Instabilities in solids and structures are ubiquitous across all length and time scales, and engineering design principles have commonly aimed at preventing instability. However, over the past two decades, engineering mechanics has undergone a paradigm shift, away from avoiding instability and toward taking advantage thereof. At the core of all instabilities—both at the microstructural scale in materials and at the macroscopic, structural level—lies a nonconvex potential energy landscape which is responsible, e.g., for phase transitions and domain switching, localization, pattern formation, or structural buckling and snapping. Deliberately driving a system close to, into, and beyond the unstable regime has been exploited to create new materials systems with superior, interesting, or extreme physical properties. Here, we review the state-of-the-art in utilizing mechanical instabilities in solids and structures at the microstructural level in order to control macroscopic (meta)material performance. After a brief theoretical review, we discuss examples of utilizing material instabilities (from phase transitions and ferroelectric switching to extreme composites) as well as examples of exploiting structural instabilities in acoustic and mechanical metamaterials.

# Exploiting Microstructural Instabilities in Solids and Structures: From Metamaterials to Structural Transitions PUBLIC ACCESS

**Dennis M. Kochmann**

**Katia Bertoldi**

Manuscript received May 4, 2017; final manuscript received September 13, 2017; published online October 17, 2017. Editor: Harry Dankowicz.

*Appl. Mech. Rev*69(5), 050801 (Oct 17, 2017) (24 pages) Paper No: AMR-17-1031; doi: 10.1115/1.4037966 History: Received May 04, 2017; Revised September 13, 2017

The long history of mechanics is rife with prominent examples of instabilities in solids and structures that have led to material failure or structural collapse, spanning all scales from atomic-scale void growth, failure, and stress-induced transformations to buckling and delamination in micro- and nano-electronics all the way up to tectonic events. Examples of causal mechanisms for mechanical instability are ubiquitous: buckling of structures, crushing of cellular solids, plastic necking, strain localization in shear or kink bands, wrinkling and crazing, void growth, fracture, and collapse. Through its more than 250-year-old history—going back to Euler's buckling studies [1–3] and involving such prominent mechanicians as Kirchhoff [4], Love [5], and the Lords Kelvin [6] and Rayleigh [7]—the theory of material and structural stability has resulted in analytical theories and numerical tools that have provided engineers with safe design guidelines. Starting with structures and advancing to continuous media within the frameworks of linear and later nonlinear elasticity as well as inelasticity, the theory of stability in solids and structures has evolved and resulted in many seminal contributions, see, e.g., Refs. [8–19] for a nonexhaustive list of classics in the field.

Over the past two decades, an exciting paradigm shift has been initiated away from traditional *design against instability* and toward novel *performance through controlled instability*: engineering mechanics is exploring the advantages of operating systems near, at, or beyond the critical point. (In)stability is instrumentalized for beneficial material or structural behavior such as, e.g., soft devices undergoing dramatic shape changes [20–22], propagating stable signals in lossy media [23] or controlling microfluidics [24], large reversible deformation of cellular solids [25,26], morphing surfaces and structures [27,28], high-damping devices and energy-absorbing technologies from nanotubes to macrodevices [29–32], acoustic wave guides and metamaterials [33–36], composites with extreme viscoelastic performance [37–40], or materials with actively controllable physical properties [41,42]. Especially at the macroscopic level, structural instability and the associated large deformation of soft matter have produced many multifunctional devices that exploit buckling, snapping, and creazing to result in beneficial acoustic or mechanical performance, see Ref. [43] for a recent survey. Like in structures, instability at the material level stems from non-(quasi)convex energy landscapes [44,45]; here, however, instability is not tied to multiple stable or metastable structural deformation modes but involves multiple stable microstructural configurations such as those in phase transitions and phase transformations [46], domain switching and domain patterning [47–49], deformation twinning [50,51], or strain localization, shear banding, and patterning in plasticity [52]. From a mathematical standpoint, non-(quasi)convex energetic potentials entail instability of a homogeneous state of deformation, thus leading to energy-minimizing sequences which, physically, translate into complex microstructural patterns, see, e.g., Refs. [49] and [52–56].

