Fundamental Defects of Classical Solid Mechanics

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Physical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

12 (2012), 10

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

Introduction

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

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

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

Solid Mechanics Unimathematical Test Fundamental Metasciences System

Solid mechanics unimathematical test fundamental metasciences system in mega-overmathematics, unimechanics, and unistrength [2] is one of such systems and can efficiently, universally and adequately strategically unimathematically test solid mechanics. This system includes:

fundamental metascience of solid mechanics test philosophy, strategy, and tactic including solid mechanics test philosophy metatheory, solid mechanics test strategy metatheory, and solid mechanics test tactic metatheory;

fundamental metascience of solid mechanics consideration including solid mechanics fundamentals determination metatheory, solid mechanics approaches determination metatheory, solid mechanics methods determination metatheory, and solid mechanics conclusions determination metatheory;

fundamental metascience of solid mechanics analysis including solid mechanics subscience analysis metatheory, solid mechanics fundamentals analysis metatheory, solid mechanics approaches analysis metatheory, solid mechanics methods analysis metatheory, and solid mechanics conclusions analysis metatheory;

fundamental metascience of solid mechanics synthesis including solid mechanics fundamentals synthesis metatheory, solid mechanics approaches synthesis metatheory, solid mechanics methods synthesis metatheory, and solid mechanics conclusions synthesis metatheory;

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

fundamental metascience of solid mechanics evaluation, measurement, and estimation including solid mechanics evaluation metatheory, solid mechanics measurement metatheory, and solid mechanics estimation metatheory;

fundamental metascience of solid mechanics expression, modeling, and processing including solid mechanics expression metatheory, solid mechanics modeling metatheory, and solid mechanics processing metatheory;

fundamental metascience of solid mechanics symmetry and invariance including solid mechanics symmetry metatheory and solid mechanics invariance metatheory;

fundamental metascience of solid mechanics bounds and levels including solid mechanics bound metatheory and solid mechanics level metatheory;

fundamental metascience of solid mechanics directed test systems including solid mechanics test direction metatheory and solid mechanics test step metatheory;

fundamental metascience of solid mechanics tolerably simplest limiting, critical, and worst cases analysis and synthesis including solid mechanics tolerably simplest limiting cases analysis and synthesis metatheories, solid mechanics tolerably simplest critical cases analysis and synthesis metatheories, solid mechanics tolerably simplest worst cases analysis and synthesis metatheories, and solid mechanics tolerably simplest limiting, critical, and worst cases counterexamples building metatheories;

fundamental metascience of solid mechanics defects, mistakes, errors, reserves, reliability, and risk including solid mechanics defect metatheory, solid mechanics mistake metatheory, solid mechanics error metatheory, solid mechanics reserve metatheory, solid mechanics reliability metatheory, and solid mechanics risk metatheory;

fundamental metascience of solid mechanics test result evaluation, measurement, estimation, and conclusion including solid mechanics test result evaluation metatheory, solid mechanics test result measurement metatheory, solid mechanics test result estimation metatheory, and solid mechanics test result conclusion metatheory;

fundamental metascience of solid mechanics supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement including solid mechanics supplement metatheory, solid mechanics improvement metatheory, solid mechanics modernization metatheory, solid mechanics variation metatheory, solid mechanics modification metatheory, solid mechanics correction metatheory, solid mechanics transformation metatheory, solid mechanics generalization metatheory, and solid mechanics replacement metatheory.

The solid mechanics unimathematical test fundamental metasciences system in megamathematics [2] is universal and very efficient.

In particular, apply the solid mechanics unimathematical test fundamental metasciences system to classical solid mechanics.

Fundamental Defects of Classical Solid Mechanics

Even the very fundamentals of classical solid mechanics 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 solid mechanics 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.

But classical solid mechanics:

uses a nonuniversal dimensional stress dependent on a choice of a specific system 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,

usually proposes no simple analytic solutions very suitable for theory and practice,

cannot set and solve truly three-dimensional problems without any assumption that some dimensions of solids are relatively small, e.g. the thickness even in thick plate theory,

cannot give general power solutions to three-dimensional harmonic and biharmonic equations playing the key role in elasticity theory etc.,

considers concentrating separate stresses only but not the equivalent stresses and therefore obtains inadequate results.

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

Therefore, the very fundamentals of classical solid mechanics have a lot of obviously deep and even cardinal defects of principle. There were no fundamental solid mechanics satisfying the complex of modern requirements for inherent unity, consistency, sufficient completeness, universality, naturalness, simplicity, and convenience for practical application.

References

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

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

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