David Mercier
email: david.mercier@univ-artois.fr

Éric Lefèvre
email: eric.lefevre@univ-artois.fr

François Delmotte
email: francois.delmotte@univ-artois.fr

Belief Functions Contextual Discounting and Canonical Decompositions

Keywords: Belief Functions, Discounting, Reinforcement, Canonical Decompositions

In this article, the contextual discounting of a belief function, a classical discounting generalization, is extended and its particular link with the canonical disjunctive decomposition is highlighted. A general family of correction mechanisms allowing one to weaken the information provided by a source is then introduced, as well as the dual of this family allowing one to strengthen a belief function.

Introduction

In the Dempster-Shafer theory of belief functions [START_REF] Dempster | Upper and Lower Probabilities Induced by Multivalued Mapping[END_REF][START_REF] Shafer | A mathematical theory of evidence[END_REF], the reliability of a source of information is classically taken into account by the discounting operation [21, page 252], which transforms a belief function into a weaker, less informative one. This operation is usually important in uncertain information management [START_REF] Bloch | Defining belief functions using mathematical morphology -Applications to image fusion under imprecision[END_REF][START_REF] Delmotte | Detection of defective sources with belief functions[END_REF][START_REF] Elouedi | Assessing sensor reliability for multisensor data fusion with the transferable belief model[END_REF][START_REF] Fabre | Presentation and description of two classification methods using data fusion based on sensor management[END_REF][START_REF] Ha-Duong | Hierarchical fusion of expert opinions in the Transferable Belief Model, application to climate sensitivity[END_REF][START_REF] Klein | Hierarchical and conditional combination of belief functions induced by visual tracking[END_REF][START_REF] Mercier | Decision fusion for postal address recognition using belief functions[END_REF][START_REF] Muñoz-Salinas | Multi-camera people tracking using evidential filters[END_REF][START_REF] Milisavljević | Improving mine recognition through processing and Dempster-Shafer fusion of ground-penetrating radar data[END_REF][START_REF] Périsse | Robust Diagnostics of Stator Insulation Based on High Frequency Resonances Measurements[END_REF][START_REF] Smets | Data Fusion in the Transferable Belief Model[END_REF].

Introduced in [START_REF] Mercier | Refined modeling of sensor reliability in the belief function framework using contextual discounting[END_REF], the contextual discounting is a refinement of the discounting operation. It takes into account the fact that the reliability of a source of information can be expected to depend on the true answer of the question of interest.

For instance, in medical diagnosis, depending on his/her speciality, experience or training, a physician may be more or less competent to diagnose some types of diseases. Likewise, in target recognition, a sensor may be more capable of recognizing some types of targets while being less effective for other types.

In this contextual model, the agent in charge of the fusion process or the decision making can hold knowledge regarding the reliability of a source of information in different contexts, which forms a partition of the universe of discourse. For example, a sensor in charge of recognizing targets can be more or less reliable depending on the fact that the target is a helicopter (ℎ), an airplane ( ) or a rocket( ), subsets {ℎ}, { } and { } forming the finest partition of the universe Ω = { , , }). However, this previous model can not handle a reliability knowing that the target is a helicopter or a rocket ({ℎ, }) as well as a reliability knowing that the target is an airplane or a helicopter ({ , ℎ}). Sets { , ℎ} and {ℎ, } do not form a partition of Ω. This last step is reached in this article. It is then shown that the contextual discounting exposed previously in [START_REF] Mercier | Refined modeling of sensor reliability in the belief function framework using contextual discounting[END_REF] is a particular case of a more general correction process [START_REF] Mercier | Belief function correction mechanisms[END_REF][START_REF] Mercier | A parameterized family of belief functions correction mechanisms[END_REF] allowing the discounting of a belief function in a finer way. In particular, a simple expression of this mechanism is given in the form of disjunctive combinations. At last, the dual version of the contextual discounting, allowing one to reinforce a belief function, is also introduced. This article extends deeply a first version of this work presented in [START_REF] Mercier | Extending the contextual discounting of a belief function thanks to its canonical disjunctive decomposition[END_REF].

To develop the justifications of these mechanisms, belief functions are interpreted as expressing weighted opinions, irrespective of any underlying probability distributions, and the Transferable Belief Model [START_REF] Smets | The Transferable Belief Model[END_REF][START_REF] Smets | What is Dempster-Shafer's model? In Advances in the theory of evidence[END_REF][START_REF] Smets | The Transferable Belief Model for quantified belief representation[END_REF] is adopted.

This article is organized as follows. Background material needed on belief functions is recalled in Section 2. Contextual discounting is extended in Section 3. A discussion is next launched in Section 4. A dual reinforcement process is introduced in Section 5, and finally, Section 6 concludes this article.

Belief functions: basic concepts

Representing information

Let us consider an agent in charge of making a decision regarding the answer to a given question of interest.

Let Ω = { 1 , . . . , }, called the frame of discernment, be the finite set containing the possible answers to question .

The information held by agent regarding the answer to question can be quantified by a basic belief assignment (BBA) or a mass function Ω , defined as a function from 2 Ω to [0, 1], and verifying:

∑ ⊆Ω Ω ( ) = 1 . (1)
Function Ω describes the state of knowledge of agent regarding the answer to question belonging to Ω. By extension, it also represents an item of evidence that induces such a state of knowledge. The quantity Ω ( ) is interpreted as the part of the unit mass allocated to the hypothesis: "the answer to question is in the subset of Ω".

When there is no ambiguity, the full notation Ω will be simplified to Ω , or even .

The following definitions and notations are considered.

Definition 1 (Focal element). A subset of Ω such that ( ) > 0 is called a focal element of .
Definition 2 (Categorical BBA). A BBA with only one focal element is said to be categorical and is denoted ; we thus have ( ) = 1.

Definition 3 (Vacuous mass function).

Total ignorance is represented by the BBA Ω , called the vacuous mass function.

Definition 4. A BBA is said to be:

• dogmatic if (Ω) = 0;
• non-dogmatic if (Ω) > 0;

• normal if (∅) = 0;

• subnormal if (∅) > 0;

• simple if has no more than two focal sets, Ω being included.

Definition 5 (Negation of a BBA). Function denotes the negation of [START_REF] Dubois | A set-theoretic view of belief functions: logical operations and approximations by fuzzy sets[END_REF], defined by ( ) = ( ), for all ⊆ Ω such that is the complement of in Ω. Definition 6. The belief, plausibility, implicability and commonality functions associated with a mass function are defined, respectively, as:

( ) = ∑ ∅∕ = ⊆ ( ), (2)
( ) = ∑ ∩ ∕ =∅ ( ), (3)
( ) = ( ) + (∅) = 1 -( ), (4)
and

( ) = ∑ ⊇ ( ), (5)
for all ⊆ Ω. Functions , , and are all in one-to-one correspondence [START_REF] Smets | The application of the matrix calculus to belief functions[END_REF], and represent then the same information.

Combining pieces of information

Two BBAs 1 and 2 induced by distinct and reliable sources of information can be combined using the conjunctive rule of combination (CRC), also referred to as the unnormalized Dempster's rule of combination, defined for all ⊆ Ω by:

1 ∩ ⃝ 2 ( ) = ∑ ∩ = 1 ( ) 2 ( ) . (6)
Alternatively, if we only know that at least one of the sources is reliable, BBAs 1 and 2 can be combined using the disjunctive rule of combination (DRC), defined for all ⊆ Ω by:

1 ∪ ⃝ 2 ( ) = ∑ ∪ = 1 ( ) 2 ( ) . (7)

Marginalization and vacuous extension on a product space

A mass function defined on a product space Ω × Θ may be marginalized on Ω by transferring each mass Ω×Θ ( ) for ⊆ Ω × Θ to its projection on Ω:

Ω×Θ↓Ω ( ) = ∑ ⊆Ω×Θ, Proj( ↓Ω)= Ω×Θ ( ), (8)
for all ⊆ Ω where Proj( ↓ Ω) denotes the projection of onto Ω. Conversely, it is usually not possible to retrieve the original BBA Ω×Θ from its marginal Ω×Θ↓Ω on Ω. However, the least committed, or least informative BBA [START_REF] Smets | Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem[END_REF] such that its projection on Ω is Ω×Θ↓Ω may be computed. This defines the vacuous extension of Ω in the product space Ω × Θ, noted Ω↑Ω×Θ , and given by:

Ω↑Ω×Θ ( ) = { Ω ( ) if = × Θ, ⊆ Ω, 0 otherwise. (9)
2.4. Conditioning and ballooning extension on a product space Conditional beliefs represent knowledge that is valid provided that an hypothesis is satisfied. Let be a mass function and ⊆ Ω an hypothesis; the conditional belief function [ ] is given by:

[ ] = ∩ ⃝ . ( 10
)
If Ω×Θ is defined on the product space Ω × Θ, and is a subset of Θ, the conditional BBA Ω [ ] is defined by combining Ω×Θ with Θ↑Ω×Θ , and marginalizing the result on Ω:

Ω [ ] = ( Ω×Θ ∩ ⃝ Θ↑Ω×Θ ) ↓Ω . ( 11
)
Assume now that Ω [ ] represents the agent's beliefs on Ω conditionally on , i.e., in a context where holds. There are usually many BBAs on Ω×Θ, whose conditioning on yields Ω [ ]. Among these, the least committed one is defined for all ⊆ Ω by:

Ω [ ] ⇑Ω×Θ ( × ∪ Ω × ) = Ω [ ]( ). ( 12
)
This operation is referred to as the deconditioning or ballooning extension [START_REF] Smets | Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem[END_REF] of Ω [ ] on Ω × Θ.

Discounting

When receiving a piece of information represented by a mass function , agent may have some doubts regarding the reliability of the source that provided this information. Such metaknowledge can be taken into account using the discounting operation introduced by Shafer [21, page 252], and defined by:

= (1 -) + Ω , (13)
where

∈ [0, 1].
A discount rate equal to 1, means that the source is not reliable and the piece of information it provides cannot be taken into account, so 's knowledge remains vacuous:

Ω = 1 = Ω .
On the contrary, a null discount rate indicates that the source is fully reliable and the piece of information is entirely accepted: Ω = 0 = . In practice, however, agent usually does not know for sure whether the source is reliable or not, but has some degree of belief expressed by:

{ ℛ ({ }) = 1 - ℛ (ℛ) = , (14)
where ℛ = { , }, and standing, respectively, for "the source is reliable" and "the source is not reliable". This formalization yields expression [START_REF] Mercier | Refined modeling of sensor reliability in the belief function framework using contextual discounting[END_REF], as demonstrated by Smets in [START_REF] Smets | Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem[END_REF]Section 5.7].

