GRAND UNIFIED THEORIES

Revised October 2011 by S. Raby （Ohio State University） 16.1. Grand Unification 16.1.1. Standard Model ： An Introduction：

Assessing the range of blasting-induced cracks in the surrounding rock of deeply buried tunnels based on the unified strength theory

Blasting-induced cracks in the rock surrounding deeply buried tunnels can result in water gushing and rock mass collapse,posing significant safety risks.However,previous theoretical studies on the range of blasting-induced cracks often ignore the impact of the in-situ stress,especially that of the intermediate principal stress.The particle displacement−crack radius relationship was established in this paper by utilizing the blasthole cavity expansion equation,and theoretical analytical formulas of the stress−displacement relationship and the crack radius were derived with unified strength theory to accurately assess the range of cracks in deep surrounding rock under a blasting load.Parameter analysis showed that the crushing zone size was positively correlated with in-situ stress,intermediate principal stress,and detonation pressure,whereas negatively correlated with Poisson ratio and decoupling coefficient.The dilatancy angle-crushing zone size relationship exhibited nonmonotonic behavior.The relationships in the crushing zone and the fracture zone exhibited opposite trends under the influence of only in-situ stress or intermediate principal stress.As the in-situ stress increased from 0 to 70 MPa,the rate of change in the crack range and the attenuation rate of the peak vibration velocity gradually slowed.

Unified Control Theory from PID to ACPID

To address the challenge of achieving unified control across diverse nonlinear systems, a comprehensive control theory spanning from PID (Proportional-Integral-Derivative) to ACPID (Auto-Coupling PID) has been proposed. The primary concept is to unify all intricate factors, including internal dynamics and external bounded disturbance, into a single total disturbance. This enables the mapping of various nonlinear systems onto a linear disturbance system. Based on the theory of PID control and the characteristic equation of a critically damping system, Zeng’s stabilization rules (ZSR) and an ACPID control force based on a single speed factor have been designed. ACPID control theory is both simple and practical, with significant scientific significance and application value in the field of control engineering.

On solitary waves.Part 2 A unified perturbation theory for higher-order waves

A unified perturbation theory is developed here for calculating solitary waves of all heights by series expansion of base flow variables in powers of a small base parameter to eighteenth order for the one-parameter family of solutions in exact form, with all the coefficients determined in rational numbers. Comparative studies are pursued to investigate the effects due to changes of base parameters on （i） the accuracy of the theoretically predicted wave properties and （ii） the rate of convergence of perturbation expansion. Two important results are found by comparisons between the theoretical predictions based on a set of parameters separately adopted for expansion in turn. First, the accuracy and the convergence of the perturbation expansions, appraised versus the exact solution provided by an earlier paper [1] as the standard reference, are found to depend, quite sensitively, on changes in base parameter. The resulting variations in the solution are physically displayed in various wave properties with differences found dependent on which property （e.g. the wave amplitude, speed, its profile, excess mass, momentum, and energy）, on what range in value of the base, and on the rank of the order n in the expansion being addressed. Secondly, regarding convergence, the present perturbation series is found definitely asymptotic in nature, with the relative error δ （n） （the relative mean-square difference between successive orders n of wave elevations） reaching a minimum, δm at a specific order, n = n both depending on the base adopted, e.g. nm,α= 11-12 based on parameter α （wave amplitude）, nm,δ = 15 on δ （amplitude-speed square ratio）, and nm.ε= 17 on ε （ wave number squared）. The asymptotic range is brought to completion by the highest order of n = 18 reached in this work.

Unified Water Gravity Wave Theory and Improved Linear Wave

Based on Least Square Method, this paper presents variational principle for handling various water gravity wave theories and the unified water gravity wave theory was given. By using this variational principle of unified water wave theory, two kinds of improved linear waves were derived. The first one uses the same boundary conditions which were applied to derive 5-order Stokes wave. The second one uses the accurate boundary conditions (Eqs. 11 and 12). The two improved linear waves were compared with the existing linear wave.

Unified semi-analytical solution for elastic-plastic stress of deep circular hydraulic tunnel with support yielding

A unified semi-analytical solution is presented for elastic-plastic stress of a deep circular hydraulic tunnel with support yielding under plane strain conditions.The rock mass is assumed to be elastic-perfectly plastic and governed by the unified strength theory (UST).Different major principal stresses in different engineering situations and different support yielding conditions are both considered.The unified solution obtained in this work is a series of results,rather than one specific solution,hence it is suitable for a wide range of rock masses.In addition,parametric study is conducted to investigate the effect of intermediate principal stress.The result shows the major principal stress should be rationally chosen according to different engineering conditions.Finally,the applicability of the unified solution is discussed according to the critical pressures.

