singularities of elastic stresses and of harmonic functions at conical ...
Int. J. ht. .I. Solids Solids Structures, Srrucrures,1974, 1974,Vo!' %I. to, IO, Pl'. pp. 957-964. 95”l-964. Pergamon Pergamon Press. Press. Printed Printed in Gt. Cit. Britain. Britain.
SINGULARITIES OF SINGULARITIES OF ELASTIC ELASTIC STRESSES STRESSES AND AND OF OF HARMONIC NOTCHES HARMONIC FUNCTIONS FUNCTIONS AT AT CONICAL CONICAL NOTCHES OR OR INCLUSIONS INCLUSIONS ZDENEK ZUENEK P. P. BAZANTt BA~ANT~
and LEON LEON M. KEERt KEER$
Northwestern University, Northwestern University, Evanston, Evanston, Illinois Illinois 60201, U.S.A. USA (Received (Received 4 4 September September 1973) 1973)
Abstract-The nature Abstract-The nature of of singularities singularities at the the vertex vertex of of conical conical notches notches and and inclusions inclusions is found found for for potential theory problems of problems of problems of potential theory and and for for elastostatic elastostatic problems of torsion torsion and and of of axisymmetric axisymmetric based upon upon stress. stress. A solution solution in terms terms of spherical sphericalharmonics harmonics and and a general general numerical numerical solution solution based the used to determine power dependence upon the field equations equations are are used determine the the power dependence of of the the field quantities quantities upon the the distance distance from from the the apex apex of of the the cone. cone. Eigenvalues Eigenvalues representing representing the the exponent exponent are are computed computed for for various various values values of of cone cone angJe angle and and for for various various Poisson Poisson ratios. ratios.
I. 1. INTRODUCTION INTRODUCTION
Whereas singularities problems of potential theory Whereas singularities in two-dimensional two-dimensiona problems of potential theory and and of of elasticity elasticity have been thoroughly have already already been thoroughly examined, examined, little Iittfe is known known of of the singularities singularities in three-dimenthree-dimensional problems. Even the sional problems. the singularities singularities at conical conical notches notches and and inclusions, inclusions, the simplest simplest of of the the three-dimensional problems, seem to have three-dimensional problems, have escaped escaped attention attention and and will, therefore, therefore, be examined examined in this this study. study. Spherical Spherical coordinates coordinates will be employed employed and and will be separated separated in a manner manner which is analogous analogous to to that that introduced introduced by Knein[l Knein[l] J(upon (upon suggestion suggestion of of von van Karman), K&-man), and and later later independently indep~ndentIy developed developed by WiJliams(2], WilliamsfZ], and and Karp Karp and and Karal[3J. Karal[3]. Attention Attention will be restricted problems that restricted to to problems that either either are axisymmetric axisymmetric or are reducible reducible to one-dimensional ane-dimensional problems. The problems for a cone been investigated problems. The axisymmetric axisymmetric elasticity elasticity problems cone has recently recently been investigated but they by Thompson Thompson and and Little[4], Little[4], but they considered considered only only cones cones of of acute acute vertex vertex angle (less than than rr) rr) in which Nevertheless, it is easy to check which no singularities singularities arise. Nevertheless, check that that their their general general solution solution applies applies for for all angles, and and so it can be used herein. herein. based on spherical Two Two variants variants of of solution soIution will be used. One will be based spherical harmonics[41. harmonics[4]. The The other other will be a general genera1 numerical numerical method method which, which, after after some additional additional refinements[5], refinements[5], is applicable problems in general. applicable to three-dimensional three-dimensional singular singular problems generaI. A demonstration demonstration of of thi~ this numerical numerical method method is a second second objective objective of of this study. study. 2. POTENTIAL THEORY POTENTlAL THEORY
e,
Consider a three-dimensional three-dimensional harmonic harmonic function function u(O, ~(0, ¢, 4, r) v) in spherical spherical coordinates coordinates 8, ¢, 4, r Consider which are centered in the apex of a cone whose axis coincides with the pole. Jn analogy which centered of cone coincides In analogy method of of Knein[l], Knein[I], Williams[2], Williams[2], and and Karp Karp and and Karal[3], Karal[3], the radial radial coordinate coordinate r with the method will be separated separated by assuming assuming the harmonic harmonic function function in the form form u(f), u(@,¢, i$, r) = = r;'U(Oj r”U(O) cos cc6 k¢ I;#
t‘f Professor Professor of of Civil Civil Engineering. Engineering. t$ Professor Professor of of Civil Civil Engineering. Engineering,
957
(1) tlf
ZDEN~K P. P. BAZANT BA~ANT and and LEON LEON M. M. KEER KEER ZDENEK
958
where k must must be an integer integer in in order order to to guarantee guarantee continuity continuity of of u. U. If If this this expression expression is subsubwhere be an stituted stituted into into the the Laplace Laplace equation equation
V 2u
2
= a u + ~ (j211 + or2
r 2 08 2
I iJ2 U + ~ au + cot 8 au = 0 r2 sin 2 8 a¢2 r or r2 (}8
( 2)
variable r may may be eliminated. This This yields yields a second-order second-order ordinary differential differential equation equation for for variable be eliminated. ordinary function function U(8), U(e), d2U ho” + cot 8 ;
+
k2 A(/? + 1) - u=o sin2 I3I
(8 (0 # 0).
(3)
In In the the limit limit 8 0 -+ + 0, this this equation equation degenerates degenerates into into a condition condition ofaxisymmetry of axisymmetry: : aU/iJ8=0 aujae=o
fork=O; fork=O;
(at (at (I0 = 0).
orU=Ofork#O or U=Ofork#O
(4) (4)
Two boundary condition on be conTwo types types of of the the boundary condition on the the surface surface of of the the cone cone (8 (0 = [3) p) will will be considered: sidered : (a) (a) U = = 0
at at 8 = = [3 j? (inclusion) (inclusion)
(Sa) (5a)
(b) (b) au/ce afJ/ZfI = 0
at IJ (notch). at f)H = ,fj (notch).
(Sb) (Sb)
(Note boundary condition be reduced by a substitution of (Note that that the the boundary condition U = = const. const. may may be reduced to to U = = 0 by substitution of a new new variable variable for for u.) The boundary conditions on be specified, The boundary conditions on the the radial radial rays rays will will not not be specified, and and so so an an infinite infinite number number of of solutions solutions is naturally naturally expected. expected. That That this this is indeed indeed so is clear clear from from the the fact fact that that equations problem. Its equations (2) and and (Sa) (Sa) or or (Sb) (5b) represent represent an an eigenvalue eigenvalue problem. Its eigenstates eigenstates form form a comcomplete orthogonal boundary conditions plete orthogonal system system and, and, therefore, therefore, a solution solution for for any any specified specified boundary conditions on on the of the radial radial rays rays is a linear linear combination combination of all all eigenstates. eigenstates. However, However, in in a sufficiently sufficiently small small neighborhood of to neighborhood of the the apex apex of of the the cone, cone, the the eigenstate eigenstate that that corresponds corresponds to the the root root Je A of of the the smallest part prevails prevails (except boundary conditions smallest real real part (except in in the the special special cases cases in in which which the the boundary conditions on on radial determination of radial rays rays yield yield a zero zero coefficient coefficient for for this this eigenstate). eigenstate). Consequently, Consequently, determination of the the particular interest, because the smallest smallest Re(Je) &(A) is of of particular interest, especially especially when when [Re 1Re /1.[ 11 < I1 because the gradient gradient is then unbounded near behavior). then unbounded near the the apex apex r = = 0 (singular (singular behavior). (A) Legendre functions functions (A) Solution Solution in terms terms of of Legendre
Introducing the Introducing the new new variable variable x = = cos cos e, 0, it is readily readily recognized recognized that that equation equation (3) is the the Legendre Legendre differential differential equation equation (for (for k = 0) or or the the associated associated one one (for (for k # 0) [6]. [6]. Its Its solutions solutions are P1(cos 8) [6] (automatically satisfying are the the associated associated Legendre Legendre functions functions P,k(cos (automatically satisfying axisymmetry axisymmetry condition boundary condition /I. that condition 4). The The boundary condition (Sa) (5a) or or (Sb) (5b) requires requires finding finding such such ,I that (a) P1(cos [3) (a) PJcos B) = = 0
8)1 -
dP;(cos 0) = 0 0 ( b) dP1(cos d8 8=/3 (b) dH tI=D -
(inclusion) (inclusion)
(6a) (6a)
(notch). (notch).
