Deriving the time-dependent dyadic Green’s functions in conductive anisotropic media

A homogeneous anisotropic conductive medium, characterized by symmetric positive definite permeability and conductivity tensors, is considered in the paper. In this anisotropic medium, the electric and magnetic dyadic Green’s functions are defined as electric and magnetic fields arising from impulsive current dipoles and satisfying the time-dependent Maxwell’s equations in quasi-static approximation. A new method of deriving these dyadic Green’s functions is suggested in the paper. This method consists of several steps: equations for electric and magnetic dyadic Green’s functions are written in terms of the Fourier images; explicit formulae for the Fourier images of dyadic Green’s functions are derived using the matrix transformations and solutions of some ordinary differential equations depending on the Fourier parameters; the inverse Fourier transform is applied numerically to obtained formulae to find dyadic Green’s functions values. Using suggested method images of electric and magnetic dyadic Green’s function components are obtained in such conductive anisotropic medium as the white matter of a human brain.     

Computation of Dyadic Green's Functions for Electrodynamics in Quasi-Static Approximation with Tensor Conductivity

Homogeneous non-dispersive anisotropic materials, characterized by a positive constant permeability and a symmetric positive definite conductivity tensor, are considered in the paper. In these anisotropic materials, the electric and magnetic dyadic Green's functions are defined as electric and magnetic fields arising from impulsive current dipoles and satisfying the time-dependent Maxwell's equations in quasi-static approximation. A new method of deriving these dyadic Green's functions is suggested in the paper. This method consists of several steps: equations for electric and magnetic dyadic Green's functions are written in terms of the Fourier modes; explicit formulae for the Fourier modes of dyadic Green's functions are derived using the matrix transformations and solutions of some ordinary differential equations depending on the Fourier parameters; the inverse Fourier transform is applied to obtained formulae to find explicit formulae for dyadic Green's functions.     

改进的数字全息光弹方法理论研究

该文通过改变参考光偏振方向,对数字全息光弹法进行改进,从而可以高效、高精度的检测二维应力场.从理论上导出了该方法的实验原理:加载前后在对应的4 个不同参考光偏振位置拍摄数字全息图,对这4 对数字全息图进行傅里叶变换、滤波及反变换,可以获得4 个光强方程,数值求解这4 个光强方程,再结合应力-光学定律就可以精确获得二维应力场的2 个主应力值和主应力方向.与传统的数字全息光弹法相比,新方法具有以下优点:不需要分离等差线与等倾线条纹;不需要确定条纹级数;方法简单,不需要已有技术以外的实验装置或实验模型.     

Contact and crack problems in generalized materials and their relationships

A new term of generalized material is introduced here. This definition covers all anisotropic magneto-electro-elastic materials and one-, two- and tree-dimensional generally anisotropic quasi-crystals and hopefully, some new not yet discovered materials. We consider a half-space \(x_{3}\ge 0\), made of generalized material and subjected to arbitrary point sources or point dislocations, which can be interpreted also as electric and/or magnetic influence. General solution was obtained by using two-dimensional Fourier transform. The final results are presented as single integrals over a unit circle. Some components of the surface Green’s functions were computed in a finite form, no computation of any integral is needed. The theory of generalized functions was used. This result allows us to derive the governing integral equations for the normal and tangential contact and crack problems. We also establish certain relationships between the Fourier transforms of the kernels of the relevant integral equations. As a bonus, some interesting properties of the determinants, which might be new, were established.     

轴荷载作用下铁路交通地面振动的半解析法研究

根据波数域内分层地基波动方程的求解理论，推导铁路有砟轨道、无砟轨道与地基的耦合振动方程，得到了波数域内的统一表达形式。材料阻尼采用粘滞阻尼。利用Fourier变换，在频率。波数域内求解振动微分方程，再通过Fourier逆变换得到大地表面的振动响应。分析了轨道和地基之间的能量传递特征，简谐荷载对振动衰减的影响，并对列车轴荷载引起的地面振动进行了仿真分析。结果表明：铁路有砟轨道和无砟轨道与地基之间的动力作用有较大差别；在简谐荷载作用下，地面振动的衰减曲线出现波动；本文方法具有较大的计算域，可以模拟编组较长的列车通过时引起的地面振动。     

Computing and simulation of time-dependent electromagnetic fields in homogeneous anisotropic materials

The goal of this paper is to derive explicit formulae for initial value problems of electromagnetic radiation arising from electric currents in homogeneous non-dispersive electrically and magnetically anisotropic materials. Computing these formulae is based on matrix symbolic transformations and the inverse Fourier transform which is done numerically. The robustness of these formulae for the simulation of electric and magnetic waves is demonstrated by computed 2D and 3D images of the electric and magnetic field components generated by a pulse dipole with a fixed polarization.     

