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УДК 539.4 + 539.3 + 623.74 + 624 + 101 + 111.85
Оптимальные формы упругих тел: равнопрочные крылья самолетов и равнопрочные подземные туннели
Г.П. Черепанов
Нью-Йоркская академия наук, Нью-Йорк, 10007-2157, США
Со времен Платона (424?-347 до н.э.) и Аристотеля (3 84—322 до н.э.) считается, что форма лежит в основе не только физического мира, но и духовного, а выявление идеальных форм подобно находке алмазов в горной породе и является источником высочайшего интеллектуального наслаждения. Поэтому поиск таких форм, стремление их познать составляет силу, движущую науку и ученых, начиная с Пифагора (ок. 570 - ок. 500 до н.э.), Архимеда (ок. 287 - ок. 212 до н.э.), Эвклида (ок. 330 - ок. 260 до н.э.) и всех последующих. Во введении настоящей статьи дано краткое изложение теории форм Платона и дополнение Аристотеля к этой теории, которые лежат в основе излагаемой ниже теории упругих тел оптимальной формы. В рамках теории упругости оптимальной является такая форма тела, которая удовлетворяет принципу равнопрочности, сформулированному автором в 1963 г. Согласно этому принципу критерий безопасности, т.е. условие прочности/разрушения, выполняется одновременно в максимально большой части тела. Такое тело называется равнопрочным; оно обладает минимальным весом для заданного материала и критерия безопасности, или максимальной безопасностью для заданного материала и веса. В отличие от традиционных задач для равнопрочных тел нет теорем существования — обнаружение равнопрочных форм целиком зависит от искусства исследователя. В настоящей статье приводится каталог простейших равнопрочных тел и элементов конструкций, а именно: равнопрочный трос батискафа, ранопрочная башня (высотное здание), равнопрочная балка Галилея, равнопрочный вращающийся диск, равнопрочная тяжелая цепь, равнопрочный сосуд давления, равнопрочные оболочки, равнопрочные подземные туннели, равнопрочные перфорированные пластины и др. Найдены равнопрочные формы стреловидного крыла самолета; передняя и задняя кромки такого крыла прямолинейны в плане, а длина хорды в направлении полета зависит от назначения самолета. Равнопрочные конструкции существуют для самолетов различного назначения, от транспортной авиации до гиперзвуковых истребителей. Найдена также равнопрочная форма упругого тела с любым числом бесконечных ветвей в условиях плоской деформации и плоского напряженного состояния.
Ключевые слова: форма Платона, равнопрочный трос, равнопрочная башня, равнопрочное крыло самолета, равнопрочные оболочки, равнопрочный подземный туннель
Optimum shapes of elastic bodies: Equistrong wings of aircrafts and equistrong underground tunnels
G.P. Cherepanov
The New York Academy of Sciences, New York, 10007-2157, USA
Since the times of Plato (424?-347 BC) and Aristotle (384-322 BC), the form has been considered as a fundamental notion of not only the physical universe, but also the spiritual world. The forms and perfect shapes are like jewels in the rock—their search and discovery make up the highest delight to human beings. This is what constituted the motive and driving force of science and scientists beginning from Pythagoras (ca. 570-ca. 500 BC), Archimedes (ca. 287-ca. 212 BC), Euclid (ca. 330-ca. 260 BC) and all that came after. In the introduction to the present article, a brief account of Plato's theory of forms and Aristotle's addition to this theory are given; the theory of optimum shapes of elastic bodies can be considered as a footnote to the Plato's theory. In the framework of the theory of elasticity, the optimal shape of a body is the shape that meets the principle of equal strength or equistrength advanced by this author in 1963. According to this principle, the safety criterion like ultimate or failure stress is simultaneously satisfied in the utmost part of the body— this body or structure is called equistrong in this case. The equistrong structure has a minimum weight for a given material and safety factor, or a maximum safety factor for a given material and weight. As distinct from traditional problems, there are no existence theorems for equistrong shapes—a success in their search depends on skills of a researcher. In the present paper, a summary of common equistrong shapes and structural elements is brought out, namely: equistrong cable of bathysphere, equistrong tower or skyscraper, equistrong beam by Galileo Galilei, equistrong rotating disk, equistrong heavy chain, equistrong pressure vessels, equistrong arcs, plates and shells, equistrong underground tunnels, equistrong perforated plates, and others. A variety of the swept wings of aircrafts is found out to be equistrong; the front and rear edges of such wings are rectilinear in the plan view, and their chord in the flight direction depends on the task of an aircraft; the equistrong design exists for any task, from transport aviation to hypersonic jet fighters. Some new equistrong shapes of elastic solids with any number of infinite branches being pulled out of a body are also discovered for plane strain and plane stress.
