The dynamic response of sandwich beams with open-cell metal foam cores

The dynamic response of sandwich beams with open-cell metal foam cores

Composites: Part B 42 (2011) 1–10 Contents lists available at ScienceDirect Composites: Part B journal homepage: www.elsevier.com/locate/compositesb...

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Composites: Part B 42 (2011) 1–10

Contents lists available at ScienceDirect

Composites: Part B journal homepage: www.elsevier.com/locate/compositesb

The dynamic response of sandwich beams with open-cell metal foam cores Lin Jing a, Zhihua Wang a,b,⇑, Jianguo Ning b, Longmao Zhao a a b

Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China

a r t i c l e

i n f o

Article history: Received 30 April 2010 Received in revised form 16 August 2010 Accepted 29 September 2010 Available online 8 October 2010 Keywords: A. Foams B. Impact behavior Sandwich beam

a b s t r a c t The deformation and failure modes of dynamically loaded sandwich beams made of aluminum skins with open-cell aluminum foam cores were investigated experimentally. The dynamic compressive stress– strain curves of core materials, open-cell aluminum foam, were obtained using Split Hopkinson Pressure Bar. And then the dynamic impact tests were conducted for sandwich beams with open-cell aluminum foam cores. The photographs showing the deflected profiles of the dynamically loaded sandwich beams are exhibited. Several impact deformation modes of sandwich beams can be observed according to contrastive photographs, i.e. large inelastic deformation, face wrinkle and core shear with interfacial failure. A comparison of the measurements is made with analytical predictions, which indicates that the experimentally measured deflections agree well with predictions employing both the inscribing and circumscribing yield loci. For comparison, the quasi-static punching deformation and failure modes of sandwich beams is presented. Crown Copyright Ó 2010 Published by Elsevier Ltd. All rights reserved.

1. Introduction Metal foams can dissipate a large amount of energy due to its relative long stress plateau, which makes it widely applicable in the design of structural crashworthiness. Sandwich structure with metal foam core is of current academic and industrial interest due to its superior properties like high specific strength, high specific stiffness and high energy absorption ability, etc. In recent years, considerable studies have been conducted focusing on constitutive relationship of foam materials [1–3] and failure mechanism of sandwich structures [4–6]. While Fleck and Deshpande [7] developed an analytical model for the shock resistance of clamped sandwich beams, Xue and Hutchinson [8] conducted Finite Element simulations to investigate the response of sandwich beams subjected to impulse loadings. Similarly, Cantwell et al. investigated the high-velocity impact response of composite and FML-reinforced sandwich structures [9]. Several main collapse modes including delamination and longitudinal splitting of the composite skins have been observed. To perfectly classify the impulsive response of sandwich beams, several studies have been further conducted to investigate experimentally [10] and theoretically [11] the response of sandwich beams subject to shock loading. Foo et al. [12] have already examined the failure response of

⇑ Corresponding author at: Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China. Tel.: +86 3516010560. E-mail address: [email protected] (Z. Wang).

aluminum sandwich panels subjected to low-velocity impact. Strain-hardening behavior of the aluminum alloys and the honeycomb core density were shown to affect the impact response. On the basis of the studies so far available, Yu et al. [13] have investigated the response and failure of the sandwich beams with open-cell aluminum foam core subjected to low-velocity impact loading by using the drop weight machine. However, few experimental investigations on the structural response of open-cell aluminum foam core sandwich beams under high-velocity impact have been reported to date. Therefore, a deep insight into dynamic response of sandwich beams with open-cell aluminum foam by using metal foam projectile loading, which was proved to be a convenient experimental tool to simulate shock loading on a structure in Ref. [14], is required to design these structures with significantly enhanced energy-absorbing and shock-resistant performance. In this study, a large number of experiments were conducted to investigate the deformation and failure modes of dynamically loaded sandwich beams made of aluminum skins with open-cell aluminum foam cores. First, the dynamic compressive stress– strain curves of core material, open-cell aluminum foam, were obtained by an SHPB technique. And then the dynamic tests of sandwich beams were achieved, the deformation and failure modes of specimens are reported. The resistance to shock loading is evaluated by the permanent deflection at the mid-span of the beams for a fixed magnitude of applied impulse and mass of beam. Moreover, the comparative study between the measurements and analytical predictions is conducted. Finally, several characteristic deformation modes of sandwich beams subjected to quasi-static are discussed comparing with dynamic failure modes.

