L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses Liang Donga , Arthur Heuerb , Harold Kahnb , Zhen Lib , Vikram Deshpandec , Haydn Wadleya a Department of Materials Science and Engineering, University of Virginia, Charlottesville, Virginia, USA of Materials Science and Engineering, Case Western Reserve University, Cleveland Ohio, USA c Engineering Department, Trumpington Street, University of Cambridge, UK b Department International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. Strong cellular lattices with nitro-carburized stainless steel hollow trusses Collinear lattice structures have been fabricated from 304 stainless steel hollow tubes via an alternating collinear lay-up process followed by bonding using a vacuum brazing method. By varying the tube wall thickness from 50 to 203 lm, square and diamond lattice topologies with relative densities between 0.03 and 0.11 were manufactured in this way and their compressive mechanical responses were characterized. A low temperature nitro-carburization treatment was then performed on a second set of the collinear lattice cores at a temperature of 440 8C for 20 h. The treatment created a thin (10 to 20 lm thick) extremely hard (*12 GPa) surface layer on the interior and external surfaces of the hollow trusses. This significantly increased the compressive buckling resistance of individual trusses. Compressive strength enhancements (compared with untreated counterparts in the annealed (as brazed) condition) varied from 1.2 for thick walled tubes to 3.8 when the wall thickness was decreased to about twice the hardened layer depth. The moduli and strengths of all the lattices were found to increase with lattice relative density, and were well predicted by micromechanical models. The lowest relative density (thinnest wall) nitro-carburized hollow truss collinear lattice structures exhibited a specific compressive strength significantly higher than that of any other cellular metal reported to date. Nitro-carburized stainless steel collinear lattices therefore appear to be promising candidates for the cores of lightweight sandwich panels intended for elevated temperature and/or multifunctional applications. Keywords: Cellular lattice; Nitro-carburization; 304 stainless steel; Compressive strength 1. Introduction Low density cellular metals have been widely used for the cores of sandwich panel structures intended for load supporting applications [1 – 6]. They have also been investigated for numerous multifunctional applications such as heat exchange [7, 8], dynamic load protection [9 – 11], and for acoustic damping [12]. Many types of cellular materials have been developed to meet the needs of these various applications. They include open and closed cell foams made from various polymers, metals and ceramics [1, 13], stronger honeycomb topology materials [14, 15], as well as lattice structures with tetrahedral [16 – 18], pyramidal [18 – 23], 3D-Kagome [18, 24, 25], octet truss [26], woven metal textile [6, 27, 28] or collinear [29] topologies, and simpler to fabricate, prismatic corrugations [30, 31]. The design of an optimal sandwich structure for one of these applications begins with selection of a material system. It then distributes the allowable panel mass per unit area between its core and faces, and then chooses a core cell topology that, in combination with material properties, avoids panel failure under the intended loading environment. For example, if an edge clamped sandwich panel must support large centrally applied bending loads, it is essential that the core has sufficient through thickness compressive and transverse shear elastic moduli and strengths to maintain face sheet separation throughout loading process. The mechanical properties of a cellular material are determined by its relative density, by its parent material mechanical properties and by the cell topology. For example, a metallic lattice made from elastic–perfectly plastic trusses that fail by plastic yielding, has a compressive collapse strength, rc , given by [30]; rc ¼ Rrys q ð1Þ where rys is the yield strength of the solid material from which the lattice is made, q is the relative density of the lattice and R is a lattice topology dependent \strength coefficient". The compressive strength coefficient describes the efficiency with which an applied compressive force is resolved into forces within the lattice structure. For a square honeycomb core with low aspect ratio webs all aligned in the loading direction, R = 1. However, for tetrahedral and pyramidal lattices, the trusses are inclined to the loading direction and R = sin2x, where x as the inclination angle between the trusses and the loading direction [30]. Figure 1 shows a collinear lattice in either a square, Fig. 1a, or 458 rotated diamond orientation, Fig. 1b. The lattice topology dependent strength coefficients, R, of such collinear lattices can be found by analyzing the unit cell, Fig. 1c. For the unit cell of a square orientation collinear lattice, Fig. 1a, half of the trusses have x = 908 with the remainder having x = 08. Therefore, the effective value of sin2x = 0.5. For a unit cell of a diamond orientation lattice, Fig. 1b, all the trusses have x = 458, and the effective value of sin2x = 0.5 again. It is important to note that for a finite width diamond orientation lattice, an edge effect should be taken into consideration as some trusses are not connected to both loading faces, leading to a reduction in the load supporting area. The compressive strength of finite width samples with the diamond 1 L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses is elastic modulus of the parent material and k is a factor accounting for the rotational stiffness of the truss ends (k = 2 for built-in end conditions and 1 for pin-jointed ends). For a circular cross-section hollow tube truss, the area moment of inertia, I is a function of the tubes cross sectional geometry: International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. I¼ p 4 a a4i Þ 4 o ð4Þ where 2 ao and 2 ai are the outer and inner diameters of the tube. For thin walled tubes (where ai approaches ao ), the buckling strength can be substantially higher than that of the equivalent area solid strut since the area moment of inertia ratio of a tube to a solid counterpart of equal area is a2o þ a2i . a2o a2i If the lattice is made from a material with a non-zero strain-hardening rate, the trusses continue to deform after plastic yielding until the onset of inelastic buckling. In this case, the compressive strength of the lattice is determined by the inelastic global buckling strength of the truss, rIE , which was first predicted by Engesser tangent modulus theory [33, 34]: rIE ¼ p2 k2 IEt Al2 ð5Þ where Et is the tangent modulus of the parent material at the inelastic bifurcation stress [33]. It is determined as the modulus at the intersection point of the true stress–true strain curve and that defined by Eq. (5). The lattice compressive strength in this case is given by: 2 2 p k IEt q rc ¼ RrIE q ¼ R Al2 Fig. 1. Sketches of (a) the square and (b) diamond orientated collinear lattice structures. (c) shows a unit cell of a square collinear lattice with hollow trusses. orientation therefore depends upon their length to height (aspect) ratio since struts at the sides do not connect the top and bottom of the sample. The strength of such finite samples can be predicted by multiplying the strength given by Eq. (1) by the factor [1 ð1=S tan xÞ] where S is the length to height (aspect) ratio of the sample [32]. If the trusses are sufficiently slender, and especially if made of a low elastic modulus (high yield strain) material, they elastically buckle prior to plastic yielding, and the compressive strength of the lattice is obtained by replacing rys in Eq. (1) with the Euler (global) buckling stress: rE ¼ p2 k2 IEs Al2 The lattice strength is then given by: 2 2 p k IEs q¼R q rc ¼ RrE Al2 ð2Þ ð3Þ where I, A, and l are the area moment of inertia, the crosssectional area, and the length of the truss respectively; Es 2 ð6Þ Engesser tangent modulus