**Stability and performance of composites with negative-stiffness components**

#### Composite materials

Composites consist of two or more materials that can be combined in a variety of ways, e.g., as particulate inclusions in a matrix, as laminated phases, or as fiber-reinforced composites, to mention but a few. The composite's overall, effective properties stem from the combination of the constituents' properties and their specific arrangement in the composite. For the elastic characteristics of a composite, the Reuss and Voigt models correspond to the simple cases of laminated composite structures, and they provide lower and upper bounds on the attainable overall stiffness of the composite. For instance, for a two-phase composite, the model cases are sketched in the following graphics.In the Voigt model, the overall Young's modulus

*Ec*of the composite with two constituents A and B in the specified load direction is given by

where

*E*i denotes the stiffness of constituent

*i*, and

*V*i stands for its volume fraction in the composite. This model provides an upper bound to the composite stiffness. A lower bound is given by the analogous Reuss model which yields

Stricter bounds are known for isotropic materials in terms of the Hashin-Shtrikman relations. A comparison of all bounds on the composite's stiffness is shown in Figure 1.3 below, which marks the achievable regime of the overall stiffness when combining two materials, here with individual stiffness of 20 GPa and 100 GPa. Obviously, the overall stiffness can not surpass the stiffness of the stiffest constituent (here, 100 GPa). As a consequence, the stiffness of soft materials, e.g., can be enhanced by incorporating particles or fibers of a material of higher stiffness, but the overall stiffness encounters an upper bound in the stiffness of the stiffest constituent.

This observation is based on the assumption of positive stiffness for each constituent, which holds for conventional materials. However, if it one of the constituents has a negative stiffness, these bounds predict an anomalous behavior: in this case, the overall stiffness can be tuned to exceed the constituents' individual stiffness values by far, resulting in an ultrastiff composite material, as can be inferred from the aforementioned stiffness bounds. Let us examine the bounds on the overall stiffness of a two-phase composite that contains one constituent of negative stiffness

*E*1 < 0 and one constituent of positive stiffness

*E*2 > 0. Choose the simple case where

*V*1 =

*V*2 = 50 % and assume the particular case when

*E*1 approaches the value -

*E*2. The composite models yield bounds on the overall stiffness in this special case:

The Voigt model now simply provides a lower bound of zero stiffness. The Reuss model, however, indicates an upper bound on the composite stiffness that approaches infinity and hence exceeds the stiffness of each constituent by far. Hence, the classical bounds indicate an anomalous behavior: the common assumption that the overall stiffness is bounded by the stiffness of its stiffest constituents does not hold, when admitting negative stiffness in one constituent. Of course, these bounds are coarse and only represent extreme cases. The Hashin-Shtrikman relations provide more exact bounds on the admissible range of the overall stiffness of isotropic composites. The results, however, confirm the reported behavior, thus indicating an extreme increase in stiffness of the composite, when the stiffness properties are appropriately tuned. Figure 1.6 illustrates the overall composite moduli according to the Hashin-Shtrikman relations for one phase with a negative shear modulus. The graphic demonstrates the anomalies produced by the negative modulus of the particulate inclusions - a similar behavior can be shown for negative inclusion Young’s modulus, and for negative inclusion bulk modulus. (An analogous analysis can be performed for the overall damping capacity of a composite material containing one constituent of negative stiffness.)

The following figure illustrates the anomalies arising in a composite of a positive-stiffness matrix that contains only 5 vol.-%
of second phase particles of a varying negative bulk modulus.
Included in the graphic is the overall loss tangent (a measure of the composite's damping capacity).
Both the overall stiffness and the overall damping exhibit anomalous behavior due to the negative-stiffness inclusions. Besides, it becomes obvious that a very low volume fraction of negative-stiffness particles is sufficient to produce large mechanical anomalies of extremely high increases in stiffness and damping capacity.

