The dynamic response of clamped composite sandwich beams

Transcription

1 The dynamic response of clamped composite sandwich beams V.L. Tagarielli, V.S. Deshpande and N.A. Fleck * Cambridge University Engineering Dept. Trumpington St., Cambridge, CB2 1PZ, UK Abstract The dynamic response of composite monolithic and sandwich beams is determined by loading end-clamped beams at mid-span with metal foam projectiles. The sandwich beams comprise glass-vinylester face sheets and a PVC foam or balsa wood core. Highspeed photography is used to measure the transient transverse deflection of the beams and determine the dynamic modes of deformation. In both the monolithic and sandwich beams, the dynamic mode of deformation comprises a flexural wave that travels from the impact site towards the support. This wave is then partially reflected from the supports. The failure mechanisms in the sandwich beams include cracking in the core, face sheet delamination and tensile failure of the face sheets at the supports. The dynamic resistance of the beams is quantified by the maximum transverse deflection at the mid-span of the beams for a fixed magnitude of projectile momentum. Appropriately designed sandwich beams have a higher dynamic strength than monolithic beams of equal mass. The low density PVC foam core beam outperform the beams with high density cores: for a given beam mass the low density core beams have thicker cores which substantially increases the bending strength of these beams. Shear failure of the balsa wood results in balsa wood core beams having a low dynamic strength, especially at the higher levels of projectile momenta. Keywords: composite sandwich panels, dynamic testing, PVC foam, balsa wood. Submitted to Int J. of Impact Engineering, July 25 * Corresponding author, Tel.: ; fax: address: naf1@eng.cam.ac.uk

2 1. Introduction Sandwich beams comprising composite face sheets and a lightweight foam core are extensively used for weight efficient structures subject to quasi-static bending loads. Steeves and Fleck [1] have investigated the static three-point bending response of simply supported sandwich beams made from glass-epoxy face sheets and a polymer foam core while Tagarielli et al. [2] investigated the effect of fully clamped boundary conditions on the bending response of these beams. These studies demonstrated that these composite sandwich designs offer considerable weight savings in static loading situations. In a broad range of practical applications, structures are subjected to dynamic loading such that the applied loads far exceed the quasi-static collapse strength. The response of metallic monolithic beams and plates to shock type loading has been extensively investigated. For example, Wang and Hopkins [3] and Symmonds [4] analysed the impulsive response of clamped circular plates and beams, respectively. There exists an increasing body of theoretical work that suggests that metallic sandwich structures outperform monolithic structures in shock loading situations, see for example Fleck and Deshpande [5] and Xue and Hutchinson [6]. The superior performance of the metallic sandwich structures is due in part to fluid-structure interaction effects, as outlined by Taylor [7], and partly due to the greater bending strength of sandwich beams compared to their monolithic counterparts. The predictions of enhanced dynamic strengths of metallic sandwich beams have been confirmed by a number of recent experimental investigations. These studies subject the sandwich structures to high intensity pressure pulses using an experimental technique developed by Radford et al. [8]. In this method, metal foam projectiles are fired at the structures. The pressure pulses applied by the projectile on the structure mimic shock loading in air and in water, with peak pressures on the order of 1 MPa and loading times of approximately.1 ms. Using this technique Radford et al. [9] and Rathbun et al. [1] have confirmed the enhanced dynamic strengths of metallic sandwich beams with lattice cores while McShane et al. [11] have shown that the same conclusions holds for metallic circular sandwich plates as well. Composite sandwich beams are expected to have significant advantages over metallic beams in underwater shock situations - the acoustic mismatch between the water and the light composite face sheets is expected to enhance the fluid-structure interaction effect [12].

3 However, there is little or no experimental work on either the dynamic strength of composite sandwich beams or on the fluid-structure interaction effects in these beams. The aim of this study is to conduct an experimental investigation into the dynamic strengths of composite sandwich beams comprising PVC foam or balsa wood cores and woven glass/vinylester face sheets. This material combination is commonly used in marine applications and hence of considerable practical interest. The outline of the paper is as follows. First, the sandwich beam configurations and materials employed in this study are detailed and the mechanical properties of the constituent materials described. Next, the dynamic beam tests are described and the results presented. Finally, we use the experimental results to draw conclusions on the effect of sandwich beam geometry and core material on the dynamic strength of the composite sandwich beams. 2. Experimental Investigation Metal foam projectiles are used to dynamically load clamped sandwich beams comprising PVC foam or balsa wood cores and woven glass fibre-vinylester face sheets, as sketched in Fig. 1. The primary objectives of the experimental investigations are: (i) To investigate the dynamic deformation and failure mechanisms in these composite sandwich beams; and (ii) To develop an initial understanding of the effect of core type and sandwich beam geometry on the dynamic strength of the composite sandwich beams. Some of the sandwich and monolithic beams are chosen to have approximately equal masses. This enables us to compare the dynamic strengths of sandwich and monolithic beams of equal mass. 2.1 Sandwich specimen configuration and manufacture The sandwich and monolithic beams tested in this study were manufactured at KTH 1 by a vacuum infusion technique that allowed for the infusion of the face sheets and their assembly with the core in a single step. The face sheets comprised a quasi-isotropic woven glass fibre fabric DBLT85 (with equal amounts of fibres in the o, o ± 45 and o 9 directions) infused with vinylester resin (overall density of the composite faces was approximately 17 kgm -3 ). 1 KTH (Kungl Tekniska Högskolan) Royal Institute of Technology. Valhallavägen 79, Stockholm Sweden. Courtesy of Prof Dan Zenkert. 2

