Length of central crack Side length of wedge Number of axial splits Current crushing load Mean post-crushing load Peak load Adhesive energy per unit area of layers Radius of curvature of the frond Displacement Wall thickness of square frustum Crush speed Energy absorbed Total energy dissipated Energy required for the crush zone formation Angle of wedge Strain Semi-apical angle of square frustum Static friction coefficient Dynamic friction coefficient Tensile fracture stress Normal stress INTRODUCTION

Components made of composite materials have been recognised to possess significant crashworthy characteristics accompanied by weight and cost effectiveness, l The influence of various parameters on the crashworthy ability of composite thin-walled structures was examined in detail, but conflicting results were produced. 2~ In Ref. 5 a simple theoretical approach is proposed, based on an equation similar to the buckling load equation for a column on an elastic foundation, which can be used to predict whether a change in energy absorption of a composite tube occurs as a result of changes in geometry or material properties. Over the last decade, the authors have been extensively dealing, both experimentally and theoretically, with the crashworthy behaviour of fibreglass composite components in the form of shells of various geometries i.e. circular tubes and frusta, square tubes and frusta, hourglass cross sections, etc., subjected to axial collapse and/or bending. 6-13 The main purpose was to study their behaviour under various loading conditions, collecting important crashworthy information on the influence of structural factors such as the properties of the fibres and resins and the fibres' orientation as well as the geometry and material characteristics. In this manner, a

Energy absorption of fibreglass composite squarefrusta

271

reliable data-bank has been constructed, providing the designer with the ability to effectively propose structures of composite material with safe crashworthy features under certain loading conditions. In the present paper the crush behaviour of square frusta, made of a fibreglass/polyester composite material and subjected to static and dynamic loading with strain-rates ranging between 10-3 and 50 s-~, is reported. The effect of the specimen geometry and the loading rate on the energy absorbing capability was experimentally investigated. The collapse modes at the macroscopic scale and the determination of the microfailures during the failure process of the material tested were also observed and analysed. A theoretical analysis of the failure mechanism pertaining to the stable mode of collapse of thin-walled fibreglass composite shells subjected to axial compression, for calculation the crushing loads and the absorbing energy during collapse, is also reported. Comparison between theory and experiment was good, indicating, therefore, that the proposed theoretical model may be efficient for predicting the energy absorbing capacity of the collapsed shell.

EXPERIMENTAL The material tested was a fibreglass composite material with individual fibre diameter of 9 #m chopped strand mat with random fibre orientation in the plane of the mat. The shells were fabricated by a hand lay-up technique using pieces of fibreglass cloth (0.8 g/mm 2) and impregnating it with a polyester resin, thus providing a composite material of 72% per weight fibre content and 1.37 g/cm 3 density. Details about the fabrication of the composite shell are presented in Ref. 6. The stress-strain curve as obtained from quasi-static tension test for the material tested is shown in Fig. 1. The static axial collapse was carried out between the parallel steel platens of a SMG hydraulic press at a crosshead speed of 10mm/min or a compression strain-rate of 10 -3 S-1. The corresponding dynamic tests were performed by direct impact on a drop-hammer at velocities exceeding 1 m/s. The existing drop-hammer facility with a 75 kg falling mass from a maximum drop height of 4 m provides a maximum impact velocity of about 10 m/s. The experimental set-up and measuring devices used throughout the present tests are described in detail in Ref. 8. Load/shell shortening (displacement) curves during the crushing process were automatically measured and recorded for both types of loading. The values of the initial peak load Pmax and the energy absorbed W for the axially collapsed specimens, obtained by measuring the area under the load/displacement curve, as well as the mean post-crushing load P (defined as the ratio of energy absorbed to the total shell shortening)

272

A. G. Mamalis, D. E. Manolakos, G. A. Demosthenous, M. B. loannidis

0.20

E E
Z
v

0.1 6

-

0.12

~
C

0.08

0.0

0

0.00/.,

0.008

0.0.12

0.016

0.020

Tensite

strain

Fig. 1. Tensilestress-strain curve of compositematerial tested. and the specific energy Ws (equal to the energy absorbed per unit mass crushed, calculated as the crushed volume multiplied by the density of the material) are tabulated in Table 1. Photographs showing characteristic terminal views of the deformed specimens for all series of experiments are presented in Fig. 2. Typical micrographs of the crush zone showing the main microfailures, as obtained using a Unimet metallographic optical microscope, are also shown in Figs 3-6. For details about specimens preparation and polishing for the microscopic observations see Ref. 7.

FAILURE MECHANISM: EXPERIMENTAL OBSERVATIONS Four distinct collapse modes at macroscopic scale, designated as Mode I, II, III and IV, respectively, were observed throughout the axial static and

Energy absorption offibreglass composite squarefrusta

273

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.

