Analytic Macroelement Fundamental Science
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
Mathematical and Physical Monograph
The “Collegium” All World Academy of Sciences Publishers
Munich (Germany)
2012
Abstract
The purpose of this monograph is to give new very effective methods of solving many typical problems in science, engineering, and life.
Classical mathematics suggests no suitable methods bringing solutions general enough.
The present quantiparametrization method holds for any quantiproblem as a quantisystem of quantiparameters in general problem theory of the author. It belongs to his elastic mathematics due to his principles of new scientific thought and is based on his quantianalysis with its uninumbers, quantisets, and multiquantities.
Quantilinear-quanticombination method is a particular case of the quantiparametrization method and generalizes the eigenfunction method and the methods of orthonormal bases and of nonorthogonal fundamental solutions. For example, it gives for the first time general power solutions to harmonic and biharmonic equations basic for many typical problems.
Introduction
In classical mathematics [1], there is no sufficiently general concept of a quantitative mathematical problem. The concept of a finite or countable set of equations only completely ignores their quantities like any Cantor set [1]. They are very important by contradictory (e.g., overdetermined) problems without precise solutions. Besides that, without equations quantities, by subjoining an equation coinciding with one of the already given equations of such a set, this subjoined equation is simply ignored whereas any (even infinitely small) changing this subjoined equation alone at once makes this subjoining essential and changes the given set of equations. Therefore, the concept of a finite or countable set of equations is ill-defined [1]. Uncountable sets of equations (also with completely ignoring their quantities) are not considered in classical mathematics [1] at all.
Science, engineering, and life often need solving sets of equations. They along with known methods of their solving belong to classical mathematics based on usual sets, numbers, measures, etc. The both are not enough general to adequately model (set and solve) many typical problems.
But even by problems adequately modeled by sets of equations, known methods often bring partial solutions only which are not general enough. This holds, e.g., for harmonic and biharmonic equations along with the eigenfunction method and the methods of orthonormal bases and of nonorthogonal fundamental solutions.
General Problem
General problem type and setting theory (GPTST) in fundamental science on general problem essence [5] defines a general quantitative mathematical problem, or simply a general problem, to be a quantisystem [2-5] (former hypersystem) P which includes unknown quantisubsystems and possibly includes its general subproblems.
In particular, a general problem can be a quantiset
q(λ)Rλ{φ∈Φ r(φ)fφ[ω∈Ω s(ω)zω]} (λ∈Λ)
of indexed known quantirelations q(λ)Rλ (with their own, or individual, quantities q(λ)) [2-5] over indexed unknown quantifunctions (dependent variables), or simply unknowns (with their own, or individual, quantities r(φ)), r(φ)fφ , of indexed independent known quantivariables s(ω)zω (with their own, or individual, quantities s(ω), all of them belonging to their possibly individual vector spaces, in their initial forms without any further transformations
where
Rλ is a known relation with index λ from an index set Λ ;
fφ is an unknown function (dependent variable) with index φ from an index set Φ ;
zω is a known independent variable with index ω from an index set Ω ;
[ω∈Ω s(ω)zω]
is a quantiset of indexed quantielements s(ω)zω .
Nota bene: Own, or individual, quantities play the roles of the weights of quantiset elements. In particular, q(λ) is the own, or individual, quantity (former hyperquantity) [2-5] as a weight of relation Rλwith index λ in a quantiset
q(λ)Rλ{φ∈Φ r(φ)fφ[ω∈Ω s(ω)zω]} (λ∈Λ).
Replace all the unknown quantisubsystems, or simply unknown variables, or unknowns (in the above particular case, unknown quantifunctions), with their possible ”values” which are known quantisubsystems (in the above particular case, some known quantifunctions). Then a general problem which is a quantisystem [2-5] (former hypersystem) P including unknown quantisubsystems becomes the corresponding known quantisystem without any unknowns. To conserve the quantisystem form, let us use the same designations for these known quantisystem and quantisubsystems, too. This known quantisystem can be further estimated qualitatively and (or) quantitatively discretely (e.g., either true or false) or continuously (e.g., via absolute and relative errors [1], unierrors, reserves, reliabilities, and risks [2-5]). If a quantisystem of pseudosolutions to a general problem transforms this problem into a known true quantisystem, then this pseudosolutions quantisystem (or simply a pseudosolution by obviously using the system meta-level) is a solutions quantisystem (or simply a solution by obviously using the system meta-level) to this general problem.
In quantitative mathematical problems, namely equations and inequations are the most typical relations.
Further general problem type and setting theory (GPTST) in fundamental science on general problem essence [5] naturally defines a general pure equation problem and a general pure inequation problem.
General Pure Equation Problem
General problem type and setting theory (GPTST) in fundamental science on general problem essence [5] defines a general quantitative mathematical pure equation problem, or simply a general pure equation problem, to be a general problem that can be represented in a form in which all relations are namely equality relations.
In the left-hand sides of all the equations in a general pure equation problem, gather all the expressions available namely in the initial forms of these equations without any further transformations. The unique natural exception is changing the signs of the expressions by moving them to the other sides of the same equations. Then a general pure equation problem can be represented, in particular, as a quantiset
q(λ){Lλ{φ∈Φ r(φ)fφ[ω∈Ω s(ω)zω]} = 0} (λ∈Λ)
of indexed known quantiequations (with their own, or individual, quantities q(λ)) [2-5] in a form of vanishing operators Lλ over indexed unknown quantifunctions (dependent variables), or simply unknowns (with their own, or individual, quantities r(φ)), r(φ)fφ , of indexed independent known quantivariables s(ω)zω (with their own, or individual, quantities s(ω), all of them belonging to their possibly individual vector spaces, in their initial forms without any further transformations
where
Lλ is a known operator with index λ from an index set Λ ;
fφ is an unknown function (dependent variable) with index φ from an index set Φ ;
zω is a known independent variable with index ω from an index set Ω ;
[ω∈Ω s(ω)zω]
is a quantiset of indexed quantielements s(ω)zω .
Nota bene: Own, or individual, quantities play the roles of the weights of quantiset elements. In particular, q(λ) is the own, or individual, quantity (former hyperquantity) [2-5] as a weight of equation Lλ = 0 with index λ in a quantiset
q(λ){Lλ{φ∈Φ r(φ)fφ[ω∈Ω s(ω)zω]} = 0} (λ∈Λ).
When replacing all the unknowns (unknown functions) with their possible ”values” (some known functions), the above quantiset of quantiequations is transformed into the corresponding quantiset of formal functional quantiequalities. To conserve the quantiset form, let us use for these known functions the same designations fφ .
