Geometrical scaling effect for penetration depth of hard projectiles into concrete targets

Accepted Manuscript

Geometrical scaling effect for penetration depth of hard projectiles into concrete targets Y. Peng , H. Wu , Q. Fang , Z.M. Gong PII: DOI: Reference:

S0734-743X(18)30220-3 10.1016/j.ijimpeng.2018.05.010 IE 3108

To appear in:

International Journal of Impact Engineering

Received date: Revised date: Accepted date:

10 March 2018 21 May 2018 27 May 2018

Please cite this article as: Y. Peng , H. Wu , Q. Fang , Z.M. Gong , Geometrical scaling effect for penetration depth of hard projectiles into concrete targets, International Journal of Impact Engineering (2018), doi: 10.1016/j.ijimpeng.2018.05.010

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ACCEPTED MANUSCRIPT Highlights  Replica scaling law holds true for DOP in rigid projectile penetrations as long as the scaling is done strictly, and the coarse aggregate with invariant size could account for the non-scaling effect in DOP found in penetration tests and empirical formulae.

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 A 3D mesoscopic finite element model for concrete target is developed, and the influences of cement strength, aggregate strength, volume fraction of aggregate on the non-scaling effects of projectile penetrations are numerically studied.

 The non-scaling effect in DOP for different concrete targets with the same magnitude implied by the

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empirical formulae and the always held scaling law in the (semi-)analytical models may be unreasonable.

 A semi-analytical model for DOP prediction is proposed by further considering the non-scaling

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effect.

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ACCEPTED MANUSCRIPT Geometrical scaling effect for penetration depth of hard projectiles into concrete targets Y Penga, H Wub*, Q Fanga, Z M Gonga a. State Key Laboratory for Disaster Prevention & Mitigation of Explosion & Impact, Army Engineering University of PLA, Nanjing 210007, China

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b. Research Institute of Structural Engineering and Disaster Reduction, College of Civil Engineering, Tongji University, Shanghai, 200092, China

Abstract: Since penetration tests of concrete targets against rigid projectiles are commonly conducted in reduced geometrical scale, whether the replica scaling law holds or not is very important for extending the

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knowledge based on the small-scale experiments to the large-scale or prototype penetration scenarios. In this paper, based on the available experimental data for depth of penetration (DOP) and discussions on the empirical formulae, it is verified that the replica scaling law is satisfied for DOP in rigid projectile penetrations, as long

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as the scaling is done strictly for both projectiles and concrete targets including the coarse aggregates. And the

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coarse aggregates with invariant size (not replica-scaled) could account for the non-scaling effect in DOP found in tests and empirical formulae. To explore the non-scaling effect in DOP caused by aggregates, a 3D

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mesoscopic finite element model for concrete target is developed. Based on the parametric analyses, it indicates

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that, the magnitude of the non-scaling effect decreases with the increasing of the cement strength when the aggregate strength is fixed. While the influence of the aggregate strength on the non-scaling effect is not so

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obvious comparing with the influence of cement strength. Besides, the magnitude increases with the increasing of the volume fraction of aggregates. These conclusions imply that, the non-scaling effect in DOP for different concrete targets with the same magnitude implied by the empirical formulae, and the always held scaling law in the (semi-)analytical models, may be unreasonable. Finally, based on the numerical results, a semi-analytical model for predicting DOP is proposed, which improved our previous model by further considering the

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ACCEPTED MANUSCRIPT non-scaling effect. Keywords: geometrical scaling effect; projectile; penetration; concrete; depth of penetration 1. Introduction Penetration depth of hard projectiles into semi-infinite concrete targets, which has been investigated

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extensively, is a basic focus for both defense engineers and weapon designers. In the past few decades, studies concentrated on the penetration models with physical basis, including analytical [1-6] and semi-analytical models [7-11], were continuously performed. These studies present physical insight to the penetration process and make the underpinning mechanics increasingly clear. However, shortcomings in these models at three

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aspects as given below need to cause enough attentions and be overcome, otherwise their applications are very limited.

Firstly, most of these models for depth of penetration (DOP) are only validated by laboratory-scale tests,

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their accuracy for full-scale tests is uncertain. In the open literature, full-scale or large-scale penetration test is

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very limited, most of the penetration experiments are performed in reduced scale to decrease costs and simplify handling. Thus, the scaling laws are essential for the translation of results from subscale tests to other scales.

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This issue naturally refers to the second shortcoming of the models.

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Secondly, almost all the above-mentioned models [1-11] are dimensionally homogeneous, it implies that the replica scaling law holds for DOP in rigid projectile penetrations, namely, the full-scale projectile impact

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performance can be predicted accurately from the subscale data and vice versa. However, the experimental works in Refs. [12, 13] showed that there exist irregularity in scaling, as the semi-analytical model proposed in Refs. [7, 8] (based on the test data for projectiles with diameter d=12.9-30.5mm) overestimated and underestimated the DOP of projectiles with diameters d=6.35 mm in Ref. [12] and d=76.2 mm in Ref. [13], respectively (detailed information refers to Section 4). Besides, most of the empirical formulae established

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ACCEPTED MANUSCRIPT since World War II, e.g., as listed in Refs. [14, 15], include a term which indicates that replica scaling law seems not to be held. On the other hands, Canfield and Clator [16] presented DOP data for full-scale (5.9 kg, 76.2-mm-diameter) and 1/10 scale (0.0059 kg, 7.62-mm-diameter) armor-piercing projectiles and found no influence of scale. Such contradictory phenomenon is also found in long rod penetrations, some experimental

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works [17, 18] supported the opinion that the scaling law holds, while in some other tests [18, 19] non-scaling was also found. Actually, whether the scaling law holds or not for penetrations is still a controversial problem, and some detailed discussions can be referred in [19, 20]. There is a probability that some important parameters may be neglected when conducting the scaling tests, and therefore the irregularity in scaling occurs [19, 20].

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For instance, the effects of strain rate were widely discussed since the strain rate is impossible to keep regular in penetration tests with different geometrical scales, however, non-scaling effect cannot be attributed to strain rate effects [21].

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Thirdly, all the foregoing (semi-)analytical models [1-11] do not consider the effect of coarse aggregates

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in the inhomogeneous concrete targets on the penetration depth. However, several experimental works [22-25] have shown that the resistance against projectile penetration depends not only on the compressive strength of

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the concrete target, but also the size and strength (hardness) of the coarse aggregate. For rigid projectile

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penetrations into concrete targets, coarse aggregates may be also an important parameter which could influence the scaling. For example, resistance acting on the projectiles with the same nominal diameter of 76.2 mm into

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the two kinds of concrete targets with similar compressive strength (35.1 MPa [16] and 39 MPa [13]) shows big difference, namely, resistance parameter R=456 for 35.1 MPa target [7] and R=360 for 39 MPa target [13]. This deviation is very likely caused by the effects of coarse aggregates since it cannot be explained by the difference of the strength. Besides, this example indicates that the projectile diameter scale effect, namely, the resistance acting on the projectile varies with the sizes of projectiles [4, 13], could not explain the above difference.