1359-8368/$ - see front matter Crown Copyright Ó 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2010.09.024

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2. Dynamic compressive behaviors of aluminum foam 2.1. Material and specimen Open-cell aluminum foam was provided by Hong Bo Metallic Material Company (China) for testing. Aluminum foams with approximate relative density of 0.40 were investigated. The morphology of the foam was characterized with a conventional scanning electron microscope (SEM), as shown in Fig. 1. The foam has an open-celled structure without the typical wavy distortion of cell column or wall of a close-celled structure, and most of column surface appears to be rough [15]. The average cell sizes (i.e. column

spacing) of these foams are approximately 0.75 mm, 1.5 mm, 2.5 mm, respectively. The composition of the cell wall material is Al–Mg1.31 (wt.%). This material was made by infiltration casting process. The SEM photographs of undeformed aluminum foam material with different cell sizes are given in Fig. 1. To obtain the stress–strain relations with a large strain (exhibits three universal deformation characteristics) [1,16,17], cylindrical samples in dynamic compressive tests, 35 mm in diameter and 10 mm in height, were cut into the final geometry using an electrical discharge machine from blocks of the foam material. 2.2. Experimental results Dynamic compression tests at strain rates of up to 1200 s1 were conducted on the cylindrical samples described above at room temperature using a split Hopkinson pressure bar apparatus. A brief description of the experiment set-up is given below, with complete details given by Sathiamoorthy [18]. The striker, incident and transmitter bars consisted of 37 mm diameter aluminum bars and their lengths were 800, 2000 and 2000 mm, respectively. The end surfaces were lubricated to reduce the frictional restraint. The compressive pulse is generated by axial impact on the incident pressure bar by the striker bar. When the compressive pulse reaches the specimen, a portion of the pulse is reflected from the interface, while the remainder is transmitted through to the transmitter bar. The incident pulse and reflected waves in the incident bar are recorded by the resistance strain gauge attached at the incident bar. The transmitted wave is also recorded by the semiconductor strain attached at the transmitter bar. Dynamic compressive stress–strain curves could be obtained from the measured results. Compressive tests were also performed at a quasistatic strain rate of 103 s1 using a servo-hydraulic test machine and specimens of 35 mm in diameter and 30 mm in height. Experimental results indicate that the compressive stress– strain curve of aluminum alloy foam, under either quasi-static or dynamic compression, exhibits three universal deformation characteristics: an initial linear-elastic region; an extended plateau region where the stress increases slowly as the cells deform plastically; and a final densification as collapsed cells are compacted together. A comparison of the nominal stress–strain curves of the aluminum foam specimens under different strain rates is shown in Fig. 2. It is shown that the yield strength and flow stress of aluminum foam material increase with strain rate. The plateau stress is considered as the most important parameter of aluminum foam as all the other characteristics of structural response (e.g. energy absorption and the permanent deflection at the mid-span of the beams) depend on it. So, in this paper, an energy-based approach is proposed to calculate the effective plateau stress, through the stress–strain curves obtained from the standard uniaxial compression tests. Define energy absorption efficiency g(ea) as the energy absorbed up to a given nominal strain ea normalized by the corresponding stress value r(e) [19].

R ea

rðeÞde : rðeÞe¼ea

ecr

gðea Þ ¼

ð1Þ

Densification strain eD is the strain value corresponding to the stationary point in the energy absorption efficiency g(ea)–strain e curve (obtained from Eq. (1)) where the efficiency is a global maximum, i.e.

 dgðeÞ ¼ 0: de e¼eD

ð2Þ

The plateau stress is determined by

R ea Fig. 1. SEM photograph of the underformed aluminum foam materials.

rpl ¼

rc ðeÞde : ea  ecr

ecr

ð3Þ

L. Jing et al. / Composites: Part B 42 (2011) 1–10

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Fig. 2. Stress–strain curves and energy absorption efficiency–strain curves of open-cell foams.

3. Impact tests of sandwich beams 3.1. Experimental process In this study, aluminum foam projectiles are used to provide impact loading of the clamped monolithic and sandwich beams,

enabling the transient transverse response of the beams to be explored. Sandwich beams were made up of two thin skins adhered to an open-cell aluminum foam core by a commercially available acrylate adhesive. Drawing of the sandwich beams is shown in Fig. 3. The skin material was made of Al-2024 aluminum alloy, which was processed by annealing. Foam projectile made from

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Fig. 3. Geometry and dimension of the aluminum foam core specimen.

