Structural and Multidisciplinary Optimization

, Volume 38, Issue 6, pp 583–597

Viscoelastic material design with negative stiffness components using topology optimization

Research Paper


An application of topology optimization to design viscoelastic composite materials with elastic moduli that soften with frequency is presented. The material is a two-phase composite whose first constituent is isotropic and viscoelastic while the other is an orthotropic material with negative stiffness but stable. A concept for this material based on a lumped parameter model is used. The performance of the topology optimization approach in this context is illustrated using three examples.


Topology optimization Material design Inverse homogenization Negative stiffness materials 


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  1. Alabuzhev PM, Gritchin AA, Stepanov PT, Khon VF (1977) Some results of an investigation of a vibration protection system with stiffness correction. J Min Sci 13:338–341Google Scholar
  2. Alabuzhev PM, Grytchin AA, Kim LI, Migirenko GS, Khon VF, Stepanov PT (1989) Vibration protecting and measuring systems with quasi-zero stiffness. Taylor & Francis, New YorkGoogle Scholar
  3. Bruns TE, Tortorelli DA (2001) Topology optimization of nonlinear elastic structures and compliant mechanisms. Comput Method Appl Math 190:3443–3459MATHGoogle Scholar
  4. Diaz AR, Benard A (2003) Designing materials with prescribed elastic properties using polygonal cells. Int J Numer Methods Eng 57:301–314MATHCrossRefMathSciNetGoogle Scholar
  5. Drugan WJ (2007) Elastic composite materials having a negative-stiffness phase can be stable. Phys Rev Lett 98:Article 055502CrossRefGoogle Scholar
  6. Falnes J (1995) On non-causal impulse response functions related to propagating water waves. Appl Ocean Res 17:379–389CrossRefGoogle Scholar
  7. Guest J, Prevost J, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61:238–254MATHCrossRefMathSciNetGoogle Scholar
  8. Hsu YL, Hsu MS, Chen CT (2001) Interpreting results from topology optimization using density contours. Comput Struct 79:1049–1058CrossRefGoogle Scholar
  9. Jaglinski T, Kochmann D, Stone D, Lakes RS (2007) Composite materials with viscoelastic stiffness greater than diamond. Science 315:620–622CrossRefGoogle Scholar
  10. Lakes RS (1999) Viscoelastic solids. CRC, Boca RatonGoogle Scholar
  11. Lakes RS, Drugan WJ (2002) Dramatically stiffer elastic composite materials due to a negative stiffness phase? J Mech Phys Solids 50:979–1009MATHCrossRefGoogle Scholar
  12. Lakes RS, Lee T, Bersie A, Wang YC (2001) Extreme damping in composite materials with negative stiffness inclusions. Nature 1410:565–567CrossRefGoogle Scholar
  13. Lee C-M, Goverdovskiy VN, Temnikov AI (2007) Design of springs with negative stiffness to improve vehicle driver vibration isolation. J Sound Vib 302:865–874CrossRefGoogle Scholar
  14. Lin C-Y, Chao L-S (2000) Automated image interpretation for integrated topology and shape optimization. Struct Multidisc Optim 20:125–137CrossRefGoogle Scholar
  15. Makris N, Inaudi JA, Kelly JM (1996) Macroscopic models with complex coefficients and causality. J Eng Mech 122:566–573CrossRefGoogle Scholar
  16. Platus DL (1991) Negative-stiffness-mechanism vibration isolation systems. Proc SPIE 1619:44–54CrossRefGoogle Scholar
  17. Prasad J (2007) Design of materials with special dynamic properties using negative stiffness components. PhD dissertation, Michigan State UniversityGoogle Scholar
  18. Prasad J, Diaz AR (2006) Synthesis of bistable periodic structures using topology optimization and a genetic algorithm. J Mech Des 128:1298–1306CrossRefGoogle Scholar
  19. Prasad J, Diaz AR (2007a) Conceptual design of materials exhibiting frequency-induced softening. ASME Proceedings of DETC2007, Paper DETC2007-34299Google Scholar
  20. Prasad J, Diaz AR (2007b) Material design for frequency-induced softening using topology optimization. Proceedings of WCSMO-7, Seoul, 21–25 May 2007Google Scholar
  21. Pritz T (1998) Frequency dependences of complex moduli and complex poisson’s ratio of real solid materials. J Sound Vib 214:83–104CrossRefGoogle Scholar
  22. Pritz T (1999) Verification of local Kramers Kronig relations for complex modulus by means of fractional derivative model. J Sound Vib 228:1145–1165CrossRefGoogle Scholar
  23. Sharnoff M (1964) Validity conditions for the Kramers-Kronig relations. Am J Phys 32:40–44CrossRefMathSciNetGoogle Scholar
  24. Sigmund O (1994) Materials with prescribed constitutive parameters: an inverse homogenization problem. Int J Solids Struct 31:2313–2339MATHCrossRefMathSciNetGoogle Scholar
  25. Sigmund O (1995) Tailoring materials with prescribed elastic properties. Mech Mater 20:351–368CrossRefGoogle Scholar
  26. Sigmund O (2000) A new class of extremal composites. J Mech Phys Solids 48:397–428MATHCrossRefMathSciNetGoogle Scholar
  27. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33:401–424CrossRefGoogle Scholar
  28. Svanberg K (1987) The method of moving asymptotes. Int J Numer Methods Eng 24:359–373MATHCrossRefMathSciNetGoogle Scholar
  29. Tai K, Prasad J (2007) Target-matching test problem for multiobjective topology optimization using genetic algorithms. Struct Multidisc Optim 34:333–345CrossRefGoogle Scholar
  30. Wang YC, Lakes RS (2004a) Extreme stiffness systems due to negative stiffness elements. Am J Phys 72:40–50CrossRefGoogle Scholar
  31. Wang YC, Lakes RS (2004b) Negative stiffness induced extreme viscoelastic mechanical properties: stability and dynamics. Philos Mag 35:3785–3801CrossRefGoogle Scholar
  32. Yi Y-M, Park S-H, Youn S-K (2000) Design of microstructures of viscoelastic composites for optimal damping characteristics. Int J Solids Struct 37:4791–4810MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringMichigan State UniversityEast LansingUSA

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