theory predicts the critical stress of an axially loaded column when lateral deflection is initiated. It defines the lower bound inelastic global buckling stress of such a compressed column since the load bearing capacity of an imperfection free column could increase with increasing lateral deflection. The upper bound to the column buckling stress is defined by a reduced modulus theory [33], in which Et in Eq. (5) is replaced by the reduced modEt ulus Er ¼ pﬃﬃﬃ4E ﬃ sp ﬃﬃﬃ 2 . ð Es þ Et Þ Shanley [34] showed the critical inelastic global buckling stress to be a function of a buckling columns transverse displacement and located between the critical load predicted by the tangent modulus theory (the lower bound) and the reduced modulus (upper bound) theory. Shanley’s equation for the inelastic global buckling critical stress (rSIE ) is given by: 2 3 6 rSIE ¼ rIE 6 41 þ 7 1 7 ao 1 þ Et =Es 5 þ v0 1 Et =Es ð7Þ where rIE is the critical inelastic global buckling stress defined by Engesser tangent modulus model (Eq. (5)), 2 ao is International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses the tube outer diameter, Et is the tangent modulus at an axial strain e, and v0 is the lateral deflection. Geometry dictates that the lateral deflection and axial strain of the tube pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ are related. To a first order apprixmation, v0 = 2l 2e e2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ for k = 1 (pin-joint) end conditions and 2l e e2 for k = 2 ends. Equations (1), (6) and (7) provide a rationale for the design of high specific strength cellular structures from hollow tubes. If the strut slenderness is sufficiently low that yielding controls the strength, increasing the specific strength of the material used to fabricate the lattice increases the strut/lattice compressive strength. However, there will eventually come a point where the yield strength is so high that strut buckling intervenes, and the elastic modulus and second moment of the struts then govern the strength. For a short and very thin-walled tube loaded in compression, failure will not be caused by global buckling as described by elastic (Euler) or inelastic buckling, but rather involves local bending of the tube walls; a failure mode referred to as local buckling [35, 36]. The most important shape-parameter governing the local buckling stress of cylindrical surfaces is the tube diameter-to-wall thickness ratio Dm /t, where t is the wall thickness and Dm refers to the tube’s mean diameter: Dm ¼ a2o a2i t ð8Þ The tube length can also affect the critical stress to some extent, but in most structural applications the members are long enough that length effects can be neglected. The critical local buckling stress in the elastic range is given by: rlE ¼ Kc E s Dm =t ð9Þ 2 where Kc ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ, and m is Poisson’s ratio; for a ma3ð1 m2 Þ terial with m = 0.3, Kc therefore equals 1.2. However, numerous studies have shown that this theoretical value cannot be achieved experimentally because the buckling behavior of thin walled tubes is highly sensitive to initial imperfections. In practice Kc is determined by fitting to experimental data, and typically varies between 0.24 and 0.8 [34, 37]. In the inelastic buckling regime, Es in Eq. (9) is usually replaced by an effective modulus equal to the geometric mean of the elastic and tangent moduli to give a conservative estimate for the critical inelastic local buckling stress: pﬃﬃﬃﬃﬃﬃﬃﬃﬃ Kc E s E t rlIE ¼ ð10Þ Dm =t Hollow tube lattice design concepts has been previously investigated by fabricating collinear lattice structures from 304 stainless steel hollow tubes with both the square and the (458 rotated) diamond orientation topologies, Fig. 1a and b [29]. The relative density of such hollow truss lattices depends upon the outer (2ao ) an inner (2ai ) tube diameters, and the unit cell length, l, Fig. 1c. The relative density, q, of the lattice is the ratio of the volumes of solid material (Vs ) within a unit cell to that of the unit cell (Vc ): q ¼ Vs pða2o a2i Þ ¼ 2ao l Vc ð11Þ The hollow tube lattice therefore enables independent control of the cell size (length) and the lattice relative density by varying the dimensions of the truss (the diameter and wall thickness of the hollow truss) and/or the spacing between trusses. The use of hollow trusses substantially increases the resistance to buckling [29], thereby offsetting the usually rapid drop in lattice strength as the relative density decreased and truss buckling became the dominant mode of compressive failure. The highest specific compressive strength structures were achieved when the failure mechanism was inelastic buckling of the trusses, and the strength was then well predicted by Eq. (5) or (7), with the scaling factor R = sin2x, and a built-in node end condition (k = 2) [29]. However, while such lattices had a higher strength coefficient than other cellular metal topologies (in part a consequence of the high tangent modulus of annealed 304 stainless steel), and exhibited excellent mechanical energy absorption, the full benefit of the high second moment area of hollow struts was truncated by the low yield strength of the annealed struts. For this alloy, the solid material’s strength cannot be increased by subsequent heat treatment. Metallic foams and periodic micro-truss cellular lattice structures with hybrid struts have been recently fabricated by electroplating Ni based alloys on the metal foams [38] and electroless Ni deposition and/or Ni electroplating (depending on the relative density) on microlattices produced by the self-propagating polymer waveguide process [39] or on rapidly prototyped polymer lattice preforms [40 – 42]. This approach takes advantage of the large strength increase that can be obtained by grain size reduction into the nm-scale, and by the fact that the electrodeposited material is positioned far away from the neutral (bending) axis of the coated struts or ligaments of the cellular material. The inner polymer preform can also be subsequently removed [43, 44], leaving just the deposited coating in the form of a hollow truss lattice, which further improves the specific strength and stiffness of the lattice. However, while the electroplating approach has excellent throwing power (non-line of sight coverage), a coating thickness of only 120 nm to 3 lm has so far been applied [43]. The strength of such thin walled metallic lattices with struts of mmlength scale was then governed by elastic modes of collapse, and could not fully exploit the coating metal’s high yield strength. Recent studies have demonstrated the possibility of greatly increasing the strength of some austenitic stainless steel by developing very high (colossal) super saturations of carbon and/or nitrogen in a thin (25 – 70 lm thick) layer on its surface [45, 46]. This is achieved using low temperature \paraequilibrium" conditions to inhibit carbide formation, and can result in increase in local strength by factors of 5 or more in some alloy systems. The process has also been shown to greatly improve pitting corrosion and fatigue loading resistance [45 – 48]; both of which can be problematic for high specific surface area cellular materials in some loading environments. Since the processing tempera3 International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses tures for this hardening process are low (around 400 8C) and the wall thicknesses of tubes used can be made comparable to the interstitial element diffusion distances [49], the possibility exists of making collinear lattices of exceptional compressive strength. This approach compliments recent studies of gas-phase aluminized 3D woven and braided Ni – Cr wires that have also demonstrated a substantial increase in the strength due to on the formation of c’ precipitates within the wires [50]. A recent numerical study [51] has investigated the effect of such a surface carburization effect upon the compressive response of a pyramidal lattice made from stainless steel tubes. The numerical results confirm that carburization treatments could significantly increase the peak strength of the tubes, but would have a relatively small effect upon the deformation mode as long as the slenderness ratio is not too high. At very high slenderness ratios, the tubes collapse by an elastic buckling, and the peak stress is then insensitive to the degree of carburization. Here, we have fabricated collinear lattice structures in an identical manner to those reported in the previous study [29], and have investigated their mechanical properties before and after nitro-carburization. We show that the lattice compressive strength is increased by almost a factor of four as the tube wall thickness decreases, and investigate the mechanisms of tube collapse that are responsible for the improved mechanical response. 