A General Solution in Bipolar Coordinates to Problem Involving Elastic Dislocations

A General Solution in Bipolar Coordinates to Problem Involving Elastic DislocationsAuthor(s): T. T. Wu and J. L. NowinskiReviewed work(s):Source: SIAM Journal on Applied Mathematics, Vol. 19, No. 1 (Jul., 1970), pp. 1-19Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2099326 .Accessed: 14/05/2012 05:03

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SIAM J. APPL. MATH. Vol. 19, No. 1, July 1970

A GENERAL SOLUTION IN BIPOLAR COORDINATES TO PROBLEMS INVOLVING ELASTIC DISLOCATIONS*

T. T. WU AND J. L. NOWINSKIt

Summary. Stress and displacement fields in bipolar coordinates are -derived and expressed in terms of a stress function, the compatibility equation becoming the governing equation of the problem. Under the assumption of single-valuedness of stress, a general form of stress function is given com- posed of the fundamental part used by Koehler and an auxiliary part which helps to satisfy the boundary conditions. The total stress function yields the desired discontinuity of displacement corresponding to the edge dislocation. A general expression for the stress field thus results which can be applied to particular problems (eccentric dislocation in a circular cylinder, dislocation in a half-space, two unlike dislocations in an infinite space) by merely adjusting the values of coordinates corresponding to the boundaries. Numerical examples are solved for all three problems and graphs are given illustrating the stress fields.

Introduction. In recent years a profusion of experimental and theoretical work has been published explaining various phenomena, in particular plastic behavior of crystalline substances, by means of the theory of dislocations. The fundamental idea is that slip or glide in a crystal takes place as a result of the movement of dislocations, the latter being certain type of line singularities in the otherwise perfect crystal. Since in the neighborhood of the dislocation lines the deformations are so severe that the elastic constitutive equations cease to be well grounded, it is a common practice to ignore or consider cut away thin cores around the lines of singularities. This procedure increases the rank of connectivity of the body, and ordinarily, following Volterra, introduces a rather stringent condition that no stress is transmitted across the cylindrical surface exposed by the removal of the core.

Such a model is also analyzed in the present paper confined to two- dimensional dislocation problems in bodies bounded by cylindrical surfaces perpendicular to the planes of deformation. In a limit case one of the cylindrical boundaries may degenerate into a plane or recede to infinity. A unified approach to this class of problems is secured by using a system of bipolar coordinates as propounded, e.g., by Jeffery [1] or Coker and Filon [2]. A general solution is obtained that can be adjusted to solution of particular problems by merely inserting the constant coordinate values associated with specific boundaries.

Three types of problems are solved in detail: a dislocation in a cylinder of eccentric circular cross section, a dislocation in a semi-infinite medium, two unlike dislocations in an infinite medium. In all three cases the medium is treated as elastic, homogeneous and isotropic and the deformation as infinitesimal.

The first problem just listed was earlier analyzed by Koehler [3] also upon using the bipolar coordinates. However, his solution gives nearly infinite stresses

* Received by the editors May 13, 1969. This work is an excerpt of the first author's Doctoral dissertation submitted to the University of Delaware. The research was supported by the National Science Foundation.

t Research Center, Uniroyal Incorporated, Wayne, New Jersey, t Department of Mechanical and Aerospace Engineering, University of Delaware, Newark,

Delaware.

2 T. T. WU AND J. L. NOWINSKI

on the inner boundary of the cylinder, which is physically unsound; it also contra- dicts the usual assumption that the stress field of a dislocation is associated with a self-equilibrated system of internal forces so that the boundaries are traction free. Shivakumar [4] solved the same problem by means of conformal mapping; while his solution could properly satisfy the boundary conditions, it is valid only for a single problem and its generalization to other problems is not straightforward. This is just what permits immediately the general form of solution derived in this paper.

A dislocation in a semi-infinite medium was treated by an approximate method using an image dislocation by Koehler [3] when determining the force attracting the dislocation to the free boundary. Unfortunately, neither the outer nor the inner boundary conditions are satisfied by solution obtained in this way since the stress field of the image dislocation fails to cancel the shear stress component. On the other hand, both normal and shear stresses exist on the inner boundary and become unbounded when the inner radius decreases. Further- more, the validity of the superposition principle when applied to the case of dislocations is questionable as explained later. Head [5] also solved the problem by a special procedure, but his solution still yields stress on the inner boundary.

