scholarly journals Quaternions and Cauchy Classical Theory of Elasticity

2020 ◽  
Vol 44 (2) ◽  
pp. 67-70
Author(s):  
Marek Danielewski ◽  
Lucjan Sapa
Keyword(s):  
Classical Theory ◽  
Wave Equations ◽  
Elastic Solid ◽  
Physical Reality ◽  
Theory Of Elasticity ◽  
Multiple Wave ◽  
French Mathematician ◽  
Starting Point ◽  
Quaternion Representation ◽  
Straightforward Consequence

AbstractDeveloped by French mathematician Augustin-Louis Cauchy, the classical theory of elasticity is the starting point to show the value and the physical reality of quaternions. The classical balance equations for the isotropic, elastic crystal, demonstrate the usefulness of quaternions. The family of wave equations and the diffusion equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic solid. Using the quaternion algebra, we present the derivation of the quaternion form of the multiple wave equations. The fundamental consequences of all derived equations and relations for physics, chemistry, and future prospects are presented.

scholarly journals Foundations of the Quaternion Quantum Mechanics

Entropy ◽  
2020 ◽  
Vol 22 (12) ◽  
pp. 1424
Author(s):  
Marek Danielewski ◽  
Lucjan Sapa
Keyword(s):  
Quantum Mechanics ◽  
Wave Equations ◽  
Bohmian Mechanics ◽  
Physical Reality ◽  
French Mathematician ◽  
Elastic Continuum ◽  
The Family ◽  
General Kind ◽  
Quaternion Representation ◽  
Straightforward Consequence

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.

scholarly journals Foundations of the Quaternion Quantum Mechanics

2020 ◽  
Author(s):  
Marek Danielewski ◽  
Lucjan Sapa
Keyword(s):  
Quantum Mechanics ◽  
Wave Equations ◽  
Bohmian Mechanics ◽  
Physical Reality ◽  
French Mathematician ◽  
Elastic Continuum ◽  
The Family ◽  
General Kind ◽  
Quaternion Representation ◽  
Straightforward Consequence

We show that the quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems allowing exposing its falsity.

scholarly journals The torsion of a semi-infinite elastic solid by an elliptical stamp

1968 ◽  
Vol 9 (1) ◽  
pp. 36-45
Author(s):  
Mumtaz K. Kassir
Keyword(s):  
Field Equation ◽  
Classical Theory ◽  
Cross Sections ◽  
Plane Surface ◽  
Boundary Surface ◽  
Elastic Solid ◽  
Theory Of Elasticity ◽  
Circular Area

The problem of determining, within the limits of the classical theory of elasticity, the displacements and stresses in the interior of a semi-infinite solid (z ≧ 0) when a part of the boundary surface (z = 0) is forced to rotate through a given angle ω about an axis which is normal to the undeformed plane surface of the solid, has been discussed by several authors [7, 8, 9, 1, 11, and others]. All of this work is concerned with rotating a circular area of the boundary surface and the field equation to be solved is, essentially, J. H. Mitchell's equation for the torsion of bars of varying circular cross-sections.

Study of internal stresses in rock massif within the framework of elastic layered block models with inclusions of the hierarchical structure of the L-rank.

2021 ◽  
Author(s):  
Olga Hachay ◽  
Andrey Khachay
Keyword(s):  
Hierarchical Structure ◽  
Classical Theory ◽  
Acoustic Waves ◽  
Degrees Of Freedom ◽  
Heterogeneous Media ◽  
Theory Of Elasticity ◽  
Rock Massif ◽  
Local Theory ◽  
Internal Stresses ◽  
Non Local

<p>In recent years, new models of continuum mechanics, generalizing the classical theory of elasticity, have been intensively developed. These models are used to describe composite and statistically heterogeneous media, new structural materials, as well as in complex massifs in mine conditions. The paper presents an algorithm for the propagation of longitudinal acoustic waves in the framework of active well monitoring of elastic layered block media with inclusions of hierarchical type of L-th rank. Relations for internal stresses and strains for each hierarchical rank are obtained, which constitute the non local theory of elasticity. The essential differences between the non local theory of elasticity and the classical one and the connection between them are investigated. A characteristic feature of the theory of media with a hierarchical structure is the presence of scale parameters in explicit or implicit form. This work focuses on the study of the effects of non locality and internal degrees of freedom, reflected in internal stresses, which are not described by the classical theory of elasticity and which can be potential precursors of the development of a catastrophic process in a rock massif. Thanks to the use of a model of a layered block medium with hierarchical inclusions, it is possible, using borehole acoustic monitoring, to determine the position of the highest values ​​of internal stresses and, with less effort, to implement the method of unloading the rock massif. If it is necessary to conduct short-term predictive monitoring of geodynamic regions and determine a more accurate position of the source of a dynamic phenomenon using borehole active acoustic observations, it is necessary to use the values ​​of the tensor of internal hierarchical stresses as a monitored parameter.</p>

