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Numerical and empirical approach in predicting the penetration of a concrete target by an ogive-nosed projectile
Finite Elements in Analysis and Design 42 (2006) 1258 – 1268 www.elsevier.com/locate/finel
Numerical and empirical approach in predicting the penetration of a concrete target by an ogive-nosed projectile C.Y. Tham ∗ Computational Mechanics Division, Institute of High Performance Computing, 1 Science Park Road, #01-01 The Capricorn, Singapore Science Park II, Singapore 117528, Singapore Received 5 December 2005; accepted 17 June 2006 Available online 17 August 2006
Abstract This paper demonstrates the application of both numerical simulation and empirical equation in predicting the penetration of a concrete target by an ogive-nosed projectile. The results from the experiment performed by Gran and Frew [In-target radial stress measurements from penetration experiments into concrete by ogive-nose steel projectiles, Int. J. Impact Eng. 19 (8) (1997) 715–726] are used as a benchmark for comparison. In the numerical simulations a 3.0-caliber radius-head steel ogival-nose projectile with a mass of 2.3 kg is fired against cylindrical concrete target with a striking velocity of 315 m/s. The simulation, performed using AUTODYN 2-D, assesses three numerical schemes, namely Langrange, Euler–Lagrange coupling and smooth particles hydrodynamics SPH–Lagrange coupling, in predicting the maximum depth of penetration and the radial stress–time response of the concrete target. When assessing the three solution techniques we hypothesize that the effect of strain rate on strength for the concrete target does not adversely affect the prediction on the maximum depth of penetration and the radial stress–time response of the concrete target. In the empirical approach the penetration equation developed by Forrestal et al. [An empirical equation for penetration depth of ogive-nose projectiles into concrete targets, Int. J. Impact Eng. 15 (4) (1994) 395–405] is used to determine the maximum depth of penetration and the deceleration–time response. The deceleration–time response for the projectile using the empirical approach is compared with those obtained from the numerical simulations. Results from both the numerical and empirical approaches are consistent. The calculated depth of penetration from both approaches yield relatively good agreement with that obtained from the experiment. The numerical simulations using each of the three numerical schemes are also able to reproduce the profiles from the radial stress measurements. Simulations using the SPH numerical scheme give the best overall agreement. The good overall agreement with the experimental radial stress measurements and consistent results between both empirical and numerical approach, enhanced the confidence in engineers and ballisticians when using these two approaches in complementing full-scale testing. 䉷 2006 Elsevier B.V. All rights reserved. Keywords: Projectile; Concrete; Penetration; Hydrocode; Penetration equation; Radial stresses; Deceleration–time
1. Introduction Concrete has been used extensively as a construction material for buildings, bridges, tunnels and nuclear reactor containments. In defense applications, concrete is used as a structural material for runways, command bunkers and hardened shelters. Concrete, which was invented in mid18th century, is a heterogeneous material, an embodiment of components, from fine cement to coarse aggregates. Each of these components exhibits different strengths and modulus of ∗ Tel.: +65 6419 1235; fax: +65 6419 1280.
E-mail address: [email protected]. 0168-874X/$ - see front matter 䉷 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2006.06.011
elasticity. And when subjected to a complex system of forces these component materials deform at different rates. Concrete is known for its availability and as a cost-effective material, more importantly it is known for its high compressive strength. Despite its strength and ubiquity, this heterogeneous material suffers certain limitations—inherently brittle, very little capacity for inelastic deformation before failure and very weak in tension. Low tensile strength has long been recognized as a shortcoming of concrete. However, structural engineers have circumvented this weakness through the use of reinforced concrete. This combination of concrete and steel has also served to enhance the overall bending strength of concrete.
C.Y. Tham / Finite Elements in Analysis and Design 42 (2006) 1258 – 1268
In many defense agencies, the complex interaction of a projectile with concrete has been a subject of interest to both civil engineers and ballisticians. To civil engineers this complex interaction provides preliminary assessment on the survivability of structural elements, while to ballisticians it is an indication on the lethality of projectiles. The complex interaction of a projectile with concrete can be predicted using tools developed over the past three decades. These tools can be broadly identified as: (1) empirical equations [1,2] (2) the application of cavity expansion theory [3–5] and (3) hydrocode simulations [6,7]. In the past when computing power was modest and not easily accessible, empirical equations were relied heavily in predicting the maximum depth of penetration of geological materials. These equations have the ability to predict penetration depth with high reliability under conditions similar to those used in developing the equations. Hence even with today’s technology, empirical equations are still being used in predicting the depth of penetration. Forrestal et al. [1] developed an empirical equation for penetration depth of ogive-nose projectiles penetrating concrete targets with normal incidence. The equation contains a constant R which is obtained from depth of penetration versus striking velocity experiments. This penetration equation has since been extended to predict the depth of penetration for limestone [8] at normal impact, and the results are very encouraging. In this computing age, the proliferation of faster and cheaper computing power is beginning to allow computer codes to play a leading role in predicting complex interaction of projectile with concrete. These computer codes, also known as hydrocodes, are representation of the conservation laws for a continuum using different numerical schemes. These numerical schemes, which were developed over the last four decades, are: (1) Lagrangian, (2) Eulerian and (3) smooth particles hydrodynamics (SPH). Chen [6] performed numerical simulations, using LS-DYNA2D, to study the perforation of concrete targets by steel projectiles. In the simulations, the projectiles and the concrete targets were modeled using the Lagrangian numerical schemes. The calculations showed good correlation with experimental data. Holmquist et al. [7] developed a constitutive model for concrete. Perforation simulations using this constitutive model were performed and benchmarked against experimental test data. The perforation of the concrete targets, which were simulated using a Langrangian finite-element code, compared reasonably well with experimental data. In the past many of these simulations were performed using mainframes and supercomputers. However with today’s computing power, most hydrocodes are available in version for running on personal computers and laptops. Thus hydrocodes have now grown to become a popular tool for civil engineers and ballisticians, for complementing penetration/perforation experiments and for exploring design variations. This paper aims to provide a complete description of the interactions’ dynamics during the penetration of an ogivenose projectile against a cylindrical concrete target through hydrocode simulations. Three solution techniques, namely Langrange, Euler–Lagrange coupling and SPH–Lagrange coupling, implemented in AUTODYN-2D, a commercial
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hydrocode, are compared based on their ability in predicting the maximum depth of penetration and radial stress–time response of the concrete target. The comparison will identify factors (i.e. cell size and erosion strain), that engineers and ballisticians should consider when selecting the aforementioned solution techniques. When comparing the three solution techniques, we hypothesize that the effect of strain-rate on strength for the concrete target does not adversely affect the prediction on the maximum depth of penetration, the projectile motion and the radial stress–time response of the concrete target. Empirical calculation on the maximum depth of penetration is also performed using the penetration equation developed by Forrestal et al. [1]. Also presented in this paper are the numerical and empirical results for the deceleration–time response for the projectile. The numerical results and the empirical calculation are compared with the test data obtained by Gran and Frew [9]. The comparison will enhance the confidence of engineers and ballisticians when using these two approaches in complementing full-scale testing. 2. Numerical approach A computer program that is capable of computing strains, stresses, velocities and propagation of shock waves as a function of time and position is known as a hydrocode [10,11]. In a hydrocode simulation, the response of a continuum subjected to dynamic loading is governed by the conservation of mass, momentum and energy, and also the equation of state (EOS) and constitutive relation of the continuum. The EOS takes into account the effects of compressibility of the continuum, whereas the constitutive relation represents the continuum’s resistance to shear. In this paper the hydrocode simulations are performed using AUTODYN-2D [12], a fully integrated and interactive hydrocode developed by Century Dynamics. One unique feature of AUTODYN is that it allows different parts of a problem to be modeled with an appropriate numerical formulation available. This allows users to couple different solution techniques for a given problem. 2.1. Hydrocode model In this paper, three solution techniques, namely Lagrange, Euler–Lagrange coupling and SPH–Lagrange coupling are employed. They are each assigned to the concrete target and are compared based on their ability to predict the maximum depth of penetration and the radial stress–time response. The ogivenosed projectile’s casing and filler material is modeled using the Lagrange numerical formulation. Fig. 1 presents the mesh for the casing and filler material of the projectile. In the hydrocode model the axial-symmetry condition is imposed on the x-axis to reduce the size of the computational domain which resulted in mere 164 and 210 nodes for the casing and filler material, respectively. The steel culvert confining the concrete target is modeled using shell elements. Fig. 2 provides details on the cylindrical concrete target. The ratio of the target diameter to the projectile diameter is D/d = 27. Fig. 3 shows the initial condition for the hydrocode model. During the
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Projectile Mass ~ 2.3 kg
152.40 83.92 R13.97
Filler density = 1.60 Mg/m3
127.00
43.64 50.80 38.10
39.37
3.58 Casing Density = 7.83 Mg/m3
12.70 355.60 Dimensions are in mm Fig. 1. The mesh for the casing and filler material of the projectile.
penetration the concrete target will experience severe deformation, while the projectile, which has a striking velocity of 315 m/s, is likely to suffer only minor nose erosion. Thus, the constitutive relation for the casing and filler material is defined using an elastic-perfectly plastic yield function (i.e. von Mises strength model). Table 1 presents the model input for the casing and filler material.
Projectile
Steel Culvert (Thickness =2.54)
Concrete Target
Dimensions are in mm
Diameter = 1370
Fig. 2. The projectile and the cylindrical concrete target.
Fig. 3. Initial-condition for the hydrocode model.
