Pre-stress effect on confined ceramic armor ballistic performance

International Journal of Impact Engineering 84 (2015) 159e170

Contents lists available at ScienceDirect

International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng

Pre-stress effect on confined ceramic armor ballistic performance Runqiang Chi a, Ahmad Serjouei b, *, Idapalapati Sridhar b, *, Tan E.B. Geoffrey c a

School of Astronautics, Harbin Institute of Technology, Nangang District, Harbin 150001, China School of Mechanical & Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore c School of Materials Science and Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 31 January 2015 Received in revised form 24 May 2015 Accepted 26 May 2015 Available online 4 June 2015

A numerical technique for the simulation of pre-stressed confined ceramic targets under long rod projectile (LRP) impact, using explicit software, AUTODYN®, is proposed. The proposed model is validated by comparing the ballistic performance results with the available experimental data in the literature. Impact simulations of the confined ceramic targets with different pre-stress conditions, namely radial, axial and hydrostatic, as well as without pre-stress for benchmarking are reported. The effects of these different pre-stress conditions on the ballistic performance of the ceramic target are explored. The upper limits of transition velocity for the targets of different pre-stress conditions are obtained. It is shown that dwell of the LRP on the ceramic interface is influenced by the pressure, obtained from pre-stressing, at the initial failure position in SiC target. It is shown that pressure at the initial failure position increases linearly with the upper limit of transition velocity. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Ceramic armor Pre-stress Confinement Dwell time Transition velocity

1. Introduction Ceramics, with inherent low density, high hardness and compressive strength, have been extensively utilized in armor applications. Though ceramics are strong in compression, they suffer from disintegration when loaded under tension. It is well known that adding confinement to ceramics can increase their ballistic performance [1] by locally containing the comminuted ceramic fragments and bearing the impact loads under compression. Applying compressive pre-stress to the confined ceramics improves their ballistic limit further by increasing their strength and ductility and hindering premature failure in tension [2,3]. Long rod projectiles (LRP) are a form of kinetic energy (KE) penetrator, typically manifest as armor-piercing fin-stabilized discarding-sabot (APFSDS) projectiles, which are a common threat to armored vehicles such as tanks. Enhanced ceramic strength can force the high velocity LRP to flow radially with no significant penetration, a phenomenon called dwell or interface defeat [4]. There is a critical impact velocity (also called transition velocity) at which dwell on the interface is no longer maintained and penetration into the target occurs [5]. Lundberg et al. [5] proposed a

* Corresponding authors. E-mail addresses: [email protected] (A. Serjouei), [email protected] (I. Sridhar). http://dx.doi.org/10.1016/j.ijimpeng.2015.05.011 0734-743X/© 2015 Elsevier Ltd. All rights reserved.

method to find an interval (velocity range) for which a transition occurs from dwell to penetration in confined ceramic targets. When the LRP impact velocity is below the lower limit and above the upper limit of the transition velocity, dwell and penetration occur, respectively. When the impact velocity is between the lower and upper limit, the LRP dwells on the ceramic interface initially and then penetrates the ceramic, i.e. dwellepenetration transition. There is a unique dwellepenetration transition velocity for each combination of LRP, target material and configuration. This implies that the upper limit and lower limit of the proposed velocity range in the definition of transition velocity should be close in value. There are several experimental investigations on the ballistic performances of pre-stressed confined ceramics [2,5]. However, numerical simulation work on the pre-stressing of the ceramics, such as the works of Holmquist and Johnson [6], are scarce in the literature. Holmquist and Johnson [6] numerically investigated the effect of pre-stressing on the penetration and damage of thin and thick SiC under radial and hydrostatic pre-stress conditions, considering two pre-stress levels (values), namely small and large. However, varying levels of pre-stress and the influence of different pre-stress conditions on the ceramic ballistic performance, such as dwell time or transition velocity, were not modeled. In addition, simulation of an LRP impacting on the radially pre-stressed ceramic performed by Holmquist and Johnson [6] was not in good agreement with experimental data of shrink-fitted (radially pre-stressed) ceramics. Therefore, detailed and precise simulation works are

160

R. Chi et al. / International Journal of Impact Engineering 84 (2015) 159e170

Nomenclature f r0 , a, b, G, Gf , εmax , C, K1 , K2 , K3 , P1 , P2 , P3 , S1 , S2 , Sfmax , T * , Tf JH-1 material model constants (see Table 1) 0 0 0 ε_ 0 , r0 , A , B , C , C1 , Cr , d0 1 , d0 2 , d0 3 , d0 4 , d0 5 , K1 , G, n, S, t0 , tm JC material model constants (see Table 2)

Symbol: DC DT HEL LC LT PIFP rp V VUL

Definition Confinement inner diameter SiC target inner diameter Hugoniot elastic limit Confinement length SiC target length Pressure at the initial failure position of ceramic material Density of LRP Impact velocity Upper limit of transition velocity

Abbreviations APFSDS Armor-piercing fin-stabilized discarding-sabot EPS Effective plastic strain DOP Depth of penetration JH JohnsoneHolmquist KE Kinetic energy LRP Long rod projectile SPH Smooth particle hydrodynamics 2D Two-dimensional

needed as a supplement to numerical research on pre-stressed ceramic ballistic performance. In this paper, a method for simulating the high-velocity impact of pre-stressed SiC targets is proposed using AUTODYN® explicit finite element and/or mesh-free particle solver software. Material model constants of SiC are validated against available experimental data in the literature. Simulations of a tungsten LRP impacting SiC targets, with and without pre-stress, have been carried out in order to investigate the effects of pre-stress on the ballistic performance of SiC. Pre-stress conditions include three setups, namely radial, axial and hydrostatic, with six different pre-stress levels. Dwell time, LRP penetration rate and damage of the SiC at impact velocities of 1645 m/s and 2175 m/s (same velocities were used in the experiments of Lundberg et al. [5]), as well as the upper limit of the transition velocity, VUL for all SiC targets, are discussed to evaluate ballistic performance.