Table 1 Material properties of face and projectile materials. Material

Density (kg m3)

Young’s modulus (GPa)

Poisson ratio

Yield stress (MPa)

Face sheet Foam projectile

2700 250

72.4 1.0

0.33 0.30

75.8 1.5

Table 2 Test variables of aluminum foam core sandwich beams. Variables Average cell size, d (mm) Face thickness, h (mm)

0.75, 1.5, 2.5 0.5, 0.8, 1.0

Alporas foam was provided by Shinko Wire Company (Germany). Table 1 shows the mechanical properties of the face and projectile materials. The length and width of the beam specimen were 300 mm and 40 mm, respectively. The core thickness of sandwich specimens was fixed to be 10 mm, while face thickness and average cell size of core material were changed as shown in Table 2. Impact tests were conducted on a dynamic loading device using a special designed fixture to clamp the sandwich beams. The overall experimental device includes aerodynamic gun, laser displacement transducer, clamping device and data recorder, etc. A sketch of the experimental set-up is shown in Fig. 4. Circular cylindrical projectiles were electro-discharge machined from Alporas aluminum alloy foam, of length l = 45 mm, diameter d = 36.5 mm and relative density 8.5%. Projectiles were fired from a 37 mm diameter bore, 4.5 m long gas gun at velocities v0 of 125–187.5 m/s, providing a projectile momentum I = mv0 of up to 2.73 Ns, where m is the mass of the metallic foam projectile. Laser displacement transducer (LD 1625-200, le.Com, Germany) was applied to measure deflectiontime history of the mid-point of back face of sandwich beams.

Fig. 4. Sketch of the dynamic experimental set-up.

3.2. Results 3.2.1. Deformation and failure modes The deflection profiles of the dynamically tested sandwich beams, with open-cell aluminum foam cores, are shown in Fig. 5. Meanwhile, sandwich beams were loaded quasi-statically to a mid-span deflection equal to that of specimen D-4, and the deflected shapes are shown in Fig. 5 for the purpose of comparison. Comparing the deformation of sandwich beams under two loading conditions, it is shown that the deflection at failure in the dynamic impact tests is significantly less than that in the quasi-static cases, presumably due to the relatively heavier local deformation and damage near the impact region in the dynamic case. Moreover, similar to the monolithic beam case, the quasi-static response of the sandwich beam is dominated by longitudinal stretching with stationary plastic hinges existing at the supports and mid-span. In contrast, the profiles of the dynamically loaded beams are continuously curved due to the travelling plastic hinges. Experiment results also indicate that several deformation modes of sandwich

Fig. 5. The deflection profiles of the sandwich beams.

beams subjected to impact loading can be observed, that is, large inelastic deformation, face wrinkle and core shear with interfacial failure. It is noted that the core compression mode was not observed in the dynamic tests. Large inelastic deformation is the dominating failure mode for the sandwich beams due to its structural characteristic, as shown in Fig. 6. Although some specimens are dominated by localized failure at the central area, the global deformation is also evident. The maximum deflection occurred at the mid-span of back face of the

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Fig. 6. Large inelastic deformation mode of sandwich beam.

