The collapse response of sandwich beams with a Y-frame core subjected to distributed and local loading

Transcription

1 ARTICLE IN PRESS International Journal of Mehanial Sienes 5 (28) The ollapse response of sandwih beams with a Y-frame ore subjeted to distributed and loal loading V. Rubino, V.S. Deshpande, N.A. Flek Engineering Department, Cambridge University, Trumpington Street, Cambridge, CB2 1PZ, UK Reeived 12 November 26; reeived in revised form 23 June 27; aepted 4 July 27 Available online 25 July 27 Abstrat Sandwih beams omprising a Y-frame ore have been manufatured by assembling and brazing together pre-folded sheets made from AISI type 34 stainless steel. The ollapse responses of the Y-frame ore have been measured in out-of-plane ompression, longitudinal shear and transverse shear; and the measurements have been ompared with finite element preditions. Experiments and alulations both indiate that the ompressive response is governed by bending of the onstituent struts of the Y-frame and is sensitive to the hoie of lateral boundary onditions: the energy absorption for a no-sliding boundary ondition exeeds that for free-sliding. Under longitudinal shear, the leg of the Y-frame undergoes uniform shear prior to the onset of plasti bukling. Consequently, the longitudinal shear strength of the Y-frame muh exeeds its ompressive strength and transverse shear strength. Sandwih beams were also indented by a flat bottomed punh, and a relatively high indentation strength was observed. It is argued that this is due to the high longitudinal shear strength of the Y-frame. While finite element alulations apture the measurements to reasonable auray, a simple analytial model over-predits the indentation strength. Finally, the finite element method was used to investigate the energy absorption apaity of the sandwih beams under indentation loading. The alulations reveal that for a given tensile failure strain of the fae-sheet material, a sandwih beam with Y-frame ore has a omparable performane to that of a sandwih beam with a metal foam ore. The relative performane is, however, sensitive to the hoie of design parameter: when the indentation depth is taken as the design onstraint, the sandwih beam with a Y-frame ore outperforms the sandwih beam with the metal foam ore. r 27 Elsevier Ltd. All rights reserved. Keywords: Sandwih beams; Energy absorption; Y-frame ore; Indentation 1. Introdution Commerial and military ship strutures require adequate strength to survive impat by submerged roks, iebergs and ollisions with other vessels. Current designs are based upon either a monolithi skin with an internal stiffening and strengthening frame or upon double-hulled designs whih have minimal mehanial oupling between the inner and outer hulls. There is urrent industrial interest in determining whether signifiant enhanements in strutural stiffness, strength and energy absorption with no weight penalty an be ahieved by employing sandwih onstrution. Corresponding author. Fax: address: vsd@eng.am.a.uk (V.S. Deshpande). Over the past few years, a set of lattie materials have been devised for sandwih ores. The nodal onnetivity of these latties is suffiiently high for them to deform by the axial strething of the onstituent members under all loading states, see for example Deshpande and Flek [1]. Consequently, these materials have a higher speifi stiffness and strength than metalli foams whih deform by ell-wall bending. The best hoie of ore remains unlear for hulls designed primarily to withstand low veloity ollision: weaker sandwih ores diffuse an external transverse load over a larger portion of the hull and may thereby absorb more energy prior to hull perforation. Over the past deade or so there have been substantial hanges in ship design, see for example the review by Paik [2]. In the urrent study, we measure and analyse the performane of the Y-shaped sandwih ore, as proposed 2-743/$ - see front matter r 27 Elsevier Ltd. All rights reserved. doi:1.116/j.ijmesi

2 234 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) Pedersen et al. [7] performed a finite element investigation of the ompressive response of the Y-frame. They developed maps of the ompressive strength and energy absorption of the Y-frame with the geometri parameters of the Y-frame as axes of these maps. Their study revealed that the ompressive ollapse response of the Y-frame is sensitive to the size of the web and to the hoie of boundary onditions. No experimental investigations into the effetive properties of the Y-frame and into the indentation response of Y-frame ore sandwih beams have been reported to date. This is the prinipal aim of the urrent study Sope of study A ombined experimental, finite element and analytial investigation is presented on the indentation response of Y-frame sandwih beams. First, the manufaturing proedure for laboratory sale Y-frame sandwih beams is desribed and measurements are given for the ompressive and shear responses of the Y-frame ore. These measurements are ompared with 3D finite element simulations. Next, the measured indentation response of Y-frame sandwih beams is reported. Analytial and finite element preditions are developed and ompared with these experimental values. Additional finite element alulations on the indentation response of sandwih beams with isotropi foam ores are used to identify those properties of the ore that enhane energy absorption of the sandwih beams prior to perforation of the fae-sheets. Finally, the sensitivity of energy absorption to the dutility of the fae-sheets is explored. Fig. 1. Sketh of the Y-frame sandwih ore as used in ship hull onstrution. The ore is sandwihed between the outer and inner hull of the ship. by Shelde Shipbuilding. 1 It is manufatured by onventional welding methods of steel sheets and its topology is shown in Fig. 1. Full-sale ship ollision trials reveal that the Y-frame design is more resistant to tearing than onventional monolithi designs, see Wevers and Vredeveldt [3] and Ludolphy [4]. Likewise, the finite element simulations by Konter et al. [5] suggest that the Y-frame onfers improved perforation resistane. Naar et al. [6] have argued in broad terms that the ability of the bendinggoverned Y-frame topology to spread the impat load over a wide area, ombined with the in-plane high strething resistane of the Y-frame, gives the enhaned performane of the Y-frame sandwih onstrution over onventional single and double hull designs. However, a detailed analysis of the elasti plasti indentation behaviour of the Y-frame sandwih beam has been laking to date. 1 Royal Shelde, P.O. Box , AA Vlissingen, The Netherlands. 2. Speimen manufature Saled-down (approximately 1/1 sale) Y-frame sandwih ores of depth ¼ 44 mm were manufatured from AISI 34 stainless steel sheets of thikness g ¼.5 mm. The ross-setional dimensions of the Y-frame are given in Fig. 2a. A global o-ordinate referene frame is inluded in the figure in order to larify the various loading diretions used in the mehanial tests: x 1 is the longitudinal axis of the Y-frame, x 2 denotes the transverse diretion and x 3 is the out-of-plane diretion for the sandwih beam. The Y-frames are lose-paked along the x 2 diretion suh that adjaent Y-frames touh; the relative density of the as-manufatured Y-frame sandwih ore (that is, the ratio of effetive density of the smeared-out ore to that of the parent solid material) is 2.1%. Y-frame sandwih beams of length L ¼ 6 mm were manufatured as follows. The stainless steel sheets were omputer-numerial-ontrol (CNC) folded to form the upper part of the Y-frame and the Y-frame leg. Slots were then CNC-ut into the Y-frame web and the Y-frame leg was fitted into the upper part of the Y-frame, as skethed in Fig. 2b. The assembly was then spot welded to fae-sheets of thikness t ¼.6 or 1.2 mm. The braze alloy Ni Cr 25-P1 (wt%) was applied uniformly over all sheets of the

