scholarly journals IX. Plane stress and plane strain in bipolar co-ordinates

1921 ◽  
Vol 221 (582-593) ◽  
pp. 265-293 ◽  
Keyword(s):  
Stress Function ◽  
Three Dimensional ◽  
Linear Partial Differential Equation ◽  
Elastic Solid ◽  
Great Difficulty ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Applied Forces ◽  
Special Solutions ◽  
Single Stress

The problem of the equilibrium of an elastic solid under given applied forces is one of great difficulty and one which has attracted the attention of most of the great applied Mathematicians since the time of Euler. Unlike the kindred problems of hydrodynamics and electrostatics, it seems to be a branch of mathematical physics in which knowledge comes by the patient accumulation of special solutions rather than by the establishment of great general propositions. Nevertheless, the many and varied applications of this subject to practical affairs make it very desirable that these special solutions should be investigated, not only because of their intrinsic importance but also for the light which they often throw on the general problem. One of the most powerful methods of the mathematical physicist in the face of recalcitrant differential equations is to simplify his problem by reducing it to two dimensions. This simplification can only imperfectly be reproduced in the Nature of our three-dimensional world, but, in default of more general methods, it provides an invaluable weapon. It was shown by Airy that in the two-dimensional case the, stresses may be derived by partial differentiations from a single stress function, and it was shown later that, in the absence of body forces, this stress function satisfies the linear partial differential equation of the fourth order ∇ 4 X = 0, where ∇ 4 = ∇ 2 . ∇ 2 , and ∇ 2 is the two-dimensional Laplacian ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 .

scholarly journals The Two Dimensional Representation of Three Dimensional Interferometer Measurements

1979 ◽  
Vol 49 ◽  
pp. 19-26
Author(s):  
R.H. Frater
Keyword(s):  
Three Dimensional ◽  
General Purpose ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Dimensional Representation ◽  
General Purpose Computer ◽  
Efficient Processing ◽  
Purpose Computer

SummaryA convolution technique for the reduction of three dimensional interferometer measurements to two dimensions is described. With the addition of relatively simple hardware to a general purpose computer the technique allows fast, efficient processing of three dimensional data.

Time-based two dimensional modelling of NATM tunnelling

2002 ◽  
Vol 39 (3) ◽  
pp. 710-724 ◽  
Author(s):  
J H Shin ◽  
D M Potts
Keyword(s):  
Three Dimensional ◽  
Two Dimensions ◽  
Time Dependent ◽  
Soil Conditions ◽  
Measuring Instruments ◽  
Two Dimensional ◽  
Time Dependent Behaviour ◽  
Decomposed Granite ◽  
Granite Soil ◽  
Dependent Behaviour

A two dimensional model is commonly employed in practice for the analysis of tunnelling. Such analyses are computationally cheap and are useful for assessing the sensitivity of the problem to the construction method, studying the influence of varying soil conditions, and (or) finding appropriate locations for placing measuring instruments. However, simulating the three dimensional nature of tunnelling in two dimensions requires certain simplifications, including the use of empirical parameters to represent the construction sequence. In many cases the choice of parameter values are arbitrary and often not fully explained. In addition, the modelling methods are often only applicable for undrained or fully drained soil conditions where no time-dependent behaviour is involved during tunnel construction. In this paper an alternative two dimensional approach termed the "time-based modelling method" is proposed that can simulate both the three dimensional effects at the tunnel heading and the time-dependent behaviour during construction. It is proposed that the new approach is appropriate for the analysis of tunnelling in a relatively permeable soil and, as an example, the method is applied to the analysis of a new Austrian tunnelling method (NATM) tunnelling problem in decomposed granite soil. The results are compared with field data and excellent agreement is obtained.Key words: numerical modelling, time-dependent behaviour, NATM tunnelling, decomposed granite soil.

Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour

1995 ◽  
Vol 305 ◽  
pp. 281-305 ◽  
Author(s):  
P. C. Matthews ◽  
M. R. E. Proctor ◽  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Shear Flow ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Mean Flow ◽  
The Magnetic Field

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.

scholarly journals On an approximate solution for the bending of a beam of rectangular cross-section under any system of load, with special reference to points of concentrated or discontinuous loading

1902 ◽  
Vol 70 (459-466) ◽  
pp. 491-496
Keyword(s):  
Approximate Solution ◽  
Special Reference ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Elastic Equilibrium ◽  
Elastic Solid ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Applied Stresses

The paper investigates the elastic equilibrium of a long bar of rectangular cross-section in those cases where the problem may be treated as one of two dimensions, namely:— ( a .) When the strain being in the plane of xy , the elastic solid extends indefinitely in the direction of the applied stresses over the bounding planes y = ± b , x = ± a being the same for any two sections parallel to the plane of xy . We then have a strictly two-dimensional strain.

