Continuum Elasticity Theory Research Articles

Dielectric function and band structure of Sn1−xGex (x < 0.06) alloys on InSb

Tin-rich Sn1−xGex alloys with Ge contents up to 6% were grown pseudomorphically on InSb (001) substrates by molecular beam epitaxy at room temperature. The alloys show a germanium-like lattice and electronic structure and respond to the biaxial stress within continuum elasticity theory, which influences bands and interband optical transitions. The dielectric function of these alloys was determined from 0.16 to 4.7 eV using Fourier-transform infrared and spectroscopic ellipsometry. The E1 and E1 + Δ1 critical points decrease with the increasing Ge content with a bowing parameter similar to the one established for Ge-rich Sn1−xGex alloys. On the other hand, the inverted direct bandgap E¯0 is nearly independent of the Ge content, which requires a bowing parameter of about 0.8 eV, much lower than what has been established using photoluminescence experiments of Ge-rich relaxed Sn1−xGex alloys.

Growth mode in heteroepitaxial system from nano- and macro-theoretical viewpoints

Growth mode in semiconductor heteroepitaxial system such as InAs/GaAs is systematically investigated from nano- and macro-theoretical viewpoints, where ab initio, empirical interatomic potential, and phenomenological macroscopic theory are used. The nanoscopic theories clarify that the misfit dislocation (MD) formation energy depends on orientation such as 1.14 eV/Å for (0 0 1), 0.96 eV/Å for (1 1 0), and 0.68 eV/Å for (1 1 1)A. On the basis of these calculated results, growth mode boundary between two-dimensional with MD formation (2D-MD) and three-dimensional Stranski-Krastanov island (3D-SK) growth modes is successfully determined as functions of surface energy and degree of strain relaxation. The region of 3D-SK growth mode is the largest for (0 0 1) while (1 1 1)A has the largest 2D-MD region and that for (1 1 0) is in between. Using surface energy obtained by ab initio calculations and the degree of strain relaxation estimated by continuum elasticity theory, it is found that (0 0 1) favors 3D-SK growth mode while 2D-MD growth mode appears for (1 1 0) and (1 1 1)A. It should be noted that the (1 1 0) data stay near the growth mode boundary between 3D-SK and 2D-MD growth modes. Reflecting this, growth mode transition from 2D-MD to 3D-SK occurs for (1 1 0) due to In0.25Ga0.75As layer insertion at the interface that decreases surface energy from 50.9 meV/Å2 to 49.1 meV/Å2. Versatility of this approach is discussed by comparing with experimental findings.

Three-dimensional analytical model for the dynamic interaction of twin tunnels in a homogeneous half-space

Ground vibration and noise induced by underground railways have become a major environmental issue. Significant research has been carried out to investigate vibrations generated by a single tunnel, with less attention been paid to the dynamic interaction of twin tunnels. This paper presents a three-dimensional analytical model for the dynamic interaction of twin tunnels in a homogeneous half-space. The two tunnels are modeled using the cylindrical thin-shell theory. The soil, a homogenous half-space with two cylindrical cavities, is simulated using the elastic continuum theory. The translation between two cylindrical waves and the transformation between plane waves and cylindrical waves are introduced to satisfy the boundary conditions on the ground surface and the twin tunnel–soil interfaces. The accuracy of the proposed model is verified by comparison with numerical models. A numerical example is presented to investigate the influence of the dynamic interaction between twin tunnels on the ground vibrations.

Experimental Study of Elastic Constants of a Dense Foam with Weak Cosserat Coupling

A dense closed cell foam is studied to determine which continuum theory of elasticity is applicable. Size effects inconsistent with classical elasticity are observed. The material exhibits a characteristic length scale considerably larger, by more than a factor 6, than the largest observable structure size. The Cosserat coupling number $N$ is shown to be small, via measurements of size effects in square section bars, comparison with size effects in round section bars, and determination of warp of a square section bar in torsion. For this material, the couple stress theory is excluded and the modified couple stress theory is excluded. Theories that force constants to their thermodynamic limits do not apply to this foam. The role of other generalized continuum theories is considered.