4 Sandwich beams were manufactured using three types of sandwich cores supplied by DIAB 2 ; (i) the H1 Divinycell PVC foam core (density 1 kgm -3 ), (ii) the H25 Divinycell PVC foam core (density 25 kgm -3 ) and (iii) an end-grain balsa wood core (DIAB designation ProBalsa LD7, density 9 kgm -3 ). Typically panels 2 m 1.5 m were manufactured during the vacuum infusion process and sandwich beams of the appropriate dimensions were subsequently cut from these panels using a diamond grit saw. The face sheet and core thicknesses of the sandwich beams were chosen as follows. The choice of the face sheet thickness for sandwich panels was limited to.85 mm, 1.5 mm and 2.2 mm, corresponding to single, double and three layers, respectively of the woven glass fibre fabric. Moreover, the sandwich cores were supplied by DIAB in either 5 mm, 1 mm or 15 mm thick sheets. Based on these constraints, we designed three sets of sandwich beams, each with roughly equal areal masses m. In addition, we also manufactured and tested 1.5 mm and 2.2 mm thick monolithic beams made from two and three layers, respectively of the woven glass fibre fabric. The geometries of the monolithic and sandwich beams are detailed in Table 1. We designate the monolithic beams with the letter M while the H1, H25 and balsa wood core sandwich beams are denoted with the letters H, HD and B, respectively. Note that each set is not populated by all the types of beams. For example, no H25 core sandwich beams are present in Set 1 while Set 3 contains no balsa wood core sandwich beams or monolithic beams. The mm thick monolithic beams ( m 2.5 kgm approximately equal areal mass and are assigned Set in Table 1. ) have no corresponding sandwich beams of The clamped beam experimental set-up is sketched in Fig. 1. All beams tested in this study had a span 2L = 2 mm and width b = 3 mm. The beams were rigidly clamped over a 75 mm length thereby minimising rotation and transverse and longitudinal deflections of the beam ends. The cores of the sandwich beam were replaced by Aluminium inserts over the 75 mm clamped portion, thereby enabling high clamping pressures to be applied. In case of the monolithic beam, holes were drilled directly into the composites beams and the beams clamped in the clamping setup sketched in Fig DIAB AB, Box 21, Laholm Sweden. 3

5 2.2 Quasi-static properties of the constituent materials In order to determine the quasi-static properties of the constituent materials, uniaxial compression and tension tests were conducted on the composite face sheet material, while only uniaxial compression were performed on all core materials. The tension and compression tests on the face sheet materials were conducted in a screw driven test machine at a nominal applied strain rate of s. Rectangular specimens 12 mm 25 mm were cut from the composite sheets and Aluminium tabs glued on the specimen ends to give test specimens with a gauge length 1 mm. The tension tests were conducted by gripping the specimens via the Aluminium tabs in friction grips while a Celanese apparatus was employed to ensure proper alignment in the compression tests. The strain in the specimens was measured using foil strain gauges adhered to the specimen surfaces in both the compression and tension tests. Load was measured via the load cell of the test machine. The quasi-static uniaxial tensile and compressive nominal stress versus nominal strain responses of the woven glass/vinylester composite are plotted in Fig. 2a (single and double fabric layers). In tension, the materials exhibit an initial elastic response with a modulus E = 9GPa and 14 GPa for the single and double fabric layer materials, respectively. At a f strain of approximately.5%, matrix cracking initiates which results in a reduction in the stiffness; a linear response continues and the composites fail by tensile tearing at approximately 2.5% tensile strain giving tensile strengths σ = 33 MPa and 22 MPa for the double and single layer materials, respectively. In compression, the GFRP laminate displays an approximately linear response, with a compressive axial modulus slightly less than the tensile modulus, due to fibre waviness. The microbuckling compressive strengths are approximately 15 MPa and 2 MPa for the single-layer specimens and double layer materials and attained at about 1.6% nominal compressive strain. The composite with three fabric layers had properties similar to that of the double layer (these results are omitted in Fig. 2a for the sake of clarity). It is argued that difference in properties between the single and multiple fabric layer materials is due to differences in the fibre volume fractions in these materials. f 4

