The Wedge Paradox and a Correspondence Principle

The Wedge Paradox and a Correspondence PrincipleAuthor(s): Xanthippi Markenscoff and Michael PaukshtoSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 454, No. 1968 (Jan.8, 1998), pp. 147-154Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/53226 .

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The wedge paradox and a correspondence principle

BY XANTHIPPI MARKENSCOFF1 AND MICHAEL PAUKSHTO2 1

University of California, San Diego, Department of Applied Mechanics Engineering Sciences, La Jolla, CA 92093-0411, USA

2Saint-Petersburg State University, Institute of Mathematics and Mechanics, 198328, Box 125, St Petersburg, Russia

Received 4 February 1997; accepted 28 May 1997

The wedge paradox for stress-free boundaries is the wedge loaded by a concentrated moment at the vertex (Carothers problem). The paradox consists of the fact the for wedges smaller than a half-space the solution exists, while for wedges bigger than a half-space as shown by Sternberg-Koiter the solution does not exist. We examine the possiblity of a paradox for clamped-clamped boundary conditions. We obtain the solution for a clamped-clamped wedge loaded by a concentrated couple at an interior point and show that for wedges smaller than a half-space the limit is zero as the moment approaches the vertex (i.e. the clamped wedge absorbs the concentrated couple), for a half-space it is a constant, and for angles bigger than a half-space the limit does not exist (paradox). Moreover, from the solution for the clamped wedge, taking the limit as Poisson's ratio tends to one, the solution for the stress-free wedge is retrieved (Carothers) according to the correspondence between cavities and rigid inclusions established by Dundurs (1989) and Markenscoff (1993).

Keywords: wedge paradox; concentrated couple; inclusion/cavity; clamped boundaries

1. Introduction

The wedge paradox is the stress-free wedge loaded by a concentrated couple at the vertex. The solution to this problem was first given by Carothers (1912), but it had the disconcerting feature that for a wedge angle equal to a critical value the solution diverges at all points of the wedge. This prompted Sternberg & Koiter (1958) and Barenblatt (1979) to treat the concentrated couple as a distributed loading on the flanks of the wedge, and take the limit as the area of distribution shrinks to the vertex. The result is that for wedge angles smaller than a half-space the limit exists; for wedge angles bigger than a half-space, but smaller than a critical angle, the limit exists only for antisymmetric loadings; and for bigger than the critical angle does not exist for any loadings. Dundurs & Markenscoff (1989) looked at the concentrated couple applied at an interior point and approaching the vertex in the limit, and also reached the same conclusions. This was elaborated by Markenscoff (1994). In this paper we follow the approach of Dundurs & Markenscoff (1989) and solve by the Mellin transform the problem of a concentrated couple at an interior point of a wedge with clamped-clamped boundaries. We find that for angles smaller than a half-space the limit of the solution as the concentrated couple approaches the vertex vanishes (that is the wedge absorbs the concentrated couple), is finite for wedges equal to a half-space and diverges for angles bigger than a half-space.

Proc. R. Soc. Lond. A (1998) 454, 147-154 ? 1998 The Royal Society Printed in Great Britain 147 TX Paper



X. Markenscoff and M. Paukshto

Applying the principle of correspondence between cavities and rigid inclusions (Dundurs 1989; Markenscoff 1993; Markenscoff & Paukshto 1995) stating that the stress field of cavities may be obtained from the one of rigid inclusions by taking the limit as the Poisson's ratio v -- 1, we show the Carothers solution of the stress-free wedge is obtained from the clamped-clamped one (rigid inclusion problem) in the limit as v -+ 1.

(a) Solution of the clamped-clamped wedge loaded by a concentrated moment M at an interior point along the bisector

In order to construct the solution for the displacement field, u, of the plane problem of a clamped-clamped wedge of angle 2a loaded by a concentrated moment M at an interior point (a, 0) on the bisector (figure 1) we superpose the solutions of problems I and II as follows: problem I is the solution to a concentrated moment in an infinite two-dimensional space; problem II clears the flanks of the wedge of the dislacements produced by problem I.