Two unlike colinear dislocations in an infinite medium were analyzed among others by Koehler [3], assuming that the stress field of two dislocations existing simultaneously is equivalent to the sum of two individual stress fields. Of course, dislocations differ generally from concentrated forces and from the theoretical point of view cannot be superposed without qualification although such a procedure may sometimes give a good approximation. As regards the elastic model considered in the present paper this argument becomes even more clear. It is obvious that an infinite body with two singularities is not equivalent to a combina- tion of two infinite bodies each with a single singularity, since the former is a doubly-connected region and the latter are simply connected. As a consequence, introducing two singular surfaces, i.e., introducing two multivalued displacement fields (associated with two singularities) in an infinite body does not yield the same result as introducing one singular surface into each of the component infinite bodies (associated with one singularity). After completion of the present investigation the work of Dean and Wilson [6] was brought to our attention. Using bipolar coordinates they actually studied two unlike dislocations having the discontinuity of displacements on the segment between the dislocations. However, these authors considered the problem as a case of a single dislocation line instead of two, which is a mistaken notion since the problem is actually a problem of two unlike dislocations in an infinite medium. It is easy to see that their solution retains its validity also in the case when the discontinuity of displacements occurs outside the segment between the dislocations which is the problem considered here. Although the present solution for the stress field takes a form differing from that in [6], the numerical values are the same if the Burgers vector in the present solution is set equal to (K + 1)2T/2. Nevertheless, it should be noted again that the validity of the solution in [6] is limited to a single problem, while the present solution is fairly general.

1. General equations. For future reference and to make the paper self- contained we first briefly review the main aspects of the bipolar coordinate

ELASTIC DISLOCATIONS 3

system and the equations for the displacement and stress fields expressed in these coordinates.

As is well known, the bipolar coordinate system is defined by the trans- formation

(1.1) z = iacoth- 2'

or equivalently by

(1.2) =log - ia' z-la

where

(1.3) z=x+iy, =tc+ ifl

and ax and fi are bipolar coordinates (a= real const.) (see Fig. 1). The curves a= const. represent a family of coaxial circles with singularities (0, ?a) as

y

0

a-0~~~~~~

FIG. 1. Bipolar coordinates

limiting points; the curves /B = const. are a set of circles passing through the points (0, ?a). The system of coordinates is orthogonal. The values of the coordinate ,B vary from i to -7r on crossing the segment of the y-axis joining the singular points in the counterclockwise direction. It follows that there is a


discontinuity 27r in # when we cross 0102 and single-valued stresses and displacements should be represented by periodic functions of period 27r. The values of a vary from - oo to + oo associated respectively with (0, - a) and (0, a). The x-axis corresponds to a = 0 while the y-axis outside the segment 0102 corresponds to,B = 0.

Denote the arc elements in the directions a and ,B increasing respectively by ds, and dsp. We then easily get

(1.4) dsa ds: 1

say. Denote the Airy stress function by f = f (a, B). A longer manipulation gives

(1.5) a = -(cosh - cos a)gX - sinha -sin,8- + cosh aJf

16 a~~ ~ ~~~ ~~2 a 1+ co (1.6) - = - cos , 2-sinh x? - sin #- + cos if

(1.7) I, --(cosh ac-Jcos ,)jf a aftcofl)aaL~ where, for instance, Ca,, is the normal stress vector tangent to the curve f = const. Also

(1.8) J = cosha - cos # a

With no body forces present the Airy stress function satisfies a biharmonic equation phrasing the compatibility of the deformations in the two-dimensional case. In bipolar coordinates this becomes

(1.9) V4f = 0,

that is, a partial differential equation with constant coefficients. Equation (1.9) represents the governing equation of the problem provided Jf is considered as dependent variable instead off.

A lengthy calculation upon using Hooke's law and strain displacement relations leads to the following equations for the components of the displacement,

2puU= ; af aiP

(1. 10)

2pV= + af+ Ja

where the function P is defined by the relations,

(.1 1) ( 2 - _-1) 2

p = - 2(A + 2u) _ a2(Jf)

(1.12) JP - A + 24u ff [ad( f) a2J) Jfl dot d: + g1(a) + g2(/3) 2(0 + it)e o - o a e i j

with g, (a) and g2(ft) as integration functions to be determined through (1.1 1).


