Cauchy Relation Research Articles

Surprises in the theory of elasticity: pairwise atomic interactions, central forces, and Cauchy relations

Although the theory of elasticity has been well known for many years there are still some commonly accepted misconceptions that continue to be taught in solid-state physics courses. In this paper we review some of these misconceptions through a detailed study of the relationship between macroscopic and microscopic theories of elasticity in a simple-cubic crystal with one atom per primitive cell. The analysis is performed at different stages of complexity. In the first-neighbor approximation we find that the crystal is simply unstable. If pairwise interaction up to second neighbors is considered the crystal is stable but the elastic constants present additional symmetries known as Cauchy relations which are not observed experimentally. The reasons for this behavior are analyzed, pointing out that the invariance of the potential energy with respect to rotations implies necessarily that pairwise interactions must be central. Only if the atomic interactions involve the participation of 3-bodies or more, the forces are non-central and the resulting elastic tensor may be acceptable.

УТОЧНЁННАЯ ЛИНЕЙНАЯ ТЕОРИЯ УПРУГОСТИ С УЧЁТОМ КВАДРАТИЧНЫХ СЛАГАЕМЫХ

A variant of the linear theory is proposed, based on taking into account in Taylor expansions for stresses and in power series and the expansion of the radical into a binomial series for deformations of not only linear, but also quadratic terms. In this case, static equilibrium equations in stresses are written in the form of second-order partial differential equations. The law of pairing of tangential stresses is observed. The relationship between deformations and displacements is described by relations similar to Cauchy relations, but taking into account the second derivatives of displacements along spatial coordinates. The equations of continuity of deformations for this version of the theory of elasticity are written in the form of two groups of relations. The first group of relations establishes the continuity equations for the linear terms of the deformation components; The second group of relations establishes the continuity equations for the quadratic terms of the deformation components. The physical equations for this variant of the linear theory of elasticity are written in the same way as for the classical linear theory of elasticity. Equilibrium equations in stresses are represented by second-order partial differential equations. The equilibrium equations in displacements are written in the form of fourth-order partial differential equations. The equations of equilibrium, as well as the equations of continuity of deformations, along with other parameters – the physical constants of the medium – contain small parameters dx, dy, dz, the magnitude of which, as shown by numerical modeling, has little effect on the nature of the stress-strain state. To determine them, it is necessary, apparently, to use experimental data. The construction of resolving equations in both stresses and displacements for special cases of the stress-strain state of an elastic continuous medium is considered: uniaxial dressed-up state; uniaxial deformed state; plane strain; generalized plane stress state. The problem of the uniaxial deformed state of a continuous medium (solution in stresses) is numerically solved. The proposed version of the linear theory of elasticity extends the classical linear theory of elasticity and, with appropriate experimental justification, can lead to new qualitative effects in the calculation of elastic deformable bodies.

Cauchy Relations in Linear Elasticity: Algebraic and Physics Aspects

Cauchy Relations in Linear Elasticity: Algebraic and Physics Aspects

Effects of deposition time on structural and optical properties of ZnS and ZnS/Au thin films grown by thermal evaporation

In this work, by exploiting the thermal evaporation technique, Zinc sulfide (ZnS) thin films are grown on glass substrates at 473 K. The effects of the deposition time on the thickness, structural, morphological, and optical properties of the thin films are investigated. The samples were deposited at 4 × 10−6 mbar with different deposition times (10, 20, and 30 min) whereas the thin films are characterized by X-ray diffraction (XRD), atomic force microscopy (AFM), and ultraviolet–visible spectrophotometers (UV–Vis). XRD analysis indicates that all fabricated films have a cubic structure of β-ZnS phase. Moreover, crystallite size, lattice constant, dislocation density, and micro-strain are calculated from XRD data. According to the results, crystallite size shows an increasing trend from 7.05 to 12.28 nm as deposition time increases. The dependency of surface roughness on deposition time is also observed. Optical parameters of the films such as transmittance spectra, direct optical band gap energy, refractive index, extinction coefficient, complex dielectric constant, dispersion energy parameters, static refractive index, and static high-frequency dielectric constant are carefully determined. The films exhibit high transparency and zero absorption in the visible and infrared regions. The refractive index values can be well fitted to Cauchy's relations for extrapolation to near strong absorption region. The prepared ZnS thin film with low surface roughness, high transparency, and large optical band gap may find its application as optical interfacing and CIGS solar cells. Moreover, a thin layer of Au ∼20 nm was deposited by the thermal evaporation method on the prepared ZnS films and the optical transmittance spectra show that the bilayers possess a good transmittance ratio while the disappearance of spectral interference was observed as a result of Au deposition.

Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin\u2013Mindlin anisotropic first strain gradient elasticity

In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin–Mindlin anisotropic first strain gradient elasticity theory is presented. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. First, 3+11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and 1+6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. Second, the numerical values of the obtained quantities are computed for four representative cubic materials, namely aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using an interatomic potential (MEAM). The positive definiteness of the strain energy density is examined leading to 3 necessary and sufficient conditions for the elastic constants and 7 ones for the gradient-elastic constants in Voigt notation. Moreover, 5 lattice relations as well as 8 generalized Cauchy relations for the gradient-elastic constants are derived. Furthermore, using the normalized Voigt notation, a tensor equivalent matrix representation of the two constitutive tensors is given. A generalization of the Voigt average toward the sixth-rank constitutive tensor of the gradient-elastic constants is given in order to determine isotropic gradient-elastic constants. In addition, Mindlin’s isotropic first strain gradient elasticity theory is also considered offering through comparisons a deeper understanding of the influence of the anisotropy in a crystal as well as the increased complexity of the mathematical modeling.

Optical studies of multiferroic HoCrO3 perovskite compound for optoelectronic device applications

The multiferroic HoCrO3 orthochromite compound was prepared using the sol-gel method. The optical properties of this ceramic were characterized by infrared (IR) and UV–Vis absorption spectroscopy. The IR spectrum shows a set of absorption peaks that can be assigned to the Cr–O and O–Cr–O vibrations. From UV–Vis absorption spectrometry, the reflectance spectrum and the Tauc model, reveal a direct and wide optical band gap evaluated at 3.34 eV and 3.206 eV respectively. The high value of the Urbach energy (528 ± 13) meV shows the presence of a high density of localized states. The extinction coefficient (k) was used to estimate the evolution of the refractive index n with the wavelength. Additionally, n obeys to Cauchy relation. Furthermore, the dispersion parameters were analyzed in the bases of Wemple and Di-Domenico model. The optical constants such as the skin depth and the optical conductivity were calculated and the results are discussed. From the UV–vis absorbance spectrum attributed to the d-d transitions of Cr3+ ion in HoCrO3, a theoretical crystal-field analysis is realized which leads to a good agreement between the calculated and the experimental energy levels. Furthermore, the electronic structure was compared with that obtained for Cr3+ ion in PrCrO3 ceramics. From Racah theoretical study, Tanabe Sugano Diagram and Photoluminescence measures, a transfer charge mechanism is proposed. The achieved results confirm the importance of our compound in optoelectronic device applications.

Unsteady dynamics of a sandwich plate under the influence of a cylindrical wave in an elastic medium

The interaction of a sandwich plate with a damped cylindrical wave in the ground has been investigated. A sandwich plate is considered as a model of a barrier in the ground, described by a system of equations by V. N. Paimushin, placed in the ground dividing it into two parts. The plane problem formulation is considered. The boundary conditions correspond to the hinge attachment of the barrier, and the initial conditions are zero. A cylindrical damped wave is considered as an external influence. To describe the ground movement, the equations of the elasticity theory, the Cauchy relations and the physical principle, or equivalent displacements in potentials and the Lame equations are used. The problem is solved in a related formulation, where the movement of the plate and its surrounding media is considered together. All components of the equations of motion of the plate and media are decomposed into trigonometric series and the Laplace transform is applied to them. As the conditions for the contact of the plate and the ground, the equality of normal displacements at the boundary of the medium and the plate is assumed. It is also assumed that the pressure amplitudes and normal stresses coincide. After determining the constants from the contact conditions, the displacement values and the values of normal and tangential stresses are found, after which their originals are found.

Constitutive modeling of pH-sensitive hydrogel: Multi-physics coupling of electromagnetics with mechanics and thermodynamics

The field theory of pH-sensitive hydrogel deformation is presented coupling electro-magnetic (EM) field (external or internal) and diffusion (Poisson–Nernst–Planck and Maxwell equations) of mobile ionic species through irreversible thermodynamics. The evolution of fixed-charge in the hydrogel is modeled by Langmuir monolayer adsorption theory. The important contribution of present work is a derivation of Cauchy stress tensor constitutive relation with EM field and species diffusion through Maxwell’s stress and spherical tensors. The EM field work done is computed in the frame-invariant settings employing Poynting vector concept, which is another novel contribution of the present work. The hydrogel deformation is finally computed by mechanical equilibrium equation coupled with the diffusion equations through traction boundary conditions. The applicability of the presented framework is successfully demonstrated by several pH-sensitive hydrogel simulations (one- and two-dimensional). The constitutive relations derived here are also verified with the other published results.

