Ph. D. & Dr. Sc. Lev Gelimson's Object Unistrength Fundamental Sciences Systems as Revolutions in Object Strength Sciences

Object Unistrength Fundamental Sciences Systems as a System of Revolutions in Object Strength Sciences

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Physical Monograph

The “Collegium” All World Academy of Sciences Publishers

Munich (Germany)

12th Edition (2012)

11th Edition (2010)

10th Edition (2004)

9th Edition (2003)

8th Edition (2001)

7th Edition (2000)

6th Edition (1995)

5th Edition (1994)

4th Edition (1993)

3nd Edition (1992)

2nd Edition (1987)

1st Edition (1977)

Abstract

Universal mechanical and strength sciences discover principally new opportunities vital not only for creating safe and efficient machinery and equipment but also for predicting earthquakes, tsunamis, and other natural disasters, as well as for saving lives and property.

Unistrength includes:

material unistrength as a system of universal fundamental sciences of mechanics and strength of materials;

object unistrength as a system of universal fundamental sciences of mechanics and strength of objects and systems.

Uniphilosophy (Exclusively Constructive Creative Philosophy) Principles as a System of Revolutions in Philosophy

Fundamental principles of uniphilosophy (exclusively constructive creative philosophy) build a fundamental system of revolutions in philosophy, in particular, the following subsystems.

1. Fundamental Principles of Uniphilosophy as a Fundamental Subsystem of Revolutions in Philosophy

The fundamental subsystem of revolutions in philosophy includes the following fundamental principles of uniphilosophy:

1. Exceptional natural constructivism (with the complete absence of artificial destructiveness).

2. Free efficient creativity (exclusively practically purposeful, verified, and efficient unlimitedly free creativity, intuition, and phantasy flight).

3. Scientific optimism and duty (each urgent problem can and must be solved adequately and efficiently enough).

4. Complication utilization (creating, considering, and efficiently utilizing only necessary and useful also contradictory objects and models, as well as difficulties, problems, and other complications).

5. Symbolic feasibility (at least symbolic existence of all the necessary and useful even contradictory objects and models).

2. Advanced Principles of Uniphilosophy as an Advanced Subsystem of Revolutions in Philosophy

The advanced subsystem of revolutions in philosophy includes the following advanced principles of uniphilosophy:

1. Exclusively efficient intuitive evidence and provability (reasonable fuzziness, intuitive ideas without axiomatic rigor if necessary and useful).

2. Unrestrictedly flexible constructivism (if necessary even creating new knowledge (concepts, approaches, methods, theories, doctrines, and even sciences) to adequately set, consider, and solve urgent problems).

3. Tolerable simplicity (choosing the best in the not evidently unacceptable simplest).

4. Perfect sensitivity, or conservation laws universality (no uncompensated change in a general object conserves its universal measures).

5. Exact discrimination of noncoinciding objects and models (possibly infinitely or overinfinitely large with infinitesimal or overinfinitesimal distinctions and differences).

6. Separate similar (proportional) limiting universalizability (the reduction of objects, systems, and their models to their own similar (proportional) limits as units).

7. Collective coherent reflectivity, definability, modelablity, expressibility, evaluability, determinability, estimability, approximability, comparability, solvability, and decisionability (in particular, in truly multidimensional and multicriterial systems of the expert definition, modeling, expression, evaluation, determination, estimation, approximation, and comparison of objects, systems, and models qualities which are disproportionate and hence incommensurable and not directly comparable, as well as in truly multidimensional and multicriterial decision-making systems).

3. Some Other Principles of Uniphilosophy

Among other principles of uniphilosophy are the following:

1. Truth priority (primacy of practically verified purely scientific truths and criteria prior to commonly accepted dogmas, views, agreements, and authority, with all due respect to them).

2. Peaceful pluralism (with peaceful development of scientific and life diversity).

3. Efficient creative inheritance (efficiently using, analyzing, estimating, and developing already available knowledge and information).

4. Efficient constructive freedom (unrestrictedly free exclusively constructive and useful self-determination and activity, in particular, in knowledge and information research, creation, and development).

5. Fundamentality priority (primacy of conceptual and methodological fundamentals).

6. Knowledge efficiency (only useful quality (acceptability, adequacy, depth, accuracy, etc.) and amount (volume, completeness, etc.) of knowledge, information, data, as well as creation, analysis, synthesis, verification, testing, structuring, systematization, hierarchization, generalization, universalization, modeling, measurement, evaluation, estimation, utilization, improvement, and development of objects, models, knowledge, information, and data along with intelligent management and self-management of activity).

7. Mutual definability and generalizability (relating successive generalization of concepts in definitions with optional linear sequence in knowledge construction).

8. Efficient unificability of opposites only conditionally distinguished (such as real/potential, real/ideal, specific/abstract, exact/inexact, definitively/possibly, pure/applied, theory/experiment/practice, nature/life/science, for example, the generally inaccurate includes the accurate as the limiting particular case with the zero error).

9. Partial laws sufficiency (if there are no known more general laws).

10. Focus on discoveries and inventions (dualistic unity and harmony of academic quality and originality, discovering phenomena of essence, inventive climbing, helpful knowledge bridges, creative multilingualism, scientific art, anti-envy, learnability, teachability, and terminology development).

The Principles of Uniphysics as the System of Revolutions in the Principles of Physics

The principles of uniphysics constitute the system of scientific revolutions in the principles of physics including the following subsystems.

The Fundamental Principles of Uniphysics as the Fundamental Subsystem of Revolutions in the Principles of Physics

The fundamental subsystem of revolutions in the principles of physics includes the following fundamental principles of uniphysics:

1. Urgent problems priority, exclusiveness, and typificability (adequately setting, exhaustively solving, and efficiently using urgent problems types only (with completely avoiding unnecessary considerations) as the unique criterion of the necessity and usefulness of creating and developing new knowledge).

2. Intuitive conceptual and methodological fundamentality priority (creating and efficiently using unified knowledge foundation due to fundamental general systems including objects, models, and intuitive fuzzy principles, concepts, and methodology).

3. Philosophical, mathematical, physical, and engineering meaningfulness, synergy, and intelligence primacy (with intuitive clarity, learnability, teachability, and efficient beauty as the united duality and harmony of quality and quantity, as well as of applicability and acceptability).

4. Controllability (the step-by-step testability, verifiability, estimability, invariance, immutability, strength, stability, and reliability of data, intermediate and final results, information, and general knowledge including concepts, approaches, methods, theories, doctrines, and sciences with the possibility of their correction, comprehensive improvement, generalization, universalization, structuring, systematization, and hierarchization).

5. Creating, inventing, and discovering directionality (the focus on creating and inventing new knowledge and the know-how, as well as on reasonably discovering new phenomena and laws of nature, along with the possibility of the generalization, universalization, systematization, and hierarchization of discoveries and inventions and with the united duality of scientific and technical architecture).

The Universalizability Principles of Uniphysics as the Universalizability Subsystem of Revolutions in the Principles of Physics

The universalizability subsystem of revolutions in the principles of physics includes the following universality principles of uniphysics:

1. Free efficient physical controlability (the expressibility, universalizability, invariance, measurability, estimability, and improvability of physical quantities, models, transformations, criteria, and knowledge).

2. Free efficient quality controlability (modelability, expressibility, universalizability, invariance, measurability, estimability, and improvability including providing and efficiently using the unity of variety and diversity, multicriteriality, polymethodicity, and multivariability in universally invariantly modeling, expressing, evaluating, measuring, and estimating data processing and approximation quality (via unierrors) along with accuracy and/or acceptability certainty via unireserves, unireliabilities, and unirisks without artificial randomization in deterministic problems).

