Guy Bouchitté
email: bouchitte@univ-tln.fr

Ornella Mattei
email: mattei@math.utah.edu

Graeme W Milton
email: milton@math.utah.edu

Pierre Seppecher
email: seppecher@imath.fr

On the forces that cable webs under tension can support and how to design cable webs to channel stresses

In many applications of Structural Engineering the following question arises: given a set of forces f1, f2, . . . , fN applied at prescribed points x1, x2, . . . , xN , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x1, x2, . . . , xN in the two-and three-dimensional case. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two-dimensions we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f1, f2, . . . , fN applied at points strictly within the convex hull of x1, x2, . . . , xN . In three-dimensions we show that, by slightly perturbing f1, f2, . . . , fN , there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for distributing stress in desired ways.

Introduction

One of the main goals of Structural Engineering is to find performing structures when one incorporates into the design a specific type of material or substructure. Many materials behave quite differently under tension or compression: concrete and masonry structures are two examples of materials that perform much better under compression. Some types of structures support also only specific loadings: a wire or cable, for example, can only support tension and not compression. Here we are interested in the case where one incorporates a material that works particularly well under tension so that a cable web is expected to be representative of the most performing structure to be used. Thus, we address the following problem: given a set of forces f 1 , f 2 , . . . , f N applied at prescribed points x 1 , x 2 , . . . , x N , under what constraints on the forces does there exist a wire web (or truss structure) with all elements under tension that supports these forces? Note that the problem is identical if one is interested in the case where all elements are under compression. We are only interested in cable wires which can be modeled as a set of straight truss elements: we do not consider the case of catenary elements, see, e.g., [START_REF] Ahmad Abad | Nonlinear Analysis of cable structures under general loadings[END_REF][START_REF] Ahmadizadeh | Three-dimensional geometrically nonlinear analysis of slack cable structures[END_REF][START_REF] Andreu | A new deformable catenary element for the analysis of cable net structures[END_REF][START_REF] Such | An approach based on the catenary equation to deal with static analysis of three dimensional cable structures[END_REF].

In the two-dimensional case, a complete answer to this problem is given by Theorem 1 in [START_REF] Milton | The set of forces that ideal trusses, or wire webs, under tension can support[END_REF] in the special case where the prescribed points are vertices of a convex polygon. This theorem states that, in this case, a web exists if and only if the net torque going clockwise around any connected portion of the boundary is positive: for any sequence (x i , x i+1 , . . . , x j ) of consecutive vertices ordered clockwise (where x k is identified with x k-N ) we have:

j k=i det(x k -x i , f k ) ≥ 0. (1)
Furthermore, by using the Airy stress functions theory, a representative web is explicitly given that contains no closed loops (that is, there is no set of wires forming the boundary of a polygon). The reader is referred to [START_REF] Milton | The set of forces that ideal trusses, or wire webs, under tension can support[END_REF] for details. So the question now is: what happens when the points are not the vertices of a convex polygon or what happens in the three-dimensional case? Theorem 1.1 (see below), which is one of the main results of this paper, answers this question completely. To make the statement clear, let us first introduce some terminology that will be used throughout the paper:

• Finite web: a collection W := [x i , x j ], [x k , x l ] . . . of segments (or bars) where x 1 , . . . , x M are a finite set of points called nodes.

• Terminal nodes: the nodes X = (x 1 , . . . , x N ), N ≤ M , where the forces are applied.

• Internal nodes: the remaining nodes, if any.

• Admissible web stress state: when each bar [x i , x j ] of the web is endowed with a non negative tension σ ij , we say that σ = (σ ij , σ kl , . . . ) is an admissible web stress state on W for the loading F at X if it is in equilibrium under the action of the forces F = (f 1 , f 2 , . . . , f N ) applied at X = (x 1 , x 2 , . . . , x N ). If there exists such an admissible stress state then the web W is said to support1 F at X. • Uniloadable webs: webs which support only one loading (up to a positive multiplicative constant). Theorem 1.1 then reads as follows:

Theorem 1.1. Existence of a web under tension Let A X be the cone of displacements U = (u 1 , u 2 , . . . , u N ) at points X = (x 1 , x 2 , . . . , x N ) defined by

A X := {U ∈ (R d ) N : ∀ 1 ≤ i < j ≤ N, (u i -u j ) • (x i -x j ) ≥ 0}.
(2)

Then, the following condition inf

U∈A X F • U ≥ 0 ( 3
)
is necessary and sufficient to ensure the existence of a finite web under tension that supports the loading F at points X. In such a case, the web connecting the terminal points X pairwise supports the loading F.

Section 2 is dedicated to the proof of Theorem 1.1 and related consequences, whereas Section 3 provides insights on the mechanical meaning of the theorem with special reference to the two-dimensional case. In general, from a mechanical point of view, the statement says that the work performed by F is non negative for any (infinitesimal) 2 displacements U corresponding to a global expansion of the system of points X. Notice that condition (3) provides a characterization of the set A * X of all the loadings F at points X which can be supported by some finite web as the solution to a finite dimensional linear programming problem. Moreover, Theorem 1.1 states that a web supporting the given loading is the one that connects the terminal points pairwise (see Example 1.1).

Example 1.1. Consider a balanced set of forces f 1 ,. . . ,f N at points x 1 ,. . . ,x N that are directed radially outwards from a central point x 0 (so that f i = c i (x i -x 0 ) for some set of positive coefficients c i ). Note that the balance of forces implies that x 0 must belong to the convex hull of the points x 1 ,. . . ,x N .

x 0 x 1 x 2 x 3 x 4 x 5 f 1 f 2 f 3 f 4 f 5 (a) x 0 x 1 x 2 x 3 x 4 x 5 f 1 f 2 f 3 f 4 f 5 (b)
Figure 1: In this example the forces F are directed radially outwards from a central point x 0 and so the web that connects the terminal points X to x 0 supports such a loading. Among all the webs that can support F, Theorem 1.1 provides the web connecting the terminal points pairwise and in Example 1.1 we determine the stress state in each wire of such a web and we prove it is an equilibrium stress state.

Clearly, the web formed by the wires connecting the points x 1 ,. . . ,x N to x 0 , Figure 1(a), supports this loading:

σ i0 := c i x i -x 0
-1 is an admissible stress state on this web. But we can find another web which supports the same loading: indeed, the web that connects the points x 1 ,. . . ,x N pairwise, Figure 1(b), is suitable when endowed with the stress state

σ ij = x i -x j c i c j ( k c k ) -1
, with the equilibrium condition

f i + j =i σ ij x j -x i x j -x i = c i (x i -x 0 ) + c i k c k -1 j c j (x j -x i ) = c i (x i -x 0 ) + c i k c k -1   j f j - j c j (x i -x 0 )   = 0
being satisfied at each node x i .

Section 4 focuses on another major topic of the paper, that is, on channeling or redistributing stresses: we are interested in webs able to channel forces in a controlled way. For example, if one considers, say, a bicycle wheel or a suspension bridge, then, a desired distribution of forces is usually achieved by appropriately tightening the spokes or cables (clearly, the layout of these substructures is also essential). By contrast here we seek to distribute stress through judicious choices of the geometry of the web.

Notice that distributing stresses in wires is quite different than distributing electrical currents in conducting wires. At a junction of more than two conducting wires one cannot tell in advance (without looking at the rest of the circuit) how much flow will be channeled into the different wires (this is an advantage if one wants the current to flow where most needed, a disadvantage if one wants to control the allocation of current, as in an irrigation system). This is due to the fact that, in a conduction network, one only has to satisfy Kirchoff's law which states that the outgoing current has to balance the incoming current. By contrast, when distributing stresses, one has that at a node where four noncoplanar wires join (for the three-dimensional case, or three non collinear wires for the two-dimensional case), balance of forces implies that the tension in one wire, and the geometry of the junction, uniquely determines the tension in the other wires. Thus, having at each internal node a coordination number of four for the three-dimensional case, or three for the two-dimensional case, is important to uniquely determining the loading that a web can support. This principle underlies the construction of "pentamode materials" [START_REF] Milton | Which elasticity tensors are realizable?[END_REF], which have a diamond-like structure with a coordination number of 4 at each node of the structure. As a consequence, the stress field is essentially uniquely determined: like fluids which only support a hydrostatic loading, pentamodes only support one loading but, unlike fluids, that loading can be a combination of hydrostatic and shear forces. Pentamode materials have been studied for their use in cloaking, in particular for cloaking against sonar. They can guide the acoustic wave around an object while having little impedance mismatch at the membrane boundary with the surrounding fluid [START_REF] Norris | Acoustic cloaking theory[END_REF].

Webs of springs that support only one loading (up to a multiplicative constant) were instrumental in [START_REF] Camar-Eddine | Determination of the closure of the set of elasticity functionals[END_REF][START_REF] Vasquez | Complete characterization and synthesis of the response function of elastodynamic networks[END_REF]. The corresponding elastic energies have the form (F • U) 2 , when their terminal nodes have displacements U = (u 1 , u 2 , . . . , u N ), so that the webs are able to support only forces proportional to F. In that context, as the elastic energy can be expressed using a rank-one matrix in the form U • (F ⊗ F) • U , such webs were called "rank-one webs". In this paper, as we are interested in energies that are not necessarily quadratic, we prefer to use the definition of uniloadable webs. In Section 4 we establish that, apart from some exceptional cases, if forces f 1 , f 2 , . . . , f N at prescribed points x 1 , x 2 , . . . , x N are supported by some web, they are also supported by a uniloadable one.

The uniloadable networks we introduce here if prestressed could replace the pentamode materials in the aforementioned cloaking applications, allowing much greater flexibility in the design and economy in the number of junctions and wire elements.

One may not be interested only in uniloadable webs but more generally in webs W for which the set C W X of supported loadings is prescribed. It is clear that, for a given web W with terminal nodes at X, the set C W X is a convex cone contained in A * X . In Section 4, we show that the converse is true too in an approximate sense: given a convex cone C ⊂ A * X , one can find a sequence of webs W n such that C Wn X approaches C as n → ∞. For two-dimensional webs where the points X are the vertices of a convex polygon, a similar question was addressed in Theorem 2 of [START_REF] Milton | The set of forces that ideal trusses, or wire webs, under tension can support[END_REF].

On the existence of a web under tension

This Section is dedicated to the proof of Theorem 1.1 and it contains some mathematical technicalities. Therefore, we recommend the reader who is more interested in the mechanical interpretation of the theorem to skip to Section 3.

