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Review Article

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

[+] Author and Article Information
Dennis M. Kochmann

Department of Mechanical
and Process Engineering,
ETH Zürich,
Leonhardstr. 21,
Zürich 8092, Switzerland;
Division of Engineering and Applied Science,
California Institute of Technology,
Pasadena, CA 91125
e-mail: dmk@ethz.ch

Katia Bertoldi

John A. Paulson School of Engineering and
Applied Sciences,
Harvard University,
Cambridge, MA 02138;
Kavli Institute,
Harvard University,
Cambridge, MA 02138
e-mail: bertoldi@seas.harvard.edu

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

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.

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Figures

Grahic Jump Location
Fig. 1

Map of stable regions in the λ–μ plane of a homogeneous, isotropic, linear elastic solid. Dark-gray regions are unstable by violation of ellipticity, whereas the requirement of positive definiteness further restricts the stable region by also making the light-gray region unstable (the three key inequalities from Eqs. (18) and (23) are shown separately).

Grahic Jump Location
Fig. 5

Stiffness (Young's modulus) versus damping (loss tangent) versus mass density for a variety of natural and manmade materials; the shadowed prism highlights the desirable but challenging region of combined high stiffness and high damping

Grahic Jump Location
Fig. 2

Simple spring examples: (a) bistable mechanical system and (b) composite spring system; the energy and stiffness of system (a) is illustrated in (c); (d) linear spring reduction of system (b) where k1=ka* and (e) its viscoelastic extension; (f) composite system

Grahic Jump Location
Fig. 4

Effective viscoelastic Young's modulus and loss tangent for a Hashin–Shtrikman composite composed of metal matrix (μmat=19.2 GPa, κmat=41.6 GPa, and three values of tan δ=0.01,0.02, and  0.04) and 5 vol % ceramic inclusions (μinc=50 GPa, tan δ=0.001, and varying κinc). The shown effective modulus refers to the absolute value of the complex-valued viscoelastic Young's modulus E*=Ere+iEim.

Grahic Jump Location
Fig. 6

Stable and unstable moduli combinations for a coated spherical inclusion (stable and unstable combinations are shown as light and dark gray regions, respectively); both phases are homogeneous, isotropic, linear elastic (with μc,μi>0 for pointwise stability). Under static conditions, an infinite effective bulk modulus of the two-phase body is unstable (shown by the ω = 0 line are all moduli combinations resulting in a positive-infinite bulk modulus). Under dynamic excitation, resonance effects lead to strong stiffness variations which, with increasing excitation frequency ω shift in to the stable region (ω0=μc/ρcb). When admitting negative stiffness κi<0, these become stable at significantly lower frequencies [107].

Grahic Jump Location
Fig. 3

Effective (normalized) stiffness and damping of the linear viscoelastic composite system of Fig. 2(e) versus (normalized) spring stiffness k1 (shown for ηω=0.01 and k2>0)

Grahic Jump Location
Fig. 7

Stability map showing stable (light) and unstable (dark gray) regions of κi/μi, where κi and μi are, respectively, the bulk and shear moduli of a spherical inclusion (radius a) embedded in a concentric coating (outer radius b and moduli κc, μc). The stability limit for κi/μi depends on the applied BCs. Plotted is the effective bulk modulus κ*=pb/3ur(b) for uniform applied pressure p, resulting in a rotational-symmetric expansion of the coated-sphere composite with radial displacement field ur(r). Consequently, under both types of BCs, the solid loses stability before the effective modulus tends to +∞ with decreasing κi/μi. Note that positive-definiteness corresponds to κ/μ≥0, so that the elastic coating expands the stable regime of the inclusion phase.

Grahic Jump Location
Fig. 8

Torsional compliance (inverse stiffness) and loss tangent versus temperature of a composite composed of 1 vol % VO2 particles embedded in a pure tin matrix (damping of pure tin is included for reference). Measurements were conducted well below sample resonance at 100 Hz. Reprinted with permission from Lakes et al. [39]. Copyright 2001 by Nature Publishing Group.

Grahic Jump Location
Fig. 9

Dynamic stiffness and damping variations in viscoelastic Sn–BaTiO3 composites under harmonic loading. Adapted from Ref. [40].

Grahic Jump Location
Fig. 13

(a) Buckling-induced reversible pattern formation in a supported microscale honeycomb lattice upon rapid swelling. Depending on the geometry of the plates, buckling induces either an achiral pattern or a chiral pattern. Multiple domains with different chirality are observed, whose boundaries are highlighted by the dashed lines. The insets show magnified images of the buckled patterns within the domains (top) and at the domain boundaries (bottom). The color-coded arrows indicate the handedness of the vertices. Adapted from Ref. [166]. (b) A soft gripper made of a buckling actuator. The claws of the gripper close upon deflation of the buckling actuator and the buckling gripper picks up a piece of chalk. Scale bars: 1 cm. Adapted from Ref. [170]. (c) Buckling-induced pattern transformation in shape-memory polymer membranes comprising a hexagonal array of micron-sized circular holes results in dramatic color switching. Adapted from Ref. [171].

Grahic Jump Location
Fig. 14

(a) Snapshots of a bistable mechanical metamaterial in response to tensile loading. The system comprises an array of double-curved beams which can snap between two stable configurations. Adapted from Ref. [189]. (b) Response of an elastic sheet perforated with a square array of mutually orthogonal cuts under uniaxial tension. In the thick limit, the perforated sheet deforms in-plane and identically to a network of rotating squares (left). For sufficiently small values of thickness, mechanical instabilities triggered under uniaxial tension result in the formation of complex 3D patterns, which are affected by the loading direction (center and right). Scale bars: 6 mm. Adapted from Ref. [192].

