response of the clamps.

The transition between ELS and LLS has also been studied by varying continuously the sharing rule trough a simple power law stress redistribution function σ ∼ r -γ , where r is the distance from the broken fiber [22][23][24]. It was shown that the stress-transfer function decay exponent dramatically impacts breaking processes, ultimate strength and creep regime. In the limiting case of ELS, the breaking of fibers is a completely random nucleation process while, for LLS, it becomes a nucleation problem.

In the present study, we modify the Soft Clamp Model by replacing the infinite half space clamps by thin elastic membranes. This corresponds to the LLS limit, that is to say an infinite exponent for the stress-transfer function (γ → ∞) in [22][23][24]. We note that both in the Soft Clamp model and the Membrane model, one of the clamps can be replaced by an infinitely rigid one. For symmetry reasons, this model will have the same behavior as were the two clamps equal.

Other physically realistic models or limits could have been considered. In particular, an elastic plate described through Love-Kirchhoff theory has naturally a bending stiffness. Indeed, there is no objection to considering such a model, (and it has already been considered for instance in [25]) however, in 1D where the plate reduced to an elastic beam, it does not coincide with the LLS model precisely because of the flexural effect and continuity of the rotation. This introduced a dampened oscillatory behavior close to a crack. Because our aim is here to introduce a natural extension of LLS model in 2D that suffers no ambiguity in its definition, we will stick to the definition of the Soft Membrane limit.

Let us also mention a related model, namely that of shear delamination of a thin film from a rigid substrate as introduced by Zaiser et al. [26][27][28]. A modeling proposed in [26] shows that the in-plane displacement obeys an anisotropic Laplace equation, that can be somehow compared to the Poisson equation ruling the outof-plane displacement for the membrane under distributed load. The source term that is present in the Poisson equation is the result of the external loading and the balance equation leading to damage and crack propagation. A similar effect is naturally present in other modelings [18,19,25,29]. Since it controls the instability of crack propagation which is the hallmark of the LLS model, it is the analogy with models that have no distributed loadings that should be considered carefully.

In the present study, we show that it is possible to define and compute a toughness allowing for understanding the post-peak behavior, i.e. after the maximum load has been reached. Moreover, this limit is also shown to match that of a crack front pinning model. The simplicity of the model finally allows us to study its behavior in the continuum limit via a perturbative approach, drawing promising path from nucleation to crack propagation in brittle heterogeneous materials.

f ∆ .n ϕ = u f x f x x y a) b) z z z broken fibers = c) n = Lap[u] -1 u -1
Fig. 1. Load sharing rules in a fiber bundle between clamps that are elastic membranes. The lower membrane is infinitely stiff, whereas the upper membrane responds elastically.

A macroscopic load F is applied to the system through a homogeneous pressure onto the membrane and is shared between unbroken fibers subjected to local forces f . a) When the upper membrane is stiff, the model behaves as the ELS model. b) In one dimension, a very soft upper membrane (or rather, string) will cause the model to behave as the LLS model. c) In two dimensions, a very soft upper membrane will transfer an additional load only to fibers adjacent to already broken ones.

Soft Membrane Fiber Bundle in One Dimension

The fiber bundle initially consists of N fibers distributed homogeneously over a linear domain of size L = N , at positions labeled by integer x coordinates. To reduce the importance of edge effects, periodic boundary conditions are implemented. Each fiber is assumed to be elastic-brittle. Initially, all fibers are present. Then, through the application of load, fibers will be broken one at a time. A sequence index, t, is introduced which simply counts the number of broken fibers. For simplicity, t will be referred to as "time". Let us emphasize that the model itself does not depend on the physical time. In the elastic regime, a macroscopic load F is applied in the system, and is shared between unbroken fibers as illustrated in Fig. 1. A fiber at position x is subjected to a local force f (x, t). If the fiber is broken, then f (x, t) = 0. If not, because of elasticity, f (x, t) is proportional to the external load, f (x, t) = F ϕ(x, t), where ϕ(x, t) characterizes the load redistribution on the fiber, and that depends on the distribution of broken fibers and hence on t. Each fiber is characterized by its strength, f c (x), independent of time. At each instant, the macroscopic breaking load is the value of F such that one and only one fiber reaches its threshold, f (x, t) = f c (x), hence

F c (t) = min x f c (x) ϕ(x, t) , (1)
which corresponds to an extremal dynamics.

