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

by

© 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.

All the usual limiting state criteria for triaxial mechanical stresses are even in principle applicable (not to mention adequacy) to the simplest special cases only, usually to constantly loaded isotropic materials equally resistant to tension and compression. For the general case of an arbitrarily anisotropic material with different resistance to tensions and compressions and for any variable loads with possibly rotating the principal directions of the stress state at a point of such a material during loading, there are even no suggestions on formulating limiting state criteria for triaxial mechanical stresses and, therefore, no hints of namely universal strength laws of nature. Furthermore, even for the simplest special case of a constantly loaded isotropic material equally resistant to tension and compression, the generally accepted Tresca and Huber-von-Mises-Hencky criteria are quite nonsensitive to adding equiaxial stress states such as pressure whose significant effect on strength has been proved via the experiments by the Nobel prize winner Bridgman and prescribe to the ratio of the tensile and shear strengths the values ​​of 2 and 3^1/2 , respectively, whereas for real materials, this ratio can take different real-number values from 1 to 4. The Tresca criterion does not take into account also the intermediate principal stress at all.

Material unistrength, or the system of universal fundamental sciences of mechanics and strength of materials, introduces with a universal dimensionless mechanical stress having simple and clear physical sense via naturally transforming a conventional dimensional stress and for the first time discovers universal strength laws of nature whose very narrow and special particular cases give all the common criteria for limiting mechanical states. These laws hold even for arbitrarily anisotropic natural and artificial materials that have different resistance to tensions and compressions and for any variable loads with possibly rotating the principal directions of the stress state at a point of such a material during loading.

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 Material Unistrength

The principles of material unistrength form a system of revolutions in the strength of materials and consists of the following subsystems:

1) the stress subsystem of revolutions in the strength of materials including such principles:

the primacy of the principal directions of the stress state rather than the optional main directions of anisotropy;

the universalizability of the strength transformation of a constant dimensional principal stress via its division by the absolute value of the corresponding limiting stress of the same direction and the same sign in uniaxial stress state;

the universalizability of the synchronous strength transformation of a variable dimensional principal stress via its division by the absolute value of the corresponding limiting stress of the same direction and the same sign in uniaxial stress state at the same time;

the self-expressibility of stress danger degree via universal stresses;

the replaceability of the program (process) of a variable uniaxial principal stress via its equivalent (equidangerous) cyclicity;

the vectorial universalizability of the constant equivalent to the scalar program (process) of a variable uniaxial principal stress;

the vectorial modular universalizability of the constant equivalent to the scalar program (process) of the triaxial variable uniaxial principal stresses;

the two-dimensionality of representing three-dimensional data;

2) the limiting state criterion subsystem of revolutions in the strength of materials including such principles:

the universalizability of limiting state criteria sensitive to the actual ratio of tensile strength to shear strength and to the influence of the intermediate principal stress and of adding equiaxial stress states;

the scalar universalizability of constant limiting state criteria;

the synchronous scalar universalizability of variable limiting state criteria;

the integrated vectorial modular universalizability of variable limiting state criteria;

limiting state criterion correctibility and improvability;

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

the comprehensivity of the unified criteria for generally nonlimiting (underlimiting, limiting, and overlimiting) stress states;

the validity and usefulness of negative and imaginary equivalent stresses.

Among the other principles of material unistrength are the following:

clear physical and mathematical sense;

the efficiency of creative inheritance (refining, correcting, generalizing, and universalizing classical limiting state criteria with establishing the domains of their applicability, acceptability, adequacy, and efficiency).

Material unistrength includes:

I) the fundamental science of universal constant stresses which includes general theories of the universal scalar reductions of mechanical stresses to their own limits of the same directions and signs. For the first time, this science invents, discovers, introduces, and efficiently utilizes a universal reduced dimensionless stress by dividing a dimensional normal stress by the modulus of its own uniaxial limit of the same direction and of the sign by the absence of all the remaining stresses and under the same other conditions. This universal dimensionless stress turns to be the reciprocal to the equal own reserves both of the dimensional stress and of the dimensionless stress so that this reciprocal is taken with the common sign of these both stresses. Along with such universal dimensionless stress theory, this science also includes the general theories of universal scalar reductions of mechanical stresses to their own limits of the same directions and of the same signs namely for the main types of deformable solids and the main types of loading:

1) isotropic materials that are equally resistant to tension and compression and are constantly loaded;

2) isotropic materials which have different resistances to tension and compression and are constantly loaded;

3) orthotropic materials under such constant loading that the principal directions of the stress state coincide with the main directions of orthotropy;

4) arbitrarily anisotropic materials under any constant loading;

II) the fundamental science of universal stresses in arbitrarily anisotropic materials under any variable loads which includes:

the general theories of the universal synchronous scalar reductions of mechanical stresses to their own limits of the same directions and of the same signs at each time moment of loading;

the general theories of the universal integrated vector reductions of the total processes (programs) of the separate dimensionless stresses to their equidangerous cycles and to the corresponding universal vector dimensionless stresses. Moreover, each of the triad of the principal unistresses is not ranked by their algebraic values and keeps its number index during all the time of loading regardless of changing both the direction of this unistress and algebraically ordering the values of all the three principal unistresses;

III) the fundamental science of universal strength laws of nature, which includes:

the general theories of negative and imaginary equivalent stresses along with their modules;

the general theory of the simultaneous (instantaneous) equivalent (uniaxial) universal stress (at an arbitrary point in a solid at any given loading time moment) as a universal (defined and determined by a limit state criterion) function of the triad of the universal principal stresses so that this equivalent (uniaxial) universal stress has the same reserve as this whole triad at the same point in the solid at the same loading time moment according to this criterion;

the general theory of the integrated scalarly equivalent universal stress (at any point of a solid) as the maximum value of the simultaneous (instantaneous) equivalent (uniaxial) universal stress at this point in the solid during the whole time of loading;

the general theory of the integrated vectorially equivalent universal stress (at any point in a solid) as the modulus of the universal (defined and determined by a limit state criterion) function of the triad of the (constant) vectorial universal principal stresses each of which is uniquely determined via such an equidangerous uniaxial cycle of the corresponding universal principal stress which is not ordered by the algebraic value (as this is a case by representing the principal stresses in their space as their natural coordinate system) and has a constant number index during the whole time of loading that this cycle has the same reserve as the whole process (entire program) of the corresponding uniaxial principal stress at the same point in the solid during the whole time of loading;

the general theory of the integrated scalarly and vectorially equivalent universal stress at any point in a solid as the maximum value of the integrated scalarly equivalent universal stress and the integrated vectorially equivalent universal stress at this point in the solid during the whole time of loading.

This science also includes the general linear, piecewise linear, and nonlinear (including quadratic and further power) strength theories for the main types of deformable solids and the main types of loading;

IV) fundamental strength information processing science which includes general theories of strength information unification, modeling (including two-dimensionally representing three-dimensional data possibly in conjunction with universal limiting state criteria whose limiting surfaces may or may not be axially symmetric with respect to the main diagonal of the stress space), processing, approximation (e.g. by parts), and estimation.

Keywords: Material 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 [2-12] that classical fundamental mathematical theories, methods, and concepts [1] are insufficient for adequately solving and even considering many typical urgent problems.

Megamathematics including overmathematics [2-12] 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 [13] which are universal.

Material Strength Science Unimathematical Test Fundamental Metasciences System

Material strength science unimathematical test fundamental metasciences system in megastrength [13] is one of such systems and can efficiently, universally and adequately strategically unimathematically test any material strength science. This system includes:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The material strength science unimathematical test fundamental metasciences system in megastrength [13] is universal and very efficient.

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

Fundamental Defects of Material Strength Sciences

Even the very fundamentals of classical material strength sciences [14-23] have evident cardinal defects of principle.

Modern engineering (astronautics, aircraft-building, ship-building, deep-sea industry, power engineering, electronics, chemical industry, building, etc.) requires optimal design of structural elements. It is based on a rational control of the necessary and sufficient strength of such elements and corresponding materials. Their types are diverse: ductile materials like metals; brittle alloys and nonmetals (glass, crystalline glass, concrete, and stone); anisotropic materials (fiber-reinforced ones and other composites). They are intended for extreme exploiting conditions (variable loading, high pressure, high or low temperature, radiation, etc.).

It is universally recognized that the most effective approach to solving strength problems in modern engineering is phenomenological. It provides considering more or less adequate mathematical, mechanical, physical, etc. models of materials and structural elements instead of real ones and uses mathematical (analytic and numeric) methods in theories of differential equations, elasticity, plasticity, creep, fracture, etc. to determine the usually triaxial stress state (in the stationary case) or process (in nonstationary loading) at each point of a structural element. It remains to compare the diverse triaxial stress states at all points of a solid (structural element) with one another by the degree of danger to reach the closest critical state (initiation of yielding, fracture etc.). Therefore, it is necessary to use so-called critical (limiting, ultimate) state (process) criteria (elasticity criteria, yield criteria, failure criteria, etc.) that reduce the problems to the simplest ones dealing with uniaxial stress states only investigated enough. To measure the proximity of a real stress state (process) to the closest critical one, safety factors might be used.

The well-known critical state and process criteria are separate for diverse materials types, have nothing in common with simple and universal fundamental laws of nature, and possess evident defects. Even if a ductile material is isotropic and has equal strength in tension and compression, the criteria ignore increasing strength in uniform triaxial compression. If a material has unequal strength in tension and compression, the criteria possess obviously restricted ranges of applicability and do not allow to compare arbitrary stress states with one another. No critical process criterion for anisotropic materials under variable loading, when the directions of the principal stresses at a solid's point under consideration can arbitrarily turn, is known at all.

Unistrength includes:

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

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

All the classical criteria of limiting states for triaxial mechanical stresses, even in principle applicable (not to mention its value), only the simplest special cases, usually to the ever-laden isotropic materials that are equally resistant to tension and compression. For the general case is an arbitrarily anisotropic materials, which have different resistance to tension and compression, for any variable loads and possible rotations of the principal directions of the stress-strain state at the point of the material during the loading did not even have suggestions on the wording of the criteria of limiting states for triaxial mechanical stresses and, therefore, a hint of the universal laws of nature strength. Furthermore, even for the simplest special case is constantly loaded with an isotropic material, which is equally resistant to tension and compression, the standard criteria Cod and Huber-Mises-Hencky completely insensitive to the addition of equiaxial stress states, a significant effect on the strength of which is proved by experiments of the Nobel Prize Bridgman and prescribe with respect to the tensile shear strength values ​​of 2 and square root of 3, respectively, for real materials varied from 1 to 4. A criterion of Tresca also ignores the effect of the intermediate principal stress.

Material unistrength, or the fundamental science of the mechanics and strength of materials, enters with the clear physical meaning of a universal dimensionless mechanical stress by the natural transformation of the conventional dimensional stresses and strength and for the first time discovers the universal laws of nature so that all the classical criteria for limiting mechanical states are very narrow and particular cases of these laws. These laws hold for even arbitrarily anisotropic natural and artificial materials that have different resistances to tensions and compressions for any variable loads with possible rotations of the principal directions of the stress-strain state at the point of the material during loading.

Material Unistrength Principles

Material unistrength principles form a system of revolutions in the strength of materials and consists of the following subsystems:

1) a stress subsystem of revolutions in strength of materials including the following principles:

stress universality;

stress invariance;

self-expression of the degree of danger;

an equivalent cycle (equidangerous to a given program of variable stresses);

vector concentration of a scalar program;

the priority of the principal stresses in anisotropy;

two-dimensional representation of three-dimensional data;

2) a limiting state criterion subsystem of revolutions in strength of materials including the following principles:

sensitivity to adding equiaxial stress states;

considering the effect of the intermediate principal stress;

taking into account the actual ratio of tensile strength to shear strength;

criterion universality;

scalar criteria for constant loads;

individual synchronous scalar criteria for variable loads;

integral program vector criteria for variable loads;

criterion correctability and improvability;

3) a subsystem of revolutions in strength of materials related to limiting and nonlimiting state criteria including the following principles:

underlimiting state criteria;

overlimiting state criteria;

quasicriticality of underlimiting, limiting, and overlimiting state criteria;

validity and usefulness of negative and imaginary equivalent stresses;

Among other material unistrength principles are the following:

fundamentality;

physical and mathematical meaning;

manifold unity;

efficient inheritance (correcting, irovmping, generalizing, and universalizing the classical criteria with establishing the limits of their applicability, appropriateness, and validity).

