Florence Dupin De Saint-Cyr

Henri Prade
email: henri.prade@irit.fr

Jokes and Belief Revision *

Keywords: Humor, belief revision, surprise

The paper deals with a topic little studied in artificial intelligence: the understanding of humor. In this preliminary study, we try to identify the basic mechanism at work in quips and narrative jokes. It seems that in many cases a belief revision process is operating, leading to an unexpected conclusion through the punchline of the jest. We propose a formal modeling of jokes based on belief revision. Namely the punchline, which triggers a revision, is both surprising and explains perfectly what was reported in the beginning of the joke. This also suggests a way of ranking jokes in terms of surprise and strength of explanation, using possibilistic logic.

Introduction

Aristotle already 1 pointed out that laughing is proper to human. Humor is one of those cognitive topics, as also, e.g., aesthetic judgments, that has been little studied in AI. Presumably such topics do not look important and may seem tricky. Their understanding both involve perception, analysis and synthesis issues. In both cases, a "sender", the joker or the artist, presents an expressive content to a "receiver", the audience, supposedly capable of appreciating what is intended for them. Humor and art also have in common that they often play with, or even shake up the prohibitions and prejudices. In both cases, there seems to be a surprise effect [START_REF] Raccah | Humour et métaphore: quelques éléments d'une analogie pour la construction d'un sens inattendu. illustration sur un corpus de citations de G. B. Shaw[END_REF][START_REF] Williams | Aesthetics and the explication of surprise[END_REF] for the public when discovering what is presented. Very schematically, while art addresses first of all our sensitivity and arouses our emotions, it seems that jokes solicit above all a logical apprehension of the world (one speaks of "understanding a joke").

The question whether an intelligent system is able to understand humor remains largely open. The aim of this paper is to propose a representation of the cognitive comprehension of jokes and more precisely to present a preliminary modeling of the mechanism that triggers laughter in a joke. We first review the AI literature related to humor then briefly discuss the modeling of surprises before presenting the formalization of the mechanism based on belief revision, which is at work in most jokes2 .

Humor in AI

A large proportion of the AI literature on humor concerns the detection of humor or irony in texts, audio signals or images, or the evaluation of a level of humor. Some other works focus on the generation of puns, or the derivation of hijacked acronyms.

Marvin [START_REF] Minsky | Jokes and the logic of the cognitive unconscious[END_REF] discusses the Freudian theory of jokes [START_REF] Freud | Humour[END_REF] and highlights that beyond questions of prohibitions, humor is also a matter of knowledge about knowledge, and of inconsistency resolution. According to [START_REF] Attardo | Script theory revis(it)ed: Joke similarity and joke representation model[END_REF] there are three main theories about humor: the theory of relief (humor would help relieve nervous tension), the theory of superiority (humor would make one feel superior), and the theory of incongruity (humor consists of disobeying mental patterns and expectations). This last theory is the dominant theory which is also the perspective in which we place ourselves in this article (even if we can argue that the punchline of the joke restores somehow the consistency and calms a cognitive dissonance, which may seem to go also in the direction of the first theory). The same authors proposed the General Theory of Verbal Humor [START_REF] Attardo | Script theory revis(it)ed: Joke similarity and joke representation model[END_REF][START_REF] Raskin | The Primer of Humor Research[END_REF]. According to GTVH, the six main resources involved in jokes are (in decreasing order of importance): script opposition (SO), logical mechanism (LM), situation (SI), target (TA), narrative strategy (NS) and language (LA). Then the paper by [START_REF] Hempelmann | Computer, tell me a joke... but please make it funny: Computational humor with ontological semantics[END_REF] uses ontologies to analyze and generate jokes automatically following predefined scripts. In this paper, we mainly consider the first two resources which can be related to the idea of surprise and to the logical machinery triggered by the punchline of a joke.

We can also mention [START_REF] Stock | Getting serious about the development of computational humor[END_REF] which presents HAHACRONYM a system for producing humorous acronyms from existing acronyms (e.g., MIT becomes "Mythical Institute of Theology") which explains the series of letters in an unexpected way. Another research work by [START_REF] Shahaf | Inside jokes: Identifying humorous cartoon captions[END_REF] concerns the automatic analysis of the degree of humor associated with an image. For this aim, the authors design a learning mechanism based on a database of thousands of funny captions associated with images. The captions were annotated in pairs indicating the funnier one of the two. The characteristics used to formulate the learning task are: the unusual nature of the language used, the unpredictability of the legend, the grammatical complexity.

Lastly, [START_REF] Raccah | Humour et métaphore: quelques éléments d'une analogie pour la construction d'un sens inattendu. illustration sur un corpus de citations de G. B. Shaw[END_REF] draws a parallel between metaphor and bon mot, both of which rely on a "manipulative" effect in communication. According to him, humor requires three conditions: i) a story; ii) an intention to make people laugh; iii) that the story causes laughter. Raccah offers as an explanation of a bon mot that makes us laugh "the feeling of having fallen into an unexpected and inevitable trap: it is the punchline that makes us fall into this trap".

Surprise in AI

Surprise is triggered by the occurrence of something considered as having low possibility. The first version of the theory of possibility proposed by the English economist George [START_REF] Shackle | Decision, Order and Time in Human Affairs[END_REF] was based on the notion of the degree of surprise associated with an event, which was in fact a degree of impossibility of that event, calculated from a possibility distribution reflecting our uncertain knowledge about the current state of the world. A fact is then all the more surprising as it is less consistent with what we thought was possible. Formally, degree of surprise(A) = 1possibility(A)

In AI, the issue of surprises was discussed in [START_REF] Lorini | The cognitive structure of surprise: looking for basic principles[END_REF]. These authors have formulated two different notions of surprise: i) mismatch-based surprise which reflects the incompatibility between what we perceive and what we expected to perceive (scrutinized expectation); ii) astonishment or surprise in recognition which reflects the difficulty of accepting what we perceive because this percept is very implausible in the absolute. In both cases the degree of surprise or astonishment is associated with a probability. In the first case, the greater the probability associated with the initial expectation, the greater the surprise; in the second case, the greater the surprise, the more unlikely the perceived thing is. Lorini and Castelfranchi propose a logic of probabilistic quantified beliefs using a semantics given in [START_REF] Fagin | Reasoning about knowledge and probability[END_REF] to which a standard action operator is added as well as constructors to talk about what is being examined and the perceived data. They proposed to model the cognitive integration that follows a surprise by a belief change operator that they call update process. Thus, for these authors, the notion of surprise is first defined by a form of inconsistency with respect to explicit expectations or with respect to prior knowledge.

In our study, we propose a different approach since we use belief change theory in order to define surprise (and not to model its integration by the listener): the new information (the punchline of the joke) generates a revision of beliefs whose result is surprising in the sense that it is inconsistent with what was initially believed. Note that in this article the notion of surprise will not correspond to a surprising evolution of the world (Dupin de Saint-Cyr and Lang 2011), but rather to a surprising evolution of knowledge about the world. Thus, our work relates to belief revision rather than to update [START_REF] Winslett | Reasoning about action using a possible models approach[END_REF][START_REF] Katsuno | On the difference between updating a knowledge base and revising it[END_REF].

Modeling Jokes

In humor, according to [START_REF] Raccah | Humour et métaphore: quelques éléments d'une analogie pour la construction d'un sens inattendu. illustration sur un corpus de citations de G. B. Shaw[END_REF], there is a manipulation of what the joker wants the listener to believe, in order to make him fall into a trap. In this article on humor, we propose to translate Raccah's analysis in terms of belief revision. Thus, the manipulation of the listener will be seen as a two-phase process:

• The first phase corresponds to the description of the con-text of the joke. It leads to a revision of listener's beliefs by this intentionally incomplete description made in order to suggest a conclusion which will turn out to be false.

• The second phase corresponds to the arrival of the punchline of the joke, which on the one hand is surprising because it is incompatible or incongruent with the previous provisional conclusion. On the other hand, the punchline seems inevitable what we express by the fact that it logically explains the initial information.

The second lesson learnt from Raccah's article concerns humor intensity: according to him, "the punchline is all the more funny as the trap was unexpected and inevitable". This prompts us to propose a way to compare jokes in terms of the discrepancy between the levels of plausibility of the worlds obtained in the two phases on the one hand, and in terms of the inevitability of the explanation of the initial situation provided by the punchline on the other hand.

Notations and Background

We consider a propositional language L, where propositions are noted by lowercase Greek symbols. The symbols ⊥, ∨, ∧, ¬, →, ≡, , |= denote respectively contradiction, logical connectors "or", "and", "not", material implication, logical equivalence, classical inference, satisfaction.

In the following, we use a belief revision operator [START_REF] Alchourrón | On the logic of theory change: Partial meet contraction and revision functions[END_REF][START_REF] Katsuno | Propositional knowledge base revision and minimal change[END_REF]. The revision of a set of propositional formulas K by a piece of information ϕ is noted K • ϕ.

Note that belief revision semantics is based on sphere systems [START_REF] Grove | Two modellings for theory change[END_REF]) and epistemic entrenchments [START_REF] Gärdenfors | Knowledge in Flux[END_REF], two notions that have their exact counterpart in possibility theory (under the form of qualitative possibility distributions and necessity relations [START_REF] Dubois | Epistemic entrenchment and possibilistic logic[END_REF]). A possibility distribution is qualitative in the sense that it only plays the role of an ordinal ranking function on the set of possible worlds (contrarily to what a probability distribution would do). Possibility distributions are used here for implementing the revision operator • because what is required is simply a plausibility ordering as in sphere systems and no numerical value is neither needed nor available. However the approach presented in this paper holds for any kind of belief revision operator satifisfying Katsuno-Mendelzon postulates (1991).

Since we need to encode common sense knowledge, we use to represent a default rule: α β, with α, β ∈ L; meaning that when α is true it is more plausible that β is true than false. This kind of rule is easily translated in the possibility theory setting [START_REF] Dubois | Possibility Theory. An Approach to Computerized Processing of Uncertainty[END_REF] as the constraint N (β|α) > 0. Due to the duality between a necessity measure N and its associated possibility measure Π, i.e., N (α) = 1 -Π(¬α), the constraint N (β|α) > 0 is equivalently written as Π(α ∧ β) > Π(α ∧ ¬β) which expresses that, in context α, β being true is the normal situation (since strictly more possible than β false). This writing supposes the existence of a plausibility relation on worlds, which is represented by a possibility distribution π, i.e., a mapping from the set of interpretations of L to a linearly ordered scale bounded by 0 and 1, where 0 stands for impossible and 1 for fully possible.

