Free vibration analysis of Euler–Bernoulli curved beams using two-phase nonlocal integral models

Eringen’s nonlocal elastic model has been widely applied to address the size-dependent response of micro-/nanostructures, which is observed in experimental tests and molecular dynamics simulation. However, several recent studies have pointed out that some inconsistent results appear while applying it in the analysis of bounded structures, which indicates that it is necessary to adopt other suitable models. In this work, both the well-posed strain-driven and stress-driven two-phase local/nonlocal integral models are used to study the size effect in the free vibration of Euler–Bernoulli curved beams. The governing equations of motion and the associated boundary conditions are derived on the basis of Hamilton’s principle. The two-phase nonlocal integral relation is transformed into an equivalent differential law with two constitutive boundary conditions. Using the generalized differential quadrature method, the governing equation in terms of displacements is solved numerically. The vibration frequencies of the beam under different boundary conditions are obtained and validated by comparing with those existing results. For all boundary conditions, the nonlocal related parameters of the two types of two-phase nonlocal strategies show consistent softening and stiffening effects on vibration response, respectively. Moreover, the effect of the curvature radius of the beam is also investigated.

Covid Conversations 1: Peter Sellars

In this profoundly dialogical exchange, Peter Sellars, theatre director, researcher, and teacher, and Maria Shevtsova open out a whole array of questions on the integral relation between politics and the theatre in its multiple manifestations. These questions not only concern the damages inflicted by the present Covid-19 pandemic but also those developed by the neoliberal economics and politics of the past forty years and more. In Sellars’s view, neoliberalism has been the hotbed of social injustices, inequities, market and other forms of current enslavement, migrations, refugee and related precarities, and the havoc of the world climate in which the plight of humanity and that of the planet are indelibly interconnected. His and Shevtsova’s discussion links such vital concerns with his theatre practice, which ranges from his engagement with local communities and indigenous peoples – he details some of his work with the collective, community organization of two Los Angeles Festivals of the early 1990s – to the various forms of his music theatre in which he collaborates, in institutional structures, with highly proficient musicians, singers and dancers. The focus chosen here from his music theatre is The Indian Queen (2013), which Sellars dramaturgically invents using pieces by Henry Purcell combined with prose fragments by Nicaraguan novelist Rosario Aguilar. Peter Sellars is an internationally renowned theatre director among whose more recent productions is Mozart’s Idomeneo, premiered at the Salzburg Festival in 2019. Maria Shevtsova, Professor of Drama and Theatre Arts at Goldsmiths, University of London, is editor of New Theatre Quarterly. This conversation took place on 16 August 2020, was transcribed from the recording by Kunsang Kelden, and was edited by Maria Shevtsova.

An Inverse Model for Two-dimensional Reflectors With Scattering

We present a novel approach of modelling surface light scattering in the context of freeform optical design. Using energy conservation, we derive an integral relation between the scattered and specular distributions. This integral relation reduces to a convolution integral in the case of isotropic scattering in the plane of incidence for cylindrically and rotationally symmetric problems.

Random walk on ellipsoids method for solving elliptic and parabolic equations

A Random Walk on Ellipsoids (RWE) algorithm is developed for solving a general class of elliptic equations involving second- and zero-order derivatives. Starting with elliptic equations with constant coefficients, we derive an integral equation which relates the solution in the center of an ellipsoid with the integral of the solution over an ellipsoid defined by the structure of the coefficients of the original differential equation. This integral relation is extended to parabolic equations where a first passage time distribution and survival probability are given in explicit forms. We suggest an efficient simulation method which implements the RWE algorithm by introducing a notion of a separation sphere. We prove that the logarithmic behavior of the mean number of steps for the RWS method remains true for the RWE algorithm. Finally we show how the developed RWE algorithm can be applied to solve elliptic and parabolic equations with variable coefficients. A series of supporting computer simulations are given.

A non-field analytical method for solving problems in aero-acoustics

In 2000, a non-field analytical method for solving various problems of energy and information transport has been developed by Kulish and Lage. Based on the Laplace transform technique, this elegant method yields closed-form solutions written in the form of integral equations, which relate local values of an intensive properties such as, for instance, velocity, mass concentration, temperature with the corresponding derivative, that is, shear stress, mass flux, temperature gradient. Over the past 20 years, applied to solving numerous problems of energy and information transport, the method—now known as the method of Kulish—proved to be very efficient. In this paper—for the first time—the method is applied to problems in aeroacoustic. As a result, an integral relation between the local values of the acoustic pressure and the corresponding velocity perturbation has been derived. The said relation is valid for axisymmetric cases of planar, cylindrical and spherical geometries.

Transverse moments of TMD parton densities and ultraviolet divergences

I review some open questions relating to the large transverse momentum divergences in transverse moments of transverse momentum dependent (TMD) parton correlation functions. I also explain, in an abbreviated and summarized form, recent work that shows that the resulting violations of a commonly used integral relation are not perturbatively suppressed. I argue that this implies a need for more precise definitions for the correlation functions used to describe transverse moments.

A conformal dispersion relation: correlations from absorption

We introduce the analog of Kramers-Kronig dispersion relations for correlators of four scalar operators in an arbitrary conformal field theory. The correlator is expressed as an integral over its “absorptive part”, defined as a double discontinuity, times a theory-independent kernel which we compute explicitly. The kernel is found by resumming the data obtained by the Lorentzian inversion formula. For scalars of equal scaling dimensions, it is a remarkably simple function (elliptic integral function) of two pairs of cross-ratios. We perform various checks of the dispersion relation (generalized free fields, holographic theories at tree-level, 3D Ising model), and get perfect matching. Finally, we derive an integral relation that relates the “inverted” conformal block with the ordinary conformal block.

Mathematical modeling of well operation in the case of two-dimensional filtration in an anisotropic heterogeneous layer

Поставлена плоская (двумерная) задача о математическом моделировании работы скважины в анизотропном неоднородном пласте грунта с раздельной анизотропией и неоднородностью, когда контур питания произвольный. Рассматривается совершенная скважина, когда она полностью вскрывает пласт своей рабочей частью (фильтром). Проницаемость грунта характеризуется тензором второго ранга, компоненты которого моделируются степенной функцией координат. Гомеоморфным аффинным преобразованием координат эта задача приводится к каноническому виду, что значительно упрощает ее исследование. Получено в конечном виде аналитическое решение задачи о дебите скважины с конкретным эллиптическим контуром питания, а также в случае, когда контур питания удален в бесконечность. В случае произвольного гладкого контура питания задача о дебите редуцирована к системе сингулярного интегрального уравнения и интегрального соотношения, которая решена численно методом дискретных особенностей. Исследовано влияние на дебит анизотропии, неоднородности пласта и формы контура питания. A flat (two-dimensional) problem has been posed on the mathematical modeling of well in an anisotropic inhomogeneous reservoir of soil with separate anisotropy and heterogeneity when the power contour is arbitrary. The considered well completely opens the formation with its working part (filter). Such a well is called perfect. The permeability of the soil is characterized by a second-rank tensor whose components are modeled by a power function of the coordinates. With a homeomorphic affine transformation of coordinates, this problem is reduced to a canonical form which greatly simplifies its study. An analytical solution of the problem of well production with an elliptical power contour is obtained in the final form as well as in the case when the power contour is removed to infinity. In the general case, the problem is reduced to a system of integral equations and the integral relation. The results were obtained in the general case using the discrete singularities method. The influence on the flow rate of anisotropy, heterogeneity of the reservoir and the shape of the power contour was studied.