553

A review of asymptotic procedures in stress analysis:

known solutions and their applications

D A Hills*, D Dini, A Magadu and A M Korsunsky

Department of Engineering Science, University of Oxford, Oxford, UK

Abstract: A comprehensive review is given of the origins of asymptotic procedures in stress analysis.
Specifically, attention is focused on the use of fracture mechanics to characterize the elastic stress state
ahead of a crack tip. Analogies are then drawn between this configuration and the stress state adjacent
to the apex of a sharp V-notch. Extensions of these asymptotic procedures to bonded and slipping
contacts are then considered and it is shown that although power order singularities may be
obtained, the solutions are more complicated. Lastly, the use of nested asymptotic procedures are
considered in order to account for a small but finite radius at the tip of cracks and notches or at the
edge of slipping contacts.

Keywords: asymptotic analysis

NOTATION 1 INTRODUCTION

a crack half-length The most commonly encountered asymptotic procedure

G strain energy release rate in stress analysis is the concept of fracture mechanics and
Ki stress intensity factor (i
I, II and III) the framework it provides for the prediction of both
ðr; Þ polar coordinate set at the notch tip or wedge monotonic fracture loads and the rate of growth of
apex fatigue cracks. As this may be a familiar subject the
ui local displacement field in the i direction x or y ideas behind it will be developed in some detail, before
vðx, yÞ local displacement field in Cartesian going on to more general applications where the same
coordinates principles apply. In brief, the whole concept of asympto-
w local displacement field in the transverse tic analysis relies on being able to focus on some feature,
direction (out of the xy plane) such as the tip of a crack, or the apex of a sharp notch, or
ðx, yÞ Cartesian coordinate set z ¼ x þ iy the corner of some slipping complete contact, and to
recognize that the stress state in the neighbourhood of
ð, Þ Dundurs’ parameters
that feature may be the same (in the sense of having
ð1 , 2 Þ wedge angles
the same spatial distribution of stresses) as all other fea-
 infinitesimal distance
tures having the same local geometry: the influence of
 Kolosov’s constant ¼ 4  3 for plane strain,
remote boundaries forming the shape of the component,
ð3  Þ=ð1 þ Þ for plane stress
may be irrelevant. It follows that a simplified solution,
order of singularity
ignoring the presence of all remote boundaries so that
modulus of rigidity
the key feature under consideration is present in a
 Poisson’s ratio
semi-infinite domain, may correctly capture that stress
ð , Þ polar coordinate set at the crack tip
state. Thus, as the feature is approached in the finite
ij direct stress component in the direction ij
and semi-infinite bodies the stress states implied by an
tij shear stress component in the direction ij
elastic solution become the same (hence, within certain
ðzÞ complex stress function
limits, all crack tips exhibit the same state of stress).
’ notch internal half-angle
This has further implications for the strength of the com-
 stress function
ponent if failure originates at the feature; in brief, it
The MS was received on 23 January 2003 and was accepted after shows that a single quantity scaling the state of stress
revision for publication on 21 July 2004. may be used to correlate the failure load in any pair of
*Corresponding author: Department of Engineering Science, University
of Oxford, Parks Road, Oxford OX1 3PJ, UK. E-mail: david.hills@ components having the same local geometry. The most
eng.ox.ac.uk commonly encountered quantity is that employed in
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554 D A HILLS, D DINI, A MAGADU AND A M KORSUNSKY

