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Numerical study of size effect in concrete penetration with LDPM
Defence Technology xxx (2018) 1e10
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Numerical study of size effect in concrete penetration with LDPM Jun Feng a, *, Weiwei Sun b, Baoming Li a a b
National Key Laboratory of Transient Physics, Nanjing University of Science and Technology, Nanjing, 210094, China Department of Civil Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 June 2018 Received in revised form 15 July 2018 Accepted 19 July 2018 Available online xxx
Projectile size effect is of great importance since the scaling researches are extensively applied to concrete penetration investigations. This paper numerically deals with the projectile size effect on penetration resistance via the recently developed Lattice Discrete Particles Model (LDPM) which is featured with mesoscale constitutive laws governing the interaction between adjacent particles to account for cohesive fracture, strain hardening in compression and compaction due to pore collapse. Simulations of two different penetration tests are carried to shed some light on the size effect issue. The penetration numerical model is validated by matching the projectile deceleration curve of and predicting the depth of penetration (DOP). By constant velocity penetration simulations, the target resistance is found to be dependent on the projectile size. By best fitting numerical results of constant velocity penetration, a size effect law for target resistance is proposed and validated against literature data. Moreover, the size effect is numerically obtained in the projectile with longer extended nose part meanwhile the shorter extended nose is found to improve the DOP since the projectile nose is sharpened. © 2018 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Penetration of concrete Size effect Lattice discrete particle model Target resistance Abnormal nose projectile
1. Introduction The response of high speed penetration of concrete target by projectiles has been a research hotspot for several decades, due to its relevance for both industry and military establishments. Usually, the concrete penetration investigations are carried out in reduced geometrical scale since the ballistic tests are dangerous and moneyconsuming. Issues concerning penetration of massive concrete targets by rigid projectiles have been dealt with empirical formulae, analytical model as well as the numerical modelling [1e7]. However, almost all of these methods fail to account for the size effect of concrete structure during penetration analyses. The studies on size effect of concrete structures mainly were conducted in the area of fracture [8e10], and the understanding of the size effect in concrete penetration is still largely lacking. With the concept of non-dimensional formula based on the dynamic cavity expansion model, Chen and Li [4,5] developed DOP prediction model for penetration with extensive boundary conditions which has been widely used by researchers in concrete penetration area. Although the non-dimensional formula for DOP
* Corresponding author. E-mail address: [email protected] (J. Feng). Peer review under responsibility of China Ordnance Society
significantly contributes to the promotion of concrete penetration investigations, there might be some problem when researchers try to extend the small scale penetration mechanism to the larger (real) scale experiment scenarios. The reason of the projectile diameter scale effect on the penetration response was explained by Rosenberg and Kositski [11] that the harder aggregate may pose more resistance to the smaller diameter projectile during its movement inside the target. On the other hand, size effect theory proposed and explained by Bazant [9] indicates that the nominal structure strength decreases as the structure size increases. Forrestal et al. [12] noticed there might be size effect between the projectile diameter and the target resistance Rt since the experimental data of 76.2 mm diameter projectile penetration doesn't match the empirical formula for the target resistance. Then Bludau et al. [13] showed that the resistance to penetration of concrete targets is dependent on both the aggregate size, through the ratio of projectile diameter and aggregate size, and the strength of the aggregate material. Similar conclusions were drawn via experiments by Dancygier et al. [14]. Although the projectile size effect was reported, few efforts have been made to address this issue. Rosenberg and Kositski [11] attempted to develop a semi-empirical model for both concrete penetration and perforation by including the projectile diameter size effect. More recently, Peng et al. [15] developed a mesoscale concrete model for penetration considering both aggregate, cement
https://doi.org/10.1016/j.dt.2018.07.006 2214-9147/© 2018 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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and ITZ. Their numerical results suggest the penetration resistance is strongly dependent on the projectile relative size against the aggregate. With reference to the cavity expansion analysis, a modified penetration resistive force formulae was proposed with aggregate strength and projectile size taken into account. In the present paper, the numerical simulation of rigid projectile penetration is performed with Lattice Discrete Particle Model (LDPM) which has been proven a robust model for concrete penetration and perforation in our previous study [16e18]. The numerical study of LDPM [19] shows a good capacity of concrete size effect modelling of 3-point bending, splitting and even compression tests. Two penetration tests concerning projectile size effect are selected for LDPM numerical study. The penetration model is validated by predicting the projectile deceleration history as well as DOP where data scattering is discussed. The extensive simulations of constant velocity penetration in concrete give the target resistance of different projectile geometries. Then, the size effect law for target resistance is proposed and validated. Finally, the size effect of abnormal nose projectile penetration is numerically captured and discussed. 2. Lattice Discrete Particle Model (LDPM) introduction As a synthesis of the Confinement Shear Lattice model and discrete method, the theoretical framework of Lattice Discrete Particle Model was developed to simulate the mechanical interaction of coarse aggregate pieces embedded in a binding matrix [20e22]. The geometrical representation of concrete mesostructure is constructed by randomly introducing and distributing spherical shaped coarse aggregate particles inside the volume of interest and zero-radius aggregate particles on its surface as shown in Fig. 1(a). With Delaunay tetrahedralization of the generated particle centers, a three-dimensional domain tessellation creates a system of polyhedral cells (see Fig. 1(b)) interacting through triangular facets and corresponding lattice system. To construct the mesoscale framework, concrete mix design parameters are needed including: maximum/minimum aggregate particle size da/d0; cement content c water-to-cement ratio wt/c; and the coefficient for the classical Fuller curve nF. 2.1. LDPM kinematics The triangular facets forming the rigid polyhedral particles are assumed to be the potential material failure locations. Each facet is shared between two polyhedral particles and is characterized by a local system of reference featured by a unit normal vector n and two tangential vectors m and l as noted in Fig. 1(b). The deformation of the lattice particle system is described by the rigid body
kinematics whereas the displacement jump EuC F at each triangular facet centroid is used to define the mesoscale strain component as:
εN ¼
nT EuC F mT EuC F lT EuC F ; εM ¼ ; εL ¼ l l l
(1)
where l is the length of the tetrahedron edge. The strain definitions in Eq. (1) have been proven by Cusatis et al. [23] to be consistent with the definition of strain in classical continuum mechanics. The corresponding normal and shear stress are calculated through LDPM mesoscale constitutive laws and the equilibrium are imposed via the principle of virtual work [20].
2.2. LDPM constitutive equations In the elastic regime, the normal and shear stresses are proportional to the corresponding strains: sN ¼ EN ,εN ; sM ¼ ET ,εM ; sL ¼ ET ,εL , where EN ¼ E0 , ET ¼ aE0 , E0 is the effective normal modulus, and a is shear-normal coupling parameter. On the facets, the reversible elastic behavior is limited by a number of nonlinear boundaries which are featured by softening for pure tension and shear-tension, as well as plastic hardening for pure and shear compression.
2.2.1. Fracturing behavior For fracture behavior characterized by normal strains ðεN > 0Þ, the fracturing evolution is formulated through the effective strain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε ¼ ε2N þ aðε2M þ ε2L Þ and the effective stress s ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2N þ ðs2M þ s2L Þ=a. The tensile boundary sbt evolves exponentially as a function of the maximum effective strain of the loading history εmax ¼ maxtt ½εðtÞ
sbt ¼ s0 ðuÞexp½ H0 ðuÞhεmax ε0 ðuÞi=s0 ðuÞ
(2)
pffiffiffi where Macaulay bracket 〈x〉 ¼ maxfx; 0g, tanðuÞ ¼ εN = aεT ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi sN a=sT , and εT ¼ ε2M þ ε2L . Reaching elastic limit ε0(u), the fracturing damage decays the boundary sbc with the post peak softening modulus defined as H0 ðuÞ ¼ Ht ð2u=pÞnt , where Ht is the softening modulus in pure tension (u ¼ p/2) expressed as Ht ¼ 2E0 =ðlt =l 1Þ; character length lt ¼ 2E0 Gt =s2 ; and Gt is the mesoscale fracture energy. LDPM provides a smooth transition between pure tension and pure shear (u ¼ 0) with parabolic variation for strength given by
Fig. 1. One LDPM polyhedral cell around an aggregate particle.
