General Power Strength Sciences

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Strength Monograph

The “Collegium” All World Academy of Sciences Publishers,

Munich (Germany), 2010

1. Introduction. In technical mechanics, tensile stresses are considered positive and compressive stresses negative. In geomechanics, on the contrary, tensile stresses are considered negative and compressive stresses positive. To provide stress sign unification nesessary and very useful in mechanics at all including both technical mechanics and geomechanics, introduce pressures

p = - σ , p₁ = - σ₃ , p₂ = - σ₂ , p₃ = - σ₁ , p₁ ≥ p₂ ≥ p₃ ,

use them in geomechanics instead of stresses, consider tensile stresses positive and compressive stresses negative as in technical mechanics, and regard tensile pressures negative and compressive pressures positive as in geomechanics, which is natural.

The τ_L/σ_L ratio of shear τ_L and normal σ_L limiting stresses of materials [1–5] takes different positive values not greater than 1 often with substantial deviations from 1/2 and 3^-1/2 predefined by the most common Tresca and Huber-von Mises-Hencky criteria [1–4]. The lower (inner) (Tresca's criterion [6]) and upper (outer) (Ishlinsky's deviatoric stress criterion [7]) bounds of all the convex (by Drucker's postulate [8]) limiting surfaces are well-known [9]. Yu [10, 11] proposed his twin-shear yield criterion coinciding with Ishlinsky's deviatoric stress criterion [7], showed that all the convex limiting surfaces correspond to relations 1/2 ≤ τ_L/σ_L ≤ 2/3, and generalized these bounds for σ_Lt ≠ σ_Lc . Yu also proposed his twin-shear unified strength theory [10, 11] generalizing that criterion and fitting data τ_L/σ_L = 0.376, 0.432, 0.451, and 0.474 [12–14], as well as τ_L/σ_L = 0.727 and up to 0.82 [12, 15, 16] for materials with nonconvex limiting surfaces. Data τ_L/σ_L = 0.71 and 0.74 for steel [17, 18], τ_L/σ_L = 0.25 and 0.27 for magnesium and 0.69 for bronze [17], τ_L/σ_L = 0.40 and 0.42 for alloys and 0.67 for steel [19], τ_L/σ_L = 0.65 and up to 0.76 for iron [20], as well as up to 1 for brittle building materials [1], etc. are available, too.

In fundamental material strength sciences [21–25], general power strength sciences including general linear strength science generalizing Yu’s twin shear unified strength theory [10, 11] also fits all these and other data with scattering, e.g. via the following general strength criteria very simple.

Use the principal stresses σ₁ ≥ σ₂ ≥ σ₃ (regulated by this ordering) at a material's point 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 . If the equivalent stress σ_e in a strength criterion is a symmetric function of the principal stresses, use nonregulated σ_1n , σ_2n , and σ_3n without any predefined relations, which is very useful analytically and graphically.

In the space of the principal stresses, consider coordinate system Oxyz whose z-axis provides σ₁ = σ₂ = σ₃ . In this coordinate system, introduce 2D diagrams σ_mOσ_d of 3D (limiting) stresses and pressures due to relation

σ_d = p_d = (σ_x² + σ_y²)^1/2

in which (nota bene) σ_d = p_d (and NOT σ_d = - p_d) for convenience to provide nonnegative values only both for σ_d and for p_d , see the next figure

in which subscripts are not shown so that

σd = σ_d ,

σdL = σ_dL,

pd = p_d ,

pdL = p_dL,

σm = σ_m = (σ₁ + σ₂ + σ₃)/3,

-pm = -p_m = -(p₁ + p₂ + p₃)/3,

σmL = σ_mL ,

σttt = σ_ttt .

The curve

By no rotational symmetry: any section, σ_d‘= σ_dσ_dL‘/σ_dL

• general reserve science: n = σ_dL/σ_d

both by constant σ_m if σ_m ≤ 0

and by the constant direction to the origin if σ_m ≥ 0

Fundamental material strength sciences give whole hierarchies of universal strength laws of nature and is based on dimensionless relative (reduced) principal stresses σ_j° in constant universal stress fundamental science [21–25]. They are introduced 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°|.

2. General power strength sciences. Fundamental material strength sciences [21–25] include general power strength sciences naturally generalizing general linear strength science and possibly using moduli and radicals which both can be also nesting. Use, e.g., the homogeneous powers of the shear stresses (with clear generalizing Hosford’s criterion [1–4]):

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

which fit the above and any other data on the relation between the shear and normal limiting stresses for any materials. Additionally simply consider the influence of adding isotropic stress states, e.g. hydrostatic pressure, via adding aσ₂° (generally, any function g(σ₂°) vanishing at σ₂°) to the both last expressions of σ_e° with no influence on the limiting stresses in uniaxial tension and compression and pure shear. The obtained strength criteria fit strength test data on many artificial materials under static and variable loading [1–4, 29, 30] with average relative errors of about 10 %. The same holds for comprehensive polyaxial strength test data on natural materials very different: Dunham dolomite, Solenhofen limestone, and Mizuho trachyte [31], coarse grained dense marble [32, 33], Shirahama sandstone and Yuubari shale [34], KTB deep hole amphibolite [35], Westerly granite [36], fine-grained Rozbark sandstone [37], and Soignies limestone [38]. For these data in triaxial compression only, no complication of this form is necessary and, by the principle of tolerable simplicity [21–28], reasonable.

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 science. 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 sciences 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. 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°).