#### Negative stiffness

Negative stiffness implies the inversion of the habitual relation between force and displacement. When a deformation is applied to a solid of positive stiffness, it reacts to the imposed deformation by a restoring force that acts against the imposed deformation. If the body exhibits negative stiffness, it reacts with a force that is no longer a restoring force opposing the applied deformation, but the material reacts with a force into the same direction as the applied deformation, thereby even increasing the effect of the external deformation.A buckled column illustrates the negative-stiffness effect (Wang and Lakes, 2004). Imagine a buckled ruler held between two fingers (see Figure 1.4). Lateral displacement of the midpoint of the buckled column initially results in a restoring force, signifying positive stiffness, until a critical state is reached when the ruler suddenly snaps through to the other side; i.e., the applied deformation is amplified, the ruler experiences an inner force into the same direction as the enforced deformation. The initially stable ruler configuration reaches an unstable state (when enough pressure or displacement is applied), where a small perturbation of load or deformation is sufficient to produce the snap-through effect. Another important feature of negative stiffness becomes obvious from the ruler example: negative stiffness in the ruler can only be stabilized by means of a sufficient geometric constraint; the bent ruler is unstable by itself, only the constraint given by the rigid support of the compressing fingers stabilizes negative stiffness in the ruler. In real-world materials, negative stiffness also represents an unstable state, so that no (static) free-standing body could be made of a negative-stiffness material. However, negative stiffness would be physically admissible, given a configuration of sufficient constraint. The sufficient conditions under which a material of negative stiffness can be stable against perturbations, is of high interest and has been examined for exemplary composite models.

From the classical theory of elasticity, we know the necessary and sufficient conditions of stability for a homogeneous, linear-elastic solid. Assume a linear-elastic body with infinitesimeal stresses σ and strains ε, which is governed by Hooke’s law in the general form (with

*C*the elastic stiffness tensor)

Necessary stability conditions can be obtained from requiring the real-valuedness of the speed of traveling waves, which results in the local Legendre-Hadamard conditions. Sufficient conditions of stability for a free-standing body (subject to pure traction boundary conditions or mixed boundary conditions) can be obtained from two independent approaches, which result in either the static or the kinematic stability conditions (which, however, coincide for linear-elastic media with conservative forces only). The body stores elastic strain energy

*U*with

where

*V*is the volume of the body. For a homogeneous, isotropic, linear-elastic body, the strain energy must be non-negative for any arbitrary displacement perturbations, from which the sufficient conditions can be derived. The following table gives an overview of the necessary and sufficient conditions of stability in two and three dimensions for a uniform homogeneous body (where two dimensions assume a state of plane strain).

It is interesting to note that the necessary conditions of stability are also the sufficient conditions of a homogeneous body with completely prescribed displacement boundary conditions (pure Dirichlet boundary conditions). In this case, a negative Young modulus and a negative bulk modulus are admissible for global stability, as long as the geometric constraint is present. Figure 1.5 gives on overview of the admissible ranges for Young’s modulus, the shear modulus, the bulk modulus and for Poisson’s ratio.

The above discussion has shown that negative stiffness is only admissible so long as the body of negative stiffness is subject to sufficient geometric constraints. Particles of negative stiffness are unstable by themselves, but the above discussion hints at a possible stabilization if a sufficient geometric constraint is present. Embedding particles in a stiff matrix material provides such a constraint on the particles, so that the composite might faciliate negative stiffness to be stable; the degree of constraint required from the matrix material is a key open question that has been addressed only recently.

#### Stability Conditions

While a macroscopic body of a material of negative stiffness is unstable, negative-stiffness particles subject to sufficient constraints can be stabilized. Drugan (2007) showed that spherical and cylindrical inclusions in a thin, concentric coating material may indeed exhibit negative stiffness with the composite being stable overall, if the coating is sufficiently stiff and sufficiently thick. In realistic composite materials, inclusions are surrounded by a matrix of finite stiffness and finite dimensions. We study the dynamic model of a coated cylinder (in two dimensions) and a coated sphere (in three dimensions), where the inclusion material is assumed to exhibit negative stiffness (see the schematic graphics below). Both materials are assumed homogeneous and isotropic, body forces are neglected. Rigid body motion is excluded and displacements have to remain finite. Compatibility of stresses and displacements across the interface between inclusion and coating is required; the outer boundary of the coating is assumed stress-free.For the given geometry, we first determine the general solution representation of the displacement field in the inclusion and the coating from the governing Navier equations. Then, we impose boundary and continuity conditions, which result in a characteristic equation to be solved for all eigenfrequencies (there is an infinite set of eigenfrequencies). Stability requires that the displacement field may not increase with time; i.e., all eigenfrequencies must be real-valued (for an elastic solid) or possess non-negative imaginary parts (otherwise, the eigenfrequency would result in a displacement yield that grows exponentially with time). From the infinite set of eigenfrequencies, we can determine those ranges of the elastic moduli of inclusion and coating materials as well as of the geometry (i.e., the radius ratio for the given examples). Our results confirm that the inclusion's elastic moduli may indeed violate positive-definiteness (i.e., have negative stiffness), if the coating is sufficiently thick and sufficiently stiff. The following figure illustrates the stability limit in the plane of elastic moduli for constant shear modulus ratio (subscripts inc and mat indicate inclusion and matrix properties, respectively,

