INVESTIGATION INTO THE EFFECT OF SLENDERNESS RATIO ON RE-ENTRANT HONEYCOMB MECHANICAL PROPERTIES By Zachary Mohamed Ahmed A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Civil Engineering – Master of Science 2024 ABSTRACT Lightweight but high energy absorption performance structures are desired to satisfy demands in applications of aerospace, automotive, manufacturing, as well as packaging. Cellular structures consisting of spatially distributed mass and voids have been widely used for impact mitigation scenarios. Recent advancements in additive manufacturing have resulted in processes to precisely control the complex microstructure of these structures, leading to an improved mechanical performance. Recently, a new class of bio-inspired auxetic structure, namely the hierarchical re- entrant honeycomb (H-ReH) has been developed. The mechanical properties of the H-ReH are controlled by multiple structural parameters, in which the slenderness ratio (SR) of structural members is among the most critical ones. Based on classical beam theory, the deformation modes of each structural member are highly sensitive to the SR, which determines the overall stability and crashworthiness of the structure. It is necessary to systematically investigate and quantify the effect of SR on the mechanical properties of H-ReHs. In this study, H-ReHs with SR ratios in a wide range have been manufactured by polyjet 3D- printing technique. The mechanical properties of the printed H-ReHs have been fully characterized by quasi-static compression tests and full-field Digital Image Correlation (DIC) technique. The evolution of strain field in the structures under compression has been analyzed. The experimental results indicated that the SR is related to the transition between bending- to stretching- dominated deformation modes. The effect of SR becomes more significant when the relative density of the H-ReHs is lower. In addition, the specific energy absorption (SEA) as well as the energy absorption plateau stress have been improved by the SR at the same relative density. These findings will guide future design and material selection of lightweight but high energy absorption structures. To God, family, and friends iii ACKNOWLEDGEMENTS I would like to take the moment to acknowledge all that have supported me during my time in this program. Family and friends, my parents in particular. Adam, my brother, thank you. A special thanks to Dr. Kutay for giving me my first opportunity to do undergraduate research and then Dr. Weiyi Lu for taking the chance on me and helping me reach my goals. Further thanks to committee members Dr. Lajnef and Dr. Kodur as well as the department office. Thanks to Anqi and Fuming for helping me learn the experiments and lab. iv TABLE OF CONTENTS LIST OF TABLES……………………………………………………………………………......vi LIST OF FIGURES……………………………………………………………………………...vii INTRODUCTION………………………………………………………………………….……..1 METHODS……………………………………………………………………………….....…….7 CHAPTER 1: IMPACT OF SLENDERNESS RATIO ON ELASTICITTY……….……………13 1.1. CHARACTERIZATION OF DENSITY EFFECT………………………………….13 1.2. DISCUSSION……....……………………………………………………………….14 1.3. INFLUENCE ON DOMINANT DEFORMATION MODE…………....…………..19 CHAPTER 2: EFFECT OF SLENDERNESS RATIO ON ENERGY ABSORPTION CHARACTERISTICS…………………………………………………………………………....22 2.1.WORKING PRESSURE AND DENSIFICATION STRAIN.…..........……………..22 2.2. SPECIFIC ENERGY ABSORPTION……...….….......…………..……....……..….26 2.3. DISCUSSION......…….……....……..…….….…...………...…........……….……..29 CONCLUSIONS………………………………………………….……………………………...33 FUTURE WORK……………………………………………………………………………...…37 BIBLIOGRAPHY………………………………………………………………………………..38 v LIST OF TABLES Table 1: Comparison of commercially available additive manufacturing methods…………....….9 Table 2: Properties of additive materials used from data sheet and determined experimentally…....9 vi LIST OF FIGURES Figure 1: Cross section of (a) cattle horn, (b) palm tree and corresponding microscopy images[2- 4]………………………………………………………………………………………………......1 Figure 2: Density – Strength bubble chart of current materials[6]…...…….………….………….2 Figure 3: Cellular structures definition flow chart[8]...…….………….………….……...…….…3 Figure 4: Energy absorption performance of cellular structures from literature[4]….……...…….4 Figure 5: Bulk unit cells of [left] positive Poisson ratio honeycomb and [right] auxetic honeycomb [34,12]………………………...……………………………………….…………………………..5 Figure 6: DIC confirmation of bending-(left) and stretching-(right) dominated deformation in R- ReH and H-ReH unit cells respectively….…….………….……….….………….……………….5 Figure 7: Examples of (left) high and (right) low slenderness values.……….………..….………6 Figure 8: Schematic of (a) R-ReH and (b) H-ReH unit cells………….………….……………….7 Figure 9: Schematic of (left) H-Vert and (right) H-Hor unit cells compared to (middle) H-ReH...8 Figure 10: Stratasys J55 Printer………….………….………….………….………….….……….9 Figure 11: Testing curves of (left) R-ReH and (right) H-ReH printed using various materials…10 Figure 12: Scope of investigated samples………….………….………….…………..………….11 Figure 13: Experimental (left) setup with Instron, DIC equipment and (right) view from software...………….…………………………………………………………………………….11 Figure 14: Speckled dog bone specimen prepared for DIC………….………….………….……12 Figure 15: Average Modulus – Relative Density trend during constant 4.1 SR………....………13 Figure 16: 4.1 SR, R-ReH Stress – Strain curves at various densities…………..………….……14 Figure 17: Stress – Strain curve of all R-ReH specimens………….………………….…………15 Figure 18: Effect of SR on normalized modulus………….………….………...…….………….16 Figure 19: Stress – Strain curve of all hierarchical specimens………….…………....………….17 Figure 20: Modulus of all hierarchical specimens………….………….……………..………….17 vii Figure 21: DIC analysis of epsilon x strain field with 0.2 scale, unrotated axes for 0.36 density (top) H-Reg, (bottom) H-Vert, and (middle) H-Hor samples………….……...……...………….18 Figure 22: DIC analysis of epsilon x strain field with 0.2 scale, 60 deg rotated axes, for 0.26 density (left) 7.75 SR H-Hor and (right) 8.25 SR H-Reg samples………………...….………..….………19 Figure 23: Drawings of (left) bending [31], (middle) stretching deformation [31] and (right) DIC view of a bending stress field………….………….………….……….………….……....……….20 Figure 24: Deformation mode susceptibility in R- models………….…………….…..……...….20 Figure 25: Gibson-Ashby agreement in R- models………….………….………………….……21 Figure 26: Representative testing curve of an R-ReH specimen with DIC correlation…….……23 Figure 27: Representative testing curve of an H- specimen with DIC correlation…………...….24 Figure 28: SEA results of all R- specimens………….………….………….……….……..…….27 Figure 29: SEA results of all H- specimens………….………….………….………….…..…….28 Figure 30: Comparative energy absorption performance of R- testing group……….……….….30 Figure 31: Comparative energy absorption performance of H- testing group…………...