Formalization

In this section, we propose a static formalization of jokes in propositional logic, i.e., we consider that a joke describes a situation α and that its punchline completes this description with a piece of information β (α, β are propositions in L).

We consider that the joke is addressed to a listener whose knowledge base is a set of propositional formulas denoted K. Moreover, the listener is also characterized by her way • to revise her belief.

In the following, a (simple) joke is viewed as a story, which is a pair (<context>, <punchline>) having some further characteristics. We first define a story.

Definition 1 (simple story, cascading story). A simple story is a pair (α, β) of formulas of L. A cascading story is a tuple (ϕ 1 , . . . , ϕ n ) with n > 2 formulas in L.

The cognitive situation induced by a story is described by the knowledge base K iteratively revised by the elements of the story. The punchline of a joke is considered surprising if the result of the revision of the knowledge base K by the initial description of the situation, is contradictory to what is obtained after revision by the entire story including the punchline.

Definition 2 (surprising story). A story (α, β) is surprising with respect to a listener associated to

(K, •) if (K • α) consistent 3 and (K • α) ∪ (K • (α ∧ β)) ⊥ A cascading story (ϕ 1 , . . . , ϕ n ) is surprising with respect to (K, •) at the stage i if (K • ψ) consistent and (K • ψ) ∪ (K • (ψ ∧ ϕ i )) ⊥ with ψ = ϕ 1 ∧ ϕ 2 . . . ∧ ϕ i-1 .
Note that the above definition excludes the possibility that the listener already knows the joke. Indeed in that latter case α and β would be already in K, forbidding any surprise. Besides, we do not require that K • (α ∧ β) is consistent, including absurd punchlines in Definition 24 .

In order to understand the joke, its logic has to appear implacable once the punchline is revealed, so the punchline is both completely admissible and explains the situation. In other words, if we had known β from the beginning, it would have explained α, which we can translate as follows:

Definition 3 (revealing punchline). Given a story (α, β), its punchline β is revealing with respect to

(K, •) if (K • β) consistent 3 and K • β |= α
Laughter can be thought to relieve the tension of cognitive dissonance caused by the inconsistency between what one expected to hear and the punchline of the joke. This relief can only come once the story is understood, with the punchline playing the role of a revelation.

Definition 4 (potentially funny story). A story is potentially funny if it is surprising and its punchline is revealing.

This definition of a potentially funny story is directly inspired by [START_REF] Raccah | Humour et métaphore: quelques éléments d'une analogie pour la construction d'un sens inattendu. illustration sur un corpus de citations de G. B. Shaw[END_REF], except that we have tried to define the revealing character of the punchline rather than its inevitability (which could be understood as K • α β, denying the surprising character of β).

Observe also that the definition relies both on K and •, which may differ from a listener to another, acknowledging the fact that a joke may not be found funny by anybody.

Note that the expression "potentially funny" is used in Definition 4 because one can imagine examples of surprising stories with revealing punchline that are not necessarily funny for their listeners. For example, a scientific discovery can be surprising and revealing, just like a well-crafted detective story, or certain political revelations, for the audience concerned in each case. This remark shows that ingredients other than the above belief revision mechanism are also necessary for laughter to occur. It includes elements such as the intention to make people laugh, lack of empathy or playing with taboos.

We are now able to formalize a first example (from a French collection of "funny" stories [START_REF] Nègre | Dictionnaire des Histoires Drôles[END_REF])) expressed in natural language that we have transposed into logic. In this paper, we disregard the stage of the transition from natural language to logical representation. However, we acknowledge the fact that the listener uses default knowledge for understanding a story. Hence, in the examples, the listener knowledge is expressed by a knowledge base Σ = P ∪ ∆ consisting in a set P of propositional formulas and a set ∆ of defeasible rules of the form α β where α, β ∈ L. From the set Σ, one can compute [START_REF] Benferhat | Nonmonotonic reasoning, conditional objects and possibility theory[END_REF] a possibility distribution π Σ on worlds that satisfies the constraints induced by ∆ and assigns 0 to every world not satisfying P . For the sake of shortness, in the examples, all the revision mechanism is hidden behind the notation "Σ • α", which is computed by the conditionalization of π Σ by α (namely π Σ (ω|α) = π Σ (ω) if ω |= α, and 0 otherwise), yielding a revision operator satisfying Katsuno and Mendelzon postulates [START_REF] Benferhat | A practical approach to revising prioritized knowledge bases[END_REF].

Example 1. A man just got hit by a car. The driver gets out of the car and says, "You're lucky we're just in front of a doctor's office. Yes! Except the doctor is me!" Modeling: α = injured ∧ doctorN earby, β = injured ∧ doctorHimself We assume that the listener has the following knowledge: Besides, the punchline explains α: Σ • β injured ∧ doctorN earby. Indeed doctorN earby is the part of the story that manipulated us by leading us to think that the wounded man was lucky.

Note that Example 1 can also be seen as a cascading joke (ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 ) with ϕ 1 = injured, ϕ 2 = lucky, ϕ 3 = doctorN earby, ϕ 4 = doctorHimself . Indeed, saying "You're very lucky" after seeing that the person is injured can already provoke laughter. Then the revelation of being "in front of a doctor's office" helps to understand. Finally, the second part of the joke corresponds to what has already been analyzed. Let us note that in the first part of the cascading story, ϕ 2 is surprising but not revealing, it is ϕ 3 which explains ϕ 2 . So for a cascading joke, the surprising and the revealing effects are not necessarily simultaneous.

The following property shows the importance of the narrative order in the joke.

Proposition 1 (Never give the punchline before the end). If (α, β) is a potentially funny story for a listener knowing K, then (β, α) is not potentially funny for that listener.

Proof. Since (α, β) is potentially funny, then the punchline is revealing, that is,

K • β ⊥ and K • β |= α.
But that prevents the story (β, α) from being surprising, and therefore potentially funny. Indeed, using (KM5) and (KM6) 4 [START_REF] Katsuno | Propositional knowledge base revision and minimal change[END_REF]

, K • (β ∧ α) ≡ (K • β) ∪ {α} so K • β is consistent with K • (β ∧ α).
The following property indicates the virtue of brevity.

Proposition 2 (The shortest jokes . . .). If (α, β) is a potentially funny story then adding α ∈ L, such that α α can make the story (α ∧ α , β) not potentially funny. However, by adding information α = a that the bird is from Antarctica we can create an unsurprising situation with respect to β = p, so (b ∧ a, p) is no longer a surprising story. Another counterexample consists in adding information α that is not related to β then we can obtain a punchline that is not completely revealing: let us consider α = t, where t means "is called Tweety" then Σ • p b does not allow us to deduce that Σ • p t.

A particular case of Proposition 2, when α = β, is another way to reveal the punchline before the end. This would correspond in Example 1 to specify that the driver has just knocked down a doctor in front of that same doctor's office. In this case, the punchline "Yes! Except that I am the doctor!" is no longer funny (because we already know that).

We can notice that Proposition 2 does not impose that the punchline be as short as possible, so we can refine the definition of "potentially funny story" into an "efficient story" by imposing minimality.

Definition 5 (conciseness of the punchline). Let α, β, β ∈ L and β ≡ β , if β |= β then the story (α, β) is said to be more efficient than the story (α, β ).

In other words, the punchline is said more efficient if it is less precise and thus less specific. Here is another example from [START_REF] Nègre | Dictionnaire des Histoires Drôles[END_REF] for illustration.

Example 2. A guy and a woman walk into a doctor's office. The doctor turns to the lady and says: -If you're sick, please take off your clothes ... But the girl has manners. She is reluctant. She is looking down on the guy. So the doctor repeats: -But, madam, please get undressed! I'm a doctor. There's no indecency, here! Then she starts fidgeting and suddenly she bursts into tears. Disconcerted, the doctor asks the guy: -What's wrong with your wife? Is she always this nervous? -I don't know. I just met her in your waiting room... Suppose now that the guy replied ϕ 3 ="Sorry, we are not married". Clearly ϕ 3 |= ϕ 3 . Hence, the new punchline ϕ 3 is less efficient. Note that the new story is no more funny since (Σ • ϕ 3 ) |= reluctant, which could have been obtained if Σ would contain ¬married ¬couple.

Finally, in order to give substance to Raccah's intuition (2016) that the greater the surprise and inevitability, the funnier the joke, we may propose the use of two quantities, in order to build a partial order between jokes for the same listener: Definition 6 (ingredients for humor). Given a base Σ = ∆ ∪ P , a joke (α, β) is associated to the possibilistic levels: -a surprise level:

Incons((Σ • α) ∪ (Σ • (α ∧ β))) -a revealing level: N (α|Σ • β)
This definition assumes that the default rules ∆ of Σ are encoded in possibilistic logic [START_REF] Benferhat | Practical handling of exception-tainted rules and independence information in possibilistic logic[END_REF], where Incons(A) represents the level of inconsistency of the possibilistic knowledge base A. The second term is the conditional necessity of α knowing Σ • β, reflecting the ineluctability of the revelation.

In addition, other elements may contribute to the degree of humor: the register, the comic effects of the narrator, the mood of the listener, her ability to inhibit her emotions, the cognitive effort required to understand the joke ... Besides, we might have considered iterated revision [START_REF] Konieczny | A framework for iterated revision[END_REF] and abductive expansion [START_REF] Pagnucco | The role of abductive reasoning the process of belief revision[END_REF]. However, as we wanted to capture cognitive aspects of the understanding of joke, we have privileged the simplicity of basic revision theory where epistemic states may be drastically changed in the revision process. More precisely the main idea is to select the best epistemic state rather than carrying the whole preference ordering as in iterated revision. Indeed the surprise would not be the same if the listener had in mind all the potential explanations. In that respect, a similar argument applies to abductive expansion whose aim is to provide reasons for the new epistemic state.

Conclusion

The proposed approach highlights the key role of belief revision and surprises in the mechanism underlying jokes. It has been illustrated with a few examples. It is clear that a validation of the relevance of these ideas would require their verification on a large corpus of jokes, although one may feel that many jokes actually work this way. As far as we know, this opens a new direction in the study of jokes from an artificial intelligence point of view.

  ¬treatedRapidly injured ∧ doctorHimself ¬treatingDoctor Due to postulate (KM1) (Katsuno and Mendelzon 1991) Σ • α |= α. Moreover, the most plausible models of Σ • α are satisfying ϕ = α ∧ treatedRapidly. The most plausible models of Σ•(α∧β) satisfy ψ = doctorN earby ∧ doctorHimself ∧ ¬treatingDoctor ∧ ¬treatedRapidly. The punchline is surprising since ϕ ∧ ψ ⊥.

Proof.