6 Compression tests were performed on the H1 and H25 PVC foams and on the balsa wood at a nominal strain rate of 1-4 s -1 using a screw-driven test machine. The compressive load was measured by the load cell of the test machine, while the compressive strain was deduced from the relative displacement of the compression platens, measured via a laser extensometer. The platens were lubricated with a silicone spray to reduce friction. Cuboidal specimens of cross-section of 3 mm 3 mm and thickness of 5 mm to 16 mm were employed in these tests. For the case of the balsa wood, care was taken to ensure that the specimens did not span the glued interfaces of the plates, so that the measured properties represent true balsa wood properties. All balsa specimens were cut from blocks with a density of about 9 kgm -3 in order to ensure a consistent set of results. Figure 2b presents a summary of the static uniaxial compressive responses for the three materials, measured at a strain rate of 1-4 s -1. The figure includes the measured engineering stress versus strain curves for the Divinycell H1 foam (specimens of thickness 5 mm and 15 mm), H25 foam (specimens of thickness 5 and 1 mm) and ProBalsa LD7 balsa wood (specimens of thickness 5 and 15 mm). In all cases, the material exhibits an initial elastic regime, followed by a plateau phase and densification. The static compressive strength σ pl of the materials is arbitrarily defined as the stress at a total strain of 15%. This strength was found to be 1.7 MPa, 5.8 MPa and 4.5 MPa for the H1, H25 and balsa wood, respectively, in line with the properties reported in the manufacturer s data sheets and independent of the specimen thickness. The nominal densification strain was 8% for the H1 and the balsa wood, and about 7% for the H25 PVC foam. While balsa wood displays a flat plateau region up to densification, the PVC foams exhibit some hardening before densification. Almost no lateral expansion was observed in the plastic range in any of the compression tests. We note that these results are consistent with the findings of Deshpande and Fleck [13] for the PVC foams and Tagarielli et al. [14] for the balsa wood; readers are referred to these studies for the multi-axial properties of these core materials. 2.3 Effect of strain rate upon compressive response of core materials Recall that we have investigated the dynamic response of sandwich beams in this study. Thus, a knowledge of the strain rate sensitivity of the sandwich constituent materials is imperative to interpret the experimental results of the dynamic response of the sandwich beams. The strain rate sensitivity of the H1 and H25 Divinycell PVC foams and the ProBalsa LD7 5

7 was investigated by Tagarielli et al. [15] in a parallel study. They performed a series of uniaxial compression experiments at strain rates ranging from 1 s 4 1 to approximately 4 s -1. The high strain rate experiments were performed in a Split-Hopkinson bar apparatus comprising Magnesium bars (so as to accurately measure the dynamic strength of the soft foams and balsa wood). Their data for the compressive strain rate sensitivity of these three materials is summarised in Fig. 3. In this figure the plateau stress σ pl (defined as the material strength a total compressive strain of 15%) is plotted as a function of the applied compressive strain rate ε. The experimental data for compressive strength as a function of strain rate are adequately approximated by power-law fits of the form σ pl m ε = σ ε where ε is a reference strain rate, σ a reference stress, and m is the power law exponent. With the reference strain rate ε = 1s 1, (1), a least-squares fit to the experimental data provides the coefficients σ and m. These values are listed in Table 2 and the corresponding fits are included in Fig. 3. The compressive yield strength of the H25 PVC foam and balsa wood doubles when the strain rate is increased from quasi-static rates (1-4 s -1 ) to rates on the order of 1 3 s -1. In contrast, the H1 PVC foam displays only a small elevation in uniaxial compressive strength (about 3%) for the same increase in strain rate. 2.4 Protocol for the dynamic beam tests Alporas aluminium foam projectiles are used to provide impact loading of the clamped monolithic and sandwich beams over a central circular patch of diameter d, as shown in Fig. 1. The use of foam projectiles as a means of providing well-characterised pressure versus time has recently been developed by Radford et al. [8] and subsequently employed to investigate the dynamic response of metallic sandwich beams with lattice cores [9] and metallic circular sandwich plates with pyramidal and square-honeycomb cores [11]. The basic idea of this experimental technique is that a foam projectile exerts a rectangular pressure versus time history on a rigid target. The pressure and the duration of the pressure pulse are given by [8] ρ v p = + (2) 2 p p σ c, ε D 6