In addition to structures and materials, metamaterials—somewhere in the diffuse interface between solids and structures—have gained popularity in recent decades. These engineered media commonly have a structural architecture at smaller scales, which is hidden at the larger scale where only an effective medium with effective properties is observed. The advent of additive manufacturing has fueled the design of complex metamaterials, see, e.g., Refs. [57–59], with instabilities now occurring across multiple scales.

Here, we present a survey of exploiting instabilities in solids and structures. Since the field has grown tremendously in recent years, we limit our review to concepts that exploit microscale instabilities to effect macroscale behavior, which applies to both structures (e.g., trusses or architected metamaterials) and materials (e.g., composites and other heterogeneous solids). The effective, macroscopic properties of heterogeneous solids and metamaterials are commonly identified by averaging over a representative volume element (RVE), including asymptotic expansions [60,61], numerical tools based on periodic boundary conditions (BCs) [62–65], as well as various upper and lower bounds on effective properties, especially for linear properties such as the (visco)elastic moduli, see, e.g., Ref. [66]. Besides the quasi-static mechanical properties such as stiffness, strength, and energy absorption, a major focus of architected metamaterials has been on wave motion. Here, the effective wave propagation characteristics of periodic, microstructured media can be obtained from the dispersion relations computed at the RVE level [67,68]. Unlike in conventional continuous media, structural metamaterials such as multiscale truss lattices can be exploited to independently control elastic moduli and dispersion relations [69]. We note that if the wavelength approaches the RVE size (in particular in acoustic metamaterials), classical homogenization approaches may lose their validity, which is why new theories have been proposed for the effective property extraction in more general settings, e.g., see Refs. [70–75], which is closed under homogenization and can be linked to Bloch–Floquet analysis [76]. Alternatively, computational techniques have obtained the effective dynamic performance even in transient, nonlinear, or inelastic settings [77,78]. These concepts have been applied to both materials and structures, resulting in interesting, extreme, peculiar, or controllable effective (meta)material properties to be reviewed in the following.

We first give a concise review of the fundamental concepts of stability theory to the extent required for subsequent discussions, followed by a survey of first material and then metamaterial strategies that take advantage of instability. Finally, we discuss the state-of-the-art and point out ongoing directions and topical opportunities.

From a theoretical viewpoint, stability problems have traditionally been treated by either *energetic principles*, linking stability to the uniqueness of solutions or minima in the potential energy landscape [79–81] or alternatively by *dynamic analysis*, defining stability as bounded solutions over time in the sense of Lyapunov [82,83]. Both definitions coincide for conservative systems under small perturbations [84] but may deviate otherwise (e.g., in the presence of gyroscopic forces [85]) where only dynamic methods generally yield the correct stability conditions [86].

For a dynamical system characterized by $\u2202u/\u2202t=f(u,t,\gamma )$ with unknown variables $u\u2208\mathbb{R}n$, time $t\u2208\mathbb{R}$, control parameter $\gamma \u2208\mathbb{R}$, and generally nonlinear driving forces $f:\mathbb{R}n\xd7\mathbb{R}\xd7\mathbb{R}\u2192\mathbb{R}n$, a solution $u0(t)$ is *stable* if for any bounded perturbation $\epsilon v$ such that $u(t)=u0+\epsilon v$

uniformly for all $t>t0$ after some initial time *t*_{0}. A solution is *asymptotically stable* if

for sufficiently small $\epsilon >0$ [82,83]. For elastic solids, perturbations oftentimes occur in the form of free vibrations about an equilibrium solution $u0$. In this case, one has a harmonic perturbation

with amplitude $v\u0302$ and frequency $\omega \u2208\mathbb{R}$ (or, more generally, superpositions of harmonic solutions of the above form with different eigenfrequencies). Stability requires $Im(\omega )\u22650$ to avoid exponential growth of $v(t)$ with time. Note that for linear elasticity, we must have $\omega \u2208\mathbb{R}$ for stable waves, so that instability implies $\omega 2<0$.

