three-dimensional harmonic functions near three...

hf. J. Engng Sci., 1974, Vol. 12, pp. 221-243. Persmm Press. Printed in Great Bhin

THREE-DIMENSIONAL HARMONIC FWNCTIONS NEAR TERMINATION OR I~ERSE.CTION OF G~IENT SINGULARITY LINES: A GENERAL NUMERICAL

METHOD

ZDENEK P. BAiANTt Department of Civil Engineering, Northwestern University, Evanston, Illinois 60201, U.S.A.

Abstract-The harmonic function u near point 0 from which a single singularity ray emanates is assumed to be dominated by the term )E#‘U where r=distance from point 0, p=known constant and p=chosen function of angular spherical coordinates 0, cp, for which a partial differential equation with boundary conditions, especially those at the singularity rays, and a variational principle, are derived. Because grad U is nonsingular, a numerical solution is possible, using, e.g. the finite difference or finite element methods. This reduces the problem to finding h of the smallest real part satisfying the equation Det (A”) = 0 where A. is a large matrix whose coethcients depend linearly on it = A(/\ -t 1). In general A and AU are complex. Solutions can be obtained either by reduction to a standard matrix eigenvalue problem for p, or by successive conversions to nonhomogeneous linear equation systems. Computer studies have confirmed the feasibility of the method and have shown that highly accurate results can be obtained. Solutions for cracks and notches ending at a plane or conical surface, and for cracks ending obliquely at a halfspace surface, are presented. In these cases, A is real and the singularity is always weaker (A > p) than on the singularity line and may even disappear (A > 1). Furthermore, elastic stresses under a wedge-shaped rigid sliding stamp or at a corner of a crack edge, and also

harmonic functions at three-sided pyramidal notches, have been analyzed. Here A < p was found to occur. A simple analytical solution for one class of special cases has also been found and used to check some of the numerical results.

INTRODUCTION

IN THREE-dimensional potential theory there exists a great number of problems in which one or several lines of singularity of the potential gradient terminate or intersect. The singularity at the point of ter~ation or intersection can naturally be expected to differ from the singularity on the line or lines entering this point. Solutions to these problems are of considerable interest, e.g. for scattering of waves at a corner of a screen, for distribution of electric charge, for problems of heat conduction, potential flow, seepage etc., and perhaps most importantly, for fracture mechanics of elastic bodies. At present, however, most of these problems remain unsolved.

One problem of this type is the ~s~bution of pressure below the tip of a rigid frictionless stamp of wedge-shaped contact, pressed upon an elastic halfspace. The problem can be reduced to a potential theory problem[4], It was discussed in 1947 by Galin[5,4] and an approximate solution restricted to non-oscillating singularities and based on mapping of a hemisphere upon a circle was presented in 1957 by Rvachev [ 111. He applied the Gale&in method in the mapped domain and, using up to six terms, made rough estimates of the real values of the sin~l~ity exponent h. He concluded that the pressure near the tip behaves as rh-’ where A monotonously increases from 0 to I as the wedge angle 2a grows from 0 to 27r (r = distance from the tip). Recently Aleksandrov and Babeshko 111 used Green’s function approach to study analytically the limiting case of the wedge angle 2a tending to zero. They concluded that near the tip the pressure exhibits an oscillating sin~l~ty of the type r-l.’ cos (k In r) (k = parameter), which obviously prevails over the non-oscillating singularity r-’ found for cy +O by Rvachev 1111.

tProfessor of Civil Engineering.

221

In!. J. Engng Sci .• 1974, Vol. 12, pp. 221-243. Pergamon Press. Printed in Great Britain

THREE-DIMENSIONAL HARMONIC FUNCTIONS NEAR TERMINATION OR INTERSECTION OF GRADIENT SINGULARITY LINES: A GENERAL NUMERICAL

METHOD

ZDENEK P. BAZANTt Department of Civil Engineering, Northwestern University, Evanston, Illinois 60201, U.s.A.

Abstract-The harmonic function u near point 0 from which a single singularity ray emanates is assumed to be dominated by the term r'p"Uwhere r=distance from point 0, p=known constant and p=chosen function of angular spherical coordinates 8, cp, for which a partial differential equation with boundary conditions, especially those at the singularity rays, and a variational principle, are derived. Because grad U is nonsingular, a numerical solution is possible, using, e.g. the finite difference or finite element methods. This reduces the problem to finding A of the smallest real part satisfying the equation Det (Au) = 0 where Au is a large matrix whose coefficients depend linearly on /.t = A(A + 1). In general A and Au are complex. Solutions can be obtained either by reduction to a standard matrix eigenvalue problem for /.t, or by successive conversions to nonhomogeneous linear equation systems. Computer studies have confirmed the feasibility of the method and have shown that highly accurate results can be obtained. Solutions for cracks and notches ending at a plane or conical surface, and for cracks ending obliquely at a half space surface, are presented. In these cases, A is real and the singularity is always weaker (A > p) than on the singularity line and may even disappear (A> 1). Furthermore, elastic stresses under a wedge-shaped rigid sliding stamp or at a corner of a crack edge, and also harmonic functions at three-sided pyramidal notches, have been analyzed. Here A < P was found to occur. A simple analytical solution for one class of special cases has also been found and used to check some of the numerical results.

INTRODUCTION

IN THREE-dimensional potential theory there exists a great number of problems in which one or several lines of singularity of the potential gradient terminate or intersect. The singularity at the point of termination or intersection can naturally be expected to differ from the singularity on the line or lines entering this point. Solutions to these problems are of considerable interest, e.g. for scattering of waves at a comer of a screen, for distribution of electric charge, for problems of heat conduction, potential flow, seepage etc., and perhaps most importantly, for fracture mechanics of elastic bodies. At present, however, most of these problems remain unsolved.

One problem of this type is the distribution of pressure below the tip of a rigid frictionless stamp of wedge-shaped contact, pressed upon an elastic halfspace. The problem can be reduced to a potential theory problem[4]. It was discussed in 1947 by Galin [5, 4] and an approximate solution restricted to non-oscillating singularities and based on mapping of a hemisphere upon a circle was presented in 1957 by Rvachev [11]. He applied the Galerkin method in the mapped domain and, using up to six terms, made rough estimates of the real values of the singularity exponent A. He concluded that the pressure near the tip behaves as r A

-t where A monotonously increases from 0 to 1 as the

wedge angle 2a grows from 0 to 21T (r = distance from the tip). Recently Aleksandrov and Babeshko [1] used Green's function approach to study analytically the limiting case of the wedge angle 2a tending to zero. They concluded that near the tip the pressure exhibits an osciIIating singularity of the type ,-1.5 cos (k In r) (k = parameter), which obviously prevails over the non-osciIIating singularity r- I found for a -+0 by Rvachev [11].

tProfessor of Civil Engineering. 221

222 Z. P. BAiANT

But it seems that no complete solutions of the near singularity fields, including the singularity on the edge of the stamp near the tip, have been presented so far.

A mathematically equivalent problem for the Helmholtz reduced wave equation, consisting in wave scattering near the tip of a right-angled corner of a screen, was studied analytically by Radlow [9]. He found that near the tip the harmonic wave function is of the type Y~“~, without checking for a possible imaginary part of the exponent, and without deducing the near-tip field itself. Diffraction of elastic waves by a right-angled wedge was also studied analytically by Kraut [ 161.

The intent of the present study is to propose a general method for problems of this type. The method will consist of reduction to a two-dimensional eigenvalue problem for a function without gradient singularity, which can presumably be solved numerically with any desired accuracy.

TYPE OF PROBLEMS TO BE DISCUSSED

Using spherical coordinates 8, 4, r, the Laplace equation for the potential u(13, 4, r) reads

vzu _ a2u ; 1 a2u ; 1 a% 2 au cot 9 au a9 T* ae2

2~+2+ray+t-=o r sin e afp t- de . (1)

The problem to be discussed herein consists in finding the limiting form of the solution of this equation as r+O, assuming that the domain of the solution is an infinite space bounded by a certain surface B(B, C#I) = 0 formed by rays emanating from point 0. The boundary conditions are u = 0 on part of the surface and au/an = 0 on the remaining part, n being a normal to the surface. (The case of the boundary conditions u = constant on part of the surface is easily reduced to the present case.) No boundary conditions will be prescribed at any point on radial rays that do not lie entirely on I3 = 0, so that an infinite number of solutions can be expected. Assuming these solutions to comprise a complete system of linearly independent functions, a certain linear combination of these solutions should satisfy the additional boundary conditions on the radial rays occurring in real problems. As will be seen, a class of admissible solutions is of the form r*F(e, q) and consequently, regardless of the type of the afore-mentioned linear combination, the solution for any boundary conditions on the radial rays will be dominated in a sufficiently small neighborhood of point 0 by that solution of the present problem for which Re(h) is smallest. When &(A) < 1, grad u near point 0 becomes unbounded, i.e. the solution exhibits a singularity at point 0. However, for the given boundary conditions on the radial rays the singularity need not necessarily occur because the solutions with Re(h) < 1 need not be present in the afore-mentioned linear combination (as in the case of a homogeneous normal stress field parallel to a crack). The sufficient conditions of singular behavior, as well as the uniqueness aspects, cannot be included in the present study.

If u is required to be finite, only Re(h) >O is admissible. (However, in some physically meaningful problems, such as the case of a line load on an elastic half-space, u is not finite at point 0.)

The surface formed by radial rays and the corresponding boundary conditions are assumed to be of such a form that the surface contains a finite number of rays on which grad u is singular (unbounded). Further, it is assumed that at any finite distance L from point 0, the field within a neighborhood of the singularity ray which is sufficiently smal-

222 Z. P. BAZANT

But it seems that no complete solutions of the near singularity fields, including the singularity on the edge of the stamp near the tip, have been presented so far.

A mathematically equivalent problem for the Helmholtz reduced wave equation, consisting in wave scattering near the tip of a right-angled corner of a screen, was studied analytically by Radlow [9]. He found that near the tip the harmonic wave function is of the type rO'

25, without checking for a possible imaginary part of the exponent,

and without deducing the near-tip field itself. Diffraction of elastic waves by a right-angled wedge was also studied analytically by Kraut [16].

The intent of the present study is to propose a general method for problems of this type. The method will consist of reduction to a two-dimensional eigenvalue problem for a function without gradient singularity, which can presumably be solved numerically with any desired accuracy.

TYPE OF PROBLEMS TO BE DISCUSSED

Using spherical coordinates e, cP, r, the Laplace equation for the potential u(e, cP, r) reads

(1)

The problem to be discussed herein consists in finding the limiting form of the solution of this equation as r~O, assuming that the domain of the solution is an infinite space bounded by a certain surface B( e, cP) = 0 formed by rays emanating from point O. The boundary conditions are u = 0 on part of the surface and au/an = 0 on the remaining part, n being a normal to the surface. (The case of the boundary conditions u = constant on part of the surface is easily reduced to the present case.) No boundary conditions will be prescribed at any point on radial rays that do not lie entirely on B = 0, so that an infinite number of solutions can be expected. Assuming these solutions to comprise a complete system of linearly independent functions, a certain linear combination of these solutions should satisfy the additional boundary conditions on the radial rays occurring in real problems. As will be seen, a class of admissible solutions is of the form r), F( e, cp) and consequently, regardless of the type of the afore-mentioned linear combination, the solution for any boundary conditions on the radial rays will be dominated in a sufficiently small neighborhood of point 0 by that solution of the present problem for which Re(A) is smallest. When Re(A) < 1, grad u near point 0 becomes unbounded, i.e. the solution exhibits a singularity at point O. However, for the given boundary conditions on the radial rays the singularity need not necessarily occur because the solutions with Re(A) < 1 need not be present in the afore-mentioned linear combination (as in the case of a homogeneous normal stress field parallel to a crack). The sufficient conditions of singular behavior, as well as the uniqueness aspects, cannot be included in the present study.

If u is required to be finite, only Re(A»O is admissible. (However, in some physically meaningful problems, such as the case of a line load on an elastic half-space, u is not finite at point 0.)

The surface formed by radial rays and the corresponding boundary conditions are assumed to be of such a form that the surface contains a finite number of rays on which grad u is singular (unbounded). Further, it is assumed that at any finite distance L from point 0, the field within a neighborhood of the singularity ray which is sufficiently smal-

Three-dimensional harmonic functions 223

ler than L is known, and is of the same type at any point of the ray except at point 0. This field may be determined by solving the plane problem in a plane normal to the ray. (The latter property is verified in some special cases by the analytical solutions reported in the sequel.)

Consider the plane section normal to a singularity ray passing through point 0’ at a finite distance L from point 0. It is assumed that within a neighborhood of point 0’ which is sufficiently smaller than L, the normal plane section of the body is bounded by two straight rays O’A, O’B from point 0’, forming angle LY (Fig. 1). The boundary condition on the rays may be either u = 0 or ~~/~~ = 0 where 4 = angular coordinate in Fig. 1. The solution near point 0’ in the normal plane section may be sought in the form u - <f,(4) where rl is the distance from point O’, and p and ~,(~) are assumed to be the same at any finite distance L from point 0.

To give an example of the determination of p and f,(4), consider that.the boundary conditions are ~~~~~=O on ray O’A, and u-0 on ray O’B. Substitution of u=rY fi(#) into the Laplace equation yields a homogeneous ordinary differential equation for f,(#), whose general solution is A cos p+ +B sin ~4. The boundary conditions reduce to df,(O)/d4 = 0, f,(v - a) = 0, which yield

u-rpfi(4) where fi(~>=cosp~,p=4?r/(a-cu). (2)

The same result is obtained when rays O’A, O’B form angle 2ar and u = 0 on both rays. This technique of the determination of the net-sin~l~ity fields has been used in

plane elasticity (where it is not as trivial as here) by Knein[8] (who thanked T. von K&m&n for suggesting the basic approach), and later independently by Williams [ 141 and Karp and Karal[7]. In analogy with this technique, the variables may be separated in the three-dimensional problem as follows:

a(& 6 r) = r”f(@, 9) (3)

where n is, in general, complex and f is a function of 8 and b, only. It may be imagined as a function defined in a certain domain fl on a unit sphere about point 0. Substitution of (3) into the Laplace equation (1) allows r to be completely eliminated from the equation and results in a differential equation for f(@, 4).

It should be noted that in the case of the Helmholtz reduced wave equation, V’u + key = 0 (arising in wave scattering problems), or the Poisson equation, V*u = k, the variable r is not eliminated since the additional term k*r*f or krzmn, respectively, appears in the equation for fi However, in a sufficiently small neighborhood of point 0, this term becomes negligible, as compared with other terms in the equation (provided that f is

Fig. I. Planar case giving rise to singularity at point 0’.


ler than L is known, and is of the same type at any point of the ray except at point O. This field may be determined by solving the plane problem in a plane normal to the ray. (The latter property is verified in some special cases by the analytical solutions reported in the sequel.)

Consider the plane section normal to a singularity ray passing through point 0' at a finite distance L from point O. It is assumed that within a neighborhood of point 0' which is sufficiently smaller than L, the normal plane section of the body is bounded by two straight rays 0' A, 0' B from point 0', forming angle a (Fig. 1). The boundary condition on the rays may be either u 0 or aul aef> = 0 where ef> = angular coordinate in Fig. 1. The solution near point 0' in the normal plane section may be sought in the form u - rfM ef» where r, is the distance from point 0' , and p and It( ef» are assumed to be the same at any finite distance L from point O.

To give an example of the determination of p and Mef», consider thatthe boundary conditions are au/aef>=o on ray 0' A, and u=O on ray 0' B. Substitution of u=r~ f.(ef» into the Laplace equation yields a homogeneous ordinary differential equation for j,(ef», whose general solution is A cos pef> + B sin pef>. The boundary conditions reduce to d/1(0)/def> = 0, 11( 1T - a) = 0, which yield

(2)

The same result is obtained when rays 0' A, 0' B form angle 2a and u = 0 on both rays. This technique of the determination of the near-singularity fields has been used in

plane elasticity (where it is not as trivial as here) by Knein[8] (who thanked T. von Karman for suggesting the basic approach), and later independently by Williams[14] and Karp and Karal[7]. In analogy with this technique, the variables may be separated in the three-dimensional problem as follows:

(3)

where n is, in general, complex and 1 is a function of 8 and ef> only. It may be imagined as a function defined in a certain domain n on a unit sphere about point O. Substitution of (3) into the Laplace equation (1) allows r to be completely eliminated from the equation and results in a differential equation for 1(8, ef».