3. Applying an approach using combined bending and torsion. To correct the Tresca and von Mises criteria [1–4] via taking the τ_L/σ_L ratio of shear τ_L and normal σ_L limiting stresses into account, consider combined bending and torsion with normal σ and shear τ stresses and the principal stresses σ₁ ≥ σ₂ ≥ σ₃ :

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

The value of τ_L/σ_L is 1/2 and 3^-1/2 by the Tresca and Huber-von Mises-Hencky criteria [1–4]

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

giving two particular and one unified criteria

σ_e = (σ² + 4τ²)^1/2 ≤ σ_L , σ_e = (σ² + 3τ²)^1/2 ≤ σ_L ; σ_e = [σ² + (σ_L/τ_L)²τ²]^1/2 ≤ σ_L .

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

(σ_e° = ) σ_e/σ_L = [(σ/σ_L)² + (τ/τ_L)²]^1/2 ≤ σ_L/σ_L ( = 1).

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 Huber-von Mises-Hencky criterion. To take additional constant τ_L/σ_L of a material into account, generalize this criterion with trying to perpetuate the symmetry of the occurrences of the principal stresses. A natural possibility is to vary the factors by the sums σ₁² + σ₂² + σ₃² and σ₁σ₂ + σ₁σ₃ + σ₂σ₃ . Uniaxial tension and compression give that the first factor has to be 1. Hence additionally include unknown factor k:

σ_e = [σ₁² + σ₂² + σ₃² + k(σ₁σ₂ + σ₁σ₃ + σ₂σ₃)]^1/2 ≤ σ_L .

To determine k, use the same combined bending and torsion once more:

σ_e = [σ² + (2 - k)τ²]^1/2 ≤ σ_L , k = 2 - (σ_L/τ_L)²,

σ_e = {σ₁² + σ₂² + σ₃² + [2 - (σ_L/τ_L)²](σ₁σ₂ + σ₁σ₃ + σ₂σ₃)}^1/2 ≤ σ_L .

For the Tresca criterion [1–4], it is impossible to keep the criterion’s linear form by its generalization providing considering the true value of the τ_L/σ_L ratio. We proved this above. The cause is that the factors 1 and -1 in this criterion are unique to provide its applicability to uniaxial tension and compression, and introducing σ₂ brings nothing for our purpose at least in the particular case of combined bending and torsion (due to vanishing σ₂) and thus cannot solve, all the more, the general problem. Now it is natural to choose a quadratic form similar to that for the Huber-von Mises-Hencky criterion [1–4]:

σ_e = [(σ₁ - σ₃)² + kσ₁σ₃]^1/2 ≤ σ_L , σ_e = [σ² + (4 - k)τ²]^1/2 ≤ σ_L , k = 4 - (σ_L/τ_L)²,

σ_e ={(σ₁ - σ₃)² + [4 - (σ_L/τ_L)²]σ₁σ₃}^1/2 ≤ σ_L , σ_e ={σ₁² + σ₃² +[2 - (σ_L/τ_L)²]σ₁σ₃}^1/2 ≤ σ_L .

Experimental data on strength of ductile materials is often placed between the curves given by the Tresca and Huber-von Mises-Hencky criteria [1–4]. Expecting more criterion precision by smaller deviation of the σ_L/τ_L ratio from its pre-defined value gives the unified criterion as the linear combination

σ_e ={[(σ_L/τ_L)² - 3](σ₁ - σ₃) + [4 - (σ_L/τ_L)²][σ₁² + σ₂² + σ₃² - (σ₁σ₂ + σ₁σ₃ + σ₂σ₃)]}^1/2 ≤ σ_L .

The unified criterion using their above forms already corrected is

σ_e = [(σ_L/τ_L)² - 3]{σ₁² + σ₃² + [2 - (σ_L/τ_L)²]σ₁σ₃]}^1/2 +

[4 - (σ_L/τ_L)²]{σ₁² + σ₂² + σ₃² + [2 - (σ_L/τ_L)²](σ₁σ₂ + σ₁σ₃ + σ₂σ₃)}^1/2 ≤ σ_L .

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 [1–4]

σ_e = {[(σ₁ - σ₂)^k + (σ₂ - σ₃)^k + (σ₁ - σ₃)^k]/2}^1/k ≤ σ_L

gives the Tresca and Huber-von Mises-Hencky criteria by k = 1 and k = 2. For them, to provide these values of k using the same predefined values of the ratio σ_L/τ_L , naturally choose k = 5 - (σ_L/τ_L)². For further generalizations, introduce the notation (with apparently extending to any sets of variables and function values)

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

for any function f taking, e.g., values v and w by values a, b and c, d of its variables x, y, respectively. Then choose f[σ_L/τ_L | f(2) = 1, f(3^1/2) = 0], g[σ_L/τ_L | g(2) = 0, g(3^1/2) = 1], and h[σ_L/τ_L | h(2) = 1, h(3^1/2) = 2] instead of k = (σ_L/τ_L)² - 3, k = 4 - (σ_L/τ_L)², and k = 5 - (σ_L/τ_L)², respectively.

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.

Basic Results and Conclusions

1. The relations between the shear and normal limiting stresses for distinct materials are very different.

2. Many known strength criteria predefine the relations between the shear and normal limiting stresses and cannot fit their true relations for many materials.

3. In fundamental material strength science, general power strength sciences correctly consider and express known physical phenomena in material sciences, have clear physical sense, and fit the true relations between the shear and normal limiting stresses for many materials. This sciences generalize many known strength criteria, determine their applicability ranges, and provide independently fitting both the influence of adding isotropic stress states and the true relations between the shear and normal limiting stresses for many materials.

4. The relation between the shear and normal limiting stresses is critical for choosing a suitable form of strength criteria.

5. The quadratic forms of strength criteria are the simplest ones in general power strength sciences and provide many advantages.

6. General power strength sciences provide fundamental material strength sciences with initial strength criteria to discover the hierarchies of strength laws of nature. General power strength sciences are very suitable, allow generally representing and processing test data, and reduce time and cost expense by polyaxial strength tests.

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