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2 sinu þ s0 ðuÞ ¼ st rst
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sin2 u þ 4a cos2 u r 2st
2a cos2 u
(3)
where rst ¼ ss =st is the ratio of shear strength to tensile strength. 2.2.2. Pore collapse and subsequent compaction Normal stresses for compressive loading (εN < 0) are computed meeting the inequality sbc ðεD ; εV Þ sN 0 where sbc is the boundary function of the volumetric strain εV and the deviatoric strain εD . Beyond the mesoscale compressive yield stress sc0 , sbc models pore collapse as a linear evolution of stress for increasing volumetric strain with stiffness Hc for εV εc1 ¼ kc0 εc0 :
sbc ¼ sc0 þ hεV εc0 iHc rDV
(4)
where Hc ðrDV Þ ¼ Hc0 =ð1 þ kc2 hrDV kc1 iÞ and kc1 , kc2 are deviatoric parameters. Beyond pore collapse εV εc1 , compaction and rehardening occur. In this case one has
sbc ¼ sc1 rDV exp½ð εV εc1 ÞHc rDV =sc1 rDV
(5)
where sc1 rDV ¼ sc0 þ ðεc1 εc0 ÞHc rDV . 2.2.3. Friction due to compression-shear The incremental shear stresses rates are computed as s_ ¼ E ð_ε ε_ p Þ and s_ ¼ E ð_ε ε_ p Þ, where ε_ p ¼ l_ v4=vs , ε_ p ¼ M
T
M
M
L
T
L
L
M
M
L
l_ v4=vsL , and l is the plastic multiplier. The plastic potential is
defined as 4 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2M þ s2L sbc sN , where the nonlinear frictional
law for the shear strength is assumed to be
sbs ¼ ss þ ðm0 m∞ ÞsN0 ½1 expðsN =sN0 Þ m∞ sN
(6)
where sN0 is the transitional normal stress, m0 and m∞ are the initial and final internal friction coefficients. 2.2.4. Strain rate effect in LDPM The LDPM formulation is extended to incorporate rate dependent fracture mechanisms associated with the interpretation of thermally activated phenomena which is governed by the classical Maxwell-Boltzmann equation [24]. The crack opening rate dependent cohesive behavior can be expressed as: _ ¼ ½1 þ c1 a sin hðw=c _ 0 Þf ðwÞ where sch is the cohesive sch ðw; wÞ stress, w is the crack opening, w_ is the crack opening rate, f ðwÞ is the cohesive law under static condition, c0 is the reference crack opening rate and c1 is the strain rate coefficient. Compared to the crack opening rate, the elastic strain rate can be negligible [25], hence the LDPM effective strain can be written as _ þ s_ =E0 zw=l. _ ε_ ¼ w=l Substituting w_ by l_ε, LDPM boundary condition can be modified as:
sbt ðε; ε_ Þ ¼ Fð_εÞs0 ðuÞexp H0 ðuÞexp
hεmax ε0 ð_ε; uÞi s0 ðuÞ
(7)
where ε0 ð_ε; uÞ ¼ Fð_εÞs0 ðuÞ=E0 and the rate effect function is expressed as Fð_εÞ ¼ 1 þ c1 a sin h c ε_=l . 0
3. Numerical analysis of Sandia Lab penetration tests [27] A serious of experimental investigations on penetration in concrete targets with different projectile nose geometry, striking
3
velocities were carried out by M.J. Forrestal et al. [12,27,28] to study the response of projectile penetration in concrete target. The 76.2mm-diameter ogival nose projectiles were designed with a singlechannel acceleration data recorder. Both penetration depth and projectile deceleration data during penetration were measured and provided in details. With 3.0 and 6.0 caliber-radius-head (CRH), ogival nose projectiles were launched with striking velocities between 140 and 460 m/s to impact the concrete targets with 23 MPa compressive strength. These experiments, including the triaxial compression and penetration tests, have been numerically studied in our previous work [26] whereby the calibration and validation of the penetration model are conducted. In this section, we further analyze the size effect phenomenon with respect to the penetration test in Ref. [27].
3.1. LDPM simulation of concrete penetration 3.1.1. LDPM parameter calibration The concrete mixture proportion by weight is characterized by cement content c ¼ 310 kg=m3 , water to cement ratio wt=c ¼ 0:84, aggregate to cement ratio a=c ¼ 5:2, fuller coefficient nF ¼ 0:5 which are input information for LDPM particle generation. The nominal compressive strength was obtained from uniaxial compression test with 50.8 mm (diameter), 114 mm (height) cylinder specimens which were also used for triaxial compression (TXC) tests. Calibration of LDPM parameters are conducted with respect to hydrostatic test, triaxial compression tests with different confinement pressure and uniaxial strain compression test. The hydrostatic simulation is conducted by applying an increasing hydrostatic pressure phy on the cylinder surface. For triaxial compression simulations, the top surface is driven by a rigid top block with velocity control corresponding to the hydrostatic pressure, and the bottom surface is assigned frictional contact with a fixed block. After the lateral surface reaches the confinement pressure, the top block keeps moving with a constant velocity while the confinement is kept constant. The uniaxial strain test is modelled with the top surface moving at a low constant velocity and the lateral surface radial boundary fixed. The calibrated results are shown in Fig. 2, which compares the numerical and experimental curves for hydrostatic compression and TXC tests. In Fig. 2(b), the damaged concrete cylinders under 50 and 200 MPa confinement are comparatively plotted where comminution occurs for the triaxial response. The detailed response of TXC simulation is depicted in Fig. 2(c), where the red circles represent the loading condition transforming from hydrostatic compression to triaxial compression. With reference to Ref. [26], the LDPM parameters are calibrated as follows: normal elastic modulus E0 ¼ 16500 MPa, densified normal modulus Ed ¼ 4:2E0 , shear-normal coupling parameter a ¼ 0:25, tensile strength st ¼ 2:7MPa, yielding compressive stress sc0 ¼ 42 MPa, shear strength ss ¼ 2:0st , tensile characteristic length lt ¼ 100 mm, softening exponent nt ¼ 0:3, initial hardening modulus Hc0 ¼ 0:46E0 , transitional strain ratio kc0 ¼ 6, initial internal friction coefficient m0 ¼ 0:4, internal asymptotic friction coefficient m∞ ¼ 0, transitional stress sN0 ¼ 400MPa, deviatoric-to-volumetric strain ratio kc1 ¼ 2, and kc2 ¼ 2. Subsequent to calibration, the obtained LDPM parameters are used to simulate the response of concrete target subjected to ogival nose projectile impact. According to the penetration experiments [12], the projectiles were launched to normal impact the concrete targets and the ballistic tunnels measured after tests were almost straight within 0.6 pitch and yaw. Furthermore, abrasion on the projectile surface was also reported negligible, thus rigid projectile
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Fig. 2. LDPM calibration of tests.
assumption can be made in this numerical simulation. The calibrated LDPM parameters are utilized for concrete target modelling and a penalty interaction [29] between rigid projectile and LDPM particles is assigned. 3.1.2. Penetration simulation validation The penetration prediction results are listed in Table 1, which indicates that the depth of penetration (DOP) is within 10% error with respect to the tests data reported [27]. Moreover, the projectile deceleration during penetration is another important ballistic property needed to be validated. Fig. 3 illustrates the numerical predicted deceleration curves during the impact process in blue lines and the measured data in black lines where G is the gravitational acceleration. Firstly, the deceleration is characterized with a quick increase corresponding to the projectile nose length. After the nose entirely enters the target, the deceleration value keeps almost constant with a scatter. The large scattering of the numerical deceleration data may due to the updating interaction between LDPM aggregate particles with projectile. As explained in 2D diagram (see Fig. 4), the blue circles represent concrete aggregates in contact with the projectile surface in Fig. 4(a). As the projectile moves forward, the interacted aggregates get updated according to the relative distance between aggregates and projectile. As shown in Fig. 4(b), the aggregate in purple starts to contact with projectile while the aggregate in green becomes too far away from the projectile and no more interaction exists in between. The constant contact pair updating leads to the rough resistive force which is different from FEM results in Ref. [30]. As the projectile velocity reaches zero, the plateau is followed by a sudden drop. This phenomenon indicates that the resistance is not only determined by the impact velocity, rather there should be a material dependent constant term which was called target resistance by Forrestal et al. [12,31]. This ballistic property is fully captured by LDPM numerical prediction and both DOP and deceleration data are in agreement with the test data.
