Pyramidal lattice truss structures with hollow trusses

Transcription

1 Materials Science and Engineering A 397 (2005) Pyramidal lattice truss structures with hollow trusses Douglas T. Queheillalt, Haydn N.G. Wadley Department of Materials Science and Engineering, 116 Engineers Way, P.O. Box , University of Virginia, Charlottesville, VA , USA Received 22 November 2004; accepted 8 February 2005 Abstract A method for fabricating pyramidal lattice structures with hollow metallic trusses has been explored. A periodic diamond hole pattern sheet consisting of 304L stainless steel hollow tubes was made by an alternating collinear lay-up process followed by vacuum brazing. The array was then folded at the nodes to create a periodic pyramidal cellular metal lattice and was then bonded to face sheets by a second vacuum brazing process. The out-of-plane compression properties of this hollow truss lattice structure have been investigated and compared to a similar lattice made with the solid trusses. In both cases, the peak strength is found to be governed by inelastic truss buckling. The compressive strength of a hollow lattice with a relative density of 2.8% was approximately twice that of a solid pyramidal lattice of similar relative density. The increased strength resulted from an increase in the buckling resistance of hollow trusses because of their higher radius of gyration Elsevier B.V. All rights reserved. Keywords: Porous material; Stainless steel; Brazing; Mechanical properties 1. Introduction Lightweight sandwich panel structures consisting of low density cores and their solid face sheets are widely used in aerospace and other applications [1]. Cellular core structures including honeycombs [2,3] and prismatic forms [4 7] are often used because of their very low density, good crush resistance and high in-plane shear resistance. Recently, lattice truss cellular structures have begun to be explored as a candidate sandwich panel core material [8 11]. The lattice truss topology can be designed to efficiently support panel bending loads by aligning them so they are subject to only axial deformations [12 14]. For trusses with lattice relative densities of 1 10%, inelastic buckling then determines the strength of the trusses and the cellular materials out-of-plane compression and in-plane shear strength [11]. Diamond textile lattice truss structures have recently been constructed with hollow trusses and found to be significantly stronger than their solid truss counterparts [15]. The hollow trusses in these structures had higher second area moments than their solid truss counterparts and this was found to Corresponding author. Tel.: ; fax: address: dougq@virginia.edu (D.T. Queheillalt). increase their inelastic buckling strength. The use of hollow truss structures also provided a means for varying the cellular structures relative density without changing the cell size or truss slenderness ratio [15]. Here, we modify the process devised for diamond lattice structures and explore a method for making a pyramidal hollow truss lattice structure like that shown in Fig. 1. We describe the fabrication of a representative structure and compare its out-of-plane compressive response to that of a solid pyramidal truss of similar relative density. 2. Hollow lattice truss fabrication Solid truss pyramidal lattice structures are fabricated via a folding operation that bends a diamond perforated sheet to create a single layer of trusses arranged with a pyramidal topology [16]. Fig. 2 schematically illustrates this process. A solid truss structure with a relative density of 2.5% was made from 304L stainless steel by a similar process using the detailed procedure initially described by Sypeck and Wadley [16]. Briefly, the process consisted of punching a metal sheet to form a periodic diamond perforation pattern, folding node row by node row using a paired punch and die tool set and /$ see front matter 2005 Elsevier B.V. All rights reserved. doi: /j.msea

2 D.T. Queheillalt, H.N.G. Wadley / Materials Science and Engineering A 397(2005) Fig. 1. A pyramidal lattice truss core sandwich structure. then brazing this core to solid face sheets to form a sandwich structure. The sheet thickness, t = 0.81 mm, truss width, w = 0.81 mm and the truss length, l = 12.7 mm. The inclination angle, ω =45. This resulted in a pyramidal lattice with square cross section trusses and a measured relative density ρ = ± A similar process was used to create a layer of circular cross section hollow trusses with the trusses arranged in a similar diamond arrangement. The process began by taking two collinear 304L stainless steel tube arrays and overlaying them at 60. They were then vacuum brazed to form the diamond pattern shown in Fig. 3(a). The brazed structure was then folded node row by node row using a paired punch and die with the sheets folded to create a square base pentahedron. Fig. 4 shows a cartoon of the folding process. The punch and die were again designed so that the trusses were inclined at an angle ω =45. An example of the hollow lattice truss diamond sheet after folding to create a hollow truss pyramidal lattice is shown in Fig. 3(b). The tubes had an o.d., d o = 1.47 mm and an i.d., d i = 1.07 mm. The truss length l = 14.3 mm and the inclination angle ω =45. This structure was then brazed to 304L stainless steel face sheets. The measured relative density of the hollow truss pyramidal lattice was ± The hollow and solid truss systems used identical brazing processes for both brazing steps. The samples were placed in a vacuum furnace at a base pressure of 10 4 Torr. They were heated at 10 C/min to 550 C, held for 1 h (to volatilize the Fig. 3. Photographs of (a) the diamond hollow tube sheet prior to the folding operation and (b) the hollow lattice truss pyramidal structure after folding. binder), then heated to the brazing temperature of 1050 C. They were held for 60 min at this temperature before furnace cooling at 25 C/min to ambient. A braze alloy with a nominal composition of Ni 25.0Cr 10.0P 0.03C (wt.%) was used for the bonding process. It was applied by spraying the samples to be bonded with a mixture of the braze powder and a polymer binder. Fig. 5 shows photographs of the as fabricated solid and hollow truss pyramidal lattice sandwich structures. Fig. 2. An illustration of the folding operation used to create the single layer pyramidal truss sandwich structures. Also shown is the solid truss shaped diamond cells prior to the forming. Fig. 4. An illustration of the folding operation used to create the single layer pyramidal truss sandwich structures. Also shown is the hollow truss shaped diamond cells prior to the forming.

