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Numerical study on craters and penetration of concrete slab by ogive-nose steel projectile
Computers and Geotechnics 34 (2007) 1–9 www.elsevier.com/locate/compgeo
Numerical study on craters and penetration of concrete slab by ogive-nose steel projectile Zhi-liang Wang b
a,*
, Yong-chi Li a, R.F. Shen b, J.G. Wang
c
a Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, China Centre for Soft Ground Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 117576, Singapore c Centre for Protective Technology, National University of Singapore, 10 Kent Ridge Crescent, Singapore 117576, Singapore
Received 5 May 2006; received in revised form 4 September 2006; accepted 8 September 2006 Available online 7 November 2006
Abstract In the design of defense structures, concrete slabs are often used to provide protection against incidental dynamic loadings such as the impact of a steel projectile. In the present study, the Taylor-Chen-Kuszmaul (TCK) continuum damage model is further improved and successfully implemented into the dynamic finite element code, LS-DYNA, with erosion algorithm. The numerical predictions of impact and exit craters of concrete slab as well as the residual velocity of projectile using the newly-implemented numerical tool show good agreement with experimental observations. The performance of the modified TCK model is evaluated by comparing with the material Type 78 (Mat_Soil_Concrete) and Type 111 (Mat_Johnson_Holmquist_Concrete) available in LS-DYNA. The effect of CRH (caliberradius-head) ratio of the ogive-nose projectile on the impact crater is also investigated using the new numerical tool. Finally, the maximum penetration depth of steel projectile into a concrete slab is studied and an empirical formula is proposed. 2006 Elsevier Ltd. All rights reserved. Keywords: Concrete slab; Projectile; Penetration; Crater; Damage model; Erosion algorithm
1. Introduction Concrete is widely used as construction material for military and civilian applications. A good understanding of the response of concrete to impact or explosive loading is essential to the design and protection of fortifications. A review of previous research work reveals that many studies of concrete shelters against dynamic loadings were conducted from the early 1940s. However, most of the research work ceased shortly after World War II and was not resumed until the 1960s. The resistance of concrete against impact loading is of great interest not only to the designers of defense structures but also to the developers of weapon systems [15,2,21]. Studies have shown that short-duration high-magnitude impact or blast loading conditions significantly influence *
Corresponding author. Fax.: +86 551 3606459. E-mail address: [email protected] (Z.-l. Wang).
0266-352X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2006.09.001
defense structural response [19,23–25]. The concrete experiences various states of stress which produce different failure modes. For example, near the impacted area, the concrete slab mainly experiences compression and it can fail due to high pressure. The severe crushing-stress together with partial reflected-waves results in the formation of a crater on the impact surface. On the opposite side of the slab, the shock waves are well reflected and converted into tensile waves, which generate tensile cracking if the material tensile strength is reached and an exit crater caused by spallation can be observed [19,26]. ˚ ga˚rdh and Laine [1], Gomez and Hanchak et al. [12], A Shuka [8] experimentally investigated the penetration of reinforced concrete slabs by steel projectiles with velocities ranging from 150 to 1000 m/s. Fig. 1 presents the post-test photographs of the impact and exit craters on the front and rear surfaces of the concrete slab in one of the experiments conducted by Hanchak et al. [12]. The concrete slab in the test has an unconfined compressive strength of 140 MPa
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Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9
designed for special purposes such as erosion, effect of strain-rate, cracking, etc. Among others, the material Type 111 ‘‘Mat_Johnson_Holmquist_Concrete’’ can be used to simulate the large strains, high strain states and high pressures to which the concrete may be subjected. The material Type 78 ‘‘Mat_Soil_Concrete’’ is also widely adopted because of its own erosion algorithm [1,16]. In order to better characterize the craters and perforation of concrete slabs, Chen [4] introduced a continuum damage model (the TCK model) to explore the dynamic process of concrete penetration numerically. Huang et al. [14] afterwards modified this model for numerical analysis of concrete perforation based on the Mohr-Coulomb yield criterion. In the present study, the authors further modified the TCK damage model by incorporating the compressive damage into it. The improved model was then coded into the non-linear dynamic software, LS-DYNA, as a constitutive augmentation and implemented with erosion algorithm. The newly-implemented numerical tool was subsequently used to simulate the craters of concrete slab, the residual velocity of projectile, as well as the effect of CRH (caliber-radius-head) ratio of projectile on the impact crater. Besides, the relationship between the striking velocities and the maximum penetration depths of the ogivenose steel projectile was also investigated and an empirical formula was proposed.
2. Concrete continuum damage model
Fig. 1. Post-test photographs showing (a) impact crater; and (b) exit crater (after [12]).
and the impact velocity is 750 m/s in this test. Fig. 2 illustrates the geometric configurations of the steel projectile and the concrete slab. Based on the observation of the test samples, Hanchak et al. [12] generalized the phenomena as follows: (i) the crater range is smaller on the impact surface than on the exit surface; (ii) the depths of the craters are roughly one-third the slab thickness; and (iii) the central one-third region of the craters has a nearly circular tunnel. Conducting high-speed penetration tests is very costly and tedious. In order to decrease the number of experiments required, the proliferation of faster and cheaper computing power is beginning to allow numerical simulation to play a more important role in predicting the complex interaction of a projectile with concrete. This paper employs the explicit dynamic finite element code LSDYNA [16] to analyze how impact loadings affect concrete slabs. LS-DYNA has several built-in concrete models
Concrete is typically brittle and non-homogeneous in composition. The basic assumption of the TCK damage model is that the concrete is permeated by an array of randomly distributed cracks which grow and interact with one another under tensile loading. The damage is reflected in the degradation of the material stiffness following the equations derived by Budiansky and O’ Connell [3] for a random array of penny-shaped cracks in an isotropic elastic medium. The effective bulk modulus of the cracked solid is K 16 ð1 v2 Þ ¼1 Cd K 9 ð1 2vÞ
ð1Þ
where K and v are the original bulk modulus and Possion’s ratio for intact material, the barred quantities represent the corresponding degraded constants for damaged material. The crack density parameter Cd is assumed to be proportional to the product of N, the number of cracks of per unit volume, and r3, the cube of the average crack radius in a representative volume element. Following Grady and Kipp [9], N is expressed as a Weibull statistical distribution function activated by the bulk strain ev = (ex + ey + ez)/3, according to N ¼ kðev Þm
ð2Þ
where k and m are material constants to be determined from strain-rate dependent tensile fracture stress data.
Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9
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Fig. 2. Geometry of (a) reinforced concrete slab; and (b) ogive-nose steel projectile.
Inspection of Eq. (1) suggests that the tensile damage can be defined by Dt ¼
16 ð1 v2 Þ Cd 9 ð1 2vÞ
ð3Þ
To extend the range of crack densities, an expression based on percolation theory [6] 16 v ¼ v exp bC d 06b61 ð4Þ 9 was used for the equivalent Poisson’s ratio v. The value of b controls the unloading and reloading behaviors and relaxes the restriction of elastic unloading in the original model [20,5]. Based on kinetic energy considerations, Grady [10] derived an expression of the nominal fragment radius for dynamic fragmentation in a brittle material as follows: !2=3 pffiffiffiffiffi 1 20K IC r¼ ð5Þ 2 qC_ev max qffiffi Here, q is the mass density, C ¼ Eq, e_ v max is the maximum volumetric strain-rate experienced by the representative volume element for the entire loading process and KIC is the fracture toughness of the material.
When bulk tension occurs in the material, the crack density parameter Cd can be calculated as 2 5 m K IC ð6Þ C d ¼ kðev Þ e_ 2 v max 2 qC The formulation of the elastoplastic model for compression follows the description given in the LS-DYNA manual [11,16]. The center of the yield surface in deviatoric space is given by tensor a which has an initial value of zero. The stress difference n denotes the stress measured from the center of the yield surface and is given by [17,11] nij ¼ rij aij The evolution rate of a is given as rffiffiffi 2 2 nij _aij ¼ H 3 3 r0
ð7Þ
ð8Þ
where H is the hardening parameter and r0 denotes the yield stress which is assumed constant. The von Mises yield condition is expressed as u ¼ K2 r20 ð9Þ qffiffiffiffiffiffiffiffiffiffiffiffi 3 where K ¼ 2 nij nij is the von Mises equivalent stress.
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Based on the coupling principle of strain-rate effect in the RDA model [7,18] the compressive damage Dc of concrete is expressed as kW_ p D_ c ¼ 1 Dt
ð10Þ
where k denotes the sensitivity constant of Dc which is taken to be equal to 1.0 · 103 kg/J in this study [7,18]. The tensile damage Dt is set as the initial value of the compression damage Dc, while the plastic work Wp is computed by Z ð11Þ W p ¼ rij depij Regardless of whether concrete is in tensile or in compression, introduce the symbol D to express the damage scalar. Therefore, the degraded shear modulus G and bulk modulus Kfor a material point can be written as G ¼ Gð1 DÞ
ð12Þ
K ¼ Kð1 DÞ
Thus, the stress increments can be calculated through the generalized Hooke’s law if the strain components are known. 3. Numerical tool and erosion algorithm 3.1. Numerical tool and constitutive implementation All the numerical investigations presented in this paper were performed using the LS-DYNA, which has been developed especially for nonlinear dynamic simulations. LS-DYNA is an explicit finite element code, which uses a Lagrangian formulation. The equations of motion are integrated in time explicitly using central differences. The method requires very small time steps for a stable solution, and is thus particularly suitable for impact and crash simulations. The modified TCK model described above has been successfully implemented into LS-DYNA as a constitutive augmentation. The implementation follows general guidelines for such augmentation and employs the provided interface subroutines. The augmentation is vectorized to be consistent with the basic LS-DYNA code, and incorporates tensile and compressive damage as two different history variables. 3.2. Application of erosion algorithm In recent years, many efforts have been devoted to the development of reliable methods and algorithms for a more
realistic analysis of defense structures subjected to high dynamic loading. For example, an erosion algorithm can be adopted to solve the excessive element distortion problem and model the fracture and failure of concrete material. The element is immediately deleted when the material response in an element reaches certain critical value. The deletion process is irreversible which means that when the applied load is reversed, the deleted material will not be able to offer further resistance. This technique can be employed to capture the physical fracture and failure processes if no significant reverse loading occurs [4,26]. There may be a variety of criteria governing the ‘‘erosion’’ of the material. In the present work, two erosion criteria, tensile and compressive damage which corresponds to material fracture and failure respectively, have been incorporated into the user defined subroutine as follows: Dt P ft
ð13Þ
Dc P fc
where ft and fc denote the critical values for tensile and compressive damage, respectively. The physical separation of the concrete can be effectively simulated by using Eq. (13). When the damage reaches the critical values, the associated element is immediately removed. The two critical damages are determined by calibrating them against experimental data for a specified impact velocity. For specific set of simulations and experiments, the impact velocity is the only variable and the critical values of the tensile and compressive damage could normally be obtained by matching calculated and measured data for one impact velocity and thereafter being applied to the calculations for all other input velocities [4,21]. In the present study, such typical approach was adopted as well and the tensile and compressive damage for the present set of calculations are determined to ft = 0.50 and fc = 0.75, respectively. 4. Plain concrete perforation behavior 4.1. FE model and numerical analysis This section considers the normal impact on concrete slab to demonstrate the performance of the newly-implemented modified TCK damage model as elaborated in the preceding sections. The geometrical configuration used by Hanchak et al. [12] and shown in Fig. 2 was adopted for the present analysis. The dimensions of the reinforced concrete slab are 610 mm · 610 mm · 178 mm (see Fig. 2(a)). The interior of the slab has three orthogonal layers of steel reinforcement with diameter of 5.69 mm. The geometry of
Table 1 Material parameters of concrete for impact analysis q (kg m3) 2520.0
E (GPa) 20.68
v 0.18
r0 (MPa) 140.0
b 0.5
k (m3) 5.753e21
m 6.0
KIC (MPa m1/2) 2.747
Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9
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Fig. 3. Finite element grid for axisymmetric simulations.