Image rotation and translation measurement based on double phase-encoded joint transform correlator

An image rotation and translation measurement technology based on a double phase-encoded joint transform correlator (DPEJTC) is proposed. The reference and the target images are Fourier transformed. Then the magnitude of the Fourier-transformed reference (MFR) and target (MFT) images are multiplied with a high-pass emphasis filter and transformed from Cartesian space into polar space. Rotation between the reference and the target image is obtained by measuring the emphasized MFR and MFT in polar coordinates by the DPEJTC. The target image is rotated by the rotation angle in the inverse orientation to get the rotation-correction target image. Finally, translation between the reference and the target image is obtained through measuring the reference and the rotation-correction target image by the DPEJTC. Results based on digital computation are given to verify our proposal. A possible optical setup is suggested.     

The temperature field induced by the dynamic stresses of a transverse crack in periodically layered composites

The temperature field induced by the dynamic application of a far-field mechanical loading on a periodically layered material with an embedded transverse crack is investigated. To this end, the thermoelastically coupled elastodynamic and energy (heat) equations are solved by combining two approaches. In the first one, the dynamic representative cell method is employed for the construction of the time-dependent Green’s functions generated by the displacement jumps along the crack line. This is performed in conjunction with the application of the double finite discrete Fourier transform on the thermomechanically coupled equations. Thus the original problem for the cracked periodic composite is reduced to the problem of a domain with a single period in the transform space. The second approach is based on wave propagation analysis in composites where full thermomechanical coupling in the constituents exists. This analysis is based on the coupled elastodynamic-energy continuum equations where the transformed time-dependent displacement vector and temperature are expressed by second-order expansions, and the elastodynamic and energy equations and the various interfacial and boundary conditions are imposed in the average (integral) sense. The time-dependent thermomechanically coupled field at any observation point in the plane can be obtained by the application of the inverse transform. Results along the crack line as well as the full temperature field are given for cracks of various lengths for Mode I and Mode II deformations. In particular the temperature drops (cooling) at the vicinity of the crack’s tip and the heating zones at its surroundings are generated and discussed.     

An analytical approach for free vibration and transient response of functionally graded piezoelectric cylindrical panels subjected to impulsive loads

In this article, the free vibration and dynamic response of simply supported functionally graded piezoelectric cylindrical panel impacted by time-dependent blast pulses are analytically investigated. Using Hamilton’s principle, the equations of motion based on the first-order shear deformation theory are derived. Also, Maxwell’s electricity equation is taken as one of the governing equations. Three sets of electric surface conditions including closed circuit and two mixtures of closed and open circuit surface conditions are considered. By introducing an analytical approach and using the Fourier series expansions, the Laplace transform and Laplace inverse method, the solution of unknown variables are obtained in the real time domain based on a combination of system frequencies. Finally, the effects of various electric surface conditions, geometric parameters and the material power law index on the free vibration and transient response of functionally graded piezoelectric cylindrical panels subjected to various impulsive loads are examined in detail.     

The scattering of oblique flexural waves of a cracked conducting plate in a uniform magnetic field

Summary Following a classical plate bending theory for magneto-elastic interactions under quasistatic electromagnetic field, we consider the scattering of time harmonic flexural waves by a through crack in a conducting plate under a uniform magnetic field normal to the crack surface. It is assumed that the plate has the finite electric conductivity, and the electric and magnetic permeabilities of the free space. An incident wave giving rise to moments symmetric about the crack plane is applied in an arbitrary direction. Fourier transform method is used to solve the mixed boundary value problem which reduces to a pair of dual integral equations. These dual integral equations are further reduced to a Fredholm integral equation of the second kind. The dynamic moment intensity factor versus frequency for several values of incident angle is computed and the influence of the magnetic field on the normalized values is displayed graphically.     

Magneto-thermoelastic Response in a Functionally Graded Isotropic Unbounded Medium Under a Periodically Varying Heat Source

This paper deals with the problem of magneto-thermoelastic interactions in a functionally graded isotropic unbounded medium due to the presence of periodically varying heat sources in the context of linear theory of generalized thermoelasticity with energy dissipation (TEWED) and without energy dissipation (TEWOED) having a finite conductivity. The governing equations of generalized thermoelasticity (GN model) for a functionally graded material (FGM) under the influence of a magnetic field are established. The Laplace–Fourier double transform technique has been used to get the solution. The inversion of the Fourier transform has been done by using residual calculus, where poles of the integrand are obtained numerically in a complex domain by using Leguerre’s method and the inversion of the Laplace transformation is done numerically using a method based on a Fourier series expansion technique. Numerical estimates of the displacement, temperature, stress, and strain are obtained for a hypothetical material. The solution to the analogous problem for homogeneous isotropic materials is obtained by taking a suitable non-homogeneous parameter. Finally, the results obtained are presented graphically to show the effect of a non-homogeneous, magnetic field and damping coefficient on displacement, temperature, stress, and strain.     

