Numerical modeling of damage in various types of hypervelocity experiments

INTERNATIONAL JOURNAL OF

IMPACT ENGINEERING PERGAMON

International Journal of Impact Engineering 23 (1999) 271-281 www.elsevier.com/locate/ijimpeng

N U M E R I C A L M O D E L I N G OF D A M A G E IN VARIOUS TYPES OF H Y P E R V E L O C I T Y EXPERIMENTS A.GEILLE CEA/CESTA, P.O. Box 2, 33114 Le Barp, France

Summary--This paper reviews some applications of numerical methods to predict damage in spalling or fragmentation problems encountered in bypervelocity experiments. After a brief description of these models, some numerical simulations are performed using the HESIONE hydrocode in its Euler and Lagrange versions coupled or not to specific algorithms and some correlations between experiments and calculations are provided when available to help evaluating the limits of the numerical techniques for the given problems. © 1999 Elsevier Science Ltd. All rights reserved.

DESCRIPTION O F T H E N U M E R I C A L M O D E L S The numerical approach must be carefully adapted to the requested specifications of the problem in order to perform the most accurate calculation. In the case of damage simulations, this very general rule takes a particular importance since the quality of the estimates is very sensitive to the nature of models.

Spall Models The spall process, as we understand it in this paper, is an instantaneous process which occurs at a given time, in a specific region of the material, depending only on its thermodynamic status, the organization of unloading waves and a specific value of the negative pressure afordable by the material. The spall geometry can be predicted either by the mean of a characteristic code or using a classical drawing of shocks and releases. The usual method to model this situation in hydrocodes is to compare the pressure in the cells to a negative threshold value and decide or not the opening of a void inside the material, immediatly returning pressure to zero if the cell is concerned (Fig. 1). Some additional refinements have been implemented (Tuler-Butcher [ 1],...), correlating the response to the time of application of the release by the way o f the strain rate. Nevertheless, none of these models can be considered as damage models because of their non-progressive effect.

Damage Models In most cases, the physics involved in hypervelocity problems cannot be reduced to the sudden opening of a well-defined spall inside materials under shock. The response of the material is progressive, initiated at randomly distributed micro-cracks locations, very early during the shock and release process. Two different kinds of numerical models are available to perform the simulations. Porosity Driven Models. These models are based on the replacement o f the pressure by a spe0734-743X/99/$ - see front matter © 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 9 9 ) 0 0 0 7 9 - 2

272

A. Geille / International Journal of Impact Engineering 23 (1999) 271-281

cific variable called porosity to handle the damage process. This variable contains all the information controling the damage effects inside the cells (nucleation, growth, coalescence of voids), but no real fragment is created. The damage numerical process is initiated by a minimum value of the porosity generally well above the limit negative pressure criterion applied in the spall models. A progressive unloading of the materials is applied which is much closer to the reality than spalling models. These models, well fitted to overall estimates about damage and basic predictions concerning clouds of debris, are usually implemented inside the hydrocodes (Johnson, N.A.G.). The Johnson model [2] is widely inspired from Carroll and Holt works [3], describing the evolution of spherical microcavities uniformly dispatched inside a non-elastic matrix, starting from a very small initial porosity up to a critical value leading to rupture. The originality of the model is to provide a simple formulation of the local behaviour of materials needing a few parameters to tune, most of them having a good physical connection. N.A.G.[4, 5], is based on a stastistical approach of the evolution of defaults sizes and locations inside the materials. More than 10 parameters are required to run the model making it very powerful but also very sensitive to the tuning data which are in this case collected from post-mortem analysis of experiments. Shrapnel Models. A different approach of the fragmentation process must be applied in problems where the composition of the cloud of debris generated in the damage process is the requested result or is the main driver for a subsequent process. For this category of problems, specific models are available to predict shrapnel populations, usually disconnected from the hydrocode because of their very different nature based on statistics. The hydrocode itself becomes a kind of sophisticated preprocessor required to propagate the shocks inside the materials up to the time the direction of computational mass cells are sufficiently established. At this time which must be carefully identified, the required data (strain rate, temperature, density and kinetic variables) are transferred to the fragmentation model and the population distribution is evaluated (Grady [6], Mott [7]). According to the Grady model, each cell potentially contributing to the fragment generation process is associated to a population of debris which are solid, ductile or liquid according to the intensity of the loading. The average size of the fragments is computed using energetic considerations (surface energy for liquid droplets, dynamic fracture for solids). Starting from this value, a Poisson distribution is elaborated with respect to the fundamental conservations, before integrating the contributions to all the generating cells and construction of the debris final distribution. Target

Pressure .....