3 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) g= g= All dimensions in mm braze Fig. 2. (a) Geometry of the Y-frame sandwih ore used in this study. (b) Sketh of the manufaturing route for the saled-down Y-frame sandwih ore. assembly and the assembly was brazed together in a vauum furnae at 175 1C in a dry argon atmosphere at.3.1 mbar. Finally, the beams were water-jet ut to the required length. Tensile speimens of dog-bone geometry were ut from the as-reeived 34 stainless steel sheets and were subjeted to the same brazing yle as that used to manufature the Y-frame beam speimens. The measured true tensile stress versus logarithmi strain response at a strain rate of _ ¼ 1 4 s 1 is shown in Fig. 3. The stainless steel behaves in an elasti plasti manner with a Young s modulus E ¼ 21 GPa, a yield strength of s Y ¼ 21 MPa and linear hardening in the plasti regime with a tangent modulus of E t E2.1 GPa. 3. Compressive and shear responses of the Y-frame sandwih ore Two types of tests were onduted on the Y-frame sandwih ore: (a) out-of-plane ompression, and (b) shear in the transverse and longitudinal diretions. These loading diretions are of prinipal pratial interest for a Y-framed sandwih plate. The sandwih ore is treated as a homogeneous effetive medium and its stress versus strain behaviour is measured Compressive response The s 33 versus e 33 out-of-plane ompressive response of Y-frame sandwih speimens, of length L ¼ 14 mm, was σ (MPa) ε Fig. 3. The quasi-stati tensile stress versus strain response of the asbrazed 34 stainless steel as used to manufature the Y-frame sandwih beams. investigated for the following two sets of lateral boundary ondition (see Fig. 4): (a) no-sliding. The fae-sheets of the Y-frame were lamped to the platens of the test mahine. Consequently, the fae-sheets were prevented from relative sliding. (b) free-sliding. One fae-sheet was lamped to the stationary platen of the test mahine while linear bearings were

4 236 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) Fig. 4. Skethes of a unit ell of the Y-frame sandwih ore with the two types of uniaxial ompression boundary onditions onsidered. (a) The freesliding and (b) no-sliding boundary onditions. loated between the other fae-sheet and the moving platen of the test mahine. Consequently, the faesheets of the Y-frame sandwih ould slide past eah other during the uniaxial ompression test. In both sets of experiments, the Y-frame ore was brazed to.6 mm thik 34 stainless steel fae-sheets. These faesheets remained elasti during the tests and so the measured ompressive response of the Y-frame ore is independent of the fae-sheet properties. These two sets of boundary onditions represent the limiting ases where the supports either (a) ompletely restrit or (b) permit free-sliding of the outer fae of the ship hull with respet to the inner hull. These hoies are appropriate for large strutures omprising many Y-shaped webs and subjeted to uniform marosopi loading. Other boundary onditions, permitting relative rotation of the two faes of the Y-frame sandwih beams, are also of pratial relevane but are beyond the sope of the present investigation. The ompression experiments were performed using a srew-driven testing mahine at a nominal ompressive strain rate of approximately _ 33 ¼ 1 4 s 1. The load was measured by the load ell of the test mahine and was used to define the nominal ompressive stress s 33, while a laser extensometer measured the relative approah of the faesheets of the Y-frame sandwih and thereby gave the applied nominal ompressive strain e 33. The measured nominal ompressive stress versus strain responses of the Y-frame sandwih ores are plotted in Fig. 5a and b for the no-sliding and free-sliding boundary onditions, respetively. The response of the Y-frame ore for both boundary onditions is haraterized by an initial elasti stress response, a peak stress and subsequent softening. The peak stress of s p 33 ¼ :54 MPa for no-sliding is higher than the peak stress of s p 33 ¼ :4 MPa for the free-sliding boundary ondition. A montage of photographs of the Y-frame speimens at seleted levels of applied ompressive strain are shown in Figs. 6a and 7a for the no-sliding and free-sliding ases, respetively. These photographs show that the deformation modes differ: while plasti deformation spreads over the entire Y-frame in the no-sliding ase, the upper half of the Y-frame remains elasti in the free-sliding ase Shear response The transverse shear (t 23 g 23 ) and longitudinal shear (t 13 g 13 ) responses of the Y-frame sandwih ores were measured using a double-lap shear apparatus as shown in Fig. 8. Eah shear speimen omprised two nominally idential sandwih beams bolted to the platens of the double-lap shear apparatus. Pin-grips were used to fasten the shear rig to the test mahine. This arrangement provided an indeterminate onstraint on the deformation of the speimen in the x 3 diretion: the onstraint is intermediate between that of zero load and zero displaement in the x 3 diretion. The load ell of the test mahine was used to measure the load and a laser extensometer measured the relative displaement of the platens in order to infer the applied shear strain Longitudinal shear tests The measured longitudinal shear response (t 13 g 13 )of the Y-frame sandwih ore with aspet ratio L/E7 (L ¼ 3 mm) is plotted in Fig. 9. Two separate measurements are plotted to give an indiation of the satter in the experimental results. The Y-frame sandwih ore displays an initial elasti response followed by an almost ideally plasti stress versus strain history with a peak shear strength t p 13 1:7 MPa. A photograph of the Y-frame at a strain level of g 13 ¼.1 is inluded in Fig. 1a. The photograph shows that wrinkling ours in the leg of the Y-frame with negligible deformation of the upper half of the Y-frame. Tearing initiates at the joint between fae-sheet and leg at g 13 E.5. The wrinkling deformation mode of the Y-frame leg in Fig. 1a is expeted to be sensitive to end effets and thus