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~

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274

A. G. Mamalis, D. E. Manolakos, G. A. Demosthenous, M, B. Ioannidis

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6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
~2

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+ ..C t"q II

Energy absorption of fibreglass composite squarefrusta

275

0

L)

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0

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e~

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A

~

°

~

e4

276

A. G. Mamalis, D. E. Manolakos, G. A. Demosthenous, M. B. loannidis

(d)

50 mm (e)

(f)

Fig. 2. (d) Mode II (sp. 17); (e) Mode III (sp. 2); (f) Mode IV (sp. 1).

dynamic loading of square frusta. These modes are essentially similar to those observed in previous investigations concerning circular frusta, see Ref. 6. These collapse modes mainly depend upon the wall thickness (number of layers), the semi-apical angle, 0, the axial length, L, and the mean circumference, C, of the shell and the testing conditions. Typical load/displacement curves for each mode of collapse for static and dynamic loading of both geometries are shown in Fig. 7. Initially the shell behaves elastically and the load rises at a steady rate to a peak value, Pmax, and then drops abruptly; the magnitude of the peak load is greatly affected by the shell geometry, the material characteristics and the corners' rigidity. As deformation progresses the shape of the load/displacement curve depends on the mode of collapse; for thin-walled composite shells it was observed that the fracture behaviour of the shell appears to affect the loading stability as well as the magnitude of the crush load and the energy absorption during the crushing process. It may be assumed that at any instant the crush load must be supported by more than one structural element of the shell and furthermore, a specific element contributes more to supporting the load; see also Ref. 14.

Progressive end-crushing (Mode I)
Two different modes of failure were observed concerning the above mentioned progressive crushing mode. Mode Ia of failure, similar to a "mushrooming" failure, is characterised by progressive collapse through

Energy absorption of fibreglass composite squarefrusta

277

L,.L.. F" ///Y//////V/l

[///////////J

0=5 °

=

0

(a) Debris wedge

(c)

Ext~ fron

Lc

~a{

/-- t--4 4
(b} (d)
Fig. 3. (a) Configuration of the crush zone through the wall thickness of a 5° specimen (cross section AA' in Fig. 2(a)); (b) Micrograph showing microfailures in the crush zone (Mode Ia) for a 5 ° statically loaded specimen (sp. 4) in a cross section at the middle of the square side; (c) Configuration of the crush zone through the wall thickness of a 10° specimen (cross section AA' in Fig. 2(a)); (d) Micrograph showing microfailures in the crush zone (Mode Ia) for a 5° statically loaded specimen (sp. 9) in a cross section at the middle of the square side.

278

A. G. Mamalis, D. E. Manolakos, G. A. Demosthenous, M. B. Ioannidis

L,// , ,///////////1

i////////////////i

5o

(a)

(c)

Externol.

Debris wedge

(b)

(d)

Fig. 4. (a) Configuration of the crush zone (Mode Ia) through the wall thickness of a 15 ° specimen (cross section AA' in Fig. 2(a)); (b) Micrograph showing microfailures in the crush zone (Mode Ia) for a 15° statically loaded specimen (sp. 13) in a cross section at the middle of the square side; (c) Configuration of the crush (Mode Ib) zone through the wall thickness of a 15° specimen (cross section AA' in Fig. 2(a)); (d) Micrograph showing microfailures in the crush zone (Mode Ib) for a 15° dynamically loaded specimen (sp. 33) in a cross section at the middle of the square side.

Energy absorption of fibreglass composite squarefrusta

279

b//////

/////

// /A

(a)

,gitudinal. Lcks

zone

/ - t .../

(b)
Fig. 5. (a) Configuration of the crush zone in the corner of a 15° specimen (cross section BB' in Fig. 2(a)); (b) Micrograph showing microfailures in the crush zone (Mode Ia) for a 15 ° statically loaded specimen (sp. 13) in a cross section at the corner of the frustum.

the formation of continuous fronds which spread outwards and inwards; see Fig. 2(a) and (b) and also Refs 6-12. At the stage that the crush load reaches the peak value, Pmax, cracks form at each of the four corners accompanied by the formation of a circumferential intrawall crack at the end of the shell adjacent to the loading area leading to the sharp drop of the load, see Fig. 7(a). As deformation proceeds further, the externally

280

A. G. Mamalis, D. E. Manolakos, G. A. Demosthenous, M. B. Ioannidis

Extern frond

Debris wedge

it

Fig.

6. Micrograph showing microfailures in the crush zone (Mode Ia) for a 5 ° dynamically loaded specimen (sp. 23) in a cross section at the middle of the square side.

formed fronds curl downwards with the simultaneous development of four axial splits followed by splaying of the material strips, see Fig. 2(a) and (b); note that the splits are always located at the four corners of the narrow end of the shell probably due to local stress concentration during the very early stage of straining. Axial tears were not apparent in the internal fronds which were more continuous than their external counterparts. The post-crushing regime is characterised by the formation of two lamina bundles bent inwards and outwards due to the flexural damage which occurs at a distance from the contact surface equal to the wall thickness; they withstand the applied load and buckle when the load or the length of the lamina bundle reaches a critical value. At this stage, a triangular debris wedge of pulversied material starts to form, see Figs 3 and 4(a) and (b); its formation may be attributed to the friction between the bent bundles and the pattern of the press or the drop mass. As loading proceeds further, resulting in crushing with the subsequent formation of the internal and external fronds, normal stresses develop on the sides of the debris wedge followed by shear stresses along the same sides due to the friction at the interface between the wedge and the fronds. Note, also, that additional normal and shear stresses develop at the interface between the steel press platen or the dropped mass and the deforming shell as the

