Erik Alex Papa Quiroz
email: erikpapa@gmail.com

Antoine Soubeyran
email: antoine.soubeyran@gmail.com

Paulo Roberto

Erik A Papa Quir

Paulo R Oliveira
email: poliveir@cos.ufrj.br

Coercivity and generalized proximal algorithms: application-traveling around the world

Keywords: Proximal point methods, Inexact algorithms, Coercivity, Quasi distances, Variational rationality, Traveler problem

published or not. The documents may come

Coercivity and generalized proximal algorithms:

application-traveling around the world

Introduction

The proximal point method, introduced by [START_REF] Martinet | Régularisaton, d'inéquations variationelles par approximations successives[END_REF] to solve the minimization problem min{ f (x) : x ∈ IR n } generates a sequence {x k } given by x 0 ∈ IR n , and

x k ∈ arg min{ f (x) + (λ k /2)||x -x k-1 || 2 : x ∈ IR n }, (1.1)
where λ k is a certain positive parameter. It is well known, see [START_REF] Güler | New proximal point proximal algorithms for convex minimization[END_REF], that if f is convex in (1.1) and {λ k } satisfies +∞ k=1 (1/λ k ) = +∞, then lim k→∞ f (x k ) = inf f ; and if the optimal set is nonempty, we show that {x k } converges to an optimal solution of the problem. Furthermore, if f is quasiconvex then the sequence converges to a critical point of the problem, see [START_REF] Quiroz | Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds[END_REF][START_REF] Quiroz | Full convergence of the proximal point method for quasiconvex function on Hadamard manifolds[END_REF].

To make possible an application of the proximal point method to solve problems in Computer Theory [START_REF] Romaguera | Applications of utility functions defined on quasi-metric spaces[END_REF], Economics (consumer choice and utility functions) [START_REF] Levin | Some applications of set-valued mappings in mathematical economics[END_REF], and Behavioral Sciences [START_REF] Soubeyran | Variational rationality and the unsatisfied man: Routines and the course pursuit between aspirations, capabilities and beliefs[END_REF][START_REF] Janeiro | Motivational dynamics in the development of career attitudes among adolescents[END_REF][START_REF] Hirschi | Dynamics in Career development. Personal and organizational perspective[END_REF] we need to consider in (1.1) a non convex function f and a quasi distance q(x, x k-1 ) instead of the Euclidan norm ||xx k-1 ||. Thus, we consider the proximal point algorithm with quasi distances given by the following iteration:

x k ∈ arg min{ f (x) + (λ k /2)q 2 (x, x k-1 ) : x ∈ IR n }.

(1.2)

In 2011, [START_REF] Moreno | A proximal algorithm with quasi distance. Application to Habit's Formation[END_REF] have been studied, for the first time, the convergence of the sequence {x k } generated by the above iteration and proved that if f satisfies a Kurdyka-Lojasiewicz condition on the set of critical points and {x k } is bounded then this sequence converges to a critical point of the problem. Moreover, the authors presented an application to habit's formation and observed that the classical proximal algorithm of [START_REF] Martinet | Régularisaton, d'inéquations variationelles par approximations successives[END_REF] and [START_REF] Rockafellar | Monotone operations and the proximal point method[END_REF] is unable to procure such an application in Economics and Social Sciences.

In 2015, Bento and Soubeyran (2015b), motivated by the papers of [START_REF] Attouch | Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods[END_REF] and [START_REF] Moreno | A proximal algorithm with quasi distance. Application to Habit's Formation[END_REF] presented an inexact version of the proximal point method using quasi distance given by the following main iteration: For k = 0, 1, 2, ..., given x k ∈ IR n , find x k+1 ∈ IR n and ϕ k+1 ∈ IR n such that

f (x k+1 ) ≤ f (x k ) -λ k+1 (1 -σ ) (q(x k , x k+1 )) (1.3) ϕ k+1 ∈ ∂ f (x k+1 ) (1.4) ψ k+1 ∈ ∂q(x k , .)(x k+1 ) (1.5) ||ϕ k+1 || ≤ b ((q(x k , x k+1 )))||ψ k+1 || (1.6)
where (.) : [0, +∞) -→ [0, +∞) satisfies some adequate conditions as definided in ( 2) and (3) of Bento & Soubeyran (2015a), δ ∈ (0, 1) and ∂ f denotes the limiting subdifferential of f . Assuming that the sequence {x k } is bounded and that f satisfies a Kurdyka-Lojasiewicz inequality condition, the authors proved that the sequence converges to a critical point of f , studied the rate of convergence of the sequence and introduced an application to change dynamics in behavioral sciences. The above algorithm has been applied to routine's formation with resistance to change, following worthwhile changes, see (Bento & Soubeyran, 2015a).

In this paper, motivated from the above papers and the work of [START_REF] Solodov | A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator[END_REF], we introduce other inexact iteration that consider two types of errors, scalar and vectorial, in the following way: For k = 0, 1, 2, ..., given x k ∈ IR n , find x k+1 ∈ IR n and ϕ k+1 ∈ IR n such that

f (x k+1 ) ≤ f (x k ) - λ k+1 2 q 2 (x k , x k+1 ) + k+1 (1.7) ϕ k+1 ∈ ∂ k+1 f (x k+1 ) (1.8) ||e k+1 || ≤ σ q(x k , x k+1 ), (1.9)
where σ > 0,

e k+1 = ϕ k+1 + λ k+1 q(x k , x k+1 )ψ k+1 (1.10) ψ k+1 ∈ ∂q(x k , .)(x k+1 ),
(1.11)

and the scalar error k+1 satisfies:

∞ k=0 k+1 < +∞.
(1.12)

The error criterion (1.7) is a k+1 sufficient-decrease condition and conditions (1.8),(1.9) and (1.10) have been introduced by [START_REF] Solodov | A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator[END_REF] to find zeroes of maximal monotone operators but considering an extragradient condition.

We should compare the two algorithms: Observe that condition (1.3) in the algorithm of Bento and Soubeyran (2015b) has not a scalar error condition and so their inequality is not inexact compared with (1.7) of the proposed algorithm. Also, we consider -subdifferentials in (1.8). The condition (1.6) of Bento & Soubeyran (2015a) is more general but it does not include condition (1.10) of the proposed algorithm. However, sufficient conditions such that equation (1.10) satisfies equation (1.6) are λ k ||ψ k+1 || ≥ 1, σ < 1 and {λ k } bounded. In fact, from (1.10) and (1.9) we have

||e k+1 || 2 ≤ σ 2 q(x k , x k+1 ) 2 ≤ σ 2 (||ϕ k+1 || 2 + ||λ k+1 q(x k , x k+1 )ψ k+1 || 2 ).
As σ < 1. Then, applying Lemma 4.1 of [START_REF] Attouch | Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods[END_REF] we obtain that ||ϕ k+1 || ≤ Mλ k+1 q(x k , x k+1 )||ψ k+1 ||. Due that {λ k } is bounded then, we obtain (1.6). Thus, we may conclude that the inexact algorithm proposed by Bento & Soubeyran (2015a) and the inexact algorithm proposed in this paper are, in general, different.

