E L S E V I E R

F.nfineertng Analysis with Boundary F./emoW IS (1996) 327-331 © 1997 F_Jaev~ S¢imce Ltd

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Research Note

On the corner tensor in three-dimerAonal Hnear elasticity

S u b r a t a M u k h e r j e e

Department of Theoretical and Applied Mechanics, 220 Kimball Hall, Cornell University, Ithaca, New York 14853, USA

(Received 5 September 1996; revised version received 20 October 1996; accepted 6 November 1996)

The subject of this paper is the comer tensor C that appears in the free term in a boundary integral equation formulation for three-dimensional linear elasticity. A general corner, locally composed of piecewise flat and curved surfaces, is comidered in explicit fashion. The solid angle at the corner appears in the expression for C. A new formula for the solid angle at a general corner, in terms of line integrals, is derived in this paper. Finally, examples for cones are presented and discus,~l. 1997 © Elsevier Science Ltd

Key words: Corner tensor, boundary integral equations, linear elasticity, sofid angle.

1 INTRODUCTION

MantiS, 1 in an elegant paper, gives general formulae for the corner tensor C (he calls it the C-matrix) that appears in the boundary integral equations (BIE) of linear elasticity. 2'3 Manti~'s formula (4.5) for three dimensions (3-D) requires the evaluation of an integral, while the corresponding formula (4.11) for two dimensions (2-D) is in closed analytic form. Also, these formulae involve the internal solid angle (for 3-D) or the usual internal angle (for 2-D) at the corner. Only 3-D elasticity problems are of concern in this short paper.

Manti~ 1 then works out a 3-D example in which, locally, the surface S of a body near a corner is a pyramid, i.e. a union of n (n > 2) fiat planes. In this case, using the expression for the solid angle for a spherical polygon [see Berger 4 and Manti~ 1 eqns (5.3), (5-4)]. Manti~ derives a closed-form analytic formula (5.7) for the tensor C which is of purely algebraic form (i.e. no integrals are involved). The integrals can be eliminated because, in this case, the outward normals n to the fiat planes at the comer are piecewise constant. While one can assume the surface S locally near a corner to be a ruled surface, it need not be a union of fiat planes. The simplest example of this situation is the local region near the vertex of a cone which has no flat plane at all! Such a vertex is called an isolated boundary point. 5 In general,

327

it is assumed in this work that the local region near a corner on a surface is composed of a union of piecewise fiat and curved surfaces. Thus, it is quite possible for n ~ to change continuously, as one travels on a contour ~(x) (see Fig. 1 in MantiS').

It is important to state clearly that Manti~'s Fig. I and eqn (4.5) does include the general case. He does not, however, present any further discussion of the general case in his paper. For example, he only presents the solid angle formulae (5-3), (5.4) for the pyramid case (i.e. for a spherical polygon). The purpose of this short paper is to provide a complement to Manti~'s work by including the general case in explicit line integral form. A new formula for the solid angle, involving only contour integrals; is first derived for the general case. This formula includes contributions from discontinuities in n ~ across edges as well as from the continuous turning of this normal across curved surfaces. Some examples involving cones are considered and solid angles at the vertices of these cones are obtained from the new formula. Next, an expression for C at the vertex of a right circular cone is derived. Both the case of the usual (vertex angle < ~r) as well as the everted (vertex angle > ~r) cone are considered. It is pointed out that in 3-D problems it is very important to follow a convention in which an observer, moving on a contour, always keeps the characteristic surface IX(P) on one side

328 S. Mukherjee

(right or left). In the present paper, the characteristic surface is always kept on the left of such an observer. Please note that this is the opposite of the convention employed by MantiE. 1

2 BOUNDARY INTEGRAL EQUATION

The boundary integral equation (BIE) for linear elasticity (Rizzo, 2 see also Mukherjee 6) is the starting point of this work. This equation, with the range of indicies 1, 2, 3, is written here as:

= Is[Uij(P, Q)'ri(Q) - Tij(P, Q)ui(Q)] Cij(P) ui(P)

xdSQ (1)