Atomically stepped, pseudomorphic, corundum-phase (Al1-xGax)2O3 thin films (0 ≤ x < 0.08) grown on R-plane sapphire

Atomically smooth, pseudomorphic (Al1−xGax)2O3 thin films were grown for 0 ≤ x < 0.08 on R-plane sapphire (01.2) by pulsed laser deposition at growth temperatures up to 1000 °C. Films up to 720 nm thickness show atomically stepped surfaces with monolayer terraces, similar to the substrates prior to growth, in wide ranges of growth pressure and temperature. A careful analysis of 13 symmetric, skew-symmetric, and asymmetric X-ray peaks agrees quite well with the continuum elastic strain theory of pseudomorphic distortion in corundum heterostructures by M. Grundmann [J. Appl. Phys. 124, 185302 (2018)]. The Ga-contents x and weak deviations from the expected ratios of rhombohedral c/a lattice constants are obtained from the best fits of the spacing of the (02.4), (04.8), and (00.6), (00.12) film and substrate reflections, in reasonable agreement with chemical energy dispersive X-ray spectrometry analyses. In addition, the measured tilts of the asymmetric film lattice planes corroborate the elastic theory.

Acoustical characteristics of single-walled noncarbon nanotubes: Longitudinal and torsional waves

A study based on nonlocal continuum elastic theory is presented on the acoustic properties of single-walled noncarbon nanotubes, including boron nitride (BN), aluminum nitride (AlN), gallium nitride (GaN), aluminum phosphide (AlP), gallium phosphide (GaP), and boron (B) nanotubes. Elastic characteristics of the noncarbon nanotubes needed to describe the elastic waves propagation correctly have been defined previously. Two-dimensional densities (2D densities) for each type of noncarbon nanotubes are calculated. The values of propagation velocities of longitudinal and torsional elastic waves for proposed nanotubes with different diameter and chirality are obtained. Specific result is that in boron nanotubes sound propagates at a speed equal 54 km/s.

Why large icosahedral viruses need scaffolding proteins

While small single-stranded viral shells encapsidate their genome spontaneously, many large viruses, such as the herpes simplex virus or infectious bursal disease virus (IBDV), typically require a template, consisting of either scaffolding proteins or an inner core. Despite the proliferation of large viruses in nature, the mechanisms by which hundreds or thousands of proteins assemble to form structures with icosahedral order (IO) is completely unknown. Using continuum elasticity theory, we study the growth of large viral shells (capsids) and show that a nonspecific template not only selects the radius of the capsid, but also leads to the error-free assembly of protein subunits into capsids with universal IO. We prove that as a spherical cap grows, there is a deep potential well at the locations of disclinations that later in the assembly process will become the vertices of an icosahedron. Furthermore, we introduce a minimal model and simulate the assembly of a viral shell around a template under nonequilibrium conditions and find a perfect match between the results of continuum elasticity theory and the numerical simulations. Besides explaining available experimental results, we provide a number of predictions. Implications for other problems in spherical crystals are also discussed.

Analysis of Indentation on Layered Elastic Medium with Surface Energy Effects

This paper presents the analysis of an axisymmetric frictionless rigid punch on a layered elastic medium with the consideration of surface energy effects by adopting Gurtin-Murdoch continuum theory of surface elasticity. The indentation problem is formulated as a mixed-boundary value problem with the displacement boundary condition being imposed at the contact area. The unknown contact pressure under the indenter is then determined by employing a discretization technique with use of the displacement Green’s functions. The required Green’s functions are expressed in the form of the Hankel integral transform. The accuracy of the proposed solution scheme is verified by comparing with existing solutions. Selected numerical results on displacement and stress profiles in a layered elastic half-space are presented to demonstrate the influence of various parameters on elastic fields. It is found that the layered medium becomes stiffer and shows size-dependent behavior due to the presence of surface stresses.

Membrane Elastic Deformations Modulate Gramicidin A Transbilayer Dimerization and Lateral Clustering

Gramicidin A (gA) is a short β-helical peptide known to form conducting channels in lipid membranes because of transbilayer dimerization. The gA conducting dimer, being shorter than the lipid bilayer thickness, deforms the membrane in its vicinity, and the bilayer elastic energy contributes to the gA dimer formation energy. Likewise, membrane incorporation of a gA monomer, which is shorter than the lipid monolayer thickness, creates a void, thereby forcing surrounding lipid molecules to tilt to fill it. The energy of membrane deformation was calculated in the framework of the continuum elasticity theory, taking into account splay, tilt, lateral stretching/compression, Gaussian splay deformations, and external membrane tension. We obtained the interaction energy profiles for two gA monomers located either in the same or in the opposite monolayers. The profiles demonstrated the long-range attraction and short-range repulsion behavior of the monomers resulting from the membrane deformation. Analysis of the profile features revealed conditions under which clusters of gA monomers would not dissipate because of diffusion. The calculated dependence of the dimer formation and decay energy barriers on the membrane elastic properties was in good agreement with the available experimental data and suggested an explanation for a hitherto contentious phenomenon.

