The quasi-static and blast loading response of lattice ...

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International Journal of Impact Engineering 35 (2008) 795–810 www.elsevier.com/locate/ijimpeng

The quasi-static and blast loading response of lattice structures S. McKowna,, Y. Shena, W.K. Brookesa, C.J. Sutcliffea, W.J. Cantwella, G.S. Langdonb, G.N. Nurickb, M.D. Theobaldb a Department of Engineering, University of Liverpool, Liverpool L69 3BX, UK Blast Impact and Survivability Research Unit (BISRU), Department of Mechanical Engineering, University of Cape Town, Rondebosch 7701, South Africa

b

Received 30 March 2007; received in revised form 8 October 2007; accepted 9 October 2007 Available online 7 November 2007

Abstract A range of metallic lattice structures have been manufactured using the selective laser melting (SLM) rapid manufacturing technique. The lattice structures were based on [7451] and [01, 7451], unit-cell topologies. Initially, the structures were loaded in compression to investigate their progressive collapse behaviour and associated failure mechanisms. Tests were then undertaken at crosshead displacement rates up to 3 m/s in order to characterise the rate-dependent properties of these architectures. A series of blast tests were then undertaken on a ballistic pendulum in order to investigate the behaviour of lattice structures under these extreme loading conditions. During the compression tests, a buckling mode of failure was observed in the [01, 7451] lattice structures, whereas a stable progressive mode of collapse was evident in the [7451] structures. The yield stress of the lattice structures exhibited moderate rate sensitivity, increasing by up to 20% over the range of conditions considered. The blast resistance of the lattice structures increased with increasing yield stress and has been shown to be related to the structures specific energy-absorbing characteristics. An examination of the lattice samples indicated that the collapse mechanisms were similar following both the compression and blast tests. r 2007 Published by Elsevier Ltd. Keywords: Lattice; Rapid manufacturing; Unit cell; Blast; Strain rate

1. Introduction Currently, there is a growing interest in the potential offered by lightweight metallic foams for use in highperformance load-bearing applications. Initial testing has shown that metal foams offer a wide range of attributes including, superior sound-proofing characteristics, electromagnetic wave shielding, low thermal conductivity, low toxicity and excellent toughness. Recently, a range of aluminium foams have been developed that retain many key mechanical properties at temperatures up to 300 1C. The introduction of these metallic foams opens up many exciting new avenues for developing novel lightweight sandwich structures for use in impact and blast-resistant applications [1]. One of the major drawbacks of these systems is the gross irregularity of the cell structure within Corresponding author.

E-mail address: [email protected] (S. McKown). 0734-743X/$ - see front matter r 2007 Published by Elsevier Ltd. doi:10.1016/j.ijimpeng.2007.10.005

the foamed material. Inhomogeneity within the cell architecture likely leads to overly conservative design criteria, rendering many such systems unsuitable for use in advanced aerospace structures. Recently, a number of workers have shown that it is possible to design core architectures that offer greater strength and stiffness to weight ratios than those offered by traditional foamed materials [2–7]. Deshpande and Fleck [6] have shown that the stiffness and strength of random foams (which undergo a bending-dominated mode of deformation when loaded) scale as r2 and r3/2, respectively, where r is the relative density of the foam. In contrast, the stiffness and strength of structures whose cell walls deform in tension or compression, exhibit a linear dependence on density and offer vastly superior mechanical properties. Deshpande and Fleck [6] used an investment-casting procedure to manufacture sandwich structures based on a truss-core design with both triangulated and solid faces and showed that such structures compete very favourably with

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S. McKown et al. / International Journal of Impact Engineering 35 (2008) 795–810

their aluminium foam counterparts. Deshpande et al. [5] used a combination of experimental and theoretical techniques to study the properties of the stretchingdominated octet-truss lattice structure. The octet-truss structures were manufactured by stacking layers of prefabricated triangulated layers and tetrahedral cores together to build up a three-dimensional core. In spite of problems encountered during manufacture, the authors concluded that octet-truss type structures offer enormous potential. Zok et al. [2] analysed the mechanical behaviour of lightweight sandwich structures based on metallic textile cores and showed that ‘angle-ply’ cores whose members are oriented at [7451] offer a near-optimal configuration under bending, compression and shear loading. Wang et al. [4] studied the performance of truss panels based on the 3D Kagome´ core. The authors manufactured the truss panels, with 1.25 mm diameter members, from a Cu–2% Be alloy using an investment-casting process. The quasi-static behaviour of metallic truss panels with strut lengths of the order 101 mm have also been investigated by several authors namely: Chiras et al. [3], Wicks and Hutchinson [8], Sypeck and Wadley [9], Zok et al. [10], Rathbun et al. [11]. Tetragonal and pyramidal truss-core topologies were investigated numerically and through experiments, showing good compressive strength at low weights when compared to metallic honeycombs for example. This can give the truss panels an advantage especially when multifunctionality is a design goal. As for the dynamic behaviour of metallic truss type structures, Fleck and Deshpande [12], Radford et al. [13], Lee et al. [14] and Vaughn et al. [15] all identified dynamic deformation modes that differ significantly from the quasistatic behaviour. For example, Lee et al. [14] found that uniaxial compression loading of pyramidal truss cores produced a peak quasi-static stress of 4 MPa. This value increased by one and a half times and three times with increasing nominal strain rates of 103, 102–103 and 104 s1, respectively. The failure mode up to strain rates of 103 s1 was characterised by plastic buckling of the pyramidal trusses, with the formation of one or two plastic hinges. At strain rates beyond 104 s1, a distinct change in behaviour was observed, the deformation became localised and truss members kink to maintain compatibility with the advancing face sheet. Numerical analysis of the high-rate compression tests suggested this unique failure mode plays a more significant role than the strain rate effects in contribution to energy absorption. The evidence all pointed to microinertia having a dominant role in determining peak nominal stresses, with inertial stabilisation against buckling arising from a coupling between the axial inertia and lateral buckling of the pyramidal truss members. They conclude that the material strain rate-hardening contribution to total energy is as significant as the failure mode contribution at strains in the post-peak stress region. Vaughn et al. [15] give a detailed study of inertial stabilisation against buckling in relation to the dynamic crush resistance of all-metal truss-core sandwich structures.

