Czeslaw Wozniak

Margaret Wozniak

published or not. The documents may come

Composite shells with interlaminar imperfections

we can introduce what will be called: Mean Pressure Approximation. The mean pressure p = (r 33) acting on an arbitrary surface ~ = const can be approximated by

p+3(0) --p_3(0) p+3(0) q-p_3(0) p(O, --- + , [-a/2, a/2].
6 2 where p+3(0), p_3(0) are the known normal loadings acting on the shell boundary surfaces ~ = -6/2, = ~/2, respectively. Hence, we see that the effective form of constitutive equations (20) (under Mean Pressure Approximation) depends on the character of normal loadings acting on shell boundary surfaces; if p(0, ~) is a compression then in (20) (sg p) + = 0, if otherwise then (sg p) + = 1.

The effective constitutive equations (20) for laminated shells with initial interlaminar imperfections have an explicit form and can be used in engineering problems provided that the imperfection densities are known. (20) describe the effect of imperfections on the shell behavior in the framework of 3 + 3m parameter shell theory. The analysis of special shell problems as well as the possible generalizations of the proposed approach will be studied separately.


Cz Woiniak

Prof. Dr Cz Wolniak

Discrete and continuum modelling of delamination processes

Keywords: Verteilung yon gelockerten Bereichen yon den Trennfl/ichen yon Schichtwerkstoffen wird durch einc Dichtefunktion der Schichtentrennung modelliert, Diese Darstellnng ermSglicht die Formulierung eines ph/~nomenologischen diskreten Modells yon SchichtenCrennungsvorgiingen, Ffir vielschichtige, periodische Verbundwerkstoffe wird das diskrete

The chaotic distributions of delaminated surface elements in laminates are described by a macrodelamination density function. Using this concept a phenomenological discrete mathematical model of the interlaminar debonding processes is proposed. For multilayered periodic composites the discrete model is approximated by a certain continuum model of delamination processes.

Diskrete und kontinuierliche Modelle der Schichtentrennungsvorgiinge t~bersicht: Eine chaotische

Introduction

In this paper we deal with composites made of a number of sheets (laminae) bonded together by very thin layers of a bonding material. We shall assume that the bonding layers can be treated as surfaces, i.e. they coincide with the interfaces of adjacent sheets ; the scheme of the composite under consideration is shown in Fig. 1. We are to investigate deformation and stresses in such laminates taking into account the possibility of the debonding of laminae. The mathematical model of debonding processes in layered composites was proposed in [START_REF] Naniewicz | On a quasi-stationary model of debonding processes in layered composites[END_REF] where the loss of a permanent contact between the sheets was uniquely determined by a certain extremal value of the strain energy of the bonding material. However, in many deformation processes met in the engineering practice a chaotic distribution of certain delaminated micro-surface elements between the adjacent layers can be observed. This chaotic distribution is due to the unknown a priori nonuniform and non-local character of bonding. Moreover, the model introduced in [START_REF] Naniewicz | On a quasi-stationary model of debonding processes in layered composites[END_REF] is too involved to be the basis of engineering applications.

The main aim of this contribution is to propose and discuss two macroscopic (phenomenological) models of the debonding processes in laminates which take into account the chaotic micro-As A K., --~////\iii 1"IK j//'<Q//~/ ".41 AK "/ Fig. 1. Scheme of the composite delaminations between the interfaces of sheets. We start with the discrete macro-modelling of delamination described in terms of functions defined on the interlaminar surfaces; the leading concept here is that of a delamination density related to these surfaces. If the number of layers is big enough, then the discrete model is not plausible from the point of view of engineering applications. That is, we shall pass to the continuum model of debonding processes where the delamination density is related to the region occupied by the whole composite in its undeformed state. In the continuum modelling we apply the microloeal approach, [START_REF] Wolniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF], which takes into account some concepts of the nonstandard analysis, [START_REF]Non-standard analysis[END_REF][START_REF] Wo~niak | Nonstandard analysis in mechanics[END_REF]. The proposed models are able to describe the physical situations met in engineering, in which only partial delamination in certain parts of a composite takes place.

The time-irreversibility of the debonding processes leads to time-nonlocal governing relations of the problems considered. However, for the quasi-stationary debonding processes it is shown that after time diseretization we obtain a sequence of problems each of them being described by a certain variational inequality in a discrete model or a certain nonlinear equation in a continuum model. The solutions to these problems exist and are unique up to the time instant in which the total delamination takes place. The considerations are carried on under the following assumptions : i. The material of separate sheets is homogeneous and linear elastic, ~i. the ~hin layer s of the interlaminar bonding material are modelled as surfaces having linearelastic properties and able to transmit only restricted values of the interlaminar tractions, iii. after the delamination only the unilateral contact without friction between the adjacent laminae is possible, iv. the small deformation gradient theory can be applied.

Micro-and macro-delamination

Let ~9 stands for the regular region in [R s occupied by the undeformed composite made of S disjointed sheets A g (K = 1 ..... S) separated by the interlaminar surfaces HK (K = 1 ..... S --1) as shown in Fig. 1.

Let the composite be subjec t to a certain deformation process in the time interval [30, 3/] with the initial time instant r0 related to the undeformed and unstressed body. Using the approach applied in [START_REF] Naniewicz | On a quasi-stationary model of debonding processes in layered composites[END_REF] we assume that the displacement fields, defined for every 3 C [v0, ~/] on f2, may suffer discontinuities across the interlaminar surfaces HK and that some critical values of these discontinuities may lead to the loss of the permanent contact between the adjacent layers AK, AK+ 1 . In this case the bonding material is no more able to transmit any interlaminar forces and the only interactions across the interfaces of sheets are due to the possible mutual unilateral contact between the sheets, Define will be called the miero-delamination process. Obviously, D(~I) ~_ D(~2) for every rl < ~ due to the time irreversibility of this process.

In this contribution we are to propose a certain phenomenological model of the miero-delamination processes which will be the basis for the mathematical macro-modelling of debonding processes in multilayered composites. Let h be the minimum thickness of the laminae and B(z, r) be a ball in ER 3 with a center z E //and a radius r, such that r < h. The case d(z, ~) = 1 will be referred to as the total delamination at z E H and at the time instant v of the deformation process; from the physical point of view this case can be interpreted as the micro-delamination in a certain vicinity of z on H. The heuristic concepts of the delamination density and that of the macro-delamination process constitute the basis for the macro-modelling of debonding processes in laminates. In the subsequent section the basic assumptions and governing relations for the discrete macro-model will be formulated.

In the sequel we denote by n(z), z E 11, the unit normal to H which for every Z E Hg is outward to AK (K = 1 ..... S --1). The unit outward normal to ~D is denoted by n(x), x C ~f2. For an arbitrary vector field w : T] ~+ [Ra we introduce the notation w~-(z) ~ w(z). n(z), WT(Z) --W(Z) --n(z) W~v(Z).

Moreover, for every field ~(.) defined on A, which has well defined traces on ~A, we introduce the jump of this field across//, setting

[yJ] (z) ~ V+(z) -V-(z), z E 1I K (K = 1,..., S -1)
where y~ (z), F-(z) are the values of the pertinent traces on ~AK+I n Lr g and ~AK n HK, respectively.

3 Discrete macro-modelling Let for every time instant v E [30, ~j] the composite under consideration be subject to the known body forces b(x, v), x E A, and the surface tractions p(x, T), x E /', where F is the corresponding part of the boundary ~D of D. The internal forces will be represented by the Cauehy stress tensor field T(x, v), x E A, and the interlaminar stress vector field t(z, r), z E H. The displacement field at an arbitrary time instant v E [v0, v I] will be denoted by u(x, T), x E A. We assume that u(., v) E V where V is a linear topological space of sufficiently regular functions defined almost everywhere on A which have well defined traces on ~A, and may suffer discontinuities [u~ (z, v), z E H, across the interlaminar surfaces//. We also assume that on the part/'0 of the boundary ~D, the traces u0(x, v), x E /'0, are known for every T E [30, ~I], and that rues (/'0) > 0. In every problem under consideration the mass density ~(x) and the tensor of elastic moduli (](x) are assumed to be known for almost every x E .4. In the sequel we shall introduce the linear operator defined on V

-(Pv +

and the space V0 of the test functions

Vo:= {v~ V:vlc. =0}.