The Kelvin kernels Uij and Tij are available in many references (see, for example, Mukherjee6). Since the corner tensor C/1 arises from Ti), the expression for Tij is given below:

1 rij - 8 (1 -.):

f onOr 2v)(r,in j _ r jni) ] x [{(1 - 2v)6ij + 3r,irj} + (1 - -

(2)

In the above, S is the surface of a solid body B and ui and r; are the components of the displacement and traction vectors, respectively, on S; P and Q are source and field points, respectively, on S, and r is the Euclidian distance between P and Q. Also, v is Poisson's ratio, ni are the components of the unit outward normal to S at a field point Q, 6i2 are the components of the Kronecker delta and Or~On is the normal derivative of r at the point Q. Finally, r,/ is the spatial derivative of r with respect to the field point coordinate Xei.

3 THE CHARACTERISTIC SURFACE IN 3-D

Consider a regular region B c R 3 (as defined by

%

I

i [l(P) .3f ~

HartmannS), which is occupied by a linear elastic homogeneous isotropic material (see Fig. 1). The surface of the body is S with unit outward normal n. It is assumed that the source point P ~ S can be located at the union of a finite number of Liapunov and conical surfaces. Thus P can lie at an edge, a corner or an isolated point (e.g. at the tip of a cone) but not at a peak or cusp. For simplicity of notation, it is called a corner point in this work.

A unit spherical surface g is centred at P. The set of all half-tangents to S at P cuts out from g a connected characteristic surface II(P). Please note that, in general, fl(P) may not be a subset of S. The orientation of fI(P) is defined by the unit normal N = reQ. In the rest of the paper, reQ will be denoted simply as r. Please note that this definition of the normal to the characteristic surface is opposite to that in MantiE 1 (see his Fig. 1). The present definition of N is felt to be consistent with the use of the outward unit normal n to S in the sense that if II(P) c S, then N = n. Finally, the boundary of the characteristic surface is the closed contour 7 which is oriented by the unit tangent vectors t ~. The direction of t ~ is chosen such that an observer, traveling on the positive side of fI(P), keeps the surface on his left. The positive side of II(P) is chosen to be the outside surface of the unit sphere K. The unit outward normal to 7, tangential to the surface II(P), is denoted by n ~. This is:

n ~ = t ~ x N (3)

It is well known from rigid body motion considera- tions (see, for example, Mukherjee6), that the corner tensor C in eqn (1) is:

C(P) = - Is T(P, Q)dSQ (4)

It has been proved by Hartmann s that:

C(P) = - [ T(P,Q)dSQ (5) jta (e)

(where the minus sign is a consequence of the choice of the normal N in this paper). Further, MantiE 1 has shown that:

¢(v) 1 Jr C(P)=---~-~-I 87r(1--v) r®n~d7 (6)

where ¢(P) is the internal solid angle at the boundary point P with respect to the body B. Equation (6) remains unchanged in the present notation.

4 THE SOLID ANGLE

4.1 A contour integral f ormula

Fig. 1. The characteristic surface fI(P). It is well known from solid geometry that the solid angle subtended at the source point P by an open surface

The corner tensor 329

+xl2

m

Fig° 2.

with unit normal n is:

<I,(P) = [ r ' n d S j~¢ ---~-- Q (7)

where r, as usual, is the vector from P to a point Q E and r is the length of r. When the characteristic surface is used, one gets a very simple formula:

¢(e) = Ia( )dSQ (8)

A formula for the solid angle, when the local region near P is a pyramid, is given by Manti~ Z, eqns (5.3), (5.4). In order to consider the general case, let AB, one of the piecewis¢ smooth sections of 7 in Fig. 1, be a curve along which the normal n ~ changes continuously. The first step is to derive a formula for the amount of turning of the unit normal n ~ between A and B.