Keywords: Plato's form, equistrong cable, equistrong tower, equistrong wing of aircraft, equistrong plates and shells, equistrong underground tunnel
© Cherepanov G.P., 2015
1. Introduction. Plato's theory of forms and equistrong shapes
Plato's theory of forms set up the foundation of philosophy and made Plato the greatest philosopher of all times. According to this theory, everything is, at first, born in human mind as a form or a blueprint of perfection that may, or may not, turn into a real thing. And so, forms are primary while what we see and feel is secondary. The source of forms in human mind cannot be understood because it is of divine nature. And any knowledge as well comes from divine insight.
However, only few selected have vision of some forms, the search and discovery of which makes up the meaning and highest delight of their life. The history of science was that of success of the Plato's theory as scientists and thinkers created forms in their mind, followed by an embodiment of some of them into real things. In particular, with respect to social statecraft, Plato advanced the idea of totalitarian state ruled by a King-Philosopher as the most perfect state structure. This Plato's form was first carried out in the Soviet Union (1922-1990) created in the mind of theoreticians Marx, Lenin, and Stalin who could be called the first King-Philosopher or King-Benefactor because he created the first state of social justice destroyed by his successors. Today, this Plato's form lives well in China.
Plato's theory of forms is universal—it cannot be refuted, e.g. any materialistic viewpoint as well as any result of touching or measuring is a Plato's form because it starts on in human mind. Plato's forms do not distinguish between good and bad, between right and wrong, and they are, thus, amoral. That's why Greeks were embarrassed by this teaching first preached by Socrates, Plato's teacher—they executed Socrates "for corrupting the youth". The life of Plato was also far from being easy: he was enslaved for a while and redeemed by his noble relatives.
By Plato's theory, moral is just a temporary form of vision for common people, which is also supported by the history of humankind. Killing ill or unfit babies was a common custom in the ancient times (read the Bible, for example) while it is the utmost crime today. Moreover, according to the current genetic knowledge, this past custom served to genetically improve humans while the latter one to deteriorate them. Based on this Plato's form, humankind is now degenerating.
It should be noticed that even all those who criticized Plato's theory, as a matter of fact, offered only some additions to this theory because everything they suggested was born in mind and, hence, was a Plato's form. Empirical observation and experience as a matter and essence of Plato's forms was the most valuable addition introduced by the best Plato's student, Aristotle. This addition reconciled Plato's theory with common people and knowledge—and it saved philosophers from execution. Aristotle became the
beloved philosopher of the West. Besides, he educated his student Alexander the Great, the first and only King of Asia.
The theory of optimum shapes of elastic bodies can be considered as a footnote to the Plato-Aristotle theory. It is based on the principle of equal strength or equistrength suggested by this author in 1963. According to this principle, an utmost part of an elastic body should simultaneously reach a limiting state by loading and, hence, be destroyed or yielded, if the material is elastic-plastic. At any point in this part, the strength or yielding criterion is simultaneously met. The bodies satisfying this principle are called equi-strong. They possess minimum weight for given material and safety factor or maximum safety factor for given material and weight of the body. That's why they can be called the most perfect elastic bodies of nature.
However, the history of this approach starts with Galileo Galilei who was the first to advance the equistrong design of the beam bent by an edge transverse force (see below, in more detail). It is remarkable that this discovery was done simultaneously with the introduction of the notion of the strength and stress itself, which appeared too difficult a problem, even for Leonardo da Vinci. Like Plato, Galilei was obsessed by search of ideal forms, that is Plato's forms. Without any exaggeration, scientists like Pythagoras from South Italy, Archimedes from Sicilia, Euclid from Alexandria and all that came after were busy in search of Plato's forms. (Then, South Italy and Sicilia were Greek colonies until 212 BC when they were occupied by Roman troops one of which killed Archimedes. Alexandria was set up by Alexander the Great who captured Egypt and made it a province of his Macedonian empire.)
The design of equistrong heavy chain was next advanced in the middle ofthe 19th century (see below). In about 1900, an equistrong design of a solid spinning disk was devised by engineers at the Laval Company in Sweden—it is a particular case of a more general equistrong design taking into account the temperature field. In 1922, Bienzeno designed an equistrong head on a cylindrical pressure vessel [1].
A catalog of some equistrong designs is given below including equistrong cable, equistrong wing of aircraft, equistrong tower, equistrong turbine blade, equistrong beams, plates and shells, equistrong underground tunnel and many others. The main difficulty in finding equistrong designs is the existence problem connected with the essential nonlinearity of problems with an unknown boundary. Needless to say that the optimization technique currently used for design of engineering structures is not absolute because it is not based on a sound physical principle.
In stress concentration problems, the maximum stress should be distributed along the utmost portion of the stress concentration area. As applied to perforated elastic plates in plain strain or plain stress, the maximum stress should be one and the same along the whole surface of holes. In such equistrong elastic structures, the failure and fracturing or
yielding starts simultaneously along all surface of holes. The plates perforated by these equistrong holes have minimum weight for a given material and safety factor, or a maximum safety factor for a given material and weight, as compared to the plates with holes of a different shape.