3.2. Size effect of concrete penetration To investigate the projectile size effect on penetration resistance, projectile constant velocity penetration of LDPM simulation is further conducted in this work to obtain the resistive force acting on the projectile nose. Right now, researchers deem that the target resistance Rt depends on the ratio dp =da [11,15] where da is the maximum aggregate size identical to the notation in LDPM introduction. Combining the cavity expansion with projectile penetration in concrete, the ballistic tunnel expansion is driven by the cavity wall with radius R ¼ dp J, namely ogival nose radius. There might be a possibility that the size effect in concrete penetration is related to the ratio R=da . Moreover, penetration tests revealed the 6.0 CRH projectile penetration is characterized with lower Rt in Ref. [12], as shown in Table 2 where Rt is the averaged value of Rt . Thus, this section tries to explore the size factors which might affect the target resistance. Projectiles with 30 mm, 60 mm and 80 mm diameters are chosen herein for numerical study where common CRH values like 3.0, 4.0 and 6.0 are determined. Constant impact velocity vc ranging from 200 to 600 m/s are assigned to the hard projectiles to penetrate the thick concrete target, and their resistive forces are then obtained through LDPM simulations where Fig. 5 gives the projectile resistive forces under 400 m/s projectile impact. For each penetration simulation, the resistive force corresponding to vc ¼ 400 m/s (dash lines in Fig. 5) is calculated as the averaged values when the projectile nose part fully enters the target. According to the widely recognized cavity expansion analysis developed by Forrestal et al. [31,32], the resistive force on the projectile nose should be:
F¼
pd2p 4
Rt þ Nr0 V 2
(8)
where dp is the projectile diameter, V is the projectile impact
Table 1 Comparison of Sandia Lab penetration tests and LDPM predictions. No.
Projectile mass/kg
Striking velocity/(m$s1)
Test DOP/m [27]
LDPM Prediction/m [26]
6e2 3e1 2e2 1e1 5e3 4e4 8e2 7e1
13.043 13.037 13.085 13.158 13.080 13.119 13.061 13.064
139 200 250 284 337 379 238 379
0.24 0.42 0.62 0.76 0.96 1.18 0.58 1.25
0.26 0.45 0.67 0.79 1.06 1.22 0.60 1.29
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Fig. 3. LDPM Prediction of projectile deceleration history vs. test data.
Fig. 4. Contact update diagram for interaction between projectile and aggregate.
Table 2 Target resistance of Forrestal et al. [12]. fc/MPa
dp/mm
da/mm
CRH
Rt /MPa
23 23 39 39
76.2 76.2 76.2 76.2
9.5 9.5 9.5 9.5
3.0 3.0 6.0 6.0
166.5 162.5 356.8 267.5
velocity and N is the “nose factor” reflecting the geometrical char-
acteristics of the projectile nose, for ogival nose projectile N ¼ 8J12 , 24J
J is caliber-radius-head value; the concrete target is described by initial density r0 and target resistance Rt. This postulation assumes that the resistance is only attributed to the compressive pressure acting on the normal direction of projectile nose surface. The target resistance Rt numerically obtained can be estimated via Eq. (8) which are plotted in Fig. 6(a). It is observed that the target resistance somehow increases with the impact velocity which might be
caused by the rate effect term of the penetration resistance, as indicated by some cavity expansion models with three resistance terms, i.e., inertial term, rate effect term and static term (namely Rt) [6,33]. Since this work concentrates on the size effect rather than the resistance mechanisms, we choose to select the Rt of vc ¼ 400 m/s for further study. Also, it is interesting to find that the larger diameter projectiles tend to suffer less target resistance. But the Rt of 60 mm dp with 6.0 CRH seems to be close to the Rt of 80 mm dp with 3.0 CRH. These two cases have same ogival nose radius R, thus we prefer to believe the size effect of Rt depends on R ¼ dp J. The Rt best fit is then performed with respect to R=da as shown in Fig. 6(b). With reference to Ref. [34], the size effect law for target resistance is expressed as Rt ¼ Sfc with S proposed as:
S ¼ 82:6fc0:544
dp J 7:77da
0:43 (9)
where there is no aggregate material strength parameter involved,
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Fig. 5. Resistive force of ogival nose projectile with vc ¼ 400 m/s.
Fig. 6. Target resistance Rt for constant velocity penetration and best fit.
because the aggregate material properties is related to the concrete mechanical property which is already accounted for in the parameter fc. This controversial issue needs to be further studied. The validation is then conducted with reference to the penetration tests with fc ¼ 23 MPa and 39 MPa in Refs. [12,27]. As mentioned by Forrestal et al. [12], the empirical equation
S ¼ 82:6fc0:544 could not be applied to penetration tests with 76.2 mm projectile diameter. By extensive mesoscale numerical simulation, Peng et al. [15] gave the size dependent target resistance function with S ¼ 82:6fc0:544 ðdp =ð2:83da ÞÞ0:208 . Rosenberg and Kositski [11] summarized penetration data and empirically
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come up with the expression Rt ¼ 0:25 ln
fc 2:72
7
d
0:046 dp . These a
two models both gained success in Rt size effect prediction. For comparison, these two models are utilized herein to predict the target resistance of 76.2 mm diameter projectile penetration in 23 MPa and 39 MPa concrete. In Fig. 7(a), the proposed model is able to predict the target resistance for penetration in 23 MPa strength concrete, but the size effect is somehow exaggerated. In Fig. 7(b), the normalized target resistance Rt is defined as Rt =Rt and the size effect due to CRH change is only captured by the proposed model due to the richer factors accounted for. And this phenomenon needs to be further validated through experimental tests. 4. LDPM simulation of abnormal nose projectile penetration [35] A series of penetration test of 14.8 MPa compressive strength concrete targets impacted by abnormal nose projectiles was carried out by Chai et al. [35]. The projectiles used for penetration were made with a special nose shape, as shown in Fig. 8 the first one is ogival nose with CRH ¼ 2 while the others are featured with a cylinder at in the middle of the ogival nose part. The ogival nose projectile (ONP) is of 40 mm diameter and the abnormal nose projectile #1 (ANP1) has a cylinder of 10 mm length in the nose part while the nose shape of ANP2 is characterized with a 30 mm length cylinder. The projectiles were designed to penetrate the concrete target with 440 and 610 m/s striking velocities. With the size effect modelling ability, the LDPM is used to capture the size effect which might occur in this penetration condition. No apparent projectile deformation or erosion was reported by the authors and rigid projectile assumption is also made herein. With reference to the calibrated LDPM parameters for different strength concrete [19], parameters for fc ¼ 14.8 MPa are estimated as: normal elastic modulus E0 ¼ 21000 MPa, densified normal modulus Ed ¼ 1:0E0 , shear-normal coupling parameter a ¼ 0.25, tensile strength st ¼ 2.0 MPa, yielding compressive stress sc0 ¼ 30 MPa, shear strength ss ¼ 2:0st , tensile characteristic length lt ¼ 160 mm, softening exponent nt ¼ 0:2, initial hardening modulus Hc0 ¼ 0:54E0 , transitional strain ratio kc0 ¼ 4, initial internal friction coefficient m0 ¼ 0:4, internal asymptotic friction coefficient m∞ ¼ 0, transitional stress sN0 ¼ 600MPa, deviatoric-tovolumetric strain ratio kc1 ¼ 1 and kc2 ¼ 5. After LDMP simulation, the predicted DOP are comparatively
Fig. 8. Dimension of projectiles used for penetration test by Chai et al. [35].
