×

Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases. (English) Zbl 1328.74075

Summary: We review the theoretical bounds on the effective properties of linear elastic inhomogeneous solids (including composite materials) in the presence of constituents having non-positive-definite elastic moduli (so-called negative-stiffness phases). Using arguments of Hill and Koiter, we show that for statically stable bodies the classical displacement-based variational principles for Dirichlet and Neumann boundary problems hold but that the dual variational principle for traction boundary problems does not apply. We illustrate our findings by the example of a coated spherical inclusion whose stability conditions are obtained from the variational principles. We further show that the classical Voigt upper bound on the linear elastic moduli in multi-phase inhomogeneous bodies and composites applies and that it imposes a stability condition: overall stability requires that the effective moduli do not surpass the Voigt upper bound. This particularly implies that, while the geometric constraints among constituents in a composite can stabilize negative-stiffness phases, the stabilization is insufficient to allow for extreme overall static elastic moduli (exceeding those of the constituents). Stronger bounds on the effective elastic moduli of isotropic composites can be obtained from the Hashin-Shtrikman variational inequalities, which are also shown to hold in the presence of negative stiffness.

MSC:

74Q20 Bounds on effective properties in solid mechanics
35B35 Stability in context of PDEs
74E30 Composite and mixture properties
74B05 Classical linear elasticity
70G75 Variational methods for problems in mechanics
35Q74 PDEs in connection with mechanics of deformable solids
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[2] Benveniste, Y., A new approach to the application of Mori-Tanaka׳s theory in composite materials, Mech. Mater., 6, 2, 147-157 (1987), URL 〈http://www.sciencedirect.com/science/article/pii/0167663687900056 〉
[3] Beran, M. J.; Molyneux, J., Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media, Q. Appl. Math., 24, 107-118 (1966) · Zbl 0139.42601
[4] Berryman, J. G., Long wavelength propagation in composite elastic media. II. Ellipsoidal inclusions, J. Acoust. Soc. Am., 68, 6, 1820-1831 (1980), URL 〈http://scitation.aip.org/content/asa/journal/jasa/68/6/10.1121/1.385172〉 · Zbl 0455.73014
[5] Budiansky, B., On the elastic moduli of some heterogeneous materials, J. Mech. Phys. Solids, 13, 4, 223-227 (1965), URL 〈http://www.sciencedirect.com/science/article/pii/0022509665900116〉
[6] Castaneda, P. P.; Willis, J. R., Variational second-order estimates for nonlinear composites, Proc. Math. Phys. Eng. Sci., 455, 1985, 1799-1811 (1999), URL 〈http://www.jstor.org/stable/53446〉 · Zbl 0984.74071
[7] Cherkaev, A., Variational Methods for Structural Optimization (2000), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0956.74001
[8] Christensen, R. M.; Lo, K. H., Solutions for effective shear properties in three phase sphere and cylinder models, J. Mech. Phys. Solids, 27, 4, 315-330 (1979), URL 〈http://www.sciencedirect.com/science/article/pii/0022509679900322〉 · Zbl 0419.73007
[9] Drugan, W. J., Elastic composite materials having a negative stiffness phase can be stable, Phys. Rev. Lett., 98, January, 055502 (2007), URL 〈http://link.aps.org/doi/10.1103/PhysRevLett.98.055502〉
