Asymptotic Check for Singularity

Downloaded FromStress singularities in classical elasticityII: Asymptoticidentification

GB SinclairDepartment of Mechanical Engineering, Louisiana State University, Baton Rouge,LA 70803-6413; [email protected]

This review article ~Part II! is a sequel to an earlier one ~Part I! that dealt with means of re-moval and interpretation of stress singularities in elasticity, as well as their asymptotic andnumerical analysis. It reviews contributions to the literature that have actually effectedasymptotic identifications of possible stress singularities for specific configurations. For themost part, attention is focused on 2D elastostatic configurations with constituent materials be-ing homogeneous and isotropic. For such configurations, the following types of stress singu-larity are identified: power singularities with both real and complex exponents, logarithmicintensification of power singularities with real exponents, pure logarithmic singularities, andlog-squared singularities. These identifications are reviewed for the in-plane loading of angularelastic plates comprised of a single material in Section 2, and for such plates comprised ofmultiple materials in Section 3. In Section 4, singularity identifications are examined for theout-of-plane shear of elastic wedges comprised of single and multiple materials, and for theout-of-plane bending of elastic plates within the context of classical and higher-order theory. Areview of stress singularities identified for other geometries is given in Section 5, axisymmet-ric and 3D configurations being considered. A limited examination of the stress singularitiesidentified for other field equations is given as well in Section 5. The paper closes with anoverview of the status of singularity identification within elasticity. This Part II of the reviewhas 227 references. @DOI: 10.1115/1.1767846#

1 INTRODUCTIONThis article is a sequel to another one on stress singularitiesin classical elastostatics which considers their removal, inter-pretation, and analysis ~Sinclair @1#hereinafter referred tosimply as Part I!. Both papers share the recognition that it isan exercise in futility to perform a stress analysis withoutappreciating the presence of a singularity when one occurs.In Part I, some methods for determining when a singularity ispresent, and possibilities for dealing with it when it is, aredrawn from the literature and discussed. Here, in Part II, theliterature is reviewed for contributions that have actually ef-fected determinations of when singularities may occur.

The means by which these determinations are made isasymptotic identification. It is therefore necessary, if Part IIis to be fairly self-contained, that we recap key results at-tending the asymptotic identification of stress singularities.These are available in the literature and a description of theirdevelopment is given in Part I, Sections 4.1 and 4.2. Theparticular approach considered in some detail there is via theAiry stress function and separation of variables ~after Will-iams @2#!: There are other approaches which can lead to thesame results ~complex potentials, Mellin transforms!.

To fix ideas, we consider an angular elastic plate in exten-

sion ~Fig. 1!. The basic separable fields used to analyze suchplates are given in Williams @2# and Part I, Section 4.1. Interms of cylindrical polar coordinates r and u, the stressessr , su , and tru and displacements ur and uu in these fieldsare

sr52lrl21@c1 cos~l11 !u1c2 sin~l11 !u

1~l23 !~c3 cos~l21 !u1c4 sin~l21 !u!#

su5lrl21@c1 cos~l11 !u1c2 sin~l11 !u

1~l11 !~c3 cos~l21 !u1c4 sin~l21 !u!#

tru5lrl21@c1 sin~l11 !u2c2 cos~l11 !u

1~l21 !~c3 sin~l21 !u2c4 cos~l21 !u!# (1.1)

ur52rl

2m @c1 cos~l11 !u1c2 sin~l11 !u

1~l2k!~c3 cos~l21 !u1c4 sin~l21 !u!#

uu5rl

2m @c1 sin~l11 !u2c2 cos~l11 !u

1~l1k!~c3 sin~l21 !u2c4 cos~l21 !u!#Transmitted by Editorial Advisory Board member R. C. Benson

Appl Mech Rev vol 57, no 5, September 2004 38

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angular plate results in a homogeneous system of equations

Downlin the four constants ci . Call the coefficient matrix of thissystem A and its determinant D . Then, also entertaining thepossibility of the participation of the fields attending ~1.2!leads to the following set of conditions for the singularstresses that are possible with homogeneous boundary con-ditions. For any stress component s, as r0:s5O~rj21 cos~h ln r !!1O~rj21 sin~h ln r !! when D50

for complex l5j1ih~0,j,1 !

s5O~rl21 ln r !1O~rl21! when D50,]nD]ln

50

for n51, .. , nA2rA and real l~0,l,1 !

s5O~rl21! when D50 for real l~0,l,1 ! (1.3)]nDs5O~ ln r ! when D50,]ln

50

oaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.orgfor n51,..,nA2rA and l51 with

c121 c2

21 c220 in the stress field attending ~1.2!

s5O~cos~h ln r !!1O~sin~h ln r !! when D50

for complex l511i h

Herein, l has taken on the role of an eigenvalue of theasymptotic problem, nA is the order of the matrix A, and rAis its rank when l is an eigenvalue. For a plate made of asingle material, nA54 at most; for a bimaterial plate, nA58, and so on. The last stress in ~1.3! is not singular as r0, being bounded under this limit. However, it is unde-fined as r0. Hence, to a degree, it shares with singularstresses some of the difficulties attending interpretation as r0.

The conditions in ~1.3! apply to angular plates in exten-sion. Adaptation of ~1.3! to states of antiplane shear followsdirectly ~see Sections 4.1 and 4.2!. Adaptation of ~1.3! tobending is less direct but nonetheless fairly straightforward~see Sections 4.3 and 4.4!. Adaptation of ~1.3! to other con-figurations is discussed in Section 5.

With inhomogeneous boundary conditions, further auxil-iary fields can participate. These fields follow from a furtherdifferentiation with respect to l; stresses are given in Part I,Section 4.2. By way of example, the su stress component inthese fields is

su5rl21@~l ln2 r12 ln r2lu2!~ c1 cos~l11 !u

1 c2 sin~l11 !u!1~l11 !~ c3 cos~l21 !u

1 c4 sin~l21 !u!1O~ ln r !1O~1 !# (1.4)as r0. In ~1.4!, tildes atop constants distinguish them fromthose of ~1.1! or ~1.2!. All three sorts of field in concert leadto the following set of conditions for the singular stressesthat are possible with uniform tractions/ linear displace-ments applied. For any stress component s, as r0:

s5ord~ ln2 r !1ord~ ln r ! when D50,]nD]ln

50

for n51,..,nA2rA with

c121 c2

21 c320 in the stresses attending ~1.4!

s5ord~ ln r ! when D50,]nD]ln

50, for n51,.., nA2rA(1.5)

with c15 c25 c350 in the stresses attending ~1.4 !

s5ord~ ln r ! when D50,]nD]ln

0, for n5nA2rA

with c121 c2

21 c320 in the stresses attending ~1.2!

provided throughout ~1.5!, l51 and rArA8 , where rA8 isthe rank of the augmented matrix formed by combining Awith the nontrivial forcing vector associated with the inho-mogeneous boundary conditions. Given such a nontrivialvector, the singularities in ~1.5! occur irrespective of far-field

Appl Mech Rev vol 57, no 5, September 2004In ~1.1!, m is the shear modulus and k equals 324n forplane strain and (32n)/(11n) for plane stress, n beingPoissons ratio. Further, ci (i51,...,4) are constants and l isthe separation-of-variables parameter. This parameter may becomplex. Then the real and imaginary parts of ~1.1! eachconstitute acceptable fields which may have distinct sets ofconstants from one another. It is also possible to have auxil-iary fields participate in the asymptotic analysis. These fieldscan be generated by differentiating with respect to l as inDempsey and Sinclair @3#, and are given in Part I, Section4.2. By way of example, the su stress component in thesefields issu5r

l21@~11l ln r !~ c1 cos~l11 !u1 c2 sin~l11 !u!

1~2l111l~l11 !ln r !~ c3 cos~l21 !u

1 c4 sin~l21 !u!2lu~ c1 sin~l11 !u2 c2 cos~l11 !u

1~l11 !~ c3 sin~l21 !u2 c4 cos~l21 !u!!# (1.2)In ~1.2!, the carets atop constants serve to distinguish themfrom those of ~1.1!

Introducing the fields in ~1.1! into a set of four homoge-neous boundary conditions holding on the two edges of the

Fig. 1 Geometry and coordinates for the angular elastic plate

386 Sinclair: Stress singularities in classical elasticityIIboundary conditions. Hence the use of the ord notation in

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Downloaded Fromsingularity conditions in ~1.5! apply directly to angular platesin extension: Adaptation to other configurations is discussedin Sections 4 and 5.

In what follows, we review asymptotic analyses that em-ploy ~1.3!, ~1.5!, or their equivalents to identify stress singu-larities. We begin in Section 2 with angular elastic platesmade of a single material under in-plane loading ~ie, in ex-tension!. In Section 3, we review the singularities identifiedwhen such plates are made of multiple materials. In Section4, we consider out-of-plane shear and bending. In Section 5,we consider a variety of other circumstances: axisymmetricand 3D configurations within classical elasticity, and a lim-ited review of the effects of other field equations. Finally, inSection 6, we close with some remarks on the general char-acter of results, and the overall state of investigations intosingularity identification.

2 STRESS SINGULARITIES FOR THE IN-PLANELOADING OF AN ELASTIC PLATE MADE OF ASINGLE MATERIAL

2.1 Formulation and eigenvalue equationsHere we obtain the eigenvalue equations governing the pos-sible stress singularities that can occur at the vertex of anangular elastic plate subjected to different homogeneousboundary conditions on its edges.

To formally state the class of problems under consider-ation, we continue to employ cylindrical polar coordinates rand u with origin O at the plate vertex ~Fig. 1!. In terms ofthese coordinates, the open angular region of interest R isgiven by

R5$~r ,u!u0,r, , 0,u,f% (2.1)where f is the angle subtended at the vertex of the plate.With these geometric preliminaries in place, we can formu-late our class of problems as follows.