Dynamic stresses created by the sudden appearance of a transverse crack in periodically layered composites

The time-dependent stress field generated by the sudden appearance of a transverse crack in a periodically layered composite that is subjected to a remote loading is determined. The resulting two-dimensional elastodynamic problem is solved by combining two approaches. In the first one, the representative cell method, which has been presently generalized to dynamic problems, is employed for the construction of the time-dependent Green’s functions generated by the displacement jumps along the crack line. This is performed in conjunction with the application of the double finite discrete Fourier transform. Thus the original problem for the cracked periodic composite is reduced to the problem of a domain with a single period in the transform space. The second approach is based on a wave propagation in composites theory which has been presently generalized to admit arbitrary types of loading. This theory is based on the elastodynamic continuum equations where the transformed time-dependent displacement vector is expressed by a second-order expansion, and the equations of motion and the various interfacial and boundary conditions are imposed in the average (integral) sense. The time-dependent field in any observation point in the plane can be obtained by the application of the inverse transform. This field is valid as long as no reflected waves from external boundaries have been arrived. Results along the crack line as well as the full field are given for cracks of various lengths for Mode I, II and III deformations. In particular the dynamic magnification with respect to the static case is determined at the interface within the first unbroken stiff layer.     

On the stress and electric field produced by dislocations in anisotropic piezoelectric crystals with special attention to the stress function

The author is concerned with the stress and electric field produced by dislocations in an anisotropic piezoelectric crystal, with a proposal on the way of choosing the best triad as stress functions from among the six components of Beltrami’s stress-function tensor. A pair of constitutive equations is assumed to connect the elastic strain and electric displacement with the stress and electric field. The fundamental equations governing the field of stress functions and electric scalar potential are presented, and solved by application of the method of Fourier transform. The stress and electric field are stated in terms of the dislocation density tensor, and by means of the convolution integrals throughout the region where there exist dislocations. The expressions are converted into those for the fields of an infinitely extended straight dislocation, as well as an elliptic dislocation, by Willis’ method. The choice of three stress functions is made on the way of numerical computations so that the line integrals can be achieved by application of Cauchy’s residue theorem. As example, the field of dislocations in a gallium arsenide is evaluated.     

Normalized explicit approximate inverse preconditioning for solving 3D boundary value problems on uniprocessor and distributed systems

Normalized explicit approximate inverse matrix techniques, based on normalized approximate factorization procedures, for solving sparse linear systems resulting from the finite difference discretization of partial differential equations in three space variables are introduced. Normalized explicit preconditioned conjugate gradient schemes in conjunction with normalized approximate inverse matrix techniques are presented for solving sparse linear systems. The convergence analysis with theoretical estimates on the rate of convergence and computational complexity of the normalized explicit preconditioned conjugate gradient method are also derived. A Parallel Normalized Explicit Preconditioned Conjugate Gradient method for distributed memory systems, using message passing interface (MPI) communication library, is also given along with theoretical estimates on speedups, efficiency and computational complexity. Application of the proposed method on a three‐dimensional boundary value problem is discussed and numerical results are given for uniprocessor and multicomputer systems. Copyright © 2005 John Wiley & Sons, Ltd.     

Transmission of sound waves in a cylindrical duct with an acoustically lined muffler

The propagation of sound in an infinite rigid cylindrical duct with an inserted expansion chamber whose walls are treated with an acoustically absorbent material is investigated rigorously through the Wiener–Hopf technique. By introducing the Fourier transform for the scattered field and applying the boundary conditions in the transform domain, the problem is reduced into a modified Wiener–Hopf equation. The solution involves four sets of infinitely many constants satisfying four infinite systems of linear algebraic equations. An approximate solution of these systems is obtained by means of numerical procedures.     

Generalized Theory of Thermoviscoelasticity and a Half-Space Problem

In this work, the equations of generalized thermoviscoelasticity for a viscoelastic medium are derived. Also, uniqueness and reciprocity theorems for these equations are proved. In addition, a one-dimensional problem for a viscoelastic half space is considered. The Laplace transform technique is used to solve the problem. The solution in the transformed domain is obtained by a direct approach. The inverse transforms are obtained in an approximate analytical manner using asymptotic expansions valid for small values of time. The temperature, displacement, and stress are computed and represented graphically.     