2. Lattice fabrication and nitro-carburization Type 304 stainless steel hollow tubes (Vita Needle Corp., Needham, MA, USA) were used to fabricate the collinear lattices. The tubes were identical to those used in a previous study [29], and had the same outer diameter of 1.473 mm and wall thicknesses of 50, 101, 127 and 203 lm. The metal tubes were collinearly aligned in layers with the aid of a BN-coated stainless steel assembly tool, Fig. 2a. The orientation of the tube layers was alternated to create both the square and diamond lattice architectures. The layers were bonded by vacuum brazing using NICROBRAZ alloy 51 (Wall Colmonoy Corp., Madison Heights, MI, USA) of nominal composition Ni-25Cr-10P-0.03C (wt.%), Fig. 2b. Vacuum brazing was performed at 1 020 8C for 1 hr at a base pressure of *10–2 Pa. Additional details of the vacuum brazing process can be found in Ref. [29]. The bonded lattices were removed from the assembly tool and cut to size using electro-discharge machining, Fig. 2c and d. For the square lattice, their length L was *51.5 mm, their width W was *29.5 mm, and their height H was *16.5 mm. For the diamond lattice, L was *64 mm, W was *29.5 mm, and H was *22 mm. The center-to-center cylinder spacing of the tubes, l was kept constant at 5 mm. For the range of inner and outer tube diameters used here, this choice of strut length resulted in lattices with relative densities of 0.03, 0.06, 0.073, and Fig. 2. Collinear lattice fabrication procedure showing (a) the stacking arrangement for the hollow truss lattice structures, (b) brazing of the square and diamond orientation lattices, (c) photograph of a square lattice sample, and (d) a photograph of a diamond orientation hollow truss lattice after cutting to shape. 4 International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses 0.11 as the wall thickness increased from 50 to 203 lm. Table 1 summarizes the geometric parameters of the hollow tubes and the corresponding relative density of collinear lattice structures made from them. The lattices were then subjected to a low temperature nitro-carburization treatment performed at 440 8C for 20 h. During nitro-carburization a 0.20 slpm (standard liters per minute) NH3 + 0.20 slpm CO + 0.90 H2 + 0.90 slpm N2 gas flow was used. A surface activation process was performed before the nitro-carburization treatment to break down surface oxide layers that otherwise imped interstitial atom entry into the stainless steel. This was conducted at 440 8C under gas flows of 0.20 slpm HCl + 1.8 slpm N2 for 3 hr. All of the processes were performed at the atmospheric pressure. After nitro-carburization, 2.5 mm thick 304 stainless steel face sheets were adhesively attached to the lattices with HYSOLE-120HPTM (LoctiteBrand, Westlakes, OH, USA) high strength epoxy. Curing of the epoxy adhesive was performed at room temperature for 12 hr. Individual tubes of each wall thickness were also nitro-carburized in the same manner. A cross-sectional scanning electron microscope image of a nitro-carburized tube with a wall thickness of 101 lm is shown in Fig. 3. A 20 lm thick nitro-carburized layer can be clearly seen on the external surface of the tube. A thin region about 8 lm thick at the surface of the layer had a different microstructure to the rest of the carburized layer. A thinner, *10 lm thick, nitro-carburized layer can also be seen on the inside of the tube. This difference is attributed to the reduced reactant gas flow rates inside the tubes. A 50 lm thick 304 stainless steel tube was also subjected to the same nitro-carburization treatment and used to characterize the depth and degree of interstitial hardening in this alloy. Nano-hardness tests again indicated the case hardening depth of the treated alloy was *20 lm on the outer side and 10 lm on the inner side where gas flow was more restricted, Fig. 4a. The volume fraction of the case hardened Fig. 3. Cross-sectional SEM image of a nitro-carburized 304 stainless steel tube with a 101 lm wall thickness. The case hardened layers can be seen on both the inner and outer surfaces of the tube. Fig. 4. (a) Hardness and Young’s modulus and (b) C, N atomic concentration data for a nitro-carburized 304 stainless steel tube with a 50 lm wall thickness. Table 1. 304 stainless steel tube parameters and corresponding relative density of lattice structures. Tube diameter Relative density of lattice, (q) Outer diameter, 2ao (mm) Inner diameter, 2ai (mm) Wall thickness, t (mm) 1.473 1.473 1.473 1.473 1.067 1.219 1.270 1.373 0.203 0.127 0.102 0.050 0.110 0.073 0.060 0.030 5 International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses layers for each of the nitro-carburized tubes was calculated and is summarized in Table 2. The nitrogen and carbon composition–depth profiles were determined on cross-sectioned specimens by Auger electron spectroscopy (AES) using a PHI-680 scanning Auger microprobe (Physical Electronics, Chanhassen, MN, USA). Before the acquisition of each AES line scan, about 50 nm of material was sputtered off the cross-sectional sample by an Ar+ plasma to remove hydrocarbon contaminants and native oxides. During acquisition of the AES line scans, dynamic Ar+ sputtering was carried out to avoid carbon accumulation within the chamber. This was needed because the overlap of oxygen and Cr peaks could induce uncertainty in the absolute Cr content. The corresponding carbon and nitrogen concentration profiles are shown in Fig. 4b. During the nitro-carburizing process, the nitrogen- and carbon-containing gas species were introduced into the furnace simultaneously, and the nitrogen and carbon interstitials entered the steel simultaneously. However, nitrogen and carbon mutually increase the chemical potential of the other interstitial, and therefore the total interstitial concentration at any depth is limited. Since carbon exhibits a greater diffusivity in austenitic stainless steel than nitrogen, the result is an accumulation of carbon interstitials deeper into the steel, and an accumulation of nitrogen interstitials closer to the surface [49]. This higher nitrogen content appears to be linked to the change in appearance of the layer near the surface evident in Fig. 3. ening resulting in the Cauchy flow stress eventually reaching almost 1 GPa just before failure. Both the elastic modulus and 0.2 % offset yield strength are summarized in Table 2 for all the tube samples. The nitro-carburized tubes exhibited no measurable increase in elastic modulus, but had a substantially higher yield strength compared to the annealed tubes, this difference increased as the tube wall thickness was reduced to 50 lm. While the nitro-carburized tubes had higher yield strengths than the annealed material, their strain hardening rates and especially fracture strains were lower, and the tubes failed to attain the tensile strength of the annealed alloy; a consequence of the brittle nature of the case hardened layer. The uniaxial compressive stress versus strain responses of the nitro-carburized stainless steel hollow tubes of different wall thicknesses were also determined and are compared to those of their annealed counterparts in Fig. 6. The tubes had an aspect ratio l=2ao &3.4 (where l is the tube length = 5 mm, and 2ao the tube outer diameter = 1.473 mm); similar