Material unistrength includes:

- The fundamental science of universal voltages at constant load, which includes a general theory of negative and imaginary equivalent stresses along with their modules, as well as the universal scalar reductions of mechanical stress to their individual limits of the same directions and signs. She first gets a universal dimensionless reduced tension by dividing the normal stress on the module size of its individual limit in the same direction and sign of the absence of all the other stresses and, ceteris paribus. This is a universal dimensionless stress is given equal treatment to each other individual stocks and the size of the dimensionless stress, taken from the general sign of stress. These are the general theory of universal scalar reductions of mechanical stress to their individual limits of the same lines and symbols for the main types of deformable bodies and types of loading:

isotropic materials, which are equally resistant to tension and compression, always loaded;

isotropic materials, which have different resistance to tension and compression, always loaded;

orthotropic material under such loading is constant, that the principal directions of the stress-strain state coincide with the main directions of orthotropy;

arbitrary anisotropic materials in any permanent loads;

- The fundamental science of the universal stresses in an arbitrarily anisotropic materials for any variable loads, which includes the general theory of universal synchronous scalar reductions of mechanical stress to their individual limits of the same lines and signs at each time of loading, as well as reductions of the universal vector obtained as much as the processes (programs ) individual dimensionless stress to their ravnoopasnym cycles and the corresponding universal dimensionless stress vector;

- The fundamental science of the universal laws of nature strength, which includes the general theory:

instantaneous (uniaxial), the equivalent of the universal reduced stress at each point of the body at any time of loading as a universal (defined specific criteria for ultimate limit states) of the major triad of universal scalar reduced stress, so this is equivalent to a universal given voltage has the same stock in accordance with this criterion, that this whole triad at the same point of the body at the same time of loading;

universal constant of the equivalent reduced stress at any point in the body such as the maximum instantaneous (single axis) given the equivalent of the universal stress at this point the body for the time of loading;

variable universal equivalent reduced stress at any point in the body as a universal module (defined specific criteria for ultimate limit states) of the triad (constant) the main purpose given the stress vector, each of which is uniquely determined in the uniaxial cycle corresponding to the principal stress is not ordered by the algebraic value (as and geometric interpretations in the space of principal stresses) and having a constant number of loading all the time that this cycle has the same stock as the whole process (entire program) corresponding to the uniaxial principal stress at the same point of the body for the time of loading;

generic equivalent reduced stress at any point in the body as most of the universal constant of the equivalent reduced stress, and variable universal equivalent reduced stress at this point the body for the time of loading.

This science also includes the general linear, piecewise linear and nonlinear (including quadratic and further power) theory of strength for the main types of deformable bodies and types of loading;

- The fundamental science of the analysis of a strength information, which includes the general theory of unification, modeling (including two-dimensional representation of three-dimensional data, possibly in conjunction with the universal criteria of limiting states, the surface which may or may not be symmetrical about the main diagonal of the stress space), processing , approximation and estimation.

The system of revolutions in mechanics and strength of materials includes:

1) a unistress subsystem of material mechanics and strength revolutions including:

negative and imaginary equivalent stress;

a unistress (universal dimensionless reduced stress);

the physical meaning of a unistress as the reciprocal of the individual reserve of the stress with its sign;

a universal scalar reduction of a mechanical stress to its individual limit of the same direction and sign for a constantly loaded isotropic material with different resistances to tension and compression;

a universal scalar reduction of a mechanical stress to its individual limit of the same direction and sign for an orthotropic materials under such constant loading that the principal directions of the stress-strain state coincide with the main directions of the orthotropy;

a universal scalar reduction of a mechanical stress to its individual limit of the same direction and sign for an arbitrarily anisotropic materials under any constant loading;

a universal scalar synchronous reduction of a mechanical stress to its individual limit of the same direction and sign for arbitrarily anisotropic materials under any variable loading;

a universal integral vector reduction of the whole processes (programs) of the individual dimensionless stresses to their equidangerous cycles for arbitrarily anisotropic materials under any variable loading;

a universal mixed synchronous-integral scalar-vector reduction of the whole process (program) of the three-dimensional stress state at an arbitrarily anisotropic material under any variable loading to the equidangerous universal dimensionless stress;

2) a universal law subsystem of material strength revolutions including:

the postulate of the universality of limiting state criteria in the universal dimensionless stresses;

a constant equivalent universal reduced dimensionless stress at constant loading;

an instant (uniaxial) equivalent universal reduced dimensionless stress at variable loading;

a constantly equivalent universal reduced dimensionless stress at variable loading;

a variably equivalent universal reduced dimensionless stress at variable loading;

an equivalent universal dimensionless reduced stress at variable loading;

metatheories of testing, correcting, improving, and generalizing universal limiting state criteria in the universal dimensionless stresses;

first discovered universal strength laws of nature in the universal dimensionless reduced stresses;

3) a strength information analysis subsystem of material strength revolutions including:

two-dimensional representation of three-dimensional data;

two-dimensional representation of three-dimensional universal limiting state criteria with critical surfaces asymmetric with respect to the main diagonal of the stress space;

the tolerable simplicity principle as a metacriterion for optimally selecting limiting state criteria types;

precise measurement of data trend and scatter, in particular, via the principal, upper, and lower quantibisectors of different orders;

overproportional influence on the results of this measurement as a criterion for determining outlier points;

the definition of data boundaries, levels, and intuitive quantibisectors without outlier points;

quantigrouping data points without outliers with respect to intuitive quantibisectors;

unimathematical division of an outlier point into its different parts each of which attachable to its appropriate quantigroup of data points;

data quantigroup quantibisector with the best regarding all outlier points;

4) a safety factor subsystem of material mechanics and strength revolutions including:

discovering the fundamental nonuniqueness of the analytic expression of any limiting state criterion;

discovering the fundamental nonuniqueness of the analytic expression of the safety factor of any nonlimiting state by any limiting state criterion;

discovering the fundamental admissibility of the classical definition of the safety factor for simple (proportional) loading only;

discovering the possibility of overestimating the real safety factor via its classical definition by an order of magnitude;

discovering the universal reserve of the worst-case combination of values ​​of certain parameters when changing within the boundaries defined by the parameters individual reserves expressed via a reserve common for them.

The fundamental science of the mechanics and strength of materials opens up entirely new opportunities vital not only for creating safe and efficient machinery and equipment, but also for the prediction of earthquakes, tsunamis, and other natural disasters, save lives and property.

I. Critical (Ultimate) States

The limiting (also called critical or ultimate) stress states [15-23] are determined by a limiting criterion [15-23] whose usual form is one of the following four:

σ_e = F(σ_j[o] | j = 1, 2, 3) = σ_L (σ_e = F(σ₁ , σ₂ , σ₃) = σ_L),

σ_e = F(σ_jk | j, k = 1, 2, 3) = σ_L ,

F(σ_j[o] | j = 1, 2, 3) = 0,

F(σ_jk | j, k = 1, 2, 3) = 0.

The two first of them explicitly define, by σ_e = F, an equidangerous (so-called equivalent, or effective) uniaxial stress, σ_e , also in non-limiting stress states. The equality F = σ_L holds if and only if a stress state is limiting correspondingly to σ_L . In all these formulae, F is a function of the triple of the principal stresses σ_j ordered (σ_1o ≥ σ_2o ≥ σ_3o), which is indicated by the optional index "o", in the case of their nonsymmetrical occurrence in F , or of the normal, σ_jj , and shear, σ_jk , or τ_jk , with j ≠ k , j , k = 1, 2, 3, stresses forming a stress tensor and possibly of some material constants. A limiting stress, σ_l , can be a yield stress, σ_y , or an ultimate strength, σ_u , equal in tension and compression, or different in tension, σ_t , σ_ty , σ_tu , and in compression, σ_c , σ_cy , σ_cu > 0, possibly in directions j, k, which is indicated by the additional indices j, k.

Each limiting criterion means a concrete definition of the function F , e.g., for isotropic ductile materials (σ_L = σ_t = σ_c), the Tresca criterion of the maximum shear stress [15, 17-23]

σ_e = σ_1o - σ_3o = σ_L ,

the Huber-von-Mises-Hencky criterion of the octahedral shear stress (or distortion energy) [15, 17-23]

σ_e = σ_i ={[(σ₁₁ - σ₂₂)² + (σ₂₂ - σ₃₃)² + (σ₃₃ - σ₁₁)² + 6(σ₁₂)² + 6(σ₂₃)² + 6(σ₃₁)²]/2}^1/2 = σ_L

2with the stress intensity, σ_i , the both being yield criteria.

For isotropic brittle materials with σ_t ≠ σ_c , the failure criteria used are the Galilei criterion of the maximum normal stress [15, 17-19, 22]

-σ_c ≤ σ_j ≤ σ_t , j = 1, 2, 3,

with at least one equality, the Coulomb criterion of internal friction [15, 17-19, 22]

σ_e = σ_1o - χσ_3o = σ_t (χ = σ_t/σ_c)

and the Pisarenko-Lebedev criterion [17, 19]

σ_e = (1 - χ)σ_1o + χσ_i = σ_t

that should give [17, 19] the Huber-von-Mises-Hencky criterion by χ = 1 and the Galilei criterion by χ = 0.

For orthotropic ductile materials with equal strengths in each tension and compression, σ_tj = σ_cj (= σ_Lj), j = 1, 2, 3, and only if the principal directions of a stress state coincide with the basic orthotropy directions 1, 2, and 3, one may use the Hu-Marin criterion [15, 17-19, 22]

σ₁²/(σ_L1)² + σ₂²/(σ_L2)² + σ₃²/(σ_L3)² - σ₁σ₂/(σ_L1σ_L2) - σ₂σ₃/(σ_L2σ_L3) - σ₃σ₁/(σ_L3σ_L1) = 1;

for anisotropic materials with σ_tj = σ_cj in each direction j , the von Mises-Hill criterion [15, 17-19, 22] with material constants F, G, H, L, M, N:

F(σ₁₁ - σ₂₂)² + G(σ₂₂ - σ₃₃)² + H(σ₃₃ - σ₁₁)² + 2L(σ₁₂)² + 2M(σ₂₃)² + 2N(σ₃₁)² = 1;

for anisotropic materials possibly with σ_tj ≠ σ_cj in some direction j , the Tsai criterion [15, 17-19, 22] with material constants of the forms F_α and F_αβ by the Einstein tensor notation:

F_ασ_α + F_αβσ_ασ_β = 1 (α, β = 1, 2, ... , 6; σ_j = σ_jj , j = 1, 2, 3; σ₄ = σ₁₂ , σ₅ = σ₂₃ , σ₆ = σ₃₁).

For uniaxial cyclic loading (fatigue) of an isotropic ductile material with σ_t = σ_c, one can use the Goodman linear approximation of the Haigh diagram [15, 17-19]

σ_m/σ_L + σ_a/σ_-1 = 1

(σ_m = (σ_max + σ_min)/2, σ_a = (σ_max - σ_min)/2)

with the mean stress σ_m of a cycle with maximum σ_max and minimum σ_min stresses, its amplitude stress σ_a, and the ultimate amplitude stress σ_-1 of a symmetric cycle (σ_min = -σ_max). For a symmetric cycle of a uniaxial normal stress combined with a symmetric cycle of a shear stress (usually bending combined with torsion), one may use the elliptic relation [15, 17-19]

(σ_a/σ_-1)² + (τ_a/τ_-1)² = 1

where τ_a is the amplitude stress of the shear cycle and τ_-1 is its ultimate amplitude stress. By nonstationarily loading an arbitrarily anisitropic solid with turning directions 1, 2, 3 of the principal stresses σ₁ ≥ σ₂ ≥ σ₃ , there are no known propositions to formulate such criteria [17, 19].

II. Noncritical States

It is not sufficient to only determine a usual safety factor, n_L , as the limiting stress, σ_L , divided by the equivalent stress, σ_e . Experiments can define in this aspect a limiting surface only:

F(σ₁ , σ₂ , σ₃) = σ_L ,

F^γ(σ₁ , σ₂ , σ₃)/σ_L^γ-1 = σ_L

both equivalent by any nonzero number γ . So, instead of

σ_e = F(σ₁ , σ₂ , σ₃),

we can consider

σ_eγ = F^γ(σ₁ , σ₂ , σ₃)/σ_L^γ-1

as the equivalent stress also for nonlimiting states when n_L ≠ 1. The usual safety factor [20],

n_Lγ = σ_L/σ_eγ = σ_L^γ/F^γ(σ₁ , σ₂ , σ₃) = n_L^γ,

can then take on any positive values when choosing suitable values of γ .

The usual safety factor can give too optimistic and therefore very dangerous values. We have in example 1

(σ₁ = 250 MPa, σ₂ = 240 MPa, σ₃ = 210 MPa, σ_L = 235 MPa)

n_L = 5.9, n_Lσ₁ - σ₃/n_L = 1439 MPa >> σ_L .

In example 2, a bar with strengths in tension and in compression

σ_t = 100 MPa, σ_c = 800 MPa

is contracted and stretched by two pairs of forces independently causing the stresses

σ = σ^- + σ⁺ = -500 MPa + 400 MPa = -100 MPa,

n_L = σ_c/|σ| = 8, n_Lσ^- + σ⁺/n_L = -3950 MPa << -σ_c , σ^-/n_L + n_Lσ⁺ = 3137.5 MPa >> σ_L .

Notata bene:

1. Modern engineering requires optimal design and rational control of resistant structural elements of ductile and brittle, isotropic and anisotropic materials under extreme stationary and variable loading. This needs adequately determining the danger of the real spatial stress process at any point of a solid with respect to the closest critical (limiting, ultimate) stress process. Critical (limiting, ultimate) spatial stress processes should be obtained from critical process criteria by using strength data available in simple experiments. The dangers of real spatial stress processes should be given by measures of the proximity of a real stress process to the closest critical one.