The discounting operation (13) of a BBA is also equivalent to the disjunctive combination [START_REF] Elouedi | Assessing sensor reliability for multisensor data fusion with the transferable belief model[END_REF] of with the mass function Ω 0 defined by:

Ω 0 ( ) = ⎧  ⎨  ⎩ if = ∅ if = Ω 0 otherwise, (15)
with ∈ [0, 1] and = 1 -. Indeed:

∪ ⃝ Ω 0 ( ) = ( ) Ω 0 (∅) = ( ) = ( ), ∀ ⊂ Ω , (16)
and

∪ ⃝ Ω 0 (Ω) = (Ω) Ω 0 (∅) + Ω 0 (Ω) ∑ ⊆Ω ( ) = (Ω) + = (Ω) . (17)

Contextual Discounting based on a coarsening

Let Θ = { 1 , . . . , } be a coarsening of Ω, which means that 1 , . . . , form a partition of Ω [21, chapter 6].

Unlike [START_REF] Mercier | Decision fusion for postal address recognition using belief functions[END_REF], in the contextual model, agent is assumed to hold beliefs on the reliability of the source of information conditionally on each ℓ , ℓ ∈ {1, . . . , }:

{ ℛ [ ℓ ]({ }) = 1 -ℓ = ℓ ℛ [ ℓ ](ℛ) = ℓ . (18)
For all ℓ ∈ {1, . . . , }, ℓ + ℓ = 1, and ℓ represents the degree of belief that the source is reliable knowing that the true answer of the question of interest belongs to ℓ .

In the same way as in the discounting operation [START_REF] Mercier | Refined modeling of sensor reliability in the belief function framework using contextual discounting[END_REF], agent considers that the source can be in two states: reliable or not reliable [START_REF] Smets | Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem[END_REF][START_REF] Mercier | Refined modeling of sensor reliability in the belief function framework using contextual discounting[END_REF]:

• If the source is reliable (state ), the information Ω it provides be- comes 's knowledge. Formally, Ω [{ }] = Ω .
• If the source is not reliable (state ), the information Ω it provides is discarded, and remains in a state of ignorance:

Ω [{ }] = Ω .
The knowledge held by agent , based on the information Ω from a source as well as metaknowledge ℛ concerning the reliability of the source can then be computed by:

• Deconditioning the BBAs ℛ [ ℓ ] on the product space Ω × ℛ using [START_REF] Mercier | A parameterized family of belief functions correction mechanisms[END_REF];

• Deconditioning Ω [{ }]
on the same product space Ω × ℛ using [START_REF] Mercier | A parameterized family of belief functions correction mechanisms[END_REF] as well;

• Combining them using the CRC (6);

• Marginalizing the result on Ω using [START_REF] Fabre | Presentation and description of two classification methods using data fusion based on sensor management[END_REF].

Formally:

Ω [ Ω , ℛ ] = ( ∩ ⃝ ℓ=1 ℛ [ ℓ ] ⇑Ω×ℛ ∩ ⃝ Ω [{ }] ⇑Ω×ℛ ) ↓Ω . ( 19
)
As shown in [START_REF] Mercier | Refined modeling of sensor reliability in the belief function framework using contextual discounting[END_REF], the resulting BBA Ω , only depends on and on the vector = ( 1 , . . . , ) of discount rates. It is then denoted by Θ .

Proposition 1 ([13, Proposition 8]). Contextual discounting Θ of a BBA is equal to the disjunctive combination of with a BBA Ω 0 such that:

Ω 0 = Ω 1 ∪ ⃝ Ω 2 ∪ ⃝ . . . ∪ ⃝ Ω , (20)
where each Ω ℓ , ℓ ∈ {1, . . . , }, is defined by:

Ω ℓ ( ) = ⎧  ⎨  ⎩ ℓ if = ∅ ℓ if = ℓ 0 otherwise. (21)
Remark 1. Two special cases of this discounting operation can be considered.

• If Θ = {Ω} denotes the trivial partition of Ω in one class, combining with 0 defined by ( 15) is equivalent to combining with 0 defined by [START_REF] Pichon | Belief functions: canonical decompositions and combination rules[END_REF], so this contextual discounting operation is identical to the classical discounting operation.

• If Θ = Ω, the finest partition of Ω, this discounting is simply called contextual discounting and denoted . It is defined by the disjunctive combination of with the BBA Ω

1 ∪ ⃝ Ω 2 ∪ ⃝ . . . ∪ ⃝ Ω , where each Ω , ∈ {1, . . . , } is defined by Ω (∅) =
and Ω ({ }) = .

Canonical conjunctive and disjunctive decompositions

In [START_REF] Smets | The canonical decomposition of a weighted belief[END_REF], extending the notion of separable BBA introduced by Shafer [21, chapter 4], Smets shows that each non-dogmatic BBA can be uniquely decomposed into a conjunctive combination of generalized simple BBAs (GS-BBAs), denoted ( ) with ⊂ Ω, and defined from 2 Ω to ℝ by:

( ) : Ω → ( ) → 1 -( ) → 0 , ∀ ∈ 2 Ω ∖ { , Ω} , (22)
with ( ) ∈ (0, ∞).

The function : 2 Ω ∖ {Ω} → (0, ∞) is yet another representation of a non-dogmatic mass function and is called the conjunctive weight function.

Let us note that the higher is the weight ( ), the higher is the incertitude on .

The canonical conjunctive decomposition of a non-dogmatic BBA is then given by: =

∩ ⃝ ⊂Ω ( ) . (23)
In [START_REF] Denoeux | Conjunctive and Disjunctive Combination of Belief Functions Induced by Non Distinct Bodies of Evidence[END_REF], Denoeux introduces another decomposition: the canonical disjunctive decomposition of a subnormal BBA into negative GSBBAs (NGSBBAs), denoted ( ) with ⊃ ∅, and defined from 2 Ω to ℝ by:

( ) : ∅ → ( ) → 1 -( ) → 0 , ∀ ∈ 2 Ω ∖ {∅, } , (24)
with ( ) ∈ (0, ∞).

Every subnormal BBA can be canonically decomposed into a disjunctive combination of NGSBBAs:

= ∪ ⃝ ⊃∅ ( ) . (25)
Indeed, as remarked in [START_REF] Denoeux | Conjunctive and Disjunctive Combination of Belief Functions Induced by Non Distinct Bodies of Evidence[END_REF], the negation of a BBA can also be conjunctively decomposed as soon as is subnormal:

= ∩ ⃝ ⊂Ω ( ) (as is non-dogmatic) ⇒ = ∩ ⃝ ⊂Ω ( ) = ∪ ⃝ ⊂Ω ( ) = ∪ ⃝ ⊃∅ ( ) . (26)
The relation between functions and is then ( ) = ( ) for all ∕ = ∅, and function : 2 Ω ∖ {∅} → (0, ∞), called the disjunctive weight function, is another representation of a subnormal mass function.

Practically, functions and have the following properties [START_REF] Denoeux | Conjunctive and Disjunctive Combination of Belief Functions Induced by Non Distinct Bodies of Evidence[END_REF]:

• for all :

∏ ⊂Ω ( ) = (Ω) , (27)
• for all :

∏ ⊃∅ ( ) = (∅) , (28)
• for all subset ⊂ Ω:

1 ( ) ∩ ⃝ 2 ( ) = 1 ( ) 2 ( ) ,
• for all subset ⊃ ∅:

1 ( ) ∪ ⃝ 2 ( ) = 1 ( ) 2 ( ) ,
• function can be conveniently obtained from the commonality function as follows:

∀ ⊂ Ω, ( ) = ∏ ⊇ , (| |)∕ = (| |) ( ) ∏ ⊇ , (| |)= (| |) ( ) , (29)
where ( ) means the parity of an integer ( ( ) = 0 if is even, 1 otherwise),

• likewise, function can be computed from the implicability function as follows:

∀ ⊃ ∅, ( ) = ∏ ⊆ , (| |)∕ = (| |) ( ) ∏ ⊆ , (| |)= (| |) ( ) , (30)
For "quasi-Bayesian" BBAs, another convenient way to compute is given by the following property.

Proposition 2 ([4, Proposition 1]). Let be a BBA which focal sets are Ω, 1 , 2 , . . . , , and possibly ∅, such that the subsets verifies ∩ = ∅ for all , ∈ {1, . . . , }. The conjunctive weight function associated with is then defined by:

( ) = ⎧  ⎨  ⎩ (Ω) ( )+ (Ω) if = , (Ω) ∏ =1 (1 + ( ) (Ω) ) if = ∅ , 1 otherwise. ( 31
)
Remark 2. If has only one focal element in addition to ∅ and Ω, Proposition 2 holds as well.

The dual version of this property to compute disjunctive weights can be obtained as follows.

Proposition 3. Let be a BBA which focal sets are ∅, 1 , 2 , . . . , , and possibly Ω, such that the subsets verifies ∪ = Ω for all , ∈ {1, . . . , }. The disjunctive weight function associated with is then defined by:

( ) = ⎧  ⎨  ⎩ (∅) ( )+ (∅) if = , (∅) ∏ =1 (1 + ( ) (∅) ) if = Ω , 1 (32)
Proof 1. Focal sets of are Ω, 1 , 2 , . . . , , and possibly ∅, such that ∩ = ∅ for all , ∈ {1, . . . , }. Consequently, from Proposition 2, the conjunctive weight function associated with is given by:

( ) = ⎧  ⎨  ⎩ (Ω) ( )+ (Ω) if = , (Ω) ∏ =1 (1 + ( ) (Ω) ) if = ∅ , 1 otherwise. (33)
Then:

( ) = ( ) = ⎧  ⎨  ⎩ (Ω) ( )+ (Ω) if = , (Ω) ∏ =1 (1 + ( ) (Ω) ) if = ∅ , 1 otherwise, (34)
and, as ( ) = ( ) ∀ , Equation (32) is obtained.

