Negative Shear Modulus Research Articles

Effect of stacking order on the vibration properties of bilayer black phosphorus

The effect of the stacking order of bilayer black phosphorus (BLBP) on the vibration properties is investigated by using molecular dynamics (MD) simulation and continuum modelling. The results show that the interlayer shear effect plays a very important role in the vibration behaviours of BLBPs. In MD simulation, a negative shear modulus is found in BLBP nanoresonators with stacking orders AA and AA′, which can decrease the natural frequency of the nanoresonators. The prediction of the natural frequency of the BLBP from the orthotropic sandwich plate model (OSPM) agrees with the MD simulation results very well. Moreover, for the vibration of the high-order mode, the interlayer shear modulus of the OSPM along one direction can mainly influence the natural frequency of the vibration mode, with the antinodes strictly aligning the direction for the BLBP. The results are thus important for understanding the dynamic behaviour of multilayer orthotropic nanostructures and designing multilayer orthotropic nanostructure resonators.

Time-domain investigation of an external cloak for antiplane elastic waves

Space folding techniques based on non-monotonic transforms lead to a new class of 2D isotropic cloaks with a constant negative shear modulus and a spatially varying negative density for antiplane elastic waves. We consider an external cloak consisting of a core with positive shear modulus and density, and a shell with simultaneously negative shear modulus and density. Such a core–shell resonant system creates a virtual folded region outside the shell. To handle such negative physical parameters in the time-domain, a two-step strategy is used: (i) assuming resonant (Drude and Lorentz-types) effective parameters in the frequency-domain; (ii) returning to the time-domain by applying the formalism of the auxiliary fields. We numerically show that, at the designed central frequency, scattering of a cylindrical antiplane elastic wave incident upon a finite set of small clamped obstacles is drastically reduced after a lapse of time, when they are placed in the close neighborhood of the external cloak. However, cloaking efficiency decreases when the source is placed in the intense near field of the external cloak and moreover at short times, cloaking breaks down.

On the equivalent shear modulus of composite metamaterials

Here, we put the equivalent shear modulus of metamaterials under scrutiny. A micromorphic model for the equivalent shear modulus of metamaterials is developed. This model represents a metamaterial as a multiscale material that exhibits a microstructural strain (micro-strain) field, which is independent of the macroscopic strain (macro-strain) field of the material. The mechanisms by which the shear modulus of a metamaterial would change and depend on the material size are discussed. It is revealed that the shear modulus of a metamaterial is size and microstructure topology-dependent as long as the micro-strain field is significant and different from the macro-strain field. The conditions at which the shear modulus would be zero, negative, or positive are also defined. A unique material parameter (β) that depends on the microstructure topology is introduced to define the conditions of the material stability. It is revealed that a metamaterial with a negative shear modulus is stable as long as β2<0. The proposed micromorphic model is also employed to determine the equivalent shear modulus of composite metamaterials of the form of coated inclusions embedded in a matrix material. It is demonstrated that a composite metamaterial with a giant shear modulus can be produced by employing either one phase of a negative modulus or an interfacial layer between the inclusion and the matrix. In addition, the implementation of an interface between the inclusion and the matrix would give a composite metamaterial with a negative shear modulus.

A new two-dimensional elastic metamaterial system with multiple local resonances

A new two-dimensional periodic elastic metamaterial system is proposed, which exhibits multiple local resonances and can generate negative effective parameters. In certain frequency ranges, negative bulk modulus, negative shear modulus and negative mass density can be achieved simultaneously. Analytical study of the new metamaterial system is performed based on a simplified model to study the effects of main material and geometric parameters. Numerical analysis is further conducted to simulate elastic wave propagation in the current metamaterial. The results show clearly the existence of negative dispersion. The association of negative mass and modulus with the mode of local resonance, the eigenstate, is evaluated. In the frequency range for negative parameters, negative phase velocities of the longitudinal and shear waves can be observed in the simulation. The developed metamaterial shows strong anisotropic properties. It can support both longitudinal and shear waves in general directions but in specific directions only longitudinal wave can propagate. In the latter case, it exhibits fluid-like properties. Typical examples of the current metamaterial under different loading conditions are presented to show both the modal response and the property of elastic wave propagation in the metamaterial.