In general, we seek the planar stress components sr , su ,and tru and their companion displacements ur and uu , asfunctions of r and u throughout R, satisfying: the stressequations of equilibrium in the absence of body forces,

]sr]r

11r

]tru]u

1sr2su

r50

(2.2)1r

]su]u

1]tru]r

12tru

r50

on R; the stress-displacement relations for a linear elasticplate which is both homogeneous and isotropic,

H srsuJ 5mF 2Qk21 H 12J S ]ur]r 2 1r ]uu]u 2 urr D G (2.3)tru5mF1r ]ur]u 1 ]uu]r 2 uur G

1A definition of ord is given in Part I, Section 1.2. The essential difference between ord

and O is that, with the former, the coefficient of the related singularity cannot be zero,whereas with the latter it can.

: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 04/07/2with

Q5]ur]r

11r

]uu]u

1ur

r(2.4)

on R, wherein Q is the dilatation while m continues as theshear modulus and k continues to equal 324n for planestrain and (32n)/(11n) for plane stress, n being Poissonsratio; any one of the admissible sets of boundary conditionslisted in Table 1 on the plate edge at u50, together withanother such set on the edge at u5f or bisector at u5f/2 asappropriate, for 0,r,; and the regularity requirements atthe plate vertex,

ur5O~1 !, uu5O~1 !, as r0 (2.5)

on R. In particular, we are interested in the local behavior ofthe fields complying with the foregoing in the vicinity of theplate vertex O .

The boundary conditions of Table 1 merit some discus-sion. Conditions I and II apply on u50 or f and are theclassical conditions for a stress-free surface and one clampedto a rigid attachment. The clamped conditions also admit tointerpretation as the homogeneous complement to the condi-tions attending indentation by a rigid punch with no slippermitted. Such indentation is also sometimes associatedwith a rough or adhesive punch in the literature.

When the same conditions apply on both plate edges (u50, f), it is useful to distinguish between symmetric andantisymmetric response about the plate bisector. In the firstinstance, it is useful because the analysis can be easier byvirtue of leading to a 232 determinant for the eigenvalueequation instead of a 434. In the second instance, it is usefulbecause it can restrict the number of singular stress statespossible in a given global configuration before undertakingits global analysis. Conditions III and IV enable one to makethis distinction. For the present plate configuration, they ap-ply on u5f/2 when used in this role.2

Conditions III can also be interpreted as the homogeneouscomplement to indentation by a frictionless rigid punch. Inthis role, they apply on u50 or f and are usually adjoinedwith the condition that the normal stress not be tensile in thecontact region. That is,

su

Downlfurther constraints outside the contact region to prevent in-terpenetration.

Conditions IV can also be interpreted as those for a thinstiff reinforcement ~Rao @4#!. The reinforcement is suffi-ciently relatively stiff to prevent extension (ur50), but notso stiff as to prevent bending because of its thinness (uu0).

Conditions V extend the contact conditions of III to per-mit finite friction via Amontons law.3 Herein f has the mag-nitude of the coefficient of friction. For these conditions, inaddition to seeking to apply the contact constraint ~2.6! andany external displacement constraints, we must try to ensurethat the shear stress opposes any slipping. This may be pos-sible by selecting the sign of f appropriately.

Conditions VI apply cohesive stress-separation laws. Thusk and k8 are the stiffnesses associated with relative trans-verse and lateral displacements between material on the twosides of the ray on which the conditions are applied. Whenapplied on u50 in Fig. 1, both k and k8 are positive: on u5f , negative. In some instances it may be possible to setone or the other of these stiffnesses to zero. For example, k8can be taken as zero when loading is symmetric. In contrast,if k and k8 are let tend to infinity, Conditions II are recov-ered. In general, k and k8 are of constant magnitude in theelastic regime and should both be consistent with the elasticmoduli of the surrounding continuum.

Conditions VI can also be interpreted as those for a plateon an elastic foundation. Usually then k8 is taken as zerogiving Winkler conditions ~Winkler @6#; Oravas @7# has thatthese conditions were given earlier in Euler @8#!.

In either role, Conditions VI differ from the others inTable 1 in that a single boundary condition involves both astress and a displacement. Such mixed boundary conditionswould seem to be fairly rare in elasticity. One further in-stance occurs for the elastic angular plate reinforced by abeam columnsee Nuller @9#.

All of the foregoing boundary conditions are appliedalong radial rays emanating from the plate vertex. That is, onstraight boundaries. If instead they are applied on curvedboundaries that smoothly make tangents to the straight at thevertex, the same singular eigenvalues can be expected. Com-panion eigenfunctions differ, however. See Ting @10#.

Some further comments on the preceding formulation arealso appropriate. First, regarding the absence requirements atinfinity on R. This renders fields complying with our formu-lation nonunique. Since the principal attribute of these fieldsis the characterization of all possible responses at the platevertex, including especially all possible stress singularitiesthere, such a lack of uniqueness is to be desired rather thanregulated against. In any configuration of finite extent locallycontaining one of the configurations admitted by our formu-lation, conditions on the other boundaries in the finite geom-etry should make its solution unique.

Second, regarding dimensions. There is no length scale in

3Also termed Coulombs law in the literature. See Ch 13, Johnson @5# for conditionsunder which there is some physical support for the use of this law.

oaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.orgthe problems formulated. This is also something provided byassociated finite problems. However, this does not mean thatthere need be concern as to the dimensional consistency ofthe asymptotic analysis. To see this, observe that the fieldequations ~2.2!, ~2.3!, and ~2.4! are equidimensional in r ,and that ur and uu occur divided by r . Thus r , ur , and uucan be replaced by r/L , ur /L , and uu /L , where L is anylength scale, and leave the equations unchanged. Hence anyasymptotic solutions obtained can be regarded as being interms of r/L and thereby made dimensionally consistent.

Third, regarding material constants. These are con-strained to the physically applicable ranges, 0,m, and0

Downloaded Fromthey are independently derived largely as a check.The free-free equations ~2.9! and ~2.13! in Tables 2 and 3,

the clamped-clamped equations ~2.10! and ~2.14! in Tables 2and 3, and the clamped-free equation ~2.17! in Table 4 alleffectively appear in Williams @2# and Kitover @18#.5 Equiva-lent equations to those for frictionless contact-frictionless

5The eigenvalue equations in Kitover @18# are correct for plane strain, but appear tohave typographical errors for plane stress.

: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 04/07/2and Molski @20#. Two further frictionless contact equationsfor such conditions in combination with free and withclamped conditions can be obtained by setting f 50 in ~2.18!and ~2.19! of Table 4, respectively: These two are given inKalandiia @19#. The equation for contact with friction actingwith itself symmetrically, ~2.12!, would not appear to bereadily available in the literature; the equation when this ac-tion is antisymmetric, ~2.16!, is essentially the same as thecorresponding equation in Dempsey @21#. The contact withfriction-free equation of ~2.18! in Table 4 can be obtainedfrom Gdoutos and Theocaris @22#. It follows on setting G2in @22# to infinity to reflect a rigid punch, and q52 fbecause the friction conditions therein hold on a negativeu-edge. The contact with friction-clamped equation of ~2.19!in Table 4 is essentially given in Dempsey @21#. The equiva-lence of stress-free conditions with those for cohesive stress-separation laws as far as the foregoing eigenvalue equationsare concerned is basically argued in Sinclair @23#.

When Conditions IV are interpreted as being for a thinstiff reinforcement, eigenvalue equations for these conditionswith others and a plate of vertex angle f/2 are given in Table4. When Conditions IV act in this role on both edges of aplate of vertex angle f, the eigenvalue equation can beformed as a product of ~2.11! and ~2.15!.6

2.2 Power singularities with homogeneous boundaryconditionsFor the homogeneous boundary conditions of Table 1, theassociated eigenvalue equations of Tables 24 can give riseto stresses with power singularities when their eigenvaluesare less than onesee ~1.1!. To be in accordance with theregularity requirements ~2.5!, these eigenvalues must not beless than zero. An eigenvalue equal to zero corresponds to arigid body displacement in ~1.1! and therefore is not of in-terest because associated stresses are not singular: The samevalue leads to unbounded displacements for the fields asso-ciated with ~1.2! and therefore is not admissible. Thus theeigenvalue range for power singularities is

0,l,1 (2.20)We review eigenvalues within this range for a variety ofconfigurations in this section.

The solution of the eigenvalue equations within the sin-gular range typically cannot be done completely analytically.Accordingly it usually proceeds numerically except for a fewselect instances. The results so found are compared withthose in the literature. For all sources given in what follows,they are consistent. Thus their calculation here may beviewed as independently confirming values already deter-mined in the cited sources.

In presenting results we introduce the singularity expo-nent g defined by

6That is, by rearranging ~2.11!, ~2.15! so that they have expressions on one side ofthe5 , zero on the other, then setting the product of these expressions50. The so-obtained equation is given in Rossle @24#. This reference also gives nine further eigen-the boundedness of attendant strain energies and the usualKirchhoff argument ~see Knowles and Pucik @12#!.

On occasion, further support for the bounded displace-ment conditions derives from solving a singular problem asthe limit of a sequence of nonsingular problems, the ana-logue of the approach adopted in concentrated load problemsand for generalized functions in general in Lighthill @13#. Anexample is the plate under uniform remote tension with anelliptical hole. As the height of the hole parallel to the ten-sion goes to zero, the nonsingular stress fields can be shownto recover the inverse-square-root stress singularity of astress-free mathematically-sharp crack ~Kolossoff @14,15#and Inglis @16#!. This singularity has bounded displacements.The same is true of other singular configurations realized inthis way: see, for example, Neuber @17#.

The analysis of the class of asymptotic problems formu-lated proceeds routinely on using the approach outlined inthe Introduction here, and described in some detail in Part I,Section 4.1. This yields the eigenvalue equations set out inTables 2, 3, and 4 for symmetric, antisymmetric, and mixedconfigurations, respectively. These eigenvalue equations aretypically available in the literature as described next: Here,

Boundary conditionson u0,f

Eigenvalueequation

Equationnumber

I or VI-I or VI l sin f52sin lf ~2.9!II-II l sin f5k sin lf ~2.10!III-III cos f5cos lf ~2.11!V-V f @(12k)sin lf1(11k12l)sin f#

5(11k)(cos lf2cos f)~2.12!

Table 3. Eigenvalue equations for antisymmetric response about uf2

Boundary conditionson u0,f

Eigenvalueequation

Equationnumber

I or VI-I or VI l sin f5sin lf ~2.13!II-II l sin f52k sin lf ~2.14!III-III cos f52cos lf ~2.15!V-V f @(12k)sin lf2(11k12l)sin f#

5(11k)(cos lf1cos f)~2.16!