The system is in a state of (static) *equilibrium* when $f(u,\gamma )=0$, which may have one or more (or no) solutions. Upon perturbation, we have

*equilibrium path*. At a critical point, $\u2202f/\u2202u$ becomes singular so that there exists either no solution or a nonunique solution (point of

*bifurcation*). Note that imperfections generally lower the critical point (i.e., they reduce the critical load

*γ*) and affect the postbifurcation behavior, which is of importance both in structures (where, e.g., geometric or fabrication-induced imperfections may dominate the mechanical response) and in materials (where, e.g., thermal fluctuations lead to temperature-dependent transformation and switching kinetics).

The previously mentioned notions of stability can be applied to discrete systems and continuous solids alike. A continuous body $\Omega \u2282\mathbb{R}d$ is described by the deformation mapping $\phi :\Omega \xd7\mathbb{R}\u2192\mathbb{R}d$, so that the deformed and undeformed positions, $x$ and $X$, respectively, are linked via $x=\phi (X,t)$. Such body is governed by linear momentum balance

are the components of the first Piola-Kirchhoff stress tensor, $FiJ=\phi i,J$ denotes the deformation gradient, *D*/*Dt* is the material time derivative, *ρ*_{0} is the mass density in the reference configuration, and *W* is the strain energy density. Moreover, $\rho 0b$ is the body force field, which is assumed to be known and independent of deformation. Here and in the following, we use index notation with Einstein's summation convention and comma indices denoting partial derivatives. Note that the above applies directly to elastic media and can also be extended to inelastic continua when using a variational formulation, see, e.g., Refs. [52] and [87].

Instabilities are often investigated in the framework of incremental deformations $\phi \u02d9$ superimposed upon a given equilibrium state of finite deformation $\phi 0$. Let us consider a perturbation that takes the body to a new equilibrium configuration where Eq. (5) is still satisfied and leaves the body force density unchanged. The incremental problem is governed by

where a dot denotes a small increment in the respective quantity caused by the perturbation. Assuming that all incremental quantities are sufficiently small (i.e., infinitesimal), the constitutive equation (6) can be linearized as

with the mixed elasticity tensor given by

where $F0=Grad\phi 0$. Finally, we note that it is often convenient to investigate instabilities by formulating the incremental boundary value problem in an updated-Lagrangian formulation, where the reference configuration changes with time and is identified with the current configuration. In this case, the linearized form of the governing equation in the current configuration can be expressed as

where $u:\phi 0(\Omega )\xd7\mathbb{R}\u2192\mathbb{R}d$ represents the incremental displacement field in the spatial description, and $\rho =\rho 0\u2009detF$ denotes the current density. Employing push-forward transformations based on linear momentum, one obtains

Note that in the special case of linear elastic media, the elastic moduli are independent of deformation, so that $C0=const$ (and one often omits the superscript).

When considering the (in)stability of continuous solids, one generally differentiates between pointwise stability (which refers to material-level stability at smaller scales or localization) and structural stability (which depends on the specific macroscopic boundary value problem and corresponds to bifurcation). In the context of (in)stability in structures, analogous concepts exist for short- and long-wavelength instabilities, see, e.g., Refs. [17], [88], and [89].

The conditions of *material stability* or *pointwise stability* are local in nature and derive from the (pointwise) governing differential equations of the medium. Consider, e.g., a solid which becomes locally unstable and localizes deformation through the formation of a discontinuity. This implies a jump in the velocity gradient field, specifically $[[\phi \u02d9i,J]]=aiNJ$, where $N$ denotes the unit normal on the plane of discontinuity, and $a$ characterizes the jump. Equilibrium across the discontinuity implies that $[[P\u02d9iJNJ]]=0$. Insertion into the incremental balance equation (8) gives the condition for localization

for some choice of $a,N\u22600$. Stability requires that localization does not occur for all directions, so that pointwise/material stability is ensured by