Vacancy-induced mechanical stabilization of cubic tungsten nitride

First-principles methods are employed to determine the structural, mechanical, and thermodynamic reasons for the experimentally reported cubic WN phase. The defect-free rocksalt phase is both mechanically and thermodynamically unstable, with a negative single crystal shear modulus ${C}_{44}=\phantom{\rule{0.16em}{0ex}}\ensuremath{-}86\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ and a positive enthalpy of formation per formula unit ${H}_{f}=\phantom{\rule{0.16em}{0ex}}0.623\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ with respect to molecular nitrogen and metallic W. In contrast, WN in the NbO phase is stable, with ${C}_{44}=\phantom{\rule{0.16em}{0ex}}175\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ and ${H}_{f}=\phantom{\rule{0.16em}{0ex}}\ensuremath{-}0.839\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$. A charge distribution analysis reveals that the application of shear strain along [100] in rocksalt WN results in an increased overlap of the ${t}_{2g}$ orbitals which causes electron migration from the expanded to the shortened W-W $\ensuremath{\langle}110\ensuremath{\rangle}$ bond axes, yielding a negative shear modulus due to an energy reduction associated with new bonding states 8.1--8.7 eV below the Fermi level. A corresponding shear strain in WN in the NbO phase results in an energy increase and a positive shear modulus. The mechanical stability transition from the NaCl to the NbO phase is explored using supercell calculations of the NaCl structure containing ${C}_{v}=\phantom{\rule{0.16em}{0ex}}0%\ensuremath{-}25%$ cation and anion vacancies, while keeping the N-to-W ratio constant at unity. The structure is mechanically unstable for ${C}_{v}l\phantom{\rule{0.16em}{0ex}}5%$. At this critical vacancy concentration, the isotropic elastic modulus $E$ of cubic WN is zero, but increases steeply to $E=445\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ for ${C}_{v}=\phantom{\rule{0.16em}{0ex}}10%$, and then less steeply to $E\phantom{\rule{0.16em}{0ex}}=561\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ for ${C}_{v}=\phantom{\rule{0.16em}{0ex}}25%$. Correspondingly, the hardness estimated using Tian's model increases from 0 to 15 to 26 GPa as ${C}_{v}$ increases from 5% to 10% to 25%, indicating that a relatively small vacancy concentration stabilizes the cubic WN phase and that the large variations in reported mechanical properties of WN can be attributed to relatively small changes in ${C}_{v}$.

The structural, elastic, phonon, thermal and electronic properties of MnX (X=Ni, Pd and Pt) alloys: First-principles calculations

We present first-principles studies of structural, elastic, phonon, thermal and electronic properties of MnX. According to the calculated total energies, the predicted phase stability trends are consistent with previous experiments. For the L10 lattice, the calculated Mn–Mn exchange couplings JMn–Mns, depending on the valence electron concentration Cv, have been investigated. The dominant inter-layer nearest-neighbors coupling J1L and in-plane next-nearest-neighbor coupling J2 are responsible for the differences in energy between the two structural phases (T=0K). The negative shear modulus C′ of the paramagnetic B2 (phase) triggers the cubic to tetragonal distortion. The calculated phonon dispersions indicate that the primary driving force for the paramagnetic L10 to antiferromagnetic L10 transition is the magnetic ordering energy, rather than some entropic contribution from thermal vibrations. The Debye temperatures, anisotropy and Poisson's ratio are also assessed. The calculated results show that the collinear antiferromagnetic CuAu-I type structures are mechanically and dynamically stable, and agree with the empirical ground states.

The first-principles study of martensitic transformations in MnMe (Me = Rh, Pd) alloys at low temperatures

The structural, elastic, phonon, thermal and electronic properties of MnMe are investigated by the first-principles calculations. The predicted phase stability trends due to structural energy differences are consistent with those observed experimentally. The obtained negative shear modulus (C′ = (C11 – C12)/2) of paramagnetic B2 (PM-B2) phases triggers the cubic to tetragonal distortion. The calculated phonon dispersions without negative frequency indicate that antiferromagnetic L10 (AFM-L10) together with paramagnetic L10 (PM-L10) structures are dynamically stable, and the PM-L10 → AFM-L10 transition is caused by magnetism rather than lattice dynamics. Debye temperatures, anisotropy and Poisson's ratios are also first evaluated. The calculated results show that the collinear antiferromagnetic states (CuAu-I type) are mechanically and dynamically stable, thus are the low temperature phases.