Problem I. A concentrated moment at the origin of an infinite two-dimensional space produces displacements u0 = (uro, uo),

2poUr, = 0, 2uo- = M/(27rr), (1.1) which, if the moment is applied at (a, 0), are

Ma sin M(r - a cos ) 2Uro = 27r(r2 - 2ar cos 0 + a2)' 2u0

= 27r(r2 - 2ar cos 0 + a2) (1.2)

Problem II. The displacement field ul = (uL, Uo1) must satisfy the equations of equilibrium and boundary conditions

Url(r, ?c) = -UrO(r, ?a), u1, (r, ?ao) = -uoe(r, ca). (1.3) The total displacement field,

u - Ur(r, ) = uro(r, ) + Ur, (r, 0), v - uo(r, 0) = uo, (r, 0) + uo, (r, 0), (1.4) is the solution of a wedge loaded by a couple at (a, 0) and being clamped on the boundaries. In order to solve problem II we express the displacement in terms of Boussinesq-Papkovitch potentials l1 (r, 0), 02(r, 0), so that

2pu1 -= I() cos 0 + 02 sin ) - r cos - + sin Or '

(os00v " .r Oq$2(1.5) 2vl = -^(q52 cos 0 - i sin 0) - r cos 0 - + sin 0- ,

where we use the simplified notation ul - ur, vl = uo, and denote by v the Poisson's ratio, and by K = 3 - 4v for plane strain or k = (3 - v)/(1 + v) for plane stress, the Kolosov constant.

We next write 01 and (2 in terms of their Mellin transforms B1 and B2 as follows,

01 = I- Bi(p)sinpOr-Pdp, 02 = 2 I B2(p) cospr-P dp, (1.6) 2- J7ri i J2i

and take the Mellin transforms of equations (1.5) and (1.3). Substituting (1.6) into the Mellin transformed equations we obtain a system of two algebraic equations for B1 and B2, that we solve to find

B 4paP-l1r[E cos2 po + p sin2 a] 4B_ paP-17r[K sin2 pa + p cos2 a] (1 ( + p) ( sin 2pa + p sin 2a) ' 2 ( + p) (n sin 2pa + p sin 2a)

(1.7)

Proc. R. Soc. Lond. A (1998)

148




From (1.5), (1.6) and (1.7) the solution for u is

-M f (a\P 7r cosp7rsin(p -1) dp 47r/~U =-- . I 2

27ri JL \r sin pr a

M a /a\P [ A ] dp 27ri JL r) ( sin(2p) + p sin2aJ ap (

_47 vl M i /a P 7r cosp7rcos((p - 1)0) dp

27i JL r sin(p7r) a

M+ j B aG\ dp 27ri J sin(2pca) + p sin 2a -r) a

where

A = 7r(Q cos2 (pa) + p2 sin2 a) sin(p0) cos 0 + (, sin2 (pa) + p cos2 a) cos(p0) sin 0, B = 27r(r sin2 (pa) - p sin2 (a)) cos(pO) cos 0 - (, cos2 (pa) - p cos2 c) sin(p0) sin 0.

The contour integrals in (1.8) are evaluated by the residue theorem after investigation of the number of roots in the strip 0 < Rep < 1 as a function of the parameter 2ca, the angle of the wedge. Subsequently we obtain the limiting behavior of the solution as the moment approaches the vertex, i.e. as a -- 0. We distinguish the following three cases.

Case 1. For 0 < 2a < r, i.e. wedges smaller than a half-space and real material with -1 < v < or I < < 1. The first integrals in (1.8 a) and (1.8b) are independent of ca, have a pole at p = 1, and are respectively 0 and -uo, that is the solution of a concentrated couple of opposite sign in an infinite plane. The second integrals give no contributions, since for sin pa/2pa being a monotonically decreasing function in the interval 0 < 2pa < (r, all roots of the equation

sin(2pa) + (p/K) sin 2a = 0, Rep > 0, (1.9) have real part greater than one. Thus a-1 - 0 as a -- 0 and hence the solution tends to zero as the couple approaches the vertex.

