Eshelby (1957) who later used them to determine the strain field in the matrix for the ellipsoidal inclusion problem (Eshelby, 1959). Also Lee and Johnson (1977) ...
J. Mech. Phys. Solids, Vol. 44, No. 12, pp. 1977-1995,
Pergamon PII: SOO22-5096(%)00066-X
1996
Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-5096/96 $15.00+0.00
ESHELBY’S INCLUSION PROBLEM FOR POLYGONS AND POLYHEDRA GREGORY
J. RODIN
Department of Aerospace Engineering and Engineering Mechanics, and Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, Austin, TX 78712, U.S.A.
(Received 26 December 1995 ; in revisedform 9 May 1996)
ABSTRACT An algorithmic closed-form solution is derived for Eshelby’s problem for polygonal and polyhedral inclusions. Illustrative calculations are presented for two- and three-dimensional problems. Also it is proven that polyhedra with constant Eshelby’s tensor do not exist. Copyright 0 1996 Elsevier Science Ltd Keywords:
A. phase transformation,
A. voids and inclusions,
1.
B. elastic material
INTRODUCTION
This paper presents an algorithmic closed-form solution to a problem of classical elasticity for an infinite homogeneous body that contains a polyhedral subdomain subject to a uniform transformation strain. If the subdomain is approximated by an infinite cylinder, the problem is formulated as a two-dimensional plane strain problem for the polygonal base. Following Eshelby (1957), the subdomain is referred to as the inclusion, the material outside the inclusion as the matrix, and the entire problem as the inclusion problem. The inclusion problem is important for two reasons. First, it can be used to model many phenomena that involve thermal expansion and structural transformations in solids (Eshelby, 1957 ; Khachaturyan, 1983). Second, the inclusion problem is closely related to the elasticity problem for an inhomogeneity imbedded in an infinite matrix (Eshelby, 1957, 1959, 1961). In general, this relation is in the form of an integral equation. For the ellipsoid, that integral equation can be solved in closed-form because the strain field induced by a uniform transformation strain inside the ellipsoid is uniform. Furthermore, Eshelby (1961) conjectured that the ellipsoid does not share this remarkable property with any other bounded domain. This author became interested in the inclusion problem after reading a controversial paper by Mura et al. (1994) who claimed that certain pentagonal star-shaped domains share the remarkable property with the ellipsoid. A major implication of that claim is that Eshelby’s (1961) conjecture is false. In contrast, all other known solutions for non-ellipsoidal inclusions do not contradict Eshelby’s (1961) conjecture. For cuboids, such solutions were obtained by Faivre (1964), Sankaran and Laird (1976), Chiu 1977
1978
G. J. RODIN
(1977), and Lee and Johnson (1977). For the circular cylinder, a complete solution was recently derived by Wu and Du (1995a,b). For two-dimensional problems, solutions can be constructed using complex variable representations (Sherman, 1959). In this paper, we present an approach to the inclusion problem that can be applied to any number of polygons or polyhedra of arbitrary shape. Further, using this approach, we prove that vertex singularities in polygons and polyhedra do not vanish except for incompressible polygonal inclusions whose angles are either 7c/2 or 3rc/2. An obvious corollary of this proof is that pentagonal star-shaped domains considered by Mura et al. (1994) do not share the remarkable property with the ellipsoid and therefore the claim of Mura et al. (1994) is false. The rest of this paper is organized as follows. In Section 2, we formulate the inclusion problem in terms of the harmonic and bi-harmonic potentials. This formulation allows us to take advantage of the algorithm proposed by Waldvogel (1979) for the harmonic potential of the polyhedron. In Section 3, we present an extended version of Waldvogel’s algorithm, which is necessary and sufficient for our purposes. In Section 4, we consider several applications of this algorithm, including a proof of Eshelby’s (196 1) conjecture for polyhedra.
2.
FORMULATION
To obtain a formal statement of the inclusion problem, we denote the inclusion by o, the transformation strain by E*, and the induced strain field at a point x by E(X). The tensors E(X) and E* are related by Eshelby’s tensor S(x) such that E(X) = S(x)&*. In the usual index notation, Cartesian components of S are (Mura, 1982)
In (1) v is Poisson’s ratio,
464 =
s s
Ix-y1
’ dy>
(2)
dy.
