Serena Biella

Massimo Cametti

Tullio Caronna

Gabriella Cavallo

Alessandra Forni

Pierangelo Metrangolo
email: pierangelo.metrangolo@polimi.it

Tullio Pilati

Giuseppe Resnati
email: giuseppe.resnati@polimi.it

Dcmic " Nfmlab

Natta " Giulio

Keywords: Supramolecular halogen bonding, diiodoperfluoroalkanes, azahelicenes, X-ray structure, DFT studies halogen bonding; diiodoperfluoroalkanes, azahelicenes, X-Ray structure, DFT studies, Molecular Electronic Potential (MEP), Molecular Orbital (MO)

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

The self-assembly of 1,8-diiodoperfluorooctane with 4,7,8,11-tetraazahelicene revealed a site-selective pattern of halogen bonds in the solid state. In fact, the N•••I-C bonding formation occurred selectively on the pyridazinic/cinnolinic nitrogen atoms of the tetraazahelicene unit, which were preferred over the pyridinic/quinolinic ones.

DFT calculations predicted and explained well this site-selective halogen bonding formation. The self-assembly of 1,8-diiodoperfluorooctane with 4,7,8,11-tetraazahelicene revealed a site-selective pattern of halogen bonds in the solid state. In fact, the N•••I-C bonding formation occurred selectively on the pyridazinic/cinnolinic nitrogen atoms of the tetraazahelicene unit, which were preferred over the pyridinic/quinolinic ones.

DFT calculations predicted and explained well this site-selective halogen bonding formation.

Introduction

The search for materials endowed with novel properties is strategic for the development of technological applications, yet extremely challenging (1). One of the leading approaches to obtain new materials is self-assembly, which consists of the design of a set of components, i.e. molecular building blocks, which, assemble each to the other in a controlled fashion due to their intrinsic structural and electronic properties [START_REF] Lehn | Supramolecular Chemistry: Concepts and Perspectives[END_REF]. In order to achieve control over the desired architecture, non-covalent interactions, a broad category which encompasses forces of various nature, are used for gluing the molecular building blocks together. For this reason, non-covalent interactions are subjected to meticulous and continuous investigation by the scientific community (3). In fact, the precise understanding of such interactions is a The polarization effect exerted by strong electron-withdrawing groups renders a nearby halogen atom an excellent acceptor of electron density from an electron-rich donor. Such a type of interaction, disregarded only a few years ago, has been recently rediscovered and the term halogen bonding (4) has been introduced for describing any noncovalent interaction involving the positive region of the electrostatic potential surface of halogen atoms [START_REF]An IUPAC Task Group set up to examine the definition of halogen bonding has not yet reported[END_REF]. The general scheme D•••X-Y thus applies to halogen bonding (XB), wherein X is the halogen (Lewis acid, XB-donor), D is any electrondonor (Lewis base, XB-acceptor), and Y is carbon, halogen, nitrogen, etc. The term halogen bonding itself sheds light on the nature of XB, which possesses numerous similarities with hydrogen bonding (HB), wherein hydrogen functions as the acceptor of electron density.

In the last decade or so, XBs have been thoroughly investigated in order to acquire insight on the physical determinants that cement together XB-donors and acceptors [START_REF] Cabot | [END_REF]. Generally, XB is considered to rely on a dominant electrostatic term, but other contributions are certainly present spanning from polarization and dispersion forces to charge-transfer interactions. The iodine atom, given its lower electronegativity and larger polarizability in the halogen series, performs best as a XB-donor, provided that strong electron-withdrawing groups are present in close proximity with the halogen atom.

Scheme 1

Among the various classes of molecules that can interact via XB, linear monoand bis-iodo-perfluoroalkanes, i.e. the 1,8-diiodoperfluoroalkane 1 (Scheme 1), are among the most commonly employed XB-donors (7). Perfluorocarbon residues are strongly electron-withdrawing, and in addition, the -(CF 2 ) n -chain in a perfluoroalkane is significantly more stiff than the -(CH 2 ) n -chain in its hydrocarbon analogue and this confers an extra advantage in terms of preorganization and structural control.

On the other hand, aromatic and aliphatic nitrogen-containing rings and their N-oxide derivatives function as XB-acceptors and have been effectively employed as building blocks for the self-assembly of various X-bonded supramolecular architectures (4). However, little is known on how to control selectivity among Recently we reported an interesting case of site-selective supramolecular synthesis in X-bonded systems, wherein only the pyridinic nitrogen atoms of trans-4,4'azobipyridine were involved in the recognition process (8). In order to increase the number of cases of site-selective XB-formation in polynitrogen systems, we decided to undertake the study of the XB-acceptor behaviour of the tetraazahelicene derivative 2, which presents both pyridinic/quinolinic and pyridazinic/cinnolinic nitrogen atoms (Scheme 1).

Helicenes belong to an intriguing class of polycyclic aromatic compounds that possess a characteristic helical conformation, responsible for the their commonly used name ( 9), as well as a collection of interesting properties, such as large circular dichroism, long triplet lifetime, absorption at low-energy wavelength, and tendency to give π-π stacking. Helicenes are indeed promising molecular species for the development of light emitting devices, sensors, and NLO-active materials. The replacement of one or more carbon atoms with nitrogens, thus forming azahelicenes, may introduce the possibility for these molecules to be involved in site-specific intermolecular interactions while maintaining their properties and their distinctive helicity. For instance, monoazahelicenes have been already shown to be able to coordinate silver cations via quinolinic and isoquinolinic nitrogen atoms positioned in the internal rim of the helicenes, affording 2:1 (helicene/Ag + ) complexes in the solid state (10).

The 4,7,8,11-tetraazahelicene 2 (Scheme 1), having two different sets of nitrogen atoms (pyridinic/quinolinic and pyridazinic/cinnolinic), is an ideal candidate to study the site-selectivity of XB-formation in polynitrogen aromatic systems. For the same reason, 2 was selected also as a good candidate for testing computational methods as tools for predicting the site-selectivity of the XB, whenever a structural ambiguity is present. For all of these reasons the Molecular Electrostatic Potential (MEP) and Moleculat Orbitals (MO) analyses of the tetraazahelicene 2 were carried out by using DFT methods in order to predict the nitrogen sites that would have been involved in the XB-formation following the co-crystallization of 2 with the 1,8diiodoperfluorooctane derivative 1 (Scheme 1).

DFT Calculations

Site-selective XBs have been already reported in several cases (8,11). The four nitrogen atoms of the tetraazahelicene 2 are all supposedly able to take part in strong XBs. While the proton affinity (and likewise gas phase basicity) of pyridine is greater than that of any of the diazines, regarding some weaker electron-acceptors/Lewis acids (e.g. Li + , Na + , and K + ), both experimental measurements and theoretical calculations show pyridine to be a considerably weaker base than pyridazine (12).

This relative tendency to function as donors of electron density of different azines is consistent with other reports on their acid/base behavior and HB-acceptor ability (13).

The two pyridazinic/cinnolinic nitrogen atoms on 2 were thus anticipated to function as preferred sites for the interaction with the XB-donor 1. DFT calculations were prompted to provide more solid grounds to this hypothesis.