Table 4. Eigenvalue equations for mixed problems

Boundaryconditionson u0,f

Eigenvalueequation

Equationnumber

I or VI-II 4@k sin2 lf1l2 sin2 f#5(11k)2 ~2.17!I or VI-V 2 f @(12k)sin2 lf2l(11k12l)sin2 f#

5(11k)(sin 2lf1l sin 2f)~2.18!

II-V 2 f @k(12k)sin2 lf1l(11k12l)sin2 f#5(11k)(k sin 2lf2l sin 2f)

~2.19!contact, ~2.11! and ~2.15! in Tables 2 and 3, are given inKalandiia @19#. Equations for these conditions which are ex-actly the same as ~2.11! and ~2.15! are provided in Seweryn

Sinclair: Stress singularities in classical elasticityII 389Table 2. Eigenvalue equations for symmetric response about uf2

Appl Mech Rev vol 57, no 5, September 2004value equations. These equations are contained in, and are consistent with, those inTables 24.


Downlg512l (2.21)Then ~1.1! has stresses which behave in accordance with

s5O~r2g! as r0 (2.22)

lar for g positive, and the larger g the more singular. Thelimits on the nature of this power singularity are, from ~2.20!and ~2.21!,

0,g,1 (2.23)

Fig. 2 Singularity exponents for varying vertex angles: a! free-free and clamped-clamped ~from ~2.9!, ~2.13! and ~2.10!, ~2.14!, respec-tively!, b! frictionless contact-frictionless contact ~from ~2.11!, ~2.15!!, c! contact with friction-contact with friction ~from ~2.12!, ~2.16!!,d! clamped-free ~from ~2.17!!, e! contact-free ~from ~2.18!!, f! contact-clamped ~from ~2.19!!

390 Sinclair: Stress singularities in classical elasticityII Appl Mech Rev vol 57, no 5, September 2004where s is any stress component. That is, stresses are singu- In the event that l is complex, we have stress singularities as

oaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 04/07/2014 Terms of Use: http://asme.org/terms

Downloaded Fromin the first of ~1.3! with Re (12l)512j5g, Im l5h, and~2.23! still applying to g. Results for g satisfying ~2.23! arepresented in Fig. 2af for varying vertex angles.

Included in Fig. 2a are the singularity exponents for thefreefree plate, for both symmetric loading from ~2.9!, andantisymmetric from ~2.13!. The symmetric curve is given inFig. 1, Williams @2#. It dominates singular character if load-ing is symmetric or mixed because the antisymmetric curverealizes weaker singularities with stress-free boundary con-ditions: Of course, it cannot dominate if loading is purelyantisymmetric. The antisymmetric curve may be found inFig. 9, Rosel @25# or Fig. 3a, Seweryn and Molski @20#.

For f5360 with free-free conditions, we have the tradi-tional, mathematically-sharp, stress-free crack with itsinverse-square-root singularity for both symmetric and anti-symmetric loading (P1 , Fig. 2a!. For f5270, we have astress-free 90 reentrant corner with two possible singulari-ties, the stronger being for symmetric loading (P2 and P3 ,Fig. 2a!. For f,257.5, no further singularities are foundfor antisymmetric loading. For f5180, we have no singu-larity for symmetric loading. This is because, for this stress-free half-plane geometry, there is no discontinuity in bound-ary directions or conditions. For f

Downlin Fig. 2a. This is because free-free with symmetry has thesame eigenvalue equation as clamped-clamped with antisym-metry and k51see ~2.9! and ~2.14!. Similarly free-freewith antisymmetry is the same as clamped-clamped withsymmetry and k51see ~2.13! and ~2.10!. Thus, as k de-creases corresponding to Poissons ratio increasing, the sin-gularity for antisymmetric clamped-clamped conditions getsstronger, while that for symmetric clamped-clamped getsweaker. The trends thus evident in Fig. 2a are confirmed bysingularity exponents for clamped-clamped conditions for k51 23 and 3 in Seweryn and Molski @20#.

In Fig. 2b, singularity exponents are plotted for the fric-tionless contact-frictionless contact plate, for both symmetricloading from ~2.11!, and antisymmetric from ~2.15!. Thesetwo eigenvalue equations are the simplest of all and admit toanalytical solution. Thus for symmetric configurations,

g5222pf

~p,f,2p! (2.24)

while for antisymmetric,

g5p

f, 22

3pf S 3p2 ,f

Downloaded Fromtionless contact are given in Fig. 2a, Seweryn and Molski@20#.9

There need be no restrictions on the branches included inFig. 2e as a result of contact constraints. This is because thesign of the participation coefficient can always be such thatthe contact stress condition ~2.6! is met, and a rigid bodydisplacement can always be added so as to ensure the fric-tional shear opposes slip. That is, the local fields associatedwith the singularity exponents of Fig. 2e can potentially par-ticipate in a global problem and all auxiliary contact condi-tions be met. Whether this actually happens for the particularglobal configuration of interest needs to be checked.

For f5360 and contact with friction-free conditions,there are three singular stress fields possible. This is also thecase for frictionless contact, although this is not apparent inFig. 2e because only the most singular branch is included.For f5180, the frictionless contact case gives the singu-larity as for a tire at the edge of a pothole on an icy pavement(P7 , Fig. 2e!. For contact with friction and f5180, thesingularity that results is as for an adhering tire but withsome slip permitted. Under these conditions there is one realsingularity ~at P78 , Fig. 2e! compared to the two for an ad-hering tire with no slip. This enables the stress singularity tobe removed for conforming contact when there is contactwith friction-free conditions. No singularities are found forf,90 when f 50, f,116.6 when f 51/2. This is thetrend in general, namely, as f becomes more positive, therange of vertex angles with stress singularities decreases. Onthe other hand, varying Poissons ratio while holding f con-stant leaves the range of singular vertex angles unchanged.

In Fig. 2f, singularity exponents for the contact-clampedplate are plotted. These exponents are from ~2.19!. All sin-gular branches are shown for the representative case of con-tact with friction ( f 51/2 and k52); just the dominant sin-gularity is shown for the representative frictionless case (k52). The exponents for contact with friction would not ap-pear to be available in the open literature: The values shownin Fig. 2f are confirmed in Smallwood @32#. The exponentsfor frictionless contact are given in Fig. 5a, Seweryn andMolski @20#. For the same reasons as for Fig. 2e, there needbe no restrictions on the branches included in Fig. 2f as aresult of contact constraints.

For f5360 and contact with friction-clamped condi-tions, there are three singular stress fields possible. The sameis true for frictionless contact-clamped conditions, thoughthis is not shown in Fig. 2f. For f5180 there is but onesingularity for a given coefficient of friction. This enablessingularities to be removed when transitioning from stick toslip in contact problems. No singularities are found for f,90 when f 50, f,63.4 when f 51/2. This is the trendin general here, namely, as f becomes more positive, therange of singular vertex angles increases. Conversely, forconstant f , increasing Poissons ratio reduces the range ofsingular vertex angles.

9England @31#, Fig. 5, gives values consistent with the exponents given here for f 50

and 0

Pure logarithmic singularities have stresses which behaveas

s5O~ ln r ! a

For the pure loghomogeneous boisfaction of thecan a log singulagularities possiblto detect absentparticipation. Acbe of significant

We therefore distinguish it as the point T1 in Fig. 2a. Thesame is true for all the other logarithmic configurations listed

m power stressr, they typically

ues to complex.singularities fore obtained from

when k51, ffrom Dempsey

e fields of ~1.1!e with homoge-

l

i

,

s

Downloaded From: http://as r0 (2.29)arithmic singularities of ~2.29! under theundary conditions of Table 1, we need sat-penultimate conditions in ~1.3!. Only thenrity occur. These are the weakest stress sin-e in elasticity, and consequently the hardestan a priori appreciation of their possible

cordingly, their asymptotic identification can

in Table 5: They all represent transitions frosingularities to no stress singularities. Furthealso represent transitions from real eigenval

Local fields containing logarithmic stressall the configurations listed in Table 5 can bSinclair @34#. For the contact-clamped plate50, and f k5f*/2, fields are also available@21#. All of these fields demonstrate that thalone are, in general, incomplete for the platsition from complex to real roots. These transitions occur forany f.101.4 except f5180 and 360, and have 0,g,0.75.

It can be expected that logarithmic intensification also oc-curs for the less singular branch of g under clamped-freeconditions wherever there is a transition from complex toreal roots. For k52, R2 in Fig. 2d is an example. For loga-rithmic intensification being possible though, the rank condi-tions in the second of ~1.3! need to be checked for theseconfigurations as well.

On occasion, repeated roots occur without a transitionfrom complex to real values. This can be so if ]D/]f50 forl and f in ~2.27!. Actual examples are R3 and P4 , both fork51, in Fig. 2d. This is not obvious from the figure becausethe less-singular intersecting branch is not shown ~see,though, Fig. 14a, Seweryn and Molski @20#!. For thesepoints, however, Dempsey @33# has that the rank conditionsof ~1.3! are not satisfied and, consequently, logarithmic in-tensification is not possible.

Further configurations wherein logarithmic intensificationcan be expected are where there are transitions from complexto real eigenvalues for plates in contact with friction. Theseinclude R4 of Fig. 2c, and R5 and R6 of Fig. 2e. Again, therank conditions need to be checked to see if this is really apossibility.

Typically logarithmic intensification of stress singularitiescan be expected as stress singularities pass from being purepower singularities to oscillatory power singularities. Insome sense, the logarithmic intensification can be viewed asa transition state between the two, resulting in stresses thatare more singular than those with just power singularities,yet arguably less pathological than oscillatory singularities.We consider logarithmic singularities further in this sort ofrole next when we review their occurrence without powerterms.

