General Power Strength Sciences (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), 2

The τ_L/σ_L ratio of shear τ_L and normal σ_L limiting stresses of materials [1] can substantially deviate from 1/2 and 3^-1/2 predefined by the most common Tresca and Huber-von Mises-Hencky criteria taking values at least between 0.25 and 1. Limiting surfaces can be convex by 1/2 ≤ τ_L/σ_L ≤ 2/3 only [2].

In fundamental material strength sciences [3, 4], general power strength sciences including general linear strength theory generalizing Yu’s twin shear unified strength theory [2] also fit all these and other data with scattering, e.g. via the following general strength criteria very simple.

Use the principal stresses σ₁ ≥ σ₂ ≥ σ₃ along with limiting stress values σ_L such as yield stress σ_y or ultimate strength σ_u , 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 .

Fundamental material strength sciences give universal strength laws of nature due to introducing relative (reduced) principal stresses σ_j° via dividing each usual principal stress σ_j (j = 1, 2, 3) by the modulus (absolute value) |σ_jL| of its individual limiting value σ_jL of the same sign in the same direction by vanishing the remaining two principal stresses under the same remaining load conditions:

σ_j° = σ_j / |σ_jL| (j = 1, 2, 3).

The inequalities σ₁° ≥ σ₂° ≥ σ₃° necessary if σ_e° nonsymmetrically depends on σ₁°, σ₂°, and σ₃° always hold for an isotropic material. For an anisotropic material, reindexing σ₁°, σ₂°, and σ₃° can be necessary to provide those inequalities.

Dependently on the essence of a certain strength criterion, safety and failure areas determined by it, and their limiting surface, also consider one-sided limitations for σ_e and σ_e° and their values even imaginary if they indicate still greater reserves than vanishing σ_e . Naturally extend σ_e and σ_e° by accepting their negative values with σ_e ≤ σ_L and σ_e° ≤ 1 and use nonnegative |σ_e| and |σ_e°|.

First, generalize Hosford’s criterion via introducing

σ_e = [a₁₃(σ₁ - σ₃)^k + a₁₂(σ₁ - σ₂)^k + a₂₃(σ₂ - σ₃)^k]^1/k ≤ σ_L (k > 0).

Uniaxial limiting stresses in tension and compression give strength criteria forms

σ_e = {a(σ₁ - σ₃)^k + (1 - a)[(σ₁ - σ₂)^k + (σ₂ - σ₃)^k]}^1/k ≤ σ_L ,

σ_e° = {a(σ₁° - σ₃°)^k + (1 - a)[(σ₁° - σ₂°)^k + (σ₂° - σ₃°)^k]}^1/k ≤ 1.

Pure shear reduced limiting stresses σ₁° = τ_L/σ_Lt , σ₂° = 0 , σ₃° = -τ_L/σ_Lc give (k ≠ 1)

σ_e° = {(σ_Lt^kσ_Lc^k/τ_L^k - σ_Lt^k - σ_Lc^k)/[(σ_Lt + σ_Lc)^k - σ_Lt^k - σ_Lc^k](σ₁° - σ₃°)^k +

[(σ_Lt + σ_Lc)^k - σ_Lt^kσ_Lc^k/τ_L^k]/[(σ_Lt + σ_Lc)^k - σ_Lt^k - σ_Lc^k][(σ₁°- σ₂°)^k + (σ₂° - σ₃°)^k]}^1/k ≤ 1.

In the simplest case k = 2 and then additionally by σ_Lt = σ_Lc = σ_L , we have criteria

σ_e° = {[σ_Ltσ_Lc/(2τ_L²) - (σ_Lt² + σ_Lc²)/(2σ_Ltσ_Lc)](σ₁° - σ₃°)² +

[(σ_Lt² + σ_Lc²)/(2σ_Ltσ_Lc) - σ_Ltσ_Lc/(2τ_L²)][(σ₁°- σ₂°)² + (σ₂° - σ₃°)²]}^1/2 ≤ 1,

σ_e° = {[σ_L²/(2τ_L²) - 1](σ₁° - σ₃°)² + [2 - σ_L²/(2τ_L²)][(σ₁°- σ₂°)² + (σ₂° - σ₃°)²]}^1/2 ≤ 1.

By k = 2 and σ_Lt = σ_Lc = σ_L , we have the following critical particular cases of τ_L/σ_L and a = σ_L²/(2τ_L²) - 1:

• τ_L/σ_L = 1/2, a = 1: σ_e° = σ₁° - σ₃° ≤ 1 (Tresca’s criterion),

• τ_L/σ_L = 1/3^1/2, a = 1/2: σ_e° = {[(σ₁° - σ₃°)² + (σ₁° - σ₂°)² + (σ₂° - σ₃°)²]/2}^1/2 ≤ 1 (the Huber-von-Mises-Hencky criterion),

• τ_L/σ_L = 1/2^1/2, a = 0: σ_e° = [(σ₁° - σ₂°)² + (σ₂° - σ₃°)²]^1/2 ≤ 1.

General power strength sciences can still better than general linear strength science fit triaxial strength data in all areas and, unlike it, admit symmetric functions σ_e of σ₁ , σ₂ , σ₃ and using σ_1n , σ_2n , σ_3n with clear advantages. The initial form of power strength criteria with general homogeneous symmetric polynomials P_i(σ_1n°, σ_2n°, σ_3n°) of power i is

σ_e° = [∑_i=0^N a_iP_i(σ_1n°, σ_2n°, σ_3n°)]^1/N ≤ 1.