2. A usual stress is not a pure number, depends on the choice of physical dimensions (units) for a force and a length, is not numerically invariant by unit transformations, and alone represents no degree of the danger of itself even in stationary loading. If a solid’s material is not isotropic with equal strength in tension and compression, it is not reasonable even in stationary loading to compose functions of different stresses without their adequate weighing because of mixing their values having distinct limits and hence diverse degrees of danger. If loading is variable, the same holds even for different values of the process of a stress alone.

3. The known critical state criteria separate for diverse materials types, unlike simple and universal fundamental laws of nature, have contradictions, restricted and vague ranges of adequacy, sometimes lose physical sense, not always bring a suitable equivalent stress, and are applicable in the stationary case only. For an isotropic ductile material with equal strength in tension and compression even under stationary loading, the criteria ignore considerable strength increase in uniform triaxial compression. The only known attempts to propose a critical process criterion are reduced to very special cases of uniaxial stress cycles and of combined cyclic bending and twisting a bar. For anisotropic materials under variable loading, when the directions of the principal stresses at a solid's point under consideration can arbitrarily turn, there has been no attempt to propose a critical process criterion at all.

4. The only known measure of the proximity of a real stress process to the closest critical one is a safety factor as the ratio of a limiting stress to an equivalent stress. This could suffice only if all the principal stresses are directly proportional to a common variable parameter. Otherwise, a usual safety factor does not determine the permissible combinations of the initial data in a strength problem, can overestimate actual reserves by an order of magnitude, and is manifestly insufficient.

5. There were no fundamental mechanical and strength sciences satisfying the complex of modern requirements for inherent unity, consistency, sufficient completeness, universality, naturalness, simplicity, and convenience for practical application.

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

Revolution in Material Strength Sciences

Material megastrength fundamental sciences systems [2, 3, 6, 11, 24-41] revolutionarily replace the inadequate very fundamentals of classical material strength sciences [14-23] via adequate very fundamentals.

Material Strength Fundamental Sciences System

I. Constant Universal Stress Fundamental Science and Fundamental Science on Universal Strength Laws of Nature

There are known separate limiting criteria for different groups of materials [14-23]. But for variably loading any anisitropic material with possible turning directions 1, 2, 3 of the principal stresses σ₁ ≥ σ₂ ≥ σ₃ , there are no known propositions to formulate such criteria at all [17].

The main idea of the material strength fundamental sciences system is that limiting criteria for different materials and loading conditions have to be sufficiently universal fundamental laws of nature. But, for example, the Tresca limiting criterion

σ_e = σ₁ - σ₃ = σ_L

(σ_e the equivalent stress, σ_L the limiting stress) with the unique constant, σ_L , of the material can model the limiting surface for a certain ductile material only and in its usual form cannot be applied to any brittle one with two different limiting stresses σ_t in tension and σ_c > 0 in compression.

Let us assume that this criterion is only an expression of a certain temporarily unknown sufficiently general criterion that is now applied to a certain ductile material with σ_L . Try to determine, for such a desired criterion, its form that may not include σ_L . By dimensionality and similarity theories, it is appropriate to divide each principal stress by σ_L > 0:

σ₁/σ_L = σ₁°,

σ₂/σ_L = σ₂°,

σ₃/σ_L = σ₃°,

σ_e/σ_L = σ_e° = 1/n_L

where n_L is the reserve of σe with respect to σ_L . Then that criterion becomes pure (dimensionless)

σ_e° = σ₁° - σ₃° = 1

without evident constants of a material and holds for any ductile material independently of the specific value of σ_L . This provides further generalizing the pure criterion by generalizing σ_j° .

For a brittle material and each index j ∈ {1, 2, 3, e},

σ_j° = σ_j/σ_t if σ_j ≥ 0,

σ_j° = σ_j/σ_c if σ_j ≤ 0.

Analyzing [2, 3, 6, 11, 24-41] experimental data [14, 15, 17-23, 46-62] for many quite different ductile and brittle solids convincingly shows that this transformation independent of the limiting criteria unifies all the data and is an immediate expression of limiting criteria generalization theory itself. Moreover, in the space of the relative (reduced) principal stresses, σ₁°, σ₂°, σ₃°, and especially in the corresponding plane, those unified data evidently cluster near the limiting surfaces and curves by this pure criterion and the generalized Huber-von-Mises-Hencky limiting criterion (with the stress intensity, σ_i)

σ_e° = σ_i° = {[(σ₁° - σ₂°)² + (σ₂° - σ₃°)² + (σ₃° - σ₁°)²]/2}^1/2 = 1.

Generalizing an arbitrary limiting criterion

σ_e = F(σ₁ , σ₂ , σ₃) = σ_L

gives

σ_e° = F(σ₁°, σ₂°, σ₃°) = 1.

This criterion and its particular cases are invariant and universal in the space of σ₁°, σ₂°, σ₃° but have forms depending on the signs of σ₁ , σ₂ , σ₃ in their space.

For orthotropic materials, the principal directions, 1, 2, 3, of a stress state coinciding with the basic orthotropy directions at the same material's point, generalizing the above transformations gives

σ_j° = σ_j/σ_tj if σ_j ≥ 0,

σ_j° = σ_j/σ_cj if σ_j ≤ 0

with possible reindexing σ_j° to provide

σ₁° ≥ σ₂° ≥ σ₃°.

For any stationary loading an arbitrarily anisotropic material,

σ_j° = σ_j/|σ_Lj|

where σ_Lj is, for the usual principal stress, σ_j , its limiting value which has the direction and sign of σ_j and acts at the same material's point, the both other principal stresses vanishing, and the other loading conditions at the same point being the same.

Such recomprehending σ_tj and σ_cj further generalizes the transformation and preserves pure criteria forms unlike the von Mises-Hill criterion [1] superfluously complicated and other ones having nothing in common with sufficiently simple and universal laws of nature.

Such natural transformation is not the only for a brittle material with different strengths in tension and compression. In this case, by uniaxial cyclic loading in any principal direction, j, the limiting amplitude stress, σ_aj , can reach its peak, σ_ajmax , by a possibly nonzero mean stress, σ_m0j , of a cycle that is asymmetric in this nonzero case. Then also the stress state with σ_j = σ_m0j , j = 1, 2, 3, as opposed to the stress state with σ_j = 0, can be considered initial instead of the zero stress state. If

σ_m0j = (σ_tj - σ_cj)/2,

then a material with unequal strengths in tension and compression can be considered as one having these initial stresses -σ_m0j as a summary effect of microstresses and submicrostresses causing unequiresistibility as a phenomenological macroresult.

II. Variable Universal Stress Fundamental Science

The corresponding generalization of this transformation for variably loading at an arbitrary instant of time, t , from its interval, T = [t₀, t₁], gives

σ_j°(t) = [σ_j(t) - σ_m0j(t)]/|σ_Lj(t) - σ_m0j(t)|.

For each uniaxial stress process, σ_j(t), its reserve, n_j , is determined by the similar limiting process, n_jσ_j(t), with possibly taking damage accumulation into account. The equidangerous cycle of the relative (reduced) stresses with mean stress σ_mj° and amplitude one σ_aj° is determined by this formula. Then the constantly vectorial reduced stress

σ_j° = (σ_mj°, σ_aj°)

can be found by means of the limiting amplitude diagram. Finally, the pure criterion function universality postulate gives criterion

σ_e° = max{sup_t∈T max_ju(t) F(σ_1u°(t), σ_2u°(t), σ_3u°(t)), max_ju |F(σ_1u°, σ_2u°, σ_3u°)|} = 1,

the most dangerous, possibly depending on t , permutations of the stationary indexes, ju, of the unordered reduced principal stresses independently of

σ_1u° ≥ σ_2u° ≥ σ_3u°

being chosen.

III. Uniform Stress Fundamental Science (on Strength Criteria Generally Considering Influence of Pressure and the Intermediate Principal Stress)

The Tresca criterion and the Huber-von-Mises-Hencky criterion are not sensitive at all to hydrostatic tensions or compressions, the Tresca criterion to the intermediate principal stress, σ₂ , too, also important [15, 17-19, 22, 23].

The idea and physical sense of linearly correcting [2, 3, 11, 24-31] a critical state criterion are essentially the hypothesis on a linear influence of the principal stresses on reaching a limiting state. Namely, in limiting states, suppose the equivalent stress be no constant but a linear function of the principal stresses in accordance with equation

λ₀F(σ₁ , σ₂ , σ₃) + λ₁σ₁ + λ₂σ₂ + λ₃σ₃ = λ₄

where λ₀ , λ₁ , λ₂ , λ₃ , λ₄ are constants. The essence of a proposed approach to linearly correcting critical (limiting) criteria may be shown, e.g., in connection with the Tresca criterion. Taking the data on uniaxial tension and compression into account gives the criterion with an additional constant, x , of the material:

σ_e = σ₁ + xσ₂ - σ₃ = σ_L .

This criterion can also be obtained by taking the more specific hypothesis on the linear influence of the intermediate principal stress, σ₂ , on reaching a limiting state. The introduction of the pure (dimensionless) constant of a material, x , additional to the unique constant of a material σ_L , is justified as follows. The limiting stress, σ_L , is not sufficient to take into account the influence of the intermediate principal stress, σ₂ , and of hydrostatic tensions and compressions on reaching a limiting state. The physical sense of this constant, x , is that it is the uniaxial limiting stress, σ_L , divided by the limiting stress in hydrostatic tension, σ_ttt (σ₁ = σ₂ = σ₃ = σ_ttt). The last can be hardly determined directly but may be obtained by using the data on a third experiment with σ₂ ≠ 0, hence pure shear is not suitable. In biaxial compression and triaxial tension and compression we have, respectively:

x = 1 - σ_L/σ_cc (σ₁ = 0, σ₂ = σ₃ = -σ_cc),

x = 2 - σ_L/σ_tcc (σ₁ = σ_tcc , σ₂ = σ₃ = -σ_tcc).

Analogously correcting the Huber-von-Mises-Hencky criterion and the above general criterion gives criteria

σ_e = σ_i = {[(σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₃ - σ₁)²]/2}^1/2 + xσ₂ = σ_L ,

σ_e = F(σ₁ , σ₂ , σ₃) + xσ₂ = σ_L .

Further generalizing critical state criteria to extend strength laws hierarchies gives equations

G(σ₁°, σ₂°, σ₃°) = λ₀ + λ₁σ₁° + λ₂σ₂° + λ₃σ₃°,

G(σ₁°, σ₂°, σ₃°) = H(σ₁°, σ₂°, σ₃°)

where G, H are certain (maybe unknown unlike F) different functions of the reduced (relative) principal normal stresses and possibly of some pure constants of a solid's material [17, 19]. It is desirable to choose functions G and H having obvious physical sense and possibly least numbers of material's parameters. Their values may be selected from experimental data on uniaxial critical states and other simple ones.

IV. General Linear Strength Science

The linear form of strength criteria (for which σ_e can be expressed as a piecewise linear function of σ₁ , σ₂ , and σ₃) is the simplest one. It provides many advantages in science and engineering especially by solving complex strength problems. Moreover, for any precision measure and any nonlinear strength criterion, there are its piecewise linear approximations whose deviations from this criterion don't exceed this measure. Substantial scatter of strength test data with a certain quote of outliers is typical. That is why it is often admissible to consider piecewise linear strength criteria only.

The general linear form of strength criteria in σ_j° can be represented as [32]

σ_e° = a₀ + a₁σ₁° + a₂σ₂° + a₃σ₃° + ∑_i=1^N b_1i|c_00i + c_11iσ₁° + c_21iσ₂° + c_31iσ₃° + b_2i|c_02i + c_12iσ₁° + c_22iσ₂° + c_32iσ₃° + b_3i|c_03i + c_13iσ₁° + c_23iσ₂° + c_33iσ₃° + ... || ... | ≤ 1

where a₀ , a₁ , a₂ , a₃ , b_hi , c_0hi , c_1hi , c_2hi , c_3hi are any constants with their possible renaming and dropping unnecessary indices; h = 1, 2, ... , H are nesting levels; H and N are any nonnegative integers. If N = 0, it is the general pure linear form of strength criteria. H = 1 leads to the general linear form without nesting of moduli

σ_e° = a₀ + a₁σ₁° + a₂σ₂° + a₃σ₃° + ∑_i=1^N b_i|c_0i + c_1iσ₁° + c_2iσ₂° + c_3iσ₃°| ≤ 1.

General linear strength science along with a linear combination of the principal normal or shear stresses, as well as a lot of linear and some piecewise linear strength criteria has clear physical sense and is their natural generalization with discovering their applicability domains. It can correctly consider and express some known physical phenomena in material science, e.g. the substantial roles of σ₂ and of the relation between the normal and shear limiting stresses, general case σ_Lt ≠ σ_Lc , and Bridgman's phenomenon [14] of strength dependence on pressure.