Численное исследование термомеханических процессов при короткоимпульсном лазерном облучении полупространства.

Лазерное облучение металлических поверхностей интенсивными тепловыми источниками используется для генерации коротких зондирующих импульсов, которые распространяются внутрь тонких образцов и позволяют оценивать структуру и механические свойства последних в рамках классического акустического подхода. При кратковременном облучении поверхности конструкции источником энергии высокой плотности возникают большие тепловые напряжения и остаточные деформации. В данной статье численно исследуется осесимметричная задача о термомеханической нагрузке полупространства. При этом учитывается влияние объемных и неупругих характеристик отдельных фаз на остаточное напряженно деформированное состояние полупространства. Постановка задачи включает соотношение Коши, уравнение движения, уравнение теплопроводности, исходные условия, тепловые и механические граничные условия. Термомеханическое поведение изотропного материала описывается унифицированной моделью течения Боднера–Партома. Задача решается с помощью конечно-элементной методики. Численная реализация задачи проводится с помощью пошагового интегрирования по времени. Уравнения движения интегрируются методом Ньюмарка, а уравнение теплопроводности — неявным методом первого порядка. С помощью методики численного решения осесимметричной динамической задачи для полупространства при термомеханической нагрузке и модели течения описано остаточное напряженно-деформированное состояние. Установлено, что микроструктурные превращения, учитываемые через термотрансформационную деформацию, и зависимость неупругих характеристик материала от фазового состава существенно уменьшают остаточные неупругие деформации и способствуют появлению сжимающих напряжений. Получена трехзонная область формирования поля остаточных напряжений.

Pressure dependence of Poisson’s ratio of glassy Baltic amber studied by Brillouin scattering spectroscopy

Amber is a typical natural glass with very long aging time. Elastic properties of amber at high pressures have been studied using Brillouin scattering and a diamond anvil cell. Both longitudinal and transverse acoustic modes have been observed up to 12 GPa. The pressure dependences of longitudinal, shear, Young’s, and bulk moduli, compressibility, and Poisson’s ratio were determined. The longitudinal, shear, Young’s, bulk moduli show the remarkable increase, and compressibility shows a marked decrease with increasing pressure. However, it is found that the pressure dependence of Poisson’s ratio is very small. The mechanism of this small pressure dependence was discussed. The Cauchy type relation between longitudinal and shear moduli was examined. Its coefficients show the deviation from the Cauchy relation owing to the coexistence of different intermolecular interactions in amber.

MODELING THE BEHAVIOR OF THE PHYSICAL AND GEOMETRIC NON-LINEAR FUNCTIONAL HETEROGENEOUS MATERIALS

The work is devoted to the problem of modeling the behavior of functionally inhomogeneous materials with the properties of pseudo-elastic-plasticity under complex loads, in particular at large strains (up to 20%), when geometric nonlinearity in Cauchy relations must be taken into account. In previous works of the authors, functionally heterogeneous materials were studied in a geometrically linear formulation, which is true for small deformations (up to 7%). When predicting work with material at large deformations, it is necessary to take into account geometric nonlinearity in Cauchy relations.Studying the behavior of bodies made of functionally heterogeneous materials under unsteady load requires the development of special approaches, methods and algorithms for calculating the stress-strain state. When constructing physical relations, it is assumed that the deformation at the point is represented as the sum of the elastic component, the jump in deformation during the phase transition, plastic deformation and deformation caused by temperature changes.A physical relationship in a nonlinear setting is proposed for modeling the behavior of bodies made of functionally heterogeneous materials. Formulas are obtained that nonlinearly relate strain rates and Formulas are obtained that nonlinearly relate strain rates and displacement rates.Keywords: mathematical modeling, functional heterogeneous materials, geometric nonlinearity, spline functions, pseudo-elastic plasticity, phase transitions

Null Lagrangians in linear theories of micropolar type and few other generalizations of elasticity

In the context of linear theories of generalized elasticity including those for homogeneous micropolar media, quasicrystals, piezoelectric and piezomagnetic media, we explore the concept of null Lagrangians. For obtaining the family of null Lagrangians we employ the sufficient conditions of H. Rund. In some cases a non-zero null Lagrangian is found and the stored energy admits a split into a null Lagrangian and a remainder. However, the null Lagrangian vanishes whenever the relevant elasticity tensor obeys certain symmetry conditions which can be construed as an analogue of the Cauchy relations.