With forementioned denotations and assumptions introduced in Sec. 1 we obtain the stressstrain relations

T(x, v) = t3(x) [E(x,v)], E(x,v) = ~u(x,v); x E A, v E [v0, vl] (3.1)
for the material of the laminae and the following variational form of the equations of motion :

ftr IT(x, 3) Ev(x)] dV + f t(z, ~). Iv] (z) dA = f ~(x) [b(x, v) -ii(x, t)]. v(x) dV + f p(x, v). v(x) dA ~ v(.) E V0, v E [
f (Iv1] (z) --lull (z, i)). n(z) r(z, v) dA > 0, V v E[K, 3 E [v0, v/]. (3.4)
H So far, the approach coincides with that given in [START_REF] Naniewicz | On a quasi-stationary model of debonding processes in layered composites[END_REF]. The crucial point of the modelling is to propose an interrelation between the interlaminar bonding stresses s(z, v), the delamination density 6(z, v) and the displacement jumps ~-ul] (z, T), z E H, in the debonding process. To this aid we introduce the two following real vwlued functions: and let V* be a dual of V. Then under the condition that mes (//g \A(~)) > 0 (K : 1 ..... S -1) it can be proved that for every/E V* and every F0 such that mes (F0) > 0, there exist the unique solution u : u(.) to the variational inequality (4.1). The exact analysis of this problem will be detailed separately.

Summarizing the obtained result we conclude that the quasi-stationary debonding processes in the laminates under consideration can be described by the sequence of variational inequalities for the displacement fields u(vA) ~ u(., 3A), (A : 1, 2 .... , M) given by The problems described by (4.2) can be succesively solved provided that the totally delaminated part A(d(~A)) of the interlaminar surfaces H satisfies the condition in the above stated theorem.

The time-local approximation can be also applied to dynamical delamination processes. In this case we arrive at the sequence of dynamical problems for functions [rA, TA+I] ) r -~ U(~) E V, satisfying the time dependent variational inequalities and the pertinent initial conditions.

Continuum modelling

In many engineering problems we deal with laminates made of a large numer of repeating basic units of laminae, every unit being composed of a few sheets. Such composites have a periodic structure in the direction n(z), z E HE (K = 1 ..... S --1). Due to the large number of interfaces HK, across which the displacement field can suffer a discontinuity, the discrete models of delamination processes, introduced in See. 3 , are not plausible in the engineering applications of the theory (e.g. as a basis of numerical calculations). That is why we shall pass now to the continuum models of maero-delamination processes. The approach outlined below is based on the mierolocal homogenization approach, [START_REF] Wolniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF].

Let every basic unit of the multilayered composite have the thickness s (in the undeformed state) and made of s thin homogeneous sheets. Let Ca, ea (a = 1 ..... s) stand for the tensors of elastic moduli and the mass densities, resp., in the particular sheets belonging to an arbitrary basic unit of the laminate. The fragment of the basic unit is shown on Fig. 2 where also the local coordinates ~ = ~(x), ~ E [0, e], is indicated.

Let P~ stand for the problem that can be stated as follows : for the periodic laminate with the basic layer of thickness e (s-periodic laminate) find displacements u(., v) and stresses T(., v), v E [v0, rf], satisfying (3.1)--(3.8) under pertinent regularity conditions, provided that the material structure and all external agencies acting on the body are known.

Let pn (n = 2, 3 .... ) stand for problems obtained from p1 by formal replacing the s-periodic structure by s/n-periodic structure, all remaining entities in (3.1)-(3.8) being unchanged. Roughly speaking, we pass from the periodic composite under consideration to a certain periodic composite in which the basic units of laminae are n-times thinner, but all other quantities in the problem under consideration remain unchanged. The method of modelling proposed in [START_REF] Wolniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF] and applied below is based on the following.

Homogenization assumption. The solution to the problem p1 can be approximated for every n = 2, 3, 4 .... by the solution to the problem Pn.

The forementioned assumption implies that the periodic laminate under consideration has sufficiently thin basic units (related to all characteristic length dimensions) and hence its macromaterial properties can be approximately treated as independent of the parameter e.

The microlocal approach we are to apply, [START_REF] Wolniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF], is based on some notions and theorems of the nonstandard analysis, [START_REF]Non-standard analysis[END_REF][START_REF] Wo~niak | Nonstandard analysis in mechanics[END_REF]. In the nonstandard analysis we deal not only with the well known standard real numbers but also with the real numbers that are infinitely large and infinitely small. From the transfer principle of the nonstandard analysis it follows that if the problem ~1 can be approximated by the problem 2n for every positive integer n, then it can be also approximated by the nonstandard problem 3 ~ where co is a certain infinite positive integer, i.e. a natural number that is greater then all standard natural numbers. From a formal point of view, ~ is a problem of a composite made of an infinite number of infinitely thin basic units having the thickness s/co. It has to be emphasized that no limit passage is applied here and that the problem 2", 2~ have only formal meaning as certain approximations of the problem 21.

~n(xl

In order to formulate the second assumption of the microloeal method of modelling, we introduce the functions ha: where the summation convention with respect to a = 1, ..., s -1 and A ~ 1, ..., s holds, and where *qa(.), *dA(.), *W(.) are standard sufficiently regular unknown functions. Using the known extension principle of the nonstandard analysis, ef. [START_REF]Non-standard analysis[END_REF], we assign to every mathematical entity P the entity *F that is called standard. It means that the standard functions *q%), *dA(-), *W(.) are uniquely determined by certain functions qa(.), dA(.), w(.), resp., of the well known standard analysis. Functions w(., T), T C Iv0, zj], are called the macro-displacement fields and approximate the displacements of the composite at the time instant T. It can be observed that the terms in the microlocal approximation assumption involving q(-), da(.) are infinitely small and can be neglected if we restrict ourselves only to the evaluation of displacements. However, in the evaluation of the displacement gradients all terms in the microloeal approximation assumption are involved. The functions qa(., 3) (a = 1, ..., s --1) are referred to as mierolocal parameters ; it can be shown that they describe the effects due to the jumps of material properties between the adjacent layers, [START_REF] Wolniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF]. The functions dA(., r) describe the jumps of displacements across the interfaces of the periodic laminate; we have to notice that the known functions i~(;) are discontinuous. Setting

1 --1/~ a if a=b d~d:+..-+d s 2~b : 1/W if a:b+ 1 0 otherwise, Nv ~ (n @ v + v @ n)/2, v C [R 3
we can see that the strain tensors Ea(x, v) in the a-th material component are determined by (5.1)

The vector fields d(., v), r E [v, vl] , will be referred to as the strain incompatibilities.

The main feature of the microloeal modelling is that the non-standard problem 2o, under the constraints for displacements given by the microlocal approximation assumption implies a certain problem ~ for the macro-displacements w(-, v), microlocal parameters qa(., r) and strain incompatibilities d(., 3), v E [v0, vl]. At the same time the problem P does not involve any nonstandard entity and represents a certain homogenized model of the s-periodic laminate under consideration.

Neglecting here all calculations leading from the problem 3 ~ to the problem ~ (for the particulars cf. [START_REF] Wolniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF]) we present below only the governing relations of the problem ~. where v0(') is the known field belonging to V0. Under the pertinent regularity conditions (5.1)-(5.6) represent the continuum model for the delamination processes in the periodic multilayered composites. The term continuum model is used in order to emphasize the fact that the delamination density (~(x, 3) is related here to the region z9 of [Ra occupied by the laminate in its undeformed state. Hence the partial as well as total debonding has to be understood in the averaged form and can take place on certain subsets of zg.

Analysis of continuum models

It can be proved that the microlocal parameters qa(x, v) together with the constitutive strains Ea(x, v) and constitutive stresses can be eliminated from the governing relations listed above. From (5.1), (5.3) and (5.4) we obtain

T(X, 3) = C[~W(X, z) --Nd(x, T)], x E -(2, "c E [vo, "Q] (6.1)
where C is the constant effective tensor of elastic moduli (for the particulars cf. [START_REF] Wolniak | A nonstandard method of modelling of thermoelastic periodic composites[END_REF]). At the same time (5.5) yields

(T(x, 3) n(x))~ C 8 indt0,~](dx(x , 3)) + [1 --8(x, v)] y~vd~T(x, v), (6.2) (T(x, 3) n(X))T= [1--8(X, V)] vTdr(x, 3); X~D, 3E [V0, V/]
where we have applied the known denotation 8 ind[0.~ ] for the subdifferential of the indicator function of [0, c~] and where we have denoted

-.(x)v(x), -v(x) -

for an arbitrary vector field v(.) defined on .(2. The crucial point of our procedure is to show that (6.1) and (6. For the sake of simplicity let us confine ourselves to the quasi-stationary problems. Let us also apply the time-local approximation, following the approach described in Sec. and assume that the delamination density 6 = ~(.) ensures the coereivness of the operator Ga(.) defined on (L2(~)) a×a. Then for every / E V~ and every 1"0 with mes (F0) > 0 there exists the unique solution of (6.6) such that wA E V0, TA E 2:. More detailed analysis of this problem will be given elsewhere.

