Complete eigenfunction expansion form of the Green's function for elastic layered half-space

Summary  The complete eigenfunction expansion form of the Green's function for a 3-D elastic layered half-space in the frequency domain is derived in this paper. The expression of the Green's function presented here is an extension of that represented by the residue terms and the branch line integrals given by Lamb [1]. The present expression, however, clarifies the mathematical common frame between the residue terms and the branch line integrals with respect to the eigenfunctions and energy integrals. For the derivation, the concept of an energy integral for the improper eigenfunctions is newly developed. The improper eigenfunctions, which can be found in the wavenumbers for the branch cuts, are not in L ₂ space, so the definition of the energy integral requires some treatment. The energy integral is defined as the limit of the inner product of the improper eigenfunction and the definition function of the improper eigenfunction, for which the inner product remains finite. Via the definition of the energy integral, the kernel of the branch line integral is decomposed into the improper eigenfunction, and the complete eigenfunction expansion form of the Green's function is derived. The Green's function can thus be expressed by summation of normal modes, complex modes pointed out in [2], the integral of the improper eigenfunction and the residue at k=0 due to the singularity of the horizontal wavefunction. Received 3 May 2001; accepted for publication 23 August 2001     

A new approach to the vakonomic mechanics

The aim of this paper was to show that the Lagrange–d’Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them we consider the generalization of the Hamiltonian principle for nonholonomic systems with non-zero transpositional relations. We apply this variational principle, which takes into the account transpositional relations different from the classical ones, and we deduce the equations of motion for the nonholonomic systems with constraints that in general are nonlinear in the velocity. These equations of motion coincide, except perhaps in a zero Lebesgue measure set, with the classical differential equations deduced with the d’Alembert–Lagrange principle. We provide a new point of view on the transpositional relations for the constrained mechanical systems: the virtual variations can produce zero or non-zero transpositional relations. In particular, the independent virtual variations can produce non-zero transpositional relations. For the unconstrained mechanical systems, the virtual variations always produce zero transpositional relations. We conjecture that the existence of the nonlinear constraints in the velocity must be sought outside of the Newtonian mechanics. We illustrate our results with examples.     

A difference equation approach to statistical mechanics of complex networks

In this paper, we propose a difference equation approach to the estimation of the degree distributions in growing networks after having analyzed the disadvantages of some existing approaches. This approach can avoid logic conflicts caused by the continuum of discrete problems, and does not need the existence assumption of the stationary degree distribution in the network analysis. Using this approach, we obtain a degree distribution formula of the Poisson growth and preferential attachment network. It is rigorously shown that this network is scale-free based on the Poisson process theory and properties of F-distribution.     

Energy-wave approach in the mechanics of brittle dynamic fracture

Moscow. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 2, pp. 118–123, March–April, 1994.     

A new based error approach to approximate the inverse langevin function

In the present paper, we propose a new approximation of the inverse Langevin function. This new approximation is based on a two-step modification of the fractional formula introduced by (Cohen 1991). Our proposal is motivated by the minimization of the error between the Cohen formula and the inverse of the Langevin function. It results in two additional terms adopting a remarkable simple power and polynomial forms. The correction provides an excellent agreement with a maximum relative error equal to 0.046 % (against a maximum error of 4.94 % for the Cohen formula).     

How to compute plastic zones of heterogeneous materials: A simple approach using classical continuum and fracture mechanics

This is a short technical paper on how to use classical continuum and fracture mechanics to calculate the plastic zones caused by cracks on heterogeneous or composite materials. As an example, a sample consisting of an α-phase and β-phase is used. A crack is introduced to the sample, and stress is then applied. The plastic zone in front of the crack resulting from the applied stress is then calculated using commercial software. The concept uses two-level modeling: a global model using homogenized stiffness from a unit cell of heterogeneous material and a local model for the α-phase and β-phase. While this paper is written for general purposes, a concrete example using ferrite and martensite is also presented along with the experimental data. General agreement between the model and the experiment is observed. This method eliminates the need for a cumbersome analytical approach.     

A coupled FEM/BEM approach and its accuracy for solving crack problems in fracture mechanics

The finite element (FEM) and the boundary element methods (BEM) are well known powerful numerical techniques for solving a wide range of problems in applied science and engineering. Each method has its own advantages and disadvantages, so that it is desirable to develop a combined finite element/boundary element method approach, which makes use of their advantages and reduces their disadvantages. Several coupling techniques are proposed in the literature, but until now the incompatibility of the basic variables remains a problem to be solved. To overcome this problem, a special super-element using boundary elements based on the usual finite element technique of total potential energy minimization has been developed in this paper. The application of the most commonly used approaches in finite element method namely quarter-point elements and J-integrals techniques were examined using the proposed coupling FEM–BEM. The accuracy and efficiency of the proposed approach have been assessed for the evaluation of stress intensity factors (SIF). It was found that the FEM–BEM coupling technique gives more accurate values of the stress intensity factors with fewer degrees of freedom.     

A solution to the hydraulic fracture problem in the traveling wave form

Solutions to the system of equations describing the propagation of hydraulic fracture cracks in a porous medium are obtained in the traveling wave form. The only sought solution is the separatrix of integral curves on the “penetration depth-crack width” plane. Some necessary dependencies that should be given at the crack inlet are found for the fluid flow rate and the fluid pressure. The crack width and the fluid penetration depth are related by power laws in the limiting cases when the crack propagation processes or the fluid penetration processes are dominant.     