………31 viii INTRODUCTION Various engineering approaches have been utilized to develop engineering materials and structures by replicating the materials and structures formed by natural process with extraordinary properties. Yet, engineers have not been able to design car bumpers with the same efficiency as nature designs horns on a buffalo or airplane wings comparable to sturdy towering trees in the amazon. The blueprint for advancing the next generation of structures has ironically always been on full display in the bone structure of our bodies and in the natural world. Many characteristics present in structures such as nacre, dentin, and bone have proven to be advantageous in regard to energy absorption functions. Figure 1 shows the open cell configuration and the hierarchical characteristic present in nature. These are the features that boost performance of lightweight structures in energy absorption applications. Figure 1: Cross section of (a) cattle horn, (b) palm tree and corresponding microscopy images [2-4] Past research efforts to maximize efficient use of resources have culminated in the addition of many materials on the Ashby Density vs Modulus bubbled chart shown in Figure 2. Applications of these discoveries have led to common energy absorption structures such as sandwich panels, thin-walled tubes, and foams. For these materials, increasing strength is always accompanied by an increase in density, which is not ideal in situations where lightweight energy absorption 1 structures are needed. In addition, by satisfying these needs, the lightweight energy absorption structures will contribute to global initiatives in improving fuel efficiency, driving range, air and air quality without compromising safety. Literature suggests that reducing the weight of an automobile by 10 percent would cut fuel use by 6-8% a year [5]. Figure 2: Density – Strength bubble chart of current materials[6] Cellular structures have been developed to lower the density as they consist of voids in them, which makes the relative density (density ratio between the cellular structures and the solid base material) less than 1. Foams are a stochastic type of cellular structures, meaning that their cell dimensions and geometries are random in nature. Non-stochastic cellular structures exist in nature such as bee honeycombs. Recent improvements in additive manufacturing have resulted in processes to rapidly produce engineering non-stochastic cellular structures with complex geometries. These non-stochastic cellular structures have shown enhancement in crashworthiness for automotive industries [8], A more detailed review of current 3D printing is included further in section. Classifications of Cellular structures are shown below in Figure 3. 2 Figure 3: Cellular structures definition flow chart[8] To design and select the non-stochastic structures for targeted engineering applications, it is necessary to identify the indices for optimized energy absorption performance [9]. Therefore, the performance indices in this study include Specific Energy Absorption (SEA) and the plateau stress. SEA is defined as the area under the stress-strain curve normalized by the mass of specimen and quantifies the capacity of the structure to absorb energy in unit weight. Plateau stress is the average stress level of post yielding process of the structures at which the specimen experiences large plastic deformation at a relatively constant stress. To maximize the SEA, wide stress plateau regions and higher plateau stress are desired. To effectively group cellular structures based on their energy absorption performance, a SEA vs plateau stress bubble chart has been created as shown in Figure 4. For specific protection targets, additional constrain can be included in the bubble chart. The vertical line in Figure 4 represents safety threshold of the protection target, which is also the upper-limit of the plateau stress level. Only the structures on the left hand side of the vertical line are effective on the protection of specific targets. Although the structures on the right hand side of the bubble chart may have higher SEA, the higher stress plateau exceeds the safety threshold of the projection target, which makes the structures irrelevant for the design. 3 Figure 4: Energy absorption performance of cellular structures from literature [4] Auxetic structure has extra energy absorption capacity due to the structure’s self-support during collapse, which is achieved by the introduction of a negative Poisson’s ratio (𝜈) [10]. Physically, this characteristic is displayed by inclined struts in the cellular configuration. Also at the end of the compression, in place of termination strain is densification, in which the space between collapsed struts is exceptionally low and high energy absorption is recorded. Theoretically, the influence of the Poisson’s ratio is derived from small deformations, isotropic materials, and conservation of energy resulting in the constitutive equation below relating the stress(σ) and strain(е) matrices [11]. 𝜀ij = 1 2𝜇 ∗ 𝜎𝑖𝑗 − 𝜆 2𝜇(3𝜆 + 2𝜇) 𝜎𝑘𝑘𝛿𝑖𝑗 Actual physical representations of this characteristic is also shown below in Figure 5. Note the positive Poisson ratio is present in the first portion of figure and is negative in the following. 4 Figure 5: Bulk unit cells of (left) positive Poisson ratio honeycomb and (right) auxetic honeycomb [34,12] Another key characteristic mentioned earlier is the hierarchical structure. Designing hierarchical structures results in a scheme to boost mechanical performance, efficiency, and safety of structures becomes possible due to altered deformation mechanisms from bending dominated to stretching dominated. This feature is advantageous in situations requiring lightweight but strong materials. Dr. Chi Zhan, proved this during his studies in which he investigated deformation mechanisms between a regular honeycomb structure and one with a hierarchy introduced [12]. Shown below in Figure 6 is a replication of his results. One can see from the resulting rotated strain fields that the hierarchy sample has much more pure deformation in the direction of the rotated x axis than the bending dominated scheme shown on the left. Figure 6: DIC confirmation of bending-(left) and stretching-(right) dominated deformation in R- ReH and H-ReH unit cells respectively 5 Structurally, Euler Beam Theory shows that Slenderness Ratio (SR) is a critical design parameter in influencing performance and was subsequently chosen to be the critical parameter in this investigation [13]. SR is taken to be the ratio of the column thickness to the length, shown below are examples of a high SR on the left and a low SR on the right. Figure 7: Examples of (left) high and (right) low slenderness values 6 METHODS The structures used in this study are displayed below in Figure 8. The variation of SR in regular and hierarchical models was intended to be constant. This was achieved in the former model but not the later due to a more complicated function between available SR and