In this paper we are interested in the analysis of convergence of the proposed algorithm when f is an arbitrary coercive function (may be nonconvex, nonsmooth), that is, that does not need to satisfy the Kurdyka-Lojasiewicz property studied by Bento and Soubeyran (2015b). Observe even that a convex function does not need to satisfy the Kurdyka-Lojasiewicks condition, see for example Theorem 36 of [START_REF] Bolte | Characterizations of Lojasiewicz inequalities: Subgradient flows, talweg, convexity[END_REF]. On the other hand, we are also interested in the application of the proposed algorithm to Behavioral Sciences, in particular we study an application to traveler problem in the context of a recent theory of stay and changes human dynamics called variational rationality approach.

This paper examines the convergence of the proposed algorithm. In fact, assuming that f is a proper lower semicontinuous and coercive function we prove that the sequence generated by the proposed algorithm is well defined, bounded and if, furthermore, f is continuous on dom( f ), then all accumulation points of the sequence are generalized limiting critical points of the problem. If f is locally Lipschitz on IR n we prove that the accumulation points are Clarke critical points of the problem and if f is convex then these accumulation points are minimal points of the minimization problem.

The paper is organized as follows. In Sect. 2 we presnt the basic aspects on subdifferential and quasi distances. In Sect. 3 we present the problem and introduced the assumptions.

In Sect. 4 we analyze the convergence of the proposed algorithm for lower semicontinuous functions. In Sect. 5 we study the convergence of the algorithm for locally Lipschitz functions.

In Sect. 6 we analyze the convergence when the function is convex. In Sect. 7 we consider, as an application of the proposed algorithm, a traveler problem, in the context of a recent theory of stays and changes human dynamics, i.e., continue, stop and start doing things in behavioral sciences. That is, the variational rationality approach. Finally in Sect. 8 we give some conclusions.

Basic facts

Fréchet and limiting subdifferentials

Definition 2.1 Let f : IR n → IR ∪ {+∞} be a proper function.

(a) For each x ∈ dom( f ), the set of regular subgradients (also called Fréchet subdifferential) of f at x, denoted by ∂ f (x), is the set of vectors v ∈ IR n such that

f (y) ≥ f (x) + v, y -x + o( y -x ), where lim y→x o( y -x ) y -x = 0. Or equivalently, ∂ f (x) := v ∈ IR n : lim inf y =x, y→x f (y)-f (x)-v,y-x ||y-x|| ≥ 0 . If x / ∈ dom( f ) then ∂ f (x) = ∅. (b)
The set of general subgradients (also called limiting subdifferential) of f at x ∈ IR n , denoted by ∂ f (x), is defined as follows:

∂ f (x) := v ∈ IR n : ∃ x l → x, f (x l ) → f (x), v l ∈ ∂ f (x l ) and v l → v .
(c) For each x ∈ dom( f ), the set of -regular subgradients (also called -Fréchet subdifferential) of f at x, denoted by ∂ f (x), is the set of vectors v ∈ IR n such that

f (y) ≥ f (x) + v, y -x -+ o( y -x ), where lim y→x o( y -x ) y -x = 0. Or equivalently, ∂ f (x) := v ∈ IR n : lim inf y =x, y→x f (y)-f (x)-v,y-x ||y-x|| ≥ -. If x / ∈ dom( f ) then ∂ f (x) = ∅. (d)
The set of -general subgradients (also called -limiting subdifferential) of f at x ∈ IR n , denoted by ∂ f (x), is defined as follows:

∂ f (x) := v ∈ IR n : ∃ x l → x, f (x l ) → f (x), v l ∈ ∂ f (x l ) and v l → v .
(e) The set of generalized limiting subdifferential of f at x ∈ IR n , denoted by ∂ f (x), is defined as follows:

∂ f (x) := v ∈ IR n : ∃ x l → x, f (x l ) → f (x), v l ∈ ∂ l f (x l ), v l → v,
∂( f + g)( x) ⊂ ∂ f ( x) + ∂ g( x)
Proof See [START_REF] Mordukhovich | Variational analysis and generalized differentiation I: Basic theory[END_REF], Theorem 2.33.

Quasi distances

In this section we give the definition of quasidistance and present some properties which will be used in the following section.

Definition 2.2 [START_REF] Stojmirović | Quasi-metric spaces with measure[END_REF] Let X be a set. A mapping q : X × X → I R + is called a quasi distance if for all x, y, z ∈ X , 1. q(x, y) = q(y, x) = 0 ⇔ x = y, 2. q(x, z) ≤ q(x, y) + q(y, z).

From Definition 2.2, if q is also symmetric, that is, for all x, y ∈ X , q(x, y) = q(y, x), then q is a distance. Therefore the concept of quasi distance generalizes the concept of a distance.

Example 2.1 [START_REF] Concilio | Quasi-metric spaces and point-free geometry[END_REF] The function q

: I R × I R → I R + defined by q(x, y) = 0 if x ≤ y, 1 otherwise.
generates a quasi distance on I R.

Example 2.2 (Example 3.3 of [START_REF] Moreno | A proximal algorithm with quasi distance. Application to Habit's Formation[END_REF] For each i = 1, ..., n, we consider c - i , c + i > 0 and the function q i : I R × I R → I R + defined by

q i (x i , y i ) = c + i (y i -x i ) if y i -x i > 0, c - i (x i -y i ) if y i -x i ≤ 0, which is a quasi distance on I R. Therefore, q(x, y) = n i=1 q i (x i , y i ) is a quasi distance on I R n . On the other hand, for each z ∈ I R n it holds q(x, z) = n i=1 q i (x i , z i ) = n i=1 max c + i (z i -x i ), c - i (x i -z i ) , x ∈ I R n , Thus q( • , z) is a convex function. By the same reasoning, q(z, • ) is convex.
Consider the following condition on the quasi distance q(., .): there is a positive constant β such that

q(x, y) ≤ β||x -y||, ∀ x, y ∈ IR n . (2.1)
We note that the quasi distances of Example 2.2 verifies (2.1).

The next results are important for our study.

Proposition 2.4 Let q : IR n × IR n → IR + be a quasi distance that verifies (2.1). Then for each z ∈ IR n the functions q(z, • ) and q( • , z) are Lipschitz continuous.

Proof See Proposition 3.6 of [START_REF] Moreno | A proximal algorithm with quasi distance. Application to Habit's Formation[END_REF].

Proposition 2.5 Let z ∈ IR n . If q verifies (2.1) then q 2 (z, • ) and q 2 ( • , z) are locally Lipschitz continuous functions on IR n .

Proof See Proposition 3.7 of [START_REF] Moreno | A proximal algorithm with quasi distance. Application to Habit's Formation[END_REF].

