Basic Analytic Macroelement Methods in Axially Symmetric Elasticity

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mechanical and Physical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

8 (2008), 1

To optimize structural elements in engineering including aeronautical fatigue, analytic methods [1-5] (along with the finite element method) are used.

An analytic macroelement method (AMEM) has been created in two modifications obtained via a linear combination method and a partial method (all by the author [4, 5]). They have been applied, e.g., to solving equations and their sets in axially symmetric elasticity by piecewise smooth boundary conditions.

The essence of the power modification of the AMEM is that the obtained general power representation of the Love’s axially symmetric biharmonic stress function [2]

L(r, z) = Σ_{i= 0}^∞ Σ_{j= 0} ^∞ (-1)ⁱ⁺¹i!^-2 j!^-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 and a_1k = 0,

a_0j and a_1j being two arbitrary number sequences, 0 ≤  j < ∞,

is used by steps to exactly or approximately satisfy the boundary conditions in a problem to be solved. The displacements and stresses are determined via applying the Love’s linear differential operators [2] to L. Love has shown that the biharmonicity of L is sufficient for precisely satisfying all the equilibrium and continuity equations. The biharmonicity necessity problem is now set and positively solved. For a cylindrical body, it is shown that it is enough to consider one base to be free of loads. The power expansions of the nonzero boundary conditions lead to four infinite subsets of linear algebraic equations over the coefficients in the function L. Their general solutions by their homogeneity are linearly expressed through the sequential zeroes of two Bessel functions and two ones analogous to them. It is shown that the compatibility of the boundary conditions in a boundary-value problem is necessary and sufficient for its exact solvability. If the body has a complicated form or the conditions are discontinuous then the body is mentally cut with canonical surfaces to a few macroelements. Solutions for them are conjugated by minimizing the responsible mean square residuals, collocation ones, and minimax modulus residuals. Their linear or quadratic correction makes stress determination in a strength problem more precise. For the Lamé solution [1], it is shown that the known linear generalization is the unique exact one.

The integral modification of the AMEM in axially symmetric elasticity is obtained by applying the partial method to the set of the differential equations in elastic stresses. The both equilibrium equations together with the continuity one having the first order allow precisely expressing the normal stresses via the boundary conditions in stresses and the distribution of the shear stress. The remaining continuity equation integro-differential of the second order makes exactly solving it unreal. Instead of that, by the introduced principle of tolerable simplicity [4], for the shear stress τ(r, z), its simplest statically possible distribution ensuring the equilibrium of each body part cut off by an arbitrary coaxial cylindrical surface is determined. For a cylindrical body a ≤ r ≤ b, c ≤ z ≤ d, we have

τ(r, z) = (b² - r²)(b² - a²)^-1 τ(a, z) + (r² - a²)(b² - a²)^-1 τ(b, z) + 6 (z - c)(d - z)(d - c)^-3

{- (b² - r²)(b² - a²)^-1∫_c^d  τ(a, z’) dz’ + (b/r - (r² - a²)(b² - a²)^-1)∫_c^d τ(b, z’) dz’ +

 r^-1∫_r^b[σ_z(r’, d) - σ_z(r’, c)] r’dr’} +(d - z)(2c + d - 3z)(d - c)^-2 [τ(r, c) -

(b² - r²)(b² - a²)^-1 τ(a, c) - (r² - a²)(b² - a²)^-1 τ(b, c)] +

(z - c)(3z - c - 2d)(d - c)^-2[τ(r, d) - (b² - r²)(b² - a²)^-1 τ(a, d) - (r² - a²)(b² - a²)^-1 τ(b, d)].

The radial, tangential, and axial normal stresses are determined by the formulae

σ_r(r, z) = (a/r)² (b² - r²)(b² - a²)^-1σ_r(a, z) + (b/r)² (r² - a²)(b² - a²)^-1σ_r(b, z) +

μr^-2∫_a^rσ_z(r’, d)r’dr’- μr^-2(r² - a²)(b² - a²)^-1∫_a^bσ_z(r’, d)r’dr’ - 2^-1(1 + μ)∫_a^r(∂τ(r’, z)/∂z)dr’+

 2^-1(1 + μ)b²r^-2(r² - a²)(b² - a²)^-1∫_a^b(∂τ(r’, z)/∂z)dr’ - 2^-1(1 - μ)r^-2∫_a^r(∂τ(r’, z)/∂z)(r’)²dr’+

2^-1(1 - μ)r^-2(r² - a²)(b² - a²)^-1∫_a^b(∂τ(r’, z)/∂z)(r’)²dr’ + μr^-1∫_z^d τ(r, z’)dz’ -

μar^-2(b² - r²)(b² - a²)^-1∫_z^d τ(a, z’)dz’ - μbr^-2(r² - a²)(b² - a²)^-1∫_z^d τ(b, z’)dz’ (μ the Poisson ratio);

σ_t(r, z) = ∂(rσ_r(r, z))/∂r + ∂(rτ(r, z))/∂z;

σ_z(r, z) = σ_z(r, d) + r^-1∫_z^d(∂(rτ(r, z))/∂r)dz’

with exact or approximate generalization of the Lamé solution [1]. To estimate the accuracy, the obtained solution is substituted into the remaining equation. For its left side in the body domain, the mean value is divided by the least upper bound on the sum of the moduli of its algebraic summands, each of them being a product of functions of the initial data.

[1] Lamé G. Lecons sur la theorie mathematique de l’élasticite des corps solides. Gauthier-Villars, Paris, 1852

[2] Love A. E. H. A Treatise on the Mathematical Theory of Elasticity. Vols. I, II. Cambridge University Press, Cambridge, 1892, 1893

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

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

[5] Lev Gelimson. Discretization Errors by Determining Area, Volume, and Mass Moments of Inertia. 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, 20-22