3D simulations of concrete penetration using SPH formulation and the RHT material model

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1 3D simulations of concrete penetration using SPH formulation and the RHT material model H. Hansson Weapons and Protection, Swedish Defence Research Agency (FOI), Sweden Abstract This paper describes work performed with the RHT concrete model, which shows promising behaviour for penetration/perforation simulations in concrete. The three dimensional simulations, using both Lagrange and SPH formulations, consider cases with varying impact conditions and target configurations. The simulations that consider normal impact conditions are compared with earlier benchmark tests. Keywords: numerical simulation, concrete penetration, RHT model, SPH, smooth particle hydrodynamics, meshless. 1 Introduction It is necessary to predict penetration depths in concrete targets to determine the vulnerability of underground structures. Several empirical and numerical tools have been developed to predict penetration depths of projectiles in concrete. Previously presented 2D simulations by Hansson [1] are extended here to 3D cases. An advanced material model is used to describe the material behaviour of concrete in this study, i.e. the RHT material model developed at EMI, see Riedel [2]. The material model is then combined with Lagrange and Smooth Particle Hydrodynamics (SPH) numerical formulations to study projectile penetration in concrete targets. A benchmark penetration test set-up is used for the initial 3D models, and the simulations are then modified to consider non-normal impact and reinforced concrete targets. The use of a 3D software package allows the simulation of non-symmetric impact cases.

212 Structures Under Shock and Impact VIII 2 Benchmark tests A benchmark test series with normal impact of 152 mm projectiles was conducted in 1999.

The properties of the concrete targets are compiled in table 1, and the benchmark test data in table 2, see Hansson [1, 3, 4]. A target after perforation is shown in figure 1.

2 212 Structures Under Shock and Impact VIII 2 Benchmark tests A benchmark test series with normal impact of 152 mm projectiles was conducted in These test data are used for comparison with the numerical simulations considering normal impact conditions. The exit velocity of the projectiles was determined with a high-speed video camera. The properties of the concrete targets are compiled in table 1, and the benchmark test data in table 2, see Hansson [1, 3, 4]. A target after perforation is shown in figure 1. Table 1: Properties of the concrete targets. Target/concrete properties Diameter Length f c, mm cylinders Tensile splitting strength, 150 mm cubes E c for mm cylinders Value 2.40 m 0.75 m 92 ± 2 MPa 6.5 ± 0.2 MPa 44.5 ± 0.9 GPa Table 2: Benchmark test data. Test No. 23 No. 24 No. 25 Mean value Projectile properties Diameter 152 mm, length 552 mm, ogive radius 380 mm, total mass 46.2 ±0.1 kg, case mass 38.8 kg V Impact (m/s) ± ± ± ±0. 2 V Exit (m/s) 183 ±6 204 ±4 181 ±4 190 ±14 3 Material modelling The RHT material model is implemented in the general release of Autodyn 4.2 or higher, and the model is presented by Riedel [2]. However, some modifications to the material model were made before the implementation in Autodyn. Earlier simulations with the material model have been presented by e.g. Riedel et al. [5]. Figure 1: Target after perforation for test no. 24, front face to the left.

3 Structures Under Shock and Impact VIII 213 The material model uses three strength surfaces, i.e. elastic limit surface, failure surface and remaining strength surface for the crushed material. There is also an option to use a cap on the elastic strength surface. The material model is briefly described by Hansson [1, 3, 4], and the material parameters used are given in table 3. The cap option is activated for all simulations in this study. The values of the material parameters chosen for the RHT model were based on unconfined and confined compressive strength, tensile splitting strength and estimations based on similar concrete types. The deviatoric strength model is combined with the P-α equation of state, see Herrmann [6], to obtain the concrete response to hydrostatic pressure, see figure 2. Table 3: Common parameters for simulations with the RHT concrete model. Parameter Value Parameter Value Parameter Value G 18 GPa BQ B 1.6 f c 92 MPa D M 0.61 f t /f c D 2 1 α (for F RATE ) f s /f c 0.30 EFMIN 0.05 δ A fail 1.9 PREFACT 2 SHRATD 0.13 N fail 0.6 TENSRAT 0.80 Q COMPRAT 0.75 A piece-wise linear strain hardening model with thermal softening and strain rate dependence together with a linear equation of state are used for the steel rebars. The material s strain hardening is based on quasi-static testing of B500BT type 1 rebar steel with 16 mm diameter. The stress-strain relationship for a representative steel bar and the approximation used for the numerical model are shown in figure 3. The material model uses the same strain rate dependence and thermal softening as the Johnson and Cook [7] model for 4340 steel Pressure (MPa) Experimental data from GREAC test performed at FFI Density (g/cc) Figure 2: Hydrostatic loading and unloading path for the concrete.