The problem

Consider the optimization problem:

min{ f (x) : x ∈ IR n } (3.1)
where f : IR n → IR ∪ {±∞} is a extended proper function satisfying the following assumptions:

(H 1 ) f is a proper lower semicontinuous function (H 2 ) f is coercive, that is, lim ||x||→+∞ f (x) = +∞ (H 3 ) q : IR n × IR n → IR + is a quasi distance satisfying: there exists β > 0 such that q(x, y) ≤ β||x -y||. (3.2)
Observe that the main justification for using coercive functions, assumption (H2), instead of functions satisfying the Kurdyka-Lojasiewicz property is that in behavioral sciences we need global solutions and not only local solutions as for Kurdyka-Lojasiewicz functions. Furthermore, the Kurdyka-Lojasiewicz hypothesis can be very difficult to verify.

Inexact proximal algorithm with quasi distance (IPAQ) for lower semicontinous functions

IPAQ Initialization: Choose a quasi distance q(., .) defined on IR n , an initial point x 0 ∈ IR n , a real number σ > 0 and two positive real sequences {λ k+1 } and { k+1 } such that

+∞ k=0 k+1 < +∞. (4.1) Main Step: For k = 0, 1, 2, ..., given x k ∈ IR n , find x k+1 ∈ IR n and ϕ k+1 ∈ IR n such that f (x k+1 ) ≤ f (x k ) - λ k+1 2 q 2 (x k , x k+1 ) + k+1 (4.2) ϕ k+1 ∈ ∂ k+1 f (x k+1 ) (4.3) ||e k+1 || ≤ σ q(x k , x k+1 ) (4.4)
where

e k+1 = ϕ k+1 + λ k+1 q(x k , x k+1 )ψ k+1 , (4.5) ψ k+1 ∈ ∂q(x k , .)(x k+1 ), ∂ k+1 f is given by Definition 2.1, (d). Stop Criterion: If x k+1 = x k or 0 ∈ ∂ f (x k+1 ), then stop. Otherwise, do k ← k + 1 and
return to Main Step.

Theorem 4.1 Under assumptions (H 1 )-(H 3 ), for each k, there exists (x k+1 ,ϕ k+1 ) satisfying (4.2), (4.3) and (4.4), that is, the (IPAQ) is well defined.

Proof Given x k ∈ IR n , consider the problem min{ f (x) + λ k+1 2 q 2 (x k , x) : x ∈ IR n }.
From the lower semicontinuity of f and q 2 (x k , .) and coercivity of f , there exists x k+1 such that

f (x k+1 ) + λ k+1 2 q 2 (x k , x k+1 ) ≤ f (x) + λ k+1 2 q 2 (x k , x),
for all x ∈ IR n . Taking x = x k we get (4.2). On the other hand, from the optimality condition, see Proposition 2.2, we have

0 ∈ ∂ f (.) + λ k+1 2 q 2 (x k , .) (x k+1 )
Due to Proposition 2.1 and Proposition 2.3 we have (4.3) and (4.5) with e k+1 = 0.

Remark 4.1 The result of the above theorem is also true if we substitute the condition (3.2) in assumption (H 3 ) by the assumption that q(x, .) :

IR n → IR + is lower semicontinuous for each x ∈ IR n .
Lemma 4.1 Let {v k },{γ k }, and {β k } be nonnegative sequences of real numbers satisfying

v k+1 ≤ (1 + γ k ) v k + β k and such that ∞ k=1 β k < ∞, ∞ k=1 γ k < ∞.Then, the sequence {v k } converges.
Proof See Lemma 2, pp. 44, of [START_REF] Polyak | Introduction to optimization[END_REF].

Proposition 4.1 Under assumptions (H 1 ), (H 2 ), (H 3 ) and the sequence {x k }, generated by the (IPAQ), satisfies:

(i) { f (x k )} converges; (ii) If λ -< λ k+1 , for some λ -> 0, then q(x k , x k+1 ) converges to zero; (iii) {x k } is bounded; (iv) If λ -< λ k+1
, for some λ -> 0, and {x k j } converges to x, then {x k j +1 } converges to

x.

Proof (i) From (4.2) we have

f (x k+1 ) ≤ f (x k ) - λ k+1 2 q 2 (x k , x k+1 ) + k+1 (4.6) Lemma 4.1 and (4.1) imply that { f (x k )} converges. (ii) From (4.6) and λ -< λ k+1 , we have q 2 (x k , x k+1 ) ≤ 2 λ -f (x k ) -f (x k+1 ) + k+1 . Given that { f (x k )} converges and { k+1 } converges to zero, then taking k → +∞ we obtain the result. (iii) From (4.6) we have f (x k ) ≤ f (x 0 ) + k-1 l=0 k+1 ≤ f (x 0 ) + +∞ l=0 k+1 < +∞. This implies that x k ∈ L f (α 0 ) := {x ∈ IR n : f (x) ≤ α 0 } where α 0 = f (x 0 ) + +∞ l=0 k+1 . As L f (α 0 ) is bounded (because f is coercive) then {x k } is bounded.
(iv) From the triangular inequality property we obtain

||x k j +1 -x|| ≤ ||x k j +1 -x k j || + ||x k j -x|| ≤ βq(x k j , x k j +1 ) + ||x k j -x||,
where the last inequality is due to (3.2). As q(x k j +1 , x k j ) converges to zero the result follows.

Remark 4.2

The results (i), (ii) and (iii) of the above proposition are also true if we substitute the condition (3.2) in assumption (H 3 ) by the assumption that q(x, .) : IR n → IR + is lower semicontinuous for each x ∈ IR n . However, to obtain the item (iv) is necessary the condition (3.2). Now, define the foollowing set

U := {x ∈ IR n : f (x) ≤ lim k→+∞ f (x k )}.
Observe that due to assumptions (H 1 )-(H 2 ) this set is nonempty closed and bounded. Theorem 4.2 Under assumptions (H 1 )-(H 3 ), each accumulation point of the sequence {x k }, generated by the (IPAQ), belongs to U .

Proof Let x ∈ IR n be an accumulation point of {x k }, then there exists {x k j } such that x k j → x. As f is lower semicontinuous and

{ f (x k )} converges then f ( x) ≤ lim j→∞ f (x k j ) = lim k→∞ f (x k ). That is, x ∈ U .
Remark 4.3 The result of the above theorem is also true if we substitute the condition (3.2) in assumption (H 3 ) by the assumption that q(x, .) :

IR n → IR + is lower semicontinuous for each x ∈ IR n .
Consider the following assumption:

(H 1 ) f : IR n → IR ∪ {±∞} is proper lower semicontinuous and continuous on dom( f ).

Observe that there are functions that are continuous on their domain but are not lower semicontinuous on IR n , consider for example the function f (x) = 3, for x ∈ (1, 2] and +∞, otherwise.

Theorem 4.3 Under assumptions (H 1 ) , (H 2 ), (H 3 ), and that 0 < λ -< λ k < λ + , each accumulation point of the sequence {x k }, generated by the (IPAQ), is a generalized limiting critical point of the problem (3.1).