Given a set of forces F applied at the points X, we question the existence of a web supporting such a loading that has all the wires under tension. Recall that the equilibrium of a wire web is achieved if the tension is constant in each wire and if each node is in equilibrium. This situation admits a nice and synthetic, even if a bit abstract, formulation in terms of measures which is convenient for proving Theorem 1.1.

Given a web W , the associated measure is

W := H 1 | [xi,xj ] + H 1 | [x k ,x l ] + . . . (4)
where

H 1 | [xi,xj ]
stands for the line measure (the one dimensional Hausdorff measure) concentrated on the segment [x i , x j ]. Accordingly the stress state σ can be represented by the measure

S := σ ij x i -x j x i -x j ⊗ x i -x j x i -x j H 1 | [xi,xj ] + σ kl x k -x l x k -x l ⊗ x k -x l x k -x l H 1 | [x k ,x l ] + . . . (5)
Similarly, associated with the applied system of forces is the discrete measure

F := N i+1 f i δ xi (6)
where δ xi stands for the Dirac measure at point x i . The equilibrium condition then simply reads

∇ • S + F = 0 (7)
and the requirement that all wires be under tension is equivalent to the requirement that the measure S take values in the set of positive semi-definite symmetric matrices. We will denote with M + the set of such measures. Using this formulation, searching for a finite web boils down to finding S ∈ M + of the form (5) such that ( 7) is satisfied.

If we drop the constraint that S must be of the form (5), we are led to an interesting generalization: a generalized web W is a positive measure, and it supports the loading F if there exists some S ∈ M + absolutely continuous with respect to W satisfying the equilibrium condition [START_REF] Ekeland | Convex analysis and variational problems[END_REF]. The relation between this generalized formulation and the so-called Michell problem is provided in Section 2.3.

Proof of Theorem 1.1

Proof. To prove that condition (3) is necessary, we consider an admissible web stress S for the loading F. As S is a positive semidefinite tensor-measure with compact support, by Green's generalized formula we have (see Section 6 of [START_REF] Milton | The set of forces that ideal trusses, or wire webs, under tension can support[END_REF]):

0 ≤ S, e(u) = -∇ • S, u = F, u = F • U. ( 8
)
for all C 1 fields u such that e(u) := (∇u(x)+(∇u(x)) T )/2 is positive semidefinite. Now, let U = (u 1 , u 2 , . . . , u N ) be an element of A X . For any κ > 0, Lemma 1 (see below) provides a Lipschitz extension u :

R d → R d satisfying u(x ) = u and ∀(x, y) ∈ (R d ) 2 , ( u(x) -u(y)) • (x -y) ≥ -κ x -y 2 . ( 9
)
This extension field is differentiable a.e. and, at every point x of differentiability, inequality [START_REF] Fraternali | On the correspondence between 2D force networks and polyhedral stress functions[END_REF] implies that e( u) ≥ -κI. In order to apply Green's formula, we introduce a regularized field u η := u * ρ η , ρ η being a smooth convolution mollifier (i.e., non negative, supported in the ball of radius η, and such that ρ η = 1). It can be readily checked that the strain e(u η ) associated with u η is smooth and satisfies everywhere the inequality e(u η ) ≥ -κI. By applying [START_REF] Farkas | Theorie der einfachen Ungleichungen[END_REF] to the field u = u η + κx, we deduce that

i f i • u η (x i ) + κ i f i • x i ≥ 0.
As u η → u uniformly on every compact set, we can pass to the limits η → 0 and κ → 0 in the last inequality to obtain the desired inequality F • U ≥ 0 (see equation ( 3)).

To prove that condition (3) is sufficient, we consider F such that F • U ≥ 0 for any u ∈ A X . The linear conditions (u i -u j ) • (x i -x j ) ≥ 0 can be rewritten in the form F (i,j) • U ≥ 0, where:

F (i,j) = (f (i,j) 1 , f (i,j) 2 , . . . , f (i,j) N ) with f (i,j) =      x j -x i if = i, x i -x j if = j, 0 otherwise. (10)
Hence, thanks to Farkas Lemma [START_REF] Farkas | Theorie der einfachen Ungleichungen[END_REF], we know that F is a linear combination of the linear forms F (i,j) with non negative coefficients:

F = 1≤i<j≤N λ i,j F (i,j) , λ i,j ≥ 0. ( 11
)
It is then easy to check that the positive semidefinite symmetric tensor measure

S = 1≤i<j≤N λ i,j (x i -x j ) ⊗ (x i -x j ) x i -x j H 1 | [xi,xj ] , (12)
is a possible web stress for the loading F. Indeed, for any a and b in R d , we have

∇ • ((b -a) ⊗ (b -a)H 1 | |[a,b] ) = b -a (b -a)(δ a -δ b ), thus ∇ • S = -i f i δ xi = -F.
Let us emphasize that this web stress measure involves only the original nodes (x 1 , x 2 , . . . , x N ) as claimed in the theorem.

We now state and prove the interpolation lemma which was needed in the previous proof.

Lemma 1. Interpolation Lemma

Let A be a finite subset of R d , and u :

A → R d the field satisfying (u(x) -u(y)) • (x -y) ≥ 0, ∀(x, y) ∈ A 2 .
Then, for any κ > 0 there exists a Lipschitz extension u of u on R d satisfying

∀(x, y) ∈ (R d ) 2 , ( u(x) -u(y)) • (x -y) ≥ -κ x -y 2 . ( 13
)
Proof. As u is bounded on the bounded set A, there exists M such that, for any x ∈ A, u(x) ≤ M and x ≤ M . Moreover, as A is finite, there exists δ > 0 such that, for any distinct points x and y in A, xy ≥ δ. We set λ = 8 M 2 δ 2 and choose s such that 0 < λ s < min(κ, 1). Let us consider φ defined on A by3

φ(x) = (1 -λ s 2 )x -su(x). (14)
For any distinct points x and y in A, we have, by straightforward computation

φ(x) -φ(y) 2 = (1 -λ s 2 )(x -y) -s(u(x) -u(y)) 2 = (x -y) -s (u(x) -u(y) + λs(x -y)) 2 = x -y 2 + s 2 u(x) -u(y) + λs(x -y) 2 -2s (x -y) • (u(x) -u(y) + λs x -y 2 ≤ x -y 2 + s 2 u(x) -u(y) + λs(x -y) 2 -2λs 2 x -y 2 ≤ x -y 2 + 4s 2 u(x) 2 + u(y) 2 + (λs) 2 x 2 + (λs) 2 y 2 -2λs 2 x -y 2 ≤ x -y 2 + 16M 2 s 2 -2λ δ 2 s 2 ≤ x -y 2 . ( 15
)
Kirszbraun's Theorem [START_REF] Kirszbraun | Über die zusammenziehende und Lipschitzsche Transformationen[END_REF] proves the existence of an extension φ of φ on R d satisfying the same condition: for any (x, y)

∈ (R d ) 2 , φ(x) -φ(y) ≤ x -y . (16)
Let us now define u on R d by setting

u(x) = (1 -λ s 2 )x -φ(x) s . (17)
For any x ∈ A we have u(x) = u(x). Moreover u is a Lipschitz function which satisfies

( u(x) -u(y)) • (x -y) = s -1 (1 -λ s 2 ) x -y 2 -( φ(x) -φ(y)) • (x -y) ≥ -λ s x -y 2 ≥ -κ x -y 2 . ( 18
)

Support of the stress field

From a physical viewpoint it seems obvious that a finite web supporting the loading F at the points X = (x 1 , x 2 , . . . , x N ) should have an associated stress measure vanishing outside the convex hull of the points x 1 , x 2 , . . . , x N . This section is devoted to proving this fact which turns out to be valid also for generalized webs.

Theorem 2.1. Let F be a vector measure with compact support K and S in M + such that ∇ • S + F = 0.

Then the support of S is contained in the convex envelope co(K) of K.

Notice that, from this result, we can deduce that the support of S is contained in the subspace spanned by the vectors4 x 1 , x 2 , . . ., x N and thus that the loading forces f i belong to this subspace. Hence, we will be able to reduce our problem to this subspace and assume without loss of generality that x 1 , x 2 , . . ., x N span R d .

Proof. By Hahn-Banach separation theorem [START_REF] Ekeland | Convex analysis and variational problems[END_REF], proving that S vanishes outside the convex envelope co(K) of K reduces to checking that S vanishes on all half spaces P + m,a := {x : x • m > a} with a ∈ R and m ∈ S d-1 which do not intersect K. This verification is achieved in several steps. We first remark that, as P + m,a and K do not intersect, we have ∇ • S = 0 on P + m,a .

Step 1: Let us consider first a field S ∈ C 1 (R d , R d 2 sym
) such that S ≥ 0 and ∇ • S = 0 on P + m,a . Without loss of generality assume that m = e 1 , where e 1 is the unit vector along the x 1 -axis (where x 1 is not to be confused with the point x 1 ). Let us apply Green's formula in the half ball Ω R := {x :

x 1 > a, |x| < R}. We have ∂Ω R ( S • e 1 ) • n dH d-1 = Ω R (∇ • S) • e 1 dH d = 0, ( 19
)
where n stands for the outward normal. Dividing the boundary ∂Ω R into Σ R := {x :

x 1 = a, |x| < R} and S + R := {x : x 1 > a, |x| = R} we get - Σ R ( S • e 1 ) • e 1 dH d-1 + S + R ( S • e 1 ) • n dH d-1 = 0. ( 20
)
Thus we have

Σ R ( S • e 1 ) • e 1 dH d-1 ≤ S + R | S| dH d-1 . ( 21
)
As S ∈ L 1 , +∞ 0

( S + R | S| dH d-1
) dR < +∞ and so there exists a sequence R n → +∞ such that

S + Rn | S| dH d-1 → 0.
This implies that

lim n→∞ Σ Rn ( S • e 1 ) • e 1 dH d-1 = 0. ( 22
)
As ( S • e 1 ) • e 1 ≥ 0 we get ( S • e 1 ) • e 1 = 0 on the whole hyperplane {x : x 1 = a}. As S ≥ 0, we deduce that S • e 1 = 0 on this hyperplane.

Step 2: We remark that, if

S ≥ 0 ∈ C 1 (R d , R d 2 sym
) satisfies ∇ • S = 0 on the whole space R d , then by applying the result of step 1 to every pair (m, a) ∈ S d-1 × R d we get S = 0 everywhere. We also remark that this result holds true for any space dimension d.

