Use Of Complex Potentials Research Articles

Математичне моделювання напруженого стану в’язкопружної напiвплощини з включеннями

The application of the method of boundary integral equations is considered for studying the stress state of flat viscoelastic bodies with inclusions. The method is based on the use of complex potentials and the apparatus of generalized functions. An analytical solution of the problem is obtained for a half-plane with inclusions of arbitrary shape. For a numerical study of the change in the stress state depending on the time and geometry of the inclusions, a discrete analogue of the system of boundary-time integral equations has been developed.

Effect of the yield stresses in uniaxial tension and pure shear on the size of the plastic zone near a crack

In this paper, we make use of complex potentials for the problem of an elliptic hole subject to far field uniaxial tension to obtain the elastic stress distribution in the vicinity of a crack. Unlike the classical Westergaard stress distribution, the new stress field obtained is not axisymmetric in the crack plane. Using this new elastic stress field, it is possible to assess the effect of the particularities of yielding on the extent of the plastic zone near a crack. Analysis is done considering yielding governed by von Mises, Tresca, and Drucker yield criterion, respectively. It is demonstrated that the ratio between the yield stresses in uniaxial tension and pure shear of a material has a great influence on the size of the plastic zone around the crack. Specifically, the larger this ratio the larger is the plastic zone. Finite-element calculations confirm the theoretical predictions. Moreover, we derive new analytic relations between the length of the plastic zone, measured from the crack tip in the crack plane, and the external applied load for the case when yielding is governed by the von Mises and Tresca criterion, respectively.

Problem of Electroviscoelasticity for Multiply Connected Plates

We propose a method for the solution of the problem of electroviscoelasticity for multiply connected plates. By the method of small parameter, the problem is reduced to the recurrent sequence of problems of electroviscoelasticity that can be solved with the use of complex potentials. We propose a procedure for the determination, according to the complex potentials, of the approximate values of the principal characteristics of electroelastic state (stresses and the components of the vectors of induction and electric-field strength) at any time after the application of loading. As an example, we present the solution of the problem for a plate with elliptic hole. The results of numerical investigations reveal a strong influence of time on the values of the principal characteristics of the electroviscoelastic state.

Definition of Locked-up Stresses around a Rectilinear Welding Seam

In article one of the ways of modelling of the locked-up stresses caused by imposing of a welding seam in a plate is described. These stresses are caused by residual deformations of welding seams. In the settlement scheme the welding seam is presented in the form of an element by the thickness equal to the thickness of a plate, inserted in it with a longitudinal and cross-section tightness. The field of locked-up stresses around the welding seam experiences longitudinal and cross-section shrinkage under cooling and it is defined in work with the use of complex potentials of the theory of elasticity.

Effect of confinement on the deformation of microfluidic drops.

We study the deformation of drops squeezed between the floor and ceiling of a microchannel and subjected to a hyperbolic flow. We observe that the maximum deformation of drops depends on both the drop size and the rate of strain of the external flow and can be described with power laws with exponents 2.59±0.28 and 0.91±0.05, respectively. We develop a theoretical model to describe the deformation of squeezed drops based on the Darcy approximation for shallow geometries and the use of complex potentials. The model describes the steady-state deformation of the drops as a function of a nondimensional parameter Caδ2, where Ca is the capillary number (proportional to the strain rate and the drop size) and δ is a confinement parameter equal to the drop size divided by the channel height. For small deformations, the theoretical model predicts a linear relationship between the deformation of drops and this parameter, in good agreement with the experimental observations.

A review of the decoherent histories approach to the arrival time problem in quantum theory

We review recent progress in understanding the arrival time problem in quantum mechanics, from the point of view of the decoherent histories approach to quantum theory. We begin by discussing the arrival time problem, focussing in particular on the role of the probability current in the expected classical solution. After a brief introduction to decoherent histories we review the use of complex potentials in the construction of appropriate class operators. We then discuss the arrival time problem for a particle coupled to an environment, and review how the arrival time probability can be expressed in terms of a POVM in this case. We turn finally to the question of decoherence of the corresponding histories, and we show that this can be achieved for simple states in the case of a free particle, and for general states for a particle coupled to an environment.

Modeling of open quantum dots and wave billiards using imaginary potentials for the source and the sink

A heuristic model for particle states and current flow in open ballistic two-dimensional (2D) quantum dots/wave billiards is proposed. The model makes use of complex potentials first introduced in phenomenological nuclear inelastic scattering theory (the optical model). Here we assume that external input and output leads connecting the system to the source and the drain regions may be represented by complex potentials. In this way, a current may be set up between the two ‘pseudo-leads’. Probability densities and current flows for an open quantum dot are analyzed here numerically and the results are compared with the microwave measurements used to emulate the system. The model is of conceptual as well as practical interest. In addition to quantum billiards, it may be used as a tool per se to analyze transport in classical wave analogues, such as microwave resonators, acoustic resonators, effects of leakage on such systems, etc.

Stress-strain state of an anisotropic plate with curved cracks and thin rigid inclusions

A mixed problem of linear elasticity for an infinite anisotropic plate with cuts and thin undeformable inclusions located along arbitrary open smooth curves is solved with the use of complex potentials. Special representations of the solutions are constructed and a governing system of singular integral equations is obtained. A numerical algorithm for determining the stress-strain state of the plate, including the stress-intensity factors at the tips of cuts and rigid inclusions, is proposed. Calculation results are given.

Stress-strain state of an anisotropic plate with curvilinear cracks and thin rigid inclusions

A mixed problem of linear elasticity for an infinite anisotropic plate with cuts and thin undeformable inclusions located along arbitrary open smooth curves is solved with the use of complex potentials. Special representations of the solutions are constructed and a governing system of singular integral equations is obtained. A numerical algorithm for determining the stress-strain state of the plate, including the stress-intensity factors at the tips of cuts and rigid inclusions, is proposed. Calculation results are given.

