HAIL IMPACT

ARTICLE IN PRESS

International Journal of Impact Engineering 31 (2005) 929–944 www.elsevier.com/locate/ijimpeng

A survey of numerical models for hail impact analysis using explicit finite element codes Marco Anghileri, Luigi-M. L. Castelletti, Fabio Invernizzi, Marco Mascheroni Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa, 34–20156, Milano, Italy Received 5 April 2004; received in revised form 10 June 2004; accepted 12 June 2004 Available online 8 October 2004

Abstract Hailstone impact is an actual threat for the integrity of aircraft structures such as leading edges, and forward sections. Though the analysis of weather conditions reduces the occurrence of intersections between flight routes and hailstorm regions, sometimes the passage through a hailstorm becomes inevitable and, in such a case, it is mandatory that the aircraft structures show an appropriate level of tolerance to damages caused by a hail impact. Therefore, as experimental tests are both expensive and troublesome, it is important to develop numerical models, which eventually support the design of high-efficient and hailproof structures. Accordingly, in the present research, using LSTC LS-Dyna, three numerical models of hailstone have been developed: finite element, arbitrary Lagrangian Eulerian, and smoothed particle hydrodynamics model. Initially, these three models had been validated referring to a documented experimental test and, subsequently, used to reproduce the impact of a hailstone with the nose-lip of a nacelle intake. Advantages and disadvantages of the three hail models have been evaluated and it has been concluded that the smoothed particle hydrodynamics model is the most efficient and effective for the analysis of the event. r 2004 Elsevier Ltd. All rights reserved. Keywords: Airworthiness; Hail impact; Explicit finite element codes

Corresponding author. LAST Labs, Politecnico di Milano, Edificio G, Via Durando, 10–20158, Milano, Italy. Tel.:

+39-02-2399-7155; fax: +39-02-2399-7153. E-mail address: [email protected] (L.-M.L. Castelletti). 0734-743X/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2004.06.009

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M. Anghileri et al. / International Journal of Impact Engineering 31 (2005) 929–944

1. Introduction Hailstone impacts represent an actual threat for aircraft structures. Indeed, it is not unlikely that aircraft, flying in adverse weather conditions, pass through hailstorm regions. Even a short while in these regions, is sufficient to cause serious damage to aircraft structures such as abrasions, dents and, in some cases, perforations. The most exposed parts in this event are the leading edges of wings and tail control surfaces, the fuselage forward sections, the engine nacelles, and other accessories such as the radar antenna and the landing lights [1]. By the observation of damaged aircraft and by specific experimental tests, it has been recognised that the extent of the mentioned damage depends on the features of both the hailstone (mass, impact angle and velocity), and the impacted structures (geometry and material). The analysis of weather conditions has so far made it possible to avoid intersections between flight routes and hailstorms regions. Nevertheless, sometimes the passage through a hailstorm becomes inevitable. Therefore, the structures of modern aircraft are called by specific requirements (such as those in the European JAR) to guarantee a certain level of functionality after having been impacted by a number of hailstones. Indeed, though hail impact has been recognised as a serious problem since the early 1950s [1], recent studies concerning the consequences of the hail impact with aircraft structures are somewhat rare. Since experimental tests are expensive and difficult to perform, it is straightforward to understand the importance of developing numerical models to reproduce and analyse the consequences of hailstone impacts. Indeed, after being properly validated also referring to experimental evidences, these models might represent a powerful (efficient and effective) tool to design high-performances and hail-proof structures. In particular, the aim of this work has been to develop a numerical model of hailstone to simulate the hail impact against aircraft structures by means of a general-purpose (commercially available) explicit Finite Element (FE) code: LSTC LS-Dyna [2,3]. In particular, three different models have been investigated: the customary Lagrangian FE model, the Arbitrary Lagrangian Eulerian (ALE) model, and a model based on Smoothed Particle Hydrodynamics (SPH) method. Initially, these models have been validated referring to the results of experimental tests carried out by the British Royal Aircraft Establishment (RAE), and lately also cited in NASA technical notes [4,5]. Then, the mentioned models have been used to reproduce the damages caused by a hailstone onto the nose-lip of a turbofan engine intake. In particular, to verify the reasonability of the obtained numerical results, these were (qualitatively) compared with the photographic documentation of actual hailstone impacts occurred to an airlines aircraft. As a result, it has been possible to highlight the advantages and disadvantages of the three different hail models. Finally, the same numerical models have been also used to verify that the structure of the intake was such to avoid hailstone penetration inside the nacelle airframe when considering the impact velocity required for certification (such as JAR-E 970 and ACJ E 970). Concluding, the feasibility of the different models for simultaneous impact of a number of hailstones has been considered—and this is likely to be a further development of the present work. 2. Numerical model of the hail As already mentioned, in this research, to reproduce a hail impact, three different numerical hailstone models (Lagrangian FE, ALE and SPH) have been validated referring to an actual