Alternatively, pointwise stability conditions are obtained from a linear dynamic analysis. Linear momentum balance in a linear elastic medium is governed by Eq. (7) with Eq. (8), so that one may seek separable solutions of the incremental displacement field of traveling wave form. To this end, consider the motion of a plane wave propagating at some speed *c* in the direction characterized by normal vector $N$, so that most generally $\phi \u0307=u\u0302\u2009f(ct\u2212N\xb7X)$ with some differentiable $f:\mathbb{R}\u2192\mathbb{R}$. The linearized form of linear momentum balance, Eq. (7), now becomes

Consequently, real-valued wave speeds require that the acoustic tensor

For the special case of homogeneous, isotropic, linear elasticity, we have

where *λ* and *μ* are the Lamé moduli; let $\kappa =\lambda +(2/3)\mu $ denote the bulk modulus in three-dimensional (3D) (for two-dimensional (2D) conditions of plane strain, we have instead $\kappa =\lambda +\mu $). The longitudinal and shear wave speeds in 3D are obtained from the eigenvalue problem (14) as, respectively,

so that strong ellipticity imposes the necessary conditions of stability (in 3D) as

The conditions of *structural stability* or *global stability*, in contrast, are nonlocal and ensure stability of an overall body in dependence of the boundary conditions. Satisfying those guarantees that an arbitrary infinitesimal perturbation of the displacement field from an equilibrium state remains bounded for all time. The conditions of structural stability are derived from either energetic considerations, enforcing uniqueness of solutions [4,44,79,95] or from a dynamic approach that seeks to analyze the eigenmodes of a free vibration with respect to stability in the sense of Lyapunov [82].

Starting with a separable solution of the infinitesimal displacement field (with $\omega \u2208\mathbb{R}$ in general)

linear momentum balance for a linear elastic body (or a general body in the linearized setting), Eq. (10), becomes

Multiplication by $u\u0302i$, integration over Ω, and application of the divergence theorem lead to the eigenvalue problem

Since the integral on the left-hand side is non-negative by definition, real-valued wave speeds (i.e., $\omega \u2208\mathbb{R}$ or $\omega 2>0$), and therefore, the stability of an elastic body requires that

For a homogeneous linear elastic body with constant moduli, positive-definiteness of $C0$ is therefore a sufficient condition of stability (which is weaker than the necessary condition of ellipticity). When introducing the concept of negative stiffness in Sec. 3.1.1, we will refer to a body with nonpositive-definite elastic moduli as one having *negative stiffness*.

For a homogeneous, isotropic, linear elastic body satisfying Eq. (16), the conditions of positive-definiteness (in 3D) reduce to

which are tighter than those of strong ellipticity in Eq. (18). This is illustrated in the map of Fig. 1 by the shaded regions (positive definiteness imposing tighter restrictions on the moduli than ellipticity). As shown, the isotropic moduli can also be expressed in terms of engineering measures such as Young's modulus and Poisson's ratio

respectively, which yields the classical bounds $\u22121<\nu <1/2$ for positive-definiteness in 3D. Note that, if a body is rigidly constrained on its entire surface $\u2202\Omega $, then ellipticity and Eq. (18) become the sufficient conditions of stability—in that case, negative Young's and bulk moduli as well as Poisson's ratios outside the above bounds can be stable, see, e.g., Refs. [94] and [96].

Equation (21) can be rearranged to yield Rayleigh's quotient, viz.,

which bounds the lowest eigenfrequency from above. Since the denominator is by definition positive and stability of a linear elastic body again requires $\omega \u2208\mathbb{R}$ and thus $\omega 2\u22650$, the numerator must be positive for elastic stability. This ensures stability in the sense of dynamic, elastic systems.