the ogive-nose steel projectile features a length of 143.7 mm, a diameter of 25.4 mm and a caliber-radiushead (CRH) ratio of 3.0, as shown in Fig. 2(b). The projectile is assumed to be elastic/perfectly plastic with the following properties: mass density = 8000 kg/m3, Young’s modulus = 206.9 GPa, Possion’s ratio = 0.3 and yield stress = 1.724 GPa [4,21]. In the Hanchak’s test, care was exercised to make sure that the projectiles always impacted on the slab at the center of a reinforced steel grid so as to minimize the effects of the steel reinforcements. Huang et al. [14] also demonstrated that reinforced concrete and plain concrete have almost the same penetration resistance as the projectile typically possesses diameter much smaller than the grid size of slab reinforcement. Therefore, the slab is regarded as a plain concrete for simplification in this study. Table 1 lists
the concrete parameters used for the present impact analysis [4,14]. For axisymmetric considerations, the slab is approximated by a cylindrical block of 344.2 mm radius and 178 mm thickness. Fig. 3 shows the finite element mesh for the Hanchak’s test series, which was constructed by utilizing the preprocessor ANSYS9.0. Owing to symmetry, only one-half of the geometry is modeled using 3935 quadrilateral elements and 4120 nodes. During the analysis, the ogive-nose steel projectile is assumed to be non-deformable with velocities varying from 376 to 998 m/s. Since the concrete has much higher compressive than tensile strength, it typically experiences brittle tensile fracture first, followed by compression-shear failure when the concrete slab is subjected to impact loading. Fig. 4 presents a deformed mesh plot of the steel pro-
Fig. 4. Deformed mesh plot for 750 m/s impact velocity.
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Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9 1.60
14.00
600 V~H relation
550
12.00 Kinematic energy (kJ)
500
450
400
10.00 Internal energy
0.80
Kinematic energy
8.00
Internal energy (kJ)
Velocity (m/s)
1.20
0.40 6.00
350 0
0.05
0.1
0.15 0.2 Depth (m)
0.25
0.3
0.35 4.00 0.00
Fig. 5. Variation of projectile velocity versus penetration depth.
200.00
400.00 600.00 Time (us)
800.00
0.00
Fig. 7. Energy time history under impact velocity of 587 m/s.
jectile with 750 m/s velocity into the plain concrete slab. It focuses on the 150 mm diameter area around the projectile at 350 ls after impact. With the movement of the projectile, both tensile and compressive damage begin to evolve. Thus, spalling grows and expands continually until the penetration course stops. The crater and tunnel expansion phases are identifiable in the slab. Clearly, the defined concrete model adopted so far is capable of describing the perforation process in a qualitative manner. Fig. 5 presents the variation of velocity during the penetration. The initial impact velocity is 580 m/s and the residual velocity is about 366 m/s. It can be seen that the decrease of velocity versus depth is almost linear along the course of penetration in the slab. Fig. 6 shows a comparison between the calculated residual velocities with Hanchak’s data. The computed results are in good agreement with the experimental data. In impact and perforation, the projectile’s kinetic energy is partly converted to the slab body’s strain energy (i.e. internal energy) (see, for example, [19]). Generally,
900 800
Measured data Calculated data
700
500
r
V (m/s)
600
400 300 200 100 0 300
400
500
600 700 V s (m/s)
800
900
1000
Fig. 6. Comparison of calculated residual velocities with test data of Hanchak et al. [12].
the projectile‘s kinetic energy gradually dissipates as the slab’s strain energy increases. Fig. 7 shows the projectile’s kinetic energy and the slab’s strain energy time history curves for an impact velocity of 587 m/s. It clearly shows that when the projectile is in contact with and starts penetration into the target, the kinetic energy of the projectile gradually falls while the internal strain energy of the target concrete gradually rises correspondingly in an almost linear manner. At about 420 ls, the projectile wholly penetrates the target, and the final kinematic energy of the projectile and the increased internal strain energy of the target concrete are 5.283 kJ and 1.234 kJ, respectively.
4.2. Comparison of different models As a comparison of the performance of different constitutive models, Fig. 8(a) and (b) illustrates the numerical simulations of concrete slab under normal impact using the modified TCK damage model presented in this study and the material Type 78 available in LS-DYNA. The constitutive parameters for the model Type 78 can be referred ˚ ga˚rdh and Laine [1]. For comparison purpose, the to A numerical results using the material Type 111 reported by Holmquist et al. [13], Zhang and Li [27] are also included in Fig. 8(c) and (d), respectively. It is noticed that the modified TCK model can well simulate the perforation process of concrete, as well as the formation of impact and exit craters with acceptable degree of accuracy (see Fig. 8(a)). On the other hand, the model Type 111 fails to simulate the formation of impact and exit craters due to its lack of erosion algorithm and only a circular tunnel was formed in the simulation (see Fig. 8(c) and (d)). It is noted that similar to the modified TCK model, LS-DYNA’s material Type 78 is capable of craters and spallation simulation with its own erosion function (see Fig. 8(b)).
Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9
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Fig. 8. Comparison of numerical simulations: (a) modified TCK model; (b) Type 78 model; (c) Type 111 model (after [13]); and (d) Type 111 model (after [27]).
4.3. Effect of CRH ratio of steel projectile It is anticipated that the CRH (caliber-radius-head) ratio of the ogive-nose projectile may be an important factor influencing the behavior of concrete under impact loading. In this section, the effects of CRH values of the ogive-nose projectile are numerically investigated. In our calculations, the total length and the diameter of the projectile are assumed to be constant (see Fig. 2(b)). Three values of CRH=1.0, 2.0, 3.0 (namely CRH1, CRH2 and CRH3) are considered to explore the effect of projectile CRH ratio on the impact crater. The material parameters and erosion criteria remain the same as those adopted in the previous analyses. Fig. 9 compares the simulated results of the three values of CRH at an impact velocity of 500 m/s. It clearly demonstrates that the higher the value of CRH, the smaller the size of an impact crater created. It can be reasonably deduced that a flat headed projectile will lead to maximum impact crater. Because the shape of projectile head would be seriously distorted during the process of high-velocity penetration
through the slab, the effect of CRH on the exit crater is not discussed here. 5. Simulations of semi-infinite slab penetration If the thickness of the concrete slab is small, the projectile will easily penetrate the slab with increasing impact velocity. In order to investigate the relation of impact velocity with maximum penetration depth, a much thicker concrete slab is desirable. Fig. 10 depicts the geometries of a 1000 mm · 1000 mm plain concrete slab and an ogivenose projectile with CRH ratio of 3.0. Because the main constants such as density, Young’s modulus, Poisson’s ratio and compressive strength of the concrete used by [22,18] are quite approximate to those of this work, the same material properties as aforementioned for steel projectile and concrete are adopted. The numerical procedure and the constitutive model described in the preceding analyses are thus not repeated here. A series of analyses is performed to predict the penetration depths of the steel projectile at impact velocities varying from 560 to 800 m/s. Fig. 11 compares the numerical
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Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9
Fig. 11. Impact velocity versus maximum penetration depth.
