UDC 539.4:620.17
GENERAL LINEAR STRENGTH THEORY (Fundamentals)
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
All World Academy of Sciences “Collegium”, Munich, Germany
Strength Monograph
The “Collegium” All World Academy of Sciences Publishers,
Munich (Germany), 2009
General linear strength theory exhaustively represents all linear and piecewise linear strength criteria via suitable formulae with moduli. It discovers the applicability domains of such criteria, corrects and generalizes many of them, rationally simulates any specific feature of the strength of a given material, and is tested analytically and experimentally.
Keywords: general linear strength theory, fundamental strength science.
This article is dedicated to the memory of my dear teacher, Academician Georgy Stepanovich Pisarenko (1910 - 2001) to the 100th anniversary of his birthday
0. Introduction. The strengths of isotropic and anisotropic natural and artificial materials (soil, rock, ice, timber, concrete, glass, metals, polymers, composites, etc.) and structural elements are cardinal for our life quality and even safety (because of crashes and natural catastrophes such as earthquakes, tsunamis, volcanoes activity, etc.). Leonardo da Vinci, Galileo Galilei, and other famous scientists created and developed strength science including strength science of structural elements and material strength science as the basis of solving strength problems for structural elements. The unique G. S. Pisarenko Institute for Problems of Strength of the National Academy of Sciences of Ukraine plays a very important role in strength science development. The scientific schools of Academician G. S. Pisarenko who created and developed this Institute as its first director, of his successor Academician V. T. Troshchenko, Academician A. A. Lebedev, and their prominent colleagues are well-known.
Strength (both yield and failure) criteria [1–4] determine the corresponding whole limiting surfaces of materials under multiaxial stress states by using data on a few simple tests only. Such criteria have to fit experimental data and to be as simple as possible. It is desirable that they have physical meaning, express the so-called equivalent, or equidangerous, stress σe via unified functions of the principal stresses σ1 ≥ σ2 ≥ σ3 (regulated by this ordering) at a material's point, and consider the influence of σ2 , Bridgman's effect [5] of adding isotropic stress states, e.g. under hydrostatic pressure, and the relations between the shear τL and normal σL limiting stresses. These functions have to consider limiting stresses σL such as yield stress σy or ultimate strength σL , namely σLt in tension and σLc in compression with σLc ≥ 0 and α = σLt/σLc if 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 analytically and especially graphically. 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 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, 3] and hence no known universal strength laws of nature.
1. Fundamental strength science. Fundamental strength science [21–25] deals with the most general fundamentals of strength science and includes fundamental strength science of structural elements and fundamental material strength science. Fundamental strength science of structural elements is based on elastic mathematics [24, 26–28] with its general data processing methods, general problem solving methods, general reserve theory, and many others, as well as a system of analytic macroelement methods. Fundamental material strength science as the basis for the previous science is based on the fundamental principles of tolerable simplicity and of the unity and universality of stresses, reserves, and strength criteria. Fundamental material strength science gives whole hierarchies of universal strength laws of nature and is based on general theory 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. A relative stress is numerically invariant by any unit transformations. It is the reciprocal to a sign-preserving individual safety factor independently of choosing a limiting criterion and stress unit and expresses the degree of the danger of a stress process better than this factor, the usual individual safety factor, and this stress process itself. A passage to the relative stresses unites strength test data on different materials and raises the reliability of the results due to their clustering. The relative stresses open many new ways in strength measurement and investigation to discover mechanical and physical laws of nature.
Fundamental material strength science determines the applicability limits of known strength criteria, corrects and extends them, and gives them universal forms. This additionally verifies fundamental strength science. Using σj° , in particular these general universalization theory and method in fundamental strength science, leads to general theories of strength criteria for any isotropic and anisotropic materials under any static or variable loading conditions, and gives many advantages in solids mechanics at all. General correction theory and method for critical state (process) criteria (also nonlinear ones) use, in particular, a linear combination of the principal stresses with discovering its physical sense and practical applications, as well as generalizing many known approaches. Hence fundamental material strength science includes a number of general methods and general strength and reserve theories. General linear strength theory is one of them.
