Leila Amgoud
email: leila.amgoud@irit.fr

Victor David
email: victor.david@irit.fr

A General Setting for Gradual Semantics Dealing with Similarity

The paper discusses theoretical foundations that describe principles and processes involved in defining semantics that deal with similarity between arguments. Such semantics compute the strength of an argument on the basis of the strengths of its attackers, similarities between those attackers, and an initial weight ascribed to the argument. We define a semantics by three functions: an adjustment function that updates the strengths of attackers on the basis of their similarities, an aggregation function that computes the strength of the group of attackers, and an influence function that evaluates the impact of the group on the argument's initial weight. We propose intuitive constraints for the three functions and key rationality principles for semantics, and show how the former lead to the satisfaction of the latter. Then, we propose a broad family of semantics whose instances satisfy the principles. Finally, we analyse the existing adjustment functions and show that they violate some properties, then we propose novel ones and use them for generalizing h-Categorizer.

Introduction

Argumentation is a useful approach for solving various theoretical problems [START_REF] Simari | Argumentation in Artificial Intelligence[END_REF] and practical ones [START_REF] Atkinson | Towards Artificial Argumentation[END_REF]. It aims at increasing or decreasing acceptability of claims by supporting them with arguments. Roughly speaking, an argument is a set of premises (or statements) intended to establish a definite claim. Its strength depends on the plausibility of the premises, the nature of the link between the premises and the claim, and the prior acceptability of the claim. Generally, an argument may be weakened by other arguments that undermine one or more of its three components. Thus, evaluation of argument strength is a crucial task, and a sizeable amount of methods, called semantics, has been proposed in the literature. The very first ones are extension-based [START_REF] Dung | On the Acceptability of Arguments and its Fundamental Role in Non-Monotonic Reasoning, Logic Programming and n-Person Games[END_REF] and the recent ones are gradual semantics [START_REF] Cayrol | Graduality in Argumentation[END_REF] that quantify strength and thus ascribe a value (representing strength) to every argument.

Under gradual semantics, the strength of an argument depends on the strengths of the argument's attackers. Therefore, it may be that each attacker has an (negative) impact on the argument and contributes to decrease it strength. This property, called Counting (or strict monotony) in [START_REF] Amgoud | Acceptability Semantics for Weighted Argumentation Frameworks[END_REF], is satisfied by some existing gradual semantics like h-Categorizer [START_REF] Besnard | A logic-based theory of deductive arguments[END_REF] and its variants that are discussed in [START_REF] Amgoud | Gradual Semantics Accounting for Varied-Strength Attacks[END_REF]. Let us illustrate the property with the following example. Example 1. Consider the three graphs G 1 , G 2 , and G 3 of the Figure 1. According to those semantics, A 1 is strictly weaker than A 2 due to B 2 which further weakens A 1 . It is also strictly weaker than A 3 due to B 1 which decreases further the strength of A 1 .

As defined in the literature, the Counting property seems reasonable when the attackers of an argument are all different, i.e, not redundant. However, it may lead to inaccurate evaluations when similarities exist between them. Consider again the four graphs of the Figure 1. Example 1 (Cont) Assume that each argument B i (i = 1, . . . , 4) supports the same claim "lowering taxes" with a premise P i , where:

P 1 Better living standards for all. P 2 Improving quality of life. P 3 Better healthcare and social justice. P 4 Better working standards for all.

The two arguments B 1 and B 2 are identical or totally similar since they support the same claim with the same evidence. One of them is thus redundant, and considering both in the evaluation of A 1 is questionable. A reasonable semantics would declare A 1 , A 2 and A 3 as equally strong.

Consider now B 1 and B 3 . They are partially similar since each of them brings a new piece of evidence (eg., entertainment for B 1 , and social justice for B 3 ) in addition to the common one (better healthcare which is part of better living standards). Finally, B 4 is based on a completely different evidence, making it dissimilar to the three others. Hence, one would expect to declare A 2 (resp. A 1 , A 3 ) as stronger than A 4 since the group {B 1 , B 3 } of A 2 's attackers is weaker than the group {B 1 , B 4 } of A 4 's ones. Indeed, the former contains some redundancy which should be removed, while the latter does not (B 1 and B 4 being different).

To sum up, ignoring (total or partial) similarities would lead to inaccurate evaluations of arguments, and thus to wrong recommendations by argumentation systems. Therefore, a semantics should be able to take them into account.

B1 B2 B3 A1 (G1) B1 B3 A2 (G2) B2 B3 A3 (G3) B1 B4 A4 (G4) Figure 1: w ≡ 1, s(B 1 , B 2 ) = 1, s(B 1 , B 3 ) = s(B 2 , B 3 ) = α with 0 < α < 1, s(B 4 , B i ) = 0 for any i = 1, . . . , 3.
Two such gradual semantics, each of which extends h-Categorizer [START_REF] Besnard | A logic-based theory of deductive arguments[END_REF], have been proposed in (Amgoud et al. 2018). They differ in the way they modify the strengths of attackers on the basis of their similarities. While those semantics seem reasonable, the followed approach for defining them is not systematic as the general rules guiding the definition of a semantics in general, and the way of dealing with similarity in particular, have not been discussed. The authors proposed some properties for bridging that gap, but it turns out that both semantics violate some of them. Moreover, those properties are not sufficient for comparing the two semantics. Hence, the approach lacks theoretical foundations that describe principles and processes involved in the definition of semantics that deal with similarity.

This paper proposes such theoretical foundations. Rather than focus narrowly on a particular semantics, we propose a general setting for defining systematically semantics that consider similarities. The contributions are six-fold:

1. Clarify the process of defining semantics using three functions (adjustment, aggregation, influence). Adjustment function is responsible for modifying the strengths of attackers based on their similarities if any.

2. Identify rules for handling similarities, i.e, key properties of an adjustment function.

3. Propose principles that a gradual semantics dealing with similarity would satisfy.

4. Provide a broad family of semantics that satisfy them.

5. Analyse the existing adjustment functions and show that they violate some of the proposed properties.

6. Propose novel adjustment functions that satisfy the desirable properties, extend the h-Categorizer semantics with the latter, and show that the new semantics are instances of the novel family.

The paper is organised as follows: It starts by presenting an abstract definition of gradual semantics dealing with similarity. Then, it proposes properties that semantics should satisfy. Next, it introduces a family of semantics that satisfy the properties. Then, it discusses the properties of existing adjustment functions, proposes novel ones and extends h-Categorizer. The two last sections are devoted respectively to related work and concluding remarks.

Similarity-based Gradual Semantics

Throughout the paper, we denote by U the universe of all possible arguments, and consider an argumentation framework as a tuple made of a non-empty and finite subset of U, every argument has an initial weight that may represent different information (certainty degree of the argument's premises [START_REF] Benferhat | Argumentative inference in uncertain and inconsistent knowledge bases[END_REF], credibility degree of its source (da Costa Pereira, Tettamanzi, and Villata 2011), etc). For the sake of simplicity, those weights are elements of the unit interval [0,1]. The greater the value, the better it is for the argument. Arguments may attack each other and may be more or less similar to others. Therefore, we assume the availability of a measure that assesses the degree of similarity between any pair of arguments. Examples of such measures can be found in [START_REF] Budan | Similarity notions in bipolar abstract argumentation[END_REF][START_REF] Amgoud | Measuring Similarity between Logical Arguments[END_REF].

Definition 1. A similarity measure on a set X ⊆ f U 1 is a function s : X × X → [0, 1] such that: • ∀a ∈ X, s(a, a) = 1, • ∀a, b ∈ X, s(a, b) = s(b, a). • ∀a, b, c ∈ X, if s(a, b) = 1, then s(a, c) = s(b, c).
A value s(a, b) represents the degree of similarity between a and b according to the measure s. The values 1 and 0 respectively denote total similarity and total difference (or dissimilarity). The three conditions above respectively state that every argument is identical (totally similar) to itself, similarity is a symmetric notion, and two identical arguments are equally similar to any third argument.

Definition 2. An argumentation framework (AF) is a tuple A, w, R, s , where A ⊆ f U, w : A → [0, 1], R ⊆ A × A, and s is a similarity measure on A. Regarding evaluation of arguments, we extend the gradual semantics that were defined in [START_REF] Amgoud | Gradual Semantics Accounting for Varied-Strength Attacks[END_REF] for AFs having s ≡ 0 and in [START_REF] Cayrol | Graduality in Argumentation[END_REF] for simple AFs with w ≡ 1 and s ≡ 0.

Roughly speaking, a gradual semantics dealing with similarity proceeds in a recursive way. For any argument a, if a is not attacked, then its strength is exactly the weight w(a). Assume now that a is attacked by a 1 , . . . , a k . The semantics starts by evaluating the strength of every attacker a i , i = 1, . . . , k. Let x 1 , . . . , x k be numerical values representing those strengths. For computing the strength of a, the semantics follows a three steps process:

1. It adjusts the values x 1 , . . . , x k according to the similarities between a i , a j where i, j = 1, . . . , k. The goal is to remove redundancy among the attackers, thus the semantics weakens the attackers. Let x 1 , . . . , x k denote the adjusted values, x i being the new strength of a i .