Using complex variables, the equistrong shapes of one and two holes as well as of periodic and double-periodic systems of holes were found out by this author for perforated plates in plane stress or plane strain. This work was later continued by S. Vigdergauz, L.T. Wheeler, I.A. Kunin, L.V. Ershov, V.M. Mirsalimov, V.M. Smolsky, N.V. Bani-chuk, M.P. Savruk et al. The list of some works on equistrong structure designs is given below in chronological order [134].
The term "equistrong" as a synonym for the poorly determined notion "optimum" was coined by the present author in 1963-1966 as applied to the design of optimum engineering structures. It should be noted that it is still unavailable in common dictionaries although it has been broadly used by dozens of researchers in science and technology, see, e.g. internet sources in WorldWideScience.com, Scribd.com, issuu.com and others. Recently, it also found a broad dissemination in pop culture and game industry, in pharmacy, in insurance industry, in equestrian sports etc., see equistrong.com in internet.
2. A catalog of some equistrong structure designs
Let us provide some most common equistrong designs of structures and structural elements every engineer should know.
2.1. Equistrong cable of bathysphere
Suppose a bathysphere of weight W in the ocean depth is supported by a cable of variable cross-section area A with material ultimate strength ab. Let the z axis be a vertical line so that the bathysphere is at z = l while the day surface is at z = 0. With the tensile stress being equal to nalong the whole cable, the cross-section area of the equistrong cable is [14]
W os
A =-exp[A(/ - z)], A = -tg-. (1)
nCTb nCTb
Here h is the safety factor of the cable material, p is its mass density, g is the gravitation acceleration.
This is the minimum-weight maximum-safety design which allows one using a steel cable to reach the bottom of the Mariana Trench which is the deepest depression in the ocean on the Earth unreachable by a steel cable of any constant cross section.
2.2. Equistrong tower or skyscraper
Suppose a vertical tower or skyscraper is subject to the only force of gravity. Suppose also that the tower is structurally homogenous, i.e. the value of mass density is constant for any specific volume large as compared to linear
dimensions of rooms/offices but small as compared to linear sizes of the whole tower.
In this case, the equistrong design of such a tower or skyscraper is given by Eq. (1) where A is the horizontal cross-section area of the tower, z is the distance of this cross section from the foundation of the tower, p is the weight of the tower divided by its volume, W is the weight of the top peak-shape structure, l is the height of the tower without the peak, and ab is the maximum weight endurable by one-floor structure of the tower divided by its area in horizontal cross section [31].
Simple estimates based on this design show that equi-strong skyscrapers can safely achieve the height of a hundred miles using steel and composite materials. However, the account of ignored factors such as wind loads and quake-induced vibrations significantly reduces this prediction. Of major importance are the aerodynamic loads from hurricane winds which velocity can achieve 200 m/s.
2.3. Equistrong turbine blades
Suppose a turbine blade of length l of variable cross-section area A is attached to a cylindrical shaft that rotates at angular velocity w. Let us take into account only inertia force while ignoring the aerodynamic and gravitational loads. In this case, the equistrong turbine blade design is provided by the following equation of the cross-section area
[14]
A = ^exp^/2 - x2)], A = (2)
nab 2nab
Here x is the distance of a cross section from the shaft, p and CTb are the blade mass density and ultimate strength, and m is a concentrated mass at the end of the blade, e.g. a small portion of the blade of a constant cross section. (Everywhere below h is the safety factor, ab is the ultimate strength, and p is the mass density.)
This equistrong shape is used in aviation turbo-engines, in vanes, and in many structures with rotating blades.
2.4. Equistrong beams
Suppose the cross section of an elastic beam has two axes of symmetry which coincide with the y and z axes, and the x axis is the central axis of the beam. Let the beam be under transverse bending in the xy plane and let the shape of the cross section be given but one parameter or the scale. The most stretched fiber on the beam surface should be under one and same tensile stress in all beam cross sections. This is the condition of equistrength, and the beam that meets this condition is called equistrong [14]. Designate as h the distance of the most stretched fiber spot in a beam cross section from the xz plane which is the neutral plane of bending.
For the equistrong beam, the value of h as a function of x is determined by the following equation [14]
hM (x) =ncTb I (x), (3)
and the displacement v of the beam central axis meets the equation
d2p
dx2 Eh
(4)
Here M(x) is the bending moment, I(x) is the moment of inertia of the cross-section area about the neutral axis, and E is Young's modulus.
Equations (3) and (4) are the basic equations of the theory of equistrong beams. Let us analyze them for the case of rectangular cross section of height 2h along the y axis and width 2a along the z axis, with the beam of span l being loaded by transverse force P applied at the free edge of the beam while the other edge being clamped. This is the problem studied by Galileo Galilei.
In this case, we have
4 3
M(x) = Px, I = Jah\
(5)
The origin of the Cartesian coordinates is taken at the free edge of the beam.