As previously highlighted, XB possesses a marked electrostatic character, similarly to HB (14). The MEP analysis, which gives information on what molecules "see" when approaching one to the other, was thought to be extremely informative in order to ascertain the occurrence of a preferential attraction of the XB-donor 1 for either the pyridinic/quinolinic or pyridazinic/cinnolinic nitrogen atoms on 2, prior to the establishing the actual interaction.

A plot of the MEP of 2 onto the surface of electron density with ρ = 0.001 a.u.

is reported in Figure 1 (left). The MEP clearly indicates that the preferential sites of approaching of the iodine atoms are located on the pyridazinic/cinnolinic nitrogen atoms, around which the largest and deepest region of negative electrostatic potential (i.e., red region) is present. On the other hand, smaller and less negative regions of the MEP are located on the pyridinic/quinolinic nitrogen atoms, making them less suitable to attract the incoming diiodoperfluorinated module. For comparison, in Figure 1 the MEPs of the diazahelicenes 3 (centre) and 4 (right) are also reported.

Both show slightly more negative values of the electrostatic potential around, respectively, the pyridinic/quinolinic and the pyridazinic/cinnolinic nitrogen atoms, with respect to the corresponding values obtained for 2, indicating a mutual influence between the two sets of nitrogen atoms, when fused in a single polynitrogen heterocycle (13). Further analysis was carried out on the nature of the frontier orbitals of the tetraazahelicene 2, in particular the Highest Occupied Molecular Orbitals (HOMOs), in order to investigate the possibility of charge-transfer interactions directing, together with the electrostatic contribution addressed above, the establishing of the XB. Such an analysis (see Figure 2) revealed that the HOMO-1 (Figure 2, centre), which is almost degenerate with the HOMO, shows both larger contributions on the nitrogen atoms of the pyridazine moiety and the appropriate symmetry to interact with the incoming iodine atoms. The first MO with similar symmetry but showing larger contributions on the pyridine nitrogen atoms is HOMO-3 (Figure 2, right), which lies in energy well below the HOMO-1. The HOMO (Figure 2, left), while being delocalized over the full molecule, has nodal surfaces in correspondence of the four nitrogen atoms and therefore cannot be responsible for such interaction. In fact, according to a previous investigation aimed at determining the best frontier MOs responsible for a given reaction ( 15), the analysis of the composition and shape of the last occupied molecular orbital represents precisely the best criterion to identify such MOs, which not necessarily coincide with the HOMO.

As expected, the combined results given by the MEP computation, altogether with the MO analysis, firmly indicate that the pyridazinic/cinnolinic nitrogen atoms on 2 are the preferential sites for the eventual acceptance of XBs in a given supramolecular system.

X-Ray Diffraction Analysis

The evaporation at r.t. of a chloroform solution containing a 1:1 mixture of consistent with a weaker XB taking place [START_REF] Bondi | The sum of the VdW radii for N (155 pm) and I (198 pm) is 353 pm[END_REF]. This difference can be easily attributed to the vicinity of the two nitrogen atoms, which inhibits the formation of two geometrically ideal XBs, which would have resulted in the clashing of one of the two perfluorinated modules onto the other.

Figure 3

The X-bonded (1) 3 •(2) 2 pentamers are then tightly packed in the crystal lattice (no solvent molecules included) thanks to a cooperative set of various intermolecular interactions including HB, π-π stacking, and vdW and dispersion forces. In fact, besides the weak HB involving the pyridinic/quinolinic nitrogen atoms, the tetraazahelicene units are densely packed in an antiparallel fashion, showing π-π staking between the external pyridinic/quinolinic rings that maximize dipole-dipole interactions (Figure 4). It is worth to be reminded that perfluorocarbon (PFC)/hydrocarbon (HC) interactions are generally very weak and thus phase separation is often observed in PFC/HC mixtures (19). XB is strong and specific enough to overcome this phase separation to the point of triggering the self-assembly of PFCs and HCs into hybrid co-crystals ( 20). In the system described in this paper, XB directs the self-assembly of the building blocks 1 and 2 into well-defined (1) 3 •(2) 2 pentamers. However, the packing of these pentamers is strongly reminiscent of the low affinity the starting materials have for each other, which results in the formation of segregated layers of perfluorinated and aromatic units, as commonly observed in similar systems (Figure 5).

Conclusions

In conclusion, this study shows that the tetraazahelicene 2 can be successfully used in the XB-mediated assembly of supramolecular architectures. Indeed, the self-assembly of the 1,8-diiodoperfluorooctane 1 with the tetraazahelicene 2 produced a lattice of packed pentameric units (1) 3 •(2) 2 , the monomers of which are connected by two nonequivalent XBs. Interestingly, only the pyridazinic/cinnolinic nitrogen atoms were involved in the formation of XBs with the 1,8-diiodoperfluorooctane 1, resulting in a site-selective supramolecular synthesis mediated by XB. This selectivity parallels the relative HB-basicity of the competing pyridinic/quinolinic and pyridazinic/cinnolinic nitrogen sites towards weaker Lewis acids. In this respect, XB shows the same selectivity found for HB in related polynitrogen heteroaromatics (12,13).

In the described pentameric system (1) 3 •(2) 2 , the self-sorting process mediated by XB favours the modules' binding sites which allows for an optimal phase segregation between perfluorocarbon and hydrocarbon domains on the basis of thermodynamic considerations.

Little is known on how to predict and control selectivity among different supramolecular synthons. In the case reported in this article, DFT calculations were able to predict the observed site-selectivity based on the analysis of both the electrostatic potential and the nature of the HOMOs of the tetraazahelicene 2. These theoretical data suggest a twofold control, by electrostatic and charge-transfer interactions, on the establishing of XB in this system, which selected the pyridazinic/cinnolinic atoms, over the pyridinic/quinolinic nitrogen atoms, as preferred sites for the XB formation with 1. Such DFT approach has a general validity and, therefore, can be applied whenever the site-selectivity of the XB in polynitrogen systems is of concern.

The results described in this article demonstrates that the tetraazahelicene 2 might be used as a new tecton in supramolecular chemistry capable of being involved into a two-fold set of intermolecular interactions showing an orthogonal selectivity towards the two different set of nitrogen atoms present onto 2. Self-assembly processes using the tetraazahelicene 2 and based on simultaneous XB and HB, or XB and metal coordination, are currently in progress.

Experimental Section

Synthesis

The synthesis of the tetraazahelicene

Crystal growth and X-ray diffraction

Computational details

Geometry optimizations of the di-and tetraazahelicenes were carried out in the gas phase with the Gaussian '03 software package (23) at DFT (B3LYP) level ( 24), using the 6-311++G** basis set for all atoms. The basis set for iodine (25) was downloaded from the Basis Set Exchange site (26).