8 and lpε D τ =, (3) v p respectively, where ρ p and length l p are the projectile density and length respectively while σ c and ε D are the foam plateau strength and nominal densification strain, respectively. Thus, the pressure and the duration of the pulse can be adjusted independently by varying the projectile velocity v p and length l p. Here we conduct a series of experiments on each beam configuration by varying v p but keeping all the other foam projectile parameters fixed. Circular cylindrical projectiles of length l 4 mm and diameter d = mm were p electro-discharge machined from Alporas foam blocks of density 3 ρ p 3 kgm (projectile mass 3 m kg ). Recall that the beams have a width b = 3 mm. Thus, the p projectiles load the beams over nearly their entire width. The projectiles were fired from a 28.5 mm diameter bore, 4.5 m long gas gun at a velocity v p in the range 5 ms -1 to 35 ms -1, providing a projectile momentum per unit area I = ρ pl pv p of up to approximately 37 Ns m -2. Four to six tests at different values of I were conducted on each of the beam configurations listed in Table 1. In the tests the value of I was gradually increased until complete failure of the beams was obtained (i.e. tearing of the beams resulted in the projectile not being arrested by the beams). High-speed photography was used to observe the dynamic transverse deformation of the beams and extract the mid-span deflection versus time histories. A Hadland Imacon-79 image-converter camera was used; this is capable of taking up to 2 frames at a maximum rate of 1 7 frames s 1. Inter-frame times of 1 µs and exposure times of 2 µs were used. 3. Experimental results At least four levels of impulse I were applied to each beam configuration by varying the foam projectile velocity v p and the deformation modes and deflection time histories of the back face recorded using high speed photography. The main results from the experimental 7

9 study are summarised in Figs 4 and Fig.5 for the monolithic and sandwich beams, respectively. In these figures, the maximum back-face displacements w max at the mid-span for each beam are plotted as a function of the initial momentum I of the foam projectile. The critical impulse I beyond which face sheet tearing resulted in complete failure of the beam (and the foam projectile being not arrested by the beams) is also marked on Figs. 4 and 5. Within the scatter of the experimental data, we observe that the maximum back face deflections of the two monolithic beams (thicknesses.85 mm and 1.5 mm) are approximately equal (Fig. 4). On other hand, we clearly see in Fig. 5 that the core thickness has a profound effect on the dynamic strength of the sandwich beams (as quantified by the maximum mid-span deflections of the back faces of the sandwich beams): w max decreases with increasing core thickness for a given choice of impulse and sandwich core. For example: (i) The H2 beams outperform the H3 beams even though the H3 beams have a higher areal mass and thicker face sheets (Fig. 5a). (ii) Similarly, the HD2 beams have a higher dynamic strength compared to the HD3 beams (Fig. 5b). Both these beams have approximately the same areal mass but the HD2 beams comprise thicker cores. (iii) Again, the B2 balsa wood core beams ( c = 15 mm ) outperform the B1 ( c = 5 mm ) balsa wood core beams. However, the effect in this case cannot solely be attributed to the core thickness as the B2 beams have a higher areal mass compared to the B1 beams. 3.1 Deformation modes Monolithic beams A sequence of high-speed photographs is shown in Fig. 6a at 1 µs intervals, for the monolithic specimen M1 of Table 1 impacted at 1 v p = 84 ms. It is seen that impact occurs between frames 1 and 2, and by frame 3 the metal foam projectile has compressed by nearly 5% due to the propagation of a plastic shock wave that moves along the foam from the impacted end to the free end. After frame 3, the plastic wave within the foam projectile has arrested and the projectile and the underlying beam elements have a common velocity. Between frames 2 and 4, a flexural wave travels along the beam from the impact site towards the supports. This wave is then reflected back as seen clearly by comparing frames 9 and 1. The corresponding mid-span deflection versus time history, as inferred from the high-speed 8