(3)
0,
and
$04 =
Ix-y1
Ci,
The functions 4(x) and G(x) are known as the harmonic and bi-harmonic potentials of o respectively, and they satisfy (Mura, 1982) A&X)
= kAA$(x)
=
-4n o
XEW X+3’
The importance of these potentials to the inclusion problem was recognized by Eshelby (1957) who later used them to determine the strain field in the matrix for the ellipsoidal inclusion problem (Eshelby, 1959). Also Lee and Johnson (1977) presented
1979
Inclusion problem for polygons
their solution for the cuboidal inclusion in terms of the potentials. The potentials not only provide a compact representation for S, but also allow us to take advantage of results in the related fields of applied mathematics and mechanics. Indeed, the calculations of Eshelby (1957, 1959) are deeply rooted in classical potential theory, and the analysis of Lee and Johnson (1977) parallels that of MacMillan (1930) for the harmonic potential of the cuboid. This obvious, but apparently not completely exhausted path, is also chosen in the present paper because the celestial mechanics literature contains numerous solutions for the harmonic potentials of polyhedra [see Werner (1996) for references]. Among those solutions, the algorithmic approach proposed by Waldvogel (1979) is particularly attractive because it is simple and robust.
3.
ALGORITHM
In this section, we extend Waldvogel’s algorithm to solve Eshelby’s inclusion problem. Although we restrict our presentation to the case of an isolated polygon or polyhedra, the algorithm, without any modifications, can be applied to any finite number of polygons or polyhedra of arbitrary shape. The algorithm is implemented in three stages. First, the inclusion is assigned a set of tetrahedral elements in such a way that x, the point where the solution is evaluated, is a common vertex of all the elements. Second, the potentials and their derivatives are calculated for each element in its local coordinate system. Third, the tensor S(x) is assembled from the elemental contributions. 3.1. Geometric construction In presenting the geometric part of Waldvogel’s algorithm, we deviate slightly from the original paper in order to include the two-dimensional case and use it as a building block for the three-dimensional construction. 3.1.1. Two-dimensional construction. Let o be a p-sided polygon and x is a point in the plane of cc).For each edge, we identify the outward unit normal vector p and the unit tangent vector x such that the pair (p, x) forms a right-hand basis. The construction requires us to assign p triangles to o such that the base of each triangle is an edge of o and x is the vertex opposite to the base. For such a triangle, the local coordinate system has the origin at x, basis vectors (p, x), and the corresponding coordinates (v],5) (Fig. 1). In these coordinates, the positions of the vertices at the base are represented by the pairs (b, c+) and (b, c-). These vertices are distinguished via the inequality c+ > c-. If v+ and v- are the position vectors of the vertices at the base, the parameters b, c+ and c can be calculated as (Fig. 1) h = /I *(v’ -x)
= p*(v-- -x),
c+ = x*(v+-x)
and
c- = x*(v--x).
(5)
The outlined construction applies to both interior and exterior points of any polygon. If w is convex and x E o, then the triangles tessellate o, and the area of the polygon is given by
1980
G. J. RODIN
(b)
(a) Fig. 1. Two-dimensional geometrical parameters
construction : (a) A polygon and triangles associated with the point x. (b) The of a typical triangle. The minus sign in front of cm emphasizes that in this particular geometry cm < 0.
A = i A,, i= I
where Ai designates the area of the i-th triangle [(Fig. 2(a)]. In general, if the divergence theorem is applied to the integral SC,, div,(y - x) dA,, the following relationship can be established : A = i
ib,(c:
i= 1
-cl-)
= f sign(bi)Ai. ,= I
(6)
To this end, it is expedient to observe that sign (b,) = 1 if the outward unit normal of the i-th triangle coincides with the corresponding outward unit normal of the polygon, and sign (bJ = - 1 if these vectors are opposite to each other [(Fig. 2(b)]. If bi = 0
(a)
(b)
Fig. 2. Areas of the polygon and triangles : (a) The triangles tessellate the polygon if the polygon is convex and x is an interior point. (b) General case. For the triangle, whose base is the edge marked by the vertical bar, b -c 0 : for the other triangles b > 0.
