Quasi-brittle fracture during structural impact of AA7075-T651 aluminium plates

International Journal of Impact Engineering 37 (2010) 537–551

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

Quasi-brittle fracture during structural impact of AA7075-T651 aluminium plates Tore Børvik a, b, *, Odd Sture Hopperstad a, Ketill O. Pedersen a, c a

Structural Impact Laboratory (SIMLab), Centre for Research-based Innovation (CRI) and Department of Structural Engineering, Norwegian University of Science and Technology, Rich. Birkelands vei 1A, NO-7491 Trondheim, Norway b Norwegian Defence Estates Agency, Research & Development Department, PB 405, Sentrum, NO-0103 Oslo, Norway c SINTEF, Materials and Chemistry, Alfred Getz vei 2, NO-7465 Trondheim, Norway

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 August 2009 Received in revised form 28 October 2009 Accepted 1 November 2009 Available online 6 November 2009

The stress–strain behaviour of the aluminium alloy 7075 in T651 temper is characterized by tension and compression tests. The material was delivered as rolled plates of thickness 20 mm. Quasi-static tension tests are carried out in three in-plane directions to characterize the plastic anisotropy of the material, while the quasi-static compression tests are done in the through-thickness direction. Dynamic tensile tests are performed in a split Hopkinson tension bar to evaluate the strain-rate sensitivity of the material. Notched tensile tests are conducted to study the influence of stress triaxiality on the ductility of the material. Based on the material tests, a thermoelastic–thermoviscoplastic constitutive model and a ductile fracture criterion are determined for AA7075-T651. Plate impact tests using 20 mm diameter, 197 g mass hardened steel projectiles with blunt and ogival nose shapes are carried out in a compressed gas-gun to reveal the alloy’s resistance to ballistic impact, and both the ballistic limit velocities and the initial versus residual velocity curves are obtained. It is found that the alloy is rather brittle during impact, and severe fragmentation and delamination of the target in the impact zone are detected. All impact tests are analysed using the explicit solver of the non-linear finite element code LS-DYNA. Simulations are run with both axisymmetric and solid elements. The failure modes are seen to be reasonably well captured in the simulations, while some deviations occur between the numerical and experimental ballistic limit velocities. The latter is ascribed to the observed fragmentation and delamination of the target which are difficult to model accurately in the finite element simulations. Ó 2009 Elsevier Ltd. All rights reserved.

Keywords: Aluminium armour plates Material tests Component tests Numerical simulations

1. Introduction The need for low weight in aircraft structures has led to the development of very high-strength aluminium alloys. In particular, the 7075 alloy, which belongs to the Al–Zn–Mg–Cu series, is considered as one of the most important engineering aluminium alloys on the market today due to its high strength-to-density ratio [1]. Owing to this, the alloy has also been used in armour applications [2–7]. The goal of this study is twofold: first, to reveal some of the properties of the 7075 alloy during impact-generated loading conditions, and second, to investigate to which extent simple constitutive relations and fracture criteria can be used in finite

* Corresponding author. Norwegian University of Science and Technology, Structural Impact Laboratory (SIMLab), Centre for Research-based Innovation (CRI) and Department of Structural Engineering, Rich. Birkelands vei 1A, NO-7491 Trondheim, Norway. Tel.: þ47 73 59 46 47; fax: þ47 73 59 47 01. E-mail address: [email protected] (T. Børvik). 0734-743X/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2009.11.001

element analysis of high-strength aluminium components with complex microstructure subjected to structural impact. The fracture of aluminium alloy 7075 under quasi-static and dynamic loading conditions has been the topic of several investigations [8–13], while other studies have characterized the flow behaviour and ductility at wide ranges of strain rate and temperature [14,15]. Also the adiabatic shear banding in thick-walled AA7075-T651 cylinders under explosive loading has been investigated experimentally and numerically [16,17]. In this study, the mechanical behaviour of AA7075-T651 plates is investigated through a number of material and component tests. On the basis of the quasi-static and dynamic material tests, a thermoelastic–thermoviscoplastic constitutive model and a ductile fracture criterion are determined for AA7075-T651. Plate impact tests using blunt and ogival steel projectiles are carried out in a compressed gas-gun to determine the alloy’s resistance to ballistic impact. The alloy is found to be quite brittle, and fragmentation and delamination of the target occurred during impact. Finite element analyses of all impact tests are carried out using the explicit solver of LS-DYNA. Both axisymmetric and solid elements

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Table 1 Nominal chemical composition (weight %) of aluminium alloy 7075. Al

Si

Fe

Cu

Mn

Mg

Cr

Zn

Ti

Others

Balance

0.06

0.19

1.3

0.04

2.4

0.19

5.7

0.08

0.15

are used in the simulations. The results show that the failure modes are reasonably well captured in the simulations, while some deviations are found between the numerical and experimental ballistic limit velocities. The reason for the latter is assumed to be problems related to accurate modelling of the fragmentation and delamination processes observed in the target. 2. Material tests 2.1. Material Rolled plates of the aluminium alloy 7075 in T651 temper were studied. The plate thickness was 20 mm. Temper T651 implies that the alloy is slightly stretched and aged to peak strength, with nominal yield and tensile strengths of 505 and 570 MPa, respectively. It should be noticed that the tensile properties of AA7075T651 is reported to strongly depend on the thickness of the product (in a similar way as for rolled 5083-H116 plates [18,19]). Both the yield and tensile strength reach maximum values for a plate thickness of 20–25 mm. At larger thicknesses, a rapid and almost linear decrease in strength properties with increasing section thickness is found [1]. The main reason for this is that the cooling rate during quenching drops considerably with plate thickness and the tensile properties of the alloy are increasing functions of the average cooling rate. In this study, 20 mm thick plates were tested due to their maximum strength. The nominal chemical composition of the 7075 alloy is given in Table 1. The microstructure of the 7075 alloy is shown in Fig. 1. Three cross-sections are shown in the figure, defined by the rolling direction (X), the transverse direction (Y) and the thickness direction (Z) of the plate. It is seen that the alloy has a complex and nonrecrystallized grain structure with flat, elongated grains. 2.2. Tensile tests Quasi-static tensile tests were carried out using smooth axisymmetric specimen [20] with a gauge length of about 40 mm and a cross-section diameter of 6 mm. The tensile axis of the specimens were oriented 0 , 45 and 90 with respect to the rolling direction of the plate. Three parallel tests were carried out in each

direction at room temperature. The cross-head velocity was 1.2 mm/min, which corresponds to an average strain rate of 5  104 s1. The force and the diameter at the minimum crosssection of the specimen were measured continuously until fracture. A purpose-made measuring rig with 4 sensors was used to accurately measure the specimen diameter. The sensors were installed on a mobile frame to ensure that the diameters were measured at the minimum cross-section. The specimen diameter was measured in the thickness direction of the plate and in the transverse direction of the specimen. These diameters are denoted DZ and Dt, respectively. The true stress and the true strain were calculated as