Step 3: Going back to the case where

S ≥ 0 ∈ C 1 (R d , R d 2 sym ) satisfies ∇ • S =
η := S ρ η . Clearly S η belongs to L 1 ∩ C ∞ (R d 2
sym ), S η ≥ 0 and, for any t > a, and for η small enough we have ∇ • S η = 0 on P + m,t . By applying the result of Step 3 to S η we get S η = 0 on P + m,t , and by passing to the limit η → 0 we get S = 0 on P + m,t . The theorem is proven by passing finally to the limit t → a.

Link with the Michell problem

A very old problem in Mechanical Engineering consists in minimizing the total volume of a network of elastic bars (trusses) while the resistance to a given load remains constant. It reduces to a linear programming problem which, according to our notation, reads:

inf σ    i j |σ ij | x j -x i : f i + j =i σ ij x j -x i x j -x i = 0 , ∀i    (23)
The classical dual formulation is the following maximization problem on deformations:

sup U i f i • u i : |(u i -u j ) • (x i -x j )| ≤ x i -x j ) 2 , ∀i = j (24)
As no assumption is made on the number of bars, this belongs to the class of topological optimization problems and it is well known that, in general, no optimal solution exists. Indeed, during the optimization process, the number of bars may increase to infinity leading thus to diffuse structures. The crucial contribution of Michell [START_REF] Michell | The limits of economy of material in frame structures[END_REF] in the 1900's was to formulate a generalized version (called Michell problem) in order to take into account all possible structures which may appear in the limit. In the generalized version, attention is focused on the stress carried by the structure rather than on its geometry. Michell stated a duality principle and obtained the optimality conditions on the stress and strain tensors: they share the same eigenvectors (principal directions) and the eigenvalues of the strain tensor have a fixed absolute value. Moreover, Michell noticed that, in the two-dimensional case, when the eigenvalues of the strain tensor have opposite sign and when the eigenvector fields are smooth enough to define stream lines (called "lines of principal action"), then these lines constitute a so-called Hencky-net. This is a family of orthogonal curves which represents the limit of the families of bars through the optimization process. We refer to [START_REF] Bouchitté | Michell trusses and lines of principal action[END_REF]

sup u {< F, u >: |( u(x) -u(y)) • (x -y)| ≤ |x -y| 2 } (26)
An admissible pair (u, S) is then optimal if and only if the following extremality relation holds (see [START_REF] Bouchitté | Michell trusses and lines of principal action[END_REF])

< F, u > = ρ 0 (S) dx. (27)
We can now emphasize the link between our problem and the Michell problem described above. It turns out indeed that admissible stress states associated with a loading F in the cone A * X are solutions of the Michell problem.

Theorem 2.2. Let F be a bounded vector-measure with compact support K and S ≥ 0 in M d 2 sym such that ∇ • S + F = 0. Then i) S is a solution to the Michell problem for F, ii) any solution S to the Michell problem for F satisfies S ≥ 0.

Proof. Clearly u(x) := x is admissible for the dual problem. As -∇ • S = F, the following relation holds:

< F, x > = < -∇ • S, x > = < S, ∇x > = Tr(S). ( 28
)
If we assume that S ≥ 0, then Michell's dual energy ρ 0 (S) coincides with Tr(S) as both λ 1 (S) and λ 2 (S) are non-negative. In particular, we have ρ 0 (S) = Tr(S), hence by (28) the pair (x, S) satisfies relation ( 27). The optimality of (x, S) follows. This proves point i).

To prove point (ii), it is enough to notice that, for any other solution S of the primal problem, the couple (x, S) is optimal and thus must satisfy the optimality condition (27) so that, by (28), one has < F, x >= Tr( S) dx = ρ 0 ( S) dx.

As ρ 0 ( S) ≥ Tr( S), we deduce that ρ 0 ( S) = Tr( S) and so S ≥ 0.

Theorem 1.1 in two dimensions

To get a better insight on the mechanical interpretation of the statement of Theorem 1.1, we focus on the twodimensional case which was also analyzed in [START_REF] Milton | The set of forces that ideal trusses, or wire webs, under tension can support[END_REF]. In particular, we will show that when the points x 1 , x 2 , . . . , x N are vertices of a convex polygon, our condition (3) is a generalization of the condition (1) proved in [START_REF] Milton | The set of forces that ideal trusses, or wire webs, under tension can support[END_REF].

3.1 Mechanical interpretation of the extreme rays of the cone A X Given a set of points x 1 , x 2 , . . . , x N that are vertices numbered clockwise of a convex polygon, let us consider the following displacement field u (j,i) k defined by:

u (j,i) k = -R ⊥ (x k -x j ), for k = j, j + 1, . . . , i -1, 0 otherwise , (29)
where

R ⊥ = 0 1 -1 0 ( 30
)
is the matrix for a 90 • clockwise rotation and, if necessary, we identify k with k -N . We call u (j,i) k a "clam-shell" displacement (see 2) as it corresponds to the infinitesimal rotation between two non-overlapping subpolygons the polygon of terminal points can be divided into: by keeping fixed one of the two subpolygons, the rotation of the other opens the clam. Given any points x k and x on opposite sides of the clam (where u (j,i) k = 0, while u (j,i) = 0) we have (u

(j,i) k -u (j,i) ) • (x k -x ) = -[R ⊥ (x k -x j )] • (x k -x ) ≥ 0, (31)
where the last inequality follows from the convexity of the polygon and the clockwise numbering of the points. Thus, this clam-shell movement is an admissible displacement as it satisfies (2). This implies that F satisfies the constraints (3), that is,

0 ≤ N k=1 f k • u (j,i) k = i-1 k=j (x k -x j ) • [R ⊥ f k ], (32)
which are precisely the same as the constraints (1) that characterize A X * , that is the set of all the loadings F at X which can be supported by a finite web. Thus, in this case of the x i forming the vertices of a convex polygon, the displacements U correspond precisely (up to an infinitesimal rigid body motion) to these "clam-shell" movements, and do not include any other movements. : Given a set of points x i that form the vertices of a convex polygon, as in (a), an extremal infinitesimal movement is obtained by breaking the polygon into two non-overlapping subpolygons connected at one vertex x j , as in (b). The "clam-shell" movement then consists of fixing one subpolygon, in this example the lower triangle, and infinitesimally rotating the other subpolygon anticlockwise about the point x j , so the "clam" opens slightly, thus moving any vertex x k on the upper subpolygon away from any vertex x on the lower subpolygon.

More generally, to check the criterion (3) it suffices to check it for those U corresponding to the extreme rays U m of the cone A X . These rays are perpendicular to the "faces" of the polar cone -A X * (see Figure 3). We use an integer m = 1, 2, . . . , M to index these rays. For any given m there exist associated loadings F m h , h = 1, 2, . . . , D -1 all perpendicular to the extreme ray indexed by m. Here, for a given m, each value of h signifies a pair (i, j) = (i(h, m), j(h, m)) such that F m h = F (i,j) , and linear combinations of the F m h , h = 1, 2, . . . , D -1 with positive weights generate the "face" perpendicular to the extreme ray of the cone A X . Let U m = (u m 1 , u m 2 , . . . , u m N ) be on this extreme ray. Then, we have

F m h • U m = 0, for h = 1, 2, . . . , D -1. (33)
In particular, if F m h = F (i(h,m),j(h,m)) , then the orthogonality implies that

(u m i(h,m) -u m j(h,m) ) • (x i(h,m) -x j(h,m) ) = 0. ( 34
)
If we think of U m as corresponding to a displacement, then this restriction says that (within the infinitesimal displacements framework) there is no change in distance between x i(h,m) and x j(h,m) : the constraint is equivalent Figure 3: Schematic illustration of the cone A X and the polar cone -A * X , the negative of the dual cone A * X . Here the F (i,j) are the extreme rays of the dual cone A * X , while the U m are the extreme rays of the cone A X .

to only allowing those deformations compatible with rigid rods joining the pairs of points (x i(h,m) , x j(h,m) ) for h = 1, 2, . . . , D -1. After eliminating the trivial infinitesimal rigid body motions (translations and rotations) from U m , by requiring it to satisfy, for instance,

N i=1 u i = 0 , N i=1 x i • Au i = 0 (35)
with A any d×d antisymmetric matrix, there still must be one degree of freedom associated with the infinitesimal motion corresponding to U m . The equality (u i -u j ) • (x i -x j ) = 0 cannot hold for all i = j as this then would correspond to a trivial overall (infinitesimal) rigid body motion, which must be zero by (35). To fix the one-degree of freedom, we can impose the normalization condition that U m • X = 1. Without loss of generality this can be seen by noting that for any U ∈ A X , 0

≤ N i,j=1 (u i -u j ) • (x i -x j ) = N i,j=1 u i • u i + u j u j -2 N i=1 u i N j=1 u j = 2U • X, (36)
where, to get the last equality, we use (35). Furthermore, U • X = 0 implies that U is a rigid body motion, which we excluded. So U • X > 0, and by replacing U with U/(U • X) we see that we can assume U • X = 1.

To conclude, we showed that if the terminal points are the vertices of a convex polygon, then all the extremal displacements correspond to clam-shell movements but notice that if there is at least one terminal point inside the convex hull of the terminal points, then besides clam-shell movements there are also other types of displacements, as shown in Figure 4.

Simplifying the two-dimensional web

In two dimensions, in any web, we define a loop to be any polygon such that its edges are the wires of the web. A minimal loop is one with no other wires inside the polygon. Any web with all pairs of terminal points interconnected, as in Figure 1 (b), can then be replaced by an equivalent one with at most P minimal loops where P is the number of points x 1 , . . . , x N that lie inside the convex hull of these N -points. To show this we first place internal nodes where any pair of wires (x i , x j ) and (x r , x s ) cross. Then take any minimal loop in the network. The vertices of this loop may include a terminal point x j so long as the net force acting on the loop at x j (including f j and the forces acting on x j due to the tension in the other wires outside the loop) points outside the loop. As the wires are all under tension, the loop then is necessarily convex and exerts forces -f m at the nodes numbered clockwise around the loop. These forces necessarily satisfy (1), and the loop can be replaced by an open web. The number of minimal loops in the web is thus reduced by one. This procedure can be continued until there are at most P minimal loops, and each of these loops is non-convex and thus its vertices include at least one terminal node x i where the associated force f j points inside the loop. Video1 (see Supplementary Material) shows an example of a web whose wires connect all the terminal points pairwise (initial frame), and how to replace each closed loop with an open web so that the final frame represents an equivalent web in which there is only one minimal loop due to the presence of one point, x 4 , inside the convex hull of the terminal points. we pick 4 of the wires (those marked in blue in Figure 6(a)), and points x 1 , x 2 , x 3 and x 4 on the four wires such that the tetrahedron T formed by x 1 , x 2 , x 3 and x 4 encloses the junction, which without loss of generality can be taken to be at the origin x 0 = 0 (this requires that the four wires be chosen so that they do not all lie on one side of any plane through the origin: balance of forces at the origin ensures that at least one choice of such four wires exists). The tensions in these four wires generally do not balance. However, consider the "tensegrity network" consisting of rods from the origin to the points x 1 , x 2 , x 3 and x 4 under compression, balanced by wires along the edges of the tetrahedron T that are under tension. Example 1.1 (see Section1) gives the explicit solution for the compressive forces in the rods and the tensions in the wires in this "tensegrity network". We next superimpose this "tensegrity network" on our junction with the tensions in the "tensegrity network" scaled so after superposition the tension near the junction in one wire cancels, while the tension in the other wires remain nonnegative, as sketched in Figure 6(b). We thus obtain a web under tension where the number of wires joined to the origin is now M -1 or less. However, we typically have also created junctions, at some of the points x 1 , x 2 , x 3 and x 4 where 5 wires meet, like those in Figure 6(c). These junctions are rather special in that one wire goes straight through the junction (but typically has different tensions on opposite sides of the junction). These junctions are then locally replaced by networks in which at most four wires meet as illustrated in Figure 6(d). Lemma 2 below guarantees this can be done. The last step is to successively repeat the argument until the junction at the origin has at most 4 wires.

