Explicit expression of derivatives of elastic Green’s functions for general anisotropic materials

The analytical expressions of Green’s function and their derivatives for three-dimensional anisotropic materials are presented here. By following the Fourier integral solutions developed by Barnett [Phys. Stat. Sol. (b) 49 (1972) 741], we characterize the contour integral formulations for the derivatives into three types of integrals H, M, and N. With Cauchy’s residues theorem and the roots of a sextic equation from Stroh eigenrelation, these integrals can be solved explicitly in terms of the Stroh eigenvalues P_i (i=1,2,3) on the oblique plane whose normal is the position vector. The results of Green’s functions and stress distributions for a transversely isotropic material are discussed in this paper.     

Antiplane time-harmonic green’s functions for a circular inhomogeneity with an imperfect interface

Two-dimensional antiplane time-harmonic Green’s functions for a circular inhomogeneity with an imperfect interface are derived. Here the linear spring model with vanishing thickness is employed to characterize the imperfect interface. Explicit expressions for the displacement and the stress fields induced by time-harmonic antiplane line forces located both in the unbounded matrix and in the circular inhomogeneity are presented. When the circular frequency approaches zero, our results reduce to those for the static case. Numerical results are presented to show the influence of the frequency and the imperfection of the interface on the stress and displacement fields.     

Three-dimensional dynamic Green’s functions in transversely isotropic bi-materials

By virtue of a complete representation using two displacement potentials, an analytical derivation of the elastodynamic Green’s functions for a linear elastic transversely isotropic bi-material full-space is presented. Three-dimensional point-load Green’s functions for stresses and displacements are given in complex-plane line-integral representations. The formulation includes a complete set of transformed stress–potential and displacement–potential relations, within the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For numerical computation of the integrals, a robust and effective methodology is laid out which gives the necessary account of the presence of singularities including branch points and pole on the path of integration. As illustrations, the present Green’s functions are analytically degenerated to the special cases such as half-space, surface and full-space Green’s functions. Some typical numerical examples are also given to show the general features of the bi-material Green’s functions.     

Fer’s expansion for generalized Hamiltonian system based on Lie transformation technique

In the present paper, we present a new method for integrating the ordinary differential equation, especially for the ordinary differential equation derived from explicitly time-dependent generalized Hamiltonian dynamic system, which is based on taking a factorization of the evolution operator as an infinite product of the exponentials of Lie operators. The above process is a Lie group (algebraic) method that retains the structural intrinsic properties of the exact solution when truncated and is used to analyze the main features of the so-called Fer’s expansion. The numerical examples are presented at the end of this paper.     

Green’s functions for bending problem of a thin anisotropic plate with an elliptic hole

The Green’s functions have not been studied in open literatures for the bending problem of an anisotropic plate with an elliptic hole subjected to a normal concentrated force and a concentrated moment. In this paper, the problem is investigated and the Green’s functions are first obtained by using the complex potential approach. The techniques of conformal mapping transformation and analytic continuation are used to derive the closed-form complex stress functions. The Green’s functions obtained have some potential applications in the analysis of composite structures such as the modification of the displacement compatibility model for notched stiffened composite panels and the formulation of a new method for interlaminar stress analysis around holes of laminates.     

Two-dimensional time-harmonic dynamic Green’s functions in transversely isotropic piezoelectric solids

The two-dimensional time-harmonic dynamic Green’s functions in an infinite transversely isotropic piezoelectric solid are obtained. After introduction of a new function, the original problem is reduced to the determination of the Green’s function for the two-dimensional Helmholtz equation and that for the two-dimensional Laplace equation. The explicit expressions of all the field components are presented. It is verified that the obtained dynamic Green’s functions can reduce to the corresponding static ones by letting the circular frequency be zero. The asymptotic expansions for Green’s functions at far-field are also given.     