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2.1.1. Lagrange solution technique The concrete target is modeled using a Lagrangian mesh. Three cell sizes are examined and they are 4, 5 and 6.1 mm. And their corresponding maximum I and J -index are 301 × 101, 245 × 61 and 210 × 75, with grading in the y-direction. Fig. 4 shows the impact regions of the cylindrical target modeled with the 4, 5 and 6.1 mm cells. The contact, sliding and separation between the projectile and the concrete target are defined using the gap interaction logic. With the gap interaction logic, each surface segment is surrounded by a contact detection zone. And the radius of this detection zone is called the gap size. Any nodes entering the detection zone of a surface segment are repelled by a force proportional to the depth of penetration of the node into the detection zone. Since the concrete target will experience severe deformation during the penetration, calculations using the Lagrange solution technique will result in excessive cell distortion and tangling. To prevent the calculation from terminating prematurely due to cell distortion and tangling, an erosion logic based on an incremental geometric strain is assigned to the concrete target. The erosion logic, which is not a representation of the physical process, automatically removes the distorted cells when the incremental geometric strain of these cells exceeds a predefined setting. The compressive strength of a cell is lost when it is eroded. Thus, cells should only be eroded when they are severely distorted and its compressive strength is not likely to effect the calculation. In [12] the recommended value for the erosion is 150–200%. To ensure that the cells are severely eroded and the erosion does not effect the calculation, the erosion is set to 350%.
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Table 1 Parameters for the von Mises strength model for the casing and filler material Material
Density (g/cm3 )
Bulk modulus (kPa)
Shear modulus (kPa)
Yield stress (kPa)
Casing Filler
7.83 1.6
1.7167E + 08 2.1248E + 07
7.923E + 07 4.54E + 06
1.60E + 06 2.0E + 05
Fig. 4. Impact region of the concrete target modeled using three different cell sizes for the Lagrange solution technique.
Fig. 5. Impact region of the concrete target modeled using three different cell sizes for the Euler–Lagrange coupling.
2.1.2. Euler–Lagrange coupling In this solution technique the projectile, modeled using a Lagrangian mesh, is coupled to the concrete target modeled using an Eulerian mesh. This coupled solution technique leverage on the strength of both the Lagrangian and Eulerian solution techniques. Brown et al. [13] recommended the use of coupled Eulerian–Lagrangian methods for earth penetrating weapon applications. In an Eulerian mesh, the cells are fixed in space and material passes through them. In modeling the concrete target, three cell sizes are examined and they are 2.5, 4 and 5 mm. Their corresponding maximum I and J -index are 551×147, 344×92 and 276 × 75, with grading in the y-direction. Fig. 5 shows impact regions of the cylindrical target modeled using 2.5, 4 and 5 mm cells. The coupling of an Euler and a Lagrange solution technique requires the specification of an Euler–Lagrange interface. At the interface the Lagrange domain (i.e. the projectile) acts as a physical constraint to the Euler domain, while the Euler domain (i.e. concrete target) provides a pressure boundary condition to the Lagrange domain. This Euler–Lagrange interface is defined by a polygon constructed using a series of nodes.
In this simulation the polygon is defined using the nodes along the nose and shank of the projectile, enclosing the projectile in an anti-clockwise convention. 2.1.3. SPH–Lagrange coupling In this approach the projectile, modeled using a Lagrangian mesh, is coupled to the concrete target modeled using SPH particles. Unlike the Lagrangian solution technique, the SPH particles do not suffer from cell distortion and tangling when solving large deformation problems. However, SPH is a relatively new technique and is still undergoing continuous refinements. In modeling the concrete target the region in the concrete closest to the penetration (i.e. Y =0.100) is represented using SPH particles. Away from the penetration (i.e. Y = 100.1400 mm) the concrete is modeled using Lagrangian mesh. The SPH particles and Lagrangian cells are joined at the interface (i.e. Y = 100). The joining of SPH particles with Lagrange cells allows material with low deformation to be modeled using the Lagrange solver. Comparing with a model which relies only on SPH particles, this mesh and mesh free technique requires a much shorter
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Fig. 6. Impact region of the concrete target modeled using three different particle sizes for the SPH–Lagrange coupling.
run time thus making this approach more efficient and flexible in modeling high velocity impact problems. In modeling the penetration of the projectile into the concrete target, three particle sizes (i.e. smoothing lengths) for the SPH domain are examined and they are 2.5, 4 and 5 mm consisting of 48 900, 19 278 and 12 250 particles respectively. Fig. 6 shows impact regions of the cylindrical target modeled using 2.5, 4 and 5 mm particles. The weighing function for the SPH solution technique provided in AUTODYN and used in this simulation is the cubic B-spline. In the SPH solution technique, the resolution length scale of the calculation is controlled by the smoothing length.
is modeled using the Porous EOS in AUTODYN. The EOS describes the compaction path of the concrete using a piecewise linear function, based on five density–pressure points. The compaction path, which models the collapse and plastic flow of void and cells of the concrete from its porous state to its fully compressed state, begins with an elastic path, followed by a plastic path and extend to a fully compacted path. Unloading and reloading at any point along the paths are based on its sound speed. The five density–pressure points are estimated using the pressure versus volumetric strain relation from the uniaxial strain experiment in [9]. Table 2 shows the input for the Porous EOS and Mohr–Coulomb strength model.
2.2. Strength model 2.4. Failure model It is known from experiments [14] that concrete exhibits a critical strain-rate or strain-rate threshold above which significant increases in strength are exhibited. This apparent strainrate threshold is approximately 1 s−1 for tension and 60 s−1 for compression. In the last decade, strength models which include strain-rate effect on strength were developed and were subsequently implemented in commercial hydrocodes [7,15]. However, the application of these strength models require input parameters, which can only be obtained from the dynamic testing of concrete specimens. In the publication by Frew and Gran [9], little is known about the behavior of the concrete specimen above the critical strain-rate. Given the striking velocity of the projectile is only 315 m/s, we hypothesize that the effect of strain-rate on strength may not adversely affect the prediction on the maximum depth of penetration, projectile motion and the radial stresses in the concrete target. The concrete target resistance to shear is hence defined using the Mohr–Coulomb model in AUTODYN. The model represents the pressure-yield behavior of the concrete target with a piecewise linear function, constructed using 10 pressure-yield points. The 10 pressure-yield points for the concrete target are obtained from the pressure-yield relation in the uniaxial strain experiments performed by Gran and Frew [9]. 2.3. Equation of state The compressibility and compaction of the concrete target, often expressed as a pressure versus volumetric strain relation,
Recognizing the fact that concrete is an order of magnitude weaker in tension than in compression, the tensile failure of the concrete target is modeled using the Hydrodynamic Tensile Failure criteria. Failure is initiated when the tensile pressure in the concrete target exceeds 2.4 MPa [16] and bulk failure is assumed to have occurred. The pressure is set to zero and the internal energy is recomputed and the material is assumed to have rehealed so that negative pressure may occur in the next time-step but limited by the hydrodynamic tensile limit. This criteria can be used to model spalling and cavitation. This simple tensile criterion is an oversimplification of failure of the concrete target. In practice, the failure would be expected to be dependent on the maximum principal tensile strain in relation to the maximum compaction imparted in the direction of that strain component. 2.5. Crack softening model The tensile crack softening model has been included to model realistically the failure of concrete. Instead of experiencing failure instantaneously, concrete will gradually lose its load carrying capacity as cracks propagate though. The model, which is based on fracture mechanics, isotropically reduces the loadcarrying capacity of a cell as a function of the crack opening strain, once the failure has been initiated. The reduction follows a linear softening slope defined as a function of the local cell size and a material parameter, the fracture energy, Gf .
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Table 2 Constants for the porous EOS and the Mohr–Coulomb strength model in AUTODYN-2D Parameter (g/cm3 )
Ref. density Solid sound speed (m/s) Porous sound speed (m/s) Density #1 (g/cm3 ) Density #2 (g/cm3 ) Density #3 (g/cm3 ) Density #4 (g/cm3 ) Density #5 (g/cm3 ) Pressure #1 (kPa) Pressure #2 (kPa) Pressure #3 (kPa) Pressure #4 (kPa) Pressure #5 (kPa) Shear modulus (kPa) Pressure #1 (kPa) Pressure #2 (kPa) Pressure #3 (kPa) Pressure #4 (kPa)
Value
Parameter
Value
2.44414 4.500E + 03 4.500E + 03 2.25000 2.26554 2.34375 2.44565 2.52310 0.000E + 00 7.692E + 04 2.231E + 05 4.923E + 05 8.000E + 05 1.640E + 07 0.000E + 00 1.433E + 04 1.039E + 05 2.000E + 05
Pressure #5 (kPa) Pressure #6 (kPa) Pressure #7 (kPa) Pressure #8 (kPa) Pressure #9 (kPa) Yield stress #1 (kPa) Yield stress #2 (kPa) Yield stress #3 (kPa) Yield stress #4 (kPa) Yield stress #5 (kPa) Yield stress #6 (kPa) Yield stress #7 (kPa) Yield stress #8 (kPa) Yield stress #9 (kPa) Hydro tensile limit (kPa) Crack softening, Gf (J/m2 ) – –
3.000E + 05 4.000E + 05 5.000E + 05 6.000E + 05 7.000E + 05 2.077E + 04 4.300E + 04 1.385E + 05 2.346E + 05 3.231E + 05 3.962E + 05 4.423E + 05 5.000E + 05 5.385E + 05 −2.400E + 03 113.8 – –
The fracture energy is derived from the fracture toughness, which is obtained from [17], based on a water-to-cement ratio of 0.55. Currently the model is not available for the Eulerian solution technique because failure flag is not transported between Euler cells, so for each Euler cell there is no history of the material status, and thus crack softening of the material cannot be applied. 3. Empirical penetration equation Forrestal and co-workers developed a penetration equation for concrete for ogive-nose projectiles [1]. The equation is derived from post-test observations on the penetration of concrete and soil. The penetration of concrete can be partitioned into two distinct processes namely crater formation, which begins from the surface of the target to a depth of two projectile shank diameter (i.e. 4a), and tunneling, which continues from the depth of the crater to the final penetration depth. The penetration equation contains an empirical constant, R, referred to as the target resistance in [18], which can be obtained from a set of penetration experiments. From [18] the empirical constant, R, is given by R=
N Vs2 3 (1 + 4a N /m) exp[2a 2 (P
− 4a)N/m] − 1
,
(1)
8 − 1 242
.