ARTICLE IN PRESS M. Anghileri et al. / International Journal of Impact Engineering 31 (2005) 929–944

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experimental test carried out by the British RAE (and described in the NASA technical note D6102). 2.1. Experimental test carried out by British RAE The test considered consisted of the impact of a 25.40 mm diameter hailstone against an aluminium alloy 2014-T4 plate (Fig. 1) with an initial impact velocity of 192 m/s. The plate was a square 0.91 mm thick panel of 305 mm side-edge, which was fixed at the boundaries with blind rivets so that eventually the free target surface consisted of a square region of 200 mm side-edge. The plastic strain observed after the impact (measured with regard to Section A–A in Fig. 1) was used in the following as reference (the continue line in Fig. 2). In particular, the maximum displacement (measured in the centre of the panel) was 11.20 mm. 2.2. Lagrangian finite element model of the hailstone The Lagrangian FE approach is typical for continuum mechanics [6] and strongly recommendable for the analysis of impact events. This approach is extremely efficient when considering nonlinear problems, though it has its weak point in the excessive mesh distortions— which are rather typical in events featuring soft-bodies or fluid-like materials such as in the case considered. Following a Lagrangian approach, the FE mesh is constructed on the material and therefore, when the material undergoes large distortions, also the mesh undergoes the same large distortions, which cause an unacceptable loss in accuracy, a considerable increase in required CPU time (due to the fall of time-step value), and sometimes, a premature analysis termination.

Fig. 1. Experimental test panel.



The FE model of the hailstone reproduced its actual shape, by approximating a 12.7 mm radius sphere with hexahedral solid elements. The common eight-node three-linear isoparametric solid elements were used and, as a result of a specific sensitivity analysis, the mesh consisted of 3045 elements, with a reference characteristic length of 1.2 mm. Moving from an established Constitutive Law [7,8], the ice was modelled as an elastic plastic material with failure (*MAT 13 of LS-Dyna [2,3]). Indeed, this material model allows a plastic hardening behaviour that adequately reproduces the effect of the propagation of the micro cracks inside the ice before it crushes, reaching a fluid-like state. When the plastic failure strain is reached, all shear stress components are relaxed to zero. Furthermore, if the tensile failure pressure is reached, the material carries only hydrostatic compressive stresses as a fluid. The mechanical properties of the ice are summarised in Table 1. Obviously, as already noticed in [7,8], this is only a simplified representation of a complex material such as the ice. In fact, this model does not consider the complicated nonlinear mutual dependency between pressure and volume and the influence of the strain rate on the failure strength. However, it was showed [7,8] that this material model makes it possible to reproduce correctly the ice behaviour during a high-velocity impact and therefore it has been adopted. 2.3. Arbitrary Lagrangian Eulerian model of the hailstone The ALE approach is meant to combine the advantages of both Lagrangian and Eulerian approaches. Following a Eulerian approach, the material flows through a mesh fixed in the space. The ALE approach differs in that the Eulerian mesh can move arbitrarily and, in case, independently on the motion of the material. Consequently, it has an advantage over the pure Eulerian approach when the motion of the material covers a wide region of the space. In this case, the number of elements using the Eulerian approach should be so large to maintain a reasonable accuracy in the calculation, that the CPU time required for the analysis would be unacceptable. With regard to the ALE model of the hailstone, it is worth noticing that it was necessary to discretise not only the hailstone, but also a small surrounding region. In this way, it was possible to avoid the outflow of the hailstone from the Eulerian mesh due to the velocity of the material, which was higher than the expansion-translation velocity of the Eulerian mesh. The ALE mesh of the hailstone and the surrounding void region consisted of 5888 hexahedral elements. For these elements, it was adopted the standard ALE formulation with one integration Table 1 Mechanical properties of the hailstone used for *MAT_13 of LS-Dyna [2,3] Properties

