Close range effects on a concrete wall of an underwater explosion A numerical approach

Transcription

1 Close range effets on a onrete wall of an underwater exlosion A numerial aroah O. Loiseau, K. Cheval & B. Autrusson Institut de Radiorotetion et de Sûreté Nuléaire (IRSN), BP 17, Fontenay-Aux-Roses edex, Frane. Abstrat The onern of this aer is the study of the effets on a thik onrete wall of the detonation rodued by an underwater exlosive devie, in lose range. This work alies in artiular to the study of some industrial ools, filled with water and a few meters dee. These ools are generally build so as to ensure a ertain watertightness, this funtion being obtained for instane by overing the internal surfae with an internal liner, a few millimeters thik and made of stainless steel. Here, we fous on the ossible loss of this funtion both by the damage aused to onrete and the erforation of the liner, due to the loal effets of the exlosion indeendently of the global strutural resonse. In the aer, based on the Holmquist [1] model for the behavior of onrete under high ressure, we first determine a lear transition between lose range and far-field tye soliitations. For a given exlosive devie, defined by its mass, the lose range domain is materialized by a minimum redued radius under whih the quasi stati harateristis are signifiantly modified due to the high ressure subsequent to the detonation. In seond, we resent a numerial examle: The goal is to determine the deth of damaged onrete in relation with the exlosive devie mass and osition. This roblem is solved using the exliit dynamis ode LS- Dyna [2] and a weak ouling between an eulerian and a lagrangian omutation. This examle is ommented and the results have been omared with exerimental data. Keywords: onrete wall, underwater exlosion, lose-field, damage.

2 1 Introdution How an the damage aused by the use of exlosive against industrial strutures be assessed? This question, in general, annot be answered by real size exerimentations on atual strutures, though this kind of demonstration is the only one that is not questionable. And even in the ase of redued sale exeriments, or details studies, the ost is most of the time dissuasive, eseially onerning blast testing. Nevertheless, in many ases and artiularly in the industrial field, regulations imose the onsequenes of malevolent ations onduted with exlosive to be studied, and this an be done by various means: In the ase of far-field blast aggressions, tehnial manuals like for instane referenes [3] and [4], or simlified numerial odes like INBLAST [5] an be very useful, as they are based on many exerimental results; But in the ase of lose-field exlosive aggressions, numerial simulation is, if not the one and only, the most widesread aroah. In this aer we resent an examle of a study onerning onrete ools filled with water. The dimensions of these ools are tyially ten meters of width and deth, versus several tens meters of length. Walls are about one meter thik. The question of entred exlosions a far-field exlosion in regards to the walls has been examined reviously by a simlified aroah [6], derived from Cole s formulas [7] for blast intensity and Cagniard theory [8] for wave refletions and interations. Suh an aroah might be onsidered as a artiular ase of the one introdued by Britt [9]. In omlement to this first aroah, here we fous on the roblem of an exlosive devie thrown into the water and detonating lose to the walls, aimed at destroying loally the steel liner and ruining onrete underneath, so as to let the water go with a more or less imortant flow. Beause this aroah is omlementary to another dediated to far-field exlosions, our first question is how to distinguish far-field from lose-field, and what should be the frontier between them and the riterion used to define it. This question is answered in the first two aragrahs of this aer: First, we desribe the main non linear asets of onrete behaviour under high ressure, illustrated by the Holmquist [1] model; In seond, based on the model, a simle one-dimensional model is resented, roduing an analytial formula to define a distane under whih ertain non linear asets of onrete behaviour need to be onsidered, figuring lose-range onsequenes. The third aragrah is dediated to the numerial aroah develoed in the numerial exliit ode LS-Dyna [2] to answer the seond question: Does a numerial aroah ermit to reliably assess the damage level aused to onrete by a lose field exlosion? The global sheme, ouling an Euler omutation of the exlosive detonating in water and a Lagrange omutation for the onrete resonse analysis and its damage assessment, is resented and illustrated on an examle. Numerial results have been omared to exerimental data obtained by IRSN unublished at this time due to the sensitivity of suh real sale exerimental results.