Energy absorption of fibreglass composite squarefrusta
80

281

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curves for the various collapse modes.

formed fronds slide along this interface. The crushing load in the postcrushing region is characterised by oscillations about a mean postcrushing load, P; these oscillations start as soon as the formation of the debris wedge is completed. Regarding the microfracture mechanism of the square frusta subjected to axial loading, as far as the Mode Ia of collapse is concerned, the experimental observations made are similar to those obtained during the axial collapse of circular frusta; see also the remarks reported in Refs 6-8. Since the circumference of the shell increases as the crushing of the frusta progresses, it is evident that the size of the wedge increases during crushing. With increasing the semi-apical angle of the frustum, the position of the intrawall crack moves towards the outside edge of the shell wall increasing in this manner the thickness of the inner frond and simultaneously resulting in a positioning of the annular wedge mainly above it. On the contrary, the crack length decreases with increasing

282

A. G. Mamalis, D. E. Manolakos, G. A. Dernosthenous, M. B. loannidis

semi-apical angle; compare Figs 3(b), (d) and 4(b). From measurements of the specimens tested, the wedge angle, ~, is about 80, 70 and 60 ° for 5, 10, and 15 ° frusta, respectively. However, some distinct differences were obtained analogous to the square tubes, see Ref. 12, resulting from a microfracture mechanism as a combination of the main characteristics observed for square tubes and circular frusta; the main intrawall crack length diminishes from the centre of the square side towards the square corners, where the typical crush zone disappears, see Fig. 5. The maximum value of the crack length, Lc, which is attained at the middle of each side of the square cross-section, is almost the same as the corresponding one observed in the case of circular frusta of the same semi-apical angle and loaded under the same conditions. The combined mechanism developed for the various regions of a compressed square frustum during the crushing process is described schematically in Fig. 2(a). The load/displacement curves have several similar features with the corresponding square tubes made of the same material. However, it must be noted that a distinct difference pertaining to the post-crushing region of the curves between square tubes and frusta (with large semi-apical angle) was observed. The crushing load tended to increase as crushing progressed due to increase of the effective cross-sectional area of the frusta being crushed, compare Fig. 7(a) with Fig. 4 of Ref. 12. Dynamically obtained load/displacement curves show more severe fluctuations with troughs and peaks than the corresponding statically ones, see also Fig. 7(a). The duration of the crushing phenomena depends upon the shell geometry, material properties and the dropped mass weight and height. However, from the shape of the dynamically obtained curves at the post-crushing region, it causes difficulties as far as the possible fracture mechanism occurred, as well as the development and propagation of microcracks during dynamic loading. As reported in Ref. 12, two different fracture mechanisms may be proposed, based on observations related to crack propagation in composite materials neglecting friction forces and bending as well as microscopic observations concerning the collapse mode of shells. The Mode Ib of collapse was mainly observed for dynamically loaded specimens for higher semi-apical angles. Progressive collapse is created by successive shearing of the region near the narrow end of the shell accompanied by a number of delaminations and longitudinal cracks whilst the tube wall inverses inwards; see Figs 2(c) and 4(c) and (d). Based on the above description concerning the micromechanism occurring during the axial collapse of the square frusta as far as the collapse Mode I is concerned, the total energy dissipated may be attributed to:

Energy absorption of fibreglass composite squareJrusta

283

• the energy absorbed during the propagation of the various cracks, the development of delaminations and the axial splitting of the shell; • the energy absorbed because of microfractures and/or flexural damage and the bending of the plies of the fronds and • the energy absorbed due to the development of frictional stresses at the various regions as outlined above.

Longitudinal corner cracking (Mode II)
The longitudinal corner cracking mode is characterised by the formation of a crack at the comer of the frustum which propagates towards the narrow end of the specimen, see Fig. 2(d). A single corner started to crack, usually followed by the cracking of the diametrically opposite corner, whilst subsequently the other two corners start to collapse provided that the crushing is allowed to continue. The load/displacement curves for Mode II of collapse do not display as much energy absorbing potential as the Mode I curves; compare Fig. 7(a) and (b). While the load was developed abruptly and sometimes there was a distinctive Pmax, most often the collapse started as the load reached its maximum level and then the extent of progressive crushing was terminated at the early stage of shell shortening, see Fig. 7(b), exhibiting therefore, rather low energy absorption capability.

Mid-length collapse mode (Mode III)
Specimens following this collapse mode exhibited extensive brittle fracture with a circumferential fracture of the material, see Fig. 2(e). Fracture started at a distance from the loaded end of the specimens, approximately equal to the mid-height of the shell, and involved catastrophic failure by cracking and separation of the shell into irregular shapes, probably due to local severe shear straining of the wall of the shell. This failure mode is similar to the Euler column-buckling of very thin metallic and PVC tubes subjected to axial loading. 15 The load/displacement curves corresponding to mid-length collapse mode showed a typical pre-crushing region, but the initial elastic response was followed by a very sharp drop in load and poor post-crushing characteristics, see Fig. 7(c).