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ACCEPTED MANUSCRIPT In this paper, the existing empirical formulae, which represent the experimental data, are firstly utilized to discuss the irregularity of scaling in detail. Penetration scenarios of geometrically scaled projectiles into the same semi-infinite concrete targets are assumed and calculated to illustrate the non-scaling effect in DOP. The reason for non-scaling effect is discussed, and it is attributed to the coarse aggregates with convincing evident.

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Then, numerical simulations are carried out by means of a 3D mesoscopic model to explore the non-scaling effect for DOP in projectile penetration tests caused by coarse aggregates. The development of 3D mesoscopic model is introduced in detail. The influences of cement strength, aggregate strength, volume fraction of aggregates are numerically discussed. Finally, based on the discussions of the empirical formulae and numerical

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simulations, our previously proposed semi-analytical model [11] for DOP prediction is improved by considering the non-scaling effect. The improved model is further utilized to compare with the existing test data which have different scales.

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2. Irregularity of scaling in the empirical formulae

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Since World War II, as listed in Refs. [14, 15], there are several empirical formulae proposed by curve-fitting the DOP test data with various projectile‟s diameters. Most of them include a term which indicates

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that replica scaling law for DOP seems not to be held. At follows, three representative formulae are listed and

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discussed to illustrate this issue.

2.1 Representative empirical formulae

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2.1.1 Army corps of engineers (ACE) formula [15] Based on the experimental results prior to 1943 from the Ordnance Department of the US Army and the

Ballistic Research Laboratory (BRL), the ACE developed the formula for DOP (represented by P) as

P 3.5  104 M 0.215 1.5  ( 3 )d V0  0.5 d f c 0.5 d

(1)

where d, M, V0 are shank diameter, mass and striking velocity of the projectile, respectively. The symbol fc is 7

ACCEPTED MANUSCRIPT the compressive strength of the concrete target in Pa. 2.1.2 The modified National Defense Research Committee (NDRC) formula [15, 26] The modified NDRC formula [15, 26] is widely used and the ratio of DOP to the projectile‟s diameter is given as for for

G 1 G 1

where 1.8

N * M  V0  G  3.8  10   d fc  d 

(2a)

(2b)

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5

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P 0.5  d  2G  P  G 1  d

where N* is the projectile nose geometry factor, which equals to 0.72, 0.84, 1.0 and 1.14 for flat, hemispherical, blunt, and very sharp noses, respectively.

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2.1.3 Whiffen formula [27]

Whiffen [27] collected the study of projectile penetration into concrete targets in UK during wartime, and

V0 n P 2. 61 M d = ( 0. 5 ) ( 3 ) ( )0. 1( ) d a 533. 4 fc d

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proposed a formula to calculate the DOP as

n =

97. 51 f c 0. 25

(3a)

(3b)

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where a is the maximum aggregate size. This formula was assessed in the ranges of 5.52
ACCEPTED MANUSCRIPT projectiles into the same semi-infinite concrete targets with fc=35 MPa and maximum aggregate size a=20 mm. The schematic of four different scaled projectiles is shown in Fig. 1(a), where the far left one is set identical with the projectile in Hanchak et al. [28] (d=25.4 mm, M=0.5 kg), and the dimensions of rest projectiles are scaling increased with a scale factor of 5:1, 10:1 and 15:1, respectively. Figs. 1(b-d) show the normalized

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penetration depth (P/d) versus striking velocity predicted by the above three representative formulae. If the replica scaling law holds, the predicted P/d for these replica projectiles should overlap and fall on a single curve. However, it is obviously shown in the figures that, the predicted dimensionless penetration depth for projectiles with different scales does not collapse on a single curve, and the improved projectile performance

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M=1687.5kg

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with increased scale can be seen, which indicates that the replica scaling seems not hold.

40

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M=500kg

35

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30

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5 0.5kg times

25.4mm 127mm

25

10 times

P/d

1437mm

M=62.5kg

20 15 10 5

15 times

0 200 254mm

ACE formula M=0.5kg, d=25.4mm M=62.5kg, d=127mm M=500kg, d=254mm M=1687.5kg, d=381mm

300

400

500

600

V0 (m/s)

381mm

(a)

(b)

9

700

800

900

1000

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30

P/d

25

35 30 25

P/d

35

Modified NDRC formula M=0.5kg, d=25.4mm M=62.5kg, d=127mm M=500kg, d=254mm M=1687.5kg, d=381mm

20

20

15

15

10

10

5

5

0 200

300

400

500

600

700

800

900

0 200

1000

V0 (m/s)

Whiffen formula M=0.5kg, d=25.4mm M=62.5kg, d=127mm M=500kg, d=254mm M=1687.5kg, d=381mm

300

400

500

600

700

800

900

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40

1000

V0 (m/s)

(c)

(d)

Fig. 1 (a) Projectiles, and the corresponding dimensionless DOP into concrete targets with fc=35 MPa predicted

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by (b) ACE, (c) Modified NDRC, and (d) Whiffen (with a=20mm assumed) formulae

However, for Whiffen formula, if projectiles penetration into concrete targets with aggregate sizes also replica-scaled, namely, d/a keep invariant, then the predicted normalized penetration depth of the above

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projectiles will overlap. From this point of view, the law of replica-scaling holds. Fortunately, an excellent example can be found in the work of Canfield and Clator [16] to support our notion. In Ref. [16], authors

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performed two sets of penetration tests with full-scale (M=5.9 kg, d=76.2 mm) and replica 1/10 scale

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(M=0.0059 kg, d=7.62 mm) armor-piercing projectiles. The caliber-radius-head (CRH) of these projectiles is 1.5. They also scaled their fc=35 MPa (nominal) reinforced concrete targets by reducing the coarse aggregate

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size and the dimensions of the steel reinforcement by a factor of 10, in order to have a truly scaled target.

