CaltechAUTHORS
  A Caltech Library Service

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

Kochmann, Dennis M. and Milton, Graeme W. (2014) Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases. Journal of the Mechanics and Physics of Solids, 71 . pp. 46-63. ISSN 0022-5096. https://resolver.caltech.edu/CaltechAUTHORS:20141120-133810593

[img] PDF - Submitted Version
See Usage Policy.

1MB

Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20141120-133810593

Abstract

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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/j.jmps.2014.06.010DOIArticle
http://www.sciencedirect.com/science/article/pii/S0022509614001367PublisherArticle
https://arxiv.org/abs/1401.4142arXivDiscussion Paper
ORCID:
AuthorORCID
Kochmann, Dennis M.0000-0002-9112-6615
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.
Group:GALCIT
Funders:
Funding AgencyGrant Number
NSFCMMI-1254424
NSFDMS-1211359
Subject Keywords:Stability; Elasticity; Composite; Negative stiffness; Effective properties
Record Number:CaltechAUTHORS:20141120-133810593
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20141120-133810593
Official Citation:Dennis M. Kochmann, Graeme W. Milton, Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases, Journal of the Mechanics and Physics of Solids, Volume 71, November 2014, Pages 46-63, ISSN 0022-5096, http://dx.doi.org/10.1016/j.jmps.2014.06.010. (http://www.sciencedirect.com/science/article/pii/S0022509614001367)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:52009
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:20 Nov 2014 21:55
Last Modified:03 Oct 2019 07:38

Repository Staff Only: item control page