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.