Table 5. Configurations with logarithmic singu

Boundary conditionson u0,f Configurat

II-II f5f*

, k5V-V f5p , 2p ,

k5cos 2f2I or VI-II f5p2fkI or VI-V k5112 coII-V f5p/2, 3p

f5f k , f 5

394 Sinclair: Stress singularities in classical elasticityIIvalue.

ppliedmechanicsreviews.asmedigitalcollection.asme.orgDetails of the application of the identification process at-tending the last of ~1.3!, for the boundary conditions of Table1, may be obtained from Sinclair @34#: Results are summa-rized ibid. Every logarithmically singular configuration soidentified complied with all of the conditions in the last of~1.3!. Moreover, when situations arose during analysis inwhich some of these requirements were complied with butothers not, no logarithmic singularities were found.

Given the importance of being aware of the participationof logarithmic stress singularities, we reiterate the configura-tions so found here in Table 5. This table gives seven differ-ent sets of specifications for configurations with log singu-larities.

In Table 5, k is additionally constrained to the range forphysically applicable Poissons ratios. This is broadest forplane strain. Hence

1

demonstrate thatincomplete for th

2.4 Singularititions

ntensified if theneous boundaryd displacementsstress singulari-ccurs when theuniform traction

e can similarly

rithmically sin-ion we considerble 6. These areinclude uniform

tractions or linear displacements.The boundary conditions of Table 6 merit some explana-

Table 6. Inhomo

IdentifyingRoman numeral

Bc

I8 sII8 uV8 u

*II8 II f5p , 2p , Df0 or8

2

p

,

Appl Mech Rev vol elasticityII 395

Downloaded Fromk51, Df0 or DfV8 V f5p , 2p , Df0, k

f 50, Df0VI-VI f5f

*, 2p

I8 or VI II8 f5fk , p6fk , 2pI8 or VI-V f5p , 2p , f p0 or

f 52cot f, fp , 2II8 V f5p , 2p , Df0

f5p/2, 3p/2, k5321f 5(k21)(32k) c

: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 04/07/2the amount by which the vertex angle of an angular plate isreduced by as a result of pinching contact with a rigid inden-tor. With this interpretation, Df is positive on a negativeu-edge, and vice versa. If such contact occurs with no slip,Df850: If it occurs with slip, we have Conditions V8. Theinclusion of the possibility of Df80 is so as to replicatedisplacement discontinuities which can occur in boundaryconditions in finite element analysis ~FEA!. Such disconti-nuities occur in displacement derivatives at nodes when dis-placement shape functions are used as boundary conditionsin submodeling with FEA, a practice implemented in somestandard codes ~eg, Chapter 14, ANSYS @36# and Section7.3, ABAQUS @37#; see Sinclair and Epps @38# for furtherexplanation!.

Using the conditions in ~1.5!, instances of logarithmicstress singularities with the inhomogeneous boundary condi-tions of Table 6 can be identified. Typically by this means,logarithmic singularities in problems have been identified inthe literature as follows: Conditions I8 I in Kolossoff @15#and Dempsey @39#; Conditions I8 II in Sinclair @40#; allcombinations of other conditions in Sinclair @34# and Sinclairand Epps @38#.

The configurations so found are given in Table 7: Thereinthere are twelve different sets of specifications for configu-rations with log singularities. In this table, k continues to be

10Apparent because displacements which are linear in r can have discontinuities in

their derivatives when r0 on different u.

rities under inhomogeneous boundary conditions

cations

0Df80

0, ff*1, f 0

fk , p0 or q0 or Df80, k2f21 tan fq0, f p0 or q0, k112 cos 2f22f21 sin 2f

Df0 or f Df80, f 23f/2All of the preceding singularities for the in-plane loading ofan elastic plate occur with homogeneous boundary condi-tions on its radial edges. Here we consider what additionalsingular stress fields can be induced by inhomogeneousboundary conditions.

For applied tractions which are themselves singular, inte-rior stresses are at least likewise singular. There is also thepotential of logarithmic intensification as in ~2.26!. This canoccur if the configuration of interest shares the same singu-larity as in the applied tractions when under correspondinghomogeneous boundary conditions. This would mean log-squared singularities in the event that the applied tractionswere logarithmically singular. However, it would not seemthat either power or log singularities in applied tractions arelikely to be needed in practice.

What is more likely are nonsingular applied tractions. Ifthey are ord(rg) as r0 and g.0, then the interior stressesare also nonsingular. This is so even if they get multiplied byln r because rg ln r50 at r50 when g.0. Alternatively, ifthe applied tractions are ord(r0) as r0, we may see a tran-sition between stresses which are nonsingular for tractionsthat are ord(rg), to stresses which are singular for tractionsthat are ord(r2g). Pure logarithmic singularities are naturalcandidates for such a transition: We look for further instancesof their being induced by uniform tractions in what follows.

For applied displacements, stress singularities can also beproduced. In the first instance, these stem from prescribeddisplacements which are not continuously differentiable ~asin uu}Ar). Then, singular stresses simply match the singu-larity in displacement derivatives ~see, eg, Browning and Ju

Table 7. Configurations with logarithmic singula

Boundary conditionson u0,f Configuration specifi

I8 or VI-I f5p , 2p , q0f5f , p0 or qtion. In Conditions I8, p is an applied pressure while q is asa constant shear. In Conditions II8, Df can be interpreted asthe original Williams eigenfunctions aree problems considered in Williams @2#.

es with inhomogeneous boundary condi-

@35#!. Again, they can be logarithmically isingularity coincides with that for homogeconditions. If, on the other hand, prescribeare continuously differentiable, generally noties are produced. An apparent exception odisplacements are linear in r , the integral ofconditions in effect.10 For these conditions, wexpect pure logarithmic singularities.

Given the importance of identifying logagular configurations, henceforth in this sectinhomogeneous boundary conditions as in Tathe counterparts of those in Table 1 which

geneous boundary conditions for in-plane loading

oundaryonditions

Physicaldescription

u52p , tru5q Uniform tractionsu5rDf , ur5rDf8 Pinching with lateral constraintu5rDf , tru5 f su Pinching with friction

57, no 5, September 2004 Sinclair: Stress singularities in classicalot f, fp , 2p , fk , Df0 or Df80, k3


Downlconstrained as in ~2.30! while f*

, fk , and f k continue tobe as in ~2.31!. By suitably adjoining rigid body rotations,any combination of boundary conditions drawn from Tables1 and 6 can be realized by the combinations givenin Table 7.

A first instance of a logarithmic stress singularity in Table7 occurs for a step shear on a half-plane (f5p and q0).The full stress field is given in Kolossoff @15#. A relatedinstance occurs for a constant shear on one side of a crack(f52p). Complete fields are given in Dempsey @39#. In

0 in the local boundary conditions. This is in contrast to thelog singularities of Table 5 whose actual participation de-pends on far-field conditions.

A further instance of a logarithmic stress singularity forConditions I8 I in Table 7 occurs in Levys problem, al-though such a log field is not included in the original solutionin Levy @41#. This problem entails an angular elastic plate ofvertex angle f subjected to a uniform pressure p on one edgewhile being free of stress on the other ~Fig. 3a wherein f5f ). Levys traditional solution to the problem may be

Fig. 3 Examples of configurations with logarithmic stress singularities: a! Levys problem for a reentrant corner (f5f*

), b! pressure ona clamped acute corner (k52), c! symmetric indentation by a frictionless rigid sharp plate, d! displacement shape functions as boundaryconditions for a submodel in FEA

396 Sinclair: Stress singularities in classical elasticityII Appl Mech Rev vol 57, no 5, September 2004both of these cases, the log singularity must participate if q *found in Article 45, Timoshenko and Goodier @42#. By way

oaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 04/07/2014 Terms of Use: http://asme.org/terms

Downloaded Fromof example, the normal stress su in Levys solution, in termsof the polar coordinates r and u of Fig. 3a, may be expressedby

su52pF12 sin u cos~f2u!2u cos fsin f2f cos f G (2.32)In ~2.32!, it can be seen that su takes on the values of 2p ,0at u50,f , respectively, and that there is no logarithmic sin-gularity in su . However, also clear in ~2.32! is that the so-lution breaks down for f5f

*of ~2.31!. This breakdown for

the critical vertex angle of f*

is passed by without commentin Levy @41#. It is noted in Fillunger @43#, but perhaps is notas widely recognized today as it could be ~eg, no mention ofits existence is made in Timoshenko and Goodier @42#!.Nonetheless, it is serious and must be remedied if any physi-cal sense whatsoever is to be made of elasticity treatments ofa loaded plate which is as in Fig. 3a.

Supplementing the fields used to generate ~2.32! by thoseattending ~1.2! rectifies the situation. This is done in Demp-sey @39#. The resulting su , for example, may be expressedby

su52pF12 uf*

2csc f

*2f

*2 ~2~sin~2u2f*!

2~2u2f*

!cos f*

!ln r1~2u2f*

!~cos~2u2f*

!

2cos f*

!!G (2.33)for f5f

*. Now there is a log singularity for this vertex

angle. Complete fields are given in Dempsey @39#. A reason-able transition between ~2.32! and ~2.33! is achieved in Ting@44#.

While it was once understandable to regard the break-down in the traditional solution to Levys problem as para-doxical ~as in Sternberg and Koiter @45#!, armed with theanalytical developments of Dempsey @39# and Ting @44#, itnow would seem to be far less so. Thus here rather than termf

*in Levys problem a critical angle, we view it as a tran-

sition angle associated with a logarithmic stress state whichis transitional much as in Section 2.3.

All of the foregoing examples occur for vertex angleswhere l51 is an eigenvalue. That is, for angles where g50 in Fig. 2. Such f represent transition angles in the fol-lowing sense. As the vertex angle f in angular elastic platesincreases, there is a companion steady increase in the singu-lar character of stresses near the plate vertex ~see Fig. 2!.These stresses go from power singularities in their deriva-tives while being themselves bounded (g502), to havingpower singularities in themselves (g501). Transitionangles with transitional log singularities demark the twotypes of behavior.

We identify such transition angles with the letter Tthroughout Fig. 2. Hence for the free-free plate of Fig. 2a,

Appl Mech Rev vol 57, no 5, September 2004we have T1 , T2 , and T3 corresponding to f of f* , p, and

: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 04/07/22p.11 In addition to a logarithmic stress singularity inducedby the pressure p for f5f

*here, we have one for the

uniform shear qa generalization of Levys problem in ef-fect. Fields for this log singularity may be found in Dempsey@39#.