3. Conservation laws universalizability (in the overinfinitesimal, the infinitesimal, the finite, the infinite, and the overinfinite).

4. The universalizability of laws of nature.

The General Noncriticality Principles of Uniphysics as the General Noncriticality Subsystem of Revolutions in the Principles of Physics

The general noncriticality subsystem of revolutions in the principles of physics includes the following general noncriticality principles of uniphysics:

1. Critical and limiting relations efficiency (the efficiency of the critical and limiting relations between the determining initial parameters of a problem).

2. General noncriticality (the joint definability and determinacy of subcritical, critical, and supercritical states, processes, and phenomena in a general structured system via joint generally noncritical relations).

3. General nonlimitability (the joint definability and determinacy of underlimiting, limiting, and overlimiting states, processes, and phenomena in a general structured system via joint generally nonlimiting relations).

4. Parameters reserves separability (the separability of the proper own reserves of the independent determining initial parameters in a problem).

The Unimathematical Principles of Uniphysics as the Unimathematical Subsystem of Revolutions in the Principles of Physics

The unimathematical subsystem of revolutions in the principles of physics includes the following unimathematical principles of uniphysics:

1. Tolerable simplicity (including the necessity and possibility of the tolerably simplest acceptable analytical solutions).

2. Unimodelability, uniexpressibility, unievaluability, and unimeasurability (using the unimathematical uninumbers, perfectly sensitive uniquantities as universal measures without any absorption and violations of conservation laws in the overinfinite, the infinite, the finite, the infinitesimal, and the overinfinitesimal, as well as unioperations, unisets, uniaggregates (unicontents), and unisystems).

3. Uniestimability, uniapproximability, and uniproblem unisolvability (using the unimathematical unierrors, unireserves, unireliabilities, and unirisks, as well as uniproblem unisolving methods, theories, doctrines, and sciences).

4. Unicomputability (using the computer fundamental sciences system, the overcoming complication fundamental sciences system, and the unimathematical data processing fundamental sciences system).

The Other Principles of Uniphysics

Among the other principles of uniphysics are the following:

1. The efficiency of transparency and ergonomicity (the analytical, numerical, and graphical unity, clarity, visibility, observability, and reviewability of knowledge, information, data, conditions, and results).

2. Creative inheritance efficiency (refining, correcting, improving, generalizing, and universalizing classical results, establishing the limits of their applicability, acceptability, adequacy, and efficiency).

3. Comprehensive self-responsibility concentration (the unity and indivisibility of research, expressing, interpreting, explaning, and presenting the results with the creative and efficient utilization of routine).

The Principles of Object Unistrength

The principles of object unistrength build a system of revolutions in the strength of objects and systems which consists of the following subsystems:

1) the problem solving subsystem of fundamental innovations including such principles:

urgent typical strength problems priority;

tolerable simplicity in solving strength problems;

namely analytically solving strength problems;

namely analytically testing numerical solutions to strength problems;

the true three-dimensionality of strength problems and their solutions;

the efficiency of the critical relationships of separate independent initial parameters (with determining both the movement of the greatest equivalent stress points and failure character change);

2) the limiting state subsystem of revolutions including such principles:

the correctability and improvability of limiting state criteria;

the universality of limiting state criteria;

3) the possibly nonlimiting state subsystem of revolutions including such principles:

the universality and invariance of strength reserves;

the significance of complicated (disproportionate) loading;

the significance of the own reserves of separate independent initial parameters;

carrying capacity significance by clearly nonuniform stress distribution;

namely equivalent stress concentration significance.

Object unistrength as a system of universal fundamental sciences of mechanics and strength of objects and systems includes:

1) the fundamental science of analytical macroelement study of the stress-strain state and strength of objects and systems which includes general theories and methods of applying power and integral analytical macroelement sciences to elasticity and strength problems. These analytical sciences have significant fundamental advantages over the methods of finite elements, points, and spheres often inappropriate also because they are based on the classical least square method by Gauss and Legendre with many fundamental defects. For the first time, it becomes possible to adequately set and solve nontrivial true three-dimensional elasticity and strength problems free from any assumptions on the relative smallness of some characteristic sizes such as the thickness even in thick plate theory. The general power-law solutions to the harmonic and biharmonic equations lead to analytical solutions to such problems with the possibility of comprehensively optically and mechanically improving objects and systems. It is shown that on the basis of the obtained conception on the deformation and fracture of the bodies of canonical forms, it is possible to develop simple analytical theories and methods of strength calculation which quite acceptably take into account the characteristics of structural elements of different configurations and build a reasonable scientific basis of their design. It becomes also possible to adequately set and solve static and fatigue strength problems for spatial bodies of ductile and brittle materials including contact problems with friction and initially uncertain areas of mutual coupling and slip. There exist such relationships between separate independent initial parameters that correspond to the qualitative changes in the beginning and the nature of failure. Simple approximate analytical solutions provide generalizing and substantially refining known plate theory solutions. Scientifically well-grounded intelligent design and new technical solutions have led to many invention certificates and patents. New phenomena in mechanics and strength of objects and systems have been discovered;

2) the fundamental science of concentrating namely the equivalent stress (and not a separate component of the stress-strain state) which includes general theories and methods of namely analytically solving problems with typical stress concentrators. The stress concentration problem for a fungoid valve limiter which is a three-dimensional cylindrical body with the cyclically symmetric system of holes has been set and solved. The methods of superposition and conjugation have been developed and tested both experimentally and via solving a trial problem. These methods and their results have led to proposing and justifying the introduction of the central lock body greatly enhancing the fungoid valve limiter strength;

3) the fundamental science of the universal reserves of the strengths of objects and systems which includes general theories and methods of determining such own reserves of separate independent initial parameters that are expressed in terms of a reserve common for them. It is determined via the worst-case combination of these parameters when they change within the boundaries established by their own reserves. This is the further generalization of the universal stresses. Such a universal science also applies to completely arbitrary problems with constraints. The usual methods for determining the reserves may be acceptable only for simple (proportional) loading but can lead to multiply overestimating the actual reserves in the general case;

4) the fundamental science of tolerance to errors in calculating the stress-strain state and strength of objects and systems which includes general theories and methods of calculating errors in determining the actual stress-strain states in objects and systems, their limiting states, and the reserves of the actual states with respect to these actual limiting states;

5) the fundamental science of tolerance to the damages and violations of objects and systems which includes the corresponding general theories and methods. Some of them compare the influence of the actual macrodamages and violations on the stress-strain state and strength of objects and systems to the influence of deviating actual materials from their continuous media models via conventional phenomenological approaches. The further general theories and methods determine the limiting damages and violations similar to the actual damages and violations, as well as the reserves of the actual damages and violations with respect to their limits;

6) the fundamental sciences of the unireliabilities and unirisks of objects and systems including the general theories and uniquantitative methods of unimeasuring and uniestimating the unireliabilities and unirisks via the unireserves of objects and systems with no artificial randomization which unnecessarily complicates calculation formulas and hinders comprehensively improving objects and systems via their reliability and risk.

Keywords: Object strength science, megascience, revolution, unistrength, megamathematics, mega-overmathematics, unimathematical test fundamental metasciences system, knowledge, philosophy, strategy, tactic, analysis, synthesis, object, operation, relation, criterion, conclusion, evaluation, measurement, estimation, expression, modeling, processing, symmetry, invariance, bound, level, worst case, defect, mistake, error, reserve, reliability, risk, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, replacement.

