FINITE DISPLACEMENTS IN RHEOLOGICAL BODIES

1956 ◽  
Vol 34 (5) ◽  
pp. 498-509 ◽  
Author(s):  
A. E. Scheidegger
Keyword(s):  
Boundary Conditions ◽  
Parameter Space ◽  
Continuous Medium ◽  
Finite Strain ◽  
Equations Of Motion ◽  
Mathematical Formalism ◽  
Mathematical Representations ◽  
Finite Displacements ◽  
Earth’S Interior ◽  
Earth's Interior

The study of the behavior of continuous matter is basic in many disciplines, such as in various branches of engineering and in the study of the Earth's interior. Herein, it is evidently necessary to have a sufficiently general mathematical formalism to encompass the behavior of any type of material under any mechanical conditions. Customary "rheological" theories suffer from various drawbacks; they are either (i) restricted to too specialized "ideal" materials, or (ii) restricted to too special displacements, or (iii) restricted to too specialized mathematical representations. The present paper attempts to fill the need for a summary of the representations of the dynamics of arbitrary materials. The displacement within the continuous medium is described by three "co-ordinate" functions as functions of three "parameters" and of time. Extensive use is made of the fact that, insofar as the expression of any physical statement is concerned, "co-ordinates" and "parameters" are entirely equivalent. Formulas are deduced which enable one to express the boundary conditions, the equations of motion, and any chosen rheological condition in either parameter space or co-ordinate space. The notion of finite strain is scrutinized.

Model of vortex flow of charge in a vessel with turbine impeller

1987 ◽  
Vol 52 (8) ◽  
pp. 1888-1904
Author(s):  
Miloslav Hošťálek ◽  
Ivan Fořt
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Theoretical Data ◽  
Eddy Diffusion ◽  
Quantitative Agreement ◽  
Creeping Flow ◽  
Model Parameters ◽  
Mean Flow ◽  
Two Dimensional Flow ◽  
Vorticity Transport

A theoretical model is described of the mean two-dimensional flow of homogeneous charge in a flat-bottomed cylindrical tank with radial baffles and six-blade turbine disc impeller. The model starts from the concept of vorticity transport in the bulk of vortex liquid flow through the mechanism of eddy diffusion characterized by a constant value of turbulent (eddy) viscosity. The result of solution of the equation which is analogous to the Stokes simplification of equations of motion for creeping flow is the description of field of the stream function and of the axial and radial velocity components of mean flow in the whole charge. The results of modelling are compared with the experimental and theoretical data published by different authors, a good qualitative and quantitative agreement being stated. Advantage of the model proposed is a very simple schematization of the system volume necessary to introduce the boundary conditions (only the parts above the impeller plane of symmetry and below it are distinguished), the explicit character of the model with respect to the model parameters (model lucidity, low demands on the capacity of computer), and, in the end, the possibility to modify the given model by changing boundary conditions even for another agitating set-up with radially-axial character of flow.

scholarly journals Bootstrapping boundary-localized interactions

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Connor Behan ◽  
Lorenzo Di Pietro ◽  
Edoardo Lauria ◽  
Balt C. van Rees
Keyword(s):  
Boundary Condition ◽  
Scalar Field ◽  
Boundary Conditions ◽  
Parameter Space ◽  
Three Dimensional ◽  
Conformal Boundary ◽  
Dimensional Boundary ◽  
Dirichlet And Neumann Conditions ◽  
Boundary Fields ◽  
Boundary Dynamics

Abstract We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a ‘shadow pair’ of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a ‘kink’ in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

A Discussion on natural strain and geological structure - Fold shapes as functions of progressive strain

1976 ◽  
Vol 283 (1312) ◽  
pp. 129-138 ◽  
Keyword(s):  
Boundary Conditions ◽  
Mechanical Behaviour ◽  
Finite Strain ◽  
Geological Structure ◽  
Stages Of Development ◽  
Rock System ◽  
Frictional Properties ◽  
Natural Strain ◽  
Progressive Strain ◽  
Strain Dependent

The folding of the components (layers or texture) of a rock system is viewed as an unstable strain-dependent process. The folds undergo successive stages of development, including initiation, amplification, propagation and decay. Fold shapes are functions of (i) initial morphology, (ii) mechanical behaviour of the rock, including stiffness contrasts and frictional properties of adjacent components, (in) overall finite strain. The folded components may or may not adopt periodic waveforms, depending on (i) the relative rates of propagation versus amplification of the folds and (n) the boundary conditions of the rock system.

scholarly journals Actions, topological terms and boundaries in first-order gravity: A review

2016 ◽  
Vol 25 (04) ◽  
pp. 1630011 ◽  
Author(s):  
Alejandro Corichi ◽  
Irais Rubalcava-García ◽  
Tatjana Vukašinac
Keyword(s):  
Boundary Conditions ◽  
Symplectic Structure ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Action Principle ◽  
Internal Boundary ◽  
Four Dimensions ◽  
First Order ◽  
Conserved Charges ◽  
Boundary Terms

In this review, we consider first-order gravity in four dimensions. In particular, we focus our attention in formulations where the fundamental variables are a tetrad [Formula: see text] and a [Formula: see text] connection [Formula: see text]. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein–Hilbert–Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle sometimes involves the need for additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. In this work, we consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. We focus on the covariant Hamiltonian formalism where the phase space [Formula: see text] is given by solutions to the equations of motion. For each of the possible terms contributing to the action, we consider the well-posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. For the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of this manuscript is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, along the way, we point out and clarify some issues that have not been clearly understood in the literature.

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