Progressive folding (Mode IV)
The progressive folding mode of collapse is characterised by the formation of a series of folds or fracture hinges as the frustum crushed, see Fig. 2(f). The hinges seem to form in regions where the relative maximum deflections occurred during the elastic deformation of the shell. Fairly

284

A. G. Mamalis, D. E. Manolakos, G. A. Demosthenous, M. B. loannidis

regular sized plates were formed between the hinges and remained relatively undamaged. This mode of collapse is similar to the collapse of metal and thermoplastic tubes and frusta where plastic hinges are formed and the specimen proceeds to fold as the crushing proceeds.15 The load/ displacement curves for Mode IV showed large fluctuations and the average collapse load was relatively small, see Fig. 7(d). Note that of the four failure modes observed, the end-crushing mode (Mode I) was found to offer the highest energy absorption capability.

F A I L U R E ANALYSIS

Static axial collapse
Various crashworthy phenomena pertaining to the axial collapse of composite multi-layered shells are associated with the distribution of the absorbed energy during the crushing process. In the proposed theoretical approach, see Fig. 3(a), the following crushing phenomena were encountered: friction between the annular wedge and the fronds and between the fronds and the platen of the press, fronds bending, crack propagation and axial splitting. The theoretical model proposed in Refs l0 and 12 for the analysis of composite circular and square tubes respectively, subjected to static axial compression was modified and used to analyse the collapse mechanism and to estimate the related energy absorbed during the axial crushing of the square frusta. During the elastic deformation of the shell the load rises at a steady rate to a peak value, Pmax, see Fig. 7. At this stage, cracks of length Lc form at the four corners of the shell and propagate downwards along the frustum axis, splitting the shell wall, see Fig. 2(b); they are accompanied by the development of a circumferential main intrawall crack at the top end of the shell, whilst the related shell shortening is sl, see Fig. 7(a). It is assumed that the crack length distribution along the circumference of the frustum follows an elliptical path as shown in Fig. 2(a). The maximum crack length Lc, is attained in the middle of each side of the square cross section and it is almost equal to the corresponding one observed in the case of the equivalent circular frusta loaded under the same conditions, see Refs 6 and 7. Therefore, the associated absorbed energy which equals the external work, as obtained by measuring the area under the load/ displacement curve in the elastic regime in Fig. 7(a), is

WL~= 2. [ft. Lc" (b2/2)] • Rad -[--n. (t/2) • G. Lc = ~

emaxds

(1)

Energy absorption of fibreglass composite squarefrusta

285

where, following the Notation, Rad is the fracture energy required to fracture a unit area of the adhesive at the interface between two adjacent layers calculated by applying fracture theory,16 b2 the shell top side width, n the number of splits, G the fracture toughness and t the shell wall thickness. The energy required for the deformation mechanism regarding the history of the formation of the crush zone, see Ref. 10, equals the external work absorbed by the deforming shell in this regime, i.e.
Wtr =

j;+0

o'0" ls," (1,,/2)dO

(2)
+

Io-° a0" 1,2 " (ls2/2) dO

• 4. (b2 + s- tan 0) =

I' P ds

where, according to the Notation, a0 is the normal stress applied by the wedge to fronds, ls, the side length of the wedge inscribed to the external bent frond, 1~ the side length of the wedge inscribed to the internal bent 2 frond, el the angle formed by the height and the external side of the wedge, e2 the angle formed by the height and the internal side of the wedge, 0 the semi-apical angle of the frustum, see Figs 2(a) and 3(a); and s2 is the related shell shortening corresponding to the completion of the wedge formation, see Fig. 7(a). Since the intrawall crack propagates at a constant speed equal to the speed of the crosshead of the press, it may be assumed that the crack length Lc remains constant. Also the height of the wedge, h, equals 0-7. (t/cosO) as observed experimentally, see Figs 3 and 4. It must be noted that the position of the occurance of the main intrawall crack, moves from shell wall axis towards the outside edge of the shell wall with increasing semi-apical angle of the frustum whilst the crack length, Lc, becomes shorter. Therefore, taking into account the failure mechanism outlined above, the total dissipated energy for a crush distance, s, can be estimated as follows.

(i) Energy dissipated due to friction between the annular wedge and fronds and between fronds and platen
Wi = (#s, " (P1 + P2) + m2" (P3 + P4))" 4- b. (s - s2)

(3)

where P1 and P2 are the normal forces per unit length applied by the platen to the internal and external fronds, respectively, which result from static equilibrium, see also Ref. 10; P3 and P4 are the normal forces per unit length applied to the internal and external sides of the wedge,

286

A. G. Mamalis, D. E. Manolakos, G. A. Demosthenous, M. B. Ioannidis

respectively, Ps, is the coefficient o f friction between frond and platen, #s2 the coefficient o f friction between the wedge and the fronds and b = [(b2 - t) + 12- 0. t/rc] + s . tan0. N o t e that P3 -- 00" 1,., P4 = °'0" ls~ and ao = k . o o ,

(4) (5)

(6)

therefore, Wi = {#~, • [(Wtotaff4 • b ' s) - (0-7. t - k - o 0 / c o s 0 ) • (tan~l + tan~2 + 2. #s2)] + #~, " (0.7. t. k . a 0 / c o s 0 ) • (1/cos0q + 1/COS~Z2)}" 4" b - ( s - s2) where Wtotal is the total energy dissipated for the d e f o r m a t i o n o f the shell, k is a c o n s t a n t and 00 the tensile fracture stress o f the c o m p o s i t e material.
(ii) Energy dissipated due to fronds bending

(7)

Wii

=

{io,+0P3" (Iv,/2) dO +
+

i?
-

P4" (Is,/2) dO

I/ z [P3" (~1 + 0) + P4" (0~2 -- 0)] ds

}

• 4. b

(8)

= (2.8. t. k . 0o" b / c o s O ) . {[(0q + 0 ) / c o s ~ l ) ] • [0.35. t/(cos~l • c o s 0 ) + s
$2]

+ [(0~2 - 0)/cos0~2)]. [0.35. t/(cos0~2 • c o s 0 ) + s - s2]}.