Inclusion
problem
for pdlygons
1981
there are two distinct possibilities. First, if x belongs to the boundary of o, it is required to evaluate the limits bi -+ O+ and bi -+ O- (see Section 4). Second, if x lies on the line of an edge but does not belong to the edge, the triangle corresponding to this edge is discarded. On the basis of the geometric construction, an integral over o can be evaluated as (Fig. 1) (...)dA sw
c: rll*, = i *‘dq (. . .) d[. *=I s 0 s c,-rll*,
(7)
It is easy to see that (7) is consistent with (6). Equation (7) can also be written in the form (. . .) d/j = .f *’ drl ” “*’ (...)d[-[“*‘dq[;“*‘(...)d&j. i=l [S 0 sw s0
(8)
In geometric terms, (8) implies that integration over each triangle is substituted by integration over two right triangles. We refer to the right triangles as the twodimensional simplexes, and to the parent triangles as the two-dimensional duplexes (Fig. 3). The former lead to simpler mathematical expressions and the latter to more efficient calculations.
DUPLEX
SIMPLEX
SIMPLEX
(a)
DUPLEX
SIMPLEX
SIMPLEX
(b) Fig. 3. Formation
of two simplexes from a duplex: (a) The simplexes are added (c’ > 0, c- < 0). (b) The simplexes are subtracted (c+ > 0, c- > 0).
1982
G. J. RODIN
3.1.2. Three-dimensional construction. Now w is a polyhedron and x is arbitrary. The construction involves three steps (Fig. 4). Step 1 : For each face, determine n, the projection point of x on this face. Step 2 : For each face, construct the two-dimensional simplexes and duplexes using II as the pivot. Step 3 : Construct the three-dimensional simplexes (duplexes) such that the base of each simplex (duplex) is a two-dimensional simplex (duplex) and x is the vertex opposite to the base. For a representative duplex, the local basis is the triplet (v, p, x), where v is the outward unit normal of the parent polyhedron, and the remaining vectors are chosen according to the prescription for the two-dimensional duplex. The coordinates corresponding to the triplet (v,~, x) are (t,q, [), respectively, and the origin of the
77 (b)
4 - plane
(cl
Fig. 4. Three-dimensional construction : (a) A three-dimensional duplex formed by the corresponding twodimensional duplex (shaded area) and the point x. (b) The same duplex and its geometrical parameters. (c) A plane 5 = const. and the corresponding integration domain with respect to q and [.
Inclusion problem for polygons
1983
coordinate system is at x. In these coordinates, the vertices of the duplex are represented by the triplets (O,O,0), (a, O,O), (a, b, c+), and (a, b, c-) (Fig. 4) ; the parameters b, c+, and c- are given by (5) and a = v’(v+ -x)
= V’(V -x).
(9)
As in two dimensions, integration over w can be substituted by integration over either duplexes or simplexes :
=
$ j: job":j;"" (...)dCdt
drl
j:dc
j;‘;:Ozdq j~q’bz(...)di].
(10)
3.2. Potentials For the two- and three-dimensional simplexes, the harmonic and bi-harmonic potentials can be calculated in closed-form. We denote the three-dimensional simplex potentials by $(a, b, c), where c can be either C+ or c- ; for the two-dimensional potentials the first argument is omitted. The duplex potentials, distinguished by the hat symbol, can be calculated in terms of the potentials from (7) (8) and (10) : &a,b,c+
,c ~ ) = $(a,b,c+)-&a,b,c-),
$(a,b,c+
,c ~ ) = Il/(a,b,c+)-$(a,b,c
)
in3-D,
(11)
and
&b,c+,c-) $(b,c+ ,c
= $(b,c+)-&b,c-), ) = $(b,c+)--tj(b,c-)
in2-D.