2. It computes the strength of the group {a 1 , . . . , a k } by aggregating the values x 1 , . . . , x k .

3. It adjusts the initial weight w(a) on the basis of the strength of the group of attackers.

Example 1 (Cont) Consider the argumentation graph G 1 of the Figure 1, where

w ≡ 1, Att(A 1 ) = {B 1 , B 2 , B 3 }, Att(B i ) = ∅ for i ∈ {1, 2, 3}, s(B 1 , B 2 ) = 1, and s(B 1 , B 3 ) = α with 0 < α < 1.
A reasonable gradual semantics would assign strength 1 to each B i since it is not attacked. For A 1 , the semantics would start with the tuple (1, 1, 1), the strengths of B 1 , B 2 , B 3 , and adjusts them. Since s(B 1 , B 2 ) = 1, the semantics would for example decide to keep only one of them, say B 1 . Hence, it adjusts the score of B 2 from 1 to 0. Regarding B 3 , it keeps only its novel part compared to B 1 hence 1 -α. The adjusted values are thus (1, 0, 1 -α). The semantics computes then the strength of the group {B 1 , B 2 , B 3 } using, for instance, the sum aggregation operator and returns the value 2 -α. Finally, it evaluates the impact of the group on the initial weight of A 1 using for instance the function

f f rac (x 1 , x 2 ) = x1 1+x2 , hence the strength of A 1 = w(A1)
1+2-α = 1 3-α . Note that if the semantics ignores the similarities, it assigns the score 1 4 to A 1 and thus A 1 would be much weaker.

Each step of the process described above can be done in different ways. For instance, a semantics may adjust differently the strengths of B 1 , B 2 by weakening both arguments, may aggregate attackers differently, or may use another function than f f rac . In what follows, we define a gradual semantics in an abstract way using a tuple of three functions, called evaluation method. Definition 3. An evaluation method (EM) is a tuple M = f , g, n such that:

• f : [0, 1] × Range(g) → [0, 1],
where Range(g) denotes the co-domain of g

• g : +∞ k=0 [0, 1] k → [0, +∞[ • n : +∞ k=0 [0, 1] k × U k → [0, 1] k
Given the set of attackers of a given argument a in an AF, the function n adjusts the strength of each attacker based on its similarities with the other attackers of a, g computes the strength of the group of attackers, and f evaluates how the latter influences the initial weight of a. Note that the domains of g and n are unions because the number of attackers may vary from one argument to another. Note also that n takes as input two kinds of information: k numerical values and k arguments. Let us illustrate the need of the set of arguments. Consider the two arguments A 3 , A 4 in Figure 1. Recall that s(B 2 , B 3 ) = α > 0 and s(B 1 , B 4 ) = 0. Since each B i is not attacked, then its strength is 1. However, the function n would not alter the values of B 1 and B 4 since the latter are dissimilar, i.e., n(1, 1, B 1 , B 4 ) = (1, 1) while it modifies those of B 2 , B 3 , i.e., n(1, 1, B 2 , B 3 ) = (x, y) as there is some redundancy between the two arguments. This means that the same values (here (1, 1)) may be adjusted in different ways according to the arguments they refer to.

We propose below key properties that should be satisfied by each of the three functions f , g, n of an evaluation method. Those properties constrain the range of functions to be considered, and discard those that may exhibit irrational behaviours. Let us first recall the definition of a symmetric function. A function t is symmetric iff

t(x 1 , . . . , x k ) = t(x ρ(1) , . . . , x ρ(k) )
for any permutation ρ of the set {1, . . . , k}. Definition 4. An EM M = f , g, n is well-behaved iff the following conditions hold:

1. (a) f is increasing in the first variable, decreasing in the second one if the first variable is not equal to 0,

(b) f (x, 0) = x, (c) f (0, x) = 0, 2. (a) g() = 0, (b) g(x) = x, (c) g(x 1 , • • • , x k ) = g(x 1 , • • • , x k , 0), (d) g(x 1 , • • • , x k , y) ≤ g(x 1 , • • • , x k , z) if y ≤ z, (e) g is symmetric, 3. (a) n() = (), (b) n(x, a) = (x), (c) g(n(x 1 , • • • , x k , a 1 , • • • , a k )) ≤ g(n(x 1 , • • • , x k , b 1 , • • • , b k )) if ∀i, j ∈ {1, • • • , k} i = j, s(a i , a j ) ≥ s(b i , b j ), (d) If ∃i ∈ {1, • • • , k} s.t. x i > 0 then g(n(x 1 , • • • , x k , a 1 , • • • , a k )) > 0, (e) g(n(x 1 , • • • , x k , a 1 , • • • , a k )) ≤ g(n(y 1 , • • • , y k , a 1 , • • • , a k )) if ∀i ∈ {1, • • • , k}, x i ≤ y i , (f) n is symmetric, (g) n(x 1 , • • • , x k+1 , a 1 , • • • , a k+1 ) = (n(x 1 , • • • , x k , a 1 , • • • , a k ), x k+1 ) if ∀i ∈ {1, . . . , k}, s(a i , a k+1 ) = 0.
We say also that f , g, n are well-behaved.

Note that the functions f , g, n are defined without referring to any AF. The idea is to describe their general behaviour. The conditions (2c) and (2d) respectively state that attackers of strength 0 have no impact on their targets, and g is monotonic in that the greater the individual values, the greater their aggregation. The conditions (3a, . . ., 3g) represent the core principles for dealing with similarities. Namely, (3b) states that if a group of attackers contains only one element, then the adjusted value of the latter is equal to the initial one. (3c) states that the greater the similarity between arguments of a set, the weaker the set, and (3d) ensures that similarities do not inhibit the attack of a group of arguments. Condition (3e) states that the stronger the individual attackers, the stronger the group. ( 3g) is an independence condition. It states that an argument which is dissimilar to all elements of a group, has no effect on the adjustment of the values of those elements. Furthermore, the argument keeps its initial value.

From the condition (3c), it follows that similarities lead to a decrease in the strength of a group of attackers. Proposition 1. If g and n are well-behaved, then for all

x 1 , • • • , x k ∈ [0, 1], for all a 1 , • • • , a k ∈ U, it holds that: g(x 1 , • • • , x k ) ≥ g(n(x 1 , • • • , x k , a 1 , • • • , a k )).
From the condition (3g), it follows that if the arguments of a set are independent (their similarities are all equal to 0), then their initial values remain unchanged by n.

Proposition 2. Let x 1 , • • • , x k ∈ [0, 1] and a 1 , • • • , a k ∈ U such that for all i, j ∈ {1, . . . , k}, with i = j, s(a i , a j ) = 0. If n is well-behaved, then n(x 1 , • • • , x k , a 1 , • • • , a k ) = (x 1 , • • • , x k ).
Let us now define formally a gradual semantics that deals with similarity. It is based on an evaluation method and assigns a single value, called strength, to every argument. Definition 5. A gradual semantics S based on an evaluation method M = f , g, n is a function assigning to every AF A, w, R, s a weighting Str S : A → [0, 1] such that for every a ∈ A,

Str S (a) = f w(a), g n Str S (b 1 ), • • • , Str S (b k ), b 1 , • • • , b k , where {b 1 , • • • , b k } = Att(a). Str S (a) is the strength of a.
The above definition shows that evaluating arguments in an AF amounts to solving a system of equations, one equation per argument. The question of existence of solutions for such systems arises naturally. Note that existence of solutions means also existence of a semantics. The following result shows that if the three functions of an EM are continuous, then a solution exists for every AF. Theorem 1. If M = f , g, n is an evaluation method such that f is continuous on the second variable, g is continuous on each variable, and n is continuous on each numerical variable, then there exists a semantics S based on M.

The following result goes further by showing that a system of equations has a single solution for every AF. This is particularly the case when the evaluation method is wellbehaved and satisfies some additional constraints. This result shows there is only one semantics that is based on the EM. Theorem 2. Let M be the set of all well-behaved evaluation methods M = f , g, n such that:

• lim x2→x0 f (x 1 , x 2 ) = f (x 1 , x 0 ), ∀x 0 = 0. • lim x→x0 g(x 1 , • • • , x k , x) = g(x 1 , • • • , x k , x 0 ), ∀x 0 = 0.
• n is continuous on each numerical variable.

• λf (x 1 , λx 2 ) < f (x 1 , x 2 ), ∀λ ∈ [0, 1[, x 1 = 0. • g(n(λx 1 , • • • , λx k , b 1 , • • • , b k )) ≥ λg(n(x 1 , • • • , x k , b 1 , • • • , b k )), ∀λ ∈ [0, 1].
For any M ∈ M, for all gradual semantics S, S , if S, S are based on M, then S ≡ S .

Properties of Semantics

So far we have presented a three-step process for defining semantics; at each step a function that obeys to specific conditions is used. We have seen that none of the three (adjustment, aggregation, influence) functions refers to argumentation frameworks, making their impact on argument strength in particular and on the behaviour of gradual semantics in general not clear. This section bridges the gap by proposing principles that gradual semantics should satisfy, and relating them to the various conditions of evaluation methods.