F A

s¼ ;

3 ¼ ln

A0 A

(1)

where F is the force, A0 ¼ ðp=4ÞD20 is the initial cross-section area and D0 is the initial diameter of the gauge section. Since there might be variations in stress and strain over the cross-section, s and 3 should be considered as average values. This is particularly important for the notched tensile tests presented in the next section. The current area of the cross-section is

A ¼

p 4

DZ Dt

(2)

The plastic strain is obtained as 3p ¼ 3  s=E, where E is Young’s modulus. The strain ratio is here defined by

r ¼

3t lnðDt =D0 Þ ¼ 3Z lnðDZ =D0 Þ

(3)

where 3Z and 3t are the true strains in the thickness direction of the plate and the transverse direction of the specimen. The strain ratio is unity for isotropic materials, and represents a measure of the anisotropy of the material. Note that plastic incompressibility and negligible elastic strain have been assumed in Eq. (1). The true stress–strain curves until fracture are shown in Fig. 2. Three parallel tests are shown for each direction. It is seen that the strength level is about the same in the 0 and 90 orientations, while at 45 the strength is significantly lower. At the same time, the true strain to fracture varies markedly with specimen orientation, and the highest fracture strain is found at 45 . The specimens after fracture are shown in Fig. 3. It is found that shear fracture is the pre-dominant failure mode in all directions tested. Typical (smoothed) curves of the strain ratio r versus true strain 3 are given in Fig. 4. Since the strain ratio varies with specimen orientation and deviates significantly from unity, the plastic flow of the material is clearly anisotropic. A marked variation of strain ratio with strain is

Fig. 1. Microstructure of AA7075-T651 where X designates the rolling direction, Y the transverse direction and Z the thickness direction of the plate.

T. Børvik et al. / International Journal of Impact Engineering 37 (2010) 537–551

1.6

800 0

o

90

o

45

o

1.4

600

45o St r a i n r a t i o r

True stress [MPa]

539

400

1.2

1 90o

200

0.8 0o 0 0

0.1

0.2 0.3 True strain

0.4

0.5

Fig. 2. True stress–strain curves to fracture in uniaxial tension for three orientations of the specimen with respect to the rolling direction of the plate.

also observed in Fig. 4, which indicates a change of the plastic anisotropy, probably caused by texture development. 2.3. Notched tensile tests To investigate the influence of the stress state on the fracture strain, tensile tests at room temperature were performed using notched axisymmetric specimens [20]. The cross-head velocity was 1.2 mm/min. Specimens with two different notch radii (R ¼ 0.8 mm and R ¼ 2.0 mm) and a cross-section diameter 2a ¼ 6 mm were tested. Duplicate tests were carried out in the rolling direction of the plate. Force and diameter at minimum cross-section were measured, using the same procedure as for the tension tests with smooth specimens. True stress and true strain were calculated from Eq. (1), while the maximum stress triaxiality (i.e. the stress triaxiality at the specimen axis) was estimated from Bridgman’s analysis [21]

s*max ¼

 1 a þ ln 1 þ 3 2R

(4)

0.6 0

0.1

0.2 True strain

0.3

0.4

Fig. 4. Typical curves of the strain ratio versus True strain in uniaxial tension for three orientations of the specimen with respect to the rolling direction of the plate.

where a is the minimum radius of the specimen. Note that the stress * ¼ s =s ; where sm ¼ 1=3s : I is the triaxiality is defined by sp m ffi eq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2s : s is the von Mises equivalent stress. mean stress and seq ¼ Here, s is the Cauchy stress tensor, s ¼ s  smI is the stress deviator and I is the 2nd order identity tensor. This formula gives s*max ¼ 1.389 for R ¼ 0.8 mm and s*max ¼ 0.893 for R ¼ 2.0 mm. The smooth, axisymmetric specimen has s*max ¼ 1=3 before necking, since in this case R is infinite. These values are estimates on the initial stress triaxiality at the specimen axis. Note that the stress triaxiality will change as the specimen changes geometry with large plastic straining. Accordingly, finite element analysis is required to get more detailed information of the stress state in the specimens during plastic straining until fracture. Furthermore, the plastic anisotropy of the material is neglected in Bridgman’s analysis, which makes the picture even more complicated. The true stress–strain curves are plotted in Fig. 5 together with the results from the tension tests with smooth specimens in the rolling direction. Duplicate tests are presented. It is clearly demonstrated that increased stress triaxiality leads to increased stress level, owing to the superimposed hydrostatic tension in the

Fig. 3. Uniaxial tension specimens after fracture showing pre-dominant shear fracture in all directions tested.

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1000 R = 0.8 mm R = 2.0 mm

True st ress [MPa]

800 Smooth 600

400

200

0 0

0.04

0.08 0.12 True strain

0.16

0.2

Fig. 5. True stress–strain curves to fracture in tension for smooth and notched specimens oriented in the rolling direction of the plate.

notch, and to reduced fracture strain. The results of duplicate tests are consistent, while some irregular behaviour is seen for small strains for one of the tests at intermediate stress triaxiality. The notched tension specimens after fracture are shown in Fig. 6. Cupand-cone fracture is seen for both stress triaxiality levels, which indicates a ductile fracture mode.

2.4. Compression tests Axial compression tests were carried out on cylindrical specimens with diameter D0 equal to 10 mm and height H0 equal to 10 or 20 mm. The specimen axis was along the thickness direction of the plate. The specimens were compressed between thick hardened steel platens. Three parallel tests were performed for each H0 =D0 ratio. For the shortest specimens different lubricants were used, namely Teflon, a graphite paste and no lubrication, while for the longest specimens only the graphite paste was used. The tests were carried out at room temperature and with cross-head velocity 0.5 mm/min. The force and the nominal strain were continuously measured. Two extensometers attached to the steel platens were