(12)
3.2.1. Two-dimensional simplex potentials. For the two-dimensional simplex, the harmonic and bi-harmonic potentials at x follow from (2) (3) and (8) : $(b,c)
=
‘dq cn’b-ln(q2+[2)d[ j0 j0
(13)
and $(b,c)
=
‘dq ‘V/b- i($ + i’) ln(y* + i’) di. j0 j0
(14)
The logarithmic terms appear in (13) and (14) as a result of integration in the third direction (MacMillan, 1930). Strictly speaking, constants of integration should be included in the integrands in (13) and (14) but those constants are omitted because they disappear once the potentials are differentiated according to (1). The one-dimensional integrals in (13) and (14) are straightforward to evaluate, and the expressions for the potentials are
1984
G. J. RODIN
4(b&) =
~[3c-2batan(;)-cln(b’+c’)](15)
and
0
33b2c + 7c3 - 24b3 atan f
- 18b*c ln(b2 + c’) - 6c3 ln(b* + c’)
1. (16)
3.2.2. Three-dimensional simplex potentials. For the three-dimensional simplex, the harmonic and b&harmonic potentials at x follow from (2), (3), and (10) : $(a,b,c)
= ~~dS~~~‘~dg~~~~*+q’+r’)~‘/‘di
(17)
and
The integrals in (17) and (18) can be evaluated following Waldvogel (1979) or with Mathematics (Wolfram, 1991) : &a,b,c)
$(a, b, c) = habcd-
=-abln ;
(zj-
k a4 atan
(19)
ka2 atan(a, +;+,a,d)’
(a2 +k&
[aid)
+ k ab(3a2 + b2) In
e)’
(20) and d = dm. It is worth mentioning that Mathematics computed a instead of Ial in (19) and (20). 3.2.3. Differentiation of potentials. To obtain the tensors s and !$ the potentials must be differentiated with respect to x as prescribed in (1). This task is straightforward to carry out once we observe that the position vectors of the vertices of cv and the local basis vectors are independent of x. Then from (5) and (9) it follows that aa
-z--y ax
ah ,
$j=-“’
and
ac
-= ax
-x.
Thus the derivative with respect to x can be evaluated in the local coordinates using
g)...)= -“$(..,)-p$(. ..)-&...)
(21)
In two dimensions, the first term in the right-hand side of (21) is absent. With (21) expressions for 3 in the local coordinates can be obtained directly from (I), (I 5), (16), (19), and (20). An explicit expression for s in two dimensions is
1985
Inclusion problem for polygons
relatively simple (see Section 4). However, in three dimensions one should rely on symbolic computations. 3.3. Assembly With the provision that s has been constructed as a function of a, b, c, the tensor S at a point x is assembled in three steps. Step 1 : For each duplex, compute a, b, c+, cc, v, p, and x. Step 2 : For each duplex, compute S(a,b,c+,c-)
= S(a,b,c+)-S(a,b,c-)
and transform the components of $(a, b, c+, cP) to global coordinates. Step 3 : Add the transformed components of @a, 6, c+, c-) of the individual duplexes.
4.
APPLICATIONS
In this section, we consider several applications of the algorithm in two dimensions. These include verification of some well-known relationships as well as a parametric study of the strain field in regular polygonal inclusions. In three dimensions, applications are limited to the analysis of vertex singularities. On the basis of this analysis, in agreement with Eshelby’s (1961) conjecture, we conclude that polyhedra with constant S(x) do not exist. 4.1. Two-dimensional problems 4.1.1. Eshelby ‘s tensor of the simplex. Expressions for the local components of s in two dimensions are calculated directly from (1) (15) (16) and (21). These expressions can be simplified if we take into account that s is non-dimensional and therefore its dependence on b and c can be reduced to a function of one argument
tl = atan E 0b ’ For computations, convention :
s,, =
q,,,, =
it is convenient to represent s as a 3 x 3 matrix using Voight’s
1+2v sm2a+ _ v> 4va- 2
1
[
1 =
%lrli
=
4( 1 - v)cI- ---sm2al-2v, 2
1
&c(l -v)
SI 2 = %lci = &(I
s13
(22)
&.ql_
v>
-vcos2c(+
1
-sin4a 1 8 gsin4cr 1
icos4a+
1,
l-4v
2
1 ,
1,
ln(sec IX)
G. J. RODIN
1986
1-2vsin2a+ 2
SZI = %Ja = C&(* _ V) 1 1 g2 = &;
=
$2, = Z&
=
8n(l-v)
3-2~ [ -sin2a2
87c(l -v)
I
1
-1
1 % I = s;l[lpj =
87c(l -v) [ 1 87c(l -v)
s3, = s;li,,i =
1 8sin4a
+(I-r)cos2a-
I +cos2akcos4at
1
For the duplex, Eshelby’s
2(1 -v)a-
tensor is computed
:ln(seca)
1 -sin40! 8
1 -sin2a+ 2
ln(seccr)
1
1,
.
(23)
as
S = S(a+) -S(cr-). The angles cl+ and a- in (24) are obtained
1,
)
&(secr)
i
87t(l - v)
3-4v ___ 2
$cos4a+
3 1 _- 2 + y4cL+
-
$1 = ?Q;; =
1’ 1,
Isin4a 8
(24)
from (22) using cf and c-, respectively.