listed with the test results in Table 3 where the shot numbers are labelled with projectile name plus “1” and “2” representing the striking velocity of about 440 and 610 m/s. It is interesting to find that under similar conditions the ANP1 tends to improve the DOP meanwhile the ANP2 doesn't contribute to the DOP even though they have similar nose shape. In general, the LDPM simulation derived DOP agrees with the experimental data in terms of DOP. The numerical results of damage contour of the impact surface of shot ONP-2 is plotted in Fig. 9 whereby the left half surface is covered by the experimental picture. The front crater shape as well as size are almost the same as the test but more radial cracks are obtained by LDPM simulation. Also the cross section views of the projectile penetration process of shot ONP-2 are plotted in Figs. 10 and 11 at every 0.5 ms. The crack opening distribution of the cross section plane of the trajectory is shown in Fig. 10 where the red cracks can be considered as the pulverized concrete material. In Fig. 11, the velocity distribution of the cracks in the cross section plane shows the history of the cracks velocity during penetration
Fig. 7. Validation of proposed size effect law for target resistance.
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Table 3 Comparison of ANP penetration tests and LDPM predictions. No.
Projectile mass/kg
Striking velocity/(m$s1)
Test DOP/m [35]
LDPM prediction/m
ONP-1 ANP1-1 ANP2-1 ONP-2 ANP1-2 ANP2-2
1.41 1.41 1.41 1.40 1.41 1.41
431 422 441 608 614 604
0.384 0.409 0.427 0.686 0.696 0.689
0.386 0.402 0.417 0.682 0.692 0.685
Fig. 9. LDPM simulation of shot No. ONP-2 impact surface versus experimental result.
whereby the cracks near the projectile nose tip always have the highest velocity until the projectile stops in the target in Fig. 11(d). Similarly, the projectile constant velocity penetration in
concrete is simulated to achieve the penetration resistance. For ONP, the cavity expansion analysis can be applied to get the target resistance Rt. The best fitted curve against the LDPM simulation results are plotted in Fig. 12 where Rt ¼ 253.5 MPa is estimated. Since the aggregate information is missing in Ref. [35], we take da ¼ 9.5 mm and Eq. (9) suggests 267 MPa for target resistance which is quiet close to the best fit value. Therefore, the Forrestal model prediction for DOP of shot ONP-1 and ONP-2 are 0.388 and 0.682 m, matching the test data well. More importantly, the constant velocity (vc ¼ 200 and 600 m/s) penetration simulation results are comparatively given in Fig. 13 where the mean resistive force after the projectile nose part fully enters the target is denoted in bold dash line. It is worth noting that under same impact velocity and target condition, the mean resistive force of ANP1 seems to be smaller comparing to ONP. This can be explained that the cylinder part of ANP1 in the nose part actually extends the length of the projectile nose height and in general leading to a sharper nose shape. However, the ANP2 suffer more resistance than ONP which may attribute to the long cylinder in the nose part resulting in greater resistance stress acting on the front nose part before the cylinder. Since the front nose þ cylinder part is significantly important in dimension, the size effect of the penetration resistance will pose greater resistance stress on the front nose. This numerical result agrees with the conclusion drawn in Ref. [35]. And this interesting projectile nose shape is worthy to be further studied. 5. Conclusions The numerical simulation of rigid projectile penetration is performed with Lattice Discrete Particle Model. Extensive numerical
Fig. 10. Crack opening distribution evolution during penetration simulation of shot ONP-2.
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Fig. 11. Cracks velocity distribution evolution during penetration simulation of shot ONP-2.
simulations of penetration are conducted to explore the projectile size effect on target resistance. The main conclusions are drawn as follows:
Fig. 12. LDPM simulation results best fit with Forrestal model.
1. The LDPM model with calibrated parameters can successfully simulate the projectile penetration in 23 MPa compressive strength concrete. The size effect phenomenon, i.e., larger projectile suffers less resistance, can be captured through the LDPM simulation. 2. The size effect law for target resistance Rt depending on the ratio of ogival nose radius and maximum aggregate size is proposed which is then validated against test data. This view point might be different from existing models which believe both projectile diameter and aggregate strength should count. Thus, large amount of scaling penetration tests are needed to better understand the size effect in penetration. 3. For the abnormal nose projectile penetration in concrete, LDPM simulation prediction agrees well with the experimental data. Also, the size effect can be numerically obtained for the projectile with longer extended nose part meanwhile the shorter extended nose can improve the DOP since the projectile nose is sharpened.
Fig. 13. Resistive force on the projectile nose during constant velocity penetration in concrete.
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Acknowledgement This effort was supported by the Natural Science Foundation of Jiangsu Province (No. BK20170824) and the Fundamental Research Funds for the Central Universities (No. 30917011343). Prof. Gianluca Cusatis from Northwestern University is gratefully acknowledged for the timely helps in LDPM study. References [1] Forrestal M, Altman B, Cargile J, Hanchak S. An empirical equation for penetration depth of ogive-nose projectiles into concrete targets. Int J Impact Eng 1994;15(4):395e405. [2] Frew DJ, Forrestal MJ, Cargile JD. The effect of concrete target diameter on projectile decel- eration and penetration depth. Int J Impact Eng 2006;32(10): 1584e94. [3] Forrestal MJ, Frew DJ, Hanchak SJ, Brar NS. Penetration of grout and concrete targets with ogive-nose steel projectiles. Int J Impact Eng 1996;18(5):465e76. [4] Chen X, Li Q. Deep penetration of a non-deformable projectile with different geometrical charac- teristics. Int J Impact Eng 2002;27(6):619e37. [5] Li Q, Reid S, Wen H, Telford A. Local impact effects of hard missiles on concrete targets. Int J Impact Eng 2005;32(1):224e84. [6] Wang J, Wu H, Feng X, Pi A, Li J, Huang F. Dynamic spherical cavity expansion analysis of concrete using the bingham liquid constitutive model. Int J Impact Eng 2018;120:110e7. [7] Forquin P, Sallier L, Pontiroli C. A numerical study on the influence of free water content on the ballistic performances of plain concrete targets. Mech Mater 2015;89:176e89. [8] Bazant ZP. Size effect in blunt fracture: concrete, rock, metal. J Eng Mech 1984;110(4):518e35. [9] Bazant ZP, Planas J. Fracture and size effect in concrete and other quasibrittle materials, vol. 16. CRC press; 1997. [10] Mahmud GH, Yang Z, Hassan AM. Experimental and numerical studies of size effects of ultra high performance steel fibre reinforced concrete (uhpfrc) beams. Construct Build Mater 2013;48:1027e34. [11] Rosenberg Z, Kositski R. Modeling the penetration and perforation of concrete targets by rigid projectiles. International Journal of Protective Structures 2016;7(2):157e78. [12] Forrestal M, Frew D, Hickerson J, Rohwer T. Penetration of concrete targets with deceleration-time measurements. Int J Impact Eng 2003;28(5):479e97. [13] Bludau C, Keuser M, Kustermann A. Perforation resistance of high-strength concrete panels. ACI Struct J 2006;103(2):188. [14] Dancygier AN, Yankelevsky DZ, Jaegermann C. Response of high performance concrete plates to impact of non-deforming projectiles. Int J Impact Eng 2007;34(11):1768e79. [15] Peng Y, Wu H, Fang Q, Gong Z, Geometrical scaling effect for penetration depth of hard projec- tilesinto concrete targets, Int J Impact Eng 120 46e59. [16] Feng J, Sun W, Liu Z, Cui C, Wang X. An armour-piercing projectile penetration in a double- layered target of ultra-high-performance fiber reinforced
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Contents lists available at ScienceDirect
Defence Technology journal homepage: www.elsevier.com/locate/dt
Numerical study of size effect in concrete penetration with LDPM Jun Feng a, *, Weiwei Sun b, Baoming Li a a b
National Key Laboratory of Transient Physics, Nanjing University of Science and Technology, Nanjing, 210094, China Department of Civil Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 June 2018 Received in revised form 15 July 2018 Accepted 19 July 2018 Available online xxx