2.1. Finite element model Two-dimensional (2D) axisymmetric models of the confined SiC target and tungsten rod were created in AUTODYN® as shown in Fig. 2. The SiC target and steel tube were discretized using four node axisymmetric Lagrangian elements with one integration point. The rod was discretized using the smooth particle hydrodynamics (SPH) method. The front and rear steel plugs were composed of two parts (outer and inner parts) with same thickness of 4 mm. Lagrangian and SPH methods were used to model the outer and inner parts, respectively, as shown in Fig. 2. The two parts of the plugs were joined together at their interface using the “JOIN module” in AUTODYN®. According to the target structure in the experiments [5], the tube was also joined with front and back plugs, while the SiC target was just in contact with the confinements without joining them. The SPH particle size used for tungsten rod

2. Numerical simulations The confined pre-stressed SiC target used in the experiments performed by Lundberg et al. [5] is modeled using AUTODYN®. The SiC target in the experiments [5] is confined by a maraging steel (Mar 350) tube and two tempered steel (SIS 2541-3, comparable to AISI/SAE 4340) plugs with pre-stress. The target assembly was impacted by a cylindrical tungsten rod of 2 mm diameter and 80 mm length at different impact velocities. Sketches of the SiC target, steel confinement and tungsten rod as deployed in the numerical models are shown in Fig. 1. Both front and rear plugs are of 8 mm thickness. Both confinement inner diameter, DC , and length, LC , are 20 mm. The SiC target diameter, DT , and length, LT , are set as different values to produce the desired pre-stress. Other dimensions of the numerical models are shown in Fig. 1. The numerical models, material model constants and pre-stress modeling technique are described in the following sections.

Fig. 1. Details of tungsten LRP, confinement and SiC geometries used in the experiments [6]. Tube is made of maraging steel (Mar 350) and two plugs are made of tempered steel. The LRP impacts the front plug first before impacting the SiC.

R. Chi et al. / International Journal of Impact Engineering 84 (2015) 159e170

161

Fig. 2. Two-dimensional (2D) axi-symmetric model for the LRP and the target.

and plug inner parts is 0.125 mm, which is the same as value used by Quan et al. [7] who conducted numerical simulations (without pre-stress) for the same experiments. The Lagrangian element sizes in the SiC target and outer parts of the plugs are 0.15 mm and 0.125 mm, respectively. The reason for considering a Lagrangian-SPH model for plugs is as follows: In the Lagrangian domain, an erosion algorithm is used to remove the elements experiencing large distortions. If the plug inner part is modeled using Lagrangian method, when an element close to the SiC target is eroded, it will introduce a zero pressure void which allows SiC material at the interface to expand towards the void and lose pressure as it expands [8]. Therefore, as the strength and damage of the SiC material are pressure dependent, the pressure drop can lead to lower strength and premature failure of the SiC. In the SPH domain, there is no grid restriction and the particles are permitted to move freely under deformation. Hence, the particle erosion method is not needed. The plug inner part in contact with SiC modeled using the SPH method can maintain the existing pressure and therefore strength in the front of the SiC target. Therefore, the SPH method is more suitable for modeling the plug inner part, in contact with SiC, than the Lagrangian method. However stability analyses conducted by Swegle et al. [9] and Balsara [10] revealed that the SPH method suffers from an instability under tension. This instability manifests itself as a clustering of the particles, which resembles fracture and fragmentation, but it is in fact a numerical artifact [9]. The outer parts of the plugs are generally under tension when the SiC target is pre-stressed and impacted. Therefore, the plug outer parts will not experience abnormal behavior if modeled in the Lagrangian domain in which there is no tensile instability as with the SPH domain. Accordingly, the integrated Lagrangian-SPH plug model shown in Fig. 2 was created to avoid the disadvantages of the Lagrangian and SPH methods mentioned above. 2.2. Pre-stress modeling A numerical technique for pre-stressing SiC target was developed in AUTODYN®. The pre-stress is produced by the precompressed SiC cylinder interacting with the confinement (tube and plugs). This basic mechanism is similar to that presented by Holmquist and Johnson [6], however, the specific steps performed in AUTODYN® are different. The numerical technique to model prestress can be performed in four distinct steps as follows: In step I, numerical models of target components (SiC target, steel Mar 350 tube and two steel 4340 plugs) are created. These four components are separated from each other without any interactions between them. The initial dimension of the SiC target is larger than the confinement inner size for producing pre-stress. The radial, RR , and axial, RA , ratios are defined as SiC target initial diameter to confinement inner diameter, DT =DC , and SiC target initial length to confinement inner length, LT =LC , respectively (see Fig. 1 for the definitions of confinement geometries). Different combinations of radial and axial ratios can be selected to produce