10 photographs of Fig.6a is plotted in Fig. 6b (the numbers on the curve correspond to the frames in Fig. 6a). Maximum deflection is reached at approximately 5 µs and subsequently, there is elastic recovery due to the reflection of the flexural wave. PVC foam core sandwich beams High-speed photographic sequences (1 µs inter-frame time) of the H1 PVC foam core sandwich beams (specimen type H2) impacted at 1 v p = 24 ms are shown in Fig. 7a. The projectile impacts the beam between frames 2 and 3 and a flexural wave travelling from the impact site towards the supports is evident in frames 3 and 4. Subsequently, the beam reaches a maximum deflection and springs back due to the partial reflection of the flexural wave. Core fracture by micro-cracking is evident in frame 4 and another crack seems to initiate at the left support in frame 5. Despite the failure of the core, the composite face sheets remain intact and the projectile is arrested. The mid-span deflection versus time history measured from the high-speed photographs is plotted in Fig. 7b with the numbers on the curve corresponding to the frame numbers in Fig. 7a. The effect of impact velocity on the deformation mode of the H1 PVC foam core beam is illustrated in Fig. 8 via high-speed photographic sequences (1 µs inter-frame time) of an H2 beam impacted at 1 v p = 323 ms. Impact occurs between frames 1 and 2 and by frame 3 the projectile has fully densified. A number of cracks are seen to initiate in the core by frame 4 and extensive failure of the core due to the coalescence of these cracks is seen in frame 5. The composite faces also tear at the supports by frame 5 which results in complete failure of the beam and the projectile no being arrested by the beam. Similar deformation mechanisms were also observed for the H25 PVC foam core beams. It is worth noting here that, the observed core compression is small in all the experiments on sandwich beams performed in this study. Balsa wood core sandwich beams High-speed photographs of the balsa wood core sandwich beam (type B2) impacted at 1 v p = 14 ms by the foam projectile are shown in Fig. 9 at an inter-frame time of 1 µs. The mechanism of deformation is similar to that of the PVC foam core sandwich beams with 9

11 the propagation of a flexural wave towards the supports. In the early stages of motion, vertical cracks form within the balsa core along the rays of the balsa wood. Subsequently, these cracks kink into delamination cracks between the core and face sheets (frame 3) and then propagate along the face sheet/core interface. This delamination of the face sheets was observed in all the dynamic experiments on balsa wood core sandwich beams. This delamination dramatically reduces the shear stiffness and strength of the sandwich beams, causing a loss of the sandwich effect and hence a reduction in the dynamic strength of the balsa wood core sandwich beams. 3.2 Comparison of the dynamic strength of the sandwich beams Recall that beams in each of the 4 sets of Table 1 have approximately equal areal masses. We proceed to compare the dynamic strengths of beams within each of these sets. Such a comparison is presented in Fig. 1 where the maximum mid-span deflection of the back faces of the beams w max is plotted a function of the projectile momentum I. For the sake of clarity we do not include the actual experimental measurements. Rather, a power law relation of the form where w w max ref I = Iref n, w ref and I ref are a reference deflection and impulse, respectively and n is the power law exponent, is fit to the experimental data in Figs. 4 and 5 using a least-squares method. These best-fits are plotted in Fig. 1. Note that the choice (4) stems from the fact that it seems to adequately describe the data as seen in Figs. 4 and 5. However, we have no particular physical justification for this relation. (4) The findings from the comparisons in Fig. 1 are summarised as follows: 2 Set 1 ( m 3.5 kgm, Fig. 1a) : The H1 foam core beams (H1) outperform balsa wood core beams (B1) of approximately the same geometry. This confirms that the low shear strength of the balsa wood core adversely affects their performance. Equal mass monolithic beams have a higher dynamic strength compared to the balsa wood core sandwich beams but are weaker than the H1 core sandwich beams. 1

12 : The c = 15 mm H1 core sandwich beams (H2) have the 2 Set 2 ( m 4.3 kgm, Fig. 1b) highest dynamic strength and outperform the c = 5mm H25 foam core beams (HD1). This clearly indicates that even though the H25 foam has a higher strength, the increased core thickness of the H1 beams gives these beams a higher specific strength. The c = 15 mm balsa wood core beams (B2) also outperform the HD1 beams at low values of I but delamination failure makes the B2 beams weaker than the HD2 beams at higher impulses. 2 Set 3 ( m 6.3 kgm, Fig. 1c) they are made from the H25 core of thickness : The HD2 beams have the best performance in this set as c = 15 mm (i.e. strongest core with the maximum thickness). The HD3 and H3 beams have a comparable performance as the higher core thickness of the H3 beams compared to the HD3 beams ( c = 1 mm versus c = 5 mm ) compensates for the lower strength of the H1 foam core. Thus, we conclude that well designed sandwich beams can outperform monolithic beams of equal mass. The sandwich beams need to be designed by taking into the fact that a trade-off exists between sandwich core strength and mass as increased core strength typically means higher density cores. The increased sandwich effect in sandwich beams with low density cores can compensate for the associated low core strength (e.g. the H2 beams have a higher dynamic strength compared to the HD1 beams). However, shear failure can intervene knockdown the dynamic strength of the sandwich beams comprising cores of low shear strength (e.g. the B2 beams re weaker than the HD1 beams at high impulse levels). 4. Concluding remarks The dynamic response of composite monolithic and sandwich beams was measured by loading end-clamped beams at mid-span with metal foam projectiles. The sandwich beams comprise glass-vinylester face sheets and a PVC foam or balsa wood core. Two types of PVC foam core were employed: a low strength H1 core (density H25 core (density 3 1 kgm ) and a high strength 3 25 kgm ). High-speed photography was used to measure the transient transverse deflection of the beams and determine the dynamic modes of deformation. 11