Generalization of Vlasov’s Equations for a Cylindrical Shell to the Case of a Transversely Isotropic Material

Differential equations of the general theory of transversely isotropic cylindrical shells are obtained; in a certain sense, these equations are generalizations of Vlasov’s and Ambartsumyan’s equations. This allowed us on the basis of Novozhilov’s criterion (comparison of variability of the stress state in the principal orthogonal directions) to divide the initial equations according to Goldenweiser into approximate equations of the type of the semi-momentless theory, theory of the edge effect and flexural state, which are also generalizations of equations that describe the elementary stress states of an isotropic shell. Numerical values are found for criteria of matching of approximate equations that describe the elementary stress states in the asymptotic synthesis of the full stress state. Examples of calculations and experimental data for a shell with and without allowance for transverse shear strain are given.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 125–132, July– August, 2005.     

An elastic–plastic interface constitutive model: application to adhesive joints

An interface constitutive model which accounts for both reversible elastic, as well as irreversible inelastic separation-sliding deformations at the interface prior to failure has been developed and implemented in a finite-element computer program. A set of experiments has been conducted to determine the constitutive parameters in the interface model for adhesively-bonded components made from an aluminum alloy and a ductile polymeric adhesive. The constitutive model and the computational capability are shown to reasonably well predict the macroscopic deformed geometries and the load–displacement curves in: (a) T-peel experiments; (b) four-point bend experiments on bonded bi-layer edge-notch specimens; (c) lap-shear experiments.     

The consistent Mindlin’s thermopiezoelectric equations and the principle of virtual work

The thermal field vector, an overlooked concept of importance, is introduced for the consistency in Mindlin’s thermopiezoelectric equations. The gradient of the temperature increment is defined as the thermal field vector; an analogous to the electric and magnetic field vectors (i.e., the gradients of the electric and magnetic potentials). The thermal field vector is used to state the universal gradient equations, and then, the principle of virtual work.     

一类铁电模型的参变量变分原理及其应用

将参变量变分原理引入铁电问题。对一类借用了经典弹塑性理论中的概念和方法的多轴铁电模型建立基于Helmholtz自由能的参变量变分原理,可以有效处理传统变分原理中由非关联流动法则或屈服面不考虑材料系数变化所引起的切线模量非对称困难。相应于参变量变分原理,引入参数二次规划算法,可获得具有可靠数值稳定性的一套铁电算法。将该算法应用于一个具体的铁电模型,数值计算结果表明本文方法的有效性。     

Green’s-function method for modeling surface acoustic wave dispersion in anisotropic material systems and determination of material parameters

A Green’s-function method for modeling propagation of surface acoustic waves in anisotropic nanolayered materials is reviewed. The mathematical model, developed at NIST, provides a computationally efficient inversion algorithm for determination of the material parameters of the film, such as its elastic constants and density, from observed dispersion of the surface acoustic waves. The application of the method to a 306 nm thick TiN film having transverse isotropy on a single crystal Si substrate is discussed as an example. The errors in the values of the parameters determined by using the inversion algorithm and the question of uniqueness of the values of the parameters are discussed in detail. It is suggested that, at least in the example considered in this paper, the SAW dispersion can be used to determine any two parameters of the film, provided other parameters are known by independent measurements. In particular, the values of c₁₁ and the density of the film, obtained from the measured SAW dispersion, are the most reliable and the value of c₄₄ is the least reliable. The method is extended to account for defective bonding between the film and the interface and the effect of an intermediate layer of silica between the film and the substrate.     

On the use of generalized Hamilton׳s principle for the derivation of the equation of motion of a pipe conveying fluid

In this paper, first, some pertinent studies are reviewed, undertaken to extend Hamilton׳s principle or Lagrange׳s equations to open systems, and continua including so-called non-material volumes; then, the corresponding mathematical formulations are reproduced and discussed using a uniform nomenclature. A typical flexible pipe conveying fluid system is considered next, and it is shown that the expression given by McIver for Hamilton׳s principle remains valid even in the presence of non-material volumes; the extra term due to the flux of kinetic energy through the control surface in the expression obtained by Casetta and Pesce vanishes.     

Second-order effects and Saint Venant’s principle in the torsion problem of a nonlinear elastic rod

The torsion problem of a circular nonlinear elastic rod loaded by end moments is considered. The solution constructed by the method of successive approximations taking into account second-order effects is compared with the solution obtained by a semi-inverse method. It is shown that the dead-loading assumption breaks the symmetry of the Cauchy stress tensor in a certain region. A refined formulation of Saint Venant’s principle is proposed for the problem of determining integral strain characteristics. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 129–136, November–December, 2006.     