Projectile size effect is of great importance since the scaling researches are extensively applied to concrete penetration investigations. This paper numerically deals with the projectile size effect on penetration resistance via the recently developed Lattice Discrete Particles Model (LDPM) which is featured with mesoscale constitutive laws governing the interaction between adjacent particles to account for cohesive fracture, strain hardening in compression and compaction due to pore collapse. Simulations of two different penetration tests are carried to shed some light on the size effect issue. The penetration numerical model is validated by matching the projectile deceleration curve of and predicting the depth of penetration (DOP). By constant velocity penetration simulations, the target resistance is found to be dependent on the projectile size. By best fitting numerical results of constant velocity penetration, a size effect law for target resistance is proposed and validated against literature data. Moreover, the size effect is numerically obtained in the projectile with longer extended nose part meanwhile the shorter extended nose is found to improve the DOP since the projectile nose is sharpened. © 2018 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Penetration of concrete Size effect Lattice discrete particle model Target resistance Abnormal nose projectile
1. Introduction The response of high speed penetration of concrete target by projectiles has been a research hotspot for several decades, due to its relevance for both industry and military establishments. Usually, the concrete penetration investigations are carried out in reduced geometrical scale since the ballistic tests are dangerous and moneyconsuming. Issues concerning penetration of massive concrete targets by rigid projectiles have been dealt with empirical formulae, analytical model as well as the numerical modelling [1e7]. However, almost all of these methods fail to account for the size effect of concrete structure during penetration analyses. The studies on size effect of concrete structures mainly were conducted in the area of fracture [8e10], and the understanding of the size effect in concrete penetration is still largely lacking. With the concept of non-dimensional formula based on the dynamic cavity expansion model, Chen and Li [4,5] developed DOP prediction model for penetration with extensive boundary conditions which has been widely used by researchers in concrete penetration area. Although the non-dimensional formula for DOP
* Corresponding author. E-mail address: [email protected] (J. Feng). Peer review under responsibility of China Ordnance Society
significantly contributes to the promotion of concrete penetration investigations, there might be some problem when researchers try to extend the small scale penetration mechanism to the larger (real) scale experiment scenarios. The reason of the projectile diameter scale effect on the penetration response was explained by Rosenberg and Kositski [11] that the harder aggregate may pose more resistance to the smaller diameter projectile during its movement inside the target. On the other hand, size effect theory proposed and explained by Bazant [9] indicates that the nominal structure strength decreases as the structure size increases. Forrestal et al. [12] noticed there might be size effect between the projectile diameter and the target resistance Rt since the experimental data of 76.2 mm diameter projectile penetration doesn't match the empirical formula for the target resistance. Then Bludau et al. [13] showed that the resistance to penetration of concrete targets is dependent on both the aggregate size, through the ratio of projectile diameter and aggregate size, and the strength of the aggregate material. Similar conclusions were drawn via experiments by Dancygier et al. [14]. Although the projectile size effect was reported, few efforts have been made to address this issue. Rosenberg and Kositski [11] attempted to develop a semi-empirical model for both concrete penetration and perforation by including the projectile diameter size effect. More recently, Peng et al. [15] developed a mesoscale concrete model for penetration considering both aggregate, cement
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and ITZ. Their numerical results suggest the penetration resistance is strongly dependent on the projectile relative size against the aggregate. With reference to the cavity expansion analysis, a modified penetration resistive force formulae was proposed with aggregate strength and projectile size taken into account. In the present paper, the numerical simulation of rigid projectile penetration is performed with Lattice Discrete Particle Model (LDPM) which has been proven a robust model for concrete penetration and perforation in our previous study [16e18]. The numerical study of LDPM [19] shows a good capacity of concrete size effect modelling of 3-point bending, splitting and even compression tests. Two penetration tests concerning projectile size effect are selected for LDPM numerical study. The penetration model is validated by predicting the projectile deceleration history as well as DOP where data scattering is discussed. The extensive simulations of constant velocity penetration in concrete give the target resistance of different projectile geometries. Then, the size effect law for target resistance is proposed and validated. Finally, the size effect of abnormal nose projectile penetration is numerically captured and discussed. 2. Lattice Discrete Particle Model (LDPM) introduction As a synthesis of the Confinement Shear Lattice model and discrete method, the theoretical framework of Lattice Discrete Particle Model was developed to simulate the mechanical interaction of coarse aggregate pieces embedded in a binding matrix [20e22]. The geometrical representation of concrete mesostructure is constructed by randomly introducing and distributing spherical shaped coarse aggregate particles inside the volume of interest and zero-radius aggregate particles on its surface as shown in Fig. 1(a). With Delaunay tetrahedralization of the generated particle centers, a three-dimensional domain tessellation creates a system of polyhedral cells (see Fig. 1(b)) interacting through triangular facets and corresponding lattice system. To construct the mesoscale framework, concrete mix design parameters are needed including: maximum/minimum aggregate particle size da/d0; cement content c water-to-cement ratio wt/c; and the coefficient for the classical Fuller curve nF. 2.1. LDPM kinematics The triangular facets forming the rigid polyhedral particles are assumed to be the potential material failure locations. Each facet is shared between two polyhedral particles and is characterized by a local system of reference featured by a unit normal vector n and two tangential vectors m and l as noted in Fig. 1(b). The deformation of the lattice particle system is described by the rigid body
kinematics whereas the displacement jump EuC F at each triangular facet centroid is used to define the mesoscale strain component as:
εN ¼
nT EuC F mT EuC F lT EuC F ; εM ¼ ; εL ¼ l l l
(1)
where l is the length of the tetrahedron edge. The strain definitions in Eq. (1) have been proven by Cusatis et al. [23] to be consistent with the definition of strain in classical continuum mechanics. The corresponding normal and shear stress are calculated through LDPM mesoscale constitutive laws and the equilibrium are imposed via the principle of virtual work [20].
2.2. LDPM constitutive equations In the elastic regime, the normal and shear stresses are proportional to the corresponding strains: sN ¼ EN ,εN ; sM ¼ ET ,εM ; sL ¼ ET ,εL , where EN ¼ E0 , ET ¼ aE0 , E0 is the effective normal modulus, and a is shear-normal coupling parameter. On the facets, the reversible elastic behavior is limited by a number of nonlinear boundaries which are featured by softening for pure tension and shear-tension, as well as plastic hardening for pure and shear compression.
2.2.1. Fracturing behavior For fracture behavior characterized by normal strains ðεN > 0Þ, the fracturing evolution is formulated through the effective strain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε ¼ ε2N þ aðε2M þ ε2L Þ and the effective stress s ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2N þ ðs2M þ s2L Þ=a. The tensile boundary sbt evolves exponentially as a function of the maximum effective strain of the loading history εmax ¼ maxtt ½εðtÞ
sbt ¼ s0 ðuÞexp½ H0 ðuÞhεmax ε0 ðuÞi=s0 ðuÞ
(2)
pffiffiffi where Macaulay bracket 〈x〉 ¼ maxfx; 0g, tanðuÞ ¼ εN = aεT ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi sN a=sT , and εT ¼ ε2M þ ε2L . Reaching elastic limit ε0(u), the fracturing damage decays the boundary sbc with the post peak softening modulus defined as H0 ðuÞ ¼ Ht ð2u=pÞnt , where Ht is the softening modulus in pure tension (u ¼ p/2) expressed as Ht ¼ 2E0 =ðlt =l 1Þ; character length lt ¼ 2E0 Gt =s2 ; and Gt is the mesoscale fracture energy. LDPM provides a smooth transition between pure tension and pure shear (u ¼ 0) with parabolic variation for strength given by
Fig. 1. One LDPM polyhedral cell around an aggregate particle.