5 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) FE (ζ =.1g) FE (ζ = g) Measured.4 σ 33 (MPa) ε 33 Fig. 6. (a) Observed shape and (b) FE predition of the deformation mode of the Y-frame sandwih under uniaxial ompression (no-sliding boundary ondition). The deformed profiles are shown at three seleted values of the nominal ompressive strain e FE (ζ =.1g) FE (ζ = g) Measured.4 σ 33 (MPa) ε 33 Fig. 5. The measured and predited response of the ompressive stress versus strain response of the Y-frame sandwih ore. (a) No-sliding and (b) free-sliding boundary onditions. the shear response of the Y-frame is expeted to depend on the speimen length L. We onduted a series of longitudinal shear tests on speimens of length L in the range 45 3 mm (ore thikness ¼ 44 mm in all ases). The measured peak shear strength t p 13 (defined as the initial peak shear stress) is plotted as a funtion of the speimen aspet ratio L/ in Fig. 11. The peak shear strength is sensitive to speimen aspet ratio L/ for L/o4 but is reasonably onstant at higher aspet ratios. This onfirms that the results for L/E7, as plotted in Fig. 1a, are representative of those for a large speimen Transverse shear tests Five Y-frames, eah of length L ¼ 55 mm, were assembled adjaent to eah other along the x 2 diretion Fig. 7. (a) Observed and (b) FE predition of the deformation mode of the Y-frame sandwih under uniaxial ompression (free-sliding boundary ondition). The deformed profiles are shown at two seleted values of the nominal ompressive strain e 33. and were brazed to fae-sheets of thikness t ¼.6 mm. Sandwih beams of length L ¼ 3 mm in the x 2 diretion were thereby manufatured. A pair of speimens was used in eah double-lap shear test, again using the apparatus skethed in Fig. 8. The transverse shear (t 23 g 23 ) response was measured using a similar proedure to that desribed above for the longitudinal shear tests. This measured shear response t 23 g 23 is plotted in Fig. 12. After an initial elasti response, the Y-frame sandwih ore yields at a shear stress t Y 23 :3 MPa and subsequently displays a mildly hardening response up to a shear strain g 23 ¼.3. Thereafter, the shear stress rises sharply with inreasing shear strain. A photograph of the deformed speimen at an applied shear strain g 23 ¼.3 is shown in Fig. 13a. The low transverse shear strength of the Y-frame (ompared to that in the longitudinal diretion) is due to the formation of two plasti hinges in the Y-frame leg, one at the bottom and one at the joint between the leg and the upper half of the Y-frame. This mehanism results in only elasti deformation of the upper half of the Y-frame. It is argued that the sharp rise in shear stress for g is due to the onstraint by the double-lap shear apparatus upon normal straining in the x 3 diretion.

6 238 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) τ 13 (MPa) FE(unonstrained) ζ =.1g FE (onstrained) ζ = g measured FE (unonstrained) ζ = g γ 13 Fig. 9. The measured and predited longitudinal shear response of the Y-frame sandwih ore. Two separate measurements are plotted to indiate the variability in the measurements. FE preditions for the onstrained and unonstrained boundary onditions are inluded for two seleted values of imperfetion. Fig. 1. (a) Observed and (b) predited deformation mode of the Y-frame ore under longitudinal shear. The deformed profiles are shown at a shear strain g 13 ¼ :1. Fig. 8. Sketh of the double-lap shear rig used to test pair of Y-frame speimens in longitudinal and transverse shear. A Y-frame speimen loaded in longitudinal shear is shown here. At large shear strains the Y-frame leg strethes and onsequently the applied shear stress inreases sharply. Finite element alulations reported below larify this effet Finite element preditions Finite element alulations of the ompressive and shear responses of the Y-frame sandwih ore have been performed using the general finite element pakage ABAQUS (HKS Hibbitt, Karlsson & Sorensen, In.). The geometry of the Y-frame analysed was idential to that employed in the experimental investigation (as detailed in Fig. 2). The FE model omprised about 25, linear shell elements (S4R in ABAQUS notation) of in-plane dimension 2g, where g is the thikness of the sheet from whih Y-frames were manufatured. A mesh sensitivity study revealed that additional mesh refinement did not hange the results appreiably. All omputations reported here employed suh a mesh. The fae-sheets were treated as rigid plates, and were modelled by the analytial rigid surfae in ABAQUS. Possible ontats between any surfaes of the mesh were modelled by the hard, fritionless ontat option in ABAQUS. The stainless steel was treated as a rate-independent J2-flow theory solid with Young s modulus E ¼ 21 GPa and Poisson ratio n ¼.3. The uniaxial tensile true stress versus equivalent plasti strain urve was tabulated in ABAQUS employing the measured data plotted in Fig. 3.