(6b)
The be brought brought to The condition condition (6b) (6b) may may be to the the form form (see (see formula formula 8.733-1 8.733-l in in [6]): [6]): (I+
l)xOP;(x,)
- (3, - k + l)P:+,(x,)
= 0.
xo = cos x,, cos [3 /3 (notch) (notch)
(6c) (6~)
The been solved which which is more more expedient expedient for for numerical numerical calculation. calculation. The values values of of 1/I. have have been solved from from
Singularities Singularities of of elastic elastic stresses stresses and and of of harmonic harmonic functions functions at conical conical notches notches or inclusions inclusions
959
equations equations (6a) and (6c) (6~) numerically numerically (with the aid of of a computer), computer), using the regula reguia falsi method. method. This has been particularly particularly simple because because it was known known in advance advance (from (from the numerical numerical method method described described below) that that all ail roots roots Aare real. The The values of of Legendre Legendre function, function, including including the associated associated one, have been computed computed from from its hypergeometric hypergeometric series representarepresentation[6]. tion[6]. The The smallest smallest A-values A-values (probably (probably accurate accurate to four digits) are shown shown in Fig. 1. 1. Singularity Singularity 4
Fig. 1. Four Four smallest smallest values values of of'\ potential theory theory problems. problems. Fig. h in potential
(i.e. 1~ 742, and A< 1) is obtained obtained only only for for #3 f3 > 1f-i2, and for for kk == 0 ~axisymmetr~~ (axisymmetric state) state) in case case of of inclusion, or for k = 1 (period 271) in case of notch. For angles /3 close to z (B > 0+957~) inclusion, or for = (period 2n:) of notch. For angles f3 close to n (f3 0'95n) the the hypergeometric hypergeometric series converges converges poorly poorly and and an asymptotic asymptotic expansion/61 expansion[6] is required; required; from from follows- that that such an an expansion expansion it further further follows. such
(a) lim t. = 0, lim dii. p-"
p-"
df3
= 00 (inclusion)
. die = 1, :_t I1m $df3 = (b) lim A = = 0 (notch). (notch). /p .... " B-+x p~n
~
(7a)
(7b) Vb)
In In the the limiting limiting case case of of a Iine line inclusion inclusion (8 (f3 = a) n) the solution solution is well known[7]: known[7J: (8)
while for for a line line notch notch a homogeneous homogeneous field (A (Ie = = 1) is the the solution, solution.
ZOENEK ZDEN~K P. P. BAZANT BA~ANT and LEON LEON M. M. KEER KEER
960 960
(B) Direct numerical field equations ( B) Direct numerical method method based based on field
Interval Interval 0 8E E (0,13) (0, 8) is subdivided subdivided by-discrete by discrete nodes, nodes, uniformly uniformly spaced, spaced, and and equation equation (3) and and the boundary conditions are replaced replaced by their finite finite difference difference approximations. approximations. If the the the boundary conditions are by their If boundary conditions boundary conditions (4) (4) and and (Sa) (5a) or or (5b) (Sb) are are eliminated, eliminated. one one obtains obtains for for the the nodal nodal values values U, U; a tridiagonal tridiagonal system system of of linear linear algebraic algebraic equations equations which which can can be brought to the form: be brought to the form: n
I1
aij
Uj
flU; = 0
with with ,~l p = ;.(). E.0. + + 1): 1j; (i (i = 1, 1, 2, 2. .. .,. n). 17).
(9)
j=
where of problem is reduced where coefficients coefficients a ajj are independent independent of J.. ;1. Thus, Thus, the the problem reduced to a standard standard ij are eigenvalue problem. Standard have been used eigenvalue problem. Standard library library subroutines subroutines have been used for for its numerical numerical solution. solution, After value After determining determining p, whether whether complex complex or or real, real, the the corresponding corresponding value of of iA is computed computed as A= -! ,,/*-+'11. been programmed programmed for been proved proved A - 4+ + .,/$? il. The The solution solution has has been for complex complex )., 3., and and it has has been that that no no complex complex roots roots exist. exist. With With a step step size size fl!J A9 = 13148 fi/48 the the results results coincided coincided to to four four digits digits with (Fig. the with those those obtained obtained analytically analytically (Fig. 1). To To illustrate illustrate the the convergence, convergence, the ;.-value~ L-values for for f3l!l(J ~~A~ = = 12, 24, 48 and and 96 are are 0'46396, 0.46396, 0'46332, 0.46332, 0·46315 0.46315 and and 0·46311 O-4631 1 in the the case case of of a notch notch with finite with 13 p = = ~1T $n and and k = = 0 (for (for second-order second-order finite difference difference formulas). formulas). The The direct direct numerical numerical solution previous numerical based on solution involves involves more more algebraic algebraic operations operations than than the the previous numerical solution solution based on an but the is so small an analytical analytical approach, approach, but the cost cost of of computation computation small that that the the difference difference is undeundetectable. tectable. When When searching searching for for complex complex I"~ i,, the the direct direct numerical numerical approach approach is more more simple simple to to program, and all are program. and even even more more importantly, importantly, all computations computations are explicit explicit and and the the computer computer runs runs are are sure sure to to succeed succeed (whereas (whereas the the solution solution of of 2A from from (6a) (6a) or or (6c) (6~) requires requires tracing tracing in the the complex plane the complex plane the curves curves of of Re(),,) Re(ll) = = 0 and and the the curves curves of of Im(}.) im(A) = = 0, and and finding finding their their intersecintersections, which a computer program that run~ and tions, for for which computer program that would would not not necessitate necessitate many many runs and an an interinteraction programmer is difficult action of of the the programmer difficult to to write). write). It It seems seems that that with with small small flO A0 the the direct direct numerical numerical method IJ sufficiently method works works satisfactorily satisfactorily even even for for 13 fi quite quite close close to to 1T rt (0-97n). (O-97n). although although for for /I sufficiently because function become singular close close to to 1T 7cit must must fail, fail, because function U(!J) U(8) tends. tends. according according to to (8). (8). to to become singular at at 0 = 7r. 0= If. Physically, of ncar Physically, the the above above solutions solutions represent represent distributions distributions of temperature temperature near a uniformly uniformly heated perfectly conducting conical conical or heated perfectly conducting conical inclusion inclusion or or a non-conducting non-conducting conical inclusion; inclusion: or similar of problems, etc. similar distributions distributions of electric electric charge, charge, of of stream stream function function in flow flow or or seepage seepage problems, etc. The to distant The cases cases for for k = = I1 and and k = = 0 correspond correspond to homogeneous homogeneous distant field held with with gradients gradients parallel to parallel to # = 0, () B= = n12, n/2, and and to to e Q= = 0, respectively. respectively. The The above above solution solution also also describes describes scattering by a reflecting because near the scattering of of waves waves by reflecting or or absorbing absorbing cone, cone, because near the the singularities singularities the Helmholtz Helmholtz reduced reduced wave wave equation equation is equivalent equivalent to to Laplace Laplace equation. equation. The The same same is true true of of the the Poisson equation. equation. Poisson 3. TORSION 3. ELASTIC ELASTIC TORSION
As problem admits As is well well known known ([8], ([8], p. 326), 326), the the elasticity elasticity problem admits solutions solutions for for which which UU,r == Uuity = 0 (u denotes displacement component in the direction defined by the subscript). (U denotes displacement component in the direction defined by the subscript). These These solutions solutions correspond to the case of torsion about axis 0 = O. Displacement lit/> then satisfies correspond to the case of torsion about axis B = 0. Displacement zl+,then satisfies the the equaequation tion ([8], ([S], p. 326): 326) : V(u,
cos 41)= 0.
(110) 10)
In boundary condition In the the case case of of aa rigid rigid conical conical inclusion, inclusion, the the boundary condition on on the the surface surface of of aa cone cone is is O. problem is identical previously solved potential theory problem of 0. Thus, Thus, the the problem identical to to the the previously solved potential theory problem of an an inclusion inclusion (case {case a in Fig. Fig. I) 1) with with k = I. I. It It is seen seen that that no no singularity singularity of of stress stress occurs occurs (). (R > > 1).
lit/> U+ = =
Singularities Singularities of elastic stresses and of harmonic harmonic functions functions at conical conical notches or inclusions inclusions
961 961
In must In the the case ofa of a conical conical notch, notch, shear shear stress (J.pe CJ+~ must vanish vanish at at the cone. cone. Using Using the the expression expression for strain reduces to strain e.p8 e@,,in spherical spherical coordinates coordinates [8, [S, 9], 91, this condition condition reduces to (Juq,/ ae = ~U~/~~ = uq, urpcot cot pP
(l I) (11)
which gives (according (according to formula formula 8'733-1 8.733-l in [6]) [C;]) )'Pi+l(X) )')xpi(x) = 0 /zPi+ ,(X) - (2 + + jL)XP:(x)
(x (X = cos CQSe).
(12)
This been solved before and This case has been solved in the same manner manner as described described before and again again no stress singularity singularity was found found to to occur. occur. 4. ELASTICITY ELASTICITY
In analogy analogy with with Knein Knein’s'$ or or Williams' Williams’ approach, approach. or or equation equation (I), (l), displacements displacements UZQ, u4, U tl,r e , uq,' will be considered considered in the the form form ii, = &Jr(%),
ug = P”L$@),
=0
u 6t9=
(axisymmetry). (axisymmetry).