A two-dimensional problem for a half-space in magneto-thermoelasticity with thermal relaxation

In this work we study a two-dimensional problem in electromagneto-thermoelasticity for a half-space whose surface is subjected to a non-uniform thermal shock and is stress free in the presence of a transverse magnetic field. The problem is in the context of the theory of generalized thermoelasticity with one relaxation time. Laplace and exponential Fourier transform techniques are used to obtain the solution by a direct approach. The solution of the problem in the physical domain is obtained by using a numerical method for the inversion of the Laplace transforms based on Fourier series expansions. The distributions of the temperature, the displacement, the stress and the induced magnetic and electric fields are obtained. The numerical values of these functions are represented graphically.     

Numerical modeling of micro- and macro-behavior of viscoelastic porous materials

This paper presents a numerical method for solving the two-dimensional problem of a polygonal linear viscoelastic domain containing an arbitrary number of non-overlapping circular holes of arbitrary sizes. The solution of the problem is based on the use of the correspondence principle. The governing equation for the problem in the Laplace domain is a complex hypersingular boundary integral equation written in terms of the unknown transformed displacements on the boundaries of the holes and the exterior boundaries of the finite body. No specific physical model is involved in the governing equation, which means that the method is capable of handling a variety of viscoelastic models. A truncated complex Fourier series with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the boundaries of the holes. A truncated complex series of Chebyshev polynomials with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the straight boundaries of the finite body. A system of linear algebraic equations is formed using the overspecification method. The viscoelastic stresses and displacements are calculated through the viscoelastic analogs of the Kolosov–Muskhelishvili potentials, and an analytical inverse Laplace transform is used to provide the time domain solution. Using the concept of representative volume, the effective viscoelastic properties of an equivalent homogeneous material are then found directly from the corresponding constitutive equations for the average field values. Several examples are given to demonstrate the accuracy of the method. The results for the stresses and displacements are compared with the numerical solutions obtained by commercial finite element software (ANSYS). The results for the effective properties are compared with those obtained with the self-consistent and Mori–Tanaka schemes.     

State space solution to three-dimensional consolidation of multi-layered soils

From the governing equations of a saturated poro-elastic soil in the Cartesian coordinate system, the state space equations of Biot’s three-dimensional consolidation problems are obtained by the Laplace transform and the double Fourier transform. Transfer matrix describing the transfer relation between the state vectors for a finite layer is derived explicitly in the transform space. Based on the continuity conditions between adjacent layers and the boundary conditions, the solution for three-dimensional consolidation of multi-layered soils is derived in a transformed domain. This solution is then transferred to that in a physical domain by the inversion of the double Fourier transform and the Laplace transform. Numerical analysis is carried out for three-dimensional consolidation of single, two, and multi-layered soils. The results for single and two-layered soils are compared well with those by the finite layer method in the literature. Numerical results for three-dimensional consolidation of five-layered soils are presented in this paper as an example to illustrate the use of the solution in this paper for three-dimensional consolidation of more than two-layered soils.     

Non-linear vibration of frames by the incremental dynamic stiffness method

The dynamic stiffness method is extended to large amplitude free and forced vibrations of frames. When the steady state vibration is concerned, the time variable is replaced by the frequency parameter in the Fourier series sense and the governing partial differential equations are replaced by a set of ordinary differential equations in the spatial variables alone. The frequency-dependent shape functons are generated approximately for the spatial discretization. These shape functions are the exact solutions of a beam element subjected to mono-frequency excitation and constant axial force to minimize the spatial discretization errors. The system of ordinary differential equations is replaced by a system of non-linear algebraic equations with the Fourier coefficients of the nodal displacements as unknowns. The Fourier nodal coefficients are solved by the Newtonian algorithm in an incremental manner. When an approximate solution is available, an improved solution is obtained by solving a system of linear equations with the Fourier nodal increments as unknowns. The method is very suitable for parametric studies. When the excitation frequency is taken as a parameter, the free vibration response of various resonances can be obtained without actually computing the linear natural modes. For regular points along the response curves, the accuracy of the gradient matrix (Jacobian or tangential stiffness matrix) is secondary (cf. the modified Newtonian method). However, at the critical positions such as the turning points at resonances and the branching points at bifurcations, the gradient matrix becomes important. The minimum number of harmonic terms required is governed by the conditions of completeness and balanceability for predicting physically realistic response curves. The evaluations of the newly introduced mixed geometric matrices and their derivatives are given explicitly for the computation of the gradient matrix.