4.1.2. Milgrom-Shtrikman traces. Milgrom and Shtrikman (1992) observed that the traces SiQj and S,,,j are independent of the inclusion shape. This result follows immediately from (1) and (4) S,, = 1 For 5, the traces calculated
and
S ,,,, = 3
ifxEa.
(25)
from (23) are
with the Greek indices running 1 to 2 as opposed to the Latin indices in (25) running from 1 to 3. ff x E o it is obvious that the angles CY of the individual simplexes sum up to 27~.Thus in two dimensions, Milgrom-Shtrikman’s traces are 1
S,jrlii= Slliqi- (, _v)
Equations (25) and (26) can be reconciled taken into account.
if xEu.
once the out-of-plane
components
of S are
4.1.3. Jump conditions. Now we consider how the jump conditions across the boundary of a p-sided polygon COare realized in terms of (23). We evaluate the jump [S] at a point x that belongs to an edge distinguished by the asterisk symbol. By definition,
1987
Inclusion problem for polygons
x-bp Fig. 5. Jump conditions : Two duplexes and their geometrical parameters
[S](x) = l&l+ [S(X+b*~*)-S(x-b*p*)l.
(27)
In terms of S, (27) takes the form [S](X)
= ,Jiy+
i$,
[~i(X+b*~*)-~i(X-b*p*)l'
This expression can be simplified if we observe that for all edges, except for the one that contains x, Si(x+6*p*)
as b* + 0.
+ Si(x-b*p*)
Thus [S](x) can be evaluated as [S](x) = ,li;+ [s*(x+b*p*)-s*(x-b*p*)].
(28)
At this point, the asterisk symbol becomes obsolete because only one edge is involved in calculations, therefore the asterisk symbol is discarded. The two duplexes that are required for evaluation of (28) are shown in Fig. 5. The corresponding tensors are identified in terms of four angles, one pair for each tensor, and these angles are +
‘+(X-bC) = a (x-bp)
-U+(X+bp)
= -cc-(x+bp)
=
atan?,
= atanc.
b
The signs of the angles are defined by the inequalities b > 0, C+ > 0, and c- < 0. Also, in Fig. 5, positive (negative) angles are marked by counter-clockwise (clockwise) arrows. In the limit, the angles satisfy the relationship cc+(x-bp)-cc-(x-bp)
= -[a+(x+bp)-a-(x+bp)]
= 71,
1988
G. J. RODIN -c;
ct
Fig. 6. Two-dimensional
vertex singularities
: Two duplexes and their geometrical
parameters
that implies that all terms in (23) except for those proportional to CIcancel each other out upon summation. As a result we obtain the local representation 2(1-v)
1
p](x) = -
2(1-v)
I
2v
0
0
0
0.
0
0
(l-v)
I
This expression is easy to verify by inserting the well-known jump conditions for the potentials (Mura, 1982)
into (1). 4.1.4. Vertex singularities. For x close to a vertex, the logarithmic terms in (23) may become dominant. To analyze this situation we introduce the following notation : We assign the subscripts 1 and 2 to the edges that form the vertex ; their respective lengths are denoted by I, and I2; the position of x with respect to the vertex is prescribed in terms of the distance 6, and with respect to the edges by the angles fl, and pz (Fig. 6). For small 6, the following relations hold (Fig. 6) : b, = 6sinp,,
c: = ~cos~,,
c; = --I, +6cosp,
X-f,,
and b2 = 6sinB,,
cl = I,-Scos/I,
Xl,,
CT = -6cos/),.
The symbol X means asymptotically equal. With these expressions, the logarithmic terms in S satisfy ln(secol:)-ln(seca;)
Vln;
and
ln(secar:)-ln(secol,)Xlni;
the other terms of 3 are non-singular and therefore discarded. Consequently, Eshelby’s tensor takes the asymptotic form S(x) =:
1 87c(1 - v)
M, lni
-M,
Ini
(29)
Inclusion problem for polygons
1989
The tensors M, and M2 in (29) are represented by the same matrix in the basis (p,, x,) and (az, x2), respectively ; this matrix is
M=i[p,
!3
Ii;::].