  Indeed here is a counterexample: let Σ = {b f , p → b, p ¬f , b ∧ a p} a knowledge base expressing that: generally birds (b) fly (f ), penguins (p) are birds, generally penguins (p) don't fly, most Antarctic birds (a) are penguins. Let us imagine a story (b, p), it is both surprising since Σ • b |= b ∧ ¬p ∧ ¬a ∧ f and Σ • (b ∧ p) |= b ∧ p ∧ ¬f and the "punchline" p is revealing since Σ • p |= b.

  ϕ 1 = together, ϕ 2 = reluctant, ϕ 3 = ¬ knowEachOther Let us compute Σ•ϕ 1 , the most plausible models satisfy ψ = together∧couple∧¬reluctant. Σ•(ϕ 1 ∧ϕ 2 ) satisfies reluctant. ϕ 2 is surprising since ψ ∧ (Σ • (ϕ 1 ∧ ϕ 2 )) ⊥. Moreover the punchline explains ϕ 2 : (Σ • ϕ 3 ) |= reluctant.

This is a fully rewritten version of a previous work in French (Dupin De Saint-Cyr and Prade 2019) which contains a more extensive introduction and more references.

This condition is automatically ensured by the usual revision postulate (KM3)[START_REF] Katsuno | Propositional knowledge base revision and minimal change[END_REF]: K •α is consistent as soon as α is consistent.

However, absurd punchlines will not be revealing in the sense of the forthcoming Definition 3 due to postulates KM5 and KM6: (KM5):(K • ϕ) ∪ {ψ} |= K • (ϕ ∧ ψ) ; (KM6): if (K • ϕ) ∪ {ψ} is consistent then K • (ϕ ∧ ψ) |= (K • ϕ) ∪ {ψ}.


Nadine Guillotin-Plantard

Clémentine Prieur
email: clementine.prieur@insa-toulouse.fr

Limit theorem for random walk in weakly dependent random scenery

Keywords: Random walks, random scenery, weak dependence, limit theorem, local time. AMS Subject Classification: Primary 60F05, 60G50, 62D05, Secondary 37C30, 37E05

Let S = (S k ) k≥0 be a random walk on Z and ξ = (ξ i ) i∈Z a stationary random sequence of centered random variables, independent of S. We consider a random walk in random scenery that is the sequence of random variables (Σ n ) n≥0 where

Under a weak dependence assumption on the scenery ξ we prove a functional limit theorem generalizing Kesten and Spitzer's theorem (1979).

Introduction

Let X = (X i ) i≥1 be a sequence of independent and identically distributed random vectors with values in Z d . We write

S 0 = 0, S n = n i=1 X i for n ≥ 1
for the Z d -random walk S = (S n ) n∈N generated by the family X. Let ξ = (ξ x ) x∈Z d be a family of real random variables, independent of S. The sequence ξ plays the role of the random scenery. The random walk in random scenery (RWRS) is the process defined by

Σ n = n k=0 ξ S k , n ∈ N.
RWRS was first introduced in dimension one by Kesten and Spitzer (1979) and Borodin (1979) in order to construct new self-similar stochastic processes. Functional limit theorems for RWRS were obtained under the assumption that the random variables ξ x , x ∈ Z d are independent and identically distributed. For d = 1, Kesten and Spitzer (1979) proved that when X and ξ belong to the domains of attraction of different stable laws of indices 1 < α ≤ 2 and 0 < β ≤ 2, respectively, then there exists δ > 1 2 such that n -δ Σ [nt] t≥0 converges weakly as n → ∞ to a self-similar process with stationary increments, δ being related to α and β by δ = 1α -1 + (αβ) -1 . The case 0 < α < 1 and β arbitrary is easier; they showed then that n -1 β Σ [nt] t≥0 converges weakly, as n → ∞, to a stable process with index β. [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF] gave a method to solve the case α = 1 and β = 2 and especially, he proved that when (S n ) n∈N is a recurrent Z 2 -random walk, (n log n) -1 2 Σ [nt] t≥0 satisfies a functional central limit theorem. For an arbitrary transient Z d -random walk, n -1 2 Σ n is asymptotically normal (see [START_REF] Spitzer | Principles of random walks[END_REF] p. 53). [START_REF] Maejima | Limit theorems related to a class of operator-self-similar processes[END_REF] generalized the result of Kesten and Spitzer (1979) in the case where (ξ x ) x∈Z are i.i.d. R d -valued random variables which belong to the domain of attraction of an operator stable random vector with exponent B. If we denote by D the linear operator on R d defined by D = (1 -1 α )I + 1 α B, he proved that n -D Σ [nt] t≥0 converges weakly to an operator self similar with exponent D and having stationary increments.

One-dimensional random walks in random scenery recently arose in the study of random walks evolving on oriented versions of Z 2 (see Guillotin-Plantard and Le Ny (2007,2008)) as well as in the context of charged polymers (see [START_REF] Chen | From charged polymers to random walk in random scenery[END_REF]). The understanding of these models in the case where the orientations or the charges are not independently distributed requires functional limit theorems for Z-random walk in correlated random scenery. To our knowledge, only the case of strongly correlated stationary random sceneries has been studied by [START_REF] Lang | Strongly correlated random fields as observed by a random walker[END_REF]. In their paper, the increments of the random walk S are assumed to belong to the domain of attraction of a non-degenerate stable law of index α, 0 < α ≤ 2. They further suppose that the scenery ξ satisfies the non-central limit theorem of [START_REF] Dobrushin | Non-central limit theorems for non-linear functionals of Gaussian fields[END_REF] with a scaling factor n -d+(βk)/2 , βk < d. Under the assumption βk < α, it is proved that n -1+βk/(2α) Σ [nt] t≥0 converges weakly as n → +∞ to a self-similar process with stationary increments, which can be represented as a multiple Wiener-Itô integral of a random function. Our aim is to study the intermediary case of a stationary random scenery ξ which satisfies a weak dependence condition introduced in Dedecker et al. (2007) and to prove Kesten and Spitzer's theorem under this new assumption.

In Guillotin-Plantard & Prieur (2008) the case of a transient Z-random walk was considered and a central limit theorem for the sequence (Σ n ) n∈N was proved. In this paper the onedimensional random walk will be assumed to be recurrent.

Our paper is organized as follows: In Section 2, we introduce the dependence setting under which we work in the sequel. In Section 3 we introduce in details our model and give the main result. In Section 4 properties of the local time of the random walk are given as well as the ones of the intersection local time. Models for which we can compute bounds for our dependence coefficients are presented in Section 5. Finally, the proof of our theorem is given in the last section.

Weak dependence conditions

In this section, we recall the definition of the dependence coefficients which we will use in the sequel. They have first been introduced in Dedecker et al. (2007). Our weak dependence condition will be less restrictive than the mixing one. The reader interested in this question would find more details in Guillotin-Plantard & Prieur (2008).

On the Euclidean space R m , we define the metric

d 1 (x, y) = m i=1 |x i -y i |. Let Λ = m∈N * Λ m where Λ m is the set of Lipschitz functions f : R m → R with respect to the metric d 1 . If f ∈ Λ m , we denote by Lip(f ) := sup x =y |f (x)-f (y)| d 1 (x,y)
the Lipschitz modulus of f . The set of functions f ∈ Λ such that Lip(f ) ≤ 1 is denoted by Λ.

Definition 2.1 Let ξ be a R m -valued random variable defined on a probability space (Ω, A, P), assumed to be square integrable. For any σ-algebra M of A, we define the θ 2 -dependence coefficient

θ 2 (M, ξ) = sup{ E(f (ξ)|M) -E(f (ξ)) 2 , f ∈ Λ} . (2.1)
We now define the coefficient θ k,2 for a sequence of σ-algebras and a sequence of R-valued random variables. Definition 2.2 Let (ξ i ) i∈Z be a sequence of square integrable random variables valued in R. Let (M i ) i∈Z be a sequence of σ-algebras of A. For any k ∈ N * ∪ {∞} and n ∈ N, we define

θ k,2 (n) = max 1≤l≤k 1 l sup{θ 2 (M p , (ξ j 1 , . . . , ξ j l )), p + n ≤ j 1 < . . . < j l } and θ 2 (n) = θ ∞,2 (n) = sup k∈N * θ k,2 (n) .
Definition 2.3 Let (ξ i ) i∈Z be a sequence of square integrable random variables valued in R.

Let (M i ) i∈Z be a sequence of σ-algebras of A. The sequence (ξ i ) i∈Z is said to be θ 2 -weakly dependent with respect to

(M i ) i∈Z if θ 2 (n) -----→ n→+∞ 0.
Remark : Replacing the • 2 norm in (2.1) by the • 1 norm, we get the θ 1 dependence coefficient first introduced by Doukhan & Louhichi (1999).

Model and results

Let S = (S k ) k≥0 be a Z-random walk (S 0 = 0) whose increments (X i ) i≥1 are centered and square integrable. We denote by P X 1 the law of the random variable X 1 . For any q ∈ N * such that P(X 1 ∈ [-q, q]) is non zero, we define the probability measure on Z P q = P X 1 [-q,q] P(X 1 ∈ [-q, q]) .

The random walk S is said to satisfy the property (P) if there exists q ∈ N * such that {x ∈ Z; ∃n, P (⋆n)

q (x) > 0} = Z.
In particular, if there exists some q ∈ N * such that the random walk associated to P q is centered and aperiodic then S satisfies the property (P). For instance, the simple random walk on Z verifies (P). Let ξ = (ξ i ) i∈Z be a sequence of centered real random variables.

The sequences S and ξ are defined on a same probability space denoted by (Ω, F, P) and are assumed to be independent. We are interested in the asymptotic behaviour of the following sum

Σ n = n k=0 ξ S k .
The case where the ξ i 's are independent and identically distributed random variables with positive variance has been considered by Kesten & Spitzer (1979) and Borodin (1979). Consider a standard Brownian motion (B t ) t≥0 , denote by (L t (x)) t≥0 its corresponding local time at x ∈ R and introduce a pair of independent Brownian motions (Z + (x), Z -(x)), x ≥ 0 defined on the same probability space as (B t ) t≥0 and independent of him. The following process is well-defined for all t ≥ 0:

∆ t = ∞ 0 L t (x)dZ + (x) + ∞ 0 L t (-x)dZ -(x). (3.2)
It was proved by Kesten and Spitzer (1979) that this process has a self-similar continuous version of index 3 4 , with stationary increments. We denote D =⇒ for a convergence in the space of càdlàg functions D([0, ∞), R) endowed with the Skorohod topology. Theorem 3.1 [Kesten and Spitzer (1979)] Assume that the ξ i 's are independent and identically distributed with positive variance σ 2 > 0. Then,

1 n 3/4 Σ [nt] t≥0 D =⇒ (σ∆ t ) t≥0 . (3.3)
A simple proof of this theorem was proposed by Cadre (1995) (Section 2.5.a.) using a weak limit theorem for stochastic integrals (Theorem 1.1. in [START_REF] Kurtz | Wong-Zakai corrections, Random evolutions, and numerical schemes for SEDs[END_REF]). Applying Cadre's method, Theorem 3.1 can be extended to any scenery given by a stationary and ergodic sequence of square integrable martingale differences. Then, a natural idea is to generalize the result to any stationary and ergodic sequence ξ of square integrable random variables as it was done for the central limit theorem. Under suitable assumptions on the sequence, for instance the convergence of the series ∞ k=0 E(ξ k |M 0 ) in L 2 , the scenery ξ is equal to a martingale differences sequence modulo a coboundary term and satisfies a Donsker theorem. However, the RWRS associated to the coboundary term (if it is non zero) is not negligible. It can be proved that the L 2 -norm of this sum correctly normalized by n 3/4 converges to some positive constant.