(\3) (131
In boundary conditions In case of of a rigid conical conical inclusion, inclusion, the the boundary conditions require require that that
U = 13 fi (inclusion). (inclusion). U,o = = Uq, U, = U C’, = 0 at ()t1 = r =
(14) (14)
In case of boundary conditions of a conical conical notch, notch, the boundary conditions require require stresses (lifO ~~~and and (JrO or@to vanish. vanish. If If these stresses are related related according according to Hooke's Hooke’s law to strains strains Zoo' egg, err, E,, , and and zrO' E,@,and and if these are in turn [9, 8]) 81) and and equation equation (13) is subturn expressed expressed in terms terms of of displacements displacements lIr' I(,, Uug e (see [9, stituted, boundary conditions stituted, variable variable r disappears disappears and and the boundary conditions of of a notch notch take take the the form form oU N)Ur - (v' aufJat3 = -(1 -(l + c v' v’ + _t ;Iv’)Ur (13’cot {3)U P)U, } o/8e = o at P (notch) at 8 = /? {notch) cUr/CO ),)Uo euJ?ct = (I ( 1 - j.)U#
e
(15) (15)
where v' V’ = = vi( IT/(Ii - v). I’). v = = Poisson's Poisson’s ratio. ratio. (A) Solution Legendre functions functions Solutiou in terms terms of of’Legendre
The displacements displacements are best expressed expressed in terms terms of of the Papkovich-Neuber Papkovich-Neuber potentials, The potentials, which automa~icalIy satisfies Navier’s differential equations equations of of equilibrium. equiIibrium. This This approach approach has been automatically Navier's differential been adopted adopted by Thompson Thompson and and Little tittie [4]. Although Although they they considered considered no notches notches or or inclusions inclusions (/r > n/2), x/2), it is is easy to check check that that their their solution solution also applies applies to these these cases. According According to (13 equation 2.3l) 2.39 in [4J, [4], non-zero non-zero solutions solutions of of the the type type (13) exist, in case of of a stress-boundary stress-boundary equation condition. condition. if x{2c(x 2 -- 1)).2 x 2 }[p;.(x)f X(2C(.UZ l)R2 + i 2c(x 2c(x22 - IV. I)/? + + XZ)[Pi(X)]Z 2 1 + x{2c(x -- 1)),2 + XC2C(X2 l)J1 + 2c(x 2c(x2 -- l)A l)i + i- l}[P;.-l(XW I)[PI-,(X)]Z 2 - 1);' - 2c{2x2(X2 2c(2x2(xZ - 1)';.2 l)i;” + $ (x (x22 -- 1)(3x 1)(3X2 - 1)/i + + [x [x44 + i- 2(1 2(1 - V»),2 v)P + -t- 1]}P;.(x)P l]}P,(x)P,_,(x) A _ 1(x) = 0,
x = cos 8, .Y 0,
(16)
l/4(1 - v). \I), c = 1/4(1
In case of of displacement displacement boundary condition (14), (141, equations equations (2.32) (2.32) and and (2.33) (2.33) from from (4) f4] can can be boundary condition shown shown to to yield (1 -- [x c(l + (1 + + d)x[p;.(x)f c4xlP,t-r)12+ cAX[P"_1(X)]2 ~~X[~,_,(X)12 [x2c(I -I-2),) 2) 2
+ ...(X)P;._l(X) = -t (1 (1 -- c)}p c)]P,(x)P,_,(x) = 00 (A (2 =ft # 0). 0). (17) (17)
962
ZDENEK ZDENEK
P_ BA~ANT BAZANT and and LEON LEON M. M _KEER KEER P.
Similarly as as in in the the potential potential theory theory case, case, E. ). has has been been solved solved from from equations equations ((16) and ((17) by Similarly 16) and 17) by the regula regula falsi falsi method. method. (Legendre (Legendre functions functions have have been been evaluated evaluated from from their their hypergeohypergeothe metric series series representation[6].) representation[6].) The The results, results, which which seem seem to to be be accurate accurate to to four four digits, digits, are are metric shown in in Table Table 1I and and Fig. Fig_ 2. 2_ shown
Table 1. I_ The The smallest smallest value value of of'\ axisymmetric elastic elastic problems problems corresponding corresponding to to Fig Fig_ 2 Table h in axisymmetric
i3 B l' 1’
0-5111'
7rr 12
21T 3
3rr
51T 6
llrr 12
0-9711'
0-9793 0.9793 0-9809 0.9809 0-9830 0*9830 0-9861 0.9861 0-9911 0.9911 0-9957 0.9957 0-9988 0.9988 0-9989 0.9989
0-8387 0.8387 0-8492 OS492 0-8638 OS638 0-8856 OS856 0-9222 0.9222 0-9599 0.9599 0-9882 0.9882 0-9987 0.9987
0-6886 0.6886 0-7028 0.7028 0-7227 0.7227 0-7528 0.7528 0-8057 0.8057 0-8675 0.8675 0-9309 0.9309 0-9779 0.9779
0-5416 05416 0-5536 0.5536 0-5700 0.5700 0-5939 0.5939 0-6325 06325 0-6709 0.6709 0-6993 0.6993 0-7099 0.7099
0-4014 04014 OA091 0.4091 0-4193 0.4193 0-4334 0.4334 0-4544 0.4544 0-4728 0.4728 OA846 0.4846 0-4887 0.4887
0-2682 0.2682 0-2720 0.2720 0-2768 0.2768 0-2834 0.2834 0-2926 0.2926 0-3003 0.3003 0-3047 0.3047 0-3066 0.3066
0-1760 o-1760 0-1777 0.1777 0-1798 0.1798 0-1827 0.1827 0-1866 0.1866 0-1899 0.1899 0-1916 0.1916 0-1927 0.1927
0-9706 0.9706 0-9676 0.9676 0-9645 O-9645 0-9614 O-9614 0-9584 0.9584 0-9553 0.9553
0-8486 0.8486 0-8293 OS293 0-8089 0~8089 0-7874 O-7874 0'76411 0-764x 0-7411 0.7411
0-8334 0.8334 0-8072 0.8072 0·7781 0.7781 0'7456 0.7456 0·7093 0.7093 0·6686 0.6686
0-8798 0.8798 0-8577 OS577 0·8318 0.8318 0-8012 0.8012 0·7644 0.7644 0-7200 0.7200
0-9411 0.9411 0-9294 O-9294 0-9153 0.9153 0·8978 OS978 0-8755 OS755 0-8464 OS464
0-9854 0.9854 0-9825 0.9825 0·9790 0.9790 0·9749 0.9749 0·9694 0.9694 0·9624 O-9624
0-9983 0.9983 0-9980 0.9980 0·9977 0.9977 0·9973 0.9973 0-99711 0,997 0·9969 O-9969
Inclusions Inclusions 0 0-1 0.1 0-2 0.2 0-3 0.3 0-4 0.4 0-46 0.46 OA9 0.49 0-499 0.499 Notches Notches 0 0-1 @I 0·2 0.2 0·3 0.3 OA 0.4 0·499 0.499
.6
.6
x8
A
.8
1-
_6 .6
.4 .4
_2 .2
0 lr/2
046
041
-.
"
03
,
:--.;'"
"'o"_~
1; {3 ;;
8
/;r'
aa/4
-ir
P
Fig. The smallest smallest value value of of IIh in axisymmetric axisymmetric elastic elastic problems. problems. Fig. 2. 2. The
Singularities Singularities of elastic stresses and of harmonic harmonic functions functions at conical conical notches or inclusions inclusions
963
(B) Direct numerical solution based field equations ( B) Direct numerical s~lut~~~ based on field equ~ti~~~
The be based based on potentials, but but with The numerical numerical approach approach could could also also be on Papkovich-Neuber Papkovich-Neuber potentials, with Navier's differregard boundary conditions regard to to stress stress boundary conditions it is more more convenient convenient to to use use the the three three Navier’s differential ential equations equations of of equilibrium equilibrium in in terms terms of of displacements displacements (see (see Ref. Ref. [9]. [Q], Section Section 96, p. 141), of of which is automatically satisfied. which the the condition condition for for the the ¢-direction &direction automatically satisfied. If If expressions expressions (13) (13) are are substituted substituted into into these these equations, equations, variable variable rr disappears disappears and and one one obtains: obtains:
dZl.J
-&
+ cot %S
au. ao
+ [v“(jL - 1) - /J,- I] $_ 8 + + v"(A v”(J - l)(A + + 2)U 2)& r
i? Ue + [v[v”(r: " , a”u, + 2) (A + v,,z00 + n
V" - - 2
+ [v"O. [v”(d - 1) - A + _ au,,, aUe + [, + I) 1) A] + vV”cot cot 0-;-0 8z + .1(E, A] $+;:) + . .(A +
uO
u
[
l]cot OU OU,e = = 0, Ilcot
(18) (18)
v" ] _vI u,o = = 0, -:-:r-o U sm sm* % I
where degenerate into of where v" Y” = = (I (1 - 1')/(0'5 v)/(O.S - v). For For 0 = 0 these these equations equations degenerate into conditions conditions of axisymmetry: axisytnmetry : XJ,jd% = 0
and
u, = 0
at at 0 = = O. 0.