Values

Density (kg/m3) Elastic shear modulus (GPa) Yield strength (MPa) Hardening modulus (GPa) Bulk modulus (GPa) Plastic failure strain Tensile failure pressure (MPa)

846 3.46 10.30 6.89 8.99 0.35 4.00


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point [3]. The Eulerian mesh was prescribed to move following the hailstone mass weighted average velocity [3]. In order to characterise the ice behaviour, the same material model used for the Lagrangian FE model (Table 1) was adopted. 2.4. Smoothed particle hydrodynamics model of the hailstone The SPH method was initially developed (1997) to study astrophysical and cosmological phenomena, subsequently, it was extended to typical problems of computational fluid dynamic (CFD) and finally to continuum mechanic. The main difference between SPH and FE method regards the discretisation of the continuum. The SPH is a meshless method: the mesh is replaced with a set of particles endowed with a mass. No direct connectivity exists among the particles: the particles are the basis of an interpolatory scheme based on the kernel function that is the core of the method. The SPH model of the hailstone was developed starting from both the FE model previously described, and the real physical and mechanical hail properties [9]. An extremely regular distribution of the particles was provided to improve the accuracy in the solution. In particular, the distance between the particles was chosen equal to the value of the characteristic length of the previously described FE model. Accordingly, the model consisted of 4169 particles. Since the material model used for the FE model is not implemented for the SPH solver, it was necessary to use a different Constitutive Law [10]. Several simulations were performed: a number of material models were investigated. For each one of these material models, a number of values of the characteristic parameters were considered. The obtained results were compared with experimental data [4,5]. As a result, a SPH model able to provide a good numerical-experimental correlation was obtained [10]. The elastic plastic hydrodynamic material model (*MAT 10 of LS-Dyna [2,3]) was used for ice, and the adopted mechanical properties are provided in Table 2. This model has made it possible to represent the hailstone behaviour correctly both in the early stages of the impact, when it is characterised by a high stiffness, and in the subsequent stages, when, cracked after impact, it behaves like a fluid. The material model is characterised by a failure criterion relative to the tensile stress which works in such a way that when the tensile failure stress is reached, the deviatoric stresses component is set to zero and the material can sustain only compressive stresses. The plastic failure Table 2 Mechanical properties of the hailstone used for *MAT_10 of LS-Dyna [2,3] Properties

Values

Density (kg/m3) Elastic shear modulus (GPa) Yield strength (MPa) Plastic hardening modulus (GPa) Pressure cut-off (MPa)

846 3.46 10.30 6.89 4.00


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strain criterion was not defined because it was considered unsuitable to reproduce the failure mechanisms of ice correctly. However, the tension instability, typical of SPH method, partially supplied this lack. As the adopted material model requires an Equation of State (EOS), the polynomial EOS of the water was used [11]. 2.5. Comparisons and remarks on the different hail models

Deflection [mm]