□ Remark 3. Proof 1 also implies, from Remark 2, that if has only one focal element in addition to ∅ and Ω, Equation ( 32) is still valid.

Extending the contextual discounting

In this section, the contextual discounting operation on a coarsening is extended to any subsets of the frame of discernment, and a general formulation linked with the canonical disjunctive decomposition of a BBA is developed.

According to the previous definitions ( 22) and ( 24), BBAs ℓ , ℓ ∈ {1, . . . , }, defined in ( 21) by ℓ (∅) = ℓ and ℓ ( ℓ ) = ℓ , can be denoted ℓ ℓ or ℓ in a simple way.

From ( 20) and ( 25), the contextual discounting on a coarsening Θ = { 1 , . . . , } of Ω of a subnormal BBA is thus defined by:

Θ = ∪ ⃝ 1 ∪ ⃝ . . . ∪ ⃝ = ∪ ⃝ ⊃∅ ( ) ∪ ⃝ 1 ∪ ⃝ . . . ∪ ⃝ .
In particular:

• The classical discounting (15) of a subnormal BBA = ∪ ⃝ ( ) is defined by: = Ω (Ω) ∪ ⃝ Ω⊃ ⊃∅ ( ) ; (35)
• The contextual discounting (Remark 1) of a subnormal BBA = ∪ ⃝ ⊃∅ ( ) is defined by:

= ∪ ⃝ ∈Ω { } ({ }) ∪ ⃝ ⊂Ω,| |>1 ( ) . (36)
These discounting operations can then be viewed as particular cases of a more general correction mechanism defined by:

∪ = ∪ ⃝ ⊃∅ ( ) , (37)
where ∈ [0, 1] for all ∕ = ∅ and is the vector { } ∕ =∅ . In [START_REF] Mercier | Refined modeling of sensor reliability in the belief function framework using contextual discounting[END_REF], the interpretation of each has been given only in the case where the union of the subsets forms a partition of Ω, being interpreted as the degree of belief held by the agent regarding the fact that the source is reliable, knowing that the value searched belongs to .

Instead of considering [START_REF] Muñoz-Salinas | Multi-camera people tracking using evidential filters[END_REF], let us now suppose that agent holds beliefs regarding the reliability of the source, conditionally on each subset of Ω:

{ ℛ [ ]({ }) = 1 - = ℛ [ ](ℛ) = , (38)
where ∈ [0, 1]. In the same way as in Section 2.6, the knowledge held by agent , based on the information Ω from a source and on metaknowledge ℛ (38) regarding the reliability of this source, can be computed as follows:

Ω [ Ω , ℛ ] = ( ∩ ⃝ ⊆Ω ℛ [ ] ⇑Ω×ℛ ∩ ⃝ Ω [{ }] ⇑Ω×ℛ ) ↓Ω . (39)
Proposition 4. The BBA Ω resulting from (39) only depends on Ω and the vector = { } ⊆Ω . It is equal to the disjunctive combination of Ω with a BBA Ω 0 defined by:

Ω 0 ( ) = ∏ ∪ = ∏ ∪ = , ∀ ⊆ Ω. ( 40
) Proof 2. See Appendix A.1. □
Like in the case of contextual discounting operations considered in Section 2.6, BBA Ω 0 defined in Proposition 4 admits a simple decomposition described in the following proposition.

Proposition 5. The BBA Ω 0 defined in Proposition 4 can be rewritten as:

Ω 0 = ∪ ⃝ ⊃∅ .
Proof 3. Directly from (40) and the definition (7) of the DRC. □ From (41), the contextual discounting resulting from (39) of a subnormal BBA = ∪ ⃝ ⊃∅ ( ) is then defined by:

∪ ⃝ ⊃∅ ( ) ∪ ⃝ ⊃∅ = ∪ ⃝ ⊃∅ ( ) = ∪ . (42)
Correction mechanism ∪ is then the general formulation for a contextual discounting on any subsets of Ω.

The following theorem sums up the contextual discounting operation in its general formulation. Theorem 1. On the one hand, agent receives an information from a source .

On the other hand, agent knows that this source is reliable with a degree in different contexts of Ω, which means that the source is reliable with a degree knowing that the true answer to the question of interest belongs to . Let us note the set containing these contexts. Then, agent 's mass function is given by the contextual discounting of defined by:

= ∪ ⃝ ∈ . (43)
Moreover, if is subnormal then:

= ∪ ⃝ ⊃∅ ( ) ∪ ⃝ ∈ , = ∪ ⃝ / ∈ ( ) ∪ ⃝ ∈ ( ) . (44)
Example 1. Let us consider a two-lane road section, the question of interest concerning the number of lanes where the traffic is flowing freely. Frame of discernment Ω is equal to { 0 , 1 , 2 } with:

• 0 standing for "0 lane is free": both are blocked,

• 1 meaning "1 lane is free": one is blocked,

• 2 signifying "2 lanes are free": traffic is flowing freely on both lanes.

A source provides information regarding the traffic on this road section. For instance:

⎧ ⎨ ⎩ (∅) = .1 , ({ 0 }) = .8 , (Ω) = .1 . ( 45
)
On the other side, you know that the source is very reliable in case of heavy traffic (situation { 0 , 1 }), and less reliable when the traffic is rather light (situation { 1 , 2 }). Formally, let us suppose that { 0 , 1 } = .8 and

{ 1 , 2 } = .6.
Contextual discounting of is then given by:

= ∪ ⃝ { 0 , 1 } .8 ∪ ⃝ { 1 , 2 } .6 . (46)
From Remark 3, the disjunctive weight function associated with can be computed in the following manner:

{ ({ 0 }) = (∅) (∅)+ ({ 0 }) = 1 9 , (Ω) = (∅)(1 + ({ 0 }) (∅) ) = .1(1 + .8
.1 ) = .9 .

(47)

Contextual discounting of knowing { 0 , 1 } = .8 and { 1 , 2 } = .6 is then given by:

= ∪ ⃝ { 0 , 1 } .8 ∪ ⃝ { 1 , 2 } .6 = { 0 } 1 9 ∪ ⃝ Ω .9 ∪ ⃝ { 0 , 1 } .8 ∪ ⃝ { 1 , 2 } .6 , (48)
which can be also written by definition from [START_REF] Smets | What is Dempster-Shafer's model? In Advances in the theory of evidence[END_REF] in the following manner: As it can be observed in the previous example, the contextual discounting allows the transfer of masses on intermediate subsets between focal sets and the frame of discernment Ω. More precisely (cf Equation ( 43)), contextual discounting consists in transferring each mass to its union with subsets not precisely known by the source.

= { ∅ →1/9 { 0 } →8/9 ∪ ⃝ { ∅ →.9 Ω →.1 ∪ ⃝ { ∅ →.8 { 0 , 1 } →.2 ∪ ⃝ { ∅ →.6 { 1 , 2 } →.4 = ⎧     ⎨     ⎩ ∅ →.048 { 0 } →.384 { 0 , 1 } →.108 { 1 , 2 } →.032 Ω →.428 (
The following example illustrates a case where a sensor is totally reliable if an object is of a certain type, and not reliable for another type.

Example 2. Let us consider a sensor in charge of the recognition of two types of objects: Ω = { , }.

The sensor knows very well how to recognize objects of type , whereas it is not the case for objects of type . Which means that:

• an object is of type , the sensor will recognize it;

• an object is of type the sensor will hesitate and make mistakes.

For instance, a possible confusion matrix for such a sensor is represented in Table 1. truth ∖ decision 10 0 5 5

A contextual discounting knowing that { } = 1 and { } = 0 (S totally in context { }, not reliable in context { }) applied on a mass function provided by is given by:

= ∪ ⃝ { } 1 ∪ ⃝ { } 0 = ∪ ⃝ { } 0 . (50)
In particular:

• if ({ }) = 1 then ({ , }) = 1, • if ({ }) = 1 then ({ }) = 1.
In other words, if the source says it is an object of type : it is an object of type or an object the source does not recognize. And, in the particular case where the source says it is an object of type , and the source knows very well the other types of object: it that the object is of type . If it had been an object of type , the source would have said it, because objects of type are very well recognized by the source.

On the notion of reliability

In the simple Example 2, we have a situation where when the source decides : it is indeed , and when the source decides : the truth is or . However, the source is totally reliable for and not for . The notion of reliability introduced in this article has then to be clearly distinguished from a different notion of reliability which would be linked with the reliability of the decision made by a source.

In the contextual discounting, the definition of the reliability (38) is given conditionally on Ω by:

ℛ [ ]({ }) = , (51)
and not conditionally on results of a decision-making process:

ℛ ["The source decides A"]({ }) = . (52)
Let us remark that this second definition will imply to define the notion of conditioning on processes (even belief functions?) which is not known at present by the authors.

To be subnormal and non-dogmatic

In order to exploit simple expressions obtained with both conjunctive and disjunctive canonical decompositions, should each mass function be subnormal and non-dogmatic?

As already mentioned in [START_REF] Denoeux | Conjunctive and Disjunctive Combination of Belief Functions Induced by Non Distinct Bodies of Evidence[END_REF], it may be argued that most (if not all) pieces of information provided in real-life applications are imperfect, and then the mass on the frame of discernment should be always strictly positive.

For instance, let us consider a coin tossing and a universe equal to {ℎ , }. As remarked by Denoeux, it is absolutely not certain in practice that the coin is perfectly balanced. An appropriate BBA may then be ( ) = 0.5(1 -), ( ) = 0.5(1 -) and (Ω) = for some small > 0.

However, we can also add that when we are trying to model a complex real life problem, there are always some approximations, and then some doubts on the model are always possible, so the mass on the conflict should These two correction mechanisms can thus be seen as belonging to a general family of correction mechanisms.

