Exploiting Microstructural Dennis M. Kochmann Instabilities in Solids and Department of Mechanical and Process Engineering, ETH Z€urich, Structures: From Metamaterials Leonhardstr. 21, Z€urich 8092, Switzerland; to Structural Transitions Division of Engineering and Applied Science, California Institute of Technology, Instabilities in solids and structures are ubiquitous across all length and time scales, and Pasadena, CA 91125 engineering design principles have commonly aimed at preventing instability. However, e-mail: [email protected] 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 Katia Bertoldi instabilities—both at the microstructural scale in materials and at the macroscopic, John A. Paulson School of Engineering and structural level—lies a nonconvex potential energy landscape which is responsible, e.g., Applied Sciences, for phase transitions and domain switching, localization, pattern formation, or structural Harvard University, buckling and snapping. Deliberately driving a system close to, into, and beyond the Cambridge, MA 02138; unstable regime has been exploited to create new materials systems with superior, inter- Kavli Institute, esting, or extreme physical properties. Here, we review the state-of-the-art in utilizing Harvard University, mechanical instabilities in solids and structures at the microstructural level in order to Cambridge, MA 02138 control macroscopic (meta)material performance. After a brief theoretical review, we e-mail: [email protected] discuss examples of utilizing material instabilities (from phase transitions and ferroelec- tric switching to extreme composites) as well as examples of exploiting structural insta- bilities in acoustic and mechanical metamaterials. [DOI: 10.1115/1.4037966] 1 Introduction properties [41,42]. Especially at the macroscopic level, structural instability and the associated large deformation of soft matter The long history of mechanics is rife with prominent examples have produced many multifunctional devices that exploit buck- of instabilities in solids and structures that have led to material ling, snapping, and creazing to result in beneficial acoustic or failure or structural collapse, spanning all scales from atomic- mechanical performance, see Ref. [43] for a recent survey. Like in scale void growth, failure, and stress-induced transformations to structures, instability at the material level stems from non-(quasi)- buckling and delamination in micro- and nano-electronics all the convex energy landscapes [44,45]; here, however, instability is way up to tectonic events. Examples of causal mechanisms for not tied to multiple stable or metastable structural deformation mechanical instability are ubiquitous: buckling of structures, modes but involves multiple stable microstructural configurations crushing of cellular solids, plastic necking, strain localization in such as those in phase transitions and phase transformations [46], shear or kink bands, wrinkling and crazing, void growth, fracture, domain switching and domain patterning [47–49], deformation and collapse. Through its more than 250-year-old history—going twinning [50,51], or strain localization, shear banding, and back to Euler’s buckling studies [1–3] and involving such promi- patterning in plasticity [52]. From a mathematical standpoint, nent mechanicians as Kirchhoff [4], Love [5], and the Lords non-(quasi)convex energetic potentials entail instability of a Kelvin [6] and Rayleigh [7]—the theory of material and structural homogeneous state of deformation, thus leading to energy- stability has resulted in analytical theories and numerical tools minimizing sequences which, physically, translate into complex that have provided engineers with safe design guidelines. Starting microstructural patterns, see, e.g., Refs. [49] and [52–56]. with structures and advancing to continuous media within the In addition to structures and materials, metamaterials— frameworks of linear and later nonlinear elasticity as well as somewhere in the diffuse interface between solids and inelasticity, the theory of stability in solids and structures has structures—have gained popularity in recent decades. These engi- evolved and resulted in many seminal contributions, see, e.g., neered media commonly have a structural architecture at smaller Refs. [8–19] for a nonexhaustive list of classics in the field. scales, which is hidden at the larger scale where only an effective Over the past two decades, an exciting paradigm shift has been medium with effective properties is observed. The advent of addi- initiated away from traditional design against instability and tive manufacturing has fueled the design of complex metamateri- toward novel