General Pure Inequation Problem
General problem type and setting theory (GPTST) in fundamental science on general problem essence [5] defines a general quantitative mathematical pure inequation problem, or simply a general pure inequation problem, to be a general problem that can be represented in a form in which all relations are namely inequality relations.
In the left-hand sides of all the inequations in a general pure inequation problem, gather all the expressions available namely in the initial forms of these inequations without any further transformations. The unique natural exception is changing the signs of the expressions by moving them to the other sides of the same inequations. Then a general pure inequation problem can be represented, in particular, as a quantiset
q(λ){Lλ{φ∈Φ r(φ)fφ[ω∈Ω s(ω)zω]} Rλ 0} (λ∈Λ)
of indexed known inequality quantirelations (with their own, or individual, quantities q(λ)) [2-5] in a form of the comparison with zero of the values of operators Lλ over indexed unknown quantifunctions (dependent variables), or simply unknowns (with their own, or individual, quantities r(φ)), r(φ)fφ , of indexed independent known quantivariables s(ω)zω (with their own, or individual, quantities s(ω), all of them belonging to their possibly individual vector spaces, in their initial forms without any further transformations
where
Lλ is a known operator with index λ from an index set Λ ;
Rλ is an inequality relation (e.g., ≈ , ∼ , ≠ , < , > , ≤ , ≥) with index λ from an index set Λ ;
fφ is an unknown function (dependent variable) with index φ from an index set Φ ;
zω is a known independent variable with index ω from an index set Ω ;
[ω∈Ω s(ω)zω]
is a quantiset of indexed quantielements s(ω)zω .
Nota bene: Own, or individual, quantities play the roles of the weights of quantiset elements. In particular, q(λ) is the own, or individual, quantity (former hyperquantity) [2-5] as a weight of inequation Lλ Rλ 0 with index λ in a quantiset
q(λ){Lλ{φ∈Φ r(φ)fφ[ω∈Ω s(ω)zω]} Rλ 0} (λ∈Λ).
When replacing all the unknowns (unknown functions) with their possible ”values” (some known functions), the above quantiset of inequations is transformed into the corresponding quantiset of formal functional inequalities. To conserve the quantiset form, let us use for these known functions the same designations fφ .
By using unstrict inequality relations such as ≈ , ∼ , ≤ , ≥ , etc. only, a general pure inequations problem clearly further generalizes a general pure equations problem.
General Approximation Problem
Let us now consider a general approximation problem.
Let
Z ⊆ X × Y
be any given subset of the direct product of two sets X and Y and have a projection Z/X on X consisting of all x ∈ X really represented in Z , i.e., of all such x that for each of them there is a y ∈ Y such that
(x, y) ∈ Z .
Let further
{ y = F(x) }
where
x ∈ X
y ∈ Y
be a certain class of functions defined on X with range in Y .
Then the graph of such a function is a curve in X × Y .
The problem consists in finding (in class { y = F(x) }) functions with graphs nearest to Z in a certain reasonable sense.
To exactly fit this with a specific function
y = F(x),
the set Z ⊆ X × Y has to be included in the graph of this function:
Z ⊆ { (x, F(x)) | x ∈ X },
or, equivalently,
F(x) = y
for each
x ∈ Z/X .
But this inclusion (or equality) does not necessarily hold in the general case. Then it seems to be reasonable to estimate the error
E( F(x) =? y | x ∈ Z/X )
of the formal equality (true or false)
F(x) =? y
on this set Z/X via a certain error function E defined at least on Z/X .
To suitably construct such an error function, it seems to be reasonable to first consider two stages of its building:
1) defining local error functions to estimate errors at separate points x ;
2) defining global error functions using the values of local error functions to estimate errors on the whole set Z/X .
Possibly the simplest and most straightforward approach includes the following steps:
1) defining on Y × Y certain nonnegative functions ryy’(y, y’) generally individual for different y , y’ and, e.g., similar to a distance [1] between any two elements y, y’ of Y (but not necessarily with holding the distance axioms [1]),
2) defining certain nonnegative functions Rx(r(F(x), y)) generally individual for different x ,
3) summing (possibly including integrating) their values on Z/X , and
4) using this sum (possibly including integrals) as a nearness measure.
General Problem Settings
Nota bene: The essence of a general problem includes, in particular, its origin (source) which can give very different settings (and hence both mathematical models and results) of a general problem even if graphical interpretations seem to be very similar or almost identical. For example, in the two-dimensional case, the same graphical interpretation with a triangle corresponds to many very different general problem settings and, moreover, to many very different general problems and even their systems (sets, families, etc.). Among them are, e.g., the following with determining:
1) the point nearest to the to the set or to the quantiset (with own quantities, which is very important by coinciding straight lines) of the three straight lines including the three sides, respectively, of the given triangle by different nearness criteria;
2) the point nearest to the triangle boundary, i.e. either to the set or to the quantiset (with counting the vertices twice) of all the points of the three sides of the given triangle by different nearness criteria;
3) the incenter and/or all the three excenters [1] of the given triangle;
4) the circumference (circle containing all the three vertices) of the given triangle;
5) the gravity (mass, length, uniquantity [2-5]) center of the triangle boundary, e.g., either of the set or of the quantiset (with counting the vertices twice) of all the points of the three sides of the given triangle;
6) the gravity (mass, area, uniquantity [2-5]) center of the triangle area including its interior and either including or not including its boundary, e.g., either of the set or of the quantiset (with counting the vertices twice) of all the points of the three sides of the given triangle.
The similar holds for a tetrahedron in the three-dimensional case with natural additional possibilities (the incenter/excenters for its flat faces along with the carcass incenter/excenters for its straight edges etc.).
By curvilinearity, the usual distance from any selected point to a certain point which lies on the curve or in the curvilinear surface is not the only. It is also possible to consider the distance from the selected point to the tangent (straight line or plane, respectively, if it exists) to the curve or curvilinear surface at that certain point if this tangent is the only. Otherwise, consider a certain suitable nonnegative function of the distances from the selected point to all the tangents. Additionally, if the selected point lies on the same curve or in the same curvilinear surface, then the usual straight line distance is not the only. It is also possible to consider the curvilinear distance as the greatest lower bound of the lengths of the curves lying on that curve or in that curvilinear surface and connecting those both points (simply the length of the shortest curve lying on that curve or in that curvilinear surface and connecting the both points if it exists). The similar can hold for polygons and polyhedra. Naturally, it is also possible to consider other conditions and limitations.