*t*is the coating thickness and

*a*the inclusion radius). It becomes apparent that with increasing coating thickness, the stability limit gradually shifts to the left, thus allowing more negative stiffness in the inclusion. (The graphic compares the thin-coating approximate solution from an energetic approach to the full-coating thickness solution from the dynamic approach, which, of course, coincide in the limit of thin coatings.)

#### Means of Stabilization

Research in our group has investigated (and still investigates) various means of stabilization for phases of negative stiffness within a composite. Besides the geometric contraint enforced by the other constituent phases, there are myriad possibilities. One successful study has proved, e.g., that the gyroscopic forces in a rotating composite are sufficient to provide additional stabilization of a negative stiffness inclusion. Besides, anisotropy, the geometric arrangement, viscoelasticity (damping) and various other means of stabilization are subject to ongoing investigation.### Experimental investigations: tin-barium titanate composites

#### Negative-stiffness effect

Negative stiffness is mechanically unstable and cannot be observed in free-standing bodies. However, negative stiffness can be achieved near the transition temperature of materials that undergo a structural transition. The negative-stiffness effect then stems from the high amount of stored internal energy to be released upon perturbation. During a structural transition, the material’s atomic lattice rearranges resulting in sudden changes of the physical (and, in particular, mechanical) properties. Phase transitions are induced by applied stresses or strains, by changes in the temperatures (where the transformation occurs at a specific transformation temperature*T*c), or by changes of the electric fields, to name but a few. We have made use of the temperature-induced transformations of barium titnate in experiments. This transitional behavior commonly comes along with a hysteresis effect: undercooling and overheating are necessary in order to let the transition occur. Structural phase transitions are governed by Landau’s theory: the landscape of the free energy Ψ of a strained body has (at least) two minima at temperatures below the transition temperature and one minimum at temperatures above transition (

*Salje, 1990*). The free energy Ψ as from Landau's theory versus the strain order parameter ε at different temperature levels

*T*is sketched in the following graphic. A body in stable, mechanical equilibrium corresponds to a state of minimum potential energy. At temperatures above transition, the energy landscape is convex. Therefore, at temperatures above transition the body is in stable equilibrium in the global energy minimum (i.e., in the vertex of the energy parabola at ε = 0). As the temperature decreases below the transition temperature (

*T*less than

*T*c), the energy landscape deforms such that the energy minimum turns into a local maximum and two energy minima form (these minima are symmetric for the present example). Now, the body is in an unstable equilibrium with high internal stored energy at ε = 0. A small perturbation is sufficient to evoke a snap-through effect known as negative stiffness: the body is self-driven and seeks to reach the state of minium energy, thus creating a large inner force. Figuratively speaking, coming from high temperatures, the red ball in the figure lies in equilibrium at the minimum of the energy landscape. As the temperature is decreased below the transition temperature, the ball still lies at the same point but is now located on an energy hill. Hence, a small perturbation is sufficient to make the ball roll down the hill into one of the two energy valleys (yellow balls). The ball is self-driven; it will not offer any resistance to a small displacement away from its original state on top of the energy hill but will amplify the imposed perturbation by rolling down the hill to a stable energy minimum.

