Regarding the Ratio of Tensile Strength to Shear Strength in Fundamental Strength Sciences

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

RUAG Aerospace Services GmbH, Germany

Strength and Engineering Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

6 (2006), 3

To predict fatigue and fracture of metals [1], fundamental strength sciences [2]generalize strength criteria and fits the substantial influence of the σ_u/τ_u ratio of the ultimate normal stress σ_u to the ultimate shear stress τ_u unlike common criteria with pre-defined values of this ratio. In combined bending and torsion with a normal stress σ and a shear stress τat a point of a solid [biaxial stress state], the principal stresses σ₁ , σ₂ , σ₃ commonly ordered (σ₁ ≥ σ₂ ≥ σ₃) are

σ₁ = (σ + (σ² + 4 τ²)^1/2)/2, σ₂ = 0, σ₃ = (σ - (σ² + 4 τ²)^1/2)/2.

That pre-defined value is 2 and 3^1/2 by the Tresca and Mises criteria [1], respectively,

σ_e = σ₁  - σ₃ ≤ σ_u , σ_e = (σ₁² + σ₂² + σ₃² + σ₁σ₂ + σ₁σ₃ + σ₂σ₃)^1/2 ≤ σ_u

[σ_e the equivalent uniaxial stress] giving in our case two particular and one unified criteria

σ_e = (σ² + 4 τ²)^1/2 ≤ σ_u , σ_e = (σ² + 3 τ²)^1/2 ≤ σ_u ; σ_e = (σ² + (σ_u/τ_u)² τ²)^1/2 ≤ σ_u .

To understand the naturalness of the last one, divide all its parts by σ_u :

σ_e° = ) σ_e/σ_u = ((σ/σ_u)² + (τ/τ_u)²)^1/2 ≤ σ_u/σ_u ( = 1)

with using relative (dimensionless) stresses (with the ° sign) introduced by the author [2].

For the general case of a triaxial stress state by using its three principal directions 1, 2, and 3 with vanishing all the shear stresses, first consider the Mises criterion. To take into account the unique additional constant σ_u/τ_u of a material, include one unknown factor k:

σ_e = (σ₁² + σ₂² + σ₃² + k(σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u

Note that uniaxial tension σ₁ > 0, σ₂ = 0, σ₃= 0 gives that the factor at the sum σ₁² + σ₂² + σ₃² should be 1. To determine k , use combined bending and tension once more:

σ_e = (σ² + (2 - k) τ²))^1/2 ≤ σ_u , k = 2 - (σ_u/τ_u)²,

σ_e = (σ₁² + σ₂² + σ₃² + (2 - (σ_u/τ_u)²) (σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u .

For the Tresca criterion, it is impossible to keep the criterion’s linear form by correction because its factors 1 and -1 are unique to provide its applicability to a uniaxial tension and compression σ₁ = 0, σ₂ = 0, σ₃ < 0, and introducing σ₂brings nothing at least in the particular case of combined bending and torsion [σ₂ vanishes] and thus cannot solve, all the more, the general problem. Now it is natural to choose a similar quadratic form of a formula

σ_e = ((σ₁ - σ₃)² + kσ₁σ₃)^1/2 ≤ σ_u .

The same combined bending and torsion leads to

σ_e = (σ² + (4 - k) τ²))^1/2 ≤ σ_u , k = 4 - (σ_u/τ_u)²,

σ_e = ((σ₁ - σ₃)² + (4 - (σ_u/τ_u)²) σ₁σ₃)^1/2 ≤ σ_u , σ_e = (σ₁² + σ₃² + (2 - (σ_u/τ_u)²) σ₁σ₃)^1/2 ≤ σ_u .

Experimental data on strength of ductile materials is often placed between the curves given by the Tresca and Mises criteria. Expecting more criterion precision by smaller deviation of the σ_u/τ_uratio from its pre-defined value gives the unified criterion as the linear combination

σ_e = ((σ_u/τ_u)² - 3) (σ₁ - σ₃) +

(4 - (σ_u/τ_u)²) (σ₁² + σ₂² + σ₃² + σ₁σ₂ + σ₁σ₃ + σ₂σ₃)^1/2 ≤ σ_u .

The unified criterion using the criteria forms already corrected by the present methods is

σ_e = ((σ_u/τ_u)² - 3) (σ₁² + σ₃² + (2 - (σ_u/τ_u)²) σ₁σ₃))^1/2 +

(4 - (σ_u/τ_u)²) (σ₁² + σ₂² + σ₃² + (2 - (σ_u/τ_u)²) (σ₁σ₂ + σ₁σ₃ + σ₂σ₃))^1/2 ≤ σ_u .

Note that such a linear unification approach is not the only. To naturally extend it, a power unification approach can be also used. And the Hosford exponential unification [3]

σ_e = (((σ₁ - σ₂)^k+ (σ₂ - σ₃)^k+ (σ₁ - σ₃)^k)/2)^1/k ≤ σ_u

of the Tresca and Mises criteria gives them by k = 1 and k = 2. To provide these values of k using the same pre-defined values of the σ_u/τ_u ratio for the both criteria, naturally choose k = 5 - (σ_u/τ_u)² . For further generalizations, introduce the notation (with apparently extending for any sets of variables and function values)

f(x, y | f(a, b) = v, f(c, d) = w)

for any function taking, e.g., values v and w by values a, b and c, d of its variables x, y, respectively. Then choose instead of (σ_u/τ_u)² - 3, 4 - (σ_u/τ_u)² and k = 5 - (σ_u/τ_u)², respectively:

f(σ_u/τ_u | f(2) = 1, f(3^1/2) = 0),

g(σ_u/τ_u | g(2) = 0, g(3^1/2) = 1),

h(σ_u/τ_u | h(2) = 1, h(3^1/2) = 2).

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

[2] Lev Gelimson. General Strength Theory. Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin, Publisher Prof. Dr. habil. V. Mairanowski, 3 (2003), Berlin, 56-62

[3] Hosford W. F. A generalized isotropic yield criterion. Trans. A. S. M. E., Ser. E 2 (1972), 290-292