With the advancement of computational and experimental capabilities, multiscale investigations have gained importance and have linked stability to effective properties. In media with two (or more) relevant length scales, where a separation between micro- and macroscales may be assumed, homogenization techniques extract macroscale properties from representative unit cells at the microscale, see, e.g., Refs. [64] and [65]. Here, (pointwise) stability at the macroscale has been linked to structural stability at the microscale [13,62,97]. When including materials that violate elastic positive-definiteness such as in solids undergoing structural transitions, extra care is required as common assumptions of continuum elasticity theory may no longer hold [98,99].

Here and in the following, we generally refer to larger and smaller scales as *macro-* and *microscales*, irrespective of the particular length scales involved. The *effective* quasi-static macroscale properties are then defined as averages over an RVE, defining

For example, in linear elasticity, effective stress and strain tensors $\u2329\sigma \u232a$ and $\u2329\epsilon \u232a$ are linked by an effective modulus tensor $C*$ such that $\u2329\sigma ij\u232a=Cijkl*\u2329\epsilon kl\u232a$, see Ref. [66]. In case of linear viscoelastic behavior, harmonic stress and strain fields $\sigma (x,t)=\sigma \u0302(x)exp(i\omega t)$ and $\epsilon (x,t)=\epsilon \u0302(x)exp(i\omega t)$, respectively, with generally complex-valued amplitudes are linked through a complex-valued effective modulus tensor $C*$ such that $\u2329\sigma \u0302ij\u232a=Cijkl*\u2329\epsilon \u0302kl\u232a$, see Refs. [100] and [101]. For particular loading scenarios, one can characterize the effective material *damping* by the loss tangent $tan\u2009\delta $, defined by the phase lag *δ* in the time domain. For example, for uniaxial loading, we have $\epsilon (t)=\epsilon \u0302(t)exp(i\omega t)$ and $\sigma (t)=\sigma \u0302\u2009exp(i(\omega t\u2212\delta ))$. In case of nonlinear material behavior, the effective response depends on deformation and, e.g., the effective incremental modulus tensor is given by

As pointed out earlier, both local and global instabilities may occur in solids and structures. In recent years, both of them have been exploited to produce interesting, peculiar, extreme, or beneficial macroscopic material behavior. Most such concepts can be grouped into two categories to be explained in great detail in Secs. 3.1 and 3.2. Section 3.1 focuses on using the loss of positive-definiteness (often referred to as *negative stiffness*) as a material property. While unstable in a free-standing solid, we will discuss how nonpositive-definite phases can be stabilized in a composite and how such stable negative stiffness is utilized to control the effective, overall composite properties. By contrast, Sec. 3.2 reviews examples in which (i) structural instabilities are exploited to induce large deformations that enable the control and tuning of the effective properties of the system (without necessarily using nonpositive-definite constituents) and (ii) the rather recent concept of utilizing instability to propel stable, large-amplitude nonlinear wave motion.

The strategy of exploiting so-called *negative-stiffness* components in structures and solids is based on a simple yet powerful idea. Most engineering design principles achieve target material properties by combining constituents which—individually or jointly—contribute the properties of interest. For example, combining a lossy viscoelastic material with a stiff elastic material can result in composites offering both stiffness (i.e., the ability to carry loads) and damping (i.e., the ability to absorb vibrational energy) [102–105]. In the design of such composite systems, one commonly works with constituents having positive definite-elastic moduli—this ensures stability of each phase under arbitrary boundary conditions. A key observation was that the geometric constraints between the various phases in a composite provide a means of stabilization, so that phases with nonpositive-definite elastic moduli (phases having *negative stiffness*) may, in principle, exist and be stable if constrained. In other words, while a homogeneous linear elastic medium must have positive-definite moduli for stability under general boundary conditions, a composite may not necessarily require all of its constituents to satisfy positive-definiteness for stability. This led to a careful analysis of both stability and effective properties of such composites containing negative-stiffness phases. The full analysis of multiphase composites is mathematically involved, which is why—before analyzing complex, higher-dimensional composite materials—we first turn to mass–dashpot–spring systems as simple structural analogs that admit intuitive, closed-form solutions. In fact, spring systems have been used early on to study the presence of negative-stiffness phases, see, e.g., Ref. [106]. While the loss of stability in solids is linked to checking the above local and global conditions imposed upon the components of the incremental stiffness tensor, the stability of a spring is simply linked to the sign of its scalar stiffness. Composites, difficult to deal with in 2D/3D, in general, because of geometric effects and complex boundary conditions, are easily assembled by multiple springs in one-dimensional (1D)—like the classical Reuss and Voigt composites. Even the transition from elastic to viscoelastic composites can be discussed by adding dashpots to the spring analogs (while requiring more complex tensorial counterparts in higher-dimensional systems). For all those reasons, we introduce the basic concepts of negative stiffness, multistability, and effective properties by simple spring examples.