to that of the lattice unit cell. The collapse strengths of all the single tubes are summarized in Table 2. The nitro-carburization treatment increased the compressive strength of all the tubes, Fig. 6. However, the increase was greatest for the thinnest walled tubes; the 3. Single tube mechanical properties 3.1. Experimental characterization In order to later analyze the lattice response, the uniaxial tensile responses of both annealed and nitro-carburized 304 stainless steel tubes were first determined according to the ASTM A370-14 standard. Metal plugs were snug-fitted in both ends of the tubular specimens to permit the testing machine jaws to grip the specimens properly without crushing or creating a stress concentration; all the specimens then failed outside the grip areas. The tensile responses measured at a strain rate of 10–4 s–1 are shown in Fig. 5. The stress–strain responses of the annealed tubes with different wall thicknesses were the same up to failure at a plastic strain of *38 %. In the annealed state, the 304 stainless steel alloy had a Young’s modulus of 182 GPa, a low yield strength of 240 MPa, but exhibited substantial strain hard- Fig. 5. True tensile stress–strain responses of nitro-carburized 304 stainless steel tubes with different wall thicknesses. The response of the annealed 304 stainless steel is included for comparison. Table 2. Properties (volumetric fraction of the treated alloy, 0.2 % offset yield strength, Young’s modulus, and ultimate strength) of the nitro-carburized 304 stainless steel tubes with different wall thicknesses. Properties of the annealed tubes have been included for comparison (data in brackets; stiffness and yield strength were determined from tensile testing). Nitro-carburized/Annealed t (lm) Vtreated (%) rys (MPa) (Tension/Compression) ES (GPa) Tensile strength rUTS (MPa) Compressive strength rUTS (MPa) 6 50 61 761/812 (240) 181 (178) 830 (1 003) 1 786 (380) 102 30 523/496 (240) 185 (184) 721 (1 003) 1 119 (517) 127 24 475/428 (240) 172 (182) 728 (1 003) 1 006 (607) 203 15 375/341 (240) 176 (181) 751 (1 003) 855 (836) International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses Fig. 6. Compressive stress–strain responses of (a) annealed and (b) nitro-carburized 304 stainless steel tubes of different wall thicknesses with a length of 5 mm. Photographs showing the buckling modes of each wall thickness tube are also shown. 7 International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses 50 lm thick tube strength was increased by about a factor of 5 over that of the annealed counterpart. The compressive strength of the nitro-carburized tubes decreased with wall thickness because of the reduced volume fraction of the case hardened layers. While the 203 lm thick treated tubes had a similar collapse strength to their annealed counterparts, their yield strength was still about a factor of 2 higher. Photographs of the annealed and nitro-carburized tubes after their peak strength was achieved are shown to the right of Fig. 6a and b. The buckling mode of the annealed tubes evolved from a local inelastic buckling mode towards a global inelastic buckling mechanism as the annealed tube wall thickness increased. Figure 6a, shows the collapse mechanisms for annealed tubes (l = 5 mm) of different wall thicknesses. For the thinnest wall tubes with t = 50 lm buckling occurred by multi-lobe diamond local inelastic buckling. As the thickness was increased to t = 101 lm, collapse occurred by two-lobe diamond local inelastic buckling. Further increases in tube wall thickness (t = 127 and 203 lm) resulted in a mixed mode of two-lobe diamond local inelastic buckling and global inelastic buckling. These observations agree well with the numerically predicted collapse mechanism map of 304 stainless steel vertical tubes [36]. Examination of Fig. 6b indicates that the collapse mechanisms of the nitro-carburized tubes were multi-lobe diamond local inelastic buckling for t = 50 lm, axisymmetric local inelastic buckling for t = 101 lm, two-lobe diamond local inelastic buckling for t = 127 lm, and a mixed mode of two-lobe diamond local inelastic buckling and global inelastic buckling for t = 203 lm. tained can be greater than for typical micro-indentation (by a factor of up to two for metals) [53]. The strength of the annealed alloy in the tube center was therefore reduced by the factor 1/1.23 for the ultimate strength calculation. The same study has also shown that the hardness of an intrinsically hard material suffers a minimal indenter size effect [53], and therefore no reduction factor was used for the case hardened layer’s ultimate strength. A Ramberg– Osgood model [54] (see Appendix) was fitted to the elasto–plastic model stress–strain curve for the case hardened layer, Fig. 7a, and we simplified the analysis by assuming an ultimate strain equal to that of the annealed alloy (*0.38); even though the case hardened layer is expected to have a much smaller strain to failure. The predicted compressive stress–strain responses of the nitro-carburized vertical tubes obtained using the effective properties given in Fig. 7a, are shown in Fig. 7b. Good agreement was seen between the predicted compressive 3.2. Vertical tube buckling analysis A nitro-carburized tube can be regarded as a composite consisting of an annular region of annealed condition material sandwiched between two rings of a much harder, but identical modulus material. The inner tube (nitro-carburized) layer is taken to be 10 lm thick while that at the outer surface was 20 lm in thickness. The uniaxial compressive response of such a composite can be analyzed with a Voigt composite model which assumes equal strain in each layer, and sufficient shear strength at the interface so that no slip will occur during deformation. Such a composite model for the case hardened tube element was also successfully used in a recent numerical study of an analogous problem [51]. This simple rule of mixture approach predicts well the tensile stress–strain curve of a unidirectional fiber composite in the fiber direction, even beyond the elastic deformation region [52]. Since the case hardened layer had an average hardness of about 9 GPa, Fig. 4a, (by using the usual practice for converting hardness to strength by dividing by 3) we take its peak strength to be 3 GPa, and use an elasto–plastic stress– strain response shown in Fig. 7a to represent it. The Young’s modulus was taken to be 220 GPa; consistent with nano-indentation measurements. It is noted that the ultimate strength of the annealed alloy is about 1 000 MPa, Fig. 4a. However, the measured hardness of the annealed condition alloy at the center of the tube wall was about 3.7 GPa, corresponding to an ultimate strength about 1 233 MPa. We attribute the higher strength to the indenter size effect; for the low loads used in nano-indentation, the hardness ob8 Fig. 7. (a) Material models for the true stress–strain responses of the case hardened layers. (b) Predicted (solid curve) stress vs strain responses of the nitro-carburized tubes with different wall thicknesses. Experimental (dashed) compressive stress–strain responses were included for comparison. International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses strength (solid lines) and the measurements, up to the critical buckling stress of the samples. The analysis indicates that in all cases, the buckling strength during tube compression was attained after the initiation of plastic yielding of the case hardened metal. Careful examination of the simulated stress versus strain responses, Fig. 7b, reveals the presence of two slope changes (yield points), which were also less clearly observed in the tube compression measurements. The first slope change corresponds to the inner annealed alloy annular ring yielding while the second resulted from yield of the case hardened layers. While the lower yield points on the compressive stress–strain curves are consistent with the yield points measured during tensile tests, Fig. 5, the subsequent tensile responses were different from those in compression