Mathematical modeling of dynamic related processes based on the finite element techniques

In the process of investigation the functioning of information systems with the help of mathematical modeling, problems can arise that are solved using finite element techniques. For example, In the present work we solved the axially symmetric dynamic problem of related phenomena under microscale thermal loading. The statement of the problem includes: Cauchy relations, equations of motion, heat conduction equation, initial conditions, thermal and mechanical boundary conditions. The nonlinear behavior of a material is described by the unified model of flow. The problem is solved numerically by the time step integration method, iterative method and finite element method. Equations of the evolution for the inelastic flow model are integrated by the second-order Euler implicit method with the use of the rule of «middle point». The system of nonlinear transcendental equations obtained in each time step is solved by the method of simple iteration. To accelerate convergence, we apply the Stephensen – Aitken procedure. The equations of motion are integrated by the Newmark method, whereas the heat-conduction equation is integrated by the first-order implicit method. The problem of heat-conduction is linearized by finding the temperature-dependent thermal characteristics according to the temperature distribution in the previous time step or previous iteration. The main results obtained in the work are the following: quantitative estimations of temperature effects of thermostructural-mechanical coupling, which are caused by volumetric thermoelastic effect, dissipation of mechanical energy and hidden heat are obtained.

Порівняльний аналіз лінійного та нелінійного правил сумішей при моделюванні напруженого стану півпростору

In the present work we solve the axially symmetric problem of a half-space under thermal loading. The statement of the problem includes: Cauchy relations, equations of motion, heat conduction equation, initial conditions, thermal and mechanical boundary conditions. The thermomechanical behavior of an isotropic material is described by the Bodner–Partom unified model of flow generalized in the case of microstructure influence on inelastic characteristics of steel. To determine the parameters of the model corresponded to yield stress and yield strength the mixture rule is utilized. The problem is solved with using the finite element technique. The numerical realization of our problem is performed with the help of step-by-step time integration. Equations of the evolution for the inelastic flow model are integrated by the second-order Euler implicit method. The equations of motion are integrated by the Newmark method, whereas the heat-conduction equation is integrated by the first-order implicit method. We use quadrangular isoparametric elements. The parameters of a fine grid are chosen with the help of the criterion of practical convergence of the solutions. The stress state taking into account linear and nonlinear rules of mixtures is described.

On equivalence relations generated by Cauchy sequences in countable metric spaces

Let X be the set of all metrics on ω, and let Xcpt be the set of all metrics r on ω such that the completion of (ω,r) is compact. We define the Cauchy sequence equivalence relation Ecs on X as: rEcss iff the set of Cauchy sequences in (ω,r) is same as in (ω,s). We also denote Ecsc=Ecs↾Xcpt.We show that Ecs is a Π11-complete equivalence relation, while Ecsc is a Π30 equivalence relation. We also show that Ecsc is Borel bireducible to an orbit equivalence relation. Furthermore, we investigate the Borel reducibility between Ecsc and some benchmark equivalence relations. For instance, we show that =+ and Rω/c0 are Borel reducible to Ecsc, and E1 is not. Restrictions of Ecsc on some special invariant subsets of Xcpt are also considered.

Influence of WO3 on elastic and structural properties of barium-borate-bagasse-cassava rhizome glass system

The production of barium-borate-bagasse-cassava rhizome-WO3 glass with varying weight fractions of WO3 from x = 0 to 8 wt% was carried out successfully using a popular melt-quenching method. The measured density and molar volume of all samples increased with respect to the heavier and larger size of WO3. The amorphous nature of the studied glass system was demonstrated by X-ray diffraction technique. Ultrasonic contact technique can be used to measure the ultrasonic velocity as well as to evaluate elastic moduli. Debye temperature, softening temperature, Cauchy relation, fugacity and fractal bond connectivity were also determined. Microhardness values were investigated using a microhardness testing machine. The results showed that these parameters were dependent upon the composition and dopant concentration in the glass system. The information from ultrasonic velocities was supported by FTIR spectra showing that the addition of WO3 affected the bridging oxygen in the network structure of the studied glass.