Final remarks

The proposed mathematical models of delamination processes in laminates ~ due to their relatively simple form, may constitute the basis not only for the theoretical analysis but also for various engineering applications including the numerical solution of special problems. Using the time local approximation and the pertinent step by step procedure we can calculate, after every step, the deiamination density up to the time instant in which the solution does not exist. This procedure makes it also possible to determine the parts of the composite where the total delamination takes place in the deformation process under consideration. The analysis of such situations will be performed separately. It must be emphasized, however, that the models proposed involve four material parameters 7~T, 7T, a0, al which determine the properties of the bonding between the interfaces of adjacent sheets in laminates. Thus, for the quantitative analysis of special problems, the forementioned parameters have to be previously evaluated on the basis of experimental results combined with the heuristic assumptions introduced in this contribution.

  part of H where at the time instant 3 C [30, vi] (in the process under consideration) such debonding takes place will be denoted by D(T). From observations and experiments carried out with real laminates it follows that D(T) consists of many chaotically distributed small surface elements. Hence D(v) can be referred to as the micro-delaminated part of H at the time instant v. The mapping 1 if zCD(~), [30, ~/] ? ~ -> xv, l('), ZD,I(z) --~ 0 if zCH\D(v)

3 )

 3 where s(z, ~) represents the interlaminar stress due to the bonding material and r(z, v) is the value of the reaction to constraints [u]~ (z, ~) ~ 0, z E H, resulting from the impenetrability of adjacent laminae. Introducing the cone ~ in V0 defined by K :: {v E V0 : ~v]~ (z) > 0 for almost every z E/7} we obtain u(., ~)c [K,

1 . 1 ~

 11 The interlaminar strain energy function ~(v), v E [R a, which describes the linear elastic properties of the bonding material before possible maero-delamination (i.e. for ~(z, v) = 0) : (v) = ~ (~'N [v:~l 2 + ~T IVTI~), where yz¢, VT are the longitudinal and shear moduli, respectively. Introducing for every v E [Ra the norm Ivle -= 1/(vT) 2 + ~(v~)~, ~ > 0, and setting 8 = Y~V/YT, V ~-lull (Z, v), we obtain Using the denotations e(z, v) = ][[u~ (z, v)l~' a(z, ~) = z([ul] (z, v)) we also get a(z, v) = 0.5yTe2(Z, V). 2. The delamination density function ~E(a), a > 0~ which determines the delamination density ~(z, ~) in term of the supremum of the interlaminar strain energy at z E/7 in the time interval [v0,v). We assume that 6~.(.) is a continuous monotone function such that for some ~0, 31 (0 < a0 < ~1) we have 6~(a) = 0 if a E [0, a0], and ~(a) = 1 if a E [a~, oo). In the simplest case we can assume that ~E(') is linear in [a0, 31]. Setting Sv ~ a~(v)/~v, v E [R a, the interlaminar bonding relations will be postulated in the form s(z, ~) = [1 -~(z, ~)] SEull (z, ~), z ~ H, ~ ~ [~0, ~] (3.5) where ~(z, 3) = ~ ( sup ~(Iu] (~, ~)), ~ ~ II, T c [~0, ~A. (3.6) ~E['%,'r) It can be seen that the functional in (3.6) describes the time irreversibility of the debonding process. Alternatively, setting OD(e) --~ OZ(0.5yrea), e ~ 0, we can replace (3.6) by --We see that if the delamination density increases, then the bonding material is able to transmit smaller values of the interlaminar tractions s(z, v).Together with the initial conditions 1)--(3.5) and the conditions (3.6) or (3.7) constitute the governing relations of the discrete model for delamination processes in the layered composites under consideration. The term discrete is used to underline the fact that the possible debonding takes place on the discret set of interlaminar surfaces//K (K -----1 .... , S -1).4 Analysis of discrete modelsFor the sake of simplicity we confine ourselves to the quasi-stationary debonding processes. Moreover, applying the time-local approximation introduced in[START_REF] Naniewicz | On a quasi-stationary model of debonding processes in layered composites[END_REF] we shall assume that the governing relations (3.1) -(3.5) have to hold only for v --~ vA (A ----0, 1 ..... M) where v0, v1 ..... VM ~ 3/ constitute the finite sequence of time instants such that 3o ~ vl ~ ... ~ 3M. Then for every fixed 3A the values of (~(z, vA), z E //, in (3.5) are known, provided that the solutions to the problems described by (3.1)-(3.6) have been succesively calculated for vl, v2 .... , vA-I, ef. [1]. Every such problem, from the formal point of view, can be treated as time-independent. Setting aside the argument v and introducing the denotations a~(u, v)~ f tr [~v(x) (C[~u] (x))] dV + f [1 -~(z)] Iv] (z) • S lull (z) dA, A II\A(~) A(6) := {z E //: 6(z) ----1}, (v,/5 f e(x) b(x). v(x) dV + f p(x). v(x) dA, A F we obtain from (3.1) -(3.5) the following variational inequality a~(v--u,u)>(v-n,/)VvE ~, uE ~, (4.1) the form of which depends on the previously calculated delamination density 5(.). Using the known denotation for the Sobolev spaces define V:= {V E (L2(Q))s: VIA K ~ (HI(AK)) a (K : 1 ..... S)}

~

  (~)(v -u(3~), u(~)) => (v -u(~A), /(~)) v v c ~, u(3~) E ~ (4.2)where the delamination density d(v) = 6(., v) is determined by

  ~o -2"o,Scheme~s = e,°f~ thee [o,basics]) unit in ~ periodic composit e

  [R --> [R (a = 1, ..., s --1) and iA: [R --> [R (A ~ 1 .... , s) which are eperiodic and given by 0 if (~a+~ -~)/~o if iA(¢) = ~ -~ + ~a --e/2 if / --~+~a+e/2 if where ~l~ ~ (~a+l -~a)/e and ~a is shown to look for the solution to the problem 3 ~ in a certain class of functions which involves only standard functions as the basic unknowns. For the problem under consideration we shall introduce the Microlocal approximation assumption. The displacement fields u(., t), v ~ [v0, rf], representing a solution to the problem 2~ can be approximated by ~(., ~), such that c [0, ¢o_~] u [¢~+~, ~], ~ [~,_~, ~], ¢ < [~, ¢~+x], in Fig. 2. The main idea of the mierolocal modelling is fi(x, 3) = *w(x, v) + __1 *h~(m~)* qa(x, z) + __1 */A(co~) *dA( x, T) , g~ = ~(X), x C *t? co co

  They comprize : ]. The equations of motion ftr IT(x, 3) ~v(x)] d V = f ~[b(x, 3) -~(x, v)]. v(x) d V + f p(x, v). v(xx, 3) is called the mean stress tensor, ~ is the mean mass density defined by ~ --Vl~ + ... + ~/~ and the functional space V0 coincides with that introduced in See. 3. 2. The constitutive equations for the mean stress Ta(x, v) = Ca[E~(x, 3)], T(x, 3) = ~ ~aTa(x, v), x E zg, v E Iv0, 3/] (5.3) a=l where Ea(x, 3) is given by (5.1). 3. The stress continuity conditions [Ta(x,v)-Ta+i(x,v)]n(x)=0 (a= 1 ..... s-1), xE.Q, vE [Vo, V/] (5.4) where n(x), x E Y2, are unit vectors directed along the local coordinate ~ = ~(x) and the constitutive stress Ta(x, 3) is determined by means of (5.3). 4. The averaged interlaminar conditions 3) = ~D ( sup Id(x, ~)]~/; xCf2, vE [v0, V/] ~E[ro,T) lwhere the linear operator S and the density delamination function were introduced in Sec. 3 and where [K n is a cone in a certain linear topological space W of vector functions defined almost everywhere on Y2 by K~ := {u E W: u(x) • n(x) > 0, x E f2}. 5. The initial conditions w(x, v0) = 0, w(x, r0) = v0(x), x C ~ (5.6)

  2) imply T(x, v) : G~(x.,)(~w(x, 3)); x C £2, v e [v0, v/l, (6.[START_REF]Non-standard analysis[END_REF] where G~(x.t)(') is the monotone nonlinear operator defined on the set of all symmetric 3× 3matrices. Moreover, if 8(x, v) ~ 1, then G~(x,~)(. ) is strongly monotone. In most cases the effective moduli tensor C related to the orthonormal vector basis e~(x) with e~(x) = n(x) has a plane of elastic symmetry normal to n(x). With the denotation ~ii[E] ~ OijktEkt (all indices run over 1, 2, 3, summation convention holds) and assuming that ~,/3 run over 1, 2, we obtain the following explicit form of the mapping (6)33[~w] _ 0 3333 -~-(1 --8) YN ~:~ : A~,(8) ~'318w], where (~)+ ~ max {0, c~} and where A.~(8) are elements of the 2 × 2-matrix which is the inverse to the matrix with components ~ + C~3~3/8~,[(1 -8) YT]. Summing up, we arrive at the conclusion that the continuum model of macro-delamination processes in multilayered periodic laminates is governed by the variational equation (5.2), the nonlinear monotone constitutive equation (6.3), the constitutive relation for the delamination density .(X) [~(x, ¥) ¥) n(X))T 8(x, ~) ---8D ( sup Id(x, ~)1~), d(x, x, ¥)] YT and by the initial conditions (5.6).