They conclude that dynamic effects give rise to substantial improvements in the energy absorption capability of the truss core. Lee et al. [16] have studied a woven stainless steel textile core material loaded in compression at strain rates in the range 103–104 s1. The unit-cell size for these structures is approximately 3.24 mm along the diagonally arranged steel wires. They report the strength to have a moderate dependence on deformation rate, with a failure mode transition occurring beyond loading rates of 500 s1. The development of a shock front at high strain rates increases the significance of inertial-induced buckling and bending of struts which accounts for the failure mode transition. From these observations, they suggest the textile core behaviour falls in between that of an open-cell aluminium foam and the pyramidal truss-core structures investigated by Lee et al. [14]. So far all of the core architectures discussed involve a tooled manufacturing approach, usually with multiple steps to arrive at the final sandwich structure. Significant advances have been made recently in the process of selective laser melting (SLM), a layered manufacturing technique, by which parts of high complexity can be built from metal powder in relatively short timescales. The system uses a high beam quality fibre laser to selectively melt powder particles to form a solid material. The powder can be any material provided that it meets the following requirements; it can be melted by laser heating, the powder has an average size below 40 mm and it offers suitable flow characteristics. Stainless steel, titanium and nickel-based super-alloys can all be used. SLM can be used to manufacture lattice structures that were hitherto virtually impossible to manufacture. The behaviour of 316L stainless steel macroscopic lattice structures with submillimetre strand dimensions produced by the SLM process are reported in this paper, firstly, under quasistatic compression loads, followed by the impulsive loading response of the lattices, induced by detonating plastic explosive on the face of an attached mass. 2. Experimental procedure Six lattice structures based on stainless steel 316L were investigated in this research programme. The lattice structures were manufactured using SLM at the University of Liverpool [17]. The SLM technique was used to manufacture cube-shaped samples with an edge length of approximately 20 mm. Initially, tests were conducted on a pillar-octahedral or [01, 7451] structure, Fig. 1a, based on an octahedral unit cell with additional pillar strands along the vertical edges of the unit cell. The second geometry is referred to as octahedral or alternatively [7451], Fig. 1b, and contains a node located at the centre of a cube from which all the strands radiate out to the corners of the cube. The [01, 7451] unit cell offers a higher density than its [7451] counterpart due to the presence of the vertical pillars. In order to investigate the effect of lattice density on

ARTICLE IN PRESS S. McKown et al. / International Journal of Impact Engineering 35 (2008) 795–810

mechanical properties, two different unit-cell volumes were investigated for each lattice structure. The edge length of the unit cell in the low-density systems was 2.5 mm and in the high-density systems it was 1.5 mm. For the lattice configuration with the higher unit-cell volume (2.53 mm3), two different strand diameters were also used: 250 mm

797

(A–D) and 200 mm (E–F). Table 1 summarises the geometrical properties of the structures based on the [01, 7451] unit-cell geometry (lattices A, B and E) and those based on the [7451] unit-cell geometry (lattices C, D and F). The relative density (r*) of the lattice structure is defined here as the ratio of the measured lattice density (r) to the density of the stainless steel (rs), taken here to be 8000 kg/m3. The cross-sections of the strands in the lattice structures were elliptical in shape as shown in the scanning electron micrograph presented in Fig. 2. One series of specimens was used to investigate the quasi-static compression behaviour and a second series was used to investigate the response of the lattices to blast loading. 2.1. Compression tests on the lattice structures

Fig. 1. Schematic of the unit-cell geometry of the lattices. (a) The [01, 745] pillar-octahedral lattice (b) the [7451] octahedral lattice.

Initial compression tests were conducted on lattice structures A–D on an Instron 4024 universal test machine. A crosshead displacement rate of 0.5 mm/min. was adopted in the elastic region and 1 mm/min in the plateau and densification regions of the load–displacement curves. Load was recorded by the load cell and an extensometer attached to the upper- and lower-compression platens recorded the displacement at a rate of 10 points per second. The [01, 7451] lattice structures were loaded in the direction of the vertical pillars, i.e., in the 01 direction. Photographs were taken at regular intervals during the compression tests to identify the localised damage mechanisms that occur prior to global failure of the lattice structures. The influence of strain rate on the mechanical properties of the lattice structures was investigated by conducting tests on lattice structures E and F. Although lattices E and F, with 200 mm strand diameters, have lower densities, it is anticipated that their rate sensitivity will reflect that of the four original structures, with 250 mm strand diameters. Tests were conducted at crosshead displacement rates of 1, 10, 100 and 500 mm/min on an Instron 4024 universal test machine. The loading arrangement is shown in Fig. 3. Impact tests were conducted on the instrumented

Table 1 Unit-cell and lattice properties Load type

Lattice ID

Unit-cell geometry

Unit-cell volume (mm3)

Relative density r* (%)

Avg. strand diameter (mm)

Lattice dimensions approx. (mm)

Quasi-static compression

A

[01, 7451]

1.53

16.6

0.25

20 20 20

B C D E F

[01, 7451] [7451] [7451] [01, 7451] [7451]

2.53 2.53 1.53 2.53 2.53

6.2 5.3 12.8 3.3 2.9

0.25 0.25 0.25 0.20 0.20

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

A B C D

[01, 7451] [01, 7451] [7451] [7451]

1.53 2.53 2.53 1.53

11.6 5.4 4.6 10.3

0.25 0.25 0.25 0.25

20 20 16.5 20 20 16.5 20 20 16.5 20 20 16.5

Blast



Fig. 3. Photograph of arrangement used for compression testing the lattice structures.

