UDC 539.4:620.17
Uniform Stress Fundamental Science (on Strength Criteria Generally Considering Influence of Pressure and the Intermediate Principal Stress)
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)
2009
Linear and nonlinear general theories of considering the influence of pressure and of the intermediate principal stress on strength are created and developed. Adding any functions (which vanish at 0) either of the intermediate principal stress alone or of the sum of the principal stresses leads to considering pressure influence.
Key words: pressure dependence, intermediate principal stress influence.
0. Introduction. Strength criteria [1–4] have to determine the limiting surface using test data on the normal σL and possibly shear τL limiting stresses, consider the relation between them, the influence of the intermediate principal stress σ2 , and Bridgman's effect [5] of adding isotropic stress states, e.g. under hydrostatic pressure. It is desirable that such criteria express equivalent stress σe as a function F of the principal stresses σ1 ≥ σ2 ≥ σ3 . In limiting states only, function F has to take limiting value σL such as yield stress σy or ultimate strength σu , namely σLt in tension and σLc (σLc ≥ 0) in compression for materials with σLt ≠ σLc . Strength criteria forms as symmetric functions of the principal stresses σ1n , σ2n , and σ3n without any predefined relations and hence nonregulated are very useful because strength criteria are represented namely in the space of σ1n , σ2n , and σ3n . For anisotropic materials, additional indices are used to indicate certain directions.
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 ≤ τL/σL ≤ 2/3, and generalized these bounds for σLt ≠ σLc . Yu also proposed his twin-shear unified strength theory [10, 11] generalizing that criterion and fitting data τL/σL = 0.376, 0.432, 0.451, and 0.474 [12–14], as well as τL/σL = 0.727 and up to 0.82 [12, 15, 16] for materials with nonconvex limiting surfaces. Data τL/σL = 0.71 and 0.74 for steel [17, 18], τL/σL = 0.25 and 0.27 for magnesium and 0.69 for bronze [17], τL/σL = 0.40 and 0.42 for alloys and 0.67 for steel [19], τL/σL = 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.
Many well-known strength criteria [1–4] completely ignore both the influence of σ2 and Bridgman's effect [5] of adding isotropic stress states, which both are very essential [1–4]. There is a well-known idea to consider material pressure dependence via adding so-called “hydrostatic terms” usually simply proportional to sum σ1 + σ2 + σ3 equaling the octahedral normal stress multiplied by 3. There was no well-known idea to universally consider the influence of σ2 .
Many strength criteria are applicable to an isotropic material with σLt = σLc under static (stationary) loading only. For the general case of any anisotropic material with σLt ≠ σLc in each direction under any variable loading with changing the directions of the principal stresses, there were no known applicable strength criteria [1–4] and hence no known universal strength laws of nature.
1. Fundamental strength sciences. Fundamental strength sciences [21–25] deal with the most general fundamentals of strength sciences and include fundamental strength sciences of structural elements based on megamathematics [24, 26–28] with its general data processing sciences, general problem solving sciences, and fundamental material strength sciences. They are based on general science of dimensionless relative (reduced) principal stresses σj°. They are introduced via dividing each usual principal stress σj (j = 1, 2, 3) by the modulus (absolute value) |σjL| of its individual limiting value σjL of the same sign in the same direction by vanishing the remaining two principal stresses under the same remaining load conditions:
σj° = σj / |σjL| (j = 1, 2, 3).
The inequalities σ1° ≥ σ2° ≥ σ3° necessary if σe° 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 σe and σe° and their values even imaginary if they indicate still greater reserves than vanishing σe . Naturally extend σe and σe° by accepting their negative values with σe ≤ σL and σe° ≤ 1 and use nonnegative |σe| and |σe°|.
2. Material strength criterion pressure dependence fundamental science. This science [21–25] provides considering both the influence of σ2 and Bridgman's effect [5] of adding isotropic stress states, e.g. under hydrostatic pressure. Use, e.g., linear combinations of the principal stresses with discovering their physical sense and practical applications, as well as generalizing many known approaches.
For an isotropic material with σLt ≠ σLc , consider a typical strength criterion
σe = F(σ1 , σ2 , σ3) ≤ σL .