sandwich beam, at which bending moment is a maximum. Similar experimental results were observed by Menkes and Zhu in the literature [20,21]. Experimental results indicated that the specimens subjected to lower level impact loadings are prone to deform globally. On the contrary, those loaded by higher speed impact tend to produce a localized deformation. Fig. 7a presents the face wrinkling mode of a sandwich beam. Face wrinkling of the sandwich beam with thin face and strong core occurs due to local buckling related to the waviness of the face in the plane as well as moduli of the face and core materials [22,23]. Because the face wrinkling may occur due to the torsion and the bending, Fig. 7b shows the deflected profiles of two sides of beam. In the tests, the wrinkling phenomenon of the front face was not symmetry of the longitudinal neutral plane. So the face wrinkling mode seems due to the torsion of the beam. It is noted that the face wrinkling often occurred in the area around the point of projectile impact. When a sandwich beam is subjected to a transverse impact loading, the shear force is carried mainly by the core, and plastic collapse by core shear can result. In the tests, core shear failure in the central loading area of sandwich beams was observed. And the interfacial failure between the core and the bottom face mode usually accompanied core shear failure, as shown in Fig. 8. It can be found that all the specimens with interfacial failure present debonding in the interface. The reason can be explained reasonably that the strength of adhesive layer is lower than core shear strength [24]. In addition, the occurrence of core shear depends on the material property and specimen geometry. Unlike static tests, transverse shear failure occurred at the boundary was not observed in the studied bound. 3.2.2. The dynamic response of sandwich beams A summary of all experiments performed on the sandwich beams with open-cell metal foam cores is provided in Table 3, which includes the permanent back face transverse deflection at the mid-span. For comparison, the results of sandwich beams with closed-cell metal foam cores and monolithic beams are also listed in Table 3. All the sandwich beams have identical geometry and dimension. Monolithic beams were machined to a length of 300 mm, width of 40 mm and a nominal thickness of 2 mm. The mechanical properties of the monolithic and closed-cell core sandwich beam are available from [25]. In order to identify the deformation process of sandwich beams with open-cell aluminum core, several typical deflection-time curves of the mid-point of back face are presented as shown in Fig. 9. It is found that the sandwich beams have negligible elastic rebound as plasticity in

Fig. 7. Face wrinkling mode of sandwich beam.

Fig. 8. Core shear with interfacial failure mode.

the core damps out the elastic vibrations, and the deflection of sandwich beam decreases with the cell size of core material. Cell size of foam aluminum core can affect markedly the shock resistance of sandwich beams, although it has weak effect on the deformation and failure modes. The maximum normalized deflection W of the back face of the sandwich beam is plotted in Fig. 10 as a function of the normalized impulse I for an impulse I applied over a central patch. Results show that the normalized deflection of sandwich beam increases

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Table 3 Summary of dynamic experiments performed on monolithic and sandwich beams. Test

Beam type

Face thickness (mm)

Core thickness (mm)

Cell size

Applied impulse (Ns)

Deflection (mm)

Deformation/failure modes

D-1 D-2 D-3 D-4 D-5 D-6 D-7 D-8 D-9 D-10 D-11 M-1 M-2 M-3 M-4 M-5 C-1 C-2 C-3 C-4 C-5

Open-cell foam core Open-cell foam core Open-cell foam core Open-cell foam core Open-cell foam core Open-cell foam core Open-cell foam core Open-cell foam core Open-cell foam core Open-cell foam core Open-cell foam core Monolithic Monolithic Monolithic Monolithic Monolithic Closed-cell foam core Closed-cell foam core Closed-cell foam core Closed-cell foam core Closed-cell foam core

1.0 1.0 1.0 1.0 1.0 1.0 0.8 0.5 1.0 1.0 1.0 2 2 2 2 2 0.5 0.5 0.5 0.5 0.5

10 10 10 10 10 10 10 10 10 10 10 – – – – – 10 10 10 10 10

2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 1.5 0.75 – – – – – – – – – –

1.85 2.04 2.05 2.49 2.73 2.01 1.96 2.09 2.42 2.39 2.37 1.04 1.09 1.12 1.14 1.53 0.96 1.10 1.21 1.33 1.59

12.8 16.2 17.6 21 31 15.6 12.8 22.4 20.2 16 13.6 27.8 31.2 32.8 39.2 55.4 8.8 12 13.6 34.4 44

Large inelastic deformation Large inelastic deformation Large inelastic deformation Core shear with interfacial failure Core shear with interfacial failure Large inelastic deformation Large inelastic deformation Face wrinkling Large inelastic deformation Large inelastic deformation Large inelastic deformation – – – – – Large global deformation Large global deformation Large global deformation Core shear Core fracture

with the normalized impulse. By fitting the data points, a linear relationship of back-face deflection versus impulse can be approximated as

W ¼ a þ bI;

ð4Þ

where a and b are two empirical constants and are equal to 0.1387 I ffi ; where W is the perand 7.708, respectively. And W ¼ WL ; I ¼ Lpffiffiffiffiffiffiffiffi qr f

fY

manent deflection of the mid-point of back face sheet, L is the half length of beam, and I, qf and rfY are applied impulse per unit area, face-sheet density and yield strength, respectively. Fig. 11 indicates the relationship of the normalized deflection with face thickness and average cell size of core material, respectively. Obviously, the shock resistance of sandwich beam enhances with increasing face thickness and decreasing cell size of foam core. Moreover, the dimensionless permanent deflection of the back face of the sandwich beams and the monolithic beams are plotted in Fig. 12 as a function of the dimensionless loading impulse. It can be seen that the shock resistance of closed-cell sandwich beams outperform the open-cell sandwich beams. Here, the

Fig. 9. Deflection–time curves of the mid-point of back face of sandwich beams subjected to loading intensity I  2.39 Ns.

