Development of lightweight auxetic composites for noise reduction and vibration damping

ii Acknowledgements iii “Se a terra do Egipto é um presente do Nilo, segundo a frase de Heródoto, a do Minho é, sem dúvida, o resultado das fadigas dos seus habitantes.” Alberto Sampaio, O Minho Rural e Industrial , 1884 Acknowledgements Dedicado à minha Avó, que ao me levar pela mão à Escola do Monte, me trouxe até aqui. I wish to thank the people that were involved in this study. Every discussion, debate and share of opinions were fundamental for the completion of this work. I take this moment to greet the reader of this thesis. If it took the time to read this section with detail, it is truly an evidence of the interest that this work was able to create. For that, I salute, You, the reader. To my supervisor, José Meireles, for the sharing of experiences, guidance, sympathy, full support during this study and the chance to develop my research under a PhD scholarship. Also to Hélder Puga for the help and, especially, the comradery without whom this work would not be possible. An acknowledgement to Professor Philip Withers and Shelley Rawson from the Henry Royce Institute for their expertise and access to the X-ray microcomputed tomography. A special acknowledgement to my “partner in crime” sweet Muriel. For the ability to listen, comfort, motivate and guide me in the most difficult moments of this work. Together with my parents and brother, whom never allowed me to break down, they are the backbone of the mental strength that was needed to complete this challenge. To my family and friends, whose support I could always trust. For the good times and experiences. Without them, life would not be worth the effort that is invested in these words. This research was supported by the project iRAIL Innovation in Railway Systems and Technologies Doctoral Programme funds and by national funds through FCT – Portuguese Foundation for Science and Technology and was developed on the aim of the Doctoral grant PD/BD/114096/2015. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping STATEMENT OF INTEGRITY I hereby declare having conducted this academic work with integrity. I confirm that I have not used plagiarism or any form of undue use of information or falsification of results along the process leading to its elaboration. I further declare that I have fully acknowledged the Code of Ethical Conduct of the University of Minho. Guimarães, 2nd March 2020 Full Name: Vitor Hugo Pimenta Carneiro Signature: Abstract v Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping The future of transportation involves the reduction of emissions by the transition from individual to communal types of transport and lowering the consumption of energy. Railway, in particular, stands out from other types of transports since it consumes less energy, requires less space and produces less CO2. Further emission reduction in railway vehicles may be promoted by their weight decrease. This would not only lower the required energy in cruising regimes, but also facilitate accelerations and reduce braking/propulsion wear. Additionally, the mitigation of railway noise/vibration would increase the attractiveness of this transports on a passenger point-of-view. It may be understood that both issues could be addressed by using lightweight materials with enhanced vibration and noise damping. This study proposes the design of such a material, also considering its structural strength. It is suggested that composite materials with a negative Poisson’s ratio ( i.e. auxetics) and non-stochastic cellular solid configurations are able to display high static and dynamic mechanical properties with lightweight characteristics. A novel additive manufacturing assisted investment casting technique is developed to produce the samples. The cellular solids are scanned using X-ray microcomputed tomography for the dimensional/defect analysis and validation of the manufacturing process. Experimental and numerical analysis are used to analyze the impact of the Poisson’s ratio in the stiffness, collapse, vibration damping and sound reflection/transmission of the samples. These results are compared with current available materials to confirm the advantage of their application. It is concluded that the proposed manufacturing process is able to reproduce the intended proposed design. Indeed, the correlation between their static/dynamic mechanical properties, sound absorption/transmission and lightweight characteristics of the samples show a very desirable solution to be used in future railway vehicle fabrication. Keywords: Auxetic; Cellular Solids; Noise; Vibration. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping vi Desenvolvimento de Compósitos Auxéticos Leves para a Redução de Ruído e Amortecimento de Vibrações O futuro dos transportes está intimamente ligado à redução de emissões, pela transição de veículos individuais para coletivos e diminuição de consumos energéticos. A ferrovia, em particular, destaca--se dos outros tipos de transporte pelo seu menor consumo energético, ocupação de menor espaço geográfico e pela reduzida emissão de CO2. Eventualmente, estes impactos podem ser ainda diminuídos com a redução do peso das carruagens. Esta otimização permite, não só, baixar a energia que é necessária para o movimento do veículo, como também, facilitar acelerações e diminuir os desgastes durante os regimes de travagem/propulsão. Adicionalmente, é necessário reduzir as vibrações e o ruido ferroviário para tornar esta alternativa mais atrativa do ponto de vista do passageiro. Facilmente se entende que estes problemas podem ser mitigados com materiais leves e características otimizadas para o amortecimento estrutural e mitigação de ruido. Este estudo propõem o design de um material com estas características, considerando que as suas propriedades de resistência estrutural não podem ser negligenciadas. É proposto que materiais compósitos leves com um coeficiente de Poisson negativo ( i.e. Auxéticos) e com um configuração de sólido celular não-estocástico podem ser uma solução para obter boas propriedades mecânicas estáticas/dinâmicas e baixa densidade especifica. Para a sua fabricação é proposta uma inovadora técnica hibrida que combina a manufatura aditiva com a fundição por modelo perdido. Os sólidos celulares produzidos foram radiografados por microtomografia computorizada de raios-X para caracterizar o processo de manufatura em termos de defeitos e variações dimensionais. Foram usados ensaios experimentais e análise numérica para avaliar o impacto do coeficiente de Poisson na rigidez, colapso, amortecimento estrutural e absorção/transmissão acústica das amostras. Estes resultados são comparados com materiais disponíveis no mercado para verificar a vantagem do material desenvolvido para aplicação desejada. É concluído que o processo de manufatura desenvolvido é capaz de replicar o design desejado. A correlação entre propriedades mecânicas estáticas/dinâmicas, a sua absorção/transmissão acústica e a sua baixa densidade específica revelam que os sólidos celulares desenvolvidos são uma solução promissora para o futuro da construção de veículos ferroviários. Palavras-Chave: Auxético; Ruído; Sólidos Celulares; Vibração. Contents vii Contents Chapter 1 – Introduction 1.1. Motivation and contextualization ................................................................... 2 1.2. Historical note and state-of-the-art ....................................................... 6 1.3. Thesis organization and research questions ................................................. 29 Chapter references ............................................................................................. 30 Chapter 2 – Alloy Selection and Processing: Optimization for Static and Dynamic Mechanical Properties 2.1. Material and process selection ..................................................................... 41 2.2. Methods for alloy processing optimization .................................................... 51 2.2.1. Al alloy processing .......................................................................... 51 2.2.2. Microstructural characterization and hardness testing ...................... 52 2.2.3. Alloy dynamic mechanical properties: Vibration analysis .................. 53 2.2.4. Alloy static properties: Uniaxial tensile testing .................................. 54 2.3. Results and Discussion ................................................................................ 54 2.3.1. Alloy processing: Microstructural characterization and hardness ...... 54 2.3.2. Dynamic mechanical properties: Vibration analysis .......................... 60 2.3.3. Static mechanical properties: Tensile testing ................................... 64 2.4. Tailoring of static and dynamic mechanical properties by artificial ageing ...... 67 2.5. Conclusions on material and process selection ............................................. 68 Chapter references ............................................................................................. 69 Chapter 3 – Design and Manufacturing of Cellular Solids 3.1. Introduction to the design and manufacturing of cellular solids ..................... 75 3.2. Methodology for the design and manufacturing of cellular solids ................... 76 3.2.1. Cellular solid design ........................................................................ 76 3.2.2. Model 3d-printing ............................................................................ 79 3.2.3. Casting mold fabrication ................................................................. 79 3.2.4. Cellular solid casting ....................................................................... 80 3.2.5. X-ray microcomputed tomography of cellular solids.......................... 81 Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping viii 3.3. Results and Discussion .................................................................................. 82 3.3.1. Optimization of investment model 3d-printing parameters ................ 82 3.3.2. Optimization of casting parameters ................................................. 84 3.3.3. Characterization of the manufacturing process ................................ 86 3.3.4. Dimensional characterization of casted samples .............................. 89 3.4. Conclusions on cellular solid manufacturing ................................................... 96 Chapter references ................................................................................................ 97 Chapter 4 – Effect of the Poisson’s ratio in the Static Structural Behavior of the Designed Cellular Solids 4.1. Introduction on the static structural properties of cellular solids .................... 105 4.2. Static elastic properties: Analytical approach ................................................ 106 4.3. Methodology for static structural characterization ......................................... 113 4.3.1. Experimental quasi-static uniaxial compression ................................ 113 4.3.2. Finite element analysis: Uniaxial compression simulation ................. 113 4.3.3. 4D computed microtomography of the samples in compression ...... 116 4.4. Results and Discussion ................................................................................ 117 4.4.1. Experimental quasi-static uniaxial compression ................................ 117 4.4.2. Comparison with analytical model ................................................... 122 4.4.3. Finite element analysis: Uniaxial compression simulation ................. 124 4.4.4. 4D X-ray μCT: Validation of collapse mechanism ............................. 125 4.5. Comparison with other materials .................................................................. 127 4.6. Conclusions on static structural behavior of the cellular solids ...................... 129 Chapter references ............................................................................................. 130 Chapter 5 – Effect of the Poisson’s ratio in the Dynamic Structural Behavior of the Designed Cellular Solids 5.1. Introduction to the dynamic structural properties of cellular solids ................ 133 5.2. Methodology for dynamic structural properties of cellular solids .................... 134 5.2.1. Experimental vibration testing ......................................................... 134 5.2.2. Finite element analysis: Harmonic simulation .................................. 135 5.3. Results and Discussion ................................................................................ 137 5.3.1. Experimental vibration testing ......................................................... 137 Contents ix 5.3.2. Finite element analysis: Harmonic simulation .................................. 140 5.3.3. Modeling of dynamic behavior ......................................................... 146 5.4. Comparison with other materials .................................................................. 147 5.5. Conclusions on the dynamic structural behavior of the cellular solids ............ 148 Chapter references ............................................................................................. 149 Chapter 6 – Acoustic Behavior of the Designed Cellular Solids 6.1. Introduction on the acoustic behavior of cellular solids ................................. 151 6.2. Sample selection for optimized static/dynamic properties............................. 152 6.3. Methodology for the acoustic testing of cellular solids ................................... 154 6.3.1. Filling of the samples with polymer foams ....................................... 154 6.3.2. Vibration testing: Experimental and FEA .......................................... 155 6.3.3. Acoustic testing of the samples ....................................................... 156 6.3.4. Acoustic measurements in railway vehicles ..................................... 159 6.4. Results and Discussion ................................................................................ 161 6.4.1. Vibration testing results ................................................................... 161 6.4.2. Acoustic testing results ................................................................... 163 6.5. Comparison with other materials .................................................................. 174 6.6. Comparison with train vehicle acoustics results ............................................ 176 6.7. Conclusions on the acoustic properties of the cellular solids ......................... 181 Chapter references ............................................................................................. 182 Chapter 7 – Conclusions 7.1. Study conclusions ........................................................................................ 187 7.2. Future works ............................................................................................... 189 7.3. Scientific impact .......................................................................................... 190 Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping xvi Figure 3.13. Average sample filling using different combinations of mold and casting temperatures: (a) with vacuum; and (b) without vacuum..................................................... 86 Figure 3.14. Detail of the manufacturing process and characterization of α=-30º ................ 87 Figure 3.15. Comparison between 3d-printed and casted cells ........................................... 87 Figure 3.16. Detail of porosity and segregation within the sample ribs ................................ 88 Figure 3.17. Detail of voids in the ribs: (a) pore and (b) segregation volumes ...................... 88 Figure 3.18. Casted lattices: (a) α=-30º, (b) α=-20º, (c) α=-10º, (d) α= 0º, (e) α=10º, (f) α=20º and (g) α=30º samples........................................................................................... 89 Figure 3.19. X-ray μCT casted lattices: (a) α=-30º, (b) α=-20º, (c) α=-10º, (d) α= 0º, (e) α=10º, (f) α=20º and (g) α=30º samples .......................................................................... 89 Figure 3.20. Comparison between the dimensions of CAD and casted samples .................. 90 Figure 3.21. X-ray μCT of cells: (a) α=-30º, (b) α=-20º, (c) α=-10º; (d) α= 0º, (e) α=10º, (f) α=20º and (g) α=30º .................................................................................................. 91 Figure 3.22. X-ray computed tomography dimensional measurements compared against CAD model specifications: (a) rib angle, (b) horizontal rib length, (c) vertical rib length and (d) rib thickness. ......................................................................................................... 91 Figure 3.23. Influence of angle in the printing area during the fused filament fabrication process: oblique with (a) α=30º, (b) α=0º and (c) horizontal rib. (d) Printing area plot of for oblique ribs (t=0.6 mm) ................................................................................................ 93 Figure 3.24. Schematic illustrating the deviation in printed rib cross-section form the CAD model, and how this results from the printing process ........................................................ 94 Figure 3.25. Cross-sectional view through the µCT reconstructed volumes, showing detail of rib cross-: (a) α=-30º, (b) α=-20º, (c) α=-10º, (d) α= 0º, (e) α=10º, (f) α=20º and (g) α=30º samples ............................................................................................................................ 95 Figure 3.26. Rib cross-section: (a) rib area and moment of inertia; (b) rib roundness (R=1, means perfect circle) ...................................................................................................... 95 Figure 4.1. Honeycomb deformation mechanism: (a) initial dimensions and loading; (b) load distribution; (c) axial deformation; and (d) flexural deformation .................................... 107 Figure 4.2. Plot of the proposed model for axial (Ka) and flexural (Kb) stiffness coefficients (Note: t=0.6mm; H=4mm; and E0=70 GPa) ........................................................................ 108 Contents xvii Figure 4.3. Hinging deformation mechanism in honeycombs with: (a) negative, (b) zero and (c) positive rib angles ................................................................................................ 109 Figure 4.4. Plot of the proposed model for correction hinging coefficient (q) ........................ 109 Figure 4.5. Plot of the proposed model for total stiffness coefficient (Kt). (Note: t=0.6mm; H=4mm; E0=70GPa) .......................................................................................................... 110 Figure 4.6. Plot of the initial apparent area (A*). (Note: t=0.6mm; H=4mm; E0=70GPa) ...... 110 Figure 4.7. Plot of initial apparent length (L*). (Note: t=0.6mm; H=4mm; E0=70GPa) .......... 111 Figure 4.8. Plot of the proposed model for apparent modulus (E*). (Note: t=0.6mm; H=4mm; E0=70GPa) .......................................................................................................... 