Lemma 2. The five wires problem

Consider five wires with tensions T i > 0, directions v i and joining at the origin x = 0. Assume that the three first directions are independent while v 5 = -v 4 . Then we can replace these wires by a web in tension such that at each of its nodes, no more than 4 wires are joining.

Proof. We can assume without loss of generality that T 5 = αT 4 so that balance of forces implies that

T 1 v 1 + T 2 v 2 + T 3 v 3 + (1 -α)T 4 v 4 = 0. Set t > 0 and s > α 3 . Set also x 1 = tT 1 v 1 , x 2 = tT 2 v 2 , x 3 = tT 3 v 3 , x 4 = t sT 4 v 4 and x 5 = -t rT 4 v 4 where r := s 3s-α .
As the real parameter t can be arbitrarily chosen provided it is small enough, then we can avoid creating new nodes where a node already exists or where a wire lies. The parts of the wires which lie between the points x i and the origin are replaced by six wires [x i , x 4 ], [x i , x 5 ] (i ∈ {1, 2, 3}). When these wires have respective (positive) tensions

T i4 = r t(r + s) x i -x 4 , T i5 = s t(r + s) x i -x 5 , (37)
the web is in equilibrium. Indeed at each node x i for i ∈ {1, 2, 3} we have

T i4 x 4 -x i x i -x 4 + T i5 x 5 -x i x i -x 5 + T i v i = rs -sr t(r + s) tT 4 v 4 + r + s t(r + s) (-tT i v i ) + T i v i = 0. (38)
Moreover, at nodes x 4 and x 5 we have

3 i=1 T i4 x i -x 4 x i -x 4 + T 4 v 4 = 1 r + s rα + s -3rs T 4 v 4 = 0. (39) (a) (b) (c) (d) x x x 1 2 3
x 4 Figure 6: Steps in the replacement of a junction of many wires under given balanced tensions, with a network localized around the junction such that at most 4 wires meet at any junction in the new network, and the network still supports the same tensions in the wires meeting the network.

3 i=1 T i5 x i -x 5 x i -x 5 + T 5 v 5 = 1 r + s -s -αr + 3rs T 4 v 4 = 0. (40)

Uniloadable webs

Given a ray C ⊂ A * X , we want to determine whether there exists a uniloadable web which can support the corresponding loading λF, with λ ≥ 0. In case C does not belong to the interior of the cone A * X , then the existence of a uniloadable web is not guaranteed. When F belongs to the interior of A * X , then we prove the existence of a uniloadable web supporting such a loading (up to a multiplicative constant).

Stuck loadings

Let us start with the following remark : let F = (f 1 , . . . , f N ) belong to A * X with X = (x 1 , . . . , x N ). Then, it also belongs, for any t ≥ 0, to the admissible loading cone A * X+tF of the shifted points (x 1 + tf 1 , . . . , x N + tf N ). Indeed, taking a web W which supports F at X and adding to W all the wires [x i , x i ] leads clearly to a web supporting F at X + tF. When t < 0 things are less clear. We say that F at X is an unstuck loading if there exists ε > 0 such that ∃ε > 0, ∀t < ε, F ∈ A * X-tF , otherwise we say that F is a stuck loading. A particular case of stuck loadings, referred to as completely stuck loadings, is when there exist some k ∈ {1, . . . , N } and ε > 0 such that,

∀ 0 < t < ε, F ∈ A * X with X = (x 1 , . . . , x k-1 , x k -tf k , x k+1 , . . . , x N ).
The stuck or completely stuck conditions can only occur when the loading F is a ray on the boundary of the cone of admissible loadings A * X , as we will prove in the next subsection that any F in the interior of A * X is unstuck. For a better insight, let us consider two examples of completely stuck loadings. Our objective is to show that this is the only web supporting these forces, thus implying that the loading is "completely stuck". First move the force f 4 = [-1/2; 0] back to the origin, by attaching a wire joining the origin to x 4 , as in Figure 7(b). Then the net force acting on the origin is [-3/2; 0] and is balanced by the forces f 3 and .2, provides another example of a web with a "completely stuck loading" where the web is uniquely determined once the loading is specified.

f 4 .
The open web in Figure 7(c) supports these three forces, and the associated Airy function (up to addition of an affine function, see also [START_REF] Fraternali | On the correspondence between 2D force networks and polyhedral stress functions[END_REF]) is

φ L (x) = max{-3|x 2 |/4, 1/4 -x 1 }, (41)
where x 1 and x 2 are the coordinates of x. Now suppose we have any web supporting the four forces f 1 , f 2 , f 3 , and f 4 . Necessarily this web will be confined within the convex hull of the three points x 1 , x 2 , x 3 . To define the associated Airy stress potential, in say all of R 2 , we need to move the four forces to infinity by attaching infinite wires to the four points in the direction of infinity (the wire attached to x 1 overlaps the wire attached to x 4 left of the origin). The existence of these wires implies a discontinuity of slope of the Airy stress function across them, matching the tension in the wire. Now consider the Airy stress function in the vicinity of the point x 4 . In particular take the tangent plane at a point x 0 that approaches x 4 from the left remaining infinitesimally above the wire that joins x 4 to the origin. Similarly take the tangent plane at a point x 0 that approaches x 4 from the left remaining infinitesimally below the wire that joins x 4 to the origin. The maximum of these two tangent planes is the valley function φ V that takes the form

φ V (x) = -|x 2 |/4 + ax 1 + bx 2 + c. ( 42
)
As the web is confined to the convex hull of x 1 , x 2 , x 3 , the Airy stress function outside this convex hull can be taken to be φ L (modulo an affine function that can be set to zero without loss of generality). Convexity of the Airy stress function then implies the inequalities φ L (x i ) ≥ φ V (x i ) for i = 1, 2, 3 and φ L (x 4 ) ≤ φ V (x 4 ). Elementary calculations then show that these inequalities allow one no freedom in the choice of a, b and c and one necessarily has a = -1/2, b = c = 0. By convexity, the Airy stress function of any web supporting the four forces must be above

φ(x) = max{φ L (x), φ V (x)} = max{-3|x 2 |/4, (1/4) -x 1 , -|x 2 |/4 -x 1 /2}, (43)
and must coincide with it at the four points x 1 , . . . , x 4 . But the polyhedral nature of φ(x) means that any other candidate convex Airy stress function must cleave it in the vicinity of x 4 , which is forbidden. Thus (modulo the addition of an affine function) φ(x) given by ( 43) is the unique possibility for the Airy stress function, and the web in Figure 7(a), is the only web that can support the four forces at the four points.

φ(x) = max{2x 1 , -2x 2 + 1, -|x 2 | -(4x 1 -1 -3x 2 ) + }, (44)
where q + = max{0, q}. The proof proceeds similarly to the first example. The completely stuck nature of the webs in our two examples has been verified numerically.

In two dimensions when the terminal points X are at the vertices of a convex polygon, the existence of an open web supporting (the assumed admissible) loading F implies that such webs are never "completely stuck" with respect to the loading F. One may wonder: is a similar result true in three dimensions? The numerical example of Figure 8 shows, to the contrary, that there exist terminal points X at the vertices of a convex polyhedron, and an admissible loading F such that the associated web is "completely stuck". This example was found by starting with N = 8 terminal nodes at the vertices of a cube, taking an admissible loading F supported at these points, then moving the terminal points backwards so that for each j = 1, 2, . . . , 8, x j is replaced by x j = x j -ε j f j , where the ε j ≥ 0 are increased until the loading F at the terminals X = (x 1 , x 2 , . . . , x 8 ) is completely stuck, while keeping the terminals X as vertices of a convex polyhedron.

2 -2 1 -2 -1 0 -1 0 0 -1 1 1 -2 2 3 2 -2 1 -2 -1 0 -1 0 0 -1 1 1 -2 2
Figure 8: We start by applying the forces f i , given in Table 1, at the 8 points shown in (a), whose coordinates are given by x i in Table 1. The points are initially chosen to be the vertices of a cube. Now, move the terminal points backwards so that for each i = 1, 2, . . . , 8, x i is replaced by x i = x i -ε i f i , where the ε i ≥ 0 are increased until the loadings f i at the terminals x i , provided in Table 1, is completely stuck, while keeping x i as vertices of a convex polyhedron. The diagonal lines along the faces in figure (a) are numerical artifacts.

Unstuck loadings

The aim is to prove that, for a given loading F belonging to the interior of the one of admissible loadings A * X , there exists a uniloadable web supporting only F (up to a multiplicative constant). First, we need to prove that the cone A * X does not have an empty interior (Lemma 3). Then, we need to prove that if we perturb slightly the positions of the terminal points, the loading is not stuck (Lemma 4) and that there exists a connected web is not collinear with f (j) i for any m, with = m. Associated with F (j)m is then a uniloadable web W (j)m supporting, and only supporting, at the terminals X, the loadings λF (j)m , λ ≥ 0. By adjusting the positions of the interior nodes of the different uniloadable webs W (j)m now indexed by j = 1, 2, . . . , m we can ensure that the webs W (j)m do not have overlapping interior nodes, nor overlapping collinear wires. By superimposing these uniloadable webs, for m = 1, 2, . . . , j one obtains the web W j having the desired loading cone. It may happen that some wire pairs cross when we superimpose the webs, generating an additional interior node at the crossing point. This is still okay, as balance of at the crossing point ensures that the tension in the wire remains the same on opposite sides of the crossing point. If more than two wires cross at the same point, then we can perturb the uniloadable webs to avoid this.