The Prediction of Three-Dimensional Stress Concentration Factors

1959 ◽  
Vol 63 (585) ◽  
pp. 549-551 ◽  
Author(s):  
I. M. Allison
Keyword(s):  
Stress Concentration ◽  
Mathematical Analysis ◽  
Three Dimensional ◽  
Stress Raiser ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Torsional Loading ◽  
Loading System ◽  
Concentration Factors ◽  
Stress Concentration Factors

Two-Dimensional Stress concentration factors may be obtained more quickly and simply than the corresponding three-dimensional factors, either by experiment or mathematical analysis. It would be convenient to obtain information, for varying geometry in the two-dimensional case of a particular type of stress raiser, e.g. a shoulder, groove or hole, and use this either to predict the three-dimensional stress concentration factors or to extend the range of existing three-dimensional results. Clearly a comparison is only possible if the three-dimensional stress raiser embodies a plane of symmetry (which gives the geometry of the similar two-dimensional stress raiser), and if the loading conditions can be reproduced in both the two- and three-dimensional cases. The latter requirement restricts the correlation to the stress concentration factors obtained in tension and in bending. The three-dimensional torsional loading system has no plane of symmetry which can be simulated in two dimensions.

[Ni(cyclam)(OCOR)2], a finite molecular complex: hydrogen-bonded supramolecular aggregation in one, two and three dimensions, and coordination polymers in one and two dimensions

2001 ◽  
Vol 58 (1) ◽  
pp. 78-93 ◽  
Author(s):  
Choudhury M. Zakaria ◽  
George Ferguson ◽  
Alan J. Lough ◽  
Christopher Glidewell
Keyword(s):  
Hydrogen Bonding ◽  
Coordination Polymer ◽  
Hydrogen Bonds ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Compound I ◽  
Supramolecular Aggregation ◽  
Linear Coordination

In the complexes [Ni(cyclam)(OCOR)2] (cyclam = 1,4,8,11-tetraazacyclotetradecane), where (RCOO)− is 2-naphtho-ate [bis-(2-naphthoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (I), monoclinic P21/c, Z′ = 0.5], 3,5-dinitrobenzoate [bis-(3,5-dinitrobenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (II), triclinic P\bar 1, Z′ = 0.5], 4-nitrobenzoate [bis-(4-nitrobenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (III), monoclinic P21/n, Z′ = 0.5], 3-hydroxybenzoate [bis-(3-hydroxybenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (IV), monoclinic P21/c, Z′ = 0.5] and 4-aminobenzo-ate [bis-(4-aminobenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (V), monoclinic C2/c, Z′ = 0.5], the Ni lies on a centre of inversion with monodentate carboxylato ligands occupying trans sites. Compound (I) consists of isolated molecules. In (II) and (III), N—H...O hydrogen bonds link the complexes into chains. Compounds (IV) and (III) form two- and three-dimensional structures generated entirely by hard hydrogen bonds. The 5-hydroxyisophthalate(2−) anion forms a hydrated complex, [Ni(cyclam)(5-hydroxyisophthalate)(H2O)]·4H2O {[aqua-(5-hydroxyisophthalato)-1,4,8,11-tetraazacyclotetradecanenickel(II)] tetrahydrate, (VI), monoclinic Cc, Z′ = 1}, in which the monodentate carboxylato ligand and a water molecule occupy trans sites at Ni: extensive hydrogen bonding links the molecular aggregates into a three-dimensional framework. The terephthalate(2−) anion forms a hydrated linear coordination polymer {catena-poly[terephthalato-1,4,8,11-tetraazacyclotetradecanenickel(II)] monohydrate, (VII), monoclinic C2/c, Z′ = 0.5}. In 1,2,4,5-benzenecarboxylate tris[1,4,8,11-tetraazacyclotetradecanenickel(II)] diperchlorate hydrate (VIII), [Ni(cyclam)]3·[1,2,4,5-benzenetetracarboxylate(4−)]·[ClO4]2·-[H2O]3, there are two distinct Ni sites: [Ni(cyclam)]2+ and centrosymmetric [C10H2O8]4− units form a two-dimensional coordination polymer, whose sheets are linked by centrosymmetric [Ni(cyclam)(H2O)2]2+ cations.

Notes on problems in hexagonal aeolotropic materials

1951 ◽  
Vol 47 (2) ◽  
pp. 401-409 ◽  
Author(s):  
R. T. Shield
Keyword(s):  
General Solution ◽  
Stress Function ◽  
Harmonic Functions ◽  
Present Note ◽  
Three Dimensional ◽  
Stress Distributions ◽  
Axially Symmetric ◽  
Isotropic Materials ◽  
Stress Functions ◽  
Single Stress

Three-dimensional stress distributions in hexagonal aeolotropic materials have recently been considered by Elliott(1, 2), who obtained a general solution of the elastic equations of equilibrium in terms of two ‘harmonic’ functions, or, in the case of axially symmetric stress distributions, in terms of a single stress function. These stress functions are analogous to the stress functions employed to define stress systems in isotropic materials, and in the present note further problems in hexagonal aeolotropic media are solved, the method in each case being similar to that used for the corresponding problem in isotropic materials. Because of this similarity detailed explanations are unnecessary and only the essential steps in the working are given below.

