Fundamentals in generalized elasticity and dislocation theory of quasicrystals: Green tensor,dislocation key-formulas and dislocation loops

The present work provides fundamental quantities in generalized elasticity and dislocation theory of quasicrystals. In a clear and straightforward manner, the three-dimensional Green tensor of generalized elasticity theory and the extended displacement vector for an arbitrary extended force are derived. Next, in the framework of dislocation theory of quasicrystals, the solutions of the field equations for the extended displacement vector and the extended elastic distortion tensor are given; that is, the generalized Burgers equation for arbitrary sources and the generalized Mura–Willis formula, respectively. Moreover, important quantities of the theory of dislocations as the Eshelby stress tensor, Peach–Koehler force, stress function tensor and the interaction energy are derived for general dislocations. The application to dislocation loops gives rise to the generalized Burgers equation, where the displacement vector can be written as a sum of a line integral plus a purely geometric part. Finally, using the Green tensor, all other dislocation key-formulas for loops, known from the theory of anisotropic elasticity, like the Peach–Koehler stress formula, Mura–Willis equation, Volterra equation, stress function tensor and the interaction energy are derived for quasicrystals.     

Temperature effect on elastic modulus of thin films and nanocrystals

The stability of nanoscale devices is directly related to elasticity and the effect of temperature on the elasticity of thin films and nanocrystals. The elastic instability induced by rising temperature will cause the failure of integrated circuits and other microelectronic devices in service. The temperature effect on the elastic modulus of thin films and nanocrystals is unclear although the temperature dependence of the modulus of bulk materials has been studied for over half a century. In this paper, a theoretical model of the temperature-dependent elastic modulus of thin films and nanocrystals is developed based on the physical definition of the modulus by considering the size effect of the related cohesive energy and the thermal expansion coefficient. Moreover, the temperature effect on the modulus of Cu thin films is simulated by the molecular dynamics method. The results indicate that the elastic modulus decreases with increasing temperature and the rate of the modulus decrease increases with reducing thickness of thin films. The theoretical predictions based on the model are consistent with the results of computational simulations, semi-continuum calculations and the experimental measurements for Cu, Si thin films and Pd nanocrystals.     

Finite element analysis of the thermal residual stress distribution in the interphase of unidirectional fiber-reinforced resin matrix composites

By means of three-dimensional finite element method (FEM) which is based upon the micro-mechanical model of fiber-reinforced composites, this paper selects representative volume elements and studies the effect of the five factors, namely, cooling rate, matrix elasticity modulus, fiber elasticity modulus, interphase elasticity modulus and fiber volume fraction, on the interphase thermal residual stress and its distribution law in epoxy resin NPEF-170/unidirectional glass fiber composites. The results indicate that thermal residual stress is mainly distributed on the fiber and the matrix of neighboring interphase; the thermal residual stress on the fiber and the matrix declines as the distance to the interphase layer grows; and it tends to zero at the distance of 1.5 times the radius of the fiber away from the interphase. The increase in any of the four factors, namely, cooling rate, matrix elasticity modulus, fiber elasticity modulus, and fiber volume fraction would trigger the rise of thermal residual stress in epoxy resin NPEF-170/unidirectional glass fiber composites. The additional flexible interphase layer can eliminate and transfer thermal residual stress effectively, whose effectiveness mainly depends on the difference between interphase elasticity modulus and fiber elasticity modulus.     

From elasticity to hypoplasticity: dynamics of granular solids

"Granular elasticity," useful for calculating static stress distributions in granular media, is generalized by including the effects of slowly moving, deformed grains. The result is a hydrodynamic theory for granular solids that agrees well with models from soil mechanics.     