In order to weaken the assumptions on the field ξ we introduce a sequence (M i ) i∈Z of σ-fields of F defined by

M i = σ (ξ j , j ≤ i) , i ∈ Z .
In the sequel, the dependence coefficients will be defined with respect to the sequence of σ-fields (M i ) i∈Z .

Theorem 3.2

Assume that the following conditions are satisfied :

(A 0 )
The random walk S satisfies the property (P).

(A 1 ) ξ = {ξ i } i∈Z is a stationary sequence of square integrable random variables.

(A 2 ) θ ξ 2 (•) is bounded above by a non-negative function g(•) such that x → x 3/2 g(x) is non-increasing, ∃ 0 < ε < 1 , ∞ i=0 2 3i 2 g(2 iε ) < ∞.
Then, as n tends to infinity,

1 n 3/4 Σ [nt] t≥0 D =⇒ i∈Z E(ξ 0 ξ i ) (∆ t ) t≥0 . (3.4)
Remark : Assumptions (A 1 ) and (A 2 ) imply that

∀λ ∈ [0, 1/2[, k∈Z |k| λ |E(ξ 0 ξ k )| < +∞ (3.5)
Indeed, this sum is equal to

E(ξ 2 0 ) + 2 ∞ k=1 |k| λ |E(ξ 0 ξ k )|
and for any k ≥ 1, from Cauchy-Schwarz inequality, we get

|E(ξ 0 ξ k )| = |E ξ 0 E(ξ k |M 0 ) | ≤ ||ξ 0 || 2 θ ξ 2 (k) ≤ ||ξ 0 || 2 g(k)
The result (3.5) follows by remarking that

∞ k=1 |k| λ g(k) ≤ g(1) ∞ k=1 1 k 3/2-λ which is finite for every λ ∈ [0, 1/2[. Remark : If θ ξ 2 (n) = O (n -a
) for some positive a, condition (A 3 ) holds for a > 3/2.

Properties of the occupation times of the random walk

The random walk S = (S k ) k≥0 is defined as in the previous section and is assumed to verify the property (P). The local time of the random walk is defined for every i ∈ Z by

N n (i) = n k=0 1 {S k =i} .
The local time of self-intersection at point i of the random walk (S n ) n≥0 is defined by

α(n, i) = n k,l=0 1 {S k -S l =i} .
The stochastic properties of the sequences (N n (i)) n∈N,i∈Z and (α(n, i)) n∈N,i∈Z are well-known when the random walk S is strongly aperiodic. A random walk who satisfies the property (P) is not strongly aperiodic in general. However, a local limit theorem for the random walks satisfing (P) was proved by Cadre (2005) (ii) (a) For any p ∈ [1, +∞), there exists some constant C such that for all n ≥ 1,

E (α(n, 0) p ) ≤ Cn 3p/2 .
(b) For any m ≥ 1, for any real θ 1 , . . . , θ m , for any 0 ≤ t 1 ≤ . . . ≤ t m , the sequence

n -3/2 i∈Z m k=1 θ k N [nt k ] (i) 2 converges in distribution to R m k=1 θ k L t k (x) 2 dx
where (L t (x)) t≥0;x∈R is the local time of the real Brownian motion (B t ) t≥0 .

(iii) For every λ ∈ ]0, 1[, there exists a constant C such that for any i, j ∈ Z,

α(n, i) -α(n, j) 2 ≤ Cn (3-λ)/2 |i -j| λ .

Examples

In this section, we present examples for which we can compute upper bounds for θ 2 (n) for any n ≥ 1. We refer to Chapter 3 in Dedecker et al (2007) and references therein for more details.

Example 1: causal functions of stationary sequences

Let (E, E, Q) be a probability space. Let (ε i ) i∈Z be a stationary sequence of random variables with values in a measurable space S. Assume that there exists a real valued function H defined on a subset of S N , such that H(ε 0 , ε -1 , ε -2 , . . . , ) is defined almost surely. The stationary sequence (ξ n ) n∈Z defined by

ξ n = H(ε n , ε n-1 , ε n-2 , . . .) is called a causal function of (ε i ) i∈Z .
Assume that there exists a stationary sequence

(ε i ′ ) i∈Z distributed as (ε i ) i∈Z and in- dependent of (ε i ) i≤0 . Define ξ * n = H(ε n ′ , ε n-1 ′ , ε n-2 ′ , . . .). Clearly, ξ * n is independent of M 0 = σ(ξ i , i ≤ 0) and distributed as ξ n . Let (δ 2 (i)) i>0 be a non increasing sequence such that E (|ξ i -ξ * i | | M 0 ) 2 ≤ δ 2 (i) . (5.6)
Then the coefficient θ 2 of the sequence (ξ n ) n≥0 satisfies

θ 2 (i) ≤ δ 2 (i) . (5.7)
Let us consider the particular case where the sequence of innovations (ε i ) i∈Z is absolutely regular in the sense of [START_REF] Rozanov | Some limit theorems for random functions I[END_REF]. Then, according to Theorem 4.4.7 in [START_REF] Berbee | Random walks with stationary increments and renewal theory[END_REF], if E is rich enough, there exists (ε ′ i ) i∈Z distributed as (ε i ) i∈Z and independent of (ε i ) i≤0 such that

Q(ε i = ε ′ i for some i ≥ k | F 0 ) = 1 2 Q εk |F 0 -Q εk v ,
where εk = (ε k , ε k+1 , . . .), F 0 = σ(ε i , i ≤ 0), and • v is the variation norm. In particular if the sequence (ε i ) i∈Z is idependent and identically distributed, it suffices to take

ε ′ i = ε i for i > 0 and ε ′ i -ε ′′ i for i ≤ 0, where (ε ′′ i ) i∈Z is an independent copy of (ε i ) i∈Z .

Application to causal linear processes:

In that case, ξ n = j≥0 a j ε n-j , where (a j ) j≥0 is a sequence of real numbers. We can choose

δ 2 (i) ≥ ε 0 -ε ′ 0 2 j≥i |a j | + i-1 j=0 |a j | ε i-j -ε ′ i-j 2 .
From Proposition 2.3 in [START_REF] Merlevède | On the coupling of dependent random variables and applications. Empirical process techniques for dependent data[END_REF], we obtain that

δ 2 (i) ≤ ε 0 -ε ′ 0 2 j≥i |a j | + i-1 j=0 |a j | 2 2 β(σ(ε k ,k≤0),σ(ε k ,k≥i-j)) 0 Q 2 ε 0 (u) 1/2 du,
where

Q ε 0 is the generalized inverse of the tail function x → Q(|ε 0 | > x).

Example 2: iterated random functions

Let (ξ n ) n≥0 be a real valued stationary Markov chain, such that ξ n = F (ξ n-1 , ε n ) for some measurable function F and some independent and identically distributed sequence (ε i ) i>0 independent of ξ 0 . Let ξ * 0 be a random variable distributed as ξ 0 and independent of (ξ 0 ,

(ε i ) i>0 ). Define ξ * n = F (ξ * n-1 , ε n ) . The sequence (ξ * n ) n≥0 is distributed as (ξ n ) n≥0 and independent of ξ 0 . Let M i = σ(ξ j , 0 ≤ j ≤ i).
As in Example 1, define the sequence (δ 2 (i)) i>0 by (5.6). The coefficient θ 2 of the sequence (ξ n ) n≥0 satisfies the bound (5.7) of Example 1.

Let µ be the distribution of ξ 0 and (ξ x n ) n≥0 be the chain starting from ξ x 0 = x. With these notations, we can choose δ 2 (i) such that

δ 2 (i) ≥ ξ i -ξ * i 2 = |ξ x i -ξ y i 2 2 µ(dx)µ(dy) 1/2 .
For instance, if there exists a sequence (d 2 (i)) i≥0 of positive numbers such that

ξ x i -ξ y i 2 ≤ d 2 (i)|x -y|, then we can take δ 2 (i) = d 2 (i) ξ 0 -ξ * 0 2
. For example, in the usual case where F (x, ε 0 ) -F (y, ε 0 ) 2 ≤ κ|x -y| for some κ < 1, we can take d 2 (i) = κ i .

An important example is ξ n = f (ξ n-1 ) + ε n for some κ-Lipschitz function f . If ξ 0 has a moment of order 2, then δ 2 (i) ≤ κ i ξ 0ξ * 0 2 .

Example 3: dynamical systems on [0, 1]

Let I = [0, 1], T be a map from I to I and define X i = T i . If µ is invariant by T , the sequence (X i ) i≥0 of random variables from (I, µ) to I is strictly stationary.

For any finite measure ν on I, we use the notations ν(h) = I h(x)ν(dx). For any finite signed measure ν on I, let ν = |ν|(I) be the total variation of ν. Denote by g 1,λ the L 1 -norm with respect to the Lebesgue measure λ on I.

Covariance inequalities. In many interesting cases, one can prove that, for any BV function h and any k in

L 1 (I, µ), | Cov(h(X 0 ), k(X n ))| ≤ a n k(X n ) 1 ( h 1,λ + dh ) , (5.8)
for some nonincreasing sequence a n tending to zero as n tends to infinity.

