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. High-

speed 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 2005

* Corresponding author, Tel.: +44-1223 332650; fax: +44-1223 765046 Email address: naf1@eng.cam.ac.uk

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 100 MPa and loading times of

approximately 0.1 ms. Using this technique Radford et al. [9] and Rathbun et al. [10] 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].

2

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 KTH1 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 DBLT850 (with equal amounts of fibres in the o0 , o45± and o90 directions) infused

with vinylester resin (overall density of the composite faces was approximately 1700 kgm-3).

1 KTH (Kungl Tekniska Högskolan) Royal Institute of Technology. Valhallavägen 79, Stockholm – Sweden. Courtesy of Prof Dan Zenkert.

3

Sandwich beams were manufactured using three types of sandwich cores supplied by DIAB2;

(i) the H100 Divinycell PVC foam core (density 100 kgm-3), (ii) the H250 Divinycell PVC

foam core (density 250 kgm-3) and (iii) an end-grain balsa wood core (DIAB designation

ProBalsa LD7, density 90 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 0.85 mm, 1.5 mm and

2.20 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, 10 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 H100, H250 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 H250 core sandwich beams are present

in Set 1 while Set 3 contains no balsa wood core sandwich beams or monolithic beams. The

1.5 mm thick monolithic beams ( -22.5 kgmm ≈ ) have no corresponding sandwich beams of

approximately equal areal mass and are assigned Set 0 in Table 1.

The clamped beam experimental set-up is sketched in Fig. 1. All beams tested in this study

had a span 2 200 mmL = and width 30 mmb = . 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. 1.

2 DIAB AB, Box 201, 312 22 Laholm – Sweden.

4

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 3 110 s− − . Rectangular specimens

120 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 100 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

9GPafE = and 14 GPa for the single and double fabric layer materials, respectively. At a

strain of approximately 0.5%, 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 330 MPafσ = and 220 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 150 MPa and 200 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.

5

Compression tests were performed on the H100 and H250 PVC foams and on the balsa wood

at a nominal strain rate of 10-4 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 30 mm × 30 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 90 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 10-4 s-1. The figure includes the measured engineering

stress versus strain curves for the Divinycell H100 foam (specimens of thickness 5 mm and

15 mm), H250 foam (specimens of thickness 5 and 10 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 plσ 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 H100, H250 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 80% for the H100 and the balsa

wood, and about 70% for the H250 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 H100 and H250 Divinycell PVC foams and the ProBalsa LD7

6

was investigated by Tagarielli et al. [15] in a parallel study. They performed a series of

uniaxial compression experiments at strain rates ranging from 4 110 s− − to approximately

4000 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 plσ (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

0 0

,m

plσ εσ ε

⎛ ⎞= ⎜ ⎟⎝ ⎠

(1)

where 0ε is a reference strain rate, 0σ a reference stress, and m is the power law exponent.

With the reference strain rate 10 1sε −= , a least-squares fit to the experimental data provides

the coefficients 0σ and m. These values are listed in Table 2 and the corresponding fits are

included in Fig. 3. The compressive yield strength of the H250 PVC foam and balsa wood

doubles when the strain rate is increased from quasi-static rates (10-4 s-1) to rates on the order

of 103 s-1. In contrast, the H100 PVC foam displays only a small elevation in uniaxial

compressive strength (about 30%) 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]

2

0 ,p pc

D

vp

ρσ

ε= + (2)

7

and

p D

p

l,

vε

τ = (3)

respectively, where pρ and length pl are the projectile density and length respectively while

cσ and Dε 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 pv and length pl . Here we conduct a series of experiments on each beam

configuration by varying pv but keeping all the other foam projectile parameters fixed.

Circular cylindrical projectiles of length mm40≈pl and diameter 28 5 mmd .= were

electro-discharge machined from Alporas foam blocks of density 3kgm300 −≈pρ (projectile

mass kg105.7 3−×≈pm ). Recall that the beams have a width 30 mmb = . Thus, the

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 pv in the range 50 ms-1 to 350 ms-1,

providing a projectile momentum per unit area ppp vlI ρ=0 of up to approximately

3700 Ns m-2. Four to six tests at different values of 0I were conducted on each of the beam

configurations listed in Table 1. In the tests the value of 0I 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-790

image-converter camera was used; this is capable of taking up to 20 frames at a maximum

rate of 17 sframes10 − . Inter-frame times of 100 µs and exposure times of 20 µs were used.