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J. Feng et al. / Defence Technology xxx (2018) 1e10
2 sinu þ s0 ðuÞ ¼ st rst
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sin2 u þ 4a cos2 u r 2st
2a cos2 u
(3)
where rst ¼ ss =st is the ratio of shear strength to tensile strength. 2.2.2. Pore collapse and subsequent compaction Normal stresses for compressive loading (εN < 0) are computed meeting the inequality sbc ðεD ; εV Þ sN 0 where sbc is the boundary function of the volumetric strain εV and the deviatoric strain εD . Beyond the mesoscale compressive yield stress sc0 , sbc models pore collapse as a linear evolution of stress for increasing volumetric strain with stiffness Hc for εV εc1 ¼ kc0 εc0 :
sbc ¼ sc0 þ hεV εc0 iHc rDV
(4)
where Hc ðrDV Þ ¼ Hc0 =ð1 þ kc2 hrDV kc1 iÞ and kc1 , kc2 are deviatoric parameters. Beyond pore collapse εV εc1 , compaction and rehardening occur. In this case one has
sbc ¼ sc1 rDV exp½ð εV εc1 ÞHc rDV =sc1 rDV
(5)
where sc1 rDV ¼ sc0 þ ðεc1 εc0 ÞHc rDV . 2.2.3. Friction due to compression-shear The incremental shear stresses rates are computed as s_ ¼ E ð_ε ε_ p Þ and s_ ¼ E ð_ε ε_ p Þ, where ε_ p ¼ l_ v4=vs , ε_ p ¼ M
T
M
M
L
T
L
L
M
M
L
l_ v4=vsL , and l is the plastic multiplier. The plastic potential is
defined as 4 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2M þ s2L sbc sN , where the nonlinear frictional
law for the shear strength is assumed to be
sbs ¼ ss þ ðm0 m∞ ÞsN0 ½1 expðsN =sN0 Þ m∞ sN
(6)
where sN0 is the transitional normal stress, m0 and m∞ are the initial and final internal friction coefficients. 2.2.4. Strain rate effect in LDPM The LDPM formulation is extended to incorporate rate dependent fracture mechanisms associated with the interpretation of thermally activated phenomena which is governed by the classical Maxwell-Boltzmann equation [24]. The crack opening rate dependent cohesive behavior can be expressed as: _ ¼ ½1 þ c1 a sin hðw=c _ 0 Þf ðwÞ where sch is the cohesive sch ðw; wÞ stress, w is the crack opening, w_ is the crack opening rate, f ðwÞ is the cohesive law under static condition, c0 is the reference crack opening rate and c1 is the strain rate coefficient. Compared to the crack opening rate, the elastic strain rate can be negligible [25], hence the LDPM effective strain can be written as _ þ s_ =E0 zw=l. _ ε_ ¼ w=l Substituting w_ by l_ε, LDPM boundary condition can be modified as:
sbt ðε; ε_ Þ ¼ Fð_εÞs0 ðuÞexp H0 ðuÞexp
hεmax ε0 ð_ε; uÞi s0 ðuÞ
(7)
where ε0 ð_ε; uÞ ¼ Fð_εÞs0 ðuÞ=E0 and the rate effect function is expressed as Fð_εÞ ¼ 1 þ c1 a sin h c ε_=l . 0
3. Numerical analysis of Sandia Lab penetration tests [27] A serious of experimental investigations on penetration in concrete targets with different projectile nose geometry, striking
3
velocities were carried out by M.J. Forrestal et al. [12,27,28] to study the response of projectile penetration in concrete target. The 76.2mm-diameter ogival nose projectiles were designed with a singlechannel acceleration data recorder. Both penetration depth and projectile deceleration data during penetration were measured and provided in details. With 3.0 and 6.0 caliber-radius-head (CRH), ogival nose projectiles were launched with striking velocities between 140 and 460 m/s to impact the concrete targets with 23 MPa compressive strength. These experiments, including the triaxial compression and penetration tests, have been numerically studied in our previous work [26] whereby the calibration and validation of the penetration model are conducted. In this section, we further analyze the size effect phenomenon with respect to the penetration test in Ref. [27].
3.1. LDPM simulation of concrete penetration 3.1.1. LDPM parameter calibration The concrete mixture proportion by weight is characterized by cement content c ¼ 310 kg=m3 , water to cement ratio wt=c ¼ 0:84, aggregate to cement ratio a=c ¼ 5:2, fuller coefficient nF ¼ 0:5 which are input information for LDPM particle generation. The nominal compressive strength was obtained from uniaxial compression test with 50.8 mm (diameter), 114 mm (height) cylinder specimens which were also used for triaxial compression (TXC) tests. Calibration of LDPM parameters are conducted with respect to hydrostatic test, triaxial compression tests with different confinement pressure and uniaxial strain compression test. The hydrostatic simulation is conducted by applying an increasing hydrostatic pressure phy on the cylinder surface. For triaxial compression simulations, the top surface is driven by a rigid top block with velocity control corresponding to the hydrostatic pressure, and the bottom surface is assigned frictional contact with a fixed block. After the lateral surface reaches the confinement pressure, the top block keeps moving with a constant velocity while the confinement is kept constant. The uniaxial strain test is modelled with the top surface moving at a low constant velocity and the lateral surface radial boundary fixed. The calibrated results are shown in Fig. 2, which compares the numerical and experimental curves for hydrostatic compression and TXC tests. In Fig. 2(b), the damaged concrete cylinders under 50 and 200 MPa confinement are comparatively plotted where comminution occurs for the triaxial response. The detailed response of TXC simulation is depicted in Fig. 2(c), where the red circles represent the loading condition transforming from hydrostatic compression to triaxial compression. With reference to Ref. [26], the LDPM parameters are calibrated as follows: normal elastic modulus E0 ¼ 16500 MPa, densified normal modulus Ed ¼ 4:2E0 , shear-normal coupling parameter a ¼ 0:25, tensile strength st ¼ 2:7MPa, yielding compressive stress sc0 ¼ 42 MPa, shear strength ss ¼ 2:0st , tensile characteristic length lt ¼ 100 mm, softening exponent nt ¼ 0:3, initial hardening modulus Hc0 ¼ 0:46E0 , transitional strain ratio kc0 ¼ 6, initial internal friction coefficient m0 ¼ 0:4, internal asymptotic friction coefficient m∞ ¼ 0, transitional stress sN0 ¼ 400MPa, deviatoric-to-volumetric strain ratio kc1 ¼ 2, and kc2 ¼ 2. Subsequent to calibration, the obtained LDPM parameters are used to simulate the response of concrete target subjected to ogival nose projectile impact. According to the penetration experiments [12], the projectiles were launched to normal impact the concrete targets and the ballistic tunnels measured after tests were almost straight within 0.6 pitch and yaw. Furthermore, abrasion on the projectile surface was also reported negligible, thus rigid projectile
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Fig. 2. LDPM calibration of tests.
assumption can be made in this numerical simulation. The calibrated LDPM parameters are utilized for concrete target modelling and a penalty interaction [29] between rigid projectile and LDPM particles is assigned. 3.1.2. Penetration simulation validation The penetration prediction results are listed in Table 1, which indicates that the depth of penetration (DOP) is within 10% error with respect to the tests data reported [27]. Moreover, the projectile deceleration during penetration is another important ballistic property needed to be validated. Fig. 3 illustrates the numerical predicted deceleration curves during the impact process in blue lines and the measured data in black lines where G is the gravitational acceleration. Firstly, the deceleration is characterized with a quick increase corresponding to the projectile nose length. After the nose entirely enters the target, the deceleration value keeps almost constant with a scatter. The large scattering of the numerical deceleration data may due to the updating interaction between LDPM aggregate particles with projectile. As explained in 2D diagram (see Fig. 4), the blue circles represent concrete aggregates in contact with the projectile surface in Fig. 4(a). As the projectile moves forward, the interacted aggregates get updated according to the relative distance between aggregates and projectile. As shown in Fig. 4(b), the aggregate in purple starts to contact with projectile while the aggregate in green becomes too far away from the projectile and no more interaction exists in between. The constant contact pair updating leads to the rough resistive force which is different from FEM results in Ref. [30]. As the projectile velocity reaches zero, the plateau is followed by a sudden drop. This phenomenon indicates that the resistance is not only determined by the impact velocity, rather there should be a material dependent constant term which was called target resistance by Forrestal et al. [12,31]. This ballistic property is fully captured by LDPM numerical prediction and both DOP and deceleration data are in agreement with the test data.