Fig. 10. The normalized deflection of sandwich beams versus the normalized impulse.

Fig. 11. The normalized deflection of sandwich beams versus the face thickness and cell size.

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dimensionless maximum deflection W of the mid-span and applied impulsive ^I are given by



W ; L

ð5Þ

and

^I ¼

I qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; M rfY =qf

ð6Þ

respectively, where M is the mass per unit area. 3.2.3. Comparison with predictions Prior to the experimental investigation, we analyzed the deformation of sandwich clamped beams subject to an impulse I per unit area over the central section of the beam. The analysis provides predictions of the maximum mid-span deflection of the back face of sandwich beams. The main equations developed in theoretical analysis are summarized in Appendix A: while a simple closed form equation is available to estimate core compression, a secondorder ordinary differential equation must be solved to obtain the

Table 4 Material properties of sandwich beams. Symbol

Value

Span of beam Loading segment Face-sheet thickness of sandwich beam

2L 2a h

Tensile strength of face sheets Density of face sheet material Thickness of foam core Density of foam core Plateau stress of foam core Longitudinal tensile strength of foam core Densification strain of metal foam core

rfY qf

250 mm 36.5 mm 0.5 mm, 0.8 mm, 1.0 mm 75.8 MPa 2700 kg m3 10 mm 1134 kg m3 13.86 MPa 13.86 MPa 0.42

c

qc rn r1 eD

maximum deflections of the sandwich beams. A comparison between the experimental measurements of the permanent backface deflections of the sandwich beams at the mid-span, and the analytical predictions is included in Fig. 13. For the theoretical predictions, we employed the material properties listed in Table 4. In making these comparisons, we have assumed that the entire momentum of the foam projectile I0 is transferred instantaneously to the front face of the sandwich beam over the impacted area. Thus, we have assumed an inelastic collision between the foam projectile and the beam in which momentum is conserved but the energy loss in this impact is not explicitly accounted for: see Radford et al. [26] for a discussion on the energy audit in a metal foam impact. Fig. 13 reveals that the experimentally measured deflections agree well with predictions of W employing both the inscribing and circumscribing yield loci. It is noted that the experimental deflection exceed the theoretical prediction at the high loading intensity, which may be caused by slippage of the specimen from the clamped end. 4. Comparing with quasi-static punching tests

Fig. 12. Permanent deflection of sandwich and monolithic beams.

Fig. 13. Comparison between experimental data and predictions for the sandwich beams.

Quasi-static tests were conducted on a hydraulic universal testing machine. The specimens used in quasi-static experiment are the same structural configuration as those of impact tests. The sandwich beam is loaded in the central area by a lateral pressure via a flat-ended steel punch with the diameter d = 36.5 mm at a loading speed of 2 mm/min. A sketch of the loading device is shown in Fig. 14.

Fig. 14. Sketch of quasi-static the experimental set-up.

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Fig. 15. Several typical deformation modes of sandwich beams under quasi-static loading.

Fig. 15 presents the typical deformation and failure modes of sandwich beams under quasi-static loading. From the experimental results, the upper face wrinkle, core shear, interfacial failure between the core and the faces and the bottom face fracture modes were observed. Due to the entire loading was suffered mainly by the upper face at first, the face deformed beyond the yield strain and failed in the yield mode with the increase of loading intensity. Face wrinkling is a local elastic instability of the faces involving short wavelength elastic buckling of the upper face sheet, resisted by the underlying elastic core, the similar phenomenon was observed in the dynamic tests. As the thicknesses of faces increased, the failure mode was changed from face failure to the core shear yield mode. Although the failure mode change was observed with respect to the face thickness, the mode change was not clear since the face failure followed the core yield mode. For core shear failure mode, the partial shear was observed in the loading central area as well as transverse shear failure was occurred at the boundary, which indicated that plastic hinges were existed in these positions. Core compression mode in the studied bound was not observed obviously due to the high relative density and high compression strength of open-cell aluminum foam. Meanwhile, the experimental results show that the effect of cell size of core material on the failure mode is small, so the detailed discussion is omitted here. Moreover, interfacial failure between the core and the bottom face mode was observed in the tests. It should be noted that the inter-

