Two-dimensional plastic flow of a Bingham solid

1947 ◽  
Vol 43 (3) ◽  
pp. 383-395 ◽  
Author(s):  
J. G. Oldroyd
Keyword(s):  
Stress Tensor ◽  
Plastic Flow ◽  
Equations Of State ◽  
Variable Viscosity ◽  
Frame Of Reference ◽  
Two Dimensional ◽  
Tensor Form ◽  
Yield Value ◽  
Image Position ◽  
Strain Tensors

The rheological equations of state for an isotropic Bingham solid may be written in tensor form:whereThe notation is explained fully in a previous paper (1). The frame of reference is essentially Eulerian. Primes denote deviatoric components of the stress, strain and rate of strain tensors, pik, εik and eik (for example, ). The dilatation is denoted by Δ. The rigidity and bulk moduli μ and κ are, if not constants, essentially scalar functions of the stress tensor. The variable viscosity η involves two constants, the yield value ϑ and the reciprocal mobility η1.

Rectilinear plastic flow of a Bingham solid

1948 ◽  
Vol 44 (2) ◽  
pp. 200-213 ◽  
Author(s):  
J. G. Oldroyd
Keyword(s):  
Plastic Flow ◽  
Conformal Transformation ◽  
Point Of View ◽  
Curvilinear Coordinates ◽  
Cartesian Coordinates ◽  
Yield Value ◽  
Natural Coordinate System ◽  
Image Position ◽  
Natural Coordinate ◽  
Special Case

It is apparent from the two previous papers of the same main title (1, 2) that velocity distributions in steady rectilinear plastic flow of a Bingham solid between moving boundaries are not easy to determine, even when the boundaries are of simple shape. In the familiar case of a Newtonian liquid (which can be regarded as a special case of a Bingham solid with zero yield value) the velocity ω is a harmonic function and can be obtained by a conformal transformation of the region of flow of the typeIn order to generalize and include materials of finite yield value, in which ω is not harmonic, one must first regard the Newtonian case from a slightly different point of view. The transformationmust be regarded as defining a change of coordinates, from Cartesian coordinates x, y to the natural curvilinear coordinates for a particular problem ω, W, one of which is the velocity. For a Bingham solid in general, it is shown in the present paper that natural coordinates ω, W exist, but are not obtainable by a conformal transformation from Cartesian coordinates, except in the limit when the yield value tends to zero. In practice it may be difficult to determine the natural coordinate system in a given problem, but when it is found the velocity distribution is automatically known.

Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

1990 ◽  
Vol 48 (2) ◽  
pp. 64-65
Author(s):  
L. Reimer ◽  
R. Oelgeklaus
Keyword(s):  
Fourier Transform ◽  
Energy Loss ◽  
Electron Energy ◽  
Rotational Symmetry ◽  
Mean Free Path ◽  
Single Scattering ◽  
Specimen Thickness ◽  
Two Dimensional ◽  
Image Position ◽  
Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

Gluing Solutions

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Stress Tensor ◽  
Compact Support ◽  
Smooth Solution ◽  
Continuous Solution ◽  
Total Momentum ◽  
Frame Of Reference ◽  
Galilean Transformation ◽  
Hölder Continuous ◽  
Pressure Form ◽  
Original Frame

This chapter deals with the gluing of solutions and the relevant theorem (Theorem 12.1), which states the condition for a Hölder continuous solution to exist. By taking a Galilean transformation if necessary, the solution can be assumed to have zero total momentum. The cut off velocity and pressure form a smooth solution to the Euler-Reynolds equations with compact support when coupled to a smooth stress tensor. The proof of Theorem (12.1) proceeds by iterating Lemma (10.1) just as in the proof of Theorem (10.1). Applying another Galilean transformation to return to the original frame of reference, the theorem is obtained.

scholarly journals On local and integrated stress-tensor commutators

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Mert Besken ◽  
Jan de Boer ◽  
Grégoire Mathys
Keyword(s):  
Stress Tensor ◽  
Free Field ◽  
Virasoro Algebra ◽  
Light Sheet ◽  
Operator Product ◽  
Local Form ◽  
Two Dimensional ◽  
Conformal Embedding ◽  
General Aspects ◽  
The Right

