Negative Bulk Modulus Research Articles

Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density

We report a metamaterial which simultaneously possesses a negative bulk modulus and mass density. This metamaterial is a zinc blende structure consisting of one fcc array of bubble-contained-water spheres (BWSs) and another relatively shifted fcc array of rubber-coated-gold spheres (RGSs) in epoxy matrix. The negative bulk modulus and mass density are simultaneously derived from the coexistent monopolar resonances from the embedded BWSs and dipolar resonances from the embedded RGSs. The Poisson ratio of the metamaterial also turns negative near the resonance frequency.

Stability of elastic material with negative stiffness and negative Poisson's ratio

AbstractStability of cuboids and cylinders of an isotropic elastic material with negative stiffness under partial constraint is analyzed using an integral method and Rayleigh quotient. It is not necessary that the material exhibit a positive definite strain energy to be stable. The elastic object under partial constraint may have a negative bulk modulus K and yet be stable. A cylinder of arbitrary cross section with the lateral surface constrained and top and bottom planar surfaces is stable provided the shear modulus G > 0 and –G /3 < K < 0 or K > 0. This corresponds to an extended range of negative Poisson's ratio, –∞ < ν < –1. A cuboid is stable provided each of its surfaces is an aggregate of regions obeying fully or partially constrained boundary conditions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Composite Materials with Viscoelastic Stiffness Greater Than Diamond

We show that composite materials can exhibit a viscoelastic modulus (Young's modulus) that is far greater than that of either constituent. The modulus, but not the strength, of the composite was observed to be substantially greater than that of diamond. These composites contain bariumtitanate inclusions, which undergo a volume-change phase transformation if they are not constrained. In the composite, the inclusions are partially constrained by the surrounding metal matrix. The constraint stabilizes the negative bulk modulus (inverse compressibility) of the inclusions. This negative modulus arises from stored elastic energy in the inclusions, in contrast to periodic composite metamaterials that exhibit negative refraction by inertial resonant effects. Conventional composites with positive-stiffness constituents have aggregate properties bounded by a weighted average of constituent properties; their modulus cannot exceed that of the stiffest constituent.

On the Bulk Modulus of Open Cell Foams

Bulk properties of open cell polyurethane foam are studied in a hydrostatic compression experiment under strain control. A linear region of behaviour is observed in the stress-strain curve, followed by a non-monotonic region corresponding to a negative incremental bulk modulus. The bulk modulus in the linear region is in reasonable agreement with the value calculated from compressional Young's modulus and Poisson's ratio. The linear region of behaviour in hydrostatic compression corresponds to less than half the axial strain range observed in axial compression.

Negative incremental bulk modulus in foams

Negative incremental stiffness is known to occur in structures such as post-buckled flexible tubes and single-cell models. A single foam cell under uniaxial loading buckles and exhibits a non-monotonic S-shaped deformation curve, which is indicative of negative incremental stiffness. Negative stiffness is not observed in bulk materials. For example, individual foam cells display negative stiffness but foams tested in uniaxial compression exhibit a plateau in the stress–strain curve because the buckled cells localize in bands. This behaviour is consistent with the continuum view in which strong ellipticity and, hence, a positive shear modulus G and positive C 11 modulus are required for stability, even for a constrained object. It is hypothesized that a solid with negative bulk modulus can be stabilized by control of the surface displacement. Experimentally, foams were hydrostatically compressed by controlled injections of small volumes of water into a plastic chamber, causing volumetric deformation. A negative incremental bulk modulus was observed in a foam with 0.4-mm cell size beyond about 20% volumetric strain. A foam with large cells, 2.5–4 mm in size, was anisotropic and did not exhibit the cell buckling required for negative modulus.

Lateral shift of acoustic wave at interface between double-positive and double-negative media

When medium has a negative effective bulk modulus and density, it is called double-negative medium. On the basis of the case, we theoretically derive lateral shift of total reflection wave at interface when an acoustic wave from one medium into the other, analyze systematically the directions of lateral shift when acoustic wave from double-positive (double-negative) medium into double-negative and double-positive medium, and point out the directions of energy flux and wave vector in double-positive medium and double-negative medium.

On extending the concept of double negativity to acoustic waves

The realization of double negative electromagnetic wave media, sometimes called left-handed materials (LHMs) or metamaterials, have drawn considerable attention in the past few years. We will examine the possibility of extending the concept to acoustic waves. We will see that acoustic metamaterials require both the effective density and bulk modulus to be simultaneously negative in the sense of an effective medium. If we can find a double negative (negative density and bulk modulus) acoustic medium, it will be an acoustic analogue of Veselago’s medium in electromagnetism, and share many novel consequences such as negative refractive index and backward wave characteristics. We will give one example of such a medium.

Thermodynamics of volume-collapse transitions in cerium and related compounds

We present a non-linear elastic model of a coherent transition with discontinuous volume change in an isotropic solid. The model reproduces the anomalous thermodynamics typical of coherent equilibrium including intrinsic hysteresis (for a pressure driven experiment) and a negative bulk modulus. The novelty of the model is that the statistical mechanics solution can be easily worked out. We find that coherency leads to an infinite-range density–density interaction, which drives classical critical behavior. The pressure width of the hysteresis loop shrinks with increasing temperature, ending at a critical point at a temperature related to the shear modulus. The bulk modulus softens with a 1/2 exponent at the transition even far from the critical point. Many well known features of the phase diagram of Ce and related systems are explained by the model.