different pre-stress conditions and levels in SiC target. In other words, since the confinement inner diameter and length are constant, the initial SiC target diameter and length can control the prestress conditions and levels. In step II, an inward velocity boundary condition of 1 m/s is applied on the radial and/or axial (depending on the pre-stress type) periphery of the SiC target. The SiC target is then compressed until its edge reaches the position that is slightly smaller than the confinement inner dimensions. During compression, the SiC edges, which are perpendicular to the moving edge, are fixed. After compression, all SiC edges are fixed and the numerical model is run so that uniform pressure is obtained in the SiC target. In this step, the static damping is used in AUTODYN® to eliminate the stress oscillations, since the compression process is done dynamically. In step III, all the target components are put in their respective geometrical position as shown in Fig. 2. The numerical contact and join settings for the component interactions are applied in this step. In step IV, all boundary conditions on the SiC target are removed. Therefore, the SiC target increases in size and comes in contact with the inner surface of the confinement and a pre-stress state is achieved. 2.3. Material models and constants Johnson and Holmquist [11,12] proposed two phenomenological models (JH models) consisting of three parts (a polynomial equation of state, a strength model and a damage model) for ceramic constitutive behavior: The JH-1 model represents the material strength using linear segments, rather than a continuous curve used by the JH-2 model, and instantaneous failure behavior rather than damage-induced strength reduction used in the JH-2 model. Holmquist and Johnson [13] indicated that the JH-1 model represents SiC behavior well. In order to better represent the SiC failure in tension regions, such as the rear side of the SiC target, the principal stress tensile failure and crack softening models proposed by Clegg et al. [14] and Taylor et al.[15], as used in the other works [14,15] for the same purpose, were deployed instead of the hydrotensile failure model of JH-1. The JH-1 material model constants used for SiC are listed in Table 1. The material model constants of SiC except for principal tensile failure stress, Tf , fracture energy, Gf and maximum failure strain, εfmax , are the same as those proposed by Holmquist and Johnson [13]. The value of εfmax ¼ 1:2 in the work of Holmquist and Johnson [13] was determined by matching the numerical penetration depth in the SiC target with experimental data [5] for impact velocity, V ¼ 1645 m/s. It was observed, as shown in Fig. 3, that if εfmax ¼ 1:2 is used in the current numerical simulations, the tungsten LRP cannot penetrate into the SiC target. This is distinct from the experimental observations [5], because no pre-stress on the SiC target was considered in the numerical simulation while there was radial pre-stress in the experiments. The εfmax value was set to 0.5, using the same validating method used in the work of Holmquist and Johnson [13].

162

R. Chi et al. / International Journal of Impact Engineering 84 (2015) 159e170

Table 1 JH-1 material model constants for SiC.

Table 2 JC material model constants for tungsten alloy [16] and steel 4340 [17].

Material/Constants

SiC

Material/Constants

Tungsten alloy

Steel 4340

Density, r0 (kg/m3) Bulk modulus, K1 (GPa) Pressure constant, K2 (GPa) Pressure constant, K3 (GPa) Bulking factor, b Shear modulus, G (GPa) Hugoniot elastic limit (HEL) (GPa) Intact strength constant, S1 (GPa) Intact strength constant, P1 (GPa) Intact strength constant, S2 (GPa) Intact strength constant, P2 (GPa) Strain rate constant, C Maximum fracture strength, Sfmax (GPa) Failed strength constant, a Normalized hydrostatic tensile limit, T * (GPa) Principal tensile failure stress, Tf (GPa) Damage constant, εfmax Damage constant, P3 (GPa) Fracture energy, Gf (J/m2)

3215 220 361 0 1.0 193 11.7 7.1 2.5 12.2 10.0 0.009 1.3 0.4 0.75 1.3 0.5 99.75 37.3

Density, r0 (kg/m3)

17,600

7830

The JohnsoneCook (JC) constitutive and failure models are used for the tungsten rod [16] and steel 4340 plugs [17], various material properties and parameters are as listed in Table 2. Some of the constants were modified slightly from original values to the values of the materials used in Ref. [5]: The density, yield strength and bulk modulus of the tungsten alloy were changed to 17,600 kg/m3, 1.2 GPa and 285 GPa, respectively. The yield strength for the steel 4340 was changed to 0.75 GPa. The Von Mises constitutive model and effective plastic strain failure model used for steel Mar 350 tube [7] are listed in Table 3.

EOS

Shock

Linear

Bulk modulus, K1 (GPa) Gruneisen constant Reference temperature, t0 (K) Parameter C1 (m/s) Parameter S Specific heat, Ct (J/kg K)

285 1.54 300 4029 1.237 134

159 e 300 e e 477

Strength model

JC

JC

Shear modulus, G (GPa) Static yield strength, A0 (GPa) Strain hardening constant, B0 (GPa) Strain hardening exponent, n Strain rate constant, C0 Reference strain rate, ε_ 0 Thermal softening exponent, m Melting temperature, tm (K)

160 1.506 0.177 0.12 0.016 1 1 1723

77 0.75 0.51 0.26 0.014 1 1.03 1793

Fracture model

JC

JC

0 0.33 1.5 0 0

0.05 3.44 2.12 0.002 0.61

Damage Damage Damage Damage Damage

constant, constant, constant, constant, constant,

d01 d02 d03 d04 d05

“interface defeat” is the dwell without penetration into the target, “dwellepenetration transition” is the dwell followed by penetration. The SiC targets were numerically pre-stressed radially ðRR ¼ 1:0035; RA ¼ 1Þ identical to experimental settings [5]. As shown in Fig. 3, the DOP variations with time in the three simulations compare well with those in experimental measurements.