Spectral gap. Define the operator L from L 1 (I, λ) to L 1 (I, λ) via the equality

1 0 L(h)(x)k(x)dλ(x) = 1 0 h(x)(k • T )(x)dλ(x) where h ∈ L 1 (I, λ) and k ∈ L ∞ (I, λ).
The operator L is called the Perron-Frobenius operator of T . In many interesting cases, the spectral analysis of L in the Banach space of BV -functions equiped with the norm h v = dh + h 1,λ can be done by using the Theorem of Ionescu-Tulcea and Marinescu (see [START_REF] Lasota | On the existence of invariant measures for piecewise monotonic transformations[END_REF] and [START_REF] Hofbauer | Ergodic properties of invariant measures for piecewise monotonic transformations[END_REF]). Assume that 1 is a simple eigenvalue of L and that the rest of the spectrum is contained in a closed disk of radius strictly smaller than one. Then there exists a unique T -invariant absolutely continuous probability µ whose density f µ is BV , and

L n (h) = λ(h)f µ + Ψ n (h) with Ψ n (h) v ≤ Kρ n h v .
(5.9)

for some 0 ≤ ρ < 1 and K > 0. Assume moreover that:

I * = {f µ = 0} is
P (h)(x) = L(f µ h)(x) f µ (x) . (5.11)
It is easy to check (see for instance [START_REF] Barbour | Iterates of expanding maps[END_REF]) that (X 0 , X 1 , . . . , X n ) has the same distribution as (Y n , Y n-1 , . . . , Y 0 ) where (Y i ) i≥0 is a stationary Markov chain with invariant distribution µ and transition kernel P . Since f g ∞ ≤ f g v ≤ 2 f v g v , we infer that, taking C = 2Kγ( df µ + 1),

P n (h) = µ(h) + g n with g n ∞ ≤ Cρ n h v .
(5.12)

This estimate implies (5.8) with a n = Cρ n (see [START_REF] Dedecker | New dependence coefficients. Examples and applications to statistics[END_REF].

Expanding maps: Let ([a i , a i+1 [) 1≤i≤N be a finite partition of [0, 1[. We make the same assumptions on T as in [START_REF] Collet | Exponential inequalities for dynamical measures of expanding maps of the interval[END_REF].

1. For each 1 ≤ j ≤ N , the restriction T j of T to ]a j , a j+1 [ is strictly monotonic and can be extented to a function T j belonging to C 2 ([a j , a j+1 ]).

2. Let I n be the set where (T n ) ′ is defined. There exists A > 0 and s > 1 such that

inf x∈In |(T n ) ′ (x)| > As n .
3. The map T is topologically mixing: for any two nonempty open sets U, V , there exists

n 0 ≥ 1 such that T -n (U ) ∩ V = ∅ for all n ≥ n 0 .
If T satisfies 1., 2. and 3., then (5.9) holds. Assume furthermore that (5.10) holds (see [START_REF] Morita | Local limit theorem and distribution of periodic orbits of Lasota-Yorke transformations with infinite Markov partition[END_REF] for sufficient conditions). Then, arguing as in Example 4 in Section 7 of Dedecker & Prieur (2005), we can prove that for the Markov chain (Y i ) i≥0 and the σ-algebras

M i = σ(Y j , j ≤ i), there exists a positive constant C such that θ 2 (i) ≤ Cρ i .
6 Proof of Theorem 3.2

The proof of Theorem 3.2 is decomposed in two parts: first, we prove the convergence of the finite-dimensional distributions of the process (n -3/2 Σ [nt] ) t≥0 , then its tightness in the Skorohod space D([0, +∞[).

Proof of the convergence of the finite-dimensional distributions: Since the random variable Σ n can be rewritten as the sum

i∈Z N n (i)ξ i
where N n (i) is the local time of the random walk S at point i, it is enough to prove that for every m ≥ 1, for any real θ 1 , . . . , θ m , for any 0 ≤ t 1 ≤ t 2 ≤ . . . ≤ t m , the sequence

1 n 3/2 m k=1 θ k Σ [nt k ] = 1 n 3/2 i∈Z m k=1 θ k N [nt k ] (i) ξ i converges in distribution to the random variable i∈Z E(ξ 0 ξ i ) m k=1 θ k ∆ t k .
We only prove the convergence of one-dimensional distributions. The general case is obtained by replacing N n (i) by the linear combination θ k N [nt k ] (i) in the computations. Let G = σ(S k , k ≥ 0) be the σ-field generated by the random walk S. For any n ∈ N and any i ∈ Z, we denote by X n,i the random variable N n (i)ξ i . We first use a classical truncation argument. For any M > 0, we define :

ϕ M : R → R x → ϕ M (x) = (x ∧ M ) ∨ (-M )
and

ϕ M : R → R x → ϕ M (x) = x -ϕ M (x) .
We now prove the following Lindeberg condition :

n -3/2 i∈Z E ϕ εn 3/4 (X n,i ) 2 -----→ n→+∞ 0 . (6.13)
We have, for ε > 0 fixed, for n large enough,

n -3/2 i∈Z E ϕ εn 3/4 (X n,i ) 2 G ≤ E ξ 2 0 1 {|ξ 0 |>εn 3/4 / max j Nn(j)} G α(n,0) n 3/2 =: Σ 1 (ε, n) Let η > 0. We decompose the expectation of Σ 1 (ε, n) as the sum of Σ 1,1 (ε, n) := E 1 {n -3/4 max j Nn(j)≥η} E ξ 2 0 1 {|ξ 0 |>εn 3/4 / max j Nn(j)} G α(n, 0) n 3/2 and Σ 1,2 (ε, n) := E 1 {n -3/4 max j Nn(j)<η} E ξ 2 0 1 {|ξ 0 |>εn 3/4 / max j Nn(j)} G α(n, 0) n 3/2 . Using (ii)(a) of Proposition 4.1, Σ 1,2 (ε, n) is bounded by CE ξ 2 0 1 {|ξ 0 |>ε/η}
From assumption (A 1 ) of Theorem 3.2, for any κ > 0, there exists η 0 > 0 such that the above term is less than κ/2 for any η ≤ η 0 . We now fix η equal to η 0 . Using Cauchy-Schwarz inequality,

Σ 1,1 (ε, n) ≤ E(ξ 2 0 ) P n -3/4 max j N n (j) ≥ η 0 1/2 E α(n, 0) 2 n 3 1/2
From assumption (A 1 ) of Theorem 3.2, (i) and (ii)(a) of Proposition 4.1, it follows that Σ 1,1 (ε, n) ≤ κ/2 for n large enough.

Since ε → E Σ 1 (ε, n) is decreasing, we can find a sequence of positive numbers (ε n ) n≥1 such that ε n -----→ n→+∞ 0, and

E Σ 1 (ε n , n) -----→ n→+∞ 0 (6.14)
Let us now prove that it implies

n -3/2 E i∈Z ϕ εnn 3/4 (X n,i ) -E(ϕ εnn 3/4 (X n,i )|G) 2 -----→ n→+∞ 0 . (6.15)
For any fixed n ≥ 0, and any i ∈ Z such that N n (i) = 0, define :

V n,i = ϕ εnn 3/4 /Nn(i) (ξ i ) -E(ϕ εnn 3/4 /Nn(i) (ξ i )|G)
If N n (i) = 0, let V n,i = 0. As for any fixed n ≥ 0 and any i ∈ Z, the function

x → ϕ εnn 3/4 /Nn(i) (x)
is 1-Lipschitz, we have for any fixed path of the random walk, for all l ≥ 1, for all k ≥ 1,

θ V•,n k,2 (l) ≤ θ ξ k,2 (l) ,
where

V •,n = (V n,i ) i∈Z and ξ = (ξ i ) i∈Z . E j∈Z N n (j)V n,j 2 G = j∈Z N n (j) 2 E(V 2 n,j |G) + i∈Z j∈Z;j =i N n (i) N n (j)E(V n,i V n,j |G) ≤ j∈Z N n (j) 2 E(V 2 n,j |G) + i∈Z N n (i) 2 j∈Z;j =i |E(V n,i V n,j |G)| by remarking that N n (i) N n (j) ≤ 1 2 (N n (i) 2 + N n (j) 2
). Then for any j > i, using Cauchy-Schwarz inequality, we obtain that

|E(V n,i V n,j |G)| = E V n,i E (V n,j |M i ) G ≤ E(V 2 n,i |G) 1/2 E E(V n,j |M i ) 2 G 1/2 ≤ E(V 2 n,i |G) 1/2 θ ξ 1,2 (j -i).
Moreover, as N n (i)V n,i = ϕ εnn 3/4 (X n,i ) -E(ϕ εnn 3/4 (X n,i )|G), we get

n -3/2 E i∈Z ϕ εnn 3/4 (X n,i ) -E(ϕ εnn 3/4 (X n,i )|G) 2 ≤ E C n α(n, 0) n 3/2 , (6.16)
with

C n = sup i∈Z E(V 2 n,i |G) + 2 sup i∈Z E(V 2 n,i |G) ∞ l=1 θ ξ 1,2 (l)
. It remains to prove that the right hand term in (6.16) converges to 0 as n goes to infinity.

We have, for n large enough,

E V 2 n,i G ≤ E ξ 2 i 1 {|ξ i |>εnn 3/4 / max j Nn(j)} G , (6.17)
so using Cauchy-Schwarz inequality,

E C n α(n, 0) n 3/2 ≤ E Σ 1 (ε n , n) + 2 ∞ l=1 θ ξ 1,2 (l) E Σ 1 (ε n , n) 1/2 E α(n, 0) n 3/2 1/2
, which tends to 0 from (ii)(a) of Proposition 4.1, assumption (A 2 ) from Theorem 3.2 and (6.14).

Define Z n,i by ϕ εnn 3/4 (X n,i ) -E ϕ εnn 3/4 (X n,i )|G .

By (6.15) we conclude that, to prove Theorem 3.2, it is enough to prove it for the truncated sequence (Z n,i ) n≥0,i∈Z , that is to show that

1 n 3/4 i∈Z Z n,i D -----→ n→+∞ k E(ξ 0 ξ k ) ∆ 1 . (6.18)
The proof is now a variation on the proof of Theorem 4.1 in [START_REF] Utev | Central limit theorem for dependent random variables[END_REF]. Let

d t (X, Y ) = E(e itX |G) -E(e itY |G) .
Let η be a random variable with standard normal distribution, independent of the random walk (S k ) k≥0 . Let (X i ) i=1,..,3 be random variables such that for i = 1, ..,

3, E(X i |G) = 0 and E(X 2 i |G) is bounded. Let Y 1 , Y 2 be random variables such that E(Y i |G) = 0 and E(Y 2 i |G) is bounded for i = 1, 2.
They are assumed to be independent conditionnally to the random walk. We define

A t (X) = d t X, η E(X 2 |G) .
We first need some simple properties of d t and A t : Lemma 6.1 (Lemma 4.3 in Utev, 1990)

A t (X 1 ) ≤ 2 3 |t| 3 E(|X 1 | 3 |G), A t (Y 1 + Y 2 ) ≤ A t (Y 1 ) + A t (Y 2 ), d t (X 2 + X 3 , X 2 ) ≤ t 2 2 E(X 2 3 |G) + (E(X 2 2 |G)E(X 2 3 |G)) 1/2 , d t (ηa, ηb) ≤ t 2 2 |a 2 -b 2 |.
We next need the following lemma :

Lemma 6.2 Let 0 < ε < 1.
There exists some positive constant C(ε) such that for all a ∈ Z, for all v ∈ N * , A t n -3/4 a+v i=a+1 Z n,i is bounded by

C(ε)   |t| 3 h 2/ε n -9/4 a+v i=a+1 E |Z n,i | 3 |G + t 2   h ε-1 2 + j :2 j ≥h 1/ε 2 3j 2 g(2 jε )   n -3/2 a+v i=a+1 N n (i) 2   ,
where h is an arbitrary positive natural number and with g introduced in Assumption (A 2 ) of Theorem 3.2.