Introduction

There are many separate scientific achievements of mankind but they often bring rather unsolvable problems than really improving himan life quality. One of the reasons is that the general level of earth science is clearly insufficient to adequately solve and even consider many urgent himan problems. To provide creating and developing applicable and, moreover, adequate methods, theories, and sciences, we need their testing via universal if possible, at least applicable and, moreover, adequate test metamethods, metatheories, and metasciences whose general level has to be high enough. Mathematics as universal quantitative scientific language naturally has to play here a key role.

But classical mathematics [1] with hardened systems of axioms, intentional search for contradictions and even their purposeful creation cannot (and does not want to) regard very many problems in science, engineering, and life. This generally holds when solving valuation, estimation, discrimination, control, and optimization problems as well as in particular by measuring very inhomogeneous objects and rapidly changeable processes. It is discovered [2] that classical fundamental mathematical theories, methods, and concepts [1] are insufficient for adequately solving and even considering many typical urgent problems.

Megamathematics including overmathematics [2] based on its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides universally and adequately modeling, expressing, measuring, evaluating, and estimating general objects. This all creates the basis for many further megamathematics fundamental sciences systems developing, extending, and applying overmathematics. Among them are, in particular, science unimathematical test fundamental metasciences systems [3] which are universal.

Object Strength Science Unimathematical Test Fundamental Metasciences System

Object strength science unimathematical test fundamental metasciences system in megastrength [2] is one of such systems and can efficiently, universally and adequately strategically unimathematically test any object strength science. This system includes:

fundamental metascience of object strength science test philosophy, strategy, and tactic including object strength science test philosophy metatheory, object strength science test strategy metatheory, and object strength science test tactic metatheory;

fundamental metascience of object strength science consideration including object strength science fundamentals determination metatheory, object strength science approaches determination metatheory, object strength science methods determination metatheory, and object strength science conclusions determination metatheory;

fundamental metascience of object strength science analysis including object strength subscience analysis metatheory, object strength science fundamentals analysis metatheory, object strength science approaches analysis metatheory, object strength science methods analysis metatheory, and object strength science conclusions analysis metatheory;

fundamental metascience of object strength science synthesis including object strength science fundamentals synthesis metatheory, object strength science approaches synthesis metatheory, object strength science methods synthesis metatheory, and object strength science conclusions synthesis metatheory;

fundamental metascience of object strength science objects, operations, relations, and criteria including object strength science object metatheory, object strength science operation metatheory, object strength science relation metatheory, and object strength science criterion metatheory;

fundamental metascience of object strength science evaluation, measurement, and estimation including object strength science evaluation metatheory, object strength science measurement metatheory, and object strength science estimation metatheory;

fundamental metascience of object strength science expression, modeling, and processing including object strength science expression metatheory, object strength science modeling metatheory, and object strength science processing metatheory;

fundamental metascience of object strength science symmetry and invariance including object strength science symmetry metatheory and object strength science invariance metatheory;

fundamental metascience of object strength science bounds and levels including object strength science bound metatheory and object strength science level metatheory;

fundamental metascience of object strength science directed test systems including object strength science test direction metatheory and object strength science test step metatheory;

fundamental metascience of object strength science tolerably simplest limiting, critical, and worst cases analysis and synthesis including object strength science tolerably simplest limiting cases analysis and synthesis metatheories, object strength science tolerably simplest critical cases analysis and synthesis metatheories, object strength science tolerably simplest worst cases analysis and synthesis metatheories, and object strength science tolerably simplest limiting, critical, and worst cases counterexamples building metatheories;

fundamental metascience of object strength science defects, mistakes, errors, reserves, reliability, and risk including object strength science defect metatheory, object strength science mistake metatheory, object strength science error metatheory, object strength science reserve metatheory, object strength science reliability metatheory, and object strength science risk metatheory;

fundamental metascience of object strength science test result evaluation, measurement, estimation, and conclusion including object strength science test result evaluation metatheory, object strength science test result measurement metatheory, object strength science test result estimation metatheory, and object strength science test result conclusion metatheory;

fundamental metascience of object strength science supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement including object strength science supplement metatheory, object strength science improvement metatheory, object strength science modernization metatheory, object strength science variation metatheory, object strength science modification metatheory, object strength science correction metatheory, object strength science transformation metatheory, object strength science generalization metatheory, and object strength science replacement metatheory.

The object strength science unimathematical test fundamental metasciences system in megastrength [2] is universal and very efficient.

In particular, apply the object strength science unimathematical test fundamental metasciences system to classical object strength sciences.

Nota bene: Naturally, all the fundamental defects of classical material strength sciences discovered due to the material strength science unimathematical test fundamental metasciences system in megastrength [2] also hold in classical object strength sciences.

Fundamental Defects of Object Strength Sciences

Additionally, even the very fundamentals of classical object strength sciences have own evident lacks and shortcomings:

6. The finite element method (FEM) is regarded standard in computer aided solving problems. To be commercial, its software cannot consider nonstandard features of studied objects. There are no trials of exactly satisfying the fundamental equations of balance and deformation compatibility in the volume of each finite element. Moreover, there are no attempts even to approximately estimate pseudosolution errors of these equations in this volume. Such errors are simply distributed in it without any known law. Some chosen elementary test problems of elasticity theory with exact solutions show that FEM pseudosolutions can theoretically converge to those exact solutions to those problems only namely by suitable (a priori fully unclear) object discretization with infinitely many finite elements. To provide engineer precision only, we usually need very many sufficiently small finite elements. It is possible to hope (without any guarantee) for comprehensible results only by a huge number of finite elements and huge information amount which cannot be captured and analyzed. And even such unconvincing arguments hold for those simplest fully untypical cases only but NOT for real much more complicated problems. In practically solving them, to save human work amount, one usually provides anyone accidental object discretization with too small number of finite elements and obtains anyone "black box" result without any possibility and desire to check and test it. But it has beautiful graphic interpretation also impressing unqualified customers. They simply think that nicely presented results cannot be inadequate. Adding even one new node demands full recalculation once again that is accompanied by enormous volume of handwork which cannot be assigned by programming to the computer. Experience shows that by unsuccessful (and good luck cannot be expected in advance!) object discretization into finite elements, even skilled researchers come to absolutely unusable results inconsiderately declared as the ultimate truth actually demanding blind belief. The same also holds for the FEM fundamentals such as the absolute error, the relative error, and the least square method (LSM) [1] by Legendre and Gauss ("the king of mathematics") with producing own errors and even dozens of cardinal defects of principle, and, moreover, for the very fundamentals of classical mathematics [1]. Long-term experience also shows that a computer cannot work at all how a human thinks of it, and operationwise control with calculation check is necessary but practically impossible. It is especially dangerous that the FEM creates harmful illusion as if thanks to it, almost each mathematician or engineer is capable to successfully calculate the stress and strain states of any very complicated objects even without understanding their deformation under loadings, as well as knowledge in mathematics, strength of materials, and deformable solid mechanics. Spatial imagination only seems to suffice to break an object into finite elements. Full error! To carry out responsible strength calculation even by known norms, engineers should possess analytical mentality, big and profound knowledge, the ability to creatively and actively use them, intuition, long-term experience, even a talent. The same also holds in any computer aided solving problems, e.g., in hydrodynamics. A computer is a blind powerful calculator only and cannot think and provide human understanding but quickly gives voluminously impressive and beautifully issued illusory "soluions" to any problems with a lot of failures and catastrophes. Hence the FEM alone is unreliable but can be very useful as a supplement of analytic theories and methods if they provide testing the FEM and there is result correlation. Then the FEM adds both details and beautiful graphic interpretation.

