Negative Stiffness Inclusions Research Articles

Rayleigh waves in isotropic elastic materials with micro-voids

In this paper, we show that a general method introduced by Fu and Mielke allows to give a complete answer on the existence and uniqueness of a subsonic solution describing the propagation of surface waves in an isotropic half space modelled with the linear theory of isotropic elastic materials with micro-voids. Our result is valid for the entire class of materials admitting real wave propagation which include auxetic materials (negative Poisson’s ratio) and composite materials with negative-stiffness inclusions (negative Young’s modulus). Moreover, the used method allows to formulate a simple and complete numerical strategy for the computation of the solution.

Negative-stiffness composite systems and their coupled-field properties

Composite materials consisting of negative-stiffness inclusions in positive-stiffness matrix may exhibit anomalous effective coupled-field properties through the interactions of the positive and negative phases, giving rise to extremely large or small effective properties. In this work, effective viscoelastic properties of a continuum composite system under the effects of negative inclusion Young’s modulus ratio $$\lambda _E = E_{\text {inc}}/E_{\text {matrix}}$$ are studied with the finite element method. Furthermore, effective coupled-field properties, such as thermal expansion coefficient, dielectric constants and piezeoelectric constants, are numerically calculated under the effects of negative inclusion bulk modulus ratio $$\lambda _K=K_{\text {inc}}/K_{\text {matrix}}$$ . Stability boundaries are determined by applying small dynamic perturbation to the systems through boundary surfaces, and the system is unstable if its field variables become divergent in time. For viscoelastic composite systems containing small volume fractions, less than $$V_i = 1.5 \%$$ , the systems can be stable up to $$\lambda _E \approx -0.3$$ in 0.3 s under 10 Hz driving frequency. For $$V_i = 5.1 \%$$ case, its stability boundary is around $$\lambda _E \approx 0$$ . Larger inclusion volume fraction reduces allowable negative stiffness in the viscoelastic system. All anomalous peaks found in the coupled-field properties are in the unstable regime, except for the piezoelectric and thermal-expansion anomalies in the composite system with electrically insulated inclusions and large inclusion volume fraction $$V_i = 26.81 \%$$ . Insulated inclusions may cause charge accumulation at the inclusion–matrix interface and boundary surface effects may serve as stabilizing agents to the composite system. Since it is known that negative-stiffness composite is unstable in the purely elastic system in statics, stability enhancement found here in the negative-stiffness systems with viscoelastic and coupled-field effects may be considered as multiphysics-induced stabilization.

Sound transmission through a stiff double-panel structure periodically stabilized by negative stiffness module: Theoretical modeling

Analytic expressions are derived for models predicting the influence of periodically spaced structural links on sound transmission through a double-panel structure. The double panel has been configured to have two sets of structural links: a rib stiffener and negative stiffness component. The stiffener is identical and is spaced periodically at a distance. However, the negative element component is shifted by an amount q from the other set. The dynamic equation of the vibroacoustic of the system is formulated in terms of the space harmonics and by the principle of virtual work. The model is validated by comparing the model predictions with the existing result from the literature. Then, influences of the negative element, engineering safety, offset, and elevation angle are investigated. A new antiresonance with a huge sound transmission loss value can be engineered at the low-frequency region when these parameters are varied. In addition, the application of the stiff model is implemented for a periodic acoustic metamaterial structure. The negative stiffness inclusion can prevent wave propagation at low frequency. The periodic structure can be designed to obtain more and wider frequency bandgaps.

Acoustic response for nonlinear, coupled multiscale model containing subwavelength designed microstructure instabilities.

Nonperiodic arrangements of inclusions with incremental linear negative stiffness embedded within a host material offer the ability to achieve unique and useful material properties on the macroscale. In an effort to study such types of inclusions, the present paper develops a time-domain model to capture the nonlinear dynamic response of a heterogeneous medium containing a dilute concentration of subwavelength nonlinear inclusions embedded in a lossy, nearly incompressible medium. Each length scale is modeled via a modified Rayleigh-Plesset equation, which differs from the standard form used in bubble dynamics by accounting for inertial and viscoelastic effects of the oscillating spherical element and includes constitutive equations formulated with incremental deformations. The two length scales are coupled through the constitutive relations and viscoelastic loss for the effective medium, both dependent on the inclusion and matrix properties. The model is then applied to an example nonlinear inclusion with incremental negative linear stiffness stemming from microscale elastic instabilities embedded in a lossy, nearly incompressible host medium. The macroscopic damping performance is shown to be tunable via an externally applied hydrostatic pressure with the example system displaying over two orders of magnitude change in energy dissipation due to changes in prestrain. The numerical results for radial oscillations versus time, frequency spectra, and energy dissipation obtained from the coupled dynamic model captures the expected response for quasistatic and dynamic regimes for an example buckling inclusion for both constrained and unconstrained negative stiffness inclusions.