Through a least square regression analysis, the impact velocity versus the maximum penetration depth can be expressed as H ¼ 5:0 expð0:003 V s Þ
Fig. 9. Comparison of impact craters: (a) CRH = 1.0; (b) CRH = 2.0 and (c) CRH = 3.0.
ð14Þ
This empirical formula shows that the penetration depth exponentially increases with the impact velocity and can be used readily to determine the minimum thickness of the protective concrete slab against a projectile with a specific speed. However, it should be noted that Eq. (14) was derived based on the typical concrete properties and a CRH3 ratio projectile. Therefore, its applicability to other conditions deserves further study and must be treated with cautions. 6. Conclusions Perforation of high-velocity objects through structural concrete is one of the most challenging problems for designers in civil defense engineering. Based on the modified TCK damage model and the erosion algorithm, numerical simulations of the dynamic responses of concrete slab impacted by ogive-nose steel projectile were performed in the present study. From these studies, the following conclusions and understanding may be drawn:
Fig. 10. Schematic thick concrete slab for penetration depth analysis.
results of the current analyses with the test data reported by [22,18]. The penetration depths, predicted by the proposed concrete damage model, vary with the impact velocities and are in good agreement with the experimental data.
(1) The modified TCK model presented is capable of capturing the brittle tensile fracture as well as the plastic compressive damage of concrete medium. The model has been successfully implemented by coding it as a constitutive augmentation into the dynamic finite element software LS-DYNA, and thus provides a good alternative tool in the analysis of impact loading on concrete slab. (2) The simulations based on the user-defined material model with erosion algorithm for concrete craters and perforation show a consistent comparison with
Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9
the relevant experimental data. It indicates that the adoption of the two damage criteria is effective in this study and has superior performance against some other available models in LS-DYNA like Type 111 in terms of capturing the tensile fracture and spallation craters. (3) Numerical simulations clearly demonstrate that the size of the impact crater heavily depends on the CRH (caliber-radius-head) ratio of the projectile. In order to reduce the size of impact crater, the higher CRH value of projectile is recommended. In addition, an empirical formula has been proposed to relate the maximum penetration depth of a steel projectile into the concrete slab to the impact velocity under specific conditions, which may be of some practical value in the design of defense shelter under certain circumstance. Acknowledgements This work reported herein is partially supported by the Postdoctoral Science Foundation of China (Project No. 2004036468). The authors are much grateful to Yangtze Science Research Institute for offering good facilities for numerical analyses. References ˚ ga˚rdh L, Laine L. 3-D FE-simulation of high-velocity fragment [1] A perforation of reinforced concrete slabs. Int J Impact Eng 1999;22(9):911–22. [2] Balandin DV, Bolotnik NN, Pilkey WD. Optimal protection from impact, shock, and vibration. Toronto: Gordon and Breach Science Publisher; 2001. [3] Budiansky B, O’Connell RJ. Elastic moduli of a cracked solid. Int J Solids Struct 1976;12(2):81–97. [4] Chen EP. Simulation of concrete perforation based on a continuum damage model. In: Carpinteri A, editor. Size-scale effects in the failure mechanisms of materials and structures. E & FN Spon Press; 1996. p. 574–87. [5] Chen EP. Non-local effects on dynamic damage accumulation in brittle solids. Int J Numer Anal Met 1999;23(1):1–21. [6] Englman R, Jaeger Z. Theoretical aids for the improvement of blasting efficiencies in oil shale and rocks AP-TR-12/87. Yavne, Israel: Soreq Nuclear Research Center; 1987. [7] Furlong JR, Davis JF, Alme ML. Modeling the dynamic load/ unload behavior of ceramics under impact loading RDA-TR-00.00001. R&D Associates: Arlington; 1990. [8] Gomez JT, Shuka A. Multiple impact penetration of semi-infinite concrete. Int J Impact Eng 2001;25(10):965–79.