7 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) τ 13 p (MPa) FE Predition Measured Compression simulations The ompression simulations were performed by imposing displaement boundary onditions. In both the free and nosliding simulations, all degrees of freedom of the lower rigid plate were taken to be zero and a vertial ompressive displaement rate (in the x 3 diretion of Fig. 2b) was applied to the top plate (while onstraining this top plate against any rotations). In the no-sliding simulations, the horizontal displaements of the top plate were also held at zero. The ompressive stress versus strain response was determined from the alulated fore versus the applied L repeat measurements Fig. 11. A summary of the measured and FE preditions of the peak longitudinal shear strength of the Y-frame as a funtion of the speimen aspet ratio L/. τ 23 (MPa) FE (onstrained) measured FE (unonstrained) γ 23 Fig. 12. The measured and predited transverse shear response of the Y- frame sandwih ore. FE preditions for both the onstrained and unonstrained boundary onditions are given. vertial displaement. In order to trigger the appropriate ollapse mode in the finite element simulations, an initial imperfetion in the shape of the first elasti bukling eigenmode was introdued into the finite element model. Two levels of imperfetion magnitude were onsidered in order to gauge the imperfetion sensitivity of the ompressive response of the Y-frame: z ¼.1g and z ¼ g, where z is the amplitude of the imperfetion. The predited ompression responses are ompared with the observed behaviours in Figs. 5a and b for the no-sliding and free-sliding ases, respetively. Exellent agreement is noted. The finite element preditions of the ompressive deformation mode in the no-sliding and free-sliding ases are inluded in Figs. 6 and 7, respetively. Again, the finite element alulations apture the observed deformation modes to good auray. It is noted in passing that the peak ompressive strength of the Y-frame sandwih ore involves plasti ollapse of the stainless steel. Exploratory finite element alulations were performed with the stainless steel modelled as a linear elasti solid: muh higher peak strengths were obtained in these alulations Longitudinal shear simulations Finite element simulations of the longitudinal shear response of the Y-frame sandwih ore were onduted with two sets of boundary onditions in order to simulate the possible onstraints imposed by the double-lap shear apparatus. Loading in the longitudinal diretion was applied by presribing a displaement in the x 1 diretion to the top fae-sheet while the bottom fae-sheet (attahed to the Y-frame leg) was held motionless. Two hoies of onstraint in the out-of-plane x 3 diretion were assumed: unonstrained simulations where the tration T 3 on the top fae-sheet vanishes, and onstrained simulations where the displaements u 3 on the top fae-sheet vanishes. These two hoies of boundary onditions bound the onstraint imposed by the double-lap shear apparatus. An initial imperfetion is introdued into the FE model in order to trigger the appropriate ollapse mode. In line with the experimental observations of the deformation mode (Fig. 1a) an elasti bukling eigenmode, with wrinkles within the Y-frame leg and negligible deformation in the upper half of the Y-frame, was hosen to model the imperfetions in the Y-frame. This mode was the seventh lowest mode of an eigenvalue analysis. Maximum imperfetion magnitudes of z ¼.1g and g were hosen in order to explore the imperfetion sensitivity of the Y-frame response. A omparison between the predited t 13 g 13 response and the observed behaviour is presented in Fig. 9 for the hoie L/ ¼ 7. Consider first the unonstrained simulations. A pronouned peak in shear stress is observed at g 13 E.1 for a small imperfetion, z ¼.1g, while the shear stress inreases monotonially with inreasing shear strain when a large imperfetion is present, z ¼ g. The predited shear stress at large shear strain g is relatively insensitive to the level of imperfetion and to the

8 24 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) Fig. 13. (a) Observed shape, (b) predited shape for unonstrained boundary ondition and () predited shape for onstrained boundary ondition, of the Y-frame ore subjet to transverse shear. The deformed profiles are shown at a shear strain g 23 ¼.3. degree of straining. The observed response resembles that of the unonstrained simulation at large imperfetion, with the measured plateau strength somewhat above the predition. Seond, onsider the onstrained FE simulations. An approximately linear hardening response is predited beyond the elasti limit. Negligible sensitivity to the imperfetion magnitude is observed and hene only the z ¼ g results are plotted in Fig. 9. We onlude that the onstraint imposed by the double-lap shear apparatus is intermediate between the two limiting sets of boundary onditions onsidered here. The predited deformed mode in shear is shown in Fig. 1b alongside the experimental photograph at g 13 ¼.1. The FE model predits orretly that wrinkling ours in the Y-frame leg, with the upper half of the Y-frame undergoing only elasti deformation. A omparison between the measurements and FE preditions (unonstrained) of the longitudinal peak shear strength t p 13 of the Y-frame is shown in Fig. 11 as a funtion of the Y-frame aspet ratio L/. Consistent with the results in Fig. 9, the unonstrained FE alulations slightly under-predit the longitudinal shear flow strength over the range of aspet ratios investigated here. Both measurement and analysis reveal that the longitudinal shear flow strength plateaus out for L/X Transverse shear simulations Finite element simulations of the transverse shear response of the Y-frame were performed by onsidering the test geometry of five Y-frame setions, as shown in Fig. 13. The spaing between the Y-frames mathed that of the experiments. Again, rigid fae-sheets were tied to the Y-frame and hard fritionless ontat was modelled between all surfaes. Loading was applied by presribing a displaement in the x 2 diretion to the top fae-sheet while the bottom fae-sheet was fully onstrained. Unonstrained and onstrained simulations were performed as for the longitudinal shear ase: the displaements u 3 ¼ were speified on the top fae-sheet in the onstrained ase whereas the tration T 3 was fully relaxed in the unonstrained ase. An imperfetion in the form of the first elasti bukling eigenmode was introdued into the FE model: in this orientation, the FE preditions of the shear response are insensitive to the magnitude of the imperfetion within the range z ¼.1g g. Thus, for the sake of brevity, only results for z ¼ g are presented below. Comparisons between the measurements and FE preditions of the transverse shear response t 23 g 23 are shown in Fig. 12. The unonstrained boundary ondition adequately predits the measurements for g 23 o.3, but at higher strain levels it does not apture the stiffening imposed by the double-lap shear apparatus. In ontrast, the onstrained FE alulations predit a muh stronger response than the measurements over the full range of shear strains investigated here. However, the slope dt 23 /dg 23 of the measured response is similar to the onstrained FE preditions for g This suggests that the onstrained boundary onditions are more appropriate to model the experiments at large levels of shear strain. 4. Indentation of Y-frame sandwih beams We proeed to investigate the transverse indentation response of Y-frame sandwih beams on a rigid foundation. The problem under onsideration is skethed in Fig. 14. The beams were of length L ¼ 6 mm along the x 1 diretion and were indented in the x 3 diretion. Eah beam was onstruted from two adjaent Y-frame setions in the