The idea and physical sense of linearly correcting such a criterion are essentially the hypothesis on the linear influence of the principal stresses on reaching a limiting state. Namely, in limiting states, the equivalent stress is no constant but a linear function of the principal stresses in accordance with
λ0F(σ1 , σ2 , σ3) + λ1σ1 + λ2σ2 + λ3σ3 = λ4
where λ0 , λ1 , λ2 , λ3 , λ4 are constants (parameters of the material). This hypothesis generalizes many known approaches to obtaining limiting criteria for materials with σLt ≠ σLc by linearly considering the mean principal (normal octahedral) stress, which is obtained as its particular case by λ1 = λ2 = λ3 . Another possibility λ1 ≠ λ2 ≠ λ3 is logical because of the different roles of the principal stresses σ1 ≥ σ2 ≥ σ3 algebraically ordered, which is confirmed, e.g., by Tresca’s strength criterion and the Lode-Nadai parameter [1–4].
The essence of proposed linear correcting critical (limiting) criteria may be shown, e.g., for Tresca’s strength criterion
σe = σ1 - σ3 ≤ σL
[1–4]. Substituting
F(σ1 , σ2 , σ3) = σ1 - σ3
and taking the data on uniaxial tension
σ1 = σL , σ2 = σ3 = 0
and compression
σ1 = σ2 = 0, σ3 = - σL
into account give the criterion
σe = σ1 + aσ2 - σ3 ≤ σL
with an additional pure (dimensionless) constant a of a material. This criterion can be also obtained by taking the more specific hypothesis on the linear influence of σ2 on reaching a limiting state. Introducing a is justified because σL as the unique constant of a material is not sufficient to consider the influence of σ2 and of hydrostatic tensions and compressions on reaching a limiting state. The physical sense of a is that it is σL divided by the limiting stress in hydrostatic tension σLttt which can be hardly determined directly but can be obtained using data on a third test with σ2 ≠ 0, hence pure shear is not suitable. In biaxial compression, e.g.,
σ1 = 0, σ2 = σ3 = - σLcc ,
we have
a = 1 - σL/σLcc ,
in triaxial tension and compression
σ1 = σLtcc , σ2 = σ3 = - σLtcc ,
the result is
a = 2 - σL/σLtcc .
Analogously correcting the Huber-von Mises-Hencky strength criterion [1–4] gives
σe = σi = {[(σ1 - σ2)2 + (σ2 - σ3)2 + (σ3 - σ1)2]/2}1/2 + aσ2 ≤ σL ,
and for the above general criterion we obtain
σe = F(σ1 , σ2 , σ3) + aσ2 ≤ σL .
For brittle and anisotropic materials, these criteria in σ1°, σ2°, σ3° are, respectively,
σe° = σ1° + aσ2° - σ3° ≤ 1,
σe° = σi° = {[(σ1° - σ2°)2 + (σ2° - σ3°)2 + (σ3° - σ1°)2]/2}1/2 + aσ2° ≤ 1,
σe° = F(σ1°, σ2°, σ3°) + aσ2° ≤ 1.
For variable loading, each of the three processes stationarily indexed of the uniaxial reduced principal stresses is replaced by the corresponding equidangerous cyclic uniaxial reduced principal stress interpreted by a stationary vectorial reduced stress σj° (j = 1, 2, 3) in the reduced diagram of limiting amplitudes, which is obtained by transformations [21–25] of the diagram of limiting stresses.
This correction reduces errors in determining the limiting surfaces of a series of brittle materials [1–4, 29, 30] to 3 %. 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.
3. General linear strength science. The linear form of strength criteria (for which σe can be expressed as a piecewise linear function of σ1 , σ2 , and σ3) is the simplest one. It provides many advantages in science and engineering especially by solving complex strength problems. Moreover, for any precision measure and any nonlinear strength criterion, there are its piecewise linear approximations whose deviations from this criterion don't exceed this measure. Substantial scatter of strength test data with a certain quote of outliers is typical. That is why it is often admissible to consider piecewise linear strength criteria only.