Mechanical effect of rotating non-spherical particles on failure in compression

When non-spherical particles (grains, blocks) in a rock or geomaterial under compression are sufficiently detached to enable rolling under shear stress the moment equilibrium dictates that further increase in displacement requires reduced shear stress. Thus the rolling non-spherical particle produces an effect of apparent negative stiffness whose value is proportional to the magnitude of the compressive stress. We model geomaterials with rolling particles at a macroscale as a matrix containing inclusions with negative shear modulus. We consider two extreme types of inclusions: spherical inclusions and shear cracks with negative stiffness shear springs inside. We show that there exists a critical concentration of inclusions at which the effective shear modulus abruptly becomes negative and the geomaterial loses stability. Furthermore, there exists a special value of negative shear modulus/stiffness of inclusions (and hence the magnitude of the compressive stress) at which the critical concentration is zero, such that the first rolling particle induces the global instability. This stress is of the order of the shear modulus the geomaterial exhibits at the advanced stage of fracturing when the particles are sufficiently detached to enable the rolling.

Structural, elastic, phonon and electronic properties of a MnPd alloy

The structural, elastic, phonon and electronic properties of a MnPd alloy have been investigated using the first-principles calculation. The calculated lattice constants and electronic structure agree well with the experimental results. The microscopic mechanism of the diffusionless martensitic transition from the paramagnetic B2 (PM-B2) phase to the antiferromagnetic L10 (AFM-L10) phase through the intermediate paramagnetic L10 (PM-L10) phase has been explored theoretically. The obtained negative shear modulus C′ = (C11 — C12)/2 of the PM-B2 phase is closely related to the instability of the cubic B2 phase with respect to the tetragonal distortions. The calculated phonon dispersions for the PM-L10 and AFM-L10 phases indicate that they are dynamically stable. However, the AFM-L10 phase is energetically most favorable according to the calculated total energy order, so the PM-L10 → AFM-L10 transition is caused by the magnetism rather than the electron—phonon interaction. Additionally, the AFM-L10 state is stabilized through the formation of a pseudo gap located at the Fermi level. The calculated results show that the CuAu-I type structure in the collinear antiferromagnetic state is dynamically and mechanically stable, thus is the low temperature phase.

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.

Elastic metamaterials with negative bands

The unusual properties of a metamaterial come from special resonances supported by its resonating structure units. Guided by a previously developed effective medium theory, which links the resonances of the microstructures and the unusual properties, two types of elastic metamaterials in two dimensions were designed with different resonant structures in their building blocks that exhibit multiple negative dispersion bands with special characteristics. The first type possesses negative mass density and negative shear modulus simultaneously over a large frequency regime, which leads to a negative band for shear waves only. Mode conversion takes place at the interface of the metamaterial and the common solids. The second type is able to produce negative effective moduli in different frequency regimes within a large frequency regime of negative effective mass density. This results in a super anisotropic negative band and a negative band that only compressional is allowed. All of these unusual properties are demonstrated by simulations. This work was supported by Hong Kong RGC Grant No. 605008, HKUST604207 and KAUST start-up package.

Surface/interface effects on elastic behavior of a screw dislocation in an eccentric core–shell nanowire

The elastic behavior of a screw dislocation which is positioned inside the shell domain of an eccentric core–shell nanowire is addressed with taking into account the surface/interface stress effect. The complex potential function method in combination with the conformal mapping function is applied to solve the governing non-classical equations. The dislocation stress field and the image force acting on the dislocation are studied in detail and compared with those obtained within the classical theory of elasticity. It is shown that near the free outer surface and the inner core–shell interface, the non-classical solution for the stress field considerably differs from the classical one, while this difference practically vanishes in the bulk regions of the nanowire. It is also demonstrated that the surface with positive (negative) shear modulus applies an extra non-classical repelling (attracting) image force to the dislocation, which can change the nature of the equilibrium positions depending on the system parameters. At the same time, the non-classical solution fails when the dislocation approaches very close to the surface/interface with negative shear modulus. The effects of the core–shell eccentricity and nanowire diameter on dislocation behavior are discussed. It is shown that the non-classical surface/interface effect has a short-range character and becomes more pronounced when the nanowire diameter is smaller than 20nm.