9 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) F, δ.8 Plasti hinge λ a θ M p t.6 measured (t =.6mm) analytial (t =.6mm) FE (t =.6mm) x 3 σ F (σ Y b t).4 x1 analytial (t =1.2 mm).2 FE (t = 1.2mm) measured (t =1.2mm) Fig. 14. The assumed deformation mode for indentation of the Y-frame sandwih. The Y-frame sandwih is indented by a flat bottom punh of width a and rests on a rigid foundation. x 2 diretion, giving a width of b ¼ 115 mm. Stainless steel fae-sheets were used, of thikness t ¼.6 mm and 1.2 m. The indentation response was determined for the following mild steel indenters: (i) a flat bottomed punh of width a ¼ 5 mm, (ii) a flat bottomed punh of width a ¼ 15 mm and (iii) a irular roller of radius R ¼ 9 mm. The indenters were of length 14 mm along the x 2 diretion in order to ensure a uniform indentation over the entire 115 mm width of the beam. The indentation experiments were performed in a srewdriven test mahine at an indentation rate d _ ¼ :3mmmin 1 and the indentation fore F was measured via the load ell of the test mahine. The relative displaement d of the top and bottom fae-sheets of the Y-frame sandwih diretly beneath the indenter was measured using a laser extensometer. The measured normalised load F F=ðs Y btþ versus normalised indentation depth d d= response of the Y-frame sandwih beams are plotted in Fig. 15a and b for the R ¼ 9 mm radius roller and the a ¼ 15 mm wide flat bottom punh, respetively. In the normalisation of load, use is made of the measured yield strength of the as-brazed 34 stainless steel from whih the Y-frame sandwih beams are onstruted, s Y ¼ 21 MPa. Results are presented for both hoies of fae-sheet thikness, t ¼.6 and 1.2 mm. In all ases, the indentation response has an initial elasti regime followed by a plateau regime where the indentation fore remains approximately onstant with inreasing indentation depth. The normalised indentation fores for the sandwih beams with thiker fae-sheets (t ¼ 1.2 mm) are lower than those for the sandwih beams with the t ¼.6 mm fae-sheets. A omparison between the observed deformation modes at an indentation depth d ¼ 1 mm is shown in Fig. 16 for the indentation of the t ¼.6 mm beams by the three types of indenters F (σ Y b t) FE (t = 1.2mm) analytial (t =.6 mm) FE (t =.6mm) measured (t =.6mm) analytial (t = 1.2 mm) measured (t = 1.2mm) Fig. 15. The measured non-dimensional indentation load versus indentation depth of the Y-frame sandwih indented by (a) a roller of radius R ¼ 9 mm and (b) a flat bottom indenter of width a ¼ 15 mm. The measured responses for sandwihes with fae-sheet thiknesses t ¼.6 and 1.2 mm are inluded along with the orresponding FE and analytial preditions. desribed above. The sandwih deformation involves ore ompression, ore shear and fae-sheet strething in all ases. The measured values of peak load and plateau load (defined as the average indentation load between indentation depths.1pd/p.15) are listed in Tables 1 and 2, respetively. Results are given for the three indenter geometries and for sandwih beams of fae-sheet thikness t ¼.6 and 1.2 mm. Analytial and FE preditions are now reported and are ompared with these measurements. δ δ

10 242 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) Fig. 16. The observed deformation modes of the t ¼.6 mm Y-frame sandwih indented by (a) the R ¼ 9 mm roller, (b) the a ¼ 1 mm flat bottom indenter and () the a ¼ 15 mm wide flat bottom indenter. The deformed profiles are shown at an indentation depth d ¼ 1 mm. Table 1 A omparison between the measured, analytial preditions and FE preditions of the normalised peak indentation loads of all the sandwih beams tested in this study Fae-sheet thikness (mm) Indenter dimensions (mm) Non-dimensional peak load, F peak /s Y bt Analytial predition FE Experiment t p 13 ¼ tp 13 ¼ 1:7MPa t ¼.6 Roller R ¼ Flat bottom a ¼ Flat bottom a ¼ t ¼ 1.2 Roller R ¼ Flat bottom a ¼ Flat bottom a ¼ Analytial preditions Ashby et al. [8] have presented an analytial model for the indentation of sandwih beams omprising metal foam ores. The ontribution of the shear strength of the foam to the indentation load was negleted and only the ontributions from ompression of the ore and from bending of the fae-sheets were onsidered. We have already observed in the present study that the longitudinal shear strength of the Y-frame onsiderably exeeds the ompressive strength