The load sharing rule is what allows us to define ϕ(x, t). For instance, ELS means that all surviving fiber support the same force as shown in Fig. 1a. Overall balance requires that ϕ(x, t) = 1, and hence ϕ(x, t) = 1/(N -t) if the fiber is unbroken, and 0 else.

The membrane model is equivalent to the LLS rule in 1D when the membrane (string) is very soft. We now discuss the LLS rule in some detail. As above mentioned that balance imposes ϕ(x, t) = 1. In 1D, it is rather straightforward to arrive at the fact that for any interval of m consecutive fibers broken, an additional force of m/2N is transfered to the two surviving fibers nearest neighbor to the end of this interval. This comes in addition to the 1/N force that it always present. Note that a fiber may be neighbor of two intervals of m l and m r failed bonds respectively for left and right, and hence its load partition coefficient will be

ϕ(x, t) = 2 + m l + m r 2N (2)
The Soft Membrane model is now considered. In this case, the deflection u(x) of this string under the applied loading has to be solved. If we consider that the fibers are much stiffer than the string, if the fiber is present at position x, then u(x) = 0. Otherwise,

Lap[u](x) = -1 (3)
where Lap is the discrete laplacian Lap[u](x) = u(x + 1) -2u(x) + u(x -1). The load transferred by a membrane to its boundary writes f T = -∇u • n where n is an outer unit vector, normal to the membrane boundary (see Fig. 1). In 1D, this means that a fiber located at x receives an additional transfer

f T (x) = [u(x + 1) -u(x)] + [u(x - 1) -u(x)]
where we separated the contributions of the left and right neighbors. In this case, the load applied to a surviving fiber x (which is such that u

(x) = 0) is ϕ(x, t) = (1/N ) 1 + y n.n.x u(y) (4)
where a summation over nearest neighbors (n.n.) of x is involved. The occurrence of the second order differential operator Lap comes from a conservation (balance) equation so that the total load applied onto the system is finally transferred to the unbroken fibers. The deflection u(x) of a membrane of extremities x i and x f writes u(x) = (x -x i )(x f -x)/2 and the load transfers

f T (x i ) = f T (x f ) = (x f -x i -1)/2.
For a fiber surrounded by two intervals of m l and m r bonds, one thus recovers the very same load partition coefficient than in the 1D LLS fiber model given above (Eq.

(2). Figure 2a shows the classical 1D model load versus time t. We considered here a uniform distribution of fiber strength in [0; 1]. The critical load is reached at t = t * = 1256.Thus, the weakest link assumption is indeed far from being applicable. Figure 2b shows the redistribution function at the peak force. We see that the largest fiber load ϕ comes from an interval of 10 consecutive broken fibers.

Soft Membrane Fiber Bundle in Two and More Dimensions

The generalization of the 1D soft membrane model to two dimensions (2D) is straight forward. The system is a L × L square, so that the number of fibers is N = L 2 . A fiber location is designated either by a 2D vector x or by its integer coordinates (x, y). The question is now to define the redistribution function. The membrane deflection, u(x), obeys

u(x) = 0 Lap[u](x) = -1 if fiber at x is intact broken (5)
where Lap[u] is again a finite difference laplacian operator

Lap[u] = y n.n.x (u(y) -u(x)) (6)
The redistribution function on a surviving fiber in x is simply

ϕ(x, t) = (1/N ) 1 + y n.n.x u(y) (7)
i.e., precisely the same expression as in the 1D model. Figures 3,4 and 5 illustrate the load versus time, the displacement and local force per fiber at the critical state, and the failure time map at a late stage. It is apparent that, as expected, a critical defect is nucleated and finally grows with no possible arrest. This growth takes the form of a circular crack as seen in Fig. 4. Let us stress again that the fraction of broken fibers at the peak stress is a rather large proportion, t * /N ≈ 23%. Again the weakest link approach is inapplicable. Simultaneously, when considering the overall macroscopic force versus time, Fig. 3a, there are no fore-signs of nucleation of the critical defect.