density essentially constraining values. Note that for the hierarchical specimen, the SR in question is 2nd order slenderness, defined by 𝑙𝑡/𝑡𝑡. The first order slenderness was found to be relatively constant over the wide range of relative densities used in this study. The word density will be used interchangeably with relative density and should be taken to mean relative density. Equations for calculating density are based off the work of Dr. Zhan [12]. Figure 8: Schematic of (a) R-ReH and (b) H-ReH unit cells To investigate a wider range of SR values at a constant relative density in the hierarchical models, two new structures were created by extending the unit cell vertically and horizontally by one unit. Specifically, an extra horizontal layer was added in each inclined strut present in -Vert models and conversely one vertical layer was added in each horizontal member. These structures are displayed below in Figure 9. Note that in keeping with the unit cell ideology, the top and bottom layers of all models (including R-ReH) were printed with exactly half of the top and bottom members of the 7 cell. Otherwise, the performance of one sample could not be accurately compared with the performance of many put together. Figure 9: Schematic of (left) H-Vert and (right) H-Hor unit cells compared to (middle) H-ReH These samples were printed using a J55 Stratasys printer utilizing Polyjet technology [14]. In general, there are a few different ways the commercial additive manufacturing industry has advanced in the use of powders and liquids. Concerning powders, available printers can either create the part by heating the powders enough to cause fusion or by melting the powders completely and allowing the liquids to cure together in order to form the whole part. These processes offer the advantages of a wide available material range and quick prototyping. On the liquids side, thermoplastics are deposited from the nozzle layer by layer and allowed to cure into the whole part. While this process is very cheap, the drawback is in the resolution of the print and in clogging of the system. Another area of liquids is the use of photopolymers, used in this study. In this process, photopolymers are deposited layer by layer and then cured by UV light. These samples have the advantages of high resolution and quick curing, but technology has not yet evolved to incorporate metals. Although, recent research has enabled an 80% reduction in processing and material usage when printing cellular lattices using polyjet [15]. Critical parameters for evaluating and altering printer performance include printing direction, part orientation, aging of material, surface finish, and storage condition [16]. A table summarizing these printer qualities can be found in Table 1 and an image of the printer used in this study is provided in Figure 10. 8 Table 1: Comparison of commercially available additive manufacturing methods Figure 10: Stratasys J55 Printer The material used in this study was VeroUltraClear by Stratasys which contains a Young’s Modulus of 1400-2100 MPa and Elongation % Strain of 20-35% per the material Data Sheet [26]. Other options included ABS and VeroUltraWhite [27,29]. Selection strategy was based off ability to replicate previous results using Rigur[28]. Dog bone specimens according to ASTM standard and regular/hierarchical specimens of each material were printed and tested [30]. Corresponding Young’s Modulus, Yield Strength, and Frac Strain are tabulated and compared with materials data sheet below in Table 2: Table 2: Properties of additive materials used from data sheet and determined experimentally 9 Types of PrinterKey FeaturesResolution (Z direction)ProConRefSelective Laser Melting (SLM)Laser melts powder layer by layer20-50 micronsWide Material RangeResidual Stress17,18Electron Beam Melting (EBM)Electron Beam induces fusion of particles50-70 micronsFast ProcessRough Surface25Fused Deposition Modelling (FDM)Liquid Thermoplastics from nozzle50-200 micronsCheapClogging21,22Selective Laser Sintering (SLS)Powder bed solidified by Laser80 micronsComplex GeometryPost- Processing19,20PolyjetDeposited Photopolymers Cured by UV14 micronsQuick CuringNo Metals23,24Elongation (%)Young's Mod. (MPa)Young's Mod. (MPa)Yield Strength (MPa)Frac. Strain (%)Rigur20-351700-210074643.248.1ABS20-352100-280097663.629.4VeroUltraWhite7-122000-3000109969.232.8VeroUltraClear20-351400-210082148.211.4MaterialMaterial Data SheetCalculated Comparison of regular & hierarchical performance was performed using each of the listed materials and with the same dimensions of previously tested Rigur samples. This included 0.32/3.33 relative density/SR in the regular model and 0.34/5.6SR in the hierarchical model. VeroUltraClear was the only option that succeeded in achieving an acceptable termination strain and modulus in each model and can be seen from corresponding testing curves in Figure 11. Figure 11: Testing curves of (left) R-ReH and (right) H-ReH printed using various materials To investigate a wide variety of slenderness ratios, R-ReH specimens were created at 0.44, 0.34, and 0.24 relative density. This was the relative density target for all of the hierarchical samples as well, however due to scheduling conflicts the printed samples were left untouched for a month and a half and consequently accumulated more material in the storage condition. The extra weight brought the measured relative densities to 0.46, 0.36, and 0.26 in the hierarchical samples. Each model had 4 printed samples, three of them were tested and one kept for a backup. Results of each group were analyzed using a representative sample of the group. With this scheme, SR values in the range of 3-10 were tested, with the full range displayed below in Figure 12. 10 Figure 12: Scope of investigated samples Compression tests were conducted at 9mm/min using an Instron Model 5982. Video of the experiment was taken and Digital Image Correlation (DIC) analysis performed using equipment from Trillion and software via Aramis. The DIC allows for the capture of the strain field evolution during compression, which enhances understanding of the dominant deformation mechanisms as they occur in real time. Images of the testing setup and software are available below. Figure 13: Experimental (left) setup with Instron, and DIC Equipment and (right) view from software Given equipment paramers and proper setup, the Aramis software will use sensor and camera connections to feed live footage to the computer to apply fields on the generated mesh. The basis for generating the mesh used in this study was to initially apply a layer of flat white RUST-OLEUM 11 (commonly avaialble) spray praint. Once drying completed, specks of flat black spray paint from the same company were splattered onto the samples, as shown in the figure below. Figure 14: Speckled dog bone specimen prepared for DIC The software will generate a pattern out of the speckled pattern which it will track throughout the compression footage. End result is the ability to overlay stress and strain fields along the mesh, among other feautrues. It is improtant to produce unform size speckles throughout the sample and for that speckle size to be relatively small. These steps will improve the generated mesh in the software. Parameters used in the software to generate the mesh include the size and overlap of facets, which are the partitions the image is cut into for generating the pattern. 