Tunable Mechanical Metamaterial with Constrained Negative Stiffness for Improved Quasi‐Static and Dynamic Energy Dissipation

This paper presents the computational design, fabrication, and experimental validation of a mechanical metamaterial in which the damping of the material is significantly increased without decreasing the stiffness by embedding a small volume fraction of negative stiffness (NS) inclusions within it. Unlike other systems that dissipate energy primarily through large‐amplitude deformation of nonlinear structures, this metamaterial dissipates energy by amplifying linear strains in the viscoelastic host material. By macroscopically tuning the pre‐strain of the metamaterial via mechanical loading, the embedded NS inclusions operate about a constrained buckling instability. When further macroscopic vibrational excitation is applied, the inclusions amplify the strains of the surrounding viscoelastic medium. This results in enhanced dissipation of mechanical energy when compared to voided or neat comparison media. Microstereolithography, an emerging high‐resolution additive manufacturing (AM) technology, is employed to fabricate the deeply subwavelength inclusions which ensures broadband damping behavior. The mechanically induced broadband energy dissipation and manufacturing approach further differentiate the metamaterial from other approaches that exploit resonances, large deformations, or non‐mechanical instabilities. The computational design, fabrication, and experimental evaluation reported is the first dynamic demonstration of such a mechanically tunable NS metamaterial, potentially enabling components with integrated structural and damping capabilities.

Negative-Stiffness Inclusions as a Platform for Real-Time Tunable Phononic Metamaterials

Tunable phononic metamaterials, which change their response to elastic wave propagation upon actuation, have important applications in cloaking, wave guiding, active noise reduction, superlensing, and acoustic mirrors. Most systems either require substantial morphological changes for actuation, or exhibit only high-frequency band gaps. The authors present a framework for developing tunable phononic metamaterials with low-frequency band gaps, using negative-stiffness inclusions in an elastic matrix, and demonstrate two implementations of this concept. The designs are robust and achieve substantial band-gap manipulation, without elastic instability or significant structural changes.

Vibration damping materials with enhanced loss due to microstructural nonlinearity

One conventional approach to attenuating structure-borne waves or reduce the ringdown time of modes in structural elements is to attach damping layers or patches to vibrating components. Common patch and layer materials are specially formulated polymers or engineered polymeric composites that demonstrate elevated viscoelastic loss factors in the frequency range of interest. Recent research has shown that small volume fractions of negative stiffness inclusions embedded in a lossy material generate effective loss factors that exceed that of the host material. The ability to generate negative stiffness behavior, however, is often the result of nonlinear inclusion material response. Presented here is a multiscale model of a particulate composite material consisting of a nearly incompressible host material containing small-scale heterogeneities with a nonlinear elastic stress-strain response. We investigate the nonlinear dynamic behavior of the heterogeneous medium for small harmonic perturbations about several pre-strain states to demonstrate the influence of the microscale dynamic response on the macroscopically observable mechanical loss. Of primary interest is the energy dissipation capabilities of the composite which can be tuned using inclusion pre-strain. Loss for composites with nonlinear inclusions is compared to conventional composites containing air voids or steel inclusions. [Work supported by ONR.]

Enhanced acoustic insulation properties of composite metamaterials having embedded negative stiffness inclusions

Despite the fact that the concept of incorporating Negative Stiffness (NS) elements within mechanical systems was formulated and validated more than 40 years ago, it has only recently received consistent attention. In this work, the design of a layered mechanical metamaterial having implemented NS inclusions is presented and its acoustic wave propagation properties are modelled. A dedicated two-dimensional periodic structure theory scheme is developed in order to compute the frequency dependent damping loss factor of the metamaterial structure. The acoustic transmission properties through the modelled panel are computed within a Statistical Energy Analysis (SEA) scheme. It is demonstrated that the suggested layered metamaterial exhibits highly superior acoustic insulation performance close and above the acoustic coincidence range, thanks to the drastic increase of its structural damping properties implied by the NS elements. Additionally, the proposed configuration presents superior performance in a broadband frequency range when compared to a viscoelastic damping constraint layer of equivalent mass.