used to measure the nominal strain of the specimen. The true stress and true strain were calculated as s ¼ F=A and 3 ¼ ln(1 þ e), where A ¼ A0 exp(3), A0 ¼ ðp=4ÞD20 and e is the nominal strain. The influence of lubrication is indicated in Fig. 7a, which shows the compressive stress–strain curves obtained with graphite paste as lubricant and no lubrication at all. There seems to be little influence of friction on the stress level, while the strain to fracture is less when no lubrication is used. If this is caused by friction, or just represents scatter in the results, is not clear. The stress–strain curves obtained with Teflon as lubricant exhibited some irregularities and are not shown. The true stress–strain curves for the specimens with H0 =D0 equal to 2 are shown in Fig. 7b, where they are compared with a typical tensile stress–strain curve in the rolling direction. The stress level in through-thickness uniaxial compression is quite similar to the stress level in uniaxial tension in the rolling direction. It is noted that if incompressible plasticity is assumed, uniaxial compression through the thickness of the plate is equivalent to equibiaxial tension in the plane. Pictures of the fractured specimens are presented in Fig. 8. All the specimens failed by shear banding oriented about 45 with respect to the loading direction. The failure mode is similar for the 10 and 20 mm specimens. Some tendency of barrelling is seen for specimen 3, which was tested without lubricant. For the other specimens, barrelling seems to be negligible. 2.5. Strain-rate tests Uniaxial tension tests at high strain rates were carried out in a split Hopkinson tension bar [22]. It should be noted that the specimens used in the dynamic tests were significantly smaller, with a gauge length of 5 mm and a cross-section diameter of 2 mm, than the specimens used in the quasi-static tests [20]. All tests were done with specimens oriented along the rolling direction of the plate. In each test, the nominal values of stress s, strain e and strain rate e_ were calculated from measured data using stress wave propagation theory. In addition, the fracture area Af (i.e. the area of the minimum cross-section of the broken specimen) was calculated based on diameter measurements using a light microscope. The fracture strain was then obtained as 3f ¼ lnðA0 =Af Þ, where A0 is the initial gauge area of the specimen. The nominal stress–strain curves from the dynamic tensile tests are plotted in Fig. 9. The results are from six tests with nominal strain rate ranging from 179 s1 to 1128 s1. The difference in flow stress between tests at various strain rates is small, indicating a very limited strain-rate sensitivity of the material, but the stress level seems to be slightly lower at high strain rates than at quasi-static conditions. The

Fig. 6. Notched tension specimens after fracture showing cup-and-cone fracture.

T. Børvik et al. / International Journal of Impact Engineering 37 (2010) 537–551

a

b

800

800

600 True stress [MPa]

True stress [MPa]

600

541

400

200

400

200

Tension test (0o-direction) Compression test # 1 Compression test # 2 Compression test # 3

Graphite paste No lubricant

0

0 0

0.04

0.08 0.12 True strain

0.16

0.2

0

0.04

0.08 0.12 True strain

0.16

0.2

Fig. 7. True stress–strain curves to fracture in uniaxial compression for cylindrical specimens with (a) height H0 ¼ 10 mm and either graphite paste as lubricant or no lubrication at all and (b) height H0 ¼ 20 mm where the results are compared with the true stress–strain curve in uniaxial tension in the rolling direction of the plate. Note that absolute values of the true stress and true strain are plotted.

average fracture strain at high strain rate was estimated to 0.29 with a standard deviation of 0.03, while the corresponding fracture strain at quasi-static conditions was in the range 0.15–0.17. It thus seems that the tensile ductility in the rolling direction increases markedly with strain rate. However, since test specimens of different geometry and different measuring techniques were used under quasi-static and dynamic conditions, the comparison of the fracture strain is not trivial. Different failure modes were found at high strain rates as shown in Fig. 10. In test #3 at e_ ¼ 179 s1 shear fracture is evident (3f ¼ 0.25), while in test #4 at e_ ¼ 1128 s1 a somewhat more ductile fracture mode occurred (3f ¼ 0.34). 3. Component tests 3.1. Experimental set-up Component tests using hardened steel projectiles (20 mm diameter, 197 g mass, 52 HRC) with blunt and ogival nose shapes

(Fig. 11) were carried out in a compressed gas-gun to reveal the alloy’s resistance to ballistic impact. The projectiles were mounted in a nine-pieced serrated sabot and launched at impact velocities just below and well above the ballistic limit velocity, i.e. the critical impact velocity, of the target. The sabot pieces were stopped by a sabot trap prior to impact. Target plates with dimension 600  600 mm2 and nominal thickness of 20 mm were clamped in a 500 mm diameter circular frame and tightened with 16 bolts. The penetration event was captured by a Photron Ultima APX-RS digital high-speed video camera operating at a constant framing rate of 50.000 fps. Initial and final velocities were measured using different laser-based optical devices (shown to be accurate to within 1–2%), as well as by the high-speed camera system. Both initial and final target deformations were measured in-situ before and after each test. More details regarding the experimental set-up and the instrumentation used during testing can be found in Børvik et al. [23]. 3.2. Experimental results A total of twelve impact tests (six with blunt projectiles and six with ogival projectiles) were conducted for the 20 mm thick AA7075-T651 plates using the experimental equipment described

600

Nom inal stress [MPa]

500 400 300 Strain rate: 179 s-1 Strain rate: 431 s-1 Strain rate: 435 s-1 Strain rate: 522 s-1 Strain rate: 944 s-1 Strain rate: 1128 s-1

200 100 0 0

0.05

0.1

0.15

0.2

Nominal strain Fig. 8. Compression specimens after fracture (a) H0 =D0 ¼ 1 and (b) H0 =D0 ¼ 2.

Fig. 9. Nominal stress–strain curves from high strain-rate tests (specimens taken parallel to the rolling direction).

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Fig. 10. Broken tensile specimens after testing at high strain rates (specimens taken parallel to the rolling direction).

above. All parameters were kept constant within each test series except for the impact velocity that varied between 180 m/s and 350 m/s. Initial (vi) and residual (vr) velocities of the projectile were measured during testing. Based on these measurements, the initial versus residual velocity curves shown in Fig. 12 were obtained. Note that all velocities represent average values based on two or more measurements. The ballistic limit velocities vbl were taken as the lowest impact velocity within each test series, since there were found to be very close to the ballistic limit (see Fig. 13). The lines through the data points were determined based on a generalization of an analytical model originally proposed by Recht and Ipson [24]

  p p 1=p vr ¼ a v i  v bl

(5)

where a and p may be considered as empirical constants and vbl is the obtained ballistic limit. Note that the Recht–Ipson analytical model, with a ¼ mp =ðmp þ mpl Þ and p ¼ 2, is only valid if the plastic deformation of the projectile during impact is negligible, which was the case in these tests. Here, mp and mpl is the mass of the projectile and plug, respectively. Both a and p were fitted to the test data using the method of least squares. Fig. 12 gives experimentally obtained ballistic limits for each projectile nose shape, together with assessed values of a and p. Even though some spread is seen in these plots, the agreement between the experimental data points and the Recht–Ipson model is in general good. Figs. 14 and 15 show typical high-speed camera images of the perforation process for blunt and ogival projectiles, respectively. The perforation is mainly due to plugging for blunt projectiles [23], and a plug with height approximately equal to the plate thickness is ejected from the target. In addition, fragmentation from the rear side of the target is seen due to the low ductility of the material. No fragments from the front side of the target are observed. For ogival projectile, the perforation process starts as ductile hole growth,

which is the dominating fracture mode for pointed-nose projectiles impacting ductile materials [25,26]. However, due to the low ductility of the material, the perforation process changes into fragmentation, and a large number of fragments are ejected from both sides of the target plate. Fig. 16 shows some pictures of the frontal and rear side cavities in the plates, while Fig. 17 shows some cross-sections of sliced plates after impact. An almost conical cavity is seen on the rear side of the target independent of projectile nose shape, and on the frontal side only after impact by ogival projectiles. The outer diameter of the craters were roughly measured, and found to be about 20 and 50 mm for the frontal and rear side of the target, respectively, when impacted by blunt projectiles, and about 30 and 50 mm when impacted by ogival projectiles. The global target deformation after impact was small. The perforation process is found to be much more brittle than normally seen during perforation of ductile alloys [27–29]. The reason for this can be related to the complex microstructure of the AA7075-T651 alloy [8,30]. The complex microstructure results in local variation in properties and strain localization to soft areas, which may lead to inter-crystalline cracking, delamination and fragmentation during impact. It should finally be noticed that more energy is required to push material aside by ductile hole growth than shearing through the plate by localized plugging, which means that the ballistic limit velocity is higher for ogival than for blunt projectiles (see Fig. 12). This has also been observed in similar tests on ductile steel plates [25,26].