General Parametrization Theory
Known analytic methods [1] do not suffice by solving many quantisets of functional equations [2]. The finite-element method [3] gives numeric “solutions” only not suitable for optimization. Some general analytic methods for solving general problems [2] as general systems [4] are represented.
General parametrization theory is general theory of solving general problems in some parametric aggregate of general systems that is their quantiset in which they are distinguished by the general systems of general values of some general parameters.
For example, consider general punctiformization theory of solving continual general problems.
An eigenaggregate for a general system of general correspondences is a general system of their general domain subsystems on which each image in any correspondence is a general homogeneous linear combination [4] of a generally linearly independent general system which can be individual for that correspondence.
General Linear Combination Theory
General linear combination theory is applying general parametrization theory to an eigenaggregate for a considered general problem [2].
Nota bene: In this general problem, an eigenaggregate includes not only all the eigenfunctions [1].
Example 1. In classical mathematics [1], a linear combination is only finite and homogeneous, implicitly includes only different elements without iterations, and only usual objects, addition, and multiplication with no unification, Cartesian multiplication, general objects and operations.
Definition 2. A general homogeneous linear combination of the quantielements-bases of its basic quantiset
A = {... , αa, ... , α’a’, ...}
over the quantielements-coefficients of its coefficient quantiset
C = {... , γc, ... , γ’c’, ...}
is their general left, or right, homogeneous linear combination if
... + ca + ... + c’a’ + ... =
... + ac + ... + a’c’ + ... .
Definition 3. A general homogeneous linear quanticombination of its basic quantiset
A = {... , αa, ... , α’a’, ...}
of quantielements-bases over its coefficient quantiset
C = {... , γc, ... , γ’c’, ...}
of quantielements-coefficients is their general left, or right, homogeneous linear combination if
... + γc αa + ... + γ’c’α’a’ + ... =
... + αa γc + ... + α’a’ γ’c’ + ... .
Definition 4. A general homogeneous linear Cartesian combination of the quantielements-bases of
A = {... , αa, ... , α’a’, ...}
over the quantielements-coefficients of its coefficient quantiset
C = {... , γc, ... , γ’c’, ...}
is their general left, or right, homogeneous linear Cartesian combination if
... + c × a + ... + c’ × a’ + ... = ... + a × c + ... + a’ × c’ + ... .
Definition 5. A general homogeneous linear Cartesian quanticombination of
A = {... , αa, ... , α’a’, ...}
of quantielements-bases over its coefficient quantiset
C = {... , γc, ... , γ’c’, ...}
of quantielements-coefficients is their general left, or right, homogeneous linear Cartesian combination if
... + γc × αa + ... + γ’c’ × α’a’ + ... =
... + αa × γc + ... + α’a’ × γ’c’ + ... .
Definition 6. A general (optional: left or right) linear (optional: Cartesian) combination (or quanticombination) of A over C is the general sum/union of the corresponding general homogeneous linear combination (or quanticombination, respectively) and of some c ∈ C.
Definition 7. A quantiset A is called generally linearly dependent over C if there exists some zero-value (empty-value) homogeneous linear combination of A over C using nonzero (nonempty, respectively) general coefficients. Otherwise, A is called generally linearly independent over C.
Definition 8. A general basis B of A over C is a generally linearly independent (over C) general subset (of A) such that any a ∈ A is a general homogeneous linear combination of B over C.
Example 9. Let H be a Hilbert space [1] including a countable orthonormal basis
F = {fk | k ∈ N}
and let
f-1 = ∑k ∈ N akfk
(by ∑k ∈ N |ak|2 finite) be an essentially infinite series. Then the ordinary set {f-1} + F is linearly independent but generally linearly dependent.
Definition 10. A general homogeneous power combination over C is
∑j ∈ J cj ∏ k ∈ K(j) xkα(k)
where
cj ∈ C,
α(k) ≥ 0,
∑k ∈ K(j) α(k)
is independent of j.
Definition 11. A general power combination over C is a general sum of general homogeneous power combinations over C.
Notation 12.
y = f(j ∈ J xj) ≡ f(xj | j ∈ J).
It is proposed to use Einstein omitting a summation sign for doubled indices in tensor calculus [1] only if they are absent in one of the sides of the equality.
Remark 13. Known linearity [1] is only finite and homogeneous (a linear function
f(x) ≡ ax + b
is not a linear operator if b ≠ 0) over usual addition and multiplication but not over unification and Cartesian multiplication.
Definition 14. A general mapping is called generally (homogeneously) addition-multiplication linear if the image of any general homogeneous linear addition-multiplication combination of pre-images is the corresponding general (homogeneous, respectively) linear addition-multiplication combination of their images.
Definition 15. A general mapping is called generally (homogeneously) unification-Cartesian-multiplication linear if the image of any general homogeneous linear unification-Cartesian-multiplication combination of pre-images is the corresponding general (homogeneous, respectively) linear unification-Cartesian-multiplication combination of their images.
Remark 16. General addition and multiplication can be generally transformed into general unification and Cartesian multiplication, and vice versa.
Definition 17. An isolated (a nonisolated) general system is a general union of some general objects along with some general relations between them and no (some, respectively) other general objects.
Definition 18. A general structure of a general system is the general system of all the general relations in a given system.
Definition 19. A general state (process) of a general system is subcritical if any sufficiently small its variations conserve the general structure of that system.
Definition 20. A general state (process) of a general system is supercritical if any its variations change the general structure of that system.
Definition 21. A general state (process) of a general system is critical if some sufficiently small its variations conserve and some other ones change the general structure of that system.
Definition 22. The general measure system ME(A) of a general system A with respect to a general unit system E is the general system
(Q(Aj)/(Ej ) | j∈J)
having the same general structure.
Definition 23. A general probability system estimator is a general system estimator giving the general probability system
PU(A) = ( ME(A), ME(U) )
of A with respect to the unit general system U by a general measure system estimator ME.
Definition 24. A general nonlinearity system estimator is a general system estimator NL (L is a general system of some linear systems) replacing a general system A by its general nonlinearity system
NL(A) = inf{E (A =? L) | L ∈ L}.
The propositions are very efficient in solving many urgent problems, e.g., by regression and correlation data analysis and determining key probabilities in finance, insurance, and reinsurance.
Consider a general pure equation problem
q(λ){Lλ{φ∈Φ r(φ)fφ[ω∈Ω s(ω)zω]} = 0} (λ∈Λ).