We now turn our attention to the determination of the stress function which would provide single-valued stress and strain components. Since, as mentioned earlier, there is a discontinuity in the coordinate ,B on passing through the segment connecting the singular points, guided by (1.5H1.7) we seek the solution in the alternative forms

(1.13) Jf = fj(ca) cos nfl, Jf = q.(a) sin nfl.

Substitution into the governing equation (1.9) yields

(1.14) - 2(n + d + n - 2n2 + lfn((X) = 0.

Thus for n # 0 and 1,

(1.15) gfn(a) = An cosh (n + 1)a + Bn cosh (n - 1)a

+ Cn sinh (n + l)oc + Dn sinh (n - l)o

while for n = 0,

(1.16) fo(x) = AO cosh a + Boc cosh o + Co sinh a + Docx sinh a

and for n = 1,

(1.17) f1(cL) = A1 cosh 2a + B1 + C1 sinh 2a + Dlcx.

Consequently the general solution to the governing equation takes the form as follows:

Jf = AO cosh ax + Booa cosh ct + CO sinh cx + Doca sinh a

+ (A1 cosh 20 + B1 + C1 sinh 2a + Dlx)cos f,

+ (A' cosh 2x + B' + C' sinh 2cL + D1cx) sin#

(1.18) + L [Ancosh (n + 1)oc + B cosh(n - 1)o n=2

+ Cn sinh (n + l)ax + Dn sinh (n - 1)x] cos nfl 00

+ Z [A' cosh (n + 1) + Bn cosh (n - l)oc n = 2

+ Qsinh (n + 1)x + Dn sinh (n - 1)a] sin nfl. It is now our task to determine whether the displacements corresponding

to the foregoing stress function are multi- or single-valued, and further to derive conditions for the single-valuedness of the displacements.

In view of the equations (1.10), in addition to the stress function f, an auxiliary function P should be given to evaluate the components of the displacement u and v.

In order to describe more clearly the method of determining the displacements, let us deal with a typical term of the function Jf given by (1.18), say with

(1.:19) Jf = D' osinfl.


Upon inserting (1.19) into (1.11) and (1.12) we obtain

(1.20) g7(0) - g"(/3) - g1((X) - g2(U) = _2( + 2 o)D l

Since gl(oc) and g2(/3) are arbitrary functions restricted only by the above equation we get gl(oc) = 0. Moreover, we ignore the complementary solution of (1.20) involving two arbitrary constants of integration. Thus, finally the function JP associated with Jf from (1.19) takes the form

jp=(A +2/tD (1.21) JP =

Proceeding in a similar way with other terms of (1.18) we conclude on the basis of the principle of superposition that the general expression of the function JP corresponding to (1.18) has the form,

J A + 2Ju Bo cosh a + Do sinh o)/

+ (A1 sinh 2a + C1 cosh 2a + DI/3) sin /B

- (A' sinh 2a + Cl cosh 2a - D1) cos /3 00

(1.22) + E [An sinh (n + 1)a + Bn sinh (n - 1)x n=2

+ Cn cosh (n + 1)x + Dn cosh (n - 1)oc] sin nfl 00

- Z [An sinh (n + 1)x + Bn sinh (n - l)o n=2

+ C cosh (n + 1)x + Dn cosh (n - l)] cos n/}.

We may easily figure from (1.10) that the only terms in the general expressions for JF and JP, (1.18) and (1.22), which may possibly yield multivalued displacements are

(1.23) Jf = (Bo cosh a + Do sinh a)a + (D1 cos ,B + DI sin f3Oa and

(1.24) JP = A + (Bo cosha + Do sinha + D sin + D1 cos,),.

Upon inserting the above expressions into the equations for displacements and ensuring single-valuedness of the latter by suppressing the terms associated with the coordinate /3 of period 27t, we obtain the following relations between the coefficients:

(1.25) D'(1 - cos# cosha) + Do sin , sinh ox + (Bo + D1) sin , cosh x = 0

and

(1.26) DO(1 - cos , cosha) - DI sin # sinha - (Bo + D1) sinh ox cos P = 0.


It is sufficient in order to satisfy the above equations to pose

(1.27) Do = O, D'1 = 0, Bo + D1 = 0,

so that these relations become the conditions of the single-valuedness of the displacements.