Target
(kg/m3 )
2250
m (kg)
Projectile 2a (mm)
2.3
50.8
3
Using these geometrical properties, the last column of Table 4 tabulates the target resistance parameter R calculated using Eq. (1). The final depth of penetration, P , for an ogive-nose projectile and a concrete target presented in [18] is defined as N V12 m P= ln 1 + + 4a, (3) 2a 2 N R where V1 is defined as follows: V12 =
mV 2s − 4a 3 R . m + 4a 3 N
(4)
Based on an average value of R, the final depth of penetration, P , can be calculated using the above equations. With this average value of R the projectile’s rigid-body deceleration versus time response is calculated using the equation given in [1] with R = Sf c. 4. Results and discussion
where N is defined as follows: N=
Table 3 Geometrical properties of the concrete target and the projectile
(2)
In Eq. (1), P is the final penetration depth, is density of the concrete target, while a and m are the projectile’s radius and mass respectively. Table 3 provides the material and geometrical properties of the concrete target and the projectile.
Results for the maximum depth of penetration from the numerical simulations and the empirical calculation are compared with the data obtained from the three penetration experiments by Frew and Gran [9]. The deceleration–time response for the projectile using the numerical simulation is also compared with that obtained from the empirical approach. To assess the three numerical schemes, radial stress results from the numerical
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Table 4 Target resistance parameter, R , with corresponding penetration data from [9] Test number
Target length (m)
Striking velocity, Vs (m/s)
Pitch/yaw (deg)
Penetration depth, P (m)
R (MPa)
1 2 3
1.22 1.22 1.22
320 310 316
0.2U/0.1R 0.0/0.3L 0.4D/0.3R
0.185 0.175 0.157
423 430 524
300.00 Test Data r/rp = 1.4 (ME53) Test Data r/rp = 1.5 (ME 56)
Radial Stress (MPa)
250.00
Test Data r/rp = 1.7 (ME 51) Lagrange 6.1mm
200.00
Lagrange 5.0mm Lagrange 4.0mm
150.00 100.00 50.00 0.00 0.0 -50.00
0.5
1.0
1.5
Time (ms)
(a) 300.00
Test Data r/rp = 2.1 (ME 50) Test Data r/rp = 2.1 (ME 46)
Fig. 7. Location of the gages within the concrete target.
simulation are also compared with those obtained from the experiments. 4.1. Radial stress–time response
Radial Stress (MPa)
250.00
Test Data r/rp = 2.1 (ME 44) Lagrange 6.1mm
200.00
Lagrange 5.0mm Lagrange 4.0mm
150.00 100.00 50.00 0.00 0.0 -50.00
0.5
1.0
1.5
Time (ms)
(b) 300.00
Test Data r/rp = 2.6 (ME 57) Test Data r/rp = 2.6 (ME 48)
250.00
Lagrange 6.1mm
Radial Stress (MPa)
In the numerical simulation, the radial stress within the concrete target at normalized axial positions of z/a = 4.0 and radial positions of r/a = 1.4, 2.1 and 2.6 are recorded. Fig. 7 shows the location of these recordings. Comparing the radial stress–time response from the simulations with those from the experiments, in Figs. 8, 9 and 10, simulations using the three numerical schemes (i.e. Lagrange, Euler–Lagrange coupling and SPH–Lagrange coupling) have a longer rise time, which could be due to the effect of strain-rate on strength. Nevertheless the numerical simulations, on the whole, demonstrate overall agreement with measured radial stresses. Fig. 8 reveals that with the Lagrange numerical schemes the radial stresses drop-off at later times than the experimental test results. Johnson et al. [16] attributed this to the erosion strain. He further explained that the drop-off at later times is the result of element erosion. When an element erodes all of the stresses in that element are instantaneously set to zero. In the Euler–Lagrange and SPH–Lagrange coupling there is no erosion strain to prevent mesh tangling in the concrete target. Thus their results are able provide a much better agreement with the experimental test results.
Lagrange 4.0mm
150.00 100.00 50.00 0.00 0.0 -50.00
(c)
Lagrange 5.0mm
200.00
0.5
1.0
1.5
Time (ms)
Fig. 8. (a) Comparison of Lagrange calculations with measured stresses at r/a ≈ 1.4 and z/a = 4.0, (b) comparison of Lagrange calculations with measured stresses at r/a = 2.1 and z/a = 4.0, (c) comparison of Lagrange calculations with measured stresses at r/a = 2.6 and z/a = 4.0.
4.2. Maximum depth of penetration Table 5 compares the maximum depth of penetration from three solution techniques, namely Lagrange, Euler–Lagrange
C.Y. Tham / Finite Elements in Analysis and Design 42 (2006) 1258 – 1268
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300.00
300.00 Test Data r/rp = 1.4 (ME53) Test Data r/rp = 1.5 (ME 56)
250.00
Test Data r/rp = 1.4 (ME53) Test Data r/rp = 1.5 (ME 56) Test Data r/rp = 1.7 (ME 51) SPH 5mm SPH 4mm SPH 2.5mm
250.00
Euler 5mm
200.00
Radial Stress (MPa)
Radial Stress (MPa)
Test Data r/rp = 1.7 (ME 51) Euler 4mm Euler 2.5mm
150.00 100.00 50.00
(a)
0.5
1.0
1.5
50.00
0.5
(a)
Time (ms) 300.00
1.0 Time (ms)
150.00 100.00 50.00
0.5
1.0
Test Data r/rp = 2.1 (ME 50) Test Data r/rp = 2.1 (ME 46) Test Data r/rp = 2.1 (ME 44) SPH 5mm SPH 4mm SPH 2.5mm
250.00 Radial Stress (MPa)
200.00
200.00 150.00 100.00 50.00 0.00 0.0
1.5
-50.00
0.5
1.0
(b)
Time (ms)
Time (ms) 300.00
300.00
Test Data r/rp = 2.6 (ME 57)
Test Data r/rp = 2.6 (ME 57) Test Data r/rp = 2.6 (ME 48)
250.00
Test Data r/rp = 2.6 (ME 48)
250.00
SPH 5mm
Euler 4mm
200.00
Radial Stress (MPa)
Radial Stress (MPa)
Euler 5mm
Euler 2.5mm
150.00 100.00 50.00
0.5
1.0 Time (ms)
(c)
SPH 4mm
200.00
SPH 2.5mm
150.00 100.00 50.00 0.00 0.0
1.5
-50.00
(c)
1.5
-50.00
(b)
0.00 0.0
1.5
300.00 Test Data r/rp = 2.1 (ME 50) Test Data r/rp = 2.1 (ME 46) Test Data r/rp = 2.1 (ME 44) Euler 5mm Euler 4mm Euler 2.5mm
250.00 Radial Stress (MPa)
100.00
-50.00
-50.00
0.00 0.0
150.00
0.00 0.0
0.00 0.0
200.00
0.5
1.0
1.5
-50.00 Time (ms)
Fig. 9. (a) Comparison of Euler–Lagrange calculations with measured stresses at r/a ≈ 1.4 and z/a = 4.0, (b) comparison of Euler–Lagrange calculations with measured stresses at r/a = 2.1 and z/a = 4.0, (c) comparison of Euler–Lagrange calculations with measured stresses at r/a=2.6 and z/a=4.0.
Fig. 10. (a) Comparison of SPH–Lagrange calculations with measured stresses at r/a ≈ 1.4 and z/a=4.0, (b) comparison of SPH–Lagrange calculations with measured stresses at r/a=2.1 and z/a=4.0, (c) comparison of SPH–Lagrange calculations with measured stresses at r/a = 2.6 and z/a = 4.0.
coupling, SPH–Lagrange coupling for different cell and particle sizes. The maximum depth of penetration from the empirical calculation is also presented in Table 5. In the experiments by Frew and Gran [9], the depth of penetration recorded were 185, 175 and 157 mm. Although the effect of strain-rate is not included in the strength model, the results from the three solution techniques demonstrate reasonably good agreement with the experimental data. Except for the Euler–Lagrange coupling with cell size of 4.0 and 5.0 mm, on the whole the penetration results from numerical and empirical approach are within the range recorded in the experiments. The results from the
Euler–Lagrange coupling tend to indicate that larger cell size produces higher maximum depth of penetration. One probable explanation is the advection step (i.e. the Euler step) in the Eulerian numerical technique, which introduces numerical diffusion on the solution. Another probable explanation is the absence of the crack softening model, which reduces mesh sensitivity. The results from the Euler–Lagrange coupling indicate that to achieve penetration depth similar to the experiments, finer cell size (i.e. < 2.5 mm) is needed. On the contrary the Lagrange numerical scheme with larger cell size produces lower maximum depth of penetration. With the Lagrange numerical
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Table 5 Results from the numerical and empirical approach Approach Lagrange solution technique Cell size (mm)
Euler–Lagrange coupling Cell size (mm)
SPH–Lagrange coupling Particle size (mm)
Penetration depth, P (mm)
Normalized penetration, P /a
4.0 5.0 6.1
177.8 175.0 166.7
7.00 6.89 6.56
2.5 4.0 5.0
185.8 197.8 249.3
7.31 7.79 9.81
2.5 4.0 5.0
184.8 181.7 184.5
7.28 7.15 7.26
170.9
6.73
185.0 175.0 157.0
7.3 6.9 6.2
Empirical penetration equation Test results from Gran & Frew [9] Test number
1 2 3
scheme and the Euler–Lagrange coupling, results on penetration depth with different cell sizes tend to infers that cell size does affect the maximum depth of penetration. And thus care must be taken in selecting cell size for the concrete target when employing these two solution techniques. However for the case of the SPH–Lagrange coupling, varying the particle size from 2.5 to 5.0 mm does not seem to result in any significant change in the depth of penetration. Fig. 11 shows the damage contours of the concrete target for the SPH–Lagrange coupling, with three different particle sizes. The damage contours along the radial and axial direction for three different particle sizes are almost identical. 4.3. Deceleration of projectile The deceleration–time response for the projectile using the empirical approach is compared with those obtained from the numerical simulations. Figs. 12–14 present the projectile’s deceleration versus time profiles calculated using the empirical penetration equation in [1] and the three numerical schemes, with different cell and particle sizes. Overall the results from both the numerical and empirical approach are consistent. Fig. 12 shows that there is a correlation between results from the Lagrange solution technique and predictions using the empirical equation. The projectile’s deceleration–time response in Fig. 12 further reveals that smaller cell size, using Lagrange solution technique, produces oscillations with lower amplitude. For the case of the numerical approach using the Euler–Lagrange coupling, Fig. 13 indicates that a larger cell size for this solution technique produces result that is less consistent with calculations using the empirical equation. A larger cell size, using the Euler–Lagrange solution technique, tends to have longer pulse duration and lower maximum deceleration. This lower maximum deceleration obtained is consistent with
Fig. 11. Damage contours of the concrete target for the SPH–Lagrange coupling using three different particle sizes.