Therefore, the very fundamentals of classical object strength sciences have a lot of obviously deep and even cardinal defects of principle.

Consequently, to make classical object strength sciences adequate, its evolutionarily locally correcting, improving, and developing which can be useful are, unfortunately, fully insufficient. Classical object strength sciences need revolutionarily replacing their inadequate very fundamentals via adequate very fundamentals.

Nota bene: Naturally, if possible, any revolution in classical object strength sciences has to be based on an adequate revolution in classical material strength sciences.

Revolution in Object Strength Sciences

Object megastrength fundamental sciences systems [8-20] based on material megastrength fundamental sciences systems [2-7] revolutionarily replaces the inadequate very fundamentals of object strength sciences [1] via adequate very fundamentals.

Object megastrength fundamental sciences systems [8-20] includes the following fundamental sciences systems:

I. Object stress analysis and synthesis fundamental sciences system

II. Object stress measurement fundamental sciences system

III. Object strength estimation fundamental sciences system

IV. Object strength management fundamental sciences system

Object unistrength principles form a system of revolutions in strength of objects and systems and consists of the following subsystems:

1) a problem solution subsystem of object strength revolutions including the following principles:

urgent problems priority;

the tolerable simplicity of solutions to strength problems;

analytic solutions;

analytical, numerical, and graphoanalytical unity;

true three-dimensionality of strength problems and solutions;

critical relationship of initial parameters that determines the maximum equivalent stress point movement and destruction nature change;

2) a limiting state subsystem of object strength revolutions including the following principles:

material limiting state criteria universality;

material limiting state criteria correction and improvement;

3) a nonlimiting state subsystem of object strength revolutions including the following principles:

nonlimiting state criteria universality;

nonlimiting state criteria invariance;

considering individual reserves of initial parameters;

taking into account the complexity of loading.

Among other object unistrength principles are the following:

visibility;

- Turn-verifiability and otsenivaemost;

- The unity of the manifold (multi, mnogometodichnost and multivariate);

- The effectiveness of succession (specification, generalization and universalization of the classical results, establishing the limits of their applicability and relevance);

- The complexity of optimization;

- Natselennnost on rational discovery and invention.

In uniprochnost objects as a system of universal basic science and mechanics, strength of facilities and systems include:

- The fundamental science of the analytic makroelementnom stress-strain state and strength of facilities and systems, which includes general theories and methods of application of power and integrated analytical sciences macronutrients to problems of elasticity and strength. These analytical sciences have weighty fundamental advantages over the often inadequate based on the classical least squares method of Gauss and Legendre methods of finite elements, points and areas. First considered and solved a true three-dimensional non-trivial problem in the theory of elasticity and strength, free from assumptions about the relative smallness of the characteristic dimensions of the individual, for example in the theories of plate thickness, and even thick plates. For such problems, the general power-law solutions of the harmonic and biharmonic equations yielded analytical solutions with the possibility of an integrated opto-mechanical optimization of facilities and systems, show that on the basis of submissions received on the deformation and fracture of bodies of canonical form is possible to develop simple analytical theories and methods of strength calculation, rather adequately take into account the specific structural elements of different configurations and are the scientific principles of their rational design. And solved the problem of static and fatigue strength of spatial bodies from plastic and brittle materials, vulyuchaya contact problems with friction and initially uncertain areas of mutual coupling and slip. The existence of relationships between source parameters corresponding to qualitative changes in the initiation and the nature of failure. The simple approximate analytical solution is to allow to generalize and substantially refine the known solutions of the theory of plates and plate theory. Scientifically sound and rational design of new technical solutions that are protected by copyright certificates and patents;

- The fundamental science of concentration is the equivalent stress (and not a component of the stress-strain state), which includes general theories and methods of this analytical solution of problems with typical stress concentrators. Appendix Terry overall correction of measurement error in elektrotenzometrii stress concentration zones showed that the maximum true strain determined by the product measured at an adequate rate of animation. It depends a lot on the characteristic size divided by the distance from the hub and the size of a strain gage measuring grid. Posed and solved the problem of stress concentration in the fungal limiter valve, which is three-dimensional cylindrical body with cyclically symmetric system holes. Developed and tested experimentally on a test problem of the method of superposition and the method of conjugation. They are allowed to propose and justify the introduction of central locking body ssuschestvenno enhances the strength;

- The fundamental science of analyzing the strength of the universal stock of facilities and systems, which includes general theory and methods to account for individual stocks on certain parameters, expressed in terms of the total for them. It is installed on the worst-case combination of these parameters when they change within the boundaries defined by individual stores. This is - the further generalization of the above stresses. Such a universal science applies in a completely random problems with constraints. The usual methods for determining the reserves may be adequate only for simple (proportional) loading, and in general lead to an overestimation of the actual multiple stores;

- The fundamental science of tolerance to errors in the calculation of the stress-strain state and strength of facilities and systems, which includes general theory and methods of accounting errors in the sampling sites and systems, analysis of the stress-strain state in them, defining their limiting states and states with respect to holdings of real limit;

- The fundamental science of tolerance to damage and violations of objects and systems, which includes general theory and methods to compare the influence of real makropovrezhdeny and violations on the stress-strain state and strength of facilities and systems to the influence of this deviation from the real materials hypothetical models of continuous media with conventional phenomenological approaches, as well as the determination of critical injuries and disorders such as real and actual damages and reserves with respect to these critical violations;

- The fundamental science of reliability and risk of objects and systems, including the general theory and quantitative methods is the measurement and estimation of reliability and risk reserves facilities and systems, with no artificial randomization, which unnecessarily complicates the calculation formulas and prevents the optimization of complex objects and systems for their reliability and risk.

The system of revolutions in the mechanics and strength of objects and systems includes:

1) a stress and strength analysis subsystem of object mechanics and strength revolutions including:

setting and solving nontrivial truly three-dimensional object strength problems;

comprehensive analytical optical and mechanical optimization of objects and systems;

general theory and methods of accounting equivalent stress concentration;

2) a universal reserve subsystem of object mechanics and strength revolutions including:

expressing of individual reserves on certain parameters through a common purpose for them;

determining the boundaries of these parameters according to their individual reserves;

determining the worst-case combination of these parameters when they change within those boundaries;

determining the universal reserve via this worst-case combination;

3) a reliability and risk subsystem of object mechanics and strength revolutions including:

deterministic definition, measurement, and estimation of object and system reliability quantitatively expressed in terms of their reserve;

deterministic definition, measurement, and estimation of object and system risk quantitatively expressed in terms of their reserve;

optimization of complex objects and systems via their reliability;

optimization of complex objects and systems via their risk.

4) a subsystem of object mechanics and strength revolutions associated with discovering new object mechanics and strength phenomena including:

existence of the critical value of the ratio of the pressure on the lateral surface of the three-dimensional cylindrical glass element to the outside pressure so that the excess of this ratio leads to the abrupt displacement of the maximum equivalent stress point from the center to the edge of the central pressure-free part of the inner base of the glass element;

changing the glass element destruction nature by excessing this critical value, namely via cleavage and subsequent cracking of the segment (smaller than the hemisphere) whose base is the central pressure-free part of the inner base of the glass element instead of its radial cracking;

existence of the optimal value of the ratio of the pressure on the lateral surface of the three-dimensional cylindrical glass element to the outside pressure so that this glass element strength increases by an order of magnitude;

uniform strength along the length of the composite cylinder with a uniform contact pressure between the layers;

the optimality of the plane stress values of the contact radius and of the contact pressure between the layers of the composite cylinder so that this contact pressure is provided via the optimal tightness distribution along the length of the cylinder of finite length due to the obtained solutions to three-dimensional problems for composite cylinders combined with their realized assembly technologies;

uniform strength along the length of the heat-collected composite cylinder of finite length with optimally linearly increasing the contact radius tightness between the layers within their slip areas towards the cylinder ends;

uniform strength along the length of the press-fit composite cylinder of finite length with optimally continuously piecewise-linearly (with a constant modulus of the derivative) distributing the contact radius tightness between the layers within their slip areas;

existence of a static internal pressure which is equivalent to a cyclic internal pressure in a composite cylinder and equals the sum of the cycle average pressure and the product of the cycle amplitude pressure and the ratio of the ultimate strength to the symmetric cycle fatigue strength if this ratio is the same for all the layers materials.