4 214 Structures Under Shock and Impact VIII Engineering stress (MPa) Rebar no Numerical model Engineering strain (mm/mm) Figure 3: Stress vs. strain for B500BT type 1 bars. 4 Three dimensional numerical simulations The simulations of concrete perforation were performed with Autodyn 3D version for the Lagrange models, and version for the SPH models. Hansson [4] has presented further data from the simulations. The numerical simulations were performed with Lagrange and SPH formulations for comparison. For the Lagrange formulation it is necessary to use numerical erosion, and the value 1.5 was used for the erosion strain for Lagrange models. However, this reduces the confinement in the vicinity of the projectile. Concrete has a pressure dependent yield stress and failure type, and the calculation results are therefore likely to be influenced by the use of numerical erosion. The use of e.g. Euler and SPH formulations do not require numerical erosion. 4.1 Model geometry The general model geometry of the target is a cylinder with 1.2 m radius and 0.75 m length. However, this geometry is modified for several models, e.g. for the study of reinforced targets where a rectangular cross section with 2.0 m sides is used instead. The target length for the model can vary a small amount depending on the size chosen for the SPH nodes, e.g. 20 mm nodes can not be equally divided into 750 mm. This is assumed to have negligible effects on the overall behaviour of the models. A free surface boundary condition is applied to all target surfaces. Exceptions from these default values are given in the tables with model descriptions and results. The 3D Lagrange models used a graded element mesh. A constant element size of either 10 or 5 mm was used for the central part of the target; with the radius of 50 cm. Graded elements with the size increased from 10 to 41.1 mm, or from 5 to 23.6 mm, were used for the outer part of the Lagrange models. This limits the number of elements far away from

5 Structures Under Shock and Impact VIII 215 the impact, and thus reduces the computational time considerably and with negligible effect on the projectile penetration path. A Lagrange mesh is used for the projectile in all 3D simulations. The total length of the projectile is 559 mm divided into a cylindrical part and an ogive nose part. The cylindrical part has a length of 324 mm and a radius of 76 mm, while the ogive nose radius is 380 mm. The element mesh size is about 21.5 mm and 7.4 mm parallel to and transverse to the projectile axis, respectively. In the benchmark test, the inner part of the projectile was instrumented with accelerometers. Thus, in order to obtain the correct projectile mass for the simulations the inner part of the projectile was modelled with a fictional material. 4.2 Lagrange simulations Penetration simulations using three-dimensional models, with the use of Lagrange formulation for both target and projectile meshes, are compiled in tables 4 and 5. 3D models allow problem geometries with e.g. varying impact angle or yaw/pitch with the use of half symmetry models. Thus, a minor study of the effects of different impact conditions for the projectile trajectory was performed. Friction between projectile and target are not considered in the presented simulations, except for models B99309 and B Table 4: Effects of friction and element size on the exit velocity. Model id. B99311 B99312 B99307 B99309 Element type / size Lagrange / 10 mm Lagrange / 5 mm Symmetry ¼ Friction coefficient Energy error 1.9% 8.0% 1.9% 8.5% Exit velocity 217 m/s 205 m/s 225 m/s 213 m/s As shown, the projectile speed is around 220 m/s after penetration for the majority of models. Furthermore, as in the earlier study of the 2D case presented by Hansson [1, 4], introducing a friction coefficient significantly affects the energy error. It is also seen, that decreasing the elements side length from 10 mm to 5 mm only results in a minor increase of the exit velocity of about 8 m/s. The main advantage of 3D models is the possibility of simulating oblique impact and other cases where non-symmetric models can be used. In table 5 the different impact cases are listed together with the simulation results. From the table it is first noted that at normal impact using ¼ or ½ symmetry results in a similar exit velocity, as expected. However, the small deviation of 1 between projectile axis and velocity vector gives a rather large effect on the projectile trajectory and the exit velocity decreases by 25 m/s to 186 m/s compared to the case with normal impact. An oblique impact angle causes different pressures on the upper and lower side of the projectile and as a consequence an increasing curvature of the trajectory. At exit, the absolute value of the velocity for model B99323 was decreased to 111 m/s.