11 -

 11 The self-assembly of 1,8-diiodoperfluorooctane with4,7,8,11-tetraazahelicene revealed a site-selective pattern of halogen bonds in the solid state. In fact, the N•••I-C bonding formation occurred selectively on the pyridazinic/cinnolinic nitrogen atoms of the tetraazahelicene unit, which were preferred over the pyridinic/quinolinic ones. DFT calculations predicted and explained well this site-selective halogen bonding formation. Tetraazahelicene driven by Halogen Bonding Serena Biella a,b ; Massimo Cametti a ; Tullio Caronna c ; Gabriella Cavallo a,b ; Alessandra Forni d ; Pierangelo Metrangolo a,b *; Tullio Pilati d ; Giuseppe Resnati a,b,d *; Giancarlo Terraneo a,b

  the design of functional building blocks and for their full exploitation in materials chemistry.

  donor sites in polynitrogen-containing heterocyclic systems.

  Figure 1

compounds 1 and 2

 2 (16) yielded, after 1 day, good-quality single crystals of composition (1) 3 •(2) 2 , thus offering the opportunity for validating the prediction made by DFT calculations (see above). The single crystal X-ray diffraction analysis revealed several interesting features. As shown in the Figure3, the observed XBs are established only with the pyridazinic/cinnolinic nitrogen atoms, as effectively predicted by the DFT calculations. Each of the two adjacent N atoms, in fact, interacts diiodoperfluoroalkane modules making the tetraazahelicene to function as a bidentate XB-acceptor. The pyridinic/quinolinic nitrogen atoms, not involved in any XB, display, instead, weak H-bond interactions (N•••H-C distances of 2.61 and 2.67 Å) with adjacent tetraazahelicenes (Figure4).Of the two X-bonded diiodoperfluorooctane molecules, one bridges two symmetry-related tetraazahelicenes, while the second one functions as a monodentate XB-donor. Hence, the overall XB-directed assembly, indefinitely repeated over the crystal lattice, is constituted by a pentameric, zigzag shaped, (1) 3 •(2) 2 adduct, i.e. two tetraazahelicenes and three diiodoperfluoroalkanes (Figure3). Both the independent diiodoperfluorooctane molecules display the typical enantiomeric disorder of the perfluorocarbon chains, whereas only the iodine atoms engaged in XB are ordered (17). The two observed XBs are not equivalent. The linear bidentate XB-acceptor molecule experiences the shortest XB (N•••I distance of 2.998 Å, while 168.3° is the mean N•••I-C1(C,D) angle involving the two disordered C1C and C1D atoms), which lays on the pyridazinic/cinnolinic ring plane. Conversely, the other XB is significantly longer (distance 3.293 Å) with a less linear N•••I-C1(A,B) angle (mean value 165.1°),

Figure 4 Deleted

 4 Figure 4

Figure 5

 5 Figure 5

2 MF 9 Å 3 , 2 ,

 2932 Single crystals were obtained by dissolving at room temperature equimolar amounts of the 1,8-diiodoperfluorooctane 1 and the tetraazahelicene 2 in a vial of clear borosilicate glass, using chloroform as solvent. The open vial was introduced in a closed cylindrical glass jar containing paraffine oil. Chloroform was allowed to diffuse at room temperature and after one day the formation of orange crystals was observed. = C 60 H 20 F 48 I 6 N 8 , FW = 2526.24, triclinic, space group P-1, a = 9.897(2), b = 10.763(2), c = 20.010(3) Å, α = 96.23(2), β = 90.90(2), γ = 116.07(2), V = 1898.5Z = 1, D x = 2.210 Mg•m -3 , µ (Mo-Kα) = 2.618 mm -1 ; crystal dimensions 0.44x0.26x0.23 mm 3 , λ = 0.71073 Å (Mo-Kα radiation), graphite monochromator, Bruker SMART-APEX CCD diffractometer. Data collection: ω and φ scan mode, 2θ < 50°; 10790 collected reflections, 10790 unique [6736 with I o > 2σ(I o )], merging R = 0.02530. The structure was solved by SIR2002 (21) and refined by SHELXL (22), fullmatrix least-squares based on F o atoms were in calculated positions. The final consistency index were R = 0.0667 and Rw = 0.1488 [0.0471 and 0.1403 respectively for I o > 2σ(I o )], goodness-of-fit = 1.029. The final map ranges between -0.83 and 0.81 e•Å 3 . Detailed crystallographic data were deposited as CCDC 783798 with the Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, UK and can be obtained free of charge from http://www.ccdc.cam.ac.uk/products/csd/request/.

Scheme 1 .

 1 Scheme 1. Molecular formulae of the XB-donor 1 and acceptors 2-4 mentioned in the text. Pyridinic/quinolinic nitrogen atoms are reported in blue, while the pyridazinic/cinnolinic ones are in red.

Figure 1 .

 1 Figure 1. Plots of the B3LYP/6-311++G** electrostatic potential of the compounds 2 (left), 3 (centre), and 4 (right) mapped on the respective isosurfaces (0.001 a.u.) of electron density. Minimum values of electrostatic potential are -0.0593 and -0.0447 a.u. (on the pyridazinic/cinnolinic and pyridinic/quinolinic nitrogen atoms, respectively, of 2), -0.0527 a.u. (3), and -0.0655 a.u. (4).

Figure 2 .

 2 Figure 2. Surface plot of the B3LYP/6-311++G** HOMO (left), HOMO-1 (centre) and HOMO-3 (right) of the tetraazahelicene 2 (|isovalue|=0.02 a.u.).

Figure 3 .

 3 Figure 3. Spacefill representation of the XB-assembled pentamer (1) 3 •(2) 2 in the crystal (colour code: white: H; gray: C; blue: N; violet: I; yellow: F).

Figure 4 .

 4 Figure 4. HB and π-π staking motifs observed in the crystal structure (the second Hbonded layer of molecules of 2 is shown in red).

Figure 5 .Scheme 1 .

 51 Figure 5. Representation of the segregation into layers in the lattice of the molecular components 1 (spacefill) and 2 (capped sticks).

Figure 1 .

 1 Figure 1. Plots of the B3LYP/6-311++G** electrostatic potential of the compounds 2 (left), 3 (centre), and 4 (right) mapped on the respective isosurfaces (0.001 a.u.) of electron density. Minimum values of electrostatic potential are -0.0593 and -0.0447 a.u. (on the pyridazinic/cinnolinic and pyridinic/quinolinic nitrogen atoms, respectively, of 2), -0.0527 a.u. (3), and -0.0655 a.u. (4).

Figure 2 .

 2 Figure 2. Surface plot of the B3LYP/6-311++G** HOMO (left), HOMO-1 (centre) and HOMO-3 (right) of the tetraazahelicene 2 (|isovalue|=0.02 a.u.).

Figure 3 .

 3 Figure 3. Spacefill representation of the XB-assembled pentamer (1) 3 •(2) 2 in the crystal.

Figure 4 .

 4 Figure 4. HB and π-π staking motifs observed in the crystal structure (the second Hbonded layer of molecules of 2 is shown in black).

Figure 5 .

 5 Figure 5. Representation of the segregation into layers in the lattice of the molecular components 1 (spacefill) and 2 (capped sticks).

  2 is an optimized version of the procedure reported by R. Huisgen in Justus Liebigs Annalen der Chemie 1948, 559, 101-152.