(19) (19)
For fJ) is subdivided by discrete For numerical numerical solution, solution, interval interval (0, p) subdivided by discrete nodes nodes and and equations equations (18), (18), (19) by their (To (IQ) and and (14) (14) or or (15) (15) are are replaced replaced by their finite finite difference difference approximations. approximations. (To achieve achieve high high [10] been used.) banded system accuracy. accuracy. fourth-order fourth-order approximations approximations [lo] have have been used.) This This results results in in a banded system of band-width ]1): of n 12linear linear algebraic algebraic equations equations (of (of band-width 111: n
LA
>$/tij(i)Uj = 0 ij(A)Uj =
(i=1,2, (i = 1,2,. ... . .,,n) 31)
(20) (20)
j=l
where coefficients coefficients Aij(J.) are nonlinear nonlinear functions functions of of 1. When complex complex roots roots aA are are searched, searched, where Ai},1,) are A. When (lij and U CJ; must be considered complex; otherwise they may be treated real. (No advanA ij and must be considered complex; otherwise they may be treated as real. (No advani tage is gained gained by eliminating the boundary conditions from equation equation 20.) 20.) Equation Equation (20) (20) tage by eliminating the boundary conditions from defines a nonlinear nonlinear generalized generalized eigenvalue eigenvalue problem which cannot cannot be reduced to to the the standard standard defines problem which be reduced linear eigenvalue eigenvalue problem (equation 9). However, However, efficient efficient computer computer methods methods exist exist which which linear problem (equation could handle handle this this problem (up to to a size size of of several several thousands thousands of of equations equations without without special special could problem (up difficulties); difficulties); see see [5]. 151. Computer analyses analyses for for a subdivision subdivision with with step step Ap = a/96 (resulting in in a system system of of 192 IQ2 Computer /Y;.p = /3/96 (resulting equations) have have yielded yielded results results (Fig. (Fig. 2, Table Table I) that that coincide coincide to to four four digits digits with with the the results results of of equations) the preceding analytical solution, solution, except except for for the the case case v = 0·499 O-499 for for which which the the error error varies varies the preceding analytical between 0.0001 and and 0·08. 0.08. (This (This is because v” --+ --t 00 co for for v --+ --+0·5; 0.5; but for v = = 0'5 0.5 the the solution solution between 0·0001 because v" but for could be on differential differential equations equations of of equilibrium equilibrium for for an an incompressible incompressible material.) The The material.) could be based based on numerical solution solution is about about equally equally easy easy to to program the analytical analytical one. one. It It involves involves more more numerical program as the arithmetic operations operations but the computer computer cost cost is not not excessive, excessive, anyhow. anyhow. arithmetic but the Approaching the cone cone vertex vertex along along any any radial ray, the the stresses stresses grow grow to r-‘-j. Note Approaching the radial ray, to co as r"-J. Note that the the stress stress singularity singularity strength strength li. depends on on Poisson's Poisson’s ratio. ratio. Physically, Physically, the the axisymaxisymthat A - J1 depends metric singular singular stress stress states states obtained obtained are are excited excited by combinations of a normal normal stress stress parallel metric by combinations of parallel to axis axis 0 = = 0 at at infinity infinity and and of of equal equal biaxial normal stresses stresses perpendicular to axis axis () 8 = = 0 at at to biaxial normal perpendicular to infinity. infinity. Knowing i,A, the the eigenstates, eigenstates, when when desired, desired, can can easily easily be computed from from the the formulas formulas 2.32 2.32 be computed Knowing and 2.33 2.33 derived derived in in [4] by setting in in these these formulas formulas C; and L?; Cx,(I + + c),)p;.(x c~)~~(.~~~~ and by setting C~ = C and D~ = CXoO o)/
964 964
ZDENfX P. P. BAZANT BAZANT ZDEN~K
},p.. -1(X O) ~P,_,(A+,)
and LEON LEON M. M. KEEK KEER and
and taking taking into into account account minor minor changes changes in in notations. notations. (In (In particular, particular, kk == c, c, and
e
an == A, Ie, pf1 == xX and and #, 4>, 0 are are interchanged.,) interchanged.) According According to to these these formulas, formulas, a,, _).
Ur -
Cr (I
+ d) { xP;.Ix)
-
XOP.l(x o ) P;'-l\XO )
\ P';-l(X)j' (21)
e,
where so Xo = = cos cos jj’, /3,.Ys = cos co:::. N, C C == arbitrary arbitrary constant. constant. The The stresses stresses can can similarly similarly be be obtained obtained where (2.34)-(2.37) in in Ref. Ref. [4J. [4}. by direct direct substitution substitution from from equations equations (2.311-f-(2.37) by Knowledge of of the the field field near near the the singularity singularity makes makes it possible possible to to construct construct aa singular singular Knowledge finite element element for for the the vertex vertex region region of of aa cone, cone, and and thui thU', to to solve solve practical practical boundary boundary value value finite problems for for bodies bodies of of finite finite dimensions. dimensions. problems 5. CONCLUSION
Conical notches notches and and inclusions inclusions produce produce singularities singularities of of stress stre;,.s or or potentiat potential gradient gradient whose whose Conical strength varies varies from from 0 to to - 1 in in dependence dependence on on the the cone cone angle. angle. The The stress stress singularity singularity also also strength depends on on the the Poisson’s Poisson's ratio. ratio. The The problem problem can can be be solved solved analytically analytically in in terms terms of of Legendre Legendre depends in the the angular angular spherical spherical coorcoorfunctions. A A numerical numerical solution solution based based on on finite finite differences differences in functions. dinate is also also possible possible and and yields yields equally equally accurate accurate results. results. This This fact fact at at the the same same time time serves serves dinate of this this numerical numerical method, method, which which is applicable applicable to to a broad broad class class of of as a check check on on the the validity validity of as other problems. problems. other Acknowledgement-The support support in ~olnputin~ computing funds provided provided by Northwestern Northwestern University University is gratefully gratefully Acknuwle~~ye,trnrt-The acknowledged. acknowledged.
REFERENCES REFERENCES I. M. Knein, Zur Theorie des Druckversuchs, Abhandlungen dewiy,l. Aerodyn. ins!. Aachen 7, 1. Knein, Zur Theo& Druckversuchs, ,~~~l~n~~~~~~~~~~ 1~~8. T.H. TX. A~ichen 7, 43-62 43-67 (1927). 2. M. L. Williams, boundary conditions Williams, Stress Stress singularities singularities resulting resulting from from various various boundary conditions in angular angular corners corners of of plates in extension, Appl. lMPch., Mech., Trans. Am. Soc. Mech. Et7grs. ElIgrs. 19, 526-528 plates extension, J. Appl. Trans. .4n7. Sot. Mech. 526-528 (1952). 3. S. M. Karp behavior in the Karp and and F. C. Karal, Karat, The The elastic elastic field behavior the neighborhood neighborhood of of a crack crack of of arbitrary arbitrary Pure .4ppl. Appl. Math. Math. 15,413-421 angle, angle, Commull. C0mmu~r. Pure 15, 413-421 (1962). 4. Mech. &pi. Appl. f. R. Thompson Thompson and and R. W. Little, Little, End End effects effects in a truncated truncated semi-infinite semi-iniinite cone, cone, Quan. Quurr. 1. J. &%ch. 4. T. Math. 23, 185-195 Murk 185-195 ([970). it970). 5. Z. P. Bazant, Baiant, Three-dimensiond Three-dimension&l harmonic harmonic functions functions near near termination termination or or intersection intersection of of gradient gradient Engl/g. Sci. 12,221·-243 singularity singularity lines: lines: a general general numerical numerical method, method, into irtr. J. Eqqrg. 12, El--243 (1974). 6. I. Integrals, Series Products. Academic 1. S. Gradstein Gradstein and and I. M. Ryshik, Ryshik, Tables Tables of of~ntegrff/s, Series and and Products. Academic: Press, New York Yorh (1965). (1965). 7. V. L. t. Rvachev, Rvachev, On pressure pressure upon upon elastic elastic halfspace halfspace by a stamp stamp having having a a wedgeshape wedgeshape in plan (in RusRusii Mekhanika 959). sian), sian), Prikladnaya Prikludnay~ Matematika .~utematika Mek/tanikff 23, 23, 169-171 169-171 {} t 1959). 8. A. I. Lur'e, Problems Lur’e, Three-Dimensional spree-~itnensio~ai ~ro~~e~~.~ ill in the ti7eTheory Tlreor~of ofElastidty ~~~.st~~~ity(translated {translated from from Russian Russian by J. .I. M. Radok). Radok). Interscience, Interscience, New New York York (1969). 0969). 9. A. H. Love, 71leory ElaSlicity, 4th Edn. Love, A rl Treatise Tveari5e on DIIthe rhe Mathematical ~at~e/~ut~cai 7%eory of q/'Ekuricii~, Edn. Dover, Dover, New New York York (1944). (2944). 10. 10 L. Collatz, Collatz, The The Numerical Numerical Treatment Trentmetlt of of DUlerential D$j?erentiul Equations, Eyuntions, 3rd 3rd Edn. Edn. Springer, Springer, Berlin Berlin (1960). A6cTpaKT B A~CT~KT- - OnpcnemleTcil OnpenenaeTcn npl1pona npnpona CHHI'YJ1i1pHocTei! ctfkfrynfiprfocTeii B BepWtlHe sepufwrte KOHH'IeCKUX KoffffYecKkfxBblpe30B sbrpesoe H H BKnlO'IeHUH, BKJffO'leHM%,llnll fina 'lana'l 3a2fa9 Teoplll1 reoputr nOTeHUl1ana noTeffunania II u lln!l znfi ynpyrocTaTlI'iecKl1x ynpyrocTaTWfecKua 'la)la'l 3aixas KpY'leHl1l1 KpyseffilaII N ocecHMMeTpII'leCKOrO Hanplll!<eHHlI, O~~~MMeT~~~e~Koro Haf'@SmeHSZ% flpHMeHSnOTCl! ffpEfMetf%oTCS pemeHII5I pelf.feHBRB B BH)1e SHae cQlepH'IeCKHX C&pWfecKkfX rapMoHHK f-ap,tiofffiK If ri o6mee 06uree 'lHCnCHHOC YEfCJfeffHOepemelllle, peuientie,OCHOBaHHhle ocnoaaffHb7e Ha tfa ypaBHeHfUlx ypaaHeH~~x nO;151, none, C c l\e:IhlO ue.zbIo OnpelleJIeHHlI o~pe~eJ~e~~~ 3aBHCHMOCTH 0'[ flo):l3aBACHMOCTK OT oT CTenenH CTenerra ):Iml LVIR KOnH'IecTB KOJElYeCTB nOITl!, iTOJVl,B B 3aBHCHMOCTH 3aBHCMMOCTM OT BepmHHbl BePLUMHbl KOHyca. KOHyCa. nOAC'IHTaHO co6cTBeHHbie CYI~T~HO CO6cTBeHHbIe '3Ha'leHHlI, ~HBYBHIIR, KOTopble k-oropbie H306parualOT u3o6pamamr nOKa3aTCITh noKa3aTenb CTeneHH cTenew ):IIT5I finsi pa3Hbix pa3HblX 3Ha'ieHIIM 11 ):IITl! pa3Hbix K03Q1Q111l\UeHTOB flyaecoHa. 3Ha%eR&fiiyrITa yrna KOHyca KoHycaII ~~~~a3HbIX Ko3~~~u~e~roB llyaccowa.