3 134 D.T. Queheillalt, H.N.G. Wadley / Materials Science and Engineering A 397(2005) For hollow trusses, Fig. 6(b), with an o.d., d o, and an i.d., d i, the volume occupied by solid metal is: V s = π(do 2 d2 i )l. (3) The relative density of the cell, ρ, is the ratio of the truss volume to that of the unit cell. For the solid pyramidal lattice the relative density is: 2wt ρ = l 2 cos 2 ω sin ω. (4) For the hollow pyramidal lattice: ρ = π(d2 o d2 i ) 2 l 2 cos 2 ω sin ω. (5) The predicted relative densities were slightly lower than that measured because of the added weight of the braze alloy and non-ideal node conditions. 4. Compressive responses Fig. 5. Photographs of the brazed pyramidal truss sandwich structures fabricated from (a) solid truss and (b) hollow truss elements. 3. Unit cell densities Fig. 6 shows the unit cells of a pyramidal lattice truss structure with solid and hollow trusses. Ignoring the added weight of the braze alloy; only the truss cross sectional area and length determine the relative density of the two structures. For any regular pyramidal structure of truss length l, the unit cell volume: V c = 2 cos 2 ω sin ωl 3 (1) The solid and hollow truss structures were tested at ambient temperature in compression at a nominal strain rate of s 1. The measured load cell force was used to calculate the nominal stress applied to the structure. The nominal through thickness strain was obtained using a laser extensometer to monitor the displacements of the face sheets. The through thickness compressive nominal stress strain responses for both the solid and hollow lattice structures are shown in Fig. 7. Both exhibit characteristics typical of cellular structures including a region of nominally elastic response, yielding, plastic strain hardening to a peak strength, followed by a drop in flow stress to a plateau region and finally rapid hardening associated with core densification [17]. The solid where ω is the angle between the truss members and the base of the unit cell. For the solid trusses with a rectangular cross section, Fig. 6(a), the volume occupied by metal: V s = 4twl. (2) Fig. 6. Schematic illustrations of the pyramidal lattice unit cells for cores fabricated from (a) solid and (b) hollow trusses. Fig. 7. Through thickness compressive nominal stress strain responses for the solid and hollow lattice truss pyramidal sandwich structures.