Let us study the following basic designs of the equistrong beam:
G-design: a = const, h = h(x) offered by Galileo Galilei, A-design: h = const, a = a(x), and S-design: h = Csh0, a = Csa0, h0 = const, a0 = const, Cs = Cs(x), where Cs is the scaling parameter of the self-similar design.
Using Eqs. (3) and (5), it is easy to find the shape and weight W of equistrong beams of these designs:
2h =
h2 = 1 "max
3Px Wb ' Pl
1
a=
4 naab
Px
W = 3 Palhmax ,
(G-design),
(6)
4 nabh
f
W =
3 pPl2
4 nh^b
(A-design),
Cs =
3Px
y/3
4rlCTb a0 h0
4V33^3 f
W =-p
\2/3
(7)
(8)
nab a0 h0
a^3^3 (S-design).
From here, it follows that the G-design provides a minimum weight only for sufficiently large values of force P while the A-design gives a minimum weight for sufficiently small values of P. The S-design cannot compete with the G-design for any values of P.
A similar analysis can be done for a variety of statically determinate beams.
2.5. Heavy equistrong beam
However, some problems statically determinate for beams of a given shape appear much more complicated for equistrong beams which shape has to be found. As an ex-
ample, the problem of a heavy equistrong beam can serve. It can be shown that, in the case of simply supported edges at x = ±l and rectangular cross sections, the heavy equistrong beam of span 2l has to meet the following integral equations: for the G-design when a = const, h = h(x), t, x e (0, l)
(l - x) f h( x)dx - f h( t)( t - x)d t = ^ h2 (x), (9)
0 x 3Pg
for the A-design when h = const, a = a(x), t, x e (0, l) (l - x) f a(x)dx - f a(t)(t - x)dt = ^^ a( x). (10)
0 x 3Pg
The most stretched fiber is on the lower surface of the heavy beam at y = -h.
2.6. Equistrong wing of aircraft
The primary function of any wing is the creation of a lift force that arises at positive angles of attack as a result of air braking and slower flow of air along the lower surface of the wing as compared to the upper surface which causes a greater pressure of air on the lower surface because of Bernoulli's law. For the purpose of demonstration, let us ignore, at first, torsion torques and all other functions of wings so that the wing is represented as a beam under transverse bending by aerodynamic loads of lift forces.
For a wing of chord lz and very large span lx the lift force F is given by the following equations set up by Jou-kowski, Prandtl and Karman:
F = Ca lxLz , V/c < 1,
V1 - (V/c)2
^ 4apaV lxLz Tr/
F = x z , vjc > 1.
V(V/c)2 -1
(11)
(12)
Here V is the aircraft velocity, c is the sound speed in the air, a is the small angle of attack, pa is the air density, and Ca is the aerodynamic coefficient depending only on the shape of profile of the wing section at x = const. Equations (11) and (12) are strictly valid for thin wings of the same profile along the x axis, for very large spans, and for small angles of attack. In this case the air flow velocity vector at each point is in the plane of a corresponding wing section.
For hypersonic flows when V >> c, even the simpler law of Newton-Busemann is valid, according to which the lift is also directly proportional to the wing chord. The same direct proportionality to the wing chord is valid for transonic flows when V is close to c and Eqs. (11) and (12) don't work.
Now, let us assume that the wing section varies slightly and self-similarly along the x axis so that the size of each section of a finite span wing is determined by the chord being a function, lz = lz (x), with small derivative, l'z << 1. Let us apply Eqs. (11) and (12) to every wing section; as a result we get the following transverse load densityp(x) on the wing of a variable section area for a variety of flight
regimes
p(x) = x| a, V lPaF(x).
(13)
Here x(a, V/c) is one and same function for each wing section. For a wing of a finite span, its lift force F is found by integration of Eq. (13) over x.
The greater ratio F—, where W is the wing weight, the more perfect is the wing design. Let us find the chord variation with x that provides one and same value for the tensile stress along the most stretched fiber on the lower surface of the wing. This is the optimum design of equistrong wings.
At a wing section, the bending moment of transverse loads is as follows
x
M(x) =Jp(t)(x - t)dt, t, xe (0, lx). (14)
0
Here x = 0 is the free edge of the wing, and lx is its span.
The moment of inertia of a thin airfoil profile can be approximated as
I = 8lzh3. (15)
Here h is the distance of the most stretched fiber of the wing from the neutral line of bending, and 8 is a size-independent number specific for the profile shape and the angle of attack.
For self-similar wing sections, we have lz = Cs(x)lz o, h = Cs( x)ho. (16)
Here Cs( x) is the dimensionless scaling parameter to be found, and lz 0 and h0 are some scaling units.