Another generalization of Levys problem is included inTable 7. This occurs when the plate edge without appliedtractions is clamped rather than free. That is, for ConditionsI8 II. Typically there are four transition angles with loga-rithmic stress singularities for this type of configuration~Table 7, ~2.30! and ~2.31! for general k; T4 T7 in Fig. 2dfor k52). These angles can be less than 180 ~eg, Fig. 3b!.Fields for associated log singularities may be obtained fromSinclair @40#.

One other generalization of Levys problem is also in-cluded in Table 7. This occurs when the plate edge withoutapplied tractions is in contact. That is, for Conditions I8 V.There is a range of transition angles with logarithmic stresssingularities for this type of configuration ~Table 7!; ex-amples are distinguished as T8 T11 in Fig. 2e. Again anglescan be less than 180. Fields for associated log singularitiesmay be obtained from Sinclair @34#.

Typically, the preceding logarithmic stress singularitiesinduced by uniform tractions can instead be produced bycohesive laws. This is because cohesive law conditions canadmit rigid body translations which in turn produce uniformtractions. Thus Conditions VI are generally shown as alter-natives to Conditions I8 in Table 7. In this role, the condi-tions given on p and q in Table 7 then apply to correspond-ing uniform tractions within Conditions VI.

Turning to logarithmic stress singularities induced by in-homogeneous displacements, we first consider those attend-ing contact conditions. That is, Conditions V8 V in Table 7.For the case of an elastic angular plate being symmetricallyindented by a rigid frictionless plate with a sharp corner ~Fig.3c!, the finite rotations of one plate edge with respect to theother produce log singularities. This is so even for smallrotations (0,Df!1). Local fields can be assembled fromthose for ~1.2!. Thus with the r and u coordinates of Fig. 3c,

H srsuJ 5 4m11k Dff F2 ln r1 H 13J G (2.34)with tru50, and

ur52r

11kDf

f@~k21 !ln r21# , uu52ru

Df

f(2.35)

Evident in ~2.34! is a log singularity which must participatefor any Df0.

Asymptotically the same log singularity as in ~2.34! maybe extracted from the global problem of indentation with

11The case of f52p is not obviously a transition angle in Fig. 2a. This is because itis on a branch with g.0 only when f.2p , a range of vertex angles not included in

Sinclair: Stress singularities in classical elasticityII 397Fig. 2a. This branch can be seen in Fig. 2a, Seweryn and Molski @20#, near a5f/25p therein.


Downl48.4, Sneddon @46#.12 Such response can be expected to bethe case in other configurations wherein a plate vertex anglegets extended or compressed. That is, that there is an isolatedlogarithmic stress singularity with a coefficient proportionalto the relative amount of rotation and the elastic moduli ofthe material rotatingsee Brock @47# and references therein.

Other configurations with logarithmic stress singularitiesin response to contact conditions are identified in Table 7.These can also be viewed as transition stress states associ-ated with transition angles ~eg, T12 and T13 of Fig. 2c!.Fields can be obtained from Sinclair @34#.

Logarithmic stress singularities can be induced by inho-mogeneous displacements without contact conditions. Thisoccurs for Conditions II8 II in Table 7. For the case of astraight boundary (f5p), these are the spurious log singu-larities that can be introduced by the use of shape functionsas boundary conditions in submodeling in finite elementanalysis. An example involving four node elements is shownin Fig. 3d. Therein, log singularities at the node at O occurwhenever there is a discontinuity in the derivatives of theboundary displacements u and v . That is, whenever the con-stants are such that c18c28 or c38c48 . Fields are given inSinclair and Epps @38#. These spurious singularities whenshape functions are prescribed also occur for higher orderelements and on any smooth submodel boundary ~ibid!.

Other configurations with logarithmic stress singularitieswhen Conditions VI and II8 occur in concert are identified inTable 7. These, too, are associated with transition angles ~eg,when g50 in Fig. 2f!. Fields can be obtained from Sinclair@34#.

In closing this section we observe that most of the logsingularities identified in Table 7 stem from compliance withthe last of ~1.5! for nA54 when rA53. Consequently, theydo not require repeated roots of the eigenvalue equation.Indeed, for the most part, repeated roots are specifically ex-cluded in Table 7. Just exactly when this is done in Table 7can be determined by comparing it with Table 5, every set ofspecifications in the latter table corresponding to a repeatedroot. Moreover, when such exclusions are relaxed and re-peated roots admitted, typically ln2 r stress singularities areproduced, in accordance with the first of ~1.5!. The onlyexception is for the second set of specifications for Condi-tions VV in Table 5 because the rank requirement is notmet.

As an example of a log-squared singularity, we considersymmetric indentation by a rigid sharp plate as in Fig. 3c, butnow with lateral motion on the contacting edges completelyconstrained. That is, Conditions II8 II8 with Df0 andDf850. For f5f

*of ~2.31! and k51, l51 is a repeated

root ~see Table 5, cf Table 7!. The corresponding fields canbe assembled from those for ~1.1!, ~1.2!, and ~1.4!. Algebraicdetails can be obtained from Sinclair @34#. In terms of the rand u coordinates of Fig. 3c, the resulting fields have:

12 21There is a factor of a missing from the stresses given at the end of Section 48.4,where a is as in Fig. 87 therein.

oaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.orgH srsuJ 5 22mDff*2 sin f

*@2 cos f*~ ln

2 r12 ln r2u2!

H 12J @2~cos 2u2cos f*!~ ln r11 !22u sin 2u1f

*2 cos 2u#

tru52mDf

f*2 sin f

*

@sin 2u~2 ln r1f*2 12 !12u~cos 2u

2cos f*

!# (2.36)

ur5rDf

f*2 sin f

*

@2~cos f*

2cos 2u!ln r12u sin 2u

2f*2 cos 2u#

uu5rDf

f*2 sin f

*

@2~sin 2u22u cos f*

!ln r12u~cos 2u

2cos f*

!1f*2 sin 2u#

for k1. Other fields with log-squared singularities may beobtained from @34,40#.

3 STRESS SINGULARITIES FOR THE IN-PLANELOADING OF AN ELASTIC PLATE MADE OFMULTIPLE MATERIALS

3.1 Formulation and eigenvalue equationsHere we consider extension of the treatment presented inSection 2 to plates made up of multiple elastic sectors. Wefirst formulate this extended class of problems for homoge-neous boundary conditions. Then we outline analyticalmeans that can be used to derive companion eigenvalueequations. We stop short of actually presenting all theseequations because of their relative complexity, but do furnishreferences which contain them subsequently in Section 3.2.

To begin, we continue to use cylindrical polar coordinatesr and u with origin O to describe the entire angular region ofinterest R, with its complete vertex angle f. Now, though, Ris comprised of N subregions, Ri , i51,2,.. . ,N , and f of Nsubangles, f i ~Fig. 4!. Thus

R5 i51

NRi , f5(

i51

N

f i (3.1)

where

Ri5$~r ,u!u0,r, , u i21,u,u i% (3.2)

u i5 (i851

i

f i8

with the understanding u050. With these geometric prelimi-naries in place, we can formulate our class of compositeproblems as follows.

In general, we seek the planar stress components sr , su ,and tru and their companion displacements ur and uu , as

Appl Mech Rev vol 57, no 5, September 2004rotation of a half-space. This problem is solved in Section

398 Sinclair: Stress singularities in classical elasticityIIfunctions of r and u throughout R, satisfying: the appropri-


there. Quite frequently in singularity analysis the special caseof frictionless ( f 50) contact is treated, so we distinguish the

Downloaded Fromassociated conditions by B0 in Table 8. Conditions A and Bare the most common in singularity analysis.

Table 8. Interface conditions for in-plane loading

Identifyingletter

Matchedquantities

Additionalconditions

Physicaldescription

A su , truur , uu

Perfectly bonded

B su , truuu

tru5 f su Contact with frictionB0 su , tru

uu

tru50 Frictionless contact

C su , truur

su50 Separating lockingsurfaces

D suur , uu

ur50 Thin rigid inclusion

E su , tru su5k(uu12uu2)1 2

Adhesivestress-separation lawstru5k8(ur 2ur )

: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 04/07/2Conditions C are from Rao @48#. They do not appeal asbeing particularly physically applicable. They might be inter-preted as the conditions for a surface which is rough to thepoint of locking and thereby prohibiting slip (ur matched!,yet on the point of separation (su0). Such an interface issketched in Fig. 5a.

Conditions D are also from Rao @48#. Essentially they arethe same as conditions given in Erdogan and Gupta @49#.They are the composite counterpart of Conditions IV whenthe latter are interpreted as being for a thin rigid reinforce-ment. As such, they model a thin inclusion which is rela-tively stiff compared to its surrounding matrix: It is stiffenough to restrain extension, however it is not so stiff that itrestrains bending.

Conditions E are the composite counterparts of cohesivestress-separation laws. Thus k and k8 are the stiffnesses as-sociated with springs resisting normal and lateral separa-tion on an interface. This action for normal separation issketched in Fig. 5b where

uu15 lim

uu iuu~u.u i! (3.3)

with uu2 defined analogously. In the elastic regime, the stiff-

nesses in these laws should be chosen so that they are con-sistent with the elastic constitutive laws of the materialscomprising the interface. When this is done, the adhesiveconditions are the physically appropriate ones for a perfectlybonded interface: Conditions A are just a simplification ofthem obtained by effectively letting k and k8 instead oftheir elastic values.

Conditions E also admit to other interpretations. One is asa model of a flexibly bonded interface in studies of elasticwave interactions in Jones and Whittier @50#. Another is as amodel for an interface in a composite which permits someslip in Lene and Leguillon @51# ~for this latter interpretation,k is effectively taken to be infinite, though k8 is finite!.

As in Section 2.1, the preceding formulation is absentconditions at infinity and insists on bounded displacements.The basic reasons for these two aspects remain the same.However, we are not aware of a formal extension of thecompleteness argument for elastic fields with bounded dis-placements to composite configurations. Absent such, theregularity conditions ~2.5! must be viewed as provisional

Fig. 5 Sketches of interfaces; a! separating locking surfaces ~Con-ditions C!, b! adhesive law action ~Conditions E!