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Impact velocities in this work ranged between 305 and 824 m/s. It turned out that the normalized DOP for those two sets of experiments, in terms of P/d as a function of V0, can be represented by a single curve, as can be seen in Fig. 2. From this important work, it can be concluded that the replica scaling holds for DOP in rigid projectile penetrations, as long as the scaling is done strictly for both projectiles and concrete targets including the coarse aggregates. Also, it should be noted that, more replica scaled penetration tests like in Canfield and Clator [16] should be done to validate that whether the replica scaling holds for projectile with larger diameters 10

ACCEPTED MANUSCRIPT or target with various strengths. 12

M=0.0059kg, d=7.62mm, fc=34.6MPa 10

M=5.9kg, d=76.2mm, fc=35.1MPa

P/d

8

6

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4

Canfield and Clator, 1966 2 300

400

500

600

700

V0 (m/s)

800

900

Fig. 2 Normalized DOP for the two sets of scaled penetration experiments [16]

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Obviously, the empirical formulae e.g., ACE and Modified NDRC ant etc. which do not consider the effects of aggregates, could not explain the phenomenon in Fig. 2. We suppose that, the aggregate size in the tests which these empirical formulae derived from may be nearly identical, and the influence of coarse

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aggregates is neglected. Therefore, the term d/a in the model analysis cannot be held constant between the model and the prototype. From Figs. 1 and 2, and the preceding analysis, a postulate can be made that the

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aggregate with invariant size (not replica-scaled) could account for the non-scaling effect in DOP implied by

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those empirical formulae without considering the effects of aggregates. Meanwhile, several experimental works [22-25] also have shown that the resistance of concrete targets under projectile impact is dependent on both the

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aggregate size and the strength of the aggregate material, which further validate our point.

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2.2.2 Magnitude of the non-scaling effect when aggregate is not scaled Nevertheless, it is very necessary to quantify the magnitude of the non-scaling effect in DOP caused by

the not scaled aggregates. This would be a very important issue for the assessment of protective structure against projectiles with various dimensions, since the aggregate in the protective structures are fixed and could not be scaled with the potential attacks. Of course, it will be obviously dangerous if the dimensionally homogeneous (semi-)analytical penetration models [1-3, 5-11], as introduced in Section 1, are directly 11

ACCEPTED MANUSCRIPT employed to design the protective structures against the full-scale or large-scale penetrations. Fig. 3 exactly shows the magnitude of this non-scaling effect by giving the relative dimensionless depth of replica projectiles with different scales into identical targets (aggregate size is constant), where the normalized deep penetration depth (P/d) when d/a=0.5 is assumed as the baseline. It indicates that the magnitude of this

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effect based on the above three formulae is different, specifically P/d is nearly proportional to d0.215, d0.2 and (d/a)0.1 based on ACE, Modified NDRC and Whiffen formulae, respectively.

ACE Modified NDRC Whiffen

2.50 2.25 2.00 1.75 1.50 1.25 1.00

10

20

d/a

30

40

50

M

0

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Relative dimensionless depth

2.75

Fig. 3 Relative dimensionless DOP of replica-scaled projectiles into concrete targets with fc=constant and

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a=constant (P/d(d/a=0.5) is assumed as the baseline)

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In addition, Table 1 lists the so-called non-scaling effect term in several empirical formulae. Most of them include such a term, while there are also few exceptions such as Hughes formulae and etc. Thus, the question is,

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the magnitude implied in which formula is accepted or are they all correct but suitable for different penetration

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scenarios? Obviously, more researches, especially experiments, have to be done to give the answer. Table 1 The non-scaling effect term in the empirical formulae [15, 29, 30]

Formulae

Magnitude

Formulae

Magnitude

Modified Petry

Scaling hold

Kar

d0.2

BRL

d0.2

Haladar-Hamieh

Scaling hold

ACE

d0.215

Hughes

Scaling hold

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ACCEPTED MANUSCRIPT Modified NDRC

d0.2

Healey and Weissman

d0.2

Ammann and Whitney

d0.2

Young

d0.15 (M<181 kg); d-0.3 (M>181 kg)

Whiffen

(d/a)0.1

Berezan

f(d), exact value unknown

It should be pointed out, a part of the terms given in Fig. 3 and Table 1 are only correct when the

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penetration depth is relatively deep. Fig. 4 shows the relative dimensionless penetration depth versus striking velocity of replica projectiles shown in Fig. 1(a) into concrete targets with fc=35 MPa. The relative dimensionless penetration depth here is denoted by P/λP1=(P/λd1)/(P1/d1), where λ is the increased scaling factor to the projectile with d1=25.4 mm and λ works in the figures between 1 and 15, P1 is the penetration

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depth of the projectile with d1=25.4 mm. It indicates that, the magnitude of the non-scaling effect (represented by P/λP1) of three formulae are totally (ACE), partly dependent (Modified NDRC) and independent (Whiffen) on the striking velocity (or P/d), respectively. Also, this is a question unsolved so far. 2.0

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P /P1

1.5

50.215

M=62.5kg, d=127mm

1.4 1.3

1.7

1.1

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100

200

300

400

500

600

700

100.2 50.2

1.4 1.3 1.2 1.1 1.0

800

0.9

900 1000

0

100

200

300

400

500

600

V0 (m/s)

V0 (m/s)

(a)

(b)

AC

0

150.2

1.5

M=0.5kg, d=25.4mm

1.0

Modified NDRC

1.6

PT

1.2

1.8

100.215

M=500kg, d=254mm

1.6

M=0.5kg, d=25.4mm M=62.5kg, d=127mm M=500kg, d=254mm M=1687.5kg, d=381mm

1.9

P /P1

1.7

0.215

M 15

M=1687.5kg, d=381mm

1.8

0.9

2.0

ACE

1.9

13

700

800

900 1000

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Whiffen

M=0.5kg, d=25.4mm M=62.5kg, d=127mm M=500kg, d=254mm M=1687.5kg, d=381mm

1.9 1.8 1.7

P /P1

1.6 1.5 1.4

100.1

150.1

1.3 1.2

50.1

1.1 1.0 0

100

200

300

400

500

600

V0 (m/s)

(c)

700

800

900 1000

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0.9

Fig. 4 Relative dimensionless penetration depth of projectiles shown in Fig. 1 into concrete targets with fc=35

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MPa predicted by (a) ACE, (b) Modified NDRC, and (c) Whiffen formulae 3. Numerical simulations

Based on the foregoing analyses in Section 2, we know that the assessments on the non-scaling effect in

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DOP caused by the not scaled aggregate are needed urgently, and the experimental works or numerical simulations may be the two feasible solutions. However, the projectile penetration tests with large scale are too

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expensive and hard to handle. Thus, the numerical simulations based on a 3D mesoscopic concrete model

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would be an efficient and meaningful approach.