Maximum entropy moment system of the semiconductor Boltzmann equation using Kane’s dispersion relation

It is known that the maximum entropy moment systems of the gas-dynamical Boltzmann equation suffer from severe disadvantages which are related to the non-solvability of an underlying maximum entropy moment problem unless restrictions on the choice of the macroscopic variables are made. In this article, we show that no such difficulties appear in the semiconductor case if Kane’s dispersion relation is used for the energy band of electrons.PACS 73.50-h; 73.61.-r     

AN APPLICATION OF SCHAUDER’S FIXED POINT THEOREM TO THEEXISTENCE OF SOLUTlONS OF IMPULSIVELYDIFFERENTIAL EQUATIONS

In this paper the existence of solutions of a boundary value problem forimpulsively differential equations that is difficult to solve by the upper and lowersolution method will be proved by means of Schauder’s fixed point theorem,whichimproves some existing results.     

Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation

In this paper, in a development of the static theory derived by Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853), we establish the equations of motion for a non-linearly elastic body in plane strain with an elastic surface coating on part or all of its boundary. The equations of (linearized) incremental motions superposed on a finite static deformation are then obtained and applied to the problem of (time-harmonic) surface wave propagation on a pre-stressed incompressible isotropic elastic half-space with a thin coating on its plane boundary. The secular equation for (dispersive) wave speeds is then obtained in respect of a general form of incompressible isotropic elastic strain-energy function for the bulk material and a general energy function for the coating material. Specialization of the form of strain-energy function enables the secular equation to be cast as a quartic equation and we therefore focus on this for illustrative purposes. An explicit form for the secular equation is thereby obtained. This involves a number of material parameters, including residual stress and moment in the properties of the coating. It is shown how this equation relates to previous work on waves in a half-space with an overlying thin layer set in the classical theory of isotropic elasticity and, in particular, the significant effect of omission of the rotatory inertia term, even at small wave numbers, is emphasized. Corresponding results for a membrane-type coating, for which the bending moment, inertia and residual moment terms are absent, are also obtained. Asymptotic formulas for the wave speed at large wave number (high frequency) are derived and it is shown how these results influence the character of the wave speed throughout the range of wave number values. A bifurcation criterion is obtained from the secular equation by setting the wave speed to zero, thereby generalizing the bifurcation results of Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853) to the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the finite deformation are then described graphically. In particular, features which differ from those arising in the classical theory are highlighted.     

间隙元在钻柱瞬态动力学分析中的应用

为了描述钻柱与井壁的随机碰撞接触状态，本文构造了动力间隙元，并推导了动力间隙元的相对压缩量和迭代求解格式。把动力间隙元与梁单元相结合，采用Newmark直接积分法和冲量定理，进行了钻柱碰撞接触非线性瞬态动力学分析。经工程应用表明，构造的动力间隙元能够描述钻柱的碰撞接触状态，所设计的偏心防斜钻具已在大庆油田得到应用，取得了明显的经济和社会效益。     

时间尺度上Herglotz变分原理及其Noether定理

Herglotz变分原理的作用量是由微分方程定义的,不仅可以描述所有经典变分原理能够描述的动力学过程,还可以对经典变分原理不能适用的非保守系统或耗散系统进行变分描述．时间尺度上微积分理论提供了一种可同时研究离散系统和连续系统的有效方法．本文结合Herglotz变分原理和时间尺度微积分理论来研究时间尺度上的Herglotz变分原理及其Noether定理．首先,给出时间尺度上Lagrange系统的Herglotz变分原理．其次,根据Herglotz变分原理和Dubois-Reymond引理,推导出时间尺度上Lagrange系统的Herglotz变分问题的运动微分方程．再次,基于时间尺度上Hamilton-Herglotz作用量在群的无限小变换下的不变性,给出Noether对称性的定义并导出其Noether等式．最后,建立了时间尺度上Lagrange系统的Herglotz变分问题的Noether定理,给出了连续和离散两种情况下基于Herglotz变分问题的Noether守恒量．文末举例说明结果的应用．