3 2 Main asets of onrete behaviour under high ressure The model roosed by Holmquist et al. [1] offers a good synthesis of the main henomena ourring in onrete submitted to high level ressures, high strains or high strain rates. Phenomena are searated between hydrostati and deviatori strength asets. Damage evolutes from ontributions of these two asets of behaviour. 2.1 Pressure-volume resonse The hydrostati ressure volume relationshi an be searated into three hases, desribed for a onstant inreasing ressure: (1) A first elasti hase where the material resonse is linear, and the relation between ressure and volumetri strain µ is governed by the initial orous bulk modulus K, finishing as soon as the ressure reahes a limit value given by rush ; (2) A non linear evolution hase where, for an inreasing ressure, air voids in the material are suosed to rush, and the orosity to disaear rogressively, until the loking ressure lok and loking volumetri strain µ lok values are reahed; (3) A seond elasti but non linear hase governing the resonse of dense material obtained after omation, governed by a olynomial relation of the third order. 2.2 Strength evolution Currently for onrete under high ressure, strength evolution is diretly related to ressure. The following formula is roosed by Holmquist et al. [1]: N & ε σ ( ) + VM = f ' A 1 D + B 1 C ln (1) f ' & ε 0 A, B, C and N are non dimensional harateristi arameters, ε& the atual strain rate and ε& 0 a referene strain rate, D is the damage arameter, f ' the quasi-stati uniaxial omressive strength, the ressure and σ VM the Von Mises equivalent stress. Strength inreases until is reahed a saturation oint, designed by an uer bound limit value for σ VM : σ max. 2.3 Damage evolution In a similar manner to what is onsidered for the Johnson-Cook model [10] damage inreases as non linear lasti deformation develos. Here, the notion is extended to onsider not only equivalent lasti strain but also the volumetri lasti strain inrease. Finally, an elementary damage inrease is exressed by: dε + dµ dd = (2) f f ε + µ

4 where dε and dµ are resetively the equivalent and volumetri lasti strain f f inreases, and ε + the lasti strain to frature related to ressure. µ 3 Frontier between lose-field and far-field 3.1 Far-field and lose-field urrent models general omarison A struture submitted to a blast loading in far-field of an exlosion is generally onsidered to be at a distane of one or several orders of magnitude greater than the harateristi dimensions of the exlosive devie itself. Considering this oint, generally the whole struture or a large art is imated by the blast wave and thus needs to be modelled. In suh a ase, urrent models involve shell finite elements and the material behaviour is dedued from quasi-stati models eventually non linear. Holmquist and al. [1] model for onrete, among many others [11], enters a seond ategory of models for high ressure, high strains or strain rates, et. But, due to energy onsiderations, suh high intensity soliitations only our on very short times, and thus very short distanes. In the ase of a struture loaded by exlosion, those models are useful only in lose-field of the devie, at a distane not exeeding the order of magnitude of the exlosive devie itself. Those onsideration seems to indiate that the distintion between far and lose-field notions ould be guided not only by orders of magnitude of the devie-target distane but also, and more reisely, by the models themselves. In artiular, high ressure, high strains or strain rates models need not to be used in the ase where maximal soliitations do not imly non linear evolutions and need to be used if external fores are suh that they imly rushing and/or a variation of the nominal omressive strength. 3.2 Pressure thresholds In the following, for a simle ase of an uniaxial omression test, we determine the external ressure threshold values beyond whih the quasi-stati omressive strength is signifiantly modified by the ressure inside the material. The roblem we onsider is resented on fig. 1. In x diretion of the model is made the assumtion of lane strains. To give a generi aset to our study, this examle is onduted with the material harateristis diretly taken from Holmquist et al. [1] aer : onrete density ρ is taken equal to 2440 kg/m 3, shear modulus G equal to 15 GPa, bulk modulus K equal to 16 GPa, nominal omressive and tensile strength, f and T, are resetively equal to 48 MPa and 4 MPa, rushing and loking ressures, rush and lok, are resetively equal to 16 MPa and 800 MPa, while orresonding volumetri strains, µ rush and µ lok, are resetively equal to and 0.1; Non dimensional arameters for the desrition of the evolution of σ VM vs., are for A, B, C and N, resetively 0.79, 1.60, and 0.61, while the saturation stress σ max is equal to 336 MPa. Any other arameter an be found in Holmquist et al. [1].

