DOI QR코드

DOI QR Code

Stress analysis of a two-phase composite having a negative-stiffness inclusion in two dimensions

  • Wang, Yun-Che (Engineering Materials Program, Department of Civil Engineering, Center for Micro/Nano Science and Technology, National Cheng Kung University) ;
  • Ko, Chi-Ching (Engineering Materials Program, Department of Civil Engineering, Center for Micro/Nano Science and Technology, National Cheng Kung University)
  • 투고 : 2009.08.02
  • 심사 : 2009.08.30
  • 발행 : 2009.09.25

초록

Recent development in composites containing phase-transforming particles, such as vanadium dioxide or barium titanate, reveals the overall stiffness and viscoelastic damping of the composites may be unbounded (Lakes et al. 2001, Jaglinski et al. 2007). Negative stiffness is induced from phase transformation predicted by the Landau phase transformation theory. Although this unbounded phenomenon is theoretically supported with the composite homogenization theory, detailed stress analyses of the composites are still lacking. In this work, we analyze the stress distribution of the Hashin-Shtrikman (HS) composite and its two-dimensional variant, namely a circular inclusion in a square plate, under the assumption that the Young's modulus of the inclusion is negative. Assumption of negative stiffness is a priori in the present analysis. For stress analysis, a closed form solution for the HS model and finite element solutions for the 2D composite are presented. A static loading condition is adopted to estimate the effective modulus of the composites by the ratio of stress to average strain on the loading edges. It is found that the interfacial stresses between the circular inclusion and matrix increase dramatically when the negative stiffness is so tuned that overall stiffness is unbounded. Furthermore, it is found that stress distributions in the inclusion are not uniform, contrary to Eshelby's theorem, which states, for two-phase, infinite composites, the inclusion's stress distribution is uniform when the shape of the inclusion has higher symmetry than an ellipse. The stability of the composites is discussed from the viewpoint of deterioration of perfect interface conditions due to excessive interfacial stresses.

키워드

참고문헌

  1. Lakes, R.S. (2001), "Extreme damping in compliant composites with a negative-stiffness phase", Philos. Mag. Lett., 81(2), 95-100.
  2. Lakes, R.S., Lee, T., Bersie, A. and Wang, Y.C. (2001), "Extreme damping in composite materials with negative-stiffness inclusions", Nature, 410, 565-567. https://doi.org/10.1038/35069035
  3. Jaglinski, T., Kochmann, D., Stone, D. and Lakes, R.S. (2007), "Composite materials with viscoelastic stiffness greater than diamond", Science, 315, 620-622. https://doi.org/10.1126/science.1135837
  4. Lakes, R.S. (2001), "Extreme damping in composite materials with a negative stiffness phase", Phys. Rev. Lett., 86, 2897-2900. https://doi.org/10.1103/PhysRevLett.86.2897
  5. Wang, Y.C. and Lakes, R.S. (2001), "Extreme thermal expansion, piezoelectricity, and other coupled field properties in composites with a negative stiffness phase", J. Appl. Phys., 90, 6458. https://doi.org/10.1063/1.1413947
  6. Lakes, R.S. and Drugan, W.J. (2002), "Dramatically stiffer elastic composite materials due to a negative stiffness phase", J. Mech. Phys. Solids, 50, 979-1009. https://doi.org/10.1016/S0022-5096(01)00116-8
  7. Thompson, J.M.T. (1982), "'Paradoxical' mechanics under fluid flow", Nature, 296, 135-137. https://doi.org/10.1038/296135a0
  8. Falk, F. (1980), "Model free energy, mechanics and thermodynamics of shape memory alloys", Acta Metall., 28, 1773-1780. https://doi.org/10.1016/0001-6160(80)90030-9
  9. Drugan, W.J. (2007), "Elastic composite materials having a negative stiffness phase can be stable", Phys. Rev. Lett., 98, 055502. https://doi.org/10.1103/PhysRevLett.98.055502
  10. Kochmann, D.M. and Drugan, W.J. (2009), "Dynamic stability analysis of an elastic composite material having a negative-stiffness phase", J. Mech. Phys. Solids, 57, 1122-1138. https://doi.org/10.1016/j.jmps.2009.03.002
  11. Ernst, E. (2004), "On the Existence of Positive Eigenvalues for the Isotropic Linear Elasticity System with Negative Shear Modulus", Commun. Part. Diff. Eq., 29, 1745-1753.
  12. Shang, X.C. and Lakes, R.S. (2007), "Stability of elastic material with negative stiffness and negative Poisson's ratio", Phys. Status Solidi. B., 244, 1008-1026. https://doi.org/10.1002/pssb.200572719
  13. Lakes, R.S. and Wojciechowski, K.W. (2008), "Negative compressibility, negative Poisson's ratio, and stability", Phys. Status Solidi. B., 245, 545-551. https://doi.org/10.1002/pssb.200777708
  14. Wang, Y.C. and Lakes, R.S. (2004), "Extreme stiffness systems due to negative stiffness elements", Am. J. Phys., 72, 40-50. https://doi.org/10.1119/1.1619140
  15. Wang, Y.C. (2007), "Influences of negative stiffness on a two-dimensional hexagonal lattice cell", Philos. Mag., 87, 3671-3688.
  16. Yoshimoto, K., Jain, T.S., Van Workum, K., Nealey, P.F. and de Pablo, J.J. (2004), "Mechanical heterogeneities in model polymer glasses at small length scales", Phys. Rev. Lett., 93, 175501. https://doi.org/10.1103/PhysRevLett.93.175501
  17. Yap, H.W., Lakes, R.S. and Carpick, R.W. (2007), "Mechanical instabilities of individual multiwalled carbon nanotubes under cyclic axial compression", Nano Lett., 7, 1149-1154. https://doi.org/10.1021/nl062763b
  18. Yap, H.W., Lakes, R.S. and Carpick, R.W. (2008), "Negative stiffness and enhanced damping of individual multiwalled carbon nanotubes", Phys. Rev. B., 77, 045423. https://doi.org/10.1103/PhysRevB.77.045423
  19. Shi, G. and Tang, L. (2008), "Weak forms of generalized governing equations in theory of elasticity", Interact. Multi. Mech., 1(3), 329-337. https://doi.org/10.12989/imm.2008.1.3.329
  20. Wang, Y.C., Swadener, J.G. and Lakes, R.S. (2007), "Anomalies in stiffness and damping of a 2D discrete viscoelastic system due to negative stiffness components", Thin Solid Films, 515, 3171-3178. https://doi.org/10.1016/j.tsf.2006.01.031
  21. Thompson, J.M.T. (1982), "'Paradoxical' mechanics under fluid flow", Nature, 296, 135-137. https://doi.org/10.1038/296135a0
  22. Hill, R. (1957), "On uniqueness and stability in the theory of finite elastic strain", J. Mech. Phys. Solids, 5, 229-241. https://doi.org/10.1016/0022-5096(57)90016-9

피인용 문헌

  1. Anomalous effective viscoelastic, thermoelastic, dielectric, and piezoelectric properties of negative-stiffness composites and their stability vol.252, pp.7, 2015, https://doi.org/10.1002/pssb.201552058
  2. Stability of viscoelastic continuum with negative-stiffness inclusions in the low-frequency range 2013, https://doi.org/10.1002/pssb.201384231
  3. Negative stiffness of a buckled carbon nanotube in composite systems via molecular dynamics simulation vol.248, pp.1, 2011, https://doi.org/10.1002/pssb.201083976