It should be noted that in the case of the Helmholtz reduced wave equation, V2 u + k

2u = 0 (arising in wave scattering problems), or the Poisson equation, V2u = k, the variable r is not eliminated since the additional term er21 or krz- n

, respectively, appears in the equation for f. However, in a sufficiently small neighborhood of point 0, this term becomes negligible, as compared with other terms in the equation (provided that 1 is

8

Fig. 1. Planar case giving rise to singularity at point 0'.

224 Z. P. BAiANT

bounded and, in the case of the Poisson equation, n ~2). Thus, the solutions of the reduced wave equation and the Poisson equation exhibit the same type of singularity as the harmonic functions.

The gradient singularity at point 0 is only partly interpreted by the term r” in equation (3) since along certain radial rays there is an additional singularity which must be embedded in function fi Consequently, function f is not suitable for determination by approximate numerical methods.

REMOVING SINGULARITY ON RADIAL SURFACE RAYS

ft is desired to separate the variables in such a manner that, in any plane normal to the singularity ray, function u near the ray be of the same form as equation (3) in which fi has no gradient singularity. Considering, for the sake of brevity, the case of a single singularity ray, this may be achieved by writing

where rl = chosen continuous smooth function of 8, 4, and r, which is identical to the distance from the singularity ray in its vicinity and is nonzero everywhere except on the ray. Obviously, if p is the correct exponent for the gradient singularity on the ray, as determined from (2), U(B, #> must be a continuous smooth function of 8 and # which has (a) no gradient singularity on the singularity ray, and (b) in its vicinity is proportional to the function fi from equation (2).

Function r, may always be introduced in the form

rr = r&f% cb> (5)

where p = chosen continuous smooth function of 8 and St, which is nonzero everywhere except on the singularity ray in whose vicinity it is identical with the distance from the ray measured on the unit sphere. Thus

u(@, 4, r)= r”F(i?, cb)= rA\pDU(8, 4) 16)

where

A=PI-bp.

If the singularity ray is located at the pole, B =O, a very simple choice of p is possible:

p= 8 or p=sin 0. (8)

With the latter choice, rI represents the exact distance from the ray not only in its vicinity but everywhere in the domain. It must be warned, however, that’sin 0 cannot be used when 8 = z is part of the domain and no sin~la~ty ray exists at f3 = 71. The function p = sin k0 may be suitable when symmetric boundary conditions at 6 = rrl2k are desired.

If the singularity ray is located at (&, &), it is possible to choose, e.g.

224 Z. P. BAZANT

bounded and, in the case of the Poisson equation, n <2). Thus, the solutions of the reduced wave equation and the Poisson equation exhibit the same type of singularity as the harmonic functions.

The gradient singularity at point 0 is only partly interpreted by the term rn in equation (3) since along certain radial rays there is an additional singularity which must be embedded in function I. Consequently, function I is not suitable for determination by approximate numerical methods.

REMOVING SINGULARITY ON RADIAL SURFACE RAYS

It is desired to separate the variables in such a manner that, in any plane normal to the singularity ray, function u near the ray be of the same form as equation (3) in which II has no gradient singularity. Considering, for the sake of brevity, the case of a single singularity ray, this may be achieved by writing

(4)

where r) = chosen continuous smooth function of 8, C/>, and r, which is identical to the distance from the singularity ray in its vicinity and is nonzero everywhere except on the ray. Obviously, if p is the correct exponent for the gradient singularity on the ray, as determined from (2), U(8, c/» must be a continuous smooth function of 8 and c/> which has (a) no gradient singularity on the singularity ray, and (b) in its vicinity is proportional to the function II from equation (2).

Function r, may always he introduced in the form

(5)

where p = chosen continuous smooth function of 8 and c/> which is nonzero everywhere except on the singularity ray in whose vicinity it is identical with the distance from the ray measured on the unit sphere. Thus

(6)

where

A=n+p. (7)

If the singularity ray is located at the pole, 8 = 0, a very simple choice of p is possible:

p = 8 or p = sin e. (8)

With the latter choice, r! represents the exact distance from the ray not only in its vicinity but everywhere in the domain. It must be warned, however, that'sin 8 cannot be used when 8 = 1T is part of the domain and no singularity ray exists at 8 = 1T. The function p = sin klJ may be suitable when symmetric boundary conditions at 8 = 1T/2k are desired.

If the singularity ray is located at (8" c/>l), it is possible to choose, e.g.


p = ((0 - @,)” + [(4 - &) sin @,J’}“*. (9)

Generalization to the case of several singularity rays contained within the domain on the unit sphere is obvious. For three rays, e.g.

where

8+ =n+p,+pz+p3 (11)

p,, p2, p3 being known exponents for the three rays and pl, p2, p3 functions of the same type as pft?, 4) was before.

Now consider a singularity ray which is placed into the pole of a spherical coordinate system (e.g. ray 00’ in Fig. 2). In the plane (@, #) the pole appears as a straight line segment 8 = 0 with end points given by the angles &, qbz forming the crack or notch that creates the singularity (Fig. 3). This line segment is a part of the boundary of domain Sz in the plane (@, 4) on which function V(6, #) has to be solved. It is obvious that a certain boundary condition must be imposed at this line segment (O’@ in Fig. 3). The boundary condition may be deduced from the fact that U must be proportional to the function f, from equation (2) in the vicinity of 6 = 0. In equation (2) f, depends only on 4, i.e. is independent of 8. Hence, the condition

aUfM=O at 0=0 (12)

must be satisfied.

(b)

Fig. 2. Various cases of a notch or crack ending into a conical or planar surface. (The sphere is not the body surface and serves only for visualizing the situation in spherical coordinates.)

IJES Vol. 12. hio. 3-C


(9)

Generalization to the case of several singularity rays contained within the domain on the unit sphere is obvious. For three rays, e.g.

(10)

where

(11)

P" Pl, P3 being known exponents for the three rays and P" P2, P3 functions of the same type as p( 8, </» was before.

Now consider a singularity ray which is placed into the pole of a spherical coordinate system (e.g. ray 00' in Fig. 2). In the plane (8, </» the pole appears as a straight line segment 8 = 0 with end points given by the angles </>h </>2 forming the crack or notch that creates the singularity (Fig. 3). This line segment is a part of the boundary of domain .n in the plane (8, </» on which function U (8, </» has to be solved. It is obvious that a certain boundary condition must be imposed at this line segment (0'0' in Fig. 3). The boundary condition may be deduced from the fact that U must be proportional to the function I. from equation (2) in the vicinity of {} = O. In equation (2) I. depends only on </>, i.e. is independent of 8. Hence, the condition

au/a8=0 at8=0

must be satisfied.

(0)

Fig. 2. Various cases of a notch or crack ending into a conical or planar surface. (The sphere is not the body surface and serves only for visualizing the situation in spherical coordinates.)

lJES Vol. 12. No. 3-C

(12)

226 Z. P. BAtiNT

/ au/an =o

\ 0’

6=0

I ae

A

rp=a (P Fig. 3. Domain O’ACO’ from Fig. 2 visualized in (6, q) plane, with a nodal net used for numerical

solution.

DIFFERENTIAL EQUATrON OF THE PROBLEM AND VARIATIONAL FORMULATION

Substitution of equation (6) into equation (1) yields, after rearrangement and simplification, the partial differential equation:

p 2a2u p2 a’u ---t-

a82+sin26 ad? ( Zpp~+p’cot 8

> 1 +;+ A(h+l)p*

+,[,~cot6+~P--l)(~)2+P~]} u=o. (13)

In the special case where p = sin 0, equation (13) becomes

si~‘~~+~+(p+~)sin2~~~+{[h(h+l)-p(p+1)1sin2B+p2}~=O. (14)

The problem consists in finding a nonzero solution of equation (13) or (14) which satisfies the boundary conditions at a surface formed by the radial rays and gives the smallest positive real part of A.

It should be noted that for 8 +O, equation (14) becomes a”U/&&” + p*U = 0, which gives U-cos pr$, as is required by equation (2).

For numerical solution, the variational formulation is of particular interest. The functional associated with the Laplace equation is known to be

W= I v i(gradu)ZdV+W,5i I, [(g>‘+ (;$>‘+ (j&$-$)2] dV+W, (15)

where V = volume, W, = certain integral over the boundary surface. The functional which is associated with equation (13) or (14) may be obtained by reducing functional W to a two-dimensional one. In conformity with the separation of variables by substitution (6), the minimizing condition for W may be considered to be of the form @&‘I&‘) = 0 where S, is the variation with regard to functions of r, and 6 is the variation with regard to functions of t9 and (6. Thus, according to (15) and (7),

226 Z. P. BAUNT

BU18n-o -0' 0' ////// ///L/// //////:://,U/.

. " \ ~

8=0

8

'L~ ~ /

~

f\ ~ ~

i\ ~ \ : ,/

8=/3 A U 0 ¢=a

Fig. 3. Domain 0' ACO' from Fig. 2 visualized in (0, q;) plane, with a nodal net used for numerical solution.

DIFFERENTIAL EQUATION OF THE PROBLEM AND VARIATIONAL FORMULATION

Substitution of equation (6) into equation (1) yields, after rearrangement and simplification, the partial differential equation:

[ a (a)2 a2

]} +p Pa~COt9+(P-l) a~ +P a; U O. (13)

In the special case where p = sin 9, equation (13) becomes

The problem consists in finding a nonzero solution of equation (13) or (14) which satisfies the boundary conditions at a surface formed by the radial rays and gives the smallest positive real part of A.

It should be noted that for 9~O, equation (14) becomes a2 u/aq/+p2 u=O, which gives U~cos pef>, as is required by equation (2).

For numerical solution, the variational formulation is of particular interest. The functional associated with the Laplace equation is known to be

f 1 2 1 f [(au)2 (1 au)2 ( 1 au)2] "W= v Z(gradu)dV+"W,=Z v ar + rae + rsin9aef> dV+"W, (15)

where V = volume, "W, = certain integral over the boundary surface. The functional which is associated with equation (13) or (14) may be obtained by reducing functional "W to a two-dimensional one. In conformity with the separation of variables by substitution (6), the minimizing condition for "W may be considered to be of the form 5 (5rW') =0 where 5, is the variation with regard to functions of r, and 5 is the variation with regard to functions of e and ef>. Thus, according to (15) and (7),


where R = r*, R’ = dRldr. ~nte~ation by parts with respect to (&R) provides

where $I is the given domain on the unit sphere. Since this equation must be satisfied for any variation &R as a function of r, the minimizing condition reduces to 8 W = 0, where W is the following functional

in which F = p”U; W, is a certain integral arising from ‘IV,. Because the surface of the body in three dimensions consists of radial rays, WI represents a certain contour integral along the boundary of the domain Q. The foregoing simple derivation is, of course, not rigorous, but it is easy to verify using standard procedures that equation (13) is indeed associated with the variational principle SW = 0.

SOME CASES ADMITTING SIMPLE ANALYTICAL SOLUTIONS

Consider solutions of equation (14) which have the form

U(@, #J) = Y( 0) sin ~4. (19)

This obviously satisfies the boundary conditions

or U=O at Cp=O andat +=2(rr--cr)=7rlp (20a)

U=O at #=0 and aVla+=O at b,=1r--==/2p (20b)

that is, U = 0 at the planes OtYA and OO’B in Fig. 2 where (Y and p are related as given by equation (2). Thus, if domain R on the unit sphere is rectangular in 8 and # (Fig. 3), i.e.

function Y(t)) in (19) needs to satisfy only the boundary conditions on 8 = 0 and 8 = @. Considering that there is a free surface at 0 = /3 (a cone, and for p = 7r/2 a plane; Fig. 2), and taking equation (12) into account, one may write

au/ae=o at 8=0, U=O at e=p. (22)

Thus, it is necessary and sufficient that Y(8) fuhill the boundary conditions:

aYfae=o at t?=O, Y=O at @=p. (23)


8 f f Iv [R' F2(8rR)' + ~ (::r 8rR + r2 S~2 0 (;:r 8rR ]r2 sin 0 dO d4> dr + 8 (8r'W1)

= 0 (16)

where R = r\ Rt = dR/dr. Integration by parts with respect to (8rR)' provides

where 0 is the given domain on the unit sphere. Since this equation must be satisfied for any variation 8rR as a function of r, the minimizing condition reduces to 8W = 0, where W is the following functional

w=Jl ! [(aF)2++ (aF)2_A(A+ l)F2] sin OdOd</, + WI (18) {l 2 ao sm 0 aq>

in which F = pPU; WI is a certain integral arising from WI. Because the surface of the body in three dimensions consists of radial rays, WI represents a certain contour integral along the boundary of the domain O. The foregoing simple derivation is, of course, not rigorous, but it is easy to verify using standard procedures that equation (13) is indeed associated with the variational principle 8W = O.

SOME CASES ADMITTING SIMPLE ANALYTICAL SOLUTIONS

Consider solutions of equation (14) which have the form

U(O, 4» = Y(O) sin P4>.

This obviously satisfies the boundary conditions

U = 0 at 4> = 0 and at 4> = 2( 1T a) "'" 1T / P or

(19)

(20a)

U=o at 4>=0 and aU/a4> =0 at 4> =1T-a=1T/2p (20b)

that is, U = 0 at the planes 00' A and 00' B in Fig. 2 where a and p are related as given by equation (2). Thus, if domain 0 on the unit sphere is rectangular in 9 and 4> (Fig. 3), i.e.

(21)

function Y( 0) in (19) needs to satisfy only the boundary conditions on 0 == 0 and 0 = 13. Considering that there is a free surface at 9 = 13 (a cone, and for 13 = 1T/2 a plane; Fig. 2), and taking equation (12) into account, one may write

au/ao=o at 9=0, U=O at 9=13. (22)

Thus, it is necessary and sufficient that Y(O) fulfill the boundary conditions:

aY/ao=o at 0=0, ¥=O at 0=13. (23)

228 Z. P. BAiANT

For LY < r/2 there is a single singularity ray in this problem, the ray 00’ in Fig. 2, and for (Y 3 1r/2 there is none. (On rays OA, OB there is no singularity.) Substituting equation (19) into (14), equation (14) is found to be satisfied if

$+(2p+*)cot e%+rA(A+l)-p(p+l)lY=O. (24)

Upon introduction of a new independent variable x = sin’ 8, equation (24) may be transformed to the form

x(l-X)~+[P+l-(p+~)X]~+~(*+~)qP(P+*)Y=O (25)

which is seen to be a special form of the hypergeometric (Gauss) differential equation. The solution of this equation [ 121 leads to hypergeometric functions and it can be shown that it may be expressed in terms of elementary functions only if /3=~/2, that is if the conical surface degenerates into a plane (Fig. 2b). Then the root h having the smallest real part is obtained as

A=p+l (26)

and the corresponding solution is

~(6, 4) = Cr’+’ (sin 0)” cos 8 sin p4 (27)

where C=an arbitrary constant. Equation (27) can be proved by substitution into (14). Note that, for any angle (Y, h is greater than 1.

An interesting result is thus obtained: the gradient singularity at the ray (edge of a crack or notch, e.g. ) disappears at the surface of the body (cases in Fig. 2b, c, d).

One important case is the half-space - IT c + 6 0 whose boundary conditions at the surface (b = 0 are u = 0 on the infinite wedge 0 ~8<2pandau/&t=au/&$=Oonthe rest of the surface (Fig. 1 la). Simple analytical solutions exist [I 11 for two limiting cases: 8

u=Cln tan2 ( > for p +O (A = 0) (28)

u=Crsin8sin+ forp-+m (A=l) (29)

and, of course, for the case CY = 7~/2 in which the problem reduces to a plane one and has a well known solution yielding A = 4.