The general linear form of strength criteria in σj° can be represented as
σe° = a0 + a1σ1° + a2σ2° + a3σ3° + ∑i=1N b1i|c00i + c11iσ1° + c21iσ2° + c31iσ3° + b2i|c02i + c12iσ1° + c22iσ2° + c32iσ3° + b3i|c03i + c13iσ1° + c23iσ2° + c33iσ3° + ... || ... | ≤ 1
where a0 , a1 , a2 , a3 , bhi , c0hi , c1hi , c2hi , c3hi are any constants with their possible renaming and dropping unnecessary indices; h = 1, 2, ... , H are nesting levels; H and N are any nonnegative integers. If N = 0, it is the general pure linear form of strength criteria. H = 1 leads to the general linear form without nesting of moduli
σe° = a0 + a1σ1° + a2σ2° + a3σ3° + ∑i=1N bi|c0i + c1iσ1° + c2iσ2° + c3iσ3°| ≤ 1.
General linear strength science along with a linear combination of the principal normal or shear stresses, as well as a lot of linear and some piecewise linear strength criteria has clear physical sense and is their natural generalization with discovering their applicability domains. It can correctly consider and express some known physical phenomena in material science, e.g. the substantial roles of σ2 and of the relation between the normal and shear limiting stresses, general case σLt ≠ σLc , and Bridgman's phenomenon [5] of strength dependence on pressure.
In general linear strength science, the final general pure linear form
σe° = σ1° + aσ2° - σ3° = 1
(a is any constant) of strength criteria generalizes the universalization [21–25] of Tresca’s criterion [1–4] via considering additional isotropic stress states, e.g. hydrostatic pressure, due to including aσ2°. In fundamental material strength sciences [21–25], this final form is already known due to uniform stress fundamental science.
4. Adding “hydrostatic terms”. There are very many known straightforward attempts to correct strength criteria by adding so called hydrostatic (more precisely, isotropic) term A(σ1 + σ2 + σ3) with some factor A to simulate Bridgman's phenomenon [5] of strength dependency on pressure. This idea does not directly work. The reason is that sum σ1 + σ2 + σ3 vanishes in pure shear but does not vanish in uniaxial tension and compression. Hence using the simplest tests data on uniaxial tensions and compressions turns this factor A into zero.
The unique possibility is using a function of σ2 alone without σ1 and σ3 . This is reasonable because the distinct principal stresses σ1 ≥ σ2 ≥ σ3 are of different importance according to Lode-Nadai’s parameter [1–4]. Many experiments proved the essential influence namely of σ2 on strength. Note that additionally using σ2 alone also provides simulating Bridgman's strength dependency on pressure. Take any function f(t) with f(0) = 0, e.g. f(t) = At with any constant A. Adding f(σ2) for correcting the initial strength criterion, we obtain
σe(σ1 , σ2 , σ3) = F(σ1 , σ2 , σ3) + f(σ2) = σL
taking both σ2 and Bridgman's effect of adding isotropic stresses into account. Further generalizations [21–25] can be used if necessary. General linear strength science also leads to the simplest f(t) = At .
5. General power strength sciences. Fundamental material strength sciences [21–25] include general power strength sciences naturally further generalizing general linear strength science and possibly using moduli and radicals which both can be also nesting. 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 σe of σ1 , σ2 , σ3 and using σ1n , σ2n , σ3n with clear advantages. The initial form of power strength criteria with general homogeneous symmetric polynomials Pi(σ1n°, σ2n°, σ3n°) of power i is
σe° = [∑i=0N aiPi(σ1n°, σ2n°, σ3n°)]1/N ≤ 1.
In the unstressed state, σe° = 0 is natural and leads to a0 = 0. Case N = 2 gives form
σe° = [a1(σ1n° + σ2n° + σ3n°) + a2(σ1n°2 + σ2n°2 + σ3n°2) + b2(σ1n°σ2n° + σ1n°σ3n° + σ2n°σ3n°)]1/2 ≤ 1
which can provide a limiting surface of a paraboloidal type (due to adding a1(σ1n° + σ2n° + σ3n°) not to σe° but to σe°2) physically adequate and further generalizes the universalization of the Huber-von Mises-Hencky criterion
σe° = [σ1n°2 + σ2n°2 + σ3n°2 - (σ1n°σ2n° + σ1n°σ3n° + σ2n°σ3n°)]1/2 ≤ 1
in fundamental material strength sciences. a1(σ1n° + σ2n° + σ3n°) 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 σL with a general constant C at least by materials with σLt = σLc and hence by any materials. This is obvious due to fundamental material strength sciences with σLt° = σLc° = 1, to the nonuniversality of this approach, and to unlimited σe when σLc - σLt is very small. Using any function f(σ2) with f(0) = 0 universally works but brings asymmetry of function σe of the principal stresses.
