Material Strength Fundamental Sciences System (Essential)

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mechanical and Physical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

11 (2011), 10

UDC 539.4:620.17

Keywords: material strength fundamental sciences system, universal dimensionless mechanical stress, limiting mechanical state criterion, universal strength law of nature.

This article is dedicated to the memory of my dear teacher, Academician Georgy Stepanovich Pisarenko (1910 - 2001) to the 101st anniversary of his birthday

There are known separate limiting criteria for different groups of materials [1]. But for variably loading any anisitropic material with possible turning directions 1, 2, 3 of the principal stresses σ₁ ≥ σ₂ ≥ σ₃ , there are no known propositions to formulate such criteria at all [1].

The main idea of the material strength fundamental sciences system is that limiting criteria for different materials and loading conditions have to be sufficiently universal fundamental laws of nature. But, for example, the Tresca limiting criterion

σ_e = σ₁ - σ₃ = σ_L

(σ_e the equivalent stress, σ_L the limiting stress) with the unique constant, σ_L , of the material can model the limiting surface for a certain ductile material only and in its usual form cannot be applied to any brittle one with two different limiting stresses σ_t in tension and σ_c > 0 in compression.

Let us assume that this criterion is only an expression of a certain temporarily unknown sufficiently general criterion that is now applied to a certain ductile material with σ_L . Try to determine, for such a desired criterion, its form that may not include σ_L . By dimensionality and similarity theories, it is appropriate to divide each principal stress by σ_L > 0:

σ₁/σ_L = σ₁°,

σ₂/σ_L = σ₂°,

σ₃/σ_L = σ₃°,

σ_e/σ_L = σ_e° = 1/n_L

where n_L is the reserve of σe with respect to σ_L . Then that criterion becomes pure (dimensionless)

σ_e° = σ₁° - σ₃° = 1

without evident constants of a material and holds for any ductile material independently of the specific value of σ_L . This provides further generalizing the pure criterion by generalizing σ_j° .

For a brittle material and each index j ∈ {1, 2, 3, e},

σ_j° = σ_j/σ_t if σ_j ≥ 0,

σ_j° = σ_j/σ_c if σ_j ≤ 0.

Analyzing [2] experimental data [1] for many quite different ductile and brittle solids convincingly shows that this transformation independent of the limiting criteria unifies all the data and is an immediate expression of limiting criteria generalization theory itself. Moreover, in the space of the relative (reduced) principal stresses, σ₁°, σ₂°, σ₃°, and especially in the corresponding plane, those unified data evidently cluster near the limiting surfaces and curves by this pure criterion and the generalized Huber-von-Mises-Hencky limiting criterion (with the stress intensity, σ_i)

σ_e° = σ_i° = {[(σ₁° - σ₂°)² + (σ₂° - σ₃°)² + (σ₃° - σ₁°)²]/2}^1/2 = 1.

Generalizing an arbitrary limiting criterion

σ_e = F(σ₁ , σ₂ , σ₃) = σ_L

gives

σ_e° = F(σ₁°, σ₂°, σ₃°) = 1.

This criterion and its particular cases are invariant and universal in the space of σ₁°, σ₂°, σ₃° but have forms depending on the signs of σ₁ , σ₂ , σ₃ in their space.

For orthotropic materials, the principal directions, 1, 2, 3, of a stress state coinciding with the basic orthotropy directions at the same material's point, generalizing the above transformations gives

σ_j° = σ_j/σ_tj if σ_j ≥ 0,

σ_j° = σ_j/σ_cj if σ_j ≤ 0

with possible reindexing σ_j° to provide

σ₁° ≥ σ₂° ≥ σ₃°.

For any stationary loading an arbitrarily anisotropic material,

σ_j° = σ_j/|σ_Lj|

where σ_Lj is, for the usual principal stress, σ_j , its limiting value which has the direction and sign of σ_j and acts at the same material's point, the both other principal stresses vanishing, and the other loading conditions at the same point being the same.

Such recomprehending σ_tj and σ_cj further generalizes the transformation and preserves pure criteria forms unlike the von Mises-Hill criterion [1] superfluously complicated and other ones having nothing in common with sufficiently simple and universal laws of nature.

Such natural transformation is not the only for a brittle material with different strengths in tension and compression. In this case, by uniaxial cyclic loading in any principal direction, j, the limiting amplitude stress, σ_aj , can reach its peak, σ_ajmax , by a possibly nonzero mean stress, σ_m0j , of a cycle that is asymmetric in this nonzero case. Then also the stress state with σ_j = σ_m0j , j = 1, 2, 3, as opposed to the stress state with σ_j = 0, can be considered initial instead of the zero stress state. If

σ_m0j = (σ_tj - σ_cj)/2,

then a material with unequal strengths in tension and compression can be considered as one having these initial stresses -σ_m0j as a summary effect of microstresses and submicrostresses causing unequiresistibility as a phenomenological macroresult. The corresponding generalization of this transformation for variably loading at an arbitrary instant of time, t , from its interval, T = [t₀, t₁], gives

σ_j°(t) = [σ_j(t) - σ_m0j(t)]/|σ_Lj(t) - σ_m0j(t)|.