(iii) Energy dissipated due to crack propagation

Wiii= Rad" [ ( s - s 0 " 4 " b + 7r. Lc" b].
(iv) Energy dissipated due to axial splitting

(9)

Wiv = 4- ( t / 2 - 6 . O . t / n ) . G . s.

(lO)

Energy absorption of fibreglass composite squarefrusta

287

F r o m eqns (3)-(10) the total energy dissipated for the deformation of the shell is given as
Wtota I : W i + Wii + Wiii-[- Wiv

(ll)

whilst the total normal force applied by the platen to the shell can be calculated as
P = mtotal/S. (12)

The friction coefficients #s, and #s: were estimated according to the experimental method proposed in Ref. 17 as/~s, = 0.3 and #.~. = 0-55. The experimental value for Rad is 0-011 kJ/mm 2, for G is 0.020 kJ/mm:, for the tensile fracture stress, a0 is 0.18 k N / m m 2 and the constant, k is 0.14; see Ref. 10.
Effect of strain rate

In Ref. 18 some interesting remarks pertaining to the effect of the strainrate on the crushing behaviour of axially loaded composite tubes, are reviewed. Strain-rate, and therefore crushing speed, can influence the mechanical properties of the fibre and matrix. Matrix stiffness and failure strain can be a function of strain-rate. Therefore, the energy absorption associated with the interlaminar crack growth can be a function of crushing speed. The mechanical properties of brittle fibres are in general, insensitive to strain-rate, whilst the fracturing of the lamina bundles is not a function of crushing speed. The coefficient of friction between the composite material and the crushing surface and between the debris wedge and the fronds can be a function of crushing speed. Therefore, energy absorption capability can be influenced by changes in crushing speed. The major part of the absorbed energy during the static axial compression of a shell is dissipated as frictional work in the crushed material or at the interface between the material and the tool and this is estimated to be about 50% or more of the total work done; see Refs 2 and 10. In dynamic collapse, attention is directed towards the influence of strain-rate on the frictional work absorbed during the impact, taking into account all structural and material parameters which may contribute to it, i.e. the fibre and matrix material, the fibre diameter and orientation in the laminate, the fibre volume content as well as the conditions at the interfaces. As mentioned above, two distinct regions, where the development of frictional forces is of great importance, were identified: the fronds/wedge contact region composed of the same material and the fronds/platen

288

A. G. Mamalis, D. E. Manolakos, G. A. Demosthenous, M. B. Ioannidis

contact area composed of different materials. The coefficient of friction in these regions depends on various phenomena described extensively and in detail in previous investigations, TM~z associated with the material flow. These phenomena will finally determine the increase or the decrease of the friction coefficient. They are greatly affected by the material properties influencing the effectiveness of the collapsed component as far as its crashworthy capacity is concerned. In the present series of dynamic collapse, an estimate of the dynamic coefficients of friction at the fronds/platen interface, Pd, and fronds/wedge interface, Pd: was obtained by comparing the crushing loads experimentally obtained. The values for #d, is 0-28 and for #d2 is 0-50 which are the same with those predicted in Ref. 12 for square tubes of same material.

DISCUSSION
Failure mechanisms

As outlined above four distinct modes of collapse, Mode I, II, III and IV, were observed throughout the axial static and dynamic loading of square frusta. As far as the collapse modes of the square frusta under axial loading is concerned, it must be noted that for the statically loaded ones all the above described modes of failure were observed, whilst for the dynamically ones almost stable collapse modes were obtained, due to the geometrical characteristics of the corresponding shells. The characteristic modes of collapse observed throughout the dynamic tests can be identified and classified as stable and unstable collapse modes. However, at impact, due to the dynamic nature of the phenomenon, unstable modes of collapse lead rapidly to a complete catastrophic failure, preventing in this manner the investigation of the deformation mechanism. Thus, the only information on this topic may be drawn from the change of the corresponding load/displacement curves. On the contrary, throughout the static test series and due to the wide range of geometric combinations that were used, see Table l(a), the effect of the various geometric parameters, i.e. wall thickness, t, semi-apical angle, 0, mean circumference, C, and axial length, L, on the collapse modes was studied. Regardless of thickness, frusta with smaller semi-apical angles (0-15 °) failed following stable modes of collapse. Frusta with large semi-apical angles, i.e. 20-30 °, followed the Mode II of collapse, associated with longitudinal corner cracking. This is a low energy collapse mode and quite inappropriate for energy absorption. Specimens of the smaller semi-apical