Instability of pressure-driven gas–liquid two-layer channel flows in two and three dimensions

2018 ◽  
Vol 849 ◽  
pp. 1-34 ◽  
Author(s):  
Lennon Ó Náraigh ◽  
Peter D. M. Spelt
Keyword(s):  
Flow Pattern ◽  
Linear Theory ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Channel Flows ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Vorticity Field ◽  
Flow Pattern Map

We study unstable waves in gas–liquid two-layer channel flows driven by a pressure gradient, under stable stratification, not assumed to be set in motion impulsively. The basis of the study is direct numerical simulation (DNS) of the two-phase Navier–Stokes equations in two and three dimensions for moderately large Reynolds numbers, accompanied by a theoretical description of the dynamics in the linear regime (Orr–Sommerfeld–Squire equations). The results are compared and contrasted across a range of density ratios $r=\unicode[STIX]{x1D70C}_{liquid}/\unicode[STIX]{x1D70C}_{gas}$. Linear theory indicates that the growth rate of small-amplitude interfacial disturbances generally decreases with increasing $r$; at the same time, the cutoff wavenumbers in both streamwise and spanwise directions increase, leading to an ever-increasing range of unstable wavenumbers, albeit with diminished growth rates. The analysis also demonstrates that the most dangerous mode is two-dimensional in all cases considered. The results of a comparison between the DNS and linear theory demonstrate a consistency between the two approaches: as such, the route to a three-dimensional flow pattern is direct in these cases, i.e. through the strong influence of the linear instability. We also characterize the nonlinear behaviour of the system, and we establish that the disturbance vorticity field in two-dimensional systems is consistent with a mechanism proposed previously by Hinch (J. Fluid Mech., vol. 144, 1984, p. 463) for weakly inertial flows. A flow-pattern map constructed from two-dimensional numerical simulations is used to describe the various flow regimes observed as a function of density ratio, Reynolds number and Weber number. Corresponding simulations in three dimensions confirm that the flow-pattern map can be used to infer the fate of the interface there also, and show strong three-dimensionality in cases that exhibit violent behaviour in two dimensions, or otherwise the development of behaviour that is nearly two-dimensional behaviour possibly with the formation of a capillary ridge. The three-dimensional vorticity field is also analysed, thereby demonstrating how streamwise vorticity arises from the growth of otherwise two-dimensional modes.

scholarly journals Capillary-induced deformations of a thin elastic sheet

2016 ◽  
Vol 374 (2066) ◽  
pp. 20150169 ◽  
Author(s):  
N. D. Brubaker ◽  
J. Lega
Keyword(s):  
Finite Thickness ◽  
Three Dimensional ◽  
Two Dimensions ◽  
System Of Equations ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Drop Volume ◽  
Exactly Solvable ◽  
Encapsulation Process

We develop a three-dimensional model for capillary origami systems in which a rectangular plate has finite thickness, is allowed to stretch and undergoes small deflections. This latter constraint limits our description of the encapsulation process to its initial folding phase. We first simplify the resulting system of equations to two dimensions by assuming that the plate has infinite aspect ratio, which allows us to compare our approach to known two-dimensional capillary origami models for inextensible plates. Moreover, as this two-dimensional model is exactly solvable, we give an expression for its solution in terms of its parameters. We then turn to the full three-dimensional model in the limit of small drop volume and provide numerical simulations showing how the plate and the drop deform due to the effect of capillary forces.

Isostructurality in one and two dimensions: isostructurality of polymorphs

2004 ◽  
Vol 60 (5) ◽  
pp. 547-558 ◽  
Author(s):  
László Fábián ◽  
Alajos Kálmán
Keyword(s):  
Crystal Structures ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Cambridge Structural Database ◽  
Two Dimensional ◽  
Structural Database ◽  
Specific Packing ◽  
Packing Interactions

A set of polymorphic crystal structures was retrieved from the Cambridge Structural Database in order to estimate the frequency of isostructurality among polymorphs. Altogether, 50 structures, the polymorphs of 22 compounds, were investigated. It was found that one-, two- or three-dimensional isostructurality is exhibited by approximately half of the compounds analyzed. Among the isostructural polymorphs, the frequency of one-, two- and three-dimensional isostructurality is similar. From the examples, it appears that three-dimensional isostructurality is connected to the gradual ordering of crystal structures, while one- and two-dimensional isostructurality can often be related to specific packing interactions. The possibility of many similar interactions seems to decrease the probability of the occurrence of isostructural polymorphs. Conformational polymorphs do not exhibit isostructurality.

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