due to the influence of the brittle case hardened layers. We note that since the post-buckling behavior cannot be addressed by this simple model, the compressive stress–strain curves for the post-buckling regime are incorrectly predicted. Using the constitutive responses for the annealed and nitro-carburized tubes (in the composite cylinders approximation), Fig. 7a, with either the Engesser (Eq. (5)) or Shanley (Eq. (7)) predicted global inelastic buckling stress enables the critical buckling stress of all the tubes to be predicted as a function of the tube wall thickness. These predictions for both end clamping conditions (k = 1 and 2) are plotted against tube wall thickness and compared to the experimental data in Fig. 8. It is noted that as the single tube ends were fixed during the experiments, the k = 2 prediction should be closest to the experimental end conditions. Figure 8 also includes the local inelastic buckling predictions based upon Eq. (10) which converge with the local elastic buckling stresses for the thinnest walled nitro-carburized tube. In the annealed condition, Fig. 8a the measured buckling strengths and modes shown in the photographs accompanying Fig. 6 are consistent with the local inelastic buckling model predictions (using Kc = 0.35 which best fitted the data) for the three tubes whose wall thicknesses were less than about 140 lm. For the thickest wall tube, the buckling stress predicted by the Shanley (k = 2) global inelastic buckling model was very slightly lower than that of a local inelastic buckle. Though this predicted stress better matched the experimental data point, an annealed tube of this wall thickness is in the vicinity of the transition point between the two failure mechanisms, and the buckling failure mode is therefore defined as a mixed mode, consistent with the buckling mode evident in the photograph of Fig. 6a. A similar analysis for the nitro-carburized tubes is shown in Fig. 8b. The composite cylinder approximated stress versus strain response of the nitro-carburized tubes was first obtained, and the local or global buckling strengths were subsequently determined using corresponding buckling criteria (Eqs. (2), (5), (7), (9) and (10)). The Kc values used for local buckling stress calculation of the nitro-carburized tubes (those with walls thicker than 30 lm) were obtained as the volumetric average of the three layers by assuming Kc = 0.8 for the case hardened layer and 0.35 for the center (annealed condition) alloy. Since the two case hardened layers had a combined thickness of 30 lm, predictions for tubes with a wall thickness of 30 lm or less, were assumed to consist of only case hardened material and shading is used to denote this region on Fig. 8b. The hardness and ultimate strength of a case hardened tube with wall thickness less than 30 lm were determined from the hardness vs depth profiles from both the outer and inner sides of the wall, Fig. 4. To simplify the analysis, it was assumed that the case hardened material exhibited the elasto–plastic response shown in Fig. 7a and the inelastic buckling stress of the tube then equaled its plastic yield stress. The elastic buckling stresses were determined by Eq. (9) assuming Kc = 0.8. The buckling model predictions for the nitro-carburized tubes have been separated into two regimes located above and below a wall thickness of 30 lm. Below a wall thickness of 30 lm, the fully nitro-carburized tubes are predicted to fail by local elastic buckling at a stress that rises linearly with wall thickness to 3 GPa at a thickness of 30 lm. As the wall thickness was increased beyond 30 lm, the nitro-carburized tubes were at first equally well predicted to collapse by either inelastic local or global inelastic buckling. However, for the thickest tube wall sample, the lowest strength buckling mode was inelastic global buckling con- Fig. 8. Predicted (curves) and measured (open and black diamonds) peak compressive strength of (a) annealed and (b) nitro-carburized 304 stainless steel tubes vs. wall thickness. In (b) the Shanley predictions are denoted by a blue dashed (k = 1) or solid (k = 2) line while those of the Engesser models are shown in black. 9 L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses sistent with the photographic observations, Fig. 6b and the measured critical stresses. The compressive strength of the thickest wall tube was better predicted by the Engesser global inelastic buckling stress (k = 2) than by Shanley global inelastic buckling stress (k = 2). The Engesser model predicts the critical stress when lateral deflection begins [33]; if imperfection or defects exist, the compressive stress would begin to decrease; otherwise, the compressive stress could continue to increase till the Shanley stress is approached [34]. International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. 4. Lattice compression results 4.1. Mechanical properties The lattice structures were tested in unconfined compression in the thickness (H) direction at ambient temperature and a strain rate of 5 · 10–4 s–1. The measured load cell force was used to calculate the nominal stress applied to the face sheet area of the structures. While this provided a good estimate for the stress supported by the square oriented lattice, it underestimates the real stress in the diamond-oriented samples, Fig. 1b, since only a part of the sample top surface is connected by a strut to its bottom. The effective area of the cellular structure in the diamond oriented case was only 0.67 that of the face sheets. In the experiments to follow, we report the apparent strength using the full cross-sectional area (L W) of the sample to compute the stress. The nominal through thickness strain was measured using a laser extensometer to monitor the displacement between the face sheets; the initial thickness of the cellular structure within the face sheets was used as the gage length. Photographs of the samples were taken periodically during testing to characterize the buckling modes of both the annealed and nitro-carburized lattices. Typical compressive nominal stress versus strain responses for the nitro-carburized lattice structures with square and diamond orientations are shown in Figs. 9 and Fig. 9. Comparison of the compressive responses of the square topology annealed and nitro-carburized lattice structures with relative densities of (a) 3.0 %, (b) 6.0 %, (c) 7.3 %, and (d) 11.0 %. 10 L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses additional flow stress peaks before final rapid hardening due to core densification (truss impingement) after a plastic strain of 30 to 50 %. The initial compressive stiffness and the peak strength of the annealed and nitro-carburized collinear lattices are summarized in Tables 3 and 4 respectively. No apparent change in lattice stiffness was observed after nitro-carburization treatment. However in all cases, the International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. 10. The compressive responses of annealed lattice counterparts are included for comparison. The nitro-carburized lattices exhibited characteristics typical of many cellular metal structures [17], including a region of elastic response, plastic yield followed by hardening, achievement of a peak strength (here controlled by inelastic buckling) terminated by a large drop in flow stress and followed by a series of Fig. 10. Comparison of the compressive responses of the annealed and nitro-carburized diamond orientation lattice structures with relative densities of (a) 3.0 %, (b) 6.0 %, (c) 7.3 % and (d) 11.0 %. Table 3. Stiffness and strength of annealed 304 stainless steel collinear lattice structures. Relative density 0.030 0.060 0.073 0.110 Square topology Diamond topology Stiffness (GPa) Peak strength (MPa) Stiffness (GPa) Peak strength (MPa) 3.15 6.06 7.35 11.70 6.42 14.67 22.48 37.30 1.57 3.14 3.86 5.45 4.28 10.38 14.81 22.96 11 L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses Table 4. Stiffness and strength of nitro-carburized 304 stainless steel collinear lattice structures. Relative density International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. 