5 Figure 1. Uniaxial omression. The external loading is realized by the aliation of an inreasing fore density σ z, alied slowly so as to remain in a quasi-stati regime. As this external fore inreases, so do the internal ressure and the Von Mises equivalent stress (reresented on fig. 1). This inrease is urely linear elasti as long as the ressure remains under the rush threshold. When the ressure overasses rush, starts the hydrostati rushing regime for onrete among whih the aarent bulk modulus is lower than the urely elasti one. The shear omonent remains elasti until the loading ath meets the strength evolution urve, and at this time and for an inreasing external fore the material begins to follow a fully non linear resonse. This ressure threshold is named nl and is equal to 35.1 MPa (see fig. 1) with the arameters values given above. When the ressure asses over nl, the initial strength f has been multilied by about two and is still evolving. So one an onsider that a standard model based on a omressive strength equal to f is erroneous, only due to high ressure effets. At this oint, we onsider that high ressure models, like Holmquist et al. [1] begin to be ertinent, as do lose-field onsiderations. On the ontrary, under the nl threshold, no signifiant evolution of strength seems to aear, lassial far-field models are onsidered as suffiiently aurate. 3.3 Conlusion on lose-field delimitation In the ase of the test reresented by fig. 1, the external fore density σ z, nl orresonding to an internal ressure value nl an be dedued analytially by the following formula: 4G 4G σ z,nl = 1 + rush ( nl rush ) K 1 + K (3) 3 3 where K is an aarent bulk modulus for the rushing hase under inreasing ressure onditions:

6 K lok rush = (4) µ lok µ rush In referene to the arameters given in the aragrah above, σ z, nl is equal to MPa. This value is used as a link with solid exlosive blast eak ressure magnitude. We have onsidered that if a solid exlosive blast in water leads to a refleted ressure inferior to σ z, nl, we an onsider being in far-field. The σ z, nl being established on quasi stati onsiderations and short time dynami effets tending to lower the onsequenes of suh a loading, we ensure that lose-field is eventually larger than stritly neessary, whih is not damaging. The redued limit radius is established on the basis of Cole s formulas [7] for TNT underwater exlosions, and a refletion oeffiient equal to 2. Suh formulas are validated for a redued distane not inferior to 0.42 m/kg 1/3 (see ref. [7]). As a onlusion, onsidering for instane a 40 kg TNT devie, treating the ase of the detonation of this harge at a distane from a onrete wall inferior to 3.5 m needs to model onrete with a lose-field adated behaviour law. This distane, summarized by the redued distane 1.02 m/kg 1/3, may be onsidered as the boundary between lose-field and far-field for ressure guided onrete models. Table 1: Limit radius between lose-field and far-field. Threshold external fore density σ MPa 1 σ α z,nl Redued distane Z = 2 K with K = 52.4 MPa and α = 1.13 (arameters for TNT) Atual distane examles z, nl W = 5 kg TNT W = 40 kg TNT 4 A numerial aroah for a lose-field study 4.1 Objetives and assumtions 1.02 m/kg 1/3 1.7 m 3.5 m The main objetive is to study a ossible loss of watertightness by ruining the stainless steel liner. In an industrial ontext, henomena have to be searated so as to ensure both reditive omutations and onservative results. Based on exerimental bakground and qualitative alulations on underwater detonations in lose-field of a stainless steel liner and onrete wall, the following assumtions have been stated: Frature of the stainless steel liner is mainly driven by the dislaements and lasti strains indued into it subsequently to the reation of an eventual rater in onrete underneath or even onrete rushing only; The onsequenes in terms of rushing and damage of a omressive shok wave assing aross a thik onrete wall might be studied in 1D,

