The completeness and a new derivation of the Stroh formalism of anisotropic linear elasticity

In this paper we present a new, simpler and unified derivation of the Stroh formalism of anisotropic linear elasticity, for both nondegenerate and degenerate cases. It is based on the potential representation and Jordan canonical representation theorems. The completeness of the Stroh formalism is proved in the derivation process itself. This new approach is also extended to piezoelastic problems. Besides, we show that the eigenvalues of the fundamental elastic matrix in planar anisotropic elasticity are always distinct, except for the case of isotropy. The project supported by the National Natural Science Foundation of China (19525207 and 19891180).     

Some explicit expressions of extended Stroh formalism for two-dimensional piezoelectric anisotropic elasticity

Since the extended Stroh formalism for two-dimensional piezoelectric anisotropic elasticity preserves essential features of Stroh formalism for pure elastic materials, it becomes important to get the corresponding explicit expressions of some important matrices frequently appeared in Stroh formalism. In this paper, explicit expressions are obtained for the fundamental matrix N, material eigenvector matrices A and B, and Barnett–Lothe tensors L, S and H. Although the explicit expressions are presented under the generalized plane strain and short circuit condition, by suitable replacement of the material constants they are still valid for the other two-dimensional states. To provide a clear picture of these expressions, two typical examples are presented, which are piezoelectric ceramics with two different poling axes.     

Causal Stroh formalism for uniformly-moving dislocations in anisotropic media: Somigliana dislocations and Mach cones

In this work, Stroh’s formalism is endowed with causal properties on the basis of an analysis of the radiation condition in the Green tensor of the elastodynamic wave equation. The modified formalism is applied to dislocations moving uniformly in an anisotropic medium. In practice, accounting for causality amounts to a simple analytic continuation procedure whereby to the dislocation velocity is added an infinitesimal positive imaginary part. This device allows for a straightforward computation of velocity-dependent field expressions that are valid whatever the dislocation velocity–including supersonic regimes–without needing to consider subsonic and supersonic cases separately. As an illustration, the distortion field of a Somigliana dislocation of the Peierls–Nabarro–Eshelby-type with finite-width core is computed analytically, starting from the Green’s tensor of elastodynamics. To obtain the result in the form of a single compact expression, use of the modified Stroh formalism requires splitting the Green’s function into its reactive and radiative parts. In supersonic regimes, the solution obtained displays Mach cones, which are supported by Dirac measures in the Volterra limit. From these results, an explanation of Payton’s ‘backward’ Mach cones (Payton, 1995) is given in terms of slowness surfaces, and a simple criterion for their existence is derived. The findings are illustrated by full-field calculations from analytical formulas for a dislocation of finite width in iron, and by Huygens-type geometric constructions of Mach cones from ray surfaces.     

Use of the reciprocity theorem for axially symmetric transient problems

Summary A fundamental solution, to be used in reciprocal theorem for the solutions of axially symmetric transient problem of elastodynamics, is presented. A cylindrical cavity problem has been solved to check the formulation. The strong singularity of the resulting integral equation for this problem has been reduced to the weak form. The new formulation provides the initial velocity on the surface for a transient loading. Some differences have been introduced for the use of generalized functions.     

Calculations of the transport properties within the PAW formalism

We implemented the calculation of the transport properties within the PAW formalism in the ABINIT code [1]. This feature allows the calculation of the electrical and optical properties, including the XANES spectrum, as well as the electronic contribution to the thermal conductivity. We present here the details of the implementation and results obtained for warm dense aluminum plasma.     

Wave propagation and reflection-transmission in a stratified viscoelastic solid

The paper investigates time-harmonic wave propagation in continuously stratified solids and provides the results of a reflection-transmission process generated by a layer sandwiched between homogeneous half-spaces. The layer is continuously stratified and allows for jump discontinuities at a finite number of planes. The dissipative effects are accounted for through the classical Boltzmann law of viscoelasticity. By using displacement and traction as convenient vector variables, the governing equations are considered in a vector Volterra integral equation and the solution is determined by means of a matricant. Next the matricant is applied to determine the reflection and transmission coefficients of a layer, with a generic piecewise continuous profile of the material properties. The reflection-transmission process produced by an obliquely incident wave, is considered for horizontally-polarized waves. The low-frequency approximation is derived for the reflection and transmission coefficients. Next, the high-frequency approximation is investigated by a WKB-like procedure which involves a complex valued frequency-dependent shear modulus. The displacement solution is obtained for the forward- and the backward-propagating waves in the layer along with the reflection and transmission coefficients.     

