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Design of Frame-Like Periodic Solids for Isotropic Symmetry by Member Sizing

Published online by Cambridge University Press:  11 July 2016

K. Theerakittayakorn ,
P. Suttakul ,
P. Sam  and
P. Nanakorn
K. Theerakittayakorn
Affiliation:
School of Civil Engineering and TechnologySirindhorn International Institute of TechnologyThammasat UniversityPathumthani, Thailand
P. Suttakul
Affiliation:
School of Civil Engineering and TechnologySirindhorn International Institute of TechnologyThammasat UniversityPathumthani, Thailand
P. Sam
Affiliation:
School of Civil Engineering and TechnologySirindhorn International Institute of TechnologyThammasat UniversityPathumthani, Thailand
P. Nanakorn*
Affiliation:
School of Civil Engineering and TechnologySirindhorn International Institute of TechnologyThammasat UniversityPathumthani, Thailand
*
*Corresponding author ([email protected])
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Abstract

In this study, a methodology to design frame-like periodic solids for isotropic symmetry by appropriate sizing of unit-cell struts is presented. The methodology utilizes the closed-form effective elastic constants of 2D frame-like periodic solids with square symmetry and 3D frame-like periodic solids with cubic symmetry, derived using the homogenization method based on equivalent strain energy. By using the closed-form effective elastic constants, an equation to enforce isotropic symmetry can be analytically constructed. Thereafter, the equation can be used to determine relative unit-cell strut sizes that are required for isotropic symmetry. The methodology is tested with 2D and 3D frame-like periodic solids with some common unit-cell topologies. Satisfactory results are observed.

Keywords

Periodic solidEffective elastic propertyIsotropic symmetryClosed form
Type
Research Article
Information
Journal of Mechanics , Volume 33 , Issue 1 , February 2017 , pp. 41 - 54
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2017 