The essence of linear combination theory is that each known operator, Lλ , and each desired function, fφ , is represented as a finite or infinite linear combination of functions from a chosen linearly independent class. It is closed over all the known operators, Lλ , external with respect to the desired functions, fφ . The combination factors are numbers determinate for the known operators, Lλ , and indeterminate for the desired functions, fφ . Then these equations become linear algebraic over the indeterminate factors and are used to obtain their values.
In particular, to the homogeneous harmonic equation
∇2φ(x , y , z) = 0
over desired function φ(x , y , z) in the Cartesian coordinates, x , y , z , in the class of power functions (power series), or, equivalently, in a general pseudosolution
φ(x , y , z) = ∑i=0∞∑j=0∞∑k=0∞aijkxiyjzk ,
linear combination theory made it possible, to determine the most general solution
φ(x , y , z) = ∑i=0∞∑j=0∞∑k=0∞(-1)[i/2][i/2]!(i!j!k!)-1∑m=0[i/2](j + 2m)!(k + 2[i/2] - 2m)!{m!([i/2] - m)!}-1ai-2[i/2], j+2m , k+2[i/2]-2mxiyjzk
where
[r] = entier r
is the integral part of a real number r ,
a0jk and a1jk are two arbitrary number sequences, 0 ≤ j , k < ∞.
To the homogeneous biharmonic equation
∇2∇2L(r , z) = 0
over desired function L(r , z) in the cylindric coordinates, r , z , in the class of power functions (power series), or, equivalently, in a general pseudosolution
L(r, z) = ∑i =0∞∑j=0∞aijr2izj ,
linear combination theory allowed to determine the most general solution
L(r , z) = ∑i =0∞∑j=0∞(-1)i+1i!-2j!-1[(2i + j - 2)! i 22-2ia1, 2i+j-2 + (2i + j)! (i - 1) 2-2ia0, 2i+j]r2izj
where conventionally by k < 0 we consider
k! = 1
and
a1k = 0,
a0j and a1j are two arbitrary number sequences, 0 ≤ j < ∞.
Restructuring Theory
Restructuring a general problem is changing its general structure [4] to simplify this problem.
The essence of restructuring theory is preliminarily restructuring a general problem to assign its general subproblems appropriate roles.
Consider a general pure equation problem
q(λ){Lλ{φ∈Φ r(φ)fφ[ω∈Ω s(ω)zω]} = 0} (λ∈Λ)
as the given quantiset of equations which are its general subproblems.
In particular, select such a most complete quantisubset of the given quantiset of equations that the evident (exact or approximate by the principle of tolerable simplicity) solutions to the quantisubset can be explicitly found. These solutions to the quantisubset of equations are considered as pseudosolutions to the given quantiset of equations itself and are substituted for the desired functions, fφ , in all the equations of the given quantiset of equations, which become formal equalities (without unknown elements) that are not necessarily identities and whose unierrors are determined.
Such general division, or distribution, or partition, methods apply to a general problem with nonequicomplicated general relations via creating the most complete exactly or approximately solvable decision subsystem of relatively simpler general relations giving an explicit quasisolution [2] (of that problem) whose unierror or/and reserve [2] is/are estimated over the rest estimation subsystem.
Analytic Macroelement Science. Introduction
Structure elements of brittle materials extremely loaded are often spatial axially symmetric bodies with elastic strains only up to fracture. To rationally design such elements, both numeric and experimental methods operating with constants only are not sufficient. It is necessary to determine their stress-strain states and strength, as well as to express optimization parameters via initial data certainly in analytic form by reasonable compromise between simplicity and precision.
An analytic macroelement methods (AMEM) system includes two methods obtained by using a linear-combination method and a partial method (both by the author). They have been applied to solving equations and their sets in axially symmetric elasticity by piecewise smooth boundary conditions without volume forces and thermal effects.
Power Analytic Macroelement Theory
To optimize structural elements in engineering including aeronautical fatigue, analytic methods [1-5] (along with the finite element method) are used.
An analytic macroelement method (AMEM) has been created in two modifications obtained via a linear combination method and a partial method (all by the author [4, 5]). They have been applied, e.g., to solving equations and their sets in axially symmetric elasticity by piecewise smooth boundary conditions.
The essence of the power modification of the AMEM is that the obtained general power representation of the Love’s axially symmetric biharmonic stress function [2]
where by k < 0 we conventionally consider k! = 1 and a1k = 0,
a0j and a1j being two arbitrary number sequences, 0 ≤ j < ∞,
is used by steps to exactly or approximately satisfy the boundary conditions in a problem to be solved. The displacements and stresses are determined via applying the Love’s linear differential operators [2] to L. Love has shown that the biharmonicity of L is sufficient for precisely satisfying all the equilibrium and continuity equations. The biharmonicity necessity problem is now set and positively solved. For a cylindrical body, it is shown that it is enough to consider one base to be free of loads. The power expansions of the nonzero boundary conditions lead to four infinite subsets of linear algebraic equations over the coefficients in the function L. Their general solutions by their homogeneity are linearly expressed through the sequential zeroes of two Bessel functions and two ones analogous to them. It is shown that the compatibility of the boundary conditions in a boundary-value problem is necessary and sufficient for its exact solvability. If the body has a complicated form or the conditions are discontinuous then the body is mentally cut with canonical surfaces to a few macroelements. Solutions for them are conjugated by minimizing the responsible mean square residuals, collocation ones, and minimax modulus residuals. Their linear or quadratic correction makes stress determination in a strength problem more precise. For the Lamé solution [1], it is shown that the known linear generalization is the unique exact one.
Integral Analytic Macroelement Theory
The integral modification of the AMEM in axially symmetric elasticity is obtained by applying the partial method to the set of the differential equations in elastic stresses. The both equilibrium equations together with the continuity one having the first order allow precisely expressing the normal stresses via the boundary conditions in stresses and the distribution of the shear stress. The remaining continuity equation integro-differential of the second order makes exactly solving it unreal. Instead of that, by the introduced principle of tolerable simplicity [4], for the shear stress τ(r, z), its simplest statically possible distribution ensuring the equilibrium of each body part cut off by an arbitrary coaxial cylindrical surface is determined. For a cylindrical body a ≤ r ≤ b, c ≤ z ≤ d, we have
τ(r, z) = (b2 - r2)(b2 - a2)-1 τ(a, z) + (r2 - a2)(b2 - a2)-1 τ(b, z) + 6 (z - c)(d - z)(d - c)-3
{- (b2 - r2)(b2 - a2)-1∫cd τ(a, z’) dz’ + (b/r - (r2 - a2)(b2 - a2)-1)∫cd τ(b, z’) dz’ +
r-1∫rb[σz(r’, d) - σz(r’, c)] r’dr’} +(d - z)(2c + d - 3z)(d - c)-2 [τ(r, c) -
(b2 - r2)(b2 - a2)-1 τ(a, c) - (r2 - a2)(b2 - a2)-1 τ(b, c)] +
(z - c)(3z - c - 2d)(d - c)-2[τ(r, d) - (b2 - r2)(b2 - a2)-1 τ(a, d) - (r2 - a2)(b2 - a2)-1 τ(b, d)].