Case 2. Wedge angle 2a = 27T, i.e. the wedge is a half-space. There is one root with Rep = 1, so that (1.8) yield

M 2(sin 0 cos0 + ? cos 0 sin 0) M M2(Q - 1) cos 0 cos0 uI=- =- + - + . (1.10) r 47r/ ' 47r/r 47r/Kr

From (1.4) and (1.10) the solution for the total displacement field is

u = sin20, v = M ) cos2 0, (1.11) 47r/nKr 27r/Kr

in agreement with Fukui et al. (1967) obtained from the Airy stress function

-M (i - 1) 1

U = -0 - -sin20. 27r K K

Case 3. Wedge angle tr < 2a < 27r, i.e. wedge bigger than a half-space. In this case there is one root p of (1.9) with Rep < 1, so that aP1 -- oo as a -0 0, and

lim u = lim ul -- oo, lim v = lim v1 -- oo. (1.12) a--O a-+O a--O a--O

Thus the solution diverges as the couple approaches the vertex.


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(b) Correspondence between the clamped-clamped and free-free wedge A correspondence between cavities and rigid inclusions has been established (Dun-

durs 1989; Markenscoff 1993; Markenscoff & Paukshto 1995), in that the stress field of a cavity problem may be obtained from the solution of the corresponding rigid inclusion problem by setting in the rigid inclusion solution r -> -1 (v -> 1). (The cavity is stress free if the relative rotation of the inclusion with respect to the material vanishes.) Moreover, Markenscoff & Paukshto (1996) showed that every solution of the equations of elasticity for K = -1 satisfying u = 0 on part of the boundary

Q2* also satisfies traction-free boundary conditions on 9Q2*. Based on the above, we verify in this section that evaluating the stress derived from the integrals (1.8) for n -* -1, the Carothers-Sternberg-Koiter solution of the stress-free wedge is obtained. The limit as , -+ -1 has to be taken before the limit of the concentrated couple approaching the vertex, i.e. a -> 0. The analysis is as follows.

We distinguish three cases for the wedge angle 2a: (1) 0 < 2a < 2ao where ao is the critical angle of the Carothers problem satisfying tan2ao = 2ao; (2) 2a = 2ao; (3) 2ao < 2a < 27; and we seek the limiting behaviour of the stress as the concentrated couple M approaches the vertex, i.e. as a -> 0.

Case 1. The expressions for the displacement (1.8 a) and (1.8b) evaluated in the limit K -+ -1 and a -> 0. We note that while this limit is not the displacement field of the traction problem, the stress of the traction problem can be obtained from these by differentiation. The first integrals of equations (1.8 a) and (1.8 b) give contributions independent of n, which were previously evaluated. Taking the limit of the second integrals in (1.8 a) and (1.8 b) we have:

_ M f A(-l,p,a)sinpO cos ? + B(-1,p,a) cos0sin0 (ayP dp lim ul - --, ? -~-1 47r/ti - sin 2pa + p sin 2cv r a

where A and B are defined from (1.8). In order to evaluate (1.13) we compute the contribution of the residue at the pole p = 1, noting that

p- i 1 lim 1(1.14) p-1 - sin 2pa + p sin 2a -(cos 2a)2a + sin 2a'

and obtain

lim ul M [ (sin2 a - cos2 a) sin 0 cos 0 + (cos2 a - sin2 a) cos 0 sin 0] li--l1 - 2r [L sin 2a - 2a cos 2a

M cos 2a sin 2a ? 0 = 0. (1.15)

4/tr sin 2a - 2a cos 2a

Similarly for vl, from the second integral in (1.8 b)

li MV f C(-l,p,a) cospOcos0 -D(-l,p,a)sinpOsin0 (a\ dp (1.16 -I 47r4i J - sin 2pa + p sin 2a r a

where C and D are defined accordingly. Evaluating (1.16) by the residue theorem at the pole p = 1, and taking into account (1.14), we have

g/ 1 lim vI - (-2 sin2 cos2 0 + 2 cos2 a sin2 0) ^-I 2iur sin 2a - 2a cos 2a

M cos2 a sin2 0 - sina cos2 0 (1.17) 2,ur sin 2a - 2a cos 2a


150



The wedge paradox and a correspondence principle 151

Computing the stresses corresponding to the displacements (1.15) and (1.17), we obtain