Sinclair: Stress singularities in classical elasticityII 399ate field equations of elasticity; interface conditions on inter-nal plate edges; boundary conditions on external edges if theplate is open (f,2p), or further interface conditions if it isclosed (f52p); and regularity requirements at the platevertex. The field equations hold on Ri (i51,2,.. . ,N) and aregiven by ~2.2!, ~2.3!, and ~2.4! with m and k in ~2.3! beingreplaced by m i and k i , where m i is the shear modulus of thematerial comprising Ri and k i5324n i for plane strain, (32n i)/(11n i) for plane stress, with n i being Poissons ratioof this material. The admissible interface conditions arelisted in Table 8 and hold on u5u i with i51,2,.. . ,N21, ifthe plate is open, i50,1,.. . ,N if the plate is closed (i50 andN are for but one set of interface conditions!. The admissibleboundary conditions continue to be as in Table 1 and hold onu50,f if the plate is open. And the regularity requirementsare the same as ~2.5! but now hold on Ri , i51,2,.. . ,N .

The interface conditions of Table 8 merit comment. Con-ditions A are the traditional conditions usually assumed for aperfectly bonded interface. Conditions B are for contact withfriction governed by Amontons law. As such, to be physi-cally applicable they further require that the normal stress benowhere tensile on the interface, as in ~2.6!, and that relativelateral motion on the interface be opposed by shear tractions

Fig. 4 Geometry and coordinates for the composite angular elasticplate

Appl Mech Rev vol 57, no 5, September 2004when applied to N-material plates.


Downlconditions/interface conditions remain as in ~1.3!. Now,though, the order of the determinant involved is typicallyincreased to

nA54N (3.4)Hence the algebra involved in expanding determinants toobtain eigenvalue equations in closed form can be consider-ably more extensive. While the eigenvalues from the deter-minant could simply be numerically calculated without alge-braic expansion, it is nonetheless useful to obtain asimplified single expression for the eigenvalue equation.Such expressions are more readily used than the raw deter-minant when the analysis of further specific configurations isrequired. In addition, typically such expressions facilitatechecking by comparison with special cases/other indepen-dent algebraic analysis. To assist in obtaining them, someapproaches for helping with the algebra entailed are offeredin Dempsey and Sinclair @3# and Ying and Katz @52#.13

Once an eigenvalue equation is obtained for anN-material problem (N>2), verification is important. Thisis a key concern because of the extent of the algebra in-volved. As previously mentioned, sometimes such verifica-tion is afforded by other independent analysis. Otherwise, inaddition to the obvious check of redoing the algebra, one canalso perform numerical checks. That is, evaluate the expres-sion for the eigenvalue equation for diverse values of theparameters involved, then compare with a direct calculationof the determinant from its originating matrix. Such compari-sons need to take account of any factors removed in simpli-fying the expansion of the determinant to obtain the eigen-value equation. They should also be carried out for parametervalues which do not, in themselves, realize simplifications ofthe determinant.

Once checked, eigenvalue equations need to be solved forsingular eigenvalues. Generally this requires numericalanalysis. Such numerics are straightforward for the mostpart. The eigenvalues so computed can be verified by backsubstitution.

At this point, the entire analysis can be further checked byconsidering limiting cases. For bimaterial plates with Condi-tions A, one check is afforded by setting

m15m2 , k15k2 (3.5)Then the eigenvalues for the corresponding single materialconfiguration should result.

A second check for bimaterials is to let one of the twomaterials tend toward being rigid. Consider the fields in ~1.1!under the limit m: The displacements go to zero. Wetherefore set displacements to zero in the interface conditionsof Table 8 to recover the corresponding boundary conditionsof Table 1 for the one remaining deformable sector. Hence asm i ,

13It is also possible to employ symbolic manipulation codes to expand determinants. Atpresent this usually results in lengthy expressions for the determinant. Consequently,

such codes typically only provide an alternative to direct numerical treatment of theoriginal determinant.

oaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.orgAII, BV, B0III (3.6)CIV, DII, EVI

for R32i and i51 or 2. Again singular eigenvalues shouldmatch single material values.

A third check for bimaterials is to let one of the twomaterials become limp. Now consider the fields in ~1.1! un-der the limit m0, but first make the exchanges mc1 forc1 ,mc2 for c2 , and so on to avoid unbounded displacements.Now the stresses go to zero. We therefore set stresses to zeroin the interface conditions to recover corresponding bound-ary conditions. Hence as m i0,

AI, BI, B0I (3.7)CI, DIV, EI

for R32i and i51 or 2. Again, singular eigenvalues shouldmatch single material values.

On occasion the eigenvalue equation for a bimaterial isinsensitive as to whether m1 or m20, or vice versa.This simply means it should recover both of the eigenvalueequations for the corresponding boundary conditions in ~3.6!and ~3.7! under either limit. For example, the eigenvalueequation for the interface crack can be written as

05sin2 lpF12 4m1m2~m11m2!2 sin2 lpG (3.8)where m15m11k1m2 and m25m21k2m1 . Equation ~3.8!is insensitive as to whether m1 or m20. From ~3.6! and~3.7!, these limits correspond to AII or AI. Thus theinterface crack ~I-A-I! becomes a half-plane with II-I or I-I.Under either limit, ~3.8! recovers the product of the eigen-value equation for a clamped-free half-plane ~~2.17! for f5p) with the eigenvalue equation for the free-free half-plane ((2.9)3(2.13) for f5p).

In addition to serving as checks, the limiting cases of~3.5!, ~3.6!, and ~3.7! enable a ready first assessment of thesingular stresses involved when faced with a new bimaterialconfiguration which lacks any singularity analysis. It is alsopossible to extend the application of these types of limits toconfigurations involving more than two materials.

For the general numerical analysis of eigenvalues forother than special cases, the parameter space to be searchedis now increased significantly in dimension over that attend-ing configurations comprised of a single material. This isbecause it now includes multiple vertex angles as well asmultiple pairs of elastic moduli.

For bimaterial plates, dimensional analysis reduces thenumber of independent elastic moduli from four to three.This number can be further reduced by employing just thetwo material constants a and b defined by

HabJ 5m2S k1H 12J 1 D2m1S k2H 12J 1 D

m2~k111 !1m1~k211 !(3.9)

The a and b of ~3.9! are given in Dundurs @53#. This articleis a discussion which points out the reduction in the number

Appl Mech Rev vol 57, no 5, September 2004Analysis follows that for single material plates. The con-ditions for singularities with homogeneous boundary

400 Sinclair: Stress singularities in classical elasticityIIof independent elastic constants that can be achieved by the


Downloaded Fromintroduction of a and b into the butt joint problem treated inBogy @54#. Equivalent a and b were given earlier in Zak andWilliams @55# to reduce the number of independent elasticconstants for the specific problem of a crack terminating per-pendicular to a bimaterial interface. Dundurs @56# establishesgeneral criteria for bimaterial configurations under whichsuch reductions can be made.

To demonstrate how the reduction is effected, we considera perfectly bonded bimaterial with stress-free edges ~I-A-I!.Absent a difference in materials, this plates stresses arecompletely independent of elastic moduli, as is any singular-ity exponent ~see ~2.9! and ~2.13! for I-I!. Consequently, onlythe traditional matching conditions associated with perfectbonding can introduce any dependence on elastic moduli.Without loss of generality, we take the perfectly bonded in-terface to occur at u50. Then, from ~1.1!, the matching con-ditions result in the following sparse set of equations:

c111lc3

11c315c1

21lc321c3

2

c111lc3

12k1c315

m1m2

~c121lc3

22k2c32!

(3.10)c2

11lc412c4

15c221lc4

22c42

c211lc4

11k1c415

m1m2

~c221lc4

21k2c42!

In ~3.10!, the constants associated with the material aboveu50 and moduli m1 and k1 are distinguished with a plussign, those with material below and m2 and k2 with a minussign. Now subtracting the second of ~3.10! from the firstgives c3

1 in terms of c12 and c3

2 and the two combinations ofelastic moduli

m22m1m2~k111 !

,

m21k2m1m2~k111 !

(3.11)

Back substituting into the first of ~3.10! then gives c11 in

terms of c12 and c3

2 and the same two combinations. Andperforming the same operations on the third and fourth of~3.10! gives c4

1 and c21 each in terms of c2

2 and c42 and the

same two combinations. Thus I-A-I stresses and singularityexponents need only depend on the two combinations ofelastic moduli given in ~3.11!. While these two combinationsare closer to those used in Zak and Williams @55# than thosein Dundurs @53#, with some algebra they can be shown to beequivalent to a and b of ~3.9!.

Similar analysis establishes that moduli dependence canbe reduced to just that on a and b for bimaterials and withany of the interface conditions A, B ~and therefore B0), or C,under any combination of boundary conditions involving I,III, or IV ~Table 9!. Given the equivalence of cohesive lawconditions with stress-free conditions as far as eigenvalueequations are concerned, singular eigenvalues with Condi-tions VI in bimaterials and with interface conditions A, B, orC can also be expected to depend only on a and b.

The constants a and b have seen widespread use for suchconfigurations since Dundurs @56#, and have come to be

Appl Mech Rev vol 57, no 5, September 2004known as Dundurs parameters. They admit to physical inter-

: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 04/07/2pretation to a degree. For plane stress, substituting for m andk in terms of Youngs modulus E and Poissons ratio n gives:

a5E22E1E21E1

, bU5 n22n14E25E1

(3.12)

Thus, a is a normalized measure of the mismatch in Youngsmoduli, while b reflects the difference in Poissons ratioswhen there is no difference in Youngs moduli. Similar re-sults hold for plane strain if E is exchanged for E/(12n2).