Phenomenological and Structural Aspects of the Mechanical Response of Arteries

2000 ◽  
Author(s):  
Ray W. Ogden ◽  
Christian A. J. Schulze-Bauer
Keyword(s):  
Residual Stresses ◽  
Mechanical Response ◽  
Physiological State ◽  
Circumferential Stress ◽  
Morphological Structure ◽  
Cylindrical Tube ◽  
Single Layer ◽  
Theory Of Elasticity ◽  
Starting Point ◽  
Circular Cylindrical Tube

Abstract In this paper we present some new data from extension-inflation tests on a human iliac artery and then, on the basis of the nonlinear theory of elasticity, we examine a possible model to represent this data. The model considers the artery initially as a thick-walled circular cylindrical tube which may consist of two or more concentric layers. In order to take some account of the architecture (morphological structure), each layer of the material is regarded as consisting of two families of mechanically equivalent helical fibers symmetrically disposed with respect to the cylinder axis. The resulting material properties are then orthotropic in each layer. General formulas for the pressure and the axial load in the symmetric inflation of an extended tube are obtained. The starting point is the unloaded circular cylindrical configuration, but (in general unknown) residual stresses are included in the formulation. The model is illustrated by specializing firstly to the case of a single layer so that the consequences of the hypothesis of uniform circumferential stress in the physiological state can be examined theoretically. This enables the required residual stresses to be calculated explicitly. Secondly, the equations are specialized for the membrane approximation in order to show how certain important characteristics of the experimental data can be replicated using a relatively simple anisotropic membrane model.

Generalization of Gravitation Theory

2017 ◽  
Author(s):  
Hanoch Gutfreund ◽  
Jürgen Renn
Keyword(s):  
Empty Space ◽  
Mathematical Structure ◽  
Physical Reality ◽  
Natural Generalization ◽  
Theory Of Relativity ◽  
Total Field ◽  
Principle Of Equivalence ◽  
Field Equations ◽  
Starting Point ◽  
Consistent Extension

This chapter attempts to formulate a consistent extension of the theory of general relativity. The starting point of the general theory of relativity is the recognition of the unity of gravitation and inertia (principle of equivalence). From this principle, it follows that the properties of “empty space” were to be represented by a symmetrical tensor expressed in the theory. The principle of equivalence, however, does not give any clue as to what may be the more comprehensive mathematical structure on which to base the treatment of the total field comprising the entire physical reality. As such, this chapter considers the problem of how to find a field structure which is a natural generalization of the symmetrical tensor as well as a system of field equations for this structure which represent a natural generalization of certain equations of pure gravitation.

The Elastic Plane With a Circular Insert, Loaded by a Tangentially Directed Force

1962 ◽  
Vol 29 (2) ◽  
pp. 362-368 ◽  
Author(s):  
M. Hete´nyi ◽  
J. Dundurs
Keyword(s):  
Closed Form ◽  
Elastic Properties ◽  
Classical Theory ◽  
Theory Of Elasticity ◽  
Elastic Plane ◽  
Elementary Functions ◽  
Explicit Formulas ◽  
Concentrated Force

The problem treated is that of a plate of unlimited extent containing a circular insert and subjected to a concentrated force in the plane of the plate and in a direction tangential to the circle. The elastic properties of the insert are different from those of the plate, and a perfect bond is assumed between the two materials. The solution is exact within the classical theory of elasticity, and is in a closed form in terms of elementary functions. Explicit formulas are given for the components of stress in Cartesian co-ordinates, and also in polar co-ordinates at the circumference of the insert.

Lamb’s problem: a brief history

2019 ◽  
Vol 25 (3) ◽  
pp. 501-514
Author(s):  
Mohamad Emami ◽  
Morteza Eskandari-Ghadi
Keyword(s):  
Special Interest ◽  
Mathematical Theory ◽  
Three Dimensional ◽  
Elastic Solid ◽  
Theory Of Elasticity ◽  
Scientific Investigation ◽  
Lamb’S Problem ◽  
Solution Methods ◽  
Dimensional Version ◽  
History Of

In this review note, a historical scientific investigation is presented for Lamb’s problem in the mathematical theory of elasticity. This problem first appeared in 1904 in the pioneering paper of Professor Sir Horace Lamb (Lamb, H. On the propagation of tremors over the surface of an elastic solid. Philos Trans R Soc Lon 1904; 203: 1–42). Of special interest here are the analytical studies of the three-dimensional version of Lamb’s problem, which consists of a semi-infinite, homogeneous, isotropic elastic solid that is set in motion by the exertion of a dynamical point force applied suddenly on the surface of the domain. The objective of this paper is to offer a comprehensive introduction to Lamb’s problem for the reader, along with discussing its mathematical complexities. An account is given of the history of this ever-significant problem from its earlier stages to the more recent investigations via outlining and discussing different rigorous approaches and methods of solution that have been hitherto suggested. The limitations of different methods, if they exist, are also discussed. Eventually, various solution methods are compared considering their nature, advantages, and restrictions.

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