11 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) Table 2 A omparison between the measured and FE preditions of the normalised plateau indentation loads of all the sandwih beams tested in this study Fae-sheet thikness (mm) Indenter dimensions (mm) (see Setion 3) and we antiipate that the high shear strength of the ore ontributes to the indentation strength. An analytial model is now derived for the peak indentation fore of the Y-frame sandwih beam resting upon a rigid foundation. We emphasise that the aim of this analysis is not to provide an aurate estimate of the measurements but rather (i) to at as an upper bound on the indentation loads with the analysis of Ashby et al. [8] serving as a lower bound and (ii) demonstrate the influene of the shear strength of the ore on the indentation strength of sandwih beams. The Y-frame ore and fae-sheets are idealised as homogenous rigid, ideally plasti solids. The fae-sheets are assumed to have a tensile strength s Y while the Y-frame has a ompressive strength s p 33 in the x 3 diretion and a longitudinal shear strength t p 13. Assume that the Y-frame an ompress in the x 3 diretion without straining along the x 1 diretion, and assume that the shear and ompressive strengths of the Y-frame are deoupled. Consider the ollapse mode as skethed in Fig. 14 and employ the oordinate system shown in Fig. 14. Then, the deformation mode is written as uðx 1 ; x 3 Þ¼, (1a) 8 _y l þ a 2 x a 1 x3 ; 2 px 1p >< a 2 þ l; _y vðx 1 ; x 3 Þ¼ lx 3; a 2 px 1p a 2 ; (1b) _y l þ a 2 þ x >: 1 x3 ; a 2 lpx 1p a 2 ; where a is the width of the indenter. Here, u and v are displaements in the x 1 and x 3 diretions, respetively, is the ore thikness and l is the length of the small segments of the top fae-sheet that rotate through an angle y. The ollapse load F (per unit depth in the x 2 diretion) an be derived by a simple upper bound alulation as Z Z Fly ¼ 4M P y þ s p da þ t p 13 g 13 da, (2) A where the usual strain omponents are e 33 ¼ qv/qx 3 and g 13 ¼ qv/qx 1 +qu/qx 3. The integrals in Eq. (2) are over the A Plateau loads, F ss /s Y bt t ¼.6 Roller R ¼ Flat bottom a ¼ Flat bottom a ¼ t ¼ 1.2 Roller R ¼ Flat bottom a ¼ Flat bottom a ¼ FE Experiment The plateau indentation load F ss is defined as the average indentation load for indentation depths in the range.1pd/p.15. retangular region a 2 lpx 1p a 2 þ l and px 3p while the plasti bending moment of the fae-sheets of thikness t is given by M P s Y t 2 /4. This expression for the indentation fore redues to F ¼ s Y t 2 l þ sp 33 ðl þ aþþtp 13. (3) Now, minimise this upper bound solution for F with respet to the free parameter l to obtain the indentation load F I, where qffiffiffiffiffiffiffiffiffiffiffiffi F I ¼ 2t s Y s p 33 and the length l is rffiffiffiffiffiffi s Y l ¼ t. (5) s p 33 þ s p 33 a þ tp 13, (4) We proeed to ompare the predition (4) with the measured indentation load. Unless otherwise speified, in these omparisons we take the tensile yield strength of the fae-sheet material to be s Y ¼ 21 MPa (Fig. 3), the peak ompressive strength of the Y-frame as s p 33 ¼ :5MPa (Fig. 5) and the longitudinal shear strength as that of the L/ ¼ 7 speimen, i.e. t p 13 ¼ 1:7 MPa (Fig. 9). Comparisons between the measurements and the analytial preditions of the indentation load are given in Fig. 15a and b for indentation by the R ¼ 9 mm roller and by the flat bottom indenter of width a ¼ 15 mm, respetively. (For the roller we take a ¼ in Eq. (4).) The analytial preditions overestimate the measurements of the peak loads substantially. We attribute this disrepany to the fat that the Y-frame ore does not deform in a homogeneous manner. Eq. (4) provides an upper bound to the indentation load and it may be signifiantly higher than the true ollapse load. The analytial preditions of the normalised peak indentation load with t p 13 ¼ 1:7 MPa and tp 13 ¼ are inluded in Table 1. In similar manner to the omparisons shown in Fig. 15, the analytial preditions with t p 13 ¼ 1:7 MPa overestimate the measured loads in all ases. We attribute this disrepany to the fat that the homogenised response of the Y-frame assumed in the analytial model is inappropriate: the large gradients in strain aross the ore make it important to take the strutural features of the Y-frame into aount as will be shown in the FE analysis presented subsequently. On the other hand, a neglet of the shear strength in the analytial model (t 13 ¼ ) leads to a signifiant under-predition of the indentation strength. This onfirms our initial hypothesis that the shear strength of the Y-frame ore plays an important role in ditating the indentation strength of the Y-frame beams Finite element preditions Finite element alulations of the indentation response of the Y-frame ore sandwih beams have been performed using the finite element pakage ABAQUS. The geometry