This observation can be rationalized, in the spirit of homogenization, by considering a uniform equivalent threshold, f eq c . Considering a circular cluster of broken bonds, the load supported by the broken fibers πR 2 F is distributed over the edge, leading to a force per fiber of R(t)F/2 and hence F c (t) = 2f eq c /R(t). The radius is related to time as πR(t) 2 = t -t 0 . Thus

F c (t) = 2 √ πf eq c √ t -t 0 (8)
Figure 3b shows that this analysis holds after nucleation of the critical defect, as 1/F c (t) 2 has indeed a lower envelope obeying an affine law. The straight line fit allows to define an effective threshold equal to f eq c ≈ 0.34.

Measurement of Toughness

Having understood the late scenario of a crack propagation, one may study it directly: a specific crack pattern already present in the initial state can be designed. For instance, half of the domain (say 0 ≤ y < L/2) consists of broken fibers in the initial state. The problem is thus initially similar to the 1D case. The membrane deflexion field is u(x, y) = (1/2)y(L/2 -y) (9) and hence the force acting along the edges is the total load (1/2)L 2 F divided by the front length (on both edges) 2L or F L/4. It can also be computed as Lap[u]. In our case, for L = 200, the force acting on the front edge is 50 times higher than in the bulk. Thus most (but not all) of the activity is taking place at the front. Note, however, that as the front propagates as a whole, the overload at the front increases, and hence the diffuse damage regime is only taking place in an initial transient regime, which vanishes as L → ∞.

Periodic boundary conditions being implemented, one may stop one of the two fronts (y = L in our case) by inserting a line of fibers with very high strength. Figure 6 shows the displacement field and force acting on the fibers after some progression starting from the previous initial conditions. It is observed that a slightly roughened front progress from y = L/2 onward. Figure 7 shows the map of the time at which the fibers were broken. One can easily observe indeed the above mentioned front propagation, together with some "dust" which corresponds to the early damage of bonds being broken ahead of the front (in the present case, all bonds having a strength less than 2/L ∼ 0.01 are broken in a first stage). Note that the dynamics of the preexisting crack does not proceed by avalanches since we perform extremal dynamic simulations, i.e. each "time" step only one fiber is broken. Figure 8 shows the load versus time. One observes again clearly the first initial (diffuse) damage regime where fibers are not located at the front. The interesting regime is the front progression stage. In the continuum, if the fiber strength f ef f c were uniform, one would expect a uniform progression y = y 0 + t/L (where y 0 = L/2 in our case), so that the force exerted at the front would be (y 0 + t/L)F/2 and hence the critical load would be

F c (t) = 2f ef f c (y 0 + t/L) (10)
Hence one can define an (instantaneous) effective bond strength, f ef f c , from the critical force through F c (t)(y 0 +t/L)/2. This quantity is shown in Figure 8. The maximum value of the latter strength appears as a well-defined quantity which is the macroscopic effective strength. The histogram shown in Fig. 8 is another way of revealing the threshold. It allows us to estimate as compared to the elementary (mean field) approximation of f c = 0.5 that ignores the effect of load redistribution. It is noteworthy that this estimate is also slightly larger than the one estimated from the random nucleation case. This may be due to the fact that a somewhat larger amount of damage has been experienced in the plain 2D lattice at the onset of nucleation. Estimating this damage at this point as roughly D = 0.23 (D is the fraction of missing bonds), leads to an effective decrease of the effective strength (when averaged over the front) down to (1 -D)f ef f c ≈ 0.34 very consistent with the previous estimate. To be complete, one should also correct the above estimate by the damage level of the front type geometry, but because of the large system size, the damage is roughly D = 0.01, and does not affect our estimate.

f ef f c ≈ 0.46 (11)
The result in Eq. ( 11) is far from trivial. We may do the same same analysis in 1D where, in contrast, the result is trivial, f ef f c is the average of the threshold distribution. This is so since the fibers break sequentially as a zipper is opened.