As a final preparation we turn our attention to the boundary conditions. By hypothesis the boundaries in the present case are stress free. On a free surface a = const. the shear stress vanishes, so that by (1.7), d2Jf/(a0f/) = 0. It follows that

(1.28) J= F

on the boundary a = const., where F is a constant. The vanishing of the normal stress on the same boundary leads to a second order linear ordinary differential equation with variable coefficients. Upon inspection it is easy to find the general solution of this equation in the form

(1.29) Jf = F tanh a + A sin # + Q(cosh o cos -1),

where A and Q are arbitrary constants. Both equations (1.28) and (1.29) are the conditions which the "revised" stress function should satisfy on a free boundary ci = const.

2. Stress functions associated with dislocations. All we did in the preceding section is of general nature not limited to dislocations. In this section we are concerned with three different problems which are: (a) an eccentric dislocation in a circular cylinder, (b) a dislocation in a semi-infinite medium, and (c) two unlike dislocations in an infinite medium.

Following Volterra's approach [7] we exclude a small region surrounding the dislocation line (the singularity where Hooke's law does not hold) and consider the Burgers vector as a relative rigid body displacement of two singular faces with respect to each other. Since we assume that the deformation field is generated by dislocations, all the boundaries of the body are free from external tractions. We consider the problem as a plane strain problem but, as is well known, a mapping A* = 24y/( + ,u) converts the problem into the generalized plane stress problem.

Koehler [3] has solved problem (a) also upon using bipolar coordinates; however, his solution yields an infinite stress on the inner boundary despite the fact that he uses the same model as we do here. Despite this shortcoming we may use his Airy stress function which yields no tractions on the outer surface and add another stress function to eliminate the remaining tractions.

The solution obtained by this method is fairly general and is valid, among others, for the three problems listed above. As a matter of fact, the differences between the configurations of the bodies considered reduce simply to different values of the coordinate cx representing the boundaries.

We now turn to the derivation of the general solution which is later specialized by merely adjusting the values of ac. The Koehler stress function [3] of the form

(2.1) Jf1 = D sin , [a + -e-2(`xi)],


where D is a constant, yields the stress fields au and cy5 that vanish at a = oc so that a = oca represents a load free boundary. We also note that the first term in (2.1) yields multivalued displacements, and the constant D may be determined from the condition of the discontinuity of the displacements upon crossing the singular surface.

Assume that the body is bounded by the boundaries cL = oc and cx = ocl. Our task now is to determine the auxiliary stress function Jf2 associated with single-valued displacements and eliminating external tractions on oc = xo% intro- duced by the fundamental function Jf1. To this end, guided by (1.18), we take

00

(2.2) Jf2 = {fxn(a) cos nfl + qn(Lx) sin n,B}, n=O

where

(2.3) fn(a() = An cosh (n + 1)c + Bn cosh (n - 1)a + Cn sinh (n + 1)a

+ Dnsinh (n -),

(2.4) qn(X) = A' cosh (n + 1)x + Bn cosh (n - 1)c + Cn sinh (n + 1)a

+ D sinh (n - 1)o, (2.5) f1(oc) = A1 cosh 2cc + B1 + C1 sinh 2c + D1a,

(2.6) ql(ox) = A' cosh 2a + Bl + Cl sinh 2a + Dla, (2.7) f0(ac) = AO cosh a + Bo0x cosh a + C0 sinh a + Doca sinh a.

Let us consider two typical terms Jf2 = AO cosh ac and Jf2 = B1 cos /3. Upon using (1.5}-(1.7) we find

?aa =%pp = A0/a, uof = O, (2.8) = (2.8)

~ ~ ~ 7a = p:= B1/a, Oa* = 0,

respectively, that is a constant stress in both cases. We may, therefore, pose for instance Ao = 0.

In view of the desired single-valuedness of the displacements associated with Jf2, conditions (1.27) yield (2.9) fn(cX) = An cosh (n + 1)a + Bn cosh (n - 1)C + Cn sinh (n + 1)oc

+ D sinh (n - )o, n > 2,

qn(Oc) = A' cosh (n + 1)a + Bn cosh (n - 1)a + Cn sinh (n + 1)a

+ D sinh (n - 1)x, n > 2,

(2.11) fp(cx) = A1 cosh 20c + B1 + C1 sinh 2c -Booc,

(2.12) qj(o) = A' cosh 2c + BI + Cl sinh 2a, (2.13) fo(Ox) = Booccoshca + Cosinho(.