Structural instability emerges, e.g., in a bistable system (i.e., a structure having two stable equilibrium configurations which correspond to two local energy minima). If the energy potential is continuous, then there must also be local energy maximum which corresponds to an unstable state.

Consider, e.g., the mass–spring toy example shown in Fig. 2(a). The system consists of two linear elastic springs of stiffness *k* and a point mass *m* which is assumed to move only horizontally. The mass has two stable equilibrium positions at $x=\xb1l$, corresponding to two energy minima, while the central position *x* = 0 corresponds to a local maximum (hence, an unstable equilibrium), as shown by the total potential energy *E* plotted in Fig. 2(c). This is as a prototype of a *bistable* elastic system. The potential energy landscape is inherently nonconvex, which comes with nonlinearity in the physical constitutive behavior since $F=\u2202E/\u2202x$. Displacement-controlled loading here results in a nonmonotonous force–displacement relation, whereas load control leads to snapping from one equilibrium branch to another.

Since we restrict the mass' motion to be horizontal, the effective static stiffness $ka*$ of the overall system depends nonlinearly on the mass' position *x* via

see Fig. 2(a) for the definition of the geometric parameters. Therefore, negative effective stiffness, $ka*<0$, occurs when

but is unstable in an unconstrained system (under load control, the mass will snap through the unstable, concave region of *E* in Fig. 2(c)).

As an extension, the composite system shown in Fig. 2(b) adds in series another linear spring of constant stiffness $k2>0$. Imagine the application of a force *F*^{0}, resulting in a position *x*^{0} of the (left) mass in the composite system. Assuming that the system is stable in that configuration, we may linearize about *x*^{0} with respect to small perturbations in the load and displacement. The resulting system with incremental load–displacement relation $F\u02d9=k*x\u02d9$ is visualized in Fig. 2(d), where we redefined $k1=ka*$ for simplicity, and it yields the effective static stiffness

Stability requires $k*\u22650$ so that (assuming $k2>0$), we must also have

for stability under general boundary conditions (any $k1<0$ is unstable under load control). Note that if we add a third spring with stiffness $k3>0$ in parallel to the other two springs, see Fig. 2(f), then the effective stiffness becomes

That is, for $k2,k3>0$ negative values of *k*_{1} can indeed be stable under general loading. Apparently, infinite effective stiffness $k*\u2192\u221e$ is predicted for $k1\u2192\u2212k2$ from below, but this is necessarily unstable because $k1\u2265k1,crit>\u2212k2$ unless $k3\u2192\u221e$ (in which case, the system is infinitely stiff anyways). Note that negative values of *k*_{1} can serve to create systems with very low effective stiffness $k*$, which may be of interest for controlling the resonance frequencies of the system [107].

It is important to note that we here refer to the positive or negative *static* stiffness of the elastic system or its components, which should not be confused with the effective *dynamic* stiffness which has also been tuned to negative values in acoustic metamaterials exploiting, e.g., local resonators [108–110]. In our case, negative stiffness refers to the quasi-static load–displacement relation and is tied to the release of internal energy (such as energy release upon snapping of the bistable spring system).