12 CHAPTER 1: IMPACT OF SLENDERNESS RATIO ON ELASTICITY 1.1 CHARACTERIZATION OF DENSITY EFFECT The density effect on the modulus of regular and hierarchical samples is predictable and more pronounced in R-ReH samples. In these models, density has a positive relationship with densification strain and modulus, as shown by the control group consisting of a constant SR and varying relative density in Figure 16. Regression analysis of the modulus reveals a linear relationship with an R^2 value of 0.99, displayed in Figure 15 below. Slenderness Effect on Average Modulus R² = 0.9999 4.1 SR ) a P M ( s u l u d o M e g a r e v A 120 100 80 60 40 20 0 0.2 0.25 0.3 0.35 Relative Density 0.4 0.45 0.5 Figure 15: Average Modulus – Relative Density trend during constant 4.1 SR 13 Figure 16: 4.1 SR, R-ReH Stress – Strain curves at various densities In the hierarchical models, the density effect on the elastic modulus is slightly more pronounced at larger relative density ranges. For example, in the H-ReH models, modulus increased 73% when going from 0.26 to 0.36 relative density and 86% when going from 0.36 to 0.46 relative density. Noteworthy is the inverse relationship between relative density and the onset of the densification stage. While all tested hierarchical samples experienced densification, achieving it was quicker in the larger density samples. 1.2 DISCUSSION In the beginning of the R-ReH compression test, the specimens experienced a tip touch interaction which added extra support to the structure. It is important to note that this tip-touch interaction was only observed in specimens with relatively high SR values for that specific density. That is to say, 4.1 SR specimens do not have this interaction at .34 and .24 relative densities. SR has an inverse relationship with the strain value this occurs at and is observed as 5-15% strain for the 14 entire population. The associated stress value increases with increased SR in all but 0.24 density specimens. At 0.24 density, the SR has negligible effect on the stress but still has moderate impact on the strain. Following this event, stress increased till rotation of the joint and subsequent buckling of opposing struts drove one side of the model below the other. Figure 17: Stress – Strain curve of all R-ReH specimens R-ReH regression analysis on the elastic moduli and slenderness ratios reveals an interesting relationship with the potential SR values at a specific density. Effectively, achieving the highest modulus is a competition between the effect and range of SR at the specific relative density. As relative density decreases, the available SR range widens, which has the overall effect of outperforming the increased SR effect within more narrow ranges at higher densities. This is demonstratable by the regression in the figure below, in which elastic moduli were normalized by the smallest modulus in that density group. While the slope of the trendline at 0.44 density is 15 double that at 0.24 density, the latter was able to achieve a higher relative modulus. Modulus gain at 0.44, 0.34, and 0.24 densities were 34%, 42%, and 43%, respectively. Figure 18: Effect of SR on normalized modulus In hierarchical samples, the effect of slenderness ratio was much more enhanced at lower densities. Analysis of the SR effect in hierarchical models was completed with the understanding that at a particular relative density, the low-medium-range values of SR were comprised by the Hor-Reg- Vert samples respectively. Findings revealed that at 0.46 relative density, the structure plays more of a role than the SR ratio. In other words, the relative density becomes so large that performance variations due to the slenderness ratio are marginalized by the overall changes in the three hierarchical structures. This reasoning is supported by the green bell shape trendline on Figure 20. This relation becomes more uniform at 0.36 density and then finally at 0.26, the SR ratio overtakes priority of the structure. Results were promising an indicated an 18% increase in observed modulus at this low structure. While one might consider this a constraining feature that limits design to low relative densities, this has no effect on the potential use of the feasibility of this structure because 16 the goal is to achieve exceptional performance while being as lightweight as possible. From an investigative perspective this quality demonstrates tunability of the structure. Figure 19: Stress – Strain curve of all hierarchical specimens Figure 20: Modulus of all hierarchical specimens 17 DIC allows for the visual confirmation of the negligible effect of SR on the elastic region in middle range relative densities. Using the same scale of epsilon-x at the time point right before elastic yielding of .34 density samples, one finds no discernable features in the severity of the strain fields or stability of critical joints. This consistent occurrence despite varying SR values of 5.1, 5.55, and 5.7 in the specimens support earlier discussion on SR effect. Figure 21: DIC analysis of epsilon x strain field with 0.2 scale, unrotated axes for 0.36 density (top) H-Reg, (bottom) H-Vert, and (middle) H-Hor samples The most interesting conclusion reached from this portion of hierarchical results is gained upon a comparison of the 7.75 and 8.25 SR samples at 0.26 relative density. Visual inspection via DIC of epsilon x reveals a significant distinction in behavior at the elastic limit of the H-Hor (7.75 SR) and H-Reg (8.25 SR) samples. While both specimen groups experience stretching deformation as indicated by the blue shear bands in the rotated axes (Figure 22), the 7.75 SR strain field is noticeably darker which suggests a higher portion of overall stretching deformation. Conversely, critical joints in the 8.25 SR samples contain significantly more rotation present at yielding. This 18 is the mechanism by which decreasing SR at lower densities results in an increased elastic modulus. Figure 22: DIC analysis of epsilon x strain field with 0.2 scale, 60 deg rotated axes, for 0.26 density (left) 7.75 SR H-Hor and (right) 8.25 SR H-Reg samples 1.3 INFLUENCE ON DOMINANT DEFORMATION MODE Slenderness Ratio plays a clear and predictable role in altering deformation modes in R-ReH samples. To understand this, one can treat each of the four inclined struts in these samples according to Euler Beam theory. Noteworthy is the referenced author’s findings of observed deviations from Euler theory in Gibson-Ashby equations for metallic lattices due to a slenderness limit. Pass this slenderness limit, the author shows Timoshenko beam theory is more appropriate [32] An inclined strut in this loading case will predominantly experience stretching and/or bending deformation, as shown in schematic below. An example of bending through DIC is shown in the right image of Figure 23. 