All the parametric coefficients involved in the above expressions may be evaluated from the boundary conditions. In fact, bearing in mind that the fundamental stress function (2.1) gives no stress on the boundary cx = ocl, we insert


the auxiliary stress function (2.2) in the condition (1.28) with a = c1 and after comparing the coefficients obtain

(2.14) f'o(oc1) = Fi, fi(ai) = , fn'(?(l) = ?,

q (ox ) = 0, qn(cx1) = 0,

where F1 is an arbitrary constant and n > 2.

A similar procedure applied to (1.29) yields for the boundary oc =Lol ,

fo(cl) = F1 tanh 1 -Ql,

fi(ocl) = Qj cosh oc1,

(2.15) q1(l1) = A1,

fn((xl) = 0,

qn(c1l) = 0,

with A1 and Q1 as arbitrary constants and n > 2. Upon combining the foregoing results with (2.9) through (2.13) we finally get the system of six relations among the coefficients Bo, C0, A C An B' and Cn (n > 1). It now remains to satisfy the boundary conditions on the boundary a ocO that is eliminating the tractions generated by the fundamental stress function. This is done by using the total stress function J(f1 + f2)

(2.16) Jf, = Dsinf3[a + 2e- + E [fn(cx)cosnfl + qn(Lc)sinnf]. n=O

A similar procedure to that used for a = oc1 yields finally

(2.17) fo(xo) = FO tanh o0 -0,

(2.18) ft(oo) = Q0 cosh oco,

(2.19) q1(cx0) + D[ao + le 2(ao1)1 = Ao,

(2.20) fn(aO) = 0, n > 2,

(2.21) qn(oLO) = 0, n > 2.

Again upon combining these results with the set (2.9) through (2.13) we get a system of relations between Bo, C0, An, Bn, Cn, A', B' and C' (n > 1).

We are thus ready to determine the parametric coefficients in the representa- tions of the stress functions in terms of six arbitrary constants Fo, Qn, AO, F1, Q1, and A1.

First we turn our attention to the equations involving solely Bo, C0, A1, B1, C1, Fo, Q0, F1 and Q,. We get eight equations involving these nine unknowns so that one of the unknown quantities may be chosen arbitrarily. In other words, we may choose various forms of the auxiliary stress function which yield the same results. This gives us the possibility of restricting the auxiliary stress function to some desired particular form. An inspection of the equations suggests the choice

(2.22) Qo = 0.


This gives

(2.23) fl = 0

and in turn

(2.24) Bo = 0, Al = B1 = 0 and Cl = 0.

There remain now the relations

(2.25) C0 cosh oc = Fo and C0 sinh oc = Fo tanh Lo,

which are in fact identical. In other words, we have actually seven distinct equations for nine unknowns under the assumption (2.22). It follows that the value of one more quantity may be assumed arbitrarily. We put

(2.26) Fo = 0

which yields

(2.27) C0 = 0, F1 = 0.

Since all the quantities Bo, C0, Al, Bl, Cl, Fo, Qf, F1 and Ql vanish under assumptions (2.22) and (2.26), a considerably simpler form of the auxiliary stress function is hopefully achieved (it appears that the only equation connecting these quantities and yet unused is now satisfied identically).

A similar procedure applied for determining five unknown quantities Al, B'1, Cl, Al and Ql leads to four equations relating these quantities. Some experi- menting suggests the assumption

(2.28) Al D[1 -e-2(o -)]

2 cosh 2Lxl(tanh 2a cosh 2Lx - sinh 2Lxo)

which gives BI = 0 and all the remaining coefficients in terms of the constant D. For the coefficients An Bn C9 and Dn a system of homogeneous linear equa-

tions is obtained which has a nontrivial solution only if its principal determinant does vanish. It is easily seen that such a condition is in general problematical and consequently possible values of the coefficients are rather zeros. Again upon a similar argument we may assume that the primed nth coefficients vanish.

The above analysis indicates that the only nonvanishing parametric coefficients of the auxiliary stress function are A' and Cl; their values are

(2.29) A' - D[1 -e-2(a?-a)] 1 2(tanh 2oc, cosh 2o - sinh 2oo)'

(2.30) C'f, _ D[1 - e2(o - T)] tanh 2oc,

2(tanh 2oc, cosh 2o - sinh 2oco)

Consequently all the coefficients (2.2) through (2.7) of the trigonometric expansion of the auxiliary stress function (2.2) vanish except

(2.3 1) q1(a) = 2(tanh -

c -2. - sh ) [cosh 2A - tanh 2 sinh 2o1].