2. General linear strength theory. 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. Note that even if a strength criterion itself is pure (not piecewise) linear, it defines a piecewise linear limiting surface in σ1n , σ2n , σ3n . There are known linear strength criteria. New ones can be also proposed. The Pisarenko-Lebedev strength theory [1, 3] and other nonlinear criteria include linear components. Creating general linear strength theory generalizing all specific linear strength criteria is of great importance.
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°|.
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.
The general method of representing piecewise functions via unified (single) formulae by any real-number functions and nesting levels of moduli gives, e.g.,
max{A1 , A2 , A3} = max{max{A1 , A2} , A3} =
((A1 + A2 + |A1 - A2|)/2 + A3 + |(A1 + A2 - |A1 - A2|)/2 - A3|)/2,
min{A1 , A2 , A3} = min{min{A1 , A2} , A3} =
((A1 + A2 - |A1 - A2|)/2 + A3 - |(A1 + A2 - |A1 - A2|)/2 - A3|)/2,
e.g., for representing the Galilei-Rankine greatest normal stress criterion [1, 3]
- σLc ≤ σ3 ≤σ1 ≤ σLt
and the Mariotto-de Saint-Venant greatest normal strain criterion (ν Poisson's ratio)
σ3 - ν(σ1 + σ2) ≥ - σLc , σ1 - ν(σ2 + σ3) ≤ σLt [1, 3].
Thus Tresca’s criterion σe = σ1 - σ3 ≤ σL [1, 3] can be also represented as
σe = (|σ1n - σ2n| + |σ2n - σ3n| + |σ3n - σ1n|)/2 ≤ σL
symmetrically for σ1n , σ2n , and σ3n without their predefined ordering. Reducing the principal stresses leads to its generalizing via universal strength law of nature
σe° = σ1° - σ3° = 1
which applies to any materials and loading. In σ1 , σ2 , and σ3 , depending on their signs, we have different expressions, e.g. for an isotropic brittle material σLt ≠ σLc :
(1) 0 ≥ σ1 ≥ σ2 ≥ σ3 ; σe = σ1 - σ3 = σLc ;
(2) σ1 ≥ 0 ≥ σ3 ; σe = σ1 - ασ3 = σLt (α = σLt/σLc);
(3) σ1 ≥ σ2 ≥ σ3 ≥ 0; σe = σ1 - σ3 = σLt ,
with obvious physical sense. If the signs of all the nonzero principal stresses are identical (1, 3), then a natural material with σLt ≠ σLc is similar to the two model materials. Each of them has equal moduli (either σLt or σLc , both instead of σL in the initial Tresca criterion) of the limiting stresses in tension and compression and is used when all principal stresses are either nonnegative (1) or nonpositive (3), respectively. If there are principal stresses with distinct signs (2), the critical states of this natural material are described by the criterion that coincides with the linear approximation of Mohr’s theory [1, 3] in this case only. This determines the applicability range of that theory, which is not quite obvious [1, 3]. In fact, the method to obtain this approximation is latently based on the distinct signs of σ1 , σ2 , and σ3 . Otherwise, the proportional increase of the corresponding Mohr circle based on segment [σ3 , σ1] results in its contacts with the approximating straight line not between its contacts with the Mohr circles based on segments [-σLc , 0] and [0, σLt]. Each stress state with σ1 = ασ3 seems to be quite safe, and uniform triaxial compression does not. Such contradictions do not concern our criterion. The last is natural if the signs of the nonzero principal stresses are identical and coincides with that verified approximation if not all these signs coincide, e.g. in biaxial stress.
The unstressed state and experimental data on the simplest tests (uniaxial tension and compression, as well as pure shear) and others with σ2 ≠ 0 apply to determine the constants. The general pure linear form, general pure one-modulus linear form, general mixed one-modulus linear homogeneous form, and general mixed one-modulus linear form of criteria are exhaustively investigated.