Resonance method of measuring shear viscoelastic properties of liquid media based on excitation of torsional oscillations in tubes

Possibilities of using torsional oscillations for measuring viscoelastic properties of liquids are discussed. The theory of torsional oscillations of an elastic tube filled with the media to be investigated possessing viscosity and shear elasticity is developed. It is shown that to determine a complex shear modulus it is sufficient to determine the resonance frequency and Q-factor of torsional oscillations. An experimental installation and the results of measurements of viscoelastic modulus of glycerin and oil of one oilfield within the temperature range from −10° to 60°C are given. The experimental installation allows measuring a viscoelastic modulus within the range of acoustic logging frequencies (10–20 kHz). The obtained results are compared with the results of rheometric measurements.     

Stress fluctuations and elasticity in one- and two-component cubic crystals

The elastic constants of the Lennard-Jones system and a binary repulsive particle system (in 1:13 composition) obeying the Weeks-Chandler-Andersen (WCA) potential (WCA2-AB13 system) have been computed using the stress fluctuation formalism by equilibrium molecular dynamics simulations. The systems are under finite fixed pressures. The evolution of the elastic moduli as a function of the temperature is characterised. In both systems, the melting transition is signalled by a jump of the shear modulus, in consistence with the behaviour of the specific volume. At zero temperature, while the elasticity is totally affine for the Lennard-Jones face-centered cubic (fcc) crystal, it clearly contains a non-affine part for the AB13 cubic superlattice. The evolution of the non-affine elasticity as a function of the temperature is investigated, and its behaviour is compared to that of a typical glass-former.     

A new orthogonality relationship for orthotropic thin plate theory and its variational principle

While Hamiltonian system was led to solution of elastic theory a symplectic system-atic methodology for theory of elasticity was established and a symplectic orthogonality relationship was presented[1,2]. For two-dimensional theory of elasticity a new dual vec-tor and a new dual differential matrix were presented by putting the old dual vector[1] in a new order. It was discovered for isotropic materials that the symplectic orthogonality relationship may be decomposed into two independent and s…     

Imaging the elastic properties of coiled carbon nanotubes with atomic force microscopy

Coiled carbon nanotubes were produced catalytically by thermal decomposition of hydrocarbon gas. After deposition on a silicon substrate, the three-dimensional structure of the helix-shaped multiwalled nanotubes can be visualized with atomic force microscopy. Helical structures of both chiralities are present in the nanotube deposits. For larger coil diameters ( >170 nm), force modulation microscopy allows one to probe the local elasticity along the length of the coil. Our results agree with the classical theory of elasticity. Similar to the case of straight nanotubes, the Young modulus of coiled multiwalled nanotubes remains comparable to the very high Young modulus of hexagonal graphene sheets.     

Nichtlineare Phonon-Phonon-Wechselwirkung in elastischen Nichtleitern

For a simplified model we investigate the influence of anharmonicity of the lattice potential on the propagation of sound. In the low temperature region it is possible to describe the lattice dynamics by means of the theory of nonlinear elastic media. It is shown that three-phonon processes are possible even if the colliding phonons have the same polarization. At low temperature the lifetime of a single lattice oscillation is decisively influenced by the processL+L?L, whereL stands for longitudinal polarization. By means of temperature dependent doubletime Green functions we investigate energy shift and damping rate of impressed phonons. The temperature dependent frequency shift is interpreted as a temperature dependence of the elasticity modulus. Comparison yields satisfactory agreement between theory and experiment.     

Elliptic hole in octagonal quasicrystals

The stress potential function theory for plane elasticity of octagonal quasicrystals is developed. By introducing stress functions, a large number of basic equations involving elasticity of octagonal quasicrystals are reduced to a single partial differential equation. Furthermore, we develop the complex variable function method (Lekhnitskii method) for anisotropic elasticity theory to that for quasicrystals. With the help of conformal transformation, an exact solution for the elliptic hole of quasicrystals is presented. The solution of the Griffith crack problem, as a special case of the results, is obtained. As a consequence, the phonon stress intensity factor is derived analytically.     