In this section, hydrocode simulations are carried out through a 3D mesoscopic concrete model to explore

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the non-scaling effect in penetration tests caused by coarse aggregates. In the 3D mesoscopic model, the

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concrete is composed of the strong aggregates embedded in the relative soft cement mortar. Since the non-scaling effect is attributed to the not scaled aggregates, one may naturally speculate that the cement strength, aggregate strength, the volume fraction of aggregates will all affect the magnitude of this effect. These issues are addressed in this section.

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ACCEPTED MANUSCRIPT 3.1 Penetration into concrete targets with random aggregates 3.1.1 3D mesoscopic model for concrete target The 3D mesoscopic model for concrete target used in the present simulation is established through the following five steps: (1) mesh the concrete target with regular hexahedral solid elements as mortar matrix; (2)

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generate a 3D geometric model for an aggregate with random size and shape; (3) distribute the aggregate model randomly into the region that the concrete target covered; (4) change the material property of those elements within the geometric model distributed in Step 3 to be the coarse aggregate; (5) repeat Steps 2-4. To leave a visual impression, Fig. 5 gives a simple illustration through the 2D mesoscopic element modelling, which has

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the same procedures with the 3D model. The detailed procedures are described as follows: Steps 2-3

Cement

Geometric model for aggregate

Aggregate

(b)

Cement

2D mesoscopic model

Aggregate

Cement

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Steps 4

PT

(a)

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Step 1

(c)

(d)

Fig. 5 Illustration of the procedures for mesoscopic element modelling Step 1. Mesh the concrete target to be studied with regular hexahedral solid elements as mortar matrix and

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ACCEPTED MANUSCRIPT the elements should be fine enough. Of course, if the size of elements is too fine, the number of elements will be enormous. Thus, there is a compromise between the element size and number. Step 2. Generate a sphere with random diameter da within the segment [da_min, da_max] and it is assumed that da is uniformly distributed between da_min and da_max, where da_min and da_max are the pre-determined

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minimum and maximum aggregate size, respectively. Then, a 3D random polyhedron with more than 10 faces which represents the coarse aggregate is generated within this sphere, thus da defines the aggregate size. Generation of the polyhedron is similar to the approach described in Fang and Zhang [31], in which the detailed procedure can be found. The difference is that the polyhedron is limited within a sphere and the polyhedron is

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more round in this paper. The information of the generated polyhedron, including its volume, da and coordinates of every vertexes, is stored.

Step 3. Place the generated aggregate (polyhedron) in Step 2 with random location and orientation into the

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region that the concrete target covered. Once an aggregate is placed successfully, refresh the coordinates of its

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vertexes, calculate the normal vector of every face that goes out of the polyhedron, and store the circumscribed sphere‟s center coordinate [Cx, Cy, Cz].

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Step 4. Calculate the central position of each element in the region of [Cx±da/2, Cy±da/2, Cz±da /2], and if

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the central point of an element is located within the polyhedrons generated in Step 3, the material property of this element is changed to be coarse aggregate. In detail, for the judgments, the dot products Ai O  Vi (i=1,

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2, … Nf) are calculated, where O is the central position of an element, Ai is one vertex on the i-th face of a polyhedron, Vi is the normal vector of the i-th face that goes out of the polyhedron, and Nf is the number of faces on a polyhedron. If the Nf dot products are all negative, this element is denoted as aggregate and then check the next element; otherwise, the property of this element is still kept as cement. Check all the elements in the region [Cx±da/2, Cy±da/2, Cz±da /2] to re-identify their material property.

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ACCEPTED MANUSCRIPT Step 5. Repeat Steps 2-4 until the volume fraction of aggregate is satisfied. When placing the new aggregate (Step 3), it must be guaranteed that there are no overlaps and direct contact with the existed aggregates. At present, the condition Si>(da+da_i)/2+l1 (i=1, 2, … j) is employed to ensure this requirements, where j is the total number of the already placed aggregates, Si is the distance between the two central points of

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the new aggregate and the i-th already placed aggregate, da_i is the size of i-th aggregate and l1 is the element size.

Until now, if the grading and volume fraction of the coarse aggregates are given, the mesoscopic model of the concrete can be established. For instance, Fig. 6 shows the geometric model of aggregates randomly

PT

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M

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distributed in the region of concrete specimen, and the corresponding mesoscopic finite element model.

(a)

(b)

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Fig. 6 Concrete specimen (a) geometric model for concrete with random aggregates (5-10 mm in size), (b) the

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corresponding finite element model with cubic mesh size=0.5 mm and total 1,000,000 elements 3.1.2 Numerical simulations of projectile penetrations In this paper, the hydrocode LS_DYNA [32] is employed to simulate the projectile penetration into

concrete targets. The 3D finite element mesoscopic model of concrete target is generated by using MATLAB through the approach introduced in Section 3.1.1, and the projectile is built and mapped through ANSYS. Since only the normal penetration is considered, one quarter of the geometry is modeled to reduce the computational 17

ACCEPTED MANUSCRIPT time. Fig. 7 presents the finite element model of the projectile and concrete targets, where the mesoscopic models for concrete target is presented.

=

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+

Fig. 7 Mesoscopic finite element model for projectile (15 mm in radius) and target (120×120×225 mm3) In this section, the numerical tests are carried out by firing ogive-nosed projectiles (CRH=3) with

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diameters of 10, 20 and 30 mm and length-to-diameter ratio 3.5 at 280 m/s into concrete targets with scaled dimensions. The maximum aggregate sizes in all the three targets are 10 mm and the minimum sizes are 5 mm,

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and the volume fractions of aggregates are about 16.7%. To guarantee the aggregates keep similar, the element

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sizes of the three targets are all 1 mm. Table 2 lists the information of the projectile and target models, where λ is the increased scale factor to the projectile with d1=10 mm.

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Table 2 Information of the projectile and target models

Projectile

Target dimension

Total element

Penetration depth (mm)

Penetration depth (mm)

diameter (mm)

(mm3)

number

(mesoscopic model)

(homogeneous model)

1

10

40×40×75

121250

42.674

35.826

2

20

80×80×150

970000

86.230

70.753

3

30

120×120×225

3273750

132.930

104.800

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λ

The projectile adopts the *MAT_RIGID [32] material model, the cement and aggregate are described by 18

ACCEPTED MANUSCRIPT the *MAT_JOHNSON_HOLMQUIST_CONCRETE (HJC) model [32, 33] through different parameters. The density of the projectiles is set about 8000 kg/m3, and the mass of three projectiles is replica-scaled. The compressive strengths of cement and aggregate are 16 MPa and 120 MPa, respectively. Table 3 lists the computational parameters of the material model for cement and aggregate according to Refs. [31-34]. The

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“CONTACT ERODING SURFACE TO SURFACE” algorithm is employed between the projectile and target, and the contact stiffness scale factor is assigned as 2. The erosion algorithm *MAT ADD EROSION is brought in to solve the excessive element distortion problem, and the erosion criterion is defined by the maximum principal strain as 0.42 for both the aggregate and cement in the present simulation.