C.Y. Tham / Finite Elements in Analysis and Design 42 (2006) 1258 – 1268 10000 0 0
0.2
0.4
0.6
0.8
1
1.2
Acceleration (G)
-10000
5. Conclusions -30000 -40000
Lag (4.0mm) Lag (5.0mm) Lag (6.1mm) Equation [2]
-60000 -70000 Time (ms)
Fig. 12. Acceleration versus time: Lagrange calculations and empirical model prediction.
10000 0 0
0.2
0.4
0.6
0.8
1
1.2
-10000 Acceleration (G)
penetration equation. Comparing Fig. 14 with Figs. 12 and 13 shows that the deceleration of the projectile calculated using the SPH–Lagrange coupling is least sensitive to changes in particle size.
-20000
-50000
-20000 -30000 -40000 -50000 Eul-Lag (2.5mm) Eul-Lag (4.0mm)
-60000
Eul-Lag (5.0mm) Equation [2]
-70000 Time (ms)
Fig. 13. Acceleration versus time: Euler–Lagrange calculations and empirical model prediction.
10000 0
0
0.2
0.4
0.6
0.8
1
1.2
-10000 Acceleration (G)
1267
-20000 -30000 -40000 -50000 SPH (2.5mm) SPH (4.0mm)
-60000 -70000
SPH (5.0mm) Equation [2]
Time (ms)
Fig. 14. Acceleration versus time: SPH–Lagrange calculations and empirical model prediction.
results on the maximum depth of penetration, presented in Table 5, which perhaps infers that a larger cell size tend to have lower resistance to penetration. Among the three numerical solution techniques, the SPH–Lagrange coupling produces results that are most consistent with predictions using the empirical
Predicting the complex interaction of a projectile with concrete has long been a subject of interest to both civil engineers and ballisticians. Over the years tools were developed to predict and describe this complex interaction and they were (1) empirical equations (2) the application of cavity expansion theory and (3) hydrocode simulations. For this research both empirical equation and hydrocode simulations were used to predict the penetration of a concrete target by an ogive-nosed projectile. The results from the experiment performed by Gran and Frew [9] were used as a benchmark for comparison. For the empirical approach, the equation developed by Forrestal et al. [1] was used to determine the maximum depth of penetration and the deceleration–time response of the projectile. With the hydrocode approach, the simulations were performed using AUTODYN 2-D. Three numerical schemes, namely Lagrange, Euler–Lagrange coupling and SPH–Lagrange coupling were compared based on their prediction on the maximum depth of penetration, radial stress–time response of the concrete target and the projectile’s decelerations–time response. In predicting the maximum depth of penetration using hydrocode simulation, different cell and particle sizes were used. Although results from the Euler–Lagrange coupling with cell size of 4 and 5 mm overestimated the maximum depth of penetration. Nevertheless the results from the SPH–Lagrange coupling and the Lagrange numerical scheme, which did not include the effect of strain-rate on strength, compared reasonably well with experimental results. The results from the calculation using the penetration equation also demonstrated good agreement with the experimental results. For the radial stress prediction, the computed stresses from the three numerical schemes have longer rise time, but overall the results from these numerical schemes were able to reproduce the profile from the radial stress measurements. We postulated that the longer rise time may be due to the effect of strain-rate on strength. The deceleration–time response for the projectile based on the empirical approach was also compared with those obtained from the numerical simulations. The results from the Euler–Lagrange coupling with cell size of 4 and 5 mm were less consistent with calculations using the empirical equation, in terms of pulse duration and maximum deceleration. However the deceleration–time response from the SPH–Lagrange coupling and the Lagrange numerical scheme demonstrated better consistency with the empirical approach. Comparing the three numerical schemes used results on maximum depth of penetration, radial stress predictions and deceleration–time response showed that the simulations using the SPH numerical scheme gave the best overall agreement. In summary, the aforementioned simulation results, on the radial stress–time response, the maximum
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depth of penetration and the deceleration of the projectile, revealed that: 1. For this concrete specimen and with this striking velocity, the strain-rate effect on strength for the concrete did not adversely affect the prediction on the maximum depth of penetration, the projectile motion and the radial stress–time response of the concrete target. 2. The results from the hydrocode simulations were consistent with the empirical calculations on the maximum depth of penetration and the deceleration–time response for the projectile. 3. For this penetration simulation, the SPH–Lagrange solution technique gave the best overall agreement for the maximum depth of penetration and the radial stress–time response of the concrete target. This solution technique also yielded consistent results with empirical calculation on the deceleration–time response for the projectile. 4. Cell size, for an Euler–Lagrange coupling, can influence the prediction on the maximum depth of penetration, and hence the deceleration–time response of the projectile. With this demonstration of consistent results between numerical simulation and empirical equation, and also their good correlation with results from experiments, engineers and ballisticians now have more reasons to be confident when using both empirical and numerical approaches in complementing their full-scale testing. References [1] M.J. Forrestal, B.S. Altman, J.D. Cargile, S.J. Hanchak, An empirical equation for penetration depth of ogive-nose projectiles into concrete targets, Int. J. Impact Eng. 15 (4) (1994) 395–405. [2] C.W. Young, Penetration equations, SAND97-2426, Sandia National Laboratories, Albuquerque, NM, 1997. [3] M.J. Forrestal, D.Y. Tzou, A spherical cavity-expansion penetration model for concrete targets, Int. J. Solids Struct. 34 (31–32) (1997) 4127–4146. [4] T.L. Warren, M.R. Tabbara, Spherical cavity-expansion forcing function in Pronto3d for application to penetration problems, SAND97-1174, Sandia National Laboratories, Albuquerque, NM, 1997.
[5] T.L. Warren, A.F. Fossum, D.J. Frew, Penetration into low-strength (23 MPa) concrete: target characterization and simulations, Int. J. Impact Eng. 30 (5) (2004) 477–503. [6] E.P. Chen, Numerical simulation of perforation of concrete targets by steel rods, in: E.P. Chen, V.K. Luk (Eds.), Advances in Numerical Simulation Techniques for Penetration and Perforation of Solids, American Society of Mechanical Engineers, vol. 171, 1993, pp. 181–188. [7] T.J. Holmquist, G.R. Johnson, W.H. Cook, A computational constitutive model for concrete subjected to large strain rates and high pressures, in: Proceedings of the Fourteenth International Symposium on Ballistics, Quebec City, Canada, 1993. [8] D.J. Frew, M.J. Forrestal, S.J. Hanchak, Penetration experiments with limestone targets and ogive-nose steel projectiles, ASME J. Appl. Mech. 67 (2000) 841–845. [9] J.K. Gran, D.J. Frew, In-target radial stress measurements from penetration experiments into concrete by ogive-nose steel projectiles, Int. J. Impact Eng. 19 (8) (1997) 715–726. [10] C.E. Anderson, An overview of the theory of hydrocodes, Int. J. Impact Eng. 5 (1987) 33–59. [11] J.A. Zukas, Introduction to Hydrocodes, Elsevier, Amsterdam, 2004. [12] AUTODYN, Theory Manual, Revision 4.0, Century Dynamics Inc., 1998. [13] K.H. Brown, S.P. Burns, M.A. Christon, Coupled Eulerian–Lagrangian methods for earth penetrating weapon applications, SAND2002-1014, Sandia National Laboratories, Albuquerque, NM, 2002. [14] C.A. Ross, Fracture of concrete at high strain-rate, in: S.P. Shah (Ed.), Toughening Mechanisms in Quasi-Brittle Materials, Kluwer Academic, Boston, 1991, pp. 577–596. [15] W. Reidel, K. Thoma, S. Hiermaier, E. Schmolinske, Penetration of reinforced concrete by BETA-B-500, Numerical analysis using a new macroscopic concrete model for hydrocodes, in: Proceedings of Ninth International Symposium on Interaction of the Effects of Munitions with Structures, Berlin, 1999. [16] G.R. Johnson, S.R. Beissel, T.J. Holmquist, D.J. Frew, Computed radial stresses in a concrete target penetrated by a steel projectile, in: N. Jones, D.G. Talaslidis, C.A. Brebbia, G.D. Manolis (Eds.), Structures Under Shock and Impact V, Computational Mechanics Publications, 1998, pp. 793–806. [17] G. Prokopski, B. Langier, Effect of water/cement ratio and silica fume addition on the fracture toughness and morphology of fractured surfaces of gravel concretes, Cem. Concr. Res. 30 (2000) 1427–1433. [18] M.J. Forrestal, D.J. Frew, J.P. Hickerson, T.A. Rohwer, Penetration of concrete targets with deceleration–time measurements, Int. J. Impact Eng. 28 (2003) 479–497.