Density Functional Study on Enhancement of Modulus of Confined Fluid in Nanopores

Elastic modulus in fluid is a crucial thermodynamic property due to its close relation to the viscoelastic behaviors in dynamics and the pore size analysis in porous materials. Especially, elasticity in nanoconfined fluid, which exhibits significant deviation from that of bulk fluid, has gained a multitude of applications in industrial and engineering fields. In this work, general expressions of bulk and shear moduli for inhomogeneous fluids have been derived based on Hooke’s law. In the bulk phase, the obtained moduli reduce exactly to those obtained from thermodynamic fluctuation theory. In addition, both moduli have been expressed as functionals of fluid density functions. Besides, the Cauchy relation and Voigt notation for inhomogeneous elasticity have been first obtained. The validity of the obtained moduli has been verified through their comparison with grand canonical Monte Carlo simulation. Our further calculations suggest that the bulk modulus of confined fluid is inhomogeneous across the pore an...

Hybrid 3D mass-spring system for simulation of isotropic materials with any Poisson’s ratio

Mass-spring systems (MSS) simulating elastic materials obey constraints known in elasticity as the Cauchy relations, restricting the Poisson ratio of isotropic systems to be exactly $$\nu =1/4$$. We remind that this limitation is intrinsic to centrosymmetric spring systems (where each node is a center of symmetry), forbidding them for instance to simulate incompressible materials (with $$\nu =1/2$$). To overcome this restriction, we propose to supplement the spring deformation energy with an energy depending on the volume only, insensitive to change of shape, permitting MSS to simulate any real isotropic materials. In addition, the freedom in choosing the spring constants realizing a given elastic behavior allows to manage instabilities. The proposed hybrid model is evaluated by comparing its response to various deformation geometries with analytical model and/or finite element model. The results show that the hybrid MSS model allows to simulate any compressible isotropic elastic material and in particular the nearly incompressible (Poisson ratio $$\nu \simeq 1/2$$) biological soft tissues to which it is dedicated.

Single-Shot and Label-Free Refractive Index Dispersion of Single Nerve Fiber by Triple-Wavelength Diffraction Phase Microscopy

Nerve fibers play a critical role in the propagation of electrical impulses in the nervous system, and several studies have demonstrated their electrical and chemical properties. However, because of the lack of practical biophotonics techniques, most of the optical properties of these fibers are unknown. Herein, we propose and demonstrate a novel method of precisely measuring the dispersion of an individual nerve fiber in striatal neurons, using a single-shot label-free triple-wavelength diffraction phase interferometer and employing Cauchy coefficients of the refractive index. Compared with the current techniques of refractive index measurement, the proposed technique employs a common-path arrangement to achieve high temporal stability sensing of dispersion by a monochromatic detector. The dispersion spectra are obtained by reconstructing the triple-color wavefront of the specimen, extracting the thickness from the projected width of the fiber in the image, and then employing a Cauchy relation that is valid for the normal dispersions of transparent materials in the visible range. The feasibility of this technique was confirmed by analyzing the refractive indices of polystyrene microspheres. The technique was then applied to measure refractive index map and dispersion behavior of nerve fibers in medium spiny neurons with high temporal and spatial resolution.

The Viscosities of Partially Molten Materials Undergoing Diffusion Creep

AbstractPartially molten materials resist shearing and compaction. This resistance is described by a fourth‐rank effective viscosity tensor. When the tensor is isotropic, two scalars determine the resistance: an effective shear and an effective bulk viscosity. Here calculations are presented of the effective viscosity tensor during diffusion creep for a 2‐D tiling of hexagonal unit cells and a 3‐D tessellation of tetrakaidecahedrons (truncated octahedrons). The geometry of the melt is determined by assuming textural equilibrium. The viscosity tensor for the 2‐D tiling is isotropic but that for the 3‐D tessellation is anisotropic. Two parameters control the effect of melt on the viscosity tensor: the porosity and the dihedral angle. Calculations for both Nabarro‐Herring (volume diffusion) and Coble (surface diffusion) creep are presented. For Nabarro‐Herring creep the bulk viscosity becomes singular as the porosity vanishes. This singularity is logarithmic, a weaker singularity than typically assumed in geodynamic models. The presence of a small amount of melt (0.1% porosity) causes the effective shear viscosity to approximately halve. For Coble creep, previous modeling work has argued that a very small amount of melt may lead to a substantial, factor of 5, drop in the shear viscosity. Here a much smaller, factor of 1.4, drop is obtained for tetrakaidecahedrons. Owing to a Cauchy relation symmetry, the Coble creep bulk viscosity is a constant multiple of the shear viscosity when melt is present.