I. Object Stress Analysis and Synthesis Fundamental Sciences System

The object stress analysis and synthesis fundamental sciences system includes the following fundamental sciences.

I.1. General Stress Fundamental Sciences

Among them are:

I.1.1. Object Discretization Fundamental Science

including general theory of discretization error estimation by determining area, volume, and mass moments of inertia and inhomogeneous distribution discretization error estimation general theory.

I.1.1.1. General Theory of Discretization Error Estimation by Determining Area, Volume, and Mass Moments of Inertia

Problem Setting

By calculating area, volume, and/or mass and moments of inertia [1, 2, 3], direct integration [1] provides the exactness of their values.

In the practice, instead of that method, the following one [2, 3] is commonly used:

1) discretizing the areas, volumes, and/or masses of objects to be considered;

2) choosing suitable approximations to the areas, volumes, and masses of the elements;

3) choosing suitable approximations to the coordinates of their centroids (centers of area/volume/mass/gravity).

That is why it is necessary to estimate the order of the corresponding discretization errors.

Design Scheme Choice

Because of the well-known possibility to provide the approximation error as small as required by suitably triangulating each area practically relevant [1] even by using right triangles only whose each cathetus is parallel to one of the axes, it is sufficient to consider the area moments of inertia only of such a triangle with constant thickness. By three-dimensional objects with variable thickness, simply use such triangular ”sheets” whose numbers vary along conventional lengths and/or widths.

To show result invariance by changing the position of such a right triangle, consider a coordinate system one of whose axes contains one cathetus, the hypotenuse dividing the first quadrant at the system origin, as well as the central coordinate system with the axes parallel to the above ones.

To obtain the lower bound of the order of the corresponding discretization errors, subsequently choose all the approximations from below.

To obtain the upper bound of the order of the corresponding discretization errors, subsequently choose all the approximations from above.

All those combinations lead to the six cases as follows including the two ones with exact determination via direct integration.

Remark: Adding an adjacent rectangle to such a right triangle brings no additional absolute error and therefore decreases the relative error. Thus our choice provides strong estimates.

Case 1: Exact Determination by "Cathetus" Axes

Fig. 1

Here the composed symbol dx denotes the (infinitesimal) differential of x.

For the area moments of inertia Ix and Iz of the right triangle OAH with base a and height h about the axes 0x and 0z, direct integration brings the following exact results:

Ix = ∫0a (hxdx/a) (0.5 hx/a)2 = 0.25 h3/a3 ∫0a x3 dx = ah3/16

Iz = ∫0a (hxdx/a) x2 = h/a ∫0a x3 dx = a3h/4

Case 2: Lower-Bound Discretization Error Order

by "Cathetus" Axes

Fig. 2

Here the symbol n denotes the (finite) cardinal number of the parts of side OA and thus of the right triangle OAH due to the corresponding vertical lines whereas k varies from 0 to n - 1 indicating the ordinal number of the part containing the interval between ak/n and a(k + 1)/n.

In order to realistically lower-bound estimate the area moments of inertia Ix and Iz of the right triangle OAH (with base a and height h) about the axes 0x and 0z, use the following method:

1) replace the area of the kth (trapezoidal) part with the smaller (shown) rectangle;

2) take the (smaller) ordinate of the rectangle centroid by calculating Ix ;

3) replace the abscissa of the rectangle centroid with the smaller value ak/n by calculating Iz .

Then direct summation brings the following lower-bound estimates:

Ix = ∑0n -1(a/n)(hk/n)(0.5 hk/n)2 = 0.25 ah3/n4 ∑0n -1k3 =

0.25 ah3/n4 ((n - 1)n/2)2 = (1 - 1/n)2 ah3/16

Iz = ∑0n -1(a/n)(hk/n)(ak/n)2 = a3h/n4 ∑0n -1k3 =

a3h/n4 ((n - 1)n/2)2 = (1 - 1/n)2 a3h/4

Comparing these estimates of the moments with their exact values (see case 1) shows that realistically lower-bound estimating the area moments of inertia Ix and Iz leads to multiplying their exact values by

(1 - 1/n)2 = 1 - 2/n + 1/n2.

By n great enough (e.g., 100 by the order of value),

1/n2

is very small and can be omitted. In such a case the factor is about

1 - 2/n

and the relative error is about 2/n.

Case 3: Upper-Bound Discretization Error Order

by "Cathetus" Axes

Fig. 3

Like case 2, the symbol n denotes the (finite) cardinal number of the parts of side OA and thus of the right triangle OAH due to the corresponding vertical lines whereas k varies from 0 to n - 1 indicating the ordinal number of the part containing the interval between ak/n and a(k + 1)/n.

In order to realistically upper-bound estimate the area moments of inertia Ix and Iz of the right triangle OAH (with base a and height h) about the axes 0x and 0z, use the following method:

1) replace the area of the kth (trapezoidal) part with the greater (shown) rectangle;

2) take the (greater) ordinate of the rectangle centroid by calculating Ix ;

3) replace the abscissa of the rectangle centroid with the greater value a(k + 1)/n by calculating Iz .

Then direct summation brings the following upper-bound estimates:

Ix = ∑0n -1(a/n)(h(k + 1)/n)(0.5 h(k + 1)/n)2 =

0.25 ah3/n4 ∑0n -1(k + 1)3 =

0.25 ah3/n4 (n(n + 1)/2)2 = (1 + 1/n)2 ah3/16

Iz = ∑0n -1(a/n)(h(k + 1)/n)(a(k + 1)/n)2 = a3h/n4 ∑0n -1(k + 1)3 =

a3h/n4 (n(n + 1)/2)2 = (1 + 1/n)2 a3h/4

Comparing these estimates of the moments with their exact values (see case 1) shows that realistically upper-bound estimating the area moments of inertia Ix and Iz leads to multiplying their exact values by

(1 + 1/n)2 = 1 + 2/n + 1/n2.

By n great enough (e.g., 100 by the order of value),

1/n2

is very small and can be omitted. In such a case the factor is about

1 + 2/n

and the relative error is about 2/n.

Case 4: Exact Determination by Central Axes

Fig. 4

Like case 1, the composed symbol dx denotes the (infinitesimal) differential of x.

For the area moments of inertia Ix and Iz of the right triangle OAH with base a and height h about the central axes 0x and 0z, direct integration brings the following exact results:

Ix =

∫-2a/3a/3 h(x/a + 1/3 + 1/3)dx (-h/3 +

0.5 h(x/a + 1/3 + 1/3))2 =

0.25 h3/a3 ∫-2a/3a/3 (x3 + 2ax2/3)dx =

0.25 h3/a3 (a4/324 - 4a4/81 + 2a4/243 + 16a4/243) =

ah3/144

Iz = ∫-2a/3a/3 h(x/a + 1/3 + 1/3)dx x2 =

h/a ∫-2a/3a/3 (x3 + 2ax2/3)dx =

h/a (a4/324 - 4a4/81 + 2a4/243 + 16a4/243) =

a3h/36

To mutually test the results in case 1 and in the present case, use these central moments of inertia, the area ah/2 of the triangle, and the coordinates of the origin of the "cathetus" coordinate system in the central coordinate system:

x = - 2a/3,

z = - h/3.