Fig. 2. SEM photographs of an individual unit-cell strand (cell edge) showing (a) the elliptical cross-section and (b) the layered nature along the strand length.

drop-weight impact tower shown in Fig. 4. Here an instrumented carriage was released from heights of up to 300 mm in order to generate increased levels of crush within the samples. The variation of load with time during the impact tests was measured using a piezo-electric loadcell positioned directly underneath the base on the test rig. The velocity of the impactor during the impact event was measured using a laser-doppler velocimeter. The velocity versus time traces from the velocimeter were recorded by a dedicated computer and subsequently integrated to yield

Fig. 4. The drop-weight rig used for characterising the dynamic properties of the lattice structures.

displacement versus time traces. After testing, a number of lattice structures were examined in a Hitachi S-2460N scanning electron microscope. 2.2. Blast loading tests Blast testing was conducted on a ballistic pendulum at the University of Cape Town. A schematic of the target arrangement is shown in Fig. 5. The lattice structures were


799

Rigid back plate

Lattice structure

Polystyrene pad

Cylindrical mass Adhesive foam tape

Pendulum

Fig. 5. Schematic of the experimental arrangement for blast testing.

Table 2 Summary of the blast conditions used during testing Specimen type

Lattice ID

PE4 (g)

Impulse (N s)

Impulse/disc mass (Initial velocity) (m/s)

(A) Pillar-octahedral high density [01, 7451]

A1

1.0

1.63

18.9

497

A2 A3 A4 A5 A6

4.0 2.0 3.0 2.5 5.0

5.71 3.24 4.67 3.9 5.69

66.2 37.6 54.2 47.2 68.9

1745 990 1425 1245 1815

B1

3.0

4.71

54.6

1435

B2 B3 B4 B5 B6

2.0 1.5 1.75 1.6 2.5

3.14 2.19 3.29 2.86 4.05

36.4 25.4 38.2 34.6 49.0

960 670 1005 910 1290

C1

1.5

2.38

27.6

725

C2 C3 C4 C5 C6 C7

1.75 1.25 1.6 1.0 1.4 2.0

3.14 1.83 2.36 1.41 2.72 3.60

36.4 21.2 27.4 17.1 32.9 43.6

960 560 720 450 915 1150

D1

2.0

3.39

39.3

1035

D2 D3 D4 D5 D6

2.2 1.6 1.0 2.0 3.0

3.43 2.32 1.88 4.46 4.72

39.8 26.9 22.8 54.0 57.1

1045 710 600 1420 1505

(B) Pillar-octahedral low density [01, 7451]

(C) Octahedral low density [7451]

(D) Octahedral high density [7451]

mounted on the ballistic pendulum using adhesive foam tape. Impulsive loading was created by detonating plastic explosive in a similar fashion to that used in previous

Nominal average strain rate (s1)

experimental investigations [18–20]. The impulse applied to the lattice structure was measured from the amplitude of the pendulum swing. Prior to detonation, a cylindrical steel



plate with a mass of 86.2 g for the first set of blast results and 82.6 g for the second set was fixed on the lattice structure using foam tape. These masses were chosen based on experience from previous investigations on tube buckling [20]. A 20-mm diameter disc of PE4 explosive was distributed evenly over the face of the polystyrene pad centred on the striking mass. After detonation, the free mass was accelerated in the axial direction causing the lattice to crush uniformly. The mass of explosive, and therefore the applied impulse, was varied by changing the thickness of the explosive disc. A total of 25blast tests were performed on the four different lattice structures and the results are given in Table 2. Since the mass of the cylindrical steel plate remains constant for a given data set, the initial velocity of the mass is varied with impulse, which produces a varying degree of crush of the lattice structures. The column, defined as impulse divided by steel plate mass, in Table 2 gives the initial velocity of the steel plate. Thus the dynamic performance of the lattice structures can be characterised in terms of the impulse knowing the steel plate mass is constant, or in terms of the initial velocity of the plate by dividing impulse by plate mass. The nominal average strain rates of magnitude 103 s1 given in Table 2 are calculated from the initial velocity of the steel plate and the lattice depth. 3. Results 3.1. Quasi-static compression Typical engineering stress–strain curves following compression tests on the lattice structures are shown in Fig. 6. The lattices exhibit responses typical of those observed following compression tests on cellular structures; following the relatively sharp rise in the stress–strain curve in the elastic region, a stress plateau region extends up to the onset of densification. The relatively high degree of linearity in the elastic region suggests localised yielding of the strands (cell edges) is not as significant as that observed in metal foams [21,22]. This is likely to be a result of the presence of regular hierarchical structures in the repeating unit cells. This is in contrast with metal foams that exhibit a distribution of cell sizes and a relatively large number of defects, which facilitate localised buckling and yielding of the cell edges at low and intermediate stress levels. A small non-linear region was observed upon initial loading of some of the specimens, for example, lattice A in Fig. 6a below 2 MPa. This is attributed to a small amount of distortion of the lattice strands as they were cut from the substrate following the manufacturing process. The distorted strands at this interface bend and possibly yield upon application of the compressive load. The compressive properties of the lattice structures are reported in Table 3. From the table, it is clear that the [01, 7451] lattices offer a significantly higher modulus than the [7451] lattices, this being due to the presence of the vertical