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[9] Grady DE, Kipp ME. Continuum modelling of explosive fracture in oil shale. Int J Rock Mech Min 1980;17(3):147–57. [10] Grady DE. mechanics of fracture under high-rate stress loading. In: Bazant ZP, editor. Preprints of the William Prager symposium on mechanics of geomaterials: rocks, concrete and soils. Evanston, IL: Northwestern University; 1983. p. 149–88. [11] Hallquist JO. LS-DYNA Theoretical manual, California, USA; 1998. [12] Hanchak SJ, Forrestal MJ, Young ER, Ehrgott JQ. Perforation of concrete slabs with 48 MPa (7 ksi) and 140 MPa (20 ksi) unconfined compressive strengths. Int J Impact Eng 1992;12(1):1–7. [13] Holmquist TJ, Johnson GR, Cook WH. A computational constitutive model for concrete to large strain, high strain rate, and high pressure. In: Proceedings of the 14th international symposium on ballistics, Quebec City, Canada; 1993. p. 591–600. [14] Huang FL, Wu HJ, Jin QK, Zhang QM. A numerical simulation on the perforation of reinforced concrete slabs. Int J Impact Eng 2005;32(1-4):173–87. [15] Kennedy RP. Review of procedures for the analysis and design of concrete structures to resist missile impact effects. Nucl Eng 1976;37(2):183–203. [16] Livemore Software Technology Corporation (LSTC). LS-DYNA Keyword User’s manual, California, USA; 2003. [17] Owen DRJ, Hinton E. Finite elements in plasticity: theory and practice. Swansea, UK: Pineridge Press Limited; 1980. [18] Sun YX. Studies on penetration resistance of concrete. Ph.D. dissertation. University of Science and Technology of China, Hefei, Anhui; 2002. [19] Tai SY, Tang CC. Numerical simulation: The dynamic behavior of reinforced concrete plates under normal impact. Theor Appl Fract Mec 2006;45(2):117–27. [20] Taylor LM, Chen EP, Kuszmaul JS. Micro-crack induced damage accumulation in brittle rock under dynamic loading. Comput Method Appl M 1986;55(3):301–20. [21] Teng TL, Chu YA, Chang FA, Chin HS. Numerical analysis of oblique impact on reinforced concrete. Cement Concrete Compos 2005;27(4):481–92. [22] Wang DR. Studies on engineering analysis methods for highvelocity penetration and related numerical simulations. Ph.D. dissertation. University of Science and Technology of China, Hefei, Anhui; 2002. [23] Wang ZL, Wang JG, Li YC, Leung CF. Attenuation effect of artificial cavity on air-blast waves in an intelligent defense layer. Comput Geotech 2006;33(2):132–41. [24] Wang ZL, Li YC, Wang JG. Study of stress waves in geomedia and effect of a soil cover layer on wave attenuation using a 1-D finite difference method. Comput Geosci 2006;32(10):1535–43. [25] Wang ZL, Li YC, Wang JG. Numerical analysis of attenuation effect of EPS geofoam on stress-waves in civil defense engineering. Geotext Geomembranes 2006;24(5):265–73. [26] Xu K, Lu Y. Numerical simulation study of spallation in reinforced concrete plates subjected to blast loading. Comput Struct 2006; 84(5-6):431–8. [27] Zhang FG, Li EZ. A computational model for concrete subjected to large strains, high strain rates, and high pressures. Explos Shock Wave 2002;22(3):198–202.
Numerical study on craters and penetration of concrete slab by ogive-nose steel projectile Zhi-liang Wang b
a,*
, Yong-chi Li a, R.F. Shen b, J.G. Wang
c
a Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, China Centre for Soft Ground Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 117576, Singapore c Centre for Protective Technology, National University of Singapore, 10 Kent Ridge Crescent, Singapore 117576, Singapore
Received 5 May 2006; received in revised form 4 September 2006; accepted 8 September 2006 Available online 7 November 2006
Abstract In the design of defense structures, concrete slabs are often used to provide protection against incidental dynamic loadings such as the impact of a steel projectile. In the present study, the Taylor-Chen-Kuszmaul (TCK) continuum damage model is further improved and successfully implemented into the dynamic finite element code, LS-DYNA, with erosion algorithm. The numerical predictions of impact and exit craters of concrete slab as well as the residual velocity of projectile using the newly-implemented numerical tool show good agreement with experimental observations. The performance of the modified TCK model is evaluated by comparing with the material Type 78 (Mat_Soil_Concrete) and Type 111 (Mat_Johnson_Holmquist_Concrete) available in LS-DYNA. The effect of CRH (caliberradius-head) ratio of the ogive-nose projectile on the impact crater is also investigated using the new numerical tool. Finally, the maximum penetration depth of steel projectile into a concrete slab is studied and an empirical formula is proposed. 2006 Elsevier Ltd. All rights reserved. Keywords: Concrete slab; Projectile; Penetration; Crater; Damage model; Erosion algorithm
1. Introduction Concrete is widely used as construction material for military and civilian applications. A good understanding of the response of concrete to impact or explosive loading is essential to the design and protection of fortifications. A review of previous research work reveals that many studies of concrete shelters against dynamic loadings were conducted from the early 1940s. However, most of the research work ceased shortly after World War II and was not resumed until the 1960s. The resistance of concrete against impact loading is of great interest not only to the designers of defense structures but also to the developers of weapon systems [15,2,21]. Studies have shown that short-duration high-magnitude impact or blast loading conditions significantly influence *
Corresponding author. Fax.: +86 551 3606459. E-mail address: [email protected] (Z.-l. Wang).
0266-352X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2006.09.001
defense structural response [19,23–25]. The concrete experiences various states of stress which produce different failure modes. For example, near the impacted area, the concrete slab mainly experiences compression and it can fail due to high pressure. The severe crushing-stress together with partial reflected-waves results in the formation of a crater on the impact surface. On the opposite side of the slab, the shock waves are well reflected and converted into tensile waves, which generate tensile cracking if the material tensile strength is reached and an exit crater caused by spallation can be observed [19,26]. ˚ ga˚rdh and Laine [1], Gomez and Hanchak et al. [12], A Shuka [8] experimentally investigated the penetration of reinforced concrete slabs by steel projectiles with velocities ranging from 150 to 1000 m/s. Fig. 1 presents the post-test photographs of the impact and exit craters on the front and rear surfaces of the concrete slab in one of the experiments conducted by Hanchak et al. [12]. The concrete slab in the test has an unconfined compressive strength of 140 MPa
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Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9
designed for special purposes such as erosion, effect of strain-rate, cracking, etc. Among others, the material Type 111 ‘‘Mat_Johnson_Holmquist_Concrete’’ can be used to simulate the large strains, high strain states and high pressures to which the concrete may be subjected. The material Type 78 ‘‘Mat_Soil_Concrete’’ is also widely adopted because of its own erosion algorithm [1,16]. In order to better characterize the craters and perforation of concrete slabs, Chen [4] introduced a continuum damage model (the TCK model) to explore the dynamic process of concrete penetration numerically. Huang et al. [14] afterwards modified this model for numerical analysis of concrete perforation based on the Mohr-Coulomb yield criterion. In the present study, the authors further modified the TCK damage model by incorporating the compressive damage into it. The improved model was then coded into the non-linear dynamic software, LS-DYNA, as a constitutive augmentation and implemented with erosion algorithm. The newly-implemented numerical tool was subsequently used to simulate the craters of concrete slab, the residual velocity of projectile, as well as the effect of CRH (caliber-radius-head) ratio of projectile on the impact crater. Besides, the relationship between the striking velocities and the maximum penetration depths of the ogivenose steel projectile was also investigated and an empirical formula was proposed.