Principles are useful properties for understanding underpinnings of semantics. They have recently generated a lot of effort (eg. [START_REF] Bonzon | A Comparative Study of Ranking-Based Semantics for Abstract Argumentation[END_REF][START_REF] Amgoud | Acceptability Semantics for Weighted Argumentation Frameworks[END_REF]; Amgoud and Ben-Naim 2018; Mossakowski and Neuhaus 2018; [START_REF] Baroni | From fine-grained properties to broad principles for gradual argumentation: A principled spectrum[END_REF]). Due to space limitation, we do not recall all the existing principles, we only focus on those that are impacted by similarity, namely Reinforcement, Monotony and Neutrality from [START_REF] Amgoud | Acceptability Semantics for Weighted Argumentation Frameworks[END_REF]. We extend each of these three principles and propose a novel one, Sensitivity to Similarity. Note that other principles from the literature, like Anonymity, Directionality, Maximality, Proportionality are also suitable in settings where s ≡ 0.

The Reinforcement principle concerns strengths of attackers. It states that the stronger an attacker, the greater its impact on the strength of the argument it is attacking. The original definition does not take into consideration similarities among attackers, and hence may lead to counter-intuitive results in presence of redundancies. Assume for instance that the attacker that is strengthened is redundant with another, in this case it should be ignored by a semantics. Principle 1 (Reinforcement). A semantics S satisfies reinforcement iff for any AF A, w, R, s , for all a, b ∈ A, if Sensitivity to similarity states that the greater the similarities between attackers of an argument, the stronger the argument. Recall that similarities means existence of redundancies, and the latter should be removed by semantics. Let us show how the above principles relate to the different conditions of evaluation methods. The first result states that any semantics that is based on a well-behaved evaluation method satisfies the non-strict versions of the principles. Theorem 3. Let S be a gradual semantics based on an EM M. If M is well-behaved, then S satisfies reinforcement, monotony, neutrality and sensitivity to similarity.

• w(a) = w(b), • Att(a) \ Att(b) = {x}, Att(b) \ Att(a) = {y}, • ∀z ∈ Att(a) ∩ Att(b), s(x, z) = s(y, z), • Str S (x) ≤ Str S (
In order to guarantee the strict version of Reinforcement, the evaluation method of a semantics should not only be well-behaved but also satisfy the condition below, which is a strict version of the constraint (3e) in Definition 4.

g(n(x1, • • • , x k , a1, • • • , a k )) < g(n(y1, • • • , y k , a1, • • • , a k )) if ∀i = 1, • • • , k, xi ≤ yi and ∃i = 1, • • • , k s.t. xi < yi. (C1)
Theorem 4. Let S be a gradual semantics based on an EM M. If M is well-behaved and satisfies (C1), then S satisfies strict reinforcement.

Strict sensitivity to similarity is satisfied by a semantics when its EM is well-behaved and enjoys the property (C2).

g(n(x1, • • • , x k , a1, • • • , a k )) < g(n(x1, • • • , x k , b1, • • • , b k )) if ∀i, j = 1, • • • , k i = j, s(ai, aj) ≥ s(bi, bj) and ∃i, j = {1, • • • , k} s.t. s(ai, aj) > s(bi, bj) and (xi > 0 or xj > 0). (C2)
Theorem 5. Let S be a gradual semantics based on an EM M. If M is well-behaved and satisfies (C2), then S is strictly sensitive to similarity.

Strict monotony is satisfied by a semantics when its EM is well-behaved and enjoys the property (C3) below.

g(x1, • • • , x k , y) < g(x1, • • • , x k , z)
if y < z (C3) Theorem 6. Let S be a gradual semantics based on an evaluation method M. If M is well-behaved and satisfies (C3), then S satisfies strict monotony. Remark: It is worth mentioning that the conditions (C1), (C2) and (C3) are not part of Def. 4 since they are more demanding than their large versions. In the same way, the large versions of the principles are mandatory while the strict ones are optional and their suitability depends on the application and the type of arguments (deductive, analogical, etc).

Novel Family of Semantics

We now introduce a broad family of gradual semantics that are able to deal with similarity between arguments. Its members use evaluation methods from the set M (see Theorem 2). Recall that every EM in this set is well-behaved and satisfies some additional properties, which guarantee that the EM characterizes a single gradual semantics. Definition 6. We define by S the set of all semantics that are based on an evaluation method from M.

From Theorem 3, it follows that any member of S satisfies all the large versions of the principles. Theorem 7. Any gradual semantics S ∈ S satisfies Reinforcement, monotony, neutrality and sensitivity to similarity.

Obviously, if the evaluation method of a semantics S ∈ S satisfies in addition the three constraints (C1), (C2) and (C3), then the semantics would satisfy the strict versions of reinforcement, sensitivity to similarity and monotony. In a next section, we show that the set S is not empty and we discuss some of its instances.

Adjustment Functions

This section presents examples of adjustment functions. Their core idea is that a modified value would represent the novelty brought by an attacker to the group of attackers. This amounts at computing approximately the similarity of the attacker with the group by aggregating its similarity with every argument of the group. A second central concern when dealing with similarity is how to distribute the redundancy burden among similar arguments. Consider the case of a group of two attackers A, B such that s(A, B) = 1, the strength of A is equal to 1 and the strength of B is 0.6. The question is: where should a function n remove redundancy? There are three possible strategies:

• Conjunctive: n removes the redundancy from the weakest argument B.

• Disjunctive: n removes the redundancy from the strongest argument A.

• Compensative: n distributes the burden to both.

We present three (families of) adjustment functions, one per strategy. The first function was proposed in (Amgoud et al. 2018). It is based on the average operator and follows a compensative strategy. Definition 7 (n rs ). Let a 1 , . . . , a k ∈ U and

x 1 , . . . , x k ∈ [0, 1]. n rs (x 1 , . . . , x k , a 1 , . . . , a k ) = avg xi∈{x1,...,x k }\{x1} avg(x 1 , x i ) × (2 -s(a 1 , a i )) 2 , . . . , avg xi∈{x1,...,x k }\{x k } avg(x k , x i ) × (2 -s(a k , a i )) 2 . n rs () = () and n rs (x 1 , a 1 ) = (x 1 ) if k = 1.
Example 1 (Cont) Using the function n rs , we get:

n rs (1, 1, 1, B 1 , B 2 , B 3 ) = (avg( 1×1 2 , 1×(2-α) 2 ), avg( 1×1 2 , 1×(2-α) 2 ), avg( 1×(2-α) 2 , 1×(2-α) 2 )) = ( 3-α 4 , 3-α 4 , 2-α 2 )
. For α = 0.5, we get (0.625, 0.625,0.75). Note that the function weakens both B 1 and B 2 , which are identical (s(B 1 , B 2 ) = 1).

We show that the adjustment function n rs satisfies almost all the constraints from Def. 4 except (3g). This means that n rs modifies the values of attackers even when they are all dissimilar. Therefore, n rs is not well-behaved. Proposition 3. The following properties hold.

• n rs violates the condition (3g) of Def. 4.

• n rs satisfies the conditions (3a), . . ., (3f) of Def. 4.

• n rs is not well-behaved.

In what follows, we propose novel functions that compute the degree of similarity of an argument with a set of arguments by aggregating the pairwise similarities using the max operator. They start by rank ordering the initial scores of arguments using a fixed permutation. The new score of an argument is equal to its old value times its novelty with respect to the preceding arguments in the permutation. Definition 8

(Parameterised Function n ρ max ). Let a 1 , . . . , a k ∈ U, x 1 , • • • , x k ∈ [0, 1],
and ρ a fixed permutation on the set {1, . . . , k} such that if x ρ(i) = 0 then x ρ(i+1) = 0 ∀i < k, or i = k. n ρ max () = (), otherwise:

n ρ max (x 1 , . . . , x k , a 1 , . . . , a k ) = x ρ(1) , x ρ(2) • (1 -max(s(a ρ(1) , a ρ(2) ))), • • • , x ρ(k) •(1-max(s(a ρ(1) , a ρ(k) ), • • • , s(a ρ(k-1) , a ρ(k) ))) .
The following result shows that the parametrised functions n ρ max return values from the unit interval. Proposition 4. For all a 1 , . . . , a k ∈ U, for all x 1 , • • • , x k ∈ [0, 1], for any permutation ρ on the set {1, . . . , k}, n ρ max (x 1 , . . . , x k , a 1 , . . . , a k ) ∈ [0, 1] k . Example 1 (Cont) Consider the graph G 1 in the Figure 1. Recall that s(B 1 , B 2 ) = 1, s(B 1 , B 3 ) = s(B 2 , B 3 ) = α, 0 < α < 1, and Att(A 1 ) = {B 1 , B 2 , B 3 }. For any reasonable semantics S, Str S (B 1 ) = Str S (B 2 ) = Str S (B 3 ) = 1 since they are not attacked. Let x i = Str S (B i ). We illustrate n ρ max using two permutations. ρ min ranks arguments from the weakest argument with maximal similarity to the strongest but less similar to other attackers. ρ max ranks arguments from the strongest with minimal similarity to the weakest with more similarity. ρ min follows thus a conjunctive strategy while ρ max a disjunctive one. Hence, ρ min (x 1 , x 2 , x 3 ) = (x 1 , x 2 , x 3 ) (since B 1 , B 2 are the most similar arguments) and n ρmin max (x 1 , x 2 , x 3 , B 1 , B 2 , B 3 ) = (1, 0, 1 -α). And, ρ max (x 1 , x 2 , x 3 ) = (x 3 , x 1 , x 2 ) (as B 3 is less similar to the others) and n ρmax max (x 1 , x 2 , x 3 , B 1 , B 2 , B 3 ) = (1 -α, 0, 1). We show next that the functions n ρ max satisfy the conditions (3a,. . .,3g) of Definition 4 and those of Theorem 2. However, they violate the conditions (C1) and (C2) because the max operator considers only the greatest similarity. Hence, increasing small similarity degrees would not impact the result of n ρ max . Proposition 5. Let f , g be well-behaved functions and g satisfies the following property:

let λ ∈ [0, 1], x 1 , • • • , x k ∈ [0, 1], then g(λx 1 , • • • , λx k ) ≥ λg(x 1 , • • • , x k ).
The following properties hold:

• n ρ max is well-behaved. • n ρ max is continuous on numerical variables. • g(n ρ max (λx 1 , • • • , λx k , b 1 , • • • , b k )) ≥ λg(n ρ max (x 1 , • • • , x k , b 1 , • • • , b k )), ∀λ ∈ [0, 1]. • n ρ
max violate the conditions (C1) and (C2). From above, it follows that the functions n ρ max are used by evaluation methods of the set M, and thus by the novel family of semantics. Proposition 6. For all functions f , g that are well-behaved and satisfy the conditions of Theorem 2, it holds that f , g, n ρ max ∈ M.

Instances of Semantics

We present instances of the broad family S that extend h-Categorizer [START_REF] Besnard | A logic-based theory of deductive arguments[END_REF]. They use the wellbehaved functions f frac and g sum defined below and the previously defined adjustment functions n ρ max .

f frac (x 1 , x 2 ) = x 1 1 + x 2 g sum (x 1 , . . . , x k ) = k i=1 x i
Definition 9. Semantics S n based on the EM f frac , g sum , n is a function transforming any AF A, w, R, s into a function Str n from A to [0, 1] s.t for any a ∈ A, Str n (a) = w(a)

1 + k i=1 n Str n (b 1 ), • • • , Str n (b k ), b 1 , • • • , b k where Att(a) = {b 1 , • • • , b k }. If Att(a) = ∅, then k i=1 n Str n (b 1 ), • • • , Str n (b k ), b 1 , • • • , b k = 0.
Example 1 (Cont) Let us consider the adjustment functions n ρ max and n rs .

Str n ρ min max (A 1 ) = 1 1 + 1 + 0 + 1 -α = 1 3 -α Str n ρmax max (A 1 ) = 1 1 + 1 + 1 -α + 0 = 1 3 -α Str nrs (A 1 ) = 1 1 + 3-α 4 + 3-α 4 + 2-α 2 = 1 3.5 -α
Note that S n ρ max covers a range of semantics using different permutations. We show that those semantics are all instances of S. They thus satisfy all the (large versions of the) principles. In addition, they satisfy strict monotony, but violate the strict versions of Reinforcement and sensitivity to similarity due to the max operator. Theorem 8. For any ρ, it holds that S n ρ max ∈ S. Furthermore, S n ρ max satisfies reinforcement, (strict) monotony, neutrality and sensitivity.

The semantics S nrs satisfies all the principles except Neutrality. Note that this semantics extends h-Categorizer which satisfies Neutrality in settings where s ≡ 0. Theorem 9. The semantics S nrs satisfies all the principles except Neutrality. Furthermore, S nrs / ∈ S.

When the arguments are all distinct (i.e., similarities are equal to 0), the above semantics assign the same values to all arguments, and coincide with the weighted h-categorizer semantics that assigns to every argument a ∈ A,

Str h (a) = w(a) 1 + bi∈Att(a) Str h (b i ) (1)
Theorem 10. For any AF A, w, R, s , if s ≡ 0, then

Str max ρ ≡ Str rs ≡ Str h .
This shows that these semantics extend weighted h-Categorizer by considering similarity degrees of attackers.

Related Work

In the computational argumentation literature, there are some works on defining measures that assess to what extent two arguments are alike. [START_REF] Amgoud | Measuring Similarity between Logical Arguments[END_REF] proposed several measures for logical arguments and [START_REF] Misra | Measuring the Similarity of Sentential Arguments in Dialogue[END_REF][START_REF] Stein | Report of Dagstuhl Seminar Debating Technologies[END_REF][START_REF] Konat | Rephrase in Argument Structure[END_REF] investigated similarity between textual ones. In our work, a similarity measure is given as input, and we focus on how to integrate it in semantics.

To the best of our knowledge, there are three works that tackled the question of dealing with available similarities. The first one, [START_REF] Amgoud | Equivalence in logic-based argumentation[END_REF], removes redundancies at the level of arguments generation. It keeps only one argument among totally similar ones. This approach does not deal with partial similarity. The second work, [START_REF] Budan | Introducing analogy in abstract argumentation[END_REF][START_REF] Budan | Similarity notions in bipolar abstract argumentation[END_REF], uses similarity at the level of attacks. It forbids attacks between quite similar arguments. The third work, (Amgoud et al. 2018), integrates similarity at the level of semantics. However, the approach is not systematics. The authors proposed particular semantics rather than focusing on foundations (rules and processes involved in the definition) of semantics. The design choices are thus not clear. One of the three semantics assume similarity between an argument and a set of arguments, which is beyond the scope of our paper. We focused on similarity measures between pairs of arguments. Their two other semantics consider pairwise similarities. We have seen that one of them is not well-behaved as it may alter the values of arguments even when they are dissimilar. The third semantics is quite complex as its adjustment function cannot be isolated from the aggregation function. In [START_REF] Amgoud | An Adjustment Function for Dealing with Similarities[END_REF], the authors proposed an adjustment function and studied its properties.

Some principles for semantics were also proposed in (Amgoud et al. 2018). However, they were tailored for specific strategies of adjustment, namely disjunctive ones. This explains why they are violated by two of the semantics defined in that paper. Our principles are more general as they are compatible with any strategy followed by a function n.

Conclusion

The paper investigated theoretical foundations of gradual semantics that deal with similarity between arguments. It analysed the principles underlying the management of similarity and formalised the process of defining semantics in terms of three functions, one of which is responsible for refining the strengths of attackers in light of similarities between them. The paper proposed also a broad family of semantics, which can be instantiated in several ways. Its instances satisfy some key properties witnessing thus their well-behaviour. The paper discussed also three strategies for dealing with similarities, and proposed concrete semantics per strategy.

This work can be extended in different directions. One of them consists of studying fair adjustment functions, which remove the exact amount of redundancy. We will also define adjustment function that allow satisfaction of the strict versions of the proposed principles. Finally, we plan to apply the new semantics to some applications like analogical reasoning, and evaluation of arguments in debate platforms.

For

  a, b ∈ A, w(a) denotes the initial weight of a, the notation (b, a) ∈ R means b attacks a (b being called attacker of a), and Att(a) denotes the set of all attackers of a in an AF. The notations w ≡ 1 and s ≡ 0 respectively mean that all arguments have weight 1 and they are all dissimilar.

  y), then the following properties hold: • Str S (a) ≥ Str S (b). (Reinforcement) • If Str S (a) > 0 and Str S (x) < Str S (y), then Str S (a) > Str S (b). (Strict Reinforcement) The Monotony principle concerns the quantity of attackers. Its original definition states "the more an argument has attackers, the weaker it is". Hence, an argument A that is attacked by B and C is weaker than if it is only attacked by B. This result is inaccurate when B and C are redundant. A should have the same strength in both cases since one of the attackers should be ignored. The new version of Monotony avoids such inaccurate evaluations and states "the more an argument has dissimilar attackers, the weaker it is". Principle 2 (Monotony). A semantics S satisfies monotony iff for any AF A, w, R, s , for all a, b ∈ A, if • w(a) = w(b), • Att(a) ⊆ Att(b), • If Att(a) = ∅, then ∀x ∈ Att(b) \ Att(a), ∀y ∈ Att(a), s(x, y) = 0, then the following properties hold: • Str S (a) ≥ Str S (b). (Monotony) • If Str S (a) > 0 and ∃x ∈ Att(b) \ Att(a) such that Str S (x) > 0, then Str S (a) > Str S (b). (Strict Monotony) Neutrality states that attackers having strength equal to 0 have no impact on their targets. The new version of the principle ensures that those lifeless attackers are dissimilar to the other attackers. Principle 3 (Neutrality). A semantics S satisfies neutrality iff for any AF A, w, R, s , for all a, b ∈ A, if • w(a) = w(b), • Att(b) = Att(a) ∪ {x} with Str S (x) = 0, • If Att(a) = ∅, then ∀y ∈ Att(a), s(x, y) = 0, then Str S (a) = Str S (b).