6 216 Structures Under Shock and Impact VIII Table 5: Effects of geometry and impact angle on the exit velocity. Model id. B99320 B99321 B99323 B99311 Element type / size Lagrange / 10 mm Symmetry ½ ¼ Impact condition Normal impact Yaw=1 Impact angle 60 Normal impact Energy error 1.9% 1.9% 2.1% 1.9% Exit velocity 211 m/s 186 m/s 111 m/s 217 m/s 4.3 SPH simulations The focus of this part of the study was to evaluate the possibilities to use the SPH formulation for concrete targets in 3D penetration simulations. Simulations were performed both with one or two symmetry planes, different node sizes, different boundary conditions, with and without reinforcement, and different impact angles for the projectile. Friction between projectile and target was not considered for the SPH simulations. The SPH models with unreinforced targets are compiled in table 6. Comparing the models SPH000 and SPH204 show that for a normal impact condition the two symmetry cases give almost equal exit velocity. The advantage of the half symmetry model is the possibility to model non-normal impact events, e.g. oblique impact, as for models SPH200 and SPH203. The latter model uses a decreased node size to study the influence from this numerical parameter, see table 6. From this table it can be seen that both target size and the size of SPH nodes influence the exit velocity. The calculated damage evaluation in the target and the location of the projectile for the model SPH200 with 60 impact angle are shown in figure 4. Table 6: Unreinforced SPH models performed in the study. Model id. SPH000 SPH204 SPH200 SPH203 SPH206 Symmetry ¼ ½ Target type 2.4 m 3.0 m Target length 760 mm 750 mm 760 mm SPH node size 20 mm 12.5 mm 20 mm SPH nodes 103, , , ,452 Impact angle Energy error 5.1% 6.2% 12.2% 10.6% 6.2% Exit velocity 199 m/s 203 m/s 164 m/s 144 m/s 185 m/s Reinforcement was simulated with beam elements for front face and back face reinforcement ( 25 mm), and with truss elements for the longitudinal rebars ( 10 mm). The total number of beam and truss elements for the reinforcement was 1,812. The calculated SPH models with reinforced targets are compiled in table 7, i.e. models SPH220-SPH223 and SPH226. From this table and the previous table 6 it can be seen that the reinforcement decreases the calculated exit velocity by 15 to 30 m/s, depending on the impact angle. By

Structures Under Shock and Impact VIII 217 varying the impact point and impact angle of the projectile, and the location of reinforcement bars, an almost unlimited number of penetration cases can be

7 Structures Under Shock and Impact VIII 217 varying the impact point and impact angle of the projectile, and the location of reinforcement bars, an almost unlimited number of penetration cases can be simulated. Depending on the local geometry, and the interaction between projectile, concrete and rebars, these penetration cases are likely to result in different exit velocities. The calculated damage evaluation in the target and the status for the reinforcement for the reinforced model SPH221 with 60 impact angle are shown in figure 5. Figure 4: Calculated damage propagation in concrete target during penetration for model SPH200 at 1.2, 2.4 and 3.6 ms after impact. Table 7: Reinforced SPH models performed in the study. Model id. SPH220 SPH223 SPH221 SPH222 SPH226 Symmetry ½ Target type Rectangular with 2 m side Target length 760 mm 750 mm SPH node size 20 mm 12.5 mm SPH nodes 190, ,800 Impact angle Boundary condition Free surface Constrained target Free surface Energy error 8.3% 6.9% 9.5% 5.4% 2.7% Exit velocity 168 m/s 177 m/s 149 m/s 88 m/s 180 m/s The influence of decreased SPH node size was tested with model SPH226. However, there does not seem to be any significant difference in exit velocity between the models SPH223 and SPH226. For the unreinforced models the exit velocity decreased by approximately 20 m/s when the node size was decreased. Model SPH222 used a constrained boundary condition for confinement of the target, which allowed the surface to move within ±1 mm in the normal direction. The use of a constrained boundary condition for the target greatly decreased the exit velocity of the projectile. This is likely to be caused by the increased confinement of the target and projectile.

218 Structures Under Shock and Impact VIII Figure 5: Calculated concrete damage and reinforcement deformations for model SPH221 with target tilted 30 at 3.0 and 5.0 ms after impact.

The decrease of energy for the projectile is relatively constant for the models, with only small variations depending on the model parameters varied.