A unified intrinsic functional expansion theory for solitary waves

A new theory is developed here for evaluating solitary waves on water, with results of high accuracy uniformly valid for waves of all heights, from the highest wave with a corner crest of 120 down to very low ones of diminishing height. Solutions are sought for the Euler model by employing a unified expansion of the logarithmic hodograph in terms of a set of intrinsic component functions analytically determined to represent all the intrinsic properties of the wave entity from the wave crest to its outskirts. The unknown coefficients in the expansion are determined by minimization of the mean-square error of the solution, with the minimization optimized so as to take as few terms as needed to attain results as high in accuracy as attainable. In this regard, Stokess formula, F2= tan , relating the wave speed (the Froude number F) and the logarithmic decrement of its wave field in the outskirt, is generalized to establish a new criterion requiring (for minimizing solution error) the functional expansion to contain a finite power series in M terms of Stokess basic term (singular in ), such that 2M is just somewhat beyond unity, i.e. 2M1. This fundamental criterion is fully validated by solutions for waves of various amplitude-to-water depth ratio =a/h, especially about 0.01, at which M=10 by the criterion. In this pursuit, the class of dwarf solitary waves, defined for waves with 0.01, is discovered as a group of problems more challenging than even the highest wave. For the highest wave, a new solution is determined here to give the maximum height hst=0.8331990, and speed Fhst=1.290890, accurate to the last significant figure, which seems to be a new record.

Fundamental Issues Towards Unified Design Theory of Recycled and Natural Aggregate Concrete Components

In the past 20 years,recycled aggregate concrete(RAC),as a type of low-carbon concrete,has become a worldwide focus of research.However,the design methodology for RAC structural components remains a challenge.Consequently,demands for a unified design of natural aggregate concrete(NAC)and RAC components have been presented.Accordingly,this study analyses the necessity of a unified design theory and provides an in-depth demonstration of the strength determination,compressive constitutive relationship,and design method of concrete components.The coefficient of variation of RAC strength is found to be generally higher than that of NAC strength.The compressive and tensile strengths of RAC can be defined and determined using the same method as that used for NAC.The uniaxial compressive constitutive relationship between NAC and RAC has a unified mathematical expression.However,the elastic modulus of RAC decreases,and its brittleness exhibits an increasing trend compared with that of NAC.Finally,to unify the design formulae of RAC and NAC components for bearing capacity,modification factors for RAC components are proposed considering safety and reliability.Additionally,the feasibility of the proposed unified time-dependent design theory is demonstrated in terms of conceptual design and structural measures considering the effects of strength degradation and reinforcement corrosion.It is believed that this study enriches and develops the basic theory of concrete structures.

Unified Theory of Fundamental Interactions

Based on local gauge invariance, four different kinds of fundamental interactions in nature are unified in a theory which has Gravitational Gauge Group gauge symmetry. In this approach, gravitational field, like electromagnetic field, intermediate gauge field, and gluon field, is represented by gauge potential. Four kinds of fundamental interactions are formulated in the similar manner, and therefore can be unified in a direct or semi-direct product group. The model discussed in this paper is a renormalizable quantum model and can be regarded as an extension of the standard model to gravitational interactions, so it can be used to study quantum effects of gravitational interactions.

Unified elastoplastic finite difference and its application

Two elastoplastic constitutive models based on the unified strength the- ory （UST） are established and implemented in an explicit finite difference code, fast Lagrangian analysis of continua （FLAC/FLAC3D）, which includes an associated/non- associated flow rule, strain-hardening/softening, and solutions of singularities. Those two constitutive models are appropriate for metallic and strength-different （SD） materials, respectively. Two verification examples are used to compare the computation results and test data using the two-dimensional finite difference code FLAC and the finite element code ANSYS, and the two constitutive models proposed in this paper are verified. Two application examples, the large deformation of a prismatic bar and the strain-softening be- havior of soft rock under a complex stress state, are analyzed using the three-dimensional code FLAC3D. The two new elastoplastic constitutive models proposed in this paper can be used in bearing capacity evaluation or stability analysis of structures built of metallic or SD materials. The effect of the intermediate principal stress on metallic or SD mate- rial structures under complex stress states, including large deformation, three-dimensional and non-association problems, can be analyzed easily using the two constitutive models proposed in this paper.