REDUCTION TO A NONLINEAR EIGENVALUE PROBLEM FOR A MATRIX

Because function V( 8,d) has no gradient singularities, standard numerical methods may be applied to its determination. For complicated geometries of the domain R on the unit sphere, the most versatile approach will certainly be the finite element method. However, development of finite elements on the basis of the functional (18) exceeds the scope of this paper and will be treated separately. For domains of simpler geometries,

228 Z. P. BAZANT

For a < 7T/2 there is a single singularity ray in this problem, the ray 00' in Fig. 2, and for a ;3 7T/2 there is none. (On rays OA, OB there is no singularity.) Substituting equation (19) into (14), equation (14) is found to be satisfied if

d2 y dY d02 + (2p + I) cot O(f8+ [A(A + 1)- pep + 1)]Y = O. (24)

Upon introduction of a new independent variable x = sin2 0, equation (24) may be transformed to the form

x(1- x) d2y + [p + 1- (p +~) x] dY + A(A + 1) - pep + 1) Y = 0 (25)

dx 2 2 dx 4

which is seen to be a special form of the hypergeometric (Gauss) differential equation. The solution of this equation [12] leads to hypergeometric functions and it can be shown that it may be expressed in terms of elementary functions only if {3= 7T/2, that is if the conical surface degenerates into a plane (Fig. 2b). Then the root A having the smallest real part is obtained as

A = p+ 1 (26)

and the corresponding solution is

u(a, </» = Cr P+! (sin ay cos 0 sin p</> (27)

where C=an arbitrary constant. Equation (27) can be proved by substitution into (14). Note that, for any angle a, A is greater than 1.

An interesting result is thus obtained: the gradient singularity at the ray (edge of a crack or notch, e.g. ) disappears at the surface of the body (cases in Fig. 2b, c, d).

One important case is the half-space - 'IT";; <1>,,;; 0 whose boundary conditions at the surface </> = 0 are u = 0 on the infinite wedge 0 ,,;; a ,,;; 2{3 and au / an = au / a</> = 0 on the rest of the surface (Fig. lla). Simple analytical solutions exist[ll] for two limiting cases:

u = C In (tan~) for {3~0 (A = 0) (28)

u = Cr sin 0 sin </> for {3~7T (A = 1) (29)

and, of course, for the case a = 'IT/2 in which the problem reduces to a plane one and has a well known solution yielding A = t

REDUCTION TO A NONLINEAR EIGENVALUE PROBLEM FOR A MATRIX

Because function U( a, </» has no gradient singularities, standard numerical methods may be applied to its determination. For complicated geometries of the domain n on the unit sphere, the most versatile approach will certainly be the finite element method. However, development of finite elements on the basis of the functional (18) exceeds the scope of this paper and will be treated separately. For domains of simpler geometries,


the use of the finite difference method appears to be easier and has been adopted for all computer studies reported herein. The di~erential equation (14) or (13) is replaced by its finite difference approximation in terms of nodal values ui (i = 1,2,. . . .) and is written for all nodes of the chosen network inside the domain fI. Some of the equations involve nodal points outside the boundary for which further equations expressing the boundary conditions must be added.

Application of the finite difference method, as well as the finite element method, yields a large system of N homogeneous linear algebraic equations for values Uj of function U in the nodes j = 1,2,. = _, N,

(30)

The determinant of matrix Aij, in general complex, must vanish for a non-zero solution to exist. The problem is thus reduced to finding a root h having the smallest positive real part. The solution is complicated by the large size of matrix [Aij] needed for accurate results, and also by the fact that Aij are nonlinear functions of A (polynomials). Various methods of solution are possible.

Method A. reduction to a matrix eigenvalue problem

Noticing that equation (13) or (14) involves h solely through the term h(h + I> = ,u, matrix Aij is a linear function of p. Fu~hermore, if the finite difference method is used, p appears only in the diagonal coefficients of Ail, and if the equations expressing the boundary conditions are eliminated, ,u appears in all diagonal coefficients. Mere division of each equation by the coefficient at lu reduces the matrix A<j to the form:

Aij(h) = F&j- P&j (311

where &Ii, is a matrix which is independent of A, and &I= 1 for i= j, 6ij=O for i # j. Thus, a standard eigenvalue problem in P for a real, general (nonsymmetrical) matrix is obtained. There are two roots A which correspond to each root t.~,

h ---+I?<& (32)

If the roots A are all real, the smallest positive h evidently corresponds to the smallest JL (2 being replaced by +). But this is not true in the case of complex roots and all roots p must be checked to determine the smallest &(A).

For the solution of the eigenvalue problem obtained, standard subroutines are available [3,6, 10,151. Nevertheless, even with the contemporary electronic computers it is difficult and costly to exceed 150 equations, approximately, which usually suffices for accurate but not highly accurate results. The cost is also substantially increased by the fact that for the finite difference method, matrix Aii is non-symmetric.

The finite element method, which may be based on minimizing the functional (18}, yields symmetric matrices Atj and Kc+ However, in contrast with the finite difference method, E*_ no longer appears exclusively in the diagonal terms of Aij and inversion of a matrix of the size N X N is necessary to obtain a standard eigenvalue problem in p.

Taking advantage of the sparsity of matrix Aij, a much larger matrix can be stored in


the use of the finite difference method appears to be easier and has been adopted for all computer studies reported herein. The differential equation (14) or (13) is replaced by its finite difference approximation in terms of nodal values U. (i = 1,2, .... ) and is written for all nodes of the chosen network inside the domain n. Some of the equations involve nodal points outside the boundary for which further equations expressing the boundary conditions must be added.

Application of the finite difference method, as well as the finite element method, yields a large system of N homogeneous linear algebraic equations for values U. of function U in the nodes j :: 1,2, ... , N,

N

2:A,{A)Uj=O (i=t,2, ... N). (30) j=l

The determinant of matrix Ali> in general complex, must vanish for a non-zero solution to exist. The problem is thus reduced to finding a root A having the smallest positive real part. The solution is complicated by the large size of matrix [Aii] needed for accurate results, and also by the fact that A j are nonlinear functions of A (polynomials). Various methods of solution are possible.

Method A. Reduction to a matrix eigenvalue problem Noticing that equation (13) or (14) involves A solely through the term A{A + 1) = I-L,

matrix Aij is a linear function of I-L. Furthermore, if the finite difference method is used, I-L appears only in the diagonal coefficients of Aij, and if the equations expressing the boundary conditions are eliminated, I-L appears in all diagonal coefficients. Mere division of each equation by the coefficient at I-L reduces the matrix Ai to the form:

(31)

where KiJ is a matrix which is independent of A, and ()ij= 1 for i= i, ()ij=O for j + j. Thus, a standard eigenvalue problem in I-L for a real, general (nonsymmetrical) matrix is obtained. There are two roots A which correspond to each root /L,

(32)

If the roots A are all real, the smallest positive A evidently corresponds to the smallest I-L (± being replaced by +). But this is not true in the case of complex roots and all roots I-L must be checked to determine the smallest Re(A).

For the solution of the eigenvalue problem obtained, standard subroutines are available [3, 6, 10, 15]. Nevertheless, even with the contemporary electronic computers it is difficult and costly to exceed 150 equations, approximately, which usually suffices for accurate but not highly accurate results. The cost is also substantially increased by the fact that for the finite difference method, matrix Ai is non-symmetric.

The finite element method, which may be based on minimizing the functional (18), yields symmetric matrices Ali and K ij• However, in contrast with the finite difference method, I-L no longer appears exclusively in the diagonal terms of Au and inversion of a matrix of the size N x N is necessary to obtain a standard eigenvalue problem in I-L.

Taking advantage of the sparsity of matrix Ai, a much larger matrix can be stored in

230 Z. P. BAiANT

rapid access memory and the eigenvalue could be obtained by iteration[3,6, lo]. How- ever, this would be economical only for the eigenvalue of largest absolute value. At- tempts have been made to recast the problem in terms of the highest eigenvalue, e.g. replacing A with l/r and linearizing with respect to y. It appears, however, that this approach is impossible because there are always some spurious negative roots whose absolute value is larger than the largest positive y.

Method B. Conversion to nonhomogeneous equation systems

(a) The case of real root. In this method, which has been used with success[2] to solve buckling problems for very large elastic structural systems leading to equations of type (30), a certain value ho is first chosen and all coefficients A&lo) are evaluated. One equation, whose number will be referred to as k, is replaced by the equation iX= 1, which makes the system nonhomogeneous. Normally the matrix of the new system is nonsingular and all ui (i = 1,2, . . ., IV) may be solved using the standard subroutines for systems of linear equations. The obtained values of ui are then substituted into the original kth equation, previously replaced, and the corresponding right-hand side &A& = Q is computed. In general Q will be nonzero and the aim is to find the smallest value of A for which Q = 0. This may be done by the regula falsi method. Several values of A,, are selected and the plot of Q versus ho is imagined as shown in Fig. 6. Subsequent choices of ho are made by linear interpolation. Good accuracy and convergence is achieved only when the k’h equation is chosen in such a manner that no 1 uil is much larger than Uk.

From computer studies it has been found that the radius of convergence of this method is sometimes quite small. The interval of interest, such as 0 < A < 2, must be scanned in small steps of A, about 0.05, to obtain a good initial guess of A. Only then may the automatic improvements of A by the regula falsi method be started. This is obvious from the sharply varying slope of the plot b in Fig. 6 (although there exists cases of slowly varying slope, as is shown by curve c in Fig. 6). Much care is needed to avoid missing the smallest root. Nevertheless, if a great number of problems with the given data varying in small steps are being solved, as in the studies reported herein, an approximate value of A is usually known in advance and scanning of the large interval may be skipped. It is because of this fact that this method is usually much more economical than the general programs for eigenvalues. Moreover, in contrast with method A, this method allows handling up to one or two thousand equations without great difficulties (using either various elimination methods for banded matrices with tape storage or the sparse matrix storage in rapid access memory combined with an iterative solution).

If no estimate of A is available, computationally the most efficient procedure is a combination of methods A and B. First, method A with a low number of nodes is used to obtain an approximate value of A and the solution is then refined to a high accuracy using method B with a large number of nodes.

As an alternative to method B, it is also possible to apply the regula falsi method to the plot of the determinant of Aii versus A. But the slope of this plot usually varies even more rapidly and the convergence properties were found to be poorer than in method B.

(b) The case of complex root. When root A of the smallest real part happens to be complex, a similar method may be used. In this case ui, as well as matrix Aii depending on A, must be considered as complex. The objective is to find complex ho for which

230 z. P. BAZANT

rapid access memory and the eigenvalue could be obtained by iteration [3, 6, to]. However, this would be economical only for the eigenvalue of largest absolute value. Attempts have been made to recast the problem in terms of the highest eigenvalue, e.g. replacing A with 1/1' and linearizing with respect to y. It appears, however, that this approach is impossible because there are always some spurious negative roots whose absolute value is larger than the largest positive 1'.

Method B. Conversion to nonhomogeneous equation systems (a) The case of real root. In this method, which has been used with success [2] to

solve buckling problems for very large elastic structural systems leading to equations of type (30), a certain value Ao is first chosen and all coefficients AlAo) are evaluated. One equation, whose number will be referred to as k, is replaced by the equation Uk = 1, which makes the system nonhomogeneous. Normally the matrix of the new system is nonsingular and all Ui (i = 1,2, ... , N) may be solved using the standard subroutines for systems of linear equations. The obtained values of Ui are then substituted into the original kth equation, previously replaced, and the corresponding right-hand side !jAkjU; = Q is computed. In general Q will be nonzero and the aim is to find the smallest value of A for which Q = O. This may be done by the regula falsi method. Several values of Ao are selected and the plot of Q versus Ao is imagined as shown in Fig. 6. Subsequent choices of Ao are made by linear interpolation. Good accuracy and convergence is achieved only when the kth equation is chosen in such a manner that no I U.I is much larger than Uk.

From computer studies it has been found that the radius of convergence of this method is sometimes quite small. The interval of interest, such as 0".; A ".; 2, must be scanned in small steps of A, about 0·05, to obtain a good initial guess of A. Only then may the automatic improvements of A by the regula falsi method be started. This is obvious from the sharply varying slope of the plot b in Fig. 6 (although there exists cases of slowly varying slope, as is shown by curve c in Fig. 6). Much care is needed to avoid missing the smallest root. Nevertheless, if a great number of problems with the given data varying in small steps are being solved, as in the studies reported herein, an approximate value of A is usually known in advance and scanning of the large interval may be skipped. It is because of this fact that this method is usually much more economical than the general programs for eigenvalues. Moreover, in contrast with method A, this method allows handling up to one or two thousand equations without great difficulties (using either various elimination methods for banded matrices with tape storage or the sparse matrix storage in rapid access memory combined with an iterative solution).

If no estimate of A is available, computationally the most efficient procedure is a combination of methods A and B. First, method A with a low number of nodes is used to obtain an approximate value of A and the solution is then refined to a high accuracy using method B with a large number of nodes.

As an alternative to method B, it is also possible to apply the regula falsi method to the plot of the determinant of Aij versus A. But the slope of this plot usually varies even more rapidly and the convergence properties were found to be poorer than in method B.

(b) The case of complex root. When root A of the smallest real part happens to be complex, a similar method may be used. In this case U i, as well as matrix Aij depending on A, must be considered as complex. The objective is to find complex Ao for which


Q = 0. First, an arbitrary value of Im(AO) = X is chosen and, keeping it fixed, the plot of Re(Q) versus Re(ho) is examined, choosing various values of Re(h& The point at which Re(Q) = 0 may be found by the regula falsi method and the corresponding ma(Q) denoted as Y. The foregoing procedure is then repeated for various ~~(A~) = X and, considering the plot of Y versus X, the point at which Y = 0 (and Q = 0) is again found by the regula falsi method.

The fact that the regula falsi method must be applied in two dimensions makes the computations much more expensive than in the case of a real A. Moreover, because the matrix is complex, the running time of each equation system is longer, Nevertheless, using tape storage or compact storage of the sparse matrix, much larger systems can be handled than with method A.

~ei~ad C. Successive linearizat~u~s with respect to A If the technique presented here is applied in elasticity, Aij will usually consist of

general polynomials in A, complex or real. Method B will still be applicable, but instead of Method A successive solutions of eigenvalue problems will be required. Since Alj are, in general, analytic functions of a complex variable A, expansion in a Taylor series about a chosen complex ha is possible. The series may be truncated after the second term to achieve linearization. Equations (30) are then reduced to the form

where A ’ = ho - A, A yi = Aij (A,,), A :j = - aAii (A~)/~A. Multiplication of matrix equation (33) by the inverse of A {j then produces a standard eigenvalue problem, in general with a complex matrix. The objective is to find ho for which A’ = 0. This can be achieved by application of the regula falsi method, in the same procedure as the condition Q = 0 is achieved in method B.

This method has so far been tested only in the problems with real A as reported in the sequel. The convergence was rapid and the radius of convergence appeared to be quite large, that is the method automatically converged even if the initial guess of ho was very poor (e.g. 0.2 in the case of plot (a) in Fig. 6). This is due to the slow change of the slope in this plot.

APPLICATION AND RESULTS OF NUMERICAL STUDIES

Cracks and ~~~c~es ending at a plane or cubical surface To examine the numerical methods described above, the cases whose exact solution

is known were solved first. These are the cases shown in Fig. 2 and characterized by boundary conditions (20a) and (22). There is only one singularity ray 00’ in these problems. The domain Q is given by equation (21) and is rectangular in terms of 8 and 4, as shown in Fig. 3. The domain was subdivided by a rectangular network with steps AC#J = (2~ - (~)/2n, A8 = f+rr/(n + i). It is noteworthy that no interior nodes of the network may be placed on the boundary line @ = 0 because the finite difference equations for the nodes on this line do not involve any node with 8f 0. However, these nodes may be considered as exterior nodes of the network used in formulating the boundary condition (as has been done in computing Table 2).