For bimaterials, the ranges 0,m i, , 0

DownlFig. 7 Open bimaterial plate geometries analyzed for stress singu-larities: a! butt joint, a8) oblique butt joint, b! two plates of equal

vertex angles, c! angular plate on a half-plane

oaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.orgfrom Tables 1 and 8 are I-A-I. Williams @58# provides boththe eigenvalue equation and resulting complex singular ei-genvalue in closed form. These results are confirmed in Bogy@59#. The eigenvalue equation is equivalent to that given in~3.8!, while the associated singularity exponent is

g512 , h5

12p ln

m1m2

(3.14)

where m1 and m2 are as in ~3.8!.For the interface crack ~Fig. 6a! with clamped conditions

~II-A-II!, Theocaris and Gdoutos @60# give an eigenvalueequation and complex singular eigenvalue in closed form.Ting @61# furnishes a different expression for the imaginarypart of the complex singular eigenvalue: This latter result isconfirmed in Ballarini @62# and elsewhere.14 The singularityexponent from Ting @61# for the clamped interface crack issimilar to that for stress-free flanks. It has

g512 , h5

12p ln

k2m1k1m2

(3.15)

Thus the imaginary part differs by at most 60.175 from thatin ~3.14!.

For the interface with one flank free and the otherclamped ~I-A-II!, Theocaris and Gdoutos @60# give an eigen-value equation. Closed-form expressions for singular eigen-values are given in Ting @61#.

For the interface crack with contact with friction betweenthe crack flanks ~B-A!, Comninou @63# provides an eigen-value equation. This equation is confirmed in Dempsey andSinclair @64#. Singular eigenvalues follow by inspection andare furnished in Comninou @63#, as is the companion eigen-function. The simpler frictionless case (B0-A) is treated inthe Appendix of Comninou @65#.

For the interface crack when there is contact with frictionon the interface ahead of a stress-free crack ~I-B-I!, Gdoutosand Theocaris @22# provides an eigenvalue equation in termsof Dundurs parameters. This equation is confirmed inComninou @66#. An expression for the resulting singular ei-genvalues is given in Gdoutos and Theocaris @22#. The sim-pler frictionless case (I-B0-I) is treated in Dundurs and Lee@67#.

Finally, for the interface crack when an inextensible in-clusion is inserted into the crack ~D-A!, Dempsey @21# givesan eigenvalue equation. Closed-form expressions for singulareigenfunctions are given in Wu @68#.

We next consider kinked interface cracks. Here the geom-etry for these cracks is taken to be such that the crack stilllies between the two materials but now terminates at a kinkon their interface ~Fig. 6a8!. This geometry may be viewedas a generalization of that for the previous straight interfacecrack ~Fig. 6a!.

For the kinked interface crack ~Fig. 6a8! with stress-freeflanks ~I-A-I!, Bogy @59# furnishes the eigenvalue equationin terms of Dundurs parameters. This eigenvalue equation isconfirmed in Dempsey and Sinclair @64#. In addition, Bogy@59# provides singular eigenvalues for a variety of kinked

Appl Mech Rev vol 57, no 5, September 2004ously well appreciatedthe contribution of this genre of in-vestigation lies in the implications of the globalconfiguration analyzed, rather than singularity identification.

For the interface crack of Fig. 6a with stress-free crackflanks, the corresponding boundary and interface conditions

Fig. 6 Bimaterial crack geometries analyzed for stress singu-larities: a! interface crack, a8) interface crack ending at a kink onthe interface, b! crack ending orthogonal to an interface, b8) crackending obliquely to an interface

402 Sinclair: Stress singularities in classical elasticityII14It is also implicit in Erdogan and Gupta @49#.


Downloaded Frommined for the most part. Further numerical eigenvalues aregiven in Chen and Hasebe @69#. Theocaris and Gdoutos @60#and van Vroonhoven @70# also treat kinked interface crackswith stress-free flanks, but do not use Dundurs parameters.

For the kinked interface crack ~Fig. 6a8! with clampedconditions either on one flank ~I-A-II! or both ~II-A-II!,Theocaris and Gdoutos @60# gives eigenvalue equations andsome singular eigenvalues.

For the kinked interface crack with crack flanks perfectlybonded ~A-A!, equation ~19! of Bogy and Wang @71# is theeigenvalue equation in terms of Dundurs parameters. Thisequation is confirmed in Dempsey and Sinclair @64# and else-where. In addition, Bogy and Wang @71# provides singulareigenvalues for quite a variety of such kinked configurations.Chen and Nisitani @72# provides the associated eigenfunctionas well as further eigenvalues. Van Vroonhoven @70#, Pageau,Joseph, and Biggers @73#, and Chaudhuri, Xie, and Garala@74# also treat the same kinked configuration without usingDundurs parameters.

For the kinked interface crack when there is contact withfriction between the crack flanks ~B-A!, an eigenvalue equa-tion may be found in Dempsey and Sinclair @64# in terms ofDundurs parameters. Corresponding singular eigenvalues fora variety of such configurations are numerically determinedin Dempsey @21#. If contact with friction also occurs on theinterface ahead of the crack ~B-B!, an eigenvalue equationmay be found in Dempsey and Sinclair @64# in terms of Dun-durs parameters.

For the kinked interface crack when Conditions C ofTable 8 hold, eigenvalue equations in terms of Dundurs pa-rameters for A-C, B-C, and C-C may be found in Dempseyand Sinclair @64#. Finally, for the kinked interface crackwhen Conditions D of Table 8 hold, eigenvalue equations forA-D, B-D, C-D, and D-D are given in Dempsey @21#.

We now consider cracks terminating at an interfacerather than lying along it. The simplest such configuration iswhen the crack impinges at a right angle ~Fig. 6b!, becausethen the geometry is symmetric enabling symmetric and an-tisymmetric loading to be analyzed separately. As a conse-quence, this special case has received attention by a numberof investigators in the literature.

For a crack terminating normal to an interface ~Fig. 6b!and having stress-free flanks ~I-A-A-I!, Zak and Williams@55# furnishes an eigenvalue equation for loading which issymmetric about the crack. This equation is in terms of pa-rameters which are equivalent to those of Dundurs. It is con-firmed in Dempsey and Sinclair @64#.15 Zak and Williams@55# provides singular eigenvalues. Further singular eigen-values are given in Khrapkov @75# and Bogy @76#. The ei-genvalue equation for antisymmetric loading is given in

15There would appear to be a typographical error in the equation in Zak and Williams@55#. All that is needed to correct this error is to replace cos p with cos lp.

: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 04/07/2Dempsey and Sinclair @64#. This equation is the same as forsymmetric loading.16

For a crack terminating normal to an interface ~Fig. 6b!with contact with friction between the flanks ~B-A-A!,Comninou and Dundurs @77# furnish an eigenvalue equationin terms of Dundurs parameters. Corresponding singular ei-genvalues are provided: These are independent of the valueof the coefficient of friction.

For a stress-free crack terminating normal to an interfacewhich is itself in contact with friction ~I-B-B-I!, eigenvalueequations when loading is symmetric or antisymmetric arefurnished in Dempsey and Sinclair @64# in terms of Dundursparameters. These equations are confirmed in Wijeyewick-rema, Dundurs, and Keer @78# which in addition providessingular eigenvalues for both modes of loading and a rangeof values of the coefficient of friction. The simpler case offrictionless contact (I-B0-B0-I) is treated in Gharpuray, Dun-durs, and Keer @79#.

Finally, for a stress-free crack terminating normal to aninterface on which Conditions C or D hold, eigenvalue equa-tions are available as follows: for I-C-C-I with either sym-metric or antisymmetric loading, from Dempsey and Sinclair@64# in terms of Dundurs parameters; for I-D-D-I and eithersymmetric or antisymmetric loading, from Dempsey @21#.

For the more general instance of a crack terminating ob-liquely ~Fig. 6b8!, several investigations are available. Whenthe crack is free of stress ~I-A-A-I!, Bogy @76# furnishes theeigenvalue equation in terms of Dundurs parameters, as wellas singular eigenvalues for a variety of such configurations.Fenner @80# and Yong-Li @81# compute singular eigenvaluesdirectly from the determinant without algebraic expansion,though Fenner @80# does establish that eigenvalues dependon only two material constants. The eigenvalues in Fenner@80# and Yong-Li @81# include ones which agree closely withcorresponding values in Bogy @76# ~provided a state of planestress is assumed in Yong-Li @81#!. Wang and Chen @82#treats the same configuration: On occasion, the singular ei-genvalues in Wang and Chen @82# agree with correspondingvalues in Bogy @76#, but in some instances there are signifi-cant discrepancies between the two.

For a crack terminating obliquely at an interface ~Fig.6b8! with flanks in contact with friction ~B-A-A!, Comninouand Dundurs @77# furnishes an eigenvalue equation in termsof Dundurs parameters. Comninou and Dundurs @77# alsoprovides singular eigenvalues for varying angles of incidenceof the crack and different coefficients of friction.

For a stress-free crack terminating obliquely to an inter-face which is itself in contact with friction ~I-B-B-I!,Wijeyewickrema, Dundurs and Keer @78# furnishes an eigen-value equation in terms of Dundurs parameters. When sim-plified for special instances, this equation agrees with othersin the literature. The simpler case of frictionless contact(I-B0-B0-I) is treated in Gharpuray, Dundurs, and Keer @79#.

16Taken together, the eigenvalue equation for both symmetric and antisymmetric re-sponse is given as a simple squared term in Bogy @76#. This equation appears to havean extraneous factor of sin2 lpsee ~28! and ~18! et seq ibid. The same factor is

Sinclair: Stress singularities in classical elasticityII 403interface cracks. These eigenvalues are numerically deter-

Appl Mech Rev vol 57, no 5, September 2004present in the eigenvalue equation for when the crack terminates obliquely. It wouldnot appear to lead to errors in eigenvalues reported.


DownlThere are some further generalizations for stress-freecracks terminating at an interface ~I-A-A-I! which are ana-lyzed in the literature. If the crack flanks in Fig. 6b8 areallowed to subtend a finite angle at their tip and thus becomea reentrant corner, an analysis may be found in Tan andMeguid @83#. If the interface in Fig. 6b8 is allowed to have akink at the point where the crack terminates, an analysis maybe found in Pinsan and Zhuping @84#.

We next consider open bimaterial plates which do not, forthe most part, involve cracks ~Fig. 7!. We begin with prob-ably the simplest such configuration, the butt joint of Fig. 7a.When the outside surfaces are free of stress and the joint isperfectly bonded ~I-A-I!, Bogy @85# furnishes the eigenvalueequation in terms of Dundurs parameters. This equation isconfirmed in Dempsey and Sinclair @64#. Bogy @85# also pro-vides singular eigenvalues: These are consistent with corre-sponding values in Hein and Erdogan @86#. The ratio of theshear moduli for which a power singularity first starts toappear is given in Kubo, Ohji, and Nakai @87#.