SINGULARITIES OF SINGULARITIES OF ELASTIC ELASTIC STRESSES STRESSES AND AND OF OF HARMONIC NOTCHES HARMONIC FUNCTIONS FUNCTIONS AT AT CONICAL CONICAL NOTCHES OR OR INCLUSIONS INCLUSIONS ZDENEK ZUENEK P. P. BAZANTt BA~ANT~
and LEON LEON M. KEERt KEER$
Northwestern University, Northwestern University, Evanston, Evanston, Illinois Illinois 60201, U.S.A. USA (Received (Received 4 4 September September 1973) 1973)
Abstract-The nature Abstract-The nature of of singularities singularities at the the vertex vertex of of conical conical notches notches and and inclusions inclusions is found found for for potential theory problems of problems of problems of potential theory and and for for elastostatic elastostatic problems of torsion torsion and and of of axisymmetric axisymmetric based upon upon stress. stress. A solution solution in terms terms of spherical sphericalharmonics harmonics and and a general general numerical numerical solution solution based the used to determine power dependence upon the field equations equations are are used determine the the power dependence of of the the field quantities quantities upon the the distance distance from from the the apex apex of of the the cone. cone. Eigenvalues Eigenvalues representing representing the the exponent exponent are are computed computed for for various various values values of of cone cone angJe angle and and for for various various Poisson Poisson ratios. ratios.
I. 1. INTRODUCTION INTRODUCTION
Whereas singularities problems of potential theory Whereas singularities in two-dimensional two-dimensiona problems of potential theory and and of of elasticity elasticity have been thoroughly have already already been thoroughly examined, examined, little Iittfe is known known of of the singularities singularities in three-dimenthree-dimensional problems. Even the sional problems. the singularities singularities at conical conical notches notches and and inclusions, inclusions, the simplest simplest of of the the three-dimensional problems, seem to have three-dimensional problems, have escaped escaped attention attention and and will, therefore, therefore, be examined examined in this this study. study. Spherical Spherical coordinates coordinates will be employed employed and and will be separated separated in a manner manner which is analogous analogous to to that that introduced introduced by Knein[l Knein[l] J(upon (upon suggestion suggestion of of von van Karman), K&-man), and and later later independently indep~ndentIy developed developed by WiJliams(2], WilliamsfZ], and and Karp Karp and and Karal[3J. Karal[3]. Attention Attention will be restricted problems that restricted to to problems that either either are axisymmetric axisymmetric or are reducible reducible to one-dimensional ane-dimensional problems. The problems for a cone been investigated problems. The axisymmetric axisymmetric elasticity elasticity problems cone has recently recently been investigated but they by Thompson Thompson and and Little[4], Little[4], but they considered considered only only cones cones of of acute acute vertex vertex angle (less than than rr) rr) in which Nevertheless, it is easy to check which no singularities singularities arise. Nevertheless, check that that their their general general solution solution applies applies for for all angles, and and so it can be used herein. herein. based on spherical Two Two variants variants of of solution soIution will be used. One will be based spherical harmonics[41. harmonics[4]. The The other other will be a general genera1 numerical numerical method method which, which, after after some additional additional refinements[5], refinements[5], is applicable problems in general. applicable to three-dimensional three-dimensional singular singular problems generaI. A demonstration demonstration of of thi~ this numerical numerical method method is a second second objective objective of of this study. study. 2. POTENTIAL THEORY POTENTlAL THEORY
e,
Consider a three-dimensional three-dimensional harmonic harmonic function function u(O, ~(0, ¢, 4, r) v) in spherical spherical coordinates coordinates 8, ¢, 4, r Consider which are centered in the apex of a cone whose axis coincides with the pole. Jn analogy which centered of cone coincides In analogy method of of Knein[l], Knein[I], Williams[2], Williams[2], and and Karp Karp and and Karal[3], Karal[3], the radial radial coordinate coordinate r with the method will be separated separated by assuming assuming the harmonic harmonic function function in the form form u(f), u(@,¢, i$, r) = = r;'U(Oj r”U(O) cos cc6 k¢ I;#
t‘f Professor Professor of of Civil Civil Engineering. Engineering. t$ Professor Professor of of Civil Civil Engineering. Engineering,
957
(1) tlf
ZDEN~K P. P. BAZANT BA~ANT and and LEON LEON M. M. KEER KEER ZDENEK
958
where k must must be an integer integer in in order order to to guarantee guarantee continuity continuity of of u. U. If If this this expression expression is subsubwhere be an stituted stituted into into the the Laplace Laplace equation equation
V 2u
2
= a u + ~ (j211 + or2
r 2 08 2
I iJ2 U + ~ au + cot 8 au = 0 r2 sin 2 8 a¢2 r or r2 (}8
( 2)
variable r may may be eliminated. This This yields yields a second-order second-order ordinary differential differential equation equation for for variable be eliminated. ordinary function function U(8), U(e), d2U ho” + cot 8 ;
+
k2 A(/? + 1) - u=o sin2 I3I
(8 (0 # 0).
(3)
In In the the limit limit 8 0 -+ + 0, this this equation equation degenerates degenerates into into a condition condition ofaxisymmetry of axisymmetry: : aU/iJ8=0 aujae=o
fork=O; fork=O;
(at (at (I0 = 0).
orU=Ofork#O or U=Ofork#O
(4) (4)
Two boundary condition on be conTwo types types of of the the boundary condition on the the surface surface of of the the cone cone (8 (0 = [3) p) will will be considered: sidered : (a) (a) U = = 0
at at 8 = = [3 j? (inclusion) (inclusion)
(Sa) (5a)
(b) (b) au/ce afJ/ZfI = 0
at IJ (notch). at f)H = ,fj (notch).
(Sb) (Sb)
(Note boundary condition be reduced by a substitution of (Note that that the the boundary condition U = = const. const. may may be reduced to to U = = 0 by substitution of a new new variable variable for for u.) The boundary conditions on be specified, The boundary conditions on the the radial radial rays rays will will not not be specified, and and so so an an infinite infinite number number of of solutions solutions is naturally naturally expected. expected. That That this this is indeed indeed so is clear clear from from the the fact fact that that equations problem. Its equations (2) and and (Sa) (Sa) or or (Sb) (5b) represent represent an an eigenvalue eigenvalue problem. Its eigenstates eigenstates form form a comcomplete orthogonal boundary conditions plete orthogonal system system and, and, therefore, therefore, a solution solution for for any any specified specified boundary conditions on on the of the radial radial rays rays is a linear linear combination combination of all all eigenstates. eigenstates. However, However, in in a sufficiently sufficiently small small neighborhood of to neighborhood of the the apex apex of of the the cone, cone, the the eigenstate eigenstate that that corresponds corresponds to the the root root Je A of of the the smallest part prevails prevails (except boundary conditions smallest real real part (except in in the the special special cases cases in in which which the the boundary conditions on on radial determination of radial rays rays yield yield a zero zero coefficient coefficient for for this this eigenstate). eigenstate). Consequently, Consequently, determination of the the particular interest, because the smallest smallest Re(Je) &(A) is of of particular interest, especially especially when when [Re 1Re /1.[ 11 < I1 because the gradient gradient is then unbounded near behavior). then unbounded near the the apex apex r = = 0 (singular (singular behavior). (A) Legendre functions functions (A) Solution Solution in terms terms of of Legendre
Introducing the Introducing the new new variable variable x = = cos cos e, 0, it is readily readily recognized recognized that that equation equation (3) is the the Legendre Legendre differential differential equation equation (for (for k = 0) or or the the associated associated one one (for (for k # 0) [6]. [6]. Its Its solutions solutions are P1(cos 8) [6] (automatically satisfying are the the associated associated Legendre Legendre functions functions P,k(cos (automatically satisfying axisymmetry axisymmetry condition boundary condition /I. that condition 4). The The boundary condition (Sa) (5a) or or (Sb) (5b) requires requires finding finding such such ,I that (a) P1(cos [3) (a) PJcos B) = = 0
8)1 -
dP;(cos 0) = 0 0 ( b) dP1(cos d8 8=/3 (b) dH tI=D -
(inclusion) (inclusion)
(6a) (6a)
(notch). (notch).
(6b)
The be brought brought to The condition condition (6b) (6b) may may be to the the form form (see (see formula formula 8.733-1 8.733-l in in [6]): [6]): (I+
l)xOP;(x,)
- (3, - k + l)P:+,(x,)
= 0.
xo = cos x,, cos [3 /3 (notch) (notch)
(6c) (6~)
The been solved which which is more more expedient expedient for for numerical numerical calculation. calculation. The values values of of 1/I. have have been solved from from
Singularities Singularities of of elastic elastic stresses stresses and and of of harmonic harmonic functions functions at conical conical notches notches or inclusions inclusions
959
equations equations (6a) and (6c) (6~) numerically numerically (with the aid of of a computer), computer), using the regula reguia falsi method. method. This has been particularly particularly simple because because it was known known in advance advance (from (from the numerical numerical method method described described below) that that all ail roots roots Aare real. The The values of of Legendre Legendre function, function, including including the associated associated one, have been computed computed from from its hypergeometric hypergeometric series representarepresentation[6]. tion[6]. The The smallest smallest A-values A-values (probably (probably accurate accurate to four digits) are shown shown in Fig. 1. 1. Singularity Singularity 4
Fig. 1. Four Four smallest smallest values values of of'\ potential theory theory problems. problems. Fig. h in potential
(i.e. 1~ 742, and A< 1) is obtained obtained only only for for #3 f3 > 1f-i2, and for for kk == 0 ~axisymmetr~~ (axisymmetric state) state) in case case of of inclusion, or for k = 1 (period 271) in case of notch. For angles /3 close to z (B > 0+957~) inclusion, or for = (period 2n:) of notch. For angles f3 close to n (f3 0'95n) the the hypergeometric hypergeometric series converges converges poorly poorly and and an asymptotic asymptotic expansion/61 expansion[6] is required; required; from from follows- that that such an an expansion expansion it further further follows. such
(a) lim t. = 0, lim dii. p-"
p-"
df3
= 00 (inclusion)
. die = 1, :_t I1m $df3 = (b) lim A = = 0 (notch). (notch). /p .... " B-+x p~n
~
(7a)
(7b) Vb)
In In the the limiting limiting case case of of a Iine line inclusion inclusion (8 (f3 = a) n) the solution solution is well known[7]: known[7J: (8)
while for for a line line notch notch a homogeneous homogeneous field (A (Ie = = 1) is the the solution, solution.