Energy absorption of fibreglass composite square./rusta

289

angle did not exhibit the Mode II collapse, however some collapsed following the Mode III, which is also a low energy collapse mode. These observations are indicative of the existence of a transitional zone, with the frusta semi-apical angle ranging from 15 to 20 ° where the collapse mode changes from a stable to an unstable one. Note that circular frusta exhibit similar behaviour, for semi-apical angles greater than 20°; see Ref. 6. Frusta of large semi-apical angles showed no significant change of the collapse mode with increasing wall thickness of the shell. Frusta of 5-15 ~ semi-apical angles with wall thickness of 1.5 m m or less, statically loaded, followed the progressive folding, Mode IV of collapse. Frusta of the same apical-angles with wall thickness greater than 2.5 mm exhibited the Mode I of collapse when loaded at elevated strain rates. Frusta of semi-apical angles of 5 or l0 ° with wall thickness 1-5mm < t < 2.5mm exhibited the Mode III of collapse when subjected to static loading. On the contrary the unstable mode of collapse Mode II was observed in the case of the dynamically loaded frusta of 15° semi-apical angle and 2.6mm wall thickness. The experimentally obtained deformation modes of all specimens tested are classified in respect to the geometry factors, wall thickness/mean circumference, t/C, and axial length/mean circumference, L/C, and are presented in Fig. 8. Distinct regions, characteristic for the various deformation modes developed, and the transition boundaries from stable to unstable modes of collapse, are indicated providing, therefore, useful information about the collapse of the square frusta and their behaviour as an energy absorber. Shell instability occurs for values of the geometric factors t/C and L/C lower than critical ones, see Fig. 8; these critical values, which are almost identical for static and dynamic loading are about 0.001 for tiC and 0-4 for L/C. Note that the critical value of L/C is almost equal to the related one for circular frusta subjected to similar loading conditions, whilst the value of t/C is much higher for the square frusta as compared to the circular ones, see Ref. 8. As far as the Mode I of collapse is concerned, the initial axial length of the shell, L seems not to affect the length of the main central crack, Lc and the mechanical response of the shell due to failure, see Table 1. As outlined above, the tip of the main intrawall crack is shifted towards the outer edge of the shell wall as the frusta semi-apical angle increases, whilst the size of the wedge as well as the length of the main crack, Lc reduce, owing to the transition from Mode Ia to Mode Ib, see Figs 2(b), (c) and 4. Dynamic loading greatly affects the microfracture mechanism, the size of the debris wedge and the main crack developed because of the wedge propagation are smaller whilst the tip of the crack is closer to the outer

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A. G. Mama/is, D. E. Manolakos, G. A. Demosthenous, M. B. loannidis

1.00

MODE I V
0

MODE

III o/
I /

0.60
/

MODE
•

I
D

/B=
0 /

0.60
0

-

/ /
°

•

[]

[]
~ • •

Q

•

0.40

-

A

A

~

~

0.20

MODE II

~

I

I

0

0.01

0.02

0.03

t/¢
Fig. 8. Classification chart showing the areas of collapse modes and transition boundaries from one mode to another for composite square frusta: (D-Mode I-static; m-Mode IImpact; A-Mode II-static; A-Mode II-Impact; ©-Mode III-static; o-Mode IV-static).

wall surface for impacted shells, as compared to statically loaded ones; compare Figs 4(b) and 6. Therefore, transition from Mode Ia to Mode Ib, see Fig. 2(b) and (c), occurs earlier with increasing of strain rate, compare Table l(a) and (b). This is in agreement with the remarks reported during the loading of circular frusta at elevated strain rates, see Ref. 8. Note, that, the characteristics of the crush zone observed at the four corners of the frustum are similar to those obtained for Mode Ib crush zone, created by successive shearing. For the dynamically loaded specimens it was observed that in some regions near the crushing zone and near the four corners turn opaque white probably due to the very strong shock wave that is caused by the violent crash between the drop-mass and the shell causing instantaneously propagation of delamitation cracks accompanied with sharp drops of the crushing load. This p h e n o m e n o n is more profound for square frusta with large semiapical angles.