Then the well-known Parallel Axis Theorem [2, 3] gives for the "cathetus" coordinate system

Ix = ah3/144 + ah/2 (-h/3)2 = ah3/144 + ah3/18 = ah3/16

Iz = a3h/36 + ah/2 (-2a/3)2 = a3h/36 + 2ah3/9 = a3h/4

The both results coincide with those in case 1, quod erat demonstrandum.

Case 5: Lower-Bound Discretization Error Order

by Central Axes

Fig. 5

In order to realistically lower-bound estimate the area moments of inertia Ix and Iz of the right triangle OAH (with base a and height h) about the central axes 0x and 0z, use the following method:

1) replace the area of the kth (trapezoidal) part with the smaller (shown) rectangle;

2) take the ordinate of the rectangle centroid by calculating Ix ;

3) replace the abscissa of the rectangle centroid with the value a(k/n - 2/3) by calculating Iz .

Then direct summation brings the following lower-bound estimates:

Ix = ∑0n -1(a/n)(hk/n)(0.5 hk/n - h/3)2 =

ah3/n4 (4-1∑0n -1k3 - 3-1n∑0n -1k2 + 9-1n2∑0n -1k) =

ah3/n4 (4-1((n - 1)n/2)2 - 3-1n (n - 1)n(2n - 1)/6 +

9-1n2 (n - 1)n/2) =

ah3/n2 (16-1(n - 1)2 - 18-1(n - 1)((2n - 1) - n)) =

(1 - 1/n)2 ah3/144

Iz = ∑0n -1(a/n)(hk/n)(a(k/n - 2/3))2 =

a3h/n4 (∑0n -1k3 - (4/3)n∑0n -1k2 + (4/9)n2∑0n -1k) =

a3h/n4 (((n - 1)n/2)2 - (4/3)n (n - 1)n(2n - 1)/6 +

(4/9)n2 (n - 1)n/2) =

a3h/n2 (4-1(n - 1)2 - (2/9)(n - 1)((2n - 1) - n)) =

(1 - 1/n)2 a3h/36

Comparing these estimates of the moments with their exact values (see case 4) shows that realistically lower-bound estimating the area moments of inertia Ix and Iz leads to multiplying their exact values by

(1 - 1/n)2 = 1 - 2/n + 1/n2.

By n great enough (e.g., 100 by the order of value),

1/n2

is very small and can be omitted. In such a case the factor is about

1 - 2/n

and the relative error is about 2/n.

Case 6: Upper-Bound Discretization Error Order

by Central Axes

Fig. 6

Like case 3, the symbol n denotes the (finite) cardinal number of the parts of side OA and thus of the right triangle OAH due to the corresponding vertical lines whereas k varies from 0 to n - 1 indicating the ordinal number of the part containing the interval between a(k/n - 2/3) and a((k + 1)/n - 2/3).

In order to realistically upper-bound estimate the area moments of inertia Ix and Iz of the right triangle OAH (with base a and height h) about the central axes 0x and 0z, use the following method:

1) replace the area of the kth (trapezoidal) part with the greater (shown) rectangle;

2) take the ordinate of the rectangle centroid by calculating Ix ;

3) replace the abscissa of the rectangle centroid with value a((k + 1)/n - 2/3) by calculating Iz .

Then direct summation brings the following upper-bound estimates:

Ix = ∑0n -1(a/n)(h(k + 1)/n)(0.5 h(k + 1)/n - h/3)2 =

ah3/n4 (4-1∑0n -1(k + 1)3 - 3-1n∑0n -1(k + 1)2 +

9-1n2∑0n -1(k + 1)) =

ah3/n4 (4-1(n(n + 1)/2)2 -

3-1n n(n + 1)(2n + 1)/6 + 9-1n2 n(n + 1)/2) =

ah3/n2 (16-1(n + 1)2 - 18-1(n + 1)((2n + 1) - n)) =

(1 + 1/n)2 ah3/144

Iz = ∑0n -1(a/n)(h(k + 1)/n)(a((k + 1)/n - 2/3))2 =

a3h/n4 (∑0n -1(k + 1)3 - (4/3)n∑0n -1(k + 1)2 +

(4/9)n2∑0n -1(k + 1)) =

a3h/n4 ((n(n + 1)/2)2 - (4/3)n n(n + 1)(2n + 1)/6 +

(4/9)n2 n(n + 1)/2) =

a3h/n2 (4-1(n + 1)2 - (2/9)(n + 1)((2n + 1) - n)) =

(1 + 1/n)2 a3h/36

Comparing these estimates of the moments with their exact values (see case 4) shows that realistically upper-bound estimating the area moments of inertia Ix and Iz leads to multiplying their exact values by

(1 + 1/n)2 = 1 + 2/n + 1/n2.

By n great enough (e.g., 100 by the order of value),

1/n2

is very small and can be omitted. In such a case the factor is about

1 + 2/n

and the relative error is about 2/n.

Discussion

The obtained relative errors corresponding to the lower and upper bounds of area, volume, and mass moments of inertia about both central and noncentral axes show that by any (not necessarily consequent) method of reasonably choosing the values of the areas, volumes, and masses of elements by one-dimensional discretizing (with n great enough) the object to be considered as well as the coordinates of the centroids of the elements, the relative error is about 2/n and this value is universal.

By three-dimensional discretizing (with n1, n2, and n3 great enough, e.g., 100 by the order of value) the object to be considered, the relative error is about

2(1/n1+ 1/n2 + 1/n3)

and this value is universal, too.

If not all among n1, n2, and n3 are great enough in the above sense, the concept of the relative error is not correct at all because its value substantially depends of choosing the denominator by its definition loosing any objective sense, see [4, 5, 6]. In such a general case, the universal generalizations [4, 5, 6] of the relative error by the author are to be used along with the above factors naturally including 1/n2, too. Those methods lead to more complicated formulae.

The above considerations hold in almost any case practically relevant when there are no too essential form and mass changes within any element of discretization (otherwise such elements should be further divided to satisfy this requirement). The well-known FEM recommendation to divide intervals with possible half-sine-wave distribution law by at least 5 (parts) seems to be adequate enough.

It is very dangerous to blindwise (as a black box like many well-known FEM software implementations) discretize and estimate any object without comparisons with the results by other methods allowing subsequently testing.

It is elementary to construct counterexamples illustrating ultimately great errors. The simplest of them seems to be the family of figures between an axis and a line coinciding with the axis at all division points. This leads to zero lower-bound estimates and unbounded errors.

Conclusion

By estimating area, volume, and mass moments of inertia about both central and noncentral axes with using adequate discretization of objects to be considered, any method of reasonably choosing the values of the areas, volumes, and masses of elements, as well as the coordinates of their centroids, leads to results with relative errors small enough.

References

[1] Encyclopaedia of Mathematics. Ed. M. Hazewinkel. Vols. 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994

[2] Elementos de resistencia de materiales; por S. P. Timoshenko y D. H. Young; traducción de Jesús Ibáñez Gar. 2ª ed. Barcelona: Montaner y Simón, DL 1979. - 404 p.