A reciprocity relation between the transmitting and the receiving properties of an antenna

Summary A reciprocity relation between the transmitting and the receiving properties of an antenna is derived. In the transmitting situation a certain part of the antenna, called source domain, is capable of carrying external currents, both of the electric and the magnetic type. In the receiving situation a plane electromagnetic wave is incident upon the antenna system.Whereas the customary way of deriving reciprocity relations in antenna theory starts with considering two separate antennas, the present method assumes a single antenna only.     

A formalism for the solution of problems involving chemical reaction and concentration-dependent diffusion coefficients

By using a previously proposed novel integral transform approach, approximate analytical solutions for a certain class of concentration-dependent diffusion and reaction problems can be obtained. Three example problems of increasing complexity are presented to demonstrate the versatility of this technique. Perturbation techniques are also employed to determine asymptotic behaviour of the solutions in the limit of large or small values of some parameters.     

Uniqueness theorem,theorem of reciprocity,and eigenvalue problems in linear theory of porous piezoelectricity

The uniqueness theorem and the theorem of reciprocity in the linearized porous piezoelectricity are established under the assumption of positive definiteness of elastic and electric fields. General theorems in the linear theory of porous piezoelectric materials are proved for the quasi-static electric field approximation. The uniqueness theorem is also proved without using the positive definiteness of the elastic field. An eigenvalue problem associated with free vibrations of a porous piezoelectric body is studied using the abstract formulation. Some properties of operators are also proved. The problem of frequency shift due to small disturbances, based on an abstract formulation, is studied using a variational and operator approach. A perturbation analysis of a special case is also given.     

弹性薄板弯曲问题的双互易杂交边界点法

基于一种板的修正变分泛函,将杂交边界点法与双互易法结合,用于薄板弯曲问题的分析。该方法将问题的解分为齐次方程的通解和非齐次的特解两部分,特解采用径向基函数插值得到,而通解则使用杂交边界点法求解。在杂交边界点法用于求解通解的列式过程中,边界变量采用移动最小二乘近似,域内变量则采用基本解插值。与有限元法相比,该方法仅需要边界上离散点的信息,无论插值还是积分都不需要网格,域内点仅用来插值非齐次项,因而仍是一种纯边界类型的无网格方法。数值算例表明,本文方法能以很少的计算自由度获得与其它方法同样的计算精度,且具有前后处理简单、收敛速度快等优点,适合于求解工程中各种薄板的弯曲问题。     

Generalization of the Stroh Formalism to 3-Dimensional Anisotropic Elasticity

A method of analysis for three-dimensional anisotropic elasticity is developed, based on Stroh's two-dimensional formalism. By use of the present method a three-dimensional solution can be constructed by the corresponding two-dimensional solution. The method is applied to solve several problems concerning a point force or a dislocation loop. Existing results for these problems are recovered in a unified manner and additional new results are obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

The effect of liquid crystal structure on the rheological properties of nitrocellulose-dimethylacetamide solutions

Solutions of Pyro grade nitrocellulose (NC) in dimethyl acetamide (DMA), containing between 42.5% and 60% NC (w/w), have been studied by differential scanning calorimetry and polarised light microscopy. The results showed that NC forms a lyotropic liquid crystal structure in DMA. A transition from the liquid crystal phase to an isotropic phase occurred over the temperature range 27 °C to 67 °C, and the enthalpy of transition increased with NC concentration. Rheological properties were determined using an extrusion rheometer with a slit die. The solutions were shown to have a yield stress for flow which increased with increasing NC concentration. The solutions were also found to be thixotropic.     

Solution of Zener–Stroh arc crack problem in plane elasticity

In this paper, solution of the Zener–Stroh arc crack in plane elasticity is present. The problem is reduced to a solution of singular integral equation. After using some formulae and equations in complex variable function a closed form solution is obtained.     

Extreme Poisson's ratios and related elastic crystal properties

With the aim of understanding anisotropic crystals that possess a negative Poisson's ratio and to lay a foundation for investigating molecular mechanisms, we discuss the definition of the ratio and establish conditions on the compliance that govern its sign. We derive results on orientation averaging that are useful in the context of anisotropy and helpful in the investigation of isotropic polycrystals. We discuss α-cristobalite, a polymorph of silicon dioxide that possesses interesting negative ratio properties in single crystals and hypothetical polycrystals. In this connection, we draw attention to the transverse compliance as an alternative and simpler metric for gaging the ratio and for orientation averaging. For α-cristobalite, we arrive at new results for the directions that yield the most negative Poisson's ratio. This result should be of value in divining the underlying molecular mechanism that explains the negative values of Poisson's ratio in α-cristobalite, a crystal of tetragonal symmetry that possesses six independent elastic constants.     