The solutions were carried out for various sizes of the step. The results for LY =0, given in Fig. 7, indicate an excellent convergence of A to the exact value 1.5 resulting


Q = O. First, an arbitrary value of Im(Ao) = X is chosen and, keeping it fixed, the plot of Re( Q) versus Re(Ao) is examined, choosing various values of Re(Ao). The point at which Re(Q) 0 may be found by the regula falsi method and the corresponding Im(Q) denoted as Y. The foregoing procedure is then repeated for various 1m (Ao) = X and, considering the plot of Y versus X, the point at which Y 0 (and Q = 0) is again found by the regula falsi method.

The fact that the regula falsi method must be applied in two dimensions makes the computations much more expensive than in the case of a real A. Moreover, because the matrix is complex, the running time of each equation system is longer. Nevertheless, using tape storage or compact storage of the sparse matrix, much larger systems can be handled than with method A.

Method C. Successive linearizations with respect to A If the technique presented here is applied in elasticity, Au will usually consist of

general polynomials in A, complex or real. Method B will still be applicable, but instead of Method A successive solutions of eigenvalue problems will be required. Since Ali are, in general, analytic functions of a complex variable A, expansion in a Taylor series about a chosen complexAo is possible. The series may be truncated after the second term to achieve linearization. Equations (30) are then reduced to the form

N

L (A?;- A' A'jj)Uj = 0 (i = 1,2, ... N) (33) j-I

where A' = Ao - A, A ~j ;;:: Ai; (Ao), A~; = - aA.; (Ao)/ aA. Multiplication of matrix equation (33) by the inverse of A;j then produces a standard eigenvalue problem, in general with a complex matrix. The objective is to find Ao for which A I O. This can be achieved by application of the regula falsi method, in the same procedure as the condition Q = 0 is achieved in method B.

This method has so far been tested only in the problems with real A as reported in the sequel. The convergence was rapid and the radius of convergence appeared to be quite large, that is the method automatically converged even if the initial guess of Ao was very poor (e.g. 0·2 in the case of plot (a) in Fig. 6). This is due to the slow change of the slope in this plot.

APPLICATION AND RESULTS OF NUMERICAL STUDIES

Cracks and notches ending at a plane or conical surface

To examine the numerical methods described above, the cases whose exact solution is known were solved first. These are the cases shown in Fig. 2 and characterized by boundary conditions (20a) and (22). There is only one singularity ray 00' in these problems. The domain n is given by equation (21) and is rectangular in terms of (} and t/>, as shown in Fig. 3. The domain was subdivided by a rectangular network with steps A.t/> = (2'lT - a )/2n, A.O =!'IT I(n + !). It is noteworthy that no interior nodes of the network may be placed on the boundary line (J = 0 because the finite difference equations for the nodes on this line do not involve any node with 8;;<: O. However, these nodes may be considered as exterior nodes of the network used in formulating the boundary condition (as has been done in computing Table 2).

The solutions were carried out for various sizes of the step_ The results for a = 0, given in Fig. 7, indicate an excellent convergence of A to the exact value 1-5 resulting

232 Z. P. BAiANT

from equations (26) and (2). For 72 equations the error is O-4 per cent and for 1152 equations it is 0.04 per cent. The convergence is monotonous and from the slope of the line in Fig. 7 it may be inferred that the error = (step size)“9, that is, the convergence is almost quadratic. A similar picture is obtained for other values of CL

The solutions were also computed for non-zero values of p, that is, the cases of cracks or notches ending at the apex of a conical surface, which do not admit simple analytical solutions. The results are plotted in Fig. 4 and some of the results are tabulated in Table 1 (where three decimals are believed to be correct).

The results for p #0 were compared and agreed with one-dimensional finite difference solutions of the ordinary differential equation (24). (For (Y = 0, p = 7~/2, step A6 = p/SO gave h = 1.4998, which is only 0.013 per cent in error.) For several cases, the eigenstates, as characterized by function Y(O) in equation (19), are plotted in Fig. 5. As a check, some solutions have also been run with pp = V% and the results were identical.

It is interesting to note that at a certain value p > 7r/2, depending on (Y, A becomes less than 1-O so that grad u is singular at point 0. But the singularity is always weaker than at the edge 00’ (see Fig. 4).

Cracks ending obliquely at a halfspace surface The general method presented herein was further applied to the case shown in Fig.

8b, which is equivalent to the case in Fig. 8a where the boundary conditions are also given. There is only one singularity ray 00’ in this problem, with p = i from equation

0

a

Fig. 4. Singularity strength for cases shown in Fig. 2.

232 Z. P. BAZANT

from equations (26) and (2). For 72 equations the error is 0·4 per cent and for 1152 equations it is 0·04 per cent. The convergence is monotonous and from the slope of the line in Fig. 7 it may be inferred that the error = (step size)"9, that is, the convergence is almost quadratic. A similar picture is obtained for other values of a.

The solutions were also computed for non-zero values of {3, that is, the cases of cracks or notches ending at the apex of a conical surface, which do not admit simple analytical solutions. The results are plotted in Fig. 4 and some of the results are tabulated in Table 1 (where three decimals are believed to be correct).

The results for {3 =1=0 were compared and agreed with one-dimensional finite difference solutions of the ordinary differential equation (24). (For a = 0, {3 = 7T /2, step AO = (3/50 gave A = 1,4998, which is only 0·013 per cent in error.) For several cases, the eigenstates, as characterized by function YeO) in equation (19), are plotted in Fig. 5. As a check, some solutions have also been run with p P = Wand the results were identical.

It is interesting to note that at a certain value {3 > 7T /2, depending on a, A becomes less than 1·0 so that grad u is singular at point O. But the singularity is always weaker than at the edge 00' (see Fig. 4).

Cracks ending obliquely at a hal/space sur/ace The general method presented herein was further applied to the case shown in Fig.

8b, which is equivalent to the case in Fig. 8a where the boundary conditions are also given. There is only one singularity ray 00' in this problem, with p = ~ from equation

a

Fig. 4. Singularity strength for cases shown in Fig. 2.


Table 1. Values of A for various (Y and fl in the problem shown in Fig. 2

P P/4 37-r/8 ITI2 5~18 3aI4 7~~18 rr

a!

-r 3.031 1.846 I .250 0.888 0.641 0.455 0.250 - rrl2 3.189 1.953 I.333 0.958 0.704 0.515 0.333 _ 7Tl4 3.314 2.039 1.400 I.015 0.755 0.565 0.400

0 3.500 2.167 1.500 1.100 0.833 0642 0.500 ml8 3.631 2.257 1.571 1.162 0.890 0.699 0.571 n/4 3.806 2449 1.667 1.244 0.967 0.778 0.667

(2). The pole of the spherical coordinates is conveniently placed into this ray. Then, using p = sin 8, the problem is again governed by the differential equation (14). Ln is the domain O’ABCDO’ (or O’AECDO’) in Fig. 8 as is shown in plane (fI,q) in Fig. 10. The boundary conditions are &/arp = 0 on side DCO’ (plane of symmetry), au/&19 = 0 at the pole 0’ (side 0’6’ in Fig. lo), u = 0 on side O’A (crack surface) and u = 0 on side ABC (or AEC). The latter side appears in the plane (0, PO> as the curve defined as follows:

@=arctan(tanp/coscp); if8<Othenete+n. (34)

Formulation of the finite difference equivalents of the boundary condition u = 0 in terms of the nodal values near boundary ABC requires locating point (&, cpJ which is symmetrical with respect to plane ABC to a given outside node (&, (p,). To this end, the value of 8 which corresponds to p = cpu according to (34) is first calculated and, using spherical trigonometry, the following expressions are successively evaluated

cos * = 1 - 2(cos /3 sin (pa)*

e,= arccos[cos e,cos (e,-e)+ sin 8, sin (e,-e) cos $1

i

(35)

cpc= cp,+arcsin [sin (e,- e) sin $/sin e,].

0 o-2 04 0.6 06 I.0

Fig. 5. Plot of function Y(0) from equation (19) for some cases shown in Fig. 2.


Table I. Values of ,\ for various ex and {3 in the problem shown in Fig. 2

{3 Tr/4 h/8 Tr/2 h/8 h/4 7Tr/8 Tr

ex

-Tr 3·031 1·846 1·250 0·888 0·641 0·455 0·250 - Tr/2 3·189 1·953 1·333 0·958 0·704 0·515 0·333 - Tr/4 3·314 2·039 1·400 1-015 0·755 0·565 0·400

0 3·500 2·167 1·500 1·100 0·833 0·642 0·500 Tr/8 3·631 2·257 1·571 1·162 0·890 0·699 0·571 Tr/4 3·806 2·449 1·667 1·244 0·967 0·778 0·667

(2). The pole of the spherical coordinates is conveniently placed into this ray. Then, using p = sin it, the problem is again governed by the differential equation (14). n is the domain 0' ABCDO' (or 0' ABCDO') in Fig. 8 as is shown in plane (e, cp) in Fig. 10. The boundary conditions are au / acp = 0 on side DCO' (plane of symmetry), au / ait = 0 at the pole 0' (side 0'0' in Fig. 10), u = 0 on side 0' A (crack surface) and u = 0 on side ABC (or ABC). The latter side appears in the plane (e, cp) as the curve defined as follows:

e = arctan (tan (3/cos cp); if e < 0 then e~e + 7T. (34)

Formulation of the finite difference equivalents of the boundary condition u = 0 in terms of the nodal values near boundary ABC requires locating point (ec, CPc) which is symmetrical with respect to plane ABC to a given outside node (ea, CPa). To this end, the value of e which corresponds to cp = cpa according to (34) is first calculated and, using spherical trigonometry, the following expressions are successively evaluated

cos", = I - 2( cos {3 sin CPa)2

ec = arccos [cos ea cos (ea - e) + sin ea sin (ea - e) cos"']

cpc = cpa + arcsin [sin (ea - e) sin", /sin ee].

CD >-

8

6

4

2

o

a= -rr/2,/3=rr/4

a = 0, /3 =rr/4

Fig. 5. Plot of function Y(8) from equation (19) for some cases shown in Fig. 2.

(35)

234 Z. P. BATANT

Fig. 6. Examples of dependence of A and parameter Q upon ho in methods A and B for solving equations (30). (Curves a, b correspond to Fig. 2c, curve c to Fig. 12a.)

N = No. of equations

Fig. 7. Convergence of A to the exact value 1.5 with diminishing step size of the net, for the case in Fig. 2c.

The U-value at point (6, (PC> is then related by linear interpolation to the U-values at the nearest three interior nodes p, q, r using area coordinates in the triangle pqr, i.e. UC = (ac,,Up + acV U, + u~JJJ/~,,~~ where a,,, etc. are the areas of the triangles indicated by subscripts. The boundary condition U=O is expressed in the form U,= - U,(sin &/sin 0,)‘.

Introducing finite difference grids with 80-100 nodes, the problem has been solved for various values of p. The values of A of the smallest real part are all real and the computer results are shown as points on curve a in Fig. 9. The limiting values A = 1 and 2 for p +O and p --, r result from equation (2) for the plane problem. The value A = 1.5

234 z. P. BAZANT

2

~Q

>-.,

Fig. 6. Examples of dependence of A and parameter Q upon Ao in methods A and B for solving equations (30). (Curves a, b correspond to Fig. 2c, curve c to Fig. 12a.)

-2

'" .2 N = No. of equations

-3

-06 +0

log Vb.8b.¢

Fig. 7. Convergence of A to the exact value 1·5 with diminishing step size of the net, for the case in Fig. 2c.

The U-value at point (Be, q;J is then related by linear interpolation to the U-values at the nearest three interior nodes p, q, r using area coordinates in the triangle pqr, i.e. Uc = (acqrUp+ aerpUQ+ aeprUr)/apQr where aCQr etc. are the areas of the triangles indicated by subscripts. The boundary condition U = 0 is expressed in the form Ua = - Ue(sin (Msin BaY.

Introducing finite difference grids with 80-100 nodes, the problem has been solved for various values of {3. The values of A of the smallest real part are all real and the computer results are shown as points on curve a in Fig. 9. The limiting values A = 1 and 2 for {3 ~O and {3 ~ 7T result from equation (2) for the plane problem. The value A = 1·5


au -=O an

(b)

Fig. 8, (a) A quarterspace with boundary conditions giving rise to sin~~arity on line 00’. (b) A halfspace with a crack whose edge is not perpendicular to the surface.

x

0 R 2r

28

Fig. 9. Smallest real root A for cases shown in Fig. 1 I (curve b), Fig. 8 (curve a) and Fig. 13 (curve c).

for 2p = w agrees with the result from Fig. 4. It is noteworthy that the singularity on the ray 00’ disappears at the surface of the halfspace (Fig. 8b) for any angle /3. The eigenstates U(8, q) for two special cases are drawn in Fig. 10. For the case p = 7r/4 (Fig. 12a) it has been verified by method A that no complex roots exist. This case has also been analyzed by method B with grid Acp = r/24, A9 = ~126 (359 nodes), which yielded A = 1.131, while the coarser grid (Acp = ~112, A8 = ~113) resulted in A = 1.128 (the second and third smallest A was 2.25 and 2.95). Assuming a similar convergence pattern as in Fig. 7, the correct value can be estimated as h = 1.131+ 0.001, or A = 1 a132.

Elastic stresses under wedge-soaped stamps or at crack edge corners. Wave scatteting by corners.

Furthermore, the case of a rigid sliding stamp with wedge-shaped contact of an

Three-dimensional harmonic functions

(0)

au=o an

E 0

( b)

C C

Fig. 8. (a) A quarterspace with boundary conditions giving rise to singularity on line 00'. (b) A half space with a crack whose edge is not perpendicular to the surface.

2~------------'---------------~~--,

o

o Computed Esti mated

" 2"

2/3

Fig. 9. Smallest real root ,\ for cases shown in Fig. 11 (curve b), Fig. 8 (curve a) and Fig. 13 (curve c).

235

for 213 = 7T agrees with the result from Fig. 4. It is noteworthy that the singularity on the ray 00' disappears at the surface of the halfspace (Fig. 8b) for any angle p. The eigenstates U(6, cp) for two special cases are drawn in Fig. to. For the case 13 = 7T/4 (Fig. 12a) it has been verified by method A that no complex roots exist. This case has also been analyzed by method B with grid dcp = 7T/24, dO = 7T/26 (359 nodes), which yielded A = 1'131, while the coarser grid (dcp = 7T/12, d6 7T/13) resulted in A = 1·128 (the second and third smallest A was 2·25 and 2·95). Assuming a similar convergence pattern as in Fig. 7, the correct value can be estimated as A = 1 ·131 + 0·001, or A = 1·132.

Elastic stresses under wedge-shaped stamps or at crack edge corners. Wave scattering by corners.

Furthermore, the case of a rigid sliding stamp with wedge-shaped contact of an

236 Z. P. BAYkANT

Table 2. Nodal values of function U; (A)-Case in Fig. 10a (p = 0.5); (B&Fig. 12a (p = 0.5); (C)-Fig. 12b (p = 0.5); (D)-Fig. 14 (p = 0.5); (E&Fig. I1 for p = a/8(p = 2/3). Nodal coordinates are I& = KAB; 9, = JA9 in cases (A), (B), (D) and (E); 9, = (J - l)A9 in case (C); grid A9 = n/24, A0 = p/26. In Table (C) the left and right of Fig. 14 are interchanged. The top row of each case was obtained by

extrapolation.

236 Z. P. BAZANT

Table 2. Nodal values of function U; (A)-Case in Fig. lOa (p = 0·5); (B)-Fig. 12a (p = 0·5); (C)-Fig. 12b (p = 0·5); (D)-Fig. 14 (p = 0·5); (E)-Fig. II for f3 = 7T/8(p = 2/3). Nodal coordinates are Ok = KAO; ipj = fAip in cases (A), (B), (D) and (E); '1'; = (J -I)Aip in case (C); grid Aip = 7T/24, AIJ = 7T/26. In Table (C) the left and right of Fig. 14 are interchanged. The top row of each case was obtained by

10 II 12 13

'4 'S 16 17 I. I.