19 Figure 23: Drawings of (left) bending [31], (middle) stretching deformation [31] and (right) DIC view of a bending stress field Derivation of stretching and bending stress yields inverse relationships with cross section area and second moment of inertia respectively. This relationship is linear and one can see in the figure below that at the highest SR value, bending dominated deformation is roughly 90 times weaker than stretching deformation. Figure 24: Deformation mode susceptibility in R- models According to Gibson-Ashby cellular theory, elastic modulus for open cells under bending and stretching dominated deformation should obey the following relationships: 20 𝑂𝑝𝑒𝑛 𝐶𝑒𝑙𝑙𝑠: 𝐸∗ 𝐸𝑠 = 𝐶1(𝜌𝑅𝐷) 2 [33] 𝐵𝑒𝑛𝑑𝑖𝑛𝑔: 𝐸∗ = 𝜌𝑅𝐷𝐸𝑠 [31] 𝑆𝑡𝑟𝑒𝑡𝑐ℎ𝑖𝑛𝑔: 𝐸∗ = 1 3 𝜌𝑅𝐷 2𝐸𝑠 [31] Figure 25: Gibson-Ashby agreement in R- models 4.1 SR specimens were used as a control group to confirm these equations. While the R-ReH models are proven to be a bending-dominated structure, the relatively low SR value explains the slight favor towards stretching deformation as seen in the provided plot. 21 CHAPTER 2: EFFECT OF SLENDERNESS RATIO ON ENERGY ABSORPTION CHARACTERISTICS 2.1 WORKING PRESSURE AND DENSFICATION STRAIN The plateau region is defined as a distinct strain boundary by which deformation occurs at a relatively constant stress. Using the average stress this occurs at is a useful way to generalize the performance and will be referred to as the working pressure. In the investigated R- samples, plateau regions were identified in all densities but the highest. At this density, brittle fracture was observed at the conclusion of the tip-touch interaction, which the SR had no observed effect on. The successful ending of plateau regions in these samples was followed by densification The regular and hierarchical models exhibit varying trends in the average plateau stress, region width, densification strain, and tunability with SR. The onset of the plateau region in the regular models occurs after rotation of the makeshift joint created during the tip-touch interaction and is identified by a slip. It is noteworthy to mention that the 0.44 relative density group failed before any plateau region could develop. Within the 0.34 and 0.24 densities there is much more knowledge to be gained. Figure 26 displays critical moments during compression and their associated location on the testing curve which can be used to understand the overall behavior of specimens examined. 22 Figure 26: Representative testing curve of an R-ReH specimen with DIC correlation Plateau region widths in this study were observed to be between 15 and 50% strain in regular models. The low end comprises samples in the 0.34 density category while the high end was achieved by the 0.24 density specimens. Within the low end, the plateau region endpoint is seemingly fixed as a function of density, based on the constant strain observed at the end of the region in all 0.34 specimens. Onset of the working region and pressure in 0.34 relative density samples were influenced by the slenderness ratio effect on tip-touch strain. Similar to 0.34 density samples, the lowest density group was designed with constant SR variation, Results indicate the existence of an SR barrier at low densities by which full realization of structure’s performance is challenged. Interestingly, the alteration of the region width at 0.24 density is not associated with an increased working pressure, which from a design perspective is pure profit of extra energy absorbed as deformation simply by tuning the SR value. On the contrary, the hierarchical samples first completely develop elastically before any tip contact. Yielding is caused by local rotation in one set of critical joints. The following collapse of 23 the nearby members is the beginning of the first plateau region, which concludes when the opposing critical joints begin taking load. Collapse of these joints is the trigger of the second plateau region, during which all critical joints are non-load bearing. Densification completes the process and begins when voids between members are negligible. Reference material for this process is below in Figure 27: Figure 27: Representative testing curve of an H- specimen with DIC correlation Comparatively, all H- samples achieved full realization of the available plateau region and subsequent densification. It is observed that the slenderness ratio had no effect on the onset of densification; at a specific density all samples had a relatively constant densification strain. An increase in relative density yielded a decrease in densification strain. This effect varied no more than 5% strain over the entire testing range and is negligible. Note that in the largest relative density category, only one group formed a plateau region. It can be inferred that a certain density requirement has to be met to achieve this characteristic. From the results it seems with decreasing 24 density there is behavior shift from not exhibiting a plateau region to experiencing a relatively smooth region of working pressure. In the middle of the two extremes is the emergence of two distinct plateau regions present in 0.36 density H- specimens. The working pressure observed in second region is roughly double than the value in the first region. This characteristic cannot be fully contributed to the SR effect as it has already been discussed that the impact of SR is more prevalent at the lower densities and density must be considered. Eventually, the two plateau regions present at this density morph into one relatively smoother plateau region at the lowest investigated density. The width of these regions remains comparatively constant at a given density. Performance of the 0.26 density group delivered promising results that point towards a useful relationship between SR and working pressure. In this group the difference of SR values are 0.15 and 0.3 between high/medium SR and medium/low SR specimens respectively. This variation is directly proportional to the working pressures observed during compression and is much more favorable to performance of R- specimens. A 12% variation in SR at this density can influence a 140% gain in working pressure. In R- specimens at the lowest density, SR had no bearing on the working pressure which remained constant. The addition of the second order hierarchy introduces a direct mechanism to influence the working pressure at low relative densities, which is ideal for achieving maximum efficiency of materials. Both the R- and H- models display promising relationships between working pressure, densification, and SR. Although between the two it is the H- models that have the advantageous characteristics which hold value in lightweight energy absorption structures. At the lowest density, one is able to significantly adjust the working pressure of the structure without drawbacks in terms of plateau region width, or densification strain. Further, it is noted that the introduction of the 25 hierarchy avoided the brittle fracture present in R- structures at higher density and all H- structures achieved densification. 