Cement

Aggregate

Parameter

Cement

Aggregate

Density ρ (kg/m3)

2000

2660

Crushing pressure Pcrush (MPa)

5.3

40

Shear modulus G (GPa)

6.41

26.93

Crushing volumetric strain Ucrush

6.2e-4

1.1e-3

Normalized cohesive strength A

0.79

0.79

Locking pressure Plock (GPa)

1

1

Normalized pressure hardening B

1.6

1.6

Locking volumetric strain Ulock

0.1

0.1

0.007

0.007

Damage constant D1

0.04

0.04

0.61

0.61

Damage constant D2

1

1

16

120

Pressure constant K1(GPa)

17

17

Maximum tensile pressure T (MPa)

1.4

12

Pressure constant K2 (GPa)

38

38

EFMIN

0.01

0.01

Pressure constant K3 (GPa)

29.8

29.8

Normalized maximum strength SFMAX

7

7

EPS0

1

1

PT

Strain rate coefficient C

CE

Pressure hardening exponent N

AC

Uniaxial compressive strength fc (MPa)

M

Parameter

ED

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Table 3 Parameters of the HJC material model for cement and aggregate

Fig. 8(a) shows the von-mises stress distribution in the mesoscopic model of concrete target (λ=3) when projectile penetrating into target, and it indicates that the stress acting on the aggregates is obviously higher

19

ACCEPTED MANUSCRIPT than which on the cements. Table 2 lists the DOP of the three replica-scaled projectiles into the concrete targets with identical aggregate size. Meanwhile, numerical simulations are performed for projectiles penetration into the concrete targets which are regarded as the homogeneous material, and the corresponding result is listed in the last column of Table 2. Fig. 8(b) also shows the von-mises stress distribution in the homogeneous model of

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concrete target (120×120×180 mm3) under projectile penetrations, in which the stress is distributed relatively uniform. The objective of these comparative simulations is to check whether there is non-scaling effect or not when the simulations are performed on the homogeneous concrete. The homogeneous concrete is set as fc=40 MPa and the values of model parameters are listed in Table 4. Of course, in mesoscopic model (Fig. 7a), if the

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parameters of material model for aggregate are set identical with which for cement, the concrete will also become homogeneous. The results of these two approaches for homogeneous concrete modeling are almost

CE

PT

ED

M

identical.

AC

t=0.2ms

t=0.3ms (a)

20

t=0.4ms

t=0.2ms

t=0.3ms (b)

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ACCEPTED MANUSCRIPT

t=0.4ms

Fig. 8 The stress distributions in the concrete targets during projectile penetrations (a) mesoscopic model, (b)

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homogeneous model

Fig. 9 shows the numerical results listed in Table 2 through the relative penetration depth versus the ratio of projectile diameter to maximum aggregate size (d/a). The relative penetration depth is also denoted by P/λP1,

M

which is the same as Fig. 4, where λ is the increased scale factor, P1 is the penetration depth of the projectile

ED

with diameter d1=10 mm. If the scaling law is held, then the relative penetration depth should be kept as 1. It can be seen in Fig. 9 that, the results based on the homogeneous model for concrete targets could not reveal the

PT

non-scaling phenomenon discussed in Section 2. As discussed in Section 2.2.1, the replica-scaling law holds,

CE

thus, the relative penetration depth based on the homogeneous model should be constant (=1), and the decreased projectile performance shown in Fig. 9 could be attributed to the influence of relative element size.

AC

However, when considering the effects of aggregates in the mesoscopic model, the improved projectile performance with increased scale can be seen. This validate our postulate in Section 2.2 that the aggregate with invariant size (not replica-scaled) may account for the non-scaling effect in DOP.

21

ACCEPTED MANUSCRIPT 1.10 1.08

Mesoscopic model Homogeneous model

1.06

P /P1

1.04 1.02

Scaling law holds 1.00 0.98

1.0

1.5

2.0

2.5

3.0

3.5

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0.96 0.5

d/a

Fig. 9 Relative dimensionless DOP of replica-scaled projectiles into the targets with identical materials Furthermore, to eliminate the effect of relative element size, the results from mesoscopic model should be

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divided by the corresponding results from homogeneous model. Fig. 10 shows the modified results of the relative penetration depth through the mesoscopic model. It indicates that the non-scaling effect is about 9% for λ=3. Meanwhile, the magnitude implied in Whiffen formulae is also added in Fig. 10 for comparison. 1.14

Mesoscopic model (d/a)0.1

M

1.12 1.10

1.06

ED

P /P1

1.08

1.04

PT

1.02 1.00

0.5

1.0

1.5

2.0

2.5

3.0

3.5

d/a

CE

0.98

Fig. 10 Modified results of the relative dimensionless DOP

AC

However, since the element size in this section is very fine, the simulations with larger scale will consume

much more computational time, and thus the discussed range of the independent variable (d/a) is relatively narrow. Aiming to improve the computing efficiency, in the following section, the mesoscopic model in which the aggregates have regular shape and uniform locations is utilized to perform further discussions. Of course, it should be pointed out that the interfacial transition zone (ITZ) in concrete model is not considered in this paper.

22

ACCEPTED MANUSCRIPT If the ITZ is included, the numerical results may be different, which should be further discussed in the future. Besides, since this paper only focuses on the non-scaling effect in DOP, the present Finite Element Method (FEM) is accurate enough. However, if the scale effect for the material failure of concrete targets under projectile impacts is studied in the future, the meshfree methods proposed in Refs. [35-38] should be employed

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instead of the FEM as these methods could simulate the cracks occurred in the concrete well. 3.2 Penetration into concrete targets with regular aggregates

To investigate the non-scaling effect occurred in the penetration tests with larger scale, a mesoscopic model for concrete target with regular aggregates, as shown in Figs. 11 and 12, is employed. The aggregate in

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this section is regarded as cube with length of 10 mm, and the size of the cubic element is 2.5 mm. Thus, an aggregate in all the finite element models has 64 elements. The projectiles in Section 3.1 are still used and the striking velocity is kept as 280 m/s. The increased scale factor λ for projectile is extended to 9, namely, the

Top view

PT

ED

M

largest projectile has the diameter of 90 mm (9×10 mm) and length of 315 mm.