2.4. Validation of the numerical model 3. Results and discussions Three experiments presented in the work of Lundberg et al. [5] were simulated. Variation of depth of penetration (DOP) into the SiC targets with time has been compared with experimental measurements. The impact velocities in these three experiments were 1410 m/s, 1645 m/s and 2175 m/s. Three typical phenomena namely interface defeat, dwellepenetration transition and penetration, were observed in the simulations. In should be noted that herein the “dwell” is referred to as the radial flow of the projectile,

Three pre-stress conditions, i.e. radial, axial, and hydrostatic, with different pre-stress values were modeled numerically on confined SiC targets. The values of RR and RA for all pre-stress conditions are listed in Table 4. Two series of numerical simulations in which tungsten LRP impacted on these targets were performed. In series 1, each target was impacted at representative velocities of 1410 m/s, 1645 m/s and 2175 m/s. The variations of DOP into SiC targets with time and the SiC damage contours due to impact at certain times were obtained. For the impact velocity of 1410 m/s, in the experiments of Lundberg et al. [5] and also current simulations with or without pre-stress, no penetration (i.e. interface defeat) was observed in the targets and it was not possible to evaluate the ceramic ballistic performance solely based on dwell time and DOP into SiC. Therefore, those cases at 1410 m/s impact velocity were not included in the discussions for evaluating the target performance of different pre-stress conditions. In series 2, a large number of impact simulations on each target at different velocities with 5 m/s intervals were performed and the dwell time was checked to determine VUL for each case. Table 3 Material model constants for steel Mar 350 [5,7].

Fig. 3. Comparison between DOP (into radially pre-stressed SiC) results obtained from simulations with experimental measurements [5] for three different impact velocities.

Material/Constants

Steel Mar 350

Density, r0 (kg/m3) Bulk modulus, K (GPa) Shear modulus, G (GPa) Yield stress (GPa) Failure strain

8080 140 77 2.6 0.4

R. Chi et al. / International Journal of Impact Engineering 84 (2015) 159e170

163

Table 4 Pre-stress conditions in numerical models. Radial pre-stress

Axial pre-stress

Hydrostatic pre-stress

RR ¼ 1:00; 1:0035; 1:01; 1:015; 1:02; 1:025; 1:03 ðRA ¼ 1Þ

RA ¼ 1:00; 1:0035; 1:01; 1:015; 1:02; 1:025; 1:03 ðRR ¼ 1Þ

RR ¼ RA ¼ 1:00; 1:0035; 1:01; 1:015; 1:02; 1:025; 1:03

Two typical phenomena observed in impact simulations will be described prior to discussing the pre-stress effects on confined SiC targets performances. The simulation result for the case with radial pre-stress ðRR ¼ 1:0035; RA ¼ 1Þ and impact velocity V ¼ 2175 m/ s, shown in Fig. 4, is used as an example for descriptions. Fig. 4(a) shows the material damage contour in the SiC target. It is obvious that SiC failure does not start from the ceramic interface; however it is initiated at a distance below the impact site. This compares well with experimental observations of LaSalvia et al. [18] and Hauver et al. [19] for SiC targets under dwell/interface defeat condition. The initial failure position for this case locates around 1.35 mm below the impact site. It was observed that there are limited differences in the initial failure positions for all prestress conditions and impact velocities. The dwell time is influenced by the pressure at the initial failure position obtained due to pre-stress before impact, PIFP . In Fig. 3, the penetration rate for the case with V ¼ 2175 m/s, between 15 ms to 20 ms, is comparatively less than that for other penetration times. This deceleration can enhance the SiC target performance under impact. This is due to a phenomenon herein called inner dwell (i.e., dwell within the ceramic), shown in Fig. 4(c). Rod penetration proceeds in three stages. In the first stage (see Fig. 4(b)), the damage in the SiC material moves much faster than penetration rate of the projectile and the damage propagation stops at position A. The rod penetrates the failed SiC at a nearly same velocity until the instance of 15 ms when the rod reaches position A. In the second stage (see Fig. 4(c)), the failure in SiC material at the front of the rod cannot propagate quickly. There is only a thin layer of SiC material failed between the rod head and the intact SiC

material. The rod is retarded by the intact SiC material as if the rod is dwelling (flowing radially without much penetration) within the ceramic. In the third stage (see Fig. 4(d)), the failure of SiC material propagates from the SiC rear surface to the rod head position and the resistance of the intact SiC disappears and the penetration rate becomes almost the same as that in stage 1. The inner dwell phenomenon can be verified since the simulation results fit well with the experimental data shown in Fig. 3. However, inner dwell only exists in some cases and the mechanism pertaining to this phenomenon is not explored in this paper. 3.1. Dwell, penetration and SiC damage for different case studies Results of numerical simulation series 1 are presented in Fig. 5, 6, 7, 9, 10 and 12. In Figs. 5e7, the first column shows the pressure profiles in the SiC targets due to pre-stress, and the second and third columns show SiC damage at instances of 32 ms and 15 ms at impact velocities of 1645 m/s and 2175 m/s, respectively. The tube, plugs and rod are shown in different colors. Figs. 9, 10 and 12 show the penetration into the SiC target as a function of time for radial, axial and hydro pre-stress conditions, respectively. It should be noted that there is no available experimental data in the literature for the comparison with the current axial and hydrostatic prestress simulation results. 3.1.1. Radial pre-stress As shown in Fig. 5, the pressure in the SiC target under radial pre-stress is not uniform due to the non-uniform deformation of the confinement. The pressure on the front and rear surface of the

Fig. 4. Material damage contour at certain times in the SiC target for the case with radial pre-stress ðRR ¼ 1:0035; RA ¼ 1Þ and impact velocity V ¼ 2175 m/s.