(30)
Equation (29) can be simplified if we observe that ln(l,/Z,) is non-singular : (31) where I is a representative edge length. It is important to recognize that (31) is valid only if the second-rank transformation strain tensor is not in the nullspace of the fourth-rank tensor M, - M2 ; otherwise the non-singular terms of 3 cannot be neglected. Since algebraic operations with the tensor M, -M, involve only 3 x 3 matrices, its nullspace can be calculated without any difficulties. In doing this we choose (p,, x,) as the basis and calculate M, in this basis by rotating its components through the angle fl = - (rc- b, - f12) (Fig. 6). The results of this calculation can be summarized as follows : l l
l
l
If p # 7r/2 and /? # 3n/2, and v # l/2 the nullspace is trivial. If p = 71/2 or /3 = 3rc/2, and v # l/2 the nullspace includes second-rank tensors proportional to pI 0 p, -x1 0 xi. If p # n/2, and p # 37~/2,and v = l/2 the nullspace includes second-rank tensors that are not traceless and therefore must be discarded. If /3 = n/2 or p = 3n/2 and v = l/2 the nullspace includes all second-rank traceless tensors.
From this summary we conclude that the singularity vanishes in all vertices in polygons whose angles are equal to either 7c/2or 3rc/2 if v = l/2. If, for such polygons, v # l/2, the singularity vanishes in all vertices only if the transformation strain tensor is traceless and its eigenvectors are parallel to the edges of the polygon. This conclusion is in agreement with calculations of List and Silberstein (1966). The presence of singularities implies that S(x) is not a constant. Thus Eshelby’s (1961) conjecture holds for all polygonal inclusions except maybe for those whose angles are equal to either 7r/2 or 3x/2 and whose material is incompressible. 4.1.5. Eshelb_y’s tensor for regular polygons. We conclude our analysis of twodimensional problems with a computational example. We compute S(x) for a family of regular polygons inscribed into a unit circle. In polar coordinates with the origin at the circle center, the vertices of such polygons are prescribed by rk = 1
and
0,=271(k-;
k=
l,...,p.
P
For each polygon, we choose one hundred equally-spaced points along the direction 0 = 0 ; the first point is at the origin and the last point is 0.01 away from the vertex. Computational results for p = 3 (dotted lines), 6 (fine dashed lines), and 12 (coarse
G. J. RODIN
1990
dashed line) are shown in Fig. 7; for comparison, we also plot S(x) for the circle (solid lines). The plots are for non-zero components S,,,,, S,rZ2, &,,, S2222,and S,,,, only; the remaining components are equal to zero due to symmetry. The non-zero components are referred to the Cartesian coordinates consistent with the polar coordinates. Figure 7 provides evidence that, with increasing p, the function S(x) for polygons converges to that of the circle. Due to the logarithmic singularity, this convergence cannot be pointwise but a weak form of convergence obviously exists.
4.2. Three-dimensional problems In three dimensions, we limit ourselves to the analysis of singularities. For this purpose, it is sufficient to retain only the singular parts in the duplex potentials. From (1 l), (19), and (20) we obtain
(a)
S 0.8
1111
0.6 0.4 0.2 0
I
i ........__...._.__.......~~........................~........................~..
-0.2
1
-0.4 0
0.2
0.6
0.4
S
(b) 0.1
-0
. 2
-0
.
3
0.8
1
1122
,
.._.__.._..__.._..
j _._........._.........._
~..___._.._._.............;
~____....._.._.._________
j. _.__.._____.__.____.....~
.. . . .. . . . . .. . . . . . . .. . .. . .
1;
w
-0.4 1 0 0.2 0.4 0.6 0.8 Fig. 7. Eshelby’s tensor of p-sided regular polygons : (a) S,,,,, (b) S,,,,, (c) S,,, ,, (d) S,,,,, and (e) S,,,,. In all plots, the x-axis measures the distance from the center, the dotted line (. .) is forp = 3, the fine dashed line (- -- -) is for p = 6, the coarse dashed line (---) is for p = 12, and the solid line (--) is for the circle,
Inclusion problem for polygons
S
(c) 0.6
I
1991
2211
0 .5
--------_...........-.-/..................~........................~.........
0 .4
.........................~.........................~........................~............... :
0
(d)
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6
(e)
S 0.5
0
.
4
....
.
...
.....j. .
..
...
.
1212
[.........................;
. . . . . . . . . . . . . . . . . . . . . . . . i ._.______________i
j
0 . 35
.... ...