In a nutshell, if is subnormal, a contextual discounting of is given by: =

∪ ⃝ ⊃∅ ( ) , (56)
and, the negation of a contextual discounting of defines a dual reinforcement process:

= ∪ ⃝ ⊃∅ ( ) = ∩ ⃝ ⊂Ω ( ) . (57)
The application of this reinforcement process as well as its comparison with other correction mechanisms [START_REF] Denoeux | Relevance and Truthfulness in Information Fusion Correction and Fusion[END_REF][START_REF] Mercier | Belief function correction mechanisms[END_REF] has been left for future researches.

Conclusion future work

In this article, the contextual discounting operation of a belief function has been extended to any subsets, and a simple and practical expression, based on disjunctive combinations, to compute it has been given. This expression has highlighted the close relationship between contextual discounting and canonical disjunctive decomposition. The dual expression of this discounting, allowing one to strengthen a belief function, has also been exposed.

Future work will aim at testing it on real data. Likewise, it would also be interesting to automatically learn the coefficients of these correction mechanisms from data, as done for the classical and the contextual discounting operations [START_REF] Elouedi | Assessing sensor reliability for multisensor data fusion with the transferable belief model[END_REF][START_REF] Mercier | Refined modeling of sensor reliability in the belief function framework using contextual discounting[END_REF].

Finally, the marginalization of this BBA on Ω is given for all subsets of Ω, by:

( ) = ∑ ∪ = ⎡ ⎣ ∏ ∪ = ∏ ∪ = ⎤ ⎦ Ω ( ) . (A.4)
Let us note that the above proof has many similarities with proofs presented in [START_REF] Mercier | Refined modeling of sensor reliability in the belief function framework using contextual discounting[END_REF]Sections A.1 and A.3].

4 . 1 .

 41 About the mass transfer during a contextual discounting

Table 1 :

 1 Confusion matrix associated with source S.

Acknowledgements

The authors are very grateful to the anonymous reviewers for their numerous comments which helped to improve the clarity and the accuracy (in particular Proposition 3) of this article as well as their valuable suggestions.

This work has been financed by the French région Nord-Pas de Calais under the project CISIT (Campus International pour la Sécurité et l'Intermodalité des Transports).

also be strictly positive: in real life, the coin can fall against a book and lands on edge, which may not have been considered.

Consequently, a more appropriate BBA may be ( ) = 0.5(1 -1 )(1 -2 ), ( ) = 0.5(1 -1 )(1 -2 ), (Ω) = 1 (1 -2 ) and (∅) = 2 some small 1 , 2 > 0.

An other approach: combining discountings

Let us consider a separable BBA , which means that = ∩ ⃝ ⊂Ω

As recently exposed in [START_REF] Kallel | Combination of partially nondistinct beliefs: the cautious adaptive rule[END_REF], a classical discounting on each simple BBA ( ) can be undertaken with a discount rate ∈ [0, 1], the result being

. The discounted simple BBAs can then be conjunctively combined which yields to the following discounting operation

. This operation, restricted to separable BBAs, is different from a contextual operation. The conjunctive combination of discounted BBAs is not a discounting of the BBAs combination in general.

However, as developed in the next section, we can remark that the dual of the contextual discounting operation, which is nevertheless a reinforcement, has a close formulation.

A new reinforcement process

In a similar way, a correction mechanism for a non-dogmatic BBA can be defined from the conjunctive decomposition of as follows:

where ∀ ⊂ Ω, ∈ [0, 1], and is the vector { } ⊂Ω . The smaller is the uncertain weight, the higher is the mass on . This process allows then the reinforcement of a BBA .

Correction mechanisms ∩ (37) and ∪ (53) are related in the following way.

Let us consider a subnormal BBA , is then non-dogmatic:

Then:

Appendix A.1. Proofs of Proposition 4

For each ⊆ Ω, the deconditioning of ℛ [ ] on Ω × ℛ is given by:

With ∕ = :

Then:

or, by exchanging the roles of and :

, ∀ ⊆ Ω .

It remains to combine conjunctively

which have focal sets of the form × { } ∪ Ω × { } and × { } ∪ × ℛ, respectively, with , ⊆ Ω. The intersection of two such focal sets is:

and it can be obtained only for a particular choice of and . Then:


Gabriele Albertini

Mathias Lebihain

François Hild

Laurent Ponson

David S Kammer
email: dkammer@ethz.ch

Effective toughness of heterogeneous materials with rate-dependent fracture energy

We investigate dynamic fracture of heterogeneous materials experimentally by measuring displacement fields as a rupture propagates through a periodic array of obstacles of controlled fracture energy. Our measurements demonstrate the applicability of the classical equation of motion of cracks at a discontinuity of fracture energy: the crack speed jumps at the entrance and exit of an obstacle, as predicted by the crack-tip energy balance within the brittle fracture framework. The speed jump amplitude is governed by the fracture energy contrast and by the combination of rate-dependency of fracture energy and inertia of the medium, which allows the crack to cross a fracture energy discontinuity at constant energy release rate. This discontinuous dynamics and the rate-dependence cause higher effective toughness, which governs the coarse-grained behavior of these cracks.

Many biological materials, such as bone, nacre and tooth, have intricate microstructures which are responsible for remarkable macroscopic mechanical properties [1,2]. Carefully designed microstructures combined with advances in micro-fabrication techniques allow for the development of new materials with unprecedented properties [3][4][5][6][7][8]. Understanding how to harness smallscale heterogeneities is, however, necessary to achieve the desired macroscopic properties. For fracture properties, recent research focused either on disordered microstructures, where randomly located obstacles distort the crack front and cause toughening by collective pinning [9][START_REF] Roux | European Journal of Mechanics -A/Solids General and plenary lectures from the 5th EUROMECH Solid Mechanics Conference[END_REF][START_REF] Ponson | [END_REF][12], or on elastic heterogeneities, where compliant inclusions provide toughening by effectively reducing the energy flow into the crack tip [13,14]. However, a complete and fundamental theory for effective material resistance against fracture remains missing, and experimental observations, which are key for establishing such theoretical knowledge, are scarce.

Theoretical fracture mechanics, based on the seminal work of Griffith [15,16] states that a crack will propagate as soon as the released elastic energy per unit increment of crack length G S = -∂ l Ω, where Ω is the elastic energy in the medium and l the crack length, balances the local fracture energy Γ (i.e., the energy necessary for creating two unit surfaces). During dynamic crack propagation, the energy balance further includes inertia of the surrounding medium and possible rate-dependence of the fracture energy Γ(v), where v = l is the crack speed. Using Linear Elastic Fracture Mechanics (LEFM) theory [START_REF] Freund | Dynamic Fracture Mechanics[END_REF], one can derive the equation of motion of a crack from this energy balance by assuming steady state crack propagation in an unbounded homogeneous domain. Under these circumstances the crack has no inertia (there is no term involving l in the equation of motion) and its speed adapts abruptly to accommodate changes in fracture energy. However, it remains unclear if these idealized con-ditions are valid at discontinuities within heterogeneous materials and how they affect the coarse-grained behavior of the crack during dynamic propagation.

In this Letter, we analyze these questions in depth through the experimental investigation of crack propagation in heterogeneous media with fracture energy discontinuities. Usually, fracture mechanics experiments are based on global measurements, thus, only capture averaged quantities. In contrast, our experimental setup and simplified 2D geometry with periodic heterogeneities allows local measurements of the near-crack-tip fields, which support the uncovering of fundamental mechanisms. While the elastic energy release rate is constant as the crack faces a fracture energy discontinuity, the speed at which the crack propagates is observed to vary discontinuously. We study the amplitude of the speed jumps as the crack crosses the interface between regions of different fracture energy and show that it stems from the combination of rate-dependency of fracture energy and inertia of the medium. Rate-dependent effects result from the non-equilibrium nature of fracture problems and are prevailing in materials. Thus, rate-dependent fracture energy applies to a wide range of materials and has been observed, for instance, on rock [START_REF] Ponson | [END_REF]19], glassy polymers [20][21][22][23][24][25][26] and metals [27]. The discontinuous dynamics and the rate-dependent effects significantly affect the effective toughness of heterogeneous materials, as we will show with our experimental observations. Our experimental setup (see FIG. 1a) consists of a tapered double cantilever beam, made of multi-material 3D-printed polymers (Stratasys Objet260 Connex3), a high-speed camera (Phantom v2511) and an electromechanical testing machine (Shimadzu AG-X Plus). The matrix material is VeroClear with static fracture energy Γ M 0 ≈ 80 J/m 2 and Young's modulus E M ≈ 2.8 ± 0.2 GPa. The obstacle material is VeroWhite-DurusWhite (Γ O 0 ≈ 106 J/m 2 , E O ≈ 1.9 ± 0.2 GPa), which is tougher and more compliant. We prescribe a constant crack mouth displacement rate δ ≈ 25mm/s. Hence, the elastic energy in the system is gradually increased, until a planar crack initiates from a pre-existing notch. The elastic energy release rate at initiation is proportional to the bluntness of the notch, which we can tune to explore a range of initial crack speeds from moderate up to 350 m/s ≈ 0.4c R , where c R ≈ 800m/s is the Rayleigh wave speed. The crack propagates then dynamically through a series of periodic obstacles (see FIG. 1b). During crack propagation no additional energy is added to the system (δ is constant) and the tapered geometry causes exponentially decaying released elastic energy G S ∼ δ 2 e -l/lsys , where l sys ≈ 17.5 mm is a structural length scale directly related to the sample size [28]. Thus, the crack speed gradually decreases on average. All properties are constant through the sample thickness and the overall behavior is quasi-2D. We analyze the crack dynamics by measuring the near-tip displacement field u using Digital Image Correlation. We apply a random speckle pattern (see FIG. 1c) onto the surface of the specimen using aerosol paint. The temporal evolution of the speckle is tracked using high speed photography at 250,000 fps. The auto-correlation length of the pattern corresponds to 4-6 pixels, where the pixel size is ≈ 45µm. u (see color in FIG. 1c) is found by minimizing the difference between the pattern at a given time t mapped back to its pre-crack configuration [29]. The resulting infinitesimal strain field ε yy is depicted in FIG. 1d. An alternative approach (see FIG. 1e) is the Integrated Digital Image Correlation (IDIC) [28,30], which assumes the analytical solution for a singular crack in an infinite elastic medium -the Williams eigenfunctions expansion [31] -as basis for u [29]. The first term of the series has singular strains at the crack tip ε ij ∼ 1/ √ r, where r is the distance from the tip and its amplitude is related to the stress intensity factor K. Note that for both methods the amplitude of ε is similar. IDIC has the advantages of precisely determining the crack tip position l and directly computing K, from which, one can find the dynamic energy release rate G = K 2 E A(v) that provides a measure of the fracture energy Γ at the crack tip [START_REF] Freund | Dynamic Fracture Mechanics[END_REF]29,32]. The effects of elastic heterogeneity are minor in our setup, but give rise to an interaction between the size of the K-dominant region (r < ∼ 5mm) with the size of the heterogeneity and are discussed in [29].

