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Published November 2014 | Submitted
Journal Article Open

Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases


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.

Additional Information

© 2014 Elsevier Ltd. Received 16 January 2014, Revised 15 June 2014, Accepted 21 June 2014, Available online 5 July 2014. The authors would like to thank the reviewers for their valuable and thoughtful comments. D.M. Kochmann acknowledges support from NSF through CAREER award CMMI-1254424. G.W. Milton acknowledges NSF support through grant DMS-1211359.

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