_ u 1 9v \

M cos2 a2 sin 0 cos 0 + sin2 a2 cos 0 sin 0 r r r 90 ) r sin 2a - 2a cos 2a

2M sin 20 r2 sin 2a - 2a cos 2a'

( v v M [cos2 a sin2 0- sin2 a cos2 0 + cos2 a sin2 0 - sin2 a cos2 01 r

Or r y r2 sin 2a - 2a cos2 J 2M [cos2 a(1 - cos2 0) - cos2 0(1 - cos2 a)] r2 sin 2a - 2a cos 2a

2M cos2 a- cos2 0 M cos 2a -cos2 0

r2 sin 2a - 2a cos2 a r2 sin2 a - 2a cos2 a ' The expressions (1.18) and (1.19) coincide with Carothers (1912) solution for a stress- free wedge loaded by a concentrated couple at the vertex.

Case 2. In this case the wedge angle is equal to the critical Carothers angle satisfying

tan 2ao = 2ao. (1.20) We consider the roots of the denominator of (1.8 a), (1.8 b), i.e. the roots of

sinp(2ao) = 2p sin 2ao, (1.21) + 2 where

A+a _ 1 2

, 1 - 2v , -1' for values of w near -1.

There are two roots of (1.21) near w = -1:

P1,2 1? asW-, -1 . (1.22) pl,2 = 1 1 (W + 1) + O(|W + 1|), as -1+. (1.22)

It can be verified that for p1,2 given by (1.22), the expansion near w = -1 of the left-hand side of (1.21)

sinp(2ao) sin2ao? (cos2ao)2 /(w + 1)-sin2ao- (w + 1)(2o)2 +.. (1.23)

and of the right-hand side of (1.21)

- 2p sin2ao 0 -w[1 - (w + 1) + . ]p sin 2ao w+2

~[1-( + 1) + *. *] [1f ? / ^ (w + 1)] sin2ao0 .. + .

sin 2ao -- i ( + 1) sin 2a - (w - 1) sin 2ao + , (1.24)

satisfy (1.21). We evaluate the stress

rr = -2At (- L + rJ (1 (1.25)





by using

1 [ A(K, p, a) sinpOcos 0 + B(, p, a)cos pO a P dp 2,1u 27ri K sin 2pa + p sin 2a r a

1 _ f C(K, C p, a) cos pO- D(sK, p, a) sinpO sin (a \P dp 1

2/u27ri JL sin 2pa + p sin 2a r a

and obtain

M 1 A a P dp rr 2u 2u2r 27ri JL K sin 2pc + p sin 2a r ( a

M j [A](p-pI)(p-pP2) 1 ( 11 aPdp

r 27ri JL K sin 2poa + p sin 2ac pi - P2 P -P1 - P rp a

M B + C a\Pi 1

r I [sin 2pa + p sin2a]p J r) a

M D+E 1 (a\P2 1

r I [ sin 2pa + p sin 2ca]P2 J r a

M 1 . -- {[C(1) - A(1) + D(1)] sin cos + [C(1) - B(1) + D(1)] cos 0 sin O} r a

[ () (1/o)/ (+l)/2 (al-(l/a)//(?+l)/2 1) x - (4a(sin2a) /(w + -1) +

(1.26)

where

A = (Cp - A + D) sinpO cos 0 + (C - B + Dp) cospO sin 0,

B = [C(pi)P - A(pi) + D(pl)] sinpl0 cos 0,

C = [C(p) - B(pl) + D(pi)pl] cospl sin0,

D = [C(p2)p2 - A(p2)] sinp20 cos0,

S = [C(p2) - B(p2) + D(p2) + D(p2)p2] cosp20sin0,

and where we have used for small (w + 1) the approximation

[sin 2po - p sin 2a], = - (cos 2pcz)2o + sin 2ac

2 a 'Vcos 2a I1+-A/(w+l)) + sin2ai

-(cos 2a)2a + 2a(sin 2)2a/2 (w + 1) + sin 2a

4a (sin 2a) (w + 1), (1.27) 2

and

[es; sin 2poe + p sin 2o]^o 2aow cos 2oI ( 1 - w -/ (c + 1) ] + sin 2o [; sin 2pca + p sin 2ca] 2 +2cos2o? 1si-

