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.     

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.     

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.     

Zero-frequency elastic moduli of uniform fluids

A statistical mechanical treatment of equilibrium elasticity of a uniform fluid phase based on density functional theory is presented. Bulk expressions for the stress tensor and the zero-frequency elastic moduli tensor involving the direct correlation function are found.     

GENERALIZATION OF ESHELBY''S METHOD TO THE ANISOTROPIC ELASTICITY THEORY OF DISLOCATIONS IN QUASICRYSTALS

The general expressions of the elastic fields induced by straight dislocations in quasicrystals have been given according to Eshelby's method which was used to treat the anisotropic elasticity of dislocations in crystals. As an example, the elastic displacement vector, the stress tensor and the elastic energy density of a screw dislocation line lying on the quasiperiodic plane of decagonal quasicrystals are calculated.     

GENERALIZATION OF ESHELBY'S METHOD TO THE ANISOTROPIC ELASTICITY THEORY OF DISLOCATIONS IN QUASICRYSTALS

The general expressions of the elastic fields induced by straight dislocations in quasicrystals have been given according to Eshelby's method which was used to treat the anisotropic elasticity of dislocations in crystals. As an example, the elastic displacement vector, the stress tensor and the elastic energy density of a screw dislocation line lying on the quasiperiodic plane of decagonal quasicrystals are calculated.     

Viscoelastic theory for nematic interfaces

A complete macroscopic theory for compressible nematic-viscous fluid interfaces is developed and used to characterize the interfacial elastic, viscous, and viscoelastic material properties. The derived expression for the interfacial stress tensor includes elastic and viscous components. Surface gradients of the interfacial elastic stress tensor generates tangential Marangoni forces as well as normal forces. The latter may be present even in planar surfaces, implying that in principle static planar interfaces may accommodate pressure jumps. The asymmetric interfacial viscous stress tensor takes into account the surface nematic ordering and is given in terms of the interfacial rate of deformation and interfacial Jaumann derivative. The material function that describes the anisotropic viscoelasticity is the dynamic interfacial tension, which includes the interfacial tension and dilational viscosities. Viscous dissipation due to interfacial compressibility is described by the anisotropic dilational viscosity, and it is shown to describe the Boussinesq surface fluid appropriate for Newtonian interfaces when the director is homeotropic. Three characteristic interfacial shear viscosities are defined according to whether the surface orientation is along the velocity direction, the velocity gradient, or the unit normal. In the last case the expression reduces to the interfacial shear viscosity of the Boussinesq surface fluid. The theory provides a theoretical framework to study interfacial stability, thin liquid film stability and hydrodynamics, and any other interfacial rheology phenomena.     

Some elastic properties of solid nematics

The general approach to study the properties of the mechanical deformations of solid nematics, which are the macroscopic homogeneous elastic media having the rotational symmetry of the nematic liquid crystals is proposed. The stress tensor, the Young modulus and the Poisson ratios for the parallel and perpendicular homogeneous orientations of nematic molecules relative to the axis of external forces influence are obtained by the varying of the free energy of mechanical deformation. It is shown that these constants have the anisotropic character and the experiments for the direct measurement of five elasticity coefficients entering the free energy expression are proposed.     

Membrane stress tensor in the presence of lipid density and composition inhomogeneities

We derive the expression of the stress tensor for one- and two-component lipid membranes with density and composition inhomogeneities. We first express the membrane stress tensor as a function of the free-energy density by means of the principle of virtual work. We then apply this general result to a monolayer model which is shown to be a local version of the area-difference elasticity (ADE) model. The resulting stress tensor expression generalizes the one associated with the Helfrich model, and can be specialized to obtain the one associated with the ADE model. Our stress tensor directly gives the force exchanged through a boundary in a monolayer with density and composition inhomogeneities. Besides, it yields the force density, which is also directly obtained in covariant formalism. We apply our results to study the forces induced in a membrane by a local perturbation.     

Approximation methods in viscoelasticity theory

According to Volterra’s principle, the solution of the quasistatic problem in linear elasticity theory for an isotropic medium can be obtained from a solution of the corresponding problem of elasticity theory by substituting an integral operator with respect to time for the Poisson ratio and the subsequent interpretation of the constitutive operator function. The method of approximation uses an expansion of constitutive functions into a sum of rational operators. An evaluation is made for the accuracy of the solution, which is related to the quality of approximation. For an anisotropic medium, the constitutive function depends on several integral operators. In this case, a special method of approximation of the constitutive function is suggested which uses the introduction of “canonical” operators. A priori and a posteriori evaluations of the solution are given. Generalizations to nonhomogeneous (composite) media and nonlinear cases are indicated.     