Elastic Metamaterials with Simultaneously Negative Effective Shear Modulus and Mass Density

We propose a type of elastic metamaterial comprising fluid-solid composite inclusions which can possess a negative shear modulus and negative mass density over a large frequency region. Such a material has the unique property that only transverse waves can propagate with a negative dispersion while longitudinal waves are forbidden. This leads to many interesting phenomena such as negative refraction, which is demonstrated by using a wedge sample and a significant amount of mode conversion from transverse waves to longitudinal waves that cannot occur on the interface of two natural solids.

Extreme viscoelastic properties of composites of strongly inhomogeneous structures due to negative stiffness phases

AbstractA fractal model of structure [V. V. Novikov et al., Phys. Rev. E 63, 036120 (2001)] is generalized and applied to study the viscoelastic properties of a two‐component, strongly inhomogeneous medium of chaotic structure with one component exhibiting a negative shear modulus. It is shown that similar to the results obtained recently in frames of the Hashin–Strikman model [R. S. Lakes, Phys. Rev. Lett. 86, 2897 (2001)], the present fractal model predicts the existence of composites having effective shear and damping coefficients much higher than those of the component phases. The viscoelastic properties of the strongly inhomogeneous medium are, however, qualitatively different from the properties of the Hashin‐Strikman medium. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

On the Existence of Positive Eigenvalues for the Isotropic Linear Elasticity System with Negative Shear Modulus

In this note well-known PDE and functional analyis results are used to prove that the set of real eigenvalues for the linear isotropic elasticity with negative shear modulus is unbounded from above as well as from below.

A simple viscoelastic model for soft tissues in the frequency range 5–15 MHz

Measurements of the shear properties of soft tissues have been performed using a simple pulse–echo reflection technique. Briefly, a shear wave, generated by a shear wave transducer, propagates through a quartz rod, is reflected at the quartz–tissue interface, backpropagates through the rod, and is received by the transducer. The phase shift and the change in the magnitude of the wave with and without a tissue sample in place are measured, and the shear mechanical impedances are calculated and the complex modulus is determined. A negative shear modulus is often obtained in this frequency range, and none of the traditional linear models can explain this result. We propose a simple extension to the Voigt model for viscoelasticity to account for the effects of mass. In this model, a negative storage modulus is possible under the conditions of high frequency and long relaxation time. When the frequency is low and relaxation time is small, the new model reduces to the classic Voigt model. Based on this model, we will present the results of measurements of shear modulus, shear viscosity, and the constant term related to mass for several soft tissues. [Work supported by the US Army.]

Well-posedness of the Dirichlet problem in non-elliptic isotropic elasticity

The well-posedness of the Dirichlet boundary value problem for an isotropic homogeneous linear elastic material is established for non-elliptic equilibrium systems. Mechanical properties of elastic materials with negative shear modulus are also deduced.

Elasticity of some mantle crystal structures: 2. Rutile GeO2

The single-crystal elastic constants are determined as a function of pressure and temperature for rutile structure germanium dioxide (GeO2). The data are qualitatively similar to those of rutile TiO2 measured by Manghnani (1969). The compressibility in the c direction is less than one-half that in the a direction, the pressure derivative of the shear constant Cs = (C11 − C12)/2 is negative, and the pressure derivative of the bulk modulus ∂K/∂P has a relatively high value of about 6.2. The high ∂K/∂P values for rutile GeO2 and rutile TiO2 are not consistent with the Slater or Dugdale-MacDonald equations relating ∂K/∂P to the Gruneisen parameter. The ∂K/∂P for stishovite (which has the rutile structure) is probably high also. According to an elastic strain energy theory, the negative shear modulus derivative implies that the kinetic barrier to diffusion decreases with increasing pressure. Kinetic data on the quartz GeO2 → rutile GeO2 transformation appear to support the applicability of the theory.