3.2. Size effect of concrete penetration To investigate the projectile size effect on penetration resistance, projectile constant velocity penetration of LDPM simulation is further conducted in this work to obtain the resistive force acting on the projectile nose. Right now, researchers deem that the target resistance Rt depends on the ratio dp =da [11,15] where da is the maximum aggregate size identical to the notation in LDPM introduction. Combining the cavity expansion with projectile penetration in concrete, the ballistic tunnel expansion is driven by the cavity wall with radius R ¼ dp J, namely ogival nose radius. There might be a possibility that the size effect in concrete penetration is related to the ratio R=da . Moreover, penetration tests revealed the 6.0 CRH projectile penetration is characterized with lower Rt in Ref. [12], as shown in Table 2 where Rt is the averaged value of Rt . Thus, this section tries to explore the size factors which might affect the target resistance. Projectiles with 30 mm, 60 mm and 80 mm diameters are chosen herein for numerical study where common CRH values like 3.0, 4.0 and 6.0 are determined. Constant impact velocity vc ranging from 200 to 600 m/s are assigned to the hard projectiles to penetrate the thick concrete target, and their resistive forces are then obtained through LDPM simulations where Fig. 5 gives the projectile resistive forces under 400 m/s projectile impact. For each penetration simulation, the resistive force corresponding to vc ¼ 400 m/s (dash lines in Fig. 5) is calculated as the averaged values when the projectile nose part fully enters the target. According to the widely recognized cavity expansion analysis developed by Forrestal et al. [31,32], the resistive force on the projectile nose should be:
F¼
pd2p 4
Rt þ Nr0 V 2
(8)
where dp is the projectile diameter, V is the projectile impact
Table 1 Comparison of Sandia Lab penetration tests and LDPM predictions. No.
Projectile mass/kg
Striking velocity/(m$s1)
Test DOP/m [27]
LDPM Prediction/m [26]
6e2 3e1 2e2 1e1 5e3 4e4 8e2 7e1
13.043 13.037 13.085 13.158 13.080 13.119 13.061 13.064
139 200 250 284 337 379 238 379
0.24 0.42 0.62 0.76 0.96 1.18 0.58 1.25
0.26 0.45 0.67 0.79 1.06 1.22 0.60 1.29
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5
Fig. 3. LDPM Prediction of projectile deceleration history vs. test data.
Fig. 4. Contact update diagram for interaction between projectile and aggregate.
Table 2 Target resistance of Forrestal et al. [12]. fc/MPa
dp/mm
da/mm
CRH
Rt /MPa
23 23 39 39
76.2 76.2 76.2 76.2
9.5 9.5 9.5 9.5
3.0 3.0 6.0 6.0
166.5 162.5 356.8 267.5
velocity and N is the “nose factor” reflecting the geometrical char-
acteristics of the projectile nose, for ogival nose projectile N ¼ 8J12 , 24J
J is caliber-radius-head value; the concrete target is described by initial density r0 and target resistance Rt. This postulation assumes that the resistance is only attributed to the compressive pressure acting on the normal direction of projectile nose surface. The target resistance Rt numerically obtained can be estimated via Eq. (8) which are plotted in Fig. 6(a). It is observed that the target resistance somehow increases with the impact velocity which might be
caused by the rate effect term of the penetration resistance, as indicated by some cavity expansion models with three resistance terms, i.e., inertial term, rate effect term and static term (namely Rt) [6,33]. Since this work concentrates on the size effect rather than the resistance mechanisms, we choose to select the Rt of vc ¼ 400 m/s for further study. Also, it is interesting to find that the larger diameter projectiles tend to suffer less target resistance. But the Rt of 60 mm dp with 6.0 CRH seems to be close to the Rt of 80 mm dp with 3.0 CRH. These two cases have same ogival nose radius R, thus we prefer to believe the size effect of Rt depends on R ¼ dp J. The Rt best fit is then performed with respect to R=da as shown in Fig. 6(b). With reference to Ref. [34], the size effect law for target resistance is expressed as Rt ¼ Sfc with S proposed as:
S ¼ 82:6fc0:544
dp J 7:77da
0:43 (9)
where there is no aggregate material strength parameter involved,
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Fig. 5. Resistive force of ogival nose projectile with vc ¼ 400 m/s.
Fig. 6. Target resistance Rt for constant velocity penetration and best fit.
because the aggregate material properties is related to the concrete mechanical property which is already accounted for in the parameter fc. This controversial issue needs to be further studied. The validation is then conducted with reference to the penetration tests with fc ¼ 23 MPa and 39 MPa in Refs. [12,27]. As mentioned by Forrestal et al. [12], the empirical equation
S ¼ 82:6fc0:544 could not be applied to penetration tests with 76.2 mm projectile diameter. By extensive mesoscale numerical simulation, Peng et al. [15] gave the size dependent target resistance function with S ¼ 82:6fc0:544 ðdp =ð2:83da ÞÞ0:208 . Rosenberg and Kositski [11] summarized penetration data and empirically
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come up with the expression Rt ¼ 0:25 ln
fc 2:72
7
d
0:046 dp . These a
two models both gained success in Rt size effect prediction. For comparison, these two models are utilized herein to predict the target resistance of 76.2 mm diameter projectile penetration in 23 MPa and 39 MPa concrete. In Fig. 7(a), the proposed model is able to predict the target resistance for penetration in 23 MPa strength concrete, but the size effect is somehow exaggerated. In Fig. 7(b), the normalized target resistance Rt is defined as Rt =Rt and the size effect due to CRH change is only captured by the proposed model due to the richer factors accounted for. And this phenomenon needs to be further validated through experimental tests. 4. LDPM simulation of abnormal nose projectile penetration [35] A series of penetration test of 14.8 MPa compressive strength concrete targets impacted by abnormal nose projectiles was carried out by Chai et al. [35]. The projectiles used for penetration were made with a special nose shape, as shown in Fig. 8 the first one is ogival nose with CRH ¼ 2 while the others are featured with a cylinder at in the middle of the ogival nose part. The ogival nose projectile (ONP) is of 40 mm diameter and the abnormal nose projectile #1 (ANP1) has a cylinder of 10 mm length in the nose part while the nose shape of ANP2 is characterized with a 30 mm length cylinder. The projectiles were designed to penetrate the concrete target with 440 and 610 m/s striking velocities. With the size effect modelling ability, the LDPM is used to capture the size effect which might occur in this penetration condition. No apparent projectile deformation or erosion was reported by the authors and rigid projectile assumption is also made herein. With reference to the calibrated LDPM parameters for different strength concrete [19], parameters for fc ¼ 14.8 MPa are estimated as: normal elastic modulus E0 ¼ 21000 MPa, densified normal modulus Ed ¼ 1:0E0 , shear-normal coupling parameter a ¼ 0.25, tensile strength st ¼ 2.0 MPa, yielding compressive stress sc0 ¼ 30 MPa, shear strength ss ¼ 2:0st , tensile characteristic length lt ¼ 160 mm, softening exponent nt ¼ 0:2, initial hardening modulus Hc0 ¼ 0:54E0 , transitional strain ratio kc0 ¼ 4, initial internal friction coefficient m0 ¼ 0:4, internal asymptotic friction coefficient m∞ ¼ 0, transitional stress sN0 ¼ 600MPa, deviatoric-tovolumetric strain ratio kc1 ¼ 1 and kc2 ¼ 5. After LDMP simulation, the predicted DOP are comparatively
Fig. 8. Dimension of projectiles used for penetration test by Chai et al. [35].