This concept of negative stiffness has been utilized in order to manufacture and test materials with inclusions of negative stiffness. Several ferroelectric materials show an energy landscape near their transition temperature that is similar to the one sketched above. These materials experience the snap-through effect of negative stiffness as temperature passes the transition temperature. Since negative stiffness, however, cannot be stable in a free-standing body, particles of such a transforming material are embedded in a stiff metal matrix that represents a sufficient geometric constraint to stabilize the negative-stiffness effect. Studies on VO2 particulate composites gave evidence for such behavior (Lakes et al., 2001; Wang, 2004).

Barium titanate (BaTiO3) is a ferroelastic material that has been subject to extensive investigations (Jona and Shirane, 1962). It has been used for practical applications because of its ferroelectric and piezoelectric properties near room temperature and its high chemical and mechanical stability. Furthermore, it is easily preparable in the form of ceramic polycrystals. Single-crystal BaTiO3 has a Curie temperature of

*T*c =120°C. Above this temperature, BaTiO3 appears as its non-polar cubic phase with a perovskite-type structure. Below the Curie temperature, BaTiO3 has a tetragonal structure that results from a tetragonal distortion of the high-temperature unit cell. The change in atomic structure results in a change of physical and mechanical properties, especially in a relative volume change (in polycrystalline barium titante) that has been observed experimentally. This sudden change in the elastic properties can give rise to negative moduli, when the structural transition occurs. According to theoretical considerations of stability, this transition in constrained BaTiO3 particles can give rise to temporarily negative moduli.

#### Experimental method

Experiments of Broadband Viscoelastic Spectroscopy (BVS) were conducted using an experimental set-up that allows for the determination of viscoelastic properties of a solid material over more than eleven decades with a single apparatus (*Lakes, 2004*). The specimens of rectangular or circular cross-section with a cross-sectional area of a few square millimeters and a mean length of about 2.5 cm were clamped into a steel support rod 1 inch in diameter. The free end of the specimen was glued to a cubic magnet that carried a mirror. Torque is applied to the specimen electromagnetically: the specimen is placed between two sets of Helmholtz coils that allow for testing in bending and in torsion with the same set-up. The set-up is located in a massive brass chamber combined with an inner mu-metal magnetic shield to reduce the influence of electromagnetic noise from the environment. The chamber also minimizes environmental air flow. Deformation is captured through electronic measurement of the displacement of a laser beam. As tests in torsion and bending are run on the same set-up, two different arrangements for detecting the displacement were made.

For viscoelastic tests in bending and torsion over a wide temperature range, the BVS instrument was modified in order to allow for controled heating and cooling of the specimen. The tests were performed with the specimen inserted into a ceramic heating tube. Air flow was used to guide heated air from an electronic heater inside the heating tube to the specimen. Air flow rates were controlled by means of a mechanical valve and measured using a analog airflow meter.

Experiments were carried out in the Micromechanics Laboratory (Prof. R. S. Lakes) at the University of Wisconsin - Madison, Department of Engineering Physics.

#### Results

Frequency scans showed a broad increase in tanδ below the first resonance, from 0.01 at 1 Hz to 0.04 below the first resonance. The normalized dynamic compliance displayed the typical behavior. Anomalies in terms of two large but shallow peaks near resonance appeared in both BaTiO3-Sn and pure tin and are hence concluded not to be due to the BaTiO3 inclusions.Although temperature scans of viscoelastic spectroscopy did not yield a regular pattern of anomalies, the presence of various anomalies in loss tangent and stiffness of BaTiO3-Sn composites near the transition temperature of barium titanate was significant in bending tests. A low volume percentage of BaTiO3 (between 1 and 10 vol.-%) was sufficient to make essential anomalies appear. Appearance of anomalies depended on sample-making technique and microstructure: cast specimens showed broader and more regular humps in the loss tangent than sintered specimens; results from powder-metallurgically manufactured specimens were highly irregular. Most striking anomalies were observed using cast specimens with unplated and Ni-plated barium titanate particles. Regularity of results increased with increasing uniformity of BaTiO3 particle size. Contrary to theoretical predictions (based on assumptions of perfect interface, isotropy and all inclusions having the same properties), anomalies were observed both irregularly in loss tangent and modulus. Temperature scans in torsion did not offer any anomalies in tanδ or stiffness. This can be explained by a lack of macroscopic volume change of the specimen when tested in torsion.