AC

CE

Side view

Fig. 11 Finite element model for projectile (15 mm in radius) and target (120×120×225 mm3) with regular aggregates

23

ACCEPTED MANUSCRIPT d=60mm

d=30mm

120×120×225mm3

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240×240×450mm3

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Fig. 12 Side view of the finite element models for λ=6 and λ=3

Similar to Section 3.1.2, simulations of the concrete targets which are regarded as the homogeneous material against the projectile impact are firstly carried out. Fig. 13 gives the corresponding results of these

M

simulations. It shows that the projectile performance is decreasing with the increased scale, which is consistent with the results shown in Fig. 9. This result is also regarded as the influence of relative element size, and it is

ED

also eliminated in the following calculations. 1.06

Homogeneous model Scaling law holds

PT

1.04 1.02

AC

CE

P/(P1)

1.00 0.98 0.96 0.94 0.92 0.90

0

1

2

3

4

5

6

7

8

9

10

d/a

Fig. 13 Relative dimensionless DOP of replica-scaled projectiles into the homogeneous concrete targets Before discussions, the results from mesoscopic concrete model with random aggregates in Section 3.1.2 and results from this section should be compared. Since the aggregates in this section have regular shape and

24

ACCEPTED MANUSCRIPT uniform locations, the volume fraction of aggregates cannot be random. Thus, the results from concrete targets with 12.5% aggregates are employed for comparison. Although this volume fraction of the regular aggregates is lower than the fraction of the random aggregates (16.7%) in Section 3.1.2, the regular aggregates all have the cubic shape with length of 10 mm, which is larger than most of the random aggregates in Section 3.1.2. Fig. 14

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shows the numerical results based on the above two approaches. It indicates that, the magnitude of non-scaling effect based on the regular aggregated model is slightly higher than which from the random aggregated model in Section 3.1.2, thus the simulations based on the concrete model with regular aggregates could also be utilized to discuss the non-scaling effect. 1.14

1.10

Model with regular aggregates Model with random aggregates (d/a)0.1

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1.12

1.06 1.04 1.02 1.00

1.0

ED

0.98 0.5

M

P /P1

1.08

1.5

2.0

2.5

3.0

3.5

d/a

PT

Fig. 14 Comparisons of the relative dimensionless DOP from the mesoscopic concrete models with random and regular aggregates

CE

3.2.1 Influence of the cement strength

AC

Projectile penetrations are simulated to study the influence of the cement strength on the non-scaling effect. Four groups of concrete targets with different cements but the same aggregates are modeled. Four kinds of cements with different compressive strengths are discussed, i.e., 12, 16, 24 and 40 MPa, the corresponding computational parameters of the material model for cement are listed in Table 4 according to Refs. [31-34], respectively. The 120 MPa aggregates are adopted and the volume fractions in the four groups of concrete targets are all 29.6%. 25

ACCEPTED MANUSCRIPT Table 4 Parameters of the HJC material model for cement 12 MPa

16 MPa

24 MPa

40 MPa

Parameter

12 MPa

16 MPa

24 MPa

40 MPa

ρ (kg/m3)

2000

2000

2000

2200

SFMAX

7

7

7

7

G (GPa)

5.55

6.41

7.85

11.69

Pcrush (MPa)

4.0

5.33

8.0

13.33

A

0.79

0.79

0.79

0.79

Ucrush

5.4e-4

6.2e-4

7.6e-4

8.6e-4

B

1.6

1.6

1.6

1.6

Plock (GPa)

C

0.007

0.007

0.007

0.007

Ulock

N

0.61

0.61

0.61

0.61

D1

fc (MPa)

12

16

24

40

D2

T (MPa)

1.1

1.4

2.4

4

EFMIN

0.01

0.01

0.01

EPS0

1

1

1

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Parameter

1

1

1

0.1

0.1

0.1

0.1

0.04

0.04

0.04

0.04

1

1

1

1

K1(GPa)

17

17

17

17

0.01

K2 (GPa)

38

38

38

38

1

K3 (GPa)

29.8

29.8

29.8

29.8

M

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1

ED

Fig. 15 shows the numerical results of the relative penetration depth versus the ratio of projectile diameter to maximum aggregate size (d/a). It can be seen from the figure that the non-scaling effect exists in all the four

PT

groups of numerical tests. The most meaningful and interested thing is that the magnitude of the non-scaling

CE

effect is directly related to the strength of cement, namely, the magnitude of the non-scaling effect decreases

AC

with the increasing of the cement strength, and asymptote to the horizontal line (P/λP1=1 with scaling law holds). Meanwhile, the magnitude implied in the empirical formulae, which are discussed in Section 2, is also added in Fig. 15. It can be derived that the magnitude based on the numerical simulations is in the reasonable range. However, the empirical formulae which imply the non-scaling effect in DOP for different concrete targets with the same magnitude as listed in Table 1 may be unreasonable.

26

ACCEPTED MANUSCRIPT 1.6

Cement:12MPa Cement:16MPa Cement:24MPa Cement:40MPa (d/a)0.2 (d/a)0.1

1.5

P/(P1)

1.4 1.3

Aggregate:120MPa a=10mm

1.2 1.1 1.0

V0=280m/s 0

1

2

3

4

5

6

7

d /a

8

9

10

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0.9

Fig. 15 Influence of the cement strength on the non-scaling effect in DOP 3.2.2 Influence of the aggregate strength

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In this sub-section, numerical simulations are carried out to study the influence of the aggregate strength on the non-scaling effect. Three groups of concrete targets with different aggregates but the same cements are modeled. The computational parameters of the material model for aggregate are listed in Table 5 according to

M

Refs. [31-34] and the discussed compressive strengths are 120, 150, and 180 MPa, respectively. The volume fraction of the aggregates are kept as 29.6% and the compressive strength of cement is 24 MPa.

ρ (kg/m3)

2660

G (GPa)

ED

120 MPa

180 MPa

Parameter

120 MPa

150 MPa

180 MPa

2660

2660

SFMAX

7

7

7

26.93

30.10

32.98

Pcrush (MPa)

40

50

60

0.79

0.79

0.79

Ucrush

1.1e-3

1.2e-3

1.4e-3

B

1.6

1.6

1.6

Plock (GPa)

1

1

1

C

0.007

0.007

0.007

Ulock

0.1

0.1

0.1

N

0.61

0.61

0.61

D1

0.04

0.04

0.04

fc (MPa)

120

150

180

D2

1

1

1

T (MPa)

12

15

18

K1(GPa)

17

17

17

AC

A

150 MPa

PT

Parameter

CE

Table 5 Parameters of the HJC material model for aggregates

27

ACCEPTED MANUSCRIPT EFMIN

0.01

0.01

0.01

K2 (GPa)

38

38

38

EPS0

1

1

1

K3 (GPa)

29.8

29.8

29.8

Fig. 16 gives the simulation results for the influence of the aggregate strength on the non-scaling effect. It indicates that the magnitude of the non-scaling effect is also dependent on the strength of coarse aggregates.