pillars in the [01, 7451] structures. Due to the lack of a pronounced yield stress in the [7451] lattice structures C and D (Fig. 6), the yield stress was determined at an offset strain of 5%. The yield point of the [01, 7451] system was taken as the initial peak stress. The yield stress of the higher-density [01, 7451] lattice was approximately 2.5 times greater than its [7451] counterpart and the yield stress of the lower-density [01, 7451] lattice was approximately 3.5 times that of the corresponding [7451] system. Good repeatability of the compressive yield stress for lattices A–D is indicated by the standard deviations given in Table 3. The transition from the elastic to the plastic region in the stress–strain curves involves fundamentally different deformation mechanisms for the [7451] and [01, 7451] lattices. The [7451] system exhibits a smooth stress–strain trace, shown in Fig. 6 for lattices C and D, with no discernable yield point and a continuously increasing stress with increasing strain. In contrast, the [01, 7451] lattice exhibits an initial peak yield point associated with plastic bending of the vertical strands followed by global shear failure triggered by localised buckling instabilities in the 01 strands. A drop in stress occurs at this point, with the lower-density structure, lattice B, showing a more pronounced upper and lower yield compared to the higherdensity structure, lattice A. The stress–strain responses of the lattice structures in Fig. 6 were correlated with images taken during the compression tests, Figs. 7 and 8. The [7451] lattice, Fig. 7, does not show any visible signs of local failure, rather, global deformation progresses in a stable manner, resulting in the smooth stress–strain traces reported earlier in Fig. 6 (lattices C and D). The [01, 7451] lattice in Fig. 8 clearly shows the development of localised buckling of the pillar strands. The localisation grows until a band of buckled pillar strands propagates diagonally at 451 across the sample after a crosshead displacement of approximately 1 mm, this coinciding with the transition from the elastic to the plastic region at a strain of approximately 5%, Fig. 6 (lattices A and B). Up to approximately 6 mm displacement in the load–displacement trace in Fig. 8 the lattice material on either side of the diagonal band of buckled strands slip relative to one another, similar to the behaviour of a shear band in a solid material. Beyond 9 mm displacement, the band becomes lost in the high level of crush. To characterise the effect of the density of the lattice structures on their stiffness and the yield properties, these mechanical properties were plotted against the relative density as shown in Fig. 9, and compared with the behaviour of an ideal open-cell foam. Here the relative stiffness, Fig. 9a, and relative yield strength, Fig. 9b, of the lattices are plotted against the relative density. The opencell foam model, as described by Ashby et al. [22], in which the cell edges initially deform by bending, is given by 2 E r ¼ (1) Es rs


801

40 Lattice A Lattice D

Engineering stress (MPa)

35 30 25 20 15 10 5 0 0.1

0

0.2

0.3

0.4

0.5

0.6

0.7

Engineering strain 8 Lattice B Lattice C

Engineering stress (MPa)

7 6 5 4

Initial (Upper) Yield

3

Lower Yield

2 1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Engineering strain Fig. 6. Typical quasi-static stress–strain curves for (a) the denser lattices A and D (relative density 16% and 13%, respectively) and (b) the lower-density lattices B and C (relative density 6.2% and 5.3%, respectively).

Table 3 Averaged mechanical property results for the lattice quasi-static compression tests. Average strand diameter is 250 mm Lattice Unit-cell ID geometry [01, 7451] [01, 7451] [7451] [7451]

A B C D a

Density r (mg/m3)

Elastic modulus E (GPa)

Yield stress sy Poisson (MPa) ratio u

Energy absorbed up to 50% strain (J)

Normaliseda energy absorbed up to 50% strain (kJ/kg)

1.33 0.50 0.43 1.03

2.70 0.45 0.05 0.44

15.070.7 3.2570.2 0.9270.06 6.0770.3

68 9.6 4.6 31.6

6.4 2.6 1.4 4.0

0.38 0.41 0.35 0.38

Normalised with respect to the mass of the lattice block.

and the relative yield stress is given bu 3=2 s r ¼ 0:3 . ss rs

(2)

Following tests on an open-cell aluminium foam, Gibson [23] showed that Eqs. (1) and (2) could be successfully used to characterise the mechanical properties of such materials. The relative moduli of lattices A and B based on the [01,



Fig. 7. The quasi-static crush behaviour of the [7451] lattice. The lower-density lattice C (relative density r* of 5.5%) is shown here at increasing levels of platen displacements.

Fig. 8. The quasi-static crush behaviour of the [01, 7451] lattice. The lower-density lattice B (relative density r* of 6.4%) is shown here at increasing levels of platen displacements. A global slip plane was formed from the growth of localised buckling of [01] strands as indicated between the dashed lines at 1 mm displacement.