	Compound 2: 1 H NMR (250MHz, CDCl 3 ): δ 9.11 ( 2H, dd, J 1 =1.5Hz, J 2 = 4.2 Hz), δ
	9.00 (2H, d, J=8.75 Hz), δ 8.84 (2H, d, J=9.25Hz), δ 8.50 (2H, d, J=9 Hz), δ 7.42 (2H,
	m); 13 C NMR(63MHz): δ 152.62, 149.74, 146.04, 136.07, 132.71, 130.80, 123.17,
	120.12, 119.25; ESI Mass: calc. for C 18 H 10 N 4 282, found 283 [M+H] + .

URL: http:/mc.manuscriptcentral.com/tandf/gsch Email: suprachem@mail.cm.utexas.edu Supramolecular Chemistry

Acknowledgements

This work was funded by Fondazione Cariplo (''New-Generation Fluorinated Materials as Smart Reporter Agents in 19 F MRI'') and MIUR (''Engineering of the Self-assembly of Molecular Functional Materials via Fluorous Interactions'').


Francesco Freddi

Michel Fremond

Michel Frémond

Collisions and fractures: a predictive theory

Keywords: collision, principle of virtual work, fracture, damage

We investigate collisions of solids which can fracture. Equations of motion and constitutive laws provide a predictive theory. Assuming the collision as instantaneous, the equations of motion are derived from the principle of virtual work introducing new interior forces which describe the very large stresses and the very large contact forces resulting from the kinematic incompatibilities. They are interior volume percussion stresses and interior surface percussions both on the unknown fractures and on the colliding surface. In order to approximate these equations, we assume solids are damageable. In this point of view, it results that velocity is continuous with respect to space but its strain rate is very large in a thin region where the material is completely damaged, so approximating a fracture. When the velocity before collision is very large, the damaged zone may be large accounting for parts of the solid completely transformed into powder. The constitutive laws result from dissipative functions satisfying the second law of thermodynamics and able to model the fracturation phenomenon at the macroscopic engineering level. Representative numerical examples confirm that the model accounts for the fracturation qualitative properties.

Introduction

Collisions and fractures of solids are important engineering issues, for instance, fragmentation of solids by blasting, by collisions,... [START_REF] Fourney | Mechanisms of rock fragmentations by blasting[END_REF], [START_REF] Romero | Fragmentation du béton sous explosion[END_REF]. Besides, the demand for impact resistant design of structures crosses a wide spectrum.

The present work deals with collisions of deformable and damageable solids. In particular, we investigate the appearance of fractures due to impact phenomena. Collisions are phenomenona which occur in a very short period of time, involving large impulsive forces in the bodies. We limit the investigation of this problem at the engineering macroscopic level and derive a model by discontinuum mechanics theory, [START_REF] Frémond | Collisions[END_REF]. We suppose that collision events are very short if compared with the duration of the other phenomena. Thus, we assume the collisions are instantaneous according to [START_REF] Frémond | Non-smooth thermomechanics[END_REF] and [START_REF] Frémond | Collisions[END_REF]. A collision is characterized by a time discontinuity of the velocity field: there is a smooth velocity before collision U -and the velocity U + after collision. This collision theory applies to collision of either rigid or deformable solids, [START_REF] Frémond | Collisions[END_REF] as well as of a solid colliding with an incompressible fluid, [START_REF] Frémond | Collision of a solid with an incompressible fluid[END_REF].

As a consequence of the collision fractures may appear in the solid. A fracture is characterized by a spatial discontinuity of the velocity after collision U + . We are not interested in the evolution of a fracture, but on its formation and this phenomenon is considered as instantaneous. Hence, we introduce different dissipative potentials as well as suitable unilateral conditions on fractures and along the boundary.

The predictive theory is based on the principle of virtual work giving the equations of motion and constitutive laws relating the internal forces to pertinent quantities describing the evolution. The constitutive laws introduce interior percussion stresses 1 , and percussions which result from the kinematic incompatibilities. They are interior volume percussion stresses and interior surface percussions both on the unknowns fractures and on the colliding surfaces. The constitutive laws are derived with dissipative potentials which fulfill the second law of thermodynamics.

In order to solve numerically the resulting set of partial differential equations (reference is made to [START_REF] Bonetti | Collisions and fractures: a model in SBD[END_REF] and [START_REF] Frémond | Collisions[END_REF] for details), we assume solids are damageable and that a damage quantity β has value 1 when the material is undamaged and value 0 when is completely damaged. We suppose that the damage quantity evolves rapidly in the collision: thus we assume it is discontinuous with respect to time at collision time: β -before collision and β + after collision. In this point of view, we may assume the velocity U + is continuous with respect to space but its strain rate is very large in a layer where the damage quantity β + is almost zero, i.e., a fracture is approximated by a thin damaged zone where β + 0, (damage field is 1 where the material is not fractured). When the velocity before collision becomes very large, the damaged zone may be large accounting for parts of the solid completely transformed into powder where cohesion is totally lost.

The variational formulation of the equations of motion is discretized by the classical finite element technique. Moreover, kinematic constraints on the velocity field U are introduced in order to avoid overlapping phenomena along the contact surface and in the volume damaged zones. Finally, the discrete solution is obtained as a minimization of a non convex problem, reached through a specific numerical strategy that couples gradient descent technique [START_REF] Ciarlet | Handbook of numerical analysis[END_REF], [START_REF] Snyman | Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms[END_REF] and the Uzawa's method [START_REF] Zulehner | Analysis of iterative Methods For Saddle Point Problems: A Unified Approach[END_REF]. This scheme permits to account for the duality variables introduced by the constraints, for instance the impenetrability condition.

Several numerical simulations are proposed. Their aim is to demonstrate the capabilities of the predictive theory to describe qualitatively different failure modes occurring at collision time.

The direct approach

At collision time, the predictive theory is based on the principle of virtual work giving the equations of motion and constitutive laws relating the internal forces to pertinent quantities describing the evolution. The equations of motion introduce interior percussion stresses, and percussions which result from the kinematic incompatibilities. They are interior volume percussion stresses and interior surface percussions both on the unknowns fractures and on the colliding surfaces2 .

For the sake of simplicity, let us consider a deformable solid Ω colliding a rigid fixed obstacle on ∂Ω 1 . Due to the kinematic incompatibilities very large interior forces appear within the solid, along the contact surface and on the unknown fracture surfaces. We consider the duration of the collision very short, i.e., the time for the velocities to adapt to the presence of the obstacle is extremely short. Thus we assume the collision as instantaneous (time t is fixed).

Very large interior forces become percussion stresses Σ and percussions R. The percussions and percussion stresses can be thought as time concentrated quantities which define an atomic measure (a Dirac measure) intervening only at collision times. The different terms in the expression of the virtual work are linear functions of the virtual velocities. They involve the percussions and percussion stresses.

The system we consider at collision time is made of a solid Ω and an immobile obstacle ∂Ω 1 which are in contact.

