Double Line Integral Research Articles

FE SOLUTIONS OF NEAR-TIP FIELD FOR MODE-I CRACKS WITH RESIDUAL SURFACE TENSION

This paper presents finite element solutions of a near-tip elastic field of a straight, nano-scale crack in a two-dimensional, linear elastic, whole plane subjected to mode-I crack-face loads. The mathematical model is formulated using a continuum-based theory of classical linear elasticity together with Gurtin-Murdoch surface model to capture the role of the residual surface tension present on the crack-face material layer. The formulation finally yields a second-order, integrodifferential equation governing the crack opening displacement. A weighted residual technique along with the regularization procedure is applied to establish a weakly singular weak-form equation with the involved kernel of O(ln )r . Galerkin strategy and the finite element procedure are then employed to discretize the weak-form equation. Various types of element shape functions, generated by standard C0 -elements, standard C1 -elements, and special elements with built-in crack-tip functions, are considered. A proper quadrature rule is selected to efficiently and accurately evaluate both nearly and weakly singular double line integrals over pairs of elements resulting from the discretization and the solution of a dense system of linear algebraic equations is obtained using an efficient indirect solver. The rate of convergence of finite element solutions is fully investigated and such information is then used to conclude the influence of the residual surface tension on the behavior of the near-tip field.

The correlated linking numbers of a Brownian loop with two arbitrary curves

The standard kinetic path integral for all spatially closed Brownian paths (loops) of duration t weighted by the product mn is evaluated, where m and n are the linking numbers of the Brownian loop with two arbitrary curves in 3D space. The path integral thus indicates the extent to which these two linking numbers are correlated, ranging from the value zero for far apart curves when it is unlikely that the Brownian loop links with both, to ±infinity for nearly coincident curves. The result takes a form that loosely resembles that for the mutual inductance of two current carrying circuits in magnetostatics, a double line integral, but is also dependent on a single extra parameter, the duration t of the path. The result for the equivalent 2D problem was given previously (Hannay ).

Theory of dislocation loops in multilayered anisotropic solids with magneto-electro-elastic couplings

This article presents a general theory on dislocation loops in multilayered anisotropic solids with magneto-electro-elastic couplings. First, a novel extended point-force Green's function is developed that not only guarantees the numerical stability but also admits a direct interpretation of the image method in multilayered solids. Then, based on the new Green's function, the classical theory of dislocation loops in pure-elastic infinite solids is systematically generalized by accounting for both the surface/interface conditions and the magneto-electro-elastic couplings. Two major contributions are made in the generalization: (i) the extended displacement and stress fields induced by an arbitrary dislocation loop are derived and expressed concisely as a simple line integral along the closed dislocation curve; (ii) the interaction energy between two arbitrary dislocation loops is also derived and expressed compactly as double line integrals along the two closed dislocation curves. Based on the new line-integral expressions for multilayered solids, special configurations such as straight dislocation segments and elliptic dislocation loops are analytically studied in detail; and more strikingly, elegant line-integral expressions for dislocation loops in bi-materials or in a semi-infinite solid are obtained by simple reduction. All the expressions derived in this article are compared with available solutions in existing literature and the correctness of them is fully verified. As novel applications, the present theory is utilized to calculate the self-energy of interfacial elliptic dislocation loops efficiently, which clearly shows that both the surface/interface conditions and the multi-field couplings have a considerable influence on the self-energy of interfacial dislocation loops. The line-integral based formulae developed here for dislocation-loop problems will not only enrich the theory of magneto-electro-elasticity, but also have potential applications in three-dimensional discrete dislocation dynamics simulations associated with multilayered thin-film structures.

A non-singular theory of dislocations in anisotropic crystals

We develop a non-singular theory of three-dimensional dislocation loops in a particular version of Mindlin's anisotropic gradient elasticity with up to six length scale parameters. The theory is systematically developed as a generalization of the classical anisotropic theory in the framework of linearized incompatible elasticity. The non-singular version of all key equations of anisotropic dislocation theory are derived as line integrals, including the Burgers displacement equation with isolated solid angle, the Peach-Koehler stress equation, the Mura-Willis equation for the elastic distortion, and the Peach-Koehler force. The expression for the interaction energy between two dislocation loops as a double line integral is obtained directly, without the use of a stress function. It is shown that all the elastic fields are non-singular, and that they converge to their classical counterparts a few characteristic lengths away from the dislocation core. In practice, the non-singular fields can be obtained from the classical ones by replacing the classical (singular) anisotropic Green's tensor with the non-singular anisotropic Green's tensor derived by Lazar and Po (2015b). The elastic solution is valid for arbitrary anisotropic media. In addition to the classical anisotropic elastic constants, the non-singular Green's tensor depends on a second order symmetric tensor of length scale parameters modeling a weak non-locality, whose structure depends on the specific class of crystal symmetry. The anisotropic Helmholtz operator defined by such tensor admits a Green's function which is used as the spreading function for the Burgers vector density. As a consequence, the Burgers vector density spreads differently in different crystal structures. Two methods are proposed to determine the tensor of length scale parameters, based on independent atomistic calculations of classical and gradient elastic constants. The anisotropic non-singular theory is shown to be in good agreement with molecular statics without fitting parameters, and unlike its singular counterpart, the sign of stress components does not show reversal as the core is approached. Compared to the isotropic solution, the difference in the energy density per unit length between edge and screw dislocations is more pronounced.