4a(sin 2al ) /(w + (1.28)


152




In view of the fact that

lim - +/((~+i)/2)-0 aV/|(w + i) a)r/

= lim in () exp[(1/c)v/(w + 1)/21n(a/r)] In (a), (1.29) /((w+1)/2)_0 r

and that

C(1) = -2 sin2 a, D(1) = -2 cos2 a,

A(1) =- cos2 oa + sin2 a, B(1) = - sin2 a + cos2 a, in the limit as w -- -1, the stress arr given by (1.28) assumes the expression

32Ma2sin2a ( a\ rr = - - 2 ln ( cos 0 sin 0. (1.30) r2 r

2. Conclusion

Having clamped-clamped boundary conditions and being loaded by a concentrated couple at the vertex, the wedge has zero stress if the wedge angle is smaller than a half-space (convex), finite stresses if the wedge is a half-space and infinite stress if the wedge is non-convex. By a correspondence principle, from the clamped-clamped wedge the Carothers-Sternberg-Koiter solution for the free-free wedge is obtained. It is interesting to observe that although for wedges smaller than a half-space, the stresses of the clamped-clamped wedge vanish as M approaches the vertex, however, if the limit as Poisson's ratio v -* 1 is taken before the limit of the concentrated moment M approaches the vertex from the interior, then the (finite) Carothers stresses are recovered from the clamped-clamped wedge.

The obtained result for the clamped-clamped wedge also answers the question raised by Dundurs (1992, private communication). 'The crack absorbs the dislocation, the anti-crack (rigid line inclusion) absorbs the force, what absorbs the concentrated couple?' By 'absorbs' we mean that the singular field of the dislocation/concentrated force disappears as the dislocation/concentrated force approaches the tip of the

crack/anti-crack. The answer is that the clamped-clamped convex wedge absorbs the concentrated couple, since the limit of the stress field is zero as the couple approaches the vertex, and indeed the clamped boundaries may be viewed as two anticracks (each equivalent to distributed line loads) that can produce a couple. We thank Professor Robert V. Goldstein who asked whether the correspondence principle applies to the wedge.

References

Barenblatt, G. I. 1979 Similarity, self-similarity, and intermediate asymptotics. New York: Con- sultants Bureau. (Translated from Russian by N. Stein.)

Carothers, S. D. 1912 Plane strain in a wedge. Proc. R. Soc. Edinb. 23, 292.

Dundurs, J. 1989 Cavities vis-a-vis rigid inclusions and some related results in plane elasticity. J. Appl. Mech. ASME 56, 786-790.

Dundurs, J. & Markenscoff, X. 1989 The Sternbergkoiter conclusions and other anomalies of the concentrated couple. J. Appl. Mech. ASME 56, 240-245.


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Fukui, K., Dundurs, J. & Fukui, T. 1967 The elastic plane with a circular insert loaded by a concentrated moment. Bull. JSME 10, 9-14. (Japanese translation in Trans. J. Soc. Mech. Eng. 93, 1-10.)

Markenscoff, X. 1993 On the Dundurs correspondence between cavities and rigid inclusions. J. Appl. Mech. ASME 60, 260-264.

Markenscoff, X. 1994 Some remarks on the wedge paradox and saint venant's principle. J. Appl. Mech. ASME 61, 519-523.

Markenscoff, X. & Paukshto, M. 1995 The correspondence between cavities and rigid inclusions in three-dimensional elasticity and the cosserat spectrum. Int. J. Sol. Str. 32, 431-438.

Sternberg, E. & Koiter, W. T. 1958 The wedge under a concentrated couple: a paradox in the two-dimensional theory of elasticity. J. Appl. Mech. ASME 25, 575-581.


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The Wedge Paradox and a Correspondence Principle