It follows that the total stress function is finally represented by a relatively simple equation

Jf, = DFo + 12()e1 (232 = D[a+ 2e ? 2(tanh 2a, cosh 2xo- sinh 2xo)

(cosh 2a - tanh 20c, sinh 2a) sin,B

3. Conditions of discontinuity of displacements. As mentioned earlier we are concerned with three problems involving edge dislocations. These are illustrated in Fig. 2 as generated by a cut along the y-axis whose faces are displaced elastically by a rigid translation b along this axis and then rejoined.

y

,=a, y _ _

V

aa- s a L

4 b7 X

B. f5.87 a-O

(c) FIG. 2. Three models


In view of (2.32) it is seen that the only term of the total stress function which may yield multivalued displacements is the term Da sin /3. Let us check whether this term can suitably represent the jump of the displacements required in the three given dislocation problems. A slight manipulation yields the following equations for the displacements associated with the term Da sin /3:

UD asinhasin/ A + p cosh oa - cos /

(1(A + 2u)D sin2 Cos

+ + -coshs o-cos

and

(3.2) 2y V uD C o sin2/ _ (A + 2u)D /3sinhosin/ A + t coshL o-cos ,B i + p coshx - cos/,

Here again u and v denote the components of the displacement in the directions normal to the curve oc and /3 constant respectively. If we now describe a closed path o = L, starting from a point P on one singular face, passing around the singularity in the clockwise sense (in a counterclockwise sense for the lower singularity in the third problem) and reaching again the point P on the other singular face, we find that the increment of the displacements across the singular surface is given by

AU =-D sin D

_ asinh x sin,

(3.3) 2y A + -

cosh -cos/

+ A+ 2)I sin 2/ \ Cs P2

A + \cosh a-cosfl Jpi

A = D Cssin2/3 X

AV4= 221X + ,) 1I(cosP cosh c - cos /3 (34)(il + 2),B sinh osin ,P2

coshL - cos/3 Here the limiting values /3, and /2 are -i and i for the first and second problems, while for the third problem they are equal to 0 and i for the first half circuit and to - i and 0 for the second half circuit. By using these values in the above equations and setting oc equal to any value in the interval oc1 < oc < Lx(o(L > 0) for the first problem, 0 < o o<o for the second problem, and -ao < oc < oX 0(oc > 0) for the third problem, we get the same expressions for the increment of the displacement across the singular surface for all three problems. They are

(K (+ )Dn

(3.5) AU ( + 2D

AV= 0,

where K was defined earlier. We assume that the right-hand singular face in the first two problems is displaced upward with respect to the left-hand one by a


rigid-body translation b along the y-axis. In the third problem, the left-hand singular faces both in the upper and lower halves of the medium are displaced downward with respect to the right-hand singular faces. All these displacements represent translation b along the y-axis. Then we have for all problems.

(3.6) Au = b.

Of course, b may take different values in different problems. Applying this condition to (3.5), we are able to determine D which is

(3.7) D = 2ub (Kc + 1)n(

From what was said one can conclude that the total stress function (2.32) exactly fits the conditions of the three problems under investigation. By using the relation (3.7) we have that the total stress function becomes

J = 2jtb !x + 1e-2(a-al) + 1 - e

(3.8) (K + 1) L 2 2(tanh 2ot cosh 2ao - sinh 2ao)

(cosh 2ax - tanh 2a1 sinh 2a)1 sin /B.

4. Stress fields associated with three problems. As indicated earlier the total stress function given by (2.32) is suitable for all three problems under investigation. It is easy to obtain general expressions for the stress fields in all these problems by merely inserting (3.8) in the formulas (1.5H1.7). This gives

21tb _ . -2ca) sin: sinh cx I - e

(K + 1)ita (4.1) [1 - e-2(ao-l)](tanh 2cx cosh 2a - sinh 2cx)

tanh 2a1 cosh 2a0 - sinh 2ao

41ib e -(2a-aj) [1 -e- 2(ao-a1)] (tanh 2cx1 sinh 2a - cosh 2a) (K + 1)ia ( tanh 2cx1 cosh 2cxo - sinh 2oxI