4 .

 4 With the denotations (~v, -f tr [T(x) 8v(x)] d V, Q (v,/) = f 6b(x). v(x) dV + f p(x). v(x) dA, D F we arrive then at the sequence of the nonlinear variational equations of the form 1, 2 ..... M) with the previously calculated 8~(.). Let us observe that 8(x) = 1 implies (T(x) n(X))T = 0 and (T(x) n(x))~-~ 0. Hence if the condition 8(x) = i holds in a certain part A of (2, then the solution to (6.6) may not exist for an arbitrary system of external forces, i. e. the equilibrium equations divT(x)+Sb(x)=0, x~£2~A; T(x) n~.n(x)=p(x), x~/l~A, may not be satisfied for an arbitrary b(x) and p(x). It means that the totally delaminated part of a composite (within the continuum model of delamination) is not able to sustain certain systems of loadings. Let us introduce the spaces v =-(w(o))3, Vo =-{v v: vl o = o}, X--~ {T C (L2(~9))a×a: div T C (L~(O))a},

, T) : f ~(z --~) ~(n(~)(2) dA(g), z E HK (K = 1 ..... S -1)

				Then
	~,(z, T)	area [B(z, r) n D(v)] area [B(z, r) n//]
	represents the mean density of the micro-delamination at z E//and at a time instant v E [v0, T/].
	The phenomenological model of delamination will be based on the heuristic assumption that in
	every debonding process there exist sufficiently regular functions ~(., T), w E [30, vl], defined al-
	most everywhere on//, such that 6(., w) ~ d~(., v) for every r ~ h. The values 6(z, v) will be called
	the delamination densities at z E H and at the time instant v E Iv0, vf]. Moreover, the mapping
	[v0, v I] ) v -+ ~(., v),	(~(z, 3) E [0, 1],	z C H
	will be referred to as the macro-delamination process. It can be assumed that the micro-and macro-
	de]amination processes are interrelated by
	8(zHe
	where 0(') is a certain function which makes it possible to treat the macro-delamination density
	~(., v) as the result of a certain filtration of a chaotic distribution D(v) of micro-delaminated
	parts of H. The main feature of the macro-delamination process is that it takes into account
	what will be called the partial delamination states in which 0 < d(z, 3) < 1 for some z E T/and
	v C [vo, vj.].


published or not. The documents may come


Mathematical modelling of delamination effects in layered composites


Czeslaw Wozniak

Zbigniev F Baczynski

Margaret Wozniak

Prof. Dr Habil.-Eng Czeslaw Wozniak

Doc. Dr Habil.-Eng Zbigniew

F Baczynski

Prof. Dr Habi1.-Eng Margaret Wozniak

Modelling of Nonstationary Heat Conduction Problems in Micro-periodic Composites


Czeslaw Wozniak

Margaret Wozniak

come

Modelling the micro-dynamic behaviour of composites

with space-varying microstructure


Pierre Asselin

P Soulard

B Madebène

M Goubet

T R Huet

Robert Georges

O Pirali

Pascal Roy

The cyclic ground state structure of HF trimer revealed by far infrared jet-cooled Fourier transform spectroscopy

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The cyclic ground state structure of the HF trimer revealed by far infrared jet-cooled Fourier transform spectroscopy.

constants. These results brought definite experimental answers to the structure of such a weakly bound trimer: firstly the vibrationally averaged planarity of cyclic (HF) 3 , also supported by the very small value of the inertia defect obtained in the GS, secondly the slight weakening of the hydrogen bond in the intermolecular excited states evidenced from the center of mass separations of the HF constituents determined in the ground, v 6 =1and v 4 =1 states of (HF) 3 as well as the decrease of the fitted rotational constants upon excitation.

Finally, lower bounds of about 2 ns on  6 and  4 states lifetimes could be derived from the deconvolution of experimental linewidths. Such long lifetimes highlight the interest to probe low frequency intermolecular motions of molecular complexes to get rid of constraints related to the vibrational dynamics of coupled anharmonic vibrations at higher energy, resulting in loss of rotational information.

1-Introduction

Hydrogen bonded dimers and trimers represent the simplestprototype systems for the study of pairwise additivity aspects and non-pairwise interactions such as three-body intermolecular forces which play a crucial role in the condensed phases of matter. [1][2][3][4][5][6] In this context, considerable research has been dedicated to (HF) n oligomers in the vapor phase providing important progress in the field of ab initio methods to derive potential energy surfaces and vibrational dynamics properties of small hydrogen fluoride clusters [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] , as well as in the various spectroscopic techniques used to record spectraof clusters. Successive microwave 28,29 and infrared [30][31][32][33] (IR) studies have been reported about (HF) 2 and (DF) 2 during the 80's and 90's. The first high resolution (HR) spectra of these hydrogen bonded complexes enabled access to the dimer structure, dissociation energy, predissociation lifetimes and tunneling dynamics, and ultimately permitted to build reliable intermolecular potential energy surfaces for such four-atom clusters.4 ,49,50 In contrast to the large number of hydrogen bonded dimer studies reported up to now, only a few homomolecular trimers such as (HCCH) 3 , 51 (HCN) 3 , 52 (HF) 3 ,9 ,36 (HCl) 3 , 53 (CO 2 ) 3 , 54,55 (DF) 3 , 41 and (H 2 O) 3 56 have been studied in nearly isolated gas phase supersonic jet environments associated to near or far infrared (FIR) tunable laser spectroscopy. Apart from (HF) 3 , molecular constants could be obtained from rotationally resolved vibrational spectra. In the case of (CO 2 ) 3 , the assignment of the spectrum was facilitated by considering nuclear spin statistics. The donor stretching modes recorded for (HCl) 3 and (DF) 3 have pointed out the problems related to the analysis of higher energy bands: depending on the high density of levels and the strength of the coupling with the bath of states via intramolecular vibrational redistribution (IVR) a loss of fully resolved structure due to significant lifetime homogeneous broadening is likely to be observed, which is indeed large for (DF) 3 (4 GHz) 41 but much smaller for (HCl) 3 ( 90 MHz). 53 Nevertheless, spectral analysis from such experiments provided reliable information about the structure and dynamics of trimers. This is not the case of the hydrogen fluoride trimer and larger clusters for which available experimental IR data beyond cryogenic matrix spectra are very scarce9 ,36,41,57 . Among these, mass selected vibrational predissociation spectra about the HF stretching bands of (HF) 3 and its isotopomers9 ,36 validated a planar and cyclic equilibrium structure in the C 3h point group. Nevertheless, the fact remains that a definite experiment for deriving the vibrationally averaged structure of (HF) 3 is still missing. Previous high resolution infrared studies failed to provide structural information. IR-IR double resonance experiments within the very narrow frequency range of the CO 2 laser enabled to observe two rovibrational transitions assigned to an in-plane overtone and an out-of-plane combination of torsional modes of (HF) 3 .9No homogeneous broadening was detected for these transitions ( 15 MHz) around 950 cm -1 , in contrast with the huge linewidths ( 8 GHz) evidenced for the 3712 cm -1 HF stretching mode even at low temperature in HR jet-cooled spectra, due to IVR broadening.

The absence of permanent dipole moment for (HF) n clusters 27 eliminates the possibility of a high resolution study in themicrowave spectral region. Consequently, high resolution far IR spectroscopy remains the only alternative to obtain structural information in the ground state, 58 by a direct probing of the intermolecular modes of the trimer since the vibrational density of states isnegligibleat such low energies. Theoretical data available on (HF) 3 were used to guide our experimental strategy and to target the most favorable intermolecular probes.