A general perturbation method for inhomogeneities in anisotropic and piezoelectric solids with applications to quantum-dot nanostructures

By introducing a homogeneous piezoelectric material and its Green’s function, we present a new semi-analytical three-dimensional perturbation method for general inhomogeneity problems in anisotropic and piezoelectric solids. This method removes the limitations associated with previous analytical methods, which often ignore the anisotropic properties or the difference between the material properties of the inhomogeneity and its surrounding matrix. As an important application, the proposed theory is employed to calculate the elastic and electric fields in a truncated pyramidal InAs/GaAs quantum-dot (QD) nanostructure. Numerical results demonstrate that the anisotropy of the materials and the difference between the material constants of the QD and the matrix have a significant influence on the strain and electric fields. The relative differences of the strain and electric field inside the QD between the simplified isotropic and homogeneous model and the real anisotropic and heterogeneous one may reach 22% and 53%, respectively. The accuracy of the calculated elastic strain and electric fields is improved greatly by a second order approximate solution (OAS). Since the third OAS nearly coincides with the second one, good convergence of the iteration procedure is demonstrated. Moreover, contours of the hydrostatic strain and electric potential within and around the QD are also presented and analyzed.     

Eigenfunction expansion method to the solution of simultaneous equations and its application in mechanics

The solution of a homogeneous vector-matrix differential equation, when the operator is of linear second order, has been established by the method of eigenfunction approach. The uniqueness of the solution has been established for both the cases when the roots of the characteristic equations are distinct and when they are repeated. Finally, the theory has been applied on two problems of mechanics and the results are compared with the existing literature.     

A review of boundary and finite element methods in fracture mechanics

Reviewed in this work are the methods of finite and boundary element as applied to solve fracture mechanics problems. The former requires the discretization of the interior of the domain while the latter involves computing an integral equation over the boundary of the domain. Applications of these methods are made to two-dimensional elastic crack problems. Efficiency and accuracy of different approaches are discussed and compared by examples. The boundary element procedure employing special Green's functions for the plane crack problem is shown to be superior. The correlation between the hybrid element formulations and boundary element regions embedded into a finite element model is also given.     

An approach to the problem of closure in non-linear stochastic mechanics

An approach combining the method of moment equations and the statistical linearization technique is proposed for analysis of the response of non-linear mechanical systems to random excitation. The adaptive statistical linearization procedure is developed for obtaining a more accurate mean square of responses. For these, a Duffing oscillator and an oscillator with cubic non-linear damping subject to white noise excitation are considered. It is shown that the adaptive statistical linearization proposed yields good accurate results for both weak and strong non-linear stochastic systems.Presented at the First European Solid Mechanics Conference, September 9–13, 1991. Munich, Germany     

A new form of the coherency energy in pseudoelasticity

In some previous papers [1], [2] pseudoelasticity in tensile experiments has been treated thermodynamically under the assumption that the relevant constitutive ingredients are

A new approach to the solution of the Navier-Stokes equations

In the present paper a numerical algorithm is given for solving a standard problem in fluid dynamics, that of inviscid, irrotational, incompressible flow over an arbitrary symmetric profile. The purpose of the paper is to propose an alternative approach to solve certain fluid dynamic flows. This paper may be thought of as the first of a possible series of papers solving new and fundamental problems. In a sense, this new approach asks the question: what is the simplest and most efficient method of solving the problem considered by finite difference methods. It is believed that the following algorithm answers this question. Standard second-order finite difference techniques, such as SLOR and ADI, are used to solve numerically a mixed boundary value problem comprised of a pair of elliptic partial differential equations with constant coefficients.     

A green's function approach to the determination of internal fields

A time-domain technique is presented for computing the internal electromagnetic field within a one-dimensional medium characterized by spatially varying conductivity and permittivity profiles. A Green's operator is defined which maps the incident fields on either side of the medium to the field at an arbitrary observation point. This operator is shown to be a matrix of integral operators with kernels satisfying known partial differential equations and various other initial and boundary identities. A scheme for numerically calculating these kernels is presented along with a few examples of the calculations. Moreover, consideration of the boundary values of the Green's operator reveals a novel way for computing the reflection and transmission operators for the medium. Finally, a Green's function approach to internal fields in the presence of a phase velocity mismatch at one of the boundaries of the medium is outlined in an appendix.     

Static and dynamic analysis of the DCB problem in fracture mechanics

A finite difference scheme for treating the static and dynamic stress fields under plane-strain conditions in the DCB, is proposed. The adequacy of the scheme is established via the static solution by comparing the results obtained numerically with those obtained experimentally. Both the numerical and experimental results are also compared with data available in the literature. Discrepancies found are explained and discussed. For the numerical scheme adjusted to handle the propagating crack problem, the results represent a situation which is close to that observed experimentally; namely, an essentially constant steady state crack propagation speed from the start, with crack length at arrest and velocity values depending on the initial conditions. In addition, the velocities predicted by the analysis are shown to be in good agreement with those reported in the literature.     

A variational approach to the fracture of brittle thin films subject to out-of-plane loading

We address the problem of fracture in homogenous linear elastic thin films using a variational model. We restrict our attention to quasi-static problems assuming that kinetic effects are minimal. We focus on out-of-plane displacement of the film and investigate the effect of bending on fracture. Our analysis is based on a two-dimensional model where the thickness of the film does not need to be resolved. We derive this model through a formal asymptotic analysis. We present numerical simulations in a highly idealized setting for the purpose of verification, as well as more realistic micro-indentation experiments.