Before proving Lemma 6.2, we achieve the proof of Theorem 3.2. We can decompose

E e it 1 n 3/4 i∈Z Z n,i -E e it √ k E(ξ 0 ξ k )∆ 1 (6.19)
as the sum of I i (n), i = 1, .., 4 where

I 1 (n) = E e it 1 n 3/4 i∈Z Z n,i -E e -t 2 2n 3/2 E ( i∈Z Z n,i) 2 |G I 2 (n) = E e -t 2 2n 3/2 E ( i∈Z Z n,i) 2 |G -E e -t 2 2n 3/2 E ( i∈Z X n,i) 2 |G I 3 (n) = E e -t 2 2n 3/2 E ( i∈Z X n,i) 2 |G -E e -t 2 2n 3/2 i,j∈Z Nn(i) 2 E(ξ i ξ j ) I 4 (n) = E e -t 2 2n 3/2 i,j∈Z Nn(i) 2 E(ξ i ξ j ) -E e it √ k E(ξ 0 ξ k )∆ 1
To prove Theorem 3.2, it is enough to prove that for any i = 1, .., 4, I i (n) goes to 0 as n → +∞. Estimation of I 1 (n) : Let us denote by M n the random variable max k=0,...,n |S k |. From Lemma 6.2, we have

|I 1 (n)| ≤ E A t n -3/4 Mn i=-Mn Z n,i ≤ C(t, ε) h 2/ε n -9/4 i∈Z E(|Z n,i | 3 ) + δ(h) , with δ(h) = h ε-1 2 + j :2 j ≥h 1 ε 2 3j 2 g(2 jε ) E α(n,0) n 3/2 .
Hence using assumption (A 2 ) from Theorem 3.2 and (ii)(a) from Proposition 4.1, we get δ(h) ----→ h→+∞ 0.

On the other hand, from assumption (A 1 ) of Theorem 3.2, there exists a constant C > 0 such that

n -9/4 i∈Z E(|Z n,i | 3 ) ≤ Cε n E α(n, 0) n 3/2 (6.20)
which tends to zero as n tends to infinity, using (ii)(a) from Proposition 4.1 and the fact that

ε n -----→ n→+∞ 0. Consequently inf h≥1 h 2/ε i∈Z E(|Z n,i | 3 ) + δ(h) -----→ n→+∞ 0.
Estimation of I 2 (n) : Using that for any x, y ≥ 0, |e -xe -y | ≤ |x -y| and the fact that

Z n,i = X n,i -[ϕ εnn 3/4 (X n,i ) -E(ϕ εnn 3/4 (X n,i )|G)],
we deduce that

|I 2 (n)| ≤ t 2 n 3/2 E i∈Z ϕ εnn 3/4 (X n,i ) -E(ϕ εnn 3/4 (X n,i )|G) 2 + E i∈Z X n,i 2 1/2 E i∈Z ϕ εnn 3/4 (X n,i ) -E(ϕ εnn 3/4 (X n,i )|G) 2 1/2   Since we have E i∈Z X n,i 2 = i,j∈Z E(N n (i)N n (j))E(ξ i ξ j ) = i∈Z E(α(n, i) -α(n, 0))E(ξ 0 ξ i ) + E(α(n, 0)) i∈Z E(ξ 0 ξ i ) ≤ Cn 3/2
by combining (ii)(a)-(iii) of Proposition 4.1 and (3.5). Then, from (6.15), we deduce that I 2 (n) converges to 0.

Estimation of I 3 (n) : We have

|I 3 (n)| ≤ t 2 2n 3/2 i∈Z E(|α(n, i) -α(n, 0)|)|E(ξ 0 ξ i )| = o(
E e -t 2 2n 3/2 i,j∈Z Nn(i) 2 E(ξ i ξ j ) converges to E e -t 2 2 i∈Z E(ξ 0 ξ i ) R L 2 1 (x) dx which is equal to the characteristic function of the random variable i∈Z E(ξ 0 ξ i )∆ 1 .
Proof of Lemma 6.2: Let h ∈ N * . Let 0 < ε < 1. In the following, C, C(ε) denote constants which may vary from line to line. Let κ ε be a positive constant greater than 1 which will be precised further. Let v < κ ε h 1 ε . We have

A t n -3/4 a+v i=a+1 Z n,i ≤ 2|t| 3 3n 9/4 E   a+v i=a+1 Z n,i 3 G   ≤ 2 3n 9/4 κ 2 ε |t| 3 h 2/ε a+v i=a+1 E(|Z n,i | 3 |G) (6.21) since |x| 3 is a convex function. Let now v ≥ κ ε h 1 ε . Without loss of generality, assume that a = 0. Let δ ε = (1-ε 2 +2ε)/2. Define then m = [v ε ], B = u ∈ N : 2 -1 (v -[v δε ]) ≤ um ≤ 2 -1 v , A =    u ∈ N : 0 ≤ u ≤ v, (u+1)m i=um+1 N n (i) 2 ≤ (m/v) ε v i=1 N n (i) 2    .
Following [START_REF] Utev | Sums of random variables with ϕ-mixing[END_REF] we prove that, for 0 < ε < 1, A ∩ B is not wide for v greater than κ ε . We have indeed

|A ∩ B| = |B| -|A ∩ B| ≥ |B| -|A| ≥ v (1-ε 2 )/2 2 1 -4v -(1-ε) 2 /2 - 3 2 ,
where A denotes the complementary of the set A. We can find κ ε large enough so that |A ∩ B| be positive. Let u ∈ A ∩ B. We start from the following simple identity

Q ≡ n -3/4 v i=1 Z n,i = n -3/4 um i=1 Z n,i + n -3/4 (u+1)m i=um+1 Z n,i + n -3/4 v i=(u+1)m+1 Z n,i ≡ Q 1 + Q 2 + Q 3 . (6.22)
For any fixed n ≥ 0, and any i ∈ Z such that N n (i) = 0, define :

W n,i = ϕ εnn 3/4 /Nn(i) (ξ i ) -E ϕ εnn 3/4 /Nn(i) (ξ i ) G .
If N n (i) = 0, let W n,i = 0. As for any fixed n ≥ 0, i ∈ Z, the function

x → ϕ εnσn/Nn(i) (x) -E ϕ εnσn/Nn(i) (ξ i ) G
is 1-Lipschitz, we have for any fixed path of the random walk, for all l ≥ 1, for all k ≥ 1,

θ W•,n k,2 (l) ≤ θ ξ k,2 (
l) , where W •,n = (W n,i ) i∈Z and ξ = (ξ i ) i∈Z . Hence, arguing as for the proof of inequality (6.16), we prove that for any fixed n, any a, b ∈ N,

E | b i=a Z n,i | 2 G ≤ C b i=a N n (i) 2 (6.23) with C = 2E(ξ 2 0 ) + 2 √ 2E(ξ 2 0 ) 1/2 ∞ l=1 θ 1,2 (
l) which is finite from assumptions (A 1 ) and (A 2 ). By Lemma 6.1,

d t (Q, Q 1 + Q 3 ) = d t (Q, Q -Q 2 ) ≤ t 2 2 E(Q 2 2 |G) + E(Q 2 2 |G) 1/2 E(Q 2 |G) 1/2 . (6.24)
Using (6.24) and (6.23), we get

d t (Q, Q 1 + Q 3 ) ≤ Ct 2 v (ε-1)ε 2 n -3/2 v i=1 N n (i) 2 . (6.25)
Now, given the random variables Q 1 and Q 3 , we define two random variables g 1 and g 3 which are assumed independent conditionally to the random walk (S k ) k≥0 such that conditionnally to the random walk, the distribution of g i coincides with that of Q i , i = 1, 3. We have

d t (Q 1 + Q 3 , g 1 + g 3 ) = E((e itQ 1 -1)(e itQ 3 -1)|G) -E(e itQ 1 -1|G)E(e itQ 3 -1|G) ≤ E(|e itQ 1 -1| 2 |G) 1/2 E E e itQ 3 -1 -E(e itQ 3 -1|G) | M um ; G 2 |G ≤ 2|t|n -3/4 E | um i=1 Z n,i | 2 | G 1/2 v |t| n -3/4 v i=(u+1)m+1 N n (i) θ ξ 2 (m + 1) ≤ C t 2 v 3/2 n -3/2 v i=1 N n (i) 2 g(v ε )
by (6.23), Definition 2.2 and Assumption (A 2 ) of Theorem 3.2. Hence (6.26) where f (v) = v 3/2 g(v ε ) is non-increasing by assumption (A 2 ) of Theorem 3.2. We also have by Lemma 6.1

d t (Q 1 + Q 3 , g 1 + g 3 ) ≤ Ct 2 f (v)n -3/2 v i=1 N n (i) 2 ,
A t (g 1 + g 3 ) ≤ A t (g 1 ) + A t (g 3 ). (6.27)

Finally, still by Lemma 6.1, and using Definition 2.2, we have

d t η E(Q 2 |G), η E ((g 1 + g 3 ) 2 |G) ≤ t 2 2 E(Q 2 |G) -E (g 1 + g 3 ) 2 |G (6.28) ≤ t 2 2 E(Q 2 2 |G) + 2E(Q 1 Q 2 |G) + 2E(Q 2 Q 3 |G) + 2E(Q 1 Q 3 |G) ≤ Ct 2 v (ε-1)ε 2 + f (v) n -3/2 v i=1
N n (i) 2 . (6.29)

Combining (6.25)-(6.29), we get the following recurrent inequality :

A t n -3/4 v i=1 Z n,i ≤ A t n -3/4 um i=1 Z n,i + A t n -3/4 v i=(u+1)m+1 Z n,i + Ct 2 v (ε-1)ε 2 + f (v) n -3/2 v i=1 N n (i) 2
for v ≥ κ ε h 1 ε ≥ κ ε . We then need the following Lemma, which is a variation on Lemma 1.2. in [START_REF] Utev | Sums of random variables with ϕ-mixing[END_REF]. Lemma 6.3 For every ε ∈ ]0, 1[, denote δ ε = (1ε 2 + 2ε)/2. Let a non-decreasing sequence of non-negative numbers a(n) be specified, such that there exist non-increasing sequences of non-negative numbers ε(k), γ(k) and a sequence of naturals T (k), satisfying conditions Proof of Lemma 6.3: The proof follows essentially the same lines as the proof of Lemma 1.2. in [START_REF] Utev | Sums of random variables with ϕ-mixing[END_REF] and therefore is omitted here. △

T (k) ≤ 2 -1 (k + [k δε ]),
We now apply Lemma 6. A t n -3/4 l+i j=l+1 Z n,j n -3/2 l+i j=l+1 N n (j) 2

.