Taking the limit j → ∞ allows us to approximate C arbitrarily closely, so that C Wj X → C.

Conclusions

In this paper we provide full answer to the question as to whether, given a set of forces applied to specific points, called terminal points, there exists a web that supports such forces with all the wires under tension. Specifically, we provide a necessary and sufficient condition on the loading forces which guarantees the existence of such a web, see Theorem 1.1. Such a condition corresponds to a finite dimensional linear programming problem: if this has solution, then a web exists which is formed by the wires connecting pairwise the terminal points. Conversely, any web under tension supporting the given loading can be replaced by the web provided by the linear programming problem. The conditions related to the linear programming problem are inequalities expressed in terms of the displacements of the terminal points: they form a cone in the displacement space and the edges of the cone correspond to those displacements that satisfy the conditions as equalities. In case the terminal points form the vertices of a convex polygon, these extreme edges correspond precisely (up to an infinitesimal rigid body motion) to clamshell movements and do not include any other movements. By clam-shell movements we denote a displacement field that arises when one breaks the convex polygon connecting the terminal points into two non-overlapping subpolygons connected at one vertex and fixes one subpolygon and infinitesimally rotates the other one, so the clam opens slightly. In the case there is at least one terminal point that lies inside the convex hull of the terminal points, the extreme rays of the cone of admissible displacements correspond to types of displacements that are not simply clam-shell movements.

In practical situations one would like to have uniloadable webs, that is webs that support only one loading and all positive multiples of it: such webs allow one to channel stresses in desired ways and the superposition of them allows one to get a desired convex loading cone. To construct a uniloadable web in two-dimensions, one has to replace each closed minimal loop, that is any polygon formed by the intersection of wires that cannot be divided into subpolygons, by an open web (not containing any closed loop). This is always possible if the terminal points are positioned at the vertices of a convex polygon. If that is not the case, then there will still be minimal closed loops in a number equal to that of the terminal points which lie inside the convex hull of the terminal points.

In general, to construct uniloadable webs, one has to reduce the number of wires meeting at either a terminal point or at an internal node. The first step is to modify the web so that only one wire is attached to each terminal point. We proved that this is possible only when the given loading lies inside the cone formed by the inequality conditions associated with the dual linear programming problem. If, instead, the loading corresponds to one of the extreme rays of such a cone, then we have that for some webs this modification is impossible and we say that such webs have stuck loadings. We provided two examples of webs of this type in two dimensions, see Examples 4.1 and 4.2. Since in two dimensions we know that a web with terminal points that are vertices of a convex polygon can always be replaced by an open web, we have that such webs are never completely stuck: indeed, in both Example 4.1 and 4.2 the terminal points are not vertices of a convex polygon. This does not hold in the three-dimensional case for which one can find a web with stuck loadings which have the terminal points forming a convex polyhedron (see Figure 8). On the other hand, if the loading belongs to the interior of the cone formed by the inequalities regarding the loadings in the dual linear programming problem, then we provide a general procedure to reduce the number of wires meeting at each internal node where initially 5 or more wires meet.

  for a detailed mathematical study where optimality conditions for a generalized truss are established in a rigorous way. The generalized stress formulation derived by Michell in dimension d = 2 reads as follows: inf S ρ 0 (S) : ∇ • S + F = 0 (25) where ρ 0 (S) := |λ 1 (S)| + |λ 2 (S)| denotes the sum of the moduli of the two principal values of S. The corresponding dual problem reads:

Figure 2

 2 Figure2: Given a set of points x i that form the vertices of a convex polygon, as in (a), an extremal infinitesimal movement is obtained by breaking the polygon into two non-overlapping subpolygons connected at one vertex x j , as in (b). The "clam-shell" movement then consists of fixing one subpolygon, in this example the lower triangle, and infinitesimally rotating the other subpolygon anticlockwise about the point x j , so the "clam" opens slightly, thus moving any vertex x k on the upper subpolygon away from any vertex x on the lower subpolygon.

Figure 4 :

 4 Figure 4: Consider the following 4 points which form an arrowhead in the plane: x 1 = [0; 0], x 2 = [1; 1], x 3 = [0.5; 0], and x 4 = [1; -1].The displacements corresponding to the extreme rays of the cone A X for this special geometry can be divided into two groups: clam-shell movements and non-clam-shell movements. In this figure, the black lines represents the rigid wires and the green lines represents the deformable bonds. Clearly, (a) is an example of a non-clam-shell displacement, whereas (b) is an example of a clam-shell displacement (the rigid wire connecting the points x 1 and x 2 can rotate infinitesimally about the point x 2 while the rigid triangle formed by x 2 , x 3 , and x 4 is held fixed). Notice that the existence of non-clam-shell movements is due to the fact that the point x 3 does not belong to the convex hull of the terminal points.

Figure 5 :

 5 Figure5: In two dimensions a junction of many wires at an internal node as in (a) can be replaced by a web in which at most three wires meet at every junction. First one determines an associated convex Airy function in the vicinity of the crossing, as in (b). Then one cleaves it by a plane as in (c), and the associated web, as in (d), then has at most three wires meeting at every junction.

Example 4 . 1 .

 41 Consider forces f 1 = [-1; 0], f 2 = [3/4; 1], f 3 = [3/4; -1], and f 4 = [-1/2; 0] at the four points considered in Figure 4, that is, x 1 = [0; 0], x 2 = [1; 1], x 3 = [1; -1], and x 4 = [1/2; 0]. They are supported by the web in Figure 7(a).

1 Figure 7 :

 17 Figure7: Examples of webs with "completely stuck loadings" in two dimensions. We argue in Example 4.1 that (a) is the unique web that supports the given forces at the terminal nodes. A wire can be attached to the node x 4 , as in (b), in order to uniquely define the associated Airy stress function, up to the addition of an affine function. This Airy stress function then lies above the triangular pyramidal Airy stress function associated with the net forces applied at the three points x 1 , x 2 , and x 3 , as in (c). At the same time it must have a discontinuity in slope across the line between x 1 and x 4 corresponding to the tension in the wire joining those two points. This then uniquely determines the Airy stress function (modulo the addition of an affine function) and thus uniquely determines the web. Figure (d), as discussed in Example 4.2, provides another example of a web with a "completely stuck loading" where the web is uniquely determined once the loading is specified.

Example 4 . 2 .

 42 A second example of a web with completely stuck loading is shown in Figure7(d). Forcesf 1 = [2; 0], f 2 = [-2; 2], f 3 = [-4; -6],and f 4 = [4; 4] are applied at the points x 1 = [0; 0], x 2 = [-1/2; 1], x 3 = [-1/2; 1], and x 4 = [1; 1]. The unique web that supports them is that drawn in Figure 7(d), and the associated Airy stress potential (modulo addition of an affine function) is

3

 3

  Step 4: In order to extend this result to measures, we introduce a smooth mollifier ρ η and S

0 only on the half space P + m,a , we apply Step 1 to every pair (m, t) with t > a and we deduce that S • m = 0 on the hyperplane Σ t := {x : x • m = t}. The restriction of S to this hyperplane is tangential and divergence free. For almost every t > a, Σt | S| < +∞, by applying the result of Step 2 to R d-1 , we deduce that S vanishes on almost all hyperplanes Σ t . Hence, S vanishes on P + m,a .

By "admissible stress state" we mean an equilibrium state in which all bars are either in tension, or carrying no load. Thus, by "supporting", we mean supporting with all bars either in tension, or carrying no load.

Notice that we solve this problem within the context of infinitesimal elasticity: examples of applications of the finite deformation theory to describe the geometric nonlinearity are given, for instance, by[START_REF] Crusells-Girona | A mixed formulation for nonlinear analysis of cable structures[END_REF][START_REF] Yang | Geometrical nonlinear analysis of cable structures with a two-node cable element by generalized displacement control method[END_REF][START_REF] Thai | Nonlinear static and dynamic analysis of cable structures[END_REF].

Physically, when s is small, we can think of φ(x) as a deformation x → φ(x) associated with the displacement field u(x). The additional uniform contractive factor of (1 -λ s 2 ) is needed to account for the finite deformation corrections when u(x) -u(y) is perpendicular to xy (with say u(y) = 0 and y = 0, and u(x) orthogonal to x, the distance |x -su(x)| lengthens as s is increased).

Here we choose x 1 as the origin for identifying points and vectors.

Acknowledgments

Ornella Mattei and Graeme Milton are grateful to the National Science Foundation for support through DMS-1814854. Graeme Milton is grateful to the Université de Toulon for hosting his two week visit there in September 2017 where this work was initiated.

Channeling the stresses in a web

In this Section we address the problem of designing wire webs that can support one and only one loading, up to a positive multiplicative factor. Note that the stress is not distributed in a unique way as there are many networks, i.e. stress patterns, that work. In a given wire web, this is possible if one can determine the stress in each wire in a unique way up to an overall proportionality constant: clearly this happens if at each internal node only four non-coplanar wires for the three-dimensional case (or three non-collinear wires for the two-dimensional case) meet and at most three non-coplanar wires meet at any terminal node for the three-dimensional case (or two non-collinear wires for the two-dimensional case). Here we provide a procedure to achieve such a goal so that at each internal node the coordination number is four for the three-dimensional case, or three for the two-dimensional case, while only one wire is connected to each terminal node.

Then, it is important to uniquely determine the loading that a web can support. Let us call C W X the set of all the loadings F that the web W can support at X. Clearly C W X is a convex cone. Indeed, if the web supports the loadings F 1 and F 2 with admissible stresses S 1 and S 2 , respectively, then for any λ 1 ≥ 0 and λ 2 ≥ 0 it also supports the loading F = λ 1 F 1 + λ 2 F 2 with the associated stress λ 1 S 1 + λ 2 S 2 . Also, by definition, C W X must be a subset of the admissible loading cone A * X . Here we address the converse question: given a convex cone C ⊂ A * X can one find a web W such that C W X = C? We first focus on the case where C is reduced to a single ray and then look for what we call uniloadable webs (that is webs which support only one loading F, up to a multiplicative constant). If C is a ray in the interior of A * X , then we can prove the existence of a uniloadable web for any ray C. If C does not belong to the interior of A * X , then the existence of a uniloadable web is not guaranteed. Finally, we will answer the question in case C is not simply a ray but a convex cone. Specifically, we will give a positive answer in the following asymptotic sense: one can find a sequence of finite webs W n such that C Wn X approaches C as n → ∞. For two-dimensional webs where the points X are the vertices of a convex polygon, a similar question was addressed by Theorem 2 in [START_REF] Milton | The set of forces that ideal trusses, or wire webs, under tension can support[END_REF], and the proof given here is similar.