  Principle 4 (Sensitivity to Similarity). A semantics S is sensitive to similarity iff for any AF A, w, R, s , for all a, b ∈ A such that w(a) = w(b), if there exists a bijective function f : Att(a) → Att(b) such that: • ∀x ∈ Att(a), Str S (x) = Str S (f (x)), • ∀x, y ∈ Att(a), s(x, y) ≥ s(f (x), f (y)), then the following properties hold: • Str S (a) ≥ Str S (b). (Sensitivity) • If Str S (a) > 0 and ∃x, y ∈ Att(a) such that (Str S (x) > 0 or Str S (y) > 0) and s(x, y) > s(f (x), f (y)), then, Str S (a) > Str S (b). (Strict Sensitivity)

X ⊆ f U means X is a finite subset of U.

Acknowledgements

Support from the ANR-3IA Artificial and Natural Intelligence Toulouse Institute is gratefully acknowledged.


Jean-Benoît Kopp
email: jean-benoit.kopp@ensam.eu

Jeremie Girardot
email: jeremie.girardot@ensam.eu

J.-B Kopp

Dynamic fracture in a semicrystalline biobased polymer: an analysis of the fracture surface

Keywords: Dynamic fracture, Rapid crack propagation, Fracture surface analysis, Semi-crystalline, Polymer

The fracture behaviour of a semi-crystalline bio-based polymer was studied. Dynamic fracture tests on strip band specimens were carried out. Fracture surfaces were observed at different scales by optical and electron microscopy to describe cracking scenarios. Crack initiation, propagation and arrest zones were described. Three distinct zones are highlighted in the initiation and propagation zone: a zone with conical markings, a mist zone and a hackle zone. The conical mark zone shows a variation in the size and density of the conical marks along the propagation path. This is synonymous with local speed variation. Microcracks at the origin of the conical marks in the initiation zone seem to develop from the nucleus of the spherulites. In the propagation zone with complex roughness, the direction of the microcracks and their cracking planes are highly variable. Their propagation directions are disturbed by the heterogeneities of the material. They branch or bifurcate at the level of the spherulites. In the arrest zone, the microcracks developed upstream continue to propagate in different directions. The surface created is increasingly smoother as the energy release rate decreases. It is shown that the local velocity of the crack varies in contrast to the macroscopic speed.

Introduction

There are numerous studies in the literature on the fracture behaviour of materials and structures [START_REF] Beguelin | The influence of inertial effects on the fracture of rapidly loaded compact tension specimens. Part a: loading and fracture initiation[END_REF][START_REF] Beguelin | The influence of inertial effects on the fracture of rapidly loaded compact tension specimens. Part a: loading and fracture initiation[END_REF][START_REF] Yoffe | The moving griffith crack[END_REF][START_REF] Broberg | The propagation of a brittle crack[END_REF][START_REF] Freund | Crack propagation in an elastic solid subjected to general loading-i. Constant rate of extension[END_REF][START_REF] Cros | Experimental and numerical analysis of the impact behaviour of polycarbonate and polyurethane multilayer[END_REF][START_REF] Irwin | On the determination of ȧk relationship for birefringent polymers[END_REF][START_REF] Sheng | Two critical crack propagating velocities for pmma fracture surface[END_REF][START_REF] Rosakis | On the dynamic fracture of structural metals[END_REF][START_REF] Ponson | Crack propagation in brittle heterogeneous solids: material disorder and crack dynamics[END_REF][START_REF] Fineberg | Recent developments in dynamic fracture: some perspectives[END_REF][START_REF] Lebihain | Effective toughness of periodic heterogeneous materials: the effect of out-of-plane excursions of cracks[END_REF]; Le [START_REF] Barbenchon | Reinforcement of cellular materials with short fibres: application to a bio-based cork multi-scale foam[END_REF][START_REF] Ravi-Chandar | Dynamic fracture of nominally brittle materials[END_REF][START_REF] Sharon | Confirming the continuum theory of dynamic brittle fracture for fast cracks[END_REF]. Approaches in the analysis of the mechanism vary widely depending on the point of view. Experimenters tend to replicate the mechanism more or less rigorously in the laboratory [START_REF] Fond | Dynamic fracture surface energy and branching instabilities during rapid crack propagation in rubber toughened pmma[END_REF][START_REF] Kopp | Rapid crack propagation in pa11: an application to pipe structure[END_REF]Ravi-Chandar and Knauss 1984a;[START_REF] Kolvin | Nonlinear focusing in dynamic crack fronts and the microbranching transition[END_REF]. They describe it with ever more sophisticated tools and sometimes different interests. Some are interested in crack initiation [START_REF] Bisoffi-Sauve | Modelling mixed mode fracture of mortar joints in masonry buildings[END_REF][START_REF] Rittel | An investigation of dynamic crack initiation in pmma[END_REF], some in crack propagation [START_REF] Kopp | Rapid crack propagation in pa11: an application to pipe structure[END_REF][START_REF] Fond | Dynamic fracture surface energy and branching instabilities during rapid crack propagation in rubber toughened pmma[END_REF][START_REF] Dalmas | Damage mechanisms in the dynamic fracture of nominally brittle polymers[END_REF][START_REF] Bonamy | Dynamic crack response to a localized shear pulse perturbation in brittle amorphous materials: on crack surface roughening[END_REF], others in crack arrest [START_REF] Imachi | Dynamic crack arrest analysis by ordinary statebased peridynamics[END_REF], and a comprehensive approach is often proposed by studying the mechanism at the macroscopic scale.

As introduced in [START_REF] Guerra | Understanding fast macroscale fracture from microcrack post mortem patterns[END_REF], when a fast crack propagation is studied, some discrepancies can be observed between theoretical LEFM-based models and real measurements. Indeed, the study of local mechanisms at the microscopic scale such as micro cracks, micro-branches, and conical marks is then required and has been widely described for different kind of materials in the literature: [START_REF] Sharon | Confirming the continuum theory of dynamic brittle fracture for fast cracks[END_REF][START_REF] Arakawa | Relationships between fracture parameters and fracture surface roughness of brittle polymers[END_REF][START_REF] Johnson | On the shape and size of the fracture zones on glass fracture surfaces[END_REF]Ravi-Chandar and Knauss 1984b;[START_REF] Kies | Interpretation of fracture markings[END_REF][START_REF] Yang | On the role of the process zone in dynamic fracture[END_REF][START_REF] Ponson | Statistical aspects in crack growth phenomena: how the fluctuations reveal the failure mechanisms[END_REF][START_REF] Ponson | Two-dimensional scaling properties of experimental fracture surfaces[END_REF][START_REF] Fineberg | Instability in dynamic fracture[END_REF][START_REF] Kolvin | Nonlinear focusing in dynamic crack fronts and the microbranching transition[END_REF][START_REF] Cambonie | Effect of the porosity on the fracture surface roughness of sintered materials: From anisotropic to isotropic selfaffine scaling[END_REF]. The numerical tools that have significantly evolved over the last few decades largely complement the experimental observations : [START_REF] Bisoffi-Sauve | Modelling mixed mode fracture of mortar joints in masonry buildings[END_REF][START_REF] Valdez | Influence of fracture roughness and micro-fracturing on the mechanical response of rock joints: a discrete element approach[END_REF][START_REF] Coré | Dynamic energy release rate evaluation of rapid crack propagation in discrete element analysis[END_REF][START_REF] Nishioka | Recent developments in computational dynamic fracture mechanics[END_REF][START_REF] Imachi | Dynamic crack arrest analysis by ordinary statebased peridynamics[END_REF][START_REF] Zhou | A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials[END_REF]. They are very useful where experimental tools are limited, such as for taking into account heterogeneities [START_REF] Villette | Influence of material heterogeneities on crack propagation statistics using a fiber bundle model[END_REF], dynamic effects, micro-and macro-branches [START_REF] Bleyer | Dynamic crack propagation with a variational phase-field model: limiting speed, crack branching and velocity-toughening mechanisms[END_REF], microcracks, or for dimensioning structures.

When a cracking mechanism is studied in the laboratory, it is generally to describe the different mechanisms observable at different scales, to quantify the fracture parameters that will feed predictive models or to validate a numerical model (or method). From the microto the macro-scale, the description of the mechanisms is necessary although complex. This makes it possible to rigorously evaluate and predict the mechanisms that may lead to the collapse of a structure. Indeed, the appearance of micro-and macro-cracks is largely correlated to the microstructure and heterogeneities of the material, the mechanical stress or even the effects of the environment [START_REF] Fineberg | Recent developments in dynamic fracture: some perspectives[END_REF]. To analyse a dynamic fracture mechanism in mode I (opening mode), the formalism of linear elastic fracture mechanics (LEFM) advises a global approach with the energy release rate G or local with the stress intensity factor K . In dynamic regime and for most materials, the size of the fracture process zone remains negligible compared to the crack length. This can be very different for a supposedly ductile material when a crack propagates slowly. In this last case, the formalisms of fracture mechanics then suggest a local approach with the contour integral J [START_REF] Rice | A path independent integral and the approximate analysis of strain concentration by notches and cracks[END_REF].