Using Eqs. (13) to (16) and Eq. (3), it is easy to derive the following nonlinear integral equation for the equistrong wing of aircraft
Cs3( x) = aJ Cs( x)( x -t)d t. (17)
0
Here
^-P^TXfa, , t, x e (0, lx). (18)
n8abh0 I c )
By means of new dimensionless variables 5 and t, Eq. (17) is reduced to the following simplest form
f 3(5) = J f (T)(5-T)dT. (19)
0
Here
f (5) = Cs ^/A, 5 = xVA, (20)
T = tVA, 5, Te (0, lxJA).
The nonlinear integral equation (19) has, at least, one exact solution
f (5) = 5/V6. (21)
The problem of other possible solutions is open.
According to the solution (21), the delta wing, both front and rear edges of which are rectilinear in the plan view, is equistrong for the following coefficient Sc and the chord
magnitude lz (x) in the flight direction:
H
lz ( x) = xSc, Sc =
V
pV 2
6n8ab »o
X| a,
V
Dimensionless number Sc is different for various critical regimes of flight and the task of aircraft described by parameters a, 8 and V/c.
The equistrong design (22) can be used in a variety of delta wings, both front and rear edges of which are rectilinear in the plan view, and the chord length of which in the flight direction as well as the span of wings depends on the task of an aircraft. This equistrong design can be used for any task, from transport aviation to hypersonic jet fighters.
The solution (21) includes as well the wing designs with the front edge, z = f(x) + Cx, the rear edge, z = f(x) + Dx, and the chord length, lz = (C - D)x = Scx, wheref(x) is an arbitrary function, which provides much more opportunities for the equistrong designs of swept wings.
2.7. Equistrong shells of revolution
Let us consider the shells of revolution under internal hydrostatic pressure p, in the cases when all boundary conditions can be met by only membrane stresses in the shell. Let the middle surface of a thin-walled shell of revolution is given by the equation r = r(z) in cylindrical coordinates r and z, where z is the axis of symmetry. In this case, it is easy to find the variable thickness of an equistrong shell, the maximum stress in which being one and same everywhere in the shell.
We provide some simplest results for the thickness 2h(z) of such equistrong shells:
a. Ellipsoidal shell where 2a and 2b are main diameters
r = -V2bz - z2,0 < z < 2b, b
p 2[br + a2(1 - z/b)]2 - a4 , 2h(z) =-¡---, v 1 n , b > a,
2nab 4bAr 2 + a 4(b - z )2
2h(z)=
pr 2rlCTb
. a4(b - z)2
1 + ,4 2 ' b <
b r
(23)
(24)
(25)
b. Paraboloidal shell where r0 is the radius of curvature
at the coordinate origin r = \!2000z, z > 0,
2h ( z ) = r0 + 4 z + 4r.
r0 + 2z
2nCTb 4'
(26) (27)
c. Toroidal shell where a and d are two radiuses of the torus
2h(r ) =
pa
2nab
1 +— I, d - a < r < d + a, d > a. (28)
The general theory of equistrong plates and shells of arbitrary shape under arbitrary loads causing their extension and bending is given in [9, 10, 14].
Arcs, viaducts, cupolas, domes, and other shell structures have been approached to as membranes and broadly
used in building since ancient times. In practice, only a part of the above-mentioned membrane used to be built so that, e.g. at a section z = const, it is connected to other structural elements. In shells of positive Gaussian curvature—as the shells in Eqs. (23) to (28)—the non-membrane edge effect is fading when the distance from the joint increases; it can be lacking, at all, for some special joints.
It is worth to notice that the structural efficiency of a perfect equistrong shell, which is closed, made of strong fibers uniformly woven on a matrix, and subject to only internal pressure, does not depend on the shape of the shell and is equal to [14]
Vn = (29)
W 3pgp
Here Vin and W are the volume inside the shell and weight of the shell.
2.8. Equistrong heavy cable
Suppose an ideally flexible cable or fiber hangs between two supports at x = ±l, z = 0 where the direction of the z axis coincides with that of gravitation. In equistrong cables, the tensile stress is equal to r|CTb at all points. The curve and the cross-section area A of equistrong cables are given by the following parametric equations [14]
x = 2b| arctg//b -n|, (30)
z = a - blnch| b j, (31)
a = blnchi -1, b =
t b J pg
r ,, , fn l 1 l n L = blntgl — + —I, -<—.
| 4 2b j b 2
Here s is the length of arc so that s = 0 at x = 0, a is the maximum deflection of the cable, pg is the specific weight of the cable material, 2L is the length of the whole cable, and A is an arbitrary constant.
The equistrong design (30) to (32) is of particular interest for building suspension bridges.
2.9. Equistrong spinning disc
Suppose a thin elastic solid disc is spinning at angular velocity ro around its center. The disc thickness h(r) can be chosen to meet the condition ar =ct6 b, at each point of the disc having a thin axially symmetric rim at the edge r = b. This disc is called equistrong.
The thickness of equistrong spinning disc is as follows [14]
h(r) =
Prm 2nat
•exp
pro 2na,
-(b2 - r2)
, 0 < r < b.