On the Generalized Thermoelasticity Problem for an Infinite Fibre-Reinforced Thick Plate under Initial Stress

In this paper, the generalized thermoelasticity problem for an infinite fiber-reinforced transversely-isotropic thick plate subjected to initial stress is solved. The lower surface of the plate rests on a rigid foundation and temperature while the upper surface is thermally insulated with prescribed surface loading. The normal mode analysis is used to obtain the analytical expressions for the displacements, stresses and temperature distributions. The problem has been solved analytically using the generalized thermoelasticity theory of dual-phase-lags. Effect of phase-lags, reinforcement and initial stress on the field quantities is shown graphically. The results due to the coupled thermoelasticity theory, Lord and Shulman's theory, and Green and Naghdi's theory have been derived as limiting cases. The graphs illustrated that the initial stress, the reinforcement and phase-lags have great effects on the distributions of the field quantities.     

Features of acoustic waves in media with large porosity values in the framework of the Biot theory

The Biot theory is used to study the properties of acoustic waves in fluid-saturated porous media with large porosity values. Materials are considered for which a large elasticity modulus of the matrix (skeleton) is comparable in size to the large fluid pore saturation modulus.     

Acoustic study of the elastic and inelastic properties of high-pressure polyethylene samples with different irradiation histories

The influence of vibrational deformation amplitude ε on the dynamic elasticity modulus (Young’s modulus E) and internal friction (logarithmic decrement δ) of high-pressure polyethylene samples with different histories is studied. Acoustic measurements are made by a resonance method using the longitudinal vibrations of a composite piezoelectric vibrator at a frequency of ≈ 100 kHz. The dependences E(ε) and δ(ε) are taken at room temperature. From the acoustic data, the elasticity and microplasticity of the samples are estimated. It is found that the microplasticity remains almost unaffected upon irradiation and aging, while the elasticity modulus and breaking elongation per unit length considerably depend on the history and clearly correlated with each other. The observed effects are explained by the fact that atom-atom interaction and defects inside polymer macromolecules substantially influence the elastic modulus and breaking strength, while the inelastic microplastic strain is most likely associated with molecule-molecule interaction, which is affected by irradiation insignificantly.     

Elastic strain response in the modified phase-field-crystal model

To understand and develop new nanostructure materials with specific mechanical properties, a good knowledge of the elastic strain response is mandatory. Here we investigate the linear elasticity response in the modified phase-field-crystal(MPFC) model. The results show that two different propagation modes control the elastic interaction length and time, which determine whether the density waves can propagate or not. By quantitatively calculating the strain field, we find that the strain distribution is indeed extremely uniform in case of elasticity. Further, we present a detailed theoretical analysis for the orientation dependence and temperature dependence of shear modulus. The simulation results show that the shear modulus reveals strong anisotropy and the one-mode analysis provides a good guideline for determining elastic shear constants until the system temperature falls below a certain value.     

Elliptic holes in octagonal quasicrystals

The stress potential function theory for the plane elasticity of octagonal quasicrystals is developed. By introducing stress functions, a large number of basic equations involving the elasticity of octagonal quasicrystals are reduced to a single partial differential equation. Furthermore, we develop the complex variable function method (Lekhnitskii method) for anisotropic elasticity theory to that for quasicrystals. With the help of conformal transformation, an exact solution for the elliptic hole of quasicrystals is presented. The solution of the Griffith crack problem, as a special case of the results, is obtained. As a consequence, the phonon stress intensity factor is derived analytically.     

First-principles calculations on the elastic and thermodynamic properties of NbN

<正>The elastic and thermodynamic properties of NbN at high pressures and high temperatures are investigated by the plane-wave pseudopotential density functional theory（DFT）.The generalized gradient approximation（GGA） with the Perdew-Burke-Ernzerhof（PBE） method is used to describe the exchange-correlation energy in the present work.The calculated equilibrium lattice constant a₀,bulk modulus B₀,and the pressure derivative of bulk modulus B₀’ of NbN with rocksalt structure are in good agreement with numerous experimental and theoretical data.The elastic properties over a range of pressures from 0 to 80.4 GPa are obtained.Isotropic wave velocities and anisotropic elasticity of NbN are studied in detail.It is indicated that NbN is highly anisotropic in both longitudinal and shear-wave velocities. According to the quasi-harmonic Debye model,in which the phononic effect is considered,the relations of（V-V₀）/V₀ to the temperature and the pressure,and the relations of the heat capacity C_V and the thermal expansion coefficientαto temperature are discussed in a pressure range from 0 to 80.4 GPa and a temperature range from 0 to 2500 K.At low temperature,C_V is proportional to T³ and tends to the Dulong-Petit limit at higher temperature.We predict that the thermal expansion coefficientαof NbN is about 4.20×10^-6/K at 300 K and 0 GPa.     