(4.2) *(cosha - cos/,)sin/,

2jib J1 - e (K + 1)i.a

W

[1-- 2(e-) (tanh 2a1 cosh 2a - sinh 2cx)l tanh 2c 1 cosh 2ao - sinh 2ao 5

sinh a sin 1


and

2yb (Jep= ( +A (cos -cosh a) cosf,

(4.3) K+Ia

- 2-a) _ [1E - e2(ao-al)] (tanh 2oc1 cosh 2c - sinh 2oc) tanh 2oc, cosh 2o - sinh 2o )

The above equations yield the stress fields for each particular problem by suitably adjusting the values of Lo and oc1 associated with the boundaries. Since the determination of the stress fields in the medium is our main goal, the solution of the problem is completed.

5. Numerical examples. The derivations of the last section permit us to solve a numerical example illustrating each problem. To this end, we first non- dimensionalize the equations in the preceding section by posing

(K + 1)ra _ (K + 1)ra Caa = ub ' - ,B b

(5.1) (K + 1)ra

0*jB .5aj

Cl 5 ~ Ca

7 2. 6

5

'.5 4

3

1.0 2

3O 7 F.0.53l 4ormal 4o 0

-2

-3

-4

-5

-6

-7

FIG. 3. Nondimensional normal stress J~along constant coordinate (x versus coordinate f


Furthermore, for definiteness we assume: (a) in the first problem, that is, for a dislocation eccentrically located in a

circular cylindrical body,

(5.2) ao = 3, a1 = 0.5;

(b) in the second problem, that is, for a dislocation in a semi-infinite body,

(5.3) cXO = 3, o1 = 0;

(c) in the third problem, that is, for two unlike dislocations in an infinite body,

(5.4) Lo=3, x1=-3.

Some numerical results are illustrated in Figs. 3 through 6. (a) First problem. (i) Figures 3 and 4 present the distribution of non-

dimensional normal and shear stresses along the curves ax = const. -aa becomes a tensile stress on the left part of the curves a = const., while it is a compressive stress on the right part. Similarly, on the inner boundary, U-p# (not shown) becomes

9

a-2.5

6 au2.0

a -1.5

-2

7 --y 7r'-3 i

4 4

-4

F-5

.-7

FIG. 4. Nondimensional shear stress along constant coordinate (x versus coordinate fl


a tensile stress on the left part of the circle, and a compressive stress on the right part of the circle; near the outer boundary the situation is reversed. The nondimensional shear stress 4, reaches its maximal values for ,B equal to 0 and - 7t.

Of course, both for oc = 0.5 and o = 3, that is, on the boundaries, the normal and shear stresses 6a and 6ajB vanish.

(ii) Figures 5 and 6 display the distribution of nondimensional normal stresses along the segments of the curves , = const. (from a = 0.5 to a = 3). r becomes a tensile stress along the curves ,B = -7r/2, -7r/4 and - 37r/4, but a compressive stress along the curves /3 = r/2, 7r/4 and 37t/4; it vanishes along ,B = 0, +? . Besides, 6aa has absolute maximal values around x = 2.5. Th, changes from tension to compression along the curves : = n/4, 7r/2, 37r/4 with an increasing distance from the inner boundary, but it changes in the reverse sense along the curves /3 = -7r/4, -7r/2, - 37r/4; it reaches its maxima at inner and outer boundaries. Cap (not shown) is positive along the curves /3 = +37r/4, ? 7T,

negative along /3 = 0, ? 7r/4, but vanishes along /3 = ? i/2. It reaches its maximal values around a = 2.4.

Cgaa

8

6 euj.

-2 \ R .

-4

-6 ;'

-8

FIG. 5. Nondimensional normal stress along constant coordinate ,B versus coordinate a


40

30

20~~~~~~~~~

-20

-30~~~~~~~e.--

-40

FIG. 6. Nondimensional normal stress saa along constant coordinate ,B versus coordinate a

(b) Second problem. Qualitatively, the stress distributions both for the first and the second problem are quite similar, but quantitatively, the magnitudes of the stress components are different. On the other hand, the stresses 6a, and a- vanish on the boundaries in both cases as they should.

Bearing the numerical results in mind and what was said in the Introduction, it seems that Koehler's approximate method of images gives rather accurate results in the vicinity of the edge of the dislocation. However, the stress fields at and near the edge of the half-plane as well as far from the dislocation basically differ from the rigorous solution.