performance through controlled instability: engi- als, see, e.g., Refs. [57–59], with instabilities now occurring neering mechanics is exploring the advantages of operating sys- across multiple scales. tems near, at, or beyond the critical point. (In)stability is Here, we present a survey of exploiting instabilities in solids instrumentalized for beneficial material or structural behavior and structures. Since the field has grown tremendously in recent such as, e.g., soft devices undergoing dramatic shape changes years, we limit our review to concepts that exploit microscale [20–22], propagating stable signals in lossy media [23] or control- instabilities to effect macroscale behavior, which applies to both ling microfluidics [24], large reversible deformation of cellular structures (e.g., trusses or architected metamaterials) and materials solids [25,26], morphing surfaces and structures [27,28], high- (e.g., composites and other heterogeneous solids). The effective, damping devices and energy-absorbing technologies from macroscopic properties of heterogeneous solids and metamaterials nanotubes to macrodevices [29–32], acoustic wave guides and are commonly identified by averaging over a representative vol- metamaterials [33–36], composites with extreme viscoelastic per- ume element (RVE), including asymptotic expansions [60,61], formance [37–40], or materials with actively controllable physical numerical tools based on periodic boundary conditions (BCs) [62–65], as well as various upper and lower bounds on effective Manuscript received May 4, 2017; final manuscript received September 13, 2017; properties, especially for linear properties such as the (visco)elastic published online October 17, 2017. Editor: Harry Dankowicz. moduli, see, e.g., Ref. [66]. Besides the quasi-static mechanical Applied Mechanics Reviews Copyright VC 2017 by ASME SEPTEMBER 2017, Vol. 69 / 050801-1 Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 01/02/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use properties such as stiffness, strength, and energy absorption, a which defines an equilibrium path. At a critical point, @f=@u major focus of architected metamaterials has been on wave becomes singular so that there exists either no solution or a non- motion. Here, the effective wave propagation characteristics of unique solution (point of bifurcation). Note that imperfections periodic, microstructured media can be obtained from the disper- generally lower the critical point (i.e., they reduce the critical load sion relations computed at the RVE level [67,68]. Unlike in con- c) and affect the postbifurcation behavior, which is of importance ventional continuous media, structural metamaterials such as both in structures (where, e.g., geometric or fabrication-induced multiscale truss lattices can be exploited to independently control imperfections may dominate the mechanical response) and in elastic moduli and dispersion relations [69]. We note that if the materials (where, e.g., thermal fluctuations lead to temperature- wavelength approaches the RVE size (in particular in acoustic dependent transformation and switching kinetics). metamaterials), classical homogenization approaches may lose their validity, which is why new theories have been proposed for 2.2 Stability of Continuous Bodies. The previously men- the effective property extraction in more general settings, e.g., see tioned notions of stability can be applied to discrete systems and Refs. [70–75], which is closed under homogenization and can be continuous solids alike. A continuous body X Rd is described linked to Bloch–Floquet analysis [76]. Alternatively, computa- by the deformation mapping u : X Â R ! Rd, so that the tional techniques have obtained the effective dynamic performance deformed and undeformed positions, x and X, respectively, are even in transient, nonlinear, or inelastic settings [77,78]. These linked via x ¼ uðX; tÞ. Such body is governed by linear momen- concepts have been applied to both materials and structures, result- tum balance ing in interesting, extreme, peculiar, or controllable effective (meta)material properties to be reviewed in the following. 2 D ui We first give a concise review of the fundamental concepts of PiJ;J þ q0bi ¼ q0 in X (5) Dt2 stability theory to the extent required for subsequent discussions, followed by a survey of first material and then metamaterial strat- where egies that take advantage of instability. Finally, we discuss the state-of-the-art and point out ongoing directions and topical @W opportunities. PiJ ¼ (6) @FiJ 2 Stability Theory are the components of the first Piola-Kirchhoff stress tensor, FiJ ¼ u ; denotes the deformation gradient, D/Dt is the