Negative stiffness in ZrW2O8 inclusions as a result of thermal stress

Materials with negative stiffness, although inherently unstable in isolation, can be stabilized by external constraints, for example, by inclusion within a material with positive stiffness. We have identified ZrW2O8, a material with negative thermal expansion, as a candidate negative-stiffness material arising from its negative bulk modulus during a ferroelastic cubic–orthorhombic pressure-induced phase transition (PIPT). A hyperelastic constituent equation for this transition was developed and implemented in a finite-element model of ZrW2O8 inclusions in positive stiffness, positive thermal expansion matrices. In these matrices, thermal stress during cooling, originating from thermal expansion mismatch, would be sufficient to initiate the PIPT after small temperature drops. The subsequent progress of the PIPT depends strongly on the thermoelastic properties of the matrix, with stiff, low thermal expansion matrices stabilizing the transition state over broad temperature ranges, indicating that ZrW2O8 or materials with similar properties could be used as versatile negative-stiffness inclusion materials. The models were used to understand previous experiments on composites that include ZrW2O8.

Hierarchical Design of Negative Stiffness Metamaterials Using a Bayesian Network Classifier1

A set-based approach is presented for exploring multilevel design problems. The approach is applied to design negative stiffness metamaterials with mechanical stiffness and loss properties that surpass those of conventional composites. Negative stiffness metamaterials derive their properties from their internal structure, specifically by embedding small volume fractions of negative stiffness inclusions in a continuous host material. Achieving high stiffness and loss from these materials by design involves managing complex interdependencies among design variables across a range of length scales. Hierarchical material models are created for length scales ranging from the structure of the microscale negative stiffness inclusions to the effective properties of mesoscale metamaterials to the performance of an illustrative macroscale component. Bayesian network classifiers (BNCs) are used to map promising regions of the design space at each hierarchical modeling level, and the maps are intersected to identify sets of multilevel solutions that are likely to provide desirable system performance. The approach is particularly appropriate for highly efficient, top-down, performance-driven, multilevel design, as opposed to bottom-up, trial-and-error multilevel modeling.

Adaptation of Energy Dissipation in a Mechanical Metastable Module Excited Near Resonance

Recent studies have demonstrated that the energetic vibrations of strategically designed negative stiffness inclusions may lead to large and adaptable damping in structural/material systems. Many researchers examine these features using models of bistable elements. From the viewpoint of system integration, bistable, negative stiffness elements often interface with positive stiffness elastic members. Under such conditions, the structural/material system may exhibit coexisting metastable states. In other words, the macroscopic displacement/strain remains fixed while the reaction force may vary due to internal change, similar to a phase transition. This coexistence of metastable states is not manifested in an individual (stand-alone) bistable element. Although the static and low frequency linear dynamics of structural/material systems possessing coexisting metastable states have been explored, much remains to be understood regarding the dynamics and energy dissipation characteristics of such systems when excited near resonance, where nonlinear dynamics are more easily activated and damping design is of greater importance. Thus, to effectively elucidate the enhanced versatility of damping properties afforded by exploiting negative stiffness inclusions in structural/material systems, this research investigates a mechanical module which leverages a coexistence of metastable states: an archetypal building block for system assembly. The studies employ analytical, numerical, and experimental findings to probe how near-resonant excitation can trigger multiple dynamic states, each resulting in distinct energy dissipation features. It is shown that, for lightly damped metastable mechanical modules, the effective energy dissipation may be varied across orders of magnitude via tailoring design and excitation parameters.

Enhancement of wave damping within metamaterials having embedded negative stiffness inclusions

The wave dissipation properties of layered periodic structures are modelled by FE as well as analytical approaches. A linear oscillator incorporating a negative stiffness element and having exceptional energy dissipation properties is exhibited and incorporated within the modelled structures. The structural dynamic stability of both the oscillator and the modelled waveguides is discussed and ensured. The numerical results provide evidence of a drastic increase of several orders of magnitude for the damping ratio of the flexural waves propagating within the structures.