3. Experimental results

At least four levels of impulse 0I were applied to each beam configuration by varying the

foam projectile velocity pv and the deformation modes and deflection time histories of the

back face recorded using high speed photography. The main results from the experimental

8

study are summarised in Figs 4 and Fig.5 for the monolithic and sandwich beams,

respectively. In these figures, the maximum back-face displacements maxw at the mid-span

for each beam are plotted as a function of the initial momentum 0I of the foam projectile.

The critical impulse 0I 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 0.85 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): maxw 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 ( 15 mmc = ) outperform the B1 ( 5 mmc = )

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 100 µs intervals, for the

monolithic specimen M1 of Table 1 impacted at 184 mspv −= . It is seen that impact occurs

between frames 1 and 2, and by frame 3 the metal foam projectile has compressed by nearly

50% 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 10.

The corresponding mid-span deflection versus time history, as inferred from the high-speed

9

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 500 µs and subsequently, there

is elastic recovery due to the reflection of the flexural wave.

PVC foam core sandwich beams

High-speed photographic sequences (100 µs inter-frame time) of the H100 PVC foam core

sandwich beams (specimen type H2) impacted at 1204 mspv −= 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 H100 PVC foam core beam is

illustrated in Fig. 8 via high-speed photographic sequences (100 µs inter-frame time) of an H2

beam impacted at 1323 mspv −= . 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 H250 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 1140 mspv −= by the foam projectile are shown in Fig. 9 at an inter-frame time of 100 µs.

The mechanism of deformation is similar to that of the PVC foam core sandwich beams with

10

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. 10 where the maximum mid-span deflection of the back faces

of the beams maxw is plotted a function of the projectile momentum 0I . For the sake of clarity

we do not include the actual experimental measurements. Rather, a power law relation of the

form

max 0

ref ref

,n

w Iw I

⎛ ⎞= ⎜ ⎟⎝ ⎠

(4)

where refw and refI 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. 10. 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.

The findings from the comparisons in Fig. 10 are summarised as follows: 2Set 1 ( 3.5 kgm , Fig. 10a)m −≈ : The H100 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 H100 core sandwich beams.

11

2Set 2 ( 4.3 kgm , Fig. 10b)m −≈ : The 15 mmc = H100 core sandwich beams (H2) have the

highest dynamic strength and outperform the 5 mmc = H250 foam core beams (HD1). This

clearly indicates that even though the H250 foam has a higher strength, the increased core

thickness of the H100 beams gives these beams a higher specific strength. The 15 mmc =

balsa wood core beams (B2) also outperform the HD1 beams at low values of 0I but

delamination failure makes the B2 beams weaker than the HD2 beams at higher impulses.

2Set 3 ( 6.3 kgm , Fig. 10c)m −≈ : The HD2 beams have the best performance in this set as

they are made from the H250 core of thickness 15 mmc = (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 ( 10 mmc = versus 5 mmc = )

compensates for the lower strength of the H100 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 H100 core (density 3100 kgm− ) and a high strength

H250 core (density 3250 kgm− ). High-speed photography was used to measure the transient

transverse deflection of the beams and determine the dynamic modes of deformation.

12

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 H100

PVC foam core outperform sandwich beams with the high strength H250 PVC foam core.

This is attributed to the higher core thicknesses of the H100 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.

13

References

[1] Steeves C.A. and Fleck N.A., 2004. 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, 46:561-583.

[2] Tagarielli V.L., Fleck N.A. and Deshpande V.S., 2004. Collapse of clamped and simply

supported composite sandwich beams in three-point bending, Composites B: Engineering, 35,

523-534.

[3] Wang A. J., Hopkins H. G., 1954. On the plastic deformation of built-in circular plates

under impulsive load, Journal of Mechanics and Physics of Solids 3, 22-37.

[4] Symmonds P. S., 1954. Large plastic deformation of beams under blast type loading.

Second US National Congress of Applied Mechanics.

[5] Fleck N.A. and Deshpande V.S., 2004. The resistance of clamped sandwich beams to

shock loading, J. of Applied Mechanics, ASME, 71:386-401.

[6] Xue Z., Hutchinson J. W., 2003. Preliminary assessment of sandwich plates subject to

blast loads. International Journal of Mechanical Sciences 45, 687-705.

[7] Taylor, G. I., 1941. The scientific papers of G I Taylor, Vol III, Cambridge University

Press, 1963, The pressure and impulse of submarine explosion waves on plates, pp. 287-303.

[8] Radford, D. D., Deshpande, V. S. and Fleck, N. A., 2005. The use of metal foam

projectiles to simulate shock loading on a structure, International Journal of Impact

Engineering, 31(9):1152-1171.