Typical experiments are illustrated in FIG. 2a&b. The crack first propagates through the matrix material with propagation speed v being maximum immediately after initiation, then v gradually decreases as crack length increases. v undergoes abrupt deceleration (acceleration) as the front enters (leaves) an obstacle. Simultaneously, Γ also abruptly increases (decreases). However, the relative jumps of the dissipation rate are significantly smaller than the ones observed on crack speed. We calculate the speed in the obstacle v O and matrix v M by selecting the mean speed over 12µs before and after the obstacle boundaries. All speed jumps at material discontinuities were studied for a collection of 30 experiments with different period ∆l = w O + w M and constant obstacle density β = w O w O +w M = 1/2. Jumps as the crack enters (trapping) and leaves (untrapping) an obstacle are shown in FIG. 2c&d, respectively. Results show that the crack dynamics at the matrix/obstacle interface is independent of obstacle width and is symmetric with respect to the direction of propagation, i.e., the jumps are the same for trapping and untrapping. This implies that the crack dynamics only depends on local fracture properties.

In order to understand the jumps and their effect on effective material properties, we analyze the fracture propagation with a crack-tip energy balance. In our experiments, failure mechanisms occur at time scales 4 orders of magnitude smaller than the viscous relaxation time typical of the polymers used in this study [29] so that an elastic response of the sample can be safely assumed. Moreover, the failure mechanisms are too fast for a craze to develop [33], making the fracture process essentially brittle. Thus, we develop a theoretical model based on LEFM to interpret the experimental observations.

As the crack advances, elastic energy G S is released from the specimen and is in part dissipated as fracture energy Γ to create new surfaces and in part radiated away as elastic waves. Analyzing the near-tip fields of a steadystate dynamic crack, Freund [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] showed that the energy release rate of a dynamic crack G(l, v) is related to the energy release rate for a corresponding static crack G S (l) by g(v), a universal function of v. The crack-tip energy balance provides the equation of motion for a crack [29]

Γ(v) = G S (l)g(v) ≈ G S (l)(1 -v/c R ), (1)
which implies that within the framework of LEFM, a sub-Rayleigh crack in an infinite medium has no inertia and v adjusts instantaneously to fluctuations in Γ or G S [29]. Note that for rate-dependent materials, the fracture energy Γ(v) is not constant. We analyze the rate-dependence of the matrix and obstacle material by independently plotting Γ vs. v (see averaged data as dashed line in FIG. 3 or full data in FIG. S3 of [29]). We observe that our measurements are in good agreement with a model [25] (solid line in FIG. 3) that considers the actual dissipative mechanism taking place within the process zone. Within the matrix or obstacle material, the fracture energy follows this kinetic law. At the material boundaries, however, the rupture needs to jump from one kinetic law to the other. The jump amplitude is governed by the equation of motion (1). The jump trajectory in the Γ-v space corresponds to the right-hand side of (1), which, since G S (l) is constant across the boundary, corresponds to a diagonal line G S g(v) (arrows in FIG. 3). Thus, at a discontinuity in material property the equation of motion of a crack becomes

G S = Γ M (v M )/g(v M ) = Γ O (v O )/g(v O ) , (2)
which captures the experimentally observed velocity discontinuity at trapping and untrapping with no fitting parameter (see FIG. 2c&d). Eq. ( 2) cannot be solved explicitly. However, assuming a linear rate-dependent fracture energy Γ(v) ≈ Γ 0 + γv, for the purpose of discussion, the velocity jump becomes

v M -v O ≈ ∆Γ 0 1 -v M /c R γ + Γ M 0 /c R , (3)
where ∆Γ 0 = Γ O 0 -Γ M 0 is the jump in fracture energy. This simple result highlights that (i) the jump amplitude is the same for trapping and untrapping (FIG. 2c&d) and (ii) during trapping the velocity right after the interface is zero if v M is smaller than a critical incident velocity v c below which the obstacle causes crack arrest

v c ≈ ∆Γ 0 / γ + Γ O 0 /c R . (4)
All these features are discernible from our experimental data and are captured fairly well by the model. Eq. ( 3) as well as a parameter study of (2) (see FIG. S3 in [29]) reveal that the speed jump and v c are proportional to the toughness discontinuity ∆Γ 0 . The latter is particularly noisy because of variations of fracture properties of both matrix and obstacle material, i.e., Var

[∆Γ 0 ] = Var[Γ M 0 ] + Var[Γ O 0 ], assuming Γ O 0
and Γ M 0 are uncorrelated. In the limit of small ratedependency γ Γ 0 /c R , inertia controls the speed jumps, that are then given by v

M -v O ≈ (∆Γ 0 /Γ M 0 )(c R -v M
) and the corresponding condition for crack arrest becomes v < v c ≈ (∆Γ 0 /Γ O 0 )c R . Conversely, in the limit of large rate-dependency γ Γ 0 /c R and quasi-static propagation v c R , inertia can be neglected and the speed

jumps become constant v M -v O ≈ ∆Γ 0 /γ ≡ v c .
How does such a trapping/untrapping dynamics impact the effective fracture properties Γ of heterogeneous materials? We compute the homogenized fracture energy Γ by integrating over an interval ∆l of uninterrupted crack propagation starting at l i , the beginning of each matrix/obstacle period,

Γ(v) = 1 ∆l li+∆l li Γ v( l) d l . (5)
As Γ in each phase depends on crack speed, Γ depends on it too. Thus, we report Γ as a function of the apparent crack velocity v = ∆l/ li+∆l li v -1 dl. First, we assume ∆l l sys , i.e., a clear separation between the micro-structural scale and the specimen scale. Hence, it is possible to define intrinsic homogenized fracture properties, decoupled from the specimen size and the details of applied boundary conditions. Under this assumption, G S remains constant during the entire crack propagation. Thus, v and Γ are constant within each material phase (insets in FIG. 4a), which allows us to calculate the dissipation rate from (5)

lim ∆l/lsys→0 Γ = βΓ O (v O ) + (1 -β)Γ M (v M ) (6)
and the apparent crack speed

lim ∆l/lsys→0 v = β/v O + (1 -β)/v M -1 , (7)
with β = 1/2. Note that ( 7) is a weighted harmonic mean, which is dominated by its lower argument, v O , so v is effectively lower than the arithmetic mean

( v = βv O + (1 -β)v M ).
As a result, the apparent kinetic law Γ(v) is shifted "horizontally" towards lower speeds in comparison to Γ( v ). This leads, in practice, to a resistance to failure Γ larger than the toughness spatial average Γ = βΓ O (v) + (1 -β)Γ M (v), but lower than the obstacle toughness Γ O predicted by rateindependent theory (see FIG. 4). However, when comparing the infinite system size prediction ( 6) and (7) to our experimental measurements we observe higher effective toughness (see FIG. 4b). The interplay between the size of the heterogeneity ∆l and the 2) depicted as a gray arrow. The black dot is the corresponding homogenized state ( Γ, v) computed using ( 6) and (7). By varying G S one can derive the entire homogenized fracture energy law Γ(v) (black solid line in a&b). (b) Γ(v), measured experimentally using (5), is depicted as colored circles for a range of ∆l ≈ lsys. Colored solid lines are the theoretical solution for ∆l ≈ lsys derived using ( 5), ( 1 structural length scale l sys makes homogenization of fracture properties particularly challenging. The emerging effective toughness depends on the ratio ∆l/l sys , and ( 6) and ( 7) only represent a lower bound of Γ(v). The larger ∆l/l sys , the higher Γ(v), which can even exceed Γ O (v) of the obstacle material. This additional toughening, related to the structural problem with ∆l ≈ l sys , is quantitatively captured by the theoretical solutions for Γ(v), which we derive from ( 5) and ( 1), assuming G S ∼ e -l/lsys . Note that as we approach ∆l l sys , the experimental toughness converges towards the theoretical one; and for ∆l l sys the rupture arrests before reaching ∆l required for homogenization of fracture properties.

How do these observations translate to macroscopic measurements? While measurements from total elastic energy input (see FIG. S5 in [29]) present increased toughness compared to the matrix material, they do not exceed the obstacle material. This is because the additional toughening observed at the small scale is a "horizontal shift" of the kinetic law. However, we observe that the macroscopic fracture energy is independent of ∆l and corresponds to the average of matrix and obstacle material, which validates (6). Furthermore, crack arrest, as described by (4), may play an important role in further increasing the macroscopic toughness. Even very thin obstacles may cause the crack to arrest, which raises interesting questions of practical importance for material design. How to design flaw insensitive materials, whose resistance to crack propagation -or ability to prevent a crack to grow indefinitely -is directly proportional to the obstacle toughness but independent of its size? What are the strategies to translate this local toughening to the macro-scale and improve the mechanical integrity of structures through the use of damage-tolerant composites?

In summary, our study shows that the classical LEFM equation of motion of cracks quantitatively predicts crack dynamics at toughness discontinuities. The crack arrests if it is slower than a threshold speed that is primarily dependent on the toughness contrast and independent of the characteristic size of the microstructure (i.e., obstacle thickness), i.e., (4). When the crack penetrates the tougher/weaker obstacle, it reacts by instantaneously adapting its speed, which is mediated by the rate-dependent fracture energy combined with inertia, i.e., (3). Finally, the heterogeneous material presents an increased effective (homogenized) toughness because of high fluctuations in crack speed between obstacles and matrix, and the rate-dependent nature of the fracture energy. Direct experimental validation of (3) and ( 4) is challenging due to limited temporal resolution and fluctuations in Γ, but increased toughness contrast and focus on a single interface could provide a path to overcome these limitations.