Proof Let x ∈ IR n be an accumulation point of {x k }, then there exists {x k j } such that x k j → x and due to Proposition 4.1, (iv), x k j +1 → x. From (4.5) and the triangular inequality we have

||ϕ k j +1 || ≤ ||e k j +1 || + λ k j +1 q(x k j , x k j +1 )||ψ k j +1 || ≤ σ + λ k j +1 ||ψ k j +1 || q(x k j , x k j +1 ) (4.7)
where the last inequality comes from (4.4). As q(x k , .) is locally Lipschitz in IR n then ∂q(x k , .) is locally bounded, see Proposition 9.13 of [START_REF] Rockafellar | Variational analysis[END_REF], then from Lemma 5.1 of [START_REF] Moreno | A proximal algorithm with quasi distance. Application to Habit's Formation[END_REF], there exists M > 0 such that ||ψ k j +1 || ≤ M, ∀ j. Thus, the above inequality (4.7) implies that ||ϕ k j +1 || ≤ (σ + λ + M) q(x k j , x k j +1 ). Therefore, there exists

x k j +1 → x, with f (x k j +1 ) → f ( x) and ϕ k j +1 → 0 with ϕ k j +1 ∈ ∂ k j +1 f (x k j +1 ), then from Definition 2.1, 0 ∈ ∂ f ( x).

Inexact proximal algorithm with quasi distance (IPAQ) for lower semicontinous and locally lipschitz functions

In this section we assume that the function f :

IR n → IR ∪ {±∞} is locally Lipschitz on IR n , thus dom( f ) is open. We consider the -Clarke subdifferential ∂ • f (x) := w ∈: IR n : f • (x, v) ≥ w, v -, ∀ v ∈ IR n , (5.1)
where

f • (x, v) = lim sup t↓0 y→x f (y + tv) -f (y) t .
Now, we consider a variant of the (IPAQ). We will only present the main step and the stop criterion because the initialization step is the same as in the (IPAQ). IPAQ2

Main

Step: For k = 0, 1, 2, ..., given x k ∈ IR n , find x k+1 ∈ IR n and ϕ k+1 ∈ IR n such that

f (x k+1 ) ≤ f (x k ) - λ k+1 2 q 2 (x k , x k+1 ) + k+1 (5.2) ϕ k+1 ∈ ∂ • k+1 f (x k+1 ) (5.3) ||e k+1 || ≤ σ q(x k , x k+1 ), ( 5.4)
where

e k+1 = ϕ k+1 + λ k+1 q(x k , x k+1 )ψ k+1 (5.5)
where

ψ k+1 ∈ ∂ • q(x k , .)(x k+1 ), ∂ • k+1 f is given by (5.1). Stop Criterion: If x k+1 = x k or 0 ∈ ∂ • f (x k+1
), then stop. Otherwise, do k ← k + 1 and return to Main Step.

We assume the following assumption:

(H 1 ) f : IR n → IR ∪ {±∞} is proper lower semicontinuous and locally Lipschitz on dom( f ).

Observe that under the assumptions (H 1 ) , (H 2 ), (H 3 ), the results of Theorem 4.1, Proposition 4.1 and Theorem 4.2 are maintained.

Theorem 5.1 Under assumptions (H 1 ) , (H 2 ), (H 3 ), and 0 < λ -< λ k < λ + , each accumulation point of the sequence {x k }, generated by the (IPAQ2), is a Clarke critical point of the problem (3.1).

Proof From (5.5), e k+1 -λ k+1 q(x k , x k+1 )ψ k+1 ∈ ∂

• k+1 f (x k+1 ), so f • (x k+1 , v) ≥ e k+1 , v -λ k+1 q(x k , x k+1 ) ψ k+1 , v -k+1 , ∀v ∈ IR n .
Let x be a limit of {x k j }, which implies that {x k j +1 } converges to x. Then,

f • (x k j +1 , v) ≥ e k j +1 , v -λ k j +1 q(x k j , x k j +1 )M||v|| -k j +1 ,
where M satisfies ||ψ k j +1 || ≤ M. Taking lim sup in the above inequality and using the property that f

• (., .) is upper semicontinuous we obtain that f • ( x, v) ≥ lim sup j→+∞ f • (x k j +1 , v) ≥ 0. It follows that 0 ∈ ∂ • f ( x).
Remark 5.1 If f is a proper, lower semicontinuous and convex function and dom( f ) is open, then f satisfies the assumption (H 1 ) and thus, from Theorem 5.1, we obatin that each accumulation point of the sequence {x k }, generated by the (IPAQ2), is a minimum point of problem (3.1). However, if dom( f ) is not open then we should search another way to obtain that result.

Remark 5.2

We should observe that if f is convex and differentiable then,

∂ • f (x) ⊂ ∂ F f (x) = ∇ f (x) + B(x, 0), where ∂ F f is the Fenchel -subdifferential: ∂ F f (x) = {s ∈ IR n : f (y) ≥ f (x) + s, y -x -, for all y ∈ IR n }.
(5.6)

That is, for the convex and differentiable case, the Fenchel -subdifferential includes the Clarke -subdifferential defined in (5.1).

The above remarks motive us to study the (IPAQ) considering the Fenchel -subdifferential when f is a proper lower semicontinuous convex function.

Inexact proximal algorithm with quasi distance (IPAQ) for lower semicontinous convex functions

In this section, we complete the convergence of the method for arbitrary proper lower semicontinous convex functions. For that, we assume the following condition:

(H 1 ) f is a proper lower semicontinuous and convex function.

As in the previous section, we will only present the main step and the stop criterion because the initialization step is the same as in the (IPAQ). IPAQ3

Main

Step: For k = 0, 1, 2, ..., given x k ∈ IR n , find x k+1 ∈ IR n and ϕ k+1 ∈ IR n such that

f (x k+1 ) ≤ f (x k ) - λ k+1 2 q 2 (x k , x k+1 ) + k+1 (6.1) ϕ k+1 ∈ ∂ F k+1 f (x k+1 ) (6.2) ||e k+1 || ≤ σ q(x k , x k+1 ) (6.3)
where e k+1 = ϕ k+1 + λ k+1 q(x k , x k+1 )ψ k+1 (6.4)

ψ k+1 ∈ ∂ • q(x k , .)(x k+1 ), ∂ F k+1 f is given in (5.6). Stop Criterion: If x k+1 = x k or 0 ∈ ∂ F f (x k+1
), then stop. Otherwise, do k ← k + 1 and return to Main Step.

Theorem 6.1 Under assumptions (H 1 ) , (H 2 ), (H 3 ) and that {λ k } is bounded, each accumulation point of the sequence {x k }, generated by the (IPAQ3), is an optimal solution of the problem (3.1).