Grahic Jump Location
Fig. 15

(a) Analogously to antiferromagnetic systems—in which nearest-neighbor spins cannot align in opposite directions when arranged on a triangle—in triangular frames, the beams cannot buckle into a half sinusoid and at the same time preserve angles at joints. As a result, the system becomes frustrated. (b) Geometric frustration in periodic 2D beam lattices favors the formation of complex buckling-induced ordered patterns. Adapted from Ref. [181].

Grahic Jump Location
Fig. 10

(a) Macroscopic (continuous line) and microscopic (dashed line) onset-of-bifurcation surfaces in the principal macroscopic logarithmic strain space for a perfectly periodic neo-Hookean solid (characterized by a bulk to shear moduli ratio equal to 9.8) with a square distribution of circular voids. The eigenmode of the microscopic bifurcation instability is shown on the right. Reprinted with permission from Triantafyllidis et al. [162]. Copyright 2006 by ASME. (b) Experimental images of an elastomeric structure comprising a square array of circular holes for increasing values of the applied deformation. Note that after instability, the lateral boundaries of the sample bend inward, a clear signature of negative Poisson's ratio behavior. Adapted from Ref. [180].

Grahic Jump Location
Fig. 11

(a) Experimental images of an elastomeric structure comprising a triangular array of circular holes when compressed horizontally (top), vertically (center), and equibiaxially (bottom). Three distinct buckling-induced patterns are formed. Adapted from Ref. [174]. (b) Orthogonal side views (onto the y–z and x–z planes) for a cylindrical sample pattern with a square array of circular holes at different levels of deformation. The structure was made watertight by a thin membrane that covered the inner surface of the voids and was then loaded hydraulically. Adapted from Ref. [169]. (c) The Hoberman Twist-O is a commercial toy which comprises a rigid network of struts connected by rotating hinges and can easily collapse into a ball having a fraction of its original size. Adapted from Ref. [22]. (d) The Buckliball is inspired by this popular toy but translates the mechanism design to the structure of an elastic spherical shell—which under pneumatic actuation undergoes buckling-induced folding, opening avenues for a new class of active and reversible encapsulation systems. Adapted from Ref. [22].

Grahic Jump Location
Fig. 12

(a) Experimental snapshots during the swelling process for a square lattice made of plates sandwiched between two thin and stiff layers. Buckling induces an effective negative swelling ratio in this structure. Reprinted with permission from Liu et al. [165]. Copyright 2016 by John Wiley & Sons. (b) An instability is induced by capillary forces during evaporation of water from a swollen hydrogel membrane with micron-sized holes in a square array. Scale bars: 10 μm. Reprinted with permission from Zhu et al. [172]. Copyright 2012 by Royal Society of Chemistry (c) Slow variations in the current through several electromagnetic coils embedded in a soft cellular elastomer induce visible strain and snap-through behavior. Reprinted with permission from Tipton et al. [184]. Copyright 2012 by Royal Society of Chemistry.

Grahic Jump Location
Fig. 16

Phononic band structure for a square array of circular voids in an elastic matrix subjected to equibiaxial compression in (a) the undeformed configuration and (b) after buckling. Adapted from Ref. [218].

Grahic Jump Location
Fig. 17

Effect of deformation on the directionality of the propagating waves for a square array of circular voids in an elastic matrix subjected to equibiaxial compression. (a) and (b) Effect of deformation on the directionality of the phase velocity. (c) and (d) Effect of deformation on the directionality of the group velocity. Adapted from Ref. [218].

Grahic Jump Location
Fig. 18

Tunable acoustic metamaterial: (a) the undeformed configuration comprises resonating units dispersed into an elastomeric matrix. Each resonator consists of a metallic mass connected to the matrix through elastic beams, which form a structural coating. The black regions in the picture indicate voids in the structure. The unit cell size is A0=50.0 mm. (b) When a compressive strain ε=−0.10 is applied in the vertical direction, buckling of the beams significantly alters the effective stiffness of the structural coating. (c) Experimentally measured transmittance at different levels of applied deformation. The band gap frequency first decreases linearly as a function of ε and then it completely disappears as ε approaches 0.10. Adapted from Ref. [34].

Grahic Jump Location
Fig. 20

A chain of laterally compressed, bistable membranes coupled by nonlinear permanent magnets obeys Eq. (47) and displays unidirectional transition wave motion upon impact [266]

Grahic Jump Location
Fig. 21

Stable transition waves in a 3D-printed chain of bistable elements [23]: (a) a compressive transition wave propagates by releasing stored elastic energy, which is dissipated due to material-internal damping mechanisms. (b) A stable transition wave of constant speed v and width w is recorded (after an initial transient period).

Grahic Jump Location
Fig. 19

Illustration of domains (constant polarization p within each domain) separated by domain walls in structures and materials. Under excitation, the domain walls move at speed v. The general structure of both systems is schematically shown with an interaction potential V and multistable potential ψ.

Grahic Jump Location
Fig. 22

Periodic array of bistable rotational elements (polarization angle φ) coupled to nearest neighbors by elastic bands [287]. The combination of an excentrically attached linear spring and the action of gravity results in a tunable bistable potential ψ which can be biased as in domain switching (by tilting the whole setup by an angle α) and which can also mimic a second-order phase transition (by adjusting the position fx of the spring anchor points, switching between a bistable potential and a single-well potential).

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