Reducing the number of wires meeting at a point

In two dimensions the procedure for replacing a junction with M > 3 wires by a localized web in which at most three wires meet is straightforward and described in Section 3 of [START_REF] Milton | The set of forces that ideal trusses, or wire webs, under tension can support[END_REF]. Briefly, and as illustrated in Figure 5, one finds the associated Airy stress function in the neighborhood of the junction. This is a convex cone with flat faces with the discontinuity in slope at the edges corresponding to the tension in the wires ( Figure 5(b)). By cleaving the top of this cone, creating a polygonal face ( Figure 5(c)), one obtains an associated web (Figure 5(d)) supporting the same loading as the original junction (Figure 5(a)), but with at most three wires meeting at every junction.

In three dimensions the procedure we use for replacing a junction with M > 4 wires is more complicated. The steps are illustrated in Figure 6, where we begin in (a) with a junction where M = 6 wires meet. First 8 whose coordinates in the original configuration are given by x i (Figure 8(a)), and in the final configuration by x i (Figure 8(b)).

with all the wires under tension which supports F. Finally, we can prove that such a web is a uniloadable web (Theorem 4.1).

To prove that the cone A * X does not have an empty interior we need to introduce the space R X of infinitesimal rigid motions on X = (x 1 . . . , x N ) and its orthogonal B X , called the space of balanced loadings on X, defined by

There is no loss of generality to assume that N i=1 x i = 0 so that X also belongs to B X . Recall that A * X , the cone of admissible forces

). The notion of interior we use is relative to this subspace.

Lemma 3. The cone A *

X is a subset of B X with non empty interior. Proof. We have already noticed, as a consequence of Theorem 2.1, that there is no loss of generality in assuming that the points (x 1 , . . . x N ) span the space R d . There is no loss of generality either in assuming that it is the first points (x 1 , . . . x d+1 ) which span it.

Assume now, by contradiction, that the set A * X has empty interior. As it is convex, this means that it is contained in a lower dimension subspace: there exists U = (u 1 , . . . , u N ) = 0 ∈ B X such that F • U = 0 for all F ∈ A * X . As (x 1 , . . . x d+1 ) span R d , and there exists a unique affine function u such that u(

Clearly, for any (i, j) ∈ {1, . . . , N }, the particular loading F (i,j) = (f 1 , . . . , f N ), defined by f i = x i -x j , f j = x j -x i and f k = 0 whenever k = i and k = j, belongs to A * X (indeed a simple wire linking x i to x j is an admissible web for this particular loading). Hence U satisfies (u i -u j ) • (x i -x j ) = 0 for any pair (i, j). This condition applied to all pairs with i ≤ d + 1 and j ≤ d + 1 implies that u is a rigid motion. The same condition applied to all pairs (i, j) with i ≤ d + 1 and j > d + 1 implies that u j = u(x j ) too. Then U is a non vanishing rigid motion and this contradicts the definition of B X .

Let us now prove that all loadings in the interior of A * X are unstuck. More precisely, Lemma 4. Let F be in the interior of A * X . Then, for ε > 0 small enough, F also belongs to the interior of A * X-εF . Proof. We first remark that, for any ε ∈ R, F belongs to B X-εF . Indeed, for any antisymmetric matrix A we have

Let now U be any vector in A X-εF with U = 1. For any pair (i, j), it fulfills

which implies

Let δ := min i =j |x i -x j | be the smallest distance between the points x i and γ := 4 F δ 2 . The vector

Therefore its projection W on B X , which satisfies the same inequalities, belongs to A X .

The cone A * X is the set of all F satisfying ∀U ∈ A X , F • U ≥ 0. As F belongs to the interior of A * X , the function V → F • V must be strictly positive on the compact intersection of A X with the unit sphere. Let us call α > 0 its minimum. By homogeneity, we have for any V in A X , F • V ≥ α V . When applied to W, this inequality gives

Hence we have

For ε smaller than α 2γ X (α+ F ) , we get F•U ≥ α 2 U . This inequality, valid for any U ∈ A X-εF with U = 1, remains true on the whole cone A X-εF by homogeneity. So the lemma is proven. Lemma 5. If F is in the interior of A * X then there exists a connected web supporting F at X with a strictly positive stress state.

Proof. We first check that there exists a web W supporting F at X = (x 1 , . . . , x N ) such that all wires with non vanishing tension make a connected set. The set D of loadings F for which there exists a set I strictly included in {1, . . . , N } such that i∈I f i = 0 is a union of subspaces (of codimension d) of the space B X of balanced loading. We want to avoid the case where F is supported by two or more disconnected webs and clearly this may only happen if F ∈ D. In that case we use the following trick: as the interior of D is empty we can find G ∈ B X such that F + G and F -G belong to A * X \ D. Let W + and W -be two webs supporting respectively F + G and F -G and let S + and S -be the associated stress measures. The measure S := (S

Its support is the union of the supports of S + and S -which both are connected sets containing all the terminal nodes where the applied forces are non vanishing. Thus we get a finite connected web W with a strictly positive stress state σ that supports the loading F = (f 1 , . . . , f N ) at points X = (x 1 , . . . , x N ).

Corollary 4.1. If F is in the interior of A * X then there exists is a connected web supporting F at X with a strictly positive stress state and only one wire joining each terminal node.

Proof. Lemma 4 provides an ε > 0 small enough for F to belong to the interior of A * X-εF . Lemma 5 then provides a connected web W supporting F at X -εF with a strictly positive stress state σ. Adding to W , for those i ∈ {1, . . . , N } such that f i = 0, the wires [x i , x i + εf i ] and fixing the tension to f i in each of these wires, makes a new web supporting F at X with a strictly positive stress state. Clearly, as W is connected, the new one is connected too.

We can finally state the theorem regarding the existence of a uniloadable web for an unstuck loading. Theorem 4.1. Existence of uniloadable webs For any F in the interior of the admissible loading cone A * X , there exists a finite web W such that C W X = {λF : λ ≥ 0}.

Proof. Corollary 4.1 states that there exists a finite connected web W with a strictly positive stress state σ that supports the loading F = (f 1 , . . . , f N ) at points X = (x 1 , . . . , x N ) and such that only one wire is attached to each terminal node. This web is then modified (according to Lemma 4) into a new web W having the extra property that all the internal nodes have in two dimensions at most three coplanar wires meeting it, and in three dimensions at most four noncoplanar wires, or three coplanar wires, meeting it. This ensures that W supports only the loadings λF for λ ≥ 0. Indeed, as F is an admissible loading for W at points X, the unique wire attached to x 1 has direction f 1 . Let F = ( f 1 , . . . , f N ) be another admissible loading for W at X. Balance of forces at x 1 imposes f 1 = λf 1 . As the wire attached to x 1 is under tension for both loadings, we have λ ≥ 0. Now at each node of the web, once the (positive) tension in one of the joining wires is fixed, the balance of forces fixes the tension in all other joining wires. As the web is connected, from node to node, the tensions of all wires are fixed : F is uniquely determined and clearly F = λF.

Possible loading cones

Here we seek answer to the question: given a convex cone C ⊂ A * X can one find a web W such that C W X = C? To proceed, as sketched in Figure 4 of [START_REF] Milton | The set of forces that ideal trusses, or wire webs, under tension can support[END_REF], one approximates the convex cone C by a cone, that we will denote with C Wj X , having a finite number j of extreme rays F (j)m , m = 1, 2, . . . , j, each strictly in the interior of C and hence strictly in the interior of the admissible loading cone A * X . As we are free to make arbitrary small perturbations of the extreme rays F (j)m , we can assume that these are chosen so that at each terminal i, f (j)m i


Q Huynh

A Selmi

G Corbel

P Lacorre

J M M Millet

Atypical synergetic effect between Te-and V-substituted phosphomolybdic cesium salt and LAMOX-type phases for the oxidation of isobutane into methacrylic acid

Keywords:

Atypical synergetic effect between Te-and V-substituted phosphomolybdic cesium salt and LAMOX-type phases

for the oxidation of isobutane into methacrylic acid

Experimental

Catalysts preparation

The Te-and V-substituted phosphomolybdic cesium salt (CsTeV) was synthesized as described previously [7]. The tellurium-containing cesium salts were first prepared by dropwise addition of 0.4 mL of an aqueous solution containing 1.3 g of cesium chloride to a 140 mL aqueous solution of 8.16 g of phosphomolybdic acid and various amounts of telluric acid. After 1 h under stirring at 80 °C, the solution was dried at reduced pressure and the solid obtained, which was dried at 125 °C and calcined under a flow of air at 360 °C for 6 h. Vanadium was then introduced by maintaining the cesium salt re-hydrated at room temperature in air for 3 h, under reflux and then stirring in a solution of toluene containing vanadium acetylacetonate (V[C 5 O 2 H 7 ] 3 FLUKA 1347-99-8). The solid dried under reduced pressure at 60 °C was not calcined but directly used for the preparation of the phase mixtures or heated under the catalytic reaction flow when tested as pure phase.

The low temperature monoclinic a form of La 2 Mo 2 O 9 (LM) has been prepared by solid-solid reaction between MoO 3 and La 2 O 3 . La 2 O 3 was previously heated up to 1000 °C in order to remove adsorbed water and carbon dioxide [8]. To prepare b-La 2 Mo 1.9 V 0.1 O 9 (sample LMV1), V 2 O 5 was partially substituted to molybdenum oxide for the preparation of the ternary oxide. The starting oxides were weighted with the chosen stoichiometry and mixed by cogrinding by hand. The solid mixture was then introduced in an alumina crucible to be heat treated at 500 °C. It was then subjected to heating and grinding several times in acetone to homogenize it until a single LAMOX phase was obtained. For pure La 2 Mo 2 O 9 , two heat treatments for 15 h at 900 °C were enough, whereas for La 2- Mo 1.9 V 0.1 O 8.95 , seven heat treatments for 15 h at 925 °C were necessary. This was because the transitory formation of LaVO 4 was difficult to avoid. Two other b-La 2 Mo 1.9 V 0.1 O 9 samples have been prepared with higher-and lower-specific surface area. The sample with a higher surface (LMV2) was prepared using the thermal decomposition of [La, Mo, V] metal chelate complexes. Citric acid (CA) was used as a chelating agent in the molar ratio CA/M % 6 (M = [La] + [Mo] + [V]). A sticky gel is obtained on a hot plate by slowly evaporating the solution at 110 °C for a few hours, and by drying the product at 120 °C for about 30 min. The calcination of this gel was carried out in air at 300 °C for 5 h. The fine powder obtained, which was brown in color, was then hand ground and fired twice at 560 °C (temperature determined from TG/DT analyses performed on Q600SDT TA instruments apparatus) under pure oxygen flow (1 atm) for 1 h and 2 h, respectively. The sample with a lowerspecific surface area (LMV3) was prepared by simply calcining the LMV2 sample at 900 °C for 12 h. The phase mixtures were obtained by mixing the respective powders and hand grinding them for 5-10 min. They were designated according to their nominal composition. For example, CsTeV-LM (50-50) denotes a solid containing the same weight of La 2 Mo 2 O 9 and Cs 2 Te 0.3 (VO) 0.1 -H x PMo 12 O 40 .