Proof Let x ∈ IR n be an accumulation point of {x k }, then there exists {x k j } such that

x k j → x. As f is lower semicontinuous and { f (x k )} converges then f ( x) ≤ lim k→∞ f (x k ).
On the other hand, as f is convex, then for each x ∈ IR n we have

f (x) ≥ f (x k+1 ) + s k+1 , x -x k+1 -k+1 (6.5) for all s k+1 ∈ ∂ F k+1 f (x k+1 ). From (6.4) e k+1 -λ k+1 q(x k , x k+1 )ψ k+1 ∈ ∂ F k+1 f (x k+1 ) (6.6)
where ψ k j +1 ∈ ∂ • q(x k , .)(x k+1 ). From (6.6) and (6.5) we have

f (x) ≥ f (x k+1 ) + e k+1 , x -x k+1 -λ k+1 q(x k , x k+1 ) ψ k+1 , x -x k+1 -k+1 .(6.7)
As q(x k ,.) is locally Lipschitz in IR n then ∂q(x k ,.) is locally bounded, see Proposition 5.15 from [START_REF] Rockafellar | Variational analysis[END_REF], then from Lemma 5.1 of [START_REF] Moreno | A proximal algorithm with quasi distance. Application to Habit's Formation[END_REF], there exists M > 0 such that ||ψ k+1 || ≤ M, ∀k. Using the above fact in (6.7) and considering that {x k } and {λ k } are bounded and q(x k , x k+1 ) → 0, we have f (x) ≥ lim k→∞ f (x k+1 ) ≥ f (x¯).

Application to location theory: traveling around the world

This section provides an application to location theory in behavioral sciences (economics, decision theory, management sciences, psychology, sociology,....).

Location theory This discipline makes a clear distinction between, i) transportation problems where goods are transported from a given location (source) to another one (target) to satisfy human needs in the target location with, ii) translocation problems where individuals (not goods) move from one location to an other one to better satisfy their needs.

In mathematics and management sciences, a transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.

In economics a translocation problem can be a migration problem where an individual moves from living in one country to live in an other one, or a changing job problem where a worker moves from doing a given job in a given city to do the same or a different job in an other city, or a tourism problem when an individual or a family visits a region of his country, a city, a landscape, a foreign country.....

The variational rationality approach Our application will be examined in the context of the (VR) variational rationality approach of stop, continue and go human dynamics where, each period, individuals constantly navigate (life and move) between stays, i.e., continue to do, and changes, i.e., stop and start to do different situated activities [START_REF] Soubeyran | Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors[END_REF][START_REF] Soubeyran | Variational rationality and the unsatisfied man: Routines and the course pursuit between aspirations, capabilities and beliefs[END_REF][START_REF] Soubeyran | Variational rationality. A theory of worthwhile stay and change approach-avoidance transitions ending in traps[END_REF](Soubeyran, , 2021a, b, c, d), b, c, d). A situated activity, i.e., an action, being an activity done in a given context = environment. The VR approach is a spatial way to model stay and change human dynamics.

The traveler model The central picture of the VR approach is the psychological traveler model that makes a clear distinction between two aspects of human dynamics: weak and strong resistance to move. It offers a very general solution to the Lewin's celebrated tentative, i.e., to build a topological approach of psychology [START_REF] Lewin | A dynamic theory of personality[END_REF][START_REF] Lewin | Principles of topological psychology[END_REF][START_REF] Lewin | The conceptual representation and measurement of psychological forces[END_REF][START_REF] Lewin | Intention, will and need[END_REF]). Lewin's model of human dynamics is a psychological translocation model of how to move and live in a psychological life space. This life space includes, for each individual, his internal environment (needs, goals that help to choose how much of each need to satisfy each period, thoughts and emotions) and his external environment (landscape, objects, and persons that help or forbid to satisfy his needs).

Our application: traveling around the world Our application is a simple translocation model. It poses the question: "How to travel around the world and prepare the trip of a lifetime". It represents a special instance of the traveler model in the context of weak resistance to move (cf, the proximal algorithm). See, for instance, [START_REF] Moreno | A proximal algorithm with quasi distance. Application to Habit's Formation[END_REF], Bento and Soubeyran (2015a;2015b). This application can be contrasted with other applications of the VR approach (changing jobs, see [START_REF] Bao | Variational principles in set optimization with domination structures and application to changing jobs[END_REF]), improving the work-life balance [START_REF] Alfuraidan | Ekeland variational principle on quasi-weighted graphs. Improving the work-family balance[END_REF], transit in migration problems [START_REF] Fakhar | Variational rationality, variational principles and the existence of traps in a changing environment[END_REF] in the context of strong resistance to move (i.e., the Ekeland variational principle).

A variational rationality "world tour" model

For simplification our application concerns a tourist who wants to make a "world tour", visiting successively different countries until ending his "world tour" in a given country and decides, eventually, to live there. The problem is: when does he will stop to visit a new country.

A simple world tour model For simplification, consider an individual that wants to visit successively different locations x, y... ∈ X , one location each period. To economize notations, the space of locations will be a straight line X = IR. Consider two successive periods. Suppose that, in the previous period, this individual has visited location x ∈ X . Then, to become able to visit location y ∈ X in the current period, that is, to get access to what he wants to visit, he must travel from x to y at the beginning of the current period. That is, he must do the move m = x y. Suppose also that, as does each simplified version of the VR model, (i) the utility of visiting a location z for the first time is g(z) ∈ IR. To economize notations, this utility is net of costs to visit location z. For simplification we suppose that the utility of visiting a location for the first and second times are equal. The opposite case, which is much more realistic, has been examined in Soubeyran (2021b, c, d). (ii) the physical distance between location x and location y is d(x, y) = yx . (iii) travel costs of moving from having visited location x ∈ X in the previous period to become able to visit location y ∈ X in the current period are T (x, y) ∈ IR + . We suppose that these travel costs can be modeled as a quasi distance T (x, y) = t r (yx), y ≥ x t l (xy), y < x , where t r , t l > 0 model per unit of distance costs to travel from the right and from the left, depending of the relative position of x and y. If y = x; travelling costs are zero: T (x, x) = 0. (iv) the disutility (x, y) = T (x, y) 2 of bearing travel costs increases strongly (quadratically for simplification) with travel costs.

Payoffs to move (change, or stay).

A payoff to change P(y/x) = g(y) -ξ T (x, y) 2 , ξ > 0 models a traveler who, (i) starts from having visited for the first time a location x in the previous period; (ii) travels, in the current period, from location x to reach a different location y = x and, (iii) visits for the first time location y at the end of the current period.

A payoff to stay P(x/x) = g(x)-ξ T (x, x) 2 = g(x) models a traveler who visits location x for the second time in the current period. The term ξ > 0 measures the importance given to the disutility of travel costs. It depends strongly of the personality of the traveler.

When to start, continue or stop visiting: the VR approach

The main question we pose is: should the traveler stays, or should he changes. Thus, the following definition.

A worthwhile balance The VR approach models the difference between a payoff to change and a payoff to stay. That is, B(y/x) = P(y/x) -P(x/x) = g(y)g(x) -ξ T (x, y) 2 .

In this simple context, (i) the difference A(y/x) = g(y)g(x) provides a simple formulation of advantages to move (change rather than stay). (ii) travel costs define inconveniences to move, I (y/x) = T (x, y). Then, in this setting, a worthwhile balance is the difference between advantages to move and the disutility of inconveniences to move. That is, B(y/x) = A(y/x) -ξ I (y/x) 2 .