2. Concrete continuum damage model
Fig. 1. Post-test photographs showing (a) impact crater; and (b) exit crater (after [12]).
and the impact velocity is 750 m/s in this test. Fig. 2 illustrates the geometric configurations of the steel projectile and the concrete slab. Based on the observation of the test samples, Hanchak et al. [12] generalized the phenomena as follows: (i) the crater range is smaller on the impact surface than on the exit surface; (ii) the depths of the craters are roughly one-third the slab thickness; and (iii) the central one-third region of the craters has a nearly circular tunnel. Conducting high-speed penetration tests is very costly and tedious. In order to decrease the number of experiments required, the proliferation of faster and cheaper computing power is beginning to allow numerical simulation to play a more important role in predicting the complex interaction of a projectile with concrete. This paper employs the explicit dynamic finite element code LSDYNA [16] to analyze how impact loadings affect concrete slabs. LS-DYNA has several built-in concrete models
Concrete is typically brittle and non-homogeneous in composition. The basic assumption of the TCK damage model is that the concrete is permeated by an array of randomly distributed cracks which grow and interact with one another under tensile loading. The damage is reflected in the degradation of the material stiffness following the equations derived by Budiansky and O’ Connell [3] for a random array of penny-shaped cracks in an isotropic elastic medium. The effective bulk modulus of the cracked solid is K 16 ð1 v2 Þ ¼1 Cd K 9 ð1 2vÞ
ð1Þ
where K and v are the original bulk modulus and Possion’s ratio for intact material, the barred quantities represent the corresponding degraded constants for damaged material. The crack density parameter Cd is assumed to be proportional to the product of N, the number of cracks of per unit volume, and r3, the cube of the average crack radius in a representative volume element. Following Grady and Kipp [9], N is expressed as a Weibull statistical distribution function activated by the bulk strain ev = (ex + ey + ez)/3, according to N ¼ kðev Þm
ð2Þ
where k and m are material constants to be determined from strain-rate dependent tensile fracture stress data.
Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9
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Fig. 2. Geometry of (a) reinforced concrete slab; and (b) ogive-nose steel projectile.
Inspection of Eq. (1) suggests that the tensile damage can be defined by Dt ¼
16 ð1 v2 Þ Cd 9 ð1 2vÞ
ð3Þ
To extend the range of crack densities, an expression based on percolation theory [6] 16 v ¼ v exp bC d 06b61 ð4Þ 9 was used for the equivalent Poisson’s ratio v. The value of b controls the unloading and reloading behaviors and relaxes the restriction of elastic unloading in the original model [20,5]. Based on kinetic energy considerations, Grady [10] derived an expression of the nominal fragment radius for dynamic fragmentation in a brittle material as follows: !2=3 pffiffiffiffiffi 1 20K IC r¼ ð5Þ 2 qC_ev max qffiffi Here, q is the mass density, C ¼ Eq, e_ v max is the maximum volumetric strain-rate experienced by the representative volume element for the entire loading process and KIC is the fracture toughness of the material.
When bulk tension occurs in the material, the crack density parameter Cd can be calculated as 2 5 m K IC ð6Þ C d ¼ kðev Þ e_ 2 v max 2 qC The formulation of the elastoplastic model for compression follows the description given in the LS-DYNA manual [11,16]. The center of the yield surface in deviatoric space is given by tensor a which has an initial value of zero. The stress difference n denotes the stress measured from the center of the yield surface and is given by [17,11] nij ¼ rij aij The evolution rate of a is given as rffiffiffi 2 2 nij _aij ¼ H 3 3 r0
ð7Þ
ð8Þ
where H is the hardening parameter and r0 denotes the yield stress which is assumed constant. The von Mises yield condition is expressed as u ¼ K2 r20 ð9Þ qffiffiffiffiffiffiffiffiffiffiffiffi 3 where K ¼ 2 nij nij is the von Mises equivalent stress.
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Based on the coupling principle of strain-rate effect in the RDA model [7,18] the compressive damage Dc of concrete is expressed as kW_ p D_ c ¼ 1 Dt
ð10Þ
where k denotes the sensitivity constant of Dc which is taken to be equal to 1.0 · 103 kg/J in this study [7,18]. The tensile damage Dt is set as the initial value of the compression damage Dc, while the plastic work Wp is computed by Z ð11Þ W p ¼ rij depij Regardless of whether concrete is in tensile or in compression, introduce the symbol D to express the damage scalar. Therefore, the degraded shear modulus G and bulk modulus Kfor a material point can be written as G ¼ Gð1 DÞ
ð12Þ
K ¼ Kð1 DÞ
Thus, the stress increments can be calculated through the generalized Hooke’s law if the strain components are known. 3. Numerical tool and erosion algorithm 3.1. Numerical tool and constitutive implementation All the numerical investigations presented in this paper were performed using the LS-DYNA, which has been developed especially for nonlinear dynamic simulations. LS-DYNA is an explicit finite element code, which uses a Lagrangian formulation. The equations of motion are integrated in time explicitly using central differences. The method requires very small time steps for a stable solution, and is thus particularly suitable for impact and crash simulations. The modified TCK model described above has been successfully implemented into LS-DYNA as a constitutive augmentation. The implementation follows general guidelines for such augmentation and employs the provided interface subroutines. The augmentation is vectorized to be consistent with the basic LS-DYNA code, and incorporates tensile and compressive damage as two different history variables. 3.2. Application of erosion algorithm In recent years, many efforts have been devoted to the development of reliable methods and algorithms for a more
realistic analysis of defense structures subjected to high dynamic loading. For example, an erosion algorithm can be adopted to solve the excessive element distortion problem and model the fracture and failure of concrete material. The element is immediately deleted when the material response in an element reaches certain critical value. The deletion process is irreversible which means that when the applied load is reversed, the deleted material will not be able to offer further resistance. This technique can be employed to capture the physical fracture and failure processes if no significant reverse loading occurs [4,26]. There may be a variety of criteria governing the ‘‘erosion’’ of the material. In the present work, two erosion criteria, tensile and compressive damage which corresponds to material fracture and failure respectively, have been incorporated into the user defined subroutine as follows: Dt P ft
ð13Þ
Dc P fc
where ft and fc denote the critical values for tensile and compressive damage, respectively. The physical separation of the concrete can be effectively simulated by using Eq. (13). When the damage reaches the critical values, the associated element is immediately removed. The two critical damages are determined by calibrating them against experimental data for a specified impact velocity. For specific set of simulations and experiments, the impact velocity is the only variable and the critical values of the tensile and compressive damage could normally be obtained by matching calculated and measured data for one impact velocity and thereafter being applied to the calculations for all other input velocities [4,21]. In the present study, such typical approach was adopted as well and the tensile and compressive damage for the present set of calculations are determined to ft = 0.50 and fc = 0.75, respectively. 4. Plain concrete perforation behavior 4.1. FE model and numerical analysis This section considers the normal impact on concrete slab to demonstrate the performance of the newly-implemented modified TCK damage model as elaborated in the preceding sections. The geometrical configuration used by Hanchak et al. [12] and shown in Fig. 2 was adopted for the present analysis. The dimensions of the reinforced concrete slab are 610 mm · 610 mm · 178 mm (see Fig. 2(a)). The interior of the slab has three orthogonal layers of steel reinforcement with diameter of 5.69 mm. The geometry of
Table 1 Material parameters of concrete for impact analysis q (kg m3) 2520.0
E (GPa) 20.68
v 0.18
r0 (MPa) 140.0
b 0.5
k (m3) 5.753e21
m 6.0
KIC (MPa m1/2) 2.747
Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9
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Fig. 3. Finite element grid for axisymmetric simulations.