Mechanical properties,anisotropy and hardness of group IVA ternary spinel nitrides

In this work, new ternary cubic spinel structures are designed by the substitutional method. The structures, elasticity properties, intrinsic hardness and Debye temperature of the cubic ternary spinel nitrides are studied by first principles based on the density-functional theory. The results show that γ-CSn₂N₄, γ-SiC₂N₄, γ-GeC₂N₄ and γ-SnC₂N₄ are not mechanically stable. The elastic constants C_ij of these cubic spinel structures are obtained using the stress–strain method. Derived elastic constants, such as bulk modulus, shear modulus, Young's modulus, Poisson coefficient and brittle/ductile behaviour are estimated using Voigt–Reuss–Hill theories. The B/G value, the Poisson's ratio and anisotropic factor are calculated for eight ternary stable crystals. Based on the microscopic hardness model, we further estimate the Vickers hardness of all the stable crystals. From the calculated hardness of the stable group IVA ternary spinel nitrides by Gao's and Jiang's methods, it is observed that the stable group IVA ternary spinel nitrides are not superhard materials except for γ-CSi₂N₄. Furthermore, the Debye temperature for the eight stable crystals is also estimated.     

On the Elastic Modulus of Powders

For the design of silos and similar equipment for storage, handling and processing of particulate materials, a thorough understanding of the mechanical behaviour of powders is of great importance. Whereas strength properties of powders have been investigated by many workers, the elastic behaviour at small deformations has been much less in focus. In that respect, a simple preliminary theory of uniaxial elasticity has been derived for particles, based on a simple system of spherical and monosized particles of a homogenous and elastically isotropic material. The equation σ = E_p?_el² gives the relationship between stress, elastic modulus and elastic deformation for the unloading of a given powder at a specific consolidation stress/ compression level. Comparisons with experimental results of real powders in a uniaxial tester show surprisingly good agreement in many cases. The equation seems to describe the elasticity of powders fairly well, although it is only a preliminary derivation based on simple considerations.     

Photothermal waves for two temperature with a semiconducting medium under using a dual-phase-lag model and hydrostatic initial stress

The dual-phase-lag (DPL) model with two different time translations and Lord–Shulman (LS) theory with one relaxation time are applied to study the effect of hydrostatic initial stress on medium under the influence of two temperature parameter(a new model will be introduced using two temperature theory) and photothermal theory. We solved the thermal loading at the free surface in the semi-infinite semiconducting medium-coupled plasma waves with the effect of mechanical force during a photothermal process. The exact expressions of the considered variables are obtained using normal mode analysis also the two temperature coefficient ratios were obtained analytically. Numerical results for the field quantities are given in the physical domain and illustrated graphically under the effects of several parameters. Comparisons are made between the results of the two different models with and without two temperature parameter, and for two different values of the hydrostatic initial stress. A comparison is carried out between the considered variables as calculated from the generalized thermoelasticity based on the DPL model and the LS theory in the absence and presence of the thermoelastic and thermoelectric coupling parameters.     

Effects of Uniaxial Stress on X-ray Diffraction Spectra

Uniaxial stress is a typical drawback of solid pressure transmitting medium, at low temperature, which affects experimental spectra by shifting and broadening the diffraction peaks. Corrections to this effects have been proposed in the past only for the systematic shifts. We presented a simple model based on elasticity theory which rationalizes both peak shifts and broadening observed in X-ray diffraction. Our results are compared to the popular Singh model.