We come back, in this simple context, to the general VR formulation of a worthwhile balance, as the weighted difference between motivation and resistance to move:

B(y/x) = M(y/x) -ξ R(y/x).
Stop and go dynamics They represent a succession of moves over a succession of periods (stays or changes). Each period, a move m = x y can be,

(1) improving, if A(y/x) ≥ 0;

(2) improving enough (satisficing), if A(y/x) ≥ γ > 0;

(3) worthwhile if B(y/x) ≥ 0, worthwhile enough if B(y/x) ≥ γ > 0;

(4) not advantageous, if A(y/x) ≤ 0; A position x is, (5) a desired end x = x * , if A(y/x * ) = g(y)g(x * ) ≤ 0 for all y ∈ X . That is, if x * is a maximum of g(.). (6) a critical point x • of a differentiable utility function g(.), if ∇g(x • ) = 0; (7) an approximate critical point if ∇g(x • ) ≤ δ, δ > 0.

Two other concepts played a major role in the VR approach: (8) a stationary trap x = x * , if B(y/x * ) ≤ 0 for all y ∈ X ; (9) a variational trap x * relative to x 0 , if, i) B(x * /x 0 ) ≥ 0 and, ii) B(z/x * ) ≤ 0 for all z ∈ X .

That is, if, i) it is worthwhile to reach x * from x 0 and if, ii) it is not worthwhile to leave x * .

To save space, these two concepts will be examined elsewhere in the context of coercivity hypothesis.

Remark 1

The status of travel costs. They represent, within the VR approach, capability costs to do something. In the traveler example, they model capability costs to visit. They are spend ex ante (before visiting, as fixed costs). Being able to visit y requires to know what to visit at location y (what are the most beautiful landscapes to contemplate, what are the more interesting museums to visit) and how to visit y (what kind of quick and cheap locomotion to use to visit y, where to eat tasty food,...).

Remark 2

The status of execution costs. They represent, in this paper, costs to visit a location. They have been included in the utility function.

A VR formulation of an exact/inexact proximal algorithm

Exact proximal algorithms

An exact proximal algorithm [START_REF] Martinet | Régularisaton, d'inéquations variationelles par approximations successives[END_REF] defined proximal payoffs (y/x) = f (y) + ξ yx 2 , where f (.) : y ∈ X = IR n -→ f (y) ∈ IR is an objective function to be minimized and ξ = λ/2. The algorithm follows a succession of stages k, k + 1, ... The current stage k +1 solves the problem arg min k+1 (y/x) = f (y) + ξ k+1 yx 2 : y ∈ X , where x = x k ∈ X is the given status quo in the current stage k + 1.

The VR approach (Soubeyran, 2021a, b, c, d) makes easily a direct link between a payoff to move and a proximal payoff as follow. Given an utilty function g(.) : y ∈ X -→ g(y) ∈ IR, g * = sup {g(y) : y ∈ X } < +∞ defines an aspiration utility level. It is the maximun utility level an individual can hope to reach. In contrast, f (y) = g *g(y) ≥ 0 represents the disutility of the frustration level that an individual feels if he does not reach his best when he performs y ∈ X . Then, in the simple context of the present paper, given that inconveniences to move I (y/x) can be identified to travel costs T (x, y) (if costs to move are symmetric, equal to the Euclidean distance) a proximal payoff models a payoff to move, i.e., Q(y/x) = (y/x) = f (y) + ξ T (x, y) 2 .

Then, each current period k + 1, a proximal algorithm minimizes the total cost of moving, i.e., the sum of the disutility of travel costs T (x, y) 2 plus the disutility of frustration feelings f (y) = g *g(y) ≥ 0 coming from not having done his best at the end of the move.

Interpretation of the results of this paper

Interpretation of our new inexact proximal algorithm

Our new (IPAQ) poses two main conditions, each period k + 1 :

(i) Weak worthwhile moves. Inequality (4.18) can be written

f (x k ) -f (x k+1 ) ≥ (λ k+1 /2)q(x k , x k+1 ) 2 -k+1 ,
with k+1 > 0 and +∞ k=0 k+1 < +∞. This means that each move m k+1 = x k x k+1 must be, a) almost worthwhile ( k+1 > 0), b) and more and more worthwhile ( +∞ k=0 k+1 < +∞ implies k+1 → 0). (ii) Vanishing marginal total disutilty to move. That is, given that e k+1 = ϕ k+1 + λ k+1 q(x k , x k+1 )ψ k+1 , given in (4.5), represents the marginal total disutility to move, such marginal total disutility must be lower enough relative to costs to move, i.e., e k+1 ≤ σ q(x k , x k+1 ), (4.4).

Importance of coercivity.

An almost finite world. The coercivity hypothesis (H2) (infinity at infinity) done in this paper ( f (.) = g * -g(.) coercive) means that the utility g(y) = g *f (y) to visit a location y ∈ X far away of the origin (initial location) in the space X = IR n (n = 1, in the example) becomes strongly negative. This models a finite world with boundaries. This is a very realistic hypothesis in behavioral sciences. The quasi distance hypothesis models, within the VR approach, asymmetric costs of moving (see also the present example with traveling costs). Hypothesis (H3) means that costs to move are not too asymmetric. Hypothesis (H1)', i.e., upper semicontinuity, prevents downward jumps of the utility function g(.), but allows upward jumps.

Results: Ending in "almost zero" or zero motivation to move . This paper shows that, under the list of hypothesis given in Theorem 4.3, (H1)', (H2), (H3), each accumulation point of the sequence {x k }, generated by the (IPAQ) is a generalized limiting critical point of the problem (3.1). That is, in behavioral terms, in the context of the VR approach, a succession of almost worthwhile moves converges to a critical point where the marginal utility to move ∇g(x * ) (when g is differentiable) is zero. Thus, ending in a zero, or almost zero, motivation to move.

Interpretations of theorems 5.1 and 6.1 are similar. However, in theorem 6.1, the end point x * maximizes the utility function, given its concavity.

Conclusions

We present a general inexact proximal algorithm using quasi distances, called (IPAQ), to find critical points and minimizer points of convergent subsequences generated by the algorithm. Under the assumption of coercivity of the objective function, we analyze the cases of lower semicontinuous, locally lipschitz and convex functions and show that all accumulation points of the sequence generated by the (IPAQ) algorithm are critical points (minimizer points for the convex case) of the minimization problem.

If we consider (IPAQ) with k+1 = 0, for all k = 0, 1, ... and the algorithm does not finish (that is, x k+1 = x k for all k = 1, 2, 3, ... ) then x k+ j = x k for all j = 1, 2, 3, ... and this mean that, in our model, for the asymptotic case the tourist cannot visit the same place twice. On the other hand, if the scalar error k+1 > 0, for some k = 0, 1, ... then, for the asymptotic case, it is possible to have x k+ j 0 = x k , for some j 0 then, it is easy to prove that the point x k is an accumulation point of the sequence {x k } and therefore a critical point (minimum in the convex case) of the problem (3.1). If the algorithm finishes in a finite number of iterations, the actual point is a critical point (minimum in the convex case) of the function or an approximate critical point (approximate minimum in the convex case), that is, e k+1 ∈ ∂ k f (x k+1 ).