The results obtained after the simulations carried out using the three different hail models were compared qualitatively and quantitatively among themselves and with the experimental test data [4,5]. Considering the graphical evidence of the impact (Fig. 2), it is evident that FE mesh (Fig. 2a) undergoes large distortions and, therefore, it is straightforward to conclude that the use of the FE model seems feasible only in the early stages of the impact. On the contrary, the ALE model (Fig. 2b) gives a somewhat accurate description of the cracked hailstone also when it is characterised by a fluid-like state. Nevertheless, it is the SPH model (Fig. 2c) that reproduces the hailstone behaviour, visually, in a way closer to common experience. 5 0 -5 -10 -15 100 110 120 130 140 150 160 170 180 190 200 Position [mm] Experimental Data

Deflection [mm]

(a)

Numerical Results (FE model)

5 0 -5 -10 -15 100 110 120 130 140 150 160 170 180 190 200 Position [mm] Experimental Data

Deflection [mm]

(b)

Numerical Results (ALE model)

5 0 -5 -10 -15 100 110 120 130 140 150 160 170 180 190 200 Position [mm]

(c)

Fig. 2. Hail impact at t=0.150 10

Experimental Data Numerical Results (SPH model)

3

s and numerical-experimental correlation. (a) FE; (b) ALE; and (c) SPH models.


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The quantitative comparisons (Fig. 2) were made referring to the plastic strain of the plate measured after the experimental tests [4,5]. In Table 3 the maximum plastic strain in the centre of the panel is directly compared with the displacement measured in the test. In particular, it is worth noticing that the FE and SPH models give errors within 1%, the ALE model within 4%. The required CPU time was also monitored. Considering the FE model of the hailstone, a drastic drop in the time-step was observed after the impact (due to the large mesh distortion). Indeed, the time-step decreases also in the SPH and ALE model, but definitively less (Table 4). Using a common PC (a Pentium 4–1700 MHz CPU, and 256 Mb RAM), the required time for 0.7 ms real-time simulation ranged from a minimum of about half an hour (SPH model) to a maximum of about 18 h (FE model)—as reported in Table 4.

3. Impact of a hailstone against a turbofan engine intake The intake of a turbofan engine nacelle is one of the aircraft parts more seriously damaged when the aircraft cross a hailstorm region. Though this event is not likely to cause an air disaster, the penetration of a hailstone (at high velocity) inside the intake structure is likely to cause the loss of the engine. In fact, the electronic system that controls the engine is usually positioned inside the intake airframe. Therefore, it makes sense to analyse the consequences of a hail impact against a turbofan engine intake. In particular, in this research, considering the actual structure of a turbofan engine intake, the dents caused by the developed hail models were initially compared with the documented damages due to hail impacts against similar structures. Then, considering Table 3 Numerical-experimental correlation

Experimental test Finite element model Arbitrary Lagrangian Eulerian model Smoothed particle hydrodynamics model

Maximum displacement (mm)

Relative error (%)

11.20 11.25 11.70 11.28

0.4 4.0 0.7

Table 4 Required CPU-time Hail model

CPU-timea

Time-step Initial (s)

Finite element model Arbitrary Lagrangian Eulerian model Smoothed particle hydrodynamics model a

4.39 10 4.39 10 2.07 10

Final (s) 8 8 7

9.51 10 1.69 10 1.85 10

CPU-time required by a Pentium 4–1700 MHz CPU, and 256 MB RAM PC.

10 8 7

39 h, 9 min 14 h, 30 min 1 h, 4 min



the impact velocity prescribed for hail ingestion (JAR-E 970 and ACJ E 970), it was verified that the structure was able to protect the mentioned system. 3.1. Numerical model of the nacelle intake The FE model reproduced in detail the main features of an actual intake. In fact, it was built on the geometry of the intake of a turbofan engine developed in a preliminary design phase. As a reference on the intake dimensions, the inner diameter measured at the leading edge is 1.60 m. The FE mesh (Fig. 3) consisted of 16 parts and 63013 four-nodes shell-elements. Five of these parts are made of composite material and one, the inner barrel, adopting a sandwich technology (composite materials and aluminium honeycomb)—typical in aircraft industry as it allows high strength/weight ratios and a good noise control. Four parts are made of titanium alloy Ti-6Al-4V and the remaining of two distinct aluminium alloys: Al 2219-T62 the nose-lip, and Al 2024-T6 the other parts. In order to obtain the necessary accuracy avoiding excessive computational efforts, the shell elements had a relatively large reference length (10 mm) throughout the whole model but in the

Fig. 3. FE model of the intake: (a) full model and (b) a detail.