The system interior virtual work we choose is a linear function of the virtual strain rate and reads

T int V = - Ω\Γ Σ : E V + + V - 2 dΩ - ∂Ω 1 R • V + + V - 2 dΓ + Γ R • V + + V - 2 dΓ , ( 1
)
where E( V ) = ∇ sym V is the classical symmetric strain rate tensor, V + and V -are virtual velocity fields. The fracture Γ is oriented: thus we consider a "left" part ( V l ) and a "right" part ( V r ) of the velocity field. So, the spatial velocity discontinuity i.e., the fracture velocity of deformation, is denoted by

V = V r -V l .
Because we assume the obstacle to be immobile, velocity V on contact surface ∂Ω 1 is the gap velocity, i.e., the system deformation velocity on ∂Ω 1 .

The densities with respect to the Dirac measure are percussion stress tensors and interaction percussions between the solid and the obstacle as well as percussions between fracture surfaces. Percussions are generalized interior forces which appear when collisions occur. They are, as said earlier, usual interior forces concentrated in a very short period of time.

The virtual work of the acceleration forces is

T acc V = Ω ρ U • V + + V - 2 dΩ , ( 2
)
where ρ is the solid density and [ U ] = U + -U -is the velocity discontinuity in the collision and ρ[ U ] is the collision inertial percussion. Let us note that the actual work of the acceleration forces is equal to the variation of the kinetic energy during collision.

The equations of motion

The principle of virtual work, [START_REF] Frémond | Collisions[END_REF] ∀ V ,

T acc V = T int V ,
gives the following equations of motion

ρ( U + -U -) = divΣ, in Ω\Γ, {Σ} • N = 0, Σ • N = -R, on Γ, Σ • N = -R, on ∂Ω 1 , Σ • N = 0, on ∂Ω\∂Ω 1 . ( 3
)
where vector N represents either the normal vector to fracture Γ directed from the left toward the right or the outward normal vector to domain Ω.

The constitutive laws

The constitutive laws are defined with dissipation functions: volume dissipation function Φ, fracture dissipation function Φ s and contact dissipation function Φ 1 . The dissipative functions are

Φ(E U + + U - 2 ) = k 0 E( U + + U - 2 ) + k 1 E( U + + U - 2 ) 2 , Φ s ( U + + U - 2 ) = √ 2k 2 U + + U - 2 + k 3 U + + U - 2 + I + U + + U - 2 • N ,
where N is the normal vector to the fracture and

Φ 1 ( U + + U - 2 , U - 2 ) = I -( U + • N ),
differently here N represents the outward normal vector to domain Ω. For simplicity, we used these symbols to denote the length of a vector

| x| = √ x i x i , (4)
and the norm of a symmetric tensor

A A = A ij A ij = √ A 2 . ( 5
)
The k's are positive material constants. Volume dissipative function is a pseudo-potential of dissipation, [START_REF] Moreau | Sur les lois du frottement, de viscosité et de plasticité[END_REF], [START_REF] Halphen | Sur les matériaux standards généralisés[END_REF]. It insures a classical behaviour outside the fracture. Fracture dissipation function is not a pseudo-potential of dissipation. It implies that the behaviour is not progressive: a small collision does not produce a small fracture and a large collision does not produce a large fracture. On the contrary a small collision does not produce a fracture and a large collision produces a fracture.

There is a threshold in the behaviour which results from function

√ 2k 2 ( U + + U -)/2 .
There is a competition between smooth strain rate measured by k 0 E(

U + + U - 2
) and non smooth strain rate measured by √ 2k 2

U + + U - 2 . Thus if k 2 is large compared to k 0 it is difficult to have
fractures and if it is small, fractures appear easily. Thus parameter k 2 is a threshold for fractures to appear. Moreover, the latter function avoids the appearance of too many fractures characterized by small velocity discontinuities (square root function, with property √ a + b ≤ √ a + √ b, forbids a fracture with amplitude a + b to be replaced by two small fractures with amplitudes a and b).

The functions

I -, I + are the indicator functions of ] -∞, 0] = R -, (I -(γ) = 0, if γ ≤ 0 and I -(γ) = +∞, if γ > 0), and of [0, ∞[= R + , (I + (γ) = 0, if γ ≥ 0 and I + (γ) = +∞, if γ < 0),
(see [START_REF] Moreau | Fonctionelles convexes[END_REF]) respectively. Indicator function I + takes into account the non-interpenetration condition along the fracture

U + • N ≥ 0. (6)
Function Φ 1 introduces the non-interpenetration condition on contact surface ∂Ω 1 . Function Φ s is split in a convex part, Φ c s and a differentiable non convex part, Φ nc s . Its generalized derivative is the sum of the sub-differential of the convex part and of the extended derivative of the non convex part

∂Φ s ( U + + U - 2 ) = ∂Φ nc s ∂ U + + U -/2 ( U + + U - 2 ) + ∂Φ c s ( U + + U - 2 ).
The constitutive laws are

Σ ∈ ∂Φ(E( U + + U - 2 )), ( 7
) -R ∈ ∂Φ s ( U + + U - 2 ), on Γ, R ∈ ∂I -( U + • N ), on ∂Ω 1 , (8)
with

∂Φ s X = ∂Φ nc s ∂ X X + ∂Φ c s X =      k 2 √ 2 | X| + k 3 X | X| , if | X| > 0, R 3 , if X = 0 .
Let us note that when discontinuity ( U + + U -)/2 is null, percussion R may assume any value, this value being given by relationship (3), R = -Σ• N . In this situation, there is no fracture and the volume constitutive law ( 7) is valid and conveys the mechanical information. Fracture constitutive law ( 8) is also valid but conveys no mechanical information. The mechanical aspects of collisions have an important influence on thermal behavior but the thermal aspects have a limited influence on the mechanical behavior. Thus it is reasonable to assume the temperature does not intervene in the mechanical equations. Following [START_REF] Frémond | Collisions[END_REF], the thermal equations involving the dissipated work may be solved to get temperature T + after collision.

The direct problem

It may be proved that there are velocities after collision which minimize a functional inf

F( V ) V ∈ SBD(Ω), V = 0 outside ∂Ω 1 , (9)
over the set SBD(Ω) of the special bonded deformation velocities (those functions are smooth outside the fractures and discontinuous on the fractures) [START_REF] Ambrosio | Special functions of bounded variations and free discontinuity problems[END_REF], [START_REF] Ambrosio | Fine properties of functions with bounded deformation[END_REF], [START_REF] Attouch | Variational analysis in Sobolev and BV spaces. Application to PDE and optimization[END_REF], [START_REF] Braides | Approximation of free-discontinuity problems[END_REF]. The functional F( V ) is

F( V ) = Ω\Γ Ξ(E( V + U -))dΩ + Γ Ξ s ( V + U -)dΓ + ∂Ω 1 Ξ 1 ( V )dΓ + Ω ρ 2 ( V ) 2 -ρ V • U -dΩ,
where functions Ξ, Ξ s and Ξ 1 are defined according functions Φ, Φ s and Φ 1

Ξ(E( V + U -)) = k 0 E( V + U -) + k 1 2 E( V + U -) 2 , Ξ s ( V + U -) = 2k 2 V + U -+ k 3 V + U -+ I + V + U -• N , Ξ 1 ( V ) = I -V • N .
This problem has been analyzed in [START_REF] Bonetti | Collisions and fractures: a model in SBD[END_REF]. It was proved that problem (9) has solutions which give both the fractures (which are a priori unknown) and the velocities. Besides, in [5] a 1-D example is analyzed and closed form solutions are described.