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However, the results for concrete with 150 MPa aggregates and 180 MPa aggregates nearly overlap. The influence of the aggregate strength on the non-scaling effect seems not so obvious comparing with the influence of cement shown in Fig. 15. Frew et al. [8] also detected small difference in penetration depth versus striking velocity data for concrete targets with limestone (Mohs hardness of 3.0) or quartz-based aggregates (Mohs

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hardness of 7.0). 1.4

1.3

M

1.2

P/(P1)

Cement:24MPa a=10mm

Aggregate:120MPa Aggregate:150MPa Aggregate:180MPa Whiffen: (d/a)0.1

1.1

ED

1.0

0

1

2

V0=280m/s 3

4

5

6

7

8

9

10

d/a

PT

0.9

CE

Fig. 16 Influence of the aggregate strength on the non-scaling effect in DOP 3.2.3 Influence of the volume fraction of aggregates

AC

The objective of this sub-section is to study the influence of the volume fraction of aggregates on the non-scaling effect. The combination of 16 MPa cement and 120 MPa aggregate is adopted, and their computational parameters of the material model can be found in Tables 4 and 5. Two series of concrete targets with different volume fractions of aggregates are modeled, i.e., 12.5% and 29.6%. Fig. 17 shows the corresponding results and it indicates that the magnitude of the non-scaling effect increases with the increasing of the volume fraction. 28

ACCEPTED MANUSCRIPT 1.6

Volume fraction 29.6% Volume fraction 12.5% (d/a)0.2 (d/a)0.1

1.5

P/(P1)

1.4

V0=280m/s

1.3 1.2 1.1

a=10mm Cement:16MPa Aggregate: 120MPa

0.9

0

1

2

3

4

5

6

7

d/a

8

9

10

CR IP T

1.0

Fig. 17 Influence of the aggregates‟ volume fraction on the non-scaling effect in DOP 4. A semi-analytical formula for DOP with considering the non-scaling effect

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Previously, in Ref. [11], based on the mean resistance approach, we have proposed a simple formula for deep penetrations of projectile, in which the initial impact cratering stage was neglected. Then, to make the formula suitable for small and medium penetration, the initial cratering phase was added in Ref. [39]. Comprehensively, the formula to predict the penetration depth is expressed as

M

I0 P 2 l   d  1  0.4  2d

I0 MV0 2 M ; I0  ; N 3 Sf c d 0 d 3 N 2 N

ED



for P  l

(4a)

(4b)

PT

where l is the projectile‟s nose length, N2 is the nose shape factor, N2=1/3ψ-1/24ψ2 for ogive nosed projectile

CE

and ψ is the caliber-radius-head (CRH). The product Sfc denotes the quasi-static target resistance and S was

AC

experimentally suggested by Frew et al. [8] as

S  82.6  ( fc / 106 )0.544

(5)

Table 6 lists the information of those penetration tests to obtain Eq. (5), the symbol „?‟ in the table means

that a=9.5 mm is uncertain since the lack of information. From Table 6, it can be seen that S in Eq. (5) is determined from the projectile penetration tests [7, 8, 35] with d/a located within [2.14, 3.21]. Thus, although the Eq. (5) is widely used for concrete targets with various compressive strength in the projectile penetration

29

ACCEPTED MANUSCRIPT analyses, its application range is actually very limited. For example, the predicted DOP from penetration equation by using S in Eq. (5) gives underestimated predictions when comparing with the test data for d/a=8.02 in Ref. [13]. Thus, considering the non-scaling effect arisen from the coarse aggregate, using the dimensionally homogeneous (semi-)analytical models [1-3, 5-11] directly to design the protective structures against the

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full-scale or large-scale penetrations will be dangerous. Table 6 Information of the penetration tests from which the S in Eq. (5) was determined [7, 8, 40] Forrestal et

Forrestal et

Forrestal et

Frew et al.

Forrestal et

Forrestal et

al. [7]

al. [35]

al. [7]

al. [35]

[8]

al. [36]

al. [7]

fc (MPa)

13.5

21.6

36.2

a(mm)

4.8

4.8

9.5?

d(mm)

12.9

12.9

26.9

d/a

2.69

2.69

2.83

M(kg)

0.064

0.064

M

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Forrestal et

0.906

51

58.4

62.8

96.7

9.5

9.5

9.5

9.5?

30.5

20.3/30.5

20.3

26.9

3.21

2.14/3.21

2.14

2.83

1.6

0.478/1.62

0.478

0.904

ED

Ref.

Nevertheless, since the non-scaling effect for d/a within (2.14, 3.21) could be neglected as tested in Frew

PT

et al. [8], Eq. (5) for concrete targets with various strength could be regarded as a baseline to extent to the

CE

penetration tests with different d/a. Based on the analysis in Section 3, and assuming that the volume fractions

AC

of aggregates in concrete targets are almost the same, Eq. (5) is modified as

S  82.6  ( fc / 106 )0.544 (

d  f ( fc ,c , fc ,a ) ) 2.83a

(6)

where d/a=2.83 in Ref. [40] is the selected as the referenced ratio, f(fc,c, fc,a) is the magnitude of the non-scaling effect for S and it is related to the compressive strengths of cement fc,c and aggregate fc,a as discussed in Section 3. Since it is difficult to determine the strength of each concrete components, i.e., fc,c and fc,a, it can be simply

30

ACCEPTED MANUSCRIPT assumed that the magnitude of the non-scaling effect for S is related to the strength of concrete, namely, f(fc,c, fc,a)=T(fc). To determine T(fc), it needs experiments obviously. For example, Fig. 18 illustrates the relative resistance acted on projectile versus d/a (0.67≤d/a≤7.99) for concrete targets with the same strength of ~38 MPa. In the

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figure, the relative resistance is denoted by S/S(d/a=2.83) and S is obtained from the penetration test by using the approach in Ref. [7]. Then, T(fc=38 MPa)=0.208 is determined by fitting the test data. Thus, for concrete targets with fc=~38 MPa, the parameter S from Eq. (5) is improved by further considering the non-scaling effect and expressed as