Coercivity is very important for two points of view, theoretical and applied: At the mathematical level, a coercitivity assumption implies that for each c ∈ IR, the level set {x : f (x) ≤ c} is bounded and, therefore, the sequence generated by any descent algorithm (in particular the proximal point algorithm) is bounded. On the other hand, to obtain convergence results using a Kurdyka-Lojasiewicz (KL) inequality it is needed, see for example [START_REF] Attouch | Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods[END_REF]Bento & Soubeyran, 2015a, b;[START_REF] Moreno | A proximal algorithm with quasi distance. Application to Habit's Formation[END_REF], to take the sequence generated by the algorithm to be bounded. Then, coercivity is a condition that helps the KL condition to obtain global convergence results.

At the moment, it is not known if in the absence of (3.2), but under the assumption that the function satisfies the Kurdyka-Lojasiewic property on the set of accumulation points, is possible to obtain the same results of convergence. This may be a future work for our research. However, a result in this direction was given in the paper authored by [START_REF] Soubeyran | Generalized descent metod using w-distance. Application to emergence of habits following worthwhile moves[END_REF]. The authors, using a w-distance (and in the absence of the hypotheses (3.2)) as regularized term, proved the convergence of bounded sequences generated by a general descent method proposed by [START_REF] Attouch | Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods[END_REF].

In this paper, we introduce two new subdifferentials, the generalized limiting subdifferential ∂ f (x), see Definition 2.1,(e), and the -Clarke subdifferential in (5.1) which is different between all the -Clarke subdifferential introduced by other authors. In this sense, the study of the properties of these subdifferentials are welcome.

  Motivation to move refers to the utility of advantages to move M(y/x) = U [A(y/x)] , where U [.] : A ∈ IR + -→ U [A] ∈ IR + is strictly increasing. Resistance to move is the disutility of inconveniences to move R(y/x) = D [I (y/x)] , where D [.] : I ∈ IR + -→ D [I ] ∈ IR + is strictly increasing. In a simple linear quadratic translocation model with weak resistance to move, U [A] = A and D [I ] = I 2 .

  and l → 0 . Let f , g : IR n → IR ∪ {+∞} be proper functions such that f is locally Lipschitz at x ∈ dom( f ) ∩ dom(g) and g is a lower semicontinuous function at x. Then,

	Proposition 2.3
	Proposition 2.1 For a function f : IR n → IR ∪ {+∞} and a point x ∈ dom( f ), the subgradient sets ∂ f ( x) and ∂ f ( x) are closed, with ∂ f ( x) convex and ∂ f ( x) ⊂ ∂ f ( x).
	Proof See Rockafellar & Wets (1990), Theorem 8.6.
	Proposition 2.2 If a proper function f : IR Proof See Rockafellar & Wets (1990), Theorem 10.1.

n → IR ∪ {+∞} has a local minimum at the point x ∈ dom( f ), then 0 ∈ ∂ f ( x).

Acknowledgements

The first author is grateful to FAPERJ-CAPES by the economic support on the Pos-Doctoral Project PAPD-2011, Process no E -26/100.057/2013 -B B P. The work of the second author was supported by the French National Research Agency Grant ANR-17-EURE-0020, and by the Excellence Initiative of Aix-Marseille University -A*MIDEX. The research of the third author was partially support by CNPq (Grant 302678/2017-4). The last version of the paper was written when the first author was working as a visiting professor at IME-Federal University of Goias.


R Brault

A Germaneau
email: arnaud.germaneau@univ-poitiers.fr

P Doumalin
email: doumalin@lms.univ-poitiers.fr

J-C Dupre

M Fazzini
email: marina.fazzini@enit.fr

S Mistou
email: mistou@enit.fr

researchers and makes it freely available over the web where possible

The complex mechanical behaviour of composite materials, due to internal heterogeneity and multilayered composition imposes deeper studies. This paper presents an experimental protocol to perform volume kinematic measurements in composite materials. This work is centered on the transverse shear effects in relation with the composite materials mechanical behaviour. A lot of theories exist in literature to consider the existing warping section during high transverse shear stresses, but very few studies have been realized in the experimental way. The association of X-ray micro-computed tomography acquisitions and Digital Volume Correlation (DVC) technique allows the measurement of displacements and deformations in the whole volume of composite specimen. To elaborate the latter, composite fibres and epoxy resin are associated with metallic particles to make contrast in X-µCT acquisitions. A specific in-situ loading device is presented for three-point bending tests, which enables the visualization of transverse shear effects in composite structures.

1.

CONTEXT Up to the present numerous studies have been performed about the mechanical response of composite structures, at different scales. Among the characteristics of composite materials, transverse shear stress effects constitute an important phenomenon which must be taken into account in the design of composite structures. Structural damages or fractures can occur because of wrong evaluation of the transverse shear effects. A large number of works show that the warping of sections due to the transverse shear can be identify with various analytical theories. Kirchhoff-Love and Reissner-Mindlin have respectively proposed the classical and natural theories to consider these effects, which are now available in commercial finite element softwares. More recently, refined theories have been developed on the formulation of warping functions f(x3) (Figure 1) [START_REF] Reddy | A simple high-order theory of laminated composite plate[END_REF][START_REF] Touratier | An efficient standard plate theory[END_REF]. However, very few studies exist on the identification and the validation of these theories by the experimental way. This paper which is a part of a research project, which deals with the visualization of transverse shear effects from volume measurements, with the major interest of observing the evolution of results in the thickness. It is proposed to develop an experimental method to make 3D displacement fields measurements in composite structure volumes.

EXPERIMENTAL TECHNIQUES

This work aims to perform experimental observations within composite materials under loading. For that, a laboratory X-ray micro-tomography (X-µCT) device is employed. Furthermore, experimental measurements of displacement fields within the material must be performed from a volume measurement method. That's why it is proposed to use the Digital Volume Correlation (DVC) technique. Volume measurement techniques are very interesting 2 to perform kinematic calculations which provide a high level of results [START_REF] Bay | Digital volume correlation: three-dimensional strain mapping using x-ray tomography[END_REF][START_REF] Forsberg | Full three-dimensional strain measurements on wood exposed to three-point bending: analysis by use of digital volume correlation applied to synchrotron radiation microcomputed tomography image data[END_REF][START_REF] Germaneau | Comparison between x-ray micro-computed tomography and optical scanning tomography for full 3d strain measurement by digital volume correlation[END_REF][START_REF] Germaneau | Développement de techniques de mesure dans le volume: Photoélasticimétrie 3D par découpage optique et corrélation volumique par tomographie optique et rayons X. Application à l'étude des effets mécaniques 3D dans les structures et les biomatériaux[END_REF]. As it is made for 2D field measurement techniques where a specific speckle pattern is used to make the discretization of specimen surfaces, a 3D pattern must be made inside the composite structure, by marker inclusion, to make a 3D grey level distribution.