The radial, tangential, and axial normal stresses are determined by the formulae
σr(r, z) = (a/r)2 (b2 - r2)(b2 - a2)-1σr(a, z) + (b/r)2 (r2 - a2)(b2 - a2)-1σr(b, z) +
μr-2∫arσz(r’, d)r’dr’- μr-2(r2 - a2)(b2 - a2)-1∫abσz(r’, d)r’dr’ - 2-1(1 + μ)∫ar(∂τ(r’, z)/∂z)dr’+
2-1(1 + μ)b2r-2(r2 - a2)(b2 - a2)-1∫ab(∂τ(r’, z)/∂z)dr’ - 2-1(1 - μ)r-2∫ar(∂τ(r’, z)/∂z)(r’)2dr’+
2-1(1 - μ)r-2(r2 - a2)(b2 - a2)-1∫ab(∂τ(r’, z)/∂z)(r’)2dr’ + μr-1∫zd τ(r, z’)dz’ -
μar-2(b2 - r2)(b2 - a2)-1∫zd τ(a, z’)dz’ - μbr-2(r2 - a2)(b2 - a2)-1∫zd τ(b, z’)dz’ (μ the Poisson ratio);
σt(r, z) = ∂(rσr(r, z))/∂r + ∂(rτ(r, z))/∂z;
σz(r, z) = σz(r, d) + r-1∫zd(∂(rτ(r, z))/∂r)dz’
with exact or approximate generalization of the Lamé solution [1]. To estimate the accuracy, the obtained solution is substituted into the remaining equation. For its left side in the body domain, the mean value is divided by the least upper bound on the sum of the moduli of its algebraic summands, each of them being a product of functions of the initial data.
(where μ is Poisson's ratio)
The AMEM has made it possible, to optimize high-pressure portholes (side-lights).
Applying the Analytic Macroelement Methods
and Fundamental Strength Sciences
to Three-Dimensional Cylindrical Glass Elements
of High-Pressure Illuminators (Deep-Sea Portholes)
Physical Monograph
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich,
Germany)
The "Collegium" All World Academy of Sciences Publishers
Munich (Germany)
FourthEdition (2010)
Third Edition (2004)
Second Edition (2002)
First Edition (2000)
The effect of pressure applied to the external base and side surface of three-dimensional cylindrical glass elements on their strength and nature of failure in a specific type of illuminator working at a high pressure is analyzed. It is also shown that such universalized strength criteria in the relative stresses as the first, third, and fourth strength theories and the Pisarenko-Lebedev criterion may be used to evaluate the strength of such glass elements.
1. Structure and Calculation Diagram
In a rational structure of illuminators (for example, deep-sea portholes) used in high-pressure technology and shown in Fig. 1, loading their cylindrical glass elements is shown in Fig. 2.
Fig. 1. Rational structure of high-pressure illuminators (deep-sea portholes).
Fig. 2. Calculation diagram of loading cylindrical glass elements of high-pressure illuminators (deep-sea portholes).
The pressure of the external medium, p, on the external base,
z = h,
pressure
p2 = pa2/(a2 - a12)
on the peripheral ring-shaped part of the internal base,
z = 0,
and pressure pl on the side surface of the element in proportional loading form a self-balanced system. The optimum value of ratio
Π = pl/p
can be set by calculating the strength of the glass element.
2. Using the Analytic Macroelement Methods
To construct a closed analytical solution for its stress state, the power and integral analytic macroelement methods (AMEM) [1] were used. For example, by the power analytic macroelement method, the examined glass element with height h is hypothetically sectioned by the cylindrical surface
r = al
into the central
0 ≤ r < al
and peripheral ring-shaped
al ≤ r ≤ a
sections. The solutions to the determining equilibrium and continuity equations together with the boundary conditions over the displacements and stresses in these sections are combined. Then the resultant discrepancies [2] on their common boundary,
r = al,
0 ≤ z ≤ h,
are minimized by the least-square method, collocation one, and also by the method ensuring the minimax of the modulus. The result formulas derived for determining radial, σr, circumferential, σφ, axial, σz, and shear, τrz, stresses at every point of the examined glass element are as follows:
at 0 ≤ r < al,
σr = p{- 1/2 + [2-1(1 + m)(1 + μ) + 3/8 × (1 - μ)a12h-2 +
3/2 × (1 + μ)a12h-2a2/(a2 - a12) × ln(a/a1) - 3/8 × (3 + μ)r2h-2] ×
(1 - 2z/h) + (2 + μ)(3/2 × z2h-2- z3h-3) + 2-1r2a1-2(a4 - 2a14) ×
a-2(a2 - a12)-1[- m + 2(1 + m)z/h - 6z2h-2+ 4z3h-3]} - p1;
σφ = p{- 1/2 + [2-1(1 + m) (1 + μ) + 3/8 × (1 - μ)a12h-2 +
3/2 × (1 + μ)a12h-2a2/(a2 - a12) × ln(a/a1) - 3/8 × (1 + 3μ)r2h-2] ×
(1 - 2z/h) + (2 + μ)(3/2 × z2h-2- z3h-3) + 2-1μr2a1-2a2 ×
(a2 - a12)-1[- (1 + m)(1 - 2z/h) - 3z2h-2+ 2z3h-3]} - p1;
σz = p(- 3z2h-2+ 2z3h-3);
τrz = 3pr/h × (z/h - z2h-2);
at al ≤ r ≤ a,
σr = pa12/(a2 - a12) × {1/2 × a2r-2 + 1/2 + [2-1(1 + m) ×
(1 - μ)(1 - a2r-2) + 3/8 × (1 - μ)a12h-2(a2r-2 - 1) +
3/2 × (1 + μ)a2h-2ln(a/r) + 3/8 × (3 + μ)(r2 - a2)h-2] ×
(1 - 2z/h) + [2 + μ+ (2 - μ)a2r-2](- 3/2 × z2h-2+ z3h-3) +
[r2a-2 + 1/2 × (a - r)2(a - a1)-2a2a1-2] ×
[m - 2(1 + m)z/h + 6z2h-2- 4z3h-3]} - p1;
σφ = pa12/(a2 - a12) × {- 1/2 × a2r-2 + 1/2 + [2-1(1 + m)(1 - μ) ×
[a2r-2 - (1 + μ)(1 - μ)-1] - 3/8 × (1 - μ)a12h-2(a2r-2 + 1) +
3/8 × (1 - 5μ)a2h-2 + 3/2 × (1 + μ)a2h-2ln(a/r) +
3/8 × (1 + 3μ)r2/h2](1 - 2z/h) + [2 + μ+ (2 - μ)a2r-2] ×
(- 3/2 × z2h-2+ z3h-3) + μ/2 × (a - r)2(a - a1)-2a2a1-2] ×
[(1 + m)(1 - 2z/h) + 3z2h-2- 2z3h-3]} - p1;
σz = pa12/(a2 - a12) × (- a2a1-2 + 3z2h-2- 2z3h-3);
τrz = 3pa12/(a2 - a12) × (a2r-1h-1 - r/h)(z/h - z2h-2)
where
m is the parameter equal to
1/5,
0,
and
1/8
in the least-square method, the collocation method, and the method providing the minimax of the modulus, respectively, in minimizing the discrepancies of the conjugate solutions;
μ is Poisson's ratio.