4 D.T. Queheillalt, H.N.G. Wadley / Materials Science and Engineering A 397(2005) were performed according to ASTM E8-01. The uniaxial tensile response is shown in Fig. 9. The average Young s modulus, E s, and 0.2% offset yield strength, σ ys, were 203 GPa and 176 MPa, respectively. 5. Compressive response predictions The effective properties of a solid truss pyramidal core have been discussed in detail by Deshpande and Fleck [13].At high relative densities (low aspect ratio trusses), the strength of a truss structure made from a rigid ideally plastic material is controlled by plastic yielding. The peak compressive strength, σ pk, of a pyramidal lattice failing by plastic yielding is then given by [13]: σ pk = σ y sin 2 ω ρ. (6) A non-dimensional peak strength coefficient, Σ = σ pk / ρσ y (where σ pk is the peak compressive strength, σ y the solid materials yield stress) was determined from the data and is plotted against ρ in Fig. 10. The average value of the peak strength coefficient was 0.43 for the solid truss structure. The hollow truss structures had a value of If a solid truss pyramidal structure (with ω =45 ) is made from a rigid ideally plastic material, Σ = 0.5. Values of Σ > 0.5, therefore exploit the strain-hardening and buckling resistance characteristics of the alloy from which it is made. If a pyramidal lattice is constructed from very slender trusses, the trusses collapse by elastic buckling during axial compression. In this case, the peak compressive strength is found by replacing σ y in Eq. (6) with the elastic bifurcation stress, σ c, of the trusses [18]. The elastic bifurcation stress, Fig. 8. Photographs showing the deformation characteristics of the hollow truss pyramidal lattice at plastic strains of 2, 5, 10, 20, 30 and 40%. truss structures plastically buckled shortly after yielding (at a strain of 0.03). The hollow trusses exhibited significant post yield hardening to a strain of 0.1 before they underwent buckling. After the onset of plastic buckling, significant core softening was observed in both structures, followed by plateau region. The plateau stress of the solid truss structure was 25% of its peak value, whereas the hollow truss structure was reduced to only 75% of its peak value. Photographs of the hollow lattice truss during plastic deformation are shown in Fig. 8. Lateral deflections of the truss members began to occur at loads prior to the attainment of the peak strength. Softening coincided with the formation of a plastic hinge near the middle of the truss members. Neither truss member nor node fracture was observed in any of the compression experiments performed. Tensile tests were performed on 304L stainless steel samples that had followed the same thermal cycle used for fabrication of the brazed sandwich structures. Three tensile tests Fig. 9. Average uniaxial tensile response (true stress true strain) of 304L stainless steel and the critical inelastic buckling stress for the solid and hollow trusses.

5 136 D.T. Queheillalt, H.N.G. Wadley / Materials Science and Engineering A 397(2005) bifurcation stress of a compressively loaded column. An expression for this stress, σ c, is obtained by replacing E s by E t, (the tangent modulus of the material used for making the trusses) in Eq. (7) [18]: Fig. 10. Predicted and measured non-dimensional peak strength coefficients Σ = σ pk / ρσ y plotted against ρ for both the solid and hollow lattice truss pyramidal sandwich structures. Built in (k = 2) assumed for all predictions. σ c, for an axially loaded column is given by: σ c = π2 k 2 IE s Al 2 (7) where E s is the elastic modulus, I the second area moment of inertia of the truss and A is the truss cross sectional area [18]. Expressions for the second area moment of inertia and cross sectional area for the solid and hollow trusses are summarized in Table 1 together with the calculated values for the trusses studied here. The factor k depends on the rotational stiffness of the nodes; for a pin-joint that can freely rotate k = 1. The case k = 2 corresponds to a built in-joint, which does not rotate. In the experiments conducted here no evidence of node rotation was observed and we take k =2. If the truss material has a non zero post yield strain hardening rate, inelastic buckling strength defined by Shanley Engesser tangent modulus theory determines the lattice strength [18,19]. The peak compressive strength is then obtained by replacing σ y in Eq. (6) with the inelastic Table 1 Cross sectional area and area moment of inertia for solid and hollow truss geometries (w = 0.81 mm, t = 0.81 mm, d o =.47 mm, d i = 1.07 mm) Truss geometry Cross sectional area, A (mm 2 ) wt = Area moment of inertia, I (mm 4 ) wt 3 12 = σ c = π2 k 2 IE t Al 2. (8) The tangent modulus is defined as the slope dσ/dε of the uniaxial stress versus strain curve of the solid material at the inelastic bifurcation stress level σ c [18]. The inelastic bifurcation stress for the solid and hollow trusses can be deduced from a Ramberg Osgood fit of the solid material stress strain response as shown in Fig. 9. We also show the stress predicted by Eq. (8) (the dashed curve) calculated using the local tangent modulus obtained by differentiation of the Ramberg Osgood response. The inelastic bifurcation stress, σ c, is obtained at the intersection of these curves and for the solid truss was 170 MPa and the hollow truss was 240 MPa. Predictions for the normalized peak compressive strengths are shown in Fig. 10 for three characteristic deformation modes: plastic yielding, elastic buckling and inelastic (plastic) buckling of the trusses as a function of the core s relative density. Some ambiguity in the precise length of a truss (l) exists in practice because of the accumulation of braze alloy at the nodes, Fig. 5. We show predictions for the hollow pyramidal lattice truss using truss lengths (l) of 14.7 and 11.7 mm to account for braze material build up at the nodes. The true length appears to lie between these bounds. Minimal braze alloy build up was observed for the solid pyramidal truss and predictions for truss length (l) of 12.7 mm are shown in Fig. 10. We also assume that during both elastic and inelastic buckling, the trusses have built-in nodes at the face sheets and take k = 2 in Eqs. (7) and (8). Recall that the average value of the peak strength coefficient for solid trusses was 0.43 and 0.75 for the hollow truss structure. Some scatter of the test results was observed because they are sensitive to the accuracy of fabrication, alignment of the applied load and the end support conditions (i.e. amount of braze material at the nodes). Fig. 10 shows the predicted normalized strength coefficient for both truss types. It can be seen that the hollow pyramidal lattice has a normalized strength that is nearly twice that of a comparable density solid truss lattices. This increase is controlled by the radius of gyration, I/A, of the trusses. The radius of gyration describes the way in which the area of a cross section is distributed around its centroidal axis [18]. Increasing the value of the radius of gyration increases a columns resistance to buckling. The radius of gyration was 0.23 for the solid truss structure and 0.46 for the hollow truss structure. 6. Summary π 4 (d2 o d2 i ) = π 64 (d4 o d4 i ) = A simple hollow tube lay up and node folding process has been devised to create pyramidal lattice truss structures with hollow metal trusses. We have demonstrated the approach