7 as long as rater deth is a more ertinent arameter to be determined than diameter, to state on the liner frature; The eventual effets of a bubble onto the wall and assoiated entering jet studied in some ases of lose-field underwater detonations (see for instane Kalumuk et al. [12]) are negleted, onsidering a TNT mass range for whih omressive effets are dominant in omarison to those. 4.2 Comutation sheme The global aroah we resent is based on an Euler omutation hained with a Lagrange omutation. The Euler omutation is done for the determination of the harateristis of TNT detonation in water in terms of eak ressure, duration, et., of the blast wave rodued. Comutations are justified by the fat that in lose-field, emirial and analytial formulas are no longer relevant (see Cole [7]). The Lagrange omutation is aimed at studying the omressive damage evolution in onrete as it is assed through by a omressive shok wave, with no model adjustment to see if suh oen literature models might be reditive. The global omutation sheme is summarized on fig. 2. water stainless steel (liner) onrete (wall) exlosive most imortant ressure oint region of onrete where damage deth is evaluated with a 1D model symmetry diretions detonation oint Figure 2. Global omutation sheme. 4.3 Euler omutation results The Euler omutations onsist in the simulation of an exlosive detonating in water lose to a refletion wall. Water is modelled by a Mie-Grüneisen equation of state as available in Hallquist [2] for the LS-Dyna ode, with a density ρ equal to 1000 kg/m 3, sound seed C equal to 1647 m/s, and two non zero onstants Y 0 and S 1 resetively equal to 0.28 and Modelling detonation in the exlosive devie, lighting times are determined by a geometri wave front roagation interolation method based on Chaman-Jouguet data for TNT, i.e. a detonation front veloity D CJ equal to 7840 m/s. Other harateristis for the exlosive are its initial density ρ 0 equal to 1520 kg/m 3 and the Chaman-

8 Jouguet ressure CJ equal to 26 Ga, and also the TNT arameters of the Jones-Wilkins-Lee model as resented in Hallquist [2], A, B, ω, R 1 and R 2 resetively equal to 371 GPa, 3.23 GPa, 0.3, 4.15 and The Euler omutation domain reresented on fig. 3 is meshed with about regular hexahedron ells. Various TNT masses are tested, eah of them being ubi with a entre laed 30 m away from the refleting wall and ouying different volumes as reresented on fig. 3. Pressure is estimated in water at the shortest distane oint of the refletion wall in referene to the exlosive devie entre. The ressure rofiles obtained by omutations at this oint are reresented on fig. 3. The ressures obtained by those alulations, at a redued standoff distane at most equal to m/kg 1/3, are absolutely out of soe of the formulas reorted in Cole [7] or Swisdak [4]. Nevertheless we have heked the fat they follow the general tendenies of these formulas if interolated to very short standoff distanes. Figure 3. Euler omutation domain and ressure rofiles obtained. 4.4 Lagrange omutation results The Lagrange omutation onsists in alying to a 1D model of onrete and liner assembly the ressure obtained with the Euler omutation. This model is reresented on fig. 4. Conrete is modelled by the Holmquist et al. [1] model with the arameters given in aragrah 3.2. The stainless steel liner is modelled by linear lastiity model with no stress hardening, in order to minimize the energy dissiated by the shok when assing through the liner. This hyothesis is hosen in order to ensure onservative results again - the less energy dissiated