1
... .....+ ......__~~~~=~~~~f_.~~~.~:..: __L__---.__---T--_
i....
.
.... ...._
I i..’ 0. 3 ..:.;...I.:.i .......&i__.....-..: .:..:..:.. ,;;_l.l_* ....._L’yyv 0 . 25 .. ... ..... ...... .i 0 . 2 .......................... 0.15
0
0.2
‘!. .... ’ ..\ . .. j ‘. \ E......................... . ... . ... .._.... I............ L.,.:. t........................ 6........................j.................... :;__
0.4 Fig. 7.-Continued.
0.6
0.8
1
1992
G. J. RODIN
(32)
and cm) X~ab(3a2 $(a,b,c+,
+b2) In
(33)
where d’ = J’m. In three dimensions, the logarithmic singularity can exist both near edges and vertices. In both cases, the dimensions a and b are much smaller than a representative length edge 1. With this provision, it is straightforward to show that after differentiation of (32) and (33) according to (1) only those terms that preserve the logarithmic term in (32) and (33) influence the singular part of S. For example, the singular part of 41, in (1) is affected only by a part of the second partial derivative of $(a, b, c+ , c-) with respect to a and b :
As a result, the singular
part of S has a very simple local representation 1
S=
(34)
87r(l -v) with I 0
ML
0
0
00
4-I-30
0
-3+4v -1+4v
0
0
00
0
00
0
00
0
00
0
00
0
00
0
(35)
Intuitively, it is easy to recognize that the logarithmic singularity near an edge in three dimensions must have precisely the same structure as the one near a vertex in two dimensions. Also this can be confirmed upon comparison of the matrices in (30) and (35). For this reason, in what follows, we consider only singular solutions for points near a vertex. A case that requires the least amount of computations and deserves special attention is that for a point near a vertex formed by three orthogonal planes. If we show that the singularity does not vanish near such a vertex we can claim that Eshelby’s (1961) conjecture holds for all polyhedra. Indeed, from the analysis of two-dimensional singularities, it follows that a singularity near an edge in a polyhedron can vanish only if the edge is formed by two orthogonal faces. Thus the only class of polyhedra that may possibly have non-singular S are those that are formed by orthogonal faces.
Inclusion
Fig. 8. Analysis
of three-dimensional
problem
singularities
for polygons
1993
near a vertex formed
by three orthogonal
faces.
A vertex formed as the intersection of three orthogonal faces is shown in Fig. 8. Following the two-dimensional singularity analysis, without any loss of generality, we suppose that the three edges intersecting at the vertex have the same length 1. The global coordinate system associated with x is chosen such that the origin is at x and the basis vectors (i,, i2, ix) are perpendicular to the faces. In these coordinates, the vertex is represented by the vector (6,, &, 6,) and its distance from x is denoted by 6. We number the faces and edges following the usual convection: Face one has unit normal i,, edge one is parallel to il, etc. The total number of duplexes we have to consider is equal to six. This is because each face requires two duplexes, each associated with one of its edges. Table 1 contains the required parameters for the six duplexes, these are the dimensions a, b, c+, and c-, the vectors p, v, and x, and the asymptotic expressions for the arguments of the logarithm in (34). From Table 1, the tensor M of each duplex is obtained through appropriate rotations of the matrix in (35). As a result of these calculations an explicit expression for S follows from (34) and (35) : Table 1. Duplex parameters for evaluation of (34) and (35) Edge
a
b
I
2
S,
1 2 2
3 1
6,
Face
3 3
3 1 2
c+
c-
V
B
x
I&
1
-6,
;2
6, S [’
-1 -I _6
i, i, i i1
i, i, i,i
-i,
6, 6 6’
6: 6,
s: 6,
1 6,
-6: -1
i: i,
ii i,
i, i, -i3 -i, i2
(c’+df)/(c-+d-)
W(6-&) 2U(6-&) w(s-w W(S - 4) 21/(6 -6,)
Zl/(S-6,)
G. 3. RODIN
I994
s=
1 8741-v)
0
0
0
(I-2v)y,
(I-2v)y,
0
0
0
0
(1-2v)y,
0
(1-2V)Y,
0
0
0
0
(1-2V)Y,
(1-2V)Y,
Y.i 1'3 0
0
0
0
‘J2
Y2
0
0
0
YI
0
0
0
0
0
YI
where y, = In [l/(&S,)], i = 1,2, 3. The nullspace of S is easy to determine. l
l
It possesses the following
properties
(36)
:
For any v the dimension of the nullspace does not exceed three. Thus the singularity vanishes only for a special class of the transformation strain tensors. This confirms Eshelby’s (1961) conjecture for polyhedra. If v = l/2, the nullspace includes the six-dimensional vectors (O,O, 0, l,O, 0), (0,0,0,0, 1, 0),and (0,0,0,0,0,l) that represent second rank tensors whose normal components are equal to zero. Thus, for polyhedra whose faces are orthogonal to each other the singularity vanishes if v = l/2 and the normal components of the transformation strain tensor (in a coordinate system naturally aligned with the polyhedron) are equal to zero.