The fracture behaviour of polymeric materials has been extensively studied in the literature. Environmental effects (temperature and strain rate for example) can significantly modify the emerging behaviour of the material [START_REF] Fond | Dynamic fracture surface energy and branching instabilities during rapid crack propagation in rubber toughened pmma[END_REF][START_REF] Kopp | Rapid crack propagation in pa11: an application to pipe structure[END_REF]. By increasing the strain rate, a polymer can change from a visco-elastic plastic ductile to a elastic brittle behaviour. Studying the resistance to initiation (or slow propagation) or to rapid propagation can be significantly different for the same polymer material. The formalisms of fracture mechanics also. Homalite 100 and polymethylmethacrylate (PMMA) have regularly served as model materials for studying materials with brittle linear elastic behaviour. Their fracture facies have been widely described. Different characteristic zones exist. At crack initiation, conical markings can be observed [START_REF] Kies | Interpretation of fracture markings[END_REF][START_REF] Yang | On the role of the process zone in dynamic fracture[END_REF][START_REF] Dalmas | Damage mechanisms in the dynamic fracture of nominally brittle polymers[END_REF]. They result from the interaction between a macro crack front and micro crack fronts that develop downstream of the main crack. The analysis of these marks in terms of density and size makes it possible to estimate the crack velocity [START_REF] Amaro | Dynamic fracture in brittle amorphous materials : Dissipation mechanisms and dynamically-induced microcracking in PMMA[END_REF]. During the rapid propagation of the crack, the fracture surface is known as a hackle zone. This zone is chaotic in terms of roughness. Micro-branches, which Sharon and Fineberg [START_REF] Sharon | Microbranching instability and the dynamic fracture of brittle materials[END_REF] describe as the origin of non-trivial roughness, develop from a critical speed. Between these two zones (initiation and propagation) there is an intermediate zone, called mist. For homalite 100, it is established that in the mist zone, several small cracks propagate simultaneously and the front of the overall crack is almost straight (Ravi-Chandar and Knauss 1984a). In the hackle zone, crack growth occurs by the same physical process as in the mist zone, except that the size scale of the microfracturing increases. For PMMA, it has been observed that in the hackle zone, and for cracking speeds above about 500 m/s, the fracture surface becomes very rough with periodic microbranch-induced marks [START_REF] Fineberg | Instability in dynamic fracture[END_REF]. They generate arrest marks spaced about 1 mm apart, commonly called "ribs markings" because of their geometrical shapes. The high roughness observed in the hackle areas is generally not taken into account in the estimation of the fracture energy. It is nevertheless admitted that the higher the energy release rate, the rougher the fracture surface is (Kopp et al. 2014b[START_REF] Kopp | A self-affine geometrical model of dynamic rt-pmma fractures: implications for fracture energy measurements[END_REF][START_REF] Kopp | Rapid crack propagation in pa11: an application to pipe structure[END_REF][START_REF] Fond | Dynamic fracture surface energy and branching instabilities during rapid crack propagation in rubber toughened pmma[END_REF].

Rubber toughened polymers (Fond andSchirrer 2001, 1997;Kopp et al. 2014b) and semi-crystalline [START_REF] Kopp | Rapid crack propagation in pa11: an application to pipe structure[END_REF]) are widely used for industrial applications. Their ductility is useful for absorbing energy induced by external impact. Nevertheless, under certain conditions (loading, temperature, hygrometry, etc.) they can present the critical appearance of rapid crack propagation. Quasi-static or fatigue-type loading [START_REF] Raphael | On the role of the spherulitic microstructure in fatigue damage of pure polymer and glass-fiber reinforced semi-crystalline polyamide 6.6[END_REF] generally induces plasticity and cavitation phenomena [START_REF] Rolland | Fatigue damage mechanisms of short fiber reinforced pa66 as observed by in-situ synchrotron x-ray microtomography[END_REF]Castagnet et al. 2000b). On the contrary of a dynamic solicitation which induces brittle fractures with the appearance of micro cracks, micro-and macro-branches [START_REF] Kolvin | Nonlinear focusing in dynamic crack fronts and the microbranching transition[END_REF][START_REF] Fineberg | Instability in dynamic fracture[END_REF]. For semicrystalline polymeric materials, spherulites composed mainly of crystalline phase are heterogeneities with rather brittle behaviour [START_REF] Schultz | Microstructural aspects of failure in semicrystalline polymers[END_REF][START_REF] Cotterell | Fracture mechanisms and fracture toughness in semicrystalline polymer nanocomposites[END_REF][START_REF] Yu | Toughening of nylon 6 with a maleated core-shell impact modifier[END_REF]. The amorphous phase is quite soft and ductile. Under quasi-static loading it is assumed that the amorphous phase deforms plastically and therefore the material has a high ductility. Semicrystalline are rather resistant to crack initiation and are therefore suitable for industrial applications subject to external impacts such as the transport of a fluid in polymer pipes [START_REF] Mason | Establishing the correlation between s4 and full scale rapid crack propagation testing for polyamide-11 (pa-11) pipe. Plastic pipes XIII Nilsson F (1972) Dynamic stress-intensity factors for finite strip problems[END_REF][START_REF] Greenshields | A fluidstructure model for fast brittle fracture in plastic pipes[END_REF][START_REF] Williams | A perturbation analysis of rapid crack propagation in pressurised pipe[END_REF]Kopp et al. 2014a[START_REF] Kopp | Rapid crack propagation in pa11: an application to pipe structure[END_REF].

In this work, dynamic fracture tests were performed on strip band specimens (SBS) of a bio-based semicrystalline polymer. The material under investigation is relatively undescribed in the literature in terms of the fracture mechanism and especially in dynamic. More generally, the link between the specific microstructure of semi-crystalline polymers and cracking mechanisms at micro-scale is poorly referenced. The fracture surface were analyzed by optical and electron microscopy in different zones (initiation-propagation-arrest) in order to propose cracking scenarios related to the microstructure of semicrystalline polymer. The link between the mechanisms observed at different scales with the local and global cracking velocity and the fracture energy is discussed.

Material and methods

Material

The material studied is polyamide 11 (PA11), the Rilsan BESNO TL grade supplied by Arkema. The monomer is a castor oil derivative. It is therefore considered as a semi-crystalline bio-based polymer. The degree of crystallinity of the polymer is 22%. It was measured by differential scanning calorimetry. The strip band specimens were machined from pre-injected plates. The dynamic modulus E d was obtained by measuring the average ultrasonic velocity in the material < v u > which is equal to 2100 ± 46 m s -1 . As a dynamic modulus, it can be also taken as the 'relaxed' modulus of the material which avoid viscous effect in the estimation of the internal energy (see Eq. ( 2)). The value E d = 1620 ± 82 MPa is obtained from Eq. ( 1), where ν = 0.43 and ρ =1040 kg m -3 . For the estimation of the fracture energy in dynamic propagation regime, only E d and ν will be considered to describe the behaviour of the material [START_REF] Bradley | Viscoelastic creep crack growth: a review of fracture mechanical analyses[END_REF].

E d = ρv 2 u (1 + ν)(1 -2ν) 1 -ν (1)

Dynamic fracture test

The strip band specimen (SBS) geometry (Nilsson 1972;[START_REF] Fond | Dynamic fracture surface energy and branching instabilities during rapid crack propagation in rubber toughened pmma[END_REF] is used to study rapid crack propagation (RCP) and crack arrest in PA11 plates. This geometry allows precise control of the boundary conditions during propagation. It generates very few inertia effects induced by rapid crack propagation. An experimental device based on these studies [START_REF] Fond | Dynamic fracture surface energy and branching instabilities during rapid crack propagation in rubber toughened pmma[END_REF]Kopp et al. 2014b[START_REF] Kopp | A self-affine geometrical model of dynamic rt-pmma fractures: implications for fracture energy measurements[END_REF] has been designed. Two notched head-to-tail plates are prestressed (see Fig. 1). One plate is cracked (the plate 1). The second one ensures the symmetry of the load during propagation. The specimens are initially uniaxially prestressed to a constant deformation using a universal tensilecompression machine. The loading is imposed by moving the crosshead. Once the prestress is reached, the displacement is maintained. The system is then locked by means of nuts mounted on 4 threaded rods. The rods pass through the jaws. The nuts stop the jaws to block the movement in the axis of the rods. The flexibility of the system is then limited. The stop of the crack is generated on the plate 1. This plate is deliberately longer than the jaw such that an unloaded zone exists. The energy release rate is almost nil and therefore very significantly lower than the fracture energy of the material. The crack is stopping at this zone and to access this arrest zone, the plate was cut with a band saw. To neglect viscous effect of the material, a relaxation time of the specimen (about 15 times the pre-load time of the specimen) is considered before crack initiation. Crack initiation is then generated by the impact of a mass on a razor blade in contact with the tip of the crack. The test is carried out at ambient temperature. Six polyamide plates were cracked. The dimension of the plates is: L = 300 mm (length), H = 80 mm (height), b = 4 mm (thickness) and l n = 120 mm (notch length). The quasi-static energy release rate in mode I, noted G I 0 , is defined as a function of the stress σ zz (z is the direction of transverse crack propagation) by considering a plane stress state and the propagation of a single longitudinal crack (see Eq. 2).