(33)
Some generalizations of this solution for discs with a central hole in a temperature field are presented in [14].
2.10. Equistrong hole in plate
Suppose there is a hole in an elastic plate stretched by principal stresses, ax = p > 0 and ay = q > 0, far from the hole. The hole surface is free from loads. In 1963 the author started a search for a hole with the tangent stress on its surface being one and same at all points. It was the first inverse 2D problem of the theory of elasticity to be solved. With the help of the theory of functions of a complex variable, the sought contours of the hole were found out to form a set of self-similar ellipses [2]
2 2 x y
—r + ^ = 1
a" b~
with the following ratio of semi-axes a _ p b q
The tangent stress at on the hole surface is equal to
(34)
(35)
Here pr is the mass of the rim per unit of its length.
=n°b = - (p+q). (36)
q
This hole was called equistrong. It possesses some remarkable properties:
(i) Yielding or fracturing begins simultaneously at all points of the hole.
(ii) The weight of the plate with an equistrong hole is minimum compared to the plates with holes of any shapes of the same area and the same maximum tangent stress.
(iii) The plate with an equistrong hole has maximum safety factor as compared to the plates with holes of any shapes of the same area and the same plate weight.
The solution (34) to (36) is valid both for plane stress of thin plates, which thickness is much less than the hole size, and for plane strain of thick plates, which thickness is much greater than the hole size. Therefore, it is reasonable to assume that this equistrong shape is the same for plate of any thickness. Equation (36) serves to determine safe loadings.
(Today, we admire Archimedes who two thousand years before Newton and Leibnitz could differentiate and integrate, even without having knowledge of Arabic arithmetic we have used for a thousand years. However, Archimedes mostly appreciated his discovery that the surface area of the maximum sphere inside a cylinder is the same as that of the cylinder—and, on his vow, this diagram was engraved on his tombstone. Simplicity is divine—complicacy is from devil.)
Later, using complex variables, the equistrong shapes were found out for two holes, for periodical and double-periodical systems of holes in plates as well as for many other 2D and 3D problems, see, e.g. [3-34]. All equistrong shapes possess the same optimum properties indicated above. Some equistrong notches and configurations of special shapes were also found and used, see, e.g. [17, 20, 35].
2.11. Equistrong underground tunnel in rocks
The building of safe underground facilities, especially transport tunnels, is the most important part of civil engineering.
Suppose it is required to build an equistrong underground tunnel in the heavy elastic half-space, y < H, as we represent Earth's rock. The tunnel surface is assumed to be a cylinder which generating lines are parallel to the day surface and which cross-section is to be found. Let us ignore all outer loads on the cylinder surface. The rock is assumed to be homogeneous and isotropic, so that we arrive at a plane strain problem with unknown boundary.
The stresses in rock without a tunnel, or very far from the tunnel, are as follows
° x = Tpg (y - H), a y = Pg (y - H), t xy = 0. (37)
Here Oxy is the Cartesian system of coordinates in the cross-section plane of the tunnel with the center being at the center of symmetry, x being parallel, and y perpendicular, to the day surface at y = H, and T is the coefficient of the side or lateral thrust of rock. This coefficient is determined by the geological history and measured experimentally. Sometimes, it is taken to be equal to v(1 - v)-1 where v is the Poisson's ratio of rock. However, as a matter of fact, T is independent of v, and it can be greater than 1.
It can be shown that an equistrong shape of the tunnel cross-section with a constant tangent stress across its whole boundary does not exist. However, if a small linear deviation from a constant is accepted, then the equistrong shape exists and for deep underground tunnels is given by the set of ellipses (34) with the following ratio of semi-axes [3-5]
The tangent stress on the wall of the equistrong tunnel is equal to
at =na b =PgH (1 + T )(y - H). (39)
It slightly deviates from a constant for deep underground tunnels when H >> b > y, which is the case. Equation (39) serves to choose the optimum depth of a safe tunnel.
If the only upper half of this ellipse is used in practice, then a stress concentration emerges at and around the corner points x = ±a, which requires some special strengthening of the stress concentration area.
2.12. Optimum shapes of elastic solids with four infinite branches
Let us consider an elastic body subject to the conditions of plane strain or plane stress. In the plane, the Cartesian axes x and y are assumed to be the axes of symmetry of the body. At infinity, the body has four branches extended along the axes of symmetry and approaching asymptotically the strips of constant thickness 2h1 and 2h2 along the corresponding axes x and y.
The surface of the body is assumed to be free of tractions. At infinity, there is a state of uniform extension along
the axes of symmetry. Along the unknown boundary of the body, tangent stress is supposed to be one and same at all points. This condition provides optimum shapes of minimum stress concentration and minimum weight plates.