In general linear strength science, the final general pure linear form

σ_e° = σ₁° + aσ₂° - σ₃° = 1

(a is any constant) of strength criteria generalizes the universalization [2, 3, 11, 27-29] of Tresca’s criterion [15-20, 22] via considering additional isotropic stress states, e.g. hydrostatic pressure, due to including aσ₂°. In fundamental material strength sciences [2, 3, 11, 27-29], this final form is already known due to uniform stress fundamental science.

V. Adding “Hydrostatic Terms”

There are very many known straightforward attempts to correct strength criteria by adding so called hydrostatic (more precisely, isotropic) term A(σ₁ + σ₂ + σ₃) with some factor A to simulate Bridgman's phenomenon [14] of strength dependency on pressure. This idea does not directly work. The reason is that sum σ₁ + σ₂ + σ₃ vanishes in pure shear but does not vanish in uniaxial tension and compression. Hence using the simplest tests data on uniaxial tensions and compressions turns this factor A into zero.

The unique possibility is using a function of σ₂ alone without σ₁ and σ₃ . This is reasonable because the distinct principal stresses σ₁ ≥ σ₂ ≥ σ₃ are of different importance according to Lode-Nadai’s parameter [17, 19, 22]. Many experiments proved the essential influence namely of σ₂ on strength. Note that additionally using σ₂ alone also provides simulating Bridgman's strength dependency on pressure. Take any function f(t) with f(0) = 0, e.g. f(t) = At with any constant A. Adding f(σ₂) for correcting the initial strength criterion, we obtain

σ_e(σ₁ , σ₂ , σ₃) = F(σ₁ , σ₂ , σ₃) + f(σ₂) = σ_L

taking both σ₂ and Bridgman's effect of adding isotropic stresses into account. Further generalizations [2, 3, 11, 27-29] can be used if necessary. General linear strength science also leads to the simplest f(t) = At .

VI. General Power Strength Sciences

Fundamental material strength sciences [2, 3, 11, 27-29] include general power strength sciences naturally further generalizing general linear strength science and possibly using moduli and radicals which both can be also nesting. General power strength sciences can still better than general linear strength science fit triaxial strength data in all areas and, unlike it, admit symmetric functions σ_e of σ₁ , σ₂ , σ₃ and using σ_1n , σ_2n , σ_3n with clear advantages. The initial form of power strength criteria with general homogeneous symmetric polynomials P_i(σ_1n°, σ_2n°, σ_3n°) of power i is

σ_e° = [∑_i=0^N a_iP_i(σ_1n°, σ_2n°, σ_3n°)]^1/N ≤ 1.

In the unstressed state, σ_e° = 0 is natural and leads to a₀ = 0. Case N = 2 gives form

σ_e° = [a₁(σ_1n° + σ_2n° + σ_3n°) + a₂(σ_1n°² + σ_2n°² + σ_3n°²) + b₂(σ_1n°σ_2n° + σ_1n°σ_3n° + σ_2n°σ_3n°)]^1/2 ≤ 1

which can provide a limiting surface of a paraboloidal type (due to adding a₁(σ_1n° + σ_2n° + σ_3n°) not to σ_e° but to σ_e°²) physically adequate and further generalizes the universalization of the Huber-von Mises-Hencky criterion

σ_e° = [σ_1n°² + σ_2n°² + σ_3n°² - (σ_1n°σ_2n° + σ_1n°σ_3n° + σ_2n°σ_3n°)]^1/2 ≤ 1

in fundamental material strength sciences. a₁(σ_1n° + σ_2n° + σ_3n°) corresponds to the typical idea to consider adding isotropic stress states, e.g. under hydrostatic pressure. But it does not work at all with using strength data in uniaxial tension and compression even by replacing σ_L with a general constant C at least by materials with σ_Lt = σ_Lc and hence by any materials. This is obvious due to fundamental material strength sciences with σ_Lt° = σ_Lc° = 1, to the nonuniversality of this approach, and to unlimited σ_e when σ_Lc - σ_Lt is very small. Using any function f(σ₂) with f(0) = 0 universally works but brings asymmetry of function σ_e of the principal stresses.

Megamathematics solves these general problems with perpetuating limiting surface continuity and the symmetry of σ_e as a function of the principal stresses. Fundamental material strength sciences replace usual σ₁ + σ₂ + σ₃ and reduced σ_1n° + σ_2n° + σ_3n° “hydrostatic sums” with their continuous functions f and f° vanishing at -σ_Lc , 0, σ_Lt and -1, 0, 1, respectively. Using uniaxial tension and compression data and renaming the constants leads to

σ_e° = [σ_1n°² + σ_2n°² + σ_3n°² - a(σ_1n°σ_2n° + σ_1n°σ_3n° + σ_2n°σ_3n°) + bf°(σ_1n° + σ_2n° + σ_3n°)]^1/2 ≤ 1.

Constant a provides considering true values of τ_L/σ_L and not only predefined 3^-1/2 by a = 1. This leads by b = 0 to ellipsoidal (by -2 < a < 1) and hyperboloidal (by a > 1) limiting surfaces and to “hydrostatic” strength limited in compression and unlimited in tension with concavity everywhere, respectively. This clearly contradicts strength test data and Drucker’s postulate [17, 19, 22]. The Huber-von Mises-Hencky cylinder [17, 19, 22] lies between those limiting surfaces as their limiting case. But using b ≠ 0 with piecewise linear functions, namely

f(σ₁ + σ₂ + σ₃) = σ₁ + σ₂ + σ₃ + σ_Lc if σ₁ + σ₂ + σ₃ ≤ -σ_Lc ,

f(σ₁ + σ₂ + σ₃) = 0 if -σ_Lc ≤ σ₁ + σ₂ + σ₃ ≤ σ_Lt ,

f(σ₁ + σ₂ + σ₃) = σ₁ + σ₂ + σ₃ - σ_Lt if σ₁ + σ₂ + σ₃ ≥ σ_Lt ;

f°(σ_1n° + σ_2n° + σ_3n°) = σ_1n° + σ_2n° + σ_3n°+ 1 if σ_1n° + σ_2n° + σ_3n° ≤ -1,

f(σ_1n° + σ_2n° + σ_3n°) = 0 if -1 ≤ σ_1n° + σ_2n° + σ_3n° ≤ 1,

f(σ_1n° + σ_2n° + σ_3n°) = σ_1n° + σ_2n° + σ_3n°- 1 if σ_1n° + σ_2n° + σ_3n° ≥ 1,

transforms those types of limiting surfaces to paraboloidal. Hence this quadratic form of strength criteria realizes the idea of independently considering the influences of τ_L/σ_L and of adding an isotropic stress state, e.g. hydrostatic pressure, on σ_e , can give a limiting surface of the paraboloidal type physically adequate in all triaxial stress areas, still better fits the same strength test data, and, by the principle of tolerable simplicity [2, 3, 11, 27-29, 33-41], needs no complication. Moreover, to truly compare the complexities of different strength criteria, represent them in forms namely with symmetric functions σ_e of σ₁ , σ₂ , σ₃ because representing limiting surfaces needs σ_1n , σ_2n , σ_3n . Hence quadratic strength criteria can be even simpler than linear and especially piecewise linear strength criteria whose namely linear forms can give functions σ_e of σ₁ , σ₂ , σ₃ always nonsymmetric.

VII. Shear to Normal Stress Fundamental Science (on Strength Criteria Generally Considering Relations between the Shear and Normal Limiting Stresses)

The τ_L/σ_L ratio of shear τ_L and normal σ_L limiting stresses of materials [17, 19, 22] takes different positive values not greater than 1 often with substantial deviations from 1/2 and 3^-1/2 predefined by the most common Tresca and Huber-von-Mises-Hencky criteria [17, 19, 22, 42]. The lower (inner) (Tresca's criterion [17, 19, 22]) and upper (outer) (Ishlinsky's deviatoric stress criterion [43]) bounds of all the convex (by Drucker's postulate [44]) limiting surfaces are well-known [45]. Yu [22, 23] proposed his twin-shear yield criterion coinciding with Ishlinsky's deviatoric stress criterion [43], showed that all the convex limiting surfaces correspond to relations 1/2 ≤ τ_L/σ_L ≤ 2/3, and generalized these bounds for σ_Lt ≠ σ_Lc . Yu also proposed his twin-shear unified strength theory [22, 23] generalizing that criterion and fitting data τ_L/σ_L = 0.376, 0.432, 0.451, and 0.474 [46-48], as well as τ_L/σ_L = 0.727 and up to 0.82 [46, 49, 50] for materials with nonconvex limiting surfaces. Data τ_L/σ_L = 0.71 and 0.74 for steel [51, 52], τ_L/σ_L = 0.25 and 0.27 for magnesium and 0.69 for bronze [51], τ_L/σ_L = 0.40 and 0.42 for alloys and 0.67 for steel [53], τ_L/σ_L = 0.65 and up to 0.76 for iron [54], as well as up to 1 for brittle building materials [17, 19], etc. are available, too.

In fundamental material strength sciences [2, 3, 11, 27-29, 33-41], general power strength sciences including general linear strength science generalizing Yu’s twin shear unified strength theory [22, 23] also fit all these and other data with scattering, e.g. via the following general strength criteria very simple.

Use the principal stresses σ₁ ≥ σ₂ ≥ σ₃ (regulated by this ordering) at a material's point along with limiting stress values σ_L such as yield stress σ_y or ultimate strength σ_L , namely σ_Lt in tension and σ_Lc in compression with σ_Lc ≥ 0 and α = σ_Lt/σ_Lc if σ_Lt ≠ σ_Lc . If the equivalent stress σ_e in a strength criterion is a symmetric function of the principal stresses, use nonregulated principal stresses σ_1n , σ_2n , and σ_3n without any predefined relations, which is very useful analytically and graphically.

1. Applying general linear strength science. This science exhaustively represents all piecewise linear strength criteria via initial general linear form

σ_e° = a₀ + a₁σ₁° + a₂σ₂° + a₃σ₃° + ∑_i=1^N b_1i|c_00i + c_11iσ₁° + c_21iσ₂° + c_31iσ₃° + b_2i|c_02i + c_12iσ₁° + c_22iσ₂° + c_32iσ₃° + b_3i|c_03i + c_13iσ₁° + c_23iσ₂° + c_33iσ₃° + ... || ... | ≤ 1

where a₀ , a₁ , a₂ , a₃ , b_hi , c_0hi , c_1hi , c_2hi , c_3hi are any constants with omitting unnecessary indices and renaming constants if this is useful; h = 1, 2, ... , H are nesting levels; H and N are any nonnegative integers. By H = 1,

σ_e° = a₀ + a₁σ₁° + a₂σ₂° + a₃σ₃° + ∑_i=1^N b_i|c_0i + c_1iσ₁° + c_2iσ₂° + c_3iσ₃°| ≤ 1.

For determining the constants, use the unstressed (σ° = 0) state (σ₁° = σ₂° = σ₃° = 0), intermediate nonlimiting (0 < σ° < 1), and limiting (σ° = 1) stress states under uniaxial tension (σ₁° = σ° ≥ 0, σ₂° = σ₃° = 0), uniaxial compression (σ₁° = σ₂° = 0, σ₃° = -σ° ≤ 0), and pure shear (σ₁° = σ°τ_L/σ_Lt ≥ 0, σ₂° = 0, σ₃° = - σ°τ_L/σ_Lc ≤ 0) as standard strength tests.

1.1. Applying the general pure linear form of strength criteria. N = 0 leads to the initial and final general pure linear forms of strength criteria

σ_e° = a₀ + a₁σ₁° + a₂σ₂° + a₃σ₃° ≤ 1, σ_e° = σ₁° + aσ₂° - σ₃° ≤ 1

with any constant a, to τ_L/σ_L = 1/2 like Tresca's criterion [17, 19, 22] by σ_Lt = σ_Lc = σ_L , and generally to 1/τ_L = 1/σ_Lt + 1/σ_Lc in its universalization via σ_e°. In fundamental material strength sciences [2, 3, 11, 27-29, 33-41], this final form is already known due to general stress criteria correction science with adding a homogeneous linear combination of the principal stresses to the expression of σ_e and using the standard tests data. Allowing negative values of σ_e° brings clear generalization replacing Tresca's prism (by a = 0) with a pyramid (by a ≠ 0) physically realistic by a > 0. Hence the pure linear form of strength criteria is the simplest one which has clear physical sense, generalizes many known pure linear strength criteria, and considers Bridgman's phenomenon for pressure-dependent materials [14]. But this form leads to the predefined relation between the normal and shear strengths and to a monotonic dependence of σ_e on σ₂ with contradicting many test data [17, 19, 22].

1.2. Applying general pure one-modulus linear form of strength criteria. a₀ = a₁ = a₂ = a₃ = 0 and N = H = 1 lead to their initial and final forms

σ_e° = |σ_e°| = |c₀ + c₁σ₁° + c₂σ₂° + c₃σ₃°| ≤ 1, σ_e° = |σ_e°| = |σ₁° + aσ₂° - σ₃°| ≤ 1

and to the same relations 1/τ_L = 1/σ_Lt + 1/σ_Lc and (by σ_Lt = σ_Lc = σ_L) τ_L/σ_L = 1/2.