Impactor

Damage model Spall model

Time Impact Velocity V

Minimum pressur in damage model

i ~ Rupture in damage model y

Spall tension "~"Rupture in spall model

Fig. la. Shock-induced spalling mechanism.

Fig. lb. Spall and damage models.

Fig. 1. Fragmentation process.

A. Geille / International Journal of lmpact Engineering 23 (1999) 271-281

273

APPLICATIONS OF THE MODELS Four examples have been selected in the domains of applications where CEA/CESTA is involved to illustrate the limits of the previously described models.

Simulation of Impact using a Spall Model This impact experiment (aluminum on titanium), performed at CESTA a few years ago using a high-explosive launcher of flyer plates (velocity 5.3 km/s), provides an efficient way to make a correlation between the experiment and a numerical simulation because of the very clean aspect of the target which has been recovered after the shot. The simulation required about 20 cells per millimeter in the horizontal direction to handle proper sharpness of the shock inside the materials leading to 200000 cells for the total run. Due to the shock intensity, the Equation of State was a standard Mie-Gruneisen and the target material strength model an Elastoplastic Steinberg one. A spall tension of -37 Kbars was applied. The comparison is made at Figure 2 after 30 ~ts, and shows a very good agreement concerning the crater dimensions but also the spall shape, diameter and thickness although the numerical conditions of the simulation are very standard.

2d. Overlay of the 2 images.

Titanium Aluminum V=5,3 km/s - -~"~

6,8 mm ~ 12 mm

" ~ l

8

-

O-mm

~N,~\\N] T 35 mm

Fig. 2a. Schematic drawing.

Fig. 2c. Numerics. Fig. 2. Comparison between numerics and experiment.

274

A. Geille / International Journal of lmpact Engineering 23 (1999) 271-281

Comparison between a Damage Model and a Spali Model in a Flyer Plate Impact Calculation The Johnson porosity model and the spall one are compared in the impact of a steel disc 0.2 mm thick flying at 5500 m/s onto a steel target 0.4 m m thick after 200 ns (Fig. 3). In the two models, the number of debris is so high that most of the fragments occupy a fraction of a cell. This means that the cloud composition is directly driven by the mesh generation procedure and the filling algorithm of the code. In that sense, the 2 models are equally unphysical. Nevertheless, in the Johnson model, behind a front spall much better identified, the density of fragments is continously varying starting from the solid in the back. This model allows to handle a more realistic population of debris, with small softened fragments flying at high velocity in the front part of the cloud and bigger solid ones flying more slowly behind, which is not the case for the simple spall model. area of variable density Front spall (solid)

V = 5500 m/s

Fig. 3b. Johnson Damage model. area of constant density fragments

Steel 0.2 mm

Front spall

Steel 0.4 mm

Fig. 3a. Schematic drawing.

°~

Fig. 3c. Spall model. Fig. 3. Comparison between Johnson model and a spall model.

275

A. Geille /International Journal of lmpact Engineering 23 (1999) 271-281

Multi-bumper Shields for Space Debris Applications The shielding technique used to protect spacecraft vehicles from space debris impact is based on the fragmentation of the projectiles impacting multiple walls placed one after the other resulting in a sharp decrease of the kinetic energy applied on the final backwall. CEA/CESTA recently performed experiments using the multistage active disk launcher of aluminum projectiles (1 gram) developed at CESTA for that purpose at a velocity of 11 km/s according to the schematic drawing described at Figure 4. Two X-Ray images were taken during the flight of the projectile to check its attitude and observe the clouds of debris, and a Fabry-Perot Velocimeter recorded the velocity versus time profile on the back face of the backwall, by the mean of 4 fiber optics. 2.5mm 2mm ~ 5mm 0.7mml

I

!