[3] Pisarenko G. S. y otros. Manual de Resistencia de Materiales. Moscú: Editorial MIR, 1989

[4] Lev Gelimson. General Strength Theory. Monograph. Drukar Publishers, Sumy, 1993

[5] Lev Gelimson. Basic New Mathematics. Monograph. Drukar Publishers, Sumy, 1995

[6] Lev Gelimson. General Problem Theory. Abhandlungen der WIGB, 3 (2003), Berlin

I.1.1.2. Inhomogeneous Distribution Discretization Error Estimation General Theory

I.1.2. Analytic Macroelement Fundamental Science

including harmonic equation general power solution theory, biharmonic equation general power solution theory, general power analytic macroelement theory, and general integral analytic macroelement theory.

General power analytic macroelement theory provides precisely satisfying the fundamental equations of balance and deformation compatibility in the volume of each macroelement, minimizing analytic pseudosolution discrepancies on macroelements boundaries, and estimating the pseudosolution quality.

General integral analytic macroelement theory uses one of the deformation compatibility equations for pseudosolution quality estimation only, all other equations being precisely satisfied.

All these theories and methods have many advantages as compared to the methods of finite elements, points and spheres based on the often inadequate classical least square method (LSM) [1] by Legendre and Gauss ("the king of mathematics").

For the first time this science provides general power solutions to the harmonic equation in a three-dimensional problem of elasticity theory and to the biharmonic equation in an axisymmetric problem of elasticity theory. The same holds for both precise and approximate (with adequate quality estimation) analytic solutions to nontrivial true three-dimensional problems of elasticity theory with possible complex optical and mechanical optimization of structures and discovering new phenomena of mechanics.

I.2. Local Stress Fundamental Sciences

Among them are:

I.2.1. Equivalent Stress Concentration Fundamental Science

including general theories on namely equivalent stress concentration at circular and elliptic holes with or without pressure in them (rather than separate stress concentration).

I.2.2. Contact Stress Fundamental Science

including contact pressure inhomogeneous distribution general theories considering or ignoring friction;

bearing strength fundamental science including general bearing strength theory and general bearing strength theory by replacing plate parts with washers.

The object stress analysis and synthesis fundamental sciences system leads to reliable results (the FEM can illustrate) and provides optimizing objects for extreme conditions.

II. Object Stress Measurement Fundamental Sciences System

The object stress measurement fundamental sciences system includes the following fundamental sciences.

II.1. Inhomogeneous Distribution Measurement Fundamental Science

This fundamental science includes general theories of measuring inhomogeneous distributions, in particular very quickly variable processes.

Definition 16. A general demodulation method is applying the general linear-combination method to a general measurement problem [2, 4] to determine true inhomogeneous distributions without measurement modulations caused by measurement operators.

Example 17. Let a one-variable function p(s) be continuously distributed and integral-mean modulated on each segment of a constant length D with the measurement operator

MΔ: p(s) → p(s) = Δ-1∫s-Δ/2 s+Δ/2 p(t)dt

to be inverted. By many most important standard functions, MΔ can be replaced with multiplication by a suitable factor. For example, this factor is 1 for each linear function;

sh(0.5nΔ)/(0.5nΔ)

for exp(ns), sh ns, and ch ns;

sin(0.5nΔ)/(0.5nΔ)

for sin ns and cos ns.

II.2. Stress Concentration Measurement Fundamental Science

This fundamental science includes general theories of measuring stress concentration with recovering true maximum stresses using possibly incomplete transformed data.

The object stress measurement fundamental sciences system leads to reliable results (the FEM can illustrate) and provides optimizing objects for extreme conditions.

III. Object Strength Estimation Fundamental Sciences System

The object strength estimation fundamental sciences system includes the following fundamental sciences.

III.1. Design Error Tolerance Fundamental Science

This fundamental science includes structure discretization error general theory, structure stress analysis error general theory, critical structure stress analysis error general theory, and structure reserve analysis error tolerance general theory.

III.2. Damage Tolerance Fundamental Science

This fundamental science includes infringement and damage admissibility general theory, infringement and damage measure general theory, critical infringement and damage general theory, infringement and damage reserve general theory, and infringement and damage tolerance general theory.

They provide comparing the influence of real objects macrodamages and infringements on objects stress and strain states and strengths with the influence on them due to the deviations of real materials from their hypothetical continua models by standard phenomenological approaches, as well as determining critical objects infringements and damages similar to real ones and the reserves of real damages and infringements with respect to these critical ones.

III.3. Object Reserve Fundamental Science

This fundamental science includes general theories of object safety factor (reserve) possibly by complex nonproportional loading including general theories and methods of considering separate parameters reserves expressed via a reserve common for them all. It is determined via the worst combination of the values of these parameters by their varying within the limits of the borders defined by the individual reserves of the separate parameters. This is further generalizing a reduced pure (dimensionless) stress. This universal science is applicable to any general problem with restrictions. Usual methods of safety factor determination can be adequate at simple (proportional) loading only and generally lead to systematically overestimating true safety factors (reserves).

III.4. Object Reliability Fundamental Science

This fundamental science includes general theories and methods of quantitatively measuring, evaluating, and estimating object reliability via object safety factors (reserves) without artificial randomization which unfairly complicates calculation formulae and hence the complex optimization of objects via their reliabilities possibly by complex nonproportional loading.

III.5. Object Risk Fundamental Science

This fundamental science includes general theories and methods of quantitatively measuring, evaluating, and estimating object risk via object safety factors (reserves) without artificial randomization which unfairly complicates calculation formulae and hence the complex optimization of objects via their risks possibly by complex nonproportional loading.

The object strength estimation fundamental sciences system leads to reliable results (the FEM can illustrate) and provides optimizing objects for extreme conditions.

IV. Object Strength Management Fundamental Sciences System

The object strength management fundamental sciences system includes the following fundamental sciences.

IV.1. Unimathematical Implantation Fundamental Science

This fundamental science includes general theories on general ion implantation essence and fundamentals, as well as on implantation strengthening.

IV.2. Design Management Fundamental Science

This fundamental science includes general theories on strengthening and complexly (mechanically, physically, optically, etc.) optimizing objects, in particular combined cylinders, three-dimensional cylindric glass elements of high-pressure illuminators (deep-sea portholes), as well as compensating defocusing in their optical systems.

The object strength management fundamental sciences system leads to reliable results (the FEM can illustrate) and provides optimizing objects for extreme conditions.

The system of revolutions in the mechanics and strength of objects and systems includes:

1) the stress and strength analysis subsystem of revolutions, in particular:

setting and solving nontrivial truly three-dimensional elasticity and strength problems;

comprehensively analytically optically and mechanically improving objects and systems;

namely equivalent stress concentration determination;

2) the universal reserve subsystem of revolutions, in particular:

expressing the own reserves of individual parameters via an universal reserve common for them;

determining the boundaries of these parameters via their own reserves;

determining the worst-case combination of the values of these parameters when they change within those boundaries;

determining the universal reserve via this worst-case combination;

3) the unireliability and unirisk subsystem of fundamental innovations, in particular:

deterministically defining, unimeasuring, and unestimating objects and systems unireliabilities quantitatively expressed via their reserves;

deterministically defining, unimeasuring, and unestimating objects and systems unirisks quantitatively expressed via their reserves;

the all-round improvement of objects and systems via their unireliabilities;

the all-round improvement of objects and systems via their unirisks;