2.2 SPECIFIC ENERGY ASBORPTION Better for an engineer from a design standpoint is the ability to alter a structure in order to achieve an energy absorption goal without using extra material. In general, more mechanical energy absorbed in the structure through deformation is desired In R- designs the observed SEA varies significantly with density at the highest and lowest densities. Important to note is that the SEA values used in this discussion will refer to the value at the end of the working stage. Due to the high density, the 0.44 density group did not exhibit desirable performance. Between the two density groups that completed a full deformation process, the average SEA values achieved were comparable. The wider working region in the lower density design compensates for the decreased average plateau stress and can even outperform it, as shown by SEA values in Figure 28. 26 Figure 28: SEA results of all R- specimens On SR tunability of SEA at a given density in R- designs, there seems no clear trend that can be seen in the 0.34 and 0.24 density groups. While the lowest/highest SR at 0.34 density achieves the highest/lowest SEA values, the trend does not appear to be linear shown by the 5.7 & 6 SR samples achieving similar SEA while the 5.7 & 5.4 SR samples achieved a 15% increase in SEA. The relationship becomes even more evasive at the lowest density in which the medium SR representative sample achieved highest SEA value, followed by the max and min SR samples. Although there is significant difference in achieved values, the relationship with SR is unclear. More interesting is the analysis of SEA performance in the hierarchical models. As all compression experiments ended in densification, a complete view of the density effect is seen. On average, density has a positive linear relationship with SEA values in these specimens. At the end of the working region, a 300% gain in SEA is noted between the lowest and highest relative densities, shown by the experimental curves in Figure 29. 27 Figure 29: SEA results of all H- specimens While SR tunability in R- yielded mostly inconclusive results, there is valuable information to be gained from H- performance. In the middle density there are slight gains noticed in SR variation, although it can considered negligible and not indicative of an SR effect as there is still significant structural effect at this density. Results of the lowest density, on the other hand, were conducive towards the indication of a promising SR effect. Relationship between SR value and SEA is observed to be linear, with the capacity to double the amount of energy absorbed within a group. On the basis of SR having significant more impact at lower densities, the performance observed in this region supported using SR as a viable parameter to improve SEA. This is a significant improvement from R- patterns, which provide no clear working relationship between SR and SEA to build upon. Also advantageous is the ability of H- models to absorb more than twice the energy of R- models at the end of plateaus. Concluded from 28 this comparison is increased feasibility of H- designs over R- due to increased performance and tunability due to clarity in relationship with SR. 2.3 DISCUSSION There are many advantages present when using the slenderness ratio as a parameter to influence the energy absorption characteristics of the regular and hierarchical models investigated in this study. Results indicate there are available options in this structure to achieve desired performance with constraints on the working stress. Both models display well-suited deformation mechanisms with the hierarchical model significantly outperforming its regular comparison. In the hierarchical models, the slenderness ratio is shown to be a controlling parameter in the efficiency of the structure. Using the first bubble chart introduced in Figure 4, findings from the regular models can be updated on the chart for comparison. This new bubble chart is provided in Figure 30. As a reminder, the largest density experimented on in the regular design did not exhibit any plateau regions due to brittle fracture and is not included. Noteworthy is the already provided honeycomb results in the figure and the differences between the results in this study. Firstly, it must be assumed the referenced author used a structure with a positive Poisson’s ratio, as it is not stated. Further, there is a material difference effect that would alter the overall vertical and horizontal alignment of the group. Finally, there is no mention of the distinct point referenced authors noted the SEA value. High values present in the bubble chart may suggest SEA at densification was used, which would be misleading. Investigation of the article did not provide answers to these concerns. What can be analyzed are the slopes, associated influences, and overall trends. 29 Figure 30: Comparative energy absorption performance of the R- testing group Effects of the slenderness ratio on energy absorption characteristics present in the regular models are the following. An improved SEA with a constant plateau stress was realized in the lowest density models. This occurs by means of an expanded working region brought upon by SR impact on tip-touch interaction strain. The exact relationship nature of this is unclear however, as the highest SEA value was achieved with the middle SR value. An explanation of this could be due to the competing influence of density and slenderness at this moderate density range. At the middle density, SEA displayed a positive relationship with SR, although this was not linear based on a constant variation of SR and an uneven SEA distribution. Plateau regions in this density range were dependent on the onset tip interaction strain, as was the working pressure. At the lowest density hierarchical group used in this study, the slenderness ratio is shown to have significant relationships with specific energy absorption and working pressure. Beyond the 0.26 30 density hierarchical samples, the 0.36 group exhibited a similar trend with decreasing SR, although the effect is not as linear as the previous density. As a reminder, performance effects due to structural differences in the hierarchical models are significant in densities beyond the smallest and is shown by the hexagon and star icons on the bubble chart in Figure _. Note the 0.46 density did not follow this trend most likely as explained due to structural effects and hence has an additional asterisk in bubble legend. Figure 31: Comparative energy absorption performance of the H- testing group One finds a more intuitive and practical relationship between the plateau stress and specific energy absorption trends with slenderness ratio in hierarchical samples. The translucent additional shapes are the previous results from the regular model and the new dark green group is the hierarchical. 