Several ab initio investigations have predicted a cyclic C 3h equilibrium structure for (HF) 3 , 18,26 confirming the experimental conclusions of Lisy et al. 36 From the structures of several (HF) n oligomers optimized at the coupled cluster level, dissociation energies, harmonic vibrational frequencies and IR intensities could be determined up to n=4. The large values of the equilibrium (D e ) and corrected harmonic zero-point vibrational energy (D 0 ) dissociation energies, respectively calculated at 16 and 10.7 kcal/mol for (HF) 3 , clearly underline the presence of three intermolecular bonds to form the cyclic structure (rather than two for a chain conformation). 36,41,53,56 The character of the normal modes of (HF) n oligomers can be classified according to group theory and vibrational coordinate analysis upon a planar oblate symmetric top having a C nh structure: H-F intramolecular stretches in the 3300-3700 cm -1 range, H-F intermolecular librations in the 500-1100 cm -1 range, F-F intermolecular stretches below 350 cm -1 and ring deformations below 100 cm -1 .

Concerning (HF) 3 , a previous harmonic vibrational analysis evidenced strong IR intensities (I 6 700 km/mol, I 4  410 km/mol) for the two librational modes, the in-plane ( 6 , ′ ) and out-of-plane ( 4 , ′′ ) bending fundamentals, 18 which gives hope for their experimental observation. Large anharmonic effects in such a cluster make uncertain the reliability of the predicted harmonic frequencies for the  6 and  4 modes, respectively at 603 and 702 cm -1 , much higher than the values obtained from a 6D variational calculation (540 and 637 cm -1 ), 26 themselves higher than the only available experimental data about  6 and  4 modes in neon (argon) matrices, respectively at 477 and 590 cm -1 (446 and 560 cm -1 ). 47,48 In summary, the presence of two intense bending modes separated by about 100 cm -1 , in a spectral range where the vibrational density of states is low enough to preclude IVR broadening, is definitely favorable to record a rotationally resolved spectrum in jet-cooled conditions. We have undertaken a simultaneous study of the  6 and  4 modes using our Jet-AILES apparatus consisting in a continuous supersonic jet coupled to a high resolution Fourier Transform IR spectrometer (FTS). The multiplex advantage of FT spectroscopy could be fully exploited here to derive simultaneously ground state,  6 and  4 excited state parameters from a global least-squares fit and to bring new insights about the HF trimer in its vibrational ground state.

Section 2 gives a short description of the Jet-AILES apparatus implemented at SOLEIL and describes the spectra recorded in the mid infrared (MIR) and FIR regions relative to HF stretches and intermolecular librations, respectively. In Section 3 high level vibrational calculations relative to cyclic (HF) 3 are presented in support of the experimental data. Section 4 reports on the rotational analysis, global fit and predicted spectra of  6 and  4 bending modes. Structural data derived from the rovibrational analysis of the high resolution FIR Jet-AILES spectrum of (HF) 3 are finally discussed with the support of theoretical calculations.

2-Experimental and theoretical details 2.1. The Jet-AILES apparatus

The Jet-AILES apparatusis a continuous supersonic high pressure jet coupled to the high resolution Fourier transform spectrometer (IFS 125 HR Bruker, 0.001 cm -1 maximal resolution) associated to the AILES beamline of the synchrotron SOLEIL. It was recently described in details and only the main characteristics will be presented hereafter (Figure 1). 59,60,61 The modulated IR parallel beam of the FTS is sent into an opticalcompartment connected to the back parallel exit port. A first setof two gold coated planar (M1) and toroidal (M2) mirrorsisfocusing the IR beam into the expansion chamber orthogonally to theslit-jet expansion with a magnification factor of 1.14. The IR beam is thendirectly sent into the detector compartment in which a second set oftoroidal (M3) and planar (M4) mirrors is used to focus the IRbeam either onto an InSb detector for the HF stretches (MIR) or a bolometer for the intermolecular librations (FIR).

The expansion chamber is connected to an ensemble of Roots blowers backed by a dry primary pump delivering an effective pumping capacity of about 1800m 3 /h.The mechanical vibrations are efficiently damped on one hand by decoupling the pumping unit from the vacuum chamber with two dedicated bellows mounted on the pumping line, on the other handbymounting independently the three vacuum compartments on a heavy framework tightly fixed to the ground. The expansion chamber is isolated from the two other compartmentsby two optical windows (wedged KBr in the MIR or 60 µm thick polypropylene films in the FIR). They are set asclose as possible to the slit nozzle in order to limit the IR absorption by the residual warm gas filling the experimental chamber after expansion. The optical and detector compartments are evacuatedtogether with the FTS in the 10 -4 hPa range in order to avoidatmospheric absorption.A series of slit nozzles of various fixed lengths (30, 60 and 90 mm) andwidths (10 up to300μm) is available to investigate weakly bonded complexesand clustersin the gas phase.

The flows of HF and carrier gas are independentlyregulated by massflow controllers (Bronkhorst 0.02-1 standard liter/minutes (slm) model F-101D-1K0-RAD-33-Kand 2-100 slm model F-202AV, respectively). Paired liquid nitrogen (LN 2 ) traps connected in serieswere mounted between the Roots and the primary pumps to condensate corrosive acid vapoursand neutralized afterwards in a soda/water mixture.

Thestagnation pressure and the residual pressure were measuredusing absolute capacitance manometers (MKS bakeable Baratron5000 torr model 615A and MKS Baratron 100 torr model 626A,respectively).

The Jet-AILES MIR spectra

Hydrogen fluoride (Messer, electronic quality) was used without any further purification. Argon was preferred to helium as carrier gas to enhance the formation of small (HF) n homomolecular clusters.

As a first step, the absorption signal of the HF trimer was optimized in the HF stretching region with respect to those of the dimer and larger clusters. A series of medium resolution jet-cooled FT spectra (0.1 cm -1 , 100 co-added scans) have been recorded to test different HF:Ar dilutions, slit geometries and stagnation pressures. For these experiments, we used a tungsten lamp, a KBr beamsplitter and a InSb detector equipped with a band pass filter centered at 3650 cm -1 , with a 400 cm -1 full width at half maximum (FWHM) and positioned in the detector compartment. The strongest absorptions have been obtained with a 90 mm  100 µm slit nozzle supplied with 5 % HF diluted in a 20 slm mass flow rate of Ar. The corresponding stagnation and chamber pressures are 240 hPa and 0.68 hPa, respectively.

A strong enhancement of the number of (HF) 3 and (HF) 4 oligomers has been observed when adding CH 3 Clto the HF:Ar mixture. To date we have no firm explanation about the role of CH 3 Cl as third collision partner in the nucleation process, but as a matter of fact its presence clearly contributes to boost the formation of larger (HF) n oligomers together with the CH 3 Cl-HF dimers observed at 3777 cm -1 (Figure 2b,c). A similar effect has already been observed in N 2 O:HF supersonic expansions seeded with argon 62 . A small amount of N 2 O greatly enhanced the (HF) 2 dimer production with respect to that of N 2 O-HF. Cryogenic matrix experiments are planned to evidence the clusters formation during the codeposition of a CH 3 Cl:HF:Ar mixture.

Figure 2 displays a series of Jet-AILES spectra recorded for different conditions of dilution of the CH 3 Cl:HF:Ar mixture. Without CH 3 Cl (Fig. 2a), (HF) 2 and (HF) 3 bonded stretch bands located respectively at 3868 and 3712 cm -1 , are the only detectable absorptions with a trimer signal rather weak. The addition of CH 3 Cl shifts the size distribution of (HF) n clusters to larger n: the intensity of the 3712 and 3447 cm -1 bands of (HF) 3 and (HF) 4 clusters increases substantially and then reaches a maximum for CH 3 Cl:HF ratios respectively equal to one (Fig. 2b) and two (Fig. 2c). Broad absorptions are also observed at 3575, 3587, 3638 and 3747 cm -1 for the highest CH 3 Cl concentration (Fig. 2c).

The Jet-AILES FIR spectra

From the preliminary MIR study, the expansion conditions maximizing the absorption of the intermediate (HF) 3 cluster could be optimized with respect to (HF) 2 and (HF) 4 and consequently the 1:1:20 mixture was used (Fig. 2b).