13 The dynamic deformation mode in both the sandwich and monolithic beams comprises a flexural wave that travels from the impact site towards the support. This wave is partially reflected from the supports with the maximum deflection of the beam mid-span occurring approximately at the instant that the flexural wave reaches the supports. The failure mechanisms in the PVC sandwich beams include cracking in the core and tensile failure of the face sheets at the supports while delamination of the face sheets from the core is also observed in the balsa wood core beams. The dynamic resistance of the beams is quantified by the maximum transverse deflection at the mid-span of the beams for a fixed magnitude of projectile momentum. Appropriately designed sandwich beams have a higher dynamic strength than monolithic beams of equal mass. Moreover, we observe that well-designed sandwich beams with the low strength H1 PVC foam core outperform sandwich beams with the high strength H25 PVC foam core. This is attributed to the higher core thicknesses of the H1 core beams. By contrast, shear failure of the balsa wood results in balsa wood core beams having a low dynamic strength. The trade-off between core strength and mass to give an optimal dynamic performance of these composite beams is a topic for future work. Acknowledgements This work was financially supported by the EPRSC (Engineering and Physical Sciences Research Council) and ONR (U.S. Office of Naval Research). The authors are pleased to acknowledge Prof. Dan Zenkert (KTH) and Dr. Darren Radford (CUED) for helpful discussions. 12

14 References [1] Steeves C.A. and Fleck N.A., 24. Collapse mechanisms of sandwich beams with composite faces and a foam core, loaded in three-point bending. Part I: analytical models and minimum weight design. International Journal of Mechanical Science, 46: [2] Tagarielli V.L., Fleck N.A. and Deshpande V.S., 24. Collapse of clamped and simply supported composite sandwich beams in three-point bending, Composites B: Engineering, 35, [3] Wang A. J., Hopkins H. G., On the plastic deformation of built-in circular plates under impulsive load, Journal of Mechanics and Physics of Solids 3, [4] Symmonds P. S., Large plastic deformation of beams under blast type loading. Second US National Congress of Applied Mechanics. [5] Fleck N.A. and Deshpande V.S., 24. The resistance of clamped sandwich beams to shock loading, J. of Applied Mechanics, ASME, 71: [6] Xue Z., Hutchinson J. W., 23. Preliminary assessment of sandwich plates subject to blast loads. International Journal of Mechanical Sciences 45, [7] Taylor, G. I., The scientific papers of G I Taylor, Vol III, Cambridge University Press, 1963, The pressure and impulse of submarine explosion waves on plates, pp [8] Radford, D. D., Deshpande, V. S. and Fleck, N. A., 25. The use of metal foam projectiles to simulate shock loading on a structure, International Journal of Impact Engineering, 31(9): [9] Radford D. D., Fleck N. A., Deshpande V. S., 25. The response of clamped sandwich beams subjected to shock loading, International Journal of Impact Engineering (in print). [1] Rathbun H. J., Radford D. D., Xue Z., Yang J., Deshpande V. S., Fleck N. A., Hutchinson J. W., Zok F. W., Evans A. G., 25. A dynamic probe for validating simulations of shock loaded metallic sandwich panels, International Journal of Solids and Structures (in print). [11] McShane G. J., Radford D. D., Deshpande V. S., Fleck N. A., 25, The response of clamped sandwich plates with lattice cores subjected to shock loading, European Journal of Mechanics A:/Solids. (in print). [12] Mäkinen, K., Underwater shock loaded sandwich structures, PhD thesis, Royal Institute of Technology, Department of Aeronautics, Stockholm, Sweden. 13

15 [13] Deshpande, V.S. and Fleck, N.A., 21. Multi-axial yield behaviour of polymer foams, Acta Materialia, 49(1), [14] Tagarielli V.L., Deshpande V.S., Fleck N.A and Chen, 25. A transversely isotropic model for foams, and its application to the indentation of balsa, Int. J. Mech. Sci., 47: p [15] Tagarielli V.L., Fleck N.A. and Deshpande V.S., 25. The uniaxial stress versus strain response of PVC foams and balsa wood at low and high strain rates, to appear in Composites Part B: Engineering. 14