[10] Ericksen, J. L.; Toupin, R. A., Implications of Hadamard׳s conditions for elastic stability with respect to uniqueness theorems, Can. J. Math., 8, 432-436 (1956) · Zbl 0071.39801
[11] Francfort, G. A.; Murat, F., Homogenization and optimal bounds in linear elasticity, Arch. Rational Mech. Anal., 94, 4, 307-334 (1986), URL 〈http://dx.doi.org/10.1007/BF00280908〉 · Zbl 0604.73013
[12] Hadamard, J., Lecons sur la propagation des ondes et les équations de l׳hydrodynamique (1903), Hermann: Hermann Paris · JFM 34.0793.06
[13] Hashin, Z., The elastic moduli of heterogeneous materials, J. Appl. Mech., 29, 143-150 (1962) · Zbl 0102.17401
[14] Hashin, Z.; Shtrikman, S., A variational approach to the theory of the elastic behaviour of multiphase materials, J. Mech. Phys. Solids, 11, 127-140 (1963) · Zbl 0108.36902
[15] Hill, R., The elastic behaviour of a crystalline aggregate, Proc. Phys. Soc. A, 65, 5, 349 (1952), URL 〈http://stacks.iop.org/0370-1298/65/i=5/a=307〉
[16] Hill, R., On uniqueness and stability in the theory of finite elastic strain, J. Mech. Phys. Solids, 5, 4, 229-241 (1957), URL 〈http://www.sciencedirect.com/science/article/pii/0022509657900169〉 · Zbl 0080.18004
[17] Hill, R., Elastic properties of reinforced solidssome theoretical principles, J. Mech. Phys. Solids, 11, 5, 357-372 (1963), URL 〈http://www.sciencedirect.com/science/article/pii/002250966390036X〉 · Zbl 0114.15804
[19] Hill, R., A self-consistent mechanics of composite materials, J. Mech. Phys. Solids, 13, 4, 213-222 (1965), URL 〈http://www.sciencedirect.com/science/article/pii/0022509665900104〉
[20] Jaglinski, T.; Frascone, P.; Moore, B.; Stone, D. S.; Lakes, R. S., Internal friction due to negative stiffness in the indium-thallium martensitic phase transformation, Philos. Mag., 86, 27, 4285-4303 (2006), URL 〈http://www.tandfonline.com/doi/abs/10.1080/14786430500479738〉
[21] Jaglinski, T.; Kochmann, D.; Stone, D.; Lakes, R. S., Composite materials with viscoelastic stiffness greater than diamond, Science, 315, 5812, 620-622 (2007), URL 〈http://www.sciencemag.org/content/315/5812/620.abstract〉
[22] Jaglinski, T.; Lakes, R. S., Negative stiffness and negative Poisson׳s ratio in materials which undergo a phase transformation, (Wagg, D.; Bond, I.; Weaver, P.; Friswell, M., Adaptive Structures (2007), John Wiley & Sons, Ltd), 231-246, URL http://dx.doi.org/10.1002/9780470512067.ch8
[23] Kashdan, L.; Seepersad, C.; Haberman, M.; Wilson, P., Design, fabrication, and evaluation of negative stiffness elements using SLS, Rapid Prototyp. J., 18, 194-200 (2012), URL 〈http://www.emeraldinsight.com/journals.htm?articleid=17015115 〉
[24] Kelvin, Lord (W. Thomson), On the reflection and refraction of light, Philos. Mag., 26, 414-425 (1888) · JFM 20.1112.01
[25] Kirchhoff, G., Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, J. Reine Angew. Math., 56, 285-313 (1859), URL 〈http://www.degruyter.com/view/j/crll.1859.issue-56/crll.1859.56.285/crll.1859.56.285.xml 〉 · ERAM 056.1494cj
[26] Knops, R. J.; Stuart, C. A., Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity, Arch. Rational Mech. Anal., 86, 3, 233-249 (1984), URL http://dx.doi.org/10.1007/BF00281557 · Zbl 0589.73017
[27] Kochmann, D. M., Stability criteria for continuous and discrete elastic composites and the influence of geometry on the stability of a negative-stiffness phase, Phys. Status Solidi B, 249, 1399-1411 (2012), URL http://dx.doi.org/10.1002/pssb.201084213