ZOENEK ZDEN~K P. P. BAZANT BA~ANT and LEON LEON M. M. KEER KEER
960 960
(B) Direct numerical field equations ( B) Direct numerical method method based based on field
Interval Interval 0 8E E (0,13) (0, 8) is subdivided subdivided by-discrete by discrete nodes, nodes, uniformly uniformly spaced, spaced, and and equation equation (3) and and the boundary conditions are replaced replaced by their finite finite difference difference approximations. approximations. If the the the boundary conditions are by their If boundary conditions boundary conditions (4) (4) and and (Sa) (5a) or or (5b) (Sb) are are eliminated, eliminated. one one obtains obtains for for the the nodal nodal values values U, U; a tridiagonal tridiagonal system system of of linear linear algebraic algebraic equations equations which which can can be brought to the form: be brought to the form: n
I1
aij
Uj
flU; = 0
with with ,~l p = ;.(). E.0. + + 1): 1j; (i (i = 1, 1, 2, 2. .. .,. n). 17).
(9)
j=
where of problem is reduced where coefficients coefficients a ajj are independent independent of J.. ;1. Thus, Thus, the the problem reduced to a standard standard ij are eigenvalue problem. Standard have been used eigenvalue problem. Standard library library subroutines subroutines have been used for for its numerical numerical solution. solution, After value After determining determining p, whether whether complex complex or or real, real, the the corresponding corresponding value of of iA is computed computed as A= -! ,,/*-+'11. been programmed programmed for been proved proved A - 4+ + .,/$? il. The The solution solution has has been for complex complex )., 3., and and it has has been that that no no complex complex roots roots exist. exist. With With a step step size size fl!J A9 = 13148 fi/48 the the results results coincided coincided to to four four digits digits with (Fig. the with those those obtained obtained analytically analytically (Fig. 1). To To illustrate illustrate the the convergence, convergence, the ;.-value~ L-values for for f3l!l(J ~~A~ = = 12, 24, 48 and and 96 are are 0'46396, 0.46396, 0'46332, 0.46332, 0·46315 0.46315 and and 0·46311 O-4631 1 in the the case case of of a notch notch with finite with 13 p = = ~1T $n and and k = = 0 (for (for second-order second-order finite difference difference formulas). formulas). The The direct direct numerical numerical solution previous numerical based on solution involves involves more more algebraic algebraic operations operations than than the the previous numerical solution solution based on an but the is so small an analytical analytical approach, approach, but the cost cost of of computation computation small that that the the difference difference is undeundetectable. tectable. When When searching searching for for complex complex I"~ i,, the the direct direct numerical numerical approach approach is more more simple simple to to program, and all are program. and even even more more importantly, importantly, all computations computations are explicit explicit and and the the computer computer runs runs are are sure sure to to succeed succeed (whereas (whereas the the solution solution of of 2A from from (6a) (6a) or or (6c) (6~) requires requires tracing tracing in the the complex plane the complex plane the curves curves of of Re(),,) Re(ll) = = 0 and and the the curves curves of of Im(}.) im(A) = = 0, and and finding finding their their intersecintersections, which a computer program that run~ and tions, for for which computer program that would would not not necessitate necessitate many many runs and an an interinteraction programmer is difficult action of of the the programmer difficult to to write). write). It It seems seems that that with with small small flO A0 the the direct direct numerical numerical method IJ sufficiently method works works satisfactorily satisfactorily even even for for 13 fi quite quite close close to to 1T rt (0-97n). (O-97n). although although for for /I sufficiently because function become singular close close to to 1T 7cit must must fail, fail, because function U(!J) U(8) tends. tends. according according to to (8). (8). to to become singular at at 0 = 7r. 0= If. Physically, of ncar Physically, the the above above solutions solutions represent represent distributions distributions of temperature temperature near a uniformly uniformly heated perfectly conducting conical conical or heated perfectly conducting conical inclusion inclusion or or a non-conducting non-conducting conical inclusion; inclusion: or similar of problems, etc. similar distributions distributions of electric electric charge, charge, of of stream stream function function in flow flow or or seepage seepage problems, etc. The to distant The cases cases for for k = = I1 and and k = = 0 correspond correspond to homogeneous homogeneous distant field held with with gradients gradients parallel to parallel to # = 0, () B= = n12, n/2, and and to to e Q= = 0, respectively. respectively. The The above above solution solution also also describes describes scattering by a reflecting because near the scattering of of waves waves by reflecting or or absorbing absorbing cone, cone, because near the the singularities singularities the Helmholtz Helmholtz reduced reduced wave wave equation equation is equivalent equivalent to to Laplace Laplace equation. equation. The The same same is true true of of the the Poisson equation. equation. Poisson 3. TORSION 3. ELASTIC ELASTIC TORSION
As problem admits As is well well known known ([8], ([8], p. 326), 326), the the elasticity elasticity problem admits solutions solutions for for which which UU,r == Uuity = 0 (u denotes displacement component in the direction defined by the subscript). (U denotes displacement component in the direction defined by the subscript). These These solutions solutions correspond to the case of torsion about axis 0 = O. Displacement lit/> then satisfies correspond to the case of torsion about axis B = 0. Displacement zl+,then satisfies the the equaequation tion ([8], ([S], p. 326): 326) : V(u,
cos 41)= 0.
(110) 10)
In boundary condition In the the case case of of aa rigid rigid conical conical inclusion, inclusion, the the boundary condition on on the the surface surface of of aa cone cone is is O. problem is identical previously solved potential theory problem of 0. Thus, Thus, the the problem identical to to the the previously solved potential theory problem of an an inclusion inclusion (case {case a in Fig. Fig. I) 1) with with k = I. I. It It is seen seen that that no no singularity singularity of of stress stress occurs occurs (). (R > > 1).
lit/> U+ = =
Singularities Singularities of elastic stresses and of harmonic harmonic functions functions at conical conical notches or inclusions inclusions
961 961
In must In the the case ofa of a conical conical notch, notch, shear shear stress (J.pe CJ+~ must vanish vanish at at the cone. cone. Using Using the the expression expression for strain reduces to strain e.p8 e@,,in spherical spherical coordinates coordinates [8, [S, 9], 91, this condition condition reduces to (Juq,/ ae = ~U~/~~ = uq, urpcot cot pP
(l I) (11)
which gives (according (according to formula formula 8'733-1 8.733-l in [6]) [C;]) )'Pi+l(X) )')xpi(x) = 0 /zPi+ ,(X) - (2 + + jL)XP:(x)
(x (X = cos CQSe).
(12)
This been solved before and This case has been solved in the same manner manner as described described before and again again no stress singularity singularity was found found to to occur. occur. 4. ELASTICITY ELASTICITY
In analogy analogy with with Knein Knein’s'$ or or Williams' Williams’ approach, approach. or or equation equation (I), (l), displacements displacements UZQ, u4, U tl,r e , uq,' will be considered considered in the the form form ii, = &Jr(%),
ug = P”L$@),
=0
u 6t9=
(axisymmetry). (axisymmetry).