The authors thank Dr. Thiago Melo Grabois and Dr. Julien Scheibert for useful discussions.

FIG. 1 .

 1 FIG. 1. (a) Model heterogeneous material made of multimaterial 3D-printed polymers in a tapered double cantilever beam geometry with applied forces F . The displacement field u = (ux, uy) is measured in the area within the blue box by digital image correlation. (b) Closeup view shows two different materials in a periodic stripe geometry. The transparent material constitutes the matrix with width w M and the opaque (darker) areas are obstacles of higher fracture energy Γ O /Γ M ≈ 1.3 with width w O . (c) Closeup of crack tip at l ≈ 35mm and v ≈ 50m/s. The crack interface is slightly visible running from left to center. A random speckle pattern is applied onto the surface, which is compared to its reference pre-cracked configuration to find u. (d) Infinitesimal strain εyy = ∂yuy found by differentiating u. Approaching the crack tip, εyy diverges. (e) εyy assuming the Williams eigenfunctions as basis for u.

  FIG. 2. (a,b) Experimental results for three specimens with ∆l/lsys = 0.57. (a) v undergoes abrupt deceleration (l = {30, 40, 50}mm) and acceleration (l = {35, 45, 55}mm) when the crack front is trapped and untrapped, respectively, at the interface. (b) Discontinuities in Γ occur at trapping and untrapping with higher values within the obstacle. (c) Trapping: speed prior to entering the obstacle v M is plotted vs. speed immediately after v O . When the approaching velocity v M < vc ≈ 130 m/s the front arrests. (d) Untrapping: speed after exiting the obstacle v M is plotted vs. speed immediately before exiting v O . (c,d) Solid black line is the theoretical model (2) with ±10% variation in Γ (dotted lines).

FIG. 3 .

 3 FIG.3. Experimental results for the same specimens shown in FIG.2a&b-with same color-code. Data points represent crack speed and fracture energy at the moment of transition of material property. Γ(v) is separated in two distinct clusters corresponding to the matrix and obstacle material. Black dashed lines are the average fracture energy measurements based on 30 heterogeneous and 10 homogeneous samples[29]. Solid black lines are the rate-dependent fracture energy law[25] for the obstacle Γ O (v) and matrix Γ M (v) materials. The transition from one branch to the other is described by G S (l)g(v) -the equation of the gray arrows (1).

FIG. 4 .

 4 FIG. 4. Homogenization of fracture energy Γ vs. average velocity, v. (a) Γ assuming the scale separation condition ∆l lsys. Blue and red dots represent the state of the crack within the two materials, which are related by (2) depicted as a gray arrow. The black dot is the corresponding homogenized state ( Γ, v) computed using (6) and(7). By varying G S one can derive the entire homogenized fracture energy law Γ(v) (black solid line in a&b). (b) Γ(v), measured experimentally using(5), is depicted as colored circles for a range of ∆l ≈ lsys. Colored solid lines are the theoretical solution for ∆l ≈ lsys derived using (5), (1) [29] (theory and experiment colors correspond). (a,b) Dash-dotted line and dashed line are Γ O (v) and Γ M (v) from FIG 3.

  FIG. 4. Homogenization of fracture energy Γ vs. average velocity, v. (a) Γ assuming the scale separation condition ∆l lsys. Blue and red dots represent the state of the crack within the two materials, which are related by (2) depicted as a gray arrow. The black dot is the corresponding homogenized state ( Γ, v) computed using (6) and(7). By varying G S one can derive the entire homogenized fracture energy law Γ(v) (black solid line in a&b). (b) Γ(v), measured experimentally using(5), is depicted as colored circles for a range of ∆l ≈ lsys. Colored solid lines are the theoretical solution for ∆l ≈ lsys derived using (5), (1) [29] (theory and experiment colors correspond). (a,b) Dash-dotted line and dashed line are Γ O (v) and Γ M (v) from FIG 3.


Jean-Michel Pereira
email: jeanmichel.pereira@enpc.fr

Mohamed Rouainia
email: m.rouainia@newcastle.ac.uk

Diego Manzanal
email: diego.manzanal@enpc.fr

Combined effects of structure and partial saturation in natural soils

The mechanical behaviour of natural clays is significantly affected by their in situ or initial structure in the form of cementation or interparticle bonding. This behaviour can differ substantially from the behaviour of reconstituted clays. Suction

as well as plastic volumetric strains drive isotropic hardening/softening as this is a simple way to account for the phenomenon of volumetric collapse upon wetting and the stiffening effect that suction has on the soil skeletal response. A model that combines unsaturated and structured behavior is presented and then used to simulate stress strain behaviour observed for an unsaturated natural clay subjected to isotropic load paths. A parametric analysis is performed to observe the influence suction hardening has on mobilized strengths. It is also shown that the model can predict the maximum of collapse of unsaturated soils.

INTRODUCTION:

The majority of research and analysis in soil mechanics to date has assumed full saturated conditions, not least because this simplifies the development of analytical, physical and numerical models. Similarly, the mechanics of structured soils, whose behaviour is influenced by physico-chemical effects at the level of interparticle bonds, is relatively well understood, despite a growing appreciation of the importance of structure in natural soils. In recent years, researchers have begun to look at the modelling of both of these difficult features of real soils, namely partial saturation and structure, and to develop constitutive models that can be used to predict engineering behaviour more realistically in situations where the standard models are known to be inadequate. Saturated models, for example, cannot predict the sudden collapse of a slope due to rainfall infiltration; and standard elasto-plastic models cannot predict the irrecoverable degradation of strength and stiffness that is observed when natural soils are deformed. In saturated conditions, Terzaghi [START_REF] Terzaghi | The shearing resistance of saturated soils and the angle between the planes of shear[END_REF] suggested that the mechanical response of a soil depends only on the effective stress tensor

σ = σ -u ι (1)
where σ is the total continuum stress tensor, u is the (liquid) pore pressure, and ι is the identity tensor. In seeking a generalisation of this principle to partially saturated states, it was initially supposed that both the state of deformation and the failure strength of unsaturated soils might be determined by a suitable generalised effective stress [START_REF] Bishop | The principle of effective stress[END_REF]. It was soon realised that that could not, by itself, explain the phenomenon of collapse upon wetting [START_REF] Jennings | Limitations to the use of effective stresses in partly saturated soils[END_REF], and this led some to doubt the value of the effective stress principle in unsaturated conditions altogether. It was later recognised that surface tension forces exert a stabilising influence on the soil fabric, and contribute significantly to the strenght of unsaturated soils [START_REF] Toll | A framework for unsaturated soil behaviour[END_REF]. A multi-phase continuum analysis (Fredlund and Morgenstern, [START_REF] Fredlund | Stress State Variables for Unsaturated Soils[END_REF] shows that any three known pressures, either the total pressure p, the water pressure u L , or the air pressure u G , may be treated as a reference pressure. Thus, with u G as a reference pressure, the stress state of an unsaturated soil can be represented by the net stress tensor σ net = σu G ι, and the suction s = u G -u L . The essential point is that unsaturated soils have an additional degree of freedom compared to saturated soils, and a full description of their state requires a second, independent kinetic variable (e.g. s), regardless of whether σ net or σ is chosen as the constitutive stress tensor. In consideration of the influence of suction on strength, most previous modelling studies focused on the generalisation of the isotropic hardening variable of the underlying saturated model. The first full constitutive model for unsaturated soils [START_REF] Alonso | A constitutive model for partially saturated soils[END_REF], known as the Barcelona Basic Model (BBM), was based on the modified Cam-clay (MCC) model, whose isotropic hardening variable is the effective consolidation pressure p 0 . The BBM was formulated in the stress space (σ net , s) and it included suction hardening by generalising the definition of p 0 to account for the influence of s. Wheeler and Sivakumar [START_REF] Wheeler | An elasto-plastic critical state framework for unsaturated soil[END_REF] suggested that the specific water content v w is a more appropriate kinematic state variable than the degree of saturation S r , because vw /v (where v is the specific volume) is power conjugate to s. Houlsby [START_REF] Houlsby | The work input to an unsaturated granular material[END_REF], on the other hand, derived an alternative set of work-conjugate stress variables retaining the more natural kinematic variables ( , S r ), consisting of: i) the generalised effective stress σ , defined in terms of the equivalent pore pressure:

u = S r u L + (1 -S r )u G (2)
and ii) the modified suction ns, where n stands for porosity. Bolzon et al. [START_REF] Bolzon | Elastoplastic soil constitutive laws generalized to partially saturated states[END_REF] developed an unsaturated model, based on a generalised plasticity model for saturated soils, that was formulated in terms of (σ , s). The advantages of using σ (rather than σ net ) as a constitutive stress are now widely accepted [START_REF] Jommi | Remarks on the constitutive modelling of unsaturated soils[END_REF], and several recent studies have pursued this approach [START_REF] Loret | An effective stress elastic-plastic model for unsaturated porous media[END_REF][START_REF] Gallipoli | An elasto-plastic model for unsaturated soil incorporating the effects of suction and degree of saturation on mechanical behaviour[END_REF][START_REF] Pereira | Adaptation of existing behaviour models to unsaturated states: Application to CJS model[END_REF][START_REF] Russell | A unified bounding surface plasticity model for unsaturated soils[END_REF]. The choice of stress variables has nevertheless been the subject of much debate [START_REF] Khalili | Effective stress in unsaturated soils: Review with new evidence[END_REF][START_REF] Gens | On constitutive modelling of unsaturated soils[END_REF][START_REF] Nuth | Effective stress concept in unsaturated soils: Clarification and validation of a unified framework[END_REF].

