Generalization of the Tresca Criterion in Fundamental Strength Sciences

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)

7 (2007), 1

For an isotropic ductile material with equal strength in tension and compression, the Tresca criterion (of the maximum shear stress) [1]

σ_e = σ₁ - σ₃ = σ_l

(σ₁ , σ₂ , σ₃ the principal stresses with

σ₁ ≥ σ₂ ≥ σ₃ ,

σ_e the equivalent stress, and σ_l the uniaxial limiting stress) is commonly used. This formula includes the specific value of the only material constant s_l . Suppose this criterion to be a specific manifestation (as applied to a given material) of some unknown universal law of nature. The criterion formula itself cannot express that law, allows no generalization, and so needs a transformation. The simplest one using similarity and dimensionality theories is dividing each principal stress by the modulus σ_l of its ultimate value ±σ_l in uniaxlal state:

σ₁⁰ = σ₁/σ_l , σ₂⁰ = σ₂/σ_l , σ₃⁰ = σ₃/σ_l , σ_e⁰= σ_e/σ_l .

The transformed criterion becomes universal:

σ_e/σ_l = σ₁/σ_l - σ₃/σ_l = σ_l/σ_l = 1; σ_e⁰ = σ₁⁰ - σ₃⁰ = 1.

It has no evident material constant and so allows imparting a generalized sense (in comparison with this transformation) to the reduced (relative) principal stresses σ₁⁰, σ₂⁰, σ₃⁰ according to the specific character of the strength of any given material.

For an isotropic brittle material with unequal strengths in tension and compression, it is natural to reduce each principal stress by dividing it by the modulus of its ultimate value in the corresponding uniaxial state, that is σ_t in tension and -σ_c (σ_c > 0) in compression (j ∈ {1, 2, 3, e}):

σ_j⁰ = σ_j/σ_t if σ_j ≥ 0,

σ_j⁰ = σ_j/σ_c if σ_j ≤ 0.

The universal criterion

σ_e⁰ = σ₁⁰ - σ₃⁰ = 1

in the space of the usual principal stresses σ₁ , σ₂ , σ₃

σ_e = σ₁ - σ₃ = σ_c (0 ≥ σ₁ ≥ σ₂ ≥ σ₃),

σ_e = σ₁ - χσ₃ = σ_t (σ₁ ≥ 0 ≥ σ₃, χ = σ_t/σ_c),

σ_e = σ₁ - σ₃ = σ_t (σ₁≥ σ₂≥ σ₃ ≥ 0)

has obvious physical sense. If the signs of all the nonzero principal stresses are identical, a natural material having unequal strengths in tension and compression is similar to the two model materials. Each of them has equal moduli (either σ_t or σ_c , both instead of σ_l in the initial criterion) of the limiting stresses in tension and compression and is used when all the principal stresses are either nonnegative or nonpositive, respectively. If there are principal stresses with distinct signs, the critical states of a natural material are described by the criterion that coincides with the linear approximation of Mohr’s theory σ_e = σ₁ - χσ₃ = σ_t [2] in this case only, which determines [3] the not quite obvious applicability range of that theory.

For an orthotropic material when its basic directions coincide with the principal directions of a stress state, if such a material has limiting stresses σ_t1 , σ_t2 , σ_t3 in uniaxial tensions and σ_c1 , σ_c2 , σ_c3 in uniaxial compressions in the basic and simultaneously principal directions 1, 2, 3, then the last reduction can be naturally generalized by transformation (j ∈ {1, 2, 3, e})

σ_j⁰ = σ_j/σ_tj if σ_j ≥ 0,

σ_j⁰ = σ_j/σ_cj if σ_j ≤ 0

if necessary with renumbering the principal stresses that gives σ₁⁰ ≥ σ₂⁰ ≥ σ₃⁰.

For any anisotropic material and arbitrary static loading, the last reduction can be generalized by transformation σ_j⁰ =σ_j/|σ_lj|. Here σ_lj is the limiting value of a sole (uniaxial) principal stress s_j . That value has the direction and sign of stress s_j and acts at the same solid’s point under the same other loading conditions. This is a new generalized re-comprehension of the previous reduction if σ_tj and σ_cj mean the limiting stresses in tension and compression both in the direction of the principal stress σ_j but not indispensably in the basic directions of the anisotropic material, which are not obliged to exist. All these transformations apply to any strength criteria and are universal. And, in contrast to the Tresca criterion, the universal critical state criterion

σ_e⁰ = σ₁⁰ - σ₃⁰ = 1

in the reduced (relative) principal stresses σ₁⁰, σ₂⁰, σ₃⁰ [3, 4] always conserves its simple form like all fundamental laws of nature.

[1]      Tresca H. E. Memoire sur l'ecoulement des corps solides soumis a de fortes pressions. Comptes Rendus de l’Academie des Sciences, Paris, 1864, 59, 754-758

[2]      Mohr O. Abhandlungen aus dem Gebiete der Technischen Mechanik. W. Ernst und Sohn, Berlin, 1914

[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, 589-600

[4]      Lev Gelimson. Correcting and Further Generalizing Critical State Criteria in General Strength Theory. 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, 47-48