The regularized approach

Numerical solution of the previous problem presents a major difficulty due to the unknown positions of fracture surfaces Γ which are free discontinuities. So, we propose a regularized approach in the context of fracture damage mechanics. In particular, we introduce a spatial damage variable β ∈ [0, 1] that approximately describes the zone where the material is fractured. In fact, β = 1 represents the sound material while β = 0 is equivalent to the completed damaged state. The main idea, firstly proposed in [START_REF] Frémond | Adhérence des solides[END_REF] and [START_REF] Frémond | Damage, gradient of damage and principle of virtual power[END_REF], is based on adaptation of the principle of virtual work. In particular, we assume that damage results from microscopic motions, and includes the work of these motions in the principle of virtual work. This contribution is assumed to depend on the rate of damage and on the rate of the damage gradient. This last term is introduced to account for the local interaction of the damage at a material point on the damage of its neighborhood. So, the internal virtual work (1) is replace by

T int V , b = - Ω Σ : E V + + V - 2 dΩ - ∂Ω 1 R • V + + V - 2 dΓ - Ω B [b] + H • grad [b] dΩ , ( 10
) being [b] = b + -b -, where V + , V -and b + , b -are virtual velocities.
The first and the second terms have already been introduced and are related to the mechanical action and reactions. The third and fourth terms are unusual: B is a mechanical work, in particular the internal damage work which is responsible for the evolution of the damage during the collision and H is a flux vector responsible for the interaction of the damage at a point on the damage on its neighborhood.

The equations of motion

The principle of virtual work

∀ V , b T acc V = T int V , b , (11)
gives two sets of equations of motion. By choosing convenient virtual velocity, we get

ρ U = divΣ, in Ω, Σ • N = -R, on ∂Ω 1 , Σ • N = 0, on ∂Ω\∂Ω 1 , (12)
and div H -B = 0, in Ω,

H • N = 0, on ∂Ω, ( 13
)
where N is the outward normal to Ω. The equations ( 12)-( 13) account for macroscopic and microscopic effects.

The constitutive laws

In our problem we assume that the velocity U -, supposed for the sake of simplicity to be a rigid body velocity, and the damage β -before the collision are assigned and let the unknowns be the velocity U + and the damage β + after the collision. The constitutive laws, which have to satisfy the second law of thermodynamics, for the mechanical and damage interior forces are defined by three functions: the volume free energy Ψ, the volume dissipative function Φ and along the contact surface with the obstacle pseudo-potential of dissipation Φ s . We have to choose them in such a way that β + is zero in thin layers and 1 elsewhere and that the strain rate is very large in the thin layer and almost zero else where. We choose the volume free energy as

Ψ (ε, β, gradβ) = w (1 -β) + k 2 (gradβ) 2 + β 2 εCε + I (β) . ( 14
)
The volume dissipative function is addressed as follows

Φ E U + + U - 2 , [β] , grad [β] = c 2 [β] 2 + µ q β -+ [β] 2 E U + + U - 2 q + + λ r E U + + U - 2 r + I -([β]) + I + div U + + U - 2 . ( 15
)
The contact surface with the obstacle pseudo-potential of dissipation is

Φ s U + + U - 2 = λ s U + + U - 2 2 + I -U + N , ( 16
)
where

U N = U • N is the normal velocity.
Quantity w is the cohesion of the material, C is the classical elasticity tensor, ε the deformation tensor which does not change during collision and k is the damage coefficient which quantifies the influence of the damage at a point onto the damage of its neighborhood. The extension parameter k control the size of the transition zone between sound material and damaged material. If k is large, damage is diffuse and spread in the whole domain. Contrarily, if k is small, the damage is concentrated in thin zones which may represent fractures. The values of k can be measured with structure experiments, but not with sample experiments where the state quantities are homogeneous.

The parameters c, λ, µ, λ s characterize the microscopic and macroscopic dissipations. The exponents r ∈ ]0, 1[ and q ∈ [START_REF] Arnold | Approximation by quadrilateral finite elements[END_REF][START_REF] Ambrosio | Special functions of bounded variations and free discontinuity problems[END_REF] indicate the nature of the behaviour of the constitutive law: convex or concave. In particular, the convex term involving the strain rate E consists of the viscoplastic Norton-Hoff-Friaa, [START_REF] Friaa | La loi de Norton-Hoff généralisée en plasticité et en viscoplasticité[END_REF] potential which approximates the perfect plastic potential as q tends to one. Moreover, the effect of the concave term with power r lower than 1 is to avoid having many regions with small discontinuities and to have the strain rates very large in thin damaged zones.

The function I is the indicator function of the intervals [0, 1], (I(γ) = 0, if 0 ≤ γ ≤ 1 and

I(γ) = +∞, if γ / ∈ [0, 1]
). The indicator functions I, I -, I + take care of the internal constraints on the damage variable and on the velocity field

β ∈ [0, 1] ; [β] ≤ 0; U + N ≤ 0, on ∂Ω 1 ; div U + + U - 2 ≥ 0 in Ω . ( 17
)
The first internal constraint results from the definition of damage. The second constraint means that the solid can not mend during the collision. The other constraints are related to the velocity field and take into account impenetrability conditions: the first one is on the obstacle contact surface and the last one, following an idea of Jean Jacques Moreau [START_REF] Moreau | Principes extrêmaux pour le problème de la naissance de la cavitation[END_REF], replaces the impenetrability condition on the fractures by a volume non overlapping condition. This relationship results from mass balance relationships div U -= 0 and div U + = 0 in the non fractured domain (assuming the density to be constant) and from impenetrability condition [START_REF] Bonetti | Collisions and fractures: a model in SBD[END_REF],

U + • N ≥ 0.
The previous functions give the following set of constitutive laws

Σ = λ E U + + U - 2 r-2 E U + + U - 2 (18)
+µ

β -+ β + 2 E U + + U - 2 q-2 E U + + U - 2 -pI , B ∈ -w + 1 2 εCε + ∂I β + + c [β] + µ 2q E U + + U - 2 q + ∂I -([β]) , (19)
H = k gradβ + , ( 20
)
R ∈ λ s ( U + + U -) + ∂I -U + N N , (21)
where

-p ∈ ∂I + div U + + U - 2 . ( 22
)
The subdifferential sets ∂I + and ∂I -of the indicator functions of the positive and negative numbers R + and R -are defined as: ∂I + (0) = R -, ∂I + (γ) = 0 for γ > 0 and ∂I + (γ) = ∅ for γ < 0, ∂I -(0) = R + , ∂I -(γ) = 0 for γ < 0 and ∂I -(γ) = ∅ for γ > 0. The quantity ∂I -U + N N is the impenetrability percussion reaction. It is active only if the other interactions are not sufficient for the solid and the obstacle not to interpenetrate, i.e., when the normal velocity after the collision is 0. The quantity -p is the internal impenetrability pressure. This reaction prevents the volume overlapping. From the numerical simulations proposed in section 4 clearly outcomes that -p is active only in the fractured zones in compression or in the fragmented 3 regions, avoiding interpenetration. In fact, the undamaged portions of the solid after collision have nearly rigid body velocity, in agreement with the rigid-plastic component of the constitutive law.