12 244 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) of the beams was idential to that used in the experimental investigation (ross-setional dimensions detailed in Fig. 2) and both the fae-sheets and the Y-frame ore were modelled using linear shell elements (S4R in ABAQUS notation) with a mesh size of 2g as in Setion 3.3. The faesheets were tied to the top half and leg of the Y-frame and ontat between the surfaes were modelled using the hard, fritionless ontat option in ABAQUS. The imposed boundary onditions were as follows. All degrees of freedom (rotational and translational) of all nodes on the bottom fae-sheet were held at zero in order to simulate stiking frition between the rigid foundation and the bottom fae-sheet of the Y-frame ore sandwih. Rigid indenters, of geometry mathing those employed in the experiments, indented the beam by imposing an inreasing displaement. Contat between the outer surfae of the top fae-sheet and the rigid indenters was modelled using the fritionless ontat option as provided by ABAQUS. The material properties of the Y-frame ore sandwih sheets was taken to be the same as those used in the simulations desribed in Setion 3.3. Comparisons between the predited and measured indentation load versus displaement response are shown in Fig. 15a and b for indentation with the R ¼ 9mm diameter roller and the a ¼ 15 mm flat bottom punh, respetively. Reasonable agreement is observed. A more omplete omparison is given in Tables 1 and 2 where the measured values of the peak load and plateau load are ompared with the FE preditions. The agreement between FE preditions and measurements is within 8% in all ases A omparison of Y-frame sandwih beams with metal foam ore sandwih beams of equal mass Metal foam ore sandwih beams have been developed for lightweight strutural and energy absorption appliations, see for example Ashby et al. [8]. Here, we ompare the FE preditions of the indentation response of sandwih beams ontaining a Y-frame or a metal foam ore of equal mass. Two metal foam ore sandwih beams with 34 stainless steel fae-sheets of thikness t ¼.6 and 1.2 mm are onsidered. These beams have the same overall geometry as the Y-frame beams desribed above, i.e. length L ¼ 6 mm, width b ¼ 115 mm and ore thikness ¼ 44 mm. For the sake of brevity only the indentation of the metal foam ore beams by the R ¼ 9 mm rigid roller is addressed here. Plane strain FE alulations were performed using ABAQUS, with both the foam ore and fae-sheets modelled using four-noded plane strain elements (CPE4 in ABAQUS notation). The fae-sheets were modelled as J2 flow theory solids as for the Y-frame beams while the Deshpande and Flek [9] model was used to desribe the onstitutive response of the metal foam ore. Write s ij as the usualpdeviatori stress and the von Mises effetive stress as s e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3s ij s ij =2. Then, the isotropi yield surfae for the metal foam is speified by ^s Y ¼, (6) where the equivalent stress ^s is a homogeneous funtion of s e and mean stress s m s kk /3 aording to ^s þða=3þ 2 ½s2 e þ a2 s 2 mš. (7) The material parameter a denotes the ratio of deviatori strength to hydrostati strength, and the normalisation fator on the right hand side of relation (7) is hosen suh that ^s denotes the stress in a uniaxial tension or ompression test. Normality of plasti flow is assumed, and this implies that the plasti Poisson s ratio n p ¼ _ p 22 =_p 11 for uniaxial ompression in the one-diretion is given by 1=2 ða=3þ2 v p ¼ 1 þða=3þ 2. (8) In order to make a diret omparison with the stainless steel Y-frame sandwih ore beams, we onsider a metal foam with a relative density r ¼ :2 and made from stainless steel of yield strength s Y ¼ 21 MPa. Thus, the ore has the same overall size and mass as that of the Y-frame ore. Following Ashby et al. [8], we speify that this foam has a Young s modulus E ¼ 1 GPa, elasti Poisson s ratio n ¼.3 and a plasti Poisson s ratio n P ¼. The uniaxial ompressive stress versus plasti strain response is assumed to be given by ( Y ¼ :3 r1:5 s Y ; ^ P p D ; :3 r 1:5 s Y þ E ð^ P (9) D Þ; otherwise; where _^ p is the plasti strain rate that is work onjugate to ^s and D lnð rþ is the logarithmi densifiation strain beyond whih negligible plasti straining of the foam ours. Thus, the ompressive plateau strength of the metal foam ore is Yð^ p p D Þ:2 MPa. The boundary onditions were speified as follows. All degrees of freedom of all nodes on the bottom fae-sheet were ompletely onstrained in order to simulate stiking frition between the rigid foundation and the bottom faesheet of the metal foam ore sandwih. Loading was applied through presribed displaements of rigid R ¼ 9 mm roller. And ontat between the outer surfae of the top fae-sheet and the rigid indenter were modelled by fritionless ontat as provided by ABAQUS. The normalised indentation fore F F=ðs Y btþ versus normalised displaement d d= response of the metal foam sandwih beams with fae-sheets are plotted in Fig. 17a along with the orresponding preditions for the Y-frame sandwih from Fig. 15. Results are shown for the two fae-sheet thiknesses, t ¼.6 and 1.2 mm. In ontrast to the response of the Y-frame sandwih, the metal foam ore sandwih beam displays a monotonially inreasing indentation fore with inreasing indentation depth. At any given indentation depth, the indentation fore for the Y-frame sandwih beams is signifiantly higher than that of