For the butt joint ~Fig. 7a! with stress-free outside surfaceand contact with friction on the interface ~I-B-I!, Theocarisand Gdoutos @88# furnishes the eigenvalue equation in termsof Dundurs parameters. This equation is confirmed in Demp-sey and Sinclair @64# ~the sign of the friction coefficient hasto be changed because the friction condition is applied on anegative u-face in Theocaris and Gdoutos @88#, a positiveu-face in Dempsey and Sinclair @64#!. In addition, Theocarisand Gdoutos @88# provides singular eigenvalues for varyingcoefficients of friction.

The more general oblique butt joint here has the interfacemeet the outside free surface at an angle other than 90 ~Fig.7a8!. When the joint is perfectly bonded ~I-A-I!, Bogy @59#furnishes the eigenvalue equation in terms of Dundurs pa-rameters. This equation is confirmed in Dempsey and Sin-clair @64#. Singular eigenvalues are also provided in Bogy@59# for several angles of incidence of the interface with theoutside surface. Further singular eigenvalues are given inHein and Erdogan @86# and Rao @48#. Geometries for whicha power singularity first starts to appear are given in Rao@48#, and Kubo, Ohji, and Nakai @87#.

For the oblique butt joint ~Fig. 7a8! when the interface isin contact with friction ~I-B-I!, Theocaris and Gdoutos @88#furnishes the eigenvalue equation in terms of Dundurs pa-rameters. This equation is confirmed in Dempsey and Sin-clair @64# ~again, the sign of the friction coefficient has to bechanged!. In addition, Theocaris and Gdoutos @88# providessingular eigenvalues for several angles of incidence andvarying coefficients of friction.

A further open bimaterial geometry investigated in theliterature is that of two plates with equal vertex angles ~Fig.7b!. When the outside edges of the plates are stress free andthey are perfectly bonded along their interface ~I-A-I!, Bogy@59# furnishes the eigenvalue equation. This equation is con-firmed in Dempsey and Sinclair @64#. Singular eigenvaluesare also provided in Bogy @59# for several plate vertexangles. Further singular eigenvalues are given in Rao @48#.Geometries for which a power singularity first starts to ap-

404 Sinclair: Stress singularities in classical elasticityIIpear are given in Rao @48# and Kubo, Ohji, and Nakai @87#.

oaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.orgThe stress-free bimaterial plate, with constituent plateswith equal vertex angles ~Fig. 7b!, can have the interface incontact with friction ~I-B-I!. Theocaris and Gdoutos @88# fur-nishes the eigenvalue equation in terms of Dundurs param-eters under these conditions. This equation is confirmed inDempsey and Sinclair @64# ~again, the sign of the frictioncoefficient has to be changed!. Theocaris and Gdoutos @88#also provides singular eigenvalues for several vertex anglesand any value of the coefficient of friction.

The last open bimaterial geometry investigated quite fre-quently in the literature is that of a plate sector on a half-plane ~Fig. 7c!. When the outside edge of the plate and half-plane surface exterior to it are free of stress, and the two areperfectly bonded along their interface ~I-A-I!, Bogy @59# fur-nishes the eigenvalue equation in terms of Dundurs param-eters. This eigenvalue equation is confirmed in Gdoutos andTheocaris @22#. Singular eigenvalues when the vertex angleof the plate is 90 are provided in Bogy @59#. Singular ei-genvalues for other vertex angles are given in Hein and Er-dogan @86# and Gdoutos and Theocaris @22#.

For a plate on a half-plane ~Fig. 7c! when the plate is incontact with friction ~I-B-I!, Gdoutos and Theocaris @22# fur-nishes the eigenvalue equation in terms of Dundurs param-eters. This equation is confirmed in Comninou @66#. Singulareigenvalues are provided in Gdoutos and Theocaris @22# forplate angles of 60 and 90 and varying friction coefficients.Singular eigenvalues for some other vertex angles are givenin Theocaris and Gdoutos @89#. The simpler case of friction-less contact (I-B0-I) is treated in Rao @48# and Dundurs andLee @67#.

There are some other bimaterial plates with asymptoticanalysis in the literature. For the perfectly bonded bimaterialplate with stress-free edges ~I-A-I! and arbitrary vertexangles, an eigenvalue equation is given in Aksentian @90#. Interms of Dundurs parameters, it is furnished in Bogy @59#.This latter equation is confirmed in Dempsey and Sinclair@64#. If one or both of the edges are clamped instead, respec-tive eigenvalue equations are furnished in Dempsey and Sin-clair @64#. These equations are confirmed in Ying and Katz@52#.17 The second configuration is also investigated in Aveti-sian and Chobanian @91#. If both edges have rigid thin rein-forcements ~IV-A-IV!, the eigenvalue equation is given inDempsey and Sinclair @64# in terms of Dundurs parameters,and some eigenvalues are given in Rao @48#.

Eigenvalue equations for bimaterial plates with other in-terface conditions are available as follows. In terms of Dun-durs parameters for stress-free bimaterial plates with differ-ent interface conditions ~I-B-I and I-C-I! and arbitrary vertexangles, eigenvalue equations are given in Dempsey and Sin-clair @64#. In terms of Dundurs parameters for further closedbimaterial plates ~A-C, B-C, and C-C!, eigenvalue equationsare also given in Dempsey and Sinclair @64#. Eigenvalueequations involving Conditions II and Conditions B or C aregiven in Dempsey @21#. Eigenvalue equations involvingConditions D are also given in Dempsey @21#.

17In addition, these equations appear to be consistent with those given in Aksentian

Appl Mech Rev vol 57, no 5, September 2004@90# provided mi is taken to be Poissons number rather than Poissons ratio as statedon p 193. That is, provided mi51/n i .


Downloaded FromTurning to the trimaterial plate, the simplest of such con-figurations occurs when the plates are all comprised of thesame material. Picu and Gupta @92# treats a closed plate ofthis type with frictionless contact on its interfaces(B0-B0-B0). Singular eigenvalues are independent of elasticmoduli and are given for a range of vertex angles in Picu andGupta @92#. Other degenerate trimaterial plates wherein thereare not three distinct materials include the crack geometriesof Figs. 6b and b8 reviewed earlier.

When a trimaterial plate is actually comprised of threedistinct materials, analysis can be extensive. Nonetheless,there are some true trimaterial plates investigated in the lit-erature. For an open trimaterial plate with bonded interfacesand stress-free/clamped exterior edges ~I-A-A-I, I-A-A-II, orII-A-A-II!, Ying and Katz @52# derives eigenvalue equations.An eigenvalue equation for the first of these configurations~I-A-A-I! is given in terms of pairs of Dundurs parameters inKoguchi, Inoue, and Yada @57#, as are some resulting singu-lar eigenvalues. Further singular eigenvalues from the sameequation are presented in Inoue and Koguchi @93#. Additionalsingular eigenvalues are given in Pageau, Joseph, and Big-gers @73#, together with some singular eigenvalues for theclosed and bonded trimaterial plate ~A-A-A!. The nature ofassociated singular eigenfunctions for I-A-A-I is consideredin Pageau et al @94#.

In closing, we comment on the one remaining set of in-terface conditions in Table 8, Conditions E. With these adhe-sive stress-separation laws instead of the classical perfectly-bonded conditions, some reduction in the occurrence ofstress singularities is to be expected. This is indicated vialimiting cases with single-material plates. However, this isyet to be formally established in general.

3.3 Log singularities identified in the literatureHere we review contributions to the literature that have as-ymptotically established the possibility of logarithmic termsin stress singularities for N-material plates under in-planeloading. We start with when such singularities can occur withhomogeneous boundary conditions, then consider their oc-currence with inhomogeneous boundary conditions. We fo-cus on bimaterial plates and follow the same order of geom-etries as previously in Section 3.2.

Before beginning this review, we recap the requirementsfor logarithmic participation in bimaterial plates becausethese continue to be incorrectly stated/applied in the litera-ture. For homogeneous boundary conditions as in Table 1,conditions for logarithmic intensification of power singulari-ties are as in the second of ~1.3! with nA58 for bimaterials.For the case of pure logarithmic singularities, conditions areas in the penultimate of ~1.3! with nA58. For inhomoge-neous boundary conditions as in Table 6, conditions for alog-squared singularity are as in the first of ~1.5! for nA58.For the case of pure logarithmic singularities, conditions areas in the last two of ~1.5! for nA58. Throughout these con-ditions for bimaterials, corresponding inequalities for cs andcs are to hold on at least one Ri (i51,2), while equationsfor cs are to hold on both.

Appl Mech Rev vol 57, no 5, September 2004For a pure logarithmic singularity, conditions other than

: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 04/07/2the preceding continue to be advanced in the literature ~eg,Murakami @95# and Wijeyewickrema et al @78#!. Typicallythese have

D5]D]l

50 for l51 (3.16)

While appealing in its simplicity, ~3.16! is not sufficient for alog singularity with homogeneous boundary conditions, andit is not necessary for a log singularity with inhomogeneousboundary conditions. To remove any doubt that this is so, wefurnish some demonstrations.

As a first demonstration of ~3.16! not being sufficient withhomogeneous boundary conditions, we consider the interfacecrack ~Fig. 6a! with crack flanks perfectly bonded together.That is, Conditions A hold both ahead of and in back of thecrack tip. The determinant for this case is given in equa-tion ~25!, Bogy and Wang @71#. In terms of the eigenvalue l,this has

D52~12b2!2 sin4 lp (3.17)Clearly D of ~3.17! satisfies ~3.16!. The coefficient matrix Awhich leads to D can be assembled from ~1.1! on applyingConditions A on u50,p . Checking the rank of this matrixreveals that it drops to four when l51. Thus ~1.3! requiresthat the first four derivatives of D be zero when l51. The Dof ~3.17! has just its first three derivatives being zero whenl51. Consequently, no log singularity is possible for thisconfiguration despite the fact that ~3.16! is met. This is whatone would expect because this configuration has two per-fectly bonded half

Asymptotic Check for Singularity

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