Continuum limit: Depinning Model of Crack Propagation

We now consider the case of a penny-shape crack in the continuum limit. Accounting for the front roughness induced by the fiber strength disorder enables to rationalize the propagation stage of the discrete model. Let us consider a (rough) penny-shaped crack of equation

ρ = R(θ) = R 0 [1 + εh(θ)] (12)
in polar coordinates, where ε sets the amplitude of the radius fluctuations. The membrane deflection u(ρ, θ) obeys

∇ 2 u = -1 if ρ < R(θ) u = 0 if ρ = R(θ) (13)
At zeroth order, for ρ ≤ R 0 , the solution for a circular crack of radius R 0 writes

u 0 (ρ, θ) = 1 4 R 2 0 -ρ 2 = 1 4 R 2 0 -zz (14)
where we defined the complex variable z = ρe iθ and its conjugate z = ρe -iθ . Let us now write u = u 0 + εR 2 0 u 1 . Since ∇ 2 u 0 = -1 and u 0 (R 0 , θ) = 0, the first order contribution to the membrane deflection u 1 (ρ, θ) obeys

∇ 2 u 1 = 0 if ρ < R(θ) εR 2 u 1 (ρ, θ) = -u 0 (ρ, θ) if ρ = R(θ) (15)
The boundary condition can be rewritten explicitly as

εR 2 u 1 [R 0 (1 + εh(θ))] = -u 0 [R 0 (1 + εh(θ))] (16)
so that, at first order

u 1 (R(θ), θ) = 1 2 h(θ) . (17)
In order to evaluate u 1 inside the crack, ρ < R(θ), it is written as a Laurent series (polynomial expansion in z):

u 1 (z, z) = α 0 + N/2 n=1 α n z n + N/2 n=1 α n z n (18)
In complex variables the Laplacian operator is ∆ = 4∂ z ∂ z so that this expansion immediately satisfies the equation ∆u 1 = 0 (in other words, u 1 being the real part of an holomorphic function, it is harmonic). The restriction of this function on the circle ρ = R 0 can be identified with a Fourier series and gives an immediate solution to the boundary condition, Eq. ( 17)

u 1 (R 0 , θ) = α 0 + N/2 n=1 α n R 0 n e inθ + N/2 n=1 α n R 0 n e -inθ h(θ) = h 0 + N/2 n=1 h n e inθ + N/2 n=1 h n e -inθ (19)
The first order membrane deflection u 1 (ρ, θ) thus reads:

2u 1 (ρ, θ) = h 0 + N/2 n=1 ρ R 0 n h n e inθ + N/2 n=1 ρ R 0 n h n e -inθ (20)
The force ϕ = -∇u • n induced at the boundary can now be computed at first order

n(θ) = e ρ -εR 0 h ′ (θ) e θ ∇u(θ) = ∂u0 ∂ρ (R 0 ) + εR 0 h(θ) ∂ 2 u0 ∂ρ 2 (R 0 ) + εR 2 0 ∂u1 ∂ρ (R 0 , θ) e ρ + ε ∂u0 ∂ρ (R 0 )h ′ (θ) + εR 0 ∂u1 ∂θ (R 0 , θ) e θ ϕ [R 0 (1 + εh(θ))] = (R 0 /2) [1 + εh(θ) -εf 1 ({h} , R 0 , θ)] (21)
where e ρ and e θ are the unit vectors in polar coordinates and

f 1 ({h} , R 0 , θ) = N/2 n=-N/2 |n|h n e inθ (22)
This result is reminiscent of the effect of a rough geometry on the stress intensity factor of a mode I crack front as derived by Rice [30] in semi-infinite geometry, and more recently by Legrand et al [31] for a plate geometry:

K I [x, z = h 0 + εh(x)] = K 0 I 1 + ε 1 K 0 I ∂K I ∂z h(x) -εf (x) (23)
with

f (x) = A 1 N/2 n=-N/2 |n|h n e inx (24)
where A 1 = 1/2 in the semi infinite case and A 1 = 2 in the plate case. It is to be observed that the elastic interaction mediated by the membrane elasticity acts as a non-local line tension, stabilizing the crack front shape. This effect is quite generic, and was reported in the above references. It has also been derived by Zaiser et al [26] for the shear delamination problem. It is also encountered in wetting problems [32]. This line tension is consistent with the observation of the almost perfect circular shape of the crack observed in Fig. 4, or the straight front of Fig. 6. However, it is to be noted that the energy release rate increases as the radius of the penny shape crack grows. Thus the overall expansion of the crack is unstable (under a constant applied pressure onto the membrane), but its shape tends to remain circular due to the line tension effect. This can be read from the expression of the force redistribution in Fourier space:

ϕ n = R 0 2 [δ n + ε(1 -|n|)h n ] (25)
The fact that ϕ ±1 vanishes can be understood as the result of Galilean invariance as the first order mode in Fourier space corresponds to a mere translation of the crack which is expected not to have any effect on the crack driving force. Crack propagation is thus controlled by the interplay between local strength disorder and long-range interface tension. One recognizes here the ingredient of a depinning problem, a paradigm that has been used to discuss various problems of front propagation in physics [33] and has been successfully applied to crack propagation in recent years [34][35][36][38][39][40].

It appears in particular in this framework that the crack front roughness exhibits a scaling invariant character: when measured over a length scale ℓ the width w of the front scales as w(ℓ) ∝ ℓ ξ with an exponent ξ ≈ 0.4 [41]. Moreover the effective force f (ℓ) needed to depin a front of lateral extension ℓ follows a universal distribution of width δf (ℓ) ∝ ℓ - (1-ζ) and is bounded by a maximum value, the critical force f * [35][36][37]. The emergence of a critical threshold and of the associated size effects is potentially of great interest in the context of the discrete fiber bundle model developed in the present study. The critical value f * indeed sets an upper bound to the effective homogenized fiber strength defined in Eq. (8). Moreover, the knowledge of the finite size fluctuations of this effective strength for small defects should allow for the study of the very early stages of crack propagation, say in the spirit of the approach used in [36] for crack arrest.

Conclusion and Critical Perspective

We have formulated a soft membrane fiber bundle model that reduces to the Local Load Sharing fiber bundle model in 1D.

Failure proceeds in two steps: There is a first regime of diffuse damage which is followed by a second one corresponding to the nucleation and growth of a penny shaped crack. The latter can be characterized in terms of a mean toughness (obtained from a simple combination of the strength f ef f c and a length scale given by the fiber separation). This toughness can be measured by a direct crack propagation experiment. This toughness is also altered by the mean damage state of the material at the onset of brittle failure.

If much emphasis has been put on the damage regime and characterization of the critical defect, it may be worth investigating the late failure regime with classical tools. The above presented perturbation analysis exploiting the Poisson equation, ∇ 2 u = -1, relates this problem to a depinning crack propagation model formulated in the continuum with the same long range kernel relating the local energy release rate to the crack front morphology. This may lead to a precise estimate of the effective toughness, but its main virtue is to shed some light on the fluctuations. Indeed, this analogy leads to the prediction that the toughness fluctuation decays as a non-trivial power law of the front length. A challenge for the future is thus to derive a statistical law for the onset of crack nucleation in this soft membrane fiber bundle.

AH thanks P.C. Hemmer and S. Pradhan for countless discussions on this topic.

Fig. 2 .

 2 Fig. 2. (a) Load versus time in 1D (N = 10000 and periodic boundary conditions). The critical time is the one where the maximum force is reached, t * = 1256. (b) Local force distribution ϕ(x, t * ) at the critical load.

Fig. 3 .

 3 Fig. 3. (a) Load versus time for the Soft Membrane fiber bundle model in 2D. (b) Postcritical analysis of critical load versus time. The red line belongs to the circular crack analysis.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Membrane displacement u(x) in the 2D Soft Membrane fiber bundle at the critical load (left), and at a later stage (right). The center of the critical defect can be guessed from the late stage. The periodic boundary conditions are also visible.

Fig. 6 .

 6 Fig. 6. Displacement u(x, y) (left) and local force f (x, y) (right) after some progression.

Fig. 7 .

 7 Fig. 7. Map of failure time where the downwards front progression is easily observed.

Fig. 8 .

 8 Fig. 8. (a) Effective fiber strength f ef f c , deduced from the critical load by a multiplication of half the mean crack length. (b) histogram of the instantaneous critical load, where the asymptotic toughness is the upper bound of this distribution.