Quaternions and Cauchy Classical Theory of Elasticity

International Journal of Solids and Structures, 2014

One of the most fruitful and elegant approach (known as Kolosov-Muskhelishvili formulas) for plane isotropic elastic problems is to use two complex-valued holomorphic potentials. In this paper, the algebra of real quaternions is used in order to propose in three dimensions, an extension of the classical Muskhelishvili formulas. The starting point is the classical harmonic potential representation due to Papkovich and Neuber. Alike the classical complex formulation, two monogenic functions very similar to holomorphic functions in 2D and conserving many of interesting properties, are used in this contribution. The completeness of the potential formulation is demonstrated rigorously. Moreover, body forces, residual stress and thermal strain are taken into account as a left side term. The obtained monogenic representation is compact and a straightforward calculation shows that classical complex representation for plane problems is embedded in the presented extended formulas. Finally the classical uniqueness problem of the Papkovich-Neuber solutions is overcome for polynomial solutions by fixing explicitly linear dependencies.

2020

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 demonstra...

2018

We show that the quaternionic field theory can be rigorously derived from the classical balance equations in an isotropic ideal crystal where the momentum transport and the field energy are described by the Cauchy-Navier equation. The theory is presented in the form of the non-linear wave and Poisson equations with quaternion valued wave functions. The derived quaternionic form of the Cauchy-Navier equation couples the compression and torsion of the displacement. The wave equation has the form of the nonlinear Klein-Gordon equation and describes a spatially localized wave function that is equivalent to the particle. The derived wave equation avoids the problems of negative energy and probability. We show the self-consistent classical interpretation of wave phenomena and gravity.

International Journal of Engineering Science, 1973

The idea of deformable shell model is generalized to elastic solid continua with diatomic structures. The balance and constitutive laws are derived and the comparison of the present work is made with those of lattice theories. Finally, it is shown that a plane wave propagating in such a medium is dispersive and may have both optical and accoustical branches. 2. KINEMATICS AND BALANCE LAWS Consider an elastic and continuous medium consisting of diatomic molecules. From the mathematical point of view, such a medium may be considered as two media, coincident at an initial reference configuration &, provided that they interact with

Provides an introduction to the concept and derivation of the Quaternions. Then derives several basic algebraic relationships involving quaternions, culminating with how quaternions can be used to describe 3d rotations. Then discusses the computational advantages that quaternions have over other 3d rotation methods, and provides an example of a classical mechanics rotation problem being solved with quaternions. Then discusses Frobenius's Theorem and the potential physical implication that it has. Overall aim is to provide a good educational resource for the intuition of quaternions and their applications.

We present quaternion quantum mechanics and its ontological interpretation. The theory combines the Cauchy model of the elastic continuum with the Planck-Kleinert crystal hypothesis. In this model, the universe is an ideal elastic solid where the elementary particles are soliton-like waves. Tension induced by the compression and twisting of the continuum affects its energy density and generates the force of gravity, as density changes alters the wave speed and hence gravity could be described by an index of refraction.

Symmetry

We present a quaternion representation of quantum mechanics that allows its ontological interpretation. The correspondence between classical and quaternion quantum equations permits one to consider the universe (vacuum) as an ideal elastic solid. Elementary particles would have to be standing or soliton-like waves. Tension induced by the compression and twisting of the elastic medium would increase energy density, and as a result, generate gravity forcing and affect the wave speed. Consequently, gravity could be described by an index of refraction.

Quaternionic Quantum Wave Equation and Applications, 2019

The perception of duality nature in physics is extended to covers the idea of superposition of field and its dual field by assuming there is resultant field emerges as a superposition when a field and its dual field are combined. Quaternion formulation is used in this model. This generalization enables in constructing a general plane wave to generalize, Klein Gordon equation, Helmholtz's equation. It can be applied to electromagnetic and Meissner effect.

Acta Crystallographica Section A Foundations of Crystallography, 1996

The bridge between the molecular descriptions of crystalline configurations and the continuum theories of crystal mechanics such as linear and nonlinear elasticity is given by a natural hypothesis that goes back to Cauchy and Born, referred to as the 'Born rule'. This paper reports on an extensive investigation on the validity of the Born rule and the possibility of applying nonlinear elasticity to describe the behavior of crystalline solids. This is done by studying the phenomenon of mechanical twinning and its implications for the invariance group of the energy density of the crystal. The analysis leads to the conclusion that, in the 'generic' case, the Born rule does not hold and that nonlinear elasticity theory cannot provide an adequate model for crystal mechanics because an unphysical energy invariance is derived. However, the Born rule works and elasticity theory can be used for crystals whose twinning shears satisfy certain quite restrictive 'nongeneric' conditions. Relevant experimental data confirm these theoretical negative conclusions. It is remarkable that two very important classes of 'nongeneric' materials to which an elastic model safely applies do emerge experimentally: shape-memory alloys and materials whose crystalline structure is given by a simple Bravais lattice.

invented the quaternions in 1843, in his effort to construct hypercomplex numbers, or higher dimensional generalizations of the complex numbers. Failing to construct a generalization in three dimensions (involving "triplets") in such a way that division would be possible, he considered systems with four complex units and arrived at the quaternions. He realized that, just as multiplication by i is a rotation by 90 o in the complex plane, each one of his complex units could also be associated with a rotation in space. Vectors were introduced by Hamilton for the first time as "pure quaternions" and Vector Calculus was at first developed as part of this theory. Maxwell's Electromagnetism was first written using quaternions (see, eg. [6]).