[9] Radford D. D., Fleck N. A., Deshpande V. S., 2005. The response of clamped sandwich

beams subjected to shock loading, International Journal of Impact Engineering (in print).

[10] 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., 2005. 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., 2005, The response of

clamped sandwich plates with lattice cores subjected to shock loading, European Journal of

Mechanics A:/Solids. (in print).

[12] Mäkinen, K., 1999. Underwater shock loaded sandwich structures, PhD thesis, Royal

Institute of Technology, Department of Aeronautics, Stockholm, Sweden.

14

[13] Deshpande, V.S. and Fleck, N.A., 2001. Multi-axial yield behaviour of polymer foams,

Acta Materialia, 49(10), 1859-1866.

[14] Tagarielli V.L., Deshpande V.S., Fleck N.A and Chen, 2005. A transversely isotropic

model for foams, and its application to the indentation of balsa, Int. J. Mech. Sci., 47: p. 666-

686.

[15] Tagarielli V.L., Fleck N.A. and Deshpande V.S., 2005. 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.

15

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 H100 and H250

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 plσ (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 0I . 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) H100 PVC foam core,

(b) H250 PVC foam core and (c) balsa wood core sandwich beams, as a function of the foam

projectile momentum 0I . 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 184 mspv −= . (a) High-speed photographic sequences at an inter-frame time of

100 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 (H100 PVC foam core) impacted by the

foam projectile at a velocity 1204 mspv −= . (a) High-speed photographic sequences at an

inter-frame time of 100 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).

16

Figure 8: High-speed photographic sequence of sandwich beam specimen H2 (H100 PVC

foam core) impacted by the foam projectile at a velocity 1323 mspv −= . The inter-frame time

is 100 µs.

Figure 9: The deformation of the B2 sandwich beam (balsa wood core) impacted by the foam

projectile at a velocity 1140 mspv −= . (a) High-speed photographic sequences at an inter-

frame time of 100 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 10: Comparison between the measured maximum mid-span deflections maxw of the

inner face of sandwich and monolithic beams of equal mass. The deflections maxw are plotted

as a function of the projectile momentum 0I 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

2 200 mmL = for all the beams). The specimens labelled M denote monolithic beams, H and

HD denote sandwich beams with the H100 and H250 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

foams and balsa wood in Fig. 3. In all cases a reference strain rate 10 1sε −= is chosen.

(Reproduced from ref. [15]).

Beam designation

Face sheet thickness, h [mm]

Core thickness, c [mm]

Areal density, m[kgm-2]

Set 0, m ≈ 2.5 kgm-2 M1 0.85 - 2.55

M2 1.5 - 3.74 H1 0.85 5 3.39 Set 1,

m ≈ 3.5 kgm-2 B1 0.85 5 3.43 H2 0.85 15 4.39

HD1 0.85 5 4.14 Set 2, m ≈ 4.3 kgm-2

B2 0.85 15 4.33 H3 1.5 10 6.10

HD2 0.85 15 6.64 Set 3, m ≈ 6.3 kgm-2

HD3 1.5 5 6.35

Table 1: The materials and geometry of the sandwich beams tested in this study (the span

2 200 mmL = for all the beams). The specimens labelled M denote monolithic beams, H and

HD denote sandwich beams with the H100 and H250 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

foams and balsa wood in Fig. 3. In all cases a reference strain rate 10 1sε −= is chosen.

(Reproduced from ref. [15]).

σ 0 m

H100 1.95 0.016

H250 7.44 0.048

Balsa 5.87 0.056

I- beamclamp

metal foam projectile

core

aluminium insertvP

face sheetsh

c

2L

P

d

I- beamclamp

metal foam projectile

core

aluminium insertvP

face sheetsh

c

2L

P

d

Figure 1: Sketch of the sandwich beam geometry, loading and clamping configuration.

0

50

100

150

200

250

300

350

0 0.5 1 1.5 2 2.5 3

TensionCompression

nom

inal

stre

ss, M

Pa

nominal strain, %

(a)

double

single

0

50

100

150

200

250

300

350

0 0.5 1 1.5 2 2.5 3

TensionCompression

nom

inal

stre

ss, M

Pa

nominal strain, %

(a)

double

single

0

5

10

15

0 20 40 60 80 100

nom

inal

stre

ss, M

Pa

nominal strain, %

c = 5 mm

c = 10 mm

c = 5, 10 mm

c = 5 mm

c = 15 mm

H100

H250

Balsa

BalsaH250

(b)

0

5

10

15

0 20 40 60 80 100

nom

inal

stre

ss, M

Pa

nominal strain, %

c = 5 mm

c = 10 mm

c = 5, 10 mm

c = 5 mm

c = 15 mm

H100

H250

Balsa

BalsaH250

(b)

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 H100 and H250

PVC foams and the balsa wood used as the sandwich beam cores.