• 10 II 12 13 14 IS 16 17 I. ,.

9 10 11 12 13 14 'S Ib

" ,. ,.

· 10 11 12 13 14 ,S I. 17 ,. 0 , , , 4 S

• 7

• • 10 11 '2

" ,4 15 I.

" ,. ,. 2. 21 .. 23

extrapolation. 10 11 12 I l 1'-, ]6 17 I i4 I'~ 20 II ?<' 23

501 1001 ]499 478 951; 1434

1'~95 2481 .,974 340;5 3927 43QO 4,11.41 "i;:>7R ';(I,Q1 o<,nQB 647'" "'~?A 11C;, 7447 nOil 19" ~121) en7 ,11.,7, A4),<, 1',45,11. 19122)89 2AflO:; ,,19 1,11.09 4273 41?CI <;]74 '5MI, 602'0 fJ414 1'17AS 71?A 7440 7717 7Q5A Al'5Q 8317 flit) I ,11.<;0" ,11.<;2, 1t'lI'IJ 2096 21)40 799) ,4'54 )9;02 4)9, 4A64 ':;)31 51AA 61'11 ", ... .,4 70<)1 7lo1l'> 174'5 ,11.011 Ai'7" 84,<,1 A"'06 ,11.,<,91 A71Q ])90 1176 2181 7bl) 3064 ,535 4(1"'0 4511'> C;O)f, 5<;14 600? 6474 6921 7116 7712 ,11.041 A12? 8<;44 P,10f, ,11.,11.0<) RA3A IOQ2 1'"71 1-rq1 72'02 :U,4? 3114 ''''17 4)10 to.M? '51'Q7 <;72,11. 6"4'5 "'11,11. 111:19 7f>}A 79AA A300 a<;<;) ''.731 AP44 AAAJ 770 US) DM 1760 21R9 26f,} .171 1110 4?11 4A4} 5411 50,,9 "C;04 101l,) 1463 7A"A A711 84fHi ~M1 AP-OQ AA~O 420 644 934 17M 1100:; 7118 ?"CICI '?5f1 11'144 4446 5050 S.<,4C) "211'1 1)156 1;;>49 7 .... 1'1<; AO')" A151 ~<;I)A 1'111)(''1 1'1744

4{1Q A?l 1 ?3S 333 !on 1023 25" 51 0 7~9 l~O 334 <;33 59 141 256

_4>

114> 14"'''' II

'61 q?S 1"'\7'" q:::< 1)91

'" '1':''3 142'ii '0< 9;q 1/,72

'" 1I'?'l , 0:::21

'11 I "t7 1':'.'<4 1122 '''<;1

17<;"1

40 20t. 451 7A7 119. 16 .... 7 ::>1C11 ::>714 11A4 40]3 4"4C1 5::>76 SAA? .... 4.:;2 1)974 1411 71'1)1 1'1\4,., 1'1111 A"i17 A<; .... "i -c;o 2C;A I)C;'" 1129 1 ........ 9 ::>?"'I ::>ACll )0::;454::>0741'167 S4C1e:; ,<,Oq) ~1',4] 7]n 7e:;100 1A7? FlI110 "l7"? 1'1111

Cll 5 .... a llle:; }721 :;" .... 1'1 1042377 .... 4404 <;060 <;6RO f>?109 .<,7<;4 111'14 7".i?A nAn 7Ql1o 791'10; -1<; <;lQ 11"'5 lA]l'- ?')06 3;>07 1003 4<:;17 <;214 5799 ""}<:I 6Ud 711'" 711e:; 7C;)3 7<;1'1"

_<;1 <;",., 1;>17 )Cl3A ;>,:,<;? 111,0 4()47 4,,97 '5793 <;1'1::>1 .... 774 6ti1<; "ClOO 70"1 711'5 -4<; ",7 1,3970,,"0 277':' 1471 41n 47)0 C:;7':'''' c:;771 60"1761'5<:; "<:;11'1 "<:;71

1'1 710 1411 21e;) ?Fl.47 1<;1'16 4110 4"47 <:;104 '547\ <;740 <;01'11 <;Q5A I.\J) '50 nn 1/,1'1<; 217Fl. ;>Rl;> 3434 lQ.<,A 442? 47A7 <;1'1'54 "i716 "inl

18 1 2 2271 ,.'7(14

11l 4" ?291 2 77:: 13"2 2:''-',,, 2 7"'2 I 94~ 24C,l-; 2;'4<;1

:n'lo;. :107" 2'027 21'l>',- 2">'':1 )(,,1"

?11~ 2""1 )I2? 2?.o') ?-o-, j;Io2

J1ljl

:OJ?I 1142 'le n4 127,:> 'l1'i6 144k

:<"54 'If,74 11<\0

776 14M ;>10") ?"9A 1,::>;><; '''71 4031 479<; 44"><; 4"i09 71 "91 13?? 190;> ::>4\7 ?A54 370<; )462 If,JA '''71

-Ill 47A 1039 1<:;17 19f.] 2100 ;:>"i4A 7"<:IQ ?74Q -436 100 <;7A 981 1107 1544 1"'1'11'1 17)1'.

_9<; 213 41<; <;"Q 1'.13

1<:;2>' )9}2 429] 4>,12 /'01'.::> <;21)'0 5c:;S~ "'1'11] ~04A "'"S7 64]9 '<''594 "'72? 6P.2Z f,P.c:q I',Q]f, t-oQSI') 1546 3933 4302 4>'<;0 4077 "iZH? C)<;f,l 'iA?O AO'5? "'2'58 bIo]8 1'.591 1)71'l IiAl4 h~84 ,.,Q2h 6940 160Q 34Q4 4359 470] 50?3 "i121 5'iQ4 ':>"42 f,1')6S 621)2 6434 o<,SP.O h.<,9Q 679Z 61,59 hP.9f) F,Q}Z 1""'0 40" .. .:.4;>"i 47f,) 5071 <;)67 c;fo,)? <'1'177 o<.OA7 6?77 6441 11'5"'0 ,<,,<,94 6783 "A .. f, f,Af'1o (-,,,97 37f,J 414] 1.500 41'132 5119 "i421 5.<,7Q ">011 f,IIA fl31'l1 10,459 ,.,,,,el) o<,70? 6787 f,P.47 f,P.f\4 ""'9f, J~51 47]1 4584 4QIO "i?IO <;4A5 57J" ~060 6160 6311) M89 "617 0<,722 hR04 6"162 6R97 690~ ~OS7 4Dl 4,.,71'1 40QA S?9) '5<;59 5AOI "'n19 6711 6]Fl3 6'530 '<'f,'54 67'55 6A33 6R89 h9"J "Q34 4074 4444 41?-4 0:::0Q7 "JID ')1'.43 ">1'\1A hl'SO 6?77 "'44? 6'584 "'704 flAOI 6877 I':>QJ) "'91':>J >,474 4?(I'" 4C,10 4901 c,?I)A S4R'" S139 S9F,1'I 6112 1':>354 .... S14 6651 1,767 hAf,2 1,93<:; 6987 70tH 71"129 4355 4711 51'l17 <;,13 5"'04 Sk49 6(HI "';>69 ,.,44"> 6600 6733 I',A4" ';937 700A 705'1 70A9 ]l'IQ9

.. fl71 5187 <;474 5731'. ,)Q74 6\89 blRI "'C;'57 6702 6R32 "'94\ 7029 70Q8 7141'1 7177 7187 5)<;6 <;")) SkR7 hl16 6124 "''''11 h677 h8?3 ,;Q4B 70S4 7\40 7707 7?5<; 7?A" 779)

<;AD '<'057 6779 64,,0 /'>"'.<;1 ,.,k?;> 1'>9"3 7085 71AR 7?7? 7137 7)84 741? 7421 .... 751 "",6<i 6"">Q 61'115 6991 7l?8 7247 1)47 742A 749? 7<;37 7<;64 7<)73

"'67Q 6111-,7 71116 7lRR 7371 7436 7<;14 76lJ 7675 7719 774'3> 7754 HOR n72 741Q 7548 7660 17'54 7A)? 7A91 793 .. 7960 7Q6A

7<::50 769? 7817 79Z<i kOl7 Aon 81<;0 A19) A?\'" k?24 RO IP Bllrl A?4J '<11? "1404 A460 8">00 RC,?", fl,)32

A<;?] Rf.30 '171'" "I78c:; 1'1839 81'177 AA9Cj 8'107 QZ55 Ql07 (14) 9163 9170

AFl02 97A4 P711 r1"1~Q -l17Q 79S'" 7f,Q9 740Q 7087 "'711 1'>149 591"> ">494 <)02" 4514 41)19 141'14 291\ 23h4 17fi4 11910 '9. 87,Q 'P?::> P~71 "''''CJ) 1<.:../7 1-(\4H 7014 71:>A7 71.0f-. 709) "'74" 1)170 S961 ')"'?<; 50611 4569 4054 lS]1 ?Q."'I 21A9 11'10) I?Ofl '" R<:;<;1'l So:::,k p<;,:,~ .. "v, "31 .. "';;(lq PO">4 786'< 7bC:;0 71QO 7112 .<,7QI h43'5 604<; ~"'20 516"'\ 41)75 41')1'1 11)}" 10SI) 2465 11'163 1749 '27 A)10 1l?7? "I" I 1"I"""c. 7"3\ 7777 7S91 7 170 7112 '<'1<1'" I)4R? blOQ ..,,,Q7 5:?IoR 47(.,4 4?47 171)0 1121) 2529 1914 1784 '" 80J3 ~1\2F<. 1l'11? 7~71 77A) 7""', 7511 71;>? 7')Q4 ..,A.?l 1)<;11 6\C:;S ~7'56 511'" 4E\36 4319 171-,Q 11 J.jQ 2<;El3 1956 Ill) '50 1711 7711 nil 7" ~r H,,.." 71,09 70,?7 7410 nsS 70">6 ..,AI? I':>c:;2\ 6\1'11 ">7Q7 ')lh.., 4P.91 4)76 1874 1719 26('6 1991) 1117 h71 13,Q

," 6" .,. h34 64. '6S .o.

:!':2 3~2 397 ...

711'-." 71'<" 7,.,) 7.:.1\ 7,) .......

74"1 71 ~n

7"0" 71H~

,,917

72H9 711',P. 7000 "'1Al 6<;\3 1)192 ')Il?n 539A 4Ql0 .. 417 11'1';'> 1777 26S9 2n}6 11<;<; hAl l1R7 71109 7'1'<'1 0;926 0<.7)<; f,4A7 6]1'14 .,1l2S <=;411 490:;;1 444? 11'191 110) 2MI2 ZO)5 llf,H '" "QA", "91'19 ,,940 "'1l14 "'66P- ,,4"'2 61'5" ,:!RI? ')411 4956 4452 1904 1116 2695 2'04<; 117') h.1

6811 h79Q ,.,721 65A3 ",)79 6110 57AO 5390 4944 4447 1903 111~ 2697 Z()49 1178 h.3 "'''41 6<;CJ-S f,4AO 629.., 604S 5729 5)'51 4915 4426 lA~A 1107 26<;10 2044 1315 h.,

""'4Q 635'" 6195 59~1 51,60 52<;1<; 4869 4389 18'59 1?R<; 2673 2032 11f,8 he. 6217 6074 5R'57 So:::71 S219 4806 4336 lRI5 3250 2646 201? 115<; hAl

'5932 573? 5462 5124 47?4 4?66 17c:;7 VO;:> Z609 1984 )1)6 h72 5<;A,; 5,]2 S009 46?4 4179 11)R) 3141 Z561 IQ4Q 1112 ". <;lAO 41'174 4504 4075 1<;94 10&7 ?SO;> IQ04 1283 "6

4717 43,.,4 3952 14AQ ?979 Z431 IPSl 1('47 h2. 420) 31'110 13"<; 7R7<; 23101'1 171>.1'1 POf> ,,7

)045 12'23 ?15<; 2251 \71 .... IIS7 5.2 2139 )1-,32 \10\ 5<;4

1773 lan 2409 2975 3<;?0 4036 4<;l') 4973 '5,87 5758 60R7 63"A MOZ 67A4 6916 6Q9'5 7022 InB )~30 24)1 2QA6 ]<;)0 4047 4<;11 4QR) S1<;15 5166 6OCl2 6_372 MO'" 61810, 6917 6Q9f, 702? 1243 IR50 2443 301<; 31)61 407Q 4<; .... 3 5011 5420 57B7 6109 631lS M12 6791 (9)9 6(91) 70n 1?t.3 lR79 7478 3055 )';04 412) 4"'06 50<;2 545R 5820 61H ..... 1"1<:1 f.6)2 ",R01 6~33 7001'1 7014 17A9 1916 2524 3101 36M 417Q 4,.,10,;> 5106 'i'i0 7 5A65 617A 6 .. 45 6664 6831'. 6954 71')];> 70<;7 1122 19"'3 ~5fll 3171 37?8 4249 4731 5)72 5S70 5924 6212 6 .. 94 6709 6811 6997 7069 709) 131'16 20n 26<;2 3~4<;1 31'111 433, 4"'14 52S2 5646 5995 6299 65C;6 611'.7 ,,932 704Q 7121') 7144

~99

P2 7'0 !le3

2096 7739 3344 3909 4432 4Qt? 5147 5736 6081 63AO 6t11; 6840 7001 7116 718<; 7208 4027 454<;1 50?'" 54e;s 5"143 61A? 6416 bPS ';92B 708'" 7199 7::>67 7?A9

51,;0 5<;IH 5Q67 6301 6590 6!S 14 7034 7189 7299 n",,; 73A8 571A 6112 6440 672] 690<,3 71<;8 7311 7419 14A'5 7506

6279 6601 6879 7}]1 7305 745'5 7562 7626 7647 67R7 7059 72p.,9 1478 7625 7730 7793 71'114

1 ;.:. ~ l' , 1717 CCJ? 1:1<;- p 1 7~ 'J ?r ~4 1171 " 1 ~ : '1 ;:n2 1:,'17 " 231)"3

'41? " 7 !.!.', ~:'I:-.~

:>7"'.:

2:<"1 0'649 2"39A 0'69) ?534 2""l'6 208', 2970

?"'''''' J121 .3035 32 74

)42':'

7ZhQ 7 .. Q5 7680 7825 7928 7990 8010 77"16 79}8 8060 8162 8223 8243

fl200 8340 8440 8501') 8520 867<; A773 8A)2 88<;l'

9178 Cj?)4 9253 972R 'n48

?933 ~?~2 14~<; )t-9C 101)0 41/)P. 4;:>90 44"i? 4'10.<, 4721 4821'> 4912 4079 S027 <;0'1f. <;I'l"f, 2414 ~?:":!l ~41l" 17\6 1'127 4121 4?9~ 44S;> 4SQO 4709 4Alfl 4SQl 4Q'57 "001 "01fl 0:::019 109.-1 :,,:~.:.9 351'11} 379;> lQfl4 41SF<. 4)14 Io4'1? 4<;71 4f,77 47"'4 41'11<; 4~90 4Ql0 4Q")1 4Qf,I 1227 ;11.61 ,"'7, JPf-,<; 4('1'< 4}'<<; "'"1)1 44S<; 4<;1'.2 4':'54 4710 47'" 4R41 4P7<; 4kQf, 4QO;> .157 J571 176) JQ37 40Ql' 4::>1::> 4"10;6 44"5 45"'0 4'<''''1 47t"1Q 471'.1. 4A07 41'111'1 4Ao;f, 4Ah? 1497 ,:11'-.79 'fl57 4007 4147 4::>7? 41~1 44Al 4<;1'-.1'> 4f.1Q 4700 47'50 471'11'1 4P.\" 4f'11? 4A'F<. )61'5 17ko; .940 4079 4;>00; 4117 4417 4<:;n<; 4'>A;> 4"4Q 4704 4749 47R4 4RnQ 4"'74 4A?9 3741 3fJQ\ 4!:1;>9 41'54 4?,;7 41~Q 44<;9 4"i4!) ""10 4,.,70 47;>] 47";> 4794 4~1" 4P3n 41'114