803

1

Relative modulus (E / Es )

0.1 [0°,+/-45°] 0.01 [0°,+/-45°]

0.001

[+/-45°]

(eqn 1) y = x2

[+/-45°]

0.0001 y = 0.93x2.20 y = 0.95x2.84 0.00001 0.01

Lattice A Lattice B Lattice E Lattice C Lattice D Lattice F Steel 316L

0.1

1

Relative density (ρ*) 1

[0°,+/-45°]

Relative yield (σ / σs)

0.1

(eqn 2) y = 0.3x1.5 [0°,+/-45°] 0.01

[+/-45°] Lattice A Lattice B Lattice E Lattice C Lattice D Lattice F Steel 316L

y = 1.01x1.52 [+/-45°] 0.001 y = 1.05x1.85 0.0001 0.01

0.1 Relative density (ρ

1 *)

Fig. 9. The variation of (a) relative modulus and (b) relative yield stress with relative density of the lattices. The dotted lines in (a) and (b) represent Eqs. (1) and (2), respectively.

7451] unit-cell geometry lie close to the open-cell foam model, suggesting the modulus of the lattice structures increases with the square of the relative density. The relative modulus values of lattices C and D, i.e., the [7451] structures, fall below the open-cell foam model. The absence of axial strands in the octahedral-shaped unit cell means that the structure offers little resistance when loaded in compression. This results in global deformation with little, if any, local failure. The normalised energy per unit mass absorbed up to 50% strain during quasi-static compression of the lattices is given in Table 3. A strain limit of 50% was chosen to avoid including the energy absorption associated with a sharp rise in stress at densification. The effect of a change in the unit-cell geometry results in the [01, 7451] geometry

absorbing approximately 86% more energy than the [7451] geometry in the case of the lower-density lattices B and C, respectively. Similarly, in the case of the higherdensity lattices A and D, the [01, 7451] geometry absorbs approximately 60% more energy than the [7451] geometry. The addition of vertical [01] pillars and their associated increase in the unit-cell mechanical properties (compressive yield strength) must account for this increase in absorbed energy since the slight change in mass associated with a change in geometry has been accounted for by normalising the energy absorption with respect to the lattice mass. If the slight change in mass associated with a change in unit-cell geometry is included in the energy absorption analysis, then from Table 3, there is a 115% increase in energy absorption in the case of the higher-density lattices


804

A and D. Similarly, in the case of the low-density lattices B and C, there is a 109% increase in energy absorption. The increase in energy absorption due to an increase in the mass associated with the change in the unit-cell geometry is therefore 55% for the high-density lattices A and D, and 23% for the low-density lattices B and C. The geometry change and associated increase in mass are thus of a comparable magnitude (60%/55%) in contributing to the increase in energy absorption for the high-density lattices A and D. The lower-density lattices B and C on the other hand show the change in geometry is approximately 4 times (86%/23%) more effective in contributing to the increase in energy absorption than the associated increase in unit-cell mass. This analysis suggests just one of many optimising approaches that can be taken with the lattice structures. 3.2. Dynamic loading effects The influence of dynamic loading, up to nominal strain rates of 150 s1, on the compressive properties of the lattice structures is shown in Fig. 10. The data for the [01, 7451] geometry, lattice E, suggest that the compression strength of this lattice increases steadily with loading rate, with the value at 3 m/s (150 s1) being approximately 25% higher than its quasi-static value. Previous work on plain stainless steel samples has shown that this material does exhibit strain rate sensitivity with the yield strength increasing with strain rate [24,25]. Langdon and Schleyer [24] tested the same stainless steel as that tested here and showed that its tensile yield behaviour is moderately rate-sensitive. The data for the [7451] geometry also exhibit a similar 25% increase in compression strength over the range of rates considered. Fig. 11 shows scanning electron micrographs of [7451] and [01, 7451] lattice structures following compression tests at crosshead displacement rates between 0.2 and

100 mm/min (104 and 102 s1). The micrograph of the partially compressed [7451] lattice in Fig. 11a (i) indicates that the ligaments have rotated around the junction point during the compression test, suggesting that much of the plastic deformation has occurred at these cross-over points within the structure. At higher levels of compression, the lattice collapses completely and the ligaments stack up one on top of the other, Fig. 11a (ii). Fig. 11b shows micrographs of the [01, 7451] lattice following tests at crosshead displacement rates of 1 and 10 mm/min (104 and 103 s1). Here, it is clear that the 451 ligaments have collapsed onto each other as observed in the [7451] lattice. Closer examination indicates that the vertical pillars have buckled leaving folded ligaments that protrude from the structure. Fig. 11c shows a region of ductile tearing in a ligament on the edge of one of the samples. However, it should be noted that very few ligaments had fractured during the compression process. 3.3. Blast loading Photographs of selected blast-loaded lattices are shown in Figs. 12 and 13. The asymmetry of the crushing of some of the samples was a result of the inherent difficulty in producing a perfectly incident blast wave impinging upon the specimen. However the characteristic crush behaviour and failure modes are still discernable from the results of the blast testing of the four lattices. The percentage crush of the lattices is plotted against the impulse (with constant steel face plate mass) transferred to the lattice from the blast load in Figs. 14 and 15. The percentage crush was determined from the average of the four edge lengths following testing. The higher-density [01, 7451] structure, lattice A, failed as a results of shear band propagation at an angle of approximately 451 to the loading direction. The shear

2.0 1.8

[0º, +/-45º] [+/-45º]

Yield stress (MPa)

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0001

0.001

0.01

0.1

1

10

100

1000

Average nominal strain-rate (1/s) Fig. 10. The variation of yield stress with crosshead displacement rate for the [01, 7451] pillar-octahedral lattice E, and the [7451] octahedral lattice F structures.