Composites with Inclusions of Negative Bulk Modulus: Extreme Damping and Negative Poisson’s Ratio

The effect of a negative bulk modulus phase in elastic composites is studied. Negative bulk modulus K<0 is shown to be possible in selected unit cells. In isotropic solids, K<0 can be attained when negative Poisson’s ratio is sufficiently small, below the stability limit (for stress control) 1/4 1. Such materials, if used as inclusions, are predicted to be stable with respect to the band formation, even if they are large. Composites with spherical inclusions of negative bulk moduli are shown to exhibit negative Poisson’s ratio and anomalies in composite bulk modulus and Young’s modulus (and in the corresponding mechanical damping) but not in the shear modulus.

Vertical stability of bubble chain: Multiscale approach

Linear stability is investigated of a uniform chain of equal spherical gas bubbles rising vertically in unbounded stagnant liquid at Reynolds number Re = 50–200 and bubble spacing s > 2.6 bubble radii. The equilibrium bubble positions are questioned for their stability with respect to small displacements in the vertical direction, parallel to the chain motion. The transverse displacements are not considered, and the chain is assumed to be laterally stable. The bubbles are subjected to three kinds of forces: buoyant, viscous, inviscid. The viscous and inviscid forces have both pairwise (local) and distant (nonlocal) components. The pairwise forces are expressed by the leading-order formulas known from the literature. The distant forces are expressed as a linear superposition of the pairwise forces taken over several farther neighbours. The stability problem is addressed on three different length scales corresponding to: discrete chain (microscale), continuous chain (mesoscale), bubbly chain flow (macroscale). The relevant governing equations are derived for each scale. The microscale equations are a set of ODE’s, the Newton force laws for the individual discrete bubbles. The mesoscale equation is a PDE for bubbles continuously distributed along a line, obtained by taking the continuum limit of the microscale equations. The macroscale equations are two PDEs, the mass and momentum conservation equations, for an ensemble of noninteracting mesoscale chains rising in parallel. This transparent two-step process (micro → meso → macro) is an alternative to the usual one-step averaging, in obtaining the macroscale equations from microscale information. Here, the scale-up methodology is demonstrated for 1D motion of bubbles, but it can be used for behaviour of 2D and 3D lattices of bubbles, drops, and solids. It is found that the uniform equilibrium spacing results from a balance between the attractive and repulsive forces. On all three length scales, the equilibrium is stabilized by the viscous drag force, and destabilized by the viscous shielding force ( shielding instability). The inviscid forces are stability neutral and generate conservative oscillations and concentration waves. The stability region in the parameter plane s − Re is determined for each length scale. The stable region is relatively small on the microscale, larger on the mesoscale, and shrinks to zero on the macroscale where the bubbly chain flow is inherently unstable. The shielding instability is expected to occur typically in intermediate Re flows where the vertical bubble interactions dominate over the horizontal interactions. This new kind of instability is studied here in a great detail, likely for the first time. Its relation to the elasticity properties of bubbly suspension on different length scales is discussed too. The shielding force takes the form of a negative bulk modulus of elasticity of the bubbly mixture.

Stabilized jellium: Structureless pseudopotential model for the cohesive and surface properties of metals.

The ions in a simple metal act on the valence electrons via a pseudopotential. The long-range part is represented by the electrostatic potential from the positive background of the jellium model. The short-range part can be simulated by a constant (over the interior of the metal), chosen to stabilize the metal at its observed bulk valence-electron density. In this structureless pseudopotential model, the bulk properties of a metal depend only upon valence z and bulk density parameter ${\mathit{r}}_{\mathit{s}}$, while the surface properties depend upon ${\mathit{r}}_{\mathit{s}}$ alone (an experimental trend heretofore not understood). These properties are calculated in closed analytic form, and the anomalies of the jellium model (negative surface energy for ${\mathit{r}}_{\mathit{s}}$\ensuremath{\approxeq}2; negative bulk modulus for ${\mathit{r}}_{\mathit{s}}$\ensuremath{\approxeq}6) are found to be rectified. The new model, perhaps the simplest one viable for all ${\mathit{r}}_{\mathit{s}}$, may also be used to study interfaces, metallic clusters, vacancies, electromagnetic response, etc. A variant of the model, which simulates the effects of atomic corrugation, predicts the crystal face dependence of surface properties. This dependence is strong for the electron-density profile, but not for the surface energy, work function, and distance from the centroid of excess charge to the first lattice plane. Results are presented for metallic hydrogen as well as for Al, Pb, Zn, Mg, Ca, Li, Sr, Ba, Na, K, Rb, and Cs.

Frequency‐Dependent Exchange Correction to the Dielectric Function of the Electron Gas

AbstractThe equation of motion of the Wigner distribution function in the homogeneous electron gas is decoupled using the Hartree‐Fock prescription. After linearization, the variational procedure, used by Rajagopal and Jain and by Langreth is applied. The transverse and the longitudinal dielectric functions in the Hartree‐Fock approximation are derived, and the consistency of the equations for the charge density and the current density is examined. Several limits are calculated. The slope of the plasmon mode is substantially lowered by the exchange interactions, and the damped acoustic mode disappears for the densities that give a negative bulk modulus in the Hartree‐Fock approximation. The exchange interaction only slightly influences the dispersion of the transverse collective modes.