(\3) (131
In boundary conditions In case of of a rigid conical conical inclusion, inclusion, the the boundary conditions require require that that
U = 13 fi (inclusion). (inclusion). U,o = = Uq, U, = U C’, = 0 at ()t1 = r =
(14) (14)
In case of boundary conditions of a conical conical notch, notch, the boundary conditions require require stresses (lifO ~~~and and (JrO or@to vanish. vanish. If If these stresses are related related according according to Hooke's Hooke’s law to strains strains Zoo' egg, err, E,, , and and zrO' E,@,and and if these are in turn [9, 8]) 81) and and equation equation (13) is subturn expressed expressed in terms terms of of displacements displacements lIr' I(,, Uug e (see [9, stituted, boundary conditions stituted, variable variable r disappears disappears and and the boundary conditions of of a notch notch take take the the form form oU N)Ur - (v' aufJat3 = -(1 -(l + c v' v’ + _t ;Iv’)Ur (13’cot {3)U P)U, } o/8e = o at P (notch) at 8 = /? {notch) cUr/CO ),)Uo euJ?ct = (I ( 1 - j.)U#
e
(15) (15)
where v' V’ = = vi( IT/(Ii - v). I’). v = = Poisson's Poisson’s ratio. ratio. (A) Solution Legendre functions functions Solutiou in terms terms of of’Legendre
The displacements displacements are best expressed expressed in terms terms of of the Papkovich-Neuber Papkovich-Neuber potentials, The potentials, which automa~icalIy satisfies Navier’s differential equations equations of of equilibrium. equiIibrium. This This approach approach has been automatically Navier's differential been adopted adopted by Thompson Thompson and and Little tittie [4]. Although Although they they considered considered no notches notches or or inclusions inclusions (/r > n/2), x/2), it is is easy to check check that that their their solution solution also applies applies to these these cases. According According to (13 equation 2.3l) 2.39 in [4J, [4], non-zero non-zero solutions solutions of of the the type type (13) exist, in case of of a stress-boundary stress-boundary equation condition. condition. if x{2c(x 2 -- 1)).2 x 2 }[p;.(x)f X(2C(.UZ l)R2 + i 2c(x 2c(x22 - IV. I)/? + + XZ)[Pi(X)]Z 2 1 + x{2c(x -- 1)),2 + XC2C(X2 l)J1 + 2c(x 2c(x2 -- l)A l)i + i- l}[P;.-l(XW I)[PI-,(X)]Z 2 - 1);' - 2c{2x2(X2 2c(2x2(xZ - 1)';.2 l)i;” + $ (x (x22 -- 1)(3x 1)(3X2 - 1)/i + + [x [x44 + i- 2(1 2(1 - V»),2 v)P + -t- 1]}P;.(x)P l]}P,(x)P,_,(x) A _ 1(x) = 0,
x = cos 8, .Y 0,
(16)
l/4(1 - v). \I), c = 1/4(1
In case of of displacement displacement boundary condition (14), (141, equations equations (2.32) (2.32) and and (2.33) (2.33) from from (4) f4] can can be boundary condition shown shown to to yield (1 -- [x c(l + (1 + + d)x[p;.(x)f c4xlP,t-r)12+ cAX[P"_1(X)]2 ~~X[~,_,(X)12 [x2c(I -I-2),) 2) 2
+ ...(X)P;._l(X) = -t (1 (1 -- c)}p c)]P,(x)P,_,(x) = 00 (A (2 =ft # 0). 0). (17) (17)
962
ZDENEK ZDENEK
P_ BA~ANT BAZANT and and LEON LEON M. M _KEER KEER P.
Similarly as as in in the the potential potential theory theory case, case, E. ). has has been been solved solved from from equations equations ((16) and ((17) by Similarly 16) and 17) by the regula regula falsi falsi method. method. (Legendre (Legendre functions functions have have been been evaluated evaluated from from their their hypergeohypergeothe metric series series representation[6].) representation[6].) The The results, results, which which seem seem to to be be accurate accurate to to four four digits, digits, are are metric shown in in Table Table 1I and and Fig. Fig_ 2. 2_ shown
Table 1. I_ The The smallest smallest value value of of'\ axisymmetric elastic elastic problems problems corresponding corresponding to to Fig Fig_ 2 Table h in axisymmetric
i3 B l' 1’
0-5111'
7rr 12
21T 3
3rr
51T 6
llrr 12
0-9711'
0-9793 0.9793 0-9809 0.9809 0-9830 0*9830 0-9861 0.9861 0-9911 0.9911 0-9957 0.9957 0-9988 0.9988 0-9989 0.9989
0-8387 0.8387 0-8492 OS492 0-8638 OS638 0-8856 OS856 0-9222 0.9222 0-9599 0.9599 0-9882 0.9882 0-9987 0.9987
0-6886 0.6886 0-7028 0.7028 0-7227 0.7227 0-7528 0.7528 0-8057 0.8057 0-8675 0.8675 0-9309 0.9309 0-9779 0.9779
0-5416 05416 0-5536 0.5536 0-5700 0.5700 0-5939 0.5939 0-6325 06325 0-6709 0.6709 0-6993 0.6993 0-7099 0.7099
0-4014 04014 OA091 0.4091 0-4193 0.4193 0-4334 0.4334 0-4544 0.4544 0-4728 0.4728 OA846 0.4846 0-4887 0.4887
0-2682 0.2682 0-2720 0.2720 0-2768 0.2768 0-2834 0.2834 0-2926 0.2926 0-3003 0.3003 0-3047 0.3047 0-3066 0.3066
0-1760 o-1760 0-1777 0.1777 0-1798 0.1798 0-1827 0.1827 0-1866 0.1866 0-1899 0.1899 0-1916 0.1916 0-1927 0.1927
0-9706 0.9706 0-9676 0.9676 0-9645 O-9645 0-9614 O-9614 0-9584 0.9584 0-9553 0.9553
0-8486 0.8486 0-8293 OS293 0-8089 0~8089 0-7874 O-7874 0'76411 0-764x 0-7411 0.7411
0-8334 0.8334 0-8072 0.8072 0·7781 0.7781 0'7456 0.7456 0·7093 0.7093 0·6686 0.6686
0-8798 0.8798 0-8577 OS577 0·8318 0.8318 0-8012 0.8012 0·7644 0.7644 0-7200 0.7200
0-9411 0.9411 0-9294 O-9294 0-9153 0.9153 0·8978 OS978 0-8755 OS755 0-8464 OS464
0-9854 0.9854 0-9825 0.9825 0·9790 0.9790 0·9749 0.9749 0·9694 0.9694 0·9624 O-9624
0-9983 0.9983 0-9980 0.9980 0·9977 0.9977 0·9973 0.9973 0-99711 0,997 0·9969 O-9969
Inclusions Inclusions 0 0-1 0.1 0-2 0.2 0-3 0.3 0-4 0.4 0-46 0.46 OA9 0.49 0-499 0.499 Notches Notches 0 0-1 @I 0·2 0.2 0·3 0.3 OA 0.4 0·499 0.499
.6
.6
x8
A
.8
1-
_6 .6
.4 .4
_2 .2
0 lr/2
046
041
-.
"
03
,
:--.;'"
"'o"_~
1; {3 ;;
8
/;r'
aa/4
-ir
P
Fig. The smallest smallest value value of of IIh in axisymmetric axisymmetric elastic elastic problems. problems. Fig. 2. 2. The
Singularities Singularities of elastic stresses and of harmonic harmonic functions functions at conical conical notches or inclusions inclusions
963
(B) Direct numerical solution based field equations ( B) Direct numerical s~lut~~~ based on field equ~ti~~~
The be based based on potentials, but but with The numerical numerical approach approach could could also also be on Papkovich-Neuber Papkovich-Neuber potentials, with Navier's differregard boundary conditions regard to to stress stress boundary conditions it is more more convenient convenient to to use use the the three three Navier’s differential ential equations equations of of equilibrium equilibrium in in terms terms of of displacements displacements (see (see Ref. Ref. [9]. [Q], Section Section 96, p. 141), of of which is automatically satisfied. which the the condition condition for for the the ¢-direction &direction automatically satisfied. If If expressions expressions (13) (13) are are substituted substituted into into these these equations, equations, variable variable rr disappears disappears and and one one obtains: obtains:
dZl.J
-&
+ cot %S
au. ao
+ [v“(jL - 1) - /J,- I] $_ 8 + + v"(A v”(J - l)(A + + 2)U 2)& r
i? Ue + [v[v”(r: " , a”u, + 2) (A + v,,z00 + n
V" - - 2
+ [v"O. [v”(d - 1) - A + _ au,,, aUe + [, + I) 1) A] + vV”cot cot 0-;-0 8z + .1(E, A] $+;:) + . .(A +
uO
u
[
l]cot OU OU,e = = 0, Ilcot
(18) (18)
v" ] _vI u,o = = 0, -:-:r-o U sm sm* % I
where degenerate into of where v" Y” = = (I (1 - 1')/(0'5 v)/(O.S - v). For For 0 = 0 these these equations equations degenerate into conditions conditions of axisymmetry: axisytnmetry : XJ,jd% = 0
and
u, = 0
at at 0 = = O. 0.