3. Strength Criteria to Be Used
Using these formulae, we shall evaluate the strength of the glass element on the basis of the Galilei, Tresca, Huber-von Mises-Hencky, and Pisarenko-Lebedev strength criteria [3], all of them being universalized and represented in the relative stresses in the form of laws of nature [4 - 6]:
the author's universalization of the Galilei strength criterion
(1)
σe0 = max{|σ10|, |σ20|, |σ30|} ≤ 1,
the author's universalization of the Tresca strength criterion
(2)
σe0 = σ10 - σ30 ≤ 1,
the author's universalization of the Huber-von Mises-Hencky strength criterion
(3)
σe0 = σi0 = {[(σ10 - σ20)2 + (σ20 - σ30)2 +
(σ30 - σ10)2]/2}1/2 ≤ 1,
the author's universalization of the Pisarenko-Lebedev strength criterion taken in a form which uses their idea [3]
(4)
σe0 = (1 - χ)max{|σ10|, |σ20|, |σ30|} + χ{[(σ10 - σ20)2 +
(σ20 - σ30)2 + (σ30 - σ10)2]/2}1/2 ≤ 1
as well as the Coulomb-Mohr criterion (in its usual form, often applied to brittle materials)
(5)
σe = σ1 - χσ3 ≤ σt
where
σe is the equivalent stress;
σ1, σ2, σ3
are the main (principal) stresses ordered with providing relations
σ1 ≥ σ2 ≥ σ3;
σt, σc
are the values of the ultimate strength, σu, in uniaxial tension and compression;
(6)
σj0 = σj/|σlj|
(j = 1, 2, 3)
where σlj is, for the usual principal stress, σj , its limiting value that has the direction and sign of σj and acts at the same point of a solid, the both other principal stresses vanishing, and the other loading conditions at the same point being the same [4 - 6];
χ is the
σt/σc
ratio.
For glass K8
σt = 29 MPa,
σc = 1400 MPa
[7],
and
χ = 0.0207 << 1.
Since
1 - χ >> χ,
the equivalent stress σe is maximum at
z = 0
and
0 ≤ r ≤ al,
where σr and σφtake their maximum values and
σz = 0.
4. Using the Pisarenko-Lebedev Criterion
Let us first use the author's universalization (4) of the Pisarenko-Lebedev strength criterion in the relative stresses (6). By
Π < Π0,
where Π0 is the solution to the equation
σr(0, 0) = σt(0, 0) = 0,
determined by the expression
Π0 = - 1/2 + 2-1(1 + m)(1 + μ) + 3/8 × (1 - μ)a12h-2 +
3/2 × (1 + μ)a12h-2a2/(a2 - a12) × ln(a/a1),
we obtain
σemax = σe(0, 0) = σr(0, 0) > 0,
and at
Π ≥ Π0
(7)
σr < 0,
σe(0, 0) = - χσr(0, 0) > 0,
σemax = σe(al - 0, 0) = (1 - χ)σr(al, 0)/2 +
(1 - χ)[σr2(al, 0)/4 + τmax2]1/2 + 2-1/2χ{[σr(al, 0)/2 +
(σr2(al, 0)/4 + τmax2)1/2 - χσt(al, 0)]2 + [χσr(al, 0)/2 -
χ(σr2(al, 0)/4 + τmax2)1/2 - χσt(al, 0)]2 +
[(1 - χ)σr(al, 0)/2 + (1 + χ)(σr2(al, 0)/4 + τmax2)1/2]2}1/2,
where τmax is the maximum of τrz.
Consequently, there is such a first critical value, Π1
(Π1 < Π0),
of ratio Π that
σe(al - 0, 0) > σe(0, 0)
if
Π > Π1,
σe(al - 0, 0) = σe(0, 0)
if
Π = Π1,
σe(al - 0, 0) < σe(0, 0)
if
Π < Π1.
It is evident that if
Π > Π0 ,
then all the stresses
σr , σφ , σz
are compressive. The exact solution to the equation
σe(al - 0, 0) = σe(0, 0)
of the fourth degree over Π1 leads to an equation that is too complicated for practical application. Therefore, since the equations for the stresses are approximate, and also owing to the fact that all the numerical calculations confirm that in Eq. (7)
(8)
[σr(al, 0)/2 + (σr2(al, 0)/4 + τmax2)1/2 - χσt(al, 0)]2 +
[χσr(al, 0)/2 - χ(σr2(al, 0)/4 + τmax2)1/2 - χσt(al, 0)]2 ≈
[(1 - χ)σr(al, 0)/2 + (1 + χ)(σr2(al, 0)/4 + τmax2)1/2]2
with inaccuracy of 1...2 %, we replace in (7) the left-hand part of (8) by its right-hand side. Then we get equation
(9)
σe(al - 0, 0) = (1 - χ2)σr(al, 0)/2 +
(1 + χ2)[σr2(al, 0)/4 + τmax2]1/2
whose inaccuracy is equal to hundredths of a percent. The first critical value of Π is
Π1= [σr2(0, 0) - (1 - χ2)σr(0, 0)σr(al, 0) - χ2σr2(al, 0) -
(1 + χ2)τmax2]/[(1 + χ2)p(σr(0, 0) - σr(al, 0))]
where Π and p1 are considered as 0 by calculating the right-hand part,
corresponds to spasmodical changing the position of the most dangerous point at which failure is initiated. By
0 ≤ Π < Π1
the failure is initiated at point
(0, 0),
by
Π > Π1
at point
(al, 0),
and by
Π = Π1
the failure start probabilities at the both points coincide.