Using the modulus gives a two-sided pyramid by a ≠ 0 and nothing new by a = 0.

1.3. Applying general mixed linear homogeneous form of strength criteria. N = H = 1 and a₀ = c₀ = 0 lead to their initial form

σ_e° = a₁σ₁° + a₂σ₂° + a₃σ₃° + b|c₁σ₁° + c₂σ₂° + c₃σ₃°| ≤ 1,

to the key role of the sign of difference τ_L/σ_Lt - σ_Lc/(σ_Lt + σ_Lc) (which equals τ_L/σ_L - 1/2 by σ_Lt = σ_Lc = σ_L), to b = τ_L(σ_Lt + σ_Lc)/(σ_Ltσ_Lc) - 1, and to the final form

σ_e° = (1 - (τ_L(σ_Lt + σ_Lc)/(σ_Ltσ_Lc) - 1)|c₁|)σ₁° + aσ₂° - (1 - (τ_L(σ_Lt + σ_Lc)/(σ_Ltσ_Lc) - 1)|c₃|)σ₃° + (τ_L(σ_Lt + σ_Lc)/(σ_Ltσ_Lc) - 1)|c₁σ₁° + c₂σ₂° + c₃σ₃°| ≤ 1.

Only for materials with τ_L/σ_Lt = σ_Lc/(σ_Lt + σ_Lc) (τ_L/σ_L = 1/2 by σ_Lt = σ_Lc = σ_L), there are additional strength criteria using any c₁ and c₃ with c₁c₃ ≤ 0 and any b. Hence even for such materials, moduli can be used and give additional strength criteria.

1.4. Applying general mixed linear form of strength criteria. N = H = 1 lead to their initial form

σ_e° = a₀ + a₁σ₁° + a₂σ₂° + a₃σ₃° + b|c₀ + c₁σ₁° + c₂σ₂° + c₃σ₃°| ≤ 1

and, only for materials with τ_L/σ_Lt = σ_Lc/(σ_Lt + σ_Lc) (τ_L/σ_L = 1/2 by σ_Lt = σ_Lc = σ_L), to additional criteria

- bc₀ + (1 - bc₁)σ₁° + a₂σ₂° + (-1 - bc₃)σ₃° + b|c₀ + c₁σ₁° + c₂σ₂° + c₃σ₃°| ≤ 1.

Hence namely such materials allow more simulation possibilities than others, which expresses strength phenomena specific for such materials only.

2. Applying general power strength sciences. Fundamental material strength sciences [2, 3, 11, 27-29, 33-41] include general power strength sciences naturally generalizing general linear strength science and possibly using moduli and radicals which both can be also nesting. Use, e.g., the homogeneous powers of the shear stresses (with clear generalizing Hosford’s criterion [16, 17, 19]):

σ_e = [a₁₃(σ₁ - σ₃)^k + a₁₂(σ₁ - σ₂)^k + a₂₃(σ₂ - σ₃)^k]^1/k ≤ σ_L (k > 0).

Uniaxial limiting stresses in tension and compression give strength criteria forms

σ_e = {a(σ₁ - σ₃)^k + (1 - a)[(σ₁ - σ₂)^k + (σ₂ - σ₃)^k]}^1/k ≤ σ_L ,

σ_e° = {a(σ₁° - σ₃°)^k + (1 - a)[(σ₁° - σ₂°)^k + (σ₂° - σ₃°)^k]}^1/k ≤ 1.

Pure shear reduced limiting stresses σ₁° = τ_L/σ_Lt , σ₂° = 0 , σ₃° = -τ_L/σ_Lc give (k ≠ 1)

σ_e° = {(σ_Lt^kσ_Lc^k/τ_L^k - σ_Lt^k - σ_Lc^k)/[(σ_Lt + σ_Lc)^k - σ_Lt^k - σ_Lc^k](σ₁° - σ₃°)^k +

[(σ_Lt + σ_Lc)^k - σ_Lt^kσ_Lc^k/τ_L^k]/[(σ_Lt + σ_Lc)^k - σ_Lt^k - σ_Lc^k][(σ₁°- σ₂°)^k + (σ₂° - σ₃°)^k]}^1/k ≤ 1.

In the simplest case k = 2 and then additionally by σ_Lt = σ_Lc = σ_L , we have criteria

σ_e° = {[σ_Ltσ_Lc/(2τ_L²) - (σ_Lt² + σ_Lc²)/(2σ_Ltσ_Lc)](σ₁° - σ₃°)² +

[(σ_Lt² + σ_Lc²)/(2σ_Ltσ_Lc) - σ_Ltσ_Lc/(2τ_L²)][(σ₁°- σ₂°)² + (σ₂° - σ₃°)²]}^1/2 ≤ 1,

σ_e° = {[σ_L²/(2τ_L²) - 1](σ₁° - σ₃°)² + [2 - σ_L²/(2τ_L²)][(σ₁°- σ₂°)² + (σ₂° - σ₃°)²]}^1/2 ≤ 1

which fit the above and any other data on the relation between the shear and normal limiting stresses for any materials. Additionally simply consider the influence of adding isotropic stress states, e.g. hydrostatic pressure, via adding aσ₂° (generally, any function g(σ₂°) vanishing at σ₂°) to the both last expressions of σ_e° with no influence on the limiting stresses in uniaxial tension and compression and pure shear. The obtained strength criteria fit strength test data on many artificial materials under static and variable loading [17, 19] with average relative errors of about 10 %. The same holds for comprehensive polyaxial strength test data on natural materials very different: Dunham dolomite, Solenhofen limestone, and Mizuho trachyte [55], coarse grained dense marble [56, 57], Shirahama sandstone and Yuubari shale [58], KTB deep hole amphibolite [59], Westerly granite [60], fine-grained Rozbark sandstone [61], and Soignies limestone [62]. For these data in triaxial compression only, no complication of this form is necessary and, by the principle of tolerable simplicity [2, 3, 11, 27-29], reasonable.

3. Applying an approach using combined bending and torsion. Another idea to correct critical state criteria is as follows. In many cases practically important, not only the ultimate normal stress σ_u , but also the ultimate shear stress τ_u and, due to them both, their ratio σ_u/τ_u are available [20, 21]. For estimating the danger of complex stress states in a ductile isotropic material with equal strength in tension and compression, the yield normal stress σ_y and the yield shear stress τ_y (both to be substituted for σ_u and τ_u , respectively), as well as the Tresca criterion and the Huber-von-Mises-Hencky criterion are commonly used. But they both, as well as other known criteria with pre-defined values of the σ_u/τ_u ratio, too, give no possibility to consider such additional data and, therefore, the distinctive features of many materials. That is why it is necessary to develop methods of correcting strength criteria in order to ensure taking the values of τ_u and σ_u/τ_u , if available, into account. To begin with, consider the classical problem [17, 19] of combined bending and torsion with a normal stress σ and a shear stress τ at a point of a solid. The principal stresses σ₁ , σ₂ , σ₃ commonly ordered (σ₁ ≥ σ₂ ≥ σ₃) are [17, 19]

σ₁ = (σ + (σ² + 4τ²)^1/2)/2, σ₂ = 0, σ₃ = (σ - (σ² + 4τ²)^1/2)/2.

According to the Tresca criterion with the pre-defined value 2 of σ_u/τ_u , we obtain criterion [17, 19]

σ_e = (σ² + 4τ²)^1/2 ≤ σ_u .

According to the Huber-von-Mises-Hencky criterion with pre-defined value 3^1/2 of σ_u/τ_u , we obtain criterion [17, 19]

σ_e = (σ² + 3τ²)^1/2 ≤ σ_u .

In the both cases, it is simply impossible to take any other value of the σ_u/τ_u ratio into account even if that is available. This lack can be very important when that value substantially deviates from the pre-defined one. Additionally, it would be a good practice to unify these both particular criteria. And namely this unification problem is relatively simple and gives further ideas to solve much more complex ones. It is easy to see that the factors 4 and 3 in these both particular criteria are the corresponding pre-defined values 2 and 3^1/2 of the σ_u/τ_u ratio which are raised to the 2nd power, i.e., the unified particular criterion is

σ_e = (σ² + (σ_u/τ_u)²τ²)^1/2 ≤ σ_u .

To understand the naturalness of this simple formula, divide all its parts by σ_u :

(σ_e⁰ = ) σ_e/σ_u = ((σ/σ_u)² + (τ/τ_u)²)^1/2 ≤ σ_u/σ_u ( = 1).

Hence each normal and shear stress is divided by its limiting value, which apparently develops the idea of the strength transformation methods. Note that the last two formulae not only unify the results of using the Tresca criterion and the Huber-von-Mises-Hencky criterion, but also for the first time provide taking any value of σ_u/τ_u into account and thus completely solving problems in the simple case of combined bending and torsion [17, 19], as well as of similar biaxial stress states [17, 19].

Let us extend the present method for the general case of a triaxial stress state. It is suitable to first consider the Huber-von-Mises-Hencky criterion. To take into account the unique additional material constant σ_u/τ_u , it is natural for the first attempt to include one unknown factor k into the corresponding formula (keeping it symmetric relatively to the principal normal stresses) in form

σ_e = (σ₁² + σ₂² + σ₃² + k(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u .

Note that the factor at the sum σ₁² + σ₂² + σ₃² should be namely unit only to provide that the criterion holds for a uniaxial tension. Thus such a suitable introduction of an unknown factor seems to be unique. To determine its value by using combined bending and tension once more, in this particular case we obtain

σ_e = (σ² + (2 - k)τ²))^1/2 ≤ σ_u .

Then we obtain

k = 2 - (σ_u/τ_u)²

and finally the corrected criterion

σ_e = (σ₁² + σ₂² + σ₃² + (2 - (σ_u/τ_u)²)(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u .

Now we see that we could initially choose formula

σ_e = (((σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₁ - σ₃)²)/2 + k(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u

leading to

k = 3 - (σ_u/τ_u)²

and finally to the equivalent corrected criterion

σ_e = (((σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₁ - σ₃)²)/2 + (3 - (σ_u/τ_u)²)(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u .

Now let us consider the Tresca criterion whose correction is much more complicated. First of all, it is impossible to keep the criterion’s linear form by correction because its factors 1 and -1 are unique to provide its applicability to a uniaxial tension and compression and introducing σ₂ brings nothing at least in the particular case of combined bending and torsion (σ₂ vanishes) and thus cannot, all the more, solve the general problem. That is why some complication is now necessary. Using the obtained experience with the Huber-von-Mises-Hencky criterion, it is natural to choose a similar quadratic form of a formula

σ_e = ((σ₁ - σ₃)² + kσ₁σ₃)^1/2 ≤ σ_u

taking into account the specific distinctive features of the initial formulae of the both criteria and using one additional constant k of a material. The same combined bending and torsion leads to

σ_e = (σ² + (4 - k)τ²))^1/2 ≤ σ_u

and to

k = 4 - (σ_u/τ_u)²

and finally to

σ_e = ((σ₁ - σ₃)² + (4 - (σ_u/τ_u)²)σ₁σ₃)^1/2 ≤ σ_u

or, equivalently,

σ_e = (σ₁² + σ₃² + (2 - (σ_u/τ_u)²)σ₁σ₃)^1/2 ≤ σ_u .

This formula shows that we could initially choose

σ_e = (σ₁² + σ₃² + kσ₁σ₃)^1/2 ≤ σ_u .

It is well known [17, 19] that for many ductile materials, experimental data on strength is often placed between the curves given by the Tresca criterion and the Huber-von-Mises-Hencky criterion. It is very natural to expect: The smaller the deviation of the σ_u/τ_u ratio from its value pre-defined by a strength criterion, the more precision of it by modeling experimental data. That is why one can use the unified criterion as the linear combination of the both criteria both using their above initial forms

σ_e = ((σ_u/τ_u)² - 3)(σ₁ - σ₃) + (4 - (σ_u/τ_u)²)(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u

or already corrected by the present methods

σ_e = ((σ_u/τ_u)² - 3)(σ₁² + σ₃² + (2 - (σ_u/τ_u)²)σ₁σ₃)^1/2 +

(4 - (σ_u/τ_u)²)(σ₁² + σ₂² + σ₃² + (2 - (σ_u/τ_u)²)(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u .

Such a linear unification approach is not the only. To naturally extend it, a power unification approach can be also used.

And the Hosford exponential unification [1–4]

σ_e = {[(σ₁ - σ₂)^k + (σ₂ - σ₃)^k + (σ₁ - σ₃)^k]/2}^1/k ≤ σ_L

gives the Tresca and Huber-von Mises-Hencky criteria by k = 1 and k = 2. For them, to provide these values of k using the same predefined values of the ratio σ_L/τ_L , naturally choose k = 5 - (σ_L/τ_L)². For further generalizations, introduce the notation (with apparently extending to any sets of variables and function values)

f[x, y | f(a, b) = v, f(c, d) = w]

for any function f taking, e.g., values v and w by values a, b and c, d of its variables x, y, respectively. Then choose f[σ_L/τ_L | f(2) = 1, f(3^1/2) = 0], g[σ_L/τ_L | g(2) = 0, g(3^1/2) = 1], and h[σ_L/τ_L | h(2) = 1, h(3^1/2) = 2] instead of k = (σ_L/τ_L)² - 3, k = 4 - (σ_L/τ_L)², and k = 5 - (σ_L/τ_L)², respectively.