31 As mentioned before, there are difficulties with comparing the results added to the figure with the results present beforehand due to unknowns with material and density. However, all samples in this study were printed with the same material and known densities. Clearly between the two groups, the hierarchical specimens provide a much improved performance and ‘bang for buck’ at a similar density than the regular specimens. Most notably this occurs in the lowest relative density samples included in this study, which is very promising in the search of structures for lightweight applications. Benefits of this is the ability to meet industry and environmental concerns by utilizing lightweight structures and excelling in performance. Paying tribute to the biologically inspired features of the unit cells investigated, as the density of the cell decreases the performance by energy absorption standards increases. 32 CONCLUSIONS From the beginning, the stated goal of this study was to investigate the feasibility of using regular and hierarchical models in applications that require lightweight energy absorption structures for delicate loading scenarios such as organ protection. Unit cells chosen in this study were inspired by biological studies and contain characteristics of efficient structures in nature. Theory suggests that slenderness ratio is a critical parameter in influencing performance. Due to the differing complexities in the two schemes, the regular design models were used as a baseline from which to understand the hierarchical model. Regular honeycombs have the benefit of wide slenderness ratios available at a particular density. On the other hand, in hierarchical models the relative density severely restricts the accessible slenderness ratios. The workaround used in this study was to create two additional hierarchical unit cells, each extended once in each unit direction of the in-plane axes. This creation allowed for the analysis of a relatively significant range of slenderness ratios across densities. Intuitively, in all models the range of slenderness ratios increased with density. However, one drawback is the introduction of structural effects on the performance of these designs. This density effect was observed to be much more prevalent at the higher densities experimented in this study. Characteristics that would encourage the use of a structure in such applications include tunability in the modulus, SEA, working pressure, and densification by slenderness ratio and all without a cost in the overall working strain range. Ideal would be a relationship that exists and is also most effective and natural at lower densities. Based on this criteria, investigation reveals the hierarchical design scheme is well suited to perform in the stated applications. Samples investigated in this study were printed off of a J55 Printer from Stratasys which utilizes trademarked Polyjet technology that involves the use of photopolymers deposited at the nozzle 33 and subsequently cured under ultraviolet light. Results of this printing process show advanced quality and minimal resolution issues. Material used was VeroUltraClear which has an elongation strain of 20-35% and Young’s modulus of 1400-2100 MPa per the manufacturer data sheet. Selection of this material was concluded after a comparison of performance with VeroUltraWhite, Rigur, and ABS was based on ductility – strength performance of trial samples printed with this material and dog bone tests. Relevant discussion in the elastic region focused on the elastic modulus and tip touch interaction in the regular models. The tip touch strain was found to have increased with density and at a specific density and decreased with SR. This is followed by an increase in modulus in the elastic region. Results indicate that in the regular models, effect of slenderness ratio on modulus is more than double at the lowest density than the highest. This pays homage to the competition between density and slenderness ratio in performance. This competition has been shown to favor density at higher densities and slenderness ratio at the lowest densities. Alteration of SR ratio in this density directly increased the observed modulus by a maximum 43%. Another notable characteristic of the density effect in regular models is the significant extension of the termination/densification strain across the density range. A comparison with theory reveals a correlation between performance of the regular design and Gibson-Ashby models for open cells. Dimension analysis reveals bending strength is almost 90 times weaker than stretching strength at the highest SR values used in this investigation. While the highest regular density did not enter any working region, the lowest two densities offer valuable information. In these samples, the total plateau region range was observed between 15- 50%. Slenderness ratio was observed to play a role in this by alteration of the tip-touch interaction strain and slightly influences beyond that in the lowest density. In the middle density, working 34 pressure and specific energy absorption increased inversely with slenderness at the middle density. In comparison, SEA variation in the lowest density was significant, but not uniform and did not come with an associated increased working pressure. Effectively, the regular model has usable properties at lower densities, although nowhere near the potential. Introduction of the hierarchy is sufficient in neutralizing these weaknesses in the R- design and turning the model into a competing structure for industrial applications. Density effect in hierarchical specimens had similar effects to its counterpart, although the change in densification strain was minimal. Further, due to the additional hierarchical unit cells, there is an introduction of a structure effect which is observed to be dominant at the higher densities. This development rendered slenderness ratio effects negligible at all densities except the lowest. Here, slenderness ratio was found to have an inverse linear relationship with modulus and achieved an 18% increase across the testing range. DIC allowed for the confirmation of behavior in the elastic range. A comparison of results supports this conclusion by the uniform strain field in the hierarchical 0.36 density samples. However, at 0.26 density this comparison shows significant more buckling in the 8.25 SR sample than the 7.75 SR sample at the moment of elastic yielding. Analysis of the plastic region illuminated a strong relationship between SR and relevant properties. At the lowest densities, the structure can achieve a 65% increase in SEA and over a 100% increase in working pressure while reducing the slenderness ratio 10%. Furthermore, the plateau region associated with this is a single, relatively smooth region which reaches the highest densification strain of any model. In comparison with other researched cellular structures, it’s been demonstrated that the hierarchical design scheme with slenderness ratio at it’s basis allows the structure to improve energy absorption related characteristics and compete for use in industry. Increasing innovation seen in the current 35 additive manufacturing industry will nullify difficulties regarding print quality, material selection, and cost. As a result, the widescale use of cellular solids will increase safety & efficiency while reducing waste & environmental impact. 