G I 0 = H σ 2 zz (1 -ν 2 ) 2E (2)
A dynamic correction must be considered to estimate the energy release rate in dynamic regime, noted G I D . The average crack velocity is required. A fast camera or a conductive coating [START_REF] Cotterell | Fracture propagation in organic glasses[END_REF]Kopp et al. 2014b) can be used to access the spatio-temporal data of the crack tip during propagation. The latest generation of cameras have a completely satisfactory spatial and temporal resolution. The average velocity of the crack tip for this polymer is approximately 400 ms -1 ≈ 0.6c R . This value was obtained by optical method and resistivity measurement (see Fig. 3). The conductive layer is sprayed on one side of the sample prior to the experiment. The evolution of the resistance of the R s layer is recorded during the propagation of the fracture. An approximately constant current i is imposed with a generator G and a 10 MΩ resistor in Fig. 3 The position of the crack tip as a function of time is plotted and modeled by a linear regression f (x) = ȧx. ȧR = 400±38 m s -1 and ȧC = 390±25 m s -1 with ȧR and ȧC derived from resistivity and fast camera measurements, respectively series with the sample which has a significantly lower resistance: R s < 45 kΩ [see in (Kopp et al. 2014b)]. A National Instruments USB-6351 data acquisition is used to measure the potential difference u at a sampling rate of 1.25 MS s -1 during fracture propagation. The resistance of the conductive layer is then calculated using Ohm's law R S = u i . After calibration, the velocity of the crack tip at the sample scale (macroscopic velocity) is known at ± 38 m/s.

The Rayleigh wave velocity (c R ) in PA11 is about 698 ms -1 [START_REF] Kopp | Rapid crack propagation in pa11: an application to pipe structure[END_REF]. It is known that the dynamic correction classically used for fast propagation which gives the factor k = 1-ȧ c R , is unsatisfactory in the case of a SBS [START_REF] Broberg | The propagation of a brittle crack[END_REF] test. A numerical model adapted to this specimen geometry has been implemented in (Kopp et al. 2014b) which gives a small dynamic correction of 10% which must be considered at 0.6c R . In other words, 10% of the energy is dissipated by the structure as inertia effects when the crack propagates at 60% of the Rayleigh wave velocity in the material in the case of SBS geometry.

Two cracking configurations are shown in Fig. 2: a single crack propagation perpendicular to the loading axis (see Fig. 2a) and a main crack propagation that branches into two secondary cracks (see Fig. 2b). In dynamic regime and for this type of multiphase materials with strong heterogeneities, the energy is mainly dissipated by surface creation. An increase in the energy release rate thus induces in fine macroscopic crack branching. The energy release rate can only be estimated when a longitudinal crack propagates. In this work, only specimens with a longitudinal crack were analysed.

Fracture surface analysis

An optical microscope and a scanning electron microscope (SEM) are used to observe the fracture surfaces of the samples. Samples are taken from the cracked plates. They are embedded in acrylic resin and metallized in order to visualize them with the SEM. The apparatus used for metallization is a Cressington 108 auto. The gold layer is assumed to be of constant thickness on the surface of the sample. The optical microscope used is a Keyence VHX1000. The samples were then observed using a SEM (Zeiss EVOHD 15). An accelerated voltage of 10 keV is used with a current of 200 pA. A fracture surface was reconstructed using several observations by light microscopy (see Fig. 4). A long length of cracking in sample B3 was analysed. One can observe in Table 1 some information on the tests carried out on the specimens. Different zones can be distinguished. From left to right, one can observe a zone where the razor blade has penetrated, a so-called initiation zone (zone 1) and a propagation zone. These different zones as well as a crack arrest zone were analyzed to qualitatively describe the majority of the mechanisms observed on a microscopic scale.

Crack initiation

The initiation zone is complex. The macro-crack will indeed grow in this zone by coalescence of several micro-cracks. The energy brought to the system allowing the crack initiation is composed of the elastic energy stored in the plate and the kinetic energy generated by the impact of the mass on the razor blade. The objective of the dynamic fracture test is to establish as quickly as possible a stable state of propagation characterised by a constant energy release rate during propagation. From the point of view of crack length, this transition zone between initiation and propagation is negligible. This is the interest of a rapid propagation (i.e. dynamic fracture) test, to be able to free oneself from the crack initiation. Nevertheless, on a microscopic scale, the fracture surface reflects this abrupt transition. Several characteristic areas can be observed in Fig. 5. From left to right, we find the zone of penetration of the blade, the zone where conical marks can be observed, an intermediate zone called "mist zone" and a chaotic zone from the point of view of roughness called "hackle zone" [START_REF] Johnson | On the shape and size of the fracture zones on glass fracture surfaces[END_REF][START_REF] Ravi-Chandar | Dynamic crack-tip stresses under stress wave loading-a comparison of theory and experiment[END_REF]. The area of conical markings and the hackle zone are considered to be associated with crack initiation and propagation, respectively. In the mist zone (see Fig. 6), plastic deformation of the resin is noticeable. left to right in the initiation zone (see Fig. 7). It suggests that the macro-crack velocity was increasing very briefly until reaching the mist zone state. The area of the conical marks is known to be smooth and sometimes mirror-like on a macroscopic scale. This is confirmed by the 3D reconstruction of the fracture surface considered smooth in this area (see Fig. 8).

The additional energy generated by the impact of a mass on a razor blade allows the transition of the crack into a rapid propagation regime where the fracture surface is more chaotic. At this instant, the energy release rate is higher than the critical energy release rate associated with crack propagation in dynamic regime (G I > G I Dc ).

Crack propagation

The analysis of crack propagation is relatively simple from the point of view of external loading and spacetime data of the crack tip. The energy release rate is assumed to be constant over almost the entire length of the crack. The crack is in a quasi-permanent regime of rapid propagation. Analysis of the fracture surface is more complex. This makes the estimation of the fracture energy hazardous.

The zone of propagation of the fracture surface of sample B5 is shown in Fig. 9. The main direction of the macrocrack front is indicated by the large white The description of crack propagation through a macro-front no longer seems at all relevant at this scale. Micro-cracks seem to propagate in different directions and planes. The interaction of several micro-cracks in planes at different altitudes seems to be at the origin of the river markings. Micro branches generating arrest marks are visible. This can be interpreted as a local stick slip phenomenon (see Fig. 10). This observation probably indicates that the microscopic speed of the crack is not constant as opposed to the macroscopic speed (see Fig. 3).

Estimation of the fracture energy

The fracture energy was estimated for 6 specimens (see Table 1). For a macroscopic velocity of the order of 0.6c R , the fracture energy is not unique and varies from 9.4 to 15.6 kJm -2 . This can be explained by not taking into account the amount of surface area created by the crack in the fracture energy estimate. Only the amount of projected surface area (crack length multiplied by the

Crack arrest

Figure 11 shows a crack arrest. The large white arrow at the top indicates the main direction of propagation of the macrocrack front. The five small black arrows indicate the propagation directions of several micro-cracks. The dashed line indicates the area of crack arrest. To the right of the specimen, there is an area where the material has been plastically deformed due to the self-heating induced by the cutting of the specimen. The microcracks described in the previous section continue to propagate in different directions. However, they are less and less numerous as they approach the arrest zone. The surface is smoother than during propagation. Nevertheless, cracking planes are still present at different altitudes. Crack arrest is abrupt. The macrocrack front is no longer parabolic. Conventional rib marking known for PMMA cannot be identified. Very locally one finds conical marks (see Fig. 12) and spherulites as in the initiation zone. The energy release rate is no longer sufficient to bypass the spherulites. The crack comes to die in the heart of the spherulites without generating micro branches. The evidence of these marks suggests

Discussion

The initiation areas of the fracture surfaces were analyzed. They confirm three distinct zones: a zone with conical markings, an intermediate zone (mist) and a chaotic zone (hackle). During the static pre-stressing of the sample, micro-cracks seem to develop from the nuclei of the spherulites. This (crystalline) phase of the material is assumed to be brittle. In contrast to the amorphous phase which brings ductility to the material when it is stressed relatively "slowly". These micro-cracks coalesce and interfere with a macro-front to generate conical marks which have been observed. The appearance of these micro-cracks, often associated with cavities, is described in the literature [START_REF] Selles | Voiding mechanisms in deformed polyamide 6 observed at the nanometric scale[END_REF][START_REF] Poulet | Observations by in-situ xray z of the microstructural evolution of semi-crystalline polyamide 11 during deformation[END_REF]. The amorphous intraspherolitic phase cavitates (or micro-crack) (Castagnet et al. 2000a). When the density of micro-cracks is critical, a macro-crack is created [START_REF] Zhurkov | The micromechanics of polymer fracture[END_REF][START_REF] Zhurkov | Mechanism of submicrocrack generation in stressed polymers[END_REF]. This crack of a few mm ends up stopping in the mist zone, where plastic deformation of the matrix is found. The ductility of the amorphous phase eventually takes over the brittle (crystalline) phase of the material. This can be explained by the (quasi-static) strain rate and the degree of crystallinity (about 20%). At this moment the energy release rate is lower than the critical energy release rate allowing crack initiation (G I < G I c ). The external impact on the razor blade will provide the energy (and instability) necessary to initiate the crack in a dynamic propagation regime. This is evidenced by the observation of a chaotic zone in the surface roughness. The local strain rate induced by the rapid crack propagation probably causes a significant drop in the ductility of the amorphous phase. The amorphous and crystalline phases are then brittle. The energy release rate is this time higher than the critical energy release rate allowing the rapid propagation of the crack (G I > G I Dc ).