Constitutive models for unsaturated soils require an additional equation relating changes in s to changes in S r , just as σ is related to ˙ in saturated models. This equation describes the water retention properties of the material which are in general strongly coupled to the mechanical behaviour. For monotonic (wetting or drying) processes, a single water retention curve is sufficient for modelling purposes. In general, however, the suction-saturation relationship is hysteretic, and S r cannot be expressed simply as a function of state. The significance of hydraulic hysteresis in geotechnical problems is a subject of increasing research interest. Techniques have recently been established for the inclusion of its effects in constitutive models (see for instance [START_REF] Gallipoli | Modelling the variation of degree of saturation in a deformable unsaturated soil[END_REF][START_REF] Li | Modelling of hysteresis response for arbitrary wetting/drying paths[END_REF]). Wheeler and coworkers [START_REF] Wheeler | Coupling of hydraulic hysteresis and stressstrain behaviour in unsaturated soils[END_REF] have developed a conceptual model to explain the phenomenon of hysteresis in terms of the structure of the pore space.

Natural soils typically exhibit a significant inherent anisotropy of fabric. They are further distinguished by the presence of inter-particle bonding, which contributes to their strength and stiffness [START_REF] Leroueil | The general and congruent effects of structure in natural soils and weak rocks[END_REF]. The combined effects of fabric and bonding will be referred to using the term structure in this paper. There have been considerable advances in the constitutive modelling of natural soils to account for structure, and damage to structure, with a single yield locus elasto-plastic framework (e.g. [START_REF] Gens | Conceptual bases for a constitutive model for bonded soils and weak rocks[END_REF][START_REF] Liu | Modelling the destructuring of soils during virgin compression[END_REF]), and more recently, using the framework of kinematic hardening and bounding surface plasticity [START_REF] Rouainia | A kinematic hardening constitutive model for natural clays with loss of structure[END_REF][START_REF] Kavvadas | A constitutive model for structured soils[END_REF][START_REF] Baudet | A constitutive model for structured clays[END_REF]. These models converge on the MCC model in the limit of total destructuration. They employ an additional isotropic hardening variable, namely the degree of structure, in combination of p 0 , to model the enhanced strength of a structured soil, relative to a reconstituted soil at the same void ratio.

The combined effects of bonding and partial saturation were considered by Alonso and Gens [START_REF] Alonso | Keynote lecture: On the mechanical behaviour of arid soils[END_REF]. It was predicted that these combined effects would result in an increase in yield stress that is larger than the sum of the individual contributions. This was confirmed by Leroueil and Barbosa [START_REF] Leroueil | Combined effect of fabric, bonding and partial saturation on yielding of soils[END_REF], who conducted experiments on an unsaturated saprolite from Brazilian gneiss, artificially bonded with cement (see Fig. 1). The question of how to model this suction-structure interaction by generalizing the definition of the effective p 0 , has only recently been considered by e.g. [START_REF] Alonso | Swelling and degradation of argillaceous rocks[END_REF][START_REF] Vaunat | Bond degradation and irreversible strains in soft argillaceous rock[END_REF][START_REF] Pinyol | A constitutive model for soft clayey rocks that includes weathering effects[END_REF][START_REF] Rouainia | A constitutive model for unsaturated soils with structure degradation[END_REF][START_REF] Koliji | New basis for constitutive modelling of unsaturated aggregated soil with structure degradation[END_REF][START_REF] Buscarnera | An elastoplastic strainhardening model for soil allowing for hydraulic bonding-debonding effects[END_REF][START_REF] Yang | A constitutive model for unsaturated cemented soils under cyclic loading[END_REF]. Most of these advanced models account for a double structure, defined by micro-and macro-scale porosities.

In the following, a relatively simple model accounting for structure and partial saturation in soils is proposed. This model is aimed at capturing the coupled effects of these features on the macroscopic behaviour of natural soils.

[Fig. 1 about here.]

2 CONSTITUTIVE MODELLING:

Choice of stress variables

Attention is restricted to small deformation regime, isothermal conditions and the rateindependent behaviour. Thus, the basic elasto-plastic assumption is the additive decomposition of the strain rate, ˙ , into an elastic and a plastic part, ˙ e and ˙ p , respectively:

˙ = ˙ e + ˙ p (3)
The effective pressure is p = p-u, where p is the total pressure and u is the equivalent pore pressure. Following an energetic approach, Coussy & Dangla ([36]) proposed the following relation, accounting for the energy of the interfaces that separate the liquid and gas phases:

u = u G -S r s - 2 3 U (S r , φ) (4)
where φ is the porosity and U (S r , φ) corresponds to the interfaces energy (see Figure 2 for a schematic representation of the relationship between interfaces energy and the water retention curve) and is given by:

U (S r , φ) = 1 Sr s(S r , φ)dS r (5)
The coefficient 2/3 appearing in Equation ( 4) takes account of the effects of skeleton deformation (via porosity changes) on the water retention properties of the soil.

[Fig. 2 about here.]

2.2 Constitutive stress-strain equations

Modelling unsaturation effects

According to the framework described in [START_REF] Pereira | Adaptation of existing behaviour models to unsaturated states: Application to CJS model[END_REF], the stress defined by eq. ( 1) together with the equivalent pore pressure defined by eq. ( 4) can be used instead of Terzaghi's effective stress to extend elastoplastic models to unsaturated states.

Starting from the Modified Cam-Clay model, the constitutive stress-strain equations of the model presented in this paper are summarized hereafter. For simplicity, the model formulation is expressed in triaxial stress space. The elastic law is represented by the incremental elastic-strain/stress equations:

d e v = κ v dp p (6)
d e q = dq 3 G (7)
where e v and e q are the volumetric and shear components of the elastic strain tensor, v is the specific volume, and κ and G are material constants. The yield function is given by:

f (p , q, p 0 ) ≡ q 2 -M 2 p (p 0 -p ) = 0 (8)
where M is a material constant and p 0 represents the (generalized) hardening parameter of the model, which can be seen as an apparent consolidation pressure. It accounts not only for the (classical) mechanical hardening arising from irreversible volumetric strains but also for the suction-induced (apparent) hardening.

The remaining ingredient to complete the unsaturated elastoplastic framework is the definition of the hardening due to suction. This feature, which is a key point in the modelling of unsaturated soil behaviour, accounts for an additional role of suction on the strength of the soil (apart from its contribution to static equilibrium). This function corresponds to the well-known Loading-Collapse (LC) curve of the Barcelona Basic Model (BBM, [START_REF] Alonso | A constitutive model for partially saturated soils[END_REF]). The identification of this relation may be achieved by using a direct comparison to the LC curve of the BBM using the following relation:

p 0 = p0 + u G -u (9)
This expression is obtained by equating the total preconsolidation pressures expressed in terms of effective stress (eq. 1) and net stress σ net = σ-u G ι. p0 is the net preconsolidation pressure as defined in the BBM:

p0 = p c p * 0 p c λ(0)-κ λ(s)-κ (10)
where p c is a reference pressure and λ(s) is given by:

λ(s) = λ(0) r + (1 -r) exp(-β s) (11)
where λ(0) is the slope of the virgin compression line at saturation in the net stressspecific volume plane, r is the fraction of λ(0) corresponding to the slope of the virgin compression line of the material under high suctions in the net stress-specific volume plane and β is a material parameter linked to the rate of change of the soil compressibility with suction. Alternatively, one can also give directly an explicit functional definition of the apparent hardening due to suction:

p 0 = p * 0 l(s) (12)
where p * 0 is the effective preconsolidation pressure under saturated conditions and at the same plastic strain state than the effective preconsolidation pressure p 0 at the current unsaturated state. In this case, the identification of the l(s) function should be based on experimental data on the evolution of the apparent yield locus with respect to changes in suction (or degree of saturation).

It is worth noting that other choices are possible for the expression of the preconsolidation pressure as a function of the saturation state. However, it is easy to show [START_REF] Dangla | Plasticité et hystérésis[END_REF] that the one given in equation ( 12) ensures that

dp 0 p 0 = dp * 0 p * 0 ( 13
)
as is the case in the BBM. It should be noted that the l(s) function may be expressed in terms of the degree of saturation S r instead of the suction s. This may induce some advantages in terms of numerical modelling (concerning transitions between saturated and unsaturated conditions). In terms of consistency, this latter approach (already used by several authors, see for instance [START_REF] Jommi | Remarks on the constitutive modelling of unsaturated soils[END_REF][START_REF] Tamagnini | An extended Cam-clay model for unsaturated soils with hydraulic hysteresis[END_REF]) presents the advantage to include in the arguments of the function describing the effective preconsolidation pressure p 0 (see eq. ( 12))

two variables of the same thermodynamical nature, namely ε p v and S r which are both generalized strains. Here, two expressions are considered:

l(S r ) = 1 + k p atm s(S r ) S r -s e (14)
l(S r ) = 1 + a tanh b p atm (s(S r ) -s e ) (15)
where p atm is a reference pressure and s e is the air entry suction, and k, a and b are are material constants for a given soil. The mechanical hardening rule does not depend on the l(S r ) function (see eq. ( 13)) and can classically be expressed as follows:

dp * 0 p * 0 = v λ -κ d p v (16)

Including structure effects

As mentioned previously, structure and partial saturation contributions to the apparent yield stress are coupled in the sense that the LC curve of a cemented material is neither a simple translation of the uncemented LC curve nor an homothetic transformation of it (see Fig. 1). This point is clearly observable in Figs. 3 and4. When inspecting the LC curves in terms of yield stress normalised with respect to the saturated stress, it can be concluded that the relative contribution of suction to the yield stress diminishes for increasing cement content. The reason for this observation is that capillary forces that contribute to interparticle efforts (and thus increase soil's strength) tend to become negligible if the soil's intrinsic stiffness increases (which occurs when the cement content increases). A schematic representation of these coupled effects is presented in Fig. 5.