164

R. Chi et al. / International Journal of Impact Engineering 84 (2015) 159e170

Fig. 5. Computed results for an LRP impacting radially pre-stressed SiC target (pressure due to pre-stress and damage due to impact are presented only for SiC target).

SiC target is larger than that on its periphery and inside. The overall pressure is increased as RR increases from 1.0035 to 1.025. For the case with RR ¼ 1:03, the overall pressure is less than that in the cases of RR ¼ 1:02; 1:025. The pressure in the SiC target results from the confinement constraint controlled by two mechanisms. The expansion of the SiC target with larger RR in the radial direction produces larger tube constraint magnitude if there is only elastic deformation in the tube. However, as shown in Fig. 8, which presents the equivalent plastic strain (EPS) contours of four levels of radial pre-stress ðRR ¼ 1:01; 1:02; 1:025; 1:03Þ, the EPS of the tube concentrating at the corners increases and the tube material undergoing (plastic) failure becomes larger and hence the tube constraint on the SiC is lost. As RR increases, the latter mechanism becomes more effective than the former one and the pressure reaches a maximum value for a specific RR after which pressure decreases. Fig. 5 shows that for the cases of an impact velocity of 1645 m/s, there is less penetration in all the SiC targets under radial pre-stress than that in the SiC target without pre-stress ðRR ¼ RA ¼ 1Þ. The

penetration decreases as RR increases from 1.0035 to 1.015. This is due to the amount of dwell that occurs. It is shown in Fig. 9 that for the cases of an impact velocity of 1645 m/s, the dwell time increases as RR increase from 1.0035 to 1.015, which is consistent with the PIFP trend. There is no penetration into the SiC targets for the cases with RR ¼ 1:02; 1:025; 1:03, as shown in Fig. 9. These results demonstrate that, for the cases of an impact velocity of 1645 m/s, the least PIFP pressure under which dwell occurs is between 1 GPa and 1.2 GPa. The SiC target with larger pressure exhibits a lower penetration rate. Especially for RR ¼ 1:015, the penetration rate keeps decreasing to 0 and the final DOP is 8.4 mm. In the SiC target without pre-stress ðRR ¼ RA ¼ 1Þ, the penetration rate is not constant, implying that inner dwell phenomenon occurs within the ceramic during penetration. This phenomenon is shown in Fig. 5, for the case without pre-stress, in which there is little failed SiC material at the front of the rod. Fig. 5 shows that, for the case of an impact velocity of 2175 m/s, all the SiC targets under radial pre-stress produce almost the same penetration which is slightly less than that in the SiC target without

R. Chi et al. / International Journal of Impact Engineering 84 (2015) 159e170

165

Fig. 6. Computed results for LRP impacting onto axially pre-stressed SiC target (pressure due to pre-stress and damage due to impact are presented only for SiC target).

pre-stress. The reason for this can be seen in Fig. 9, where the rods begin to penetrate at the same time without dwell. There is no dwell for 2175 m/s as the LRP kinetic energy is much higher than the impact resistance of the pre-stressed target. The penetration rate varies inversely with the amount of the pre-stress for RR ¼ 1:01; 1:025; 1:03, but not for other cases in which inner dwell occurs. Different inner dwell time produces obvious differences in penetration, as shown in Fig. 9. The amount of SiC material damaged in the front and rear side of the SiC target is consistent with the amount of pre-stress in those respective regions. However, the damage for RR ¼ 1:02; 1:025; 1:03 is essentially the same as the difference of the overall pressure in these SiC targets is not large. 3.1.2. Axial pre-stress It is shown in Fig. 6 that the pressure in the regions adjacent to the front and rear surfaces of the SiC target under axial pre-stress is negative (under tension) and less than that in other regions. The plugs bend outward when the SiC target expands in the axial direction. Gaps appear around the shot line between the SiC target and plugs since the bending of the plugs (which are not stiffer than the tube) is relatively large. The SiC target only interacts with plugs on the periphery without gaps. Therefore, part of the SiC material close to the gaps is under tension. The tensile pressure, for the case with RA ¼ 1:03, is so large that the SiC material fails under pre-

