Analysis of Speckle Photographs by Subtracting Phase Functions of Digital Fourier Transforms

This paper presents a method for measuring displacements and strain in digital speckle photography that is an alternative to currently used correlation techniques. The method is analogous to heterodyne speckle photogrammetry wherein optical Fourier transforms were taken of individually recorded specklegrams and combined in a heterodyne interferometer where an electronic phase meter measured the phase differences between the two transforms. Here, digital photographs are recorded and Fourier transformed so that their phase functions can be subtracted and fitted to a linear function of the transform coordinates. The effect of different recording and processing parameters is investigated. It is found that incoherent speckles give better results than those formed by coherent laser light. In addition, image correlation is used to process an identical data set so that comparison of the two methods can be made.     

Dynamics of a truncated elastic cone

The problem of determining the nonstationary wave field of an elastic truncated cone with nonzero dead weight is formulated in terms of wave functions. The Laplace transform with respect to time and an integral transform with respect to time polar angle are used to reduce the problem to a one-dimensional vector problem in the transform domain. The transforms of the wave functions are expanded into series in inverse powers of the Laplace transform parameter, which makes it possible to study the wave process at the initial instants of interaction. A method is proposed to solve the problem for an elastic cone doubly truncated by spherical surfaces     

General solutions to a class of time fractional partial differential equations

A class of time fractional partial differential equations is considered, which in- cludes a time fractional diffusion equation, a time fractional reaction-diffusion equation, a time fractional advection-diffusion equation, and their corresponding integer-order partial differential equations. The fundamental solutions to the Cauchy problem in a whole-space domain and the signaling problem in a half-space domain are obtained by using Fourier- Laplace transforms and their inverse transforms. The appropriate structures of the Green functions are provided. On the other hand, the solutions in the form of a series to the initial and boundary value problems in a bounded-space domain are derived by the sine- Laplace or cosine-Laplace transforms. Two examples are presented to show applications of the present technique.     

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.     

Solution for the thermoelastic problem in vertically inhomogeneous media

The fundamental transient-thermoelastic problem with body forces and a heat source in vertically inhomogeneous media is investigated by a method presented in this paper. The basic equations in Fourier transforms and Laplace transform are obtained in the form of two sets of first order linear ordinary differential equations inz, Eq. (7). Furthermore, forN-layered media, the general solution in the transformed spaces of thej-th layer is given for fully connected interface between layers, Eq. (11). Finally, under general condition, a closed-form solution for the quasi-static transient displacements, stresses, temperature in the body can be obtained by the convolution theorems for the two integral transforms. In the final solution, the Green's functions can be expressed in terms of Hankel transforms of order zero and unity as well as inverse Laplace transform, and come out rather neatly. Comprehensive Institute of Geotechnical Investigation and Surveying, Ministry of Urban and Rural Construction and Environmental Protection     

Nonlinear Behaviour of Structures Using the Volterra Series—Signal Processing and Testing Methods

To extend modal analysis to nonlinear structures, and adopting the Volterra series as a mathematical framework, we present some new routes together with progress on signal processing. The closed form expressions of higher-order transfer function on the other hand would permit one to obtain eigenvalues of various orders and eigenvectors. Existing signal processing analyzers are initially devoted to linear systems. Programs treating input and output signals of systems are tailored for one-time (or frequency) variable functions. Nonlinear systems can indeed be analyzed by one-dimensional (direct or inverse) Fourier transforms. However, the experimenter rapidly discovers their limitation when dealing with coupling phenomena that require functions with many time (or frequency) variables. In this framework, multidimensional Fourier transforms are necessary.     

Methods of interconversion between linear viscoelastic material functions. Part II—an approximate analytical method

A new, very simple approximate interconversion method is proposed and verified by examples. This technique, employing the slope of the source function on logarithmic scales, is found to substantially enhance the accuracy compared to existing approximate methods. The new method is based on the characteristic mathematical properties of the narrow-band weight functions involved in the interrelationships between broad-band material functions. With the material functions represented locally by a power law, they are interrelated in terms of adjustment factors expressed through the local, log–log slope of the given (source) function. A number of existing approximate interconversion methods are also tested and compared with the new method. In Part I (Park and Schapery, 1998) , an efficient numerical interconversion method, based on a Prony (exponential ) series representation of both the source and target functions, was presented ; such a series representation is not needed here. The new method, when applied to the prediction of broad-band time-dependent functions from Laplace or Fourier transforms, is an approximate method of transform inversion that is applicable to functions which are not necessarily viscoelastic material functions.     