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nose-lip region, where a locally finer mesh was built (2.5 mm). The Belytschko–Tsay formulation (that is the default formulation for LSTC LS-Dyna) was adopted. As a remark, it is worth noticing that the sandwich panels were not modelled in detail, but equivalent shell elements were used. Such a model made it easier to change the thickness of the honeycomb core, considering that, in a preliminary design phase, changes in thickness are somewhat usual. Furthermore, it was observed that the sandwich panels were not directly involved in the hail impact and, therefore, an accurate model of the part was not deemed necessary. The parts of the intake made up with metallic material were modelled using the elastic piecewise (isotropic) linear plasticity material model (*MAT_24 of LS-Dyna [2,3]), using for the mechanical properties of the materials customary values (also indicated in the MIL handbooks). Appropriate Cowper–Symond coefficients were defined to model the effect of high strain rate deformation. Also, for the nose-lip material, the residual stress was considered. The composite material used in manufacturing the inner and outer barrel, a carbon fibre reinforced plastic (CFRP) woven with resin volume fraction of 42%, was modelled with a specific material model (*MAT_54 of LSDyna [2,3])—already validated [12]. Concluding, it is important to notice that the riveted junctions were not modelled also because, since the FE model was developed on a preliminary design, the characteristics (number, placement, and mechanical properties) of the riveted junctions were not available. Contacts among the parts were carefully modelled. Furthermore, in order to reproduce the actual constraint that the airframe of the engine imposes to the intake, the inner and outer flanges of the aft part of the intake (in the FE model) were fixed. 3.2. Comparison with documented damages due to hail impacts In order to verify the accuracy of the hail models developed in the first phase of the research, initially, simulations reproducing the impact against a nacelle nose-lip were performed using velocities close to those of documented accidents. In particular, the accident occurred to the Douglas DC-9-81 (MD 81) of the Scandinavian airlines system (SAS) was considered [13]. The aircraft, flying at a true-air speed (TAS) of about 140 m/s in the middle of a heavy hailstorm characterised by hailstones with diameters up to 5 mm, reported serious dents in the nacelle leading edge. The largest dents were within 50 and 70 mm in diameter and 5 mm deep (Fig. 5). In analogy with this accident, the impact of a hailstone at 140 m/ s was reproduced using the described FE model of a turbofan engine intake. This model does not represent specifically the engine intake of a MD 81, but it reproduces the intake of an aircraft of similar dimensions. The three hail models previously investigated were used. Also, since the hailstones are characterised by extremely various dimensions, to reproduce the most common conditions in which an aircraft can incur, it was decided to perform simulations using two different dimensions for the radius of hailstone: 12.70 mm (the same previously used) and 21.35 mm (the dimension of a golf ball)—the model of which was obtained from the former by scaling the dimensions. The most critical conditions were considered: the hailstone impacted the most prominent point of the intake lower section, in a direction parallel to the nacelle axis. As a result of the analyses, it was observed that in none of the considered cases the hailstones caused the failure of the nose-lip and penetrated inside the intake structure