facial failure mode may be caused by the friction and interfacial strength. If delamination between the face and the crushed foam core is essentially a result of core fracture, the friction may be a dominant factor; instead the interfacial strength could be a dominant factor if the specimen with interfacial failure presents debonding in the interface. In our tests, the specimens occurred interfacial failure when the interfacial strength is lower than the core shear strength [24]. More surprising failure mode of the bottom face fracture was occurred owing to crack extension of core and increase of loading intensity. The measured load–displacement curves of the static punch test of the sandwich beams with different face thickness are given in Fig. 16. Experimental results show that the load-carrying capacity of sandwich beam increases with face-sheet thickness, but the effect may become weak when the face-sheet thickness exceed a certain value. It is noted that the deformation and failure mode of sandwich beam can affect markedly its load-carrying capacity. 5. Concluding remarks The main aim of this study is presenting the deformation modes of sandwich beams with open-cell aluminum foam cores under dynamic impact loading. The dynamic compression nominal stress– strain curves of core material were obtained by split Hopkinson pressure bar apparatus. Using a special clamped fixture, a large number of sandwich beams have been loaded, and the typical failure modes are discussed in Section 3. In the dynamic tests, large inelastic deformation, face wrinkle and core shear with interfacial failure modes were observed. And the profiles of the dynamically loaded beams are continuously curved due to the travelling plastic hinges. A comparison of the measurements is made with analytical predictions, which indicates that the experimentally measured deflections agree well with predictions employing both the inscribing and circumscribing yield loci. Comparing with dynamic experiments, sandwich beams subjected to quasi-static punch tests show more deformation and failure modes, i.e. the upper face wrinkle, core shear, interfacial failure between the core and the faces and the bottom face fracture. It should be noted that the specimen may slip slightly from the fixed ends, especially in the dynamic tests. But the effect on the deformation and failure modes was neglected in this study. Acknowledgments

Fig. 16. Load–displacement curves of the sandwich beams in the quasi-static tests.

This work is supported by the National Natural Science Foundation of China (Grant Nos. 90716005, 10802055, 10972153),

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Postdoctoral Science Foundation of China, the Natural Science Foundation of Shanxi Province (Grant No. 2007021005) and Program for the Homecomings Foundation and the Top Young Academic Leaders of Higher Learning Institutions of Shanxi. The financial contribution is gratefully acknowledged.

as well. Based on the stresses distribution, circumferential bending moment M and membrane force N in both cases can be calculated by

  N nr M n2 r ¼ ¼ 1  ;  M0  þ hÞ  þr N0 r   þ 2h 4hð1

  N r for 0 6   6 ; N0 r þ 2h ðA:2aÞ

Appendix A. Summary of the analytical model Based on the analytical model of Qiu et al. [27], a modified model is developed for the response of clamped sandwich beams subjected to impulse loading over a central loading patch. In the mode, a new yield criterion is adopted for the cross section of the sandwich beam, where both plastic bending and stretching have been taken into account, and the strength of the cellular core is considered as well. Consider a clamped sandwich beam of span 2L with identical face-sheets of thickness h and a core of thickness c, see Fig. 1. The face-sheets are made from a rigid ideally-plastic solid of yield strength rfY, density qf. The core is taken to be a compressible solid of density qc. In line with the measured stress versus strain response of cellular materials, it is assumed that the core compresses at a constant stress rn in a direction transverse to the beam length with no lateral expansion up to a densification strain eD, beyond densification the core is treated as rigid. The axial or longitudinal tensile strength of core is taken to be r1. An incident shock wave provides an impulse I to the front face sheet over a central section of length 2a. The response of sandwich structure is divided into three sequential stages. A.1. Stage I: fluid–structure interaction phase Similar to Fleck and Deshpande [7], we neglected this phase and assumed that the entire impulse I of the shock was transmitted to the front face of the sandwich beam over a central patch of length 2a. A.2. Stage II: core compression phase The impulse I imparts a velocity to the front face over the central segment of length 2a, and the front face has a velocity v0 = I/ (qfh) while the rest of the beam is stationary. Here, any bending effects are neglected and this phase reduces to the one dimension core compression problem. The degree of core compression is obtained by equating the initial kinetic energy of the front face sheet to the sum of the final kinetic energy of sandwich beam and plastic work dissipated in compressing the core by a compressive strain ec.