ACKNOWLEDGEMENTS I am grateful to Professor Dimitris Lagoudas (Texas A&M University) for helpful discussions, to Dr Altha Rodin (UT Austin) for editorial comments, and to Dr Robert Werner (UT Austin) for educating me about the celestial mechanics literature. Special thanks go to Professor Toshio Mura (Northwestern University) for inspiration. This work was sponsored by NSF through the grant MSM-9114856 and by the ALCOA foundation.
REFERENCES Chiu, Y. P. (1977) On the stress field due to initial strains in cuboid surrounded
by an infinite elastic space. J. Appl. Me& 44, 587-590. Eshelby, J. D. (1957) The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. Sot. (London) A241, 376-396. Eshelby, J. D. (1959) The elastic field outside an ellipsoidal inclusion. Proc. Ro)~. Sot. (London) A252,561&569. Eshelby, J. D. (1961) Elastic inclusions and inhomogeneities. Progress in Solid Mechanics (ed. I.N. Sneddon and R. Hill), Vol. 2, pp. 89-140. North-Holland, Amsterdam. Faivre, G. (1964) DCformations de cohCrence d’un prkcipite quadratique. Phvs. Stat. Sol. 35, 249-259. Khachaturyan, A. G. (1983) Theory ofStructural Trunsfllrmations in Solids. Wiley, New York. Lee, J. K. and Johnson, W. C. (1977) Elastic strain energy and interactions of thin square plates which have undergone a simple shear. SU. Metall. l&477-484. List, R. D. and Silberstein, J. P. 0. (1966) Two-dimensional elastic inclusion problems. Proc. Cumh. Phil. Sot. Muth. Phys. Sci. 62, 303-31 I MacMillan, W. D. (1930) The Theory ofthe Potential. Dover Publications.
Inclusion problem for polygons
1995
Milgrom, M. and Shtrikman, S. (1992) The energy of inclusions in linear media exact shapeindependent relations. J. Mech. Phys. Solids 40,927-931. Mura, T. (1982) Micromechanics of Defects in Solids. Martinus Nijhoff Publishers. Mura, T., Shodja, H. M., Lin, T. Y., Safadi, A. and Makkawy, A. (1994) The determination of the elastic field of a pentagonal star shaped inclusion. Bull. Tech. Univ. Istanbul 47, 267280. Sankaran, R. and Laird, C. (1976) Deformation field of a misfitting inclusion. J. Mech. Phys. Solids 24, 251-262. Sherman, D. I. (1959) On the Problem of Plain Strain Non-homogeneous Media. Non-hornogeneity in Elasticity and Plasticity. Pergamon Press, Oxford. Waldvogel, J. (1979) The newtonian potential of homogeneous polyhedra. ZAMP 30, 388% 398. Werner, R. A. (1996) Polyhedron gravitation. PhD Dissertation. Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX 78712, U.S.A. Wolfram, S. (1991) Mathematics. Addison-Wesley. Wu, L. and Du, S. Y. (1995a) The elastic field caused by a circular cylindrical inclusion-Part I : Inside the region .x: + xs < a*, - n) < x3 < cc where the circular cylindrical inclusion is expressed by x: +xz > a’, -h > x3 > h. J. Appl. Mech. 62, 579-584. Wu, L. and Du, S. Y. (1995b) The elastic field caused by a circular cylindrical inclusion-Part II : Inside the region x: +x: > a’, - CC < xj < CCIwhere the circular cylindrical inclusion is expressed by x: +x: > a’, -h 3 xJ 3 h. J. Appl. Mech. 62, 585-589.
