Linear Combination Method in Three-Dimensional Elasticity

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

RUAG Aerospace Services GmbH, Germany

Mechanical and Physical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

7 (2007), 1

In engineering including aeronautical fatigue, it is often necessary to optimize structural elements via analytic methods [1-4] (along with the FEM). Let us consider a general problem (an arbitrary set of functional equations) which perhaps can be represented in the form

F_α[_β∈_Β f_β[_γ∈_Γ z_γ]] r_α 0 (α∈Α)

where

F_α are known functions (α∈Α);

f_β  are desired functions (β∈Β);

z_γ  are independent variables (γ∈Γ);

Α, Β, Γ are corresponding sets of indexes;

[_α∈_Α a_α] is a system of indexed elements, a_α , in particular:

[_β∈_Β f_β] is a system of indexed desired functions,

[_γ∈_Γ z_γ] is a system of indexed independent variables;

r_α is a general relation.

The essence of the linear combination method is that each known function, F_α , and each desired one, f_β , is represented as a finite or infinite linear combination of functions from a chosen linearly independent class. It is closed over all the known functions, F_α , external with respect to the desired functions, f_β . The combination factors are numbers determinate for the known functions, F_α , and indeterminate for the desired functions, f_β . Then the equations of the set become linear algebraic over the indeterminate factors and are used to obtain their values.

In particular, to the homogeneous harmonic equation

∇²φ(x, y, z) = 0

over desired function φ(x, y, z) in the Cartesian coordinates, x, y, z, in the class of power functions (power series), or, equivalently, in a general pseudosolution [2]

L(r, z) = ∑_{i= 0}^∞∑_{j= 0}^∞∑_{k= 0}^∞a_ijkxⁱy^jz^k,

the linear combination method made it possible, to determine the most general solution

φ(x, y, z) = ∑_{i= 0} ^∞∑_{j= 0} ^∞∑_{k= 0} ^∞ (-1)^[i/2] [i/2]!(i!j!k!)^-1×

∑_{l= 0} ^[i/2] (j + 2l)!(k + 2[i/2] - 2l)! (l! ([i/2] - l)!)^-1×

 a_{i- 2[i/2], j + 2l, k + 2[i/2] - 2l}  xⁱy^jz^k

where

[m] = entier m

is the integral part of a real number m,

a_0jk and a_1jk are two arbitrary number sequences,

0 ≤ j < ∞, 0 ≤ k < ∞.

To the homogeneous biharmonic equation

∇²∇²L(r, z) = 0

over desired function L(r, z) in the cylindrical coordinates, r, z, in the class of power functions (power series), or, equivalently, in a general pseudosolution [2]

L(r, z) = ∑_{i= 0} ^∞ ∑_{j= 0} ^∞a_ijr²ⁱz^j,

the same method allows determining the most general solution

L(r, z) = ∑_{i= 0} ^∞a_{j= 0}^∞ (-1)^{i+ 1}i!^-2j!^-1[(2i + j - 2)!i 2^{2 - 2i}a_{1, 2i + j - 2}  + (2i + j)!(i - 1)2^-2ia_{0, 2i + j}]r²ⁱz^j

where by k < 0 we conventionally consider

k! = 1,

a_1k= 0,

a_0j and a_1j are two arbitrary number sequences,

0 ≤ j < ∞.

The both general solutions lead to the power modification of an analytic macroelement method (AMEM) [2-4] very suitable for three-dimensional problems in mechanics, strength, and fatigue [1-4].

[1] Handbuch Struktur-Berechnung. Prof. Dr.-Ing. L. Schwarmann. Industrie-Ausschuss-Struktur-Berechnungsunterlagen, Bremen, 1998

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

[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. Theory of Measuring Stress Concentration. In: Review of Aeronautical Fatigue Investigations in Germany During the Period May 2005 to April 2007, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2007-042 Technical Report, Aeronautical fatigue, ICAF2007, EADS Innovation Works Germany, 2007, 53-54