For each uniaxial stress process, σ_j(t), its reserve, n_j , is determined by the similar limiting process, n_jσ_j(t), with possibly taking damage accumulation into account. The equidangerous cycle of the relative (reduced) stresses with mean stress σ_mj° and amplitude one σ_aj° is determined by this formula. Then the constantly vectorial reduced stress

σ_j° = (σ_mj°, σ_aj°)

can be found by means of the limiting amplitude diagram. Finally, the pure criterion function universality postulate gives criterion

σ_e° = max{sup_t∈T max_ju(t) F(σ_1u°(t), σ_2u°(t), σ_3u°(t)), max_ju |F(σ_1u°, σ_2u°, σ_3u°)|} = 1,

the most dangerous, possibly depending on t , permutations of the stationary indexes, ju, of the unordered reduced principal stresses independently of

σ_1u° ≥ σ_2u° ≥ σ_3u°

being chosen.

Fundamental mechanical and strength sciences introduce a universal dimensionless mechanical stress having clear physical sense by natural transformation of a usual dimensional pressure and for the first time discover universal strength laws of nature for which all classical limiting mechanical state criteria are very narrow special cases only. These laws hold for any anisotropic natural or artificial material with also different strengths in tension and compression under any variable loading and possibly rotating the principal directions of the stress state at a material point during loading time. Fundamental mechanical and strength sciences open essentially new vital possibilities not only for creating safe and resource-saving machines and constructions, but also for predicting earthquakes, tsunamis, and other natural cataclysms, saving people and property.

The material strength fundamental sciences system includes:

constant universal stress fundamental science which includes general theories of negative and imaginary equivalent stresses along with their modules (absolute values) and also of universal scalar reductions of mechanical stresses to their individual limits of the same directions and signs. This science introduces a universal dimensionless stress via dividing a usual dimensional stress by the module of its individual limit of the same direction and a sign in the absence of all the remaining stresses and under the same other loading conditions. This universal stress appears to be the inverse of the common value of the individual reserves of the both stresses with their common sign. These theories are general theories of universal scalar reductions of mechanical stresses to their individual limits of the same directions and signs for the following basic types of the materials of the deformable solids and types of their loading:

isotropic materials with equal strength in tension and compression under constant loading;

isotropic materials with different strengths in tension and compression under constant loading;

orthotropic materials when its basic orthotropy directions coincide with the principal directions of a stress state under constant loading;

anisotropic materials under constant loading;

variable universal stress fundamental science which includes general theories of universal synchronous scalar reductions of mechanical stresses to their individual limits of the same directions and signs at every moment of time and also of universal vector reductions of the received whole processes (programs) of the individual dimensionless stresses to their equidangerous cycles and to the corresponding universal vector dimensionless stresses;

fundamental science on universal strength laws of nature including general theories of:

– an instant (monoaxial) equivalent universal reduced stress at each point of a material at any moment of loading as a universal (defined by a specific limiting criterion) function of the triad of the principal scalar universal reduced stress so that this equivalent universal reduced stress has just the same reserve according to this criterion as this whole triad at the same point of a material at the same moment of loading time;

– a constantly equivalent universal reduced stress at any point of a material as the maximum of such instant (monoaxial) equivalent universal reduced stresses at this point of this material for the whole time of loading;

– a variably equivalent universal reduced stress at any point of a material as the module of the universal (defined by a specific limiting criterion) function of the triad of the (constant) principal vectorial universal reduced stresses each of which is unequivocally defined by such a monoaxial cycle of the corresponding reduced principal stress unordered by algebraic value (as well as at geometric interpretation in the space of the principal stresses) and having a constant number during the whole time of loading that this cycle has just the same reserve as the whole process (the whole program) of the corresponding monoaxial principal stress at the same point of a material for the whole time of loading;

– an equivalent universal reduced stress at any point of a material as the maximum of the constantly equivalent universal reduced stress and the variably equivalent universal reduced stress at this point of this material for the whole time of loading;

uniform stress fundamental science including intermediate principal stress general theory, principal stresses sum general theory, and nonlinear general theories;

general linear strength science including general pure linear criterion theory, general pure one-modulus linear criterion theory, general mixed linear homogeneous criterion theory, and general mixed linear nonhomogeneous criterion theory;

general power strength sciences including general power normal stress fundamental sciences and general power shear stress fundamental sciences;

shear to normal stress fundamental science including general theories of:

applying general linear strength science;

applying general power strength sciences;

applying an approach using combined bending and torsion;

general reserve fundamental science including general additive reserve theory and general multiplicative reserve theory.

The material strength fundamental sciences system is universal and very efficient.

References

[1] Pisarenko G. S., Lebedev A. A. Deformation and Strength of Materials at Complex Stress State. Izd. Naukova Dumka, Kiev, 1976, 416 pp. (Russ.).

[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2004, 496 pp.