Object Stress Analysis and Synthesis Fundamental Sciences System (Essential)

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), 5

UDC 539.4:620.17

Keywords: object stress analysis and synthesis fundamental sciences system, finite element method, stress and strain states analysis, analytic macroelement science, theory, discretization error, area, volume, mass moment of inertia, inhomogeneous distribution, biharmonic equation, general power solution, deformation compatibility, discrepancy, nontrivial true three-dimensional problem, elasticity theory, axisymmetric problem, complex optimization, equivalent stress concentration, circular, elliptic hole, contact stress, friction, bearing strength, washer.

This article is dedicated to the memory of my dear teacher, Academician Georgy Stepanovich Pisarenko (1910 - 2001) to the 101st anniversary of his birthday

The finite element method (FEM) is regarded standard in calculating the stress and strain states of objects. 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, obtains anyone "black box" result without any possibility and desire to check and test it but with beautiful graphic interpretation also impressing unqualified customers simply thinking 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 principal mistakes, 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. A computer is a blind powerful calculator 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.

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

General Stress Fundamental Sciences

Among them are:

object discretization fundamental science including general theory of discretization error estimation by determining area, volume, and mass moments of inertia and inhomogeneous distribution discretization error estimation general theory;

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

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

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

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

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

Local Stress Fundamental Sciences

Among them are:

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

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

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

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

References

[1] Encyclopaedia of Mathematics / Managing editor M. Hazewinkel. Dordrecht etc., 1988-1994.

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