For the FIR experiments, we used a 6 µm thick Mylar beamsplitter and a liquid He cooled Si-bolometer detector equipped with a polyethylene filter limiting the spectral range to 50-650 cm -1 . Survey Jet-AILES FIR spectra have been firstly recorded at 0.05 and 0.01 cm -1 medium resolutions (Figures 3a and3b) to localize the band centers of the bending fundamental bands of (HF) 3 . Secondly, globar and synchrotron radiation (SR) sources have been tested at higher resolution (0.005 cm -1 ) to compare their corresponding signal-to-noise (S/N) ratios. In the investigated spectral range, it appears that the globar source still keeps a slight advantage at this resolution. The S/N rather favors the synchrotron radiation for lower frequencies (below 300 cm -1 ) and/or higher resolutions up to 0.001 cm -1 . Note finally that the choice of the globar source for these experiments owed to the fact that the SR radiation saturated the Si-bolometer for apertures larger than 1.3 mm even for the lowest available stage of gain preamplification. A reduction of the gain preamplification is planned in further experiments.

The Jet-AILES FIR spectrum used for the rovibrational analysis of both bending intermolecular modes of cyclic (HF) 3 is displayed in Figure 3c. It represents the averaging of 700 scans recorded at 0.005 cm -1 resolution which required a maximal iris aperture of 3.15 mm corresponding to an IRspot diameter of about3.6 mm focused at the center of the slit, 3 mm downstream from the exit plane of the nozzle.

Two fully resolved bands are observed at about 494 and 602 cm -1 . On the basis of cryogenic matrix experiments 47,48 and variational calculations, 26 they are assigned to the inplane ( 6 , ′ ) and out-of-plane ( 4 , ′′ ) bending fundamental bands. Expectedly, calculated harmonic vibrational frequencies fall largely above gas phase values while the neon matrix observed ones are red shifted by about 2-4 % supporting futher the attribution.

The line position accuracy is a function of the S/N ratio and was estimated to be 0.0005 cm -1 for the highest resolution spectrum.

Computational details

Second order Møller-Plesset calculations (MP2) were carried out using the Gaussian09 software package. [START_REF] Frisch | Gaussian 09[END_REF] Explicitly correlated coupled clusters (CCSD(T)-F12) calculations were carried out using the Molpro2010 package. [START_REF] Werner Andp | MOLPRO, a package of ab initio programs[END_REF] The frozen-core approximation was used throughout. Dunning and coworkers [START_REF] Dunning | [END_REF]66 augmented correlation consistent basis set aug-cc-pVTZwas used (denoted hereafter AVTZ). All geometries optimizations were performed using"verytight" convergence criteria.

At the MP2 level, anharmonic correction on frequencies and geometrical parameters were calculated using thesecond-order vibrational perturbation theory (VPT2) as implemented in Gaussian09. 67 Anharmonic correction have been extrapolated to the CCSD(T) level using eq (1) where  and  are the harmonic and anharmonic frequencies,respectively.

 CCSD (T) =  CCSD (T) -( MP 2 - MP 2 )
(1)

Theoretical results

As prototype of hydrogen bond complexes, a lot of theoretical studies have been realized on small (HF) n clusters and more particularly on (HF) 3 , the simplest complex to analyze nonpairwise additivity. However most of those studies have neglected anharmonicity and only focused on equilibrium structure and harmonic frequencies (see ref [77] for a partial review of earlier work on harmonic frequencies). Xantheas 68 computed VPT2 anharmonic frequencies for the trimer using MP2/aug-cc-pVDZ high order energy derivative, and compared its results with those of Chaban et al 69 obtained using a correlation corrected vibrational self-consistent field approach on 2D grids potential energy surface (PES) computed at MP2/TZV level of theory. He concluded that both methods yield similar results, but the ab initio level used is too low to reach spectroscopic accuracy. Quack and Suhm built several semi-empirical PES for HF trimer based on the sum of manybody terms. [START_REF] Suhm | HF Dampf. Habilitationsschrift[END_REF][START_REF] Quack | [END_REF] Three body terms were determined from ab initio calculations only while two body terms were based on ab initio calculations but then empirically adjusted. Diffusion Quantum Monte-Carlo (DQMC) calculations have been performed by Suhm [START_REF] Suhm | HF Dampf. Habilitationsschrift[END_REF] for a few modes of the trimer on the SNB+HF3BL PES. Wang and Carrington 26 analyzed the same PES and the improved SO3+HF3BG PES using an elegant 6D variational approach which couples all the bending modes of the trimer. 26 The results of both studies are reported in Table 1. As they differ from our experimental results by several tens of cm -1 , we decided to theoretically study (HF) 3 using an approach which showed its efficiency on the water trimer, 70 by firstly computing the harmonic frequencies at a high level of theory and then reporting on those frequencies the anharmonic correction obtained from VPT2 at a lower level of theory.

We chose to use the CCSD(T)-F12 approach combined with the AVTZ basis set. As the convergence to complete basis set (CBS) limit is faster with the CCSD(T)-F12 approach than with CCSD(T), harmonic frequencies should be close to the CCSD(T)/CBS ones. The anharmonic corrections were computed at MP2/AVTZ level. As Wang and Carrington 26 proved the absence of tunneling effect within the trimer, we can reasonably rely on the anharmonic correction obtained by the VPT2 approach. Both harmonic and anharmonic frequencies are reported in Table 1.

Due to the C 3h symmetry of the complex (see Figure 4), only four vibrational modes are IR active: the asymmetric HF stretching 5 , the in-plane 6 and out-of-plane 4 bendings, and the asymmetric intermolecular stretching 7 . Our anharmonic frequencies are in excellent agreement with all observed gas phase frequencies of the trimer, which are reproduced with a maximum deviation of 2 cm -1 . Because of its low intensity, the  7 mode could not be observed in the gas phase. However our calculated frequency of 173 cm -1 also agrees well with the neon matrix value of 167 cm -1 , taking into account that the neon matrix lowers (HF) 3 frequencies by a few wavenumbers with respect to the gas phase (Table 1).

We tried to understand whether the shift between Wang's frequencies 26 and experiment were due to neglecting the coupling with stretching modes or came from an error on the PES used. The anharmonic frequencies 6 and 4 computed by Suhm (410 (10) and 527(10) cm -1 ) are far from the observed values, mainly due to errors on the two body terms of the SNB+HF3BL PES. However these DQMC calculations which take into account all the vibrational couplings can be used to estimate the error on the fundamentals computed by Wang in neglecting the coupling with the intra-and inter-molecular stretchings. 6 and 4 are both overestimated by 27(10) cm -1 using the SNB+HF3BL PES. From our VPT2 calculation, the coupling between stretchings and 6 or 4 is respectively equal to -51and -46 cm -1 . If we report those corrections on the frequencies of the SO3+HF3BG PES, we found 489 [513 (10)] cm -1 for 6 and 591 [610 (10)] cm -1 for 4 , using respectively VPT2 [DQMC] corrections. The good agreement between corrected SO3+HF3BG PES and observed frequencies seems to confirm that the error made by the 6D variational approach mainly comes from neglecting the coupling with stretching modes, and not from the PES used. FromIR-IR double resonance spectroscopy, Kolenbrander et al 9 observed two bands of the trimer at 940.5 and 954.4 cm -1 ,assigned respectively to the 2 6 overtone and the ( 4 +  8 ) combination, assuming in-plane and out-of-plane bending modes uncoupled. Wang et al. 26 proposed that one of them could correspond to the ( 6 +  8 ) combination computed at 956 and 983cm -1 . From our calculated frequencies, we list the combination bands ( +  ) and overtones (2 ) which fall in the 900-1000 cm -1 range. Keeping only the IR active ones by symmetry ( ′ and ′′ ) only three transitions are remaining at 926 ( 2 +  7 ),948 (2 6 ) and950 ( 4 +  8 ) cm -1 . The ( 6 +  8 ) combination is discarded because it falls below 900 cm -1 .The ( 2 +  7 ) combination band, never computed in earlier works has a frequency much lower than the observed frequencies and is discarded as well. As our calculations overestimate 6 by 2 cm -1 and underestimate 4 by 2 cm -1 with respect to observed frequencies, we can reasonably conclude that the corrected computed frequencies for 2 6 and ( 4 +  8 ) nicely match experimental ones observed by Kolenbrander. Moreover, an excellent agreement is obtained between computed and experimental anharmonic shifts, namely the (2 6 ) -2( 6 ) difference, respectively equal to -46 and -49.0 cm -1 .