Let us note that the second law of thermodynamics equivalent to the following inequality is satisfied for all admissible values of r and q, [START_REF] Frémond | Collisions[END_REF] Σ

: E U + + U - 2 + B [β] + H • grad [β] -[Ψ] ≥ µ β -+ β + 2 E U + + U - 2 q (23) +λ E U + + U - 2 r -w [β] + c [β] 2 + µ 2q [β] E U + + U - 2 q + ∂I -([β]) [β] +∂I β + [β] + k gradβ + • grad [β] ≥ 0 .

A simplified problem: the equation

The principle of virtual work and a proper use of the constitutive laws lead to two sets of equations and permit to compute the velocity and damage of the body after the collision depending on the incoming velocity and the damage state before the collision.

For simplicity, we assume the body undamaged before the collision, that its velocity is a rigid body velocity and that it is not deformed before collision

β -= 1, E( U -) = 0, ε = 0 . ( 24
)
Moreover, assuming [START_REF] Zulehner | Analysis of iterative Methods For Saddle Point Problems: A Unified Approach[END_REF] it results

∂I(β

+ ) + ∂I -(β + -1) = ∂I(β + ) . ( 25
)
So, the equations needed to find U + and β + are

ρ U + -divΣ( U + ) = ρ U -, in Ω, ( 26
)
Σ • N + λ s ( U + + U -) + ∂I + (U + N ) N 0, on ∂Ω 1 , ( 27
) Σ • N = 0, on ∂Ω\∂Ω 1 , ( 28
)
cβ + -k β + + ∂I(β + ) w + c - µ 2q E U + 2 q , in Ω, ( 29
)
∂β + ∂N = 0, on ∂Ω. ( 30
)
3 The difference between fractured and fragmented states consists in the fact that in the first case cracks are thin and well separated while in the second case the rupture is completely diffused and well defined cracks cannot be appreciated.

Numerical examples

In this section three numerical experiments are presented. Velocity U -is given and velocity U + is computed. Note that there is no time step because collision is instantaneous. Let us emphasize that we do not try to compare our results with those of actual experiments (even if the last example shows that the theory describes at the qualitative level an actual experiment) but we want to illustrate the capability of the proposed model to describe different failure modes.

After collision the material may present three different states: not fractured, fractured and fragmented. In details, the not fractured region is represented by the undamaged zone (β = 1), the fractured state presents completed damaged materials with an opening mode (i.e., div U + > 0) or in compression regime (i.e., div U + = 0) while the fragmented regions are diffused damaged zones mainly due to very high strain rate.

The discrete solutions, obtained via finite element method, involves the resolution of a nonlinear non convex minimization problem. The solution is obtained by an iterative calculation with a sequence of decreasing parameter q → 1 starting from a quadratic convex term (q = 2). Specially, the calculation can be stopped at any iterative step and restarted from the previous solution even with an updated parameter q. This technique demonstrates to considerably speed up numerical convergence. Kinematical constraints are introduced by duality via the well known technique of the Lagrange multipliers. An ad-hoc iterative Uzawa algorithm has been developed for the resolution of the resulting saddle point problem due to the constraint inequalities.

Moreover, we restrict our attention to the case of a 2-dimensional problems in plane strain condition of a single deformable body coming in contact with a rigid, fixed obstacle. The discretization is composed by quadrilateral finite element characterized by linear displacement and constant pressure inside the element, i.e. the so called Q1 -P 0 finite element [START_REF] Braides | Mixed and Hybrid Finite Element Methods[END_REF], [START_REF] Arnold | Approximation by quadrilateral finite elements[END_REF] We also report a mesh sensitivity analysis in order to prove mesh objectivity of the predictive theory and of the numerical strategy adopted.

Falling Bar

We consider the case of a slender rectangular bar colliding three rigid supports. Prior the collision, the bar is assumed to be undamaged (β -= 1) and falls with a rigid body vertical descending velocity (see Figure 1). The predictive theory gives the velocity and the damage after collision, U + ( x) and β + ( x), depending on falling velocity U -. Two representative cases have been analyzed: all the parameters are fixed except the density of the material thus an heavy material and a light material, having the same resistance, have been considered (a ratio between the material densities equal to 10 has been adopted).

Figure 1: Bar falling onto three rigid supports. The length of the bar is L = 4000 mm and ratio L/h = 20 and l/h = 0.8 have been assumed. The other parameters have the following value: µ = 5.e2 M P a, λ = 1.e2 M P a, λ s = 1.e2 M P a/mm ρ = 3.e -5 N/mm 3 or 3.e -6N/mm 3 , p = 0.5, q = 1.01, c = 1.e3 M P a s, k = 0.1 M P a mm 2 , w = 2.e3 M P a.

Figure 2 shows the damage field β for different incoming velocities for the case of the heavy material. From Figure 2 clearly outcomes that the failure process can be classified with three states depending on the amount of imparted energy proportional to ( U -) 2 : not fractured, fractured and fragmented states with rather sharp transitions in between. For a value of U -smaller than 1 m/s, no fracture appears in the solid. For 1.5625 ≤ U -≤ 3.125 m/s well defined fractures appear in the solids. It may be noted that the fractured portions of solid in compression are larger than those in traction. Intermediate incoming velocity value 6.25 ≤ U -≤ 12.5 m/s outlines the coexistence of the fractured and fragmented states. In fact, if fragmented regions appear after collision above the rigid supports, well defined fractures are still present inside the solid. Differently, high values of the impact velocity do not allow for the creation of well defined fractures and only fragmentation due to the large percussion transmitted between the solid and the rigid obstacles is present. It is important to note that the fragmented parts of the solids do not change significantly passing from U -= 25 m/s to U -= 50 m/s even if the kinetic energy becomes four times higher. This fact is shown by the representation of the horizontal velocity along two straight horizontal lines located at different positions reported in Figure 4. Strong discontinuities in the velocity behaviour are localized in the fractured regions while the remaining pieces of the solid undergo a nearly rigid body motions. Moreover, the fractures in opening mode are outlined by the representation of the divergence of the velocity field reported in Figure 5. Similar considerations maybe done for the case of U -= 3.125 m/s reported in Figures 6 and7 except by the fact that the bar breaks into seven pieces.

In Figure 8 is represented the damage field β for different incoming velocities for the case of the light material. In Figures 9-12 are reported the velocity fields and div U + for two different values of falling velocity: U -= 6.25, 12.5 m/s. The main differences with respect to the heavy material case can be summarized as follow:

• because of minor weight of the body, rebounds occur on the obstacles. This a general property of collisions both theoretical and experimental, [START_REF] Frémond | Collisions[END_REF];

• the central crack is completely different in this case: it is induced by rebound of the central portion;

• for high velocity values the light body breaks in several parts while the heavy material is cut into two elements.

Mesh objectivity

In this example we show the lack of mesh sensitivity of the predictive theory. The case of the heavy bar falling at a velocity of 5 m/s is solved with different meshes to investigate whether or not the solutions converge. The computations are carried out for three structured meshes with different element sizes h, h/2 and h/4 starting from the initial value h = 20 mm. The damage field evaluated for the three meshes is reported in Figure 13 showing no mesh sensitivity with regard to the global solution of the problem and convergence to a solution is clearly represented. The velocity field after collision presents negligible differences.