16 Figure Captions Figure 1: Sketch of the sandwich beam geometry, loading and clamping configuration. Figure 2: The quasi-static responses of the sandwich beam constituent materials. (a) The uniaxial compressive and tensile responses of the single and double layer woven glass/vinylester laminates. (b) The uniaxial compressive response of the H1 and H25 PVC foams and the balsa wood used as the sandwich beam cores. Figure 3: Summary of the strain rate sensitivity of the PVC foams and balsa wood under uniaxial compression (data from ref. [15]). The plateau stress σ pl (defined as the flow stress at a total strain of 15%) is plotted against the applied strain rate. Figure 4: The measured maximum deflection at the mid-span of the single and double layer woven glass/vinylester monolithic composite beams, as a function of the foam projectile momentum I. Fits to the experimental data using the power law relation (4) are also included. Figure 5. The measured maximum deflection at the mid-span of the (a) H1 PVC foam core, (b) H25 PVC foam core and (c) balsa wood core sandwich beams, as a function of the foam projectile momentum I. Fits to the experimental data using the power law relation (4) are also included. Figure 6: The deformation of the M1 monolithic beam impacted by the foam projectile at a velocity 1 v p = 84 ms. (a) High-speed photographic sequences at an inter-frame time of 1 µs. (b) The mid-span deflection versus time history extracted from the photographs in (a). Note that the numbers on the curve correspond to the frames in (a). Figure 7: The deformation of the H2 sandwich beam (H1 PVC foam core) impacted by the foam projectile at a velocity 1 v p = 24 ms. (a) High-speed photographic sequences at an inter-frame time of 1 µs. (b) The mid-span deflection versus time history extracted from the photographs in (a). Note that the numbers on the curve correspond to the frames in (a). 15

17 Figure 8: High-speed photographic sequence of sandwich beam specimen H2 (H1 PVC foam core) impacted by the foam projectile at a velocity 1 v p = 323 ms. The inter-frame time is 1 µs. Figure 9: The deformation of the B2 sandwich beam (balsa wood core) impacted by the foam projectile at a velocity 1 v p = 14 ms. (a) High-speed photographic sequences at an interframe time of 1 µs. (b) The mid-span deflection versus time history extracted from the photographs in (a). Note that the numbers on the curve correspond to the frames in (a). Figure 1: Comparison between the measured maximum mid-span deflections w max of the inner face of sandwich and monolithic beams of equal mass. The deflections w max are plotted as a function of the projectile momentum I for the beams in (a) Set 1, (b) Set 2 and (c) Set 3 of Table 1. For the sake of clarity, only the power law fits to the experimental data in Figs. 4 and 5 are included here. Table Captions Table 1: The materials and geometry of the sandwich beams tested in this study (the span 2L = 2mm for all the beams). The specimens labelled M denote monolithic beams, H and HD denote sandwich beams with the H1 and H25 PVC foam cores, respectively and the specimens labelled B are the sandwich beams with a balsa wood core. The specimens are divided into 4 sets, with beams in each set having approximately equal areal masses. Table 2: The power-law coefficients (1) employed to fit the experimental data for the PVC 1 foams and balsa wood in Fig. 3. In all cases a reference strain rate ε = 1s is chosen. (Reproduced from ref. [15]). 16

18 Set, m 2.5 kgm -2 Set 1, m 3.5 kgm -2 Set 2, m 4.3 kgm -2 Set 3, m 6.3 kgm -2 Beam designation Face sheet thickness, h [mm] Core thickness, c [mm] Areal density, m [kgm -2 ] M M H B H HD B H HD HD Table 1: The materials and geometry of the sandwich beams tested in this study (the span 2L = 2mm for all the beams). The specimens labelled M denote monolithic beams, H and HD denote sandwich beams with the H1 and H25 PVC foam cores, respectively and the specimens labelled B are the sandwich beams with a balsa wood core. The specimens are divided into 4 sets, with beams in each set having approximately equal areal masses.

19 σ m H H Balsa Table 2: The power-law coefficients (1) employed to fit the experimental data for the PVC 1 foams and balsa wood in Fig. 3. In all cases a reference strain rate ε = 1s is chosen. (Reproduced from ref. [15]).

20 metal foam projectile aluminium insert P v P d c core h 2L face sheets I- beam clamp Figure 1: Sketch of the sandwich beam geometry, loading and clamping configuration.