stress. The simulated results for the case with RA ¼ 1:03 are not shown in Figs. 6 and 10. It is obvious in Fig. 6 that the variation of pressure with RA in the region close to the front and rear surface is different from that in other regions. The tensile pressure in the region close to the front and rear surface for RA ¼ 1:01 is the least and increases as RA increases from 1.01 to 1.025. The amount of the tensile pressure is proportional to the gap size between the SiC target and plugs controlled by two mechanisms. Firstly, the plastic deformation on the periphery of the plugs results in release of the constraint, on the SiC, provided by the plugs and diminishes the gap size in the axial direction. Secondly, the plastic deformation in the outer portion of the plugs, which produces material damage, diminishes the outward bending resistance (in axial direction) of the plugs resulting in gap size increase in the axial direction. As RA increases, the plastic deformation in both regions mentioned above increases. As RA increases from 1.0035 to 1.01, the amount of tensile pressure decreases because the former mechanism is more effective than the latter one to influence the gap size. As RA increases from 1.01 to 1.025, the latter mechanism becomes more effective and the amount of tensile pressure increases. The pressure on the initial SiC material failure position, PIFP which is close to the SiC front surface varies in the same trend. The pressure in other region increases as RA increases from 1.0035 to 1.02 and it is slightly less for RA ¼ 1:025 than that for RA ¼ 1:02.

166

R. Chi et al. / International Journal of Impact Engineering 84 (2015) 159e170

Fig. 7. Computed results for LRP impacting onto hydrostatically pre-stressed SiC target (pressure due to pre-stress and damage due to impact are presented only for SiC target).

The overall pressure due to axial pre-stress in all the cases and their differences are less than that for radial pre-stress. This is because the plugs cannot provide constraint for the expanding SiC target, as strong as that provided by the tube. It is shown in Fig. 10 that for the cases of an impact velocity of 1645 m/s the dwell time in all axial pre-stress cases is almost the same and is almost equal to the dwell time for the case without prestress ðRR ¼ RA ¼ 1Þ, since the axial pre-stress produces small

pressures of little differences. In addition, the dwell time is not consistent with PIFP variation. A possible reason is that the pressure difference, in the SiC targets of different axial pre-stress ratios, is not so large and there is a small error in numerical simulations. Complex simulations, like the one in the current work, include different steps in transforming a physical problem into a discrete model and solving it through finite difference or finite element methods. Numerical errors may occur in each stage due to any of

Fig. 8. Effective plastic strain (EPS) contour for radially pre-stressed targets.

R. Chi et al. / International Journal of Impact Engineering 84 (2015) 159e170

167

Fig. 9. Penetration depth into radially pre-stressed SiC targets for impact velocities of (a) 1645 m/s and (b) 2175 m/s obtained from simulations (experimental data [5] for radially pre-stressed target of RR ¼ 1:0035 are included as well for comparison).

the discretization, interface modeling conditions, material and geometric non-linearities, parameters involved in the solution algorithm and finding exact error is a difficult task. These errors might have contributed to the minor inconsistency for the dwell time with the pressure on the initial SiC material failure position, PIFP . It is shown in Fig. 10 that for the cases of an impact velocity of 2175 m/s the dwell time is zero and inner dwell occurs in all cases from about 15:5 ms to 17:5 ms. The amount of the damage on the rear side of the SiC target, shown in Fig. 6, is almost consistent with the tensile pressure at velocities of 1645 m/s and 2175 m/s. In conclusion, axial pre-stress cannot improve significantly the ballistic performance of the target with the current geometry and material system. 3.1.3. Hydrostatic pre-stress It is shown in Fig. 7 that the pressure due to hydrostatic prestress in the SiC target is approximately uniform for RR ¼ RA ¼ 1:0035, but not for other cases. The deformation of both the tube and plugs controls the pressure. For the case with RR ¼ RA ¼ 1:0035, the deformation of the confinement at its corners in axial and radial directions is not large and it differs only slightly. It provides the SiC target similar constraint in radial and axial directions. Since the tube is stiffer than the plugs, as RR and RA increases the constraint (pressure) of the tube on the SiC is more effective than that of the plugs. The pressure in the front and rear side of the SiC target is much higher than that on its circumference. The pressure in regions close to the front and rear SiC surface is less for RR ¼ RA ¼ 1:0035 compared to the one for

RR ¼ RA ¼ 1:01  1:03 due to larger effect of radial pre-stress in the latter cases. For the case with RR ¼ RA ¼ 1:0035, the pressure is approximately equal to the sum of that due to individual radial and axial pre-stress since the tube and plugs are under elastic deformation. The typical pressure enhancement mechanism for RR ¼ RA ¼ 1:01  1:03 is illustrated in Fig. 11, which shows the target models around SiC corners after hydrostatically pre-stressing with RR ¼ RA ¼ 1:025 (Fig. 11(a)), in comparison with that for radially pre-stressing wit RR ¼ 1:025; RA ¼ 1:00 (Fig. 11(b)). For the case with RR ¼ 1:025; RA ¼ 1:00, the SiC target corners squeeze the tube mostly in the radial direction and severe shear failure occurs. For the case with RR ¼ RA ¼ 1:025, the movement of the front and rear surface of the SiC target in the axial direction contributes to avoid severe shear failure. Thus, it maintains a stronger load on the tube, thereby producing larger pressure. The pressure enhancement by this mechanism is much larger than the pressure reduction by the axial pre-stress. The variation trend of overall pressure in the hydrostatically pre-stressed SiC target with increasing RR and RA can be explained in the same way as described for a radially pre-stressed target. Figs. 7 and 12 show that for the case of an impact velocity of 1645 m/s, the rod penetrates the SiC target only when RR ¼ RA ¼ 1:035; 1:01 and the larger initial pressure results in longer dwell time. The inner dwell occurs in these two cases and the increase in penetration depth stops at 55:7 ms for RR ¼ RA ¼ 1:01. For the occurrence of dwell, a minimum value of PIFP is needed, below which dwellepenetration transition or penetration occurs. If PIFP is high enough, dwell is prolonged and interface defeat occurs as in the case of RR ¼ RA ¼ 1:015, as shown

Fig. 10. Penetration depth into axially pre-stressed SiC targets for impact velocities of (a) 1645 m/s and (b) 2175 m/s.