Tailored heterogeneity increases overall stability regime of composites having a negative-stiffness inclusion

Recent theoretical and experimental results have shown the possibility of enormous increases in composite material overall elastic stiffness, damping, thermal expansion, piezoelectricity, etc., when the composite contains a tuned non-positive-definite (i.e., negative stiffness) constituent. For such composite materials to have practical utility, they must be stable. Recent research has shown they can be, for a limited range of constituent negative stiffness. This research has treated linear elastic composite materials with homogeneous phases, via the energy method and full dynamic stability analyses.In the present work, we first show how to analyze the composites previously treated by the comprehensive but simpler static stability approach, obtaining closed-form results. We then employ this approach to show that permitting heterogeneity of the positive-definite phase can substantially increase the range of constituent negative stiffness while maintaining overall composite stability. We first treat the positive-definite phase heterogeneity as piecewise homogeneous, and then treat it as continuously-varying. In the continuously-varying heterogeneity case, we seek the radially optimal distribution of the elastic moduli in the coatings, under constant coating average moduli constraint, to permit the most negative possible inclusion stiffness while maintaining overall composite stability. This is accomplished for three coating cases: constant bulk modulus but arbitrarily radially-varying shear modulus; constant shear modulus but arbitrarily radially-varying bulk modulus; and both moduli arbitrarily radially varying. We find the optimal coatings to be: a heterogeneous one with shear modulus being a specific continuously decreasing function of radius for the first case; a homogeneous one for the second case; and a heterogeneous one with both moduli being either Dirac-delta or Heaviside-step decreasing functions of radius for the last case (if the coating moduli are unrestricted in magnitude or have upper limits, respectively). The results show a substantial increase in the permissible inclusion negative stiffness range is provided by coating heterogeneity, while maintaining overall composite stability. Such an increased range of constituent negative stiffness provides an enlarged tuning parameter range for the development of novel, high-performance composite materials.

Nonlinearity parameters B/A and C/A and wave equation for heterogeneous media containing negative stiffness inclusions

This work considers nonlinear propagation in a medium consisting of a low volume fraction of metamaterial inclusions dispersed in a fluid-like material. The metamaterial inclusions of interest are assumed to possess non-monotonic stress-strain constitutive relations, which results in regimes of negative stiffness. For modeling purposes, the constitutive relation for these inclusions is approximated with an expansion to third order in volume strain with coefficients that can be tuned with the geometry of the metamaterial structure and ambient pressure. A far-reaching goal of this research is to model the hysteretic response of the heterogeneous medium resulting from metamaterial inclusion snapping events and the associated effect on acoustic disturbances that cycle through regimes of both positive and negative stiffness. As an initial step, results are presented here for small but finite-amplitude disturbances limited to local regions of the constitutive relation. For this case, the quadratic and cubic nonlinearity parameters B/A and C/A, respectively, as traditionally defined for fluids are obtained. An evolution equation with both quadratic and cubic nonlinearity is also obtained. Numerical solutions of the evolution equation illustrate nonlinear waveform distortion as a function of the volume fraction and constitutive behavior of the inclusions. [Work supported by ARL:UT McKinney Fellowship in Acoustics.]

Stability of viscoelastic continuum with negative‐stiffness inclusions in the low‐frequency range

High damping and high stiffness (HDHS) viscoelastic materials, which exhibit desired energy-dissipating capability and structural integrity, may be achieved by using negative-stiffness inclusions, such as ferroelastic particles, in composite materials. However, stability has long been questioned for such composites to be realistic. Wang and Ko (pp. 2070–2079) examine a two-dimensional, two-phase, negative-stiffness viscoelastic composite under partial displacement loading by time–domain analysis with the finite element method. It was found that force responses may be stable or unstable, depending on the amount of negative stiffness. A small amount of negative stiffness in the viscoelastic composites may not deteriorate their stability, but is insufficient to trigger HDHS. Through examining the existing stability criteria of three-dimensional composites, the softening anomaly in the overall viscoelastic stiffness may be in the stability range, indicating stable damping enhancement. However, simultaneously enhancing damping and stiffness violates ellipticity requirements, hence the system is unstable/metastable. The cover image shows stress distributions around the inclusion of the two-dimensional composite in the background and stiffness/damping anomalies, as well as divergent force responses, deviating from expected sinusoidal responses.

Stability of viscoelastic continuum with negative‐stiffness inclusions in the low‐frequency range

The stability of viscoelastic composites with negative stiffness (NS) inclusions and their associated extreme mechanical properties are studied with time‐domain finite element analysis and composite theory. In the finite element analysis, the stability of the plane‐strain, two‐phase NS composite is evaluated by monitoring the stress divergence under uniaxial straining. In addition, 2D and 3D two‐phase NS composites are studied by using composite theories and the elastic–viscoelastic correspondence principle. Effective stiffness and damping anomalies are observed, and the stability of the composites is evaluated. In the low frequency regime, stability conditions for purely elastic systems are suitable to estimate the stability of the viscoelastic composites. This is verified with the finite element analysis. For the 2D problem, the finite element analysis and composite theory agree well with each other in terms of the overall properties and stability. For the 3D problem, only the results obtained from the composite theory are presented. Our analysis shows that the extreme mechanical properties for both of the 2D and 3D cases are located near the strong ellipticity boundary. When the matrix modulus is suitably chosen to satisfy all stability conditions, the effective stiffness softening anomaly (i.e., damping enhancement) may be in the stability range. The effective stiffness enhancement locates in the non‐elliptic range of the negative inclusion bulk modulus parameter space.