listed with the test results in Table 3 where the shot numbers are labelled with projectile name plus “1” and “2” representing the striking velocity of about 440 and 610 m/s. It is interesting to find that under similar conditions the ANP1 tends to improve the DOP meanwhile the ANP2 doesn't contribute to the DOP even though they have similar nose shape. In general, the LDPM simulation derived DOP agrees with the experimental data in terms of DOP. The numerical results of damage contour of the impact surface of shot ONP-2 is plotted in Fig. 9 whereby the left half surface is covered by the experimental picture. The front crater shape as well as size are almost the same as the test but more radial cracks are obtained by LDPM simulation. Also the cross section views of the projectile penetration process of shot ONP-2 are plotted in Figs. 10 and 11 at every 0.5 ms. The crack opening distribution of the cross section plane of the trajectory is shown in Fig. 10 where the red cracks can be considered as the pulverized concrete material. In Fig. 11, the velocity distribution of the cracks in the cross section plane shows the history of the cracks velocity during penetration
Fig. 7. Validation of proposed size effect law for target resistance.
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Table 3 Comparison of ANP penetration tests and LDPM predictions. No.
Projectile mass/kg
Striking velocity/(m$s1)
Test DOP/m [35]
LDPM prediction/m
ONP-1 ANP1-1 ANP2-1 ONP-2 ANP1-2 ANP2-2
1.41 1.41 1.41 1.40 1.41 1.41
431 422 441 608 614 604
0.384 0.409 0.427 0.686 0.696 0.689
0.386 0.402 0.417 0.682 0.692 0.685
Fig. 9. LDPM simulation of shot No. ONP-2 impact surface versus experimental result.
whereby the cracks near the projectile nose tip always have the highest velocity until the projectile stops in the target in Fig. 11(d). Similarly, the projectile constant velocity penetration in
concrete is simulated to achieve the penetration resistance. For ONP, the cavity expansion analysis can be applied to get the target resistance Rt. The best fitted curve against the LDPM simulation results are plotted in Fig. 12 where Rt ¼ 253.5 MPa is estimated. Since the aggregate information is missing in Ref. [35], we take da ¼ 9.5 mm and Eq. (9) suggests 267 MPa for target resistance which is quiet close to the best fit value. Therefore, the Forrestal model prediction for DOP of shot ONP-1 and ONP-2 are 0.388 and 0.682 m, matching the test data well. More importantly, the constant velocity (vc ¼ 200 and 600 m/s) penetration simulation results are comparatively given in Fig. 13 where the mean resistive force after the projectile nose part fully enters the target is denoted in bold dash line. It is worth noting that under same impact velocity and target condition, the mean resistive force of ANP1 seems to be smaller comparing to ONP. This can be explained that the cylinder part of ANP1 in the nose part actually extends the length of the projectile nose height and in general leading to a sharper nose shape. However, the ANP2 suffer more resistance than ONP which may attribute to the long cylinder in the nose part resulting in greater resistance stress acting on the front nose part before the cylinder. Since the front nose þ cylinder part is significantly important in dimension, the size effect of the penetration resistance will pose greater resistance stress on the front nose. This numerical result agrees with the conclusion drawn in Ref. [35]. And this interesting projectile nose shape is worthy to be further studied. 5. Conclusions The numerical simulation of rigid projectile penetration is performed with Lattice Discrete Particle Model. Extensive numerical
Fig. 10. Crack opening distribution evolution during penetration simulation of shot ONP-2.
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Fig. 11. Cracks velocity distribution evolution during penetration simulation of shot ONP-2.
simulations of penetration are conducted to explore the projectile size effect on target resistance. The main conclusions are drawn as follows:
Fig. 12. LDPM simulation results best fit with Forrestal model.
1. The LDPM model with calibrated parameters can successfully simulate the projectile penetration in 23 MPa compressive strength concrete. The size effect phenomenon, i.e., larger projectile suffers less resistance, can be captured through the LDPM simulation. 2. The size effect law for target resistance Rt depending on the ratio of ogival nose radius and maximum aggregate size is proposed which is then validated against test data. This view point might be different from existing models which believe both projectile diameter and aggregate strength should count. Thus, large amount of scaling penetration tests are needed to better understand the size effect in penetration. 3. For the abnormal nose projectile penetration in concrete, LDPM simulation prediction agrees well with the experimental data. Also, the size effect can be numerically obtained for the projectile with longer extended nose part meanwhile the shorter extended nose can improve the DOP since the projectile nose is sharpened.
Fig. 13. Resistive force on the projectile nose during constant velocity penetration in concrete.
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Acknowledgement This effort was supported by the Natural Science Foundation of Jiangsu Province (No. BK20170824) and the Fundamental Research Funds for the Central Universities (No. 30917011343). Prof. Gianluca Cusatis from Northwestern University is gratefully acknowledged for the timely helps in LDPM study. References [1] Forrestal M, Altman B, Cargile J, Hanchak S. An empirical equation for penetration depth of ogive-nose projectiles into concrete targets. Int J Impact Eng 1994;15(4):395e405. [2] Frew DJ, Forrestal MJ, Cargile JD. The effect of concrete target diameter on projectile decel- eration and penetration depth. Int J Impact Eng 2006;32(10): 1584e94. [3] Forrestal MJ, Frew DJ, Hanchak SJ, Brar NS. Penetration of grout and concrete targets with ogive-nose steel projectiles. Int J Impact Eng 1996;18(5):465e76. [4] Chen X, Li Q. Deep penetration of a non-deformable projectile with different geometrical charac- teristics. Int J Impact Eng 2002;27(6):619e37. [5] Li Q, Reid S, Wen H, Telford A. Local impact effects of hard missiles on concrete targets. Int J Impact Eng 2005;32(1):224e84. [6] Wang J, Wu H, Feng X, Pi A, Li J, Huang F. Dynamic spherical cavity expansion analysis of concrete using the bingham liquid constitutive model. Int J Impact Eng 2018;120:110e7. [7] Forquin P, Sallier L, Pontiroli C. A numerical study on the influence of free water content on the ballistic performances of plain concrete targets. Mech Mater 2015;89:176e89. [8] Bazant ZP. Size effect in blunt fracture: concrete, rock, metal. J Eng Mech 1984;110(4):518e35. [9] Bazant ZP, Planas J. Fracture and size effect in concrete and other quasibrittle materials, vol. 16. CRC press; 1997. [10] Mahmud GH, Yang Z, Hassan AM. Experimental and numerical studies of size effects of ultra high performance steel fibre reinforced concrete (uhpfrc) beams. Construct Build Mater 2013;48:1027e34. [11] Rosenberg Z, Kositski R. Modeling the penetration and perforation of concrete targets by rigid projectiles. International Journal of Protective Structures 2016;7(2):157e78. [12] Forrestal M, Frew D, Hickerson J, Rohwer T. Penetration of concrete targets with deceleration-time measurements. Int J Impact Eng 2003;28(5):479e97. [13] Bludau C, Keuser M, Kustermann A. Perforation resistance of high-strength concrete panels. ACI Struct J 2006;103(2):188. [14] Dancygier AN, Yankelevsky DZ, Jaegermann C. Response of high performance concrete plates to impact of non-deforming projectiles. Int J Impact Eng 2007;34(11):1768e79. [15] Peng Y, Wu H, Fang Q, Gong Z, Geometrical scaling effect for penetration depth of hard projec- tilesinto concrete targets, Int J Impact Eng 120 46e59. [16] Feng J, Sun W, Liu Z, Cui C, Wang X. An armour-piercing projectile penetration in a double- layered target of ultra-high-performance fiber reinforced
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Please cite this article in press as: Feng J, et al., Numerical study of size effect in concrete penetration with LDPM, Defence Technology (2018), https://doi.org/10.1016/j.dt.2018.07.006