Energy absorption offibreglass composite squarefrusta

291

Energy absorbing characteristics
In Fig. 7 the crushing load/shell shortening curves for the various collapse modes of statically and dynamically loaded shells are presented. In general, the end-crushing mode Mode I seems to offer the highest energy absorption capability in relation to the other three collapse modes. Initially the shell behaves elastically until the initial maximum value of the load is reached Pmax. It is greatly affected by the wall thickness (number of layers), the semi-apical angle and the main circumference of the shells, see Table 1. For constant semi-apical angle the peak load, Pmax increases with increasing wall thickness whilst for the same number of layers, i.e. for constant wall thickness, decreases with increasing semi-apical angle, but increases with increasing mean circumference. Note that the above mentioned observations were made on frusta with the same kind of triggering mechanism, i.e. tapered top of the frustum properly polished free of microdefects. The mean post-crushing load and the energy absorbed increase considerably with increasing thickness or number of layers of wrapped glass mat. However, due to the fact that the radius of curvature of the internal frond, r i increases with increasing semi-apical angle whilst the external frond is constrained to deform through a smaller radius r0, see Figs 3 and 4, the total energy dissipated depends directly upon the semi-apical angle; see Table 1 and the similar remarks concerning the axial loading of circular frusta reported in Refs 6-8. Therefore, for a given slenderness ratio, t/C, the specific energy decreases as the semi-apical angle increases, see Table 1. Note also that there is no effect on the increase of specific energy for specimens with number of layers greater than three; see Table 1 for details. This observation is in agreement with those reported in Refs 6 and 8 concerning circular frusta statically and dynamically loaded, respectively. For a given angle the mean post-crushing load, P increases with increasing slenderness ratio, t/C whilst the mean load tends to increase as the semi-apical angle increases with the slenderness ratio kept constant, see Table 1. For dynamically loaded frusta the effect of the strain-rate on the specific energy and the mean post-crushing load seems to be almost negligible, see Table l(b). For specimens with three or more layers, peaks and valleys of the load occur for approximately the same displacement, indicating therefore, that the specific energy remains practically constant. For relatively thick shells (four to six layers), the valleys of the load approach zero probably due to the fact that a secondary deformation mechanism develops after the intense cracking and delamination. For the frusta subjected to static and dynamic loading, it is evident that with increasing geometrical factor t/C, i.e. by increasing wall thickness or decreasing mean circumference, the specific energy absorbed increases.

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A. G. Mamalis, D. E. Manolakos, G. A, Demosthenous, M. B. loannidis

From the experimental results, it is indicated that static tests for the same shell geometries develop higher values of specific energy than these obtained in dynamic testing probably due to the higher values of the static friction coefficients, #s, between the wedge and the fronds and between the platen surface and the fronds (about 10-15% of the related dynamic ones); this increase in the crashworthy ability of the shell was about 5-15%. This is in agreement with those referred to in Ref. 8 concerning the crashworthy characteristics of circular frusta at elevated strain rates. As can also be observed, the increase of the specific energy for the circular frusta statically loaded is greater than for the corresponded square ones for all slenderness ratios, whilst for the dynamically ones the specific energy seems to be influenced by the geometrical differences of the shells. The mean post-crushing load, P, the energy absorbed, W, and the specific energy, Ws, are well predicted theoretically by the proposed analysis within +12%, see Table 1. According to the proposed theoretical analysis the distribution of the dissipated energy of the crush shell due to the four main energy sources was estimated as: • energy of friction between the annular wedge and the fronds and between the fronds and the platen, i.e. estimated to be about 45, 47-5 and 50% of the total one for the 5, 10 and 15° square frusta, respectively; • energy due to fronds bending, i.e. about 48, 46 and 44%, respectively; • energy due to crack propagation, i.e. about 6, 5-5 and 5%, respectively; • energy due to axial splitting at the four corners of the shell, i.e. about 1% for all specimens tested. Note that the contribution of the frictional conditions between wedge/ fronds and fronds/platen to the energy absorbing capability is more significant than the other ones. As discussed above, it mainly depends upon the friction coefficients #~,, #d, and #~2, #d2 which are affected by the surface conditions at the interfaces between composite material/platen or drop mass and composite material/debris wedge, respectively. Classifying the geometries tested from the energy absorbing capacity point of view, the geometry of a 5° square frustum seems to be the more efficient, see Table 1. This is in agreement with those observed for circular frusta axially loaded and reported in Refs 6 and 8.

CONCLUSIONS Summarising the main features of the results reported above pertaining to the crashworthy characteristics of composite square tubes and frusta

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293

subjected to axial static and dynamic loading, the following conclusions may be drawn: (a) Four collapse modes were observed; the stable progressive collapse mode, Mode I, associated with large amounts of crush energy, resulting, therefore, in a high crashworthy capacity of the structural component, the longitudinal corner cracking, Mode II, showing successive formations of cracks at the shell corners, the mid-length collapse mode, Mode III, featuring a brittle fracture involving catastrophic failure, and finally a progressive folding mode, Mode IV, with the formation of sharp hinges. (b) Regarding the microfracture mechanism of square frusta subjected to axial loading, as far as the Mode Ia of collapse is concerned, the experimental observations resulted to the conclusion that the microfracture mechanism may be determined as a combination of the main characteristics observed for square tubes and circular frusta. In general this mechanism is similar for statically and dynamically loaded shells, respectively. The only differences encountered are associated with the shape of the wedge and the microcracking development. The transition from Mode Ia to Mode Ib occurs earlier for the dynamically loaded frusta. (c) The post-crushing regions of the load/displacement curves for the statically and dynamically loaded shells which followed Mode I of collapse, revealed distinct differences; the dynamically obtained curves are highly serrated probably due to the impact nature of loading and the inertia response of the load cell, whilst the static curves show the formation of typical peaks and valleys with a very narrow fluctuation amplitude. (d) As far as the behaviour of square frusta under static and dynamic loading is concerned, the variation of characteristic parameters, i.e. specific energy and mean post-crushing load, is similar to that observed during the axial collapse of circular frusta at elevated strain rates, i.e. specific energy decreases with increasing semi-apical angle and mean-post crushing load increases with increasing wall thickness. Note also that under dynamic loading and for specimens with number of layers greater than three, no significant effect on the increase of the specific energy was observed. During dynamic testing lower values by about 5-15% were obtained for both the parameters mentioned above than those predicted in static testing for same shell geometries probably due to the lower values of the dynamic coefficients, between the wedge/fronds and fronds/platen interface. (e) Square frusta with large mean circumference were more prone to catastrophic failure. Shell buckling instability is prevented by ensuring that the tiC and L/C ratios are above a critical value. These values seem to