A specific in-situ loading device is proposed in order to perform measurements in the three-point bending testing configuration. Bending case is chosen to impose shear stresses, which allowing the warping visualization. The new device allows bending loadings to analyze other 3D mechanical effects linked with transverse mechanical response. Also, tests are carried out at the scale of composite structures, which is new in comparison with previous volume measurement studies. Figure 2 presents an example of displacement field obtained by DVC during bending tests.

RESULTS

Different kinds of results are used in this study. Experiments from DVC are confronted to the analytical values, calculated with different warping functions. A numerical modelling is also used to constitute the reference results of the work. All results are analysed to study the transverse shear effects in the thickness direction. Also, numerical modelling and theoretical theories are discussed to determine their validity, regarding of the mechanical properties and the specimen geometry. For metrology aspects, the global error is determined to validate the DVC parameters and the use of metallic particles in the composite structures.

4.

Figure 1 -

 1 Figure 1-Analytical theories for transverse shear effects analysis

Figure 2 -

 2 Figure 2 -Longitudinal displacement field measured by X-µCT and DVC


D Busca

M Fazzini
email: marina.fazzini@enit.fr

B Lorrain

S Mistou

M Karama

M L Pastor

HIGH-SPEED DIGITAL IMAGE CORRELATION: APPLICATION TO BI-AXIAL FATIGUE

To study the behavior of a carbon fiber reinforced plastic laminate under a bi-axial stress state, a new type of cruciform specimen has been developed. This specimen is loaded bi-axially under static and cyclic fatigue loadings. The experiments are monitored simultaneously using Digital Image Correlation with two High-speed cameras, and infrared thermography. A comparison between the measurements and finite elements model is used to validate the design of the sample. .

1.

INTRODUCTION

Composite materials are increasingly used in industry. Many experiments have been made to develop models to allow very complex structure sizing. But only few experiments have been done to characterize the behaviour of those materials under complex loading, such as biaxial planar loading, and even less dealing about fatigue. The biaxial testing machine owned by the LGP laboratory allows running experiments on biaxial cruciform specimens. The issue of those specimens is that the stress repartition is not constant over the specimen. It is why the Digital Image Correlation technique seems to be appropriate to monitor those tests, since it allows measuring an important area on the specimen with good accuracy. Thermography is as well appropriate to monitor fatigue testing as it permits to measure temperature elevation due to damage and hysteretic phenomenon. In the following study, a new kind of specimen is created to try to better respond to the needs of biaxial testing and to fit with the manufacturing process. Then, this specimen is tested during static and cyclic loading.

DESIGN OF THE CRUCIFORM SPECIMEN

Material and manufacturing process

The material used in this study is a carbon/epoxy composite laminate. The carbon part is a 12K balanced woven fabric. This fabric is made by Hexcel and is referenced as HexForce ® 48302. The resin part is an epoxy resin made by Sicomin and referenced as SR1710. The hardener used is referenced as SD 8824.

The coupons are made using the LRI (Liquid Resin Infusion) process. All the manufacturing part was made in the premises of Nimitech Etudes. The stacking sequence was made only with 0° plies (as the fabric is balanc ed on the 1 and 2 directions, we have the same properties on the x and y axis, e.g. the main axis of the specimen).

Finite Element Model

All the design part was made using the finite elements software ABAQUS 6.10. As many different designs have already been tried in [START_REF] Escárpita | Modified cruciform specimen for bi-axial testing of fibre reinforced composites[END_REF] and [START_REF] Smits | Design of a cruciform specimen for biaxial testing of fiber reinforced composite laminates[END_REF], those have been used as a base for the design of the new cruciform specimens. The design criterions were: -Stress concentration nowhere else than at the center of the specimen -Failure should occur in the central zone due to bi-axial stress state -

The central zone must be as flat as possible to allow DIC -Adaptability to the LRI process -Easy reproducibility -Adaptation to the machine (mechanical jaws)

The elements that have been modified during the design procedure were: -Thickness of the arms and of the central zone -Width of the arms -Size and shape of the corners between the arms -Size and shape of the central zone

The model chosen is a 3D model, using one element per ply in the thickness. The zone considered to be taken in the jaws is considered as a rigid body. All the calculations are run on 1/8 th of the total specimen (2 symmetries due to the shape of the specimen, 1 due to the shape through the thickness and the stacking sequence).

EXPERIMENTAL SETUP

Bi-axial tensile test machine

The experiments were made on an Instron 8800 (Fig. 1). It has four cylinders which can be controlled independently for fatigue or static loading. They can handle a 100kN effort and have a maximum travel of 40mm. This machine is driven by the WaveMatrix and Console FT softwares, which allow complete static and cyclic programming of the machine. This machine has mechanic jaws.

Figure 1-bi-axial test machine and specimen

High-speed DIC and infrared thermography

DIC is a widely spread way to measure a full displacement field on an extended surface during mechanical testing. It has been used for example in [START_REF] Brault | Bi-axial fatigue analysis by stereo-correlation measurement[END_REF] to monitor fatigue testing of cruciform specimens under bi-axial loading, or in [START_REF] Lamkanfi | Strain distribution in cruciform specimens subjected to biaxial loading conditions -Part 1 -Two-dimentional versus 3-dimentional finite element model[END_REF] for static loading of cruciform specimens.

The digital image correlation is made by the ARAMIS-V6.3.0-5 software. A set of two rapid cameras was used to take the pictures. Those cameras are referenced as Photron Fastcam SA1.1, used with the Photron Fastcam Viewer V3.0 1.08E software. Those cameras are capable of 5400 frames per second at 1 megapixel resolution.

Thermography is a good way to monitor fatigue testing. As developed in [START_REF]Approches analytique et expérimentale de l'endommagement par fatigue d'un composite carbone/epoxy thèse de l'Université Toulouse[END_REF], it allows reaching the fatigue limit of materials. The camera used is a Flir jade III retrofitted Titanium, with a thermic resolution of 20 mK at 30°C.

Conditions of the experiments

All the experiments were conducted in the LGP laboratory. Since the machine is vertical, it allows putting simultaneously the two cameras for the DIC setup on one side of the specimen, and the infrared camera on the other side of it. To ensure constant lightning conditions to maximise the accuracy of the DIC measurements, a spotlight was used to lighten the specimen's surface. All the shutters and windows of the room have remained closed for the whole duration of the test.

Both the thermography and the DIC cameras were trigged automatically by signal sent by the machine at different steps of the experiment. High-speed cameras allow taking pictures at precise steps of the loading cycle.

Thermographic images were taken regularly after a certain amount of cycles to measure the increase of the temperature during the test, due to hysteretic or damage effects. Pictures for the DIC were taken after the same amount of cycles as with the thermographic setup. They were taken around the maximum loading value. Several experiments have been made to ensure that pictures were taken at the maximum loading point is reached. The value of the effort and the displacement of the cylinder is known for every picture taken in DIC, as well as the number of cycles. The number of cycles is known as well for the pictures taken with the thermography. This way, the evolution in the elastic behaviour can be measured and compared with tests run on one axis. Beside, a correlation between damage, change in the elastic behaviour and local warming of the specimen has been attempted.

4.