the ogive-nose steel projectile features a length of 143.7 mm, a diameter of 25.4 mm and a caliber-radiushead (CRH) ratio of 3.0, as shown in Fig. 2(b). The projectile is assumed to be elastic/perfectly plastic with the following properties: mass density = 8000 kg/m3, Young’s modulus = 206.9 GPa, Possion’s ratio = 0.3 and yield stress = 1.724 GPa [4,21]. In the Hanchak’s test, care was exercised to make sure that the projectiles always impacted on the slab at the center of a reinforced steel grid so as to minimize the effects of the steel reinforcements. Huang et al. [14] also demonstrated that reinforced concrete and plain concrete have almost the same penetration resistance as the projectile typically possesses diameter much smaller than the grid size of slab reinforcement. Therefore, the slab is regarded as a plain concrete for simplification in this study. Table 1 lists
the concrete parameters used for the present impact analysis [4,14]. For axisymmetric considerations, the slab is approximated by a cylindrical block of 344.2 mm radius and 178 mm thickness. Fig. 3 shows the finite element mesh for the Hanchak’s test series, which was constructed by utilizing the preprocessor ANSYS9.0. Owing to symmetry, only one-half of the geometry is modeled using 3935 quadrilateral elements and 4120 nodes. During the analysis, the ogive-nose steel projectile is assumed to be non-deformable with velocities varying from 376 to 998 m/s. Since the concrete has much higher compressive than tensile strength, it typically experiences brittle tensile fracture first, followed by compression-shear failure when the concrete slab is subjected to impact loading. Fig. 4 presents a deformed mesh plot of the steel pro-
Fig. 4. Deformed mesh plot for 750 m/s impact velocity.
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Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9 1.60
14.00
600 V~H relation
550
12.00 Kinematic energy (kJ)
500
450
400
10.00 Internal energy
0.80
Kinematic energy
8.00
Internal energy (kJ)
Velocity (m/s)
1.20
0.40 6.00
350 0
0.05
0.1
0.15 0.2 Depth (m)
0.25
0.3
0.35 4.00 0.00
Fig. 5. Variation of projectile velocity versus penetration depth.
200.00
400.00 600.00 Time (us)
800.00
0.00
Fig. 7. Energy time history under impact velocity of 587 m/s.
jectile with 750 m/s velocity into the plain concrete slab. It focuses on the 150 mm diameter area around the projectile at 350 ls after impact. With the movement of the projectile, both tensile and compressive damage begin to evolve. Thus, spalling grows and expands continually until the penetration course stops. The crater and tunnel expansion phases are identifiable in the slab. Clearly, the defined concrete model adopted so far is capable of describing the perforation process in a qualitative manner. Fig. 5 presents the variation of velocity during the penetration. The initial impact velocity is 580 m/s and the residual velocity is about 366 m/s. It can be seen that the decrease of velocity versus depth is almost linear along the course of penetration in the slab. Fig. 6 shows a comparison between the calculated residual velocities with Hanchak’s data. The computed results are in good agreement with the experimental data. In impact and perforation, the projectile’s kinetic energy is partly converted to the slab body’s strain energy (i.e. internal energy) (see, for example, [19]). Generally,
900 800
Measured data Calculated data
700
500
r
V (m/s)
600
400 300 200 100 0 300
400
500
600 700 V s (m/s)
800
900
1000
Fig. 6. Comparison of calculated residual velocities with test data of Hanchak et al. [12].
the projectile‘s kinetic energy gradually dissipates as the slab’s strain energy increases. Fig. 7 shows the projectile’s kinetic energy and the slab’s strain energy time history curves for an impact velocity of 587 m/s. It clearly shows that when the projectile is in contact with and starts penetration into the target, the kinetic energy of the projectile gradually falls while the internal strain energy of the target concrete gradually rises correspondingly in an almost linear manner. At about 420 ls, the projectile wholly penetrates the target, and the final kinematic energy of the projectile and the increased internal strain energy of the target concrete are 5.283 kJ and 1.234 kJ, respectively.
4.2. Comparison of different models As a comparison of the performance of different constitutive models, Fig. 8(a) and (b) illustrates the numerical simulations of concrete slab under normal impact using the modified TCK damage model presented in this study and the material Type 78 available in LS-DYNA. The constitutive parameters for the model Type 78 can be referred ˚ ga˚rdh and Laine [1]. For comparison purpose, the to A numerical results using the material Type 111 reported by Holmquist et al. [13], Zhang and Li [27] are also included in Fig. 8(c) and (d), respectively. It is noticed that the modified TCK model can well simulate the perforation process of concrete, as well as the formation of impact and exit craters with acceptable degree of accuracy (see Fig. 8(a)). On the other hand, the model Type 111 fails to simulate the formation of impact and exit craters due to its lack of erosion algorithm and only a circular tunnel was formed in the simulation (see Fig. 8(c) and (d)). It is noted that similar to the modified TCK model, LS-DYNA’s material Type 78 is capable of craters and spallation simulation with its own erosion function (see Fig. 8(b)).
Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9
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Fig. 8. Comparison of numerical simulations: (a) modified TCK model; (b) Type 78 model; (c) Type 111 model (after [13]); and (d) Type 111 model (after [27]).
4.3. Effect of CRH ratio of steel projectile It is anticipated that the CRH (caliber-radius-head) ratio of the ogive-nose projectile may be an important factor influencing the behavior of concrete under impact loading. In this section, the effects of CRH values of the ogive-nose projectile are numerically investigated. In our calculations, the total length and the diameter of the projectile are assumed to be constant (see Fig. 2(b)). Three values of CRH=1.0, 2.0, 3.0 (namely CRH1, CRH2 and CRH3) are considered to explore the effect of projectile CRH ratio on the impact crater. The material parameters and erosion criteria remain the same as those adopted in the previous analyses. Fig. 9 compares the simulated results of the three values of CRH at an impact velocity of 500 m/s. It clearly demonstrates that the higher the value of CRH, the smaller the size of an impact crater created. It can be reasonably deduced that a flat headed projectile will lead to maximum impact crater. Because the shape of projectile head would be seriously distorted during the process of high-velocity penetration
through the slab, the effect of CRH on the exit crater is not discussed here. 5. Simulations of semi-infinite slab penetration If the thickness of the concrete slab is small, the projectile will easily penetrate the slab with increasing impact velocity. In order to investigate the relation of impact velocity with maximum penetration depth, a much thicker concrete slab is desirable. Fig. 10 depicts the geometries of a 1000 mm · 1000 mm plain concrete slab and an ogivenose projectile with CRH ratio of 3.0. Because the main constants such as density, Young’s modulus, Poisson’s ratio and compressive strength of the concrete used by [22,18] are quite approximate to those of this work, the same material properties as aforementioned for steel projectile and concrete are adopted. The numerical procedure and the constitutive model described in the preceding analyses are thus not repeated here. A series of analyses is performed to predict the penetration depths of the steel projectile at impact velocities varying from 560 to 800 m/s. Fig. 11 compares the numerical
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Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9
Fig. 11. Impact velocity versus maximum penetration depth.
Through a least square regression analysis, the impact velocity versus the maximum penetration depth can be expressed as H ¼ 5:0 expð0:003 V s Þ
Fig. 9. Comparison of impact craters: (a) CRH = 1.0; (b) CRH = 2.0 and (c) CRH = 3.0.
ð14Þ
This empirical formula shows that the penetration depth exponentially increases with the impact velocity and can be used readily to determine the minimum thickness of the protective concrete slab against a projectile with a specific speed. However, it should be noted that Eq. (14) was derived based on the typical concrete properties and a CRH3 ratio projectile. Therefore, its applicability to other conditions deserves further study and must be treated with cautions. 6. Conclusions Perforation of high-velocity objects through structural concrete is one of the most challenging problems for designers in civil defense engineering. Based on the modified TCK damage model and the erosion algorithm, numerical simulations of the dynamic responses of concrete slab impacted by ogive-nose steel projectile were performed in the present study. From these studies, the following conclusions and understanding may be drawn:
Fig. 10. Schematic thick concrete slab for penetration depth analysis.
results of the current analyses with the test data reported by [22,18]. The penetration depths, predicted by the proposed concrete damage model, vary with the impact velocities and are in good agreement with the experimental data.
(1) The modified TCK model presented is capable of capturing the brittle tensile fracture as well as the plastic compressive damage of concrete medium. The model has been successfully implemented by coding it as a constitutive augmentation into the dynamic finite element software LS-DYNA, and thus provides a good alternative tool in the analysis of impact loading on concrete slab. (2) The simulations based on the user-defined material model with erosion algorithm for concrete craters and perforation show a consistent comparison with
Z.-l. Wang et al. / Computers and Geotechnics 34 (2007) 1–9
the relevant experimental data. It indicates that the adoption of the two damage criteria is effective in this study and has superior performance against some other available models in LS-DYNA like Type 111 in terms of capturing the tensile fracture and spallation craters. (3) Numerical simulations clearly demonstrate that the size of the impact crater heavily depends on the CRH (caliber-radius-head) ratio of the projectile. In order to reduce the size of impact crater, the higher CRH value of projectile is recommended. In addition, an empirical formula has been proposed to relate the maximum penetration depth of a steel projectile into the concrete slab to the impact velocity under specific conditions, which may be of some practical value in the design of defense shelter under certain circumstance. Acknowledgements This work reported herein is partially supported by the Postdoctoral Science Foundation of China (Project No. 2004036468). The authors are much grateful to Yangtze Science Research Institute for offering good facilities for numerical analyses. References ˚ ga˚rdh L, Laine L. 3-D FE-simulation of high-velocity fragment [1] A perforation of reinforced concrete slabs. Int J Impact Eng 1999;22(9):911–22. [2] Balandin DV, Bolotnik NN, Pilkey WD. Optimal protection from impact, shock, and vibration. Toronto: Gordon and Breach Science Publisher; 2001. [3] Budiansky B, O’Connell RJ. Elastic moduli of a cracked solid. Int J Solids Struct 1976;12(2):81–97. [4] Chen EP. Simulation of concrete perforation based on a continuum damage model. In: Carpinteri A, editor. Size-scale effects in the failure mechanisms of materials and structures. E & FN Spon Press; 1996. p. 574–87. [5] Chen EP. Non-local effects on dynamic damage accumulation in brittle solids. Int J Numer Anal Met 1999;23(1):1–21. [6] Englman R, Jaeger Z. Theoretical aids for the improvement of blasting efficiencies in oil shale and rocks AP-TR-12/87. Yavne, Israel: Soreq Nuclear Research Center; 1987. [7] Furlong JR, Davis JF, Alme ML. Modeling the dynamic load/ unload behavior of ceramics under impact loading RDA-TR-00.00001. R&D Associates: Arlington; 1990. [8] Gomez JT, Shuka A. Multiple impact penetration of semi-infinite concrete. Int J Impact Eng 2001;25(10):965–79.
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