0.030 0.060 0.073 0.110 Square topology Diamond topology Stiffness (GPa) Peak strength (MPa) Stiffness (GPa) Peak strength (MPa) 3.28 6.80 7.83 12.05 23.65 28.52 35.70 42.75 1.59 4.08 4.51 5.88 16.20 22.39 25.17 28.00 compressive strength of the nitro-carburized lattices was significantly higher than annealed lattice counterparts, and the square lattice had a higher peak strength than its diamond counterpart of identical relative density. 4.2. Buckling mechanisms A sequence of photographs showing the progressive collapse of square collinear lattices with a relative density of 3 % in both the annealed and nitro-carburized conditions are shown in Fig. 11. In both cases the vertical tubes had a free length of 15 mm and were bonded to the face sheets through which the compressive load was applied with a thick layer of adhesive. Four rows of horizontal tubes crossed the vertical trusses, but two were embedded in the adhesive bond with the face sheets. The unit cell length (and width) was 5 mm. Examination of Fig. 11a shows that the q ¼ 0:03 annealed square lattice (whose tube walls were 50 lm in thickness) failed by cooperative local inelastic buckling of the trusses parallel to the loading direction. This was initiated mid-way along the vertical tube trusses at a strain of slightly less than 1.4 %, and had propagated across the entire row of vertical struts when the strain had reached 5 %. The tubes collapsed by a local inelastic buckling mode at a strain corresponding to the first peak in strength seen in the stress versus strain response of this lattice, Fig. 9a. As deformation progressed beyond the peak load to a strain of 10 %, the sample compression was accommodated by continued buckling in the center row of cells. Local inelastic buckling then initiated near the center of the vertical trusses in the bottom row of cells, and had again spread across the entire sample once the strain reached 20 %. The onset of this second cooperative buckling process coincided with the second peak in the stress versus strain response, Fig. 9a. A similar peak in strength, but at a strain of 30 % was observed when the third row of trusses buckled. The deformation sequence of a square nitro-carburized collinear lattice of the same relative density is shown in Fig. 11b. Initiation of local inelastic buckling again coincided with the first peak in strength at a strain of 0.6 %, Fig. 9a and usually midway along the vertical tubes (within the mid-plane cell row). However, in this case, the location of the buckling sites was near the nodal region where a horizontal tube crossed the sample. Like the annealed sample, further compression was accommodated by compression of the center row of cells until a strain of 15 % was reached, whereupon the vertical trusses of either the top or bottom row of cells began to buckle by a similar mode. As the 12 strain continued to increase towards 30 %, the deformation of the newly buckled trusses caused the height of the originally horizontal tubes to be reduced causing an inclination of the two center horizontal tubes. Some of the buckled regions also underwent fracture. Figure 11 shows that neither the annealed or carburized square lattices exhibited a lateral Poisson strain during compression. Annealed square lattices with relative densities of 0.06 and 0.073 exhibited similar deformation characteristics to the q ¼ 0:03 sample during compressive loading, Fig. 12a. However, as the relative density was increased to 0.11, the failure mode changed from local inelastic buckling to global buckling of the vertical tube struts, Fig. 13a. In Fig. 13a, views of the side (angle I) and the front (angle II) of the sample are shown at a strain of 11 % where the first peak in strength was observed, Fig. 9d. Further straining to a plastic strain of 25 % was accommodated by cooperative rotation of the top and bottom vertical trusses, with an associated lateral displacement. No fracture of individual trusses was observed in any of the annealed lattice structures. As the relative density of the square nitro-carburized lattice increased to q ¼ 0:06 and then 0.073, a mixed mode of buckling developed with both local and global inelastic buckling components, Fig. 12b. The peak in strength, Fig. 9b and c, coincided with global inelastic buckling of the vertical tubes and was followed by local inelastic buckling of the vertical trusses as deformation progressed. The mechanism of collapse in the q ¼ 0:11 nitro-carburized square lattice was global plastic buckling, Fig. 13b of the vertical tube struts, and was very similar to that seen in the annealed sample. However, the peak in strength of the nitro-carburized lattices occurred at a lower strain than annealed counterparts, Fig. 9d. No fracture of individual trusses was observed in the square nitro-carburized lattices with relative densities of 0.06, 0.073 and 0.11. Photographs of diamond lattice structures with a relative density of 0.03 taken during compressive loading are shown in Fig. 14a at several strains. Analogous results for a density of 0.06 are shown in Fig. 12c. The annealed diamond lattices with q ¼ 0.03, 0.06 and 0.073 all failed by inelastic local buckling of individual trusses between nodes at the first peak in strength, Fig. 10a, b and c. As deformation progressed, locally buckled trusses continued to buckle until a zigzag layer of diamond unit cells collapsed and trusses of adjacent layers made contact. This was accompanied by a rapid drop in flow stress. An adjacent layer of diamond unit cells then began to buckle and was accompanied by a second peak in strength and rapid drop in flow stress after- L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses The peak compressive strengths of nitro-carburized and annealed lattice structures are shown in Fig. 16a for the square lattices and for the diamond orientations in Fig. 16b. It is clear that the nitro-carburzied process has significantly increased the lattice peak strength. The ratio of the nitrocarburized to annealed peak strengths (strength improvement factor), is also plotted on Fig. 16a and b. For thinnest wall thickness lattices, almost four-fold increases in International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. wards. Such a process was repeated until the lattice had completely collapsed. The annealed lattice core with q ¼ 0:11 deformed in a similar way, but the individual trusses failed by mixed mode of local and global buckling, Fig. 15a. The nitro-carburized diamond lattice cores exhibited similar deformation characteristics as their annealed lattice counterparts, but peak in strength occurred at a smaller strain. Fig. 11. Photographs showing the deformation characteristics of q = 3.0 % annealed (a) and nitro-carburized (b) square orientation lattice structures at different plastic strains. 13 L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses strength were observed. These figures also show that the strength ratio decreased with increasing relative density; a consequence of the decreasing cross sectional area fraction of the case hardened layer. 5. Lattice buckling analysis International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. 