111 Figure 4.9. Plot of the proposed model for correction hinging coefficient (p). (Note: t=0.6mm; H=4mm; E0=70GPa).......................................................................................... 112 Figure 4.10. Plot of the proposed model for Poisson’s ratio (ν). (Note: t=0.6mm; H=4mm; E0=70GPa)......................................................................................................................... 112 Figure 4.11. Compression testing: (a) Experimental apparatus and (b) Poisson’s ratio calculation with axial and transverse strain gauges ............................................................. 113 Figure 4.12. Representation of FEA model with boundary conditions .................................. 114 Figure 4.13. Representation of model meshing .................................................................. 114 Figure 4.14. FEA input values for rib thickness (q) .............................................................. 115 Figure 4.15. Plot of FEA input stress-strain material model ................................................. 115 Figure 4.16. Graphical representation of the adopted coupled structural numerical analysis ............................................................................................................................. 116 Figure 4.17. Average apparent stress-strain curves for the teste samples………………………. 117 Figure 4.18. Sample (α=-30°) in compression testing: (a) After plastic collapse stress, (b) during compression plateau and (c) during densification ……………………………………….. 118 Figure 4.19. Average Poisson’s ratio (ν) of the tested samples in uniaxial compression ....... 118 Figure 4.20. Average apparent modulus (E*) of the tested samples in uniaxial compression ...................................................................................................................... 119 Figure 4.21. Average plastic collapse stress (σP) of the tested samples in uniaxial compression ...................................................................................................................... 119 Figure 4.22. Deformation due to plastic collapse (ε=0.1) of the samples with (a) -30°, (b) -20°, (c) 0°, (d) 20° and (e) 30° ................................................................................. 120 Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping xviii Figure 4.23. Comparison of the suggested analytical apparent modulus (E*) with experimental results and other published models ............................................................... 122 Figure 4.24. Comparison of the suggested analytical Poisson’s ratio (ν) with experimental results and other published models ............................................................... 123 Figure 4.25. FEA results for apparent modulus (E*) ............................................................ 124 Figure 4.26. FEA results for plastic collapse stress (σP) ....................................................... 124 Figure 4.27. Graphical representation of FEA in samples with: (a) negative, α=-30°, (b) positive, α=30° and (c) zero, α=0° Poisson’s ratio ............................................................ 125 Figure 4.28. 4D X-ray μCT of the auxetic sample (α=-30º): (a) undeformed – ε=0; (b) elastic deformation – ε=0.01; (b) post-collapse – ε=0.04……………………………………………………..126 Figure 4.29. 4D X-ray μCT of the positive Poisson’s ratio sample (α=30º): (a) undeformed – ε=0; (b) elastic deformation – ε=0.01; (b) post-collapse – ε=0.04 ................ 126 Figure 4.30. 4D X-ray μCT of the near zero Poisson’s ratio sample (α=0º): (a) undeformed – ε=0; (b) elastic deformation – ε=0.01; (b) post-collapse – ε=0.04 ................ 127 Figure 4.31. Comparison of specific modulus (E*/E0) to specific density (ρ*/ρ0) of the manufactured samples and other published data ............................................................... 128 Figure 4.32. Comparison of specific strength (σ*/σ0) to specific density (ρ*/ρ0) of the manufactured samples and other published data……………………………………………………….129 Figure 5.1. Vibration testing: (a) Experimental apparatus and (b) example of damping ratio calculation……………………………………………………………………………………………………134 Figure 5.2. Developed Application Builder environment ...................................................... 135 Figure 5.3. FEA model: (a) different bodies and (b) meshing ............................................... 136 Figure 5.4. Optimization for (a) dynamic modulus and (b) damping ratio ............................ 137 Figure 5.5. Sample frequency response function (FRF) from the vibration testing ................ 137 Figure 5.6. Eigenmodes of the samples: (a) 1st mode-Torsion and (b) 2nd mode-Flexion .... 138 Figure 5.7. Average damping ratios (ξ) from the vibration testing ........................................ 138 Figure 5.8. Flexural component (F) and volume variation (V) in the samples due to the different rib angles……………………………………………………………………………………………….140 Figure 5.9. Theoretical representation of the vibrating system ............................................. 141 Figure 5.10. Relative weight of the samples in the overall dynamic system (i.e. experimental apparatus) .................................................................................................... 143 Contents xix Figure 5.11. FEA results for samples with rib angles: (a) α=-30°; (b) α=-20°; (c) α=-10°; (d) α=0°; (e) α=10°; (f) α=20°; (g) α=30° ...................................................................... 144 Figure 5.12. Comparison on static experimental and dynamic FEA apparent modulus ........ 145 Figure 5.13. Comparison between experimental and FEA damping ratios……………………….146 Figure 5.14. Analytical model for relative damping variation (ξ/ξMAX) due to the variation in rib angle ........................................................................................................................ 146 Figure 5.15. Comparison between this study results and other published data .................... 147 Figure 6.1. Correlation on the damping ratio and (a) apparent modulus and (b) plastic collapse stress ................................................................................................................... 152 Figure 6.2. Geometry selection index (GSi) for the tested rib angles……………………………….153 Figure 6.3. Sample foaming: (a) initial sample; (b-e) detail of foam filling the interior of the sample ........................................................................................................................ 154 Figure 6.4. Examples of final samples filled with (a) flexible and (b) rigid foams .................. 155 Figure 6.5. Impedance tube and sample assembly for sound absorption testing ................. 156 Figure 6.6. Detail of incident and reflected waves during Sound Absorption testing ............. 157 Figure 6.7. Detail of impedance tube assembly for Transmission loss testing…………………..158 Figure 6.8. Incident, reflected and transmitted waves during Transmission Loss testing ...... 158 Figure 6.9. Example of measurement using Spectralissime ................................................ 159 Figure 6.10. Behringer C-1U microphone calibration settings ............................................. 160 Figure 6.11. Sample frequency response function (FRF) from the vibration testing .............. 161 Figure 6.12. Average damping ratios (ξ) from the vibration testing ...................................... 162 Figure 6.13. FEA results for samples with (a) flexible and (b) rigid foam fillings ................... 162 Figure 6.14. Average damping ratios (ξ) from FEA routines ................................................ 163 Figure 6.15. Noise progression by (a) external and (b) internal sources .............................. 164 Figure 6.16. Results from acoustic reflection testing: (a) Sound absorption coefficient and (b) acoustic impedance ............................................................................................... 164 Figure 6.17. JCAL model in (a) unfilled and (b) foam filled samples .................................... 166 Figure 6.18. . Influence of visco-inertial parameters in the sound absorption coefficient of porous media (JCAL model): (a) Ф - porosity; and (b) σ - air flow resistivity ..................... 166 Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping xx Figure 6.19. Influence of visco-inertial parameters in the sound absorption coefficient of porous media (JCAL model): (a) α∞ - high frequency limit of the tortuosity; and (b) Λ - viscous characteristic length .............................................................................................. 167 Figure 6.20. Influence of visco-thermal parameters in the sound absorption coefficient of porous media (JCAL model): (a) Λ’ - thermal characteristic length; and (b) k’0 - static thermal permeability .......................................................................................................... 168 Figure 6.21. Influence of changing poroelasticity parameters in the sound absorption coefficient of a generic porous media: (a) E - elastic modulus and (b) ρ - density................. 171 Figure 6.22. Combination of changing parameters in the sound absorption of porous media due to effect of: (a) rigid structure (JCAL model) and (b) poroelasticity (Biot’s theory) .......... 171 Figure 6.23. Results from acoustic transmission loss ......................................................... 172 Figure 6.24. Sound absorption coefficient comparison: designed cellular solids vs other Al foams ............................................................................................................................ 174 Figure 6.25. Transmission loss comparison: designed cellular solids vs materials used in train vehicle fabrication ...................................................................................................... 175 Figure 6.26. Transmission loss comparison: designed cellular solids vs other suggested solutions for noise damping ............................................................................................... 175 Figure 6.27. Interior noise in High-speed trains .................................................................. 176 Figure 6.28. Interior noise in Inter-city trains ...................................................................... 177 Figure 6.29. Interior noise in Light-rail ................................................................................ 177 Figure 6.30. Identification of critical frequencies in High-speed trains ................................. 178 Figure 6.31. Identification of critical frequencies in Inter-city trains ..................................... 178 Figure 6.32. Identification of critical frequencies in Light-rail trains ..................................... 179 Figure 6.33. Comparison between experimental tests and critical frequencies in Highspeed trains: (a) Sound absorption coefficient and (b) Transmission loss ............................ 179 Figure 6.34. Comparison between experimental tests and critical frequencies in Intercity: (a) Sound absorption coefficient and (b) Transmission loss .......................................... 180 Figure 6.35. Comparison between experimental tests and critical frequencies in Lightrail: (a) Sound absorption coefficient and (b) Transmission loss .......................................... 180 Figure 6.36. Hybrid cellular solid concept: combination of unfilled/foam filled portions ....... 181 Contents xxi List of Tables Table 1.1. Sources of noise and vibration in railway transportation ..................................... 3 Table 1.2. Thermodynamic limits of the Poisson’s ratio for isotropic elastic bodies .............. 8 Table 1.3. Published works concerning indentation resistance of auxetics ........................... 11 Table 1.4. Equations for non-dimensional fracture displacements ....................................... 12 Table 1.5. Brief progression of published works concerning the vibration/acoustic propagation in auxetics ...................................................................................................... 15 Table 1.6. Naturally occurring auxetic materials ................................................................. 16 Table 1.7. Poisson’s ratio of reentrant structures ............................................................... 18 Table 1.8. Published works on reentrant auxetic structures ................................................ 19 Table 1.9. Published works concerning chiral auxetic structures ......................................... 21 Table 1.10. Poisson’s ratio of Nodule fibril models ............................................................. 23 Table 1.11. Published works concerning Nodule fibril auxetic structures ............................. 23 Table 1.12. Published works concerning Rotating rigid auxetic structures ........................... 24 Table 1.13. Published works concerning inclusion based auxetic structures ........................ 25 Table 2.1. Identification of relevant mechanical properties .................................................. 42 Table 2.2. Shapes for different metal manufacturing processes .......................................... 45 Table 2.3. Mold-metal interface reactions in the investment casting of magnesium alloys .... 49 Table 2.4. Aluminum alloy (A356) chemical composition .................................................... 52 Table 3.1. 3d-printing (fused filament fabrication) parameters ............................................ 79 Table 3.2. X-ray μCT testing parameters ............................................................................ 82 Table 3.3. Relation between nozzle diameter, layer height, printing time and the successful printing of the models (t=0.6 mm and v=2 mm) ................................................ 84 Table 3.4. Statistical significance of the rib dimensions ...................................................... 92 Table 4.1. Theoretical models for the elastic properties of honeycomb lattices .................... 106 Table 4.2. FEA input for material mechanical properties ..................................................... 116 Table 4.3. 4D X-ray μCT testing parameters ....................................................................... 117 Table 4.4. Suggested deformation models for the compression of the designed samples .... 121 Table 4.5. Correlation of the experimental values with the theoretical models ..................... 123 Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping xxii Table 4.6. Materials that were analyzed for comparison ..................................................... 127 Table 5.1. Input parameters for the different bodies ........................................................... 136 Table 5.2. Suggested models for the variation in damping ratios during the vibration testing ............................................................................................................................... 139 Table 5.3. Other publication for comparison with this study results ..................................... 147 Table 6.1. Proportion of polyurethane foam reagents.......................................................... 154 Table 6.2. General characteristics of the manufactured foams ............................................ 155 Table 6.3. Ambient properties inside anechoic camber during testing ................................. 157 Table 6.4. Behringer C-1U microphone characteristics ....................................................... 160 Table 6.5. Detail of experimental noise measurements in railway transports ....................... 161 Table 6.6. Expected variable changes in foam filled samples .............................................. 173 Table 7.1. Scientific outputs in written documents .............................................................. 191 Table 7.2. Dissemination of results and conclusions in scientific events .............................. 191 Chapter 1 Introduction 1 Chapter 1 – Introduction “Philosophy is written in that great book which ever lies before our eyes - I mean the universe - but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written.” Galileo Galilei – translated from Il Saggiatore , 1623 Experiment, postulate and refute hypotheses are basic procedures to understand our surroundings. To do so, it is mandatory to create ingenious ways to perform such experiments and devise conclusions. These are frequently the terms for evolution in the domains of “natural philosophy” and engineering. This document presents a study on mechanical engineering, showing the current developments of its field, and a research methodology for the production of new knowledge. The concepts to be developed in this work are intended to be submitted as a proof of the candidate suitability to be considered for a Doctor of Philosophy degree. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 2 1.1. Motivation and contextualization Curiosity and necessity are basic propulsion mechanisms for “artificial” evolution. It is the human nature to defy nature itself, either by sheer fascination of being able to contradict established "natural laws” or only by pure survival instinct. Based on these two established principles, pioneering minds that did not conform their status quo , were able to originally develop technology and change the world around them. One can now fly the skies, travel to space and communicate globally. For that we have the inherited “responsibility” of contributing to this technological and, consequently, social advance. Being able to participate in this evolution is a privilege, and the opportunity to develop new knowledge in the form of technology is the basic motivation for the proposed study. Especially, the current trend of studying, synthesizing metamaterials and engineering composites for the benefit of society, mankind and nature itself. This mindset is included in the philosophy and policy that contemplates the implications of technology in the future of our society. For instance, the European Union has established one of its goals the reduction of oil and fossil fuel consumption, by decreasing carbon emissions in transports by 60% in 2050 [1]. One key aspect to reach this objective is the transition of 50% road passengers to rail and waterborne transports by the same year [2]. Railway is particularly suitable for this task: it consumes less energy, requires less space and produces less CO2 than any other form of transportation [3]. Current trends in the railway industry to reduce emissions and economic viability include: (i) lower vehicle weight; (ii) lower wear on braking/propulsion systems; (iii) reduce maintenance costs; and (iv) faster accelerations to reduce journey times [4]. These measures, even though are effective routes to benefit railway transports, do not appeal directly to the passengers. Solutions have to be found to promote the passenger shifting to railway. It is fundamental to determine the factors that are currently dissuading passenger from this transportation, as well as, actions that would increase its attractiveness. Noise, for instance, is a major reason for public opposition to rail transportation in different European regions. In fact, rail transport is the second principal source of environmental noise in Europe [5]. This is particularly problematic, since Noise is identified by the World Health Organization as the second causer of disease in the European Union, right after air pollution [6]. The noise that is generated by rail transportation, not only affects the neighboring population and residences, but also the passengers Chapter 1 Introduction 3 themselves. It is estimated that a reduction in railway noise by at least half of the affected population could generate savings up to 2255M€ during the years 2015-2035 [5]. In conclusion, the design of rail vehicles, either by fuel consumption or noise perspectives, is a social, economic and environmental problem. Population and their political representatives urge the railway industry to become quieter, less prone to vibration and less pollutant. Considering these common sources of structural vibration/noise in railway transportation have been thoroughly studied, being identified in Table 1.1 [7]. Table 1.1. Sources of noise and vibration in railway transportation. Type Description Rolling noise Interaction of the wheel with the rail: unevenness in contacting faces. Track vibration Vibration with rails acting as waveguides to structural oscillation. Wheel vibration Vibration of wheels in resonance. Aerodynamic noise Turbulent air boundary layers that sorrounds moving vehicles. Curve squeal Contacts between rail and self-excited resonant wheel. Impact noise Impact between rail and wheel due to discontinuities in track or wheel. Bridge noise Vibration of the rails transmitted through bridge radiating noise. Ground vibration Transmission of vibration from the rail to the ground. Equipment noise Operating equipment noise ( e.g. HVAC, converters, motors, etc). Relatively to the interior vibration and noise, those that affect passengers, it is known that it is mainly originated by exterior sources according to Fig. 1.1(a). Such exterior dynamic excitations usually arise from the sources described in Table 1.1. They are transmitted to the interior of the vehicle by structural vibration and airborne transmission. Structural vibration, however, may also be converted into noise, leading to structure-borne noise transmission. The noise transmitted to the interior of the vehicle interacts with the noise that is generated within the coach ( e.g. HVAC and other interior equipment, passengers, etc) and is then reflected in the vehicle walls (Fig. 1.1(b)) [8]. Thus, noise mitigation is directly correlated with the ability of the materials that are used in vehicle manufacturing to display high values of structural and noise damping. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 10 The relative increase of indentation resistance is supported by the classical Theory of Elasticity, in which for isotropic solids, is presented in terms of hardness (H) by Eq. 1.4 [41]. The referred equation relates the hardness of a given material, as a function of Young’s modulus (E), Poisson’s ratio (ν) and a constant that represents the type of indentation contact (γ). The latter may assume the values of 1 or 1/3, respectively, for uniform pressure distribution or Hertzian type of contact. The graphical representation of the referred dependency may be observed in Fig.1.8, where it is clear that there is a significant increase in hardness for lower values of Poisson’s ratio. 𝐻 𝛼 [ 𝐸 (1−𝜈2)]𝛾 (1.4) Figure 1.8 Hardness (H) dependence on the Poisson’s ratio (ν) and Young’s modulus (E). Experimental and theoretical studies have demonstrated that auxetic materials may be advantageous to enhance indentation resistance. Some of these works are presented in Table 1.3. This counterintuitive behavior is also predicted to be relevant in terms of fracture resistance. According to finite element analysis (FEA) mode I fracture analysis (Fig. 1.9) it has been predicted that “normal” bulk solid (Fig. 1.9 (a)) has a wider plastic affected zone compared with an auxetic material (Fig. 1.9 (c)) with a normalized Young’s modulus [52]. This effect is attribute to local stiffness reduction, justified by the results using artificial reduction of the Young’s modulus shown in Fig. 1.9 (b). Additionally, their widening effect in tension may have a relevant role in a “closing” effect around the crack tips [32]. Chapter 1 Introduction 11 Table 1.3 Published works concerning indentation resistance of auxetics. Year Lead Author Ref. Description 1993 R.S. Lakes [42] Reentrant copper foam is shown to have smaller indentation radius and higher yield strength when compared with conventional foams with the same relative density; 1994 K.L. Alderson [43] Negative Poisson’s ratio Ultra High Molecular Weight Polyethylene (UHMWPE) is shown to have superior hardness than samples with positive Poisson’s ratio; 1998 N. Chan [44] Auxetic polyurethane foams present better indentation behavior and increased resilience when compared with conventional polyurethane foam samples; 2000 K.L. Alderson [45] Indentation resistance in auxetic UHMWPE is enhanced at low loads when compared with sintered and compression molded samples; A. Lowe [46] Auxetic polyurethane foams show lower values of localized pressure when indented with human buttocks, in comparison with conventional foams with the same density; 2011 V.L. Coenen [47] Enhanced indentation resistance and smaller localized damage areas in auxetic Carbon Fiber laminates, when compared with positive Poisson’s ratio samples; 2012 I.I. Argatov [48] Friction contact between indentor, elastic medium increases and relative indentation compliance decreases as the values of Poisson’s ratio tends to -1; 2015 L.L. Hu [49] Indentation resistance changes with cell angles in reentrant structures with similar density; 2016 D. Photiou [50] Finite element analysis is used to prove the increase in conic indentation resistance in auxetic materials, due to shear stiffening and miotic response of the elastic solid. 2019 L.I. Hu [51] Indentation of auxetic lattices under large plastic deformations. In such regime, indentation resistance in auxetics is not always higher than regular lattices. Figure 1.9 Von Mises stress near the crack tip and affected Plastic zone (adapted from [52]). The auxetic plastic affected zone, although may be smaller, this may not be the case of specific local values of stress or strain. For the diverse fracture modes, the value of Poisson’s ratio is a fundamental constant to determine the displacement fields around the crack tip. Eqs. 1.5 to 1.13 in Table 1.4 [53] determine the variation of non-dimensionalized cartesian displacement fields (Ux, Uy and Uz), for a normalized Young’s modulus and angle (θ) of the classic fracture model (Fig. 1.10(a)) are shown in Table 1.4 and plotted in Fig. 1.10(b) to (f). According to the plots presented in Figs. 1.10 (b) to (f), it may be observed that auxetic behavior may be interesting for fracture resistance in particular situations. In tensile scenarios (Mode I – Figs. 1.10 (b) and (c)), it seems that auxetic behavior is beneficial in plane strain conditions. However, in-plane shear Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 12 fracture (Mode II – Figs. 1.10 (d) and (e)), auxetic behavior seem to be only hazardous in elevated yy - direction displacements in plane stress conditions. Finally, in out-of-plane shear fracture (Mode III – Fig. 1.10 (f)), auxetic characteristics seem to be advantageous. Table 1.4 Equations for non-dimensional fracture displacements. Displacement Field Ux Uy Mode I Plane Strain 2(1+𝜈)cos(𝜃2 ⁄)[1−2𝜈+sin2(𝜃2 ⁄)] (1.5) 2(1+𝜈)sin(𝜃2 ⁄)[2(1−𝜈)−cos2(𝜃2 ⁄)] (1.6) Plane Stress 2cos(𝜃2 ⁄)[1−𝜈+(1+𝜈) sin2(𝜃2 ⁄)] (1.7) 2sin(𝜃2 ⁄)[2−(1+𝜈)cos2(𝜃2 ⁄)] (1.8) Mode II Plane Strain 2(1+𝜈)sin(𝜃2 ⁄)[2(1−𝜈)−cos2(𝜃2 ⁄)] (1.9) −2(1+𝜈)cos(𝜃2 ⁄)[1−2𝜈−sin2(𝜃2 ⁄)] (1.10) Plane Stress 2sin(𝜃2 ⁄)[2+(1+𝜈)cos2(𝜃2 ⁄)] (1.11) −2cos(𝜃2 ⁄)[1−𝜈−(1+𝜈) sin2(𝜃2 ⁄)] (1.12) Mode III Uz 4(1+𝜈)sin(𝜃2 ⁄) (1.13) Figure 1.10 Representation of the non-dimensional displacements from the conditions and equations in Table 3: (a) Basic fracture schematics; (b) Mode I - Plane-Strain; (c) Mode I – Plane-Stress; (d) Mode II – Plane Strain; (e) Mode II – Plane Stress; (f) Mode III. Chapter 1 Introduction 13 Considering the dynamic behavior and properties, materials with a negative Poisson’s ratio are also characterized by a peculiar behavior. It has been shown that auxetics subjected to an incident of a longitudinal wave (Ai) (Fig. 1.11(a)) may exhibit an enhanced capacity to reflect waves (Ar) for a widespan of incident angles (γ) (Fig. 1.11(b)) [54]. Figure 1.11. Surface reflection: (a) wave reflection scheme; and (b) dependence on the Poisson’s ratio and incidence angle (adapted from [54]). Concerning the propagation of Rayleigh waves, it has been shown that auxetic materials possess enhanced damping properties in comparison with “normal” materials for normalized displacements (y) as a function of wavelength (λ) (Fig. 1.12) [54]. Figure 1.12. Propagation of Rayleigh waves depending on the Poisson’s ratio and normalized displacement (adapted [54]). On the propagation of solitary waves, it is known that extreme values of Poisson’s ratio, near -1 or 0.5, generate an increase in wave amplitude (Fig. 1.13(a)) and velocity (Fig. 1.13(b)). Studies show that minimum amplitude and velocity are generated for the Poisson’s ratios, respectively, -0.2 and 0. A compromise near these values, on the auxetic domain, is convenient for practical applications [55]. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 14 Figure 1.13. Solitary wave: (a) amplitude and (b) velocity dependence on the Poisson’s ratio (adapted from [55]). In terms of lattices with auxetic effect, it has been proven that this kind of structures possess a direction angle (ф) dependent band gap at a wide variety of frequencies (ω) (Fig. 1.14 (a)). Band gaps are not observed in regular honeycomb lattices (Fig. 1.14 (b)), being useful in practical applications [56]. Figure 1.14. Representation of band-gaps in (a) auxetic and (b) “normal” materials (adapted from [56]). Sandwich panels with auxetic core have also been studied and investigated, being suggested that they may generate enhanced transmission loss, especially at high frequencies [57]. Additionally, it has been shown that auxetic fillings promotes changes of smaller root mean square wave velocity propagation (Fig.1.15) [58]. Figure 1.15. Root mean velocity propagation dependence on the Poisson’s ratio (adapted from [58]). Chapter 1 Introduction 15 Considering the development and study of auxetic materials with enhanced vibration and noise damping, a fundamental evolution on this matter is presented in Table 1.5. Due to their attractive characteristics, such materials seem to be an interesting opportunity as base material for a novel generation of railway vehicles. There has been a recent effort to use such materials in practical applications, however, this task has been proven harder than it may initially seem. The main reason is that, although auxetic isotropic materials have been proved theoretically, there seems to be a lack of natural auxetics. Table 1.5. Brief progression of published works concerning the vibration/acoustic propagation in auxetics. Year Lead Author Ref Description 1988 A. Lipsett [54] Study concerning negative Poisson’s ratios in dynamic elasticity; 1989 C.P. Chen [59] Experimental analysis of the dynamic behavior of auxetic polyester foams; 1991 B. Howell [60] Report on the acoustic damping of auxetic polyurethane foams; 1994 [61] Study concerning the sound absorption of auxetic polyurethane foams; 2000 F. Scarpa [57] Influence of cell geometry on the Poisson’s ratio and vibration properties of sandwich plates; 2003 M. Ruzzene [56] Existence of band-gaps in periodic auxetic lattices; 2010 P. Kolat [55] Propagation velocity and amplitude of solitary waves in auxetic plates; 2013 W.I. Azoti [62] Natural frequencies and loss factor in sandwich composites with auxetic layers; 2014 F. Agnese [63] Use of star shaped inclusions in turbine blades to enhance structural damping; 2014 T.C. Lim [64] Elastic wave propagation velocity in auxetic solids; 2015 T. Strek [58] Effect of auxetic structure and filler in the dynamic behavior of auxetic sandwich panels. 2017 N.D. Duc [65] Analytical deduction of vibration response on sandwich panels with auxetic cores. 2017 X. Chen [66] Modeling of the dynamic behavior on thin layered plates including auxetic layers. 2018 L. Ma [67] Increase in stiffness and damping capacity in auxetic corrugated sandwich panels. 2018 M. Mazloomi [68] Auxetic 2D lattices reduce the radiated sound power level. 2018 H. Reda [69] Analysis of acoustic behavior and bandgaps in pre-deformed auxetic scaffolds. 2019 D. Qi [70] Acoustic bandgaps in novel hybrid anti-chiral auxetic structures. The first reference (1882) of experimental evidence of this behavior may be found in the works of Woldemar Voigt when testing Iron Pyrite monocrystals [71]. Historically, it has been shown that naturally occurring auxetic behavior (Table 1.6) is frequently associate to materials and systems with extreme anisotropy. In this last condition, it has been shown that the Poisson’s ratio may assume extreme values and are not constrained by the values deduced for isotropic conditions [72]. Counterintuitively to most common assumptions, it has also been shown that roughly 70% of cubic elemental metals possess auxetic behavior, at least, in one preponderant direction [73]. A summary of relevant naturally occurring auxetic materials is presented in Table 1.6. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 16 Table 1.6 Naturally occurring auxetic materials. Year Ref. Poisson’s ratio Material 1882 [71] ν < 0.000 (C12 = - 4.740) Iron Pyrite Monocrystals 1892 [74] ν = - 0.143 1945 [75] ν < 0.000 (C12 = -4.400) 1947 [76] ν < 0 (-5.290 < C12 < - 4.400) 1955 [77] ν < 0.000 (C12 = -4.640) 1969 [78] - 0.100 < ν < - 0.094 Low porosity dry granite 1970 [79] ν < 0.000 Cat Skin* 1973 [80] - 0.266 < ν < - 0.222 Sandstone** 1974 [81] ν < 0.000 Unsaturated Clay 1976 [82] ν = - 0.300 Porous Rocks 1981 [83] ν = - 0.073 α - Quartz 1982 [84] ν = - 0.070 Cancellous Bone 1992 [85] ν = - 0.175 α - Cristobalite [86] - 0.133 < ν < - 0.191 1995 [87] - 0.025 < ν < - 0.053 Carbon Nitride 1999 [88] ν = - 0.075 Monocrystalline Zinc (Basal Plane) 2014 [89] ν = - 3.000 Copy Paper*** ν = - 0.300 Paperboard*** Cotton Paper*** 2015 [90] ν = - 1.400 Human Achilles Tendons 2018 [91] ν = - 0.74 Liquid Crystal Elastomers * Volume increase in tension. ** Calculated from original data. *** Thickness (ou-of-plane) direction. Given the absence of naturally occurring isotropic auxetics and the attractive properties that such materials could offer in practical applications, there has been a substantial effort to produce artificial auxetics. The most simple and common example (1985) is the design of mechanisms [92] (Fig. 1.16 (a)) and structures with protruding ribs [93] (Fig. 1.16 (b)), implying that such systems will expand in tension (Fig. 1.16 (c) [94]). Interestingly, there is an evidence of early report [95] (1966) of deforming nonprotruding into protruding ribs by plastic deformation (Fig. 1.16 (d)), although there is no evidence of using that effect to obtain auxetic behavior. Chapter 1 Introduction 17 Figure 1.16. Reentrant based auxetic (a) mechanism [92]; (b) structure [93]; and (c) expansion [94]. (d) Possible unidentified reentrant structure [95]. In 1987, the most common three-dimensional isotropic auxetic micromechanical design has been idealized, being shaped in the form of a reentrant tetrakaidecahedron (Fig. 1.17 (a)) [29]. The presented model has been used to propose the manufacturing of bulk polymer auxetics based on the molecular network of (1,4)-reflexyne (Fig. 1.17 (b)) [36]. Figure 1.17. Auxetics: (a) Reentrant tetrakaidecahedron model [29] and (b)molecular network of (1,4)- reflexyne [36]. In terms of geometric dependence of the Poisson’s ratio presented by these structures, reentrant models (Fig. 1.18) based on rib flexure may be described as a function of unit cell angle (θ), vertical length (l) and horizontal length (h) as presented by Eqs. 1.14 and 1.15 (Table 1.7) [10]. A more detailed deformation behavior may be achieved considering the unit cell thickness (t), depth (b), base material Young’s (E) and Shear (G) moduli (Table 1.7). These constants are able to further describe rib flexure Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 18 (Kf– Eq. 1.18), stretching (Ks – Eq. 1.19) and hinging (Kh – Eq. 1.20) stiffness, using Eqs. 1.16 and 1.17 [96]. In the case of thick walled cellular lattices the Poisson’s ratio may be described by Eq. 1.21 [97]. Figure 1.18 Two-dimensional reentrant model [96]. Table 1.7 Poisson’s ratio of reentrant structures. Ref. Lead Author Poisson’s ratio [10] L. Gibson 𝜈𝑥𝑦=sin(𝜃)[ℎ+𝑙sin(𝜃)] 𝑙cos2(𝜃) (1.14) 𝜈𝑦𝑥=𝑙cos2(𝜃) sin(𝜃)[ℎ+𝑙sin(𝜃)] (1.15) [96] I.G. Masters 𝜈𝑥𝑦=−sin(𝜃)(ℎ𝑙 ⁄+sin(𝜃)) [ 1 𝐾𝑠−1 𝐾𝑓−1 𝐾ℎ cos2(𝜃) 𝐾𝑓+cos2(𝜃) 𝐾ℎ+2ℎ𝑙 ⁄+sin2(𝜃) 𝐾𝑠 ] (1.16) 𝜈𝑦𝑥=−sin(𝜃)cos(𝜃)[1 𝐾𝑓+1 𝐾ℎ−1 𝐾𝑠] (ℎ𝑙 ⁄+sin(𝜃))[sin2(𝜃) 𝐾𝑓cos(𝜃)+sin2(𝜃) 𝐾ℎcos(𝜃)+cos(𝜃) 𝐾𝑠] (1.17) 𝐾𝑓=𝐸𝑏𝑡3 𝑙3 (1.18) 𝐾𝑠=𝐸𝑏𝑡 𝑙 (1.19) 𝐾ℎ=𝐺𝑏𝑡 𝑙 (1.20) [97] Malek 𝜈= cos(𝜃) (ℎ𝑙+sin(𝜃))sin2(𝜃)[1+(1.4+1.5𝜈)(𝑡𝑙)2 1+(2.4+1.5𝜈+cot2(𝜃))(𝑡𝑙)2] (1.21) Reentrant structures have been analyzed and developed into variations, e.g. star-shaped [98] and arrowheads [99]. A fundamental reference to the evolution of this research may be found in Table 1.8. Chapter 1 Introduction 19 Table 1.8. Published works on reentrant auxetic structures. Year Lead Author Ref Description 1985 A. Kolpakov [93] Fine-celled structures with protruding ribs shown negative Poisson’s ratios; R.F. Almgren [92] Reentrant structure composed by rods, hinges and springs has a Poisson’s ratio = -1; 1990 T.L. Warren [100] Linear elastic Poisson’s ratio of reentrant structures in dependent on the reticulation angle; 1991 K.E. Evans [36] (1,4)-reflexyne groups are presented as a possibility to obtain molecular scale auxetics; 1997 P. Theocaris [101] Reentrant (star-shaped) lattices shown an initial rib-angle dependent auxetic behavior; 2004 H. Wan [102] Poisson’s ratio depends on cell geometry and is not constant in large deformations; 2005 J.N. Grima [98] Connected star-shapes are analyzed by the inclusion of non-convex structures; 2009 C. Lira [103] Influence of geometry and material in the Poisson’s ratio of multi reentrant structures; 2011 W. Miller [104] Study on the out-of-plane buckling strength in reentrant honeycombs; 2012 P. Soman [105] Combination of reentrant and non-reentrant honeycombs to obtain variable Poisson’s ratios; 2013 Y. Sun [106] Stiffness of hierarchical honeycombs (ribs composed by auxetic structures); L. Yang [107] Role of geometry and material in the Poisson’s ratio of 3D reentrant structures; 2014 M.S. Rad [108] Theoretical and finite element analysis of 3D reentrant structures; 2015 A. Spagnoli [99] Analysis of the auxetic behavior of arrowhead reentrant structures; 2016 Z.X. Lu [109] Stiff variation of auxetic reentrant structure with connected protruding ribs; R. Brighenti [110] Analysis of the non-linear behavior of arrowhead reentrant structures; T.C. Lim [111] Geometrical analysis of novel double-arrowhead reentrant auxetic structures; X. Hou [112] Study concerning the dynamic crushing behavior of reentrant honeycombs; 2017 M.F. Fu [113] Combination of rhombic and reentrant structures to elevate stiffness and buckling strength. 