EXPRESSION OF THE ELASTIC ENERGY IN TWO-DIMENSIONAL QUASICRYSTALS

The application of group theory to elasticity in two-dimensional (2D)quasicrystals is presented. The expression of elastic energy as a function of gradients of the phonon and phason fields has been derived to quadratic order. The phonon response to an external stress is isotropic, but the response of phason field is anisotropic for eightfold and twelvefold symmetries.     

Green’s tensor for crystals of the hexagonal system

Expressions for components of the Green’s tensor of the basic equation of the elasticity theory for hexagonal system crystals have been obtained using the Lifshitz-Rozentsveig method. A problem is in principle reduced to finding the roots of a sixth-order algebraic equation. They are either complex or purely imaginary for all known hcp metals. In both cases, the desired components of the Green’s tensor are calculated exactly in contrast to metals of the cubic system. A limiting transition to the isotropic approximation is shown.     

A thermal-mechanical constitutive model for b-HMX single crystal and cohesive interface under dynamic high pressure loading

Due to the significant thermal-mechanical effects during hot spot formation in PBX explosives,a thermodynamic constitutive model has been constructed for HMX anisotropic single crystal subjected to dynamic impact loading. The crystal plasticity model based on dislocation dynamics theory was employed to describe the anisotropic plastic behavior along the preferential slip systems. A modified equation of state (EOS) was introduced into the constitutive equations through the decomposing stress tensor and the n...     

On microcontinuum field theories: the Eshelby stress tensor and incompatibility conditions

We investigate linear theories of incompatible micromorphic elasticity, incompatible microstretch elasticity, incompatible micropolar elasticity and the incompatible dilatation theory of elasticity (elasticity with voids). The incompatibility conditions and Bianchi identities are derived and discussed. The Eshelby stress tensor (static energy momentum) is calculated for such inhomogeneous media with microstructure. Its divergence gives the driving forces for dislocations, disclinations, point defects and inhomogeneities which are called configurational forces.     

Young’s modulus and Poisson’s ratio for seven-constant tetragonal crystals and nano/microtubes

In the paper, the elasticity theory was applied to consider the mechanical properties of rectilinearly anisotropic seven-constant tetragonal crystals and their cylindrically anisotropic nano/microtubes with and with no chiral angle, being the angle between the crystallographic symmetry axis and elongated tube axis. Pt is found that the number of crystals with negative Poisson’s ratio is the least for rectilinear anisotropy and is much larger for curvilinear anisotropy. With a nonzero chiral angle, all nano/microtubes can have negative Poisson’s ratio. The elastic problem on axial tension of cylindrical nano/microtubes is solved for radially inhomogeneous stresses: three normal stresses and one shear stress.     

Sound transmission prediction by 3-D elasticity theory

The problem of sound transmission through layered panel structures is studied with the exact theory of three-dimensional (3-D) elasticity. The exact solution to the 3-D elasticity equations is obtained by the use of the Fourier spectral method. Based on this analytical solution, a transfer matrix is derived that relates the spectral displacements and stresses on the one surface of the panel to those on the opposite panel surface. The transfer matrix is then used to develop the analytical solutions for sound reflection and transmission coefficients. Explicit, concise expressions are obtained for the analytical solutions of the acoustic transmission and reflection coefficients under the general conditions of layered anisotropic panels. Examples are given for both single-layer and sandwich panels. Predictions on sound transmission from the 3-D elasticity theory are compared with available data from other methods, and the results are discussed.     

A generalization of the relativistic equilibrium equations for a non-rotating star

The problem of elastomechanical equilibrium for a static, spherically symmetric star composed of an elastic material is analyzed. A suitable formulation of relativistic elasticity theory is used, and the second order equilibrium equations are found. It is shown that the equilibrium conditions with anisotropic pressure introducedad hoc by some authors are in fact the dynamical conditions for a relativistic elastic material. The corresponding first order equations for the components of the metric and of the energy-momentum tensor reduce to the Tolman-Oppenheimer-Volkhoff equations if the material exhibits no shape-rigidity. Two interesting classes of solutions are discussed.     

Probability distributions for the stress tensor in conformal field theories

The vacuum state—or any other state of finite energy—is not an eigenstate of any smeared (averaged) local quantum field. The outcomes (spectral values) of repeated measurements of that averaged local quantum field are therefore distributed according to a non-trivial probability distribution. In this paper, we study probability distributions for the smeared stress tensor in two-dimensional conformal quantum field theory. We first provide a new general method for this task based on the famous conformal welding problem in complex analysis. Secondly, we extend the known moment generating function method of Fewster, Ford and Roman. Our analysis provides new explicit probability distributions for the smeared stress tensor in the vacuum for various infinite classes of smearing functions. All of these turn out to be given in the end by a shifted Gamma distribution, pointing, perhaps, at a distinguished role of this distribution in the problem at hand.