1q97 4\?C 42;1::\ 431" 44?"" 4511 4'>P.4 I.f-,49 4704 47<;0 47A'" 4A)7 4F13~ 4"'S\ 4.A5S ¥ 4217 4370 4414 4491l, 4574 4"41 4700 475\ 4794 4f1..?Q 4P-S6 4F17"> 4AA7 4k91

1.:<2<; 4418 4"'0' 45RI'l 464Q 4111 47"'641'11 4A"i1 4Rp.,e; 4910 4971'1 4Q)9 4Q41 4527 41'.0<; 4",7S 47]9 .. 790<, 41'147 4~9) 4Q?f'I 4Q'iR 4qRI 4Q9,A; <;1'1(11'1 S('I17 4051 472? 47",,,, 4<l4C:; 4R9Q 494'5 4QA'" <;/);>1 <;049 501\ 501'17 "i09,., <;099 47Q2 41'\S7 4QI7 4971 <;07n c:;no<,4 <;10? <;114 <;1"'1511'11519'" "'?O") 520R

1(1 lAJo'IBDA '" .2057 snl.:. 51"1"'9 <;12'0 0:::1"'''' <;21'l" 5;>"'2 '"i?72 <;?Cj7 5)1'" 53)1'1 "i'38 <;141 '>19"'- ~24Cj ~?96 ,>,,0 <;177 'i41n <;411'1 '54';1 5479 549? <;499 0;<;0? "414 C;4~2 550'; ,,>0;4'" 'iSA) ""'12 Sf,lP. 5.<,5'1 567f<. <;688 "''''QS Sf<.97 <;,;"? "717 <;759 0:::790<, 1:,8::>8 5F<.C,6 C;,AAI'l 0;900 S916 5921 "iQ31 59)6

"02Cl """,7 0<.101 1,110 .... 1<:;6 "'17'" 1'.19'" 6?10 ";>20 .<,::>;>t> ,;22A tl4S? 1'-.41l1 h<;I'l'; ~t;?R "'<i47 [<.C;61 6<;7'; ,;"'8S "'''''Oil 6<;92

....9P.l 7001 7017 703\ 704? 70'50 70')<:; 70S'" 1605 7'<'18 7'<'1? 7f,44 165) 7661 7f,I,S 71,66

R460 "'417 fl.4"R p.,49" p,<;O? A"'Ot> 1"">01 Q74Q 9751 Q7C:;1 97'5A


Fig. 10. Function U computed for the cases shown in Fig. 8 (p”= d& -&

8

n/2 _ _.-_--.-

arbitrary angle, 2& pressed upon the surface of an isotropic elastic half-space (Fig. 1 la), has been studied. As is well known[4], this problem of elasticity may be reduced, with the help of Papkovich-Neuber potentials, to a potential theory problem, V% = 0. Introducing Cartesian coordinates x, y, z and considering the half-space z SO or 0 s cp ES V, the boundary conditions at the halfspace surface are

G( = 0 on 9 and ~~1~2 = 0 outside 9 (34)

where 5 = the wedge-shaped domain below the stamp (OO’AO” in Fig. 1 la). Also, at infinity, grad u is required to vanish. After solving for u, the displacements ux, uy, uz in Cartesian coordinates x, y, z at any point of the half-space z?=O are given as[4]:

u,=(l-2V) CC du

zdz-z$$ u,=(l-2~) _ au

avdz-z$, au

z&=2(1-v)u-z;j;, (37)

where z, = Poisson’s ratio of the material. The normal stress below the stamp is a, = ( cW/C?Z)E/( 1 -t Y) where E = Young’s modulus.

This problem is equivalent to the case of an infinite elastic body containing a sharp crack whose edge forms a sharp corner of an arbitrary angle 2,&I (Fig. 1 lb). The body is subjected at infinity to uniformly dist~buted normal stresses pe~endicul~ to the crack


(]

o

1T/4

(]

u=r'92 U~ rr/2

31'12 11"12

4> Fig. 10. Function U computed for the cases shown in Fig. 8 (p' = V sin 0).

arbitrary angle, 2{3, pressed upon the surface of an isotropic elastic half-space (Fig. IIa), has been studied. As is well known [4], this problem of elasticity may be reduced, with the help of Papkovich-Neuber potentials, to a potential theory problem, ~u = O. Introducing cartesian coordinates x, y, z and considering the half-space z;=;: 0 or o .=:; cp .=:; 'IT, the boundary conditions at the halfspace surface are

u == 0 on <fJ and au! az = 0 outside <fJ (36)

where '!f::::: the wedge-shaped domain below the stamp (00' AO" in Fig. Ita). Also, at infinity, grad u is required to vanish. After solving for u, the displacements Ux, Uy, uz in cartesian coordinates x, y, z at any point of the half-space z;=;:O are given as[4J:

l~ au au l~ au au au u == (I - 2 v) - dz - z - u == (1 - 2 v) - dz - z - U, ::::: 2( 1 - v) u - z-x z ax ax' Y , ay ay' az' (37)

where v::::: Poisson's ratio of the material. The normal stress below the stamp is CT,= (au/az)E/(1 + v) where E::::: Young's modulus.

This problem is equivalent to the case of an infinite elastic body containing a sharp crack whose edge forms a sharp corner of an arbitrary angle 2f3 (Fig. lIb). The body is subjected at infinity to uniformly distributed normal stresses perpendicular to the crack

238 Z. P. BA2ANT

(b)

Fig. 11. (a) Wedge stamp contact (unshaded) upon elastic halfspace or, alternatively, a screen, scattering waves. (b) Crack corner in an infinite elastic space. (The sphere is not the body

surface and serves only for visualizing the situation in spherical coordinates.)

plane. Furthermore, the near field in this problem happens to be the same as in the scattering of waves by a sharp corner of a screen[9].

Although in Fig. 11 there are two singularity rays, 00’ and 00”, each with p = : from equation (Z), the problem may be reduced to one with a single singularity ray, taking advantage of the fact that the solution II or F must be symmetrical with regard to plane OAK‘ in Fig. 1 lb. The formulation of the problem then differs from the previous case (Fig. 8) solely by the boundary condition on side ABC (or AEC) which reads:

a[(sin @)“U]/&r = 0 (38)

where p = i and n is the normal to the side AEC (normal on the sphere, not in the plane (6, cp)). The finite difference formulation of the boundary condition is the same as in the previous case (Fig. 8, equations 34, 35), except that Ua= U, (sin &/sin &)“.

Computer results for grids with 80-100 nodes are shown by the points on curve b in Fig. 9. Note that the points adjacent to A = O-5 lit a smooth curve passed through the well-known value of X = O-5 for 2p = 71: The point A = 0 for /3 +O follows from equation (28), and h = I for /3+~ results from equation (29). The eigenstates U(0, cp) are shown for two special cases by lines of equal U-value in Fig. 12. It is worth noting that the computed variation of U along the top side 0’6’ of Fig. 12a or b is proportional to sin (e/2), as is required by equation (2).

The results prove that the singularity at the sharp concave (2p<7r) corner 0 of a crack edge (Fig. 1 lb) in an elastic body is more severe (h<p) than at a straight crack edge (p = f). This fact is of interest for fracture mechanics of brittle elastic materials. It shows that at a concave corner of the crack edge of a planar crack the crack will propagate faster or at a lower stress than at a straight edge. Obviously, this would tend to straighten the crack edge. A plane crack with a straight edge is thus more stable than a plane crack with zig-zag edge.

All roots have been computed by method B, restricting consideration to real h. However, the case 2p = 7r/2 has also been computed by method A; the result was identical and it was found that no complex roots A exist in this case (similar to cases 2p = r and ~3 = a).

238 Z. P. BAZANT

(a)

Fig. 11. (a) Wedge stamp contact (unshaded) upon elastic halfspace or, alternatively, a screen, scattering waves. (b) Crack corner in an infinite elastic space. (The sphere is not the body

surface and serves only for visualizing the situation in spherical coordinates.)

plane. Furthermore, the near field in this problem happens to be the same as in the scattering of waves by a sharp corner of a screen [9].

Although in Fig. 11 there are two singularity rays, 00' and 00", each with p = ~ from equation (2), the problem may be reduced to one with a single singularity ray, taking advantage of the fact that the solution u or F must be symmetrical with regard to plane OABC in Fig. lIb. The formulation of the problem then differs from the previous case (Fig. 8) solely by the boundary condition on side ABC (or ABC) which reads:

a [(sin (;YU]/an = 0 (38)

where p = ~ and n is the normal to the side ABC (normal on the sphere, not in the plane (8, <p». The finite difference formulation of the boundary condition is the same as in the previous case (Fig. 8, equations 34, 35), except that Ua Uc (sin fJc/sin (JaY.

Computer results for grids with 80-100 nodes are shown by the points on curve bin Fig. 9. Note that the points adjacent to A = 0·5 fit a smooth curve passed through the well-known value of A = 0·5 for 2f3 = Tr. The point A = 0 for f3 ~O follows from equation (28), and A ::: 1 for f3 ~ Tr results from equation (29). The eigenstates U( (J, <p) are shown for two special cases by lines of equal U-value in Fig. 12. It is worth noting that the computed variation of U along the top side 0'0' of Fig. 12a or b is proportional to sin (9/2), as is required by equation (2).

The results prove that the singularity at the sharp concave (2f3< Tr) corner 0 of a crack edge (Fig. 11b) in an elastic body is more severe (A<p) than at a straight crack edge (p = n. This fact is of interest for fracture mechanics of brittle elastic materials. It shows that at a concave corner of the crack edge of a planar crack the crack will propagate faster or at a lower stress than at a straight edge. Obviously, this would tend to straighten the crack edge. A plane crack with a straight edge is thus more stable than a plane crack with zig-zag edge.

All roots have been computed by method B, restricting consideration to real A. However, the case 2f3 = Tr /2 has also been computed by method A; the result was identical and it was found that no complex roots A exist in this case (similar to cases 2f3 = 7r and f3 = Tr).


Fig. 12. Function U computed for the case shown in Fig. 11 (p” = V%%).

All points on curve b in Fig. 9 have first been computed with grids Aq = ~/12,A8 = 7r/13 (SO-100 nodes). For the case of a right-angled corner (2j3 = r/2; Fig. 12a) this grid (having 86 nodes, not counting those outside the domain) yielded A = O-2905. This case has then also been computed with a grid Aq = 7~124, A8 = ~126 (having 359 nodes), which has resulted in h = O-2947 (using method B). This differs by 0.0043 (i.e. 1.46 per cent) from the coarser grid. Nearly the same difference, namely OW44, is found between the grids with the same spacing in Fig. 7. Because the result for a grid with 359 nodes would be in Fig. 7 about OW15 below the exact value, it can be inferred that the correct value probably is A = 0.2947 + OW15, i.e.

A = 0.2% (for 2p = 7r/2). (39)

(However, Radlow [9] deduced for this case analytically the value Re(h) = 0.25, whose difference from the present result is obviously much greater than the probable value of numerical error, and so reexamination of the analysis in[9] seems to be in order.) The second and third smallest A, as found by method A with 86 nodes for p = 4, are 1.43 and 2.03. It was also checked that higher p does not give smaller h.

The case 2p = 3~/2 (Fig. 12b) has also been computed with the finer grid, which has furnished A =0*8148; assuming similar convergence behavior as before, the correct

8

Three-dimensional harmonic functions

,

lau=oan 311"/4 L-..l.----L---L_L-.J...----L---.L---.JL--L-...r-~---.Jc

(b) 0 0'

11"14

8 U=\ 2

11"/2

Fig. 12. Function U computed for the case shown in Fig. 11 (p' = Ysin -lJ).

239

All points on curve b in Fig. 9 have first been computed with grids Acp = 7T/12,A(J = 7T/13 (80-100 nodes). For the case of a right-angled corner (2(3 = 7T/2; Fig. 12a) this grid (having 86 nodes, not counting those outside the domain) yielded A = 0·2905. This case has then also been computed with a grid Acp = 7T/24, A(J = 7T/26 (having 359 nodes), which has resulted in A = 0·2947 (using method B). This differs by 0·0043 (i.e. 1·46 per cent) from the coarser grid. Nearly the same difference, namely 0·0044, is found between the grids with the same spacing in Fig. 7. Because the result for a grid with 359 nodes would be in Fig. 7 about 0·0015 below the exact value, it can be inferred that the correct value probably is A = 0·2947 + 0·0015, i.e.

A = 0·2% (for 2(3 = 7T /2). (39)

(However, Radlow [9] deduced for this case analytically the value Re(A) = 0·25, whose difference from the present result is obviously much greater than the probable value of numerical error, and so reexamination of the analysis in[9] seems to be in order.) The second and third smallest A, as found by method A with 86 nodes for p = tare 1·43 and 2·03. It was also checked that higher p does not give smaller A.

The case 2(3 = 37T/2 (Fig. 12b) has also been computed with the finer grid, which has furnished A =0·8148; assuming similar convergence behavior as before, the correct

240 Z. P. BAiANT

value is A =0~815+0~001 =0816. For 2p = 7r/4, the finer grid yielded A = 0.2057 + 0.0015 = 0.207.

Rvachev’s results for real roots[l I], obtained by a mapping technique and the Galerkin method, differ considerably (by as much as 0.2) from curve b in Fig. 9. This is probably due to numerical error because Rvachev used only six unknown parameters.

For p -0, Aleksandrov and Babeshko [l] found A to be complex, with &(A) = - O.S, which overrides the value A =0 indicated by equation (28) and shown in Fig. 9. However, no cases with very small p have yet been run to verify this result. This will be postponed until the finite element formulation is completed (because refinement of the nodal network near side OA in Fig. 10 will probably be desirable for high accuracy).

Singularity at the apex of pyramidal notches with three equal sides The cases of pyramidal notches shown in Fig. 13 have 3 singularity rays and possess

one plane of symmetry through each of the three edges, so that only the domain O’ABCDO’ needs to be considered. This domain includes only one singularity ray and the formulation of the problem is the same as for the cases shown in Fig. 11, discussed previously, except for two differences. First, angle 2a of the planes forming each radial edge depends on the apex angle 2p of each side, namely sin (Y = l/(2 cos p), and the value of p is then given by equation (2). Second, the location of the right boundary in the (8,(p) plane is different, as is shown in Fig. 14 for the case of the right-angled pyramidal notch (2/3 = r/2).

This case was analyzed only by method B, assuming A to be real. The smallest roots computed with networks of about 80 nodes are given by the points on curve c in Fig. 9 and the eigenstate U for one case is shown in Fig. 14. Note that the cases 2/3 >2~/3 represent a convex solid pyramid rather than a concave notch. For 2p=2~/3 the pyramid degenerates into a planar surface ((Y =O), for which A = 1. The case p +O should give the same value as for curve 6. (According to the result in[l], the values in Fig. 9 will possibly be overriden by some complex A at p -0.) The right-angled pyramidal notch (Figs. 14 and 13a) has also been computed with a finer grid AL\CP = n/24, A0 = ~-126, which has yielded A = 0.4533. Assuming similar convergence behavior as before, the correct value is A = O-4533 + OW15 = O-455. For 2p = 7r/4, the finer grid yielded A = 0.2288 + 0@015 = 0.230 (a = 32.765”, p = 0.61127).

Fig. 13. Pyramidal notches with three sides of equal apex angle. (The sphere is not the body surface and serves only for visualizing the situation in spherical coordinates.)