805

Fig. 11. (a) Scanning electron micrographs of the [7451] lattice F tested at a crosshead displacement rate of 0.2 mm/min (104 s1). (b) Scanning electron micrographs of the [01, 7451] lattice E tested at crosshead displacement rates of (i) 1 mm/min (104 s1) and (ii) 10 mm/min (103 s1). (c) Scanning electron micrographs showing the fracture of a ligament in the [01, 7451] lattice E tested at crosshead displacement rate of 10 mm/min (103 s1).

bands were initiated by buckling failure of the vertical pillars at the top edge of the samples (Figs. 12a and b). This is consistent with the failure mode observed during the quasi-static tests. The lower-density [01, 7451] lattice B also showed evidence of buckling of the pillars, although this is difficult to visualise due to the high degree of crush in many of these specimens, Figs. 12c and d. In contrast, the octahedral lattices [7451] C and D both showed a

concertina-type collapse with no shear band formation (Figs. 13a–d), this again being similar to the quasi-static behaviour. The presence of the vertical pillars allows the [01, 7451] lattices to withstand a higher impulse for a given level of crush. The [7451] lattices undergo global stable progressive crushing with no localised buckling of the pillars. The post-test analysis of the blast-loaded samples therefore



Fig. 12. Blast-loaded specimens based on the [01, 7451] pillar-octahedral unit-cell geometry: (a) specimen A3 (approximate 12% crush); (b) specimen A4 (approximate 35% crush); (c) specimen B3 (approximate 8% crush) and (d) specimen B4 (approximate 55% crush).

Fig. 13. Blast-loaded specimens based on the [745o] octahedral unit-cell geometry: (a) specimen C3 (approximate 10% crush); (b) specimen C1 % (approximate 50% crush); (c) specimen D4 (approximate 0% crush) and (d) specimen D1 (approximate 45% crush).


0.9 Impulse (Ns) for 50% crush of lattice

6

Lattice A Lattice B

0.8 0.7 0.6 Percent Crush

807

0.5 0.4 0.3 0.2 0.1

[0°,+/-45°] (1.33)

5 4

[+/-45°] (1.03)

3

[0°,+/-45°] (0.5)

Lattice A Lattice B Lattice C Lattice D

[+/-45°] (0.43)

2 1 0

0 0

1

2

3

4

5

0

6

4

2

6

8

10

12

14

16

Quasi-static Yield strength (MPa)

Impulse (Ns) Fig. 14. The variation of the degree of crush versus impulse for lattice structures A and B.

Fig. 16. Relationship between the quasi-static compressive yield and the measured impulse from the blast load to crush the lattices by 50%. The numbers in parenthesis indicate the lattice density in mg/m3.

0.9 6 Impulse (Ns) for 50% crush of lattice

Lattice C Lattice D

0.8

Percent Crush

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3 Impulse (Ns)

4

5

6

Fig. 15. The variation of the degree of crush versus impulse following for lattice structures C and D.

indicates that the crush behaviour under these conditions produces the same deformation mechanisms as the quasistatic results. The relationship between the quasi-static yield stress and blast resistance of the lattices is shown in Fig. 16. Here, the impulse (with constant steel face plate mass) measured to compress the lattices by 50% increases as a function of the quasi-static yield stress (and density). The relationship is close to linear over the range of lattice densities tested in this study, and shows that an increase in quasi-static yield stress gives a corresponding increase in blast energy absorption of the lattice. Fig. 17 shows the variation of the impulse to achieve 50% crush with the specific energyabsorbing capacity of the lattice structure. Once again there is a linear relationship between these two, suggesting that the quasi-static properties can be used to characterise the blast resistance of these lattice structures. It is interesting to note that both families of lattice structure

5.5

[0°,+/-45°] (0.9)

5 4.5 4

[+/-45°] (0.82)

3.5

Lattice A Lattice B Lattice C Lattice D

[0°,+/-45°] (0.46)

3 [+/-45°] (0.35)

2.5 2 1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

Specific energy absorption (kJ/kg) Fig. 17. Relationship between the specific energy absorption under quasistatic conditions and the measured impulse from the blast load to crush the lattices by 50%.

(i.e., the [01, 7451] and [7451] lattices) fall on one trendline. From the measured impulse values in Table 2 and the free mass value of the ballistic pendulum, it is evident that the nominal average strain rates, _, are in the range 450–1815 s1, approximately 106 times greater than the strain rates produced during the quasi-static crush tests. It is interesting to note, therefore, that the same failure modes are evident in the blast-loaded and quasi-statically crushed lattice specimens. The higher strain rates will produce an increase in the yield stress of the lattice. By assuming the Cowper–Symonds relationship [25], given by Eq. (3), an estimate of the increase in yield strength due to the impulsive load was made: 1=q sdy _ ¼1þ , (3) D sy