Finally, consider the viscoelastic system obtained from replacing the elastic springs by viscoelastic spring–dashpot combinations, see Fig. 2(e). In case of linear dashpots (i.e., velocity-proportional damping) with viscosities $\eta i>0$, the effective complex-valued *quasi-static* stiffness of the simple composite under harmonic excitation with frequency *ω* (inertial effects are neglected) is by the correspondence principle [100]

The resulting effective stiffness (assuming $k2>0$) is

and the effective damping is characterized by

Figure 3 illustrates the effective stiffness and damping and demonstrates the impact of negative values of *k*_{1} (assuming $k2>0$). Recall that stability requires (31) (which remains unaffected by the addition of viscosity). Results in Fig. 3 indicate that, as *k*_{1} approaches the stability limit ($k1\u21920$ from above), significant increases in damping can be achieved, while the effective stiffness is reduced considerably. The same was shown for more complex two-phase composites consisting of negative-stiffness inclusions embedded in a stabilizing matrix phase [101]. Note that these analyses ignore inertial effects, which can easily be added and provide altered results that also depend on the masses and may produce local resonance effects [107].

The previously mentioned concept of utilizing mechanical systems near instability to create an effective negative stiffness was first utilized in structures, e.g., for vibration isolation [29,111]. Constrained snap-through instabilities in precompressed, buckled beams (as a more practical implementation of the scenario in Fig. 2(a)) were used to reduce the stiffness of elastic suspensions and to achieve low natural frequencies, see also Ref. [31]. Applications ranged from vibration isolation for nano-instrumentation [112] to vibration reduction in vehicles [113] and structures [32], to seismic protection technologies [114]. Negative incremental stiffness has also been reported in foams [115] and structures with interlocked elements [116]. Further examples of structures exploiting constrained bistable, negative-stiffness elements can be found in Refs. [117–120] with nonlinear extensions in Ref. [121]. The concept of operating a stabilized mechanical system close to a critical point had also been found in biological systems such as myofibrils [122], muscles [123], and hair cell walls [124], where—among others—the high, controllable sensitivity near the critical point is exploited.

At the material level, negative stiffness implies nonpositive-definite (incremental) elastic moduli $C$, which may result from constrained material instabilities (e.g., from phase transformations). For example, the potential energy of materials undergoing second-order phase transformations is often described by Landau's theory [46]: having only a single well at high temperature, the potential energy turns into a multiwelled landscape below the transformation temperature. Upon cooling through the transition temperature, the initially stable high-temperature energy minimum turns into a local maximum below the transformation point which becomes unstable but, if sufficiently constrained, may be stabilized to display nonpositive-definite moduli [40].

Early approaches to take advantage of the negative-stiffness concept in materials were based on theoretical predictions of stiffness and damping in composites. For example, by evaluating the property bounds of Hashin–Shtrikman composites [125], nonpositive-definite phases were shown to produce extreme viscoelastic properties [38]. Similar to later predictions based on fractal composite models [126], those investigations predicted extreme variations in the composite's effective stiffness and damping due to the presence of negative-stiffness phases (where the term *extreme* stands for effective properties that surpass those of each constituent).

As an example, Fig. 4 illustrates the effective dynamic Young's modulus and the effective damping obtained from evaluating the Hashin–Shtrikman lower bound for a composite composed of a metal matrix (with moduli $\mu mat=19.2$ GPa, $\kappa mat=41.6$ GPa, and three values of $tan\u2009\delta =0.01$, 0.02, and 0.04) with 5 vol % ceramic inclusions ($\mu inc=50$ GPa and $tan\u2009\delta =0.001$) whose bulk modulus $\kappa inc$ is varied from positive to negative values. For $\kappa inc>0$, the few inclusions have little impact on the effective response. For $\kappa inc\u2248\u221224$ GPa, strong variations in Young's modulus and loss tangent are predicted (these are even more remarkable as only 5 vol % inclusions are responsible for the shown changes in the overall properties). As shown later in Refs. [127] and [128], composite bounds also predicted the piezo-, pyro-, or thermomechanical coupling coefficients to assume extreme values in particulate composites with negative-stiffness inclusions in a viscoelastic matrix, similar to Fig. 4. Composites were further shown to serve as waveguides with exceptional damping due to negative-stiffness phases [129].