Referring to Fig. 2, using, for simplicity of notation m - n ~, and noting that both m and m + Am are unit vectors, one has:

cos(A~b) = m. (m + Am) = 1 + m- Am

= 1 - [ A m I sin(A~b/2)

Expanding both sides of the above equation in a Taylor series about A~b = 0, one gets:

1 - (A~b)2/2 + O(A~)) 4 = 1 -- IAml

x [(A~p)/2 - O(~k~)) 3]

so that, to second order, A~b =IAm I. Further, from geometrical considerations, one can deduce that the contribution/9 to ~(P) from the continuously turning

APB, BPC, C P D are p lanes

m|" ffi Nh.l

m 0 = m , rl~ I 1 r I

Fig. 3. The contour 7, in general, is composed of segments along which the normal at remains constant and others along

which it varies continuously.

normal on segment AB of 7 is:

= sgn(m × dm.r)Idml (9)

where the signum function is defined as follows: sgn(x) =-I for x<0, sgn(x)=0 for x=0 and sgu(x) = 1 for x > O.

The general situation, where the contour 7 is the union of segments along some of which m remains constant and others along which it varies continuously, is depicted in Fig. 3. The vertices vi are numbered consecutively in a counter-clockwise sense. In Fig. 3, m7 = mi-l, m + = mi, m0 = m. and r.+! = rl. The solid angle at P is given by the formula:

~(e) = 2~r - 3j - 3c (10)

where 3j and 3c, respectively, arc the contributions to • (P) from the jumps and the continuous changes in the normal m as one moves around the contour "y. These quantities are:

3J = ~ A, -- ~ sen(m,: x m + • r,)arccos

× (m~-.m +) (II)

~c = ~'-~/~c, = E I~k sgn(m × dm.r)]dm[ (12)

where the first summation is carried out across all the corners in 7 and the second across all the segments of 7 along which m changes continuously. Please note that eqn (11) is Manti~'s eqns (5.3), (5.4) (in the present notation) while eqn (12) is new.

4 .2 E x a m p ~ for cones

4.2.1 Example 1 - - A cahe with a spherical base Figure 4(a) shows a fight circular cone intersecting a unit sphere. The base of the cone is the characteristic surface [Z(P) and 2a is its vertex angle. The sofid angle at the vertex P is to be determined.

Equation (10) is used to determine the solid angle 4~(P). In this example, m-= n ~, where n ~ is the unit outward normal to the cone at a point Q. Here

m = e~, d m = e0 cos a dO,

m x din. r - cos c~ d0

so that

sgn(m x din. r) Idml -- sgn(cos cos cdd0 = cos c~ dO Therefore, from eqn (12),

I2" 3c = cos a dO = 2~r cos a

The above expression is quite interesting. Referring to Fig. 4(a), for a ~ 0, the contour 7 shrinks to a point and the normal m spins one revolution about the z-axis. Hence, 3c --~ 21r. For a = ~r/2, the cone becomes a flat

330 S. Mukherjee

¢-,

X

x--r * i N e o ~ y=r sin~ ain0 z-"-r c o ~

e.=i cos~ c~e +j c ~ sine .k siM eo--.4 ~0 +J c~O (er ,e~ ,eo) is a lillht-handcd vector

(a)

(b) CKa. .~ ,

B C

(c)

Fig. 4. (a) Example 1; (b) example 2; (c) example 3.

8<¢t

disc. Now the normal m does not rotate but merely translates parallel to itself as it moves around 7. So, ~c=0.

Finally, from eqn (10), with/~j = 0,

~(P) = 2~r(1 - cos a) (13)

The above result can be easily verified from the surface integral formula for the solid angle, eqn (8). This result is true for the entire range 0 < ~ < ~r. For a > 7r/2 (the everted cone), the characteristic surface is larger than a hemi-sphere and sgn(cos ~) = - 1 since cos ~ < 0. Now, however, ] cos a] = - cos a and the result given in eqn (13) remains unchanged. Referring to Fig. 4(a), it is important to remember that one must always move on the contour 7 on the positive side of FI(P) while keeping II(P) on one's left.

