Generalization of the Huber-von-Mises-Henky 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)

8 (2008), 1

For an isotropic ductile material with equal strength in tension and compression, the Huber-von-Mises-Henkycriterion (of the potential energy of distortion) [1-3]

σ_e = {[(σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₃ - σ₁)²]/2}^1/2 = 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. To generalize this criterion, divide (using ideas and approaches in similarity and dimensionality theories) each principal stress by the modulus σ_l of its ultimate value ±σ_l in uniaxlal state [4, 5]:

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

The transformed criterion becomes universal:

σ_e⁰ = {[(σ₁⁰ - σ₂⁰)² + (σ₂⁰ - σ₃⁰)² + (σ₃⁰ - σ₁⁰)²]/2}^1/2 = 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, reduce each principal stress by dividing it by the modulus of its ultimate value in the corresponding uniaxial state, that is s_tin 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.

In this case, the universal criterion in the space of the usual principal stresses σ₁ , σ₂ , σ₃ is

σ_e ={[(σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₃ - σ₁)²]/2}^1/2 = σ_c (0 ≥ σ₁ ≥ σ₂ ≥ σ₃);

σ_e = σ_i ={[(σ₁ - χσ₂)² + χ² (σ₂ - σ₃)² + (χσ₃ - σ₁)²]/2}^1/2 = σ_t (σ₁ ≥ 0 ≥ σ₂ ≥ σ₃, χ= σ_t/σ_c);

σ_e ={[(σ₁ - σ₂)² + (σ₂ - χσ₃)² + (χσ₃ - σ₁)²]/2}^1/2 = σ_t (σ₁ ≥ σ₂ ≥ 0 ≥ σ₃);

σ_e ={[(σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₃ - σ₁)²]/2}^1/2 = σ_t (σ₁ ≥ σ₂ ≥ σ₃ ≥ 0).

It gives a limiting surface symmetric with respect to axis σ₁ = σ₂ = σ₃ and consisting of the two semi-infinite von Mises cylinders with radii (2/3)^1/2σ_t by σ₁ + σ₂ + σ₃ ≥ σ_t , (2/3)^1/2σ_c by σ₁ + σ₂ + σ₃ ≤ -σ_c and of the frustum of a cone (connecting them) by -σ_c ≤ σ₁ + σ₂ + σ₃ ≤ σ_t .

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

with no renumbering σ₁ , σ₂ , σ₃ that gives σ₁⁰ ≥ σ₂⁰ ≥ σ₃⁰.

σ_lj = σ_tj = σ_cj (j = 1, 2, 3),

we obtain the Hu-Marin criterion (as a particular case) but with σ_e⁰

σ_e⁰ = [σ₁²/(σ_l1)² + σ₂²/(σ_l2)² + σ₃²/(σ_l3)² - σ₁σ₂/(σ_l1σ_l2) - σ₂σ₃/(σ_l2σ_l3) - σ₃σ₁/(σ_l3σ_l1)]^1/2 = 1

and (by σ_tj ≠ σ_cj in some direction j) its generalization, e.g., by σ₁ ≥ 0, σ₂ ≤ 0, σ₃ ≤ 0,

σ_e⁰ = [σ₁²/(σ_t1)² + σ₂²/(σ_c2)² + σ₃²/(σ_c3)² - σ₁σ₂/(σ_t1σ_c2) - σ₂σ₃/(σ_c2σ_c3) - σ₃σ₁/(σ_c3σ_t1)]^1/2 = 1.

For any anisotropic material and arbitrary static loading, the last reduction is also generalized:

σ_j⁰ = σ_j/|σ_lj|.

By this transformation, σ_lj is the limiting value of a sole (uniaxial) principal stress σ_j . That value has the direction and sign of σ_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 σ_j but not indispensably in the basic directions of the anisotropic material which are not obliged to exist. All these universal transformations apply to any strength criteria. And, in contrast to the Huber-von-Mises-Henky criterion, the universal critical state criterion in σ₁⁰, σ₂⁰, σ₃⁰ [4, 5] always conserves its simple form like all fundamental laws of nature.

[1]     Huber M. T. Die spezifische Formänderungsarbeit als Maß der Anstrengung eines Materials. Czasopismo Techniczne, Lemberg (Lwow), 1904

[2]     von Mises R. Mechanik der festen Körper im plastisch-deformablen Zustand. Nachrichten der Gesellschaft der Wissenschaften, Göttingen, 1913

[3]     Henky H. Zur Theorie plastischer Deformationen. Zeitschrift angewandter Mathematik und Mechanik, 1924

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

[5]      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