Unified Field Theory

In the Einstein field equations, the geometry or the curvature of space-time defined as depended on the distribution of mass and energy principally resides on the left-hand side is set identical to a non-geometrical tensorial representation of matter on the right-hand side. In one or another form, general relativity accords a direct geometrical significance only to the gravitational field while the other physical fields are not of space time. They reside only in space time. Less well known, though of comparable importance is Einstein’s dissatisfaction with the fundamental asymmetry between gravitational and non-gravitational fields and his contributions to develop a completely relativistic geometrical field theory of all fundamental interactions, a unified field theory. Of special note in this context and equally significant is Einstein’s demand to replace the symmetrical tensor field by a non-symmetrical one and to drop the condition gik = gki for the field components. Historically, many other attempts were made too, to extend the general theory of relativity’s geometrization of gravitation to non-gravitational interactions, in particular, to electromagnetism. Still, progress has been very slow. It is the purpose of this publication to provide a unified field theory in which the gravitational field, the electromagnetic field and other fields are only different components or manifestations of the same unified field by mathematizing the relationship between cause and effect under conditions of general theory of relativity.

The Universal Grand Unified Theory Based on Life and Thought

The universe is life, which includes galaxies, living beings, society and man-made worlds. The universe is unified in life and thought. The universal grand unified theory is a theory to completely unify all phenomena in the universe with a fundamental mode （structural mode, operational mode, generation mode and evolution mode）. The book of changes of China： The great ultimate - two states - four images - eight hexagrams are life science and thought science theories, which represent the fundamental mode of universal unification, i.e. represent l） structural mode, operational mode, generation mode and evolution mode of life; 2） the four states that life has simultaneously： 0 dimension, ] dimension, 2 dimension, 3 dimension, and these four states have different structural and operational modes respectively; 3） the breathing and four seasonal trajectories of celestial bodies; 4） three generation modes of life： generation by transformation, generation by augmentation, and self-generation; 5） five dynamics systems inside life; 6） the sublimation and degration of life in the universe, the unification of religion and science; 7） the classification mode of the universal being; 8） the derivation mode of life; 9） the transformation between different forms of life in the universe; 10） the unification of six grand forces in the universe. All life systems in the universe： galaxy, organism, society, man-made worlds as well as cognitive science, natural science, social science are all in conform with the mode of the book of changes.

New Asphalt Pavement Failure Criterion Based on Unified Strength Theory

This study aims to introduce an appropriate analytical method for asphalt pavement based upon unified strength theory （UST）. The traditional maximum shear stress strength theory （MSST） cannot describe the marked difference between tension strength and compressive strength or variable intermediate principal stress, which significantly affects the geotechnical materials. Our studies try to find a new asphalt pavement failure criterion that considers the influence of both tension-compression strength ratio and intermediate principal stress of asphalt mixture. In order to select a suitable theory on pavement material, the UST is introduced and compared with the traditional theory. Results show that the tension-compression strength ratio of asphalt mixture, which is used as a material parameter, dramatically affects the stress and stress distribution law in pavement; the pavement stress level increases dramatically after considering the intermediate principal stresses. Therefore, the UST which considers both tension-compression strength ratio and intermediate principal stress is more in line with the material characteristics of asvhalt pavement.

Calculation of the Canonical Rate Constant for the Nonadiabatic Trapping Model Based on Unified Statistical Theory:A Test on The Exchange Reaction H_2+H

A new approach was employed to calculate the canonical (thermal) rate constant basedon unified statistical theory. All information for the calculation was obtained from ab initio meth-ods. The flux integral for any point of reaction coordinate was calculated by counting the numberof quantum states and applied to determine the dividing surfaces along the intrinsic reaction coor-dinate (IRC). The classical exchange reaction H2+H, as an example, was investigated. The IRC forthe reaction has been traced and detailed information of IRC was carried out at the QCISD/6-311 G** level .The calculated rate constants are well consistent with the experimental results.

A NON-DUALISTIC UNIFIED FIELD THEORY OF GRAVITATION,ELECTROMAGNETISM AND SPIN

The wisdom of classicalunified field theories in the conceptual framework of Weyl, Eddington, Einstein and Schrodinger has often been doubted and in particular there does not appear to be any empirical reason why the Einstein-Maxwell (E-MJ theory needs to be geometrized. The crux of the matter is, however not whether the E-M theory is aesthetically satisfactory but whether it answers all the modern questions within the classical context. In particular, the E-M theory does not provide a classical platform from which the Dirac equation can be derived in the way Schrodinger's equation is derived from classical mechanics via the energy equation and the Correspondence Principle. The present paper presents a non-dualistic unified field theory (UFT) in the said conceptual framework as propounded by M. A. Tonnelat. By allowing the metric form ds2=g,dx dx and the non-degenerate two-form F=(1/2t)rdx dx to enter symmetrically into the theory we obtain a UFT which contains Einstein's General Relativity and the Born-Infeld electrodynamics as special cases. Above all, it is shown that the Dirac equation describing the electron in an 'external' gravito-electromagnetic field can be derived from the non-dualistic Einstein equation by a simple factorization if the Correspondence Principle is assumed.