4) the phenomenon discovery subsystem of fundamental innovations, in particular:

the existence of such a critical value of the ratio of the pressure on the lateral surface of a three-dimensional cylindrical glass element to the outside pressure that the excess of this value leads to the abrupt displacement of the highest equivalent stress point from the center to the edge of the pressure-free central part of the inner base of the glass element;

changing the destruction nature of such a three-dimensional cylindrical glass element by excessing this critical value from the cleavage and subsequent cracking of a segment (smaller than the hemisphere) whose base is the pressure-free central part of the inner base of the glass element instead of radially cracking the whole glass element;

the existence of the best value of the ratio of the pressure on the lateral surface of a three-dimensional cylindrical glass element to the outside pressure so that this value provides multiply increasing the strength and bearing capacity of the glass element in comparison with the case of the pressure-free lateral surface of the glass element;

the constant strength of a combined cylinder along its length by the uniform contact pressure between the layers of the cylinder;

the optimality of such values of the coupling radius and of the contact pressure between the layers of a combined cylinder that these values correspond to its plane stress and this contact pressure constant along the cylinder length is provided via the best change in the radial tightness between the cylinder layers along the cylinder length so that this best change law is determined via the obtained solutions to the three-dimensional problems for a combined cylinder of finite length for real technologies to build such a cylinder;

the constant strength of a heat-collected combined cylinder along its length via the best linear increase (in the directions to the both cylinder ends) in the radial tightness between the cylinder layers along the cylinder length on the both cylinder end zones of mutually axially sliding the cylinder layers;

the constant strength of a pressure-assembled combined cylinder along its length via the best continuous piecewise-linear radial tightness between the cylinder layers along the cylinder length so that the modulus of the radial tightness distribution derivative along the cylinder length is constant;

the existence of the constant internal pressure equivalent to a cyclic internal pressure in a cylinder and equal to the average pressure of the cycle with adding the amplitude pressure of the cycle multiplied by the ratio of the ultimate strength and the fatigue strength in a symmetric cycle;

the existence of the constant internal pressure equivalent to a cyclic internal pressure in a combined cylinder and equal to the average pressure of the cycle with adding the amplitude pressure of the cycle multiplied by the ratio of the ultimate strength and the fatigue strength of the cylinder layers materials in a symmetric cycle if this ratio is the same for all these materials.

References

[1] Encyclopaedia of Mathematics / Managing editor M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994

[1] Lamé G. Lecons sur la theorie mathematique de l’élasticite des corps solides. Gauthier-Villars, Paris, 1852

[2] Love A. E. H. A Treatise on the Mathematical Theory of Elasticity. Vols. I, II. Cambridge University Press, Cambridge, 1892, 1893

13. S. P. Timoshenko, J. N. Goodier ( 1970) Theory of Elasticity. McGraw-Hill, N. Y.

14. R. Hill (1950) The Mathematical Theory of Plasticity. Clarendon Press, Oxford Engineering Series, Oxford

15. A. L. Nadai (1950) Theory of Flow and Fracture of Solids. McGraw-Hill, N. Y.

16. H. Liebowitz (Ed.) (1968) Fracture. An Advanced Treatise. Academic Press, N. Y. and London

[4] Encyclopaedia of Optimization, Ed. C. A. Floudas, P. M. Pardalos, Vols. 1 to 6, Kluwer Academic Publ., Dordrecht, 2001

1. W. Prager (1970) Optimization of structural design. J. Optimiz. Theory and Appl. 6, No. 1, 1-21

11. O. C. Zienkiewicz, Y. K. Cheung (1967) The Finite Elements Method in Structural and Continuum Mechanics. McGraw-Hill, N. Y.

[3] O. C. Zienkiewicz, R. L. Taylor. Finite Element Method. Volumes 1 to 3. Butterworth Heinemann Publ., London, 2000

Bruhn, E. F.: Analysis and Design of Flight Vehicle Structures. Jacobs Publishing, Inc., Indianapolis (IN), 1973

Military Handbook. Metallic Materials and Elements for Aerospace Vehicle Structures. MIL-HDBK-5H, 1998

[3] Handbuch Struktur-Berechnung. Prof. Dr.-Ing. L. Schwarmann. Industrie-Ausschuss-Struktur-Berechnungsunterlagen, Bremen, 1998

[4] Lev Gelimson. Elastic Mathematics. General Strength Theory. The “Collegium” All World Academy of Sciences Publishers, Munich, 2004

[5] Lev Gelimson. Discretization Errors by Determining Area, Volume, and Mass Moments of Inertia. In: Review of Aeronautical Fatigue Investigations in Germany During the Period May 2005 to April 2007, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2007-042 Technical Report, Aeronautical fatigue, ICAF2007, EADS Innovation Works Germany, 2007, 20-22

13. Timoshenko, S. P.: Theory of Elasticity, 3rd ed. McGraw-Hill, New York, 1970

14. Hertz, H.: Über die Berührung fester elastischer Körper. J. reine und angewandte Matematik, 92 (1882), 156-171

2. Lev Gelimson (1977) Composite cylinder having autofretted outer layer under cyclic loading [In Russian]. Institute for Compressor Engineering Publishers, Sumy, 70-77

Lev Gelimson. General Analytic Methods. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 260-261

3. L. G. Gelimson, A. A. Kaminskii, and I. B. Karintsev (1985) On strength optimization of flat-parallel deep-sea portholes [In Russian]. Dynamics and Strength of Machines 41, 108-114

9. Lev Gelimson (1992) Generalization of Analytic Methods for Solving Strength Problems [In Russian]. Drukar Publishers, Sumy.

[16] Lev Gelimson. General Strength Theory. Drukar Publ., Sumy, 1993

10. Lev Gelimson (1993) Analytic Macroelement Method in Axially Symmetric Elasticity [In Russian]. International Scientific and Technical Conference “Glass Technology and Quality”. Mathematical Methods in Applying Glass Materials. Theses of Reports, 104-106

[20] Lev Gelimson. Basic New Mathematics. Drukar Publ., Sumy, 1995

[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2004, 496 pp.

[3] Lev Gelimson. Science Unimathematical Test Fundamental Metasciences Systems. Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 12 (2012), 1.

[18] P. W. Bridgman. Collected Experimental Papers. Vols. 1 to 7. Harvard University Press Publ., Cambridge (Massachusetts), 1964

Lev Gelimson. Equivalent Stress Concentration Factor. In: Review of Aeronautical Fatigue Investigations in Germany During the Period March 2003 to May 2005, Ed. Dr. Claudio Dalle Donne, SC/IRT/LG-MT-2005-039 Technical Report, Aeronautical fatigue, ICAF2007, EADS Corporate Research Center Germany, 2005, 30-32

Lev Gelimson. Maximum Rivet Contact Pressure. In: Review of Aeronautical Fatigue Investigations in Germany During the Period March 2003 to May 2005, Ed. Dr. Claudio Dalle Donne, SC/IRT/LG-MT-2005-039 Technical Report, Aeronautical fatigue, ICAF2007, EADS Corporate Research Center Germany, 2005, 32-33

Lev Gelimson. General Reserve Theory. In: Review of Aeronautical Fatigue Investigations in Germany During the Period March 2003 to May 2005, Ed. Dr. Claudio Dalle Donne, SC/IRT/LG-MT-2005-039 Technical Report, Aeronautical fatigue, ICAF2007, EADS Corporate Research Center Germany, 2005, 55-56

Lev Gelimson. Discretization Errors by Determining Area, Volume, and Mass Moments of Inertia. In: Review of Aeronautical Fatigue Investigations in Germany During the Period May 2005 to April 2007, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2007-042 Technical Report, Aeronautical fatigue, ICAF2007, EADS Innovation Works Germany, 2007, 20-22

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