Extended Lagrangian formalism and the corresponding energy relations

In this paper an extended Lagrangian formalism for the rheonomic systems with the nonstationary constraints is formulated, with the aim to examine more completely the energy relations for such systems in any generalized coordinates, which in this case always refer to some moving frame of reference. Introducing new quantities, which change according to the law τ_a=φ_a(t), it is demonstrated that these quantities determine the position of this moving reference frame with respect to an immobile one. In the transition to the generalized coordinates qⁱ they are taken as the additional generalized coordinates q^a=τ_a, whose dependence on time is given a priori. In this way the position of the considered mechanical system relative to this immobile frame of reference is determined completely.Based on this and using the corresponding d'Alembert–Lagrange's principle, an extended system of the Lagrangian equations is obtained. It is demonstrated that they give the same equations of motion qⁱ=qⁱ(t) as in the usual Lagrangian formulation, but substantially different energy relations. Namely, in this formulation two different types of the energy change law dE/dt and the corresponding conservation laws are obtained, which are more general than in the usual formulation. So, under certain conditions the energy conservation law has the form E=T+U+P=const, where the last term, so-called rheonomic potential expresses the influence of the nonstationary constraints.Afterwards, a detailed analysis of the obtained results and their connection with the usual formulation of mechanics are given. It is demonstrated that so formulated energy relations are in full accordance with the corresponding ones in the usual vector formulation, when they are expressed in terms of the rheonomic potential. Finally, the obtained results are illustrated by several simple, but characteristic examples.     

The Smoluchowski equation for colloidal suspensions developed and analyzed through the GENERIC formalism

The Smoluchowski equation (SE) and the mechanical stress tensor for the over-damped dynamics of colloidal particles is derived directly at the pair distribution level starting from a thermodynamic basis using the general equations for equilibrium non-equilibrium reversible and irreversible coupling (GENERIC) formalism. Within the GENERIC formalism, the effect of the non-trivial convection due to hydrodynamic interactions is incorporated for the first time. The method generates a thermodynamically valid set of transport equations for the colloidal dispersion, thus properly identifying the extra stress due to the presence of the colloids. The derivation connects a formal entropy expansion to the many-body terms that arise in both the transport equation and the stress tensor, thus unifying their origin and providing a systematic path forward for improvement in the theory. The analysis identifies the thermodynamically valid stress expression, thus clarifying a long-standing problem in the literature that arises when separate derivations are performed for the transport equations and the stress tensor. The results of previous investigators are analyzed within this framework. Comparison with alternate methods of deriving the many-body Smoluchowski equation provide new insight into the nature of the many-body terms.     

Applications of reciprocity,electronic holography and shearography in statics

This paper illustrates the use of reciprocity in electronic-holography experiments to determine unknown forces or couples acting on a mechanical structure. Electronic holography and electronic shearography are simple, fast and contact-free methods for the measurement of deformation fields or gradients of deformation fields, both for static and sinusoidal loadings. From a first measurement, the deformation field of the structure for a known force is determined. In a second experiment, the deformation field for the unknown quantity is measured. Reciprocity is then used to determine the unknown quantity. These optical methods are useful tools for the understanding of reciprocity.     

A reciprocity theorem in linear gradient elasticity and the corresponding Saint-Venant principle

In many practical applications of nanotechnology and in microelectromechanical devices, typical structural components are in the form of beams, plates, shells and membranes. When the scale of such components is very small, the material microstructural lengths become important and strain gradient elasticity can provide useful material modelling. In addition, small scale beams and bars can be used as test specimens for measuring the lengths that enter the constitutive equations of gradient elasticity. It is then useful to be able to apply approximate solutions for the extension, shear and flexure of slender bodies. Such approach requires the existence of some form of the Saint-Venant principle. The present work presents a statement of the Saint-Venant principle in the context of linear strain gradient elasticity. A reciprocity theorem analogous to Betti’s theorem in classic elasticity is provided first, together with necessary restrictions on the constitutive equations and the body forces. It is shown that the order of magnitude of displacements are in accord with the Sternberg’s statement of the Saint-Venant principle. The cases of stretching, shearing and bending of a beam were examined in detail, using two-dimensional finite elements. The numerical examples confirmed the theoretical results.