13 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) F (σ Y b t) W σ Y b t R the orresponding metal foam ore beam. The energy absorption W of the sandwih beams is defined as WðdÞ ¼ metal foam (t = 1.2mm) δ Z d Y-frame (t = 1.2 mm) metal foam (t =.6mm) Y-frame (t =.6mm) Y-frame (t=.6 mm) Y-frame (t=1.2 mm) Metal Foam (t=.6 mm) Metal Foam (t=1.2 mm) Fig. 17. A omparison between the normalised (a) indentation load versus indentation depth and (b) energy absorption versus indention depth responses of the Y-frame and metal foam sandwih ore beams. The sandwih beams have 34 stainless steel fae-sheets of thikness t ¼.6 and 1.2 mm. The metal foam ore is made from stainless steel and has a relative density equal to that of the Y-frame ore. F dd, (1) with the normalised energy absorption W W=ðs Y brtþ. This normalised energy absorption is plotted against the normalised indentation depth d for the metal foam and Y-frame sandwih beams in Fig. 17b. As expeted, the Y-frame sandwih beams signifiantly outperform their metal foam ounterparts in that that they absorb more energy for any given indentation depth. In the above omparison, it is lear that the Y-frame sandwih beams outperform the equivalent metal foam δ ore beams. However, the possibility of fae-sheet tearing has been negleted in the omparison. In most pratial irumstanes failure of the beam is ditated by tearing of the fae-sheets rather than by a limiting indentation depth. It is instrutive to ross-plot the normalised energy absorption W against the maximum prinipal strain e max in the fae-sheets in order to determine whih beam absorbs more energy prior to the initiation of tearing in the fae-sheets, as haraterised by the e max. The FE preditions of W versus e max for the t ¼.6 and 1.2 mm Y-frame ore and metal foam ore sandwih beams are shown in Fig. 18a. These urves are determined as follows. For a given indentation depth d, the energy absorption W is determined via Eq. (1) while e max determined by examining the through-thikness average strains in the fae-sheets. These two values give one point on the W versus e max urve in Fig. 18a. This proedure has been repeated for a large number of values of d in order to onstrut the urves shown in Fig. 18a. The trajetories plotted in Fig. 18a are limited to the regime do:2, onsistent with the data presented in Fig. 17. It is onluded from Fig. 18a that the metal foam ore provides for a larger value of absorbed energy W than the Y-frame, for any assumed value of e max. The strain onentrations for the foam ore and the Y-frame are ompared in Fig. 18b and, respetively. Severe strain onentrations are present at the joints of the Y-frame, and these lead to its redued energy absorption apaity. An alternative approah is to assume that the maximum fae-sheet strain ditates the limiting strain for both types of ore. When this assumption is made, the energy absorption apaities of the Y-frame and metal foam ore sandwih beams are similar. 5. Conluding remarks Y-frame sandwih beams made from AISI type 34 stainless steel sheets have been manufatured by a folding, slotting and brazing tehnique. The out-of-plane ompressive and transverse shear responses are governed by plasti bending of the onstituent struts of the Y-frame. Under longitudinal shear the Y-frame leg undergoes uniform straining prior to the onset of plasti wrinkling. Thus, the Y-frame has a low ompressive and transverse shear strength but a high longitudinal shear strength. These measured ompressive and shear responses of the Y-frame sandwih ore are in good agreement with three-dimensional finite element preditions. The Y-frame sandwih beams are designed to withstand low veloity impats by spreading the load over a large area and thus preventing the tearing of the sandwih beam fae-sheets. In this study, we have investigated the transverse indentation response of Y-frame sandwih beams resting upon a rigid foundation. The high longitudinal shear strength of the Y-frame inreases the indentation strength of the Y-frame beams substantially: while an analytial upper bound analysis for this indentation proess over-predits the indentation strength, the

14 246 ARTICLE IN PRESS V. Rubino et al. / International Journal of Mehanial Sienes 5 (28) Y-frame (t=.6 mm) W σ Y b t R.3 Metal Foam.2 (t=.6 mm) Metal Foam (t=1.2mm).1 Y-frame (t=1.2 mm) ε max Max prinipal plasti strain Max prinipal plasti strain Fig. 18. (a) The normalised energy absorption versus maximum prinipal strain in the fae-sheets of the Y-frame and metal foam ore sandwih beams onsidered in Fig. 16. The deformed profiles of sandwih beams with t ¼.6 mm fae-sheets at an indentation depth d ¼ 1 mm are shown (b) for the metal foam ore and () for the Y-frame ore. Contours of maximum prinipal strain are inluded in these deformed profiles. three-dimensional finite element simulations apture the response to a high level of auray. These finite element simulations demonstrate that for any given indentation depth, the Y-frame sandwih beams absorb signifiantly more energy than metal foam ore sandwih beams of equal mass. However, the Y-frame indues large strain onentrations in the top fae-sheet that inreases the likelihood of tearing of the fae-sheet. Aknowledgments This work was supported by the Netherlands Institute for Metal Researh, projet no. MC1.3163, The Optimal Design of Y-ore sandwih strutures. Referenes [1] Deshpande VS, Flek NA. Collapse of truss ore sandwih beams in 3- point bending. International Journal of Solids and Strutures 21; 38(36 37): [2] Paik JK. Innovative strutural designs of tankers against ship ollisions and grounding: a reent state-of-the-art review. Marine Tehnology 23;4(1): [3] Wevers LJ, Vredeveldt AW. Full sale ship ollision experiments TNO-report 98-CMC-R1725. The Netherlands, [4] Ludolphy, H. The unsinkable ship development of the Y-shape support web. In: Proeedings of the 2nd international onferene on ollision and grounding of ships, Copenhagen, Denmark, 21. [5] Konter A, Broekhuijsen J, Vredeveldt A. A quantitative assessment of the fators ontributing to the auray of ship ollision preditions with the finite element method. Tokyo, Japan: International Conferene of Ship Collisions and Grounding; 24. [6] Naar H, Kujala P, Simonsen BC, Ludolphy H. Comparison of the rashworthiness of various bottom and side strutures. Marine Strutures 22;15(4 5): [7] Pedersen CBW, Deshpande VS, Flek NA. Compressive response of the Y-shaped sandwih ore. European Journal of Mehanis A/Solids 24;25: [8] Ashby M, Evans A, Flek N, Gibson L, Wadley H. Metal foams: a design guide. Butterworth Heinemann; 2. [9] Deshpande VS, Flek NA. Isotropi onstitutive models for metalli foams. Journal of Mehanis and Physis of Solids 2;48(6 7):