Numerical and empirical approach in predicting the penetration of a concrete target by an ogive-nosed projectile C.Y. Tham ∗ Computational Mechanics Division, Institute of High Performance Computing, 1 Science Park Road, #01-01 The Capricorn, Singapore Science Park II, Singapore 117528, Singapore Received 5 December 2005; accepted 17 June 2006 Available online 17 August 2006
Abstract This paper demonstrates the application of both numerical simulation and empirical equation in predicting the penetration of a concrete target by an ogive-nosed projectile. The results from the experiment performed by Gran and Frew [In-target radial stress measurements from penetration experiments into concrete by ogive-nose steel projectiles, Int. J. Impact Eng. 19 (8) (1997) 715–726] are used as a benchmark for comparison. In the numerical simulations a 3.0-caliber radius-head steel ogival-nose projectile with a mass of 2.3 kg is fired against cylindrical concrete target with a striking velocity of 315 m/s. The simulation, performed using AUTODYN 2-D, assesses three numerical schemes, namely Langrange, Euler–Lagrange coupling and smooth particles hydrodynamics SPH–Lagrange coupling, in predicting the maximum depth of penetration and the radial stress–time response of the concrete target. When assessing the three solution techniques we hypothesize that the effect of strain rate on strength for the concrete target does not adversely affect the prediction on the maximum depth of penetration and the radial stress–time response of the concrete target. In the empirical approach the penetration equation developed by Forrestal et al. [An empirical equation for penetration depth of ogive-nose projectiles into concrete targets, Int. J. Impact Eng. 15 (4) (1994) 395–405] is used to determine the maximum depth of penetration and the deceleration–time response. The deceleration–time response for the projectile using the empirical approach is compared with those obtained from the numerical simulations. Results from both the numerical and empirical approaches are consistent. The calculated depth of penetration from both approaches yield relatively good agreement with that obtained from the experiment. The numerical simulations using each of the three numerical schemes are also able to reproduce the profiles from the radial stress measurements. Simulations using the SPH numerical scheme give the best overall agreement. The good overall agreement with the experimental radial stress measurements and consistent results between both empirical and numerical approach, enhanced the confidence in engineers and ballisticians when using these two approaches in complementing full-scale testing. 䉷 2006 Elsevier B.V. All rights reserved. Keywords: Projectile; Concrete; Penetration; Hydrocode; Penetration equation; Radial stresses; Deceleration–time
1. Introduction Concrete has been used extensively as a construction material for buildings, bridges, tunnels and nuclear reactor containments. In defense applications, concrete is used as a structural material for runways, command bunkers and hardened shelters. Concrete, which was invented in mid18th century, is a heterogeneous material, an embodiment of components, from fine cement to coarse aggregates. Each of these components exhibits different strengths and modulus of ∗ Tel.: +65 6419 1235; fax: +65 6419 1280.
E-mail address: [email protected]. 0168-874X/$ - see front matter 䉷 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2006.06.011
elasticity. And when subjected to a complex system of forces these component materials deform at different rates. Concrete is known for its availability and as a cost-effective material, more importantly it is known for its high compressive strength. Despite its strength and ubiquity, this heterogeneous material suffers certain limitations—inherently brittle, very little capacity for inelastic deformation before failure and very weak in tension. Low tensile strength has long been recognized as a shortcoming of concrete. However, structural engineers have circumvented this weakness through the use of reinforced concrete. This combination of concrete and steel has also served to enhance the overall bending strength of concrete.
C.Y. Tham / Finite Elements in Analysis and Design 42 (2006) 1258 – 1268
In many defense agencies, the complex interaction of a projectile with concrete has been a subject of interest to both civil engineers and ballisticians. To civil engineers this complex interaction provides preliminary assessment on the survivability of structural elements, while to ballisticians it is an indication on the lethality of projectiles. The complex interaction of a projectile with concrete can be predicted using tools developed over the past three decades. These tools can be broadly identified as: (1) empirical equations [1,2] (2) the application of cavity expansion theory [3–5] and (3) hydrocode simulations [6,7]. In the past when computing power was modest and not easily accessible, empirical equations were relied heavily in predicting the maximum depth of penetration of geological materials. These equations have the ability to predict penetration depth with high reliability under conditions similar to those used in developing the equations. Hence even with today’s technology, empirical equations are still being used in predicting the depth of penetration. Forrestal et al. [1] developed an empirical equation for penetration depth of ogive-nose projectiles penetrating concrete targets with normal incidence. The equation contains a constant R which is obtained from depth of penetration versus striking velocity experiments. This penetration equation has since been extended to predict the depth of penetration for limestone [8] at normal impact, and the results are very encouraging. In this computing age, the proliferation of faster and cheaper computing power is beginning to allow computer codes to play a leading role in predicting complex interaction of projectile with concrete. These computer codes, also known as hydrocodes, are representation of the conservation laws for a continuum using different numerical schemes. These numerical schemes, which were developed over the last four decades, are: (1) Lagrangian, (2) Eulerian and (3) smooth particles hydrodynamics (SPH). Chen [6] performed numerical simulations, using LS-DYNA2D, to study the perforation of concrete targets by steel projectiles. In the simulations, the projectiles and the concrete targets were modeled using the Lagrangian numerical schemes. The calculations showed good correlation with experimental data. Holmquist et al. [7] developed a constitutive model for concrete. Perforation simulations using this constitutive model were performed and benchmarked against experimental test data. The perforation of the concrete targets, which were simulated using a Langrangian finite-element code, compared reasonably well with experimental data. In the past many of these simulations were performed using mainframes and supercomputers. However with today’s computing power, most hydrocodes are available in version for running on personal computers and laptops. Thus hydrocodes have now grown to become a popular tool for civil engineers and ballisticians, for complementing penetration/perforation experiments and for exploring design variations. This paper aims to provide a complete description of the interactions’ dynamics during the penetration of an ogivenose projectile against a cylindrical concrete target through hydrocode simulations. Three solution techniques, namely Langrange, Euler–Lagrange coupling and SPH–Lagrange coupling, implemented in AUTODYN-2D, a commercial
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hydrocode, are compared based on their ability in predicting the maximum depth of penetration and radial stress–time response of the concrete target. The comparison will identify factors (i.e. cell size and erosion strain), that engineers and ballisticians should consider when selecting the aforementioned solution techniques. When comparing the three solution techniques, we hypothesize that the effect of strain-rate on strength for the concrete target does not adversely affect the prediction on the maximum depth of penetration, the projectile motion and the radial stress–time response of the concrete target. Empirical calculation on the maximum depth of penetration is also performed using the penetration equation developed by Forrestal et al. [1]. Also presented in this paper are the numerical and empirical results for the deceleration–time response for the projectile. The numerical results and the empirical calculation are compared with the test data obtained by Gran and Frew [9]. The comparison will enhance the confidence of engineers and ballisticians when using these two approaches in complementing full-scale testing. 2. Numerical approach A computer program that is capable of computing strains, stresses, velocities and propagation of shock waves as a function of time and position is known as a hydrocode [10,11]. In a hydrocode simulation, the response of a continuum subjected to dynamic loading is governed by the conservation of mass, momentum and energy, and also the equation of state (EOS) and constitutive relation of the continuum. The EOS takes into account the effects of compressibility of the continuum, whereas the constitutive relation represents the continuum’s resistance to shear. In this paper the hydrocode simulations are performed using AUTODYN-2D [12], a fully integrated and interactive hydrocode developed by Century Dynamics. One unique feature of AUTODYN is that it allows different parts of a problem to be modeled with an appropriate numerical formulation available. This allows users to couple different solution techniques for a given problem. 2.1. Hydrocode model In this paper, three solution techniques, namely Lagrange, Euler–Lagrange coupling and SPH–Lagrange coupling are employed. They are each assigned to the concrete target and are compared based on their ability to predict the maximum depth of penetration and the radial stress–time response. The ogivenosed projectile’s casing and filler material is modeled using the Lagrange numerical formulation. Fig. 1 presents the mesh for the casing and filler material of the projectile. In the hydrocode model the axial-symmetry condition is imposed on the x-axis to reduce the size of the computational domain which resulted in mere 164 and 210 nodes for the casing and filler material, respectively. The steel culvert confining the concrete target is modeled using shell elements. Fig. 2 provides details on the cylindrical concrete target. The ratio of the target diameter to the projectile diameter is D/d = 27. Fig. 3 shows the initial condition for the hydrocode model. During the
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Projectile Mass ~ 2.3 kg
152.40 83.92 R13.97
Filler density = 1.60 Mg/m3
127.00
43.64 50.80 38.10
39.37
3.58 Casing Density = 7.83 Mg/m3
12.70 355.60 Dimensions are in mm Fig. 1. The mesh for the casing and filler material of the projectile.
penetration the concrete target will experience severe deformation, while the projectile, which has a striking velocity of 315 m/s, is likely to suffer only minor nose erosion. Thus, the constitutive relation for the casing and filler material is defined using an elastic-perfectly plastic yield function (i.e. von Mises strength model). Table 1 presents the model input for the casing and filler material.
Projectile
Steel Culvert (Thickness =2.54)
Concrete Target
Dimensions are in mm
Diameter = 1370
Fig. 2. The projectile and the cylindrical concrete target.
Fig. 3. Initial-condition for the hydrocode model.