Applying Lemma 6.3 yields the statement of Lemma 6.2.

Proof of the tightness: By Theorem 13.5 of [START_REF] Billingsley | Convergence of probability measures, 2nd edition[END_REF], it suffices to prove that there exists K > 0 such that for all t, t Using Cauchy-Schwarz inequality, it is enough to prove that there exists K > 0 such that for all t 1 ≤ t, for all n ≥ 1,

E |Σ [nt] -Σ [nt 1 ] | 2 ≤ Kn

Proposition 4 . 1

 41 (i) The sequence n -3/4 max i∈Z N n (i) converges in probability to 0.

  an interval, and there exists γ > 0 such that f µ > γ -1 on I * . (5.10) Without loss of generality assume that I * = I (otherwise, take the restriction to I * in what follows). Define now the Markov kernel associated to T by

  1), by combining (iii) of Proposition 4.1 and (3.5). Estimation of I 4 (n) : From (ii)(b) of Proposition 4.1, we know that the sequence

  a(k) ≤ max k 0 ≤s≤k (a(T (s)) + γ(s)) for any k ≥ k 0 with an arbitrary k 0 ∈ N * . Then a(n) ≤ a(n 0 ) + 2 k 0 ≤2 j ≤n γ(2 j ) ,for any n ≥ k 0 , where one can take n 0 = 2 c with c > 2-δε 1-δε .

3 above with ⋆ k 0 = κ ε h 1 ε⋆ 2 +

 12 , for k ≥ k 0 , T (k) = max {u k m k , ku k m km k }where u k and m k are defined from k as u and m from v (see the proof of A ∩ B not wide), ⋆ c < ln(κε) ln(2)(we may need to enlarge κ ε ),⋆ for s ≥ k 0 , γ(s) = C t 2 s ε(ε-1) f (s) , ⋆ for s ≥ k 0 , a(s) = sup l∈Z max k 0 ≤i≤s

31 ) 2 =

 312 Since ab ≤ (a 2 + b 2 )/2 for any real a, b, we haveE |Σ [nt] -Σ [nt 1 ] | 2 ≤ i∈Z |E(ξ 0 ξ i )| j∈Z E (N [nt] (j) -N [nt 1 ] (j)) CE α [nt] -[nt 1 ] -1, 0 ≤ Cn 3/2 |tt 1 | 3/2using (3.5), the Markov property for the random walk S and (ii)-(a) of Proposition 4.1. △

  1 , t 2 ∈ [0, T ], T < ∞, s.t. t 1 ≤ t ≤ t 2 , for all n ≥ 1, E |Σ [nt] -Σ [nt 1 ] | • |Σ [nt 2 ] -Σ [nt] | ≤ Kn 3/2 |t 2t 1 |

	3 2 .	(6.30)


Hervice Kouefouet

Jeanne Sandrine

Takam Mabekou

Joseph Albert Mukam Fotsing

Pierre Kisito Talla

Rostand Moutou Pitti

M-THETA METHOD ON MODE I AND II FAILURE OF TWO CAMEROONIAN HARDWOODS IN BENDING

Keywords: Mode I and II failure, Mtheta method, Finite element analysis, SENB specimen I

This work deals with the numerical simulation on bending test to characterize two Cameroonian hardwoods under mode I and II loading for different crack lengths. The finite element analysis for fracture in orthotropic medium is developed. The algorithm of fracture is introduced in a finite element software Cast3M. According to the Mtheta method, the calculation of the stress intensity factors and the energy release rate for pure mode I and II fracture are deduced using a SENB (Single Edge Notch Bending) specimen. The path independence of Mθ-method on the specimen is confirmed.

INTRODUCTION

The wood species lovoa trichilioides and triplochiton scleroxylon are subject of an increasingly high exploitation and exportation each year in Cameroon and belong to the first five types of hardwoods exploited in the country and used in the various ways [1], [START_REF] Kouefouet | Probabilistic approach of the failure of lovoa trichilioides and triplochiton scleroxylon[END_REF]. They are very little used for construction, but important for socioeconomic function of the country.

Often used in an empirical way, Cameroonian woods are the source of many security and dimension problems, for instance when they are used for the construction of bridges and heavy roof frames. In this context, wood material has a special interest to revive construction sector. Furthermore, due to the economic cost, the design of structure is required not for oversized dimension, but to respect the limit range of the security. Therefore, fracture behavior of wood is important to be more investigated to better understand the crack process in the structure [3].

In this regard, many work have been done to evaluate the characteristics of fracture and their consequences on wood structure facing crack problems. Reference [4] investigated in 2002 the influence of the physical and mechanical characteristics on the behavior of wood during machining. The determination of fracture energy and fracture process zone length in Mode I fracture of European spruce wood were reported by [START_REF] Aicher | Process zone length and fracture energy of spruce wood in Mode-I from size effect[END_REF]. Reference [START_REF] Alami | Fracture energy of wood and root burl wood of thuya (Tetraclinis articulata)[END_REF] work on fracture energy of wood and root burl wood of Thuya. The mode I fracture of tropical woods using grid method has been developed by [START_REF] Odounga | Mode I fracture of tropical woods using grid method[END_REF]. The Influence of loading rate on the mode II fracture toughness of vinyl ester GRP where investigated by [START_REF] Brambleby | Influence of loading rate on the mode II fracture toughness of vinyl ester GRP[END_REF] in 2017. In this work bending simulation used the Mtheta method introduced by [9]- [START_REF] Pitti | Stress Intensity Factors for Viscoelastic Axisymmetric Problems Applied to Wood[END_REF] in the plane constrain for linear elastic fracture mechanics to model crack growth on two Cameroonian hardwoods.

II.

LOCAL MECHANICAL FIELDS IN ORTHOTROPIC MEDIUM FOR PLANE CRACK II.1.

FIELDS OF STRESSES AND DISPLACEMENTS

We considered a thin solid with a constant width subjected to the applied forces in his contour as shown in Fig. 1. The stresses in the z direction is neglected and are too small in front of

.

                         
The compatibility equation become with Airy's stress function the equation [START_REF] Aicher | Process zone length and fracture energy of spruce wood in Mode-I from size effect[END_REF]

             
By setting a change of variables [START_REF] Boulazreg | Modélisation de la fissuration fragile par une formulation mixte hybride ; évaluation des coefficients de concentration de contraintes et du taux de restitution d'énergie[END_REF], [START_REF] Dubois | Modélisation du comportement mécanique des milieux viscoélastiques fissurés : application au matériau bois[END_REF] as follow below,  

i i i Z x y Z x y Z Z i x y                   1 1 2 2 2. (7) e Z Z          
The combination of equation ( 5), ( 6) and [START_REF] Odounga | Mode I fracture of tropical woods using grid method[END_REF]

                     
The elements 11 12 22 33 , , , S S S S are the elastic compliance tensor components for an orthotropic symmetry. Considering the orthotropic directions (L, R, T), the elastic compliance tensor components are directly related to the elastic orthotropic characteristics for an opening mode solicitation in (R, L) plane with the crack oriented in the L direction of fiber. In the plane stress their expression is stated in the form are respectively the shear module and Poisson coefficient in the considering direction x and y. From ( 5) and considering the equilibrium equations, the field of stresses is stated as:

    2 2 1 1 2 2 1 2 1 1 2 2 2. '( ) '( ) 2. '( ) '( ) (10) 2.
'( ) '( )

x y xy e z z e z z e z z                                 
Where ϕ' and χ' denote the derivative.

By combining [START_REF] Pitti | A new M-integral parameter for mixed-mode crack growth in orthotropic viscoelastic material[END_REF] and ( 4), we obtain the deformation and by integration we deduced the displacement as follow:

      1 1 1 2 2 2 1 1 2 2 2 22 11 12 12 2. ( ) ( ) (11) 2.
( ) ( ) , 1; 2

i i i i i u
e p z p z u e q z q z S with p S S and q S i

                       II.2. RELATION BETWEEN STRESS INTENSITY FACTOR,

DISPLACEMENTS AND STRESSES

From Airy's function, let set [START_REF] Boulazreg | Modélisation de la fissuration fragile par une formulation mixte hybride ; évaluation des coefficients de concentration de contraintes et du taux de restitution d'énergie[END_REF] in order to deduce the relation between the stress intensity factor and stress.

()

n N n n n N n n i Z A Z Z B Z With Z x iy re                    1 2 1 2 (12 ) n n n n n n A a ia a B b ib          Fig. 2. Crack orientation.
Taking the surface area of the crack free in tension (Fig. 2), for ϴ = ± π permit to deduce that 0 [START_REF] Dubois | Modélisation du comportement mécanique des milieux viscoélastiques fissurés : application au matériau bois[END_REF] y xy   Therefore ( 10) and ( 13) imply that,

          1 1 2 2 1 1 2 2 1 '( ) 1 '( ) 1 '( ) 1 '( ) (14) , 1; 2 ,
y xy i j j j i z z z z with i j Z xe for                              
For each values of ϴ, ( 13) and ( 14) can be presented as follow:

                1 1 2 2 1 1 2 2 1 1 1 1 0 (15) 1 1 1 1 0 n i n i n n n n n i n i n n n n A A e B B e A A e B B e                                   
From [START_REF] Rice | A path independent integral and the approximate analysis of strain concentration by notches and cracks[END_REF] the relation between

n n
A and B more over between

1 1 2 2 , , (15 )
n n n n a b and a b a Is obtained by imposing that deformation energy should be finite at the vicinity of the crack, therefore n = 1 [START_REF] Boulazreg | Modélisation de la fissuration fragile par une formulation mixte hybride ; évaluation des coefficients de concentration de contraintes et du taux de restitution d'énergie[END_REF], [START_REF] Dubois | Modélisation du comportement mécanique des milieux viscoélastiques fissurés : application au matériau bois[END_REF]. By comparing to the isotropic medium, the stress intensity factor is related to the field of stresses by [START_REF] Pitti | Mixed mode fracture separation in viscoelastic orthotropic media: numerical and analytical approach by the Mθv-integral[END_REF]. 16) proves that the singular field of stresses is dependent on the characteristic of materials in contrary to the isotropic case. Also, the stress intensity factor is related to the field of displacement by:

2 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 1 2 2 1 2 1 1 2 1 ( ) . ( ) 1 2 2 ( ) ( ) 1 1 1 1 2 2 ( ) . 2 I II x I II y I xy K K e e r r K K e e r r K e r                                                                                                  1 2 2 1 2 2 1 2 1 (16) () 1 1 1 2 II K e r                                                                cos( ) sin( ) , 1;2 (16 )
j j i j a        Equation (
        1 2 2 2 1 1 2 2 1 1 1 2 1 2 1 2 2 2 1 1 2 2 1 1 1 2 1 2 1 1 2 ( ) . 2 ( ) . 2 2 (17) 1 1 2 ( ) . 2 ( ) . 2 2 x I II y I II r r u K e p p K e p p r r u K e q q K e q q                                                                            j j
p and q are defined as stated in [START_REF] Pitti | Stress Intensity Factors for Viscoelastic Axisymmetric Problems Applied to Wood[END_REF].