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A. G. Mamalis, D. E. Manolakos, G. A. Demosthenous, M. B. Ioannidis

be almost equal for dynamic and static tests, respectively. The t/C ratio overestimates the corresponding one of the circular tubes and frusta under axial loading. (f) The crashworthy characteristics of the square frusta statically loaded underestimate the corresponding ones concerning circular frusta whilst there is no significant variation of them for the dynamically loaded ones. (g) The theoretical model proposed predicts well the mean post-crushing load and the energy absorbed; theoretical results are in good agreement with the experimental measurements to within + 12%. (h) From the theoretical calculations, supported also by experimental observations, it is evident that the contribution of the frictional conditions between wedge/fronds and fronds/platen to the energy absorbing capability of the shells, is the most significant factor affecting of as compared to the other ones. (i) The geometry of a 5 ° square frustum seems to be the more efficient from the energy absorbing capacity point of view for both testing conditions.

REFERENCES 1. Thornton, P. H. & Jeryan, R. A., Crash energy management in composite automotive structures. Int. J. Impact Engineering, 7 0987) 167. 2. Fairfull, A. H. & Hull, D., Energy absorption of polymer matrix composite structures: friction effects. Structural Failure for International Symposium on Structural Failure, MIT, June 1988. 3. Schmueser, D. W. & Wickliffe, L. E., Impact energy absorption of continuous fiber composite tubes. J. Eng. Mat. Trans. ASME, 72 (1987) 72. 4. Berry, J. & Hull, D., Effect of speed on progressive crushing of epoxy-glass cloth tubes. 3rd Int. Conf. Mech. Prop. High Rates of Strain, Oxford (1984) 463. 5. Farley, G. L. & Jones, R. M., Analogy for the effect of material and geometrical variables on energy absorption capability of composite tubes. J. Composite Materials, 26(1) (1992) 78. 6. Mamalis, A. G., Manolakos, D. E., Viegelahn, G. L., Demosthenous, G. A & Sin Min Yap, On the axial crumpling of fibre-reinforced composite thinwalled conical shells. Int. J. Vehicle Design, 12 (1991) 450. 7. Mamalis, A. G., Manolakos, D. E., Viegelahn, G. L., Sin Min Yap & Demosthenous, G. A., Microscopic failure of thin-walled fibre-reinforced composite frusta under static axial collapse. Int. J. Vehicle Design, 12 (1991) 557. 8. Mamalis, A. G., Manolakos, D. E., Demosthenous, G. A. & Ioannidis, M. B., Axial collapse of thin-walled fibreglass composite tubular components at elevated strain rates. Composites Engineering, 3 (1994).

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9. Mamalis, A. G., Manolakos, D. E. & Viegelahn, G. L., Crashworthy behaviour of thin-walled tubes of fibreglass composite material subjected to axial loading. J. Composite Materials, 24 (1990) 72. 10. Mamalis, A. G., Manolakos, D. E., Demosthenous, G. A. & Ioannidis, M. B., Analysis of failure mechanisms observed in axial collapse of thinwalled circular fibreglass composite tubes. Thin-Walled Structures, 24 (1995) 335. 11. Mamalis, A. G., Manolakos, D. E., Demosthenous, G. A. & loannidis, M. B., The static and dynamic collapse of fibreglass composite automotive frame rails. Composite Structures, 30 (1995). 12. Mamalis, A. G., Manolakos, D. E., Demosthenous, G. A. & Ioannidis, M. B., The static and dynamic axial crumbling of thin-walled fibreglass composite square tubes. Composites Engineering, 5 (1995). 13. Mamalis, A. G., Manolakos, D. E., Demosthenous, G. A. & loannidis, M. B., On the bending of automotive fibre-reinforced composite thinwalled structures. Composites, 25 (1994). 14. Czaplicki, M. J., Robertson, R. E. & Thorton, P. H., Comparison of bevel and tulip triggered pultruded tubes for energy absorption. Composites Science and Technology, 40 (1991) 31. 15. Mamalis, A. G., Manolakos, D. E. & Viegelahn, G. L., Deformation characteristics of crashworthy components. Fortschritt-Berichte der VDI-Z, Reihe 18, Nr. 62, Dusseldorf, Germany, 1989, pp. 337. 16. Kendall, K., Interfacial cracking of a composite. J. Material Science, I 1 (1976) 638 and 1263. 17. Yuan, Y. B. & Viegelahn, G. L., Modelling of crushing behaviour fibreglass! vinylester tubes. Developments in Mechanics, Vol. 16. Proceedings of the twenty-second Midwestern Mechanics Conference, University of MissouriRolla, Missouri (199l). 18. Farley, G. L & Jones, R. M, Crushing characteristics of continuous fibrereinforced composite tubes. J. Composite Materials, 26(1) (1992) 37.