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2.1.1. Lagrange solution technique The concrete target is modeled using a Lagrangian mesh. Three cell sizes are examined and they are 4, 5 and 6.1 mm. And their corresponding maximum I and J -index are 301 × 101, 245 × 61 and 210 × 75, with grading in the y-direction. Fig. 4 shows the impact regions of the cylindrical target modeled with the 4, 5 and 6.1 mm cells. The contact, sliding and separation between the projectile and the concrete target are defined using the gap interaction logic. With the gap interaction logic, each surface segment is surrounded by a contact detection zone. And the radius of this detection zone is called the gap size. Any nodes entering the detection zone of a surface segment are repelled by a force proportional to the depth of penetration of the node into the detection zone. Since the concrete target will experience severe deformation during the penetration, calculations using the Lagrange solution technique will result in excessive cell distortion and tangling. To prevent the calculation from terminating prematurely due to cell distortion and tangling, an erosion logic based on an incremental geometric strain is assigned to the concrete target. The erosion logic, which is not a representation of the physical process, automatically removes the distorted cells when the incremental geometric strain of these cells exceeds a predefined setting. The compressive strength of a cell is lost when it is eroded. Thus, cells should only be eroded when they are severely distorted and its compressive strength is not likely to effect the calculation. In [12] the recommended value for the erosion is 150–200%. To ensure that the cells are severely eroded and the erosion does not effect the calculation, the erosion is set to 350%.
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Table 1 Parameters for the von Mises strength model for the casing and filler material Material
Density (g/cm3 )
Bulk modulus (kPa)
Shear modulus (kPa)
Yield stress (kPa)
Casing Filler
7.83 1.6
1.7167E + 08 2.1248E + 07
7.923E + 07 4.54E + 06
1.60E + 06 2.0E + 05
Fig. 4. Impact region of the concrete target modeled using three different cell sizes for the Lagrange solution technique.
Fig. 5. Impact region of the concrete target modeled using three different cell sizes for the Euler–Lagrange coupling.
2.1.2. Euler–Lagrange coupling In this solution technique the projectile, modeled using a Lagrangian mesh, is coupled to the concrete target modeled using an Eulerian mesh. This coupled solution technique leverage on the strength of both the Lagrangian and Eulerian solution techniques. Brown et al. [13] recommended the use of coupled Eulerian–Lagrangian methods for earth penetrating weapon applications. In an Eulerian mesh, the cells are fixed in space and material passes through them. In modeling the concrete target, three cell sizes are examined and they are 2.5, 4 and 5 mm. Their corresponding maximum I and J -index are 551×147, 344×92 and 276 × 75, with grading in the y-direction. Fig. 5 shows impact regions of the cylindrical target modeled using 2.5, 4 and 5 mm cells. The coupling of an Euler and a Lagrange solution technique requires the specification of an Euler–Lagrange interface. At the interface the Lagrange domain (i.e. the projectile) acts as a physical constraint to the Euler domain, while the Euler domain (i.e. concrete target) provides a pressure boundary condition to the Lagrange domain. This Euler–Lagrange interface is defined by a polygon constructed using a series of nodes.
In this simulation the polygon is defined using the nodes along the nose and shank of the projectile, enclosing the projectile in an anti-clockwise convention. 2.1.3. SPH–Lagrange coupling In this approach the projectile, modeled using a Lagrangian mesh, is coupled to the concrete target modeled using SPH particles. Unlike the Lagrangian solution technique, the SPH particles do not suffer from cell distortion and tangling when solving large deformation problems. However, SPH is a relatively new technique and is still undergoing continuous refinements. In modeling the concrete target the region in the concrete closest to the penetration (i.e. Y =0.100) is represented using SPH particles. Away from the penetration (i.e. Y = 100.1400 mm) the concrete is modeled using Lagrangian mesh. The SPH particles and Lagrangian cells are joined at the interface (i.e. Y = 100). The joining of SPH particles with Lagrange cells allows material with low deformation to be modeled using the Lagrange solver. Comparing with a model which relies only on SPH particles, this mesh and mesh free technique requires a much shorter
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Fig. 6. Impact region of the concrete target modeled using three different particle sizes for the SPH–Lagrange coupling.
run time thus making this approach more efficient and flexible in modeling high velocity impact problems. In modeling the penetration of the projectile into the concrete target, three particle sizes (i.e. smoothing lengths) for the SPH domain are examined and they are 2.5, 4 and 5 mm consisting of 48 900, 19 278 and 12 250 particles respectively. Fig. 6 shows impact regions of the cylindrical target modeled using 2.5, 4 and 5 mm particles. The weighing function for the SPH solution technique provided in AUTODYN and used in this simulation is the cubic B-spline. In the SPH solution technique, the resolution length scale of the calculation is controlled by the smoothing length.
is modeled using the Porous EOS in AUTODYN. The EOS describes the compaction path of the concrete using a piecewise linear function, based on five density–pressure points. The compaction path, which models the collapse and plastic flow of void and cells of the concrete from its porous state to its fully compressed state, begins with an elastic path, followed by a plastic path and extend to a fully compacted path. Unloading and reloading at any point along the paths are based on its sound speed. The five density–pressure points are estimated using the pressure versus volumetric strain relation from the uniaxial strain experiment in [9]. Table 2 shows the input for the Porous EOS and Mohr–Coulomb strength model.
2.2. Strength model 2.4. Failure model It is known from experiments [14] that concrete exhibits a critical strain-rate or strain-rate threshold above which significant increases in strength are exhibited. This apparent strainrate threshold is approximately 1 s−1 for tension and 60 s−1 for compression. In the last decade, strength models which include strain-rate effect on strength were developed and were subsequently implemented in commercial hydrocodes [7,15]. However, the application of these strength models require input parameters, which can only be obtained from the dynamic testing of concrete specimens. In the publication by Frew and Gran [9], little is known about the behavior of the concrete specimen above the critical strain-rate. Given the striking velocity of the projectile is only 315 m/s, we hypothesize that the effect of strain-rate on strength may not adversely affect the prediction on the maximum depth of penetration, projectile motion and the radial stresses in the concrete target. The concrete target resistance to shear is hence defined using the Mohr–Coulomb model in AUTODYN. The model represents the pressure-yield behavior of the concrete target with a piecewise linear function, constructed using 10 pressure-yield points. The 10 pressure-yield points for the concrete target are obtained from the pressure-yield relation in the uniaxial strain experiments performed by Gran and Frew [9]. 2.3. Equation of state The compressibility and compaction of the concrete target, often expressed as a pressure versus volumetric strain relation,
Recognizing the fact that concrete is an order of magnitude weaker in tension than in compression, the tensile failure of the concrete target is modeled using the Hydrodynamic Tensile Failure criteria. Failure is initiated when the tensile pressure in the concrete target exceeds 2.4 MPa [16] and bulk failure is assumed to have occurred. The pressure is set to zero and the internal energy is recomputed and the material is assumed to have rehealed so that negative pressure may occur in the next time-step but limited by the hydrodynamic tensile limit. This criteria can be used to model spalling and cavitation. This simple tensile criterion is an oversimplification of failure of the concrete target. In practice, the failure would be expected to be dependent on the maximum principal tensile strain in relation to the maximum compaction imparted in the direction of that strain component. 2.5. Crack softening model The tensile crack softening model has been included to model realistically the failure of concrete. Instead of experiencing failure instantaneously, concrete will gradually lose its load carrying capacity as cracks propagate though. The model, which is based on fracture mechanics, isotropically reduces the loadcarrying capacity of a cell as a function of the crack opening strain, once the failure has been initiated. The reduction follows a linear softening slope defined as a function of the local cell size and a material parameter, the fracture energy, Gf .
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Table 2 Constants for the porous EOS and the Mohr–Coulomb strength model in AUTODYN-2D Parameter (g/cm3 )
Ref. density Solid sound speed (m/s) Porous sound speed (m/s) Density #1 (g/cm3 ) Density #2 (g/cm3 ) Density #3 (g/cm3 ) Density #4 (g/cm3 ) Density #5 (g/cm3 ) Pressure #1 (kPa) Pressure #2 (kPa) Pressure #3 (kPa) Pressure #4 (kPa) Pressure #5 (kPa) Shear modulus (kPa) Pressure #1 (kPa) Pressure #2 (kPa) Pressure #3 (kPa) Pressure #4 (kPa)
Value
Parameter
Value
2.44414 4.500E + 03 4.500E + 03 2.25000 2.26554 2.34375 2.44565 2.52310 0.000E + 00 7.692E + 04 2.231E + 05 4.923E + 05 8.000E + 05 1.640E + 07 0.000E + 00 1.433E + 04 1.039E + 05 2.000E + 05
Pressure #5 (kPa) Pressure #6 (kPa) Pressure #7 (kPa) Pressure #8 (kPa) Pressure #9 (kPa) Yield stress #1 (kPa) Yield stress #2 (kPa) Yield stress #3 (kPa) Yield stress #4 (kPa) Yield stress #5 (kPa) Yield stress #6 (kPa) Yield stress #7 (kPa) Yield stress #8 (kPa) Yield stress #9 (kPa) Hydro tensile limit (kPa) Crack softening, Gf (J/m2 ) – –
3.000E + 05 4.000E + 05 5.000E + 05 6.000E + 05 7.000E + 05 2.077E + 04 4.300E + 04 1.385E + 05 2.346E + 05 3.231E + 05 3.962E + 05 4.423E + 05 5.000E + 05 5.385E + 05 −2.400E + 03 113.8 – –
The fracture energy is derived from the fracture toughness, which is obtained from [17], based on a water-to-cement ratio of 0.55. Currently the model is not available for the Eulerian solution technique because failure flag is not transported between Euler cells, so for each Euler cell there is no history of the material status, and thus crack softening of the material cannot be applied. 3. Empirical penetration equation Forrestal and co-workers developed a penetration equation for concrete for ogive-nose projectiles [1]. The equation is derived from post-test observations on the penetration of concrete and soil. The penetration of concrete can be partitioned into two distinct processes namely crater formation, which begins from the surface of the target to a depth of two projectile shank diameter (i.e. 4a), and tunneling, which continues from the depth of the crater to the final penetration depth. The penetration equation contains an empirical constant, R, referred to as the target resistance in [18], which can be obtained from a set of penetration experiments. From [18] the empirical constant, R, is given by R=
N Vs2 3 (1 + 4a N /m) exp[2a 2 (P
− 4a)N/m] − 1
,
(1)
8 − 1 242
.