with sdy and sy being the respective dynamic and quasistatic yield strengths. The values of D and q are assumed to be similar to the solid 316L material, which are found to be an average of 1704.5 s1 and 5.2 from [25] for 2-mm-thick rolled 316L stainless steel. It is known that the manufacturing process (laser melting from 316L powder) may cause the mechanical properties to differ from the traditionally rolled 316L stainless steel, but in the absence of dynamic test data this assumption seems reasonable. Inputting the strain rates into Eq. (3) indicates that the blast loading increased the yield strength of the lattices by 50 to 100% of the quasi-static magnitude. This effect is important because it enables the lattices to absorb up to twice the amount of energy for a given strain level (or damage tolerance) although the effect of the increased peak load must be considered when designing lattice structures for resistance to blast loading. However, this assumes that the form of the stress–strain curve obtained during the quasi-static crush tests is similar to that which would be obtained at strain rates of the order of 102 to 103 s1. This remains an open topic for investigation in further studies. 4. Discussion At this stage it is worth drawing a comparison of these results with those of Lee et al. [14], in which pyramidal truss cores were loaded in compression at similar deformation rates to those investigated in this study. The geometry of the pyramidal unit cell is similar to the half-symmetry of the [7451] unit cell. They observed a one and one-half fold increase in peak stress at intermediate strain rates, and a three-fold increase in peak stress at the high rates compared to quasi-static peak stresses. Inertial stabilisation [15] of truss members due to coupling of the plastic wave along the truss- and buckling-induced lateral motion were shown to account for a deformation mode change at the high strain rates. In comparison, a modest 25% increase in strength was observed at intermediate strain rates for the [7451] and [01, 7451] lattices suggesting that inertial stabilisation is not significant in these microlattices. A possible explanation for this observation is attributed to the different boundary conditions at the lattice-loading plate interface in this study and those of Lee et al. [14]. In the former, the lattice strands are not bonded to a face sheet at the loading plate; therefore, the boundary conditions for individual strands are not fixed at both ends. This allows for lateral movement of the strand nodal points along the face of the compression platens in both the [7451] and [01, 7451] lattices, working against any lateral confinement of the strands by the plastic wave. In fact, the [7451] lattices were shown to deform by plastic bending (Fig. 7) only, as opposed to buckling seen in the later observations of the fully constrained pyramidal truss cores with similar 451 angled struts. A second factor also thought to influence the modest strength increase is variations in the strand cross-section (Fig. 2) and lack of straightness of individual strands being

significantly larger in the present SLM produced lattices as compared to the truss structures studied by Lee et al. [14,16]. This suggests the imperfection factor in any sort of buckling analysis of the [01, 7451] lattice will have a large relative value, and thus reduce the inertial effects on strengthening as the impact velocity increases. Continued improvements in the SLM manufacturing process, have already led to circularised strand cross-sections; this will help to reduce the imperfection factor. Repeating the tests with optimised lattice strands and constrained boundary conditions, i.e., lattice attached to face plates, should show an improvement in strength at increasing loading rates. The high strain rate blast tests showed no observational evidence of a change in deformation mode of the [7451] and [01, 7451] lattices (comparing Figs. 7, 8, 12 and 13), but a suggested doubling of yield strength. This is contrary to Lee et al. [16] who observed a change in deformation mode of their woven textile cores in the transition to high strain rates. The characteristic shear banding seen at low strain rates, similar to that of Fig. 12, was replaced by a shock front of crushed material parallel to the load face. To gain insight into the high strain rate behaviour of the [7451] and [01, 7451] lattices, the work of Tan et al. and others [26–28] was investigated. Tan et al. [26] present results from dynamic compression tests on aluminium foams performed over a wide range of loading rates. Three types of behaviour were described for aluminium foam, based on nominal loading rate (i.e., initial velocity): (1) quasi-static response, (2) dynamic response and (3) dynamic shock response. Dynamic response occurs at sub-critical velocities and relatively small enhancements in the force–time history were observed due to microinertia effects (since the aluminium foam base material was almost strain rate insensitive, the influence of strain rate could be neglected). At super-critical velocities, a ‘shock wave’ (that is, a narrow band with a high deformation and velocity gradient travelling through the foam) was observed to form, with fully compressed material left in its wake and undisturbed material ahead of it. Using thermodynamic considerations, and assuming a rigid perfectly plastic locking idealisation, Eq. (4) is derived in Tan et al. [27] to calculate the critical velocity required for the shock wave to form in an open cell foam, and this is used in [26] where it was shown to provide good agreement with experimental observations for the dynamic compression of aluminium foams: V CR ¼ ð2C n sys =rs Þ1=2

1=4 r 1=2 D , rs

(4)

where r is the density of the foam, rs the density of the base material, sys the quasi-static yield strength of the base material and eD the densification strain of the foam. The Cn term was found from experiment. It was equal to the gradient of the graph where sqs cr =sys is plotted against ðr=rs Þ3=2 , where sqs is the quasi-static plastic collapse stress cr of the foam. Tan et al. [26] calculated this to be 0.522 for a


small cell aluminium foam. However, by combining Eq. (4) with this definition for Cn, Eq. (4) can be expressed in a much simpler form: 1=2 V CR ¼ ð2sqs . cr D =rÞ

(5)

Eq. (5) is a convenient form as all of the required parameters for lattices examined in this paper were either measured or found from the quasi-static compression experiments. The critical velocities for the lattices types A, B, C and D are calculated from Eq. (5) to be 126, 95, 55 and 90.8 m/s respectively. These values are all considerably greater than the initial velocities generated in the blast loading tests for that lattice type, which is evident when the results from Table 2 are compared with the critical velocities given above. From 1D shock analysis [28] the downstream stress, sD relationship to plastic strength, sqs cr and the hydrodynamic term, are given by sD sqs cr þ

rV 2 . D

(6)