~I

/i

ll_Fabry_Perot

I

Fabry-Perot, 2 fibers

t

projectile

I

6~ r n m ~-gb- Fabry-Perot . y I Second wall l backwall aluminum I aluminum

~ 60 mm First wall aluminum

~ ~

I

X-Ray picture number 1 projectile in flight

X-Ray picture number 2 second cloud of debris

Fig. 4. Schematic drawing of the experiment and diagnostics. Experimental Results (Fig. 5). The X-ray image of the projectile in balistic flight shows a perfect shape and attitude with no evidence of fracturation or melting before the impact. The X-ray image of the second cloud of debris (snapped a few microseconds before impact on the backwall) needed a reconstruction process because of the very low density of the fragments hardly visible on the original image and not compatible with the printing resolution of this document. The FabryPerot record suggests a complete perforation of the backwall since the recorded limit velocity is higher than 1000 rn/s. 1200.0 . . . . . . . . . . . . . . . . . . . . . . .

1000.0 800.0

Backwall

600.0 400.0 200.0

0.0 Fig. 5a. First X-Ray picture. Fig. 5b. Second X-Ray picture. Fig. 5. Experimental results.

5 5 7'5 9'5 1i5

Time ([ts) Fig. 5c. Fabry-Perot velocity profile (m/s).

5

A. Geille /International Journal of Impact Engineering 23 (1999) 271-281

276

Numerics (Fig. 6). The numerical simulation of this experiment is driven by the 2 clouds of debris generated by the impact of the initial projectile on the 2 intermediate walls. To keep reasonable the cost of this calculation requiring a very refined grid, we selected a spall model to handle damage of the different walls. The comparison with experiment is very good concerning the shape and depth of the cloud, and the limit velocity measured by the Fabry-Perot. The main difference is located in the risetime of the velocity profile which is around 5 ~ts in the simulation and about 60 in the experiment. This discrepancy illustrates the importance of the fragmentation process to handle proper dislocation of materials and predict the content of each small impact contributing to the acceleration of the backwall. The limit velocity is accurately predicted because it depends only on the correct conservation of momentum and kinetic energy, and, at late time, all the impacts have been collected. A more accurate description of the velocity profile would be obtained using a real damage model, at a more expensive cost. I

i

!

Fig. 6a. Impact of the first cloud on the second wall.

3000

i

2000

c'ou i

Fig. 6b. Clouds of debris at time of X-Ray snapshot.

i

i

1000

0.0

Backwall

-1000

-2000

I

I

5

15

I

I

I

25 35 45 Time Gts) Fig. 6c. Velocity profiles (m/s) versus time. Fig. 6. Results of the numerical simulation.

A. Geille / International Journal of Impact Engineering 23 (1999) 271-281

277

Fragmentation of a Steel Tube under High Explosive Loading The problem of dynamic fracture of tubes and other axisymetrical hollow objects under internal explosive loading is very similar to the previous one in the sense that it needs also a complete understanding of the generation of schrapnels and debris. Some recent experiment has been performed at CESTA [8, 9] to evaluate the debris population generated by a steel cylinder 3 mm thick filled with high-explosive, a Fabry-Perot interferometer recording the velocity of expansion of the cylinder (Fig. 7). Experiments. The collected debris consists of a large distribution (about 800 fragments) of small to medium-size debris (below 2 mm) rather spherical, combined with some elongated strips of metal having the same thickness as the original tube. The shock-induced debris generated at early time of the simulation are combined with shear effects which can be attributed to the dynamic fracture induced by the mechanical limits of the steel while it is expanding.

A y

x

J

10 mm Collector (polyethylene)

lO~am \

,

# \

z

.-

. - "' ~ ~ 70 mm

1

"

,'~------ Steel tube (thickness 3 mm) ", x\

High explosive (diameter 10 mm)

\

",,

" %

Fig. 7. Schematic drawing of the experiment. Modeling. The modeling task has be separated in two different analysis to handle the different fragmentation regimes. The first simulation was performed using the Lagrange scheme of HESIONE coupled to a separated shrapnel generation algorithm as described previously and the second one a full euler run including a spall model. The 2 simulations were performed in 2D planar geometry, which means that the length of the cylinder is supposed to be infinite for the simulation purpose. The results concerning fragmentation populations are given per centimeter of the tube supposed infinite along its main axis. Figure 8 shows the main results concerning the prediction of fragments size and also a correlation between numerical and experimental free surface velocity.