From [START_REF] Dubois | Modélisation du comportement mécanique des milieux viscoélastiques fissurés : application au matériau bois[END_REF], the crack opening intensity factor and the stress intensity factor are related. Their expressions are given by the relation ( 18):

( ) . ( ) ( ) . ( ) (18)
I I I II II II K C K and K C K      
Where C I and C II [START_REF] Malki | Influence du chargement sur la propagation en fatigue de fissures courtes dans un acier de construction navale[END_REF] are the reduced elastic compliances for orthotropic behavior develop by [START_REF] Dubois | Modélisation du comportement mécanique des milieux viscoélastiques fissurés : application au matériau bois[END_REF] and cited by [START_REF] Pitti | Mixed mode fracture separation in viscoelastic orthotropic media: numerical and analytical approach by the Mθv-integral[END_REF] and [START_REF] Meite | Caractérisation des paramètres de fissuration par un couplage corrélation d'images et éléments finis[END_REF], which derive from the crack opening displacement vectors x u and y u for each mode of failure considered (Mode I and Mode II).

2 1 1 2 2 1 1 2 1 2 4. 4. ( 19
) I II q q p p C e i and C e i                           III.

MTHETA-METHOD IN PROPAGATION

J-integral [START_REF] Rice | A path independent integral and the approximate analysis of strain concentration by notches and cracks[END_REF], [START_REF] Rice | A path independent integral and the approximate analysis of strain concentration by notches and cracks[END_REF], is one of the widely applied in rate-independent quasi-static fracture analysis, as a parameter characterizing crack tip field. It could be used for linear-elastic as well as elastic-plastic material behavior and the definition is given by ( 20):

(20)

x xy yx y u v u v J G W dy dx x x x x                                       Fig. 3. Contour of integration.
Г is an arbitrary curvilinear contour oriented by its normal vector n surrounding the crack type in the external failure zone, W is the strain energy density, σ ij is the component stress and u i is the displacement vector. The integral form supposes that the crack is oriented in the x direction Fig. 3. However, energetic criterions necessitate a mixed-mode separation in order to isolate open and shear cinematic effects. In this regard, the generalization of the M-integral to orthotropic material has been developed by [START_REF] Pitti | A new M-integral parameter for mixed-mode crack growth in orthotropic viscoelastic material[END_REF]. M-integral takes the following notation: Finite element analysis (FEA) is employed to transform this form in terms of a surface integral by introducing a θ scalar field which is continuous and differentiable for θ =1 and θ = 0 inside and outside the ring as it is shown in Fig. 4. The vector field  r must be continuous and differentiable on the considered S domain. Then, the general modeling form of integral is given by [START_REF] Pitti | Mixed mode fracture separation in viscoelastic orthotropic media: numerical and analytical approach by the Mθv-integral[END_REF], [START_REF] Meite | Caractérisation des paramètres de fissuration par un couplage corrélation d'images et éléments finis[END_REF].

  , , , 1 . . (22) 2 u v ij i k ij k i k j S M v u dS           
where the virtual field v is given by the Sih's singular form presented in equation [START_REF] Meite | Caractérisation des paramètres de fissuration par un couplage corrélation d'images et éléments finis[END_REF].

According to Dubois cited by [START_REF] Pitti | Mixed mode fracture separation in viscoelastic orthotropic media: numerical and analytical approach by the Mθv-integral[END_REF], [START_REF] Meite | Caractérisation des paramètres de fissuration par un couplage corrélation d'images et éléments finis[END_REF], the superposition principle enables to write: . . . . (23) [START_REF] Brambleby | Influence of loading rate on the mode II fracture toughness of vinyl ester GRP[END_REF] 8

I I II II I II K K K K M C C   
where C I and C II are the reduced elastic compliances in mode I and mode II defining in equation ( 19), K I and K II are respectively the real stress intensity factors for opening mode and shear mode. Replacing virtual and real fields, the Mθ-integral is assimilated to the energy release rate as follow:

I II I II I II I II M K K M K K K and K C C         1; 0) 8. ( 0; 1) (25)
Expression (25) inside (24), compute the energy release rate part for each mode.

IV. GEOMETRY OF SENB SIMULATION SPECIMEN

The geometry of the specimen is firstly generated according to the SENB specimen. This specimen is adapted to model easily in three point bending the evolution of crack. The obtained specimen dimensions are presented in Fig. 5 with the following values base on [START_REF] Lahlou | Procedure for the numerical modeling of the specimen (SENB) using CAST3M calculation code[END_REF] work. Fig. 6. Finite element mesh of specimen: (i) Mode I deformation shape, (ii) Mode II deformation shape, (iii) Open mode virtual displacement, (iv) Shear mode virtual displacement, (v) Circular mesh around crack tip.

The geometry is calculated using a plane stress assumption, using a unitary loading force corresponding to the determination of stress intensity factor and the energy release rate values. This finite element mesh in which a circular discretization around the crack tip (Fig. 6 (v)), allows to easily define the θ vector field and the integration crown using parametric elements. Mθ-integral performed the virtual finite element displacement fields (Fig. 6 (iii) and

(iv)) and deformation (Fig. 6 (i) and (ii)) for open and shear mode respectively. The computation is limited for an elastic response according to 2-D formulation and the characteristics of the hardwoods are taken as follow:

     
The hardwoods Lovoa trichilioides properties are cited in (28) with the same components define previously.

V.

RESULTS AND DISCUSSIONS

The simplified numerical routine (flow diagram) used to compute the fracture parameters based on Mθ-integral is sketched in Fig. 7. The Fig. 8 below evaluates the path independence of the methods on the specimen according to crown. By considering different crack orientations, it is observed the unvarying results versus different crowns for energy release rate evolution. Across singularity in the crack tip or near zero the mechanical fields cause a disturbance through the integration process for the crown 0, which is the origin of the perturbation. Nevertheless, there is a great stability with the increase of crown sizes. This satisfied the theoretical assumption of path independence in agreement with [START_REF] Malki | Influence du chargement sur la propagation en fatigue de fissures courtes dans un acier de construction navale[END_REF] work.

From the result, in what follows crown six, will be used to performed calculations of the stress intensity factors and energy release rate. obtained by [START_REF] Benguediab | Modélisation par éléments finis du comportement d'un polymère semi cristallin : Polyéthylène à haute densité[END_REF], [START_REF] Fischer-Cripps | Introduction to contact mechanics[END_REF]. Hence the total energy of the two wood is reduced to the released strain energy consequences of the decrement and induced their brittle fracture. This type of result would occur in a "fixed-grips" apparatus with the applied load and the apparatus clamped into position. The same arguments can be exactly being applied for a "dead-weight" loading, where the fracture surface energy corresponds to a decrease in potential energy of the loading system [START_REF] Fischer-Cripps | Introduction to contact mechanics[END_REF]. The apparent drop in mode II toughness of stress intensity factor and energy release rate (Fig. 9 (iii), (iv)) and (Fig. 10 (c),(d)) observed as crack length increases for the two hardwoods were obtained by [START_REF] Brambleby | Influence of loading rate on the mode II fracture toughness of vinyl ester GRP[END_REF] on four-point bending end notched flexure (4ENF) tests. The causes of this behavior is not yet known but, [START_REF] Brambleby | Influence of loading rate on the mode II fracture toughness of vinyl ester GRP[END_REF] observed that the crack tends to open under large displacements and tended to migrate away from the mid-thickness of the specimen and towards face of tension. They remark that mode I fracture in addition induce to the intended mode II fracture for the opening of the crack and the measured of fracture toughness is modify by the mixed mode. In this respect, this study reveals the decreasing of the energies for the both two modes of fracture; which imply future investigation of the effect of mixed mode to overcome some conclusion on these behaviors.

VI. CONCLUSION AND PERSPECTIVE

In this study, the evolution of energy release rate and stress intensity factors of two woods species have been investigated on bending. The finite element mesh of the two hardwoods in different modes were presented. The independently computation results of the stress intensity factor and the energy release rate for mode I and mode II fracture are proved. The pathindependence of Mθ-integral result have been showed. It has been observed the quasi decrement of energy release rate and stress intensity factors of the both two hardwoods species. The future investigation will be consecrated to study mixes mode fracture on the two hardwoods to evaluate his effect and finally it will be important to make an experimental setup result to compare with the numerical results.

Fig. 1 .

 1 Fig. 1. Solid in a plane stress.

  xy  satisfies the biharmonic equation (3) for isotropic medium. For orthotropic case let consider the orthotropic principal planes and (4) give the relation between stress and strain.

Fig. 4 .

 4 Fig. 4. Domain of integration.

  K II can be obtained independently, let perform two distinct calculations from the Mϴ-

Fig. 5 .

 5 Fig. 5. Dimension of specimen.

Fig. 7 .

 7 Fig. 7. Flow diagram on computing fracture parameters.

Fig. 8 .

 8 Fig. 8. Path independence on triplochiton scleroxylon and lovoa trichilioides: (a), (b) Mode I and (c), (d) Mode II.

Fig. 9 .

 9 Fig. 9. (i), (iii) Energy release rate and (ii), (iv) Stress intensity factor versus crack growth position for lovoa trichilioides in mode I and II.

Fig. 10 .

 10 Fig. 10. (a), (c) Energy release rate and (b), (d) Stress intensity factor versus crack growth position for triplochiton scleroxylon in mode I and II.