168

R. Chi et al. / International Journal of Impact Engineering 84 (2015) 159e170

Fig. 11. (a) Shear failure prevention in the tube material due to the pressure enhancement mechanism of the hydrostatically pre-stressed ðRR ¼ RA ¼ 1:025Þ SiC (b) shear failure occurred in the tube material as it is squeezed by the corners of radially pre-stressed ðRR ¼ 1:025; RA ¼ 1:00Þ SiC.

in Fig. 12(a). For the cases of an impact velocity of 2175 m/s, there is no dwell in all hydrostatic pre-stress cases, which is similar to the phenomenon observed for radial and axial pre-stress cases. There is an obvious inner dwell for the case with RR ¼ RA ¼ 1:0035, which is similar to that for the radial pre-stress with RR ¼ 1:0035. In other cases, the penetration rate differs slightly. Fig. 7 shows that the damage of SiC material for the case with RR ¼ RA ¼ 1:0035 is the largest, which is consistent with the pressure contour in the SiC and differs slightly from the damage in the cases of RR ¼ RA ¼ 1:015  1:03.

value is shown in Fig. 14. Although the variation of PIFP with increasing RR or RA was discussed in the previous section, Fig. 14 provides quantitative data for further consideration. The maximum values of PIFP are 1.196 GPa and 1.264 GPa due to radial pre-stress of RR ¼ 1:025 and hydrostatic pre-stress of RR ¼ RA ¼ 1:02, respectively. These maximum pressure values are

3.2. The upper limit of transition velocity The simulation results in the previous section indicate that the dwell time, penetration rate and SiC damage are consistent with the amount of pressure for some of the pre-stress conditions. However, it is not possible to determine the effect of pressure on the pre-stressed SiC target ballistic performance by only using the impact simulation results at certain velocities if there is interface defeat, penetration or small pressure difference among various cases. Fig. 13 shows the variation of the upper limit of transition velocity, VUL with the pre-stress value for all pre-stress conditions listed in Table 4. The VUL can be considered as a measure for ballistic performance of the SiC targets as there is a unique VUL value for each pre-stress condition. The variation of PIFP with the pre-stress

Fig. 13. Variation of upper limit of transition velocity, VUL with pre-stress value.

Fig. 12. Penetration depth into hydrostatically pre-stressed SiC targets for impact velocities of (a) 1645 m/s and (b) 2175 m/s.

R. Chi et al. / International Journal of Impact Engineering 84 (2015) 159e170

much higher than the maximum value of PIFP due to the axial prestress of RA ¼ 1:01. The sum of PIFP due to the individual radial and axial pre-stress is also shown in Fig. 14. This sum, for the case when individual RR and RA are both 1.0035, is approximately equal to PIFP due to the hydrostatic pre-stress, but is less than the PIFP due to the radial pre-stress. The reason for this was discussed in section 3.1.3. The sum of PIFP due to the individual radial and axial pre-stresses, for the case when individual RR and RA are 1.01e1.025, is obviously less than the PIFP due to the hydrostatic pre-stress. This is due to the pressure enhancement mechanism under hydrostatic prestress presented in section 3.1.3. The variation of VUL (shown in Fig. 13) with RR or RA shows, in general, a similar trend to that of PIFP (shown in Fig. 14). The maximum values of VUL is 1890 m/s when RR ¼ 1:025 and 1905 m/s when RR ¼ RA ¼ 1:02, for the radial and hydrostatic pre-stresses, respectively. These maximum values of VUL are respectively 7.39% and 8.24% higher than that for the case without pre-stress ðRR ¼ RA ¼ 1Þ. The maximum value of VUL for axial pre-stress is 1780 m/s, which occurs when RA ¼ 1:015, but not when RA ¼ 1:01 for which the maximum value of PIFP is found. This slight mismatching is believed to be due to numerical error in the simulations. Both Figs. 13 and 14 imply that VUL variation with the pre-stress ratio is consistent with that of the PIFP trend. The relationship between VUL and PIFP is presented in Fig. 15 in which VUL for all prestress conditions is shown as a function of PIFP . It can be seen that the VUL increases linearly as the PIFP increases.

4. Summary and conclusions In this paper, a numerical modeling technique for ceramic armor pre-stressing using AUTODYN® has been proposed. Computed results for the responses of confined SiC targets of radial, axial and hydrostatic pre-stress, as well as without pre-stress conditions, subjected to LRP impact have been presented. The effects of different pre-stress conditions and levels on the ballistic performances of SiC targets have been discussed. The results of this study are summarized as follows: (1) The radial and hydrostatic pre-stress improve the ballistic performance of the confined SiC target significantly. As the plug material is not stiff enough in the target considered in this study, the axial pre-stress slightly influences the ballistic performance of the confined SiC target.