A. Geille /International Journal of lmpact Engineering 23 (1999) 271-281

278

SIMULATION 2

SIMULATION 1

1

HESIONE 2D HYDROCODE LAGRANGE CALCULATION

EULER CALCULATION (spall model)

Free surface velocity 1500.0 "~1200 • 0

-

900.0 600.0 > 300.0

.-.

J

Strain rate

1.0e+06 Experiment

/

A

8.0e+05

& 6.0e+05 [\lk/ Numerics

.= 4.0e+05

I'J"'

ff~ 2.0e+05

00' J . . . . "0.0 2.0 4.0 6.0 8.0 10.0 Time (gs)

O.Oe+O~

.0 2.5 5.0 7.5 Time (~ts)

10

SHRAPNEL ALGORITHM

S,442¢+OS1 S.Og~i*OS 4.74~e+05 43S3t+OS 4o44l*o9 3~44t*o9 29S4*+OS 2.Z~S*+09

r

2000[

,

,

STS2t+OS

1500

1000

a.~*4St+OS

......... ii!ii!i

246.+09 I e.~61*+oe 5.4691.08 -1S27e÷08 -S.O24t*O8

500

I

a. STO..O8

00

1000

2000

3000

..........

Fig. 8a. Debris size (mm)

Fig. 8b. Surface fragmentation at 2.5 gs Fig. 8. Schematic diagram of the numerical simulations.

Comments about simulation 1. The velocity and strain rate time profiles have been evaluated in the vicinity of the free surface of the tube. Concerning the velocity profile, the agreement between Fabry-Perot recorded data and calculations is good, despite of some oscillations in the Lagrange run. The strain rate profile exhibits a peak value which is very high (about 106s-1) for a short period of time (<>on Figure 8). At this time, the first shock reaches the free surface of the steel tube and consequently initiates the shock-induced phase of fragmentation. The second period, much longer, concerns constraints induced by the expansion of the tube under the effect of the explosive pusher and is characterized by a slowly decreasing rate starting from 105 s -1 (<~B>>area on the strain rate profile).

A. Geille /International Journal of lmpact Engineering 23 (1999) 271-281

279

Consequence for the shrapnel populations. As mentioned in the model description part of this paper, the accuracy of the Grady fragmentation analysis depends strongly on the instant selected to stop the hydrocode calculation and activate the method. This time should be long enough to be sure that the interaction between shocks and unloading waves are terminated inside the material and that debris generating cells are in a full balistic flight towards the target. No rigorous criterion can be recommended for its selection which is very dependent on the problem to solve. The observation of the time variation of the thermodynamic and kinetic variables as the simulation proceeds is the only way to minimize the errors. Another possible way is to switch when the geometry of the material has expended a given time its original value. To illustrate the sensitivity of the results versus this selection, we performed the Grady analysis at 2 different instants: 1 and 10 gs. Thermodynamic and kinetic variables have been extracted from HESIONE Lagrange hydrocode run at these instants, and transferred to the shrapnel generation algorithm. The resulting populations are given at Figure 9 in a combined histogram of fragment sizes. The 1 Its simulation does not fully satisfy the previously described requirements since it is performed in the vicinity of the peak shock. The two distributions exhibit an overestimate of the small debris population and a very good fit of the highest range sizes, compared to the experimental results. This trend is general applying the Grady formulation at early time, when the strain rate is too high and not fully stabilized. For the smallest debris, it seems reasonable to assume that most of them have not been collected or identified in the experiment. Another possible explanation of the discrepancy is located in the numerical simulation which is performed in 2D, as the real experiment is at least locally 3D. 150 ~

,

,

,

,

t= 1 gs analysis t=

10 Its analysis

100

O

E Z

50

experiment 0 0

1000

2000

3000

4000

5000

size of debris (gm) Fig. 9. Distribution of fragments (histogram).