Catalysts characterization

The pure phases and physical-phase mixtures have been characterized by different techniques. X-ray powder diffraction (XRD) patterns were collected at room temperature with a Brüker D5005 diffractometer using the Ni-filtered Cu-Ka radiations. They were recorded with 0.02°(2h) steps in the angular range of 3-80 with 1s counting time per step.

Chemical analysis was determined by atomic emission for Mo, V, Te, and P, using an induced plasma technique, and for Cs, using an air-acetylene flame on a SPECTROLAME-ICP from SPECTRO. Specific surface areas were measured by the BET method on a Micromeritics ASAP 2020 instrument by adsorption of liquid nitrogen on the samples after desorption at 250 °C for 3 h under secondary vacuum. The precision of the measurements was 0.1 m 2 g À1 .

Raman spectra were recorded using a DILOR XY spectrophotometer equipped with an intensified photodiode array detector. The emission line at 514.5 nm from an ArC ion laser (Spectra Physics, Model 2016) was used for excitation at low power (about 1 mW). The particle size and morphology of the solids (fresh, after reaction conditions) were examined through a Hitachi S800 scanning electron microscope (25 kV). Powdered specimens were spread on the SEM slabs and sputtered with gold.

Catalytic testing

The oxidation of isobutane was performed in a fixed bed reactor operating at atmospheric pressure under conditions described elsewhere [5]. The catalytic properties were determined between 310 and 360 °C in a conventional flow reactor with a catalyst amount of either 0.5, 1 or 2 g. Two feedstock compositions with different O 2 :C 4 H 10 ratios and dilutions into N 2 (O 2 :C 4 H 10 :H 2 O:N 2 = 30.3:22.4:10:37.3 or 10:10:10:70) but the same water content and gas hour space velocity (2500 h À1 ) have been used to test the catalysts. The gases at the outlet of the reactor (isobutane, isobutene, CO, and CO 2 ) were analyzed on line with an on-line gas chromatograph with Porapak-Q and CP-Molsieve columns. The organic substrates, methacrolein (MA), acetic acid (AcA), methacrylic acid (MAA), acrylic acid (Acry), and acetone (Ace) were condensed during the reaction and analyzed off-line [5]. The catalytic data were collected after at least 8 h in the reaction conditions and for at least 48 h. No change in catalytic properties was observed during this time. Blank tests without catalyst have been performed to show that homogeneous reaction was negligible in the range of temperature studied. Carbon balances have been calculated and were satisfactory for all the runs to within ±2%. The catalyst was recovered after a rapid quenching from the reaction temperature to room temperature with a concomitant shift of the reaction gas flow to a nitrogen flow.

Results

The chemical analyzes of the pure phases (CsTeV, LM, and LMV) were in good agreement with the desired stoichiometries and the presence of single phases was detected by X-ray powder diffraction. The CsTeV and LM phase mixtures with different relative phase ratios have first been tested as catalysts for the partial oxidation of isobutane into methacrylic acid. The results obtained are presented in Table 1 and in Fig. 1. The CsTeV phase was very active with a high selectivity to MAA, whereas the pure LM phase was very poorly active and mostly selective to isobutene. When the LM content increased up to 50% in the CsTeV-LM phase mixtures, the conversion slightly increased or remained constant whereas the selectivity to MMA increased passing through a maximum for about 20% of LM. Isobutene, which was observed as a product on the LM phase, was not observed on the phase mixtures similarly to the CsTeV phase. At high LM content both the activity and the selectivity to MAA and MA decreased. This behavior of the catalysts was observed independently of the reaction temperature (Table 1). The apparent activation energies for the conversion of isobutane remained approximately the same for the phase mixtures as for the CsTeV catalyst (E a = 76 ± 3 kcal/mol), which tend to show that the same reaction mechanism was still present, and that it was the number of active sites that changed.

The partially substituted lanthanum molybdate (LMV, 5 mol% of vanadium) tested with a diluted feedstock appeared as poorly active as the pure LM phase but very selective to isobutene (Table 2). This phase, obtained with different specific surface areas, has been tested in mixture with the CsTeV phase in different conditions used in the patents (Table 2). First, it can be observed that in conditions closed but not exactly the same as those used for mixture with LM, a synergetic effect was observed as previously with the LM phase. The best yields in MA and MAA have been obtained with the most diluted charge (iBu/O 2 /H 2 O/N 2 = 10/10/10/ 70). In this case, the activity of the phase mixtures remained high although it has a tendency to decrease with the LMV1 content. However, the selectivity to MAA and MA was stronger (Fig. 2). The CsTeV/LMV1 (50/50) catalyst tested in such conditions was among the most efficient catalysts for that reaction compared to those proposed in the patent literature [START_REF] Imai | Japanese Patent 62132832[END_REF][10][11]. The mixtures with the LMV2 and LMV3 phases presenting respectively a higher and a lower specific surface area (LMV2 with 11.8 m 2 g À1 and LMV3 with 0.5 m 2 g À1 ) were less active although more active than the CsTeV pure sample with equivalent mass. A synergetic effect was taking place but was much less intense. In the case of the phase mixture with LMV3, a large amount of CsTeV phase was not

Table 2

Catalytic performances with the mechanical mixtures of the CsTeV and LAMOX (LM or LMV) phases (50/50) at 360 °C with contact time: 6 s, mass of catalyst: 2 g, and different compositions of the reaction mixture. LMV1 specific surface area 6.0 m 2 g À1 , LMV2 specific surface area 11.8 m 2 g À1 , LMV3 specific surface area 0.5 m 2 g À1 ; AcA: acetic acid, MAA: methacrylic acid, MA: methacrolein. supported and sintered whereas in the case of the LMV2, the supporting effect should be less efficient since the relative particle sizes of the two phases got closer and were not optimal. Furthermore in this last case a large part of the LMV2 particles were not coated and were active to oxidize the MAA and MA formed on the CsTeV phase to CO and CO 2 . This explains why the selectivity to MAA and MA strongly decreased (Table 2). In all cases, the catalysts properties appeared very stable with time in reactions conditions. The phase mixtures have been characterized by X-ray diffraction before and after the catalytic testing. The patterns did not show any modification of the phase compositions or crystallinity after the co-grinding and testing. As an example, the X-ray diffraction patterns of the CsTeV-LM (50/50) phase mixture recorded before and after catalytic test are given in Fig. 3. The samples have also been studied by the Raman spectroscopy to try to detect surface modifications. The spectra recorded before and after catalytic testing were comparable and exhibited the same bands (Fig. 4). These bands corresponded to those characteristic of the Keggintype alkali salt (991 cm À1 : m s Mo = O d , 974 cm À1 : m as Mo = O d 962 cm À1 : m s P-O 880 cm À1 : m as Mo-O b -Mo) and to new peaks at 1024, 1008, 844, 790, 732, 676, 628, 545, 495, 306, 283, and 183 cm À1 attributed to capped heteropolyanions bonds vibrations [5]. After catalytic testing, the catalysts should be more reduced (change in color from green to darker) and Mo 5+ -Mo 6+ charge transfers in the Keggin unit enhanced. Since these charge transfers induced a Raman resonance, the bands related to the capped anions appeared stronger. The LM phase peaks were not detected in the spectra. This may certainly be related to the covering of the LM particles by the CsTeV ones.

Catalyst

Discussion

Synergetic effects between phases in heterogeneous oxidation catalysis may have different origins and it is difficult to unambiguously attribute it to one of those [12]. In our case, the characterization made by XRD and Raman spectroscopy after catalytic test did not show any chemical reaction between the phases. Furthermore, a bifunctional catalysis, with one phase catalyzing the oxidative dehydrogenation of isobutane and the other one catalyzing the oxidation of the formed isobutene cannot be retained as the main origin since the LM phase is very selective to isobutene but almost inactive. It restricts the possibilities of explanation to the formation of a coherent interface between the phases leading to a supporting effect.

It has been observed that the specific surface areas of heteropolycompounds when tested in oxidation reactions strongly decreased after testing. Such sintering effect, which is taking place systematically during the first hours on stream when the reaction temperature exceeds 300 °C, explains the strong decrease in the activity of this type of catalysts generally observed when the catalysts start to be tested. This phenomenon was also taking place for the pure CsTeV phase in the oxidation of isobutane during the first hours on stream (Fig. 5). The evolution after catalytic testing of the specific surface areas of the CsTeV and LM phases and their 50/50 phase mixture has been compared (Table 3). First, it can be observed that the specific surface area of the phase mixture prepared by co-grinding corresponded approximately to the average of those of the pure phases. Furthermore, only a small decrease in the specific surface area is observed for the phase mixture after catalytic testing, whereas the surface area of the pure CsTeV compound was divided by 2. This result explains why the phase mixtures have an activity comparable to that of the pure CsTeV phase with up to 50% of LM. Indeed, if we hypothesize that the specific surface area of the phase mixture after catalytic test corresponds to the weighted sum of the specific surface area of the constituting phases as it is the case before catalysis, and that the surface of the LM phase in the phase mixture was not modified in the reaction conditions as it is the case when it is tested alone, the surface area of the CsTeV phase in the phase mixture after catalytic test can be evaluated and the intrinsic rate of isobutane conversion per square meter of CsTeV phase calculated. It appears that it is approximately the same as that calculated for the pure CsTeV phase (Table 4). The selectivity set is also comparable, which is in agreement with the hypothesis since it is the same active sites in comparable number that are exposed to the reactants.

To support the obtained results, a CsTeV-LM (70-30) sample has been characterized by scanning electron microscopy (Fig. 6). The comparison of the SEM micrographs recorded before and after catalytic testing clearly showed large particles covered with very small spherical-shaped particles. The first one should correspond to the LM phase and the second to the CsTeV phase. No modification of the morphology was observed after catalytic testing, which tends to confirm the hypothesis.

