Strength Information Fundamental Sciences System

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Strength Monograph

The “Collegium” All World Academy of Sciences Publishers

Munich (Germany)

2010

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 Analysis, Synthesis, and Unification. Use the principal stresses [1-9] 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, tensile stresses are considered positive and compressive stresses negative. In geomechanics, 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 [8, 10-12]. Consider the space of the principal stresses with coordinate system Oσ₁σ₂σ₃ [1]. First, rotate this coordinate system about axis Oσ₃ by the Euler precession angle [13] 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 [13] 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 [13] 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 [5-9] to selecting appropriate limiting strength criteria due to the principle of tolerable simplicity.

4. Strength Data Processing. Apply overmathematics [10-12, 14] and fundamental science of data processing [15] to limiting strength data.

5. Strength Data Approximation. Apply fundamental approximation science [16] to limiting strength data, namely to selecting appropriate limiting strength criteria due to the principle of tolerable simplicity.

6. Strength Data Estimation. Apply fundamental estimation science [17-19] to strength data, namely to comparing them with limiting strength data.

7. Solving Strength Problems. Apply fundamental material strength sciences [5-9] and fundamental science of solving general problems [20, 21] to solving strength problems.

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

4. Fundamental science 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.

References

[1] Filonenko-Borodich M. M. Mechanical Strength Theories. – Moscow State University Publishers, Moscow, 1961. – 92 pp. – In Russian.

[2] Pisarenko G. S., Yakovlev A. P., Matveev V. V. Handbook on Strength of Materials. – Kiev, Naukova Dumka, 1975. – 704 pp. – In Russian. – The same: in Spanish, – Moscow, 1979, – 696 pp. – The same in Spanish, 1985. – 696 pp. – The same in Spanish. – Moscow, 1989. – 694 pp. – The same in Portugese. – Moscow, 1985. – 682 pp. – The same in French. – Moscow, 1979. – 874 pp. – The same in French. – Moscow, 1985. – 874 pp.

[3] Pisarenko G. S., Lebedev A. A. Deformation and Strength of Materials at a Complex Stress State. – Kiev, Naukova Dumka, 1976. – 416 pp. – In Russian.

[4] Lebedev A. A., Koval'chuk B. I., Giginyak F. F., Lamashevsky V. P. Mechanical Properties of Structural Materials at a Complex Stress State. Handbook. – Kiev, Naukova Dumka, 1983. – 366 pp. – In Russian.

[5] Lev Gelimson. Generalization of Analytic Methods for Solving Strength Problems. – Sumy, Drukar Publishers, 1992. – 20 pp. – In Russian.

[6] Lev Gelimson. General Strength Theory. – Sumy, Drukar Publishers, 1993. – 64 pp.

[7] Lev Gelimson. General strength theory. Dedicated to Academician G. S. Pisarenko // Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin. – 2003. – 3. – P. 56 – 62.

[8] Lev Gelimson. Elastic Mathematics. General Strength Theory. – Munich, The “Collegium” All World Academy of Sciences Publishers, 2004. – 496 pp.

[9] Lev Gelimson. Providing helicopter fatigue strength: Unit loads [Fundamental strength sciences]. Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets, Proc. of the 23rd ICAF Symposium, 2005, Hamburg, Vol. II, pp. 589 – 600.

[10] Lev Gelimson. Basic New Mathematics. Drukar Publishers, Sumy, 1995. – 48 pp.

[11] Lev Gelimson. Providing Helicopter Fatigue Strength: Flight Conditions. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 405-416

[12] Lev Gelimson. Fundamental Science of Data Modeling. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

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

[14] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[15] Lev Gelimson. Fundamental Science of Data Processing. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[16] Lev Gelimson. Fundamental Science of Approximation. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[17] Lev Gelimson. General estimation theory. Transactions of the Ukraine Glass Institute, 1 (1994), 214-221. The same with its translation into Japanase. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2001

[18] Lev Gelimson. General Estimation Theory. The “Collegium” All-World Academy of Sciences Publishers, Munich (Germany), 2004. The same with its translation into Japanase. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[19] Lev Gelimson. Fundamental Science of Estimation. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[20] Lev Gelimson. General problem theory // Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin. – 2003. – 3. – P. 26 – 32.

[21] Lev Gelimson. Fundamental Science of Solving General Problems. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010