(19) (19)
For fJ) is subdivided by discrete For numerical numerical solution, solution, interval interval (0, p) subdivided by discrete nodes nodes and and equations equations (18), (18), (19) by their (To (IQ) and and (14) (14) or or (15) (15) are are replaced replaced by their finite finite difference difference approximations. approximations. (To achieve achieve high high [10] been used.) banded system accuracy. accuracy. fourth-order fourth-order approximations approximations [lo] have have been used.) This This results results in in a banded system of band-width ]1): of n 12linear linear algebraic algebraic equations equations (of (of band-width 111: n
LA
>$/tij(i)Uj = 0 ij(A)Uj =
(i=1,2, (i = 1,2,. ... . .,,n) 31)
(20) (20)
j=l
where coefficients coefficients Aij(J.) are nonlinear nonlinear functions functions of of 1. When complex complex roots roots aA are are searched, searched, where Ai},1,) are A. When (lij and U CJ; must be considered complex; otherwise they may be treated real. (No advanA ij and must be considered complex; otherwise they may be treated as real. (No advani tage is gained gained by eliminating the boundary conditions from equation equation 20.) 20.) Equation Equation (20) (20) tage by eliminating the boundary conditions from defines a nonlinear nonlinear generalized generalized eigenvalue eigenvalue problem which cannot cannot be reduced to to the the standard standard defines problem which be reduced linear eigenvalue eigenvalue problem (equation 9). However, However, efficient efficient computer computer methods methods exist exist which which linear problem (equation could handle handle this this problem (up to to a size size of of several several thousands thousands of of equations equations without without special special could problem (up difficulties); difficulties); see see [5]. 151. Computer analyses analyses for for a subdivision subdivision with with step step Ap = a/96 (resulting in in a system system of of 192 IQ2 Computer /Y;.p = /3/96 (resulting equations) have have yielded yielded results results (Fig. (Fig. 2, Table Table I) that that coincide coincide to to four four digits digits with with the the results results of of equations) the preceding analytical solution, solution, except except for for the the case case v = 0·499 O-499 for for which which the the error error varies varies the preceding analytical between 0.0001 and and 0·08. 0.08. (This (This is because v” --+ --t 00 co for for v --+ --+0·5; 0.5; but for v = = 0'5 0.5 the the solution solution between 0·0001 because v" but for could be on differential differential equations equations of of equilibrium equilibrium for for an an incompressible incompressible material.) The The material.) could be based based on numerical solution solution is about about equally equally easy easy to to program the analytical analytical one. one. It It involves involves more more numerical program as the arithmetic operations operations but the computer computer cost cost is not not excessive, excessive, anyhow. anyhow. arithmetic but the Approaching the cone cone vertex vertex along along any any radial ray, the the stresses stresses grow grow to r-‘-j. Note Approaching the radial ray, to co as r"-J. Note that the the stress stress singularity singularity strength strength li. depends on on Poisson's Poisson’s ratio. ratio. Physically, Physically, the the axisymaxisymthat A - J1 depends metric singular singular stress stress states states obtained obtained are are excited excited by combinations of a normal normal stress stress parallel metric by combinations of parallel to axis axis 0 = = 0 at at infinity infinity and and of of equal equal biaxial normal stresses stresses perpendicular to axis axis () 8 = = 0 at at to biaxial normal perpendicular to infinity. infinity. Knowing i,A, the the eigenstates, eigenstates, when when desired, desired, can can easily easily be computed from from the the formulas formulas 2.32 2.32 be computed Knowing and 2.33 2.33 derived derived in in [4] by setting in in these these formulas formulas C; and L?; Cx,(I + + c),)p;.(x c~)~~(.~~~~ and by setting C~ = C and D~ = CXoO o)/
964 964
ZDENfX P. P. BAZANT BAZANT ZDEN~K
},p.. -1(X O) ~P,_,(A+,)
and LEON LEON M. M. KEEK KEER and
and taking taking into into account account minor minor changes changes in in notations. notations. (In (In particular, particular, kk == c, c, and
e
an == A, Ie, pf1 == xX and and #, 4>, 0 are are interchanged.,) interchanged.) According According to to these these formulas, formulas, a,, _).
Ur -
Cr (I
+ d) { xP;.Ix)
-
XOP.l(x o ) P;'-l\XO )
\ P';-l(X)j' (21)
e,
where so Xo = = cos cos jj’, /3,.Ys = cos co:::. N, C C == arbitrary arbitrary constant. constant. The The stresses stresses can can similarly similarly be be obtained obtained where (2.34)-(2.37) in in Ref. Ref. [4J. [4}. by direct direct substitution substitution from from equations equations (2.311-f-(2.37) by Knowledge of of the the field field near near the the singularity singularity makes makes it possible possible to to construct construct aa singular singular Knowledge finite element element for for the the vertex vertex region region of of aa cone, cone, and and thui thU', to to solve solve practical practical boundary boundary value value finite problems for for bodies bodies of of finite finite dimensions. dimensions. problems 5. CONCLUSION
Conical notches notches and and inclusions inclusions produce produce singularities singularities of of stress stre;,.s or or potentiat potential gradient gradient whose whose Conical strength varies varies from from 0 to to - 1 in in dependence dependence on on the the cone cone angle. angle. The The stress stress singularity singularity also also strength depends on on the the Poisson’s Poisson's ratio. ratio. The The problem problem can can be be solved solved analytically analytically in in terms terms of of Legendre Legendre depends in the the angular angular spherical spherical coorcoorfunctions. A A numerical numerical solution solution based based on on finite finite differences differences in functions. dinate is also also possible possible and and yields yields equally equally accurate accurate results. results. This This fact fact at at the the same same time time serves serves dinate of this this numerical numerical method, method, which which is applicable applicable to to a broad broad class class of of as a check check on on the the validity validity of as other problems. problems. other Acknowledgement-The support support in ~olnputin~ computing funds provided provided by Northwestern Northwestern University University is gratefully gratefully Acknuwle~~ye,trnrt-The acknowledged. acknowledged.
REFERENCES REFERENCES I. M. Knein, Zur Theorie des Druckversuchs, Abhandlungen dewiy,l. Aerodyn. ins!. Aachen 7, 1. Knein, Zur Theo& Druckversuchs, ,~~~l~n~~~~~~~~~~ 1~~8. T.H. TX. A~ichen 7, 43-62 43-67 (1927). 2. M. L. Williams, boundary conditions Williams, Stress Stress singularities singularities resulting resulting from from various various boundary conditions in angular angular corners corners of of plates in extension, Appl. lMPch., Mech., Trans. Am. Soc. Mech. Et7grs. ElIgrs. 19, 526-528 plates extension, J. Appl. Trans. .4n7. Sot. Mech. 526-528 (1952). 3. S. M. Karp behavior in the Karp and and F. C. Karal, Karat, The The elastic elastic field behavior the neighborhood neighborhood of of a crack crack of of arbitrary arbitrary Pure .4ppl. Appl. Math. Math. 15,413-421 angle, angle, Commull. C0mmu~r. Pure 15, 413-421 (1962). 4. Mech. &pi. Appl. f. R. Thompson Thompson and and R. W. Little, Little, End End effects effects in a truncated truncated semi-infinite semi-iniinite cone, cone, Quan. Quurr. 1. J. &%ch. 4. T. Math. 23, 185-195 Murk 185-195 ([970). it970). 5. Z. P. Bazant, Baiant, Three-dimensiond Three-dimension&l harmonic harmonic functions functions near near termination termination or or intersection intersection of of gradient gradient Engl/g. Sci. 12,221·-243 singularity singularity lines: lines: a general general numerical numerical method, method, into irtr. J. Eqqrg. 12, El--243 (1974). 6. I. Integrals, Series Products. Academic 1. S. Gradstein Gradstein and and I. M. Ryshik, Ryshik, Tables Tables of of~ntegrff/s, Series and and Products. Academic: Press, New York Yorh (1965). (1965). 7. V. L. t. Rvachev, Rvachev, On pressure pressure upon upon elastic elastic halfspace halfspace by a stamp stamp having having a a wedgeshape wedgeshape in plan (in RusRusii Mekhanika 959). sian), sian), Prikladnaya Prikludnay~ Matematika .~utematika Mek/tanikff 23, 23, 169-171 169-171 {} t 1959). 8. A. I. Lur'e, Problems Lur’e, Three-Dimensional spree-~itnensio~ai ~ro~~e~~.~ ill in the ti7eTheory Tlreor~of ofElastidty ~~~.st~~~ity(translated {translated from from Russian Russian by J. .I. M. Radok). Radok). Interscience, Interscience, New New York York (1969). 0969). 9. A. H. Love, 71leory ElaSlicity, 4th Edn. Love, A rl Treatise Tveari5e on DIIthe rhe Mathematical ~at~e/~ut~cai 7%eory of q/'Ekuricii~, Edn. Dover, Dover, New New York York (1944). (2944). 10. 10 L. Collatz, Collatz, The The Numerical Numerical Treatment Trentmetlt of of DUlerential D$j?erentiul Equations, Eyuntions, 3rd 3rd Edn. Edn. Springer, Springer, Berlin Berlin (1960). A6cTpaKT B A~CT~KT- - OnpcnemleTcil OnpenenaeTcn npl1pona npnpona CHHI'YJ1i1pHocTei! ctfkfrynfiprfocTeii B BepWtlHe sepufwrte KOHH'IeCKUX KoffffYecKkfxBblpe30B sbrpesoe H H BKnlO'IeHUH, BKJffO'leHM%,llnll fina 'lana'l 3a2fa9 Teoplll1 reoputr nOTeHUl1ana noTeffunania II u lln!l znfi ynpyrocTaTlI'iecKl1x ynpyrocTaTWfecKua 'la)la'l 3aixas KpY'leHl1l1 KpyseffilaII N ocecHMMeTpII'leCKOrO Hanplll!<eHHlI, O~~~MMeT~~~e~Koro Haf'@SmeHSZ% flpHMeHSnOTCl! ffpEfMetf%oTCS pemeHII5I pelf.feHBRB B BH)1e SHae cQlepH'IeCKHX C&pWfecKkfX rapMoHHK f-ap,tiofffiK If ri o6mee 06uree 'lHCnCHHOC YEfCJfeffHOepemelllle, peuientie,OCHOBaHHhle ocnoaaffHb7e Ha tfa ypaBHeHfUlx ypaaHeH~~x nO;151, none, C c l\e:IhlO ue.zbIo OnpelleJIeHHlI o~pe~eJ~e~~~ 3aBHCHMOCTH 0'[ flo):l3aBACHMOCTK OT oT CTenenH CTenerra ):Iml LVIR KOnH'IecTB KOJElYeCTB nOITl!, iTOJVl,B B 3aBHCHMOCTH 3aBHCMMOCTM OT BepmHHbl BePLUMHbl KOHyca. KOHyCa. nOAC'IHTaHO co6cTBeHHbie CYI~T~HO CO6cTBeHHbIe '3Ha'leHHlI, ~HBYBHIIR, KOTopble k-oropbie H306parualOT u3o6pamamr nOKa3aTCITh noKa3aTenb CTeneHH cTenew ):IIT5I finsi pa3Hbix pa3HblX 3Ha'ieHIIM 11 ):IITl! pa3Hbix K03Q1Q111l\UeHTOB flyaecoHa. 3Ha%eR&fiiyrITa yrna KOHyca KoHycaII ~~~~a3HbIX Ko3~~~u~e~roB llyaccowa.