And in general linear strength science [32], for a material with strengths σ_Lt in tension and σ_Lc (σ_Lc ≥ 0) in compression, sign(τ_L/σ_Lt - σ_Lc/(σ_Lt + σ_Lc)) plays the key role. If σ_Lt = σ_Lc = σ_L , then it is sign(τ_L/σ_L - 1/2). General power strength sciences generalizing that science can fit any relations between the shear and normal limiting stresses for materials with convex and nonconvex limiting surfaces.

VIII. General Reserve Fundamental Science

The main idea [2, 3, 6, 11, 25, 27-29] to realistically determine the reserve of a system under consideration is separately taking the reserves of its original parameters into account, each of these reserves being expressed via a common additional number. It is obtained from the condition that, by the worst realizable combination of the values of these parameters arbitrarily modified within the bounds determined by the corresponding reserves, the state of at least one element of the system becomes limiting, no element of it being in an overlimiting state. This is a further generalization of the universalization methods for critical state criteria. In a general problem, for any function of an arbitrary set of variables, where (α) means that index α ∈ Α is optional,

z = f[_α∈Α z_α], Z = f[_α∈Α Z_α], z_(α) ∈ Z_(α).

The genuine values of the independent variables, z_α , and of the dependent one, z , usually deviate from their values calculated. Those should belong to their admissible sets (domains), [Z_(α)], if the problem has certain limitations like strength criteria in strength problems. If

f[_α∈Α [Z_α]] ⊆ [Z],

the problem has been already solved. Otherwise, it is necessary to determine such a combination of the restrictions, Z_α , of the admissible sets, [Z_α], that the inclusion

f[_α∈Α Z_α] ⊆ [Z]

is true. For the existence of the numeric measures of those restrictions, or the reserves of the independent variables, it is sufficient that, for any α∈Α, [Z_(α)] is included into a certain Hilbert space, L_(α). It has the norm, ||z_(α)||_(α), of each element, z_(α), and the scalar product, (z_(α), z’_(α))_(α), of each pair of elements, z_(α) and z’_(α). The additive approach to obtaining reserves develops, generalizes, and extends the relative error in a certain sense. That naturally determines the neighborhood, Z_(α)(δ_(α), z_0(α)), of set Z_(α) with respect to element z_0(α) ∈ L_(α) with error δ_(α) ≥ 0 as the set of all z’_(α) ∈ L_(α) with

||z’_(α) - z_(α)||_(α) ≤ δ_(α)||z_(α) - z_0(α)||_(α).

The additive reserve of set Z_(α) by set [Z_(α)] with respect to element z_0(α) is defined as

n_a(α) = 1 + sup{δ_(α) ≥ 0: Z_(α)(δ_(α), z_0(α)) ⊆ [Z_(α)]}.

The multiplicative approach to obtaining reserves develops, generalizes, and extends the reserve factor in a certain sense. That gives the neighborhood,

Z_(α)(n_(α)exp(iφ_(α)), z_0(α)),

of set Z_(α) with respect to element z_0(α) ∈ L_(α) as the set of all z’_(α) ∈ L_(α). Here

0 ≤ φ_(α) ≤ π, i² = -1, n_(α)^-1||z_(α) - z_0(α)||_(α) ≤ ||z’_(α) - z_0(α)||_(α) ≤ n_(α)||z_(α) - z_0(α)||_(α)

where n_(α) ≥ 1 is a multiplicative reserve,

arccos[(z’_(α) - z_0(α), z_(α) - z_0(α))_(α)/(||z’_(α) - z_0(α)||_(α) ||z_(α) - z_0(α)||_(α))] ≤ φ_(α).

If the dimensionality of space L_(α) is at least two, then two independent parameters n(α) and φ_(α) can be used, but possibly with relation φ(α)(n(α)). If L_(α) is one-dimensional, then φ_(α) = 0. The multiplicative reserve of set Z_(α) by set [Z_(α)] with respect to element z_0(α) is defined as

n_m(α) = sup{n_(α) ≥ 1: Z(_α)(n_(α)exp(iφ_(α)), z_0(α)) ⊆ [Z_(α)]}.

By any of the both approaches, reserves n_α can be expressed via different nondecreasing functions of an additive reserve, n_fa , or a multiplicative one, n_fm , respectively, the both being common for reserves n_α and determined by the condition that there is an element z ∈ Z in a limiting state by the worst realizable combination of all z_α :

n_fa = sup{n ≥ 1: f[_α∈Α Z_α(n_α(n), z_0α)] ⊆ [Z]},

n_fm = sup{n ≥ 1: f[_α∈Α Z_α(n_α(n)exp(iφ_α(n_α(n))), z_0α) ⊆ [Z]}.

For simply (proportionally) loading, the multiplicative reserve is obtained from the condition

F(n_fmσ₁ , n_fmσ₂ , n_fmσ₃) = σ_L .

In the simplest case of the equal reserves of all z_α , in example 1 we obtain n_fa = 1.423, n_fm = 1.5 and in example 2 n_fmt = 1.25, n_fmc = 2 in tension and compression, respectively. These realistic reserves are significantly less than the usual ones too optimistic.

Fundamental mechanical and strength sciences introduce a universal dimensionless mechanical stress having clear physical sense by natural transformation of a usual dimensional pressure and for the first time discover universal strength laws of nature for which all classical limiting mechanical state criteria are very narrow special cases only. These laws hold for any anisotropic natural or artificial material with also different strengths in tension and compression under any variable loading and possibly rotating the principal directions of the stress state at a material point during loading time. Fundamental mechanical and strength sciences open essentially new vital possibilities not only for creating safe and resource-saving machines and constructions, but also for predicting earthquakes, tsunamis, and other natural cataclysms, saving people and property.

The material strength fundamental sciences system includes:

constant universal stress fundamental science which includes general theories of negative and imaginary equivalent stresses along with their modules (absolute values) and also of universal scalar reductions of mechanical stresses to their individual limits of the same directions and signs. This science introduces a universal dimensionless stress via dividing a usual dimensional stress by the module of its individual limit of the same direction and a sign in the absence of all the remaining stresses and under the same other loading conditions. This universal stress appears to be the inverse of the common value of the individual reserves of the both stresses with their common sign. These theories are general theories of universal scalar reductions of mechanical stresses to their individual limits of the same directions and signs for the following basic types of the materials of the deformable solids and types of their loading:

isotropic materials with equal strength in tension and compression under constant loading;

isotropic materials with different strengths in tension and compression under constant loading;

orthotropic materials when its basic orthotropy directions coincide with the principal directions of a stress state under constant loading;

anisotropic materials under constant loading;

variable universal stress fundamental science which includes general theories of universal synchronous scalar reductions of mechanical stresses to their individual limits of the same directions and signs at every moment of time and also of universal vector reductions of the received whole processes (programs) of the individual dimensionless stresses to their equidangerous cycles and to the corresponding universal vector dimensionless stresses;

fundamental science on universal strength laws of nature including general theories of:

– an instant (monoaxial) equivalent universal reduced stress at each point of a material at any moment of loading as a universal (defined by a specific limiting criterion) function of the triad of the principal scalar universal reduced stress so that this equivalent universal reduced stress has just the same reserve according to this criterion as this whole triad at the same point of a material at the same moment of loading time;

– a constantly equivalent universal reduced stress at any point of a material as the maximum of such instant (monoaxial) equivalent universal reduced stresses at this point of this material for the whole time of loading;

– a variably equivalent universal reduced stress at any point of a material as the module of the universal (defined by a specific limiting criterion) function of the triad of the (constant) principal vectorial universal reduced stresses each of which is unequivocally defined by such a monoaxial cycle of the corresponding reduced principal stress unordered by algebraic value (as well as at geometric interpretation in the space of the principal stresses) and having a constant number during the whole time of loading that this cycle has just the same reserve as the whole process (the whole program) of the corresponding monoaxial principal stress at the same point of a material for the whole time of loading;

– an equivalent universal reduced stress at any point of a material as the maximum of the constantly equivalent universal reduced stress and the variably equivalent universal reduced stress at this point of this material for the whole time of loading;

uniform stress fundamental science including intermediate principal stress general theory, principal stresses sum general theory, and nonlinear general theories;

general linear strength science including general pure linear criterion theory, general pure one-modulus linear criterion theory, general mixed linear homogeneous criterion theory, and general mixed linear nonhomogeneous criterion theory;

general power strength sciences including general power normal stress fundamental sciences and general power shear stress fundamental sciences;

shear to normal stress fundamental science including general theories of:

applying general linear strength science;

applying general power strength sciences;

applying an approach using combined bending and torsion;

general reserve fundamental science including general additive reserve theory and general multiplicative reserve theory.

The material strength fundamental sciences system is universal and very efficient.

Material Strength Fundamental Sciences System. Basic Results and Conclusions

1. Modern engineering requires optimal design and rational control of resistant structural elements of ductile and brittle, isotropic and anisotropic materials under extreme stationary and variable loading. This needs adequately determining the danger of the real spatial stress process at any point of a solid with respect to the closest critical (limiting, ultimate) stress process. Critical (limiting, ultimate) spatial stress processes should be obtained from critical process criteria by using strength data available in simple experiments. The dangers of real spatial stress processes should be given by measures of the proximity of a real stress process to the closest critical one. Strength criteria have to express the equivalent stress both by limiting and nonlimiting states via unified functions of the principal stresses and to take the influence of the intermediate principal stress, of an additional uniform stress state, and of the relation between the shear and normal strengths into account. These functions have to take their limiting values on the corresponding limiting surfaces only.

2. A usual stress is not a pure number, depends on the choice of physical dimensions (units) for a force and a length, is not numerically invariant by unit transformations, and alone represents no degree of the danger of itself even in stationary loading. If a solid’s material is not isotropic with equal strength in tension and compression, it is not reasonable even in stationary loading to compose functions of different stresses without their adequate weighing because of mixing their values having distinct limits and hence diverse degrees of danger. If loading is variable, the same holds even for different values of the process of a stress alone.

3. The known critical state criteria separate for diverse materials types, unlike simple and universal fundamental laws of nature, have contradictions, restricted and vague ranges of adequacy, sometimes lose physical sense, not always bring a suitable equivalent stress, and are applicable in the stationary case only. For an isotropic ductile material with equal strength in tension and compression even under stationary loading, the criteria ignore considerable strength increase in uniform triaxial compression. The only known attempts to propose a critical process criterion are reduced to very special cases of uniaxial stress cycles and of combined cyclic bending and twisting a bar. For anisotropic materials under variable loading, when the directions of the principal stresses at a solid's point under consideration can arbitrarily turn, there has been no attempt to propose a critical process criterion at all.

4. The only known measure of the proximity of a real stress process to the closest critical one is a safety factor as the ratio of a limiting stress to an equivalent stress. This could suffice only if all the principal stresses are directly proportional to a common variable parameter. Otherwise, a usual safety factor does not determine the permissible combinations of the initial data in a strength problem, can overestimate actual reserves by an order of magnitude, and is manifestly insufficient.

5. There were no fundamental mechanical and strength sciences satisfying the complex of modern requirements for inherent unity, consistency, sufficient completeness, universality, naturalness, simplicity, and convenience for practical application.

6. Fundamental mechanical and strength sciences being developed are based on the principle of tolerable simplicity specified analytically and the postulate stating the essential universality of critical process criterion functions. Fundamental mechanical and strength sciences include a transformation method for usual stresses, a generalization method for critical state (process) criteria, a correction method for critical state (process) criteria, and a generalization method for safety factors.

7. The transformation method for usual stresses is (essentially) reducing them to relative ones. A relative stress stationarily reduced is a number-value function of time, which is introduced as a usual stress divided by the modulus of its limiting value of the same sign in the same direction by vanishing all the other stresses at the same solid's point at the same time instant by the same other conditions. A relative stress nonstationarily reduced is a stationary vector of real-number length, whose abscissa and ordinate are the mean and amplitude relative stress values, respectively, of a cycle equidangerous to the corresponding usual stress process stationarily indexed. A relative stress is numerically invariant by any unit transformations. It is the reciprocal to a sign-preserving individual safety factor independently of choosing a limiting criterion and stress unit and expresses the degree of the danger of a stress process better than this factor, the usual individual safety factor, and this stress process itself. A passage to the relative stresses unites the experimental data on the strengths of different materials and raises the reliability of the results by their clustering. The relative stresses open many new ways in strength measurement and investigation to discover mechanical and physical laws of nature.

