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Journal Club Theme of June 2012: Composites with Tunable Extreme Properties

Dennis M. Kochmann's picture

The past two months’ blogs have highlighted novel concepts in the design of phononic composites and artificially designed materials (often referred to as metamaterials) with carefully designed acoustic wave propagation or attenuation characteristics. This month focuses on a different class of recently proposed composites whose properties have the potential to reach extreme values of viscoelastic stiffness and damping, piezoelectricity, pyroelectricity, or thermal expansion. This concept of tunable composites is based on the paradigm of including one constituent in a mechanically metastable state, and it is exactly this (in)stability that, if appropriately tuned, can give rise to overall composite performance that greatly exceeds conventional behavior.

Violating Positive-Definiteness and the Question of Stability

About a decade ago Rod Lakes and coworkers proposed in a series of conceptual papers (see e.g. Lakes, 2001a, 2001b; Lakes et al., 2001, a detailed updated list of publications can be found here) to include so-called ­­negative stiffness elements in a composite material, where ­­negative stiffness refers to non-positive-definite elastic moduli (examples included viscoelastic elements with negative spring stiffness values, or solids whose elasticity tensor violates positive-definiteness thus admitting e.g. negative bulk or Young modulus values). Such negative stiffness is evidently unstable: the principles of thermodynamics dictate that e.g. a free-standing homogeneous linear elastic solid (with mixed Neumann/Dirichlet boundary conditions) must possess positive-definite elastic moduli and hence cannot have a negative bulk or Young modulus. Note that negative Poisson’s ratio is indeed permitted and has been realized by careful structural material design; this approach is pursued in the field of auxetic materials (see the previous blog by Katia Bertoldi, or more information here). In fact, positive definiteness of an isotropic solid in three dimensions only restricts Poisson’s ratio to lie between -1 and 0.5. However, negative bulk, shear or Young moduli are normally forbidden in a homogeneous linear elastic solid. Of course, in the context of e.g. soft materials, the incremental elastic constants change inhomogeneously with finite deformation; here, point-wise stability (i.e., incremental stability against infinitesimal perturbations) requires point-wise strong ellipticity of the incremental modulus tensor only (this is the necessary condition of stability, and e.g. Poisson's ratio is allowed to assume any value less than 0.5 or larger than 1). The sufficient condition of (global) stability then follows from Hill’s energy criterion (Hill, 1957) and may locally admit violation of positive-definiteness. However, in a homogeneous linear elastic solid with free surface, its elastic moduli must be positive-definite for overall stability and thus negative bulk, shear or Young moduli are forbidden. Interestingly, this restriction no longer holds if the solid is embedded in another material which enforces a geometric constraint. It could be shown that the ranges of admissible elastic moduli of linear elastic solids are weakened considerably when embedded e.g. in a stiff coating or in a stiff matrix in a composite (e.g., Drugan, 2007; Kochmann and Drugan, 2009, 2011, 2012). Each constituent in an elastic composite must possess strong ellipticity of its elasticities for overall stability (to ensure point-wise real-valued wave speeds); this rules out a negative shear modulus which is always unstable (and would result e.g. in microstructure formation). But not every composite phase must obey elastic positive-definiteness as has been shown for various cases of two-phase solids and composites. Global stability of e.g. a particle-matrix composite enforces weaker stability conditions on the inclusion elastic moduli than positive-definiteness: this allows for locally stable negative (incremental) bulk or Young modulus values in the inclusion phase.

Material Systems with Metastable Phases

Now that there is hope to stabilize negative values of some of the elastic moduli of constituents in a composite, why should one care? There are several reasons. Metastable states or negative stiffness can be found in material systems and structures that undergo instabilities; e.g. prestressed elastic systems can experience snapping behavior (see e.g. Douglas Holmes’ blog). The close-to-snapping states correspond mathematically to negative incremental stiffness which can be utilized in a composite system (see Lee et al., 2007 and Kashdan et al., 2012 for examples); a similar effect was found in buckled carbon nanotubes (Yap et al., 2007, 2008); electrostatic negative stiffness was studied as well. Our focus here is on composite materials. The structural transition in many phase-transforming ceramics and metals is accompanied by full softening of the (visco)elastic moduli that is indicative of the underlying material instability upon transformation. Systems of perovskite piezoceramics such as barium titanate as well as geomaterials such as quartz show this behavior upon temperature changes near their respective transition temperatures. Experiments have indeed confirmed the pronounced softening of e.g. barium titanate (Dong et al., 2010) near transformation, displaying close to vanishing elastic moduli. The above stability results indicate that in the presence of a stiff matrix material, this phenomenon can be expected to result in moduli whose softening does not stop at zero: the matrix stabilizes an otherwise unstable transition state. This is of great practical importance for many material systems with transforming and non-transforming constituents whose interplay can affect the transformation behavior (e.g., precipitates in shape memory alloys, or particle-reinforced composites with pierzoceramic inclusions).