9 in the liner the more energy to damage onrete underneath. The main harateristis for the liner stainless steel are a density of 7830 kg/m 3, a Young s modulus of 210 GPa, Poisson s ratio equal to 0.3 and a dynami yield strength equal to 230 MPa. Figure 4. Lagrange omutations damage assessment results. On fig. 4 the damage arameter reresented is the Holmquist et al. [1] damage arameter as seified in LS-Dyna [2]. Damage values on the oosite side of the wall, with reset to the exlosive osition, must not be onsidered as reliable; This is due to the fat that no tension failure has been introdued in the model, and thus there is no material ejetion ourring. The analysis fouses on the shok entering side of the wall, and we have onsidered here that the maximum deth of a rater orresonds to the length of material where damage is equal to unity. Then deth is used to assess the ossible frature of the liner and subsequent loss of the watertightness funtion. The final results is a funtion of rater deth in onrete related to the mass of the exlosive devie. These results have been validated by omarison with exerimental data and the method has roven to be reliable and onservative. The exloitation of this aroah annot be fully illustrated, eseially onerning ratial onlusions, due to the general sensitivity of these onlusions. 5 Conlusion A omlete aroah used in an industrial ontext for the assessment of damage in a onrete ool wall aused by the detonation of an exlosive devie in losefield has been exosed. At first, the meaning of lose-field in this ase has been desribed in details in relation with the non linear loal model used for onrete. A lear limit standoff distane has been established based on analytial simle

10 models and onsiderations on the behaviour arameters variations due to high ressures. In seond, an aroah based again on simle models and LS-Dyna omutations has been illustrated on an examle. Referenes [1] Holmquist, T. J., Johnson, G. R., Cook, W. H., A omutational onstitutive model for onrete subjeted to large strains, high strain rates, and high ressures, in Proeedings of the 14 th International Symosium on Ballistis, Quebe, Canada, [2] Hallquist, J. O., LS-Dyna Keyword User s Manual Ver 960 Vol. 1&2, Livermore Software Tehnology Cororation, Livermore, USA, [3] US Deartment of the Army, Strutures to Resist the Effets of Aidental Exlosions, Army: TM5-1300, Navy: NAVFAC P-397, Air Fore: AFR 88-2, Deartment of the Army, Navy and Air Fore, Washington DC, USA, [4] Swisdak, M. M., Exlosion effets and roerties, Part II Exlosion effets in water, NSWC/WOL/TR , Naval Surfae Weaons Center - White Oak Laboratory, Silver Sring, USA, [5] Protor, J. F., Internal Blast Damage Mehanisms Comuter Program, NOL/TR , Naval Ordonane Laboratory, Silver Sring MD, USA, [6] Delmaire-Sizes, F., Autrusson, B., Niaud, A., Brohard, D., Guerin, J.-M., Trameçon, A., Behavior of a retangular reinfored onrete ool filled with water and loaded by an internal exlosion, in Proeedings of the annual PVP ASME Conferene, Seattle, USA, [7] Cole, R. H., Underwater Exlosions, Prineton University Press, Prineton, USA, [8] Cagniard, L., Réflexion et réfration des ondes séismiques rogressives, Ghautiers-Villars, Paris, Frane, [9] Britt, J. R., Aliation of Cagniard Theory to refletion and refration of exlosion shok waves in soil and water, in Proeedings of the 9 th International Symosium on Military Aliations of Blast Simulations, Oxford, UK, [10] Johnson, G. R., Cook, W. H., Frature harateristis of three metals subjeted to various strains, strain rates, temeratures and ressures, Journal of Engineering Frature Mehanis, Vol. 21, No. 1, , [11] Field, J. E., Proud, W. G., Walley, S. M., Goldrein, H. T., Review of exerimental tehniques for high rate deformation and shok studies, in New Exerimental Methods in Material Dynamis and Imat, Nowaki, W. K. and Kleazko, J. R. (Eds.), IPPT PAN and CoE AMAS, Warsaw, Poland, [12] Kalumuk, K. M., Chahine, G. L., Duraiswami, R., Analysis of the resonse of a deformable struture to underwater exlosion bubble loading using a fully ouled fluid-struture interation roedure, in Proeedings of the 66th Shok and Vibration Symosium, Biloxi MS, USA, 1995.