Strength Information Fundamental Sciences System (Essential)

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Strength and Engineering Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

10 (2010), 3

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.

Use the principal stresses σ₁ ≥ σ₂ ≥ σ₃ (regulated by this ordering) [1, 2] 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 [1, 2]. 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.

Apply fundamental science of data modeling [3, 4]. 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 [1, 5] ψ = π/4 = 45° with cos φ = sin φ = 2^-1/2 and 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 [1, 5] 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 [1, 5] 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

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

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.

References

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

[2] Lev Gelimson. Providing helicopter fatigue strength: Unit loads. 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, pp. 589 – 600.

[3] 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

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

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