8. The transformation method for critical process criteria is essentially a passage in an initial critical state criterion to the relative stresses. First they are reduced stationarily, then nonstationarily with further using the modulus of the vector value of a criterion function by rules of vector algebra. Equalizing to unit the maximum of the two values of the criterion function by choosing the most dangerous permutation of the stress indexes in the both cases and the most dangerous instant of time for the criterion function of the relative stresses reduced stationarily gives the corresponding critical process criterion. It naturally generalizes the physical sense of the initial critical state criterion and expresses a relative equivalent stress, being the reciprocal to the result reserve (safety) factor, as a function of the individual reserve (safety) factors. They are taken with the signs (in stationary reducing) or the vector directions (in nonstationary reducing) of the corresponding relative stresses. Each general critical process criterion has a universal form like laws of nature and applies to any solid arbitrarily loaded. Critical process criteria in the relative stresses determine the applicability ranges of the known yield and failure criteria and extend them to universal strength laws of nature, which are verified by many known theoretical and experimental data and systematize them. Critical process criteria in the relative stresses can be naturally extended to transcritical criteria in the relative stresses naturally discriminating subcritical, critical, and supercritical stress processes.

9. The correction method for critical process criteria is essentially based on the natural hypotheses that in critical state, the value of a critical state criterion function of the stresses either usual or relative is equal to the value of another function of the same nature. In particular, the last function might be a certain linear function of the principal stresses either usual or relative. This method first allows to take into account the well-known considerable influence (established by Bridgman) of uniform triaxial compressions on reaching critical states. The method first establishes a considerable influence of uniform triaxial tensions on reaching critical states and allows to indirectly determine such limiting states and processes.

10. The generalization method for safety factors as a deep extension of the generalization method for limiting criteria is essentially worst-case measuring the proximity of a real process to the closest critical one and is not restricted to stress problems with stress processes. In any problem, for all input and output parameters, using such natural methods as additive and multiplicative ones, the ranges of the parameters are expressed through their design values and individual reserves being introduced. Applying worst-case approach to functional dependences expressing the output parameters through the input ones, the corresponding dependences correlating their individual reserves by the worst realizable combinations of the parameters values in the parameters ranges determined by these reserves are obtained. These reserves and their dependences realistically estimate tolerable deviations of the actual values of the parameters from their design values. It is possible to introduce some additional correlations and attach a priori values to some reserves. In particular, a result reserve can be expressed through the other ones as some functions of one real number, which makes it possible, to realistically determine all the other reserves, given a value of the result reserve. This allows to rationally control the authentic reserves of structures.

11. Fundamental mechanical and strength sciences first created as a source to discover many strength laws of nature are based on the principle, ideas, postulate, and hypotheses that generalize many well-known theoretical propositions and experimental data. These fundamental mechanical and strength sciences give simple (as far as it is possible in complicated problems to be solved) and practically convenient final results. They are very suitable, allow generally representing and processing measurement data, reduce time and cost expense, and bring many further advantages in mutual completion of difficultly obtained experimental data on the strength of different materials in spatial stress states under arbitrary variable loading. Fundamental mechanical and strength sciences create scientific foundations for rational control of the strength of modern materials and structural elements intended for extreme loading conditions. These sciences make clear physical sense, cover all stages in solving strength problems, develop many well-known results as manifestations of the discovered laws of nature, and found a new promising scientific trend of great importance in solid mechanics.

General Linear Strength Science. Additional Basic Results and Conclusions

1. The relation between the shear and normal strengths is a key one for choosing a suitable form of linear strength criteria. The critical value of the shear strength equals the doubled harmonic mean value of the normal strengths in tension and in compression. For a material with equal strength in tension and compression, this critical value is a half of normal strength. Materials namely with such a critical relation allow more simulation possibilities than others, which is especially interesting from the scientific point of view. This cannot be accident and therefore expresses strength phenomena specific for such materials which can be considered as the simplest ones from the mechanical point of view. This result is unexpected and very interesting from the scientific point of view. Note that Tresca's criterion as the historically first one explicitly considering shear stresses is namely linear and holds for isotropic materials with equal strength in tension and compression and the above value of shear strength only.

2. General linear strength science can correctly consider and express some known physical phenomena in material sciences. Among them are the following:

the substantial role of the intermediate principal stress;

the substantial role of the relation between the normal and shear strengths;

possibly different strengths in tension and compression;

Bridgman's phenomenon of the pressure dependence of strength.

3. General linear strength science is tested analytically. It naturally generalizes any linear strength criteria and can approximate any nonlinear ones. Therefore, general linear strength theory can precisely or approximately represent any strength criterion and discover its applicability domain.

4. General linear strength science is tested experimentally. There are many tests for a lot of linear and some piecewise linear strength criteria, as well as many nonlinear ones. If available experimental data correlate with a certain strength criterion, then the same holds for its precise or approximate representation via general linear strength science. Many such examples for the Galilei, Tresca, Coulomb-Mohr, Huber-von Mises-Hencky and other criteria are given in general linear strength science. And the exhaustive character of general linear strength science gives it many additional possibilities to precisely or aproximately express these and many other strength theories and criteria not only linear.

5. In particular, general linear strength science expresses and generalizes the Yu twin-shear unified strength theory. This piecewise linear strength theory generalizes many other ones. It determines the lower (inner) (Tresca's criterion) and upper (outer) (Yu's twin-shear yield criterion) bounds of all the convex (by Drucker's postulate) limiting surfaces. The Yu theory can also express nonconvex limiting surfaces and fit data for such materials. Naturally, the same holds for its representation in general linear strength science which gives many additional possibilities due to its exhaustion. If available experimental data do not correlate with a certain strength criterion, then general linear strength science can give the best linear or piecewise linear approximation to the data via an applicable strength criterion.

6. General linear strength science has many additional possibilities to rationally simulate practically any specific feature of the strength of a given material such as a certain character of the dependence of the equivalent stress on the intermediate principal stress, the availability of the extremums of this dependence, their positions, etc.

7. General linear strength science provides fundamental mechanical and strength sciences with initial criteria to discover the hierarchies of strength laws of nature. General linear strength science is based on the principle, ideas, postulate, and hypotheses that generalize many well known theoretical propositions and experimental data. This science gives simple (as far as it is possible in complicated problems to be solved) and practically convenient final results. It is very suitable, allows generally representing and processing measurement data, reduces time and cost expense, and brings many further advantages in mutual completion of experimental data (difficultly obtained) on the strength of different materials in polyaxial stress states. General linear strength science creates scientific foundations for rational control of the strengths of modern materials and structural elements intended for extreme loading conditions. It makes clear physical sense, develops many well known results as manifestations of the discovered laws of nature, and founds a new promising scientific trend of great importance in solid mechanics.

Uniform Stress Fundamental Science. Additional Basic Results and Conclusions

1. Both the influence of the intermediate principal stress on material strength and pressure dependence of material strength (Bridgman’s phenomenon) are very essential.

2. Many well-known strength criteria ignore both the influence of the intermediate principal stress on material strength and pressure dependence of material strength (Bridgman’s phenomenon).

3. To consider both the influence of the intermediate principal stress on material strength and pressure dependence of material strength (Bridgman’s phenomenon), strength criteria correction theory and methods in fundamental material strength science correct strength criteria via adding a function (vanishing at zero) of the intermediate principal stress to the expression of the equivalent stress in these criteria. Even the simplest linear function provides fitting available strength test data both on some artificial materials (under static and variable loading) and comprehensive polyaxial strength test data on many natural materials very different. General linear strength science in fundamental material strength sciences leads to the same linear function. But using these methods cannot provide the symmetry of the equivalent stress in these criteria as a function of the principal stresses.

4. To keep the symmetry of the equivalent stress in strength criteria as a function of the principal stresses by correcting strength criteria to provide considering both the influence of the intermediate principal stress on material strength and pressure dependence of material strength (Bridgman’s phenomenon), adding a symmetric function of the principal stresses can be useful. The most straightforward idea is adding a “hydrostatic term” as a function of the sum of the principal stresses. But this idea cannot work directly because using strength test data on uniaxial tension and compression leads to vanishing this function.

5. Megamathematics and fundamental material strength sciences provide suitable piecewise linear and other transformations of the sum of the principal stresses with independently fitting the standard tests data including the relation between the shear and normal limiting stresses and considering both the influence of the intermediate principal stress on material strength and pressure dependence of material strength (Bridgman’s phenomenon). This provides fitting both available strength test data on some artificial materials (under static and variable loading) and comprehensive polyaxial strength test data on many natural materials very different.

Shear to Normal Stress Fundamental Science. Additional Basic Results and Conclusions

1. The relations between the shear and normal limiting stresses for distinct materials are very different.

2. Many known strength criteria predefine the relations between the shear and normal limiting stresses and cannot fit their true relations for many materials.

3. In fundamental material strength sciences, general linear strength science and its natural generalization, namely general power strength sciences, correctly consider and express known physical phenomena in material sciences, have clear physical sense, use single formulae possibly with moduli, and fit the true relations between the shear and normal limiting stresses for many materials. These sciences generalize many known strength criteria, determine their applicability ranges, and provide independently fitting both the influence of adding isotropic stress states and the true relations between the shear and normal limiting stresses for many materials.

4. The relation between the shear and normal limiting stresses is critical for choosing a suitable form of strength criteria.

5. The linear, piecewise linear with one modulus, and quadratic forms of strength criteria are the simplest ones and provide many advantages.

6. General power strength sciences including general linear strength science provide fundamental material strength sciences with initial strength criteria to discover the hierarchies of strength laws of nature. These theories are very suitable, allow generally representing and processing test data, and reduce time and cost expense by polyaxial strength tests.

Strength Information Fundamental Sciences System of Strength Data Unification, Modeling, Analysis, Processing, Approximation, and Estimation

The strength information fundamental sciences system includes fundamental sciences of its analysis, synthesis, and unification; modeling (including two-dimensional representation of three-dimensional data possibly together with universal limiting criteria whose surfaces can be or not to be symmetric with respect to the principal diagonal of the space of the principal stresses); processing; approximation; and estimation.

1. Strength Data Unification. Use the principal stresses [15-19, 22] σ₁ ≥ σ₂ ≥ σ₃ (regulated by this ordering) at a material's point along with limiting stress values σ_L such as yield stress σ_y or ultimate strength σ_L , namely σ_Lt in tension and σ_Lc in compression with σ_Lc ≥ 0 and α = σ_Lt/σ_Lc if σ_Lt ≠ σ_Lc . If the equivalent stress σ_e in a strength criterion is a symmetric function of the principal stresses, use nonregulated σ_1n , σ_2n , and σ_3n without any predefined relations, which is very useful analytically and graphically.

In technical mechanics [15-19, 22], tensile stresses are considered positive and compressive stresses negative. In geomechanics [22], on the contrary, tensile stresses are considered negative and compressive stresses positive. To provide stress sign unification necessary and very useful in mechanics at all including both technical mechanics and geomechanics, introduce pressures

p = - σ , p₁ = - σ₃ , p₂ = - σ₂ , p₃ = - σ₁ , p₁ ≥ p₂ ≥ p₃ ,

use them in geomechanics instead of stresses, consider tensile stresses positive and compressive stresses negative as in technical mechanics, and regard tensile pressures negative and compressive pressures positive as in geomechanics, which is natural.

2. Strength Data Modeling. Apply fundamental science of data modeling [41]. Consider the space of the principal stresses with coordinate system Oσ₁σ₂σ₃ [42]. First, rotate this coordinate system about axis Oσ₃ by the Euler precession angle [42] ψ = π/4 = 45° with cos φ = sin φ = 2^-1/2and obtain coordinate system Oσ₁'σ₂'σ₃ such that transformed axis Oσ₁' becomes the bisectrix (bisector) between the previous axes Oσ₁ and Oσ₂ . Secondly, rotate this coordinate system Oσ₁'σ₂'σ₃ in plane Oσ₁'σ₃ by the Euler nutation angle [42] φ = arccos 3^-1/2 with cos φ = 3^-1/2 and sin φ = 3^-1/2 and obtain coordinate system Oσ₁''σ₂'σ₃' such that transformed axis Oσ₃' becomes the spatial diagonal σ₁ = σ₂ = σ₃ of the first octant of the initial coordinate system Oσ₁σ₂σ₃ . Use no intrinsic rotation, i.e., the Euler intrinsic rotation angle [42] θ = 0 with cos φ = 1 and sin φ = 0. Finally, rename the axes, namely, Oσ₁'' to Oσ_x , Oσ₂' to Oσ_y , and Oσ₂₃' to Oσ_z so that namely z-axis provides σ₁ = σ₂ = σ₃ . The transformation formulae for using this coordinate system Oσ_xσ_yσ_z instead of the initial coordinate system Oσ₁σ₂σ₃ are

σ_x = 6^-1/2σ₁ + 6^-1/2σ₂ - (2/3)^-1/2σ₃ , σ_y = - 2^-1/2σ₁ + 2^-1/2σ₂ , σ_z = 3^-1/2σ₁ + 3^-1/2σ₂ + 3^-1/2σ₃ .