240 Z. P. BAZANT

value is A =0-815+0-001 =0-816. For 2{3 = 1T/4, the finer grid yielded A = 0-2057 +0-0015 = 0-207_

Rvachev's results for real roots[11], obtained by a mapping technique and the Galerkin method, differ considerably (by as much as 0-2) from curve b in Fig_ 9_ This is probably due to numerical error because Rvachev used only six unknown parameters.

For (3~0, Aleksandrov and Babeshko[l] found A to be complex, with Re(A) = -0·5, which overrides the value A = 0 indicated by equation (28) and shown in Fig. 9. However, no cases with very small {3 have yet been run to verify this result. This will be postponed until the finite element formulation is completed (because refinement of the nodal network near side OA in Fig. 10 will probably be desirable for high accuracy).

Singularity at the apex of pyramidal notches with three equal sides The cases of pyramidal notches shown in Fig. 13 have 3 singularity rays and possess

one plane of symmetry through each of the three edges, so that only the domain 0' ABCDO' needs to be considered. This domain includes only one singularity ray and the formulation of the problem is the same as for the cases shown in Fig. 11, discussed previously, except for two differences. First, angle 2a of the planes forming each radial edge depends on the apex angle 2{3 of each side, namely sin a = 1/(2 cos (3), and the value of p is then given by equation (2). Second, the location of the right boundary in the (O,'P) plane is different, as is shown in Fig. 14 for the case of the right-angled pyramidal notch (2{3 = 1T /2).

This case was analyzed only by method B, assuming A to be real. The smallest roots computed with networks of about 80 nodes are given by the points on curve c in Fig. 9 and the eigenstate U for one case is shown in Fig. 14. Note that the cases 2{3 > 21T/3 represent a convex solid pyramid rather than a concave notch. For 2{3=21T/3 the pyramid degenerates into a planar surface (a =0), for which A = 1. The case {3~0 should give the same value as for curve b. (According to the result in[I], the values in Fig. 9 will possibly be overriden by some complex A at (3 ~O.) The right-angled pyramidal notch (Figs. 14 and 13a) has also been computed with a finer grid ~'P = 1T/24, ~O = 1T/26, which has yielded A = 0-4533. Assuming similar convergence behavior as before, the correct value is A = 0·4533 + 0·0015 = 0·455. For 2{3 = 1T /4, the finer grid yielded A = 0-2288 + 0·0015 = 0-230 (a = 32-765°, p = 0·61127).

0'

Fig. 13. Pyramidal notches with three sides of equal apex angle. (The sphere is not the body surface and serves only for visualizing the situation in spherical coordinates.)

241

e

Fig. 14. Function U computed for the case shown in Fig. 13~1 (p” = (sin 8)“‘).

Development of singular finite elements and extensions to elasticity It should be noted that knowledge of the dominant near-singularity field is all that is

needed for the development of finite elements comprising the singularity, following the path shown, e.g. by Walsh [ 131. Table 2 gives sufficient information for formulation of such elements for the most important cases. This brings within reach the numerical solutions by the finite element method for three-dimensional domains with arbitrary boundaries.

The general method presented can also be applied to problems of three-dimensional elasticity which are not reducible to potential theory problems. Here expressions of the form (6) or (10) augmented by rigid-body displacement terms for the neighborhood of the ray must be introduced for each of the three displacement components. The analysis may be based either on Navier’s differential equations of equilibrium in spherical coordinates or the Papkovich-Neuber potentials. In either case the problem is reduced to three simultaneous second order partial differential equations in 0, cp for three functions with nonsingular gradients. Applying a numerical method, the problem is again reduced to the system of algebraic equations (30). But its matrix is about three times larger than for the same nodal network in a potential theory problem, has a wider band, and is less sparse. This will make computations more costly.

Application of the numerical method to singularities in plane elasticity There is a number of unanalyzed singular problems in plane elasticity, especially in

combinations of cracks, notches and bonded dissimilar materials. Analytical solutions are complicated and a great effort is being made to solve various cases one by one, as can be seen in the recent literature. The present method of numerical solution can be useful in all these problems, too. The technique of Knein[8] and Williams [13] reduces the plane problem to an ordinary rather than a partial differential equation. The system of equations (30) arising in numerical solution is therefore of a much smaller size, and usually also narrowly banded, which substantially simplifies computations according to methods A, I3 or C outlined previously.

UES “0, 12. N”. 3-D


0'

B

12

c 3"./4 0':----'---'--.".-'-/::-4-.l....----'1--.. .L/

2=---'---'---=3,,--'/4

¢

Fig. 14. Function U computed for the case shown in Fig. 13a (p" = (sin (/)2/3).

Development of singular finite elements and extensions to elasticity It should be noted that knowledge of the dominant near-singularity field is all that is

needed for the development of finite elements comprising the singularity, following the path shown, e.g. by Walsh [13]. Table 2 gives sufficient information for formulation of such elements for the most important cases. This brings within reach the numerical solutions by the finite element method for three-dimensional domains with arbitrary boundaries.

The general method presented can also be applied to problems of three-dimensional elasticity which are not reducible to potential theory problems. Here expressions of the form (6) or (10) augmented by rigid-body displacement terms for the neighborhood of the ray must be introduced for each of the three displacement components. The analysis may be based either on Navier's differential equations of equilibrium in spherical coordinates or the Papkovich-Neuber potentials. In either case the problem is reduced to three simultaneous second order partial differential equations in 8, 'P for three functions with nonsingular gradients. Applying a numerical method, the problem is again reduced to the system of algebraic equations (30). But its matrix is about three times larger than for the same nodal network in a potential theory problem, has a wider band, and is less sparse. This will make computations more costly.

Application of the numerical method to singularities in plane elasticity There is a number of unanalyzed singular problems in plane elasticity, especially in

combinations of cracks, notches and bonded dissimilar materials. Analytical solutions are complicated and a great effort is being made to solve various cases one by one, as can be seen in the recent literature. The present method of numerical solution can be useful in all these problems, too. The technique of Knein[8] and Williams [13] reduces the plane problem to an ordinary rather than a partial differential equation. The system of equations (30) arising in numerical solution is therefore of a much smaller size, and usually also narrowly banded, which substantially simplifies computations according to methods A, B or C outlined previously.

IJES Vol. 12, No. 3-D

242 Z. P. BAZANT

CONCLUSION

Three-dimensional harmonic functions near termination or intersection of gradient singularity lines can be determined by the general numerical method presented herein. Knowledge of the near field will enable the development of finite elements comprising the singularity.

Acknowledgement-The author wishes to express his gratitude to his friend Dr. Leon M. Keer, Professor at Northwestern University, who called his attention to the present problem and helped by providing precious critical remarks in frequent discussions during the progress of the work. The computing funds were furnished by Northwestern University.

REFERENCES [l] V. M. ALEKSANDROV and V. A. BABESHKO, Prikl. Mat. Mekh. 36,88 (1972). Translated in J. appl.

Math. Mech., 78 (1972). PI r31

[41

PI bl

[71 [81 [91

1101 illI WI u31 [141 WI

Z. P. BAZANT and M. CHRISTENSEN, Int. _J. Solids Strut. 8, 327 (1972). L. COLLATZ, Eigenwerraufgaben mit Technischen Anwendungen. Akademische Verlagsgesellschaft Geest & Portig (1%3). L. A. GALIN, Contact Problems in the Theory of Elasticity (in Russian). Gostechteorizdat (1953) Translated by H. MOSS, Dept. of Math., North Carolina State College, Raleigh (l%l). L. A. GALIN, Dokl. Akad. Nauk SSSR 58, (1947). V. N. FADDEEVA, Computational Methods of Linear Algebra (Translated from Russian) Dover (1959). S. N. KARP and F. C. KARAL, Commun. pure Appl. Math. 15, 413 (1962). M. KNEIN, Abhandlungen aus dem Aerodyn. Inst an der T. H. Aachen 7, 43 (1927). J. RADLOW, Arch. ration. Me&. Analysis 19, 62 (1%5). A. RALSTON, A First Course in Numerical Analysis. (Chapter 10). McGraw-Hill (1%5). V. L. RVACHEV, Prikl. Mat. Mekh. 23, 169 (1959). I. M. RYSHIK and I. S. GRADSTEIN, Tables of Series, Products and Integrals, VEB (1957). P. F. WALSCH, Int. .I. Solids Struct. 7, 1333 (1971). M. L. WILLIAMS, J. appl. Mech. 19, 526 (1952). J. H. WILKINSON and C. REINSCH, Handbook for Automatic Computation- Vol. II, Linear Algebra, Springer (1971).

[I61 E. A. KRAUT, Bull. Seism. Sot. Am. 58, 1083, 1097 (1968).

(Received 18 January 1973)

R&m&-La fonction harmonique U au voisinage du point 0, a partir duquel part un rayon a singularite, est supposCe &e dominte par le terme r*p’LJ oh r = distance du point 0, p = constante connue, et p une fonction choisie des coordonntes spheriques angulaires 8, cp telles que u - (pr)” au voisinage du rayon pour tout r fini. U est une fonction inconnue de 8, cp, pour laquelle une equation ditferentielle partielle avec les conditions aux limites, en particulier celles sur les rayons singuliers, et un principe variationnnel sont deduits. Parce que grad U est non singulier, une solution numerique est possible, utilisant, par exemple, les methodes des differences finies ou des elements finis. Ceci ram&e le probltme a celui de trouver un A de partie rtelle la plus petite et satisfaisant a l’tquation Det (A,j) = 0, oh A,i est une grande matrice dont les coefficients dependent linCairement du p = h(A + 1). En general A et A, sont complexes. Des solutions peuvent &tre obtenues soit en se ramenant a un probleme ordinaire de valeur propre de matrice pour /.L, soit par des conversions successives a des systemes d’equations lineaires non homogenes. Des etudes par ordinateur ont confumees la faisabilitt de cette methode et ont montrees que des resultats de grande precision peuvent etre obtenus. Des solutions pour des fissures et des entailles aboutissant a un plan, ou a une surface conique, et pour des fissures aboutissant obliquement a la surface dun demi-espace, sont present&es. Dans ces cas, A est reel et la singularite est toujours plus faible (h > p) que sur la ligne singulitre et peut meme disparaitre (A > 1).

De plus, des contraintes Clastiques sous l’effet dun poincon rigide, en forme de coin et glissant, ou a l’angle du bord dune fissure ont Cte analysees, ainsi que des fonctions harmoniques pour des entailles pyramidales a trois c&es. Ici on avait A <p.

Une solution analytique simple pour une classe de cas particuliers a tgalement et& trouvee. On l’a utilise pour verifier quelques uns des rtsultats numbriques.

Znsammenfassung-Es wird angenommen, dass die harmonische Funktion u nahe dem Punkt 0. von dem ein einzelner Singularititsstrahl ausgeht, durch den Ausdruck r*p’U beherrscht wird, wo r = Abstand vom Put&t 0, p = bekannte Konstante und p = gewlhlte Funktion von winkeligen Kugelkoordinaten 0, cp

242 Z. P. BAZANT

CONCLUSION

Three-dimensional harmonic functions near termination or intersection of gradient singularity lines can be determined by the general numerical method presented herein. Knowledge of the near field will enable the development of finite elements comprising the singularity.

Acknowledgement-The author wishes to express his gratitude to his friend Dr. Leon M. Keer, Professor at Northwestern University, who called his attention to the present problem and helped by providing precious critical remarks in frequent discussions during the progress of the work. The computing funds were furnished by Northwestern University.

REFERENCES [I] V. M. ALEKSANDROV and V. A. BABESHKO, Prikl. Mat. Mekh. 36, 88 (1972). Translated in J. appl.

Math. Mech., 78 (1972). [2) Z. P. BAZANT and M. CHRISTENSEN, Int. J. Solids Struc. 8, 327 (1972). [3) L. COLLATZ, Eigenwertaufgaben mit Technischen Anwendungen. Akademische Verlagsgesellschaft

Geest & Portig (963). [4) L. A. GALIN, Contact Problems in the Theory of Elasticity (in Russian). Gostechteorizdat (1953)

Translated by H. MOSS, Dept. of Math., North Carolina State College, Raleigh (1961). [5) L. A. GALIN, Dokl. Akad. Nauk SSSR 58, (1947). [6) V. N. F ADDEEV A, Computational Methods of Linear Algebra (Translated from Russian) Dover

(1959). [7) S. N. KARP and F. C. KARAL, Commun. pure Appl. Math. 15, 413 (1962). [8) M. KNEIN, Abhandlungen aus dem Aerodyn. Inst an der T. H. Aachen 7,43 (1927). [9) J. RADLOW, Arch. ration. Mech. Analysis 19,62 (1965).

[10) A. RALSTON, A First Course in Numerical Analysis. (Chapter 10). McGraw-Hill (1965). [11) V. L. RVACHEV, Prikl. Mat. Mekh. 23, 169 (1959). [12) I. M. RYSHIK and I. S. GRADSTEIN, Tables of Series, Products and Integrals, VEB (1957). [13) P. F. WALSCH, Int. J. Solids Struct. 7, 1333 (1971). [14) M. L. WILLIAMS, J. appl. Mech. 19, 526 (1952). [15] J. H. WILKINSON and C. REINSCH, Handbook for Automatic Computation- Vol. II, Linear Algebra,

Springer (1971). [16] E. A. KRAUT, Bull. Seism. Soc. Am. 58, 1083, 1097 (1968).

(Received 18 January 1973)

Resume-La fonction harmonique U au voisinage du point 0, a partir duquel part un rayon a singularite, est supposee ctre dominee par Ie terme r·pPU ou r = distance du point 0, p = constante connue, et p une fonction choisie des coordonnees spheriques angulaires Ii, cp telles que u - (pr)" au voisinage du rayon pour tout r fini. U est une fonction inconnue de Ii, cp, pour laquelle une equation differentielle partielle avec les conditions aux Iimites, en particulier celles sur les rayons singuliers, et un principe variationnnel sont deduits. Parce que grad U est non singulier, une solution numerique est possible, utilisant, par exemple, les methodes des differences finies ou des elements finis. Ceci ramene Ie probleme a celui de trouver un A de partie reelle la plus petite et satisfaisant a I'equation Det (Ai;) = 0, ou A.; est une grande matrice dont les coefficients dependent lineairement du IL = A(A + 1). En general A et Ai; sont complexes. Des solutions peuvent etre obtenues soit en se ramenant a un probleme ordinaire de valeur propre de matrice pour IJ., soit par des conversions successives a des systemes d'equations lineaires non homogenes. Des etudes par ordinateur ont confirmees la faisabilite de cette methode et ont montrees que des resultats de grande precision peuvent etre obtenus. Des solutions pour des fissures et des entailles aboutissant a un plan, ou a une surface conique, et pour des fissures aboutissant obliquement a la surface d'un demi-espace, sont presentees. Dans ces cas, A est reel et la singularite est toujours plus faible (A > p) que sur la ligne singuliere et peut meme disparaitre (A > 1).

De plUS, des contraintes elastiques sous l'effet d'un poin~on rigide, en forme de coin et glissant, ou a I'angle du bord d'une fissure ont ete analysees, ainsi que des fonctions harmoniques pour des entailles pyramidales a trois cotes. lei on avait A < p.

Une solution analytique simple pour une cIasse de cas particuliers a egalement ete trouvee. On I'a utilise pour verifier quelques uns des resultats numeriques.

Zusammenfassung-Es wird angenommen, dass die harmonische Funktion u nahe dem Punkt O. von dem ein einzelner Singularitiitsstrahl ausgeht, durch den Ausdruck r·pPU beherrscht wird, wo r = Abstand vom Punkt 0, p = bekannte Konstante und p = gewahlte Funktion von winkeligen Kugelkoordinaten Ii, cp


so dass u - (pr)” nahe dem Strahl filr jedes endliche r. U ist eine unbekannte Fur&ion von 0, cp, fiir die eine

partielle Differentialgleichung mit Grenzbedingungen, besonders die an den Singularitatsstrahlen, und ein Variationsprinzip abgeleitet werden. Weil Gradient LJ nicht-singuliir ist, ist eine numerische Lissung miiglich, verwendend, zum Beispiel, die den Netzverfahren oder die Methode der endlichen Elementen. Dies reduziert das Problem auf das Finden von A des kleinsten reellen Teiles, der die Gleichung Det (Aij) = 0 befriedigt, wo A, eine grosse Matrize ist, deren Koeffizienten linear van p = h(A f 1) abhlngen. Im Allgemeinen sind A und Acj komplex.