This support effect explains the catalytic results obtained. The decrease in active phase mass in the catalysts is balanced by the absence of sintering of the later phase. The positive effect on the selectivity to MAA and MA is more difficult to explain. it may be hypothesized that the small size of the active CsTeV particles allows better thermal exchanges with the gas phase avoiding hot spot on the catalysts, thus avoiding over-oxidation. The lanthanum molybdate may also be involved. Although it is not very active, it is very selective to isobutene which is an intermediate of the reaction; this may have a direct positive effect on the selectivity since isobutene is very selectively transformed into MAA and MA over the CsTeV catalysts, but also an indirect effect since isobutene oxidation will help to maintain the polyoxometallate phase in a more reduced state, which has been shown to be beneficial to the selectivity in MAA and MA [13]. With such synergetic effect, there

Table 4

Intrinsic rate of isobutane conversion on the heteropolycompound (CsTeV) and the (50-50) phase mixture with lanthanum molybdate (CsTeV + LM). For the calculations, specific surface areas measured after catalytic test have been considered with the hypothesis that the surface area of the LM phase was not modified in the reaction conditions. Contact time: 4. should be an optimal ratio between the LM and CsTeV particle sizes and in the case of the low surface area LMV sample, a lot of CsTeV particles in the 50/50 phase mixture are not mainly stuck to the LMV surface and sintered.

A question that remains is what is specific to the LM phase to allow this support effect, which has never been observed with other phases.

The structure of b-La 2 Mo 2 O 9 is rather difficult to describe as usual in terms of cationic coordination polyhedra. This is due to the fact that two out of three oxide sites are only partially occupied (O2 et O3). The structure can better be described in terms of anionic coordination anti-polyhedra [14] (Fig. 7a). The site O1 fully occupied is surrounded by four cations, 3 La 3+ + 1 Mo 6+ . These [O1La 3 Mo] anti-tetrahedra are connected to each other by La corners forming a 3D network in green in Fig. 7a. Atoms in O2 and O3 are located in the channels formed by this network. As it can be seen, Mo and its substituent (V) point out toward inside the channels. When the structure is observed along the (1 1 1) direction (see Fig. 7b andc), the complex arrangement of La 3+ cations is built up from the interweaving of hexagonal sublattices, one of which is highlighted as larger blue spheres in Fig. 7b. The LMV phase crystallizes in this b structural type with room temperature cubic parameter 0.71458 nm [15].

The structure of he CsTeV compound can be described as an arrangement of (PMo The surface of the a-Keggin assembly exhibits four cavities defined by eight-membered (Mo 4 O 4 ) rings where the (VO) 2+ and Te 4+ cations are located. One of these cavities (or two opposite to produce idealized point symmetry) is occupied. The oxidation state of vanadium and tellurium cations has been demonstrated to be 4+ and the pyramidal geometries of the sites are likewise consistent with Te 4+ and (VO) 2+ centers with the stereochemically active lone pair and the vanadyl oxygen, respectively, occupying the vacant apical site (in green in Fig. 8a) [5]. To balance charges in the capped anions, molybdenum cations are reduced. But it has not been possible to determine the extent of reduction of Mo in the capped anions neither whether the anions are single or bi-capped or both.

The mixed proton and cesium salt crystallized with a cubic cell with cell parameter equal to 1.1824 nm [5]. The structure of the cubic cesium salt in the (1 1 1) plane is presented in Fig. 8b with the P arrays in this plane. P atoms being in central position in the Keggin anions, this array can be considered that of the Keggin anions.

Interestingly, when La arrays in the LMV phase and the P (Keggin-type anions) arrays in the cesium phosphomolybdic salt structures are superimposed, a good fit was obtained (Fig. 9). Presumably OH surface basic sites bound to the La cations are substituted by phosphomolybdic anions leading to strong and coherent boundaries between lattices of the two phases and to the stabilization of the salt particles at the surface of the lanthanum molybdate. The ''covering rate", defined as the total number of matching atoms over the total number of atoms in the P and La arrays, is larger in the (1 1 1) plane (50%) than in the (1 1 0) and in the (1 0 0) planes (40% and 28.6%, respectively). The misfits between the arrays have been calculated in terms of difference of unit cell surface determined by the La and P arrays. These misfits were of about 8%, which is not small but could still lead to efficient boundary conditions [16]. The phosphomolybdic particles are much smaller than the lanthanum molybdate particles and the surface contact is limited which should allow relaxing the misfit accumulated. Furthermore, the structure of the salt could easily absorb the local lattice strain generated by the misfit.

The lanthanum molybdate would thus play a role of support for the polyoxometallate active-phase particles, avoiding their sintering in the reaction conditions and consequently maintaining their high catalytic activity. Although it is not clear yet if it is because of the intrinsic properties of the lanthanum molybdate or if it is related to the supporting effect, the selectivity of the catalysts appeared improved. It is also the case for the stability of the catalyst, which was already high for the pure active phase [5].

Conclusion

Several conclusions can be drawn from this study:

1. Very efficient catalysts for the oxidation of isobutane can be prepared by mechanically mixing a phosphomolybdic polyoxometallate with cesium tellurium and vanadyl as counter-cat-ions, and a lanthanum molybdate either pure or with molybdenum partially substituted by vanadium, which is poorly active and mainly selective to isobutene. 2. The synergetic effect observed between the two phases has been explained by a support effect. The heteropolycompound particles, which are much smaller than the lanthanum molybdate ones, are evenly adsorbed on the laters avoiding their sintering, which occurs during the first hours under stream. With up to 50% of lanthanum molybdate in the phase mixtures, the activity of the catalyst remained almost the same as that of the pure phosphomolybdic polyoxometallate whereas the selectivity to MAA and MA slightly increased. The observed catalytic properties also appeared very stable with time. 3. The support effect avoiding the sintering of the active phase particles can tentatively be explained by the existence of coherent phase boundaries between the two phases and more specifically the La array in the lanthanum molybdate and the Keggin unit array in the polyoxometallate salt. These fits should help for the development of a stronger bonding between the particles enhancing the anchorage of the polyoxometallate particles at the surface of the lanthanum molybdate. This type of synergetic effect had never been reported in catalytic oxidation reaction for which most of the time it is the spreading of one phase on the other which is at the origin of a synergetic effect.

Fig. 1 .

 1 Fig. 1. Variation of the conversion of isobutane (j) and of the selectivity in MAA + MA (h), MAA (Ç), MA (s), AcA (M), and CO x (O) as a function of the wt.% of LM in the CsTeV-LM phase mixtures. The reaction was run at 369 °C; feedstock composition: O 2 /C 4 H 10 /H 2 O/N 2 = 27/14/10/50; total flow = 30 mL min À1 .

Fig. 2 .

 2 Fig. 2. Variation of the conversion of isobutane (j) and of the selectivity in MAA + MA (h), MAA (Ç), MA (s), AcA (M), and CO x (O) as a function of the wt.% of LMV1 (LMV with 6 m 2 g À1 ) in the CsTeV-LMV phase mixtures. The reaction was run at 360 °C; feedstock composition: O 2 /C 4 H 10 /H 2 O/N 2 = 10/10/10/70; total flow = 30 mL min À1 .

Fig. 3 .

 3 Fig. 3. X-ray diffraction patterns of the CsTeV-LM (50/50) phase mixture before (a) and after (b) catalytic test. .: La 2 Mo 2 O 9 . The patterns have been indexed up to 32°(2h).

Fig. 4 .

 4 Fig. 4. Raman spectra of the CsTeV-LM (80/20) compound recorded before (a) and after (b) catalytic test at 360 °C.

Fig. 5 .

 5 Fig. 5. Evolution of conversion as a function of time on stream for the CsTeV catalyst tested in standard conditions at 350 °C.

Fig. 6 .

 6 Fig. 6. MEB images of the CsTeV/LM (70/30) phase mixture before (left) and after (right) catalytic test in standard conditions at 350 °C.

Fig. 7 .

 7 Fig. 7. Crystal structure of b-La 2 Mo 2 O 9 (and of La 2 Mo 1.9 V 0.1 O 8.95 ): (a) projection along (1 0 0) of the structural framework built up from [O 1 La 3 Mo(V) in green] antitetrahedra (La corners in blue, Mo(V) in pink) with partially occupied sites O 2 (faded yellow) and O 3 (faded red) inserted in the tunnels; (b) projection of the cationic arrangement along (1 1 1) threefold symmetry axis, with a highlighted La 3+ hexagonal sublattice; (c) isolated La hexagonal array.

  12 O 40 ) anions capped with Te and VO cations with Cs and protons as counter-cations. The (PMo 12 O 40 ) anions have the well-known a-Keggin structure composed of a central tetrahedral PO 4 surrounded by 12 edge-sharing metal-oxygen MoO 6 .

Fig. 8 .

 8 Fig. 8. (a) Structure of the CsTeV compound, (a) (PMo 12 O 40 ) Keggin-type heteropolyanions capped by vanadyl cations (green), (b) anionic arrangement in the (1 1 1) plane of the cubic cesium salt. The capping cations have been omitted for the clarity of the figure; (c) Isolated P hexagonal array.

Fig. 9 .

 9 Fig. 9. Fit between the P array in CsTeV (red dots and dotted lines) and the La array in LMV (blue circles and dashed lines) along the 3 main cubic directions of both lattices: (a) along (1 1 1) with covering rate 50% and misfit 8.74%; (b) along (1 1 0) with covering rate 40% and misfit 8.74%; (c) along (1 0 0) with covering rate 28.6% and misfit 8.70%. Note that the P lattice is either slightly expanded (a and b) or slightly contracted (c) relative to the La lattice.

Table 1

 1 Catalytic performances with the mechanical mixtures of the CsTeV and LM phases with contact time: 4.8 s, mass of catalyst: 2 g, composition of the reaction mixture: iBu/O 2 /H 2 O/N 2 = 27/13.5/10/49.5; AcA: acetic acid, MAA: methacrylic acid, MA: methacrolein.

	Catalyst	T (°C)	Conversion (%)	Selectivity (%)

Table 3

 3 BET surface area of the compounds before and after catalytic testing.

	Compound	Before testing (m 2 g À1 )	After testing (m 2 g À1 )
	CsTeV	22.0	11.4
	LM	5.2	5.0
	CsTeV-LM (50/50)	14.5 (13.6) a	13.2

a Theoretical value.

Q. Huynh et al. / Journal of Catalysis 266 (2009) 64-70

Q. Huynh et al. / Journal of Catalysis 266 (2009) 64-70