(Fig. 4)—like in the documented case. Furthermore, it was noticed that the three hail models produced similar dents quantitatively (Tables 5 and 6) and qualitatively (Fig. 4). Despite the constructive differences and the lack of relevant data on the actual hail impacts, the damages numerically obtained are definitively similar to the actual dents in dimension and in appearance (Fig. 5). 3.3. Simulations considering the required actual impact velocity After verifying the numerical model of the hail with regard to actual hail impacts, simulations were performed with an impact velocity of 185 m/s—which, for the considered aircraft, is the hail ingestion velocity indicated by the JAR/ACJ-E 970 requirements for certification of the intake. When considering the impact of the 12.70 mm radius hailstone (Fig. 6), it is straightforward to observe that the FE and SPH hailstone models produce similar results with respect to the residual plastic strain. On the contrary, the impact of the ALE model of the hailstone is less

Fig. 4. Dents due to hailstones of 12.70 mm (left) and 21.35 mm (right) radius. (a) FE; (b) ALE; and (c) SPH models.


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Table 5 Dents due to a 12.70 mm radius hailstone Hail model

Diameter (mm)

Depth (mm)

Finite element model Arbitrary Lagrangian Eulerian model Smoothed particle hydrodynamics model

38 39 40

3 4 4

Hail model

Diameter (mm)

Depth (mm)

Finite element model Arbitrary Lagrangian Eulerian model Smoothed particle hydrodynamics model

118 120 121

10 9 10

Table 6 Dents due to a 21.35 mm radius hailstone

severe—so that, eventually, in contrast with the two other models, no collapses in the nacelle were observed. This occurrence is somewhat typical in Lagrangian/Eulerian coupled analyses and, basically, depends on the definition of the coupling between Lagrangian and Eulerian meshes. The CPU-time required to run the simulations using the SPH model of hailstone was considerably smaller than the runtime of the other two models—as shown in Table 7. When considering the impact of the 21.35 mm radius hailstone (Fig. 7), the results showed that the FE model of the hail is not suitable for the analysis of the event. In fact, after having impacted the nose-lip and penetrated inside the airframe, the distortion of the hailstone mesh was such to cause a premature termination of the simulation. As a remark, it is worth noticing that, in some cases, an erosion criterion could be defined to avoid excessive mesh distortion. Nevertheless, it was not used here because, to be effective, it produced damages, which are far from the evidences collected in real hailstone impacts. Instead, using the ALE and SPH models of the hailstone, the simulation reached a normal termination. In both cases, the hailstone, after breaking the nose-lip, penetrated inside the airframe impacting the tube and the bulkhead. 3.4. Final remarks Although all the developed hail models are appropriate for the analysis of the impact of hailstone with compliant structures, the SPH model results in being the most suitable for the impact with the intake of a turbofan engine. In fact, this model provides accurate solutions and requires small CPU-time—these features make the SPH model a reliable and effective design tool. Furthermore, this model has made it possible to analyse the penetration of the hailstone inside the airframe and eventually it seems the only hail model suitable to analyse also the simultaneous



Fig. 5. Dents on the nose-lip of turbofan engine intakes.


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Fig. 6. Impact of the 12.70 mm radius hailstone. (a) FE; (b) ALE; and (c) SPH models.

Table 7 Required CPU-time Hail model

CPU-timea

Time-step Initial (s)

Finite element model Arbitrary Lagrangian Eulerian model Smoothed particle hydrodynamics model a

4.39 10 4.39 10 2.07 10

Final (s) 8 8 7

CPU-time referred to a Pentium 4–1700 MHz CPU, and 256 MB RAM PC.

1.27 10 1.23 10 1.98 10

9 8 7

39 h, 9 min 14 h, 30 min 1 h, 4 min



Fig. 7. Impact of the 21.35 mm radius hailstone. (a) FE; (b) ALE; and (c) SPH models.

impact of a number of hailstones. As a further development of the present work, the feasibility of the different models for simultaneous impact of a number of hailstones could be considered. Among the models evaluated, the SPH model of the hailstone proves to be the most suitable to analyse this phenomenon, though further investigation is recommended. Concluding, as a result of this research the intake design was modified.