36 FUTURE WORK Further development of this subject would include a wide scale approach pushing towards expanding experimental approach, clarifying implementation barriers and model validation. Most uses of energy absorption structures are experiencing dynamic loading; hence it is prudent to expand the experimental approach. Further is the actual implementation, in which multiple unit cells are put together to form the overall desired part. Experimentally validating results with the bulk unit cell would be beneficial to this research. Another area of validation that would be useful is computational. Confirmation of the slenderness effect from Finite Element modeling would enhance the overall view of the investigation. As referenced in Section 2.3, material effects can lead to confusion during comparison of results with current literature. To that end, compiling a consistent set of results would help further the development of this area of research. 37 BIBLIOGRAPHY Tertuliano, O., Greer, J. The nanocomposite nature of bone drives its strength and damage 1. resistance. Nature Materials, 1195–1202 (2016). 2. Animal Wellness – More about Diallo and his condition, Virginia Zoo in Norfolk Dept of Animal & Food Sciences, Breeds of Livestock: Ankole-Watusi Cattle, Oklahoma 3. State University Ha, Lu, A review of recent research on bio-inspired structures and materials for energy 4. absorption applications, Composites Part B: Engineering, 2020 5. Wang, L., Di, C., Liu, Q., Song, L. and Jiao, F., Energy saving and emission reduction of study on lightening of dump truck carriage, IOP Conference Series: Earth and Environmental Science, vol. 512, no. 1, pp. 6–13. doi: 10.1088/1755-1315/512/1/012003, 2020. Fleck, Deshpande, Ashby, Micro-architectured materials: past, present and future, 6. Proceedings of the Royal Society A, Mathematical, Physical and Engineering Sciences, 2010 NASA: Materials and Manufacturing Additive Manufacturing Pioneering Affordable 7. Aerospace Manufacturing, George c. Marshall Space Flight Center, 2015 Wenjin Tao Ming C Leu, Design of Lattice Structure for Additive Manufacturing, 8. Proceedings of ISFA2016, 2016 International Symposium on Flexible Automation 9. Sun, Gibson, Gordaninejad, Suhr, Energy absorption capability of nanocomposites: A review, Composites Science and Technology, Volume 69, Issue 14, November 2009, Pages 2392- 2409 10. Mir, Najabat Ali, Sami, Ansari, Review of Mechanics and Applications of Auxetic Structures, Advances in Materials Science and Engineering, 2014 Chen, Saleeb, Part A, Studies in Applied Mechanics 37, Constitutive Equations for 11. Engineering Materials, Elasticity and Modeling C. Zhan, M. Li, R. McCoy, L. Zhao, and W. Lu, Composite Structures, vol. 290, p. 115550, 12. Jun. 2022, 13. Bauchau, O.A., Craig, J.I. (2009). Euler-Bernoulli beam theory. In: Bauchau, O.A., Craig, J.I. (eds) Structural Analysis. Solid Mechanics and Its Applications, vol 163. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2516-6_5 J55 Prime 3D Printer, Polyjet, Stratasys USA & Canada, J55™ Prime Full Color 3D Printer 14. | Stratasys 38 15. Liu, Song, Huang, Maximizing mechanical properties and minimizing support material of Polyjet fabricated 3D lattice structures, Additive Manufacturing, Volume 35, October 2020, 101257 Brandt et al, 3D Printing of polymer composites with material jetting: Mechanical and 16. fractographic analysis, Additive Manufacturing, Volume 36, December 2020, 101558 17. Jeng et al, A state-of-the-art review on types, design, optimization, and additive manufacturing of cellular structures, International Journal of Advanced Manufacturing Technology Lauwers et al, Selective laser melting of iron-based powder, Journal of Materials 18. Processing Technology, Volume 149, Issues 1-3, Pages 616-622 Haq et al, 3D Printing – A review of processes, materials and applications in industry 4.0, 19. Sustainable Operations and Computers, Volume 3, 2022, Pages 33-42 Poulikakos et al, All-inkjet-printed flexible electronics fabrication on a polymer substrate laser sintering of metal nanoparticles, low-temperature high-resolution selective 20. by Nanotechnology, Volume 18, Number 34 Hui et al, 3D Printing of polymer matrix composites: A review and prospective, 21. Composites Part B: Engineering, Volume 110, Pages 442-458 22. Recent Advancements of Micro-Lattice Structures: Application, Manufacturing Methods, Mechanical Properties, Topologies and Challenges, Shah, Arabian Journal for Science and Engineering Tran et al, PolyJet 3D Printing of Composite Materials: Experimental and Modelling 23. Approach, 2nd Asia-Pacific International Conference on Additive Manufacturing Patpatiya et al. A review of polyjet 3D printing of polymers and multi-material structures. 24. Institution of Mechanical Engineers, Part C: Journal of mechanical Engineering Science, 2022 25. Lausmaa et al, Characterization and comparison of materials produced by Electron Beam Melting (EBM) of two different Ti–6Al–4V powder fractions, Journal of Materials Processing Technology VeroUltra 26. J55 https://www.stratasys.com/en/materials/materials-catalog/polyjet-materials/veroultraclear/ Stratasys Material Sheet, Data for Stratasys, Biocompatible Digital ABS Plus Material Data Sheet, Stratasys USA & Canada, 27. https://www.stratasys.com/en/materials/materials-catalog/polyjet-materials/digital-abs-plus/ PolyJet Materials 28. https://support.stratasys.com/en/materials/polyjet/rigur-durus EN Data Sheet, – Stratasys USA & Canada, 39 VeroUltraWhite 29. USA & https://www.stratasys.com/en/materials/materials-catalog/polyjet-materials/veroultra/ Performance Properties, Stratasys Canada, Standard Test Method for Tensile Properties of Plastics, American Society of Testing 30. Materials, ASTM D638-22 Zhong et al, Current Opinion in Solid State and Materials Science, Volume 27, Issue 3, 31. 2023 S.P. Timoshenko, On the correction for shear of the differential equation for transverse 32. vibrations of prismatic bars, Phil. Mag. 41 (245) (1921) 744–746. Gibson, I., and Ashby, M. F., 1982, Proceedings of the royal society of London. A. 33. Mathematical and physical sciences, 382(1782), pp. 43-59 40