Compressive response and buckling of graphene nanoribbons

We examine the mechanical response of single layer graphene nanoribbons (GNR) under constant compressive loads through molecular dynamics simulations. Compressive stress-strain curves are presented for GNRs of various lengths and widths. The dependence of GNR’s buckling resistance on its size, aspect ratio, and chiral angle is discussed and approximate corresponding relations are provided. A single master curve describing the dependence of the critical buckling stress of GNRs on their aspect ratio is presented. Our findings were compared to the continuum elasticity theories for wide plates and wide columns. In the large width limit, the response of the GNRs agrees with the predictions of the wide plates theory and thus, with that of wide graphenes. In the small width limit, the behavior of graphene nanoribbons deviates from that of periodic graphenes due to various edge related effects which govern the stiffness and the stability of the graphene membranes, but it qualitatively agrees with the theory of wide columns. In order to assess the effect of thermal fluctuations on the critical buckling stress a wide range of temperatures is examined. The findings of the current study could provide important insights regarding the feasibility and the evaluation of the performance of graphene-based devices.

Quantitative Characterization of the Nanoscale Local Lattice Strain Induced by Sr Dopants in La_{1.92}Sr_{0.08}CuO_{4}.

The nanometer scale lattice deformation brought about by the dopants in the high temperature superconducting cuprate La_{2-x}Sr_{x}CuO_{4} (x=0.08) was investigated by measuring the associated x-ray diffuse scattering around multiple Bragg peaks. A characteristic diffuse scattering pattern was observed, which can be well described by continuum elastic theory. With the fitted dipole force parameters, the acoustic-type lattice deformation pattern was reconstructed and found to be of similar size to lattice thermal vibration at 7K. Our results address the long-term concern of dopant introduced local lattice inhomogeneity, and show that the associated nanometer scale lattice deformation is marginal and cannot, alone, be responsible for the patched variation in the spectral gaps observed with scanning tunneling microscopy in the cuprates.

Treatise on Winkler Modulus of Subgrade Reaction and Its Estimation for Improved Soil–Structure Interaction Analysis

The paper pertains to the development of a new relationship expressing the Winkler modulus of subgrade reaction as a function of elastic parameters such as the modulus of elasticity and Poisson’s ratio of the foundation soil, and the relative rigidity of the foundation soil and the beam resting over the same. It ensures that the maximum values of beam deflection and bending moment computed by using both the theory of elastic continuum and lumped parameter modeling are either identical or very close to each other. In this respect the developed expression can be construed to be superior to those proposed by Biot and Vesic, which respectively predict correct values for only one of the quantities (either maximum bending moment or maximum deflection). In addition, the proposed model for subgrade modulus is applicable to multiple load conditions as well unlike the other two approaches as reported.

Wrinkling patterns of tantalum films on modulus-gradient compliant substrates

Wrinkling patterns are ubiquitous in nature and they are beneficial for a wide range of practical applications. Here we report on the wrinkling patterns in tantalum films sputter deposited on modulus-gradient compliant polydimethylsiloxane substrates. It is found that the sputtering process generates intrinsic and thermal compressive stresses, which are relieved by formation of locally ordered (herringbone) but overall disordered (labyrinth) wrinkling. The wrinkle sizes are strongly dependent on the substrate stiffness and film thickness. The formation mechanisms and evolutional behaviors of the wrinkling patterns have been discussed and analyzed in detail based on the continuum elasticity theory. Some potential applications of gradient wrinkling including surface hydrophobicity and measurement of substrate modulus are also presented in this paper.

Effect of surface on the dissociation of perfect dislocations into Shockley partials describing the herringbone Au(1 1 1) surface reconstruction

In the framework of the continuum elastic theory, the equilibrium positions of Shockley partial dislocations have been determined as a function of their distance from the free surface. It is found that the dissociation width decreases with the decreasing depth, except for a depth range very close to the free surface for which the dissociation width is enlarged. A similar behaviour is also predicted when Shockley dislocation pairs are regularly arranged, whatever the wavelength. These results derived from the elastic theory are compared to STM observations of the reconstructed (1 1 1) surface in gold, which is usually described by a Shockley dislocations network.

Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: Applications to sandwich beams and auxetics

The present work is devoted to the modelling of strongly size-dependent bending, buckling and vibration phenomena of 2D triangular lattices with the aid of a simplified first strain gradient elasticity continuum theory. As a start, the corresponding generalized Bernoulli–Euler and Timoshenko sandwich beam models are derived. The effective elastic moduli corresponding to the classical theory of elasticity are defined by means of a computational homogenization technique. The two additional length scale parameters involved in the models, in turn, are validated by matching the lattice response in benchmark problems for static bending and free vibrations calibrating the strain energy and inertia gradient parameters, respectively. It is demonstrated as well that the higher-order material parameters do not depend on the problem type, boundary conditions or the specific beam formulation. From the application point of view, it is first shown that the bending rigidity, critical buckling load and eigenfrequencies strongly depend on the lattice microstructure and these dependencies are captured by the generalized Bernoulli–Euler beam model. The relevance of the Timoshenko beam model is then addressed in the context of thick beams and sandwich beams. Applications to auxetic strut lattices demonstrate a significant increase in the stiffness of the metamaterial combined with a clear decrease in mass. Furthermore, with the introduced generalized beam finite elements, essential savings in the computational costs in computational structural analysis can be achieved. For engineering applications of architectured materials or structures with a microstructure utilizing triangular lattices, generalized mechanical properties are finally provided in a form of a design table for a wide range of mass densities.

Nematic liquid crystal in a cylindrical sample: Theoretical analysis of the electrical response.

The electrical responses of a nematic liquid crystal sample confined between two cylindrical surfaces are investigated in the framework of elastic continuum theory. The responses are the result of the molecular reorientation induced by both the applied electric field and the cylindrical geometry of the sample. The nematic medium is considered as a parallel RC circuit since the capacitance and the resistance are under the same difference of potential. The electrical properties, including the total electric current, are determined from the molecular reorientation of the director. The elastic anisotropy has been shown to influence substantially the profile of the electrical current, capacitance, and resistance characterizing the equivalent circuit for the medium.

Dynamic behavior of nematic liquid crystal mixtures with quantum dots in electric fields.

The dynamic behavior of a mixture consisting of liquid crystalline 4-cyano-4’-pentylbiphenyl (5CB) and CdSe/ZnS quantum dots in electric fields was theoretically studied. The model was based on elastic continuum theory considering the interaction of the nematic molecules with the surrounding molecules, with the quantum dots and with the electric field. Experimental data obtained by dynamic measurements on a sample containing 0.89% (mass fraction) of CdSe/ZnS quantum dots revealed a decrease of the relaxation time compared to pure 5CB.

Mathematical Models of Protein Induced Membrane Deformation

I will describe recent advances we have made in using fast, continuum elasticity theory to describe membrane deformations around proteins. I will show that our calculations match the deformations predicted from all atom molecular dynamics simulations for two proteins: gramicidin (a small antibiotic ion channel) and nhTMEM16 (a member of the calcium activated chloride channel family). Our calculations reveal that nhTMEM16 produces large distortions in the membrane potentially related to its ability to scramble lipids from one leaflet to the other. This hypothesis is supported by atomistic simulations in which we observe lipids flipping from one leaflet to the other. Experiments to test the lipid flipping mechanism in a mammalian TMEM16 family member will also be discussed.

Dynamic behavior of a nematic liquid crystal with added carbon nanotubes in an electric field.

The dynamic behavior of a nematic liquid crystal with added carbon nanotubes (CNTs) in an electric field was analyzed. A theoretical model based on elastic continuum theory was developed and the relaxation times of nematic liquid crystals with CNTs were evaluated. Experiments made with single-walled carbon nanotubes dispersed in nematic 4-cyano-4’-pentylbiphenyl (5CB) indicated a significant difference of the relaxation time when compared to pure liquid crystal. We also noticed that the relaxation time when the field is switched off depends on how long the field was applied. It is shorter when the field is switched off immediately after application and longer when the field was applied for at least one hour.

Vibrational coupling in plasmonic molecules

Plasmon hybridization theory, inspired by molecular orbital theory, has been extremely successful in describing the near-field coupling in clusters of plasmonic nanoparticles, also known as plasmonic molecules. However, the vibrational modes of plasmonic molecules have been virtually unexplored. By designing precisely configured plasmonic molecules of varying complexity and probing them at the individual plasmonic molecule level, intramolecular coupling of acoustic modes, mediated by the underlying substrate, is observed. The strength of this coupling can be manipulated through the configuration of the plasmonic molecules. Surprisingly, classical continuum elastic theory fails to account for the experimental trends, which are well described by a simple coupled oscillator picture that assumes the vibrational coupling is mediated by coherent phonons with low energies. These findings provide a route to the systematic optical control of the gigahertz response of metallic nanostructures, opening the door to new optomechanical device strategies.