The optimum shapes of such equistrong bodies were found to be determined by the following parametric equations [29]
x = h2 + — h1 lnctg—,0< 6<n, (40)
n 2 2
y = h h2lnctgjf^ -—j. (41)
It is noteworthy that the optimum shapes in the case under consideration do not depend on the applied stress at infinity. Equations (40) and (41) provide the equistrong shape of the body only for x > 0, y > 0; the rest of the shape is built using the axes of symmetry x and y.
For very small values of h1 /h2 or h2 /h1, Eqs. (40) and (41) provide the shape of the equistrong jointing of a halfplane with a strip.
Equtions (40) and (41) as well as Eqs. (34) to (36) and Eqs. (37) to (39) are also valid for linear viscoelastic materials due to the viscoelastic analogy [33]. Namely, the stresses and unknown boundaries in corresponding problems for viscoelastic materials are the same as for elastic materials; however, the displacements are different [33]. Keeping this in mind, Eqs. (40) and (41) provide also the shape of polymer fluids being solidified in the process of pulling-out.
3. Optimum shape of elastic bodies with n infinite branches
Now, suppose that an elastic body subject to the conditions of plane strain or plane stress has any number n infinite branches extended along n axes of symmetry, at 6 = k(2n)/n, in the complex plane, z = re6, and approaching asymptotically n strips of the constant thickness 2h when r ^ «> (k = 0, 1, 2, ..., n - 1; n = 2, 3, 4, ...).
The surface of the body is assumed to be free of tractions. At infinity, there is a state of uniform extension by tensile stress p along n axes of symmetry. Along the unknown boundary of the body, the tangent stress at is the same at all points of the boundary.
Let us find the shape of this equistrong body using the general representation equations by Kolosov-Muskheli-shvili:
ax + ay = at + an = 4ReO(z), z = x + iy = re16, (42)
ay - ax + 2iTxy = e~lla (an - at + 2iTtn ) =
= 2[ zO'( z) + T( z)].
Here ax, ay, t^, an, at and Ttn are stresses, O(z) and T(z) are analytic functions, t and n are the Cartesian rectangular coordinates associated with the unknown boundary contour L. Namely, t is the direction of a tangent to L and n is an
outer normal, a is the angle between the x and t axes (in the direction of rotation from x to t).
According to Eqs. (42), the boundary value problem in the z plane may be formulated as follows
4ReO(z) = p, z e L, (43)
2e2ia[zO'(z) + z)] = -p, ze L. (44)
From Eqs. (42) and (43), it follows that at = p along L because an = 0 along L.
According to Eqs. (42), 4O(z) = p for z ^ From here, the limited solution to the Dirichlet problem, Eq. (43), is as follows
0( z ) = -4 p.
(45)
From here and Eq. (44), we find
2e2iaT(z) = -p, z e L. (46)
According to Eq. (42), the following condition has to be met at infinity:
z ) = - 1 pe
-2(k -1)Pz
(47)
for y - xtg(k - 1)P| < |h sec(k- 1)P|, where z ^ - k = 1, 2, 3, ..., n, P = 2n/ n.
The solution to this boundary value problem is invariant with respect to the following group of transformations, r = r, 6' = 6 + P; the symmetry operations form a group with n elements. This group property allows us to find the exact solution to this problem.
Let us introduce the parametric plane of the auxiliary complex plane Z by means of the analytic function z = ro(Z) conformally mapping the interior of the unit circle of the Z plane onto the elastic domain with the boundary contour L of the z plane. The following correspondence of points of the conformal mapping is accepted:
ro(0) = 0, ro(Z) ^ ~ (48)
when Z ^ exp[(k - 1)Pi], k = 1, 2, ..., n.
The elastic domain, kP < 6 < (k + 1)P, of the physical plane corresponds to the sector, |Z| < 1, kP < argZ < (k + 1)P, of the Z plane, and the symmetry lines of the physical plane coincide with those of the parametric plane.
For an increment of z along contour L, we have dz =|dz | eia (49)
For the respective increment of Z along unit circle, we find dZ = iZ|dZ|. (50)
From Eqs. (49) and (50), it follows that
ia = iZro' (Z)
|»' (01'
Substituting e'a in Eq. (46) by Eq. (51) provides
Z2»' (Z)v(O = »Û),|C| = 1,
where
(51)
(52)
(53)
V(Z) = -T(ro(Z)).
p
The nonlinear boundary value problem (Eq. (52)) serves to determine two unknown functions ro(Z) and ^(Z) ana-
lytic inside the unit circle and satisfying the following conditions
»(Z) ^ -, ) ^-e"2(k -1)Pi (54)
when Z^-e(k -1)Pi', k = 1, 2, ..., n.