Fractional Trigonometry and the Spiral Functions

This paper develops two related fractional trigonometries based on the multi-valued fractional generalization of the exponential function, the R-function. The trigonometries contain the traditional trigonometric functions as proper subsets. Also developed are relationships between the R-function and the new fractional trigonometric functions. Laplace transforms are derived for the new functions and are used to generate solution sets for various classes of fractional differential equations. Because of the fractional character of the R-function, several new trigonometric functions are required to augment the traditional sine, cosine, etc. functions. Fractional generalizations of the Euler equation are derived. As a result of the fractional trigonometry a new set of phase plane functions, the Spiral functions, that contain the circular functions as a subset, is identified. These Spiral functions display many new symmetries.     

Response of periodic structures due to moving loadsRéponse de structures périodiques à des charges mobiles

A method is proposed to calculate the response of periodic structures subjected to moving loads. It is based on the Floquet decomposition which allows the restriction of the analysis for the overall system to a generic cell. The main contribution of the approach presented hereafter is that the response is directly deduced from transfer functions in the space-wavenumber domain calculated in an unbounded generic cell. Moreover, the equivalence of this new solution with the response of invariant structures obtained using Fourier transforms is established. To cite this article: H. Chebli et al., C. R. Mecanique 334 (2006).     

Fractional Trigonometry and the Spiral Functions

This paper develops two related fractional trigonometries based on the multi-valued fractional generalization of the exponential function, the R-function. The trigonometries contain the traditional trigonometric functions as proper subsets. Also developed are relationships between the R-function and the new fractional trigonometric functions. Laplace transforms are derived for the new functions and are used to generate solution sets for various classes of fractional differential equations. Because of the fractional character of the R-function, several new trigonometric functions are required to augment the traditional sine, cosine, etc. functions. Fractional generalizations of the Euler equation are derived. As a result of the fractional trigonometry a new set of phase plane functions, the Spiral functions, that contain the circular functions as a subset, is identified. These Spiral functions display many new symmetries.     

Unsteady axisymmetric deformation of an elastic thick-walled sphere under the action of volume forces

This paper considers a homogeneous isotropic elastic body bounded by concentric spheres and acted upon by axisymmetric unsteady volume forces. Displacement fields are determined using series expansions in Legendre and Gegenbauer polynomials, Laplace transforms in time, and integral representations with kernels in the form of Green’s functions. Explicit formulas for the Green’s functions are constructed that allow accurate determination of the originals. Examples of the calculations are presented.     

Axially symmetric thermal stress of an external circular crack under general thermal conditions

Summary The paper presents a solution for the linear thermoelastic problem of determining axisymmetric stress and displacement fields in an isotropic elastic solid of infinite extent weakened by an external circular crack under general mechanical loadings and general thermal conditions. The mechanical loadings and thermal conditions applied on the crack faces are axisymmetric, being non-symmetric about the crack plane. In similar lines of [7], equations of equilibrium of an elastic solid conducting heat have been solved using Hankel transforms and Abel operators of the first kind. Expressions for stress, displacement, temperature and heat flux functions are obtained in terms of Abel transforms of the first kind of the jumps of stress, displacement, temperature and heat flux at the crack plane. Two types of thermal conditions, that is, general surface temperatures and general heat flux on faces of the crack are considered. In both the cases, closed form solutions have been obtained for the unknown functions solving Abel type of integral equations. Explicit expressions for stresses, displacements, temperature fields, stress intensity factors have been obtained. Two special cases of thermal conditions in which: (i) crack faces are subjected to constant non-symmetric temperatures over a circular ring area, (ii) crack faces are subjected to constant non-symmetric heat flux over a circular ring area, have been considered. In some special cases, results have been compared with those from the literature.     

A new look at 2-D time-domain elastodynamic Green''s functions for general anisotropic solids

2-D time-domain elastodynamic displacement Green's functions for general anisotropic solids are obtained by a new method. This method is based on the use of a cosine transform with respect to time and exponential Fourier transforms with respect to both spatial coordinates. By use of a change of variables and the homogeneity and symmetry of the problem, the inverse transforms are reduced to an integral which can be evaluated by a simple use of redidue calculus. The solutions are expressed in terms of three wave fields. The field inside a wavefront corresponds to a complex root of a polynomial of order six with real coefficients. A simple relation between the spatial and time derivatives is found, and is used to reduce the corresponding stresses to a form that is directly applicable to the boundary element method. Numerical implementations are explained in some detail and are demonstrated by three examples.     