2018 V.H. Carneiro [114] Minimum number of cells in 2D auxetic lattices to have bulk mechanical properties. 2019 D. Xiao [115] Dynamic compression of 2D auxetic reentrant lattices. [116] Experimental and FEA compression of 2D auxetic lattices. Z. Dong [117] Auxetic behavior from reentrant models is generated by the expansive/contractive effect of protruding ribs in tension/compression. This effect, however, has been also shown to be present in diamond shape unit cell structures (Fig. 1.19 (a)) with absent ribs. This model is known as the missing rib model (Fig. 1.19 (b)) [118]. Their value of Poisson’s ratio is described by Eq. 1.22. The deformation behavior of the missing rib model, is geometrically similar to chiral structures (Fig. 1.20). The latter are characterized by their lack of symmetric reflection and by rotating their ribs around a central node, which generates a rotational effect that permit their auxetic behavior (Fig.1.19 (a)). Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 26 In terms of transposing the referred models to specific applications, there have been developed manufacturing procedures to produce “artificial” auxetic materials. The most common routine is the conversion of positive Poisson’s ratio to auxetic foams (Fig. 1.26) by the transformation of regular ribs (Fig. 1.28 – microscopies on the left [175]) into protruding ribs [176] (Fig. 1.28 – microscopies in the right [175]). This procedure is based on a thermo/chemicomechanical process and applicable to both polymer and metal based foams. Fig. 1.29 shows the basic procedures to transform regular foams into auxetic. Figure 1.28. Transformation of “normal” into auxetic foam (microscopies adapted from [175]). Figure 1.29 Basic foam transformation procedure. When processing metallic foams to convert them into auxetics, the overall process is fairly simple: it implies a triaxial compression, where the regular tetrakaidecahedron cells are deformed in the plastic Chapter 1 Introduction 27 regime to acquire a protruding shape [177,178]. In the case of polymeric foams, this process is more complex. The conversion mechanism is based in shape memory characteristics. Additionally to the triaxial compression, a chemical route (by immersion in acetone) [176][145] or a thermic route [175,179,180] (exposing the samples at temperature higher than the base material glass transition temperature [181]) have to be used. These exposition times and temperatures influence directly the success of the conversion and the value of the Poisson’s ratio of converted foams [176]. Additionally, it is currently suggested the inclusion of Styrene-Acrylonitrile (SAN) copolymer particle to improve shape memory characteristics [182,183]. Due to the base foam manufacturing process, cellular foam structures are randomly generated by packed nearly polyhedral cells [184]. Thus, there is a current trend to produce non-stochastic cellular structures with a negative Poisson’s ratio [185]. The latter have the advantage of possessing periodic shapes with uniform properties that may be tunable for specific applications by tailoring the foam structure. Such structures may be produced by additive manufacturing techniques, such as 3D polymer printing (Fig. 1.30 (a)) [186] or Selective Electron Beam Melting (SEBM) (Fig. 1.30 (b)) [187]. Figure 1.30. Three-dimensional reentrant: (a) polymeric – 3D printed [186]; and (b) metallic – SEBM [187]. Polymeric non-stochastic 3D printed structures have the advantage of being easily produced and fairly economic. However, for the application that this study is aimed and for structures in general, it presents low relative mechanical strength. As for the metallic scaffolds that may be produced by SEBM, they present higher mechanical strength. However, they are characterized by a high cost and low rate of production, difficulty in microstructure control and, generally, usually use base materials ( e.g. Ti6Al4V and 316L Steel alloys [188]) that are not lightweight. Another technique currently being used and developed is related with the manufacturing of nonstochastic scaffolds by investment casting (Fig. 1.31) [189,190]. This method has some inherent Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 28 advantages, such as: good microstructural control, relative ease in processing and the possibility to easily use lightweight alloys. Figure 1.31. Casting of Reentrant structures [190]. A current trend on auxetic material is the use of foam filling of metal structures [191] to enhance mechanical properties (Fig.1.32 [192]) such as toughness and crashworthiness [193]. Even though, most cases present a positive Poisson’s ratio structure filled with an auxetic foam, it is suggested in this study that using a negative Poisson’s ratio structures filled with expanded foam may be interesting to explore due to the stiffer auxetic phase. Figure 1.32 Tube filled with (a) “normal”; and (b) auxetic foam [192]. Interpreting the relevant studies performed by researchers in the last years, it may be observed that there is an interesting opportunity to investigate the benefits of using auxetic scaffolding structures filled with polymeric foam for lightweight structural applications with enhanced damping capacity. Chapter 1 Introduction 29 1.3. Thesis organization and research questions. Considering the previous sections, it is established that there is a trend in the design of cellular solids for applications in the transportation industry. Their successful integration in the fabrication of railway vehicles is dependent on the current developments of these materials and their ability to display attractive characteristics for this industry. The referred trends on the evolution of materials for transportation industries, particularly in railway vehicles, imposes a question: What should be the design and manufacturing processes of cellular solids for the fabrication of lightweight vehicles with high structural vibration/noise damping without compromising structural strength? This is defined as the broad research question of this study. It is able to define a boundary of current knowledge, raising awareness of contemporary trends in the scientific community and technological advances. Additionally, its answer would have a potential interest to society. Given the evidences that are presented in section 1.2, the author suggests the following hypothesis: the design of non-stochastic cellular solids with auxetic behavior ( i.e. negative Poisson’s ratio) manufactured by investment casting. If successful, the hypothesis would fulfill the objective: to design a lightweight material with high specific structural strength and enhanced vibration/noise damping capacity for future railway vehicle fabrication. Accordingly, each individual chapter is presented in a detailed research paper format and describes the steps that have been taken to approach the main research question: - Chapter 2 is dedicated to the selection and optimization of the base material that comprises the solid phase of the proposed cellular solids suggested in Chapter 1. Such selection and optimization process are performed to optimize the static and dynamic mechanical properties of the base material. - Chapter 3 details the design of the lightweight cellular solids and the development of a route for their manufacturing. The parameters of the suggested manufacturing route are optimized to allow the successful fabrication of the design with the base material selected in Chapter 2. Manufactured samples are analyzed in terms of defects and dimensions to correlate their final shape/geometry with the initial design. - Chapter 4 shows the static mechanical testing and characterization of the samples that have been designed and manufactured according to Chapter 3. Their static mechanical properties are compared with the specific properties of other materials to estimate if the designed cellular solids are a viable Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 30 solution and suitable to the load bearing applications intended in Chapter 1. Their failing mechanisms are also presented by in-situ experimental observation techniques and further detailed by FEA. - Chapter 5 is devoted to the vibration testing of the samples that have been designed and manufactured according to Chapter 3. The proposed designs are investigated in terms of damping capacity by the calculation of their damping ratios. The experimental results and the sample harmonic behavior are further detailed by FEA. Damping capacity is compared with other current materials to understand if the dynamic properties are interesting for the application defined in Chapter 1. - Chapter 6 combines the results of Chapters 4 and 5. The static and dynamic structural behaviors of the samples are compared to identify the design with the best combination of load bearing and vibration damping capacity. The selected design is posteriorly impregnated with polymeric foams to determine the acoustic properties of the cellular solids. Their acoustic reflection and transmission are compared with current solutions to estimate if the designed composites are an interesting solution for the application defined in Chapter 1. - Chapter 7 presents the overall conclusions of the study. The initial research question (Chapter 1) is compared with the results and discussion of the previous chapters (2 to 6) and, therefore, determine if the adopted hypothesis is able to fulfill the established objective of this study. Considerations on future works and scientific impacts are also suggested. Chapter references [1] P. Vos, Railway Noise in Europe, International Union of Railways, Paris, 2016. [2] V.P. 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Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 42 Figure 2.1 Selection of material and process (adapted from [1]). Considering the different classes of materials, it is known they have inherent contrasting properties. For instance, the high damping properties ( e.g. loss factor) and low density that characterize polymeric materials may seem a solution for the proposed problem. However, it is also known that in unfilled states, such materials possess lower mechanical strength than metallic and ceramic materials [2]. There is a need to see their fundamental mechanical properties and if they should be maximized or minimized. According to Table 2.1, it was defined that the base material that composes the lightweight cellular solid must possess high damping, yield strength and stiffness ( i.e. Young’s modulus). Table 2.1 Identification of relevant mechanical properties. Dimensionless Dimensional Property Damping Capacity Specific Density Yield Strength Young’s Modulus Objective Elevate Lower Elevate Elevate Relation Damping Capacity / Specific Density Yield Strength x Young’s Modulus Axis (Fig. 2.2) Vertical Horizontal To facilitate this decision process, the density values were evaluated in terms of specific density. This allows the sub-division of the four magnitudes into two distinct classes: dimensionless (damping capacity and specific density) and dimensional (Yield strength and Young’s modulus). According to Table 2.1, the damping capacity was defined as being divided by those of specific density, as these results Chapter 2 Alloy Selection and Processing: Optimization for Static and Dynamic Mechanical Properties 43 would generate the most beneficial combination of these dimensionless magnitudes for the proposed work. A similar methodology was used for the dimensional magnitudes (Table 2.1), however, a multiplication operation was used to correlate the values of Yield strength and Young’s modulus due to the intent of maximizing both. It is worth mentioning there is no ponderation between the magnitudes. Considering this method, and using bibliography concerning the mechanical properties of the different classes of materials (compiling the data from [3–28]), it is possible to plot the Ashby diagram in Fig. 2.2. Figure 2.2 Ashby diagram comparing properties of the different material classes (compilation of data from [3–28]). According to Fig.2.2 it may be visualized that the best compromise, in a bulk material to be used for the established purposes (see section 1.3 – Chapter 1) must be the one who displays the coordinates that have maximum values on both axis. Therefore, it may be inferred that the best compromise in terms of bulk materials are polymer and metal based. In fact, it may be stated that there is even a significant interception between their domains. Thus, a constraint must be imposed to determine a final selection between the two materials classes [29]. It may be seen that the Young’s modulus and Yield strength are already characteristics of the material in energetic terms and may be roughly used to describe strain energy states ( e.g. resilience) Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 44 [30]. After some thought, it may be seen that in physical terms, the system also has to be characterized in terms of mass. Since this is directly related to the density values that were already analyzed, an estimation may be drawn by the proportionality to the volume of the system. Comparing the latter, when considering metallic and polymeric materials, it may be observed that metals generally displays higher values of Yield strength (σ Y ). Thus, according to Fig. 2.3, it may be concluded that for a constant load value (Fmax) a polymeric material would need a larger area ( i.e. higher volume for constant thickness) to prevent yielding. The ability to support a given load in a lower volume of material is an important feature in most mechanical applications, and thus, this motivates the selection of a metallic material. Figure 2.3 Metals vs Polymers: material volume that would be necessary to support the same maximum load (Fmax). Having selected the material class, it is fundamental to establish a route to transform the bulk material into the cellular solid final shape. According to Table 2.2, each manufacturing technique may reveal certain abilities to produce specific shapes and are not suited to others [29]. Considering that the intended geometry is mostly correlated with a 3D Hollow structure, it can be concluded that the manufacturing technique must be selected between casting, powder methods, electro-machining or additive manufacturing. An important feature of the developed composites is the minimum section thickness that is possible to be manufactured by the referred processes. Such approach helps the reduction of the overall composite relative density and, consequently, contribute to its lightweight characteristics. Fig. 2.4 shows the generic range of section thickness for the different manufacturing processes [29] that are able to produce 3D Hollow shapes (according to Table 2.2). Chapter 2 Alloy Selection and Processing: Optimization for Static and Dynamic Mechanical Properties 45 Table 2.2 Shapes for different metal manufacturing processes (adapted from [29]). Circular Prismatic Non-circular Prismatic Flat Sheet Dished Sheet 3D Solid 3D Hollow Sand Casting     Die Casting     Investment Casting     Low Pressure Casting     Forging    Extrusion   Sheet Forming     Powder Methods     Electro-Machining      Conventional Machining      Additive Manufacturing*       * Added by the author given the current development of this technique. Figure 2.4 Section thickness for different manufacturing processes (adapted from [29]). Fig. 2.4 shows that the minimum values of thickness may be obtained by Electro Machining (EM). However, due to its working principles, it may be easily seen that the complex scaffolding of the composite matrix (Fig. 2.1) is not possible to be manufactured by this technique. Even though, some authors discuss Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 46 the obtainment complex structures by this technique, these possess an opening in one of the Cartesian planes that is not allowable in this scaffold design [31,32]. Fig.2.5 shows a possible route to obtain an approximate design by EM, in which: the bulk material (Fig.2.5 (a)) is subjected to successive wire cuts (Fig. 2.5 (b)) and drilling (Fig. 2.5 (c)) procedures to manufacture honeycombs (Fig. 2.5 (d)). This may be applied to different Cartesian planes to achieve a quasi-open cell three-dimensional configuration (Fig. 2.5 (e)). It may be observed that, due to the design constraint of not machining one plane, the geometry is not fully open and, thus, platforms are generated in the basis of each cell. In conclusion, the intended design is not fully replicated by the use of EM. Figure 2.5 Suggested route to manufacture the proposed design by EM: (a) Bulk materials; (b-d) Wire cutting and drilling; (e) Transition for other planes. Additionally, due to the complexity of the machining routines and the fact that each individual cell must be first drilled for the EM process (Fig.2.5), the overall process is a rather costly and extremely inefficient. Additionally, by exploring different manufacturing techniques, the author was able to produce thin sections (≈ 0.3x0.3 mm2) by vacuum investment casting. Thus, it was concluded that investment casting is the most suitable process, being selected to produce the intended cellular solids. In fact, investment casting is a well-known technique to produce complex components with intricate and netshaped details [33]. Once the process is selected, a new iteration concerning the selection of base material may be performed. Such process is handled on a much more specific approach: (i) given that metallic materials and investment casting were selected, it is necessary to define a specific alloy. (ii) This selection process Chapter 2 Alloy Selection and Processing: Optimization for Static and Dynamic Mechanical Properties 47 was performed by the analysis of the fundamental bulk mechanical and process parameters that are the focus of this study: Yield strength (σ Y ), density (ρ), damping ratio ( ξ ) and castability ( Ci ). While, yield strength, density and damping ratio, are related to the ability of the final cellular solid to display high strength, lightweight and damping characteristics, the role of their castability is more complex. Castability is able to directly influence the success to obtain the complex scaffolds by investment casting. Moreover, castability may be defined as “the ability of an alloy to form a dimensionally accurate castings of acceptable soundness and integrity” [34]. I.e. the success of the alloy in to be casted with good quality in geometric and metallurgical terms. Considering these prepositions, a bibliographical search (from the compilation of the data presented in references [3,4,6,10–12,15,17–27,35–37]) was performed to determine the average values of the referred properties for the most common alloys that may be used in investment casting (Fig. 2.6). This average values and the correspondent standard deviations are fundamental tools that allow a direct comparison between the alloy properties. The values of castability were taken directly from the published data, however given that they were presented in different scales, the published values were converted in to a percentage (%) scale. Figure 2.6 Comparison between some castable alloy properties: (a) Yield strength (σ Y ); (b) Damping ratio ( ξ ); (c) Density ( ρ ); and (d) Castability ( Ci ). Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 48 Observing Fig.2.6, it may be seen that each individual alloy presents elevated characteristics on certain properties and a relatively poor performance on others. It is shown that, for instance, cast iron shows elevated values of yield strength and damping, however it has a high density. Whereas, Mg alloys have a low density and high damping, but have a low relative values of castability. To overcome the difficulty of this selection process, a protocol was established to determine the alloy that is most suitable for the referred application and process. Considering that each property has the same importance ( i.e. no weighting criterion was used), a Material Selection index (MSi) was established, according to Eq.2.1. Using this approach, the values that are intended to be enhanced were compared with the maximum values of each variable. For example, the individual values of yield strength (σ y ) compared with the maximum observed yield value (σ ymax ) by their ratio (σ y / σ yma x). As for the density, the value for each alloy is compared with the minimum value to generate a relative comparison between them. The result is a dimensionless value that represents the overall role of the selected properties in the alloy performance for this particular study. 𝑀𝑆𝑖=𝜎𝑦 𝜎𝑦𝑚𝑎𝑥 ·𝜉 𝜉𝑚𝑎𝑥 ·𝜌𝑚𝑖𝑛 𝜌·𝐶𝑖 𝐶𝑖𝑚𝑎𝑥 (2.1) The results of this approach are shown in Fig. 2.7, in which it is determined that the metals that display the best compromise for the proposed application are Mgand Al-based alloys. Comparing the results, it is shown that these light alloys have an overall favorable combination of properties relatively to Cu alloys and Cast Iron. Figure 2.7 Material selection index (MSi) according to the established criteria. Chapter 2 Alloy Selection and Processing: Optimization for Static and Dynamic Mechanical Properties 49 Given the overall interpretation of Figs. 2.6 and 2.7, it is apparent that Mg alloys are the most appropriate solution to be used in this project. However, it is known that the use of Mg alloys in investment casting has inherent problems that may compromise a successful casting [33,38–41]. Fig. 2.8 shows an example of the reactions that occur between Mg alloy melts and mold interfaces (Table 2.3), in which it may be observed a bad surface finishing as a consequence of a violent exothermal reaction [33]. Additionally, this reaction is known to difficult the melt flow and reduce metal fluidity [30]. Given the role that these factors have in the overall success of the manufacturing process, it is concluded that Mg alloys are not applicable to this study. Thus, Al alloys are selected to manufacture the metal matrix scaffold that composes the intended cellular solid. Figure 2.8 Optical Microscopy of sectioned A356 (Al alloy) and AZ91D (Mg alloy) investment casting samples. Table 2.3 Mold-metal interface reactions in the investment casting of magnesium alloys [40,41]. Reaction Description 1 2Mg (l) + SiO2 (s) → 2MgO (s) + Si (s) 2 4Mg (l) + SiO2 (s) → 2MgO (s) + Mg2Si (s) 3 MgO (s) + SiO2 (s) → MgSiO3 (s) 4 2MgO (s) + SiO2 (s) → MgSiO4 (s) Fig. 2.9 represents the most common binary and ternary aluminum alloys [10]. It may be observed that the combination of alloying elements has a predominant role in the processing of aluminum alloys and it is known these alloyed with silicon (Al-Si) are preferable for casting. However, it would be interesting, for material tailoring purposes to allow that material to be hardened and change their strength/damping, i.e. allow ageor work-hardening to be performed. Given that the thin rib configuration that comprises the metallic cellular solid, it may be easily understood that it would be impossible to Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 50 perform work-hardening. Additionally, elements like copper (Cu) and magnesium (Mg) allow the heat treatment of aluminum alloys [42] and are also present in casting alloys such as Al-Si-Mg and Al-Si-Cu Thus, they are able to be both casted and heat-treated. Finally, Al-Si-Mg alloys are selected to perform this study and produce the metal matrix of the proposed composite due to their common use in investment casting processes. Figure 2.9 Principal aluminum alloys and processes (adapted from [10]). AlSi7Mg0.3, typically referred as A356, is a classic casting hypoeutectic aluminum alloy. It is usually employed in transportation industries [43] ( e.g. automotive, railway and aerospace) for its capability to be heat treated, casted in ceramic block to produce thin walled/ complex geometries [44] and overall specific mechanical properties. However, in its as-cast condition, this alloy is characterized by poor mechanical properties ( e.g. low yield/UTS strength and extension) and, therefore, must be heat treated [45] (generally by T6, i.e. solution and quenching followed by artificial ageing). As this alloy comprises a good part of the overall casted products in bulk [45–49] or composite [50–53] forms, there is a continuous research dedicated to its development and characterization. There has been significant research concerning the A356 α-Al primary dendrites [43,44,54,55], eutectic Si [56–59], main secondary phases (β - Al5Fe2Si, and π - Al8FeMg3Si6) [60–63] and heat treatments, such as solution [59,64,65] and ageing [66–70]. Additionally, there are numerous studies concerning the effects of melt treatment by chemical or physical processes at a microstructural level. Most chemical processes are based on common grain refinement using Titanium Diboride (TiB2) and modification with Strontium (Sr) [71–73] master alloys, or the more expensive combined treatment by Scandium (Sc) [74,75]. Chapter 2 Alloy Selection and Processing: Optimization for Static and Dynamic Mechanical Properties 51 Recently, physical processes based on ultrasonic vibration have been employed to promote the referred refinement and modification of the matrix morphology [46,76,77]. Several studies have also been published to characterize the influence of the referred processing parameters in the overall mechanical properties of this alloy [78–80]. Fundamentally, there is a strong emphasis on its load bearing capability under quasi-static loading, either by simple hardness testing [75,81,82] of the casted alloy and/or posterior tensile testing [83–85]. Recent studies show a concern on the damping properties (mainly using dynamic mechanical analysis) of this alloy [43,86]. Given the importance of this characteristic in the reduction of noise/vibration and its implication in practical problems, such as high resonance response, vibration fatigue and comfort issues on the industrial application of A356 alloys, there seems to still be a lack of data on this matter. Even though, it is also known that there is a general inverse proportionality between static and dynamic ( e.g. vibration damping) mechanical properties [43,86] this correlation has not been directly studied in this alloy. From a practical designer point-of-view, this chapter intends to relate the dependence of static and damping properties of investment casted A356 alloys and the ageing heat treatment as a route to tailor specific damping ratios and/or yield strengths. The results are used to determine the processing parameters to be used in the design of products that fulfill the properties imposed by the ISO 3522 and ASTM B686 standards. 2.2 Methods for alloy processing optimization 2.2.1. Al alloy processing A356 alloy study was received in the form of 12.5 kg ingots with the chemical composition shown in Table 2.4. Melting of 0.3 kg of alloy was carried out in an electrical resistance furnace and maintained in a SiC crucible at 720°C during 30 min for homogenization. Master alloys of grain refiner (0.2%wt – Al5Ti1B) and eutectic Si modifier (0.3%wt - Al10Sr) were added and the melt was kept isothermic for 20 min, being stirred every 5 min to prevent particle sedimentation (a total of 5 stirring routines). It is known that the addition of eutectic Si modifier generates an increase in porosity due to the decrease in superficial tension [87,88]. Thus, the regular degassing process was inverted being performed after the addition of the master alloy in the melt. The alloy cooled down being degassed for 5 min at 700 ± 5°C with Argon, until a pouring temperature of 700 ± 3°C was reached. The melt was poured into pre-heated (250 ± 2°C) ceramic blocks with six rectangular openings (25×4×140 mm), corresponding to the primitive Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 58 Figure 2.16 Details of π - β and Mg2Si transformation during solution treatment: (a) as-cast, (b) 2-h, (c) 4-h, and (d) 8-h solution at 540 °C. Figure 2.17 Hardness of the samples for different heat treatments [75,81,82]. Chapter 2 Alloy Selection and Processing: Optimization for Static and Dynamic Mechanical Properties 59 During the ageing treatment, the supersaturated α-Al (αSS) has its solute Mg and Si atoms clustered and precipitated (Eq. (2.5)) in GuinierPreston ( GP – Mg2Si3Al6) zones along the < 100 > direction. These primordial precipitates decompose into nano-sized coherent Widmanstätten needleshaped β ’’ (Mg5Si6) and semi-coherent rod-like β ’ (Mg1.8Si) metastable strengthening phases [89]. Such precipitates are nucleated in the < 100 > direction, preferably in dislocation locations. Finally, a dissolved rod/plate-shape equilibrium β(Mg2Si) FCC phase is precipitated due to minor atomic rearrangement in the matrix [70]. 𝛼𝑆𝑆 → 𝐺𝑃 𝑧𝑜𝑛𝑒𝑠 → 𝛽′ → 𝛽′′ → 𝛽(𝑀𝑔2𝑆𝑖) (2.5) The initial stages of artificial ageing are characterized by low hardness values (Fig 2.17). However, according to Fig. 2.18 and due to the clustering of solute atoms in GP zones and the precipitation of β ’’ , the overall initial nucleation and growth of shearable precipitates promotes an elevation in hardness [67]. When peak-age is reached, the nucleation and growth of β ’’ is superseded by the precipitation and coarsening of non-shearable β ’ and β(Mg2Si) [93] (Fig.2.18). The dislocation mechanism is changed to an Orowan mechanism and further ageing will imply a reduction in hardness. Figure 2.18 Generic progression of hardness and precipitation during artificial ageing. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 60 The increasing hardness in Fig. 2.17 suggests that the methodology was able to avoid a dislocation mechanism transition. Experimental results propose that before dislocation transition ( i.e. Peak Ageing), the relation between hardness (H) and ageing time (At) follows the linear regression (Fig. 2.17) described in Eq. (2.6). Additionally, it may be observed that the experimental hardness values are highly relatable with other studies [75,81,82], suggesting a successful processing of the samples. 𝐻 =72.2+0.04𝐴𝑡 (2.6) 2.3.2. Dynamic mechanical properties: Vibration analysis Fig.2.19 shows an example of the resulting FRF plots for the different test conditions. It may be observed that the performed heat-treatment is able to successfully change the overall stiffness and damping of the samples. These results, particularly the resonance peaks, were used to estimate the damping ratio of the samples by the half-power bandwidth method. Figure 2.19 Example of FRF curves obtained for the tested specimens. Fig. 2.20 shows the damping ratios (ζ) that were determined during the vibration testing. It is apparent that the solution treatment and the initial phase of artificial ageing are able to enhance the damping capacity of the A356 alloy relatively to its as-cast state. However, as the ageing treatment progresses, there is a decrease in damping capacity. The experimental results suggest that there is a Chapter 2 Alloy Selection and Processing: Optimization for Static and Dynamic Mechanical Properties 61 linear function that describes the lowering of the damping ratio (ζ) as the ageing time (At) progresses, according to Eq. (2.7). 𝜉 =0.0042−3×10−6𝐴𝑡 (2.7) Figure 2.20 Damping ratios for different heat treatments. Damping is the ability of a material to dissipate energy under cyclic loading, being closely related with the microstructural internal friction and changed by the material processing. The damping mechanisms are known to be originated by thermoelastic, magnetic, viscous and defect effects. However, due to the presented particular processing characteristics and isothermal (room temperature) testing, only the latter is relevant as an internal friction mechanism. The defect damping is closely related to point (vacancies), line (dislocations), surface (grain boundaries) and bulk (micropores and inclusions) defects [43,94]. Nevertheless, it is known that the internal friction in this particular alloy is regulated by a static hysteretic damping [26] due to dislocation and grain boundary damping [43,94]. Hence, it is suggested that the particular origin of the predominant damping mechanisms are related with the dislocation and pinning in (Figs. 2.21 and 2.22): (i) grain boundaries between α-Al/α-Al, α-Al/eutectic Si, α-Al/β-Al5Fe2Si and α-Al/ π - Al8FeMg3Si6 pairs; (ii) interactions between dissolved Mg, Si and α-Al in pre-aged states (solution hardening) and α-Al/{β’’, β’, β(Mg2Si)} in aged states (precipitation hardening). Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 62 Figure 2.21 Damping changes due to microstructural transformation during solution treatment. Figure 2.22 Damping changes due to microstructural transformation during ageing treatment. Fig. 2.21 represents a schematic evolution of the microstructure during solution treatment. Initially, the fundamental constituents of the microstructure are α-Al grains (with small quantities of dissolved Mg and Si), eutectic Si and secondary phases (needle-shaped β-Al5Fe2Si, Mg2Si and scriptshaped π -Al8FeMg3Si6 – Figs. 2.12 and 2.16). During the solution treatment, the eutectic Si is transformed to a spheroidized morphology, migrating and changing the interface balance. It is know that Si migration from the dendrite arms results in the formation of particle-free zones [89] and changes the eutectic Si shape/size. This implies a decrease in the grain boundary pinning effect and an elevation in internal friction due to boundary sliding [86,94]. Schoek [95,96] showed that a spheroid inclusion with a radius (ai) in a matrix of volume (V) with a determined Poisson's ratio (ν), generates an inverse quality factor (Q−1) by interface relaxation and anelastic strain when a shear stress ( p13 ) in its respective component (𝑝 13 ) is applied (Eq. 2.8). Thus, it is suggested that an overall increase in eutectic Si diameter (Fig. 2.15) enhances the interface damping. Chapter 2 Alloy Selection and Processing: Optimization for Static and Dynamic Mechanical Properties 63 𝑄−1 =1 𝑝1328(1−𝜈) 3𝜋(2−𝜈)1 𝑉∑𝑎13(𝑝13     2)𝑖 (2.8) During solution, however, there is also dissolution of Mg2Si in the form of Mg and Si elements in the α-Al. Additionally, there is further migration of Mg atoms due to the dissolution of the π -Al8FeMg3Si6 phase into fine needle shape β-Al5Fe2Si [91,97] (Fig. 2.16). This supersaturation of α-Al enhances pinning and reduces the overall dislocations by solution hardening [93], as it may be observed by the hardness increase in Fig. 2.17. However, due to the scattered dispersions of the atoms, there is no significant reduction in the overall solution treated system damping. Consequently, it may be determined that after solution treatment the maximum value of damping is reached and the most relevant damping mechanism is proportioned by the interface interaction. Fig. 2.22 represents the overall microstructural transformation in A356 alloys during ageing treatment. As the intermetallic compounds, eutectic Si and grain boundaries remain stable due to the relatively low artificial ageing temperatures, it is suggested that the damping is changed due to precipitation hardening. Alloy hardening until peak-age promotes the inhibition of dislocations, being this shown by the increase of hardness values in Fig. 2.17. Phase I in Fig. 2.22 represents an initial stage of ageing treatment, where the overall energy levels are not enough to cluster Mg and Si atoms and form GP zones. Studies suggest that GP zones are initially formed after 40 min of ageing treatment [98]. This hypothesis is in accordance with the experimental values of hardness (Fig. 2.17) and damping ratio (Fig. 2.20) for low ageing times. As the ageing treatment progresses, Si and Mg atoms cluster into GP zones, initiating phase II (Fig. 2.22), in which the metastable coherent β ’’ nucleate and grow in the dislocation lines with a < 100 > direction. There is an increase in pinning points and a reduction of the overall distance between them, being the dislocation mechanism governed by shearing [99]. According to the Granato-Lücke theory [100], the strain independent internal friction (δ 0 ) is proportional to the dislocation density (ρ) and the mean length between weak pinning points (Ld) (Eq. (2.9)) [101]. The independent strain amplitude approach is applied in this study due to low stresses (σMax≈30 MPa, by classic Timoshenko beam theory) generated by the hammer strike. 𝛿0~𝜌𝐿𝑑4 (2.9) Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 64 Therefore, it is suggested that the precipitation of β’’ metastable phases is able to lower the damping by the obstruction of dislocations. This hypothesis is supported by the progressive hardness increase during the ageing treatment (Fig. 2.17). As the nucleation and growth of metastable β ’’ increases its volume fraction, the precipitation hardness tends to peak-age [68]. However, some semi-coherent β ’ and equilibrium β(Mg2Si) may simultaneously precipitate from β ’’ (phase III - Fig. 2.22). The nucleation processes between metastable precipitates are not exclusive and, thus, there is concurrent growth between the metaand stable-phases [69]. Due to this effect the damping values tend to a minimum, being lower than as-cast samples (Fig. 2.20). 2.3.3. Static mechanical properties: Tensile testing Fig.2.23 displays an example of stress-strain plots for the tested samples. From the average slopes in the elastic region, it was possible to determine the Young’s modulus of the samples as the treatment progresses (Fig.2.24). It is shown that the increase of stiffness is able to be correlated with the shifting in resonance frequency (Fig.2.19). Figure 2.23 Example of tensile stress-strain plots for the different test conditions. Chapter 2 Alloy Selection and Processing: Optimization for Static and Dynamic Mechanical Properties 65 Figure 2.24 Average Young’s modulus for the different test conditions. Fig. 2.25 shows the yield strength (σ Y ) of as-cast, solution treated and different artificial aged samples. It may be observed that the solution treatment was able to elevate the alloy yield strength, while the artificial ageing treatment has a very pronounced effect in the enhancement of this variable. Figure 2.25 Yield strength for different heat treatments [83–85]. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 66 According to the experimental results, it is suggested that the dependence between yield strength and ageing time for A356 alloy poured in ceramic block follows the linear function described by Eq. 2.10. Additionally, it may be noticed that this data is fairly correlated and follows the tendency of other published experimental results on the same alloy [83–85]. 𝜎𝑌=97.72+0.19𝐴𝑡 (2.10) The yield strength (σ Y ) of Al-Si-Mg ( e.g. A356) alloys may be expressed as an addition of solid solution (σ SS ) and precipitation (σ PPT ) strengthening components to an initial pure aluminum strength (σ i ), as shown in Eq. (2.11) [102]. 𝜎𝑌=𝜎𝑖+𝜎𝑆𝑆 +𝜎𝑃𝑃𝑇 (2.11) The solution strengthening term (σ SS ) is dependent on the alloy intrinsic solid solution strength (σ 0SS ), proportion of solute released (α) and relative volume fraction of precipitates (fr) (Eq. 2.12) [102]. The σ 0SS term includes the influence of eutectic Si and intermetallic compounds, thus, the overall solution strengthening (σ SS ) is increased by solution treatment since: (i) Si spheroidization increases yield strength [56,64]; (ii) πβ transformation is beneficial to reduce microstructural stress concentration [92]; and (iii) precipitation is residual (fr~0) during solution treatment [93]. This increase in yield strength by solution strengthening is observed in the experimental results of Fig. 2.25. 𝜎𝑆𝑆 =𝜎0𝑆𝑆(1−𝛼𝑓𝑟)2 3 (2.12) According to the same relation (Eq. (2.12)), due to the nucleation and growth of precipitates during artificial ageing and no variation in σ 0SS , the solution strengthening may decrease. However, this effect has a relatively small role when compared with the increase in aged samples precipitation strengthening [102] as shown by the experimental results. Precipitation strengthening (σ PPT – Eq. (2.13)) before peak-age is fundamentally dependent on the volume fraction of precipitates (fr) and mean precipitate radius (r), for r smaller than the transition radius (rtrans). The constant c1 is related to the Taylor factor, Burgers vector, precipitate shear modulus and interfacial energy [102]. However, according to the relation that is represented in Fig.2.26, when the precipitate reach the transition radius (rtrans), the overall dislocation is changed into a bypassing (Orowan) mechanism. The precipitation is then be estimated by Eq.2.14. 𝜎𝑃𝑃𝑇 =𝑐1𝑓𝑟1 2〈𝑟〉1 2 → 〈𝑟〉<𝑟𝑡𝑟𝑎𝑛𝑠 (2.13) Chapter 2 Alloy Selection and Processing: Optimization for Static and Dynamic Mechanical Properties 67 𝜎𝑃𝑃𝑇 =𝑓𝑟1 2 𝑟 → 〈𝑟〉>𝑟𝑡𝑟𝑎𝑛𝑠 (2.14) Figure 2.26 Shearing vs Bypassing competing strengthening mechanism. As the ageing time progresses, GP zones (r < 2 nm [67,102]) followed by β ’’ (2 <r < 7 nm [102]) precipitates, are nucleated and grown (Fig. 2.22 – Phase I to II). This increases the mean precipitate radius (r) and the overall yield strength (σ Y ) (Eq. (2.25)). Some β ’ (7 <r < 15 nm [66,102]) and β(Mg2Si) (15 < r < 500 nm [102,103]) with r larger than rtrans may also nucleate ( i.e. lowering σ PPT by Orowan dislocation – phase II in Fig. 2.22). However, their presence is residual before peak-age and does not contribute significantly to the mean radius (< r >) value. Overall, the experimental results (Fig. 2.25) show the described transformations and, consequently, an enhancement of yield strength by precipitation strengthening. 2.4. Tailoring of static and dynamic mechanical properties by artificial ageing The microstructural changes in A356 alloys during ageing treatment (Section 2.3.1) occur in an inverse proportionality between the experimental damping ratio (ζ - Section 2.3.2) and yield strength (σ Y - Section 2.3.3). These variables are correlated in Fig. 2.27, showing that under-aged samples (8–64 min ageing times) present similar damping ratios and yield strengths, meaning that the precipitation is residual. However, as the ageing time increases, the damping ratios tend to decrease as the yield strength increases on a linear function relation. It is suggested, from a mechanical design point-of-view, that these Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 74 [87] C.M. Dinnis, A.K. Dahle, J.A. Taylor, M.O. Otte, The influence of strontium on porosity formation in Al-Si alloys, Metall. Mater. Trans. A. 35 (2004) 3531–3541. [88] K. Nogita, A.K. Dahle, Mechanism of porosity formation and eutectic modification in hypoeutectic Al–Si alloys, J. Jpn. Inst. Light Met. 54 (2004) 440–446 [89] R. Chen, Q. Xu, Z. Jia, B. Liu, Precipitation behavior and hardening effects of Si-containing dispersoids in Al–7Si–Mg alloy during solution treatment, Mater. Des. 90 (2016) 1059–1068. [90] J. Davids, ASM Specialty Handbook, Aluminium and Aluminum Alloys, ASM international, Russel Township (1993). [91] J.Y. Yao, J.A. Taylor, Characterisation of intermetallic particles formed during solution treatment of an Al–7Si–0.4Mg–0.12Fe alloy, J. Alloys Compd. 519 (2012) 60–66. [92] X. Cao, J. Campbell, Morphology of β-Al5FeSi phase in Al-Si cast alloys, Mater. Trans. 47 (2006) 1303–1312. [93] A. Bahrami, Modeling of Precipitation Sequence and Ageing Kinetics in Al-Mg-Si Alloys (PhD thesis), Delft University of Technology, Delft, 2010. [94] J. Zhang, R.J. Perez, C.R. Wong, E.J. Lavernia, Effects of secondary phases on the damping behaviour of metals, alloys and metal matrix composites, Mater. Sci. Eng. R Rep. 13 (1994) 325– 389. [95] G. Schoeck, Internal friction due to precipitation, Phys. Status Solidi B. 32 (1969) 651–658. [96] G. Schoeck, E. Bisogni, Internal Friction in AlAg Alloys, Phys. Status Solidi B. 32 (1969) 31–40. [97] J.A. Taylor, D.H. StJohn, J. Barresi, M.J. Couper, Influence of Mg content on the microstructure and solid solution chemistry of Al-7% Si-Mg casting alloys during solution treatment, in: Trans Tech Publ, 2000: pp. 277–282. [98] S. Liu, K. Li, J. Lu, G. Sha, J. Wang, M. Yang, G. Ji, M. Song, J. Wang, Y. Du, On the atomic model of Guinier-Preston zones in Al-Mg-Si-Cu alloys, J. Alloys Compd. 745 (2018) 644-650. [99] G. Guo, Q. Wang, G. Wang, Y. Rong, A Brief Review of Precipitation Hardening Models for Aluminum Alloys, in: 2nd World Congr. Integr. Comput. Mater. Eng., John Wiley & Sons, Inc., 2013: pp. 249– 254. [100] A. Granato, K. Lücke, Theory of mechanical damping due to dislocations, J. Appl. Phys. 27 (1956) 583–593. [101] H. Puga, V.H. Carneiro, J. Barbosa, D. Soares, Effect of grain and secondary phase morphologies in the mechanical and damping behavior of Al7075 alloys, Met. Mater. Int. 22 (2016) 863–871. [102] L.J. Colley, M.A. Wells, W.J. Poole, Microstructure–yield strength models for heat treatment of Al– Si–Mg casting alloys II: modelling microstructure and yield strength evolution, Can. Metall. Q. 53 (2014) 138–150. [103] G. Ran, J.E. Zhou, Q.G. Wang, Precipitates and tensile fracture mechanism in a sand cast A356 aluminum alloy, J. Mater. Process. Technol. 207 (2008) 46–52. Chapter 3 I Design and Manufacturing of Cellular Solids 75 Chapter 3 – Design and Manufacturing of Cellular Solids “Well, I think the curves of the four pillars of the monument, as the calculations have provided them … Give it a great sense of force and beauty” Gustave Eiffel in interview to the newspaper Le Temps , 1887. 3.1 Introduction to the design and manufacturing of cellular solids Cellular structures, as a form of composite materials, are a classic route to obtain high values of static and dynamic specific mechanical properties [1,2]. This characteristic has always been advantageous in many fields and applications, from diverse areas such as packaging, the transportation industry ( e.g. , railway [3–5], aeronautic [6–8], etc), and medical implants [9–11]). This class of materials, when using a metallic matrix, are classically manufactured by foam blowing agents/gas injection [12– 17] and casting with space holders with leachable [18–22] or non-leachable [23–25] particles. Other techniques may also be used such as powder metallurgy [26,27] or wire weaving [28–30]. These techniques are able to be produced in an array of specific densities in both open-cell [26,28,31,32] and closed-cell [15,33,34] configurations, both being commonly found in stochastic nature. Recent studies have explored additive manufacturing ( e.g. , selective metal laser sintering or electron beam melting) [35–39] and metal casting [40,41] to produce non-stochastic cellular materials. These have the advantage of allowing in situ geometric and dimensional control, thus, they are able to display tailored properties [42]. However, the referred processes are known to have inherent problems: the difficult microstructural control in additively manufactured metals [43–45] and the casting [46] of thin-ribbed cellular materials. Vacuum assisted high pressure die [47–49] and low pressure casting Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 76 [50,51] are able to produce thin-walled and/or high detailed structures, however, they are not suited for the manufacturing of intricate three-dimensional complex geometries. This chapter shows an additive manufacturing assisted investment casting technique to manufacture different designs of cellular solids with diverse Poisson’s ratios. Although there are already studies that use similar techniques ( e.g. , 3D printing of sand molds [52–56], wax and PLA/ABS [57,58]), the proposed study recurs to vacuum as an auxiliary technique for the filling of thin-rib and high aspect ratio complex three-dimensional metallic scaffolds. Additionally, vacuum is also used to promote the sanity of the resultant samples, as it is known to reduce oxide inclusion [59,60] and porosity [61,62]. The manufactured samples are also inspected in terms of final dimensions by their comparison to the designed CAD models. For such purpose, the final samples are subjected to X-ray microcomputed tomography (μ-CT). This technique is known as a non-destructive technique that allows a threedimensional scanning of these structures, being able to execute metrological and defect characterizations. Such technique has successfully been used in to characterize cellular solids in similar cellular solids [63– 67]. In this study, μ-CT is employed to characterize the true rib and sample dimensions and defects. 3.2 Methodology for the design and manufacturing of cellular solids 3.2.1. Cellular solid design Non-stochastic cellular solids may be represented by an assembly of elementary cellular units (Fig. 3.1 (a)). Such units may be assembled on a single planar two-dimensional matrix, forming a lattice (Fig. 3.1 (b)). Otherwise, they can also assume a three-dimensional matrix configuration, when elements are added to a third cartesian direction (Fig. 3.1 (c)). Figure 3.1 Non-stochastic cellular solids: (a) unitary cell - unidimensional (m); (b) two-dimensional (m x n); and (c) three-dimensional (l x m x n). Chapter 3 I Design and Manufacturing of Cellular Solids 77 Auxetic re-entrant cells were reported in Chapter 1 as having a beneficial increase in static and dynamic mechanical properties. Considering that such configuration facilitates the casting process relatively to other auxetic geometries ( e.g. 3D Chiral, Rotating Geometries, etc), this was the selected model to perform the study. However, the minimum amount of cells ( i.e. l x m x n) that must be used to eliminate any cell size/distribution effect is still a matter of discussion in the field [68]. Fig. 3.2 (a) shows the frequency of cell per row in published works (compilation of data [69– 74,68,75–116]) on re-entrant auxetics. It may be observed that 2D cells have been used with a higher number of cells per row. When 3D cells are used, it is not common to use more than ten cells per row. Figure 3.2 Auxetic lattice cell counting (compilation of data on [69–74,68,75–116]): (a) frequency count; and (b) total cell number. However, since these cells are assembled in a three-dimensional matrix, they may display higher total cell number (Fig. 3.2 (b)). It may be observed that most published studies use less than 400 cells, either in 2D or 3D configurations, according to Fig. 3.2 (b). This may be attributed to the considerable difficulty in the manufacturing and numerical simulation of cellular solids with high cell numbers. A lattice geometry was designed with 1D struts and defined as an array of repeating unit cells. Individual cells (Fig. 3.3 (a)) are defined by two horizontal and four vertical ribs with a square crosssection. The rib horizontal and vertical lengths, and thickness (h, l and t respectively) were varied to optimize the manufacturing process, while the vertical rib angle (α) was varied to produce cells with different Poisson’s ratios (ν). Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 78 Figure 3.3 Lattice design process: (a) unitary cell; (b) twoand (c) three-dimensional lattice. Initially, the individual cells may be assembled into a two-dimensional lattice (Fig. 3.3 (b)), combining their common ribs. These lattices are repeated in the three Cartesian axis to produce a threedimensional (Fig. 3.3 (c)) lattice matrix. To guarantee that size effects do not interfere with the mechanical properties of the samples, they were designed using a basic three-dimensional unitary cell assembly, in which the cells are assembled in a matrix (9x9x8) configuration. This assures that they comply with the cells per row and a higher total cell number that are used in other publications (Fig. 3.2). Sample CADs may be observed in Fig. 3.4. Skins ( i.e. plates) on the top and bottom faces enclose the cellular core. Figure 3.4 CAD models: (a) α=-30º, (b) α=-20º, (c) α=-10º, (d) α= 0º, (e) α=10º, (f) α=20º and (g) α=30º samples. Chapter 3 I Design and Manufacturing of Cellular Solids 79 3.2.2. Model 3D-printing The lattice investment models were 3D-printed in PLA (Fig. 3.5) by fused filament fabrication. A BCN3D Sigma printer was used to produce these prototypes that were subsequently used to produce the ceramic casting mold. The design of the samples ( e.g. rib angles, span and orientation) allows 3d-printing without requiring support materials, which benefits the simplicity and speed of the process. Figure 3.5 3d-printing of investment PLA model. Printing parameters are shown in Table 3.1. The extrusion/bed temperatures (T) and printing speed (Ps) were selected in accordance with equipment manufacturer guidelines and kept constant. Nozzle diameters (N) and layer heights (Lh) were optimized to provide a successful print ( i.e. all details are printed) whilst minimizing printing time. Table 3.1 3d-printing (fused filament fabrication) parameters. Material PLA N - Nozzle diameter (mm) 0.3; 0.4; 0.6; 1 Lh - Layer height (mm) 0.05; 0.10; 0.15; 0.20; 0.30; 0.40 Ps - Printing speed (mm/s) 10 T - Temperature (ºC) Extrusion 210 Bed 40 3.2.3. Casting mold fabrication The 3d-printed polymer investment models were subsequently infiltrated with liquid plaster (30%wt distilled water and 70%wt GoldStar Omega+ plaster) to produce ceramic molds. The molds were subjected to the thermal cycle represented in as Fig.3.6. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 80 Figure 3.6 Thermal cycle and PLA weight during ceramic mold fabrication. This thermal cycle allows the complete cure of the plaster, while assuring a good accuracy on the investment model shape and dimension. During the first stage of the curing cycle (isothermal at 300ºC for 3h) the PLA model begins to evaporate. At the end of the second stage (ramp from 300ºC to 730ºC in 5h), only trace residues of the PLA model may be found (≈0.12%wt). Thus, during the third and fourth stages (respectively, isothermal at 730ºC for 6h and cooling to 250 ºC), the PLA sample is eliminated and all details are embossed in the ceramic mold. 3.2.4. Cellular solid casting A356 alloy (24g) was cut from a primary fusion ingot (Fig. 3.7 (a)). After being cleaned and dried, the alloy was inserted in a SiC crucible, along with added Al5Ti1B (0.05g – 0.2%wt) and Al10Sr (0.07g – 0.3%wt) master alloys, respectively added to promote grain refinement and eutectic Si modification. The maximum material for the lattice is 21g, corresponding to 87.5% of total crucible capacity. The remaining 3g (12.5% crucible capacity) corresponds to the casting gating system that assures sufficient metalostatic pressure during the liquid metal filling. Chapter 3 I Design and Manufacturing of Cellular Solids 81 Figure 3.7 Casting process: (a) A356 and Al5Ti1B/Al10Sr alloys being (b) cast into the mold. The alloy was melted (700ºC) inside an Indutherm MC15+ induction casting furnace in vacuum (P=-1 bar) and kept isothermal at 700±2ºC for 3 minutes for homogenization, while the imposed induction magnetic stirring prevented nucleant particle sedimentation. After this period, the melt was cast (Fig. 3.7 (b)) into the ceramic mold (250±5ºC). Following a 10 min solidification period, the ceramic mold was submerged in water to separate the plaster from the casted sample. The produced samples were subjected to a T6 heat treatment: solution (540°C – 480 min), water-quench and artificial ageing (160°C – 500 min), according to the alloy optimization that is reported in Chapter 2 (section 2.4). Due to the inverse proportionality between static strength and damping properties, this treatment was tuned to an optimized compromise between maximum yield strength and damping [113] while complying with the ISO 3522 standard. 3.2.5. X-ray microcomputed tomography of cellular solids X-ray microcomputed tomography ( µ -CT) imaging was performed using a Zeiss Versa XRM-520 µ -CT scanner controlled using XRM scout-and-scan control system. Samples were mounted on a polystyrene wedge to minimize cone-beam artefacts (one sample of each geometry was imaged). The imaging settings were optimized for each sample and are summarized in Table 3.2. Data was reconstructed using XRM reconstruction software, then segmentation and visualization was performed using Avizo standard. Cross-sections through the reconstructed 3D volume were exported as 2D tiff images to allow measurement of rib dimensions. The overall dimensions of the samples and the ribs were measured using the micrographs using Fiji [117]. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 82 Table 3.2. X-ray μ-CT testing parameters. Sample Low resolution High resolution α=-30º α=-20º α=-10º α=0º α=10º α=20º α=30º α=-30º Cell α=-30º Ribs Cast PLA Voxel (μm) 20.9 28.4 26.5 27.2 28.4 56.7 6.3 2.3 Exposure time (s) 4 6 4 1 6 60 Source-Sample (mm) 130 192 165 190 192 50 120 Sample-Detector (mm) 80 36 45 36 220 60 Voltage (kV) 140 100 140 Power (W) 10 9 10 Objective 0.39x 4x Projections 2401 3001 2401 Filter LE3 The whole samples were scanned in a low-resolution mode (Fig. 3.8) to measure the overall rib dimensions and visualization of exterior defects. Posteriorly, high resolution scans were performed in cellular units to detail the exterior defects and quantify porosity. A final high resolution scan was performed in a horizontal and a vertical rib to characterize porosity. Test parameters are detailed in Table 3.2. Figure 3.8 Volumes to characterize by μ-CT. 3.3 Results and Discussion 3.3.1. Optimization of investment model 3d-printing parameters Printing parameters have a key role in the shape of the samples and investment model printing process. Fig. 3.9 shows the 3D-printer ability to produce the samples using diverse nozzles. Smaller nozzles are able to produce ribs with lower thickness, which is desirable to minimize specific density. Rib thicknesses less than 0.6 mm could not be produced with the tested nozzles. Chapter 3 I Design and Manufacturing of Cellular Solids 83 Figure 3.9 Relationship between rib thickness (t) and alloy material mass, for nozzle diameters of: (a) 1 mm, (b) 0.6 mm, (c) 0.4 and 0.3 mm. Note – Solid lines: crucible limit; Dashed line: Printing failure. These printing failures occurred due to under-extrusion of printing filament (Fig. 3.10 (a)), caused by the wearing of filament in the extrusion gear (Fig. 3.10 (b)). Filament wear is caused by the change in feeding in the extrusion/retraction cycles needed to deploy reduced amounts of PLA in each individual rib [118]. In conclusion, the minimum rib thickness that was achieved in the described process is 0.6 mm, using nozzle diameters lower than 0.4 mm. Figure 3.10 Printing failure: (a) whole sample failure by under-extrusion and (b) extrusion failure resulting from filament wear. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 90 µ -CT measurements (Fig. 3.20 (a)), demonstrate that the casted sample height (hS) is higher (+0.2 mm to +0.4 mm) than the CAD model. Rib angle impacts the dimensional variations of the casted sample width and depth (WS and DS - Fig. 3.20 (b)) and negative/positive rib angles produce positive/negative dimensional variations (-0.26 mm to 0.54 mm). According to Fig. 3.20 (c), these dimensional variations change the sample specific density (ρ * /ρ0). Given that the casted samples display a tendency to have higher dimensions (Figs. 3.20 (a) and (b)) than the CAD models, the sample volume is increased. Thus, there is mostly a decrease in density (-0.008 to 0.001). Figure 3.20 Comparison between the dimensions of CAD and casted samples. Fig. 3.21 shows examples of casted samples periodic cellular units. Samples with negative angles (Fig.3.21 (a) to (c)) have an inverted honeycomb configuration. This characteristic is related with cellular lattices with negative Poisson’s ratios ( i.e. auxetic behavior) [56]. Fig.3.21 (d) has a square-shaped cell (α=0º), where the ribs may be described by near horizontal and vertical configurations. The samples with positive rib angles (Fig.3.21 (e) to (g)) have a classic honeycomb ( i.e. hexagonal) configuration with a positive Poisson’s ratio [119]. Chapter 3 I Design and Manufacturing of Cellular Solids 91 Figure 3.21 X-ray μ-CT of cells: (a) α=-30º, (b) α=-20º, (c) α=-10º, (d) α= 0º, (e) α=10º, (f) α=20º and (g) α=30º. Fig. 3.22 shows that the rib angle has an impact on the overall shape of the cell and in the ability of the studied manufacturing process to replicate the CAD samples. True rib angles (αTRUE) display an angular error (from -1.40º to 0.73º - Fig. 3.22 (a)). However, these errors are small in comparison with the deviations of other linear dimension. Figure 3.22 X-ray computed tomography dimensional measurements compared against CAD model specifications: (a) rib angle, (b) horizontal rib length, (c) vertical rib length and (d) rib thickness. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 92 Figs. 3.22 (b) and (c), show that the horizontal and vertical rib lengths have a more pronounced dimensional variations. Indicating a prominent role in the dimensional variations of the casted samples (Fig.3.19). Horizontal ribs (h - Fig.3.22 (b)) and vertical ribs (l - Fig. 3.22 (c)) have dimensional range of, -0.50% to 3.75% and 0.04% to ≈10% respectively. Such changes are more prominent in samples with wider angles, either negative or positive. This is attributed to the different bending stiffness of the ribs during the printing process. It is known that in the 3D-printing, thin inclined struts without support ( e.g. α=-30º or α=30º) are subjected to: (i) bending due to the collision with the rigid nozzle [120,121]; and (ii) coalescence higher surface tension [122]. Ribs with wider angles ( e.g. α=-30º or α=30º) are subjected to bending while the nozzle is extruding. Ribs with near-zero angles (α≈0º) are prone to axial loads [121]. Given the matrix configuration of these periodic cellular units, the changes in Figs. 3.22 (b) and (c) are responsible for the increase in the dimensions of samples with wider angles (Fig. 3.22). Although rib thickness (t – Fig.3.22 (d)) has a relatively low impact on the exterior dimensions of cast samples (Figs. 3.20 (a) and (b)), it changes the sample specific density (Fig. 3.20 (c)). There is a tendency for the samples to have lower values of rib thickness than the CAD models. This is more noticeable when the rib angle is approximates zero (α≈0º - Fig. 3.20 (d)). Such change in thickness geometry impacts on rib cross-sectional area. The rib dimensions that are displayed in Fig. 11 were subjected non-parametric statistical analysis to analyze their significance (Table 3.4) using an one-way ANOVA on Ranks (Kruskal-Wallis test). Table 3.4. Statistical significance of the rib dimensions. α -30 -20 -10 0 10 20 30 h -30 -20 -10 0 10 20 30 -30 ●● ●● ●● ●● ●● ●● -30 ●● ●● ●● ●● -20 ●● ●● ●● ●● ●● -20 ●● -10 ●● ●● ●● ●● -10 ●● 0 ● ●● ●● 0 ●● 10 ●● ●● 10 ●● 20 ●● 20 ● 30 30 l -30 -20 -10 0 10 20 30 t -30 -20 -10 0 10 20 30 -30 ●● ●● ●● ●● -30 ●● ●● ●● ●● ●● ●● -20 ●● -20 ●● ●● ●● -10 ●● -10 ●● ●● ●● 0 ●● 0 ●● 10 ●● 10 ●● 20 ● 20 ●● 30 30 No significant difference. ● p<0.05 Significant difference. ●● p<0.01 Chapter 3 I Design and Manufacturing of Cellular Solids 93 In terms of angular dimensions (αTRUE– Fig. 3.22 (a)) each sampling group is supposed to display different rib angles, thus, it is expected that there are significant differences between them. Table 3.4 shows that there is a difference between the groups, although by comparing samples with α=0º and α=10º, this difference is less significant (p<0.05) than other possible combinations (p<0.01). When different groups are compared in terms of rib lengths (h and l), it is desirable that there are no significant differences due to the constant value of these dimensions ( i.e. h=4mm and l=2mm). Table 3.4 shows there is a tendency for the different groups not to be significantly different. However, this is not true for the wider rib angles (α=-30º and α=30º), in which there is a significant statistical difference (typically p<0.05) between these angles and the other groups. The statistical analysis of rib thickness (t) does not seem to follow a pattern. In fact, according to Table 3.4, most groups show a significant difference with each other. This may be attributed to the overall variations in the 3D-printing process of the PLA investment model. Given that the layer height is constant (Lh=0.2 mm), ribs with wider angles (Fig.3.23 (a)) have less layers than samples with lower rib angles (Fig.3.23 (b)). Since the 3D-printing process follows the sample Cartesian planes, the layer deposition in inclined ribs occurs in a tilted cross-section ( e.g. Fig.3.23 (a)), generally referred as a staircase surface [123]. These variations, however, are not applicable to the printing process of horizontal ribs (Fig. 3.23 (c)) in which the printing area is more elevated and constant. All these factors generate significant statistical differences in rib thickness. Figure 3.23 Influence of angle in the printing area during the fused filament fabrication process: vertical with (a) α=30º, (b) α=0º and (c) horizontal ribs. (d) Printing area plot of for oblique ribs (t=0.6 mm). Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 94 Consequently, the printing area (PA - Fig.3.23 (c)) of each layer changes with the rib angle, according to Eq. 3.1. Due to the discrete segmentation process during the G-code programming, this also occurs in metal-based additive manufacturing [57]. It is known that there is a deterioration of strut geometry in inclined angles relatively to the build platform [124–126]. However, in metal additive manufacturing techniques this issue is mainly attributed to the dissipation of heat flux through the neighboring sintering powder in oblique beams [127]. 𝑃 𝐴=𝑡2 cos(𝛼) (3.1) Given that the printer positioning resolution in the XX /YY axes is 12.5 μm, the reduction in printing area in near-zero rib angle (α≈0º) increases possible errors (Figs. 3.22 (d) and 3.23 (c)). Due to the equipment resolution and circular shape of the nozzle, the printed model presents with a quasicircular cross-section instead of the square cross-section in the CAD (Fig 3.24). Since extrusion cannot be stopped instantly, the constant movement of the extrusion head and filament viscoelastic effects [128], the nozzle leaves protrusion defects in some areas, as observed from µ -CT imaging (Fig. 3.21 and 3.25). Figure 3.24 Schematic illustrating the deviation in printed rib cross-section form the CAD model, and how this results from the printing process (Top view). Chapter 3 I Design and Manufacturing of Cellular Solids 95 Figure 3.25 Cross-sectional view through the µCT reconstructed volumes, showing detail of rib cross-: (a) α=-30º, (b) α=-20º, (c) α=-10º, (d) α= 0º, (e) α=10º, (f) α=20º and (g) α=30º samples. Fig. 3.26 (a) shows a reduction in the rib areas, especially in near-zero rib angles (α≈0º), as demonstrated by the model in Figs. 3.23 and 3.24. Furthermore, this changes the rib moment of inertia by: (i) the reduction in rib thickness (t – Fig. 3.22 (d)); and (ii) change in cross-section from a square (I■=t4/12) to a circular (I●=πt4/64) shape. Figure 3.26 Rib cross-section: (a) rib area and moment of inertia; (b) rib roundness (R=1, means perfect circle). The circular shape of the ribs may be observed in the values of roundness (Eq. 3.2) presented in Fig. 3.26 (b), where AR is the rib area and P is the perimeter. Given the high roundness values (R=0.89 to 0.93), the cross-sectional shaped may be considered a circular geometry. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 96 𝑅 = 4𝜋𝐴𝑅 𝑃2 (3.2) 3.4 Conclusions on cellular solid manufacturing This chapter details the design and reports a route to producing low-cost non-stochastic cellular lattices by an additive manufacturing assisted investment casting technique. Overall, the process and the involved techniques are detailed to characterize such manufacturing route. The following conclusions are drawn: (i) Optimization of the fabrication parameters (nozzle diameter and layer height) is essential to produce the thin-ribbed and low specific density PLA investment models that are used to produce ceramic casting molds. (ii) It is shown that an optimized combination of different mould and casting temperatures must be employed to ensure a complete filling of the samples. A minimum temperature combination of 250°C for the mould and 700°C for the casting must be used to guarantee the complete filling of the samples. It is suggested that in near complete casting parameters, the most important factor is the casting temperature. Additionally, it is shown that the vacuum assistance is needed to ensure the success of the procedure. (iii) Even though the casting process occurs in vacuum, μ-CT shows that there is still porosity in the samples. However, the volume of these pores is extremely low and it is proposed that this does not significantly affect the mechanical properties of the samples. Additionally, horizontal ribs may display segregation due to the meeting of opposing liquid aluminum streams. (iv) µ -CT demonstrated that linear rib dimensions such as horizontal and vertical rib lengths may deviate from CAD models. Final dimensions may be up to 10% than original CAD dimensions, and these deviations are more pronounced as rib angle is increased. This corresponds with an observed increase in the sample volume, which is beneficial to further reduce the specific density of the casted samples. (v) The ribs of casted samples present a circular shape and the process is unable to produce square cross-sections as defined in the CAD model, due to the shape and size of the extrusion nozzle. 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The transition from the negative to positive Poisson’s ratio generates a change in the overall collapse mechanism of the samples. An analytical model is proposed to describe the elastic behavior in these composites. Finite element analysis is used to analyze and describe the Poisson’s ratio dependent collapse mechanism and the overall mechanical behavior of the samples. 4.2 Static elastic properties – Analytical approach Even though there have been developments in the manufacturing of these composites and their mechanical properties [17,18], there still no unified theory that describes their elastic behavior. In the early 1980’s Gibson et al [19,20] determined Eqs. 1 and 2 (Table 4.1) to calculate the fundamental elastic properties of honeycombs. It is known that the latter have a good correlation with flexure dominated deformation, i.e. honeycombs with reduced wall thickness and low specific densities. Table 4.1 Theoretical models for the elastic properties of honeycomb lattices. Author Ref. E* Apparent Modulus ν Poisson’s ratio Gibson (1982) [19,20] (𝑡𝑙)3𝐸𝑠cos(𝛼) (ℎ𝑙+sin(𝛼))sin2(𝛼) (1) cos2(𝛼) (ℎ𝑙+sin(𝛼))sin(𝛼) (2) Masters (1996) [21] (𝑏(ℎ𝑙+sin(𝛼))[sin2(𝛼) 𝐾𝑓cos(𝛼)+sin2(𝛼) 𝐾ℎcos(𝛼)+cos(𝛼) 𝐾𝑠])−1 (3) OBS: 𝐾𝑓=𝐸𝑆𝑏𝑡3 𝑙3 (5) 𝐾ℎ=𝐺𝑆𝑏𝑡 𝑙 (6) 𝐾𝑠=𝐸𝑆𝑏𝑡 𝑙 (7) −sin(𝛼)cos(𝛼)[1 𝐾𝑓+1 𝐾ℎ−1 𝐾𝑠] (ℎ𝑙+sin(𝛼))[sin2(𝛼) 𝐾𝑓cos(𝛼)+sin2(𝛼) 𝐾ℎcos(𝛼)+cos(𝛼) 𝐾𝑠] (4) Malek (2015) [22] 𝐸𝑠(𝑡𝑙𝑏)3cos(𝛼) (ℎ𝑙+sin(𝛼))sin2(𝛼)[1 1+(2.4+1.5𝜈+cot2(𝛼))(𝑡𝑙𝑏)2] (8) OBS: 𝑙𝑏=𝑙− 𝑡 2cos(𝛼) (10) cos(𝛼) (ℎ𝑙+sin(𝛼))sin2(𝛼)[1+(1.4+1.5𝜈)(𝑡𝑙𝑏)2 1+(2.4+1.5𝜈+cot2(𝛼))(𝑡𝑙𝑏)2] (9) Hedayati (2016) [23] (𝑡𝑙)3l sin(𝛼) ℎ(cos(𝛼)+1)(sin2(𝛼))(𝑡𝑙)2+cos2(𝛼) (11) 𝑙 sin2(𝛼)cos(𝛼)(𝑙2−𝑡2) (𝑡2sin2(𝛼)+𝑙2cos2(𝛼))(ℎ+𝑙cos(𝛼)) (12) To address these limitations, in the 1990’s Masters et al [21] included the effect of rib/wall stretching and hinging, interpreting these effects of honeycomb cell geometry in the overall cell stiffness. The fundamental elastic properties are determined by Eqs. 3 and 4 (Table 4.1), being supported by the calculation of the stretching, hinging and flexure stiffness constants (respectively Ks, Kh and Kf – Eqs. 5 to 7 in Table 4.1). Chapter 4 I Effect of the Poisson’s ratio in the Static Structural Behavior of the Designed Cellular Solids 107 Recently, Malek et al [22] determined Eqs. 8 and 9 (Table 4.1) to compensate the effect of thick walled honeycombs and further approximated the values of the theoretical models to the experimental results. A similar approach was performed by Hedayati et al [23] , in which Eqs. 11 and 12 (Table 4.1) are deduced to model the elastic properties of thick honeycombs. These models are frequently adopted in two-dimensional honeycombs. However, due to the recent approaches in the manufacturing of three-dimensional honeycombs, there is a need to further analyze and adapt them to this context. Each cell in twoor three-dimensions may be divided into an Euler-Bernoulli beam system, as displayed in Fig. 4.1. It may be observed that the deformation behavior is dependent on the geometry of the cell, i.e. rib horizontal length (h), vertical length (l), thickness (t) and angle (α). In two dimensional approaches, the width (b) of the lattice can be also considered to influence the elastic properties. However, this aspect may be simplified in three-dimensional cells if the width is equal to the rib thickness (b=t). Figure 4.1 Honeycomb deformation mechanism: (a) initial dimensions and loading; (b) load distribution; (c) axial deformation; and (d) flexural deformation. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 108 If a load (F) is applied in the structural system (Fig. 4.1 (b), axial (Fa – Fig. 4.1 (c)) and flexural (Fb – Fig. 4.1 (d)) components are introduced and applied in the ribs. Overall, the vertical (ΔYa and ΔYb) and horizontal (ΔXa and ΔXb) deformations in the ribs are proportional to the loading by axial and flexural stiffness coefficients (respectively, Ka and Kb – Eqs. 13 and 14). Given that these stiffness coefficients are not dimensionless magnitudes, they are dependent not only on the geometry of the cell, but also on the fundamental elastic modulus of the base material (E0). 𝐾𝑎=𝜋 𝑡2 𝐸0 2 (cos(𝛼))2 𝐻 (13) 𝐾𝑏=3𝜋 𝑡4 𝐸0 8 (sin(𝛼))2 𝐻3 (14) Fig. 4.2 represents the plots of Eqs.13 and 14. As it would be expected, when α=0º, the axial stiffness coefficient (Ka) tends to a maximum as the cell tends to a square configuration and there are no flexure solicitations. Thus, for the same angle, the flexure stiffness coefficient tends to a minimum. Figure 4.2 Plot of the proposed model for axial (Ka) and flexural (Kb) stiffness coefficients (Note: t=0.6mm; H=4mm; and E0=70 GPa). According to Masters et al [21] honeycomb deformation model (Table 4.1), it is fundamental to consider a rib hinging effect to determine their elastic properties. Considering to the referred model, this hinging effect may be compensated by the inclusion of a hinging stiffness coefficient (Kh – Eq. 6), that is dependent on the cell with (b), thickness (l), length (l) and base material shear modulus (G0). The present study considers that this hinging effect is also dependent on the rib angle (α). According to this hypothesis, in situations where the rib angle is negative (Fig. 4.3 (a)), zero (Fig. 4.3 (b)) and positive (Fig. 4.3 (c)), the hinging effect is substantially different. Fig. 4.3 shows that in negative rib Chapter 4 I Effect of the Poisson’s ratio in the Static Structural Behavior of the Designed Cellular Solids 109 angles, there is a double closing effect, while in positive values, there seems to be a single closing effects. Thus, honeycombs with negative rib angles ( i.e. auxetic honeycombs) display must display a higher hinging stiffness coefficient. Figure 4.3 Hinging deformation mechanism in honeycombs with: (a) negative, (b) zero and (c) positive rib angles. To compensate for this rib angle dependent hinging stiffness variation, a hinging coefficient (q – Eq. 15, Fig.4.4) is introduced in the reformulated hinging stiffness coefficient (KH – Eq. 16). 𝑞=tan(𝛼) (15) 𝐾ℎ=2 𝑡2 𝐺0 𝐻 𝑞 (16) Figure 4.4 Plot of the proposed model for correction hinging coefficient (q). . All the stiffness coefficients (Ka, Kb and Kh) may be correlated by a simple addition operation of springs in series to calculate a total stiffness coefficient (KT), according to Eq. 17. Fig. 4.5 represents the plot of KT, showing that the maximum stiffness value occurs near α=0º. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 110 𝐾𝑇=1 1 𝐾𝑎+1 𝐾𝑏+1 𝐾𝐻 (17) Figure 4.5 Plot of the proposed model for total stiffness coefficient (Kt). (Note: t=0.6mm; H=4mm; E0=70GPa). The apparent elastic modulus of the modeled honeycombs (E*) may be determined using Hooke’s law (Eq. 18). In which A* (Eq. 19 – Fig.4.6) and L* (Eq. 20 – Fig.4.7) are, respectively, the initial honeycomb apparent resistant area and height. Fig. 4.5 shows that, for the experimentally tested rib angles (α=-30º to 30º), the total stiffness coefficient (KT) is fairly symmetric for an axis that passes through α=0º. Thus, overall, the different values of apparent modulus for symmetrical rib angles are unbalanced by different apparent areas (A*). 𝐸∗=𝐾𝑇 𝐿∗ 𝐴∗ (18) 𝐴=[𝐻(1+sin(𝛼))]2 (19) 𝐿=𝐻 cos(𝛼) (20) Figure 4.6 Plot of the initial apparent area (A*). (Note: t=0.6mm; H=4mm; E0=70GPa). Chapter 4 I Effect of the Poisson’s ratio in the Static Structural Behavior of the Designed Cellular Solids 111 Figure 4.7. Plot of the initial apparent length (L*). (Note: t=0.6mm; H=4mm; E0=70GPa). Fig. 4.8 shows the plot of Eq.18. According to the analytical model, the maximum values of apparent modulus are expected in the vicinity of α=0º. This is due to the high values of the axial stiffness coefficient in this angle range (Fig.4.2) and is supported by other analytical honeycomb theories [19–23]. Figure 4.8 Plot of the proposed model for apparent modulus (E*). (Note: t=0.6mm; H=4mm; E0=70GPa). From Fig. 4.1, it is possible to observe that the Poisson’s ratio (ν) of the honeycombs is generated by the referred axial and horizontal deformations (ΔYa, ΔYb, ΔXa and ΔXb). It is suggested that the imposed deformations may be determined by Eqs. 21 to 24. 𝛥𝑌𝑎𝑥𝑖𝑎𝑙=2 𝐹 (cos(𝛼)2) 𝐻 𝜋 𝑡2 𝐸0 (21) 𝛥𝑌𝑏𝑒𝑛𝑑𝑖𝑛𝑔=8 𝐹 (sin(𝛼)2) 𝐻3 3𝜋 𝑡4 𝐸0 (22) 𝛥𝑋𝑎𝑥𝑖𝑎𝑙=2 𝐹 cos(𝛼)sin(𝛼) 𝐻 𝜋 𝑡2 𝐸0 (23) 𝛥𝑋𝑏𝑒𝑛𝑑𝑖𝑛𝑔=8 𝐹 cos(𝛼)sin(𝛼) 𝐻3 3𝜋 𝑡4 𝐸0 (24) Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 112 According to the suggested model, the Poisson’s ratio (ν) of the modeled honeycombs may be determined by Eq. 25, in which p (Eq. 26) is a correction factor to compensate hinging effect in the ribs. Fig. 4.9 displays the plot of the correction factor (p) due to the hinging effect shown in Fig.4.3, in which it is shown that for negative values it tends to reduce the axial deformation by bending (ΔYb), while the opposite happens to positive rib angles. 𝜈=−𝜀𝑥 𝜀𝑦= − 𝛥𝑋𝑎𝑥𝑖𝑎𝑙+𝛥𝑋𝑏𝑒𝑛𝑑𝑖𝑛𝑔 𝐴∗𝐿∗ 𝛥𝑌𝑎𝑥𝑖𝑎𝑙+2𝛥𝑌𝑏𝑒𝑛𝑑𝑖𝑛𝑔 𝑝 (25) 𝑝=cos(𝛼−𝜋2) + 1 (26) Fig.4.9 Plot of the proposed model for correction hinging coefficient (p). (Note: t=0.6mm; H=4mm; E0=70GPa). Fig.4.10 shows the plot of Eq.25, in which it may be observed that negative rib angles generate negative values of Poisson’s ratio ( i.e. auxetic behavior), while positive rib angles imply positive values of this elastic constant. In transition regimes, the plot assumes an asymptote shape that is characteristic in other published honeycomb deformation models. Figure 4.10 Plot of the proposed model for Poisson’s ratio ( ν ). (Note: t=0.6mm; H=4mm; E0=70GPa). Chapter 4 I Effect of the Poisson’s ratio in the Static Structural Behavior of the Designed Cellular Solids 113 4.3 Methodology for static structural characterization 4.3.1. Experimental quasi-static uniaxial compression Samples were subjected to uniaxial compression testing using an INSTRON 8874 universal testing equipment. Five specimens for each rib angle were placed between two steel clamps (Fig. 4.11 (a)) and pre-stressed with a 50N load. A 0.01 mm/s displacement was imposed in the upper clamp to compress the samples while the loads were measured by the equipment load-cell. Axial deformations were measured by placing a mechanical strain gauge in one of the sample faces. The Poisson’s ratio of the samples was measured using an additional transverse mechanical strain gauge (Fig. 4.11 (b)). Additional samples (three for each tested rib angle) were also tested without strain gauges and video recorded to monitor their global and local deformation behavior. Figure 4.11 Compression testing: (a) Experimental apparatus and (b) Poisson’s ratio calculation with axial and transverse strain gauges. 4.3.2. Finite element analysis – Uniaxial compression simulation Finite element analysis (FEA) was performed in parametrized models to determine the fundamental mechanical behavior and collapse mechanism with more detail that may be obtained in the experimental compression tests. Initially, a model that represents the manufactured lattices was designed to study them in structural simulations. According to Fig. 4.12, an eighth (1/8) of a samples was modeled in ANSYS 17. The lattice ribs were modeled as unidimensional beam elements ( BEAM188 ), with an upper top two-dimensional rigid plate ( SHELL181 ). Symmetry was imposed in the YX, ZX and YZ mid-sample planes to simplify the simulations. A compression displacement of 1 mm is imposed in the top plate to simulate the experimental compression testing. This reduces the overall simulation processing time/complexity and helps with convergence, while representing the deformation behavior that is observed in the experimental tests. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 114 Figure 4.12 Representation of FEA model with boundary conditons. According to Fig. 4.13, it may be observed that an additional two dimensional fixed rigid plate is added to the bottom of the model. This plate was found to facilitate the conversion of the simulation allowing the contacting ribs to slide in the ZX symmetry plane. Thus, while the contact between the ribs and the top plate was imposed as Bonded, the contact with the lower plate is modeled as Frictionless. The overall mesh for the adopted Beam ( BEAM188 ) and Plate ( SHELL181 ) elements has a 0.2 mm size. It was observed that this size is able to generate a good convergence of the results, while maintaining acceptable discretization of the model and low processing time. Figure 4.13 Representation of model meshing. The model was designed with the linear dimensions (according to Chapter 3 – Fig.3.3), given that these dimensions of the casted samples do not present significant changes relatively to the CAD models. However, according to the conclusions of the referred chapter, this is not true for the rib thickness and shape. Thus, the beam moment of inertia in the simulations were corrected to a circular shape and parametrized in terms of rib thickness in accordance to the values of Fig.14. Chapter 4 I Effect of the Poisson’s ratio in the Static Structural Behavior of the Designed Cellular Solids 115 Figure 4.14 FEA input values for rib thickness. Material properties input was based in the optimization of heat treated A356 alloy that is established in Chapter 2. The stress-strain behavior of the alloy is presented in Fig.4.15. Figure 4.15 Plot of FEA input stress-strain material model. According to the referred figure, the alloy behavior was modeled using a bilinear isotropic strain hardening formulation. In both elastic and plastic regimes, the FEA input has a considerable approximation to the experimental curve. Basic mechanical properties (E, ν and ET) are shown in Table 4.2. Their difference consists in the approach for the transition period in the elasto-plastic regime. While the real material has a smooth transition between the two domains that is initiated by the yield strength (σ Y ), in the FEA routine this transition is instantaneous. The difference, however, is not significant due to the overall high correlation of both functions (Fig.4.15) and the use of BEAM188 elements, where there is no stress/strain variation through the rib thickness. Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 122 of the ribs. This event is also highly localized and sequential cell rows during the compression plateau, being verified in the instability of the apparent stress-strain curve in Fig. 4.17. After the transition, the samples are characterized by a positive Poisson’s ratio. Thus, during the compression in the linear elastic regime, their cross-sectional area tends to increase. Without the densification effect that is displayed in the samples with auxetic behavior there is a significant reduction in the apparent modulus (Fig. 4.20). Their plastic collapse is well distributed along the sample faces. This significantly reduces the collapse stress (Fig. 4.21), however, it allows a smooth transition between the elastic and plastic domains. 4.4.2. Comparison with analytical model Figs. 4.23 and 4.24 represent the plotting of the analytical model for honeycomb elastic properties that is suggested in this study, experimental and published classic honeycomb deformation models [19–23] (Table 4.1). All analytical models are compared with experimental results presented in section 4.4.1 on three-dimensional aluminum honeycombs (h=4 mm, l=2 mm, t=0.6 mm, E0=71 GPa and G0=27 GPa). Figure 4.23 Comparison of the suggested analytical apparent modulus (E*) with experimental results and other published models. Chapter 4 I Effect of the Poisson’s ratio in the Static Structural Behavior of the Designed Cellular Solids 123 Figure 4.24 Comparison of the suggested analytical Poisson’s ratio ( ν ) with experimental results and other published models. It may be observed that the suggested model is able display a good description of the experimental apparent modulus (E*) and Poisson’s ratio (ν) in comparison with other analytical models. While most models are fairly symmetrical between negative and positive rib angles, the suggested model is able to adapt itself to the changes in hinging stiffness. A final validation of the suggested analytical model may be observed in Table 4.5, in which the plot are fitted with the experimental results. It is shown that the presented model is able to display a significant higher correlation to the experimental data, relatively to the other honeycomb deformation models. Table 4.5 Correlation of the experimental values with the theoretical models. Model E* Apparent modulus correlation (R2) ν Poisson’s ratio correlation (R2) Analytical (This study) 0.71 0.94 Gibson et al [19,20] 0.20 0.01 Masters et al [21] 0.41 0.64 Malek et al [22] 0.13 0.60 Hedayati et al [23] 0.12 0.01 Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 124 4.4.3. Finite element analysis – Uniaxial compression simulation While discrete values of rib angles were analyzed in the experimental approach, the performed FEA analysis was used to obtain a continuous analysis of the influence of rib angles. By the monitoring of the instant values of reaction loading and plate displacement it was possible to determine the influence of rib angle on the apparent modulus and plastic collapse stress (Figs. 4.25 and 4.26). Figure 4.25 FEA results for apparent modulus (E*). Figure 4.26 FEA results for plastic collapse stress (σ P ). It may be observed by the high correlation values, that the FEA results are able to fully describe the behavior that is expressed in the experimental results. In conclusion, it is suggested that there is no variation in the tendency displayed by the discrete data of experimental testing. Additionally, Figs. 4.25 Chapter 4 I Effect of the Poisson’s ratio in the Static Structural Behavior of the Designed Cellular Solids 125 and 4.26 are able to validate the numerical analysis. Given the validation of the FEA routines, they may be used to analyze the collapse mechanism of the samples. In section 4.4.1 (Table 4.4), it is hypothesized that the collapse mechanism of the manufactured lattices is dependent on the sample Poisson’s ratio. Fig. 4.27 are represents the deformations in the XX axis in samples with different Poisson’s ratios. Figure 4.27 Graphical representation of FEA in samples with: (a) negative, α=-30°, (b) positive, α=30° and (c) zero, α=0° Poisson’s ratio. It is suggested in Fig. 4.27 (a) that samples with a negative Poisson’s ratio ( i.e. auxetic behavior) collapse by the deformation of a single row of cells. Given the negative Poisson’s ratio, the collapsing row reduces its width and a local densification effect may be observed. In samples with a positive Poisson’s ratio (Fig. 4.27 (b)), given that the sample tends to bulge, such collapse is well distributed along the samples lateral face. This behavior is characteristic of regular materials in compression. Transition regimes, in which the Poisson’s ratio is near zero (Fig. 4.27 (c)), the collapse occurs in a hybrid regime. The buckling occurs in a particular row and is localized, however, there is no densification in the sample. The deformations that are observed in the FEA simulation results are able to validate the hypothesis that is suggested in Table 4.4. 4.4.4. 4D X-ray μCT – Validation of collapse mechanism 4D X-ray μ-CT was done in three samples in compression, according to the experimental uniaxial compression and FEA described in sections 4.4.1 and 4.4.3. This analysis is intended as an in-situ Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 126 approach for a final validation of the Poisson’s ratio dependent collapse mechanism proposed in section 4.4.1 (Table 4.4). It may be seen that an initially undeformed auxetic scaffold ( i.e. α=-30º - Fig. 4.28 (a)) is compressed, it starts to contract its exterior faces in the elastic regime (ε=0.01 - Fig. 4.28 (b)). After its collapse (ε=0.04 – Fig. 4.28 (c)), one of the rows starts to densify and the deformation is localized according to the model suggested in Table 4.4. Figure 4.28 4D X-ray μ-CT of the auxetic sample (α=-30º): (a) undeformed – ε=0; (b) elastic deformation – ε=0.01; (b) post-collapse – ε=0.04. In samples with hexagonal honeycomb cells (Fig. 4.29 (a)), that are solicited to deformations in the elastic regime (Fig. 4.29 (b)) it may be observed that they bulge as they are compressed, i.e. they display a positive Poisson’s ratio. The collapse of these samples occurs by a well distributed deformation in the faces of the samples (Fig. 4.29 (c)), in accordance with the model proposed in Table 4.4. Figure 4.29 4D X-ray μ-CT of the positive Poisson’s ratio sample (α=30º): (a) undeformed – ε=0; (b) elastic deformation – ε=0.01; (b) post-collapse – ε=0.04. In samples with a transition Poisson’s ratio regime (Fig.30), in which the value of this constant approximates zero (α=0º), the collapse mechanism is a hybrid of the results in Figs. 4.28 and 4.29. Chapter 4 I Effect of the Poisson’s ratio in the Static Structural Behavior of the Designed Cellular Solids 127 Figure 4.30 4D X-ray μCT of the near zero Poisson’s ratio sample (α=0º): (a) undeformed – ε=0; (b) elastic deformation – ε=0.01; (b) post-collapse – ε=0.04. Using X-ray μ-CT, it may be seen that when an initially unloaded sample (Fig. 4.30 (a)) is compressed in the elastic regime, the deformation is distributed along the faces (Fig. 4.30 (b)). However, it is hardly noticeable due to the near zero Poisson’s ratio. Posteriorly to the plastic collapse stress, the sample starts to bulge in localized cell rows, according to Fig. 4.30 (c)). Once again, these results are able to validate the hypotheses that was initially established in Table 4.4. 4.5. Comparison with other materials To compare the performance of the designed cellular solids, their fundamental static mechanical properties ( i.e. load and deformation bearing capacity) were compared with other current solutions. For that comparison, the specific modulus and strength (E*/E0 and σ*/σ0) of published data (Table 4.6) were compared with the experimental results that were determined in this study. Given that only cellular solids are analyzed, the referred properties are also correlated in terms of specific density ( ρ */ ρ 0). Table 4.6 Materials that were analyzed for comparison. Material Method Poisson’s ratio ( - ) Ref. Al alloy Foaming Positive [24–27] SLM [28] Cast Negative [29] 316 L SLM Positive [30] Ti-6Al-4V [31,32] EBM [33–35] Negative [18] Ti Powder metallurgy Positive [36] Mg [37] Ni Electrodeposition [38] Maraging steel Powder bed fusion [39] Polyurethane Thermomechanical Negative [40] Polymethylmethacrylate Laser cutting [41] Development of Lightweight Auxetic Composites for Noise Reduction and Vibration Damping 128 Fig. 4.31 shows the plot of the specific modulus (E*/E0) relatively to the specific density ( ρ */ ρ 0) of the other published data [18,24–27,29–31,33–36,38–41] and the experimental results of this study. Given that it is desirable to obtain cellular solids with high stiffness ( i.e. E*/E0) and low weight ( i.e. ρ*/ρ0), the slopes for each plot are also determined. According to Fig. 4.31 it may be observed that there is a tendency for the cellular solids with a negative Poisson’s ratio ( ν <0) to display higher slope values. This implies that these samples are inherently stiffer for the same specific densities, as initially proposed in the study. Figure 4.31 Comparison of specific modulus (E*/E0) to specific density (ρ*/ρ0) of the manufactured samples and other published data. Comparing the slopes of other published data with the experimental results of this study, it may be also concluded that there is a tendency for the designed samples to display higher stiffness values for the same specific densities. The same observation is also verified in when the samples are compared to other published data [18,24–29,31–33,35,37–39] in terms of specific strength (Fig. 4.32). It is shown that the designed cellular solids tend to display higher values of specific strength for the same specific density. This is more prominent in samples with a negative Poisson’s ratio. In conclusion, it is proposed that the designed composites are able to display a good relation between the static mechanical properties and specific density. Thus, it is suggested that they may be an interesting solution for high specific mechanical properties in structural applications. Chapter 4 I Effect of the Poisson’s ratio in the Static Structural Behavior of the Designed Cellular Solids 129 Figure 4.32 Comparison of specific strength (σ*/ σ 0) to specific density (ρ*/ρ0) of the manufactured samples and other published data. 4.6. Conclusions on static structural behavior of the cellular solids This chapter explores the influence of the rib angle and Poisson’s ratio in the overall static mechanical properties of three-dimensional metallic cellular solids. Uniaxial compression was performed in samples with different geometries to determine the dependence of apparent modulus and plastic collapse stress to the auxetic and non auxetic deformation behaviors. The following conclusions are drawn: (i) A formulation intended to describe the elastic properties of twoand three-dimensional honeycomb lattices is shown to correct the rib angle dependent hinging stiffness. This is fundamental to enhance the correlation between analytical models with experimental results. The suggested model and formulation are a good approximation to determine the elastic properties of these structures based on their initial geometry and base material. (ii) A deformation model that depends on the Poisson’s ratio was proposed to describe the static mechanical properties of cellular materials. It is shown that the densification of auxetic samples generates a combination of higher values of apparent modulus and plastic collapse stress; (iii) Finite element analysis is performed to detail the Poisson’s ratio dependent collapse of the samples. The numerical results are validated by the experimental testing and validate the proposed collapse mechanism. 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