Comments about simulation 2. The Euler simulation provides only global results and does not allow to quantify debris. Nevertheless, the snapshot of the fragmentation process given at Figure 10 after 10 gs confirms the previous results concerning the fragmentation of the ring in a qualitative way. Large fragments supporting (low) positive pressures can be attributed to the dynamic fracture regime, and small fragments surrounded by fully unloaded areas to the shock-induced regime.

280

A. Geille /International Journal of lmpact Engineering 23 (1999) 271-281

PRE3510H 1.348e+08 I

and thick strips nic fracture regime)

1.036¢+08

7.248e+07 4.231e+07

-2.103e+07

~dium-size debris bly one or the other

-5.21~e+07

the 2 r e g i m e s )

1.014e+07

t'~ -8.33ge+07 -....-

-1.145e+08 -1.457e+08 -1.769e+05 -2,080e+08

1all spall-induced debris

-2.?04t+08 -3.01Se+08 -3.327e+08 -3.639e+08 -3.950e+08 -4.262¢+08 -4.574e+08

Fig. 10. Euler s i m u l a t i o n o f fragmentation @ t = 10 Its.

One question is still unsolved about the numerical or physical origin of the radial cracks observed. According to the 1D expanding hypothesis, all the cells located at the same radius are identical and must support the same tensions. Consequently, they must crack at the same time which does not happen. We must consider that some numerical noise initiates the very first cracks at particular locations. The initial crack location is not physical but the subsequent behaviour of the materials under the effect of the unloading process is physically driven. In the real material, the fracturation process is also randomly initiated, due to pre-existing micro-cracks or impurities trapped in the materials. This unexpected numerical noise looks to be an excellent way to introduce a random component in the simulation.

CONCLUSION Three different levels of numerical response to a given problem of damage inside shock-loaded materials have been summarized and several applications to experiments performed, allowing a gradual approach to solve the damage problem. The spall models predict fairly well the basic process caused by unloading waves inside materials but do not allow any sophisticated prediction of fragmentation. The damage models based on porosity control implemented inside hydrocodes allow a more realistic representation of the debris clouds and reasonable predictions of effects. The specific models usually disconnected from hydrocodes potentially provide the best results for microscopic predictions of the behaviour of materials, fragmentation aspects, and are the only available models to transport accurately fragments on a long distance of flight.

Acknowledgments--The author wishes to ackowledge the profitable discussions on these topics with R.Tockheim of Standford Research International and G.Talabart and R.Courchinoux from CEA.

A. Geille / International Journal of Impact Engineering 23 (1999) 271-281

281

REFERENCES 1. F.R.Tuler, B.M.Butcher, A criterion for the time dependence of dynamic fracture, Int. J. Fract. Mech. (4), pp.431437 (1968). 2. J.N. Johnson, Dynamic fracture and spallation in ductile solids, J.AppI.Phys., 52(4) (1981). 3. M.M. Carroll, A.C.Holt, Static and dynamic pore-collapse relations for ductile porous materials, J.Appl.Phys., 43(4), pp1626-1636 (1972). 4. D.R.Curran• L.Seaman• D.A. Sh•ckey• C•mputati•na• pp 4814-4826 (1976).

m•de•s f•r ducti•e and britt•e fracture• J.Appl.Phys.• 47( • • )•

5. Lee Davison, Dennis Grady, Mohsen Shahinpoor, High-Pressure Shock Compression of solids 11, Dynamic fracture and fragmentation, Springer (1996). 6. D.E. Grady, Particle size statistics in dynamic fragmentation, J.AppLPhys., 68(12) (1990). 7. N.F.Mott, Fragmentation of Shell Cases. Proc.R.Soc., London A. 189, pp.300-308 (1947). 8. Daniel Schirmann, Alan Burnham, LMJ/NIF Project, Target area development, Tasks 97-3, Lawrence Livermore National Laboratory (1997). 9.

D.Schirmann, L.Bianchi, R.Courchinoux, C.Cordillot, C.Dubern, A.Fornier, A.Geille, F.Jequier, J.C.Gommr, G.Sibille and J.P.Thrbault, Engineering physics inside the LMJ target chamber, Proc. Symposium on Fusion Engineering, San Diego (1997).