Falling annular disk

The proposed model has been applied to the ideal case of an annular disk falling with a rigid vertical velocity and colliding with a fixed rigid floor. The annular disk has been decomposed with 20000 finite elements. Figures 14 shows the damage field, the divergence of the velocity field U + after the collision and the horizontal and the vertical components of the velocity field. We only underline the fact that two failure zones appear in the ring after collision. A first major fragmented zone is above the contact surface and involves the entire thickness of the annulus (see Figure 15) while in the opposite part a well defined fracture is produced along the vertical direction due to a small but conspicuous discontinuity of the horizontal velocity field as outlined in Figure 16.

Impact simulation

As a third example we reproduced qualitatively the impact tests reported in [START_REF] Zineddin | Dynamic response and behavior of reinforced concrete slabs under impact loading[END_REF]. They illustrated different impact tests on reinforced concrete slabs with the machine reported in Figure 17. The aim of this example is to show the capability of the proposed model to simulate actual experiments even if we do not compare numerical results with experimental evidences. First of all, we consider a simplified 2D example composed of 18000 finite elements. Left and right extremities are clamped while the colliding mass is replaced by an external percussion applied on the upper surface of the slab. Moreover, we do not take into account the metallic reinforcements. Figure 18 shows the damage field, the divergence of the velocity field U + after the collision and the horizontal and vertical components of the velocity field. Different failures are outlined from the numerical result in accordance with actual experiment. In fact, minor and major cracks are evidenced by the map of the velocity field reported in Figure 19.

Conclusions

In this predictive theory for collisions and fractures only macroscopic quantities are involved. The equations of motion are derived from the principle of virtual work where new interior forces are introduce to describe the very large stresses and the very large contact forces resulting from the cinematic incompatibilities. The theory of collisions and fracture of solids outlined in this paper is consistent from the mechanical point of view and has good mathematical formulations to which adapted numerical methods may be applied. In the adopted collision theory the position of solid does not change but the velocity changes instantaneously. The acceleration at collision time is the velocity discontinuity. The constitutive laws which appear in this non smooth situation sum up the very sophisticated and rapid phenomena which occur in collision. Some numerical simulations have been performed via the classical finite element method, concerning a solid colliding against a rigid fixed obstacle or a clamped solid subjected to high velocity impact. The numerical results outlined multiple failure modes. Let us also stress that few parameters are involved, less than 9, to predict at the engineering level complex solid fracturation. x and U + y of the velocity field after collision. A zoom of the horizontal component at the top of the disk is shown in Figure 16.

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 x and U + y of the velocity field after collision.

Figure 2 :

 2 Figure 2: Damage in a heavy bar falling at different velocities. The blue zone is not damaged. The thin red zones are damaged. They account for fractures. When the falling velocity is very large, the damaged zones become completely fragmented domains.In Figure3the velocity field U + is reported for the case U -= 1.5625 m/s. The bar is broken into five pieces after the collision. A central block stops above the central obstacle while instantaneous rigid rotations in the other block are induced by fracture creations.

Figure 3 :

 3 Figure 3: Heavy bar: velocity field U + after the collision for U -= 1.5625 m/s. The bar is broken into five pieces which do not bounce on the obstacles.

Figure 4 :

 4 Figure 4: Heavy bar: horizontal velocity U + x along two straight horizontal lines located at different positions for U -= 1.5625 m/s. The vertical distances y are measured from the bottom of the beam with thickness h = 0.2 m. The strong discontinuities in the velocity values outlined the fractures while the remaining pieces of the solids undergo a nearly rigid body motions. Note the null velocity of the central piece on the central obstacle.

Figure 5 :

 5 Figure 5: Heavy bar: divergence of the velocity vector after the collision for U -= 1.5625 m/s. The opening fractures are represented by zone with high values of div U + .

Figure 6 :

 6 Figure 6: Heavy bar: velocity U + after the collision for U -= 3.125 m/s. The bar is broken into seven pieces.

Figure 7 :

 7 Figure 7: Heavy bar: divergence of the velocity vector after the collision for U -= 3.125. The opening fractures are represented by zone with high values of div U + .

Figure 8 :

 8 Figure 8: Damage in a light bar falling at different velocities (The damage scale depends on the velocity). The blue zone is not damaged. The thin red zones are damaged. They account for fractures. When the falling velocity is very large, the damaged zones become completely fragmented domains.

Figure 9 :

 9 Figure 9: Light bar: velocity U + after the collision for U -= 6.25 m/s. The bar is broken into four pieces which bounce on the obstacles.

Figure 10 :

 10 Figure 10: Light bar: divergence of the velocity vector after the collision for U -= 6.25 m/s. The opening fractures are represented by zone with high values of div( U + ).

Figure 11 :

 11 Figure 11: Light bar: velocity U + after the collision for U -= 12.5 m/s. The bar is broken into four pieces.

Figure 12 :

 12 Figure 12: Light bar: divergence of the velocity vector after the collision for U -= 12.5 m/s. The fractures are represented by zone with high values of div U + .

Figure 13 :

 13 Figure 13: Heavy bar: damage field for three meshes after the collision for U -= 5 m/s.

Figure 14 :

 14 Figure14: Damage field β + and divergence of the velocity field div U + after collision. The adopted parameters have the following value: µ = 8.e1 M P a, λ = 1.e1 M P a, λ s = 2.e3 M P a/mm, 3.e -6N/mm 3 , p = 0.5, q = 1.01, c = 1.e3 M P a s, k = 0.1 M P a mm 2 , w = 1.e1 M P a.

Figure 15 :

 15 Figure 15: Horizontal and vertical components U +x and U + y of the velocity field after collision. A zoom of the horizontal component at the top of the disk is shown in Figure16.

Figure 16 :

 16 Figure 16: Zoom of the horizontal component at the top of the disk.

Figure 17 :

 17 Figure17: Precision impact testing system proposed in[START_REF] Zineddin | Dynamic response and behavior of reinforced concrete slabs under impact loading[END_REF].

Figure 18 :

 18 Figure18: Damage β + and velocity field U + after collision. The adopted parameters have the following value: µ = 5.e2 M P a, λ = 1.e1 M P a, ρ = 2.5e -5 N/mm 3 , p = 0.5, q = 1.01, c = 3.e3 M P a s, k = 0.1 M P a mm 2 , w = 1.e1 M P a.

Figure 19 :

 19 Figure 19: Horizontal and vertical components U +x and U + y of the velocity field after collision.

Percussions are intended here as the variation of stress state or forces occurring during collision.

It is usual to derive the equation of motion through the principle of virtual power,[START_REF] Frémond | Non-smooth thermomechanics[END_REF]. This principle can be used when all the quantities have densities with respect to the Lebesgue measure. When they have densities with respect to the atomic measure, as in the present situation, the principle of virtual power is advantageously replaced by the principle of virtual work. The principle of virtual work we use here, is not to be confused with the principle of virtual power where the velocities are understood as small displacements.