Abstract We discuss some general aspects of commutators of local operators in Lorentzian CFTs, which can be obtained from a suitable analytic continuation of the Euclidean operator product expansion (OPE). Commutators only make sense as distributions, and care has to be taken to extract the right distribution from the OPE. We provide explicit computations in two and four-dimensional CFTs, focusing mainly on commutators of components of the stress-tensor. We rederive several familiar results, such as the canonical commutation relations of free field theory, the local form of the Poincaré algebra, and the Virasoro algebra of two-dimensional CFT. We then consider commutators of light-ray operators built from the stress-tensor. Using simplifying features of the light sheet limit in four-dimensional CFT we provide a direct computation of the BMS algebra formed by a specific set of light-ray operators in theories with no light scalar conformal primaries. In four-dimensional CFT we define a new infinite set of light-ray operators constructed from the stress-tensor, which all have well-defined matrix elements. These are a direct generalization of the two-dimensional Virasoro light-ray operators that are obtained from a conformal embedding of Minkowski space in the Lorentzian cylinder. They obey Hermiticity conditions similar to their two-dimensional analogues, and also share the property that a semi-infinite subset annihilates the vacuum.

scholarly journals Existence and uniqueness in the theory of bending of elastic plates

1986 ◽  
Vol 29 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Christian Constanda
Keyword(s):  
Existence And Uniqueness ◽  
Integral Equation Method ◽  
Fundamental Solutions ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Equilibrium Equations ◽  
Dimensional Theory ◽  
Neumann Problems ◽  
Image Position

Kirchhoff's kinematic hypothesis that leads to an approximate two-dimensional theory of bending of elastic plates consists in assuming that the displacements have the form [1]In general, the Dirichlet and Neumann problems for the equilibrium equations obtained on the basis of (1.1) cannot be solved by the boundary integral equation method both inside and outside a bounded domain because the corresponding matrix of fundamental solutions does not vanish at infinity [2]. However, as we show in this paper, the method is still applicable if the asymptotic behaviour of the solution is suitably restricted.

Piezotronics and piezo-phototronics in two-dimensional materials

MRS Bulletin ◽  
2018 ◽  
Vol 43 (12) ◽  
pp. 959-964 ◽  
Author(s):  
Yudong Liu ◽  
Erlin Tresna Nurlianti Wahyudin ◽  
Jr-Hau He ◽  
Junyi Zhai
Keyword(s):  
Two Dimensional ◽  
Two Dimensional Materials ◽  
Image Position

Abstract

Weak solutions for generalized stationary Oldroyd-B fluid with a diffusive stress

2016 ◽  
Vol 23 (4) ◽  
pp. 469-475
Author(s):  
Hafedh Bousbih ◽  
Mohamed Majdoub
Keyword(s):  
Stress Tensor ◽  
Weak Solutions ◽  
Fluid Model ◽  
Three Dimensional ◽  
Shear Thickening ◽  
Bounded Domains ◽  
Two Dimensional ◽  
Existence Of Weak Solutions ◽  
Elastic Part ◽  
Viscosity Function

AbstractThis paper focuses on the analysis of the stationary case of incompressible viscoelastic generalized Oldroyd-B fluids derived in [2] by Bejaoui and Majdoub. The studied model is different from the classical Oldroyd-B fluid model in having a viscosity function which is shear-rate depending, and a diffusive stress added to the equation of the elastic part of the stress tensor. Under some conditions on the viscosity stress tensor and for a large class of models, we prove the existence of weak solutions in both two-dimensional and three-dimensional bounded domains for shear-thickening flows.

The number of limit cycles of certain polynomial differential equations

1984 ◽  
Vol 98 (3-4) ◽  
pp. 215-239 ◽  
Author(s):  
T. R. Blows ◽  
N. G. Lloyd
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Equations ◽  
Image Position ◽  
Cubic Systems

SynopsisTwo-dimensional differential systemsare considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

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