General linear strength theory is exhaustive because it represents any piecewise linear strength criterion. Representing pure linear strength criteria is trivial. Straightforwardly representing twin-shear unified strength theory [10, 11]
σe = σ1 - α(bσ2 + σ3)/(1 + b) ≤ σLt if σ2 ≤ (σ1 + ασ3)/(1 + α),
σe = (σ1 + bσ2)/(1 + b) - ασ3 ≤ σLt if σ2 ≥ (σ1 + ασ3)/(1 + α)
(which is piecewise linear strength theory, b is a constant) generalizing Yu’s twin-shear strength theory [10] gives (as a very special case of the general mixed one-modulus linear homogeneous form in general linear strength theory)
σe = ((2 + b)σ1 + (1 - α)bσ2 - α(2 + b)σ3 + b|σ1 - (1 + α)σ2 + ασ3|)/(2 + 2b) ≤ σLt .
The relation between the shear and normal limiting stresses is a key one for choosing suitable forms of linear strength criteria. Relation 1/τL = 1/σLt + 1/σLc is critical. For a material with σLt = σLc = σL , the critical value of τL is σL/2. Materials with such a critical relation allow additional simulation possibilities.
General linear strength theory 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 theory, the general pure linear form σe° = σ1° + aσ2° - σ3° = 1 (a is any constant) of strength criteria generalizes the universalization of Tresca’s criterion [1, 3] via considering additional isotropic stress states, e.g., hydrostatic pressure due to including aσ2°. This form fits 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 giving a pyramidal limiting surface adequate in this area is necessary and, by the principle of tolerable simplicity [21–28], reasonable. To better model the remaining areas, using one modulus suffices. In geomechanics, compressive stresses are considered positive and tensile stresses negative, which is very unsuitable for unifying strength science and can lead to confusion. It is better to replace stresses in geomechanics with pressures p1 ≥ p2 ≥ p3 by transformation formulae p1 = - σ3 , p2 = - σ2 , p3 = - σ1 with providing σ1 ≥ σ2 ≥ σ3 and using the same numeric data. Because of changing the indices in these formulae, formally sharing results can lead to mistakes, especially by nonsymmetric functions σe of σ1 , σ2 , σ3 .
3. General power strength theory. Fundamental material strength science [21–25] includes general power strength theory naturally further generalizing general linear strength theory and possibly using moduli and radicals which both can be also nesting. Use, e.g., the homogeneous powers of the shear stresses:
σe = [a13(σ1 - σ3)k + a12(σ1 - σ2)k + a23(σ2 - σ3)k]1/k ≤ σL .
General power strength theory can still better than general linear strength theory 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 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 strength science. 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 science with σLt° = σLc° = 1, to the nonuniversality of this approach, and to unlimited σe when σLc - σLt is very small. Using any function g(σ2) with g(0) = 0 universally works but brings asymmetry of function σe of the principal stresses.
Elastic mathematics solves these general problems with perpetuating limiting surface continuity and the symmetry of σe as a function of the principal stresses. Fundamental material strength science replaces 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. The general method of representing any linear and nonlinear real-number functions piecewise defined via unified formulae using moduli is very useful.
2. The relative stresses open many new ways in strength measurement and investigation to discover mechanical and physical laws of nature.
3. The linear form of strength criteria as piecewise linear functions of the principal stresses is the simplest one and provides many advantages.
4. General linear strength theory in fundamental material strength science has clear physical sense, uses single formulae possibly with moduli, exhaustively generalizes all the specific linear strength criteria, approximates nonlinear strength laws of nature, and correctly considers and expresses some known physical phenomena in material science.
5. The relation between the shear and normal strengths is critical for choosing a suitable form of linear strength criteria.
6. General linear strength theory is tested analytically and experimentally both directly and indirectly. It fits available strength test data both for some artificial materials (under static and nonstationary loading) and comprehensive polyaxial strength test data on many natural materials very different.
7. General linear strength theory provides fundamental material strength science with initial strength criteria to discover the hierarchies of strength laws of nature. It is very suitable, allows generally representing and processing test data, and reduces time and cost expense by polyaxial strength tests.
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