21 (a) 35 3 nominal stress, MPa double single 5 Tension Compression nominal strain, % (b) 15 nominal stress, MPa 1 5 H25 c = 1 mm Balsa c = 15 mm H25 c = 5 mm Balsa c = 5 mm H1 c = 5, 1 mm nominal strain, % Figure 2: The quasi-static responses of the sandwich beam constituent materials. (a) The uniaxial compressive and tensile responses of the single and double layer woven glass/vinylester laminates. (b) The uniaxial compressive response of the H1 and H25 PVC foams and the balsa wood used as the sandwich beam cores.

22 1 H25 σ pl, MPa Balsa H ε, s 1 Figure 3: Summary of the strain rate sensitivity of the PVC foams and balsa wood under uniaxial compression (data from ref. [15]). The plateau stress σ pl (defined as the flow stress at a total strain of 15%) is plotted against the applied strain rate.

23 maximum deflection w, mm max M1 M2 tearing M1, h =.85 mm M2, h = 1.5 mm impulse per unit area I, Nsm -2 Figure 4: The measured maximum deflection at the mid-span of the single and double layer woven glass/vinylester monolithic composite beams, as a function of the foam projectile momentum I. Fits to the experimental data using the power law relation (4) are also included.

24 (a) 25 maximum deflection w, mm max H2 H1 tearing H3 H1, h =.85 mm, c = 5 mm H2, h =.85 mm, c = 15 mm H3, h = 1.5 mm, c = 1 mm impulse per unit area I, Nsm -2 Figure 5. The measured maximum deflection at the mid-span of the (a) H1 PVC foam core, (b) H25 PVC foam core and (c) balsa wood core sandwich beams, as a function of the foam projectile momentum I. Fits to the experimental data using the power law relation (4) are also included.

25 (b) maximum deflection w, mm max 2 tearing HD1 15 HD3 HD2 1 5 HD1, h =.85 mm, c = 5 mm HD2, h =.85 mm, c = 15 mm HD3, h = 1.5 mm, c = 5 mm impulse per unit area I, Nsm -2 Figure 5. continued.

26 (c) 3 tearing maximum deflection w, mm max 25 B B2 5 B1, h =.85 mm, c = 5 mm B2, h =.85 mm, c = 15 mm impulse per unit area I, Nsm -2 Figure 5. Continued.

27 (a) travelling flexural wave 1 2 travelling flexural wave Figure 6: The deformation of the M1 monolithic beam impacted by the foam projectile at a velocity 1 v p = 84 ms. (a) High-speed photographic sequences at an inter-frame time of 1 µs. (b) The mid-span deflection versus time history extracted from the photographs in (a). Note that the numbers on the curve correspond to the frames in (a).

28 (b) mid-span deflection w, mm time, µs Figure 6. Continued.

29 (a) cracking Figure 7: The deformation of the H2 sandwich beam (H1 PVC foam core) impacted by the foam projectile at a velocity 1 v p = 24 ms. (a) High-speed photographic sequences at an inter-frame time of 1 µs. (b) The mid-span deflection versus time history extracted from the photographs in (a). Note that the numbers on the curve correspond to the frames in (a).

30 (b) 2 mid-span deflection w, mm time, µs Figure 7. Continued.

31 cracks Face tearing 5 6 Figure 8: High-speed photographic sequence of sandwich beam specimen H2 (H1 PVC foam core) impacted by the foam projectile at a velocity 1 v p = 323 ms. The inter-frame time is 1 µs.

32 (a) face debonding Figure 9: The deformation of the B2 sandwich beam (balsa wood core) impacted by the foam projectile at a velocity 1 v p = 14 ms. (a) High-speed photographic sequences at an interframe time of 1 µs. (b) The mid-span deflection versus time history extracted from the photographs in (a). Note that the numbers on the curve correspond to the frames in (a).

33 (b) mid-span deflection w, mm time, µs Figure 9. Continued.

34 (a) 3 tearing maximum deflection w, mm max M2 B1 H1 Set 1 m 3.5 kgm impulse per unit area I, Nsm -2 Figure 1: Comparison between the measured maximum mid-span deflections w max of the inner face of sandwich and monolithic beams of equal mass. The deflections w max are plotted as a function of the projectile momentum I for the beams in (a) Set 1, (b) Set 2 and (c) Set 3 of Table 1. For the sake of clarity, only the power law fits to the experimental data in Figs. 4 and 5 are included here.

35 (b) 25 maximum deflection w, mm max HD1 B2 H2 tearing Set 2 m 4.3 kgm impulse per unit area I, Nsm -2 Figure 1. Continued.

36 (c) 25 maximum deflection w, mm max H3 HD3 tearing HD2 Set 3 m 6.3 kgm impulse per unit area I, Nsm -2 Figure 1. Continued.