Target
(kg/m3 )
2250
m (kg)
Projectile 2a (mm)
2.3
50.8
3
Using these geometrical properties, the last column of Table 4 tabulates the target resistance parameter R calculated using Eq. (1). The final depth of penetration, P , for an ogive-nose projectile and a concrete target presented in [18] is defined as N V12 m P= ln 1 + + 4a, (3) 2a 2 N R where V1 is defined as follows: V12 =
mV 2s − 4a 3 R . m + 4a 3 N
(4)
Based on an average value of R, the final depth of penetration, P , can be calculated using the above equations. With this average value of R the projectile’s rigid-body deceleration versus time response is calculated using the equation given in [1] with R = Sf c. 4. Results and discussion
where N is defined as follows: N=
Table 3 Geometrical properties of the concrete target and the projectile
(2)
In Eq. (1), P is the final penetration depth, is density of the concrete target, while a and m are the projectile’s radius and mass respectively. Table 3 provides the material and geometrical properties of the concrete target and the projectile.
Results for the maximum depth of penetration from the numerical simulations and the empirical calculation are compared with the data obtained from the three penetration experiments by Frew and Gran [9]. The deceleration–time response for the projectile using the numerical simulation is also compared with that obtained from the empirical approach. To assess the three numerical schemes, radial stress results from the numerical
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Table 4 Target resistance parameter, R , with corresponding penetration data from [9] Test number
Target length (m)
Striking velocity, Vs (m/s)
Pitch/yaw (deg)
Penetration depth, P (m)
R (MPa)
1 2 3
1.22 1.22 1.22
320 310 316
0.2U/0.1R 0.0/0.3L 0.4D/0.3R
0.185 0.175 0.157
423 430 524
300.00 Test Data r/rp = 1.4 (ME53) Test Data r/rp = 1.5 (ME 56)
Radial Stress (MPa)
250.00
Test Data r/rp = 1.7 (ME 51) Lagrange 6.1mm
200.00
Lagrange 5.0mm Lagrange 4.0mm
150.00 100.00 50.00 0.00 0.0 -50.00
0.5
1.0
1.5
Time (ms)
(a) 300.00
Test Data r/rp = 2.1 (ME 50) Test Data r/rp = 2.1 (ME 46)
Fig. 7. Location of the gages within the concrete target.
simulation are also compared with those obtained from the experiments. 4.1. Radial stress–time response
Radial Stress (MPa)
250.00
Test Data r/rp = 2.1 (ME 44) Lagrange 6.1mm
200.00
Lagrange 5.0mm Lagrange 4.0mm
150.00 100.00 50.00 0.00 0.0 -50.00
0.5
1.0
1.5
Time (ms)
(b) 300.00
Test Data r/rp = 2.6 (ME 57) Test Data r/rp = 2.6 (ME 48)
250.00
Lagrange 6.1mm
Radial Stress (MPa)
In the numerical simulation, the radial stress within the concrete target at normalized axial positions of z/a = 4.0 and radial positions of r/a = 1.4, 2.1 and 2.6 are recorded. Fig. 7 shows the location of these recordings. Comparing the radial stress–time response from the simulations with those from the experiments, in Figs. 8, 9 and 10, simulations using the three numerical schemes (i.e. Lagrange, Euler–Lagrange coupling and SPH–Lagrange coupling) have a longer rise time, which could be due to the effect of strain-rate on strength. Nevertheless the numerical simulations, on the whole, demonstrate overall agreement with measured radial stresses. Fig. 8 reveals that with the Lagrange numerical schemes the radial stresses drop-off at later times than the experimental test results. Johnson et al. [16] attributed this to the erosion strain. He further explained that the drop-off at later times is the result of element erosion. When an element erodes all of the stresses in that element are instantaneously set to zero. In the Euler–Lagrange and SPH–Lagrange coupling there is no erosion strain to prevent mesh tangling in the concrete target. Thus their results are able provide a much better agreement with the experimental test results.
Lagrange 4.0mm
150.00 100.00 50.00 0.00 0.0 -50.00
(c)
Lagrange 5.0mm
200.00
0.5
1.0
1.5
Time (ms)
Fig. 8. (a) Comparison of Lagrange calculations with measured stresses at r/a ≈ 1.4 and z/a = 4.0, (b) comparison of Lagrange calculations with measured stresses at r/a = 2.1 and z/a = 4.0, (c) comparison of Lagrange calculations with measured stresses at r/a = 2.6 and z/a = 4.0.
4.2. Maximum depth of penetration Table 5 compares the maximum depth of penetration from three solution techniques, namely Lagrange, Euler–Lagrange
C.Y. Tham / Finite Elements in Analysis and Design 42 (2006) 1258 – 1268
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300.00
300.00 Test Data r/rp = 1.4 (ME53) Test Data r/rp = 1.5 (ME 56)
250.00
Test Data r/rp = 1.4 (ME53) Test Data r/rp = 1.5 (ME 56) Test Data r/rp = 1.7 (ME 51) SPH 5mm SPH 4mm SPH 2.5mm
250.00
Euler 5mm
200.00
Radial Stress (MPa)
Radial Stress (MPa)
Test Data r/rp = 1.7 (ME 51) Euler 4mm Euler 2.5mm
150.00 100.00 50.00
(a)
0.5
1.0
1.5
50.00
0.5
(a)
Time (ms) 300.00
1.0 Time (ms)
150.00 100.00 50.00
0.5
1.0
Test Data r/rp = 2.1 (ME 50) Test Data r/rp = 2.1 (ME 46) Test Data r/rp = 2.1 (ME 44) SPH 5mm SPH 4mm SPH 2.5mm
250.00 Radial Stress (MPa)
200.00
200.00 150.00 100.00 50.00 0.00 0.0
1.5
-50.00
0.5
1.0
(b)
Time (ms)
Time (ms) 300.00
300.00
Test Data r/rp = 2.6 (ME 57)
Test Data r/rp = 2.6 (ME 57) Test Data r/rp = 2.6 (ME 48)
250.00
Test Data r/rp = 2.6 (ME 48)
250.00
SPH 5mm
Euler 4mm
200.00
Radial Stress (MPa)
Radial Stress (MPa)
Euler 5mm
Euler 2.5mm
150.00 100.00 50.00
0.5
1.0 Time (ms)
(c)
SPH 4mm
200.00
SPH 2.5mm
150.00 100.00 50.00 0.00 0.0
1.5
-50.00
(c)
1.5
-50.00
(b)
0.00 0.0
1.5
300.00 Test Data r/rp = 2.1 (ME 50) Test Data r/rp = 2.1 (ME 46) Test Data r/rp = 2.1 (ME 44) Euler 5mm Euler 4mm Euler 2.5mm
250.00 Radial Stress (MPa)
100.00
-50.00
-50.00
0.00 0.0
150.00
0.00 0.0
0.00 0.0
200.00
0.5
1.0
1.5
-50.00 Time (ms)
Fig. 9. (a) Comparison of Euler–Lagrange calculations with measured stresses at r/a ≈ 1.4 and z/a = 4.0, (b) comparison of Euler–Lagrange calculations with measured stresses at r/a = 2.1 and z/a = 4.0, (c) comparison of Euler–Lagrange calculations with measured stresses at r/a=2.6 and z/a=4.0.
Fig. 10. (a) Comparison of SPH–Lagrange calculations with measured stresses at r/a ≈ 1.4 and z/a=4.0, (b) comparison of SPH–Lagrange calculations with measured stresses at r/a=2.1 and z/a=4.0, (c) comparison of SPH–Lagrange calculations with measured stresses at r/a = 2.6 and z/a = 4.0.
coupling, SPH–Lagrange coupling for different cell and particle sizes. The maximum depth of penetration from the empirical calculation is also presented in Table 5. In the experiments by Frew and Gran [9], the depth of penetration recorded were 185, 175 and 157 mm. Although the effect of strain-rate is not included in the strength model, the results from the three solution techniques demonstrate reasonably good agreement with the experimental data. Except for the Euler–Lagrange coupling with cell size of 4.0 and 5.0 mm, on the whole the penetration results from numerical and empirical approach are within the range recorded in the experiments. The results from the
Euler–Lagrange coupling tend to indicate that larger cell size produces higher maximum depth of penetration. One probable explanation is the advection step (i.e. the Euler step) in the Eulerian numerical technique, which introduces numerical diffusion on the solution. Another probable explanation is the absence of the crack softening model, which reduces mesh sensitivity. The results from the Euler–Lagrange coupling indicate that to achieve penetration depth similar to the experiments, finer cell size (i.e. < 2.5 mm) is needed. On the contrary the Lagrange numerical scheme with larger cell size produces lower maximum depth of penetration. With the Lagrange numerical
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Table 5 Results from the numerical and empirical approach Approach Lagrange solution technique Cell size (mm)
Euler–Lagrange coupling Cell size (mm)
SPH–Lagrange coupling Particle size (mm)
Penetration depth, P (mm)
Normalized penetration, P /a
4.0 5.0 6.1
177.8 175.0 166.7
7.00 6.89 6.56
2.5 4.0 5.0
185.8 197.8 249.3
7.31 7.79 9.81
2.5 4.0 5.0
184.8 181.7 184.5
7.28 7.15 7.26
170.9
6.73
185.0 175.0 157.0
7.3 6.9 6.2
Empirical penetration equation Test results from Gran & Frew [9] Test number
1 2 3
scheme and the Euler–Lagrange coupling, results on penetration depth with different cell sizes tend to infers that cell size does affect the maximum depth of penetration. And thus care must be taken in selecting cell size for the concrete target when employing these two solution techniques. However for the case of the SPH–Lagrange coupling, varying the particle size from 2.5 to 5.0 mm does not seem to result in any significant change in the depth of penetration. Fig. 11 shows the damage contours of the concrete target for the SPH–Lagrange coupling, with three different particle sizes. The damage contours along the radial and axial direction for three different particle sizes are almost identical. 4.3. Deceleration of projectile The deceleration–time response for the projectile using the empirical approach is compared with those obtained from the numerical simulations. Figs. 12–14 present the projectile’s deceleration versus time profiles calculated using the empirical penetration equation in [1] and the three numerical schemes, with different cell and particle sizes. Overall the results from both the numerical and empirical approach are consistent. Fig. 12 shows that there is a correlation between results from the Lagrange solution technique and predictions using the empirical equation. The projectile’s deceleration–time response in Fig. 12 further reveals that smaller cell size, using Lagrange solution technique, produces oscillations with lower amplitude. For the case of the numerical approach using the Euler–Lagrange coupling, Fig. 13 indicates that a larger cell size for this solution technique produces result that is less consistent with calculations using the empirical equation. A larger cell size, using the Euler–Lagrange solution technique, tends to have longer pulse duration and lower maximum deceleration. This lower maximum deceleration obtained is consistent with
Fig. 11. Damage contours of the concrete target for the SPH–Lagrange coupling using three different particle sizes.