In the unstressed state, σ_e° = 0 is natural and leads to a₀ = 0. Case N = 2 gives form

σ_e° = [a₁(σ_1n° + σ_2n° + σ_3n°) + a₂(σ_1n°² + σ_2n°² + σ_3n°²) +

b₂(σ_1n°σ_2n° + σ_1n°σ_3n° + σ_2n°σ_3n°)]^1/2 ≤ 1

further generalizing the universalization of the Huber-von-Mises-Hencky criterion

σ_e° = [σ_1n°² + σ_2n°² + σ_3n°² - (σ_1n°σ_2n° + σ_1n°σ_3n° + σ_2n°σ_3n°)]^1/2 ≤ 1

in fundamental strength sciences. a₁(σ_1n° + σ_2n° + σ_3n°) corresponds to the typical idea to consider adding isotropic stress states, e.g. under hydrostatic pressure. But it does not work at all with using strength data in uniaxial tension and compression even by replacing σ_L with a general constant C at least by materials with σ_Lt = σ_Lc and hence by any materials. This is obvious due to fundamental material strength science with σ_Lt° = σ_Lc° = 1, to the nonuniversality of this approach, and to unlimited σ_e when σ_Lc - σ_Lt is very small. Using any function g(σ₂) with g(0) = 0 universally works but brings asymmetry of function σ_e of the principal stresses.

Elastic mathematics [24, 26–28] solves these general problems with perpetuating limiting surface continuity and the symmetry of σ_e as a function of the principal stresses. Fundamental material strength science replaces usual σ₁ + σ₂ + σ₃ and reduced σ_1n° + σ_2n° + σ_3n° “hydrostatic sums” with their continuous functions f and f° vanishing at -σ_Lc , 0, σ_Lt and -1, 0, 1, respectively. Using uniaxial tension and compression data and renaming the constants leads to

σ_e° = [σ_1n°² + σ_2n°² + σ_3n°² - a(σ_1n°σ_2n° + σ_1n°σ_3n° + σ_2n°σ_3n°) +

bf°(σ_1n° + σ_2n° + σ_3n°)]^1/2 ≤ 1.

Constant a provides considering true values of τ_L/σ_L and not only predefined 3^-1/2 by a = 1. This leads by b = 0 to ellipsoidal (by -2 < a < 1) and hyperboloidal (by a > 1) limiting surfaces and to “hydrostatic” strength limited in compression and unlimited in tension with concavity everywhere, respectively. This clearly contradicts strength test data and Drucker’s postulate [8]. The Huber-von-Mises-Hencky cylinder [1–4] lies between those limiting surfaces as their limiting case, see Figure.

Figure

But using b ≠ 0 with piecewise linear functions, namely

f(σ₁ + σ₂ + σ₃) = σ₁ + σ₂ + σ₃ + σ_Lc if σ₁ + σ₂ + σ₃ ≤ -σ_Lc ,

f(σ₁ + σ₂ + σ₃) = 0 if -σ_Lc ≤ σ₁ + σ₂ + σ₃ ≤ σ_Lt ,

f(σ₁ + σ₂ + σ₃) = σ₁ + σ₂ + σ₃ - σ_Lt if σ₁ + σ₂ + σ₃ ≥ σ_Lt ;

f°(σ_1n° + σ_2n° + σ_3n°) = σ_1n° + σ_2n° + σ_3n°+ 1 if σ_1n° + σ_2n° + σ_3n° ≤ -1,

f(σ_1n° + σ_2n° + σ_3n°) = 0 if -1 ≤ σ_1n° + σ_2n° + σ_3n° ≤ 1,

f(σ_1n° + σ_2n° + σ_3n°) = σ_1n° + σ_2n° + σ_3n°- 1 if σ_1n° + σ_2n° + σ_3n° ≥ 1,

transforms those types of limiting surfaces to paraboloidal. Hence this quadratic form of strength criteria realizes the idea of independently considering the influences of τ_L/σ_L and of adding an isotropic stress state, e.g. hydrostatic pressure, on σ_e , can give a limiting surface of the paraboloidal type physically adequate in all triaxial stress areas, still better fits the same strength test data, and, by the principle of tolerable simplicity [21–28], needs no complication. Moreover, to truly compare the complexities of different strength criteria, represent them in forms namely with symmetric functions σ_e of σ₁ , σ₂ , σ₃ because representing limiting surfaces needs σ_1n , σ_2n , σ_3n . Hence quadratic strength criteria can be even simpler than linear and especially piecewise linear strength criteria whose namely linear forms can give functions σ_e of σ₁ , σ₂ , σ₃ nonsymmetric only.

It is very important that using f(σ₁ + σ₂ + σ₃) and f°(σ_1n° + σ_2n° + σ_3n°) has no influence on uniaxial tension and compression as well as on pure shear and hence on the value of the τ_L/σ_L ratio fully determined by a strength criterion with vanishing the factors by f(σ₁ + σ₂ + σ₃) and f°(σ_1n° + σ_2n° + σ_3n°).

And in general linear strength science [21–25], for a material with strengths σ_Lt in tension and σ_Lc (σ_Lc ≥ 0) in compression, sign(τ_L/σ_Lt - σ_Lc/(σ_Lt + σ_Lc)) plays the key role. If σ_Lt = σ_Lc = σ_L , then it is sign(τ_L/σ_L - 1/2). General power strength sciences generalizing that science can fit any relations between the shear and normal limiting stresses for materials with convex and nonconvex limiting surfaces.

References

[1] Lev Gelimson. Strength Criteria Generally Considering Relations Between the Shear and Normal Limiting Stresses. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[2] Yu M. H. Advances in strength theories for materials under complex stress state in the 20th Century // Appl. Mech. Rev. – 2002. – 55. – No. 3. – P. 169 – 218.

[3] 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.

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

[5] 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