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4. Conclusions Hailstone impacts represent an actual threat for aircraft structures and, therefore, it is important to develop numerical models to design hail-proof and high-efficiency structures. In particular, using LSTC LS-Dyna, three numerical models of hailstone have been developed and investigated in this research with regard to actual hail impact events: Lagrangian FE, ALE and SPH model. Initially, a close correlation with experimental data was obtained for each one of the three models so that, subsequently, it was possible to analyse the hail impact against an engine nacelle with a certain confidence. However, before considering the case of interest, in order to verify the hailstone models, hail impacts against a nacelle nose-lip were simulated using velocities close to the ones of a documented case, so that it could be possible to make a comparison between numerical and actual damages. Despite the constructive differences and the lack of further impact data, the numerical damages were definitively similar to the real dents (in dimensions and appearance). Then, simulations using the impact velocity prescribed for the certification of an engine intake (JAR-E 970 and ACJ E 970) were carried out. As a result, a direct comparison between the three hail models was possible. In particular, the FE model proves to be suitable to reproduce (only) the early stages of the impact when the distortion of the mesh is not such to cause inaccuracy, drastic drops in time-step or premature analysis termination. An erosion criterion could be a solution to avoid excessive mesh distortion. It was not introduced because, to be effective, it produced damages far from the evidences collected after real hailstone impacts. On the contrary, the timestep when using the ALE model remains roughly constant throughout all the simulation. Unfortunately, the obtained results suffer the typical problems arising from the coupling of solvers based on different approaches. The SPH model proves to be the most effective. In fact, this model grants a close numerical-experimental correlation and the required CPU-time is considerably smaller than the one required by the two other models. Furthermore, the SPH model effectively reproduces the hailstone behaviour also when the hailstone cracks after impact and, therefore, it seems the most suitable to analyse the simultaneous impact of a number of hailstones. Concluding, it is worth noticing that guided by the results of the analyses performed, the intake design was modified and certified. This can be regarded as a further confirmation of the usefulness of an appropriate numerical model as a powerful design tool and as a means to considerably reduce the number of pre-certification experimental tests.

References [1] Souter RK, Emerson JB. Summary of available hail literature and the effect of hail on aircraft in flight. Washington: NACA technical note 2734; 1952. p. 1–33. [2] Hallquist JO. LS-DYNA theoretical manual, LSTC, May 1998. [3] Hallquist JO. LS-DYNA keyword user’s manual, LSTC, vol.1–2, March 2001. [4] Thomson RG, Hayduk RJ. An improved analytical treatment of the denting of thin sheets by hail. NASA technical note D-6102, Washington, DC, January, 1971, p. 1–36.



[5] Thomson RG, Hayduk RJ. An analytical evaluation of the denting of airplane surfaces by hail. NASA technical note D-5363, Washington, DC, August 1969, p. 1–34. [6] Belytschko T, Liu WK, Moran B. Non linear finite elements for continua and structures. New York: Wiley; 2000. [7] Kim H, Kedward KT. Experimental and numerical analysis correlation of hail ice impacting composite structures. AIAA-99-1366, 1999, p. 1416–26. [8] Kim H, Kedward KT. Modeling hail ice impacts and predicting impact damage initiation in composite structures. Am Inst Aeronautics Astronautics 2000;38(7):1278–88. [9] Schulson EM. The structure and mechanical behavior of ice, Army Research Office. J Met 1999;51(2):21–7. [10] Anghileri M, Castelletti L-ML, Invernizzi F, Mascheroni M. A numerical model for hail impact analysis. 34th European rotorcraft forum, September 2004. [11] Brockman RA, Held TW. Explicit finite element method for transparency impact analysis. University of Dayton Research Institute, 1991. [12] Anghileri M, Castelletti L-ML, Lanzi L, Mentuccia F. Composite materials and bird-strike analysis using explicit finite element commercial codes. Eighth International Conference on Structures Under Shock and Impact (SUSI). 2004. [13] Accident Investigation Board Finland, Aircraft Damage in Hailstorm West of Helsinki on 21.7.2001. Investigation report B 5/2001 L , p. 1–31.

HAIL IMPACT

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