4. Constitutive relation and fracture criterion A modified version of the Johnson–Cook constitutive relation (or the MJC model) was chosen in a first attempt to model the target material [20,31]. Thus, the constitutive behaviour of the alloy is assumed to be isotropic. The von Mises equivalent stress is expressed as



seq ¼ A þ B3neq Fig. 11. Geometry and dimensions (in mm) for blunt and ogival (CRH ¼ 3) projectiles.



1 þ 3_ *eq

C 

1  T *m



(6)

where 3eq is the equivalent plastic strain and A, B, n, C and m are material constants. The dimensionless plastic strain rate is given by

T. Børvik et al. / International Journal of Impact Engineering 37 (2010) 537–551

350

b

Experimental data Best fit ( a = 0.89 and p = 2.15)

300

Residual velocity [m/s]

Residual velocity [m/s]

a

250 200 150 100 50

350

543

Experimental data Best fit ( a = 0.87 and p = 2.54)

300 250 200 150 100 50

Blunt projectiles

vbl = 183.8 m/s

0 150

200

250

300

350

Ogival projectiles

vbl = 208.7 m/s

0 400

150

Initial velocity [m/s]

200

250

300

350

400

Initial velocity [m/s]

Fig. 12. Initial versus residual velocity curves for 20 mm thick AA7075-T651 plates impacted by (a) blunt and (b) ogival projectiles.

Fig. 13. High-speed camera images showing the position of the projectile after testing at the lowest impact velocity within each test series. These velocities were taken as the ballistic limits of the target: (a) blunt projectile (vi ¼ 183.8 m/s, vr ¼ 0 m/s) and (b) ogival projectile (vi ¼ 208.7 m/s, vr ¼ 0 m/s).

3_ *eq ¼ 3_ eq =3_ 0 ;

where 3_ 0 is a user-defined reference strain rate. The homologous temperature is defined as T * ¼ ðT  Tr Þ=ðTm  Tr Þ, where T is the absolute temperature, Tr is the room temperature and Tm is the melting temperature. The temperature change due to adiabatic heating is calculated as

DT ¼

Z3eq 0

c

seq d3eq rCp

(7)

where r is the material density, Cp is the specific heat and c is the Taylor–Quinney coefficient that represents the proportion of plastic work converted into heat. Fracture is modelled using a criterion proposed by Cockcroft and Latham (CL) [32]

W ¼

Z3eq

hs1 id3eq  Wcr

(8)

0

where s1 is the major principal stress, Cs1 D ¼ s1 when s1  0 and Cs1 D ¼ 0 when s1 < 0. It is seen that fracture cannot occur when there is no tensile stress operating. The critical value of W, denoted Wcr, can be determined from one simple uniaxial tensile test. It was shown by Dey et al. [33–35] that the oneparameter CL criterion gives equally good results as the fiveparameter MJC fracture criterion in LS-DYNA simulations of perforation of steel plates under various stress states using different projectile nose shapes. However, it should be noted that owing to the anisotropy of the material, the quasi-brittle behaviour of the alloy and the uncertainty in the calibration of

Fig. 14. Perforation of the 20 mm thick AA7075-T651 target plate by a 20 mm diameter, 197 g mass blunt nose projectile (vi ¼ 199.8 m/s, vr ¼ 60.8 m/s). The given times (in ms) refer to the first image taken by the high-speed camera system.

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Fig. 15. Perforation of the 20 mm thick AA7075-T651 target plate by a 20 mm diameter, 197 g mass ogival nose projectile (vi ¼ 277.7 m/s, vr ¼ 186.2 m/s). The given times (in ms) refer to the first image taken by the high-speed camera system.

the CL criterion, Wcr should not be regarded as a material characteristic. Note also that this simple one-parameter criterion cannot be used to describe all phenomena taking place in this very complex material during structural impact, and is mainly intended for design. If more details are desired, a more sophisticated fracture criterion is required. The model is coupled with an element kill algorithm that erodes the elements when W reaches its critical value Wcr. The constitutive relation and fracture criteria have been implemented in the standard version of LS-DYNA as material model 107. Based on the uniaxial tension tests, the hardening parameters A, B and n were determined by best fits to the experimental data using the method of least squares. Since the alloy exhibited insignificant rate sensitivity, the material parameter C was given a very small positive value, while 3_ 0 was taken equal to the strain rate in the quasi-static tensile tests. Owing to the lack of tensile test data at elevated temperatures, the material parameter m was set to unity, implying a linear decrease of the flow stress with increasing temperature. The fracture parameter Wcr

was calibrated by use of the uniaxial tension test, since in this R 3f case we simply have Wcr ¼ 0 sd3p . Similar results would have been obtained if stress–strain data from the notched tensile tests were used to determine Wcr. However, if data from compression tests were used the results would have been very different since the compression specimens fail in shear at a rather low macroscopic strain. As will be shown in Section 6, the local strain at failure in the compression test seems to be significantly larger than the measured macroscopic failure strain. Owing to this uncertainty, the compression test was not used for the determination of Wcr. Physical constants were given nominal values for aluminium alloys provided in the literature. The material parameters for AA7075-T651 obtained from tensile tests in the 0 direction are given in Table 2. These data will be applied in the bulk of the numerical simulations to be presented in Section 5. Hardening and fracture parameters determined from tensile tests in the 45 and 90 directions are given in Table 3. The material constants for the hardened steel projectiles can be found in Børvik et al. [20].

Fig. 16. A selection of plots showing the frontal and rear side cavities in the plates after impact by (a) blunt and (b) ogival projectiles.

T. Børvik et al. / International Journal of Impact Engineering 37 (2010) 537–551

545

Table 3 Hardening and fracture parameters for the AA7075-T651 target determined from tensile tests in the 45 and 90 directions. Parameters valid for the 45 direction

Fig. 17. Some cross-sections of sliced plates after impact by blunt and ogival projectiles.