1

10

10-4 10-3 10-2 10-1 100 101 102 103

H250

Balsa

H100

,

MPaplσ

1, sε −

1

10

10-4 10-3 10-2 10-1 100 101 102 103

H250

Balsa

H100

,

MPaplσ

1, sε −

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 plσ (defined as the flow stress

at a total strain of 15%) is plotted against the applied strain rate.

0

5

10

15

20

25

0 500 1000 1500 2000 2500

M1, h = 0.85 mm

M2, h = 1.5 mm

impulse per unit area I0, Nsm-2

max

imum

def

lect

ion

wm

ax ,

mm

tearing

M1

M2

0

5

10

15

20

25

0 500 1000 1500 2000 2500

M1, h = 0.85 mm

M2, h = 1.5 mm

impulse per unit area I0, Nsm-2

max

imum

def

lect

ion

wm

ax ,

mm

tearing

M1

M2

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 0I . Fits to the experimental data using the power law relation (4) are also

included.

0

5

10

15

20

25

0 500 1000 1500 2000 2500 3000 3500 4000

H1, h = 0.85 mm, c = 5 mmH2, h = 0.85 mm, c = 15 mmH3, h = 1.5 mm, c = 10 mm

impulse per unit area I0, Nsm-2

max

imum

def

lect

ion

wm

ax ,

mm

tearing

H1

H2 H3

(a)

0

5

10

15

20

25

0 500 1000 1500 2000 2500 3000 3500 4000

H1, h = 0.85 mm, c = 5 mmH2, h = 0.85 mm, c = 15 mmH3, h = 1.5 mm, c = 10 mm

impulse per unit area I0, Nsm-2

max

imum

def

lect

ion

wm

ax ,

mm

tearing

H1

H2 H3

(a)

Figure 5. The measured maximum deflection at the mid-span of the (a) H100 PVC foam core,

(b) H250 PVC foam core and (c) balsa wood core sandwich beams, as a function of the foam

projectile momentum 0I . Fits to the experimental data using the power law relation (4) are

also included.

0

5

10

15

20

0 500 1000 1500 2000 2500 3000 3500 4000

HD1, h = 0.85 mm, c = 5 mmHD2, h = 0.85 mm, c = 15 mmHD3, h = 1.5 mm, c = 5 mm

impulse per unit area I0, Nsm-2

max

imum

def

lect

ion

wm

ax ,

mm

tearing

HD3

HD1

HD2

(b)

0

5

10

15

20

0 500 1000 1500 2000 2500 3000 3500 4000

HD1, h = 0.85 mm, c = 5 mmHD2, h = 0.85 mm, c = 15 mmHD3, h = 1.5 mm, c = 5 mm

impulse per unit area I0, Nsm-2

max

imum

def

lect

ion

wm

ax ,

mm

tearing

HD3

HD1

HD2

(b)

Figure 5. continued.

0

5

10

15

20

25

30

0 500 1000 1500 2000 2500

B1, h = 0.85 mm, c = 5 mm

B2, h = 0.85 mm, c = 15 mm

impulse per unit area I0, Nsm-2

max

imum

def

lect

ion

wm

ax ,

mm

tearing

B1

B2

(c)

0

5

10

15

20

25

30

0 500 1000 1500 2000 2500

B1, h = 0.85 mm, c = 5 mm

B2, h = 0.85 mm, c = 15 mm

impulse per unit area I0, Nsm-2

max

imum

def

lect

ion

wm

ax ,

mm

tearing

B1

B2

(c)

Figure 5. Continued.

1 2

travelling flexural wave

4

6

7 8

9 10

3

travelling flexural wave

5

(a)

1 2

travelling flexural wave

4

6

7 8

9 10

3

travelling flexural wave

5

1 2

travelling flexural wave

4

6

7 8

9 10

3

travelling flexural wave

5

(a)

Figure 6: The deformation of the M1 monolithic beam impacted by the foam projectile at a

velocity 184 mspv −= . (a) High-speed photographic sequences at an inter-frame time of

100 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).