[28] Kochmann, D. M.; Drugan, W. J., Dynamic stability analysis of an elastic composite material having a negative-stiffness phase, J. Mech. Phys. Solids, 57, 7, 1122-1138 (2009), URL 〈http://www.sciencedirect.com/science/article/pii/S0022509609000301〉 · Zbl 1173.74309
[29] Kochmann, D. M.; Drugan, W. J., Analytical stability conditions for elastic composite materials with a non-positive-definite phase, Proc. R. Soc. A, 468, 2230-2254 (2012), URL 〈http://rspa.royalsocietypublishing.org/content/early/2012/03/14/rspa.2011.0546.abstract〉 · Zbl 1371.74013
[30] Koiter, W., Energy criterion of stability for continuous elastic bodies, Proc. K. Ned. Acad. Wet. B, 68, 178-202 (1965) · Zbl 0146.21103
[31] Lakes, R. S., Extreme damping in compliant composites with a negative-stiffness phase, Philos. Mag. Lett., 81, 2, 95-100 (2001), URL 〈http://www.tandfonline.com/doi/abs/10.1080/09500830010015332〉
[32] Lakes, R. S., Extreme damping in composite materials with a negative stiffness phase, Phys. Rev. Lett., 86, 2897-2900 (2001), URL 〈http://link.aps.org/doi/10.1103/PhysRevLett.86.2897〉
[33] Lakes, R. S.; Drugan, W. J., Dramatically stiffer elastic composite materials due to a negative stiffness phase?, J. Mech. Phys. Solids, 50, 5, 979-1009 (2002), URL 〈http://www.sciencedirect.com/science/article/pii/S0022509601001168〉 · Zbl 1032.74004
[34] Lakes, R. S.; Lee, T.; Bersie, A.; Wang, Y., Extreme damping in composite materials with negative-stiffness inclusions, Nature, 410, 6828, 565-567 (2001), URL 〈http://www.ncbi.nlm.nih.gov/pubmed/11279490〉
[35] Lee, C.-M.; Goverdovskiy, V., A multi-stage high-speed railroad vibration isolation system with negative stiffness, J. Sound Vib., 331, 4, 914-921 (2012), URL 〈http://www.sciencedirect.com/science/article/pii/S0022460X11007541〉
[36] Lee, C.-M.; Goverdovskiy, V.; Temnikov, A., Design of springs with negative stiffness to improve vehicle driver vibration isolation, J. Sound Vib., 302, 4-5, 865-874 (2007), URL 〈http://www.sciencedirect.com/science/article/pii/S0022460X07000235〉
[37] McCoy, J. J., On the displacement field in an elastic medium with random variations of material properties, Recent Adv. Eng. Sci., 5, 235-254 (1970)
[38] Milton, G. W., Bounds on the electromagnetic, elastic, and other properties of two-component composites, Phys. Rev. Lett., 46, 542-545 (1981), URL 〈http://link.aps.org/doi/10.1103/PhysRevLett.46.542〉
[39] Milton, G. W., Modelling the properties of composites by laminates, (Ericksen, J.; Kinderlehrer, D.; Kohn, R.; Lions, J.-L., Homogenization and Effective Moduli of Materials and Media. The IMA Volumes in Mathematics and its Applications, vol. 1 (1986), Springer: Springer New York), 150-174, URL 〈http://dx.doi.org/10.1007/978-1-4613-8646-9_7〉 · Zbl 0631.73011
[40] Milton, G. W., Theory of Composites (2002), Cambridge University Press: Cambridge University Press Cambridge, England · Zbl 0993.74002
[41] Milton, G. W., Universal bounds on the electrical and elastic response of two-phase bodies and their application to bounding the volume fraction from boundary measurements, J. Mech. Phys. Solids, 60, 1, 139-155 (2012), URL 〈http://www.sciencedirect.com/science/article/pii/S0022509611001748〉 · Zbl 1244.74050
[42] Milton, G. W.; Phan-Thien, N., New bounds on effective elastic moduli of two-component materials, Proc. R. Soc. Lond. A, 380, 1779, 305-331 (1982), URL 〈http://rspa.royalsocietypublishing.org/content/380/1779/305.abstract〉 · Zbl 0497.73016