4.2.2 Example 2 - - A quarter o f a pair o f cones In this example, shown in Fig. 4(b), the body B is one quarter of the region between two cones with vertex angles 2a and 2~5, respectively. The base ABCD is the characteristic surface. Here, the contour 7 contains segments along which n ~ remains constant, and seg- ments along which it changes continuously, as well as comers. Following the same approach as in the previous example, one finds that on the segment BC, /~c, = 0r/2) cos a. On the segment DA, on which d0 < 0,

one has

m = -e¢ , dm = -e0 cos 6 d0,

Idml = Icos6 d0l, m x d m . r = cos~ d0

Next, using eqn (12), one gets

= ~/2 sgn(cos ~ d0)l cosb d0l

= Jrf/2 cos6 d0 = -(7r/2) COS~

At each of the four comers A, B, C and D, one gets, from eqn (11), /~j, = lr/2 for i = 1 ..... 4. Finally, from eqn (10),

• (P) = 0r/2)(cos 8 - cos a) (14)

As before, the above result is true for 0 < a < ~r, 0 < 8 < ~r, with 8 < a. Also, eqn (14) can be easily verified from the surface integral formula (8).

4.2.3 Example 3 - - the region between a pair o f cones This time the body B is the entire region between the two cones in Fig. 4(b) together with the characteristic surface II(P). It is multiply connected. The appropriate slit contour 7 for this example is shown in Fig. 4(e). Now,

/~c = 2P(cos a - cos/5)

It is interesting to note that the comers A, B, C and D at

The corner tensor 331

the ends of the abutting contours each contribute ~j, = lr/2 so that ~I = 2~r. Finally, from eqn (10)

¢ (P) = 21r(cos 6 - cos a) (15)

a result that can be easily verified from eqn (8).

Example 1 of the cone with a spherical base [see Fig. Ha)] is considered again. Application of eqn (16), with CF(P) = 0 and ~(P) given by eqn (13), leads to the following expressions for the non-zero components of the comer tensor C:

$ THE C O R N E R T E N S O R AT A GENERAL C O R N E R

Equation (6) is now wr i t t en in mo reex p h c i t f o rm , at a general comer, as follows:

C(P) = ~ - - ~ I + (~(e) (16)

where cI,(P) is given by ¢qn (I0) and

(~(P) = (~F(P) + Cc(P) (17)

The quantities (~F(P) and (~c (P) are the contributions to C from segments of the contour ~/with fixed (constant) and continuously varying m, respectively. From Manti~ 1 [see also vqn (6) and Fig. 3 of the present paper], these a r e :

1 CF(P) = 8~'(1 - v) E [ ( r i + l - ri) x mi] ® m i (18)

1 L (~C = - E S l r ( l _ v) , r ® m d7 (19)

where the summations in eqn (18) are carried out over the appropriate segments of 7.

Cll = (722 = 1(1 -- COSa) sin 2 a cos a

8 ( 1 - u )

1 (1 - cos a) + sin2 a COS a C33 = ~ 4(1 - v)

(20)

(21)

ACKNOWL E DGE M E NT

The author sincerely appreciates several useful com- ments regarding this manuscript from Dr V. MantiS.

R E F E R E N C E S

1. Manti/S, V. A new formula for the C-matrix in the Somigliana identity, J. Elasticity, 1993, 33, 191-202.

2. Rizzo, F. J. An integral equation approach to boundary value problems of classical elastostatics. Quart. J. Appl. Math., 1967, 25, 83-95.

3. Cruse, T. A. Numerical solutions in three dimensional elastostatics. Int. J. Solids. Structures, 1969, $, 1259-74.

4. Berger, M. G/a~tr/e, 2nd ed. CEDIC/Nathan, Paris, 1979. 5. Hartmann, F. The Somigliaua identity on piecewise

smooth surfaces. J. Elasticity, 1981, 11, 403-23. 6. Mukherj~, S. Boundary Element Methoda in Creep and

Fracture. Elsevier Applied Science, London, 1982.