Static and Dynamic Green’s Functions in Peridynamics

We derive the static and dynamic Green’s functions for one-, two- and three-dimensional infinite domains within the formalism of peridynamics, making use of Fourier transforms and Laplace transforms. Noting that the one-dimensional and three-dimensional cases have been previously studied by other researchers, in this paper, we develop a method to obtain convergent solutions from the divergent integrals, so that the Green’s functions can be uniformly expressed as conventional solutions plus Dirac functions, and convergent nonlocal integrals. Thus, the Green’s functions for the two-dimensional domain are newly obtained, and those for the one and three dimensions are expressed in forms different from the previous expressions in the literature. We also prove that the peridynamic Green’s functions always degenerate into the corresponding classical counterparts of linear elasticity as the nonlocal length tends to zero. The static solutions for a single point load and the dynamic solutions for a time-dependent point load are analyzed. It is analytically shown that for static loading, the nonlocal effect is limited to the neighborhood of the loading point, and the displacement field far away from the loading point approaches the classical solution. For dynamic loading, due to peridynamic nonlinear dispersion relations, the propagation of waves given by the peridynamic solutions is dispersive. The Green’s functions may be used to solve other more complicated problems, and applied to systems that have long-range interactions between material points.     

Mixed Boundary Value Problem in a Non-homogeneous Medium Under Steady Distribution of Temperature

This paper is concerned with a mixed boundary value problem of a non-homogeneous medium under steady distribution of temperature whose elastic constants are exponential functions of y. By using Fourier cosine transforms the mixed boundary value problem of heat conduction is reduced to a Fredholm integral equation of the second kind. Then the elastic problem of the non-homogeneous semi-infinite half-plane under distribution of load over a plane face is solved. The influence of the non-homogeneity of the material on the thermal stress distribution is illustrated graphically.     

Dynamic Green’s Functions for an Anisotropic Multilayered Poroelastic Half-Space

The dynamic responses of an anisotropic multilayered poroelastic half-space to a point load or a fluid source are studied based on Stroh formalism and Fourier transforms. Taking the boundary conditions and the continuity of the materials into consideration, the three-dimensional Green’s functions of generalized concentrated forces (force and fluid source) applied at the free surface, interface and in the interior of a layer are derived in the Fourier transformed domain, respectively. The actual solutions in the frequency domain can further be acquired by inverting the Fourier transform. Finally, numerical examples are carried out to verify the presented theory and discuss the Green’s fields due to three cases of a concentrated force or a fluid source applied at three different locations for an anisotropic multilayered poroelastic half-space.

     

Unsteady flow of viscoelastic fluid between two cylinders using fractional Maxwell model

The unsteady flow of an incompressible fractional Maxwell fluid between two infinite coaxial cylinders is studied by means of integral transforms.The motion of the fluid is due to the inner cylinder that applies a time dependent torsional shear to the fluid.The exact solutions for velocity and shear stress are presented in series form in terms of some generalized functions.They can easily be particularized to give similar solutions for Maxwell and Newtonian fluids.Finally,the influence of pertinent parameters on the fluid motion,as well as a comparison between models,is highlighted by graphical illustrations.     

A certain boundary value problem and its applications

Summary In the present paper a linear second order partial differential system arising in mathematical physics is solved with the help of Laplace transforms, involving more general boundary conditions, in terms of Mathieu functions. Four physical problems are presented and it is indicated that their solutions can be deduced as a particular case.     

Electromechanical impact of a crack in a functionally graded piezoelectric medium

The Mode-I transient response of a functionally graded piezoelectric medium is solved for a through crack under the in-plane mechanical and electric impact. Integral transforms and dislocation density functions are employed to reduce the problem to singular integral equations. Numerical results display the effects of the loading combination parameter λ and the material parameter βa on the dynamic stress intensity factor and electric displacement intensity factor. The energy density factor criterion is applied to obtain the maximum of the minimum energy density factor and the direction of crack initiation.     

Interaction of plane elastic nonstationary waves with an elastic inclusion under complete adhesion

We solve the problem on the interaction of plane elastic nonstationary waves with a thin elastic strip-shaped inclusion. The inclusion is contained in an unbounded body (matrix) which in under conditions of plane strain. It is assumed that the condition of perfect adhesion between the inclusion and the matrix is satisfied. Because of the small thickness of the inclusion we assume that the bending and shear displacements at any inclusion point coincide with the displacements of the corresponding points of its midplane. The displacements on the midplane itself are found from the corresponding equations of the theory of plates. The statement of the boundary conditions for these equations takes into account the forces and moments acting on the inclusion edges from the matrix. The solution method is based on representing the displacements in the space of Laplace transforms as a discontinuous solution of the Lame’ equations for the plane strain with subsequent determining the transforms of the unknown jumps from integral equations. The passage to the original functions is performed numerically by methods based on replacement of the Mellin integral by the Fourier series. As a result, we obtain approximate formulas for calculating the stress intensity factors for the inclusion. These formulas are used to study the time dependence of the stress intensity factors and the influence of the inclusion rigidity on their values. We also study the possibility of treating inclusions of high rigidity as absolutely rigid inclusions.