5. Numerical simulations 5.1. Finite element models All impact tests were analysed using the explicit solver of the non-linear finite element code LS-DYNA [36]. Two different sets of material constants for the constitutive relation and fracture criterion described in Section 4 were used in the simulations. This was done to check the possible effect of anisotropy on the predictions. The first set (Set 1) is given in Table 2 and is entirely based on uniaxial tensile tests in the rolling direction. The second set (Set 2) is based on the uniaxial tension tests in the 45 direction, and thus the hardening and fracture parameters are given in Table 3. Set 2 gives lower strength and higher ductility of the material compared with Set 1 (Fig. 2). The other parameters are equal in the two sets. All simulations were run using axisymmetric and solid elements. The geometries of the targets and projectiles in the finite element models were identical to those used in the experimental tests. For 2D axisymmetric conditions, the mesh consisted of 4-node axisymmetric elements with one integration point and stiffnessbased hourglass control. Contact between the various parts was modelled using an automatic 2D penalty formulation without friction. For the targets impacted by blunt projectiles, a fixed mesh with an element size of 0.2  0.2 mm2 in the impact region was

Parameters valid for the 90 direction

A (MPa)

B (MPa)

n

Wcr (MPa)

A (MPa)

B (MPa)

n

Wcr (MPa)

426

339

0.31

292

478

414

0.38

164

chosen. This gave a total of 100 elements over the target thickness. To reduce the computational time, which is affected both by the element size and the number of elements, the mesh was coarsened towards the fully clamped boundary using transition zones. Also a coarser mesh (having an element size of 0.4  0.4 mm2) was made to study the mesh size effect for blunt projectiles. For targets impacted by ogival projectiles, r-adaptive remeshing with a predefined time interval between adaptive refinements was used. Here, a coarser mesh with an element size of about 0.3  0.3 mm2 was chosen. It is not possible to coarsen the mesh towards the boundary when remeshing is applied. For the projectiles, an element length typically between 0.1 and 1 mm was used in the various parts. These mesh sizes resulted in about 15 000 axisymmetric elements in simulations with blunt projectiles, and about 55 000 elements in simulations with ogival projectiles. An example of an axisymmetric finite element mesh used in 2D simulations of blunt projectile impact is given in Fig. 18. Some mesh size sensitivity must be expected in these simulations involving shear localization and softening, but earlier studies on steel and aluminium targets (see [23,25,26,28,35]) have shown that the chosen element sizes give reasonable results. For 3D conditions, 8-node constant-stress solid elements with one integration point and stiffness-based hourglass control were applied. Contact was now modelled using an eroding surface-tosurface algorithm available for SMP/MPP simulations. Independent of projectile nose shape, a fixed element mesh was used. Two models with different mesh sizes were generated. The fine model, having nearly 8 times more elements than the coarse model, was too time-consuming and it was only used for some mesh size sensitivity studies. To save computational time, the 3D model was coarsened towards the fully clamped boundary using tetrahedral elements in the transition zone. The element size in the impact region of the coarse model was equal to 0.5  0.5  0.8 mm3, giving 25 elements through the thickness, while only 7 elements were used over the target thickness in the global part of the plate. Even so, the coarse 3D model resulted in about 330 000 elements and 850 000 nodes. Due to the size of the numerical model, all 3D simulations were run using SMP (Shared Memory Parallel). An example of a solid element mesh used in 3D simulations is shown in Fig. 19. A limited study on the mesh size sensitivity in these simulations will be described in Section 5.2.

5.2. Numerical results Numerical results for 2D axisymmetric conditions are presented first. Based on a large number of simulations, using the finite element models presented in Section 5.1, the initial versus

Table 2 Material parameters for the AA7075-T651 target (where the hardening and fracture parameters are determined from tensile tests in the 0 direction.). Elastic constants and density

Yield stress and strain hardening

Strain-rate hardening

Temperature softening and adiabatic heating

Fracture parameter

E (GPa)

n

r (kg/m3)

A (MPa)

B (MPa)

n

3_ 0 ðs1 Þ

C

Tr (K)

Tm (K)

m

Cp (J/kg/K)

c

a (K1)

Wcr (MPa)

70

0.3

2700

520

477

0.52

5  104

0.001

293

893

1

910

0.9

2.3$105

106

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Fig. 18. 2D axisymmetric element mesh used in simulations of blunt and ogival projectile impact.

Fig. 19. 3D solid element mesh used in simulations of blunt and ogival projectile impact.

residual velocity curves shown in Fig. 20 were obtained. Since the material is rather anisotropic (see Fig. 2), it was decided to run the simulations with two different sets of material parameters (Set 1 and Set 2) to get an indication of the effect of anisotropy on the perforation resistance.

350

b

Numerical result (Set 1; vbl = 155 m/s, a = 0.84, p = 2.28) Numerical result (Set 2; vbl = 152 m/s, a = 0.80, p = 2.25)

300

Residual velocity [m/s]

Residual velocity [m/s]

a

As seen from Fig. 20, the predicted initial versus residual velocity curves are almost unaffected by the chosen parameter set when the target is impacted by blunt projectiles. The ballistic limit velocity is estimated to around 155 m/s, i.e. an underestimation of about 20% compared to the experimental data. This conservative estimate seems reasonable taking the complexity of the problem

250 200 150 100 50

350

Numerical result (Set 1; vbl = 244 m/s, a = 0.77, p = 3.52) Numerical result (Set 2; vbl = 276 m/s, a = 0.91, p = 2.32)

300 250 200 150 100

vbl = 244 m/s

50 Blunt projectiles

vbl = 152 - 155 m/s

0 150

200

250

300

Initial velocity [m/s]

350

400

vbl = 277 m/s Ogival projectiles

0 150

200

250

300

350

400

Initial velocity [m/s]

Fig. 20. Predicted initial versus residual velocity curves for 20 mm thick AA7075-T651 plates impacted by (a) blunt and (b) ogival projectiles using 2D axisymmetric elements. The lines through the data points are best fits to the numerical results.

T. Børvik et al. / International Journal of Impact Engineering 37 (2010) 537–551

547

Fig. 21. Perforation of 20 mm thick AA7075-T651 plates by (a) blunt and (b) ogival projectiles from typical simulations using 2D axisymmetric elements and material parameters based on tension tests from the rolling direction (Set 1). Plotted as fringe levels of accumulated plastic strain in the range 0 (light gray) to 0.5 (dark gray). The axisymmetric model has been reflected about a plane to better show the perforation process.

and the simplicity of the models into account. When the target is impacted by ogival projectiles, the predicted initial versus residual velocity curves seem unaffected by the parameter set at the highest impact velocities. As the impact velocity is decreased, the projectile is stopped at a ballistic limit of 277 m/s for Set 2 (based on tensile tests in the 45 direction), giving an overestimation of about 30%

350

b

Numerical result (Set 1; vbl = 185 m/s, a = 0.87, p = 2.32) Numerical result (Set 2; vbl = 195 m/s, a = 0.85, p = 2.26)

300

Residual velocity [m/s]

Residual velocity [m/s]

a

250 200 vbl = 185 m/s

150 100

compared to the experimental data. For Set 1 (based on tensile tests in the 0 direction) the ballistic limit is significantly lower and equals 244 m/s, i.e. an overestimation of about 15% compared to the experimental data. It is further seen that some data points lie below the fitted initial versus residual velocity curve for Set 1. For ogival projectiles the perforation resistance is more dependent on the

vbl = 195 m/s

50

350

Numerical result (Set 1; vbl = 271 m/s, a = 1.00, p = 2.01) Numerical result (Set 2; vbl = 269 m/s, a = 1.00, p = 2.01)

300 250 200 150 100

vbl = 269 m/s

50 Blunt projectiles

0 150

200

250

300

Initial velocity [m/s]

350

400

vbl = 271 m/s Ogival projectiles

0 150

200

250

300

350

400

Initial velocity [m/s]

Fig. 22. Predicted initial versus residual velocity curves for 20 mm thick AA7075-T651 plates by (a) blunt and (b) ogival projectiles using 3D constant-stress solid elements. The lines trough the data points are best fits to the numerical results.