ec ¼

I2   n c2 h 2r

þq  h ; þq  2h

  þ ð4  2nÞh  N 2nh M 2nh½2 ¼1 ¼ ;    N0  4hð1 þ hÞ þ r r þ 2h M0

  N r   6 1; for 6 r þ 2h N0  ðA:2bÞ

where n is a constant and 0 6 n 6 1. Eliminating n, the corresponding yield locus is expressed as

     2  2 M N  þ 2hÞ ðr N  þ   ¼ 1 when 0 6 M  4r N   þ hÞ  þr  2 N0  hð1 0 0 r 6 ; r þ 2h h  i2 N     þ ð1  r  2  Þ  ð1 þ 2hÞ N0 ðr þ 2hÞ M  þ ¼0 M   þ hÞ  þr  4hð1 0   N r 6   6 1; when   r þ 2h N

ðA:3aÞ

ðA:3bÞ

0

where N0 and M0 are the plastic collapse values of the longitudinal force and bending moment, respectively. We approximate this yield locus by either a circumscribing or inscribing square such that

jMj ¼ M 0

and=or jNj ¼ N0 ;

ðA:4aÞ

and

jMj ¼ fM 0 jNj ¼ fN0 ;

ðA:4bÞ

8 pffiffiffiffiffiffiffiffiffiffi < 1þ4s1 1 2 ð1 þ hÞ  r 2 6 0 8h  2  n þ2hÞ 2s1 ffi where f ¼ pffiffiffiffiffiffiffiffiffiffi , s2 ¼ , and s1 ¼ 4r ðrhð1þ   r  2n hÞþ n : s22 þ4s3 s2 2  r 2 > 0 8h ð1 þ hÞ 2

  n Þð3r  n þ8hÞ ð1r  2  n þ2hÞ ðr



hÞ þ 1; s3 ¼ 1  ð2ð1þ2  : r þ2hÞ n

We proceed to give the formulae for the normalized maximum deflection at mid-span of the back face sheet of the sandwich beam

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a22 a23 W 64 I2 r2  a1 a3 ; 1 þ w  2Þ 1 þ ¼ ða1 a2 þ w W¼ L 3 ða1 a3 þ w 1 þ w  2 Þ2 ðA:5Þ

ðA:1Þ

pffiffiffiffiffiffiffiffiffiffiffiffi  ¼ h=c, c ¼ c=L, q  n ¼ wrn =rfY ,  ¼ qc =qf , I ¼ I=ðL rfY qf Þ, r where h and w = 1.3 is the ratio of dynamic plateau stress and the static plateau stress of foam material. From Fig. 2, it can be seen that the plateau stress of foam material is an approximate constant at the strain rate of 103 s1. A.3. Stage III: beam bending and stretching phase The shape of the yield surface of a sandwich beam in (N, M) space depends upon the relative strength and thickness of the faces and the core. With the deflection increasing, effect of membrane force becomes critical. When the deflection is beyond the initial thickness of the panel, the membrane effect even dominates the structural response. Therefore, in this paper, a new inscribing yield criterion is adopted for the cross section of the sandwich beam [28], where both plastic bending and stretching have been taken into account, and the strength of the cellular core is considered

 1 at the end of phase I of where the non-dimensional deflection w the motion ends is given by

 1 ¼ a1 a3 w

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 16 a22 22 r I 1 : 1þ 3 a21

ðA:6Þ

 2 during phase II can be obThe non-dimensional deflection w tained by

2 ¼ w

Z

T 2

T 1

8a22 a3Ir  dt; r þ nðtÞ

ðA:7Þ

where  nðtÞ is the differential equation

ðr þ nÞ€nðtÞ ¼ 6a22 ;

ðA:8Þ

with initial conditions

  1 _ T Þ ¼ 3 a þ w nðT 1 Þ ¼ r; andnð : 1 1 a3 4Ir

ðA:9Þ

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The termination condition for phase II of the motion, gives the time T 2 as

nðT 2 Þ ¼ 1  r:

ðA:10Þ

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