In the form given in Eq. (5), the critical velocity is defined in [26] as the velocity above which the hydrodynamic term in Eq. (6) is 200% of the quasi-static plastic collapse stress of the cellular material. Others, including Deshpande and Fleck [29] and Lee et al. [16], have set this contribution as low as 20% when defining the critical velocity at which the shock wave develops (VCR). When set at this much lower threshold value of 20%, critical velocities of 40, 30, 17 and 29 m/s are calculated for the lattices types A–D, respectively. The initial velocities calculated from the blast tests are in many cases higher than these VCR values, yet it is evident from inspection of the lattices that no shock wave develops in any of the blast tests. It should also be noted that Eqs. (4)–(6) do not consider the influence of strain rate effects, as the plastic collapse stress used is the quasi-static one. This may be appropriate for aluminium foams which show little rate sensitivity, but for a moderately ratesensitive material like stainless steel, comparing the hydrodynamic term contribution to the quasi-static plastic collapse stress to obtain a critical velocity may not be appropriate. From the strain rates given in Table 2, it has been shown that the plastic flow stress increases between 50 and 100% (depending upon the impulse magnitude) during the blast loading tests. Next, the percentage contribution of the hydrodynamic term at which the shock wave was considered to have developed was varied until all of the critical velocities calculated were above those applied in the blast tests. Interestingly, the contribution of the hydrodynamic term had to be increased to 130% of the quasi-static plastic collapse stress. This value of 130% must be considered a lower-bound value, as no shock wave was developed to define the actual critical velocity. As shown previously, rather than attributing the increase in strength of the lattice to shock wave enhancements, the nominal structural strain rates given in Table 2 suggest that strain rate enhancement

809

is responsible. The shock response of the lattice structures is a topic of interest for future studies. 5. Concluding remarks The geometry of the unit cells investigated in this study was not subject to any optimisation; rather, they were selected because of a combination of manufacturing issues and their relative simplicity to generate computer models for the manufacturing process. Despite the lack of optimisation, the quasi-static and blast loading analysis shows the mechanical property behaviour of the lattices is predictable, and depending upon the unit-cell geometry, the progressive damage and failure modes can be selected accordingly for the intended application. This offers the possibility of extending the functional ability of SLM metal lattice structures to load-bearing applications, for which metal foams are not favoured due to their inhomogeneity. Optimisation of the unit cell could improve the relative mechanical property to relative density relationship to increase the competitiveness of SLM produced metal lattices with open and closed cell metal foams. However, even this non-optimised study of quasi-static and blast loading with different unit-cell geometries and densities highlighting the possibilities of using metal lattices in highperformance areas such as aerospace sandwich structures or space-based applications. For example, the recoverable elastic energy seen in one particular unit-cell configuration of this study could offer vibration control in a structure as well as load-bearing and impact protection. Another important feature of the blast loading response is the apparent doubling of the yield strength of the lattice, thought to be mainly due to the rate sensitivity of the parent metal and the high strain rates involved. This will improve the energy absorption of the lattice under high loading rates but also increase the peak forces involved. This will be considered more fully in a future study. Microinertia effects have been shown elsewhere to enhance the strength of truss type cores at increasing deformation loading rates. Here, however the lateral stabilisation against buckling of lattice strands does not occur, and so inertial effects are of less significance. A relatively large imperfection factor, lateral deformation of the lattices and unconstrained strands at the loading faces are thought to all counter inertial stabilisation of the lattice strands, and thus reduce the strengthening effect at increasing loading rates. Continual improvement in SLM manufacturing has already produced circularised lattice strand cross-sections and improved axial linearity. This should lead to an improvement in the dynamic behaviour with constrained lattices, attached to a face sheet for example, rather than unconstrained samples as investigated in this study. At the highest strain rates (103 s1), the deformation mode remained unchanged from the quasi-static and intermediate rates. Rather than attributing the increase in strength of the lattice to shock wave enhancements, the nominal structural strain rates suggest that strain rate



enhancement is responsible as the initial velocities are lower than the critical velocity for shock wave development, as predicted by Tan et al. [26]. Acknowledgements The authors wish to acknowledge the Engineering and Physical Sciences Research Council for their support of this study through grant number EP/C009525/1, the National Research Foundation (NRF South Africa) and the 1851 Royal Commission. References [1] Hanssen AG, Enstock L, Lanseth M. Close-range blast loading of aluminium foam panels. Int J Impact Eng 2002;27(6):593–618. [2] Zok FW, Rathbun HJ, Wei Z, Evans AG. Design of metallic textile core sandwich panels. Int J Solids Struct 2003;40(21):5707–22. [3] Chiras S, Mumm DR, Evans AG, Wicks N, Hutchinson JW, Dharmasena K, et al. The structural performance of near-optimized truss core panels. Int J Solids Struct 2002;39(15):4093–115. [4] Wang J, Evans AG, Dharmasena K, Wadley HNG. On the performance of truss panels with Kagome´ cores. Int J Solids Struct 2003;40:6981–8. [5] Deshpande VS, Fleck NA, Ashby MF. Effective properties of the octet-truss lattice material. J Mech Phys Solids 2001;49(8):1747–69. [6] Deshpande VS, Fleck NA. Collapse of truss core sandwich beams in 3-point bending. Int J Solids Struct 2001;38(36–37):6275–305. [7] Deshpande VS, Ashby MF, Fleck NA. Foam topology: bending versus stretching dominated architectures. Acta Mater 2001;49(6):1035–40. [8] Wicks N, Hutchinson JW. Performance of sandwich plates with truss cores. Mech Mater 2004;36:739–51. [9] Sypeck DJ, Wadley HNG. Cellular metal truss core sandwich structures. Adv Eng Mater 2002;4(10):759–64. [10] Zok FW, Waltner SA, Wei Z, Rathbun HJ, McMeeking RM, Evans AG. A protocol for characterizing the structural performance of metallic sandwich panels: application to pyramidal truss cores. Int J Solids Struct 2004;41:6249–71. [11] Rathbun HJ, Wei Z, He MY, Zok FW, Evans AG, Sypeck DJ, et al. Measurement and simulation of the performance of a lightweight metallic sandwich structure with a tetrahedral truss core. J Appl Mech 2004;71:368–74. [12] Fleck NA, Deshpande VS. The resistance of clamped sandwich beams to shock loading. J Appl Mech 2004;71:386–401.

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