[43] Moore, B.; Jaglinski, T.; Stone, D. S.; Lakes, R. S., Negative incremental bulk modulus in foams, Philos. Mag. Lett., 86, 10, 651-659 (2006), URL 〈http://www.tandfonline.com/doi/abs/10.1080/09500830600957340〉
[44] Mori, T.; Tanaka, K., Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall., 21, 5, 571-574 (1973), URL 〈http://www.sciencedirect.com/science/article/pii/0001616073900643〉
[45] Nemat-Nasser, S.; Hori, M., Micromechanics: Overall Properties of Heterogeneous Materials (1993), North-Holland: North-Holland Amsterdam · Zbl 0924.73006
[46] Norris, A. N., A differential scheme for the effective moduli of composites, Mech. Mater., 4, 1, 1-16 (1985), URL 〈http://www.sciencedirect.com/science/article/pii/016766368590002X〉
[47] Paul, B., Prediction of elastic constants of multiphase materials, Trans. Metall. Soc. AIME, 218, 36-41 (1960)
[48] Ponte Castaneda, P., The effective mechanical properties of nonlinear isotropic composites, J. Mech. Phys. Solids, 39, 1, 45-71 (1991), URL 〈http://www.sciencedirect.com/science/article/pii/002250969190030R〉 · Zbl 0734.73052
[49] Roscoe, R., The viscosity of suspensions of rigid spheres, Br. J. Appl. Phys., 3, 8, 267 (1952), URL 〈http://stacks.iop.org/0508-3443/3/i=8/a=306〉
[50] Roscoe, R., Isotropic composites with elastic or viscoelastic phasesgeneral bounds for the moduli and solutions for special geometries, Rheol. Acta, 12, 3, 404-411 (1973), URL http://dx.doi.org/10.1007/BF01502992
[51] Talbot, D. R.S.; Willis, J. R., Variational principles for inhomogeneous non-linear media, IMA J. Appl. Math., 35, 1, 39-54 (1985), URL 〈http://imamat.oxfordjournals.org/content/35/1/39.abstract〉 · Zbl 0588.73025
[53] Torquato, S., Random Heterogeneous Materials: Microstructure and Macroscopic Properties (2002), Springer: Springer New York · Zbl 0988.74001
[54] Walpole, L., On bounds for the overall elastic moduli of inhomogeneous systems, J. Mech. Phys. Solids, 14, 3, 151-162 (1966), URL 〈http://www.sciencedirect.com/science/article/pii/0022509666900354〉 · Zbl 0139.18701
[55] Wang, Y.; Lakes, R. S., Extreme thermal expansion, piezoelectricity, and other coupled field properties in composites with a negative stiffness phase, J. Appl. Phys., 90, 12, 6458-6465 (2001), URL 〈http://link.aip.org/link/?JAP/90/6458/1〉
[56] Wang, Y.-C.; Lakes, R. S., Stable extremely-high-damping discrete viscoelastic systems due to negative stiffness elements, Appl. Phys. Lett., 84, 22, 4451-4453 (2004), URL 〈http://link.aip.org/link/?APL/84/4451/1〉
[57] Wang, Y. C.; Lakes, R. S., Composites with inclusions of negative bulk modulusextreme damping and negative Poisson׳s ratio, J. Compos. Mater., 39, 18, 1645-1657 (2005), URL 〈http://jcm.sagepub.com/content/39/18/1645.abstract〉
[58] Wojnar, C. S.; Kochmann, D. M., A negative-stiffness phase in elastic composites can produce stable extreme effective dynamic but not static stiffness. 〈http://www.tandfonline.com/doi/abs/10.1080/14786435.2013.857795?journalCode=tphm20#.U772tbEkzzo〉, Philos. Mag., 94, 532-555 (2014)
[59] Wojnar, C. S.; Kochmann, D. M., Stability of extreme static and dynamic bulk moduli of an elastic two-phase composite due to a non-positive-definite phase, Phys. Status Solidi B, 251, 397-405 (2014), URL 〈http://onlinelibrary.wiley.com/doi/10.1002/pssb.201384241/abstract〉
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.