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Fig. 23. Perforation of 20 mm thick AA7075-T651 plates by (a) blunt and (b) ogival projectiles from typical simulations using 3D constant-stress solid elements and material parameters based on tension tests from the rolling direction (Set 1). Plotted as fringe levels of accumulated plastic strain in the range 0 (light gray) to 0.5 (dark gray). The 3D model has been sliced through the centre to better show the perforation process.

parameter set. The perforation process was (in a similar way as for steels [25,26]) quite ductile for Set 2 (Wcr ¼ 292 MPa) and more brittle with considerable fragmentation for Set 1 (Wcr ¼ 106 MPa). Ogival projectiles compress the material in the plane of the plate and open a hole with nearly the diameter of the shank. This

350 300

Residual velocity [m/s]

b 350

Numerical result (3D - fine mesh) Numerical result (3D - coarse mesh) Numerical result (2D - coarse mesh) Fit ( a = 0.87 and p = 2.32)

250 200 150

vblc (2D) = 180 m/s

100

vblf (3D) = 181 m/s

50

vblc (3D) = 185 m/s Blunt projectiles

0 150

200

250

300

Initial velocity [m/s]

Numerical result (3D - fine mesh) Numerical result (3D - coarse mesh) Fit ( a = 1 and p = 2)

300

Residual velocity [m/s]

a

deformation mechanism is known as ductile hole growth, and is the basic assumption for the cavity-expansion theory [19]. When the impact velocity or the ductility is high, the deformation mode is dominated by triaxial compression. Thus, the cracking and fragmentation process is delayed and partly prevented. However, when

350

400

250 200 150 100

vblf = 268 m/s

50

vblc = 271 m/s Ogival projectiles

0 150

200

250

300

350

400

Initial velocity [m/s]

Fig. 24. Limited mesh size sensitivity study for (a) blunt and (b) ogival projectiles using a fine and a coarse mesh and material parameters based on tension tests from the rolling direction (Set 1).

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549

Fig. 25. Micrographs showing the deformed microstructures and failure modes in (a) uniaxial tension in the rolling direction and (b) in through-thickness compression.

the impact velocity or the ductility is low, the compression is reduced, cracking and fragmentation on the rear side of the target occurs, and the perforation resistance to the target plate drops. This was clearly seen in the simulations. For blunt projectiles, the perforation process is controlled by adiabatic shear localization, and less cracking and fragmentation take place. Fig. 21 shows some typical plots of the perforation process when 20 mm thick AA7075-T651 plates are impacted by blunt or ogival projectiles using 2D axisymmetric elements and material parameters based on tensile data in the 0 direction (Set 1). The quasibrittle behaviour during perforation seen experimentally (Figs. 14 and 15) is partly captured in the simulations. Further, if the predicted cavities in the targets after perforation are compared to the corresponding experimental cavities (Figs. 16 and 17), also these are well predicted (e.g. the spalling of the front face is only seen for ogival projectiles). These plots illustrate that the fragmentation of the target material can be reasonably well simulated using a rather simple numerical model. Numerical results for 3D conditions are presented next. Again, based on a number of simulations using the finite element models presented in Section 5.1, the initial versus residual velocity curves in Fig. 22 were constructed. Also here the two different sets of material parameters were used in the simulations. Fig. 22 shows that when the target is impacted by blunt projectiles, some spread in the initial versus residual velocity curves is obtained when the material constants are varied. For Set 1 an almost perfect fit to the experimental data is obtained. Thus, 3D simulations seem more reliable than 2D axisymmetric simulations for this particular case. For ogival projectiles the results are more similar to the 2D axisymmetric simulations. Using Set 1, an overestimation of the ballistic limit by about 30% is obtained. The ballistic limit is slightly higher for Set 2, owing to the increased ductility. Fig. 23 shows some typical plots of the perforation process when 20 mm thick AA7075-T651 plates are impacted by blunt and ogival projectiles using solid elements and material parameters obtained from the tension tests in the 0 direction (Set 1). Again, the quasibrittle behaviour during perforation seen experimentally (Figs. 14 and 15) is partly captured in the simulation. Note also that the 3D plots of the perforation process are very similar to the 2D axisymmetric plots in Fig. 21. Thus, the qualitative agreement between experimental tests, 2D axisymmetric simulations and 3D simulations is good, even though there are some quantitative deviations. The reason for this seems to be that the FE models are

not able to fully capture the quasi-brittle fracture behaviour of the alloy, especially during impact by ogival projectiles, and the predictions tend to overestimate the ballistic capacity of the target plates. Finally, a limited mesh size sensitivity study was carried out. For the axisymmetric models involving blunt projectiles, a coarse mesh with 50 elements through the thickness was used in addition to the baseline mesh with 100 elements. For the 3D models coarse and fine meshes having respectively 25 and 50 elements over the target thickness were used. Material parameters obtained from the tension tests in the 0 direction (Set 1) were adopted. The results are plotted in Fig. 24. Within the limitations of these simulations, no significant mesh size sensitivity was obtained for the 3D case. However, when comparing the ballistic limits obtained by the two meshes for axisymmetric conditions and blunt projectiles, a 16% increase in capacity was observed when coarsening the mesh. Thus, 2D axisymmetric models seem to be more mesh size sensitive than corresponding 3D models for ballistic impact. 6. Discussion and conclusions The materials tests showed that the AA7075-T651 plates are anisotropic in flow stress, plastic flow and ductility (or strain to fracture). The tensile flow stress is markedly higher in the rolling and transverse directions than in the 45 direction. On the contrary, the fracture strain obtained from the tensile tests is highest in the 45 direction and lowest in the rolling direction, where the specimens failed in shear before incipient necking. The tensile tests with smooth and notched specimens showed that the tensile ductility decreases with increasing stress triaxiality, indicating a ductile fracture mode. The through-thickness compression tests exhibited shear-dominated failure and an overall fracture strain that was at the same level as for uniaxial tension in the rolling direction. It thus seems like there is a change in fracture mode as the stress triaxiality reduces and turns negative. Similar fracture behaviour for aluminium alloys has been found by Bao and Wierzbicki [37] for aluminium alloy AA2024-T351. They proposed a fracture locus in the equivalent strain and stress triaxiality space for this alloy based on experimental and numerical results. The fracture locus has three distinct branches and possible slope discontinuities in the transition regime. They conclude that for large triaxialities void growth is the dominant failure mode, while shear failure is dominant for negative stress triaxialities. For low stress triaxialities between these two

