Analytical stability conditions for elastic composite materials with a non-positive-definite phase

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 468, Issue 2144, 2012, Pages 2230-2254

  Kochmann, D M ^a   Drugan, W J ^b  

^a California Institute of Technology   (United States)

^b University of Wisconsin Madison   (United States)

Author keywords

Composite material; Elasticity; Negative stiffness; Stability

Indexed keywords

COMPOSITE MATERIALS; CONVERGENCE OF NUMERICAL METHODS; ELASTIC MODULI; ELASTICITY; FINITE ELEMENT METHOD; NUMERICAL METHODS; THICKNESS MEASUREMENT;

CLOSED FORM; COATED CYLINDER; COATING THICKNESS; COMPLEX MATRICES; COMPOSITE STABILITY; DYNAMIC DEFORMATION MODE; EFFECTIVE MODULI; ENERGY METHOD; FINITE-ELEMENT; MATERIAL PROPERTY; MULTIPHASE MATERIALS; NEGATIVE STIFFNESS; STABILITY ANALYSIS; STABILITY CONDITION; STABILITY RESULTS; THREE DIMENSIONS; TWO-DIMENSION;

STABILITY;

EID: 84864590848     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2011.0546     Document Type: Conference Paper

References (33)

1. Drugan, W. J. 2007 Elastic composite materials having a negative stiffness phase can be stable. Phys. Rev. Lett. 98, 055502.
2. Hashin, Z. 1972 Analysis of composite materials-a survey. J. Appl. Mech. 50, 481-505. (doi:10.1115/1.3167081)
3. Hashin, Z. & Shtrikman, S. 1963 A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11, 127-140. (doi:10.1016/0022-5096 (63) 90060-7)
4. Hill, R. 1957 On uniqueness and stability in the theory of finite elastic strain. J. Mech. Phys. Sol. 5, 229-241. (doi:10.1016/0022-5096 (57) 90016-9)
5. Hill, R. 1963 Elastic properties of reinforced solids-some theoretical principles. J. Mech. Phys. Solids 11, 357-372. (doi:10.1016/0022-5096 (63) 90036-X)
6. Hill, R. 1972 On constitutive macrovariables for heterogeneous solids at finite strain. Proc. R. Soc. Lond. A 326, 131-147. (doi:10.1098/rspa.1972.0001)
7. Jaglinski, T., Kochmann, D., Stone, D. & Lakes, R. S. 2007 Composite materials with viscoelastic stiffness greater than diamond. Science 315, 620-622. (doi:10.1126/science.1135837) (Pubitemid 46214701)
8. Kelvin, Lord, (Thomson, W.) 1888 On the reflection and refraction of light. Philos. Mag. 26, 414-425.
9. Kirchhoff, G. 1859 Überdas Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes. J. Reine Angew. Math. 56, 285-313. (doi:10.1515/crll.1859.56.285)
10. Kochmann, D. M. In press. Stability criteria for elastic composites and the influence of geometry on the stability of a negative-stiffness phase. Phys. Stat. Sol. (doi:10.1002/pssb.201084213)
11. Kochmann, D. M. & Drugan, W. J. 2009 Dynamic stability analysis of an elastic composite material having a negative-stiffness phase. J. Mech. Phys. Solids 57, 1122-1138. (doi:10.1016/j.jmps. 2009.03.002)
12. Kochmann, D. M. & Drugan, W. J. 2011 Infinitely-stiff composite via a rotation-stabilized negative-stiffness phase. Appl. Phys. Lett. 99, 011909.
13. Koiter, W. T. 1965 Energy criterion of stability for continuous elastic bodies I and II. Proc. K. Ned. Akad. Wet. B-Phys. Sci. 68, 178-202.
14. Lakes, R. 1987 Foam structures with a negative Poisson's ratio. Science 235, 1038-1040. (doi:10.1126/science.235.4792.1038)
15. Lakes, R. S. 2001 Extreme damping in compliant composites with a negative stiffness phase. Philos. Mag. Lett. 81, 95-100. (doi:10.1080/09500830010015332) (Pubitemid 34590569)
16. Lakes, R. S. & Drugan, W. J. 2002 Dramatically stiffer elastic composite materials due to a negative stiffness phase? J. Mech. Phys. Solids 50, 979-1009. (doi:10.1016/S0022-5096 (01) 00116-8) (Pubitemid 34247625)
17. Liu, Z., Chan, C. T. & Sheng, P. 2005 Analytic model of phononic crystals with local resonances. Phys. Rev. B 71, 014103. (doi:10.1103/PhysRevB. 71.014103) (Pubitemid 40486264)
18. Lyapunov, A. M. 1966 Stability of motion. New York, NY: Academic Press.
19. Mandel, J. 1962 Ondes plastique dans un milieu indéfini à trois dimensions. J. Mech. 1, 3-30.
20. Mandel, J. 1966 Conditions de stabilité et postulat de Drucker. In Rheology and solid mechanics (eds J. Kravtchenko & M. Sirieys), pp. 58-68. Berlin, Germany: Springer.
21. Mary, T. A., Evans, J. S. O., Vogt, T. & Sleight, A. W. 1996. Negative thermal expansion from 0.3 to 1050 Kelvin in ZrW2O8. Science 5, 90-92. (Pubitemid 126557294)
22. Milton, G. W. 2001 Theory of composites. Cambridge, UK: Cambridge University Press.
23. Morrey, C. B. 1952 Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2, 25-53.
24. Olver, F. W. J. 1970 Bessel functions of integer order. In Handbook of mathematical functions (eds M. Abramowitz & I. A. Stegun), NewYork, NY: Dover Inc.
25. Paul, B. 1960 Prediction of elastic constants of multi-phase materials. Trans. AIME 218, 36-41.
26. Pearson, C. E. 1956 General theory of elastic stability. Quart. Appl. Math. 14, 133-144.
27. Reuss, A. 1929 Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Z. Angew. Math. Mech. 9, 49-58. (doi:10.1002/zamm. 19290090104)
28. Shelby, R. A., Smith, D. R. & Schultz, S. 2001 Experimental verification of a negative index of refraction. Science 292, 77-79. (doi:10.1126/science.1058847) (Pubitemid 32289127)
29. Voigt, W. 1889 Über die Beziehung zwischen den beiden Elastizitätskonstanten isotroper Körper. Wied. Ann. 38, 573-587.
30. Wang, Y. C. & Lakes, R. S. 2001 Extreme thermal expansion, piezoelectricity, and other coupled field properties in composites with a negative stiffness phase. J. Appl. Phys. 90, 6458-6465. (doi:10.1063/1.1413947) (Pubitemid 34046823)
31. Wang, Y.-C. & Lakes, R. 2004 Negative stiffness-induced extreme viscoelastic mechanical properties: stability and dynamics. Phil. Mag. 84, 3785-3801. (doi:10.1080/14786430420 00282702)
32. Wang, Y.-C. & Lakes, R. 2006 Stability of negative stiffness viscoelastic systems. Quart. Appl. Math. 63, 34-55.
33. Ziegler, H. 1953 Linear elastic stability. Zeitschr. Angewandt. Math. Phys. (ZAMP) 4, 89-121. (doi:10.1007/BF02067575)