On the grounds of the good results obtained in reporting anharmonic correction from VPT2 computed at MP2/AVTZ level on CCSD(T)-F12/AVTZ harmonic frequencies, the same approach was applied to the structural parameters. The CCSD(T)-F12/AVTZ equilibrium parameters are reported in Figure 4 and should be associated to the equilibrium rotational constants, i.e. = 0.25422 cm -1 and = =0.12711 cm -1 . In Table 2

Rovibrational analysis

Experimental spectra of the cyclic hydrogen fluoride trimer were analyzed within the oblate symmetric rovibrator approximation including centrifugal distortion constants , and . Ground state rotational energy levels are given by: 71

, = + 1 + - 2 - 2 ( + 1) 2 - + 1 2 - 4 (2)
For the  6 perpendicular band,rotational energy levels are calculated from the following expression which includes the first order rotation-vibration interaction: 71

 6 , , =  6 + ′ + 1 + ′ -′ 2 -2 ′ -′ 2 + 1 2 -′ + 1 2 -′ 4 ( 3
)
where  is the first-order Coriolis constant for the degenerate vibration  6 . Transitions follow the selection rules ∆ = 0, ±1 ; ∆ = ±1 ; ∆ = ±1 and ∆ -= 0. Rotational branches for the  6 perpendicular band will be designated by

( , ) , ( , ) , and ( , )
, where the superscripts , correspond to ∆ = -1, +1 and the capital letters , , to ∆ = -1,0,1,respectively.

For the  4 parallel band, rotational energy levels are calculated from the following expression: 71

 4 , , =  4 + ′ + 1 + ′ -′ 2 -′ 2 + 1 2 -′ + 1 2 -′ 4 (4)
lines and 7 ( , ) lines. The highest rotational quantum number corresponds to 19 for transitions and 17 for transitions.

The jet-cooled out-of-plane bending  band

The fundamental band  4 of (HF) 3 is displayed in Figure 6a. Its rovibrational structure between 594 and 609 cm -1 shows the characteristic structure of a parallel band with a branch fully resolved and a resolved substructure for the and transitions. 110 lines have been assigned, namely 47 ( , ) lines, 28 ( , ) lines and 35 ( , ) lines with highest values of equal to 14, 21 and 12 for and transitions, respectively.

Determination of molecular parameters

Ground state and upper states parameters of (HF) 3 have been obtained by a global fitting procedure of the 245 IR transitions from both  6 and  4 bands. The simultaneous analysis of bending bands allows to reduce the correlations between several pairs of parameters, such as and ′ , and ′ , ′ and , and so to determine them independently.

Regarding the global fit, all the rotational parameters were set free to test the robustness of our attributions. The ground state constant is the only one associated with a large standard deviation, likely due to correlation effects. In the following, this parameter is therefore fixed to its ab initio value. Finally, 16 rotational parameters (4 for the ground state and 6 for each upper state) were adjusted to reproduce the 245 values with a root mean square (RMS) value of 0.00131 cm -1 .The Coriolis coupling constant deduced from the fit is undetermined and very small, which indicates a very little vibrational angular momentum in the excited  6 band. The results of the least squares optimization are presented in Table II. All assigned lines, together with the observed minus calculated wavenumbers are provided as Electronic Supplementary Information in Table S1. Figures 5b and6b display a global view of simulated spectra of both bending in-plane  6 and out-of-plane  4 bands issued from our best fit.

All the simulated spectra were plotted using the PGOPHER program. [START_REF] Pgopher | a program for Simulating Rotational Structure[END_REF] A rotational temperature ( ) of 30 K was found to provide the best agreement between the observed and calculated relative intensities of the rotational , ,

, , branches of both perpendicular and parallel bands. While the 25 K value of obtained for the recent Jet-AILES spectra of naphthalene seeded in He is probably due to the 423 K nozzle temperature necessary to PAH vaporization 60 , the jet-cooled conditions used here (5 % HF seeded in a CH 3 Cl:Ar mixture) represents our best compromise to maximize the production of small HF clusters using a heavy carrier gas instead He and to obtain a reasonably low value of .

The quality of the fit is illustrated by the comparison between observed and simulated spectra for a small portion of ( , ) transitions of the  6 band (Figure 7) and for the ( , ) transitions of the  4 band (Figure 8).

Discussion

Vibrationally averaged structural properties, which have been limited to a very few cyclic trimers so far, can be extracted from the results of the present rovibrational analysis of (HF) 3 .

To check the consistency of our fit with respect to the state-of-the-art knowledge about the (HF) 3 structure, it is meaningful to verify the planarity of this trimer by exploiting indirect information obtainable from rotational parameters. Firstly, we used the relation between the distortion constants used by Han et al. 53 to verify the planarity of (HF) 3 . Indeed, an oblate symmetric top complex of heteronuclear diatomic molecules with a ℎ ( ≥ 3) symmetry is planar and obeys the relation: 75

+ 2 3 + 2 = 0 (5)
Fitted ground state distortion constants in Table 2 and the ab initio value of (0.98 x 10 -6 cm -1 ) have being substituted into the eq (5) which yields (-1.3  4.0) x 10 -7 cm -1 and clearly shows that the resulting value is close to zero.

Secondly, we calculated the inertial defects of the complex in the ground and upper states from the following relation:

 = ħ 4 ( 1 - 2 ) (6)
From the ground state rotational constants reported in Table 2, the inertial defect is found to be equal to -0.12(2) amuÅ 2 , close enough to zero to be consistent with a planar cyclic geometry. For the excited states, the inertial defects show significant differences: for the inplane bending  6 mode, = 0.04(1) amuÅ 2 and for the out-of-plane  4 mode,  =-0.56(2) amuÅ 2 . These results are quite consistent with the motions of the intermolecular bending mode. The inertial defect is close to zero in the  6 mode excitation because the in-plane bending motion remains in the plane of the cyclic(HF) 3 . On the other hand, the relatively high value of the inertial defect for the out-of-plane  4 mode is consistent with a large amplitude motion out of plane of the trimer.

The rotational parameters derived from our spectroscopic fit clearly indicate that (HF) 3 is vibrationally averaged planar. The fitted constants can then be used to determine the center of mass separations of the HF constituents of the trimer, from the formula of the moment of inertia i.e.

3 2 + 2 = ℎ 8 2 (7)
We assumed that the intramolecular HF bond length remains unchanged with respect to the monomer ground state values, i.e. ( )=0.926 Å. In any case, the distance is only slightly sensitive to the value of ( ) and does not degrade the precision in determining center of mass separations. In the same manner, we derived the separation in the  6 and  4 states from ′ fitted constants. The small increase of about 1% observed in indicates a slight weakening of the hydrogen bond in the intermolecular states which is consistent with the negative sign of  and  experimental values, respectively equal to 2.1% and 1.9 % for  6 and  4 . Calculated rotational constants (Table 2) do not reach experimental accuracy but the rms deviation from experiment is smaller than 0.4% in the ground state and for  6 and about 0.9% for  4 . Moreover, relative  and  calculated values are of same sign as experimental ones with similar variations (2% and 1.3 % for  6 and  4 , respectively). Table 3 gathers the vibrationally averaged center of mass separations and inertial defects for (HF) 3 .

The narrow linewidths of the low frequency bending intermolecular transitions observed in (HF) 3 sharply contrast with the large broadenings evidenced in the donor acid stretch region of (DF) 3 41 and (HF) 3 . 36 collisionless spectra of a low frequency mode in a stiff hydrogen-bonded system. Nevertheless our rough estimateis comparable to more accurate linewidths (74 MHz) measured for the and transitions of the librational band of (HF) 4 around 710 cm - 1 using a pulsed slit jet diode laser spectrometer. 57

6-Conclusion

In the absence of permanent dipole moment and in the presence of extensive IVR homogeneous broadening in (HF) 3 , we faced an unusual case where far infrared spectroscopy could provide structural information even for the ground state in contrast to microwave and MIR spectroscopies.

Taking advantage of the association between high resolution broadband FTS and jetcooled conditions, the Jet-AILES spectrum recorded in thefar infrared range at a rotational temperature of 30 K enabled to perform the simultaneous rotational analysis of two intermolecular  6 and  4 bending fundamentals of the cyclic HF trimer within the oblate symmetric rovibrator model. The 15 fitted molecular parameters, including band centers, rotational and quartic centrifugal distortion constants in both ground and upper states have been used to bring definite proofs to the vibrationally averaged planarity of the complex, (i) the consistency of the relationship between centrifugal distortion constants with the criterion for the planarity of the GS structure (ii) the expected value of inertia defects, very small in the GS and in the v 6 = 1 state of the in-plane bending motion, and much larger in the v 4 = 1 state of the large amplitude motion out of the plane of the trimer. On the other hand, the small increase of the center of mass separation of the HF constituents (GS value, = 2.600(1) Å) correlates well with the decrease of fitted rotational constantsupon excitation which indicates a slight weakening of the hydrogen bond in the bending states.