C.Y. Tham / Finite Elements in Analysis and Design 42 (2006) 1258 – 1268 10000 0 0
0.2
0.4
0.6
0.8
1
1.2
Acceleration (G)
-10000
5. Conclusions -30000 -40000
Lag (4.0mm) Lag (5.0mm) Lag (6.1mm) Equation [2]
-60000 -70000 Time (ms)
Fig. 12. Acceleration versus time: Lagrange calculations and empirical model prediction.
10000 0 0
0.2
0.4
0.6
0.8
1
1.2
-10000 Acceleration (G)
penetration equation. Comparing Fig. 14 with Figs. 12 and 13 shows that the deceleration of the projectile calculated using the SPH–Lagrange coupling is least sensitive to changes in particle size.
-20000
-50000
-20000 -30000 -40000 -50000 Eul-Lag (2.5mm) Eul-Lag (4.0mm)
-60000
Eul-Lag (5.0mm) Equation [2]
-70000 Time (ms)
Fig. 13. Acceleration versus time: Euler–Lagrange calculations and empirical model prediction.
10000 0
0
0.2
0.4
0.6
0.8
1
1.2
-10000 Acceleration (G)
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-20000 -30000 -40000 -50000 SPH (2.5mm) SPH (4.0mm)
-60000 -70000
SPH (5.0mm) Equation [2]
Time (ms)
Fig. 14. Acceleration versus time: SPH–Lagrange calculations and empirical model prediction.
results on the maximum depth of penetration, presented in Table 5, which perhaps infers that a larger cell size tend to have lower resistance to penetration. Among the three numerical solution techniques, the SPH–Lagrange coupling produces results that are most consistent with predictions using the empirical
Predicting the complex interaction of a projectile with concrete has long been a subject of interest to both civil engineers and ballisticians. Over the years tools were developed to predict and describe this complex interaction and they were (1) empirical equations (2) the application of cavity expansion theory and (3) hydrocode simulations. For this research both empirical equation and hydrocode simulations were used to predict the penetration of a concrete target by an ogive-nosed projectile. The results from the experiment performed by Gran and Frew [9] were used as a benchmark for comparison. For the empirical approach, the equation developed by Forrestal et al. [1] was used to determine the maximum depth of penetration and the deceleration–time response of the projectile. With the hydrocode approach, the simulations were performed using AUTODYN 2-D. Three numerical schemes, namely Lagrange, Euler–Lagrange coupling and SPH–Lagrange coupling were compared based on their prediction on the maximum depth of penetration, radial stress–time response of the concrete target and the projectile’s decelerations–time response. In predicting the maximum depth of penetration using hydrocode simulation, different cell and particle sizes were used. Although results from the Euler–Lagrange coupling with cell size of 4 and 5 mm overestimated the maximum depth of penetration. Nevertheless the results from the SPH–Lagrange coupling and the Lagrange numerical scheme, which did not include the effect of strain-rate on strength, compared reasonably well with experimental results. The results from the calculation using the penetration equation also demonstrated good agreement with the experimental results. For the radial stress prediction, the computed stresses from the three numerical schemes have longer rise time, but overall the results from these numerical schemes were able to reproduce the profile from the radial stress measurements. We postulated that the longer rise time may be due to the effect of strain-rate on strength. The deceleration–time response for the projectile based on the empirical approach was also compared with those obtained from the numerical simulations. The results from the Euler–Lagrange coupling with cell size of 4 and 5 mm were less consistent with calculations using the empirical equation, in terms of pulse duration and maximum deceleration. However the deceleration–time response from the SPH–Lagrange coupling and the Lagrange numerical scheme demonstrated better consistency with the empirical approach. Comparing the three numerical schemes used results on maximum depth of penetration, radial stress predictions and deceleration–time response showed that the simulations using the SPH numerical scheme gave the best overall agreement. In summary, the aforementioned simulation results, on the radial stress–time response, the maximum
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depth of penetration and the deceleration of the projectile, revealed that: 1. For this concrete specimen and with this striking velocity, the strain-rate effect on strength for the concrete did not adversely affect the prediction on the maximum depth of penetration, the projectile motion and the radial stress–time response of the concrete target. 2. The results from the hydrocode simulations were consistent with the empirical calculations on the maximum depth of penetration and the deceleration–time response for the projectile. 3. For this penetration simulation, the SPH–Lagrange solution technique gave the best overall agreement for the maximum depth of penetration and the radial stress–time response of the concrete target. This solution technique also yielded consistent results with empirical calculation on the deceleration–time response for the projectile. 4. Cell size, for an Euler–Lagrange coupling, can influence the prediction on the maximum depth of penetration, and hence the deceleration–time response of the projectile. With this demonstration of consistent results between numerical simulation and empirical equation, and also their good correlation with results from experiments, engineers and ballisticians now have more reasons to be confident when using both empirical and numerical approaches in complementing their full-scale testing. References [1] M.J. Forrestal, B.S. Altman, J.D. Cargile, S.J. Hanchak, An empirical equation for penetration depth of ogive-nose projectiles into concrete targets, Int. J. Impact Eng. 15 (4) (1994) 395–405. [2] C.W. Young, Penetration equations, SAND97-2426, Sandia National Laboratories, Albuquerque, NM, 1997. [3] M.J. Forrestal, D.Y. Tzou, A spherical cavity-expansion penetration model for concrete targets, Int. J. Solids Struct. 34 (31–32) (1997) 4127–4146. [4] T.L. Warren, M.R. Tabbara, Spherical cavity-expansion forcing function in Pronto3d for application to penetration problems, SAND97-1174, Sandia National Laboratories, Albuquerque, NM, 1997.
[5] T.L. Warren, A.F. Fossum, D.J. Frew, Penetration into low-strength (23 MPa) concrete: target characterization and simulations, Int. J. Impact Eng. 30 (5) (2004) 477–503. [6] E.P. Chen, Numerical simulation of perforation of concrete targets by steel rods, in: E.P. Chen, V.K. Luk (Eds.), Advances in Numerical Simulation Techniques for Penetration and Perforation of Solids, American Society of Mechanical Engineers, vol. 171, 1993, pp. 181–188. [7] T.J. Holmquist, G.R. Johnson, W.H. Cook, A computational constitutive model for concrete subjected to large strain rates and high pressures, in: Proceedings of the Fourteenth International Symposium on Ballistics, Quebec City, Canada, 1993. [8] D.J. Frew, M.J. Forrestal, S.J. Hanchak, Penetration experiments with limestone targets and ogive-nose steel projectiles, ASME J. Appl. Mech. 67 (2000) 841–845. [9] J.K. Gran, D.J. Frew, In-target radial stress measurements from penetration experiments into concrete by ogive-nose steel projectiles, Int. J. Impact Eng. 19 (8) (1997) 715–726. [10] C.E. Anderson, An overview of the theory of hydrocodes, Int. J. Impact Eng. 5 (1987) 33–59. [11] J.A. Zukas, Introduction to Hydrocodes, Elsevier, Amsterdam, 2004. [12] AUTODYN, Theory Manual, Revision 4.0, Century Dynamics Inc., 1998. [13] K.H. Brown, S.P. Burns, M.A. Christon, Coupled Eulerian–Lagrangian methods for earth penetrating weapon applications, SAND2002-1014, Sandia National Laboratories, Albuquerque, NM, 2002. [14] C.A. Ross, Fracture of concrete at high strain-rate, in: S.P. Shah (Ed.), Toughening Mechanisms in Quasi-Brittle Materials, Kluwer Academic, Boston, 1991, pp. 577–596. [15] W. Reidel, K. Thoma, S. Hiermaier, E. Schmolinske, Penetration of reinforced concrete by BETA-B-500, Numerical analysis using a new macroscopic concrete model for hydrocodes, in: Proceedings of Ninth International Symposium on Interaction of the Effects of Munitions with Structures, Berlin, 1999. [16] G.R. Johnson, S.R. Beissel, T.J. Holmquist, D.J. Frew, Computed radial stresses in a concrete target penetrated by a steel projectile, in: N. Jones, D.G. Talaslidis, C.A. Brebbia, G.D. Manolis (Eds.), Structures Under Shock and Impact V, Computational Mechanics Publications, 1998, pp. 793–806. [17] G. Prokopski, B. Langier, Effect of water/cement ratio and silica fume addition on the fracture toughness and morphology of fractured surfaces of gravel concretes, Cem. Concr. Res. 30 (2000) 1427–1433. [18] M.J. Forrestal, D.J. Frew, J.P. Hickerson, T.A. Rohwer, Penetration of concrete targets with deceleration–time measurements, Int. J. Impact Eng. 28 (2003) 479–497.