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regimes, fracture may develop as a combination of shear and void growth modes. It is also recommended to study the recent papers by Xue and Wierzbicki on fracture initiation and propagation in ductile materials [38,39]. Owing to the strong anisotropy of the material with respect to tensile ductility and the lack of reliable data from shear tests, it was not attempted to establish a fracture locus for the AA7075-T651 alloy in the current study. However, some first results from a metallurgical study [40] are given in Fig. 25, which presents micrographs showing the fracture modes in uniaxial tension in the rolling direction and in through-thickness compression. At the macroscopic level, the fracture mode in the uniaxial tension test in the rolling direction was shear failure oriented about 45 with the tensile axis. At the microscopic level, it appears that the crack propagation is partly inter-crystalline and partly trans-crystalline to accommodate the 45 orientation of the shear failure. The intercrystalline crack growth is assumed to occur in the precipitate-free zones along the grain boundaries of the alloy, verified by the facets in the fracture surface and that the facets contain a high density of small dimples [40]. Also in the through-thickness compression test the macroscopic fracture mode was shear failure in the 45 direction with respect to the compression axis. The micrograph reveals that the crack propagates along a distinct shear band, while other shear bands not leading to fracture are found close to the main fracture surface in the deformed sample. It is further seen from the deformation of the fibrous grain structure that the shear strain carried by the shear band is large and significantly higher than the overall compressive strain in the sample, cf. Fig. 7. As already mentioned, Yang et al. [16,17] observed multiple shear bands in thick-walled 7075 cylinders under external explosive loading. Here microcracks developed within the adiabatic shear bands, and at a critical crack length catastrophic failure occurred. The impact tests on the 20 mm thick rolled AA7075-T651 plates using blunt and ogival projectiles showed that the alloy behaves like a quasi-brittle material under impact-generated loading conditions. The result is fragmentation and conical crater formation on the rear side of the target for blunt projectiles and on both sides for ogival projectiles. The reason for the different behaviour is presumably that the dominant failure modes are different, namely plugging by adiabatic shear banding for blunt projectiles and ductile hole growth for ogival projectiles. The plugging failure leads to significantly lower ballistic capacity than failure by ductile hole growth, which is similar to what was found for 12 mm thick targets made of high-strength Weldox steels with good ductility [25,26]. However, owing to the severe fragmentation for the ogival projectile, the relative difference in ballistic limits was less for AA7075-T651. It is also interesting to compare these results to similar tests with ogival projectiles in 20 mm thick AA5083-H116 aluminium plates [27]. Even though the yield stress of AA5083H116 is only about 240 MPa, or less than half the yield stress of AA7075-T651, the ballistic limit was found to be 244 m/s, or about 20% higher than the ballistic limit found in this study for AA7075T651. This is in some conflict with earlier observations [41], where it was found that strength is a more important characteristic than ductility in the perforation resistance of steel plates. The decrease in ballistic limit compared to the softer AA5083-H116 should be attributed to the quasi-brittle behaviour of AA7075-T651. Simulations with 2D axisymmetric and 3D solid models gave fracture modes similar to those observed in the experiments, but the accuracy of the predicted ballistic limits varied somewhat. The axisymmetric model underestimated the ballistic limit for blunt projectiles, while the 3D model was rather accurate. It was shown by a limited mesh sensitivity study that the axisymmetric simulations were much more sensitive to mesh size than the 3D simulations for blunt projectiles. For ogival projectiles both models overestimated the ballistic limit. This was attributed to the extensive fragmentation

observed experimentally in this case, which was clearly underestimated in the numerical simulations. The influence of using two different calibrations of the material parameters, based respectively on tensile tests in the rolling direction and the 45 direction, was rather limited for the 3D models. Under axisymmetric conditions, a somewhat strange behaviour was observed for ogival projectile when the impact velocity was close to the ballistic limit where extensive fragmentation occurred. The reason for this is attributed to the constraint put on the fragmentation and crater formation from the assumption of axisymmetric conditions. Note that when an element fails under axisymmetric conditions, a ring (‘‘doughnut’’) of material is eroded. Thus, axisymmetric models are more sensitive to element erosion than 3D models when fragmentation is possible. In the experiments, the fragmentation is not an axisymmetric process, and thus the behaviour of the targets is over-constrained in the axisymmetric simulations. The sensitivity study further showed that the 3D models seemed to exhibit little mesh size dependency for both types of projectiles. As seen in previous studies for other materials, axisymmetric models used for simulation of plugging are mesh size sensitive. The constitutive relation and the fracture criterion used in the finite element analyses are both isotropic, and thus the influence of anisotropy found in flow stress, plastic flow and ductility is neglected. Also, no attempt was made to model the shear fracture which was the dominant failure mode for negative stress triaxialities in the material tests. However, one major goal of this study has been to investigate whether simple models can be used with reasonable accuracy for computer-aided design of high-strength aluminium structures subjected to structural impact. The results show that even if the quasi-brittle behaviour of the high-strength alloy leads to fragmentation, crater formation and some delamination during impact, a reasonable estimate of the ballistic limit for blunt projectile may be obtained using these rather simple models. For ogival projectiles, on the other hand, the prediction of the ballistic limit is less accurate and non-conservative.

Acknowledgement The financial support of this work from the Structural Impact Laboratory (SIMLab), Centre for Research-based Innovation (CRI) at the Norwegian University of Science and Technology (NTNU), is gratefully acknowledged. The authors would also like to express their gratitude to Heidi Aunehaugen for her participation in the various experimental programmes.

References [1] Hatch JE. Aluminium: properties and physical metallurgy. Metals Park, Ohio: American Society for Metals; 1984. [2] Forrestal MJ, Luk VK, Rosenberg Z, Brar NS. Penetration of 7075-T651 aluminum targets with ogival-nose projectiles. Int J Solids Struct 1992;29:1729–36. [3] Vlot A. Impact loading on fibre metal laminates. Int J Impact Eng 1996;18:291–307. [4] Anderson Jr CE, Dannemann KA. Deformation and damage of two aluminum alloys from ballistic impact. In: Furnish M, Thadhani NN, Horie Y, editors. Shock compression of condensed matter-2001, vol 620. Melville, NY: American Institute of Physics; 2002. p. 1298–301. [5] Gooch WA, Burkins MS, Squillacioti RJ. Ballistic testing of commercial aluminum alloys and alternative processing techniques to increase the availability of aluminum armor. Proceedings of the 23rd International Symposium on Ballistics, Spain, 2007. [6] Demir T, Ubeyli M, Yildirim RO. Investigation on the ballistic impact behavior of various alloys against 7.62 mm armor piercing projectile. Mater Des 2008;29:2009–16. [7] Børvik T, Aunehaugen H, Hopperstad OS. Impact behaviour of the highstrength aluminium alloy AA7075-T651. Paper submitted to 9th International Conference on the Mechanical and Physical behaviour of Materials under Dynamic Loading (DYMAT2009), Brussels, Belgium; 7–11, September 2009. [8] Zinkham RE. Anisotropy and thickness effects in fracture of 7075-T6 and -T651 aluminum alloy. Eng Fract Mech 1968;1:275–89.