Ab initio vibrational analyses about cyclic (HF) 3 reproduced remarkably all the observed frequencies and gave an additional argument for the assignment of intermolecular bending fundamentals. It has been clearly proved that the shifts between and computed frequencies previously obtained by a 6D variational approach are due to neglecting the coupling with stretching modes. Also they greatly contributed to the spectral analysis in providing reliable rotational values, namely about the rotational constant to start the initial prediction, and the quartic constant, kept fixed in the global fit with 15 free parameters. Finally, the assignment of several unidentified bands in the Jet-AILES MIR spectrum, correlated with the strongest absorption signal of cyclic (HF) 4 , will be presented in a forthcoming paper. including assignments of the patterns with J = K and J = K+1: a) Jet-cooled spectrum recorded at 0.005 cm -1 resolution, (b) spectrum simulated at T rot = 30 K.

q Q(J,K)

  suggests significant longer lifetimes in the v = 1 and v 4 = 1 upper states. The observation of these transition linewidths provides lower bounds for such excited statelifetime = 1 2 where  is the full width at half maximum (FWHM) of a transition. Experimental linewidths of 114(6) and 126(9) MHz are deduced from the spectra, for  6 and  4 respectively. Our experimental conditions are such that the observed linewidths are largely dominated by the instrumental width of the interferometer, equal to 0.0035 cm -1 (105 MHz) accounting for boxcar apodization. A deconvolution of several intense ( , ) transitions of the  6 band with the boxcar function set to 105 MHzyields upper bounds of 70 MHz for non-instrumental widths and consequently lower bounds of 2 ns for excited state  6 and  4 lifetimes, consistent with expectations for sub-Doppler, almost

Figure 1 :

 1 Figure 1: Schematic view of Jet-AILES set-up.

Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 2345 Figure2: Jet-FTIR spectra of the HF stretch mode of small (HF) n clusters recorded at 0.1 cm -1 resolution for different dilutions of the CH 3 Cl:HF:Ar mixture. The intense and narrow bandsobserved in the 3700-3900 cm -1 range, separated by about 40 cm -1 , are rotational P(J) lines of the HF monomer.

Figure 6 :

 6 Figure 6: Global view of the Jet-AILES  4 spectrum recorded at 0.005 cm -1 resolution (a) and our best simulation at = 30 K (b).

Figure 7 :

 7 Figure 7: Portion of the r R(J,K) branch of the bending in-plane  6 band of cyclic (HF) 3 including assignments of the patterns r R(J -n, K -2n): a) Jet-cooled spectrum recorded at 0.005 cm -1 resolution, (b) spectrum simulated at T rot = 30 K.

1 )Figure 8 :

 18 Figure 8: Portion of the q Q(J,K) branch of the bending out-of-plane  4 band of cyclic (HF) 3

Table 1 :

 1 Harmonic and [anharmonic] frequencies of cyclic (HF) 3 (cm -1 ). Harmonic intensities are reported in parenthesis (km/mol). D e and D 0 energies with respect to 3 HF molecules are also reported (cm -1 ).

	Mode Sym.	Ab Initio		Empirically refined PES	Exp.
		MP2/AVTZ	CCSD(T)-F12/AVTZ		SNB+HF3BL a	SO-3+HF3BG b	Gas	Ne Matrix
	   E'	3832 [3668] (701)	3877 [3713]	3912 c		3884 f	[3712] g [3706] i
	   A'	3712 [3560] (0)	3765 [3612]	3855 c		3807 f
	   A'	988 [775] (0)	977 [764]	843 c	[695] e 980 f [830] e
	   A"	702 [612] (378)	690 [600]	621 c [527(10)] d [554] e 719 f [637] e	[602] h [590] i
	   E'	609 [498] (348)	608 [497]	509 c [410(10)] d [437] e 625 f [540] e	[495] h [477] i
	   E"	507 [427] (0)	495 [415]	421 c [355(10)] d [371] e 500 f [439] e
	   A'	216 [183] (0)	226 [193]	180 c		228 f
	   E'	195 [166] (26)	203 [173]	181 c		209 f	[167] i
	
	D e	5409	5545		5068 c	5391 f
	D 0	3642 [3793]	3755 [3896]	3495 c	[3620] c	3495 f [3645] f

a Ref

[76]

and Ref

[77]

. b Ref

[23]

and Ref

[77]

. c Ref

[76]

. d Approximate fully coupled DQMC calculations. Ref

[76]

. e 6D variational calculations. Ref

[26]

. f Ref

[77]

. g Ref

[36]

. h This work. i Ref

[48]

.

Table 2 :

 2 Molecular parameters (cm -1 ) for the ground state,  6 and  4 states of cyclic (HF) 3 .

	Equ.	GS	 6	 4
	R COM (Å) [2.551] 2.600(1)[2.603] 2.629(1)[2.630] 2.619(1)[2.624]
	(amuÅ 2 )	-0.12(2)	0.04(1)	-0.56(2)

Table 3 :

 3 Vibrationally averaged HF's units center-of-mass separation and inertial defects for (HF)3 . CCSD(T)-F12/AVTZ values are reported in squared brackets.

* Author to whom correspondance should be addressed. Electronic mail:pierre.asselin@upmc.fr.

Acknowledgements

The authors are grateful to SOLEIL and the AILES staff for providing technical support under the proposal 20110648.

Transitions follow the selection rules ∆ = 0, ±1 ; ∆ = 0.Rotational branches for the  4 parallel band will be designated by ( , ) , ( , ) and ( , ) where the superscript correspond to ∆ = 0. For both perpendicular and parallel bands, each branch is composed of a number of sub-bands with ≥ . In what follows, the notation  = ( ′ -)/ is used to designate the change in the rotational constant (= , ) in an excited state.

The statistical weights for (HF) 3 depends on nuclear statistics for a complex which belongs to the 3ℎ symmetry group. The nuclear spins relevant to the (HF) 3 complex are = 1 2 and = 1 2 . Statistical weights have been determined in the same manner as for (C 2 H 2 ) 3 (which possesses two hydrogen atoms of spin 1 2 ) [START_REF] Weber | [END_REF] , one obtains for the GS state = 48 for = 3 and = 40 otherwise.

The SPFIT/SPCAT suite of programs, developed by Pickett, 73 was used for the determination of the molecular spectroscopic parameters.

In contrast with previous infrared studies performed on cyclic trimers, the simultaneous detection of two fully resolved bands of vibration-rotation leads to a complete determination of the ground state constants , , , and as well as the excited state constants ′ , ′ , ′ , ′ and ′ . In the following section, the rotational analysis for each band are presented separately, together with the results derived from the global fit including the complete set of assigned lines belonging to both intermolecular bending modes.

The jet-cooled in-plane bending  band

The rovibrational structure of the fundamental band  6 of (HF) 3 , which extends from 488 to 499 cm -1 , is displayed in Figure 5a. A preliminary assignment was done by assuming a strictly planar ground state structure for the oblate top perpendicular band contour ( = = 2 ). Starting from a prediction based on calculated ground state rotational and quartic centrifugal distortion rotational constants, the clusters of intense lines (see insert in Fig. 3) spaced from 0.26 to 0.34 cm -1 on the low frequency side, and from 0.22 to 0.16 cm -1 on the high frequency side were assigned to ( , ) and ( , ) transitions with ≥ , respectively. The frequency spacing increases on the ( , ) side and decreases on the ( , ) side which is explained by the differences between successive transitions of the type + 1, + 1 and ( , ) , or ( , ) and ( -1, -1) . These differences include the sum of a J-dependentterm almost equal to 2 ′ and K-dependent terms whose sign changes according to ∆ = ±1.

The substructure within each cluster displays a characteristic pattern composed of typically four ( , ) or ( , ) lines equally spaced in frequency whatever the value of .

Such a pattern was assigned for a series of ( -, -2 ) or ( -, -2 ) lines with = 0 -3.

Concerning the ( , )

, lines, these transitions are expected to be significantly less intense than ( , ) and ( , ) ones. They are blended for large values of and but we could nevertheless identify some of them close to the band center for small values of . 135 lines have been finally assigned from which 62 ( , ) lines, 62 ( , ) lines, 4 ( , )