Tunable Performance

The goal of composite design is to arrive at new materials that combine the beneficial properties of their ingredient materials. Unfortunately, classical composite bounds hint at the fact that the overall properties cannot surpass those of the individual constituents (see e.g. the Reuss-Voigt bounds or the stronger Hashin-Shtrikman bounds for isotropic media). Combining materials of different elastic moduli will result in a new material with moduli somewhere in between the original values. However, Lakes and Drugan (2002) reported that the overall moduli of a composite (made of homogeneous linear elastic materials) might increase by far beyond the moduli of their constituents, if one of the constituent materials possesses negative values of some of its elasticities (and if the elastic moduli and composite arrangement are appropriately tuned). This concept was explored in a series of models which showed that not only can the incorporation of such normally unstable elements achieve high (visco)elastic moduli but also can one expect anomalously high increases in the composite’s damping capacity (Lakes, 2001a, 2001b; Lakes et al., 2001). For simplicity, we have restricted our discussion here to the viscoelastic properties; however, one can argue analogously to predict extreme increases in e.g. the piezoelectric or pyroelectric properties as well as thermal expansion (Wang and Lakes, 2001). Triggering the anomalous performance by temperature-, pressure- or otherwise induced transformation mechanisms facilitates tunability of the material properties. While the stabilization of extreme properties has not been reported possible in static systems, dynamic systems have been successfully explored to exhibit the predicted response.

Realization and Controlability

This concept has resulted in a number of interesting experiments in which in particular high dynamic viscoelastic moduli and/or high damping were achieved (by dynamic moduli, we refer to the absolute values of the dynamic moduli; whereas damping is measured in terms of the time-domain phase lag between stress and strain response, given by the ratio of loss to storage modulus in a viscoelastic solid). Such systems could be beneficial in vibration-isolation devices that required high damping and high stiffness at the same time (high damping in soft materials such as polymers or rubbers is ubiquitous but these materials suffer from rather low mechanical stiffness and strength compared to structural metals and ceramics whose excellent stress-strain response for engineering applications is compromised by their poor vibration damping capabilities). The aforementioned design paradigm promises to combine both beneficial properties in one system: high stiffness and high damping. Examples have been given, among others, for phase-transforming inclusions of barium titanate (Jaglinski et al., 2007) or vanadium dioxide (Lakes et al., 2001) in stiff matrices; in all cases strong anomalous increases in dynamic stiffness and/or damping have been detected experimentally. These experiments have confirmed the extreme impact on the overall composite properties and promise interesting further developments. As both stability and performance of such materials still pose many open questions, they are subject to ongoing research in many groups.


Dennis, thank you for your posting on metamaterials.

Even though I do not have any expertise in the metamaterials, especially materials that exhibit unique material properties such as negative Poisson's ratio, I have recently found a paper (Nicolaou and Motter, Nat. Mater., in press) that reports how to design materials (based on atomic lattice structure) for tailoring the compressibility, particularly negative compressibility. The design of atomic structure for metamaterial that possesses the negative compressibility is based on molecular dynamics simulation. The paper by Nicolaou and Motter may be also of interest and relevant to the journal club.

Dennis M. Kochmann's picture

Dear Kilho,

Thank you for pointing out this interesting reference!

The design of materials on the atomistic and molecular levels has indeed been of great interest for a while (particularly incited by modern and ever improving techniques of materials synthesis and fabrication), including for example the work from the Evans group on negative Poisson's ratio materials by molecular network design. 

This paper by Nicolaou et al. is indeed very interesting as it reports a molecular-level design with potential to yield materials with overall stable negative compressibility, where the reported negative compressibility refers to a spontaneous stress-induced transformation that does not imply negative incremental elastic moduli. Such stress-induced transformation softening has been observed experimentally in solids undergoing structural transitions, commonly as pronounced transformation softening (see e.g. Ding et al.), similar to the temperature-induced softening observed e.g. in barium titanate (Dong et al.) which is the effect that was utilized to achieve high dynamic stiffness and damping in the above examples.

Again, thanks for bringing this to our attention!


Amir Shakouri's picture


Dear Dennis,


It is very nice to have you here and thanks a lot for sharing your perspective on the on-going research in the field of metamaterials. I am particularly inspired by your contributions on "negative (incremental) elastic moduli". But, then this question came to my mind that if we consider that the stiffness of a solid comes from the bondings between the atoms which are often acting as some sorts of a generalized spring, what would be the atomistic interpretation for a negative elastic modulus?


Best Regards,


Dear. Amir,

When you assume that the lattice structure is modeled as harmonic spring network (i.e. inter-atomic potential is regarded as harmonic potential), the negative incremental stiffness cannot be found for such lattice structure. In other words, the harmonic system cannot possess the unstable equilibrium (and also negative incremental stiffness). On the other hand, in general, the lattice structure cannot be modeled as a simple harmonic system due to the fact that inter-atomic interaction is usually anharmonic. The negative incremental stiffness of a metamaterial is attributed to the unstable equilibrium due to anharmonic potential. This issue has been well described in a paper by Nicolaou, et. al., who showed that the negative incremental stiffness originates from anharmonic potential such as a cubic potential that leads to bifurcation in the mechanical process.  