Liisungen kiinnen entweder durch Reduktion auf ein gewiihnlicher Matrizeneigenwertproblem fiir p erhalten werden, oder durch aufeinanderfolgende Umwandlungen auf nicht-homogene lineare Gleichungssysteme. Untersuchungen an Rechenanlagen bestatigten die Durchfiihrbarkeit der Methode und zeigten, dass sehr genaue Resultate erhalten werden kiinnen. Es werden Lijsungen fib Risse und Kerben vorgelegt, die an einer Ebene oder Kegeloberthiche enden, und fur R&se, die quer an einer H~braumo~rfl~~he enden. In diesen Fallen ist A reel1 und die Singular&at ist immer schwacher (A > p) als an der die Sin~larit~tslinie und kann sogar verschwinden (h > I). Weiter wurden elastische Spannungen unter einem keilformigen starren gleiten- den Stempel oder an der Ecke einer Risskante analysiert, wie such harmonische Funktionen an dreiseitigen pyramidischen Kerben. Es wurde gefunden, dass A >p hier vorkommt. Es wurde such eine einfache analytische Liisung fur eine Klasse von Spezialflllen gefunden und wurde verwendet, urn einige der numerischen Resultate zu priifen.

Sommario-Si assume the la funz~one armonica u, vicino al punto 0 da cui emana un raggio con singolarit~

singola, sia dominata dal termine r*p’U dove r b la distanza dal punto 0, p t una costante nota e p t una funzione scelta delle coordinate sferiche angolari 6 e cp tale the u - (pr)” vicino al raggio per ogni valore

finito di r. U & una funzione non nota di tl e cp per la quale vengono derivati un’equazione a differenziali

parziali con condizioni allo strato limite, specialmente quelle sui raggi con singolarita, ed un principio var-

iazionale. Poiche grad U non i: singolare, una soluzione numerica i: possibile, usando per esempio il metodo delle differenze finite o quell0 degli elementi finiti. Cih riduce ii problema a trovare il valore di h della piu piccola parte reale the soddisfa l’equazione Det (As) = 0 dove Aii & una grande matrice i cui coefficienti di-

pendono linearmente da p = h(h + 1). In generale, h et A,, sono numeri complessi. Le soluzioni possono

venir ottenute o per riduzione al problema de1 valore speciale di una matrice normale per F, 0 per conversioni successive a sistemi di equazioni iineari non omogenee. Degli studi effettuati su calcolatori hanno confermato la praticabilita di quest0 metodo e hanno dimostrato the si possono ottenere risultati molto accurati. Vengono presentate soluzioni per fenditure e intagli terminanti su una superficie piana o conica, e per fenditure ter-

minanti obliquamente sulla superficie di un semispazio. In questi casi A e reale e la singolarita e sempre piu

debole (A > p) the sulla linea di singolarita e pub per&o scomparire (4 > 1). Sono state analizzate inoltre le sollecitazioni elastiche sotto uno stamp0 scorrevole rigid0 a forma di cuneo o su un angolo de1 bordo delta

fenditura, come pure le funzioni armoniche su in&&i piramidali a tre facce. I? stata inoltre trovata una

semplice soluzione analitica per una classe di casi speciali, e la soluzione e stata usata per controllare alcuni dei risultati numerici.

A6crpaHT - ~-~E~WITO, YTO rapMoHHrecKan ~~I)YHKUUR u ~6nis3 TO’IKM 0, or KOTO~O~O rwrynaercs OATH

oco6eHHocrHbrii nys, nonsepxetia K ~~C~O~CTBY wtena r”ppU, rAe r = paccToaHife 0~ TO~KU 0, p = w3-

B~~CTH* ~OCT~SHH~~~, p = Bbl6op ~ytt~rtnn OT yrnoebrx e@epwiecKnx KoopAaHaT 8, + raK, ‘ITO u _ (~r)~

8651333 JIyYa AJIK nro6oro XOHeYHOrO 3HaVeHWSi Y. i@U 3TOM u eCTb He~3B~THa~ @yHKIuia OT f?, (b, AJnl

KOTOpOrO AaH BbtBOA ‘IaCTHOrO ~~~~~HuU~bHOrO ypaBHeHn5l C rnaHU’iHbiMA YCJIOBFfSiMii, B ‘iaCTHOCTII,

AnR Tex Ha OC06eHHOCTHbIX AyYaX, a TaKxe BbiBOAMTCfl BapRaunOHHbIn npnHunn. TaK KaK grad u eCTb

HeOCO6eHHas, B03MOxHO nOJIyWTb WCAeHHOe nellIeHue, nnnMelnnl HanpuMep MeTOAbl KOHeYHblX

pa3AWiuk NJ’IH KOHe’iHblX 3AeMeHTOB. nO3TOMy npo6neMa CBOAHTCs K HaXOmAHellm x AIIX HauMeHbmeti

peaJIbHOti YaCTU, yAOBJIeTBOnatomen ypaBHeHneM Det (Al,) = 0, rae ,‘tij - 6onbmas MaTpiina, K03(t)-

&illeHTbI KOTOpOn 3aBWCsT JIuHefiHO OT p = h(A -t 1). BooGme h H Ai,- KOMnAeKCHblMn. PemeHRa

MO2KHO nOffyWTb IfAu npaBeAeHUeM npO6neMbI CO~TBeHHOrO 3Ha’ieHnlf CTaHAapTHO~ MaTnnubr OTHOCU-

Te~bHO~,UA~nOC~eAOBaTe~~bHbiMunpeo6pa3oaaH~~~KK CuCTeMaM HeOAHOpOAHbtX AKKetiHbtX ypaBHeHnk.

M3 ItCCJIeAOBaHnti KOMnbZOTenOM nOATBepmAeHa nnaKTu%CK~ OCymeCTBJIaeMOCTb MeTOAa, a TaKHCe

B03MOIKHOCTb nOny’ieHna BbICOKOTO’IHbIX pe3yAbTaTOB. npeACTaBJTeHb1 pemeHAK AJIa TpemnH H HaAne308,

0rpaHawiBaromsxcn y nJlOCKOCTu unu KOHWeCKOir nOBepXHOCTn, H AJlR TpeumH, orpaserm3atoqcrxcs KOCH~

y nOAynpOCTpaHCTBeHHOfi nOBepXHOCTH. ho 3THX CflyVaRX h eCTb peaJIbHOe, a OCO6eHHOCTb BcerAa 6onee cna6oe (ii >p), YeM 3Ha'feHHe et? Ha OCO6eHHOCTHOR AnHUH, BOSMOXHO u riCYe3aHAe ee. (;\ > 1). KpoMe Tot-o,

nnOaHaJm3HpOBaHbi ynpyrne HanpSKeHns nOA KflnH-06pa?HOM, W(eCTKUM, CKOJlb3amiiM mTaMnOM, $iJIA a0

yrAe Knaa T~mnHbI, A rap~oH~~~Kae ~yHL@iS TPeXrnaHHbrX n~paM~~aNbHblX HaArEZSOS. 3AeCb yCTaE,Oa-

AeHO VT0 h<p. nOny’IeH0 u nnoCTOe aH~PTRYeCKOe pemeHne Ana OAHOrO KRaCCa YaCTHblX CAyqaeB,

nnnMeHeHHOe AAs ITpOBepKn HeKOTOpbtX %iCAeHHbIX pe3yAbTaTOB.


so dass u - (pr)' nabe dem Strahl fiir jedes endliche r. U ist eine unbekannte Funktion von 6, '1', fOr die eine partielle Differentialgleichung mit Grenzbedingungen, besonders die an den Singularitatsstrahlen. und ein Variationsprinzip abgeleitet werden. Weil Gradient U nicht-singular ist, ist eine numerische Losung moglich, verwendend, zum Beispiel, die den Netzverfahren oder die Methode der endlichen Elementen. Dies reduziert das Problem auf das Finden von A des kleinsten reellen Teiles, der die Gleichung Det (A) "" 0 befriedigt, wo Au eine grosse Matrize ist, deren Koeffizienten linear von /.L "" A(A + 1) abhlingen. 1m AIlgemeinen sind A und Ai) komplex.

Losungen konnen entweder durch Reduktion auf ein gewohnlicher Matrizeneigenwertproblem fur /.L erhalten werden, oder durch aufeinanderfolgende Umwandlungen auf nicht-homogene lineare Gleichungssysteme. Untersuchungen an Rechenanlagen bestatigten die Durchfiihrbarkeit der Methode und zeigten. dass sehr genaue Resultate erhalten werden konnen. Es werden Losungen fUr Risse und Kerben vorgelegt, die an einer Ebene oder Kegeloberflache enden, und fiir Risse, die quer an einer Halbraumoberflliche enden. In diesen Flillen ist A reell und die Singularitlit ist immer schwlicher (A > p) als an der die Singularitatslinie und kann sogar verschwinden (A > 1). Weiter wurden elastische Spannungen unter einem keilformigen starren gJeitenden Stempel oder an der Ecke einer Risskante analysiert, wie auch harmonische Funktionen an dreiseitigen pyramidischen Kerben. Es wurde gefunden, dass A > P hier vorkommt. Es wurde auch eine einfache analytische Losung fiir eine Klasse von Spezialfallen gefunden und wurde verwendet, urn einige der numerischen Resultate zu priifen.

Sommario--Si assume che la funzione armonica u, vicino al punto 0 da cui emana un raggio con singolariHt singola, sia dorninata dal termine r'pPU dove r e la distanza daI punto 0, p e una costante nota e p e una funzione scelta delle coordinate sferiche angolari 6 e 'I' tale che u - (pr)P vicino al raggio per ogni valore finito di r. U e una funzione non nota di 6 e 'I' per la quale vengono derivati un'equazione a differenziali parziali con condizioni allo strato limite, special mente quelle sui raggi con singolarita, ed un principio variazionale. PoichC grad U non e singolare, una soluzione numerica e possibile. usando per esempio iI metodo delle differenze finite 0 quello degli elementi finiti. Cio riduce iI problema a trovare il val ore di A della piil piccola parte reale che soddisfa l'equazione Det (Ai) = 0 dove A) e una grande matrice i cui coefficienti dipendono Iinearmente da /.L = A(A + l}. In generale, A et Aij sono numeri complessi. Le soluzioni possono venir ottenute 0 per riduzione al problema del valore speciale di una matrice normale per /.L, 0 per conversioni successive a sistemi di equazioni lineari non omogenee. Degli studi effettuati su calcolatori hanno confermato la praticabilita di questo metodo e hanno dimostrato che si possono ottenere risultati molto accurati. Vengono presentate soluzioni per fenditure e intagli terminanti su una superficie piana 0 conica, e per fenditure terminanti obliquamente sulla superficie di un semispazio. In questi casi A e rea Ie e la singolarita e sempre piil debole (A > p) che sulla linea di singolarita e puo perfino scomparire (A > n. Sono state analizzate inoltre Ie sollecitazioni elastiche sotto uno stampo scorrevole rigido a forma di cuneo 0 su un angolo del bordo della fenditura, come pure Ie funzioni armoniche su intagli pirarnidaIi a tre facce. E stata inoltre trovata una semplice soluzione analitica per una classe di casi speciali, e la soluzione e stata usata per controllare alcuni dei risultati numerici.

A6CTpaKT flPHHlITO, 'ITO rapMOHH'IeCKali <PYHKUHlI U B6J1H3 TO'lKH O. OT KOToporo IBJlY'IaeTCli OAHH oc06eHHoCTHblH JlY'I, rroABeplKeHa K rocnoACTBY '1J1eHa r'pP U, rAe r paccTollHHe OT TO'lKH 0, P = H3-BecTHoe nocTollHHoe, p = Bbl60p <PYHKlJ,HH OT yrJlOBblX c<pepH'IecKHx KoopAHHaT fJ. </> TaK, 'ITO U - (pr)P B6J1H3 JlY'Ia .QJJ1I 1II060ro KOHe'lHOrO 3Ha'leHHlI r. npH 3TOM U ecTb HeH3BecTHall <PYHKI{HlI OT fJ, </>, AJllI KOToporo L\aH BblBOA '1aCTHOro AH<p<l>epeHI{HMbHOro ypaBHeHHlI c rpaHH'lHblMH YCJlOBHlIMH, B '1aCTHOCTH, AJllI Tex Ha oc06eHHocTHbiX Jly'lax. a TaKlKe BblBO,llHTCH BapHaUHOHHblH npHHlJ,Hn. TaK KaK grad U eCTb Heoc06eHHall, B03MOlKHO nOJlY'IHTb '1HCJleHHOe peweHHe, npHMeHlIli HanpHMep MeTOAbl KOHe'lHbIX pa3L\H'IHIl: HJlH KOHe'lHbIX 3J1eMeHTOB. nOJTOMY np06J1eMa CBOL\HTCli K HaXOlK,lIHeHlO ,\ ,lIfllI HaHMeHbweil: peMbHoH '1aCTH, YAoBJleTBOpalOlL\eH ypaBHeHHeM Det (Ai) = 0, rJ\e Ai) - 60JlbWali MaTpHl{a. KOJ<p<pHueHTbl KOTOPOil: 3aBHCliT JlHHeHHO OT ft = ,\(,\ + J). Bo06we ,\ HA,} - KOMnJleKCHbIMH. PeweHHlI MOlKHO rrOJlY'IlITb HJlH rrpHBeL\eHHeM rrp06J1eMbl co6cTBeHHoro 3Ha'leHHlI CTaH,lIapTHoil: MaTpHlJ,hl OTHOCHTeJlhHO ft, HJlH nOCJleAOBaTeJlhHhIMH npeo6pa30BaHHlIMHK CHCTeMaM HeO,llHOpoJ\HhIX JlHHeHHhlx ypaBHeHHiI:. 113 HCCJleL\OBaHHIl: KOMIThlOTepOM rrOATSCplKL\eHa rrpaKTH'IecKlllI OCYlL\eCTBJlHeMOCTb MeTO));a, a TaKlKe B03MOlKHOCTh rrOJlY'leHHlI BblCOKOTO'lHhIX pe3YJlbTaTOB. flpe));CTaBJleHhl peweHHH AJllI TpelL\HH H HaApe'30B, OrpaHH'IHBalOlL\HXCli y nJlOCKOCTH HJlH KOHH'IeCKoil: nOBepXHOCTH, H AJllI TpelL\HH, OrpaHH'IHBalOlL\HXCH KOCHO y nOJlynpocTpaHCTBeHHOH nOBepXHOCTH. Bo JTHX cJlY'IaliX ,\ eCTh peMhHoe, a oc06eHHOCTh BCer,lla 60Jlee cJla60e (,\ > p), '1eM 3Ha'leHHe ee Ha oc06eHHocTHoil: flHHHH, B03MOlKHO H HC'Ie3aHHe ee (,\ > 1). KpoMe TOro, npOaHaJIlf3HpOBaHhi ynpyrHe HanplllKeHHH rro)); KJlHH-06pa3HoM, lKeCTKHM, CKOJlh311lL\HM WTaMnOM, HJlH BO yrJle Kpall TpelL\HHbl. H rapMOHH'IecKHe <PYHI{HH TpexrpaHHhlX !lHpaMH));aJlhHhIX HaApe30B. 3));eCb YCTaHOBJleHO 'ITO ,\ < p. flOJlY'IeHO H npOCToe aHaJIHTH'IecKoe peweHHe llJlH O,llHOrO KJlaCCa qaCTHblX CJJyqaeB, rrpHMeHeHHoe J\JlH npOBepKH HeKOTopblX '1HCJleHHbIX pe3YJlbTaTOB.

three-dimensional harmonic functions near three...

Documents