Stability, resonance and stress concentration in prestressed piezoelectric crystals containing a crack

I start with the theory of small deformations and fields superposed on large static deformations and fields, given by Eringen and Maugin ( Electrodynamics of Continua, Vol. 1, Foundations and Solid Media. Springer, New York (1990); Section 7.14). The considered body is an elastic dielectric and the quasi-static approximation is used. In the paper are given conditions in which the stability of the equilibrium state of a prestressed piezoelectric crystal is assured. Also established are conditions in which non-uniqueness, internal instability or resonance can occur. Finally, results are presented concerning the asymptotic behavior of the incremental fields near the tips of a crack in a prestressed piezoelectric crystal. It is proven that the stress singularity is the same as in an initial stress free body. The analysis is based on the use of complex potentials in order to represent the incremental fields.

Determination of the stress state of a circular disk with a crack

With the use of complex potentials from the solution of a contact problem for slits in a multiply-connected region, a solution is found for a problem of the theory of elasticity for an isotropic circular disk with an arbitrary radial crack. The case of an edge crack is among the cases for which a solution is found. The types of loading examined are uniform tension on an outside edge, internal pressure on the edges of cracks, and concentrated forces at arbitrary points of a disk. The unknown coefficients in the complex potentials are found from the boundary conditions on the outside edge of the disk by the series method, the colocation method, or the least squares method. Detailed numerical studies are conducted to determine the effect of the geometric characteristics and the points of application of concentrated forces on the character of the stress distribution and the stress intensity factor. A comparative characteristic of the methods used to find the coefficients is presented.

Study of the p(p, n) Δ++ reaction in nuclei

The p(p, n) Δ ++ reaction has been studied theoretically from threshold to 5.5 GeV/ c beam momenta. A formalism has been written which includes the effect of various channels in the pp scattering on the pp→ n Δ ++ channel through the use of complex potentials in the continuum states. The scattering wave functions for them are written directly in terms of the scattering amplitudes. Calculations have been done for various differential and total cross-sections for the pp → n Δ ++ reaction. The calculated results using the one-pion-exchange transition potential with Λ π ≈ 1 GeV/ c agree remarkably well with all the available experimental data in the energy region considered. The “distortion” is found to essentially weaken the transition potential in the large momentum domain. The effect of rho-exchange is also calculated.

The problem of a Dugdale crack in a plate under normal bending

The plane problem of a Dugdale crack in a plate under remote normal bending is investigated. By use of complex potentials an exact closed form solution is given and the corresponding crack deformation is discussed.

Absorption cross sections and the use of complex potentials in coupled-channels models.

We give a simple and transparent derivation of a result obtained recently by Kim, Udagawa, and Tamura for the loss of flux occurring in the coupled-channels scattering problem when a complex potential is used. The loss of flux, or absorption cross section, consists of three components: absorption from the elastic channel, absorption from the nonelastic channels, and absorption occurring during the elastic-nonelastic transitions. Some discussion is given of these results and their application to fusion reactions.

Analysis of forces in three phase gas insulated cables

A three phase gas insulated cable, like these used in electrical power transmission or SF6 substations, has been analysed for steady state symmetrical currents. All conductors have zero thickness and are non ferrous, i.e. μ=μ0. Magnetic fields and forces have been calculated by using two dimensional complex potentials. It may be of practical importance that the forces found are significantly different than these published elsewhere in which simplified models with filamentary phase conductors were used. It is also shown how effective is in such calculations the use of complex potential.

Use of complex potentials in solving boundary-value problems of the nonlinear theory of hereditary elasticity

Use of complex potentials in solving boundary-value problems of the nonlinear theory of hereditary elasticity

The use of complex potentials in nuclear physics

Derivations of optical potentials are given for several simple, physically interesting reaction models, to illustrate the content of more general derivations. The use of complex coupling potentials in direct-reaction calculations is seen to be natural. A time-dependent analysis is also given.

A model for the study of the interference of the direct and compound-nucleus reaction mechanisms

A model is proposed for the investigation of the interference of direct and compound-nucleus reaction mechanisms in the inelastic scattering of nucleons. The continuum eigenfunction is considered to consist of a sum of two types of components: Product functions represent the nonlocalized open-channel or scattering components. Closed-channel or compound-nucleus components are represented by localized functions. The direct reaction mechanism is characterized by the set of matrix elements of the nucleon-nucleus interaction that couple two open-channel components. The compound-nucleus reaction mechanism is characterized by the matrix elements that couple open- and closed-channel components. The formal development of the above model is essentially a simple application of Feshbach's unified theory of nuclear reactions. However an additional “surface-transition model” is employed, in which the coupling of the various open- and closed-channel components is restricted to occur at one value of the nucleon-nucleus separation. Within this model, the effective Schrödinger equation is completely solvable and an analytic expression for the collision matrix is derived. This expression indicates clearly the dynamical interference between the reaction mechanisms; namely, it indicates the nonlinear manner in which the two mechanisms contribute to the collision matrix. The collision matrix is further investigated in two situations. The case of an isolated compound-nucleus state is considered and the departures from the form of a direct term plus Breit-Wigner term are examined. The case of a dense set of overlapping compound-nucleus states is also treated. Here the energy-averaged collision function is compared to that which would arise in the absence of any compound mechanism. Insight is gained into the relation between such energy averages and the use of complex potentials in reaction theory. Implications of the dynamical interference for studies of statistical fluctuations is noted.