σemax decreases with increase in
Π ∈ [0, Π1],
the fracture pressure,
pf = σtp/σemax,
where σemax is the running value proportional to p,
increases, and there is also the limiting strength in compression, σc .
Hencethere is a second critical (and optimal) value, Π2
(Π2 > Π1),
of Π, which maximizes pf and minimizes σemax. Differentiation of Eq. (9) in respect to
σr(al, 0)
leads to its single critical value ensuring the required extremum. In this case, the best value of the ratio
p1/p
is
Π2 = 3/(4χ) × al/h
giving the maximum fracture pressure,
pfmax = σt/[3/4 × χ1/2al/h + χ/p × σr(0, 0) | Π=0 - χ2 - χμ/2].
If Π becomes greater than Π2, then the most dangerous point spasmodically jumps to the center of the external base with
σemax = σe(0, h) = χ2[p1 + σr(0, 0) | Π=0] +
χ[1 - χ(2 - μ/2)]p.
Note that the Galilei strength criterion (1) gives analogous results:
Π1 = p-1{σr(0, 0) | Π=0 - τmax2/[σr(0, 0) - σr(al, 0)]};
σe(al - 0, 0) = σr(al, 0)/2 + [σr2(al, 0)/4 + τmax2]1/2;
σe(0, h) = χ[p1 + σr(0, 0) | Π=0 - μ/2p].
in the Relative Stresses
and the Coulomb-Mohr Criterion
The author's universalizations (2) of the Tresca strength criterion and (3) of the Huber-von Mises-Hencky strength criterion, both in the relative stresses (6), also by using (8) give the coinciding results:
Π1 = [σr2(0, 0) - (1 - χ)σr(0, 0)σr(al, 0) - χσr2(al, 0) -
(1 + χ)τmax2]/{(1 + χ)p[σr(0, 0) - σr(al, 0)]};
σe(al - 0, 0) = (1 - χ)σr(al, 0)/2 +
(1 + χ)[σr2(al, 0)/4 + τmax2]1/2;
Π2 = 8τmax2/χ/(1 + χ)/p/{[σr(0, 0) +
σr(al, 0)] | Π=0 - (1 + μ/2)p}
exists only if
[σr(0, 0) + σr(al, 0)] | Π=0 > (1 + μ/2)p;
σe(0, h) = χ[p1 + σr(0, 0) | Π=0 - (1 + μ/2)p];
there exists an optimum value, Πopt, of ratio Π:
Πopt = 3/4 × (1 - χ)χ-1/2al/h + p-1σr(al, 0) | Π=0.
The Coulomb-Mohr criterion (5) gives for the internal base of the glass element the same results as the last both theories do but is unfit for the external one with σr < 0 and for the spatial solid at all. For example, we have
σe(0, h) = χ[p1 + σr(0, 0) | Π=0 - (1 + μ/2)p] < 0
and it is inadmissible to compare this with the positive value of σt by the criterion.
6. Failure Nature and Pressure
The resultant data make it possible to predict the nature and pressure of failure of the glass element.
At
Π < Π1
it should be expected that bent circular glass sheets will fail by radial cracking initiated at point
(0, 0).
In the case
Π > Π1
failure should start at the point
(al - 0, 0),
and if the normal to the surface of the first crack at this point is assumed to be parallel to the vector σl, the crack should extend from this point under the angle
π + 0.5 arctan 2τmax/σr(al, 0)
to the axis 0r cutting off the central part of the internal base of the glass element initially along the conical surface with
z > 0
for a tip and with a negative angle of inclination of the generating line to the base,
0.5 arctan 2τmax/σr(al, 0) = α
to the negative direction of the axis 0r.
Geometrical interpretation of the results of minimizing conjugation discrepancies via the least-square method is shown in Fig. 3 for glass K8 and arbitrary values of Πand
h : al : a = 2 : 1 : 11/6
(i.e., the glass element is assumed to be geometrically similar to that with the dimensions
h = 60 mm;
al = 30 mm;
a = 55 mm).
Fig. 3. Effect of side surface compression of a glass element on the position of the most dangerous point (encircled in schemes), the failure pressure, pf, the magnitude and orientation of the highest main (greatest principal) stress, σ1 (vector), and the probable direction of the first crack (angle α) in accordance with the Galilei strength criterion (line 1), as well as the Tresca, Huber-von Mises-Hencky, and Pisarenko-Lebedev strength criteria (line 2), all of them being universalized by the method [4 - 6] and represented in the relative stresses.
According to all the mentioned criteria, at
0 ≤ Π < Π1 = 0.338
the fracture pressure slowly increases and then increases more rapidly together with Πfrom 42.0 to 82.3 MPa, and radial cracking with initiation in the center of the internal base of the glass element is expected. When the critical ratio
p1/p = Π1 = 0.338
is exceeded, failure initiates at the edge of the central part of this base. At
Π > Π1 = 0.338
the angle of splitting, -α, decreases with Π increasing from 45° initially rapidly and then more slowly to 8° at
Π = 3.
According to all the mentioned criteria besides the first strength theory, pf increases at a reducing rate to 255 MPa at
Π = 2
and to a maximum value of 272 MPa at
Π = Π2= 2.83,
and then very slowly decreases losing less than 1 MPa to
Π = 3,
see line 2.
According to the Galilei strength criterion (1), see line 1, at
0 ≤ Π < 0.338,
the results are identical with those obtained previously, the relationship remains unchanged up to the values
Π1= 0.354
at which failure starts at the point
(al - 0, 0),
the angles of splitting are the same, pf increases at a higher rate than that predicted by the other mentioned criteria (a difference is up to 10 % at
Π ≤ 1
and then rapidly increases), and by an almost linear law to 461 MPa at
Π2 = 2.45,
and then rapidly decreases in accordance with a hyperbolic law to 390 MPa at
Π = 3.
After reaching the maximum pf , failure is initiated at the point
(0, h).
An increase of Π above Π2 reduces pf and has no meaning for practice.
Experimental verification of the determined dependences shows the following results.