Elastic composite with negative stiffness inclusions in antiplane strain

An elastic composite consisting of an isotropic matrix with isotropic cylindrical inclusions of negative shear modulus is considered in the antiplane strain approximation. We show that the effective moduli of a composite with negative stiffness inclusions are defined uniquely and can be determined by using the conventional elastic solution where the positive moduli are replaced with the negative ones. We compute the effective shear modulus of the composite with interacting negative stiffness inclusions using the differential self-consistent method. We found that for concentrations of inclusions below a certain critical value the composite is stable and has positive effective shear modulus that is higher than that of the matrix. Thus negative stiffness inclusions stiffen the matrix, as has been suggested by R. Lakes. However, as soon as the concentration of inclusion exceeds the critical value the effective modulus of the composite abruptly drops to a negative value and the composite becomes unstable. Correspondingly, the value of the negative effective stiffness depends upon the stabilising device. The critical concentration depends upon the value of the negative modulus of inclusions; there exists a value of negative shear modulus at which the critical concentration is zero, i.e. the composite is always unstable.

Stress analysis of a two-phase composite having a negative-stiffness inclusion in two dimensions

Recent development in composites containing phase-transforming particles, such as vanadium dioxide or barium titanate, reveals the overall stiffness and viscoelastic damping of the composites may be unbounded (Lakes et al. 2001, Jaglinski et al. 2007). Negative stiffness is induced from phase transformation predicted by the Landau phase transformation theory. Although this unbounded phenomenon is theoretically supported with the composite homogenization theory, detailed stress analyses of the composites are still lacking. In this work, we analyze the stress distribution of the Hashin-Shtrikman (HS) composite and its two-dimensional variant, namely a circular inclusion in a square plate, under the assumption that the Young`s modulus of the inclusion is negative. Assumption of negative stiffness is a priori in the present analysis. For stress analysis, a closed form solution for the HS model and finite element solutions for the 2D composite are presented. A static loading condition is adopted to estimate the effective modulus of the composites by the ratio of stress to average strain on the loading edges. It is found that the interfacial stresses between the circular inclusion and matrix increase dramatically when the negative stiffness is so tuned that overall stiffness is unbounded. Furthermore, it is found that stress distributions in the inclusion are not uniform, contrary to Eshelby`s theorem, which states, for two-phase, infinite composites, the inclusion`s stress distribution is uniform when the shape of the inclusion has higher symmetry than an ellipse. The stability of the composites is discussed from the viewpoint of deterioration of perfect interface conditions due to excessive interfacial stresses.

Negative stiffness metamaterial elements for enhanced material damping capacity.

Recent research has shown that the introduction of small quantities (1%–2% by volume) of constrained negative stiffness inclusions into a viscoelastic host material leads to an effective energy absorbing behavior greatly exceeding that of conventional materials [Lakes et al., Nature (London) 410, 565–567 (2001)]. The root cause of this increased absorptive capacity is the ability of the negative stiffness materials to do work on the viscoelastic host when perturbed by an incident disturbance such as a sound wave or vibrational motion [Koutsawa et al., Int. J. Mech. Mater. Des. 5, 29–44 (2007)]. A buckled beam constrained to an S‐shaped configuration has a strain energy density at a local maximum and will accordingly display negative stiffness. This negative stiffness behavior is non‐transient and broadband in nature. The buckled beam structure is therefore a compelling candidate for the construction of new metamaterials for noise reduction, anechoic coatings, and transducer backing materials. This work pr...

A Concept for a Material That Softens With Frequency

This work presents design concepts to synthesize composite materials with special dynamic properties, namely, materials that soften at high frequencies. Such dynamic properties are achieved through the use of a two-phase material that has inclusions of a viscoelastic material of negative elastic modulus in a typical matrix phase that has a positive elastic modulus. A possible realization of the negative-stiffness inclusion phase is presented. A numerical homogenization technique is used to compute the average viscoelastic properties of the composite. The method and the properties of a composite material designed with it are demonstrated through an example.