5.1. Lattice strength The lattice strengths can be predicted by analyzing the unit cell shown in Fig. 1c. For the unit cell of a square lattice, Eq. (1) was used to predict the strengths, where rys was taken as either the yield strength of the solid material for the plastic yielding failure mechanism, or replaced by Eqs. (2), (5), (7), (9) and (10), for the corresponding local and global buckling mechanisms. Since the single vertical tube properties have already been analyzed in Section 3.2, the square lattice responses can be simply obtained by multiplying the single vertical tube properties, the lattice relative density and the topology dependent strength coefficient, R = 0.5 for the square lattice. For a diamond lattice unit cell, the tube elements have an inclined angle, x = 458. When an end clamped inclined tube is placed under compression loading, the bending moments at the ends introduce shear forces, and the total force, F applied along the vertical direction is given by [55]: 1 F ¼ FA sin x þ FS cos x ¼ pﬃﬃﬃ ðFA þ FS Þ 2 ð12Þ 12I where FS is shear force at the ends given by FS ¼ FA 2 . Al The diamond lattice unit cell compressive strength is therefore; FA FS FA 3p 4 4 rc ¼ 1þ 1þ ¼ D d 4ao l FA 4ao l 16l2 1 3p 4 4 ¼ qrA 1 þ ð13Þ D d 2 16l2 where rA is the axial collapse strength of a single tube defined by Eqs. (2), (5), (7), (9) and (10), for the corresponding local or global buckling mechanisms. Equation 13 can be used to predict the compressive strength of a diamond lattice without an edge effect. However, it is important to recall that the diamond lattice test samples had a low L/H (aspect) ratio which reduced their apparent strength by a factor of 0.67. The stress predicted by Eq. (13) is therefore multiplied by 0.67 to make comparisons with the experimental measurements. The compressive strength of the diamond lattice test samples is therefore given by: 3p 4 4 qrA rc ¼ 0:33 1 þ ð14aÞ D d 16l2 It is noted here that Eq. (14a) was obtained by assuming clamped (k = 2) ends; for pin-jointed inclined tubes (k = 1), bending moment at the ends is negligible, and the predicted compressive strength of the diamond lattice test samples would then be given by: rc ¼ 0:33 qrA ð14bÞ Fig. 12. Photographs showing the deformation characteristics of q = 6.0 % annealed ((a) and (c)) and nitro-carburzied ((b) and (d)) collinear lattice structures at a plastic strain of 15 %. 14 L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses 5.2. Comparison of predictions with measurements International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. The predicted strength (normalized by the product of the annealed alloy yield strength and colinear lattice relative density) is plotted in Fig. 17a and c for the normalized lat- tice. The predicted strengths have been reduced by the aspect ratio dependent factor (0.67) for the diamond oriented test samples used here. The plastic yielding mechanism predicts a structural efficiency parameter that is independent of relative density, and is shown by a broken line in both fig- Fig. 13. Photographs showing the deformation characteristics of q = 11.0 % annealed and nitro-carburized square orientation lattice structures at different plastic strains. 15 L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses elastic buckling. As the relative density increases local inelastic buckling can occur (when q > 0:02). As the relative density increased, the predicted local inelastic buckling strength eventually exceeds the Engesser and Shanley (k = 2) predicted inelastic buckling strengths. The measured compressive strength coefficients obtained by normalizing the (finite aspect ratio) sample compressive strength by the product of the annealed stainless steels yield strength and colinear lattice relative density International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. ures. The critical stress to activate the local buckling mechanisms is denoted by red lines in the two figures. It can be seen that the strength predicted by the Engesser and by the Shanley (global) inelastic buckling mechanisms significantly exceeded the plastic yielding predicted strength. The predicted global inelastic buckling strength of cellular structures with built-in (k = 2) nodes also significantly exceeded that of the pin-jointed case. At low relative densities, q < 0:02, core collapse is predicted to occur by local Fig. 14. Photographs showing the deformation characteristics of q = 3.0 % annealed and nitro-carburized diamond orientation lattice structures at different plastic strains. 16 International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses are summarized in Table 5, and are overlaid on the predictions shown in Fig. 17a and c. The normalized strength of the annealed lattices increased from 0.6 to 1.4 as their relative density was increased from 0.03 to 0.11. In all cases these measured strengths exceeded the predicted strengths for plastic yielding consistent with the observation of a yield point in the stress–strain responses before attainment of the peak strength, Figs. 9 and 10. At low relative densities ( q < 0:08) the peak strengths of both lattice oreintations are consistent with model predictions for failure by local inelastic buckling of the trusses, as observed during the q > 0:08; they are close tests. For relative densities above to the Engesser model predictions for global inelastic buckling with a built-in end node condition. These predicted mechanisms correlate well with the optical observations of the buckling failure mechanisms, Figs. 11 and 12. Their normalized strengths increased with relative density for the samples studied here (with fixed strut lengths) because the increase in density was achieved by an increase in tube wall thickness which increased the tubes resistance to buckling more rapidly than its increase of lattice density. Note that examination of Fig. 17a and c appears to indicate that the diamond lattice structures have lower strength coefficients than square lattice counterparts. However, this is a consequence of the sample (aspect ratio dependent) knock-down factor of 0.67 that was applied to the predictions of the diamond lattices. It also evident that the measured diamond lattice strengths are slightly lower than the model predictions for q > 0:03. However, the global inelastic buckling (k = 2 case) and the local buckling model predictions presented in Fig. 17c were obtained by assuming the inclined tube struts have a built-in end condition, and transverse displacement was constrained at both ends. However, the actual compression tests were performed without lateral confinement, and small transverse displacements of the nodes occurred during deformation. In addition, the adhesive nodes have a rotational resistance that lay between k = 1 and 2. The bending moments at the strut ends are therefore lower than the model predictions when k = 2 was assumed. The predicted strength coefficients of the nitro-carburized collinear lattices (again normalized by the product of the annealed alloy yield strength and colinear lattice relative density) and reduced by a factor of 0.67 for the diamond oriented samples, are shown in Fig. 17b and d. The predictions of the models have been seperated into two parts with q ¼ 0:018 as the boundary (which corresponds to a truss tube wall thickness of 30 lm). When q < 0:018, Fig. 15. Photographs showing the deformation characteristics of q = 11.0 % annealed and nitro-carburized diamond orientation lattice structures at different plastic strains. 17 International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses the individual trusses are fully nitro-carburized, and the stress–strain relationship was modeled as elasto–plastic, Fig. 7a, with a yield strength of 3 GPa. The inelastic global buckling stress then coincided with the plastic yielding stress. However, in this regime, the local elastic buckling stress is predicted to dominate the response at low density. The stress to activate this increased with relative density until it reached the global inelastic buckling stress at q 0:016. q > 0:018, the compressive stress–strain curves of For individual tubes were modeled using the Voight approximation and the mechanical properties are summarized in Fig. 7. The local inelastic buckling model, as well as the Engesser and Shanley inelastic global buckling models predicted nonlinear decreases of the critical strength with increasing relative density. The local inelastic buckling model predictions are lower than the Engesser and Shanley inelastic buckling model predictions (k = 2), until q approached 0.07; thereafter the local buckling model predictions began to increase in a nonlinear manner with increasing relative density, and exceeded the Shanley inelastic buckling (k = 2) predictions at q 0:09. The sample test data have been overlaid on the predictions. After the nitro-carburization treatement, the measured compressive strength coefficients