Dennis M. Kochmann's picture

Dear Amir,

Great to see you here, and many thanks for your post!

On the atomic or molecular level one can, in principle, indeed find negative incremental stiffness when one considers complex periodic networks as was done e.g. in the context of auxetic materials (see my links above). If you consider the representative unit cell shown in the figure below and assume that each line is a harmonic spring (with e.g. red springs being more compliant than the stiff black springs), then volumetric compression runs into a bistable state before the unit cell exhibits a snapping effect (the red springs snap inside). Around the point of snapping, the system has negative compressibility. Of course, there is still a question of stability (the system will go unstable and snap right before its effective compressibility becomes negative but a stiff matrix phase can be used to constrain amounts of negative effective incremental stiffness). The paper by Nicolaou and Motter also uses a structural solution to achieve negative stiffness. Besides, phase transformations in solids can give rise to pronounced softening of the elastic modul (during such transformation, when the material is constrained within a stiff matrix, it is theorized that the inclusion phase may indeed exhibit negative incremental moduli during transformation).

All the best,




negative compressibility cell

Amir Shakouri's picture


Dear Dennis,

Dear Kilho,


Thanks for replying to my post.

I am not sure that the two answers I recieved from you are actually the same but, I think, I got a main point from Dennis's explanations. After all, I really need to go through the details of the papers you recommended to find out more.




Mike Ciavarella's picture

dear Dennis

 I do not want to sound harsch,particularly being in germany now with a humboldt fellowship, but I am not convinced these line of research is leading really to new "materials".  Can you convince us that it is not just playing with mathematical theorems?  I read you make examples, but then you refer to many open questions, rather than real applications..... Anyway, I am sure other people would pose you this question, so I take the blame for doing it!


Michele Ciavarella, Politecnico di BARI

Dennis M. Kochmann's picture

Dear Mike,

I appreciate your honest critique, and I must admit that I was expecting such response sooner or later (because it always comes up when discussing this topic). Your point is well taken that so far there has not been a practical application in the sense that a new material could be fabricated for technological applications that posseses the described extreme properties. All experiments have been proof-of-concept, no negative-stiffness metamaterial is on the market yet but:

1) The concept of using "negative stiffness" to achieve high damping and stiffness has indeed been used in the past in a structural context for various technological applications, e.g. for vibration insulation of optical tables and scientific devices. I agree that these are not actual materials but structures, but on the one hand this shows that the concept itself is indeed applicable, and on the other hand with the advent of scalable nanomanufacturing the frontier between structures and materials tends to vanish at tremendous pace (which will open new pathways in the future to design architectured metamaterials).

2) At least in my view, it is neither surprising nor discouraging to not see any practical applications yet. The topic is very complex and intricate (and, as you say, many open questions remain concerning both composite performance and stability), which means we'll need more time to explore. These examples may sound too high-level but: after predicting materials with negative refractive index in 1967, Veselago had to wait more than 30 years until the first such metamaterial was actually fabricated; similarly, since the first materials with negative Poisson's ratio where fabricated by Lakes and coworkers in 1987 as an academic proof-of-concept example, those materials have transformed into a whole field of research with myriad practical applications.

So, I'm optimistic that, given time and more fundamental research (both theoretical and experimental), this concept will eventually transform into new materials, as you say. Of course, this is only my opinion -- and I'm very curious to hear from others in this field of research about their perspectives!

All the best,



P.S.: Enjoy your time in Germany (I'm in California)!

nicoguaro's picture

Dear Mike,

Talking about applications in metamaterials I think there are a wide field. This company in New Zealand is working on it: .


I cite some patents in this topic:

Boeing International (2010). Lens for Scanning Angle Enhancement of Phased Array Antennas. US2010079354 (A1).

Duke University (2010). Metamaterial for surfaces and waveguides. WO2010021736 (A2).

Hewlett Packard (2009). Metamaterial structures for light processing and methods of processing light. CN101595609 (A).

Lockheed Martin (2010). Horn antenna, waveguide or apparatus including index dielectric material WO2010039340 (A1)

Los Alamos National Security Laboratory (2010). Dynamical frequency tuning of electric and magnetic metamaterial response. WO2009048616 (A2).

Lucent Technology (2010). Conductive polymer metamaterials. US2010086750 (A1).

Rayspan Corporation (2010). Metamaterial loaded antenna devices). WO2010033865 (A2)


As an example of a theoretical "game" I want to mention Laser development. Laser is a device that was first completely developed from the theoretical point of view and decades later appeared as a functional one.



Nicolás Guarín


I can imagine people describing Newton's law of universal gravitation or Einstein's theory of relativity as mathematical curiosities with no practical applications when they were developed.

My GPS, however, begs to differ.


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