The difference in behaviour between crack initiation and rapid crack propagation for this type of material is considerable. Initiating a crack requires an energy release rate that is often very high compared to that which the material would require to sustain rapid crack propagation. This is due to a significant difference between the resistance to slow propagation (initiation) and rapid propagation of the material: G I c > G I Dc . These significant differences have been reported for other industrial materials [START_REF] Fond | Dynamic fracture surface energy and branching instabilities during rapid crack propagation in rubber toughened pmma[END_REF]. In the area of rapid propagation, the propagation velocity of the macrocrack front is relatively constant ( ȧ ≈0.6c R ). This critical velocity is a priori the limit at which the crack branches due to inertia effects [START_REF] Yoffe | The moving griffith crack[END_REF]. The specimens studied show a longitudinal crack. Analysis of the fracture surface reveals a complex roughness. Numerous micro-cracks, micro-branches and arrest marks are observed. They appear in different crack planes. A single macro-crack front seems to be no longer identifiable. The material seeks to dissipate a significant amount of energy (G I >> G I Dc ) by cracking mostly in the amorphous phase. Spherulites, i.e. the crystalline phase, are heterogeneities that disturb the local stress field of the microcrack tips. This induces micro-branches and microcracking phenomena in different planes in large numbers. This allows the material (i.e. the individual cracks) to dissipate in fine a large amount of energy. One could postulate that the material dissipates as much or more energy with a single crack with many micro-cracks and micro-branches than with macro-branches. This is probably why a longitudinal macro-crack could be generated in several samples despite G I >> G I Dc .

The fact that G I >> G I Dc during fast propagation induces a maximum macroscopic speed (0.6c R ) and a high roughness. The estimation of the fracture energy is thus complex. G I D is not unique at 0.6c R (see Table 1). This estimation is made by applying the LEFM formalism. The quantity of projected surface area (the thickness times the crack length) is considered. With the observation of the fracture surface, it turns out that this consideration is erroneous. It would also appear that the local crack velocity is less than 0.6c R but probably higher than the critical micro-branching velocity. A fine estimate of the amount of surface area created by the crack improves the estimate of G I D (Kopp et al. 2014b). However, it is difficult to access the amount of sub-surface area created by the micro-branches.

A relevant avenue of study would be to limit the energy release rate during rapid propagation. For a large number of industrial polymer materials, there is a significant difference between resistance to crack initiation and resistance to rapid crack propagation. It is sufficient to note the numerous studies on the influence of the strain rate on the behaviour of the material. For many semi-crystalline materials, an increase in the strain rate leads to a drop in ductility. For these materials, the study of cracking mechanisms is complex. They must be dissociated: initiation-propagationarrest. When the energy release rate is limited (in the crack arrest zones), the surfaces observed on a macroscopic scale are smooth and the cracking speeds seem to vary (conical marks). A suitable fracture test setup is suggested. Two minimum loading levels would be required along the crack length. One level where the energy release rate is higher than the crack initi-ation resistance (G I > G I c ). Another loading level where the energy release rate is close to the minimum fracture energy allowing dynamic material fracture (G I ≈ G I Dc ). This would make it possible to limit the micro-branching mechanisms which generate complex roughnesses difficult to measure and to apprehend by the formalism of the LEFM. With a variable zone loading in terms of energy release rate, materials with antagonistic fracture behaviour (ductile-fragile) that depend significantly on the strain rate could be rigorously studied. The surfaces created by the crack would be smooth. The velocity of the macro-crack would vary. The micro-cracking and macro-cracking mechanisms would be limited. It would therefore be possible to give a fine analysis of the fracture behaviour (initiation-propagation-arrest) for many materials as is the case for one of the most studied brittle materials, PMMA.

Conclusions

The work described in this article focuses on the qualitative analysis of fracture surface. The cracks were generated by a dynamic fracture test. Strip band specimens known to generate very few inertia effects were used. The material studied is a semi-crystalline industrial biobased polymer: PA11. Its complex fracture behaviour ultimately resembles that of a large number of polymers used in industry. There is a significant difference in fracture behaviour between initiation and rapid propagation. This was evidenced by the analysis of the fracture surfaces. The mechanisms observed in the initiation and propagation zones are relatively well known. The influence of the highly heterogeneous microstructure of a semi-crystalline and the cracking regime induces a signature different from those described for the well-known homalite 100 and PMMA. The random organization of spherulites in the material confers relatively singular small-scale fracture mechanisms. The estimation of relevant material parameters is therefore complicated. Nevertheless, these parameters are indispensable in the development of predictive behaviour models used to dimension structures with respect to cracking mechanisms. A new set-up is suggested as a perspective to this work to further study the resistance to rapid crack propagation of this kind of material.

Fig. 1

 1 Fig. 1 Strip band specimen loading system

Fig. 2

 2 Fig. 2 Fast camera images obtained with a sampling rate of 10 4 frames per second to visualize the rapid propagation of cracks in a PA11 strip specimen. The time t 0 is chosen as the reference point where the crack starts to propagate in dynamic regime. The time between two images is 10 -4 s. There is a longitudinal crack without macro branches a-(sample B3) and a main crack macro branch b-(sample B9)

4

  Observation by light microscopy of the fracture surface of sample B3. Zone 1 represents the crack initiation zone. The arrow indicates the direction of propagation 3 Results

Fig. 5

 5 Fig. 5 Several zones can be identified (from left to right): the razor blade penetration zone-the conical mark zone-the intermediate "mist" zone-the "hackle" zone. The arrow indicates the direction of propagation

Fig. 6

 6 Fig. 6 Observation by electron microscopy of the mist zone of the fracture surface of sample B4. Several zones can be identified (from top to bottom): the smooth zone with the appearance of spherulites (end of the zone of conical marks)-the mist zone with plastic deformation of the resin-a stop mark of the macrocrack in dotted line-the hackle zone. The long arrow on the left indicates the direction of propagation

Fig. 7

 7 Fig. 7 Observation by electron microscopy of the conical marks of the initiation zone of the fracture surface of sample B3. The nuclei of the spherulites are identifiable. The arrow indicates the direction of propagation

Fig. 9

 9 Fig. 9 Observation by electron microscopy of the propagation zone of the fracture surface of sample B5. The large white arrow at the top indicates the main direction of propagation of the macrocrack front. The three small white arrows indicate the propagation directions of several microcracks. Characteristic river patterns in multi-phase polymers with high heterogeneity can be observed

Fig. 10

 10 Fig. 10 Observation by electron microscopy of the microcracks in the propagation zone of the fracture surface of sample B5. The arrow indicates the direction of propagation of the microcrack front. Several parabolic arrest marks of the micro-crack can be observed. The dashed line indicates the boundary between two crack planes at two different altitudes

Fig. 11

 11 Fig. 11 Observation by electron microscopy of the arrest zone of a crack on the fracture surface of sample B7. The large white arrow at the top indicates the main direction of propagation of the macrocrack front. The five small black arrows indicate the directions of propagation of several micro-cracks. The dashed line indicates the arrest zone of the crack. To the right of the specimen is an area where the material has been plastically deformed as a result of cutting the specimen

Fig. 12

 12 Fig. 12 Observation by electron microscopy of the arrest zone of a crack on the fracture surface of sample B7. Conical markings can be observed. The continuous black line indicates the arrest mark of the macro-crack

Table 1

 1 Estimation of the fracture energy G I D (kJm -2 ), the quasi-static energy release rate G I 0 (kJm -2 ) as a function of the crack velocity ȧ ((ms -1 )), the temperature T ( • C) and the displacement δ (mm) imposed during pre-loading

	Sample	δ	T	ȧ	ȧ/c R	G I 0	G I D
	B3	2	19	390	0.55	10.4	9.4
	B4	2	19	420	0.60	10.1	9.1
	B5	2.3	20	400	0.57	12.2	11.0
	B6	2.5	21	410	0.59	15.5	14.0
	B7	2.5	20	380	0.54	16.2	14.6
	B8	2.7	23	390	0.55	17.3	15.6
	B9	2.9	21	440	0.63	NA	NA

Acknowledgements

The authors would like to warmly thank the participation in this work of V. Honno and A. Bradu who did a Bachelor internship on the subject and J. Bega for his technical support on microscopic analyses.

Compliance with ethical standards

Conflicts of interest The authors declare that there are no conflicts of interest regarding the publication of this paper.