T. Børvik et al. / International Journal of Impact Engineering 37 (2010) 537–551 [9] Rizal S, Homma H. Dimple fracture under short pulse loading. Int J Impact Eng 2000;24:69–83. [10] Rizal S, Homma H. Effects of loading rate and stress-state on dimple fracture. Mat Sci Res Int 2000;6:123–30. [11] Rizal S, Homma H, Nazer M, Kishida E. Experimental approach to dimple fracture mechanisms under short pulse loading. Eng Fract Mech 2002;69:1377–90. [12] El-Magd E, Brodmann M. Influence of precipitates on ductile fracture of aluminium alloy AA7075 at high strain rates. Mater Sci Eng 2001;A30:143–50. [13] Loya JA, Ferna´ndez-Sa´ez J. Three-dimensional effects on the dynamic fracture determination of Al 7075-T651 using TPB specimen. Int J Solids Struct 2008;45:2203–19. [14] El-Magd E, Abouridouane M. Characterization, modelling and simulation of deformation and fracture behaviour of the light-weight wrought alloys under high strain rate loading. Int J Impact Eng 2006;32:741–58. [15] Lee W-S, Sue W-C, Lin C-F, Wu C-J. The strain rate and temperature dependence of the dynamic impact properties of 7075 alloy. J Mater Proc Tech 2000;100:116–22. [16] Yang Y, Zeng Y, Li DH, Li M. Damage and fracture mechanism of aluminium alloy thick-walled cylinder under external explosive loading. Mat Sci Eng 2008;A49:378–84. [17] Yang Y, Zeng Y, Gao ZW. Numerical and experimental studies of self-organization of shear bands in 7075 aluminium alloy. Mat Sci Eng 2008;A49:291–302. [18] Clausen AH, Børvik T, Hopperstad OS, Benallal A. Flow and fracture characteristics of aluminium alloy AA5083-H116 as function of strain rate, temperature and triaxiality. Mater Sci Eng 2004;A36:260–72. [19] Børvik T, Forrestal MJ, Hopperstad OS, Warren T, Langseth M. Perforation of AA5083-H116 aluminium plates with conical-nose steel projectiles – calculations. Int J Impact Eng 2009;36:426–37. [20] Børvik T, Hopperstad OS, Berstad T, Langseth M. A computational model of viscoplasticity and ductile damage for impact and penetration. Eur J Mech – A/ Solids 2001;5:685–712. [21] Hill R. The mathematical theory of plasticity. London: Oxford University Press; 1950. [22] Clausen AH, Auestad T, Berstad T, Børvik T, Langseth M. High-temperature tests on aluminium in a split-Hopkinson bar – experimental set-up and numerical predictions. J De Physique IV 2006;134:603–8. [23] Børvik T, Hopperstad OS, Langseth M, Malo KA. Effect of target thickness in blunt projectile penetration of Weldox 460 E steel plates. Int J Impact Eng 2003;28:413–64. [24] Recht RF, Ipson TW. Ballistic perforation dynamics. J Appl Mech 1963;30:384–90. [25] Børvik T, Langseth M, Hopperstad OS, Malo KA. Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with blunt, hemispherical and conical noses, part I: experimental study. Int J Impact Eng 2002;27:19–35.

551

[26] Dey S, Børvik T, Hopperstad OS, Leinum JR, Langseth M. The effect of target strength on the perforation of steel plates using three different projectile nose shapes. Int J Impact Eng 2004;30:1005–38. [27] Børvik T, Clausen AH, Hopperstad OS, Langseth M. Perforation of AA5083H116 aluminium plates with conical-nose steel projectiles – experimental study. Int J Impact Eng 2004;30:367–84. [28] Børvik T, Clausen AH, Eriksson M, Berstad T, Hopperstad OS, Langseth M. Experimental and numerical study on the perforation of AA6005-T6 panels. Int J Impact Eng 2005;32:35–64. [29] Børvik T, Forrestal MJ, Warren TL. Perforation of 5083-H116 aluminum armor plates with ogive-nose rods and 7.62 mm APM2 bullets. Exp Mechanics, in press, doi:10.1007/s11340-009-9262-5. [30] Unwin PNT, Lorimer GW, Nicholson RB. The origin of the grain boundary precipitate free zone. Acta Metallurgica 1969;17:1363–77. [31] Johnson GR, Cook WH. A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In: Seventh International Symposium on Ballistics. The Netherlands: The Hague. p. 1–7; 1983. [32] Cockcroft MG, Latham DJ. Ductility and workability of metals. J Inst Metals 1968;96:33–9. [33] Dey S. High-strength steel plates subjected to projectile impact. Ph.D.-thesis 2004:38 (ISBN 82-471-6282-2). Department of Structural Engineering, Norwegian University of Science and Technology, Norway, 2004. [34] Dey S, Børvik T, Hopperstad OS, Langseth M. On the influence of fracture criterion in projectile impact of steel plates. Comput Mater Sci 2006; 38:176–91. [35] Dey S, Børvik T, Teng X, Wierzbicki T, Hopperstad OS. On the ballistic resistance of double-layered steel plates: an experimental and numerical investigation. Int J Solids Struct 2007;44:6701–23. [36] LSTC. LS-DYNA keyword user’s manual, version 971. Livermore, California, USA: Livermore Software Technology Corporation; May, 2007. [37] Bao Y, Wierzbicki T. On fracture locus in the equivalent strain and stress triaxiality space. Int J Mech Sci 2004;46:81–98. [38] Xue L, Wierzbicki T. Ductile fracture and propagation modeling using damage plasticity theory. Eng Fract Mech 2008;75:3276–93. [39] Xue L, Wierzbicki T. Numerical simulation of fracture mode transition in ductile plates. Int J Solids Struct 2009;46:1423–35. [40] Pedersen KO, Hopperstad OS, Børvik T. Fracture mechanisms in aluminium alloy AA7075-T651 under quasi-static and dynamic loading conditions, in press. [41] Børvik T, Dey S, Clausen AH. Perforation resistance of five different highstrength steel plates subjected to small-arms projectiles. Int J Impact Eng 2009;36:948–64.