Let us introduce new functions F'(Z) and G'(Z) : F (Z) = Z V (Z)v (Z) -»' (Z), (55)
G(Z) = Z 2»U)v(Q+»' (C). (56)
By means of these functions, the boundary value problem (Eq. (52)) is reduced to the following Dirichlet problems:
Re F'(Z) = 0, | Z | = 1, (57)
ImG'(Z) = 0, | Z | = 1. (58)
On the basis of Eqs. (54) to (56), F'(Z) and G'(Z) are infinite at n points of the unit circle at
Z = Zk =e(k"1№, k = 1,2,3,..., n. (59)
We continue F'(Z) and G'(Z) analytically onto the entire Z plane by means of the following functional equations
■f 1N
f 1N
F'(Z) = -F' Z , G'(Z) = G'
z
(60)
According to Eqs. (60), the boundary value conditions of the Dirichlet problems, Eqs. (57) and (58), are satisfied.
On the basis of Eqs. (59) and (60), F'(Z) and G'(Z) are analytical and limited everywhere throughout the Z plane except for n points Z = Zk where these functions may have single-valued singularities.
We seek the solution having first-order poles at these exceptional points as follows:
F' (Z) = E
k =1
g' (z )=i
k =1
f
N
C-Ck Bk
1 C-Ck
C-Ck 1/C-Ck
+ iF0,
+ Go
(61)
Here Ak and Bk are some complex constants, and F0 and G0 are some positive constants
The boundary conditions, Eqs. (57) and (58), and Eqs. (60) are satisfied by any values of these constants. From Eqs. (55) and (56) we have
2ro' (Z) = G (Z) - F' (Z), F (Z) + G (Z)
v(Z) = -
2Z2» (Z )
(62) (63)
According to Eqs. (54), we have
Z V(Z) +1 ^ 0 when Z^Zi. From here and Eq. (56), it follows that
Bi = 0, G (Z) = G0.
From here, the solution, Eqs. (62) and (63), becomes 2ro'(Z) = G0 - F'(Z), F (Z) + G0
v(Z) = ■
(64)
(65)
2Z2»' (Z )
On the basis of the group symmetry of this problem, the complex constants Ak in Eq. (61) can be written as follows
Ak = ae(k -1№. (67)
Here a is a real number to be found.
Let us check this result by another way. First, put Z = = t ^ 1 in Eqs. (61) and (66) where t is real; we have 2o>'(t) = G0 + iF0 - Ax!(t -1). From here, it follows that F0 = 0 and A1 is real because m'(t) is real. Now, along an arbitrary line of symmetry, we have
z = rZk , Z = tZk , _
' Zk = eiP(k-1). (68)
— = G - Ak Ak
dz=Go -zk (t -1) Halt -Sk
From here, it follows that Eq. (67) is valid, again.
Integrating ro'(Z) over the infinitesimal semi-circle with the center at the pole, Z = Zk, provides aniZk which should be equal to 2hiZk on the physical plane. As a result, we get
a = —. (69)
n
The function ^(Z) is analytic at Z = 0. From here and Eqs. (61), (66) and (67), it follows that
2 nh. (70)
n
Taking into account Eqs. (61) and (66) to (70), we can write the final solution in the following shape
Go =-
®'(Z)=n h E
n k=1
(
1 -
Z k
Z = J3(k -1) , Z k = e ,
¥(Z) =
Go -m'(Z) Z2ro'(Z) .
Because of Eq. (70) and the following equations
0 = 0,
n E - e
k=1
n (
E 1
k=1
Z
V1
1
(71)
(72)
(73)
(74)
, =1+ZI7-+ 0(Z2) when Z ^ 0
Z k J k=1 Z k
\ /
the function ^(Z) in Eq. (72) is limited and analytic at Z = 0.
By integrating Eq. (71), we find out the function con-formally mapping the interior of the unit circle onto the elastic domain
®(Z) = -- h EZ k in
n k=1
f
1 i
\
, Zk = eiP(k-1). (75)
The parametric equation of the boundary of the domain is
■ = — h EZk in(1 -
it-ip( k -1)
),
(76)
n k=1
0 < t < 2n, n = 2, 3, 4, ... .
Equations (75) and (76) provide the exact solution for the equistrong shapes of n elastic bodies. For n = 2, we have
2M 1+Z
z = — h in-.
n 1 -Z
(77)
In this case, the equistrong body is a strip of thickness
2h.
For n = 4, we get Eqs. (40) and (41) for h1 = h2 = h, see the drawing in [30].
Drawing other beautiful equistrong shapes is granted to the respected reader. Their number is infinite, and their shape does not depend on tensile load p so that each body is equistrong for any load but safe only when p <nab.
4. Conclusion
By a few touches, this author tried to show an unexplored, alluring area where art and science are closely interconnected. The quest for objects satisfying some requirements put forward beforehand has always been, and will always be, thorny. No theorems of existence, and no expectation of them. However, everybody dreams of perfect objects, Plato's forms, and succeeds sometimes in discovering them.
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nocTynana b peaaKUHro 04.06.2015 r.
Ceedenun 06 aemope
Genady P. Cherepanov, Prof., Hon. Life Member, The New York Academy of Sciences, USA, [email protected]