0

5

10

15

0 200 400 600 800 1000

mid

-spa

n de

flect

ion

w, m

m

time, µs

(b)

1

2

3

4 5 6

7

8

9

10

0

5

10

15

0 200 400 600 800 1000

mid

-spa

n de

flect

ion

w, m

m

time, µs

(b)

1

2

3

4 5 6

7

8

9

10

Figure 6. Continued.

5 6

7 8

3 4

21

(a)

cracking

5 6

7 8

3 4

21

(a)

cracking

Figure 7: The deformation of the H2 sandwich beam (H100 PVC foam core) impacted by the

foam projectile at a velocity 1204 mspv −= . (a) High-speed photographic sequences at an

inter-frame time of 100 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).

0

5

10

15

20

0 200 400 600 800 1000

mid

-spa

n de

flect

ion

w, m

m

time, µs

(b)

2

3

4

5 6

7

8

0

5

10

15

20

0 200 400 600 800 1000

mid

-spa

n de

flect

ion

w, m

m

time, µs

(b)

2

3

4

5 6

7

8

Figure 7. Continued.

1 2

5

3

6

Face tearing

4 cracks

1 2

5

3

6

Face tearing

4 cracks

Figure 8: High-speed photographic sequence of sandwich beam specimen H2 (H100 PVC

foam core) impacted by the foam projectile at a velocity 1323 mspv −= . The inter-frame time

is 100 µs.

1 2

6

7 8

3 4

5

face debonding

(a)

1 2

6

7 8

3 4

5

face debonding

(a)

Figure 9: The deformation of the B2 sandwich beam (balsa wood core) impacted by the foam

projectile at a velocity 1140 mspv −= . (a) High-speed photographic sequences at an inter-

frame time of 100 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).

0

5

10

15

20

0 200 400 600 800 1000

mid

-spa

n de

flect

ion

w, m

m

time, µs

(b)

5

2

3

4

6 78

0

5

10

15

20

0 200 400 600 800 1000

mid

-spa

n de

flect

ion

w, m

m

time, µs

(b)

5

2

3

4

6 78

Figure 9. Continued.

tearing

0

5

10

15

20

25

30

0 500 1000 1500 2000 2500

impulse per unit area I0, Nsm-2

max

imum

def

lect

ion

wm

ax ,

mm

(a)

M2 H1

B1

Set 1m 3.5 kgm-2≈

tearing

0

5

10

15

20

25

30

0 500 1000 1500 2000 2500

impulse per unit area I0, Nsm-2

max

imum

def

lect

ion

wm

ax ,

mm

(a)

M2 H1

B1

Set 1m 3.5 kgm-2

Set 1m 3.5 kgm-2≈

Figure 10: Comparison between the measured maximum mid-span deflections maxw of the

inner face of sandwich and monolithic beams of equal mass. The deflections maxw are plotted

as a function of the projectile momentum 0I 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.

0

5

10

15

20

25

0 500 1000 1500 2000 2500 3000

max

imum

def

lect

ion

w max

, m

m

impulse per unit area I0, Nsm-2

tearing

(b)

HD1H2

B2

Set 2m 4.3 kgm-2≈

0

5

10

15

20

25

0 500 1000 1500 2000 2500 3000

max

imum

def

lect

ion

w max

, m

m

impulse per unit area I0, Nsm-2

tearing

(b)

HD1H2

B2

Set 2m 4.3 kgm-2

0

5

10

15

20

25

0 500 1000 1500 2000 2500 3000

max

imum

def

lect

ion

w max

, m

m

impulse per unit area I0, Nsm-2

tearing

(b)

HD1H2

B2

Set 2m 4.3 kgm-2

Set 2m 4.3 kgm-2≈

Figure 10. Continued.

0

5

10

15

20

25

0 500 1000 1500 2000 2500 3000 3500 4000

max

imum

def

lect

ion

wm

ax ,

mm

impulse per unit area I0, Nsm-2

tearing

(c)

HD2

H3

HD3

Set 3m 6.3 kgm-2≈

0

5

10

15

20

25

0 500 1000 1500 2000 2500 3000 3500 4000

max

imum

def

lect

ion

wm

ax ,

mm

impulse per unit area I0, Nsm-2

tearing

(c)

HD2

H3

HD3

Set 3m 6.3 kgm-2

0

5

10

15

20

25

0 500 1000 1500 2000 2500 3000 3500 4000

max

imum

def

lect

ion

wm

ax ,

mm

impulse per unit area I0, Nsm-2

tearing

(c)

HD2

H3

HD3

Set 3m 6.3 kgm-2

Set 3m 6.3 kgm-2≈

Figure 10. Continued.