In any half-plane starting at the diagonal axis Oσ_z and containing it and for any strength data point (σ_x , σ_y , σ_z) situated in this half-plane, introduce axis Oσ_m identified with axis Oσ_z , take axis Oσ_d in this plane with

σ_d = p_d = (σ_x² + σ_y²)^1/2

and place the corresponding diagram point (σ_m , σ_d = p_d). Here (nota bene) σ_d = p_d for using the first and second quadrants only (and NOT σ_d = - p_d) to provide nonnegative values only both for σ_d and for p_d , and introduce 2D diagram σ_mOσ_d of 3D both nonlimiting and limiting stresses and pressures, see the next figure

in which subscripts are not shown so that

σd = σ_d ,

σdL = σ_dL ,

pd = p_d ,

pdL = p_dL,

σm = σ_m = (σ₁ + σ₂ + σ₃)/3,

-pm = -p_m = -(p₁ + p₂ + p₃)/3,

σmL = σ_mL ,

σttt = σ_ttt .

The curve in this figure shows the intersection of the limiting surface of a certain limiting strength criterion with this half-plane starting at the diagonal axis Oσ_z and containing it.

If this limiting surface has a rotational symmetry about axis Oσ_z , then the choice of this half-plane has no influence on the results and, in particular, on this diagram.

By no rotational symmetry of this limiting surface, select any certain half-plane starting at the diagonal axis Oσ_z , containing it, and building any definite angle η (0 ≤ η < 2π) in the anticlockwise direction with axis Oσ_x , and denote this half-plane as the η-half-plane for which we shall create a unified end diagram by the following algorithm:

1. For any certain η'-half-plane starting at the diagonal axis Oσ_z , containing it, and building any definite angle η' (0 ≤ η' < 2π) in the anticlockwise direction with axis Oσ_x , create the corresponding η'-diagram in this η'-half-plane. In this η'-diagram, the curve shows the intersection of the limiting surface of the limiting strength criterion under consideration with this η'-half-plane. For any (either limiting or nonlimiting) η'-half-plane strength data point [σ_x(η'), σ_y(η'), σ_z(η') = σ_m(η')] initially situated in this η'-plane and own for it, consequently define and determine

σ_d(η') = p_d(η') = {[σ_x(η')]² + [σ_y(η')]²}^1/2 ,

limiting value σ_dL(η') of σ_d(η') either by the constant direction to (or from) the origin O if σ_m(η') ≥ 0 or by constant value σ_m(η') if σ_m(η') ≤ 0, the both definition and determination approaches coinciding if σ_m(η') = 0,

reserve

n(η') = σ_dL(η')/σ_d(η')

of this η'-diagram point [σ_m(η'), σ_d(η') = p_d(η')] also placed in this η'-diagram.

2. In particular, for the η-half-plane for which we shall create the unified end η-diagram, the curve in this η-diagram shows the intersection of the limiting surface of the limiting strength criterion under consideration with this η-half-plane. For any (either limiting or nonlimiting) η-half-plane strength data point [σ_x(η), σ_y(η), σ_z(η) = σ_m(η)] initially situated in this η-half-plane and own for it, consequently define and determine

σ_d(η) = p_d(η) = {[σ_x(η)]² + [σ_y(η)]²}^1/2 ,

limiting value σ_dL(η) of σ_d(η) either by the constant direction to (or from) the origin O if σ_m(η) ≥ 0 or by constant value σ_m(η) if σ_m(η) ≤ 0, the both definition and determination approaches coinciding if σ_m(η) = 0,

reserve

n(η) = σ_dL(η)/σ_d(η)

of this η-diagram point [σ_m(η), σ_d(η) = p_d(η)] also placed in this η-diagram.

3. For any (either limiting or nonlimiting) η'-half-plane (η’ ≠ η) strength data point [σ_x(η'), σ_y(η'), σ_z(η') = σ_m(η')] (initially NOT situated in this η-half-plane and NOT own for it) in which this point is initially situated and own for which, consider namely the corresponding η'-half-plane in which this point is initially situated and own for which, as well as for the corresponding η'-diagram point [σ_m(η'), σ_d(η') = p_d(η')], define and determine a corresponding additional (either limiting or nonlimiting) η-half-plane strength data point [σ_x(η), σ_y(η), σ_z(η) = σ_m(η)] in this η-half-plane initially NOT situated in this η-half-plane and NOT own for it, as well as for the corresponding additional η-diagram point [σ_m(η), σ_d(η) = p_d(η)] initially NOT situated in this η-diagram and NOT own for it. Namely, take σ_z(η) = σ_m(η) = σ_z(η') = σ_m(η') and n(η) = n(η'). To provide the last equality, first in this η'-diagram in this η'-half-plane, consequently take σ_d(η'), the corresponding values σ_dL(η'), and then n(η'). Secondly, consider this η-diagram in this η-half-plane, take n(η) = n(η'), σ_z(η) = σ_m(η) = σ_z(η') = σ_m(η'), then (by value σ_z(η) = σ_m(η) in this η-diagram) determine σ_dL(η), further

σ_d(η) = p_d(η) = σ_dL(η)/n(η),

place desired additional η-diagram point [σ_m(η), σ_d(η) = p_d(η)] and desired additional η-half-plane strength data point [σ_d(η) cos η , σ_d(η) sin η , σ_z(η) = σ_m(η)].

Placing all the spatial strength data points in one half-plane and in one two-dimensional diagram brings very many advantages by strength data analysis, comparing, processing, approximation, and estimation.

3. Strength Data Analysis. Apply fundamental material strength sciences [2, 3, 6, 11, 24-41] to selecting appropriate limiting strength criteria due to the principle of tolerable simplicity [2-12].

4. Strength Data Processing. Apply overmathematics [2-12], unimathematical modeling fundamental sciences system [63], and unimathematical data processing fundamental sciences system [64] to limiting strength data.

5. Strength Data Approximation. Apply unimathematical approximation fundamental sciences system [65] to limiting strength data, namely to selecting appropriate limiting strength criteria due to the principle of tolerable simplicity [2-12].

6. Strength Data Estimation. Apply unimathematical estimation fundamental sciences system [66] to strength data, namely to comparing them with limiting strength data.

7. Solving Strength Problems. Apply fundamental material strength sciences [2, 3, 6, 11, 24-41] and fundamental science of solving general problem fundamental sciences system [67] to solving strength problems.

The System of Revolutions in Mechanics and Strength of Materials

The system of revolutions in mechanics and strength of materials includes:

1) the stress universalization subsystem of revolutions, in particular:

stress universalization;

the physical sense of a universal dimensionless reduced stress as the reciprocal of its own reserve with the sign of the stress;

the universal scalar reduction of a mechanical stress via its division by the modulus of the stress limit with the same directions and the same sign for a constantly loaded isotropic material with different resistances to tension and compression;

the universal scalar reduction of a mechanical stress via its division by the modulus of the stress limit with the same directions and the same sign for a constantly loaded orthotropic material with different resistances to tensions and compressions if the principal directions of the stress state coincide with the main directions of orthotropy;

the universal scalar reduction of a mechanical stress via its division by the modulus of the stress limit with the same directions and the same sign for a constantly loaded arbitrarily anisotropic material with different resistances to tensions and compressions;

the universal synchronous scalar reduction of a mechanical stress via its division by the modulus of the stress limit with the same directions and the same sign for a variably loaded arbitrarily anisotropic material with different resistances to tensions and compressions;

the universal integrated vectorial reduction of the whole process (program) of a mechanical stress for an arbitrarily anisotropic material and for any variable loading so that the abscissa and the ordinate of such a vector equal the average universal stress and the amplitude universal stress, respectively, of the universal stress cycle equidangerous to the whole process (program) of the mechanical stress;

the universal mixed synchronous-integrated scalar-vectorial reduction of the triad of the whole processes (programs) of mechanical stresses for an arbitrarily anisotropic material and for any variable loading to the equidangerous universal dimensionless stress;

2) the limiting state criteria universalization subsystem of revolutions, in particular:

the admissibility, usability, and efficiency of negative and imaginary equivalent stresses along with their moduli;

the postulate of the universality of limiting state criteria in the universal stresses;

a constant equivalent (uniaxial) universal stress at constant loading;

an instant equivalent (uniaxial) universal stress at variable loading;

a scalarly equivalent (uniaxial) universal stress at variable loading;

a vectorially equivalent (uniaxial) universal stress at variable loading;

an equivalent (uniaxial) universal stress at variable loading;

the metatheories of testing, correcting, improving, generalizing, and universalizing limiting state criteria in the universal stresses;

the first discovered universal strength laws of nature in the universal stresses;

3) the strength information processing subsystem of principal innovations, in particular:

two-dimensionally representing three-dimensional data;

two-dimensionally representing universal limiting state criteria whose limiting surfaces may be not axially symmetric with respect to the main diagonal of the stress space;

the principle of tolerable simplicity as the best choice metacriterion for the types of limiting state criteria;

exactly unimeasuring strength data directedness and scatter including power mean and determinable via the main, upper, and lower unibisectors of different orders;

the clearly overproportional impact on the results of this unimeasuring as the criterion for determining outlier points;

determining strength data boundaries, levels, and intuitive unibisectors without outlier points;

unigrouping strength data without outlier points relatively to intuitive unibisectors;

unimathematically dividing any point into any parts separately attachable to appropriate strength data unigroups;

strength data unigroup unibisectors with the best regarding all the outlier points;

4) the reserve (safety factor) subsystem of revolutions, in particular:

discovering the fundamental nonuniqueness of the analytic expression of any limiting state criterion;

discovering the fundamental nonuniqueness of the analytical expression of the safety factor of any nonlimiting state via any limiting state criterion;

discovering the fundamental admissibility of the classical definition of the safety factor for simple (proportional) loading only;

discovering the principal possibility of multiply overestimating the actual reserve via the classical definition of the safety factor;

discovering the universal reserve via the worst-case combination of the values ​​of the separate independent determining parameters varying within the boundaries determined by the separate parameters reserves expressed via the universal reserve common for them.

Unimathematics, unimetrology, unimechanics, and unistrength lead to the following hierarchy of the laws of nature, in particular, the laws of strength of materials:

1) universal laws. For example, the universal law of strength of materials (the general mechanical state at an arbitrary point of a solid under loading is determined by a general relation between the universal state parameters including the principal stresses divided by the moduli of their own uniaxial limiting values ​​of the same directions and the same signs);

2) overgeneral laws, in particular, for a certain type of general relations such as a certain possibly nonlimiting relationship, say, a certain strength criterion;

3) general laws such as additionally for of a certain type of generally transforming the dimensional principal stresses into the dimensionless principal stresses;

4) subgeneral laws such as additionally for a particular type of loading, say, constant, cyclic, etc.;

5) certain laws such as additionally for a particular type of material anisotropy, for example, orthotropic;

6) peculiar laws such as additionally for a certain type of mutually orienting the principal stresses and material anisotropy, for example, for coinciding the principal directions of the stress-strain state and the main directions of material orthotropy;

7) particular laws such as additionally for a certain type of material nonequiresistibility to tensions and compressions, for example, for an equiresistible material;

8) special laws such as additionally for a particular type of the stress-strain state limitability, for example, yielding;

9), specific laws such as additionally for a particular choice of the possible nonlimitability of the stress-strain state, for example, sublimiting, limiting, or overlimiting;

10) individual laws such as additionally for a certain material (but under any loading of the selected type).

Basic Results and Conclusions

1. Well-known strength criteria, as well as concepts, methods, and approaches to strength data unification, modeling, analysis, processing, approximation, and estimation, have many principal shortcomings.

2. The fundamental sciences system of strength data unification, modeling, analysis, processing, approximation, and estimation provides 2D modeling of 3D strength data along with adequate strength data unification, modeling, analysis, processing, approximation, and estimation.

3. Fundamental sciences of strength data unification, modeling, analysis, processing, approximation, and estimation provide fitting the given strength data with fundamental strength laws of nature.

4. The fundamental sciences system of strength data unification, modeling, analysis, processing, approximation, and estimation is very suitable, allows generally representing and processing strength test data, and reduces time and cost expense by polyaxial strength tests.

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L. B. Tsvik. Triaxial Stress and Strength of Single-Layered and Multilayered High Pressure Vessels with Branch Pipes [In Russian]. Dr. Sc. Dissertation, Irkutsk, 2001

L. B. Tsvik. Computational Mechanics of Structural Elements Deformation and the Finite Element Method [In Russian]. GUPS Publishers, Irkutsk, 2005

L. B. Tsvik. Computer Technology, Stress and Strain Fields Modeling [In Russian]. Irkutsk State Technical University Publishing House, Irkutsk, 2005

In this paper the problem. Dry to handle. for 1978 is a fact - the destruction, where a rigid scheme of stress ..

L. B. Tsvik, P. G. Pimshteyn, E. G. Borsuk. Experimental studies of the stress-strain state of a multilayered cylinder with a monolithic input [In Russian]. Strength Problems, 1978, 4, 74-77

In the work 67 - tity of rooms specimen for mechanical testing in conditions of mild stress.

L. B. Tsvik. Specimen for mechanical testing of structural steel under cyclic loading [In Russian]. Track Facilities Problems in Eastern Siberia. Scientific papers of IrGUPS, 2005, 3, 200-203