(c) Third problem. The distribution of nondimensional normal and shear stresses is here similar to Figs. 3 to 6, respectively. (i) Along the curves a = const., Qxx appears to be a tension on the left part of the curves - = const. located above the x-axis, while it is a compression on the right part; an opposite situation occurs along the curves a = const., located below the x-axis. -xx vanishes on the x-axis and on the boundaries. The maxima of 6a occur at # = ? ic/2.


4# changes from tension to compression and vice versa in a manner similar to ( It reaches its maximum values around a = +? /2, but vanishes along the x-axis.

6e along the curves a = const. is positive between f = -/2 and it/2 and negative outside this segment. It has positive maximum at ft = + 7t.

(ii) Along the segments of the curves ft = const. located between the boundaries, a changes from tension to compression for f = 7t/4, 7r/2, 37r/4, with an increasing distance from the boundary a = -3; it changes in a reverse sense for f = - 7t/4, - 7r/2, - 37r/4. The absolute maximal values occur around a = + 2.4. The distribution of 6,, along curves ft = 7r/4 and 37t/4 as well as ft = 7-/4 and - 37r/4 is identical. jaa vanishes along the curves t = 0 and + 7t.

With regard to 6p this stress component changes from tension to compression with an increasing distance from a = -3 for ft = 7t/4, 7t/2, 37t/4 and vice versa for ft = - 7t/4, - 7t/2, - 37t/4 along the same segments as mentioned previously. It has absolute maximal values at both boundaries, while on t = 0 + 7r, it identically vanishes.

On ft = ? 37t/4 and + 7t, ka is positive and has two maxima around oc =-2.5 and 2.5, respectively. However, on ft = 0 and +7t/4, ?a: is negative, and has two minima at c = + 2.5. Moreover on ft = 0 it vanishes at. a = 0 as well as on the boundaries.

Acknowledgment. The authors express their gratitude for the reviewer of this paper for drawing their attention to the treatise of A. Seeger on the Theorie der Gitterfehlstellen published in Handbuch der Physik, Vol. VII, Part 1, S. Flugge, ed., Springer-Verlag, Berlin, 1955. In this work on pp. 561-562, results concerning two out of three problems solved in the present paper (eccentric edge dislocation in circular cylinder; edge dislocation in a half-infinite medium) are quoted from a thesis of H.-D. Dietze. It is worthwhile noticing that, although as Dietze's models serve dislocations with removed cores, his solutions, given in terms of Airy's stress functions, fail to have the inner boundaries of the dislocations free from stress. This can be easily checked, e.g., by evaluating the normal stress a,. for 0 = 0 and x = xa + a, y = 0 in Fig. 93, and for 0 = 0 and x = 4 + a, y = 0 in Fig. 95. Actually, the major difficulty in finding stress functions for models other than the concentric cylinders stems from the elimination of tractions on the inner boundary. This remark also concerns solutions of other authors who have studied similar problems.

Perhaps as another feature of the present method, that should be mentioned, is its generality; the latter enables one to solve the above two problems plus one extra problem at the same time (two unlike dislocations in an infinite medium) by using a unified approach in bipolar coordinates.

REFERENCES

[1] G. B. JEFFERY, Plane stress and plane strain in bipolar coordinates, Phil. Trans. Roy. Soc. London Ser. A., 221 (1920), pp. 265-293.

[2] E. G. COKER AND L. N. G. FILON, A Treatise on Photoelasticity, Cambridge University Press, Cambridge, England, 1957.

[3] J. S. KOEHLER, On the dislocation theory of plastic deformation, Phys. Rev., 60 (1941), Pp. 397-410.


[4] P. N. SHIVAKUMAR, Dislocation of isotropic cylinders of eccentric circular cross-sections, Quart. J. Mech. Appl. Math., 16 (1963), pp. 129-136.

[5] A. K. HEAD, Edge dislocations in inhomogeneous media, Proc. Phys. Soc., 66 (1953), pp. 793-801. [6] W. R. DEAN AND A. H. WILSON, A note on the theory of dislocations in metals, Proc. Camb. Phil.

Soc., 43 (1947), pp. 205---212. [7] V. VOLTERRA, Sur l'equilibre des corps Olastiques multiplement connexes, Ann. Sci. Ecolc Norm.

Sup. Ser. 3, 24 (1907), pp. 401-517.

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