6 D.T. Queheillalt, H.N.G. Wadley / Materials Science and Engineering A 397(2005) using 1.47 mm diameter 304L stainless steel tubes with a wall thickness of 200 m. This process results in a lattice, which can be bonded to face sheets to form a metallic sandwich structure with a pyramidal topology. Similar relative density pyramidal truss structures have been fabricated with solid trusses and the compressive mechanical properties of the two systems have been computed. The use of the hollow trusses as ligaments of pyramidal lattice truss structures increases the peak compressive collapse strength by increasing the inelastic buckling resistance of the hollow tubes. The outof-plane compressive strength of a hollow pyramidal lattice with a relative density of 2.8% is shown to be approximately twice that of similar relative density solid truss structures. Acknowledgements We are very grateful to Vikram Deshpande (Cambridge University, UK) for helpful discussions. This work has been performed as part of the Topologically Structured Materials: Blast and Multifunctional Implementations program conducted by a consortium of Universities consisting of Harvard University, Cambridge University, the University of California at Santa Barbara and the University of Virginia. We are grateful for the many helpful discussions with our colleagues in these institutions. The Office of Naval Research (ONR), monitored by Dr. Steve Fishman, funds the consortiums work under grant number N References [1] T. Bitzer, Honeycomb Technology: Materials, Design, Manufacturing, Applications and Testing, Chapman & Hall, [2] R.M. Christensen, J. Mech. Phys. Solids 34 (1986) 563. [3] B. Kim, R.M. Christensen, Int. J. Mech. Sci. 42 (2000) 657. [4] W. Ko, NASA Technical Paper 1562, [5] T.S. Lok, Q.H. Cheng, J. Sound Vib. 229 (2000) 311. [6] T.S. Lok, Q.H. Cheng, Comp. Struct. 79 (2001) 301. [7] T.S. Lok, Q.H. Cheng, J. Sound Vib. 245 (2001) 63. [8] A.G. Evans, J.W. Hutchinson, M.F. Ashby, Curr. Opin. Solid State Mater. Sci. 3 (1998) 288. [9] A.G. Evans, J.W. Hutchinson, M.F. Ashby, Prog. Mater. Sci. 43 (1999) 171. [10] A.G. Evans, J.W. Hutchinson, N.A. Fleck, M.F. Ashby, H.N.G. Wadley, Prog. Mater. Sci. 46 (2001) 309. [11] H.N.G. Wadley, N.A. Fleck, A.G. Evans, Compos. Sci. Technol. 63 (2003) [12] N. Wicks, J.W. Hutchinson, Int. J. Solids Struct. 38 (2001) [13] V.S. Deshpande, N.A. Fleck, Int. J. Solids Struct. 38 (2001) [14] J.C. Wallach, L.J. Gibson, Int, J. Solids Struct. 38 (2001) [15] D.T. Queheillalt, H.N.G. Wadley, Acta Mater. 53 (2005) 303. [16] D.J. Sypeck, H.N.G. Wadley, Adv. Eng. Mater. 4 (2002) 759. [17] M.F. Ashby, A.G. Evans, N.A. Fleck, L.J. Gibson, J.W. Hutchinson, H.N.G. Wadley, Metal Foams: A Design Guide, Butterworth Heinemann, United Kingdom, [18] J.M. Gere, S.P. Timoshenko, Mechanics of Materials, PWS Engineering, Boston, [19] F.R. Shanley, Mechanics of Materials, McGraw-Hill, New York, 1967.