increased by a factor of between 2 and 4; increasing from 1.1 to 3.2 as their relative density was decreased, (i. e. with increasing tube wall thickness), Fig. 17b and c. At low relative density, they were substantially stronger than their annealed counterparts. However, this difference decreased as the tube wall thickness increased. The measured compressive strength coefficients agree well with the model predictions, and the optically observed failure mechanisms. Lattices failed by local inelastic buckling of trusses at low relative density (q ¼ 0:03), and a mixed mode of local and global inelastic buckling of truss members at intermediate relative densities (q = 0.06 and 0.073). At a relative density q ¼ 0:11 collapse was by global inelastic buckling of trusses. Again, the nodal rotational resistance of the tested samples lay somewhere between k = 1 and 2, consistent with the observed constrained node rotation. The rapid rise in lattice strength as the tube wall thickness was decreased is a direct result of the increase in effective strength of the nitro-carburized tube walls which delayed the onset of the inelstic buckling operative in the samples tested here. 6. Concluding remarks Fig. 16. Peak compressive strength of the nitro-carburized and annealed lattice structures as a function of relative density. Model predictions of peak compressive strength are provided for comparison. Peak strength improvement factors of nitro-carburized lattices over the annealed lattice counterparts are also summarized. 18 Collinear lattice structures with 304 stainless steel hollow trusses were fabricated with an alternating collinear lay-up process and metallurgically bonded by a vacuum brazing method. The dimensions of the cylinders, the wall thickness of hollow trusses and spacing between the trusses enabled independent control of the cell size and the relative density of the structure. Square and diamond topologies with relative densities from 0.03 to 0.11 were manufactured. Low temperature nitro-carburization treatment was then performed on the collinear lattice cores at a temperature of 440 8C for 20 hrs. The nitro-carburization treatment created a thin but extremely hard surface layer on the truss members, which significantly improved the collapse strength of the individual trusses, and enabled a compressive strength enhancement by a factor ranging from 1.2 to 3.8 compared with the annealed lattice counterparts. Inelastic buckling of the trusses was the dominant failure mode; however, such buckling might be delayed by filling tubes with lightweight polymer or metal foams: though it is unclear if this additional mass might be better deployed by increasing the tube wall thickness. The compressive strength of the annealed and nitro-carburized lattices are compared with other types of stainless steel cellular structures such as woven textiles [28], pyramidal lattices [19, 20] and square honeycomb cores [14], in Fig. 18. The predicted diamond lattice strengths in this case have not been multiplied by the sample aspect ratio dependent factor of 0.67 so they represent the response of large aspect ratio (wide) samples in which almost all the struts are in contact with both the upper and lower face sheets. The experimental data were therefore scaled by 1/0.67 to represent the wide sample strength limit. Examination of L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses The nitro-carburized square and diamond lattices exibited significantly improved strength coefficients over the annealed lattice counterparts. At low relative density where the wall thickness becomes comparable to the depth of the case hardened layers, they are superior to all other topolo- International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. the data shows that the annealed lattices are substantially stronger than woven textiles and their strength factors (R), and are very competitive with the pyramidal and square honeycomb structures made from annealed 304 stainless steel. Fig. 17. Predicted and measured peak compressive strength coefficients of collinear lattice structures vs. relative density, (a) square annealed lattice, (b) square nitro-carburized lattice, (c) diamond annealed lattice and (d) diamond nitro-carburized lattice. Note the difference in scales between the separate figures. Table 5. Comparison of the compressive strength coefficients of nitro-carburized collinear lattices made from 304 stainless steel hollow tubes with the annealed lattice counterparts. Compressive strength coefficient (R) Relative density Nitro-carburized square Annealed square Nitro-carburized diamond Annealed diamond 0.030 0.060 0.073 0.110 3.28 1.98 2.04 1.62 0.89 1.02 1.28 1.41 2.25 1.55 1.44 1.06 0.59 0.72 0.85 0.87 19 L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015 For personal use only. gies. For example, at a relative density of 0.03, the nitrocarburized hollow truss structures are 6 times stronger than a square honeycomb and 7 time stronger than a pyramidal truss structure constructed from annealed 304 stainless steel. Low relative density nitro-carburized hollow truss lattice structures made by the alternating collinear lay up process appear to exhibit a higher specific compressive strength than any other cellular metal topologies reported to date. It is noted that other topologies reported in Fig. 18 could, in principle also be nitro-carburized and their strength improved. It is also noted that the superior corrosion and fatigue resistance of the nitro-carburized stainless steel may have beneficial consequences for some applica- Fig. 18. (a) The compressive strength vs. density of various 304 stainless steel cellular structures. (b) Comparison of the peak compressive strength coefficients of nitro-carburized hollow truss lattices with annealed lattice counterparts and other types of 304 stainless steel cellular structure topologies as a function of the relative density. 20 tions. The nitro-carburized 304 stainless steel collinear lattices therefore appear to be promising candidates for lightweight sandwich cores intended for elevated temperature and multifunctional applications. We are grateful to the DARPA MCMA program (Grant Number W91CRB-10-1-005) for the financial support of this work. The program was overseen by Dr Judah Goldwasser of DARPA. References [1] J. Gibson, M.F. Ashby: Cellular Solids: Structure and Properties, 2nd ed. Cambridge University Press, Cambridge (1997). DOI:10.1017/cbo9781139878326 [2] V.S. Deshpande, N.A. Fleck: Int. J. Solids Struct. 38 (2001) 6275. DOI:10.1016/S0020-7683(01)00103-2 [3] N. Wicks, J.W. Hutchinson: Int. J. Solids Struct. 38 (2001) 5165. 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DOI:10.3139/146.101594 (Received June 17, 2015; accepted September 3, 2015) Correspondence address Liang Dong Department of Materials Science and Engineering University of Virginia Charlottesville, VA, 22903 USA Tel.: +1-434-982-5678 E-mail: [email protected] Bibliography DOI 10.3139/146.111312 Int. J. Mater. Res. (formerly Z. Metallkd.) 106 (2015) E; page 1 – 21 # Carl Hanser Verlag GmbH & Co. KG ISSN 1862-5282 Appendix Modified Ramberg–Osgood fitting The true tensile stress–strain curve has been separated in two parts and fitted separately. Annealed condition (1) r < r0:2 , n = 28, E0 = 182 GPa, r0:2 = 240 MPa, Ramberg–Osgood fitting n r r e¼ þ 0:002 E0 r0:2 (2) r > r0:2 , a modified Ramberg–Osgood fitting [53] is applied by moving both vertical and horizontal axes from the origin to a selected point which enables a smooth transition between the two fitting curves. e1 ¼ 0:0025, r1 ¼ 235:6 MPa, E1 = 6.745 GPa, n = 2.05, eu ¼ 0:365, ru = 1 000 MPa, a modified Ramberg–Osgood fitting r r1 ru r1 r r1 n þ eu e1 e e1 ¼ E1 E1 ru r1 Case hardened layer stress–strain (1) r < r0:2 , n = 3.5, E0 = 220 GPa, r0:2 = 1 500 MPa, Ramberg–Osgood fitting n r r þ 0:002 e¼ E0 r0:2 (2) r > r0:2 , e1 ¼ 0:01, r1 ¼ 1851 MPa, E1 = 1.41 GPa, n ¼ 15, eu ¼ 0:38, ru ¼ 3000 MPa, a modified Ramberg– Osgood fitting r r1 ru r1 r r1 n þ eu e1 e e1 ¼ E1 E1 ru r1 21

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