1.5

fc=38.15MPa

1.4

1.2

1.0

fc=39MPa

fc=36.2MPa

0.9

(7)

Gomez and Shukla [12] Forrestal et al. [7] Forrestal et al. [13] (d/2.83a)-0.208

M

1.1

ED

Relative resistance

1.3

d 0.208 ) 2.83a

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S  82.6  ( fc / 106 )0.544 (

0.8

0

1

2

3

4

5

6

7

8

9

d/a

PT

0.7

CE

Fig. 18 Relative resistance acted on projectile versus d/a for concrete targets with fc=~38 MPa Fig. 19 illustrates the comparisons between the test data of DOP as well as the predictions by the present

AC

modified formula Eqs. (4) and (7), as well as the formula proposed by Forrestal et al. [7, 8], respectively. It indicates that, the predictions of the proposed model agree much better with the test data. However, it should be pointed out that, since the related tests with wide range of d/a are very limited, even the formula expressed in Eqs. (4) and (7) for fc=~38MPa concrete targets is still need more penetration tests for further validations.

31

ACCEPTED MANUSCRIPT 0.16 0.14 0.12

1.1

Test data [12] Eqs. (4) and (7) Forrestal et al. [7, 8]

1.0 0.9 0.8

P (m)

0.10

P (m)

Test data [13] Eqs. (4) and (7) Forrestal et al. [7, 8]

0.08

0.7 0.6

0.06

0.5 M=0.015kg d=6.35mm a=9.5mm

0.04 0.02

0.3

fc=38.15MPa 200

300

400

500

600

0.2 200

700

250

300

350

400

450

500

CR IP T

0.00 100

M=12.92kg d=75.95mm a=9.5mm fc=39MPa

0.4

V0 (m/s)

V0 (m/s)

(a)

(b)

Fig. 19 Experimental and predicted DOP (a) d/a=0.67 (b) d/a=7.99

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5. Conclusion

To explore geometrical scaling effect for rigid penetration into concrete targets, based on the empirical formulae, available penetration tests and 3D mesoscopic finite element model, a series analyses and discussions

M

are carried out, the main works and conclusions are listed as follows:

(1) Based on available experimental data, it is found that replica scaling law holds for rigid projectile

ED

penetrations, as long as the scaling is done strictly for both projectiles and concrete targets. The non-scaling

PT

effect implied in empirical formulae is caused by the aggregate with invariant size (not replica-scaled). Thus, the dimensionally homogeneous (semi-)analytical penetration models, which only validated by laboratory-scale

CE

tests, can not be employed to design the protective structures (aggregate size fixed) against the full-scale or

AC

large-scale penetrations.

(2) By developing a 3D mesoscopic finite element model for concrete target, the parametric influences on

the non-scaling effect are numerically discussed. It is derived that the magnitude of non-scaling effect decreases with the increasing of the cement strength when the aggregate strength is fixed. While the influence of the aggregate strength on the non-scaling effect is not so obvious comparing with the influence of cement. Besides, the magnitude increases with the increasing of the volume fraction of aggregates. All these conclusions indicate 32

ACCEPTED MANUSCRIPT that, the non-scaling effect in DOP for different concrete targets with the same magnitude implied by the empirical formulae and the always held scaling law in the (semi-)analytical models may be unreasonable. (3) By further considering the non-scaling effect, a semi-analytical model for DOP prediction is proposed base on our previous model, which show good predicted accuracy with the available test data.

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(4) More importantly, large amounts of experimental works on the scaling law through replica scaled penetration tests and more penetration test with large scale are urgently needed. Acknowledgment

The project was supported by the National Natural Science Foundations of China (51522813, 51578542,

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51438003). Reference

[1] Luk V K, Forrestal M J. Penetration into semi-infinite reinforced-concrete targets with spherical and

M

ogival nose projectiles. Int J Impact Eng 1987; 6: 291-301.

Struct 1997; 34: 4127-46.

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[2] Forrestal M J, Tzou D Y. A spherical cavity-expansion penetration model for concrete targets. Int J Solids

PT

[3] He T, Wen H M, Guo X J. A spherical cavity expansion model for penetration of ogival-nosed projectiles

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into concrete targets with shear-dilatancy. Acta Mech Sin 2011; 27: 1001-12. [4] Feng J, Li W B, Wang X M, Song M L, Ren H Q, Li W B. Dynamic spherical cavity expansion analysis of

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rate-dependent concrete material with scale effect. Int J Impact Eng 2015; 84: 24-37.

[5] Warren T L, Forquin P. Penetration of common ordinary strength water saturated concrete targets by rigid ogive-nosed steel projectiles. Int J Impact Eng 2016; 90: 37-45. [6] Kong X Z, Wu H, Fang Q, Peng Y. Rigid and eroding projectile penetration into concrete targets based on an extended dynamic cavity expansion model. Int J Impact Eng 2017; 100: 13-22.

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ACCEPTED MANUSCRIPT [7] Forrestal M J, Altman B S, Cargile J D. An empirical equation for penetration depth of ogive-nose projectiles into concrete targets. Int J Impact Eng 1994; 15(4): 395-405. [8] Frew D J, Hanchak S J, Green M L, Forrestal M J. Penetration of concrete targets with ogive-nose steel rods. Int J Impact Eng 1998; 21(6): 489-97.

non-deformable projectile. Int J Impact Eng 2003; 28(1): 93-116.

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[9] Li Q M, Chen X W. Dimensionless formulae for penetration depth of concrete target impacted by a

[10] Wu H, Fang Q, Zhang Y D, Gong Z M. Semi-theoretical analyses of the concrete plate perforated by a rigid projectile. Acta Mech Sin 2012; 28(6): 1630-43.

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[11] Peng Y, Wu H, Fang Q, Gong Z M, Kong X Z. A note on the deep penetration and perforation of hard projectiles into thick targets. Int J Impact Eng 2015; 85: 37-44.

[12] Gomez J T, Shukla A. Multiple impact penetration of semi-infinite concrete. Int J Impact Eng 2001; 25:

M

965-79.

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[13] Forrestal M J, Frew D J, Hickerson J P, Rohwer T A. Penetration of concrete targets with deceleration-time measurements. Int J Impact Eng 2003; 28(5): 479-97.

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[14] Kennedy R P. A review of procedures for the analysis and design of concrete structures to resist missile

CE

impact effects. Nucl Eng Des 1976; 37: 183-203. [15] Li Q M, Reid S R, Wen H M, Telford A R. Local impact effects of hard missiles on concrete targets. Int J

AC

Impact Eng 2005; 32(1): 224-84.

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