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References

1. Gibson, L. J. and Ashby, M. F., Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge (1997).CrossRefGoogle Scholar
2. Suquet, P. M., “Elements of Homogenization for Inelastic Solid Mechanics,” Homogenization Techniques for Composite Media, 272, pp. 193–278 (1987).Google Scholar
3. Alderson, A. and Alderson, K. L., “Auxetic Materials,” Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 221, pp. 565–575 (2007).Google Scholar
4. Ahmadi, S., et al., “Additively Manufactured Open-Cell Porous Biomaterials Made from Six Different Space-Filling Unit Cells: The Mechanical and Morphological Properties,” Materials, 8, pp. 1871–1896 (2015).Google Scholar
5. Hedayati, R., Sadighi, M., Mohammadi-Aghdam, M. and Zadpoor, A. A., “Mechanical Properties of Regular Porous Biomaterials Made from Truncated Cube Repeating Unit Cells: Analytical Solutions and Computational Models,” Materials Science and Engineering: C, 60, pp. 163–183 (2016).Google Scholar
6. Heinl, P., Müller, L., Körner, C., Singer, R. F. and Müller, F. A., “Cellular Ti–6al–4v Structures with Interconnected Macro Porosity for Bone Implants Fabricated by Selective Electron Beam Melting,” Acta Biomaterialia, 4, pp. 1536–1544 (2008).CrossRefGoogle ScholarPubMed
7. Murr, L. E., et al., “Next-Generation Biomedical Implants Using Additive Manufacturing of Complex Cellular and Functional Mesh Arrays,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368, pp. 1999–2032 (2010).CrossRefGoogle ScholarPubMed
8. Evans, K. E. and Alderson, A., “Auxetic Materials: Functional Materials and Structures from Lateral Thinking!,” Advanced Materials, 12, pp. 617–628 (2000).Google Scholar
9. Yang, W., Li, Z. M., Shi, W., Xie, B. H. and Yang, M. B., “On Auxetic Materials,” Journal of Materials Science, 39, pp. 3269–3279 (2004).CrossRefGoogle Scholar
10. Bettini, P., et al., “Composite Chiral Structures for Morphing Airfoils: Numerical Analyses and Development of a Manufacturing Process,” Composites Part B: Engineering, 41, pp. 133–147 (2010).Google Scholar
11. Bornengo, D., Scarpa, F. and Remillat, C., “Evaluation of Hexagonal Chiral Structure for Morphine Airfoil Concept,” Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 219, pp. 185–192 (2005).Google Scholar
12. Spadoni, A. and Ruzzene, M., “Static Aeroelastic Response of Chiral-Core Airfoils,” Journal of Intelligent Material Systems and Structures, 18, pp. 1067–1075 (2007).Google Scholar
13. Heo, H., Ju, J. and Kim, D.-M., “Compliant Cellular Structures: Application to a Passive Morphing Airfoil,” Composite Structures, 106, pp. 560–569 (2013).Google Scholar
14. Olympio, K. R. and Gandhi, F., “Flexible Skins for Morphing Aircraft Using Cellular Honeycomb Cores,” Journal of Intelligent Material Systems and Structures, 21, pp. 1719–1735 (2010).Google Scholar
15. Srikantha Phani, A., Woodhouse, J. and Fleck, N. A., “Wave Propagation in Two-Dimensional Periodic Lattices,” Journal of the Acoustical Society of America, 119, pp. 1995–2005 (2006).Google Scholar
16. Fleck, N. A. and Qiu, X., “The Damage Tolerance of Elastic-Brittle, Two-Dimensional Isotropic Lattices,” Journal of the Mechanics and Physics of Solids, 55, pp. 562–588 (2007).CrossRefGoogle Scholar
17. Srikantha Phani, A. and Fleck, N. A., “Elastic Boundary Layers in Two-Dimensional Isotropic Lattices,” Journal of Applied Mechanics, Transactions ASME, 75, pp. 0210201–0210208 (2008).Google Scholar
18. Neves, M. M., Rodrigues, H. and Guedes, J. M., “Optimal Design of Periodic Linear Elastic Microstructures,” Computers and Structures, 76, pp. 421–429 (2000).CrossRefGoogle Scholar
19. Challis, V. J., Roberts, A. P. and Wilkins, A. H., “Design of Three Dimensional Isotropic Microstructures for Maximized Stiffness and Conductivity,” International Journal of Solids and Structures, 45, pp. 4130–4146 (2008).CrossRefGoogle Scholar
20. Guth, D. C., Luersen, M. A. and Muñoz-Rojas, P. A., “Optimization of Periodic Truss Materials Including Constitutive Symmetry Constraints,” Materialwissenschaft und Werkstofftechnik, 43, pp. 447–456 (2012).CrossRefGoogle Scholar
21. Guth, D. C., Luersen, M. A. and Muñoz-Rojas, P. A., “Optimization of Three-Dimensional Truss-Like Periodic Materials Considering Isotropy Constraints,” Structural and Multidisciplinary Optimization, pp. 1–13 (2015).Google Scholar
22. Shan, S., Kang, S. H., Zhao, Z., Fang, L. and Bertoldi, K., “Design of Planar Isotropic Negative Poisson's Ratio Structures,” Extreme Mechanics Letters, 4, pp. 96–102 (2015).Google Scholar
23. Shufrin, I., Pasternak, E. and Dyskin, A. V., “Planar Isotropic Structures with Negative Poisson's Ratio,” International Journal of Solids and Structures, 49, pp. 2239–2253 (2012).Google Scholar
24. Grima, J. N., et al., “On the Auxetic Properties of Generic Rotating Rigid Triangles,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 468, pp. 810–830 (2012).Google Scholar
25. Sam, P., “Symbolic-Numerical Object-Oriented Finite Element Programming,” Master's Thesis, Thammasat University, Pathumthani, Thailand (2015).Google Scholar
26. Zhang, W., Wang, F., Dai, G. and Sun, S., “Topology Optimal Design of Material Microstructures Using Strain Energy-Based Method,” Chinese Journal of Aeronautics, 20, pp. 320–326 (2007).Google Scholar
27. Zhang, W., Dai, G., Wang, F., Sun, S. and Bassir, H., “Using Strain Energy-Based Prediction of Effective Elastic Properties in Topology Optimization of Material Microstructures,” Acta Mechanica Sinica/Lixue Xuebao, 23, pp. 77–89 (2007).CrossRefGoogle Scholar
28. Dai, G. and Zhang, W., “Cell Size Effect Analysis of the Effective Young's Modulus of Sandwich Core,” Computational Materials Science, 46, pp. 744–748 (2009).Google Scholar
29. Dai, G. and Zhang, W., “Size Effects of Effective Young's Modulus for Periodic Cellular Materials,” Science in China, Series G: Physics, Mechanics and Astronomy, 52, pp. 1262–1270 (2009).Google Scholar
30. Michel, J. C., Moulinec, H. and Suquet, P., “Effective Properties of Composite Materials with Periodic Microstructure: A Computational Approach,” Computer Methods in Applied Mechanics and Engineering, 172, pp. 109–143 (1999).Google Scholar
31. Nemat-Nasser, S. and Hori, M., Micromechanics: Overall Properties of Heterogeneous Materials, Second Edition (North-Holland Series in Applied Mathematics and Mechanics), North Holland, Amsterdam (1999).Google Scholar
32. Drago, A. and Pindera, M.-J., “Micro-Macromechanical Analysis of Heterogeneous Materials: Macroscopically Homogeneous Vs Periodic Microstructures,” Composites Science and Technology, 67, pp. 1243–1263 (2007).Google Scholar
33. Pindera, M.-J., Khatam, H., Drago, A. S. and Bansal, Y., “Micromechanics of Spatially Uniform Heterogeneous Media: A Critical Review and Emerging Approaches,” Composites Part B: Engineering, 40, pp. 349–378 (2009).CrossRefGoogle Scholar
34. Sigmund, O., “Materials with Prescribed Constitutive Parameters: An Inverse Homogenization Problem,” International Journal of Solids and Structures, 31, pp. 2313–2329 (1994).Google Scholar
35. Sigmund, O., “Tailoring Materials with Prescribed Elastic Properties,” Mechanics of Materials, 20, pp. 351–368 (1995).Google Scholar
36. Xia, Z., Zhang, Y. and Ellyin, F., “A Unified Periodical Boundary Conditions for Representative Volume Elements of Composites and Applications,” International Journal of Solids and Structures, 40, pp. 1907–1921 (2003).CrossRefGoogle Scholar
37. Xia, Z., Zhou, C., Yong, Q. and Wang, X., “On Selection of Repeated Unit Cell Model and Application of Unified Periodic Boundary Conditions in Micro-Mechanical Analysis of Composites,” International Journal of Solids and Structures, 43, pp. 266–278 (2006).Google Scholar
38. Bower, A. F., Applied Mechanics of Solids, CRC Press, Boca Raton (2010).Google Scholar