A cycle-dependence of the anomalies was found; existing anomalies often changed or gradually vanished with increasing number of thermal cycles. This effect can be explained by a weakening particle constraint with increasing thermal cycling due to excessive deformation, plastic yielding and debonding at the particle-matrix interface when the transition in barium titanate occurs.

An increase in tanδ by an average factor of 1.2 over a broad temperature range near the transition temperature (100°- 160°C) was observed with many of the tested specimens, showing an interesting increase similar to that predicted by composite theory. This effect usually decreased in height with increasing number of thermal cycles. Several tested specimens displayed a remarkable increase in stiffness, by approximate factors of up to 2. Ultimately, an extreme mechanical anomaly was observed from various several specimens with a BaTiO3 content of up to 10 vol.-%, which exhibi stiffness increases of several orders of magnitude (surpassing the Young modulus of diamond by far) and strong anomalies in the loss tangent. Thus, the negative-stiffness effect in the barium-titante inclusions gives rise to an extreme increase in the dynamic, oveall Young's modulus and an anomaly in the damping capacity, which greatly exceed the properties of each individual constituent (hence, these values can only be explained in terms of negative stiffness, according to classical composite theory). The following figure demonstrates such extreme increase in stiffness of a tin-barium titanate composite during the structural transition of the barium titanate particles.

From: T. Jaglinski, et al., Composite Materials with Viscoelastic Stiffness Greater than Diamond, *Science* **315**, 620-622 (2007).

##### For further reading:

- J.-B. le Graverend, C. S. Wojnar, D. M. Kochmann.
Broadband Electromechanical Spectroscopy: characterizing the dynamic mechanical
response of viscoelastic materials under temperature and electric-field control in a vacuum environment,
*J. Mater. Sci.***50**(2015), 3656-3685. - C. S. Wojnar, J.-B. le Graverend, D. M. Kochmann.
Broadband control of the viscoelasticity of ferroelectrics via domain switching,
*Appl. Phys. Lett.***105**(2014), 162912. - F. Fritzen, D. M. Kochmann.
Material instability-induced extreme damping in composites: a computational
study,
*Int. J. Solids Struct.***54**(2014), 4101-4112. - D. M. Kochmann, G. W. Milton.
Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases,
*J. Mech. Phys. Solids***71**(2014), 46-63. - C. S. Wojnar, D. M. Kochmann.
A negative-stiffness phase in elastic composites can produce stable
extreme effective dynamic but not static stiffness,
*Philos. Mag.***94**(2014), 532-555. - D. M. Kochmann.
Stable extreme damping in viscoelastic two-phase composites with
non-positive-definite phases close to the loss of stability,
*Mech. Res. Comm.***58**(2014), 36-45. - S. Krödel, T. Delpero, A. Bergamini, P. Ermanni, D. M. Kochmann.
3D auxetic microlattices with independently-controllable acoustic
band gaps and quasi-static elastic moduli,
*Adv. Eng. Mater.***16**(2014), 357–363. - C. S. Wojnar, D. M. Kochmann.
Stability of extreme static and dynamic bulk moduli of an elastic two-phase composite due to a non-positive-definite phase,
*Phys. Stat. Solidi B***251**(2014), 397-405. - D. M. Kochmann, W. J. Drugan.
Analytical stability conditions for elastic composite materials with a non-positive-definite phase,
*Proc. R. Soc. A***468**(2012), 2230-2254. - D. M. Kochmann, W. J. Drugan,
**Infinitely-stiff composite via a rotation-stabilized negative-stiffness phase**,*Appl. Phys. Lett.***99**(2011), 011909. - D. M. Kochmann, W. J. Drugan,
**Dynamic stability analysis of an elastic composite materials having a negative-stiffness phase**,*J. Mech. Phys. Solids***57**(2009), 1122-1138. - T. M. Jaglinski, D. M. Kochmann, D. S. Stone, R. S. Lakes,
**Composite Materials with Viscoelastic Stiffness Greater Than Diamond**,*Science***315**(2007), 620-622.