Radiant fluxes from various off-axis point sources incident on a circular disk

A general multidomain integral formula is presented for calculating fluxes of radiation striking a circular disk from various off-axis point source types embedded in an attenuating or nonattenuating medium. This formula is expressed by double line integrals of radiant intensity and sine functions with respect to the polar and horizontal angles determining the angular distribution of the emitted radiation. The formula reduces to single line integral expressions when radiation does not depend on the horizontal angle and is directly applicable for calculating fluxes of revolutional symmetry around the optical axis of the source perpendicular to the disk. The applicability of this reduced formula is tested by computing radiant fluxes from Lambertian and Gaussian point sources using a simple numerical procedure for single integrals. The computed data are illustrated graphically, tabulated, and validated using OSLO. Finally, the accuracy, similarity, and applicability of the results provided by the integral formula and the OSLO program are analyzed. Numerical results have shown the effectiveness of the presented formulas for calculating radiant fluxes from various on- and off-axis point sources passing through a nonattenuating or attenuating homogeneous isotropic media and incident on a circular disk perpendicular to optical axes of these sources. Practical applications of these formulas include optical sensing and metrology, optical coupling, fiber optic for biomedical measurements, and creative lighting design.

Calculation of the solid angle subtended by a cylinder at a point

An analytical formula for calculating the solid angle subtended by a cylinder at a point has been derived from the general solid angle equation. The formula is expressed by double line integrals and by single integrals of simple elementary functions. These functions were then integrated and the formula was represented by products of some elementary functions and the incomplete Lagrange–Jacobi elliptic integrals of the third kind. The final formula was used to calculate representative values of solid angles and compare them with literature data.

Decoherence and Vacuum Fluctuationsa

The interference pattern of coherent electrons is effected by coupling to the quantized electromagnetic field. The amplitudes of the interference maxima are changed by a factor which depends upon a double line integral of the photon two-point function around the closed path of the electrons. The interference pattern is sensitive to shifts in the vacuum fluctuations in regions from which the electrons are excluded. Thus this effect combines aspects of both the Casimir and the Aharonov-Bohm effects. The coupling to the quantized electromagnetic field tends to decrease the amplitude of the interference oscillations, and hence is a form of decoherence. The contributions due to photon emission and to vacuum fluctuations may be separately identified. It is to be expected that photon emission leads to decoherence, as it can reveal which path an electron takes. It is less obvious that vacuum fluctuations also can cause decoherence. What is directly observable is a shift in the fluctuations due, for example, to the presence of a conducting plate. In the case of electrons moving parallel to conducting boundaries, the dominant decohering influence is that of the vacuum fluctuations. The shift in the interference amplitudes can be of the order of a few percent, so experimental verification of this effect may be possible. The possibility of using this effect to probe the interior of matter, e.g., to determine the electrical conductivity of a rod by means of electrons encircling it is discussed. (Presented at the Conference on Fundamental Problems in Quantum Theory, University of Maryland, Baltimore County, June 18-22, 1994.)

On the Estimation of the Generalized Force due to Random Pressure and on Necessary Modes

Under certain conditions of major practical interest, the double surface integral giving the generalized force due to a random pressure field for a mode of vibration can be reduced to the product of two double line integrals. For sinusoidal mode shapes, the double line integral further reduces to a single integral, easily evaluated for any correlation function. It is also pointed out that for cylindrical structures, lacking a boundary condition of zero deflection for the circumferential modes, the modes cos (nθ) and sin (nθ) must both be considered.

A flux theorem

In the literature1)2) some formulae are known which give the total flux incident on an area originating from a surface source. These formulae are derived again by aid of a flux theorem which transforms the original double surface integral into a double line integral over the boundaries of both areas.