8 218 Structures Under Shock and Impact VIII Figure 5: Calculated concrete damage and reinforcement deformations for model SPH221 with target tilted 30 at 3.0 and 5.0 ms after impact. 5 Discussion and summary In general the simulations gave reasonable results for the different simulation cases. The decrease of energy for the projectile is relatively constant for the models, with only small variations depending on the model parameters varied. However, to be able to predict warhead performance against hardened structures of different types it is necessary to verify the use of these types of models for other projectile geometries, target configurations and materials. 3D simulations using the SPH formulation for the concrete targets, and the RHT concrete model results in a realistic interaction between projectile and target. Both Euler and SPH formulations make it possible to retain both the

9 Structures Under Shock and Impact VIII 219 material of the target and confinement around the projectile during the simulation. With these methods it is possible to start to identify problems regarding the material descriptions. For the simulations performed using a Lagrangian element formulation it is difficult to distinguish between errors caused by numerical problems, i.e. from distorted elements and erosion, and errors caused by the material models. It is clear that the 3D simulations can give additional and valuable information considering projectile penetration, where the 2D restriction to symmetric cases causes limitations. The use of 3D models allows oblique impact, as well as simulations with varying yaw and pitch, and must be considered necessary to predict the interaction between target and projectile. Simulations show that initial yaw and/or pitch of the projectile will result in increased stresses in the projectile during the penetration phase due to lateral forces. It is also important to note that a projectile without yaw or pitch and with a normal impact angle normally starts to rotate in the target. This is caused by inhomogeneities in the concrete and unsymmetric failure initiation in the target. Considering the concrete material description, the RHT model is promising. The strength is modelled by three surfaces, representing elastic limit, failure surface and finally a residual strength of the damaged material under compression. Furthermore, apart from pressure, the strength surfaces are dependent on the strain rate and the tri-axial stress state. However, to be able to predict penetration depths in concrete it is important that correct material properties are determined for the material used. This requires extensive laboratory testing at high pressures and deformation rates of the concrete type used. This material data should then be used to evaluate the material models and determine the values for the material parameters to be used in the models. Enhancement of the material models with statistical strength or failure strain is likely to improve the behaviour of several types of numerical models. In many cases the material cannot be considered homogeneous when fracturing and fragmentation are considered. Further improvement of the damage evolution models and determination of the residual strength of concrete are two major areas that should be considered for future research. These modifications to the material model will in the future enhance the probability to predict effects from penetrating warheads with large length to diameter ratios, dual charge warheads and multiple impact attacks on hardened concrete structures. During the penetration of a reinforced concrete target it is likely that rebars prohibit radial expansion and cracking, which results in a reduced penetration depth in relatively small targets. For large concrete targets it is therefore likely that the reduction of the penetration depths may be small for a reinforced target, when compared to an unreinforced target. The penetration depth is normally decreased when the size of the target is increased, especially if unconfined and unreinforced concrete targets are used. To obtain a lower limit for the penetration depths it is necessary to use unrealistically large concrete targets, therefore it is recommended that confined targets should be used for penetration tests. This ensures that the boundary conditions are known and it also reduces the radial expansion of relatively small targets. This study showed that the choice of

10 220 Structures Under Shock and Impact VIII suitable boundary conditions may be one of the most important considerations for the numerical models. This also applies to the design of penetration experiments if good quality data is to be obtained. For these ballistic experiments it is important that the boundary conditions are well defined and that the influence of the target size is known. References [1] Hansson, H., Modelling of concrete perforation. Proc. of Structures under Shock and Impact VII, WIT Press Southampton, pp , [2] Riedel, W., Beton unter dynamishen lasten, Meso- und makromechanische modelle und ihre parameter, EMI-Bericht 6/00, EMI Freiburg, [3] Hansson, H., Numerical simulation of concrete penetration with Euler and Lagrange formulation, FOI-R SE, FOI Tumba, [4] Hansson, H., 2D and 3D Simulations of Concrete Penetration using the RHT Material Model, FOI-R SE, FOI Tumba, [5] Riedel, W., Thoma K, Hiermaier S, Schmolinske E, Penetration of Reinforced Concrete by BETA-B-500. Numerical Analysis using a New Macroscopic Concrete Model for Hydrocodes. Proc. of 9 th Int. Symp. on Interaction of the Effects of Munitions with Structures, pp , [6] Herrmann, W., Constitutive Equation for the Dynamic Compaction of Ductile Porous Materials. Journal of Applied Physics, Vol 40, No 6, pp , [7] Johnson, G. R. and Cook, W.H., A constitutive model and data for metals subjected to large strains, high strain rates, and high temperatures. Proc. of 7 th Int. Symp. on Ballistics. The Netherlands, pp

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