Megamathematics solves these general problems with perpetuating limiting surface continuity and the symmetry of σe as a function of the principal stresses. Fundamental material strength sciences replace usual σ1 + σ2 + σ3 and reduced σ1n° + σ2n° + σ3n° “hydrostatic sums” with their continuous functions f and f° vanishing at -σLc , 0, σLt and -1, 0, 1, respectively. Using uniaxial tension and compression data and renaming the constants leads to
σe° = [σ1n°2 + σ2n°2 + σ3n°2 - a(σ1n°σ2n° + σ1n°σ3n° + σ2n°σ3n°) + bf°(σ1n° + σ2n° + σ3n°)]1/2 ≤ 1.
Constant a provides considering true values of τL/σL 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, 3] 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 + σLc if σ1 + σ2 + σ3 ≤ -σLc ,
f(σ1 + σ2 + σ3) = 0 if -σLc ≤ σ1 + σ2 + σ3 ≤ σLt ,
f(σ1 + σ2 + σ3) = σ1 + σ2 + σ3 - σLt if σ1 + σ2 + σ3 ≥ σLt ;
f°(σ1n° + σ2n° + σ3n°) = σ1n° + σ2n° + σ3n°+ 1 if σ1n° + σ2n° + σ3n° ≤ -1,
f(σ1n° + σ2n° + σ3n°) = 0 if -1 ≤ σ1n° + σ2n° + σ3n° ≤ 1,
f(σ1n° + σ2n° + σ3n°) = σ1n° + σ2n° + σ3n°- 1 if σ1n° + σ2n° + σ3n° ≥ 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 τL/σL and of adding an isotropic stress state, e.g. hydrostatic pressure, on σe , 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 σe of σ1 , σ2 , σ3 because representing limiting surfaces needs σ1n , σ2n , σ3n . Hence quadratic strength criteria can be even simpler than linear and especially piecewise linear strength criteria whose namely linear forms can give functions σe of σ1 , σ2 , σ3 always nonsymmetric.
Basic Results and Conclusions
1. Both the influence of the intermediate principal stress on material strength and pressure dependence of material strength (Bridgman’s phenomenon) are very essential.
2. Many well-known strength criteria ignore both the influence of the intermediate principal stress on material strength and pressure dependence of material strength (Bridgman’s phenomenon).
3. To consider both the influence of the intermediate principal stress on material strength and pressure dependence of material strength (Bridgman’s phenomenon), strength criteria correction theory and methods in fundamental material strength science correct strength criteria via adding a function (vanishing at zero) of the intermediate principal stress to the expression of the equivalent stress in these criteria. Even the simplest linear function provides fitting available strength test data both on some artificial materials (under static and variable loading) and comprehensive polyaxial strength test data on many natural materials very different. General linear strength theory in fundamental material strength science leads to the same linear function. But using these methods cannot provide the symmetry of the equivalent stress in these criteria as a function of the principal stresses.
4. To keep the symmetry of the equivalent stress in strength criteria as a function of the principal stresses by correcting strength criteria to provide considering both the influence of the intermediate principal stress on material strength and pressure dependence of material strength (Bridgman’s phenomenon), adding a symmetric function of the principal stresses can be useful. The most straightforward idea is adding a “hydrostatic term” as a function of the sum of the principal stresses. But this idea cannot work directly because using strength test data on uniaxial tension and compression leads to vanishing this function.
5. Elastic mathematics and fundamental material strength science provide suitable piecewise linear and other transformations of the sum of the principal stresses with independently fitting the standard tests data including the relation between the shear and normal limiting stresses and considering both the influence of the intermediate principal stress on material strength and pressure dependence of material strength (Bridgman’s phenomenon). This provides fitting both available strength test data on some artificial materials (under static and variable loading) and comprehensive polyaxial strength test data on many natural materials very different.
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