A NEW MODELLING METHOD FOR EVALUATING EXTERNAL DISTURBING POTENTIAL BASED ON THEORY OF UNIFIED REPRESENTATION OF GRAVITATIONAL FIELD

For a special use a new modelling method of evaluating external disturbing potential is presented in this paper.Being different from classical methods in physical geodesy this method is grounded upon the theory of unified representation of gravitational field.The models created in this way are particularly satisfactory for a high_speed computation of gravitational field in low altitude because they take account of topographic effects and have their kernel functions with simple structure and weak singularity.

Remarks to Gravitation Theories

Gravitation is one of the most significant phenomena which attract the attention of humankind for ages. There are twomost influential theories of the Gravitation at the current time being (summer 2019), namely, the Isaac Newton’s Theory ofGravitation and Albert Einstein’s General Theory of Relativity, which includes an explanation of the gravitation. However, both ofthem are incommensurate — so far — with plenty of quantum theories of the World. In this contribution, there is outlined a strategyfor unifying the current approaches to the Gravitation. Namely, that the gravitation/gravity should be considered as a consequence ofinteraction — a “pinning effect” — of the Spacetime with the non-integral spin of the Fermions, to saturate/add/ the missing part ofthem. It might induce a deformation/curvature of the Spacetime and known effects on moving material objects, as well aselectromagnetic radiation, including light, and other types of radiation, and properties/parameters of transformations of fields,described in the General Theory of Relativity and another. It is Gravitation which modifies/creates Spacetime curvature and itsfeatures and not vice versa.

Unified Understanding of Cosmological Energy Fields: Exploring Gravitation, Repulsion, and More

First, we develop a unitary process that can explain all four physical activations: electromagnetic wave, electromagnetic force, repulsion, and attraction. Second, it is clarified that cosmological and physical entities relate to different levels of existence in the universe and, hence, cannot be considered under the same paradigm. We know that biological entities use physical substances as executants;analogously, physical entities use cosmological substances as executants. Therefore, we have introduced new terms for the cosmological substances used as executants in physical activities. Third, this study introduces the primary elements of the cosmological world, such as visibility, forcibility, fullness, and hollowness, and defines them according to their attributes. This study explains how different combinations and placements of primary elements create different cosmological fields. These fields are used in all physical activations. Finally, we explain the entity model and how all physical activations occur. This study concludes that all physical activations use primary elements and follow the same universal law. Therefore, this study addresses the untouched subject of the creation of repulsion and attraction (gravitation). Furthermore, it addresses several cosmic mysteries that are yet to be resolved.

An Outline of the Grand Unified Theory of Gauge Fields

This paper attempts to propose a grand unified guiding principle of gauge fields from the mathematical and physical picture of fiber bundles: it is believed that our universe may have more fundamental interactions than the four fundamental interactions, and the gauge fields of these fundamental interactions are just a unified gauge potential on the fiber bundle manifold or the components connected to the bottom manifold, that is, our universe;these components can meet the transformation of gauge potential, and even can be transformed from a fundamental interaction gauge potential to another fundamental interaction gauge potential, and can be summarized into a unified equation, namely the expression of the generalized gauge equation, corresponding to the gauge transformation invariance;so gauge transformation invariance is a necessary condition to unify field theory, but quantization of field is not a necessary condition;the four (or more) fundamental interaction fields of the universe are unified into a universal gauge field defined by the connection of the principal fiber bundle on the cosmic base manifold.

A New Unified Electro-Gravity Theory for the Electron, and the Fundamental Origin of the Fine Structure Constant and the Casimir Effect

A rigorous model for the electron is presented by generalizing the Coulomb’s Law or Gauss’s Law of electrostatics, using a unified theory of electricity and gravity. The permittivity of the free-space is allowed to be variable, dependent on the energy density associated with the electric field at a given location, employing generalized concepts of gravity and mass/energy density. The electric field becomes a non-linear function of the source charge, where the concept of the energy density needs to be properly defined. Stable solutions are derived for a spherically symmetric, surface-charge distribution of an elementary charge. This is implemented by assuming that the gravitational field and its equivalent permittivity function is proportional to the energy density, as a simple first-order approximation, with the constant of proportionality, referred to as the Unified Electro-Gravity (UEG) constant. The stable solution with the lowest mass/energy is assumed to represent a “static” electron without any spin. Further, assuming that the mass/energy of a static electron is half of the total mass/energy of an electron including its spin contribution, the required UEG constant is estimated. More fundamentally, the lowest stable mass of a static elementary charged particle, its associated classical radius, and the UEG constant are related to each other by a dimensionless constant, independent of any specific value of the charge or mass of the particle. This dimensionless constant is numerologically found to be closely related to the fine structure constant. This possible origin of the fine structure constant is further strengthened by applying the proposed theory to successfully model the Casimir effect, from which approximately the same above relationship between the UEG constant, electron’s mass and classical radius, and the fine structure constant, emerges.