The combination of high stiffness and high damping is of particular technological interest as those properties are naturally exclusive [130], owing to the distinct responsible microscale mechanisms (high stiffness favors perfect crystallinity, whereas high damping requires high mobility of defects or related mechanisms of internal friction). In fact, plotting the stiffness versus damping of many natural and manmade materials (see Fig. 5) revealed a seemingly natural upper bound on the combined figure of merit of stiffness times damping, $E\xd7tan\u2009\delta $, for all known materials [131]; the region of high stiffness and damping in Fig. 5 remained empty. The negative-stiffness strategy offered opportunities to achieve composites having both high stiffness and high damping (with optimal combinations sought, e.g., via topology optimization [132]), which can enter the empty parameter range, as discussed in Secs. 3.1.3 and 3.1.4.

Following those early estimates based on composite bounds, more theoretical studies of specific composites followed with the aim to contrast extreme property predictions with stability restrictions. After all, the question had remained whether or not extreme composite properties due to negative-stiffness phases could be stable under general loading/boundary conditions.

Coated cylinders and spheres are prototypical examples of two-phase composite bodies with Lamé-type solutions in small strains. A first continuum-mechanics analysis of both 2D and 3D configurations (i.e., coated cylinders and spheres) showed that negative stiffness in the inclusion may indeed lead to extreme overall increases in the overall bulk modulus in case of linear and nonlinear elastic composites [133]. Dynamic stability analysis using the same two-phase composite bodies within linear elasticity further revealed that nonpositive-definite inclusions can indeed be stable if sufficiently constrained, e.g., when embedded in a stiff coating or matrix [98,134]. Specifically, for homogeneous, isotropic linear elastic phases, it was shown that a negative inclusion bulk modulus can be stable if the surrounding coating is sufficiently stiff and thick.

The global stability conditions were derived in the two ways described in Sec. 2. As shown in Ref. [98], one can solve the dynamic, linear elastic eigenvalue problem for macroscopic bodies Ω to determine the infinite set of eigenfrequencies (or at least the lowest few). That is, one solves Eq. (20) with suitable boundary conditions (e.g., pure traction boundary conditions over $\u2202\Omega $ as the weakest constraint) for the eigenfrequencies *ω _{i}*. For linear elastic bodies, stability requires that $\omega i\u2208\mathbb{R}$. Therefore, once the lowest nonzero eigenfrequency

*ω*

_{0}is known as a function of elastic moduli and geometry, stability conditions on the elastic moduli can be established. Alternatively, stability conditions were derived by evaluating Eq. (22) for specific composite geometries [134]. Furthermore, by exploiting Rayleigh's coefficient (25), finite element analysis was conveniently used as an inexpensive alternative [94] (which requires checking the positive-definiteness of the global stiffness matrix). As could be expected for conservative systems (such as linear elastic media), all the three approaches led to the same conclusion, viz., that nonpositive-definite inclusion phases can be stable if sufficiently constrained. For example, Fig. 6 shows the stable and unstable moduli combinations for the shown coated-sphere two-phase body as light and dark regions, respectively, clearly indicating that the inclusion's bulk modulus, $\kappa i$, can be negative and stable if the coating's bulk modulus, $\kappa c$, is sufficiently high [107].

For conservative linear systems, one can exploit the equivalence of static and dynamic stability, following Hill [79] and Koiter [84], which led to simplified stability analyses. For example, for rotational-symmetric bodies such as coated cylinders or coated spheres, the lowest (nonzero) eigenfrequency typically corresponds to a rotational-symmetric deformation mode and is therefore linked to the effective bulk modulus of the two-phase body [96]. Therefore, the stability limit can alternatively be obtained by computing $\u2329\kappa \u232a$ as a function of constituent moduli and geometry and enforcing $\u2329\kappa \u232a\u22650$ for stability (in close analogy to the spring example of Sec. 3.1.1). Based on this approach, new closed-form stability bounds were derived for 2D and 3D composite bodies [96].