[Fig. 3 To reproduce these observations, and following [START_REF] Rouainia | A kinematic hardening constitutive model for natural clays with loss of structure[END_REF], a scalar parameter R ≥ 1 is introduced as a measure of the degree of structure of the material. This parameter is supposed to degrade from its initial value to unity as the material deforms and loses its initial structure. Starting from the saturated net preconsolidation pressure of the equivalent remoulded or uncemented soil p * 0 , the apparent preconsolidation pressure of the structured soil is R p * 0 at saturation, and for a given value of suction it is given by:

p0 = p c R p * 0 p c λ(0)-κ λ(s)-κ (17)
Other possibilities exist. For instance, if the alternative choices to the LC curve of BBM hereinabove mentioned (equations ( 14) and ( 15)) are used, the following expressions for the effective preconsolidation pressure are obtained (see Fig. 6):

p 0 (s) = R p * 0 l(S r ) = R p * 0 1 + k p atm (s S r -s e ) (18)
p 0 (s) = R p * 0 1 + a tanh b p atm (s -s e ) (19)
The rate of degradation of the soil structure is also assumed to be saturation dependent and given by:

dR = -k(S r ) R -1 λ(0) -κ dε d (20)
where dε d is a measure of the incremental irreversible deformation (including both volumetric and shear strains):

dε d = (1 -A)(d p v ) 2 + A(d p q ) 2 (21)
with A a material constant. Without further indication on the effective couplings between structure and partial saturation effects, the structure degradation function, k(S r ), is assumed to have the following form:

k(S r ) = k 0 p * 0 p 0 ψ (22)
where k 0 is a material parameter representative of structure degradation under saturated conditions and ψ is a material parameter that determines the effect of partial saturation on the rate of degradation via the ratio between the saturated and unsaturated preconsolidation pressures. The modified preconsolidation stress as defined in Equations ( 9) and ( 17) (or alternatively in equations ( 18) or ( 19)) is then included in the yield function (eq. ( 8)). Invoking the consistency condition gives:

∂f ∂p dp + ∂f ∂q dq + ∂f ∂p 0 dp 0 = 0 (23)
Expanding the last term in [START_REF] Liu | Modelling the destructuring of soils during virgin compression[END_REF] gives:

∂f ∂p dp + ∂f ∂q dq + ∂f ∂p 0 ∂p 0 ∂ p v d p v + ∂f ∂p 0 ∂p 0 ∂s ds + ∂f ∂p 0 ∂p 0 ∂R dR = 0 (24)
Accounting for the hardening function [START_REF] Pereira | Adaptation of existing behaviour models to unsaturated states: Application to CJS model[END_REF], the plastic deformation increment may formally be written as:

d p = Λ 1 dσ + Λ 2 dsι + Λ 3 dRι (25)
which shows the three possible (and coupled) mechanisms involved in the generation of plastic strains where Λ 1 , Λ 2 and Λ 3 are hardening functions.

[Fig. 6 about here.]

Modelling the water retention properties

In a first step, the water retention properties of the soil will be modelled using Brooks & Corey's model (see [START_REF] Brooks | Hydraulic properties of porous media[END_REF]):

S r = s e s α ( 26
)
where s e and α are model parameters related to the air entry value and the slope of the retention curve in a (ln(S r ), ln(s)) plane. This equation does not include hysteresis effects. It can be seen as the description a main branch (either drying or wetting) of the water retention curve if monotonic changes of water content are to be accounted for.

Furthermore, eq. ( 26) assumes no coupled effects of soil deformation (porosity changes for instance) on the water retention curve.

Identification of model parameters

Identification of the parameters of the model is a key point. The aim of the present model is to add as less as possible new parameters with respect to the Barcelona Basic Model (BBM). The existing parameters originating from Modified Cam-Clay Model (MCC) are:

λ, κ, M , N (or initial void ratio) and p * 0 . In addition, the parameters coming from BBM to account for unsaturation effects are: β, p c and r. The full description of these latter effects requires the definition of the water retention properties, which is not included in the original BBM. This is done here by using 2 additional parameters: s e and α. Finally, structure effects are described using 4 parameters: R 0 , k 0 , A and ψ, where R 0 is the initial degree of structure. These parameters are present in the original contribution of Rouainia & Muir Wood [START_REF] Rouainia | A kinematic hardening constitutive model for natural clays with loss of structure[END_REF]. It should be noted that particular attention has been taken in order to avoid including specific parameters devoted to the coupling effects between saturation and structure although these coupling effects are included in the present model formulation through the dependence (on R and s) of the hardening functions Λ 1 , Λ 2 and Λ 3 in (Eq. ( 25)).

Among the parameters newly introduced with respect to BBM, R 0 has a straightforward physical meaning and can be deduced from common laboratory tests performed on structured and remoulded soils. The others (k 0 , A and ψ) will require an estimation based on a comparison of model predictions with experimental measurements.

One of the main difficulties is the discrimination between the effects of structure and partial saturation on the yield locus in the (p , s) (or equivalently (p, s)) plane. In practice, experimental data (typically isotropic compressions) on saturated material on both intact and reconstituted samples would be sufficient to characterize the structure effects on the studied soil without the coupled influence of partial saturation effects (remember that the scalar parameter describing the degree of structure has been defined in this study as an apparent over-consolidation of the intact material with respect to the remoulded material, both at saturated state). One of the advantages of this procedure is that it involves simple tests on saturated reconstituted specimens which are more easily reproducible.

To close the identification of the overall hardening function, a common procedure is then required which consists in determining the yield point of intact samples at various levels of suction. Doing so will of course give an idea of the couplings that may appear between partial saturation and structuration effects (see Figs. 3 and4 and the corresponding discussion). In complement, a schematic representation of the coupled effects between structure and partial saturation is given in Fig. 7.

[Fig. 7 about here.]

MODEL PERFORMANCE:

The proposed constitutive model has been implemented into a constitutive driver in order to test its capabilities at the stress point level. Some simulations of theoretical tests and comparisons with available experimental data are presented hereafter. The model parameters used in this section are summarised in Table 1. Figure 9 highlights the role of the initial degree of structure on the simulations. Soils presenting a higher initial degree of structure are characterized by a higher apparent yield point and an increasing trend to brittle behaviour. At high stresses, corresponding to considerable degradation of the initial structure, the curves of initially structured soils (R 0 = {2, 3, 4, 5}) tend to converge towards that of unstructured materials (R 0 = 1).

Illustration of model capabilities

[Fig. 9 about here.]

Confrontation to experimental results

Isotropic compression tests performed on specimens (sampled at 8 m depth) of a residual soil from São Paolo, Brazil (see [START_REF] Machado | Geotechnical characteristics of an unsaturated soil deposit at Sao Carlos, Brazil[END_REF]) are simulated. The main drying branch of the water retention curve has been given by the authors. A good fit is obtained using s e = 0.55 kPa and α = 8.57 (see Fig. 10). Simulation results using the proposed model are presented in Fig. 11. The model is able to capture the increase in elastic stiffness due to its formulation using an effective stress framework (a higher suction means a higher effective stress and thus a higher bulk stiffness as predicted by eq. ( 6), see [START_REF] Pereira | A general framework for constitutive models adaptation to unsaturated states[END_REF][START_REF] Alonso | A microstructurally based effective stress for unsaturated soils[END_REF] for more details).

A reasonably good prediction can be observed. From this figure, one can see that the proposed model is able to reproduce the maximum of collapse which is typical of natural and compacted unsaturated soils (see among others [START_REF] Bolzon | Elastoplastic soil constitutive laws generalized to partially saturated states[END_REF][START_REF] Castelblanco | Some aspects of the compression and collapse behaviour of an unsaturated natural loess[END_REF][START_REF] Vecchia | A fully coupled elastic-plastic hydromechanical model for compacted soils accounting for clay activity[END_REF][START_REF] Georgiadis | Three-Dimensional Constitutive Model for Partially and Fully Saturated Soils[END_REF]). This can indeed be seen from the increasing and then decreasing vertical distance between the saturated and any of the unsaturated compression curve in Fig. 11. This feature arises from the chosen coupling between suction and structure contributions to the yield limit (see Eq. ( 17)).

[Fig. 10 Fig. 3: Coupled effects of suction and cement ratio (data from [START_REF] Leroueil | Combined effect of fabric, bonding and partial saturation on yielding of soils[END_REF]). Curves represent a best fit using a power law.

  about here.] [Fig. 4 about here.] [Fig. 5 about here.]

Figures 8 and 9

 9 Figures 8 and 9 illustrate the influence of initial suction and degree of structure on the

Fig. 1 : 1 Fig. 2 :

 112 Fig.1: Evolution of the yield stress as a function of suction in the case of an artificially cemented soil (after[START_REF] Leroueil | Combined effect of fabric, bonding and partial saturation on yielding of soils[END_REF]).

  Yield stress normalised with respect to saturated state

Fig. 4 : 1 Fig. 5 :Fig. 6 :Fig. 7 :

 41567 Fig. 4: Coupled effects of suction and cement ratio on normalised yield stress (data fromCurves represent a best fit using a power law.

Fig. 8 :

 8 Fig. 8: Influence of initial suction value on oedometer response of a virtual unstructured soil: constant suction compressions (s = {1, 50, 100, 200, 500} kPa, R 0 = 1).

Fig. 9 :Fig. 10 :

 910 Fig.9: Influence of the initial degree of structure on oedometer response of a virtual soil: constant suction compressions (s = 100 kPa, R 0 = {1, 2, 3, 4, 5}).

Fig. 11 :

 11 Fig. 11: Isotropic compression of residual soil from São Paolo (8 m) at different suction values (experimental data from [40]).

Table 1 :

 1 Model parameters. Evolution of the yield stress as a function of suction in the case of an artificially cemented soil (after[START_REF] Leroueil | Combined effect of fabric, bonding and partial saturation on yielding of soils[END_REF]). . . . . . . . . . . . . . . . . . . . . 26 2 Water retention curve and interface energy. . . . . . . . . . . . . . . . . . Schematic representation of isotropic behaviour of saturated and unsaturated structured soils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8 Influence of initial suction value on oedometer response of a virtual unstructured soil: constant suction compressions (s = {1, 50, 100, 200, 500} kPa, R 0 = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 9 Influence of the initial degree of structure on oedometer response of a virtual soil: constant suction compressions (s = 100 kPa, R 0 = {1, 2, 3, 4, 5}). 34 10 Water retention curve of residual soil from São Paolo (8 m): experimental data from [40] and simulation using Brooks & Corey's model. . . . . . . . 35 11 Isotropic compression of residual soil from São Paolo (8 m) at different suction values (experimental data from [40]). . . . . . . . . . . . . . . . . 36

	about here.]
	[Fig. 11 about here.]
	[Table 1 about here.]