by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
RUAG Aerospace Services GmbH, Germany
Second Edition (2006)
First Edition (2002)
Strength and Engineering Journal
of the “Collegium” All World Academy of Sciences
Munich (Germany)
2 (2002), 1
For predicting fatigue and fracture of composites [1], general strength theory [2] generalizes strength criteria and fits the substantial influence [3] of the intermediate normal stress via introducing additional material constants. Suppose in limiting states, the equivalent stress se be no constant limiting stress σl (in tension σt and in compression σc) but a linear function of the ordered principal stresses σ1 ≥ σ2 ≥ σ3 via equation
λ0F(σ1, σ2, σ3) + λ1σ1 + λ2σ2 + λ3σ3 = λ4 [λ0 , λ1 , λ2 , λ3 , λ4 constants]
where F is the function in general limiting criterion σe = F(σ1 , σ2 , σ3) = σl . For the Tresca criterion with F(σ1 , σ2 , σ3) = σ1 - σ3 , using data on uniaxial tension and compression gives the criterion with an additional constant x of the material:
σe = σ1 + xσ2 - σ3 = σl .
The physical sense of this constant x is that it is the uniaxial limiting stress σl divided by the limiting stress in hydrostatic tension σttt . The last can be hardly determined directly but can be obtained by using the data on a third experiment with σ2 ≠ 0, hence pure shear is not suitable. In biaxial compression, σ1 = 0, σ2 = σ3 = -σcc we have x = 1 - σl /σcc , in triaxial tension and compression σ1 = σtcc , σ2 = σ3 = -σtcc the result is x = 2 - σl /σtcc .
Analogously correcting the Mises criterion and that general limiting criterion gives criteria
σe = σi = {[(σ1 - σ2)2 + (σ2 - σ3)2 + (σ3 - σ1)2]/2}1/2 + xσ2 = σl ,
σe = F(σ1, σ2, σ3) + xσ2 = σl .
Further generalizing method to extend strength laws hierarchies [2] gives equations
G(σ10, σ20, σ30) = λ0 + λ1σ10 + λ2σ20 + λ3σ30, G(σ10, σ20, σ30) = H(σ10, σ20, σ30)
where G, H are certain (maybe unknown unlike F) different functions of the reduced (relative) principal normal stresses σ10, σ20, σ30 and possibly of some pure constants of a material. For example, using the data on uniaxial tension and compression gives the structure
σe = F(σ1 , σ2 , σ3)/(1 + λ) + λ(σ1 - σ3)/(1 + λ) + xσ2 = σl
having two additional constants λ and x of a material. The experimental data on hypothetical uniform triaxial tension σ1 = σ2 = σ3 = σttt and torsion σ1 = τl , σ2 = 0, σ3 = - τl gives criterion
σe = (2τl - σl)F(σ1 , σ2 , σ3)/[2τl - F(τl , 0, - τl)] +
[σl - F(τl , 0, - τl)](σ1- σ3)/[2τl - F(τl , 0, - τl)] + σlσ2/σttt = σl .
Let the potential energy of deformation, multiplied by the maximum shear stress, be a certain linear combination of the principal stresses. Using uniaxial tension, uniaxial and uniform biaxial compression σ1 = 0, σ2 = σ3 = -σcc to the criterion in the reduced principal stresses
(σ10)2 + (σ20)2 + (σ30)2 - 2μ[(σ10)2(σ20)2 + (σ20)2(σ30)2 + (σ10)2(σ30)2] =
1 + [1 - 2(1 - μ)(σcc0)2](σ20)2/(σ10 - σ30) [μ the Poisson ratio, σcc0 = σcc/σc].
For the squared maximum shear stress and the maximum shear stress multiplied by the octahedral shear stress, respectively, we obtain criteria
σ10 - σ30 = 1 + (1 - σcc0)σ20/(σ10 - σ30),
σi0 = 1 + (1 - σcc0)σ20/(σ10 - σ30).
By designation σcc0 = σcc/σc (the reduced limiting stress in uniform biaxial tension), these two criteria give σtt0 + σcc0 = 2 and can be adequate only under this condition. The analogous assumption for the maximum shear stress multiplied by the squared octahedral shear stress leads to the criterion [with its necessary condition (σtt0)2 + (σcc0)2 = 2]:
(σi0)2 = 1 + [1 - (σcc0)2]σ20/(σ10 - σ30).
Let the squared maximum shear stress in critical states of an isotropic material with equal strength in tension and compression be equal to a certain linear combination of the limiting uniaxial stress and the principal stresses. Four experimental data on uniaxial and uniform biaxial tensions and compressions then give criterion
(σtt0 + σcc0)(1 - σtt0σcc0)(σ10 - σ30) + (2σtt0σcc0 - σtt0 - σcc0)(σ10 - σ30)2 +
(σcc0 - σtt0)(1 - σtt0)(1 - σcc0)σ20 = σtt0σcc0(2 - σtt0 - σcc0).
The analogous hypothesis as applied to the squared octahedral shear stress or the potential energy of distortion by the same experimental data leads to criterion
(σtt0 + σcc0)(1 - σtt0σcc0)(σ10 - σ30) + (2σtt0σcc0 - σtt0 - σcc0)(σi0)2 +
(σcc0 - σtt0)(1 - σtt0)(1 - σcc0)σ20 = σtt0σcc0(2 - σtt0 - σcc0).
[1] Handbuch Struktur-Berechnung. Prof. Dr.-Ing. L. Schwarmann. Industrie-Ausschuss-Struktur-Berechnungsunterlagen, Bremen, 1998
[2] Lev Gelimson. General Strength Theory. Abhandl. der Wissenschaftlichen Gesellschaft zu Berlin, Publisher Prof. Dr. habil. V. Mairanowski, 3 (2003), Berlin, 56-62
[3] Bridgman P. W. Collected Experimental Papers. Vols. 1 to 7. Harvard University Press Publ., Cambridge(Massachusetts), 1964