169

Fig. 15. Linear relationship between upper limit of transition velocity, VUL and pressure at the initial failure position in SiC, PIFP .

(2) The ballistic performance of the confined SiC target with radial and hydrostatic pre-stress does not increase monotonically with increasing RR and RA ; an optimal value exists for the pre-stress value related to the maximum level of ballistic performance. The SiC target with hydrostatic prestress has better ballistic performance than that with radial pre-stress, except for the cases in which both plugs and tube deform elastically. (3) Dwell at the ceramic interface is influenced by the pressure at the initial failure position in the SiC target, PIFP which increases linearly with the upper limit of transition velocity, VUL . (4) The inner dwell occurring during LRP penetration, as well as the interface dwell, contribute significantly to the ballistic performance of the confined SiC target. Further studies are needed to understand the inner dwell mechanism and the effect of pre-stress on it. Acknowledgment Ahmad Serjouei thanks Nanyang Technological University for the financial support in the form of graduate scholarship. Authors thanks Temasek Laboratories @ Nanyang Technological University (TL@NTU) for the financial support through the project number TL9013103084-02. Runqiang Chi thanks Fundamental Research Funds for the Central Universities for the financial support through the grant number HIT.NSRIF.2015029. References

Fig. 14. Variation of PIFP with pre-stress value.

[1] Weber K, El-Raheb M, Hohler V. Experimental investigation on the ballistic performance of layered AlN ceramic stacks. Reinecke WG. Proceedings of the 18th International Symposium on Ballistics. San Antonio, TX: Technomic; 1999. p. 1247e54. [2] Meyer HW, Abeln T, Bingert S, Bruchey WJ, Brannon RM, Chhabildas LC, et al. Crack behavior of ballistically impacted ceramic. AIP Conf Proc 2000;505(1): 1109e12. [3] Han C, Sun CT. A study of pre-stress effect on static and dynamic contact failure of brittle materials. Int J Impact Eng 2000;24(6e7):597e611. [4] Holmquist TJ, Anderson Jr CE, Behner T, Orphal DL. Mechanics of dwell and post-dwell penetration. Adv Appl Ceram 2010;109(8):467e79. € m R, Lundberg B. Impact of metallic projectiles on ceramic [5] Lundberg P, Renstro targets: transition between interface defeat and penetration. Int J Impact Eng 2000;24(3):259e75. [6] Holmquist TJ, Johnson GR. Modeling prestressed ceramic and its effect on ballistic performance. Int J Impact Eng 2005;31:113e27. [7] Quan X, Clegg RA, Cowler MS, Birnbaum NK, Hayhurst CJ. Numerical simulation of long rods impacting silicon carbide targets using JH-1 model. Int J Impact Eng 2006;33(1e12):634e44.

170

R. Chi et al. / International Journal of Impact Engineering 84 (2015) 159e170

[8] Johnson GR, Stryk RA, Beissel SR, Holmquist TJ. An algorithm to automatically convert distorted finite elements into meshless particles during dynamic deformation. Int J Impact Eng 2002;27(10):997e1013. [9] Swegle JW, Hicks DL, Attaway SW. Smoothed particle hydrodynamics stability analysis. J Comput Phys 1995;116(1):123e34. [10] Balsara DD. Von Neumann stability analysis of smoothed particle hydrodynamicsdsuggestions for optimal algorithms. J Comput Phys 1995;121(2): 357e72. [11] Johnson GR, Holmquist TJ. A computational constitutive model for brittle materials subjected to large strains, high strain rates and high pressures. In: Meyers MA, Murr LE, Staudhammer KP, editors. Shock-wave and high-strainrate phenomena in materials. NY: Marcel Dekker Inc.; 1992. p. 1075e81. [12] Johnson GR, Holmquist TJ. An improved computational constitutive model for brittle materials. AIP Conf Proc 1994;309(1):981e4. [13] Holmquist TJ, Johnson GR. Response of silicon carbide to high velocity impact. J Appl Phys 2002;91(9):5858e66. [14] Clegg RA, Hayhurst CJ. Numerical modeling of the compressive and tensile response of brittle materials under high pressure dynamic loading. AIP Conf Proc 2000;505(1):321e4.

[15] Taylor EA, Tsembelis K, Hayhurst CJ, Kay L, Burchell MJ. Hydrocode modeling of hypervelocity impact on brittle materials: depth of penetration and conchoidal diameter. Int J Impact Eng 1999;23(1, Part 2):895e904. [16] Lee JK. Analysis of multi-layered materials under high velocity impact using CTH. Department of aeronautics and astronautics engineering, Master of science dissertation. Ohio: Air University; 2008. [17] Johnson GR, Cook WH. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng Fract Mech 1985;21(1):31e48. [18] LaSalvia JC, Normandia MJ, Miller HT, MacKenzie DE. Sphere impact induced damage in ceramics: I. Armor-grade SiC and TiB2. Advances in ceramic armor: a collection of papers presented at the 29th international conference on advanced ceramics and composites, January 23e28, 2005, Cocoa Beach, Florida, ceramic engineering and science proceedings. John Wiley & Sons, Inc.; 2008. p. 170e81. [19] Hauver GE, Netherwood PH, Benck RF, Rapacki EJ. Interface defeat of long-rod projectiles by ceramic armor. Aberdeen Proving Ground, MD, USA: Army Research Laboratory; 2005. p. 98.