At
p1 = 0,
a glass element made of glass K8 with
h = 45 mm,
al = 30 mm,
a = 50 mm,
failed at
pf = 30 MPa
with the formation of a system of cracks similar to the radial system with initiation in the vicinity of the center of the internal base.
The calculated values of pf are
pf = 29.0 Mpa,
pf = 33.0 MPa,
and
pf = 30.4 MPa
bythe least-square method, collocation one, and the method providing the minimax of the modulus, respectively, in minimizing conjugation discrepancies.
A large number of experiments with elements made of the K8 glass with dimensions
h = 60 mm;
al = 30 mm;
a = 55 mm,
at
p1 = p
(Π = 1)
always resulted in splitting of the central part at the internal base with its subsequent cracking. The splitting angle was approximately 25-30° (its calculated value is 23°).
pf was 150-200 MPa at one or several loading cycles. Its calculated value is 166 MPa by the least-square method of minimizing conjugation discrepancies and slightly higher by the two other methods.
1. Some new relationships governing the effect of the pressure applied to the side surface of a three-dimensional cylindrical glass element on its strength and failure were investigated. When reaching a specific critical value of the ratio of this pressure to the external pressure, the positions of the points at which failure initiates change abruptly and the appearance of failure changes from a radial crack to the cleavage of the central part of the element from the side of its internal base.
2. Application of pressure to the side surface of the cylindrical glass element with proportions typical for illuminators very essentially increases its strength (approximately four times if this pressure is equal to the external pressure, and six or seven times if it is two or three times higher than the external pressure).
3. The Galilei strength criterion and especially the Tresca, Huber-von Mises-Hencky, and Pisarenko-Lebedev strength criteria universalized by the author, expressed in the relative stresses, and applied to the strengths of the three-dimensional cylindrical glass elements of illuminators lead to similar results experimentally confirmed and give relationships governing the failure of these elements.
1. Lev Gelimson. Analytic Macroelement Method in Axially Symmetric Elasticity. International Scientific and Technical Conference "Glass Technology and Quality". Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 104-106
2. L. B. Tsvik. Discrepancy of conjugation of displacements and stresses in problems of conjugation and contact of elastic solids. Dokl. Akad. Nauk SSSR, 268 (1983), No. 3, p. 570-574
3. G. S. Pisarenko and A. A. Lebedev. Deformation and Strength of Materials in a Multiaxial Stress State [in Russian], Naukova Dumka, Kiev (1976)
4. Lev Gelimson. Generalization of Analytic Methods for Solving Strength Problems [In Russian]. Drukar Publishers, Sumy, 1992
5. Lev Gelimson. General Strength Theory. Drukar Publishers, Sumy, 1993
6. Lev Gelimson. Generalization Method for Limiting Criteria. International Scientific and Technical Conference "Glass Technology and Quality". Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 98-100
7. I. I. D'yachkov, A. L. Kvitka, Yu. V. Komyagin, and V. S. Morganyuk. Development of efficient illuminator. Report 1 and 2. Probl. Prochn., No. 11, 104-109 (1985); No.12, 90-94 (1985)
Remark 11. Applying the previous methods leads to an analytlc macroelement methods (AMEM) system giving, unlike the finite element method [3], analytic quastsolutions.
Definition 12. The power analytic macroelement method is an application of the general linear-combination method to a linear general problem with generally subdividing a general domain into its several subdomains and generally minimizing and correcting all the general conjugation residuals (as absolute errors) between the corresponding subquasisolutions on the general boundaries of those subdomains.
Remark 13. Applying M of AMEM permits discovering new phenamena in general problems:
1) A general problem can limit the degree of a power representation of its general pseudosolution [2] from above.
2) A (whole) general type [4] of natural general problems can be generally overdetermined.
3) The general overdetermination of a general type [4] of natural general problems can be multiple.
Definition 14. The integral analytic macroelement method is applying the general restructurization method to a general system of nonequicomplicated general relations by the principle of tolerable simplicity [4].
Remark 15. Applying PM of AMEM permits discovering new phenamena in general problems:
1) Continuously varying a general system can lead to spasmodically varying its general structure.
2) A critical relation [4] in a general system can exist and bifurcate.
3) A critical relation in a general system can be invariant when a general system varies.
4) There can exist total critical relations forming the general boundary subsystem of the general system of critical relations.
5) The skips of the general structure of a general system can be both successed and reversive.
6) There can exist the determining general parameter and the equivalent general parameter for a general system.
7) A rational control by the determining general parameter of a general system can raise its equivalent general parameter by an order.
8) A coincidence of a rational control and of a critical relation can depend on that relation.
9) An equivalent general parameter can be uniform if a determining general parameter is (symmetrically or asymmetrically) nonuniform.
10) A general state can be initial out of the general center of a general system.
11) A general state of a general center of a general system is initial for some critical relation.
12) The general measure of a changed-structured general subsystem of a general system can be invariant.
13) The total general measure of all the changed-structured general subsystems with variable general measures can be invariant.
14) The well-posedness and the ill-posedness [1] of a general system are relative.
General problem type and setting theory (GPTST) in fundamental science on general problem essence is very efficient by solving many urgent (including contradictory) problems.
References
[1] Encyclopaedia of Mathematics. Ed. M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994
[1] Lamé G. Lecons sur la theorie mathematique de l’élasticite des corps solides. Gauthier-Villars, Paris, 1852
[2] Love A. E. H. A Treatise on the Mathematical Theory of Elasticity. Vols. I, II. Cambridge University Press, Cambridge, 1892, 1893
[3] Handbuch Struktur-Berechnung. Prof. Dr.-Ing. L. Schwarmann. Industrie-Ausschuss-Struktur-Berechnungsunterlagen, Bremen, 1998
[4] Lev Gelimson. Elastic Mathematics. General Strength Theory. The “Collegium” All World Academy of Sciences Publishers, Munich, 2004
[5] Lev Gelimson. Discretization Errors by Determining Area, Volume, and Mass Moments of Inertia. In: Review of Aeronautical Fatigue Investigations in Germany During the Period May 2005 to April 2007, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2007-042 Technical Report, Aeronautical fatigue, ICAF2007, EADS Innovation Works Germany, 2007, 20-22
[3] O. C. Zienkiewicz, R. L. Taylor. Finite Element Method. Volumes 1 to 3. Butterworth Heinemann Publ., London, 2000
[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The ”Collegium” International Academy of Sciences Publishers, Munich, 2004
[3] Lev Gelimson. Providing Helicopter Fatigue Strength: Flight Conditions. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 405-416
[4] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010
[5] Lev Gelimson. General Problem Fundamental Sciences System. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2011