linear elastic fracture mechanics, viz. the stress intensity Wanhill [3] and Gdoutos [4]. Many elementary
factor. approaches to the subject fail to take a strong enough
Recent developments in the applications of asymptotic account of why the subject hinges on the stress intensity
analysis have included the prediction of the fracture factor, and recent texts at the first year postgraduate
strength of notched components and fretting contacts. level, taking a more rigorous approach, include another
These will be described, together with the use of nested by Gdoutos [5], one by Kanninen and Popelar [6] and
asymptotic solutions, to permit the effects of various another by Broberg [7]. A particularly rigorous and
local variations in the geometry to be gauged, and in appropriate approach, in the present context, is the one
which the authors have been particularly involved. taken by Aliabadi and Rooke [8]. It is not possible to
develop the subject ab initio here, but the key elements
in the technique will be given.
2 FRACTURE MECHANICS Consider a crack, of half-length a, in an infinite
plane, shown in Fig. 1a. Suppose, for the time being,
There are many undergraduate textbooks on fracture that it is subjected to remote tension, 0 , in both the x
mechanics, including those by Broek [1, 2], Ewalds and and y directions. A solution is required to the boundary

Fig. 1 (a) Crack in an infinite plane. (b) Local coordinates at the crack tip

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 A REVIEW OF ASYMPTOTIC PROCEDURES IN STRESS ANALYSIS 555

value problem This solution is far more than the asymptotic stress state
8 9 8 9 relevant to the case of uniform remote loading. Provided
< xx = < 0=
that the value of KI is chosen correctly, equation (7) gives
yy ! Limit jx; yj ! 1 ð1Þ
: ; : 0; the stress at the apex of any crack, subject to remote
txy 0
loading, which simply tends to open the crack ( yy > 0,
   
yy 0 txy ¼ 0 on  ¼ 0, > 0), and is known in the jargon as
! Limit j yj ! 0, jxj < a ð2Þ mode I loading. As part of the analysis for a wedge or
txy 0
notch (section 3), this solution, i.e. equation (7), will be
This problem was first solved by Westergaard [9], who shown to correspond to a special case [equation (33)].
showed that the solution was given by A second, independent, solution may be found by an
0z analogous method (see Gdoutos [5] for details) for the
ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi ð3Þ
z  a2
2 case when the crack in an infinite plane is subject to
remote shear, txy ¼ t0 , i.e.
where ðzÞ is a complex stress function and
8 9 8 9
z ¼ x þ iy ð4Þ < xx >
> = <0>
> =
The corresponding stresses are then defined as yy ! 0 Limit jx; yj ! 1 ð10Þ
>
: >
; : >
> ;
txy t0
xx ¼ Re½ ðzÞ  y Im½ 0 ðzÞ
yy ¼ Re½ ðzÞ þ y Im½ 0 ðzÞ The corresponding potential is given by
0
txy ¼ y Re½ ðzÞ it0 z
ðzÞ ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi ð11Þ
ð5Þ z 2  a2
where 0 ðzÞ is the first-order derivative of ðzÞ. The local This complex stress function is connected to the stress
displacement field is given by field by the equations
1
2 uðx, yÞ ¼ Re½ ðzÞ  y Im½ ðzÞ xx ¼ 2 Re½ ðzÞ  y Im½ 0 ðzÞ
2
1 yy ¼ y Im½ 0 ðzÞ
2 vðx, yÞ ¼ Im½ ðzÞ  y Re½ ðzÞ
2 txy ¼  Im½ ðzÞ  y Re½ 0 ðzÞ
ð6Þ
ð12Þ
where
ðzÞ is the complex conjugate of ðzÞ, is the
modulus of rigidity,  is Kolosov’s constant [¼ 3  4 while the local displacement field is given by
in plane strain and ð3   Þ=ð1 þ  Þ in plane stress] and
 is Poisson’s ratio. þ1
2 uðx; yÞ ¼ Re½ ðzÞ  y Im½ ðzÞ
Now a new local polar coordinate set,  ¼ z  a, is 2
set up where  ð , Þ is positioned at the crack tip (see 1
2 vðx; yÞ ¼ Im½ ðzÞ  y Re½ ðzÞ
Fig. 1b). The stress function may be written in the 2
form ðÞ ¼ 0 ð þ aÞ=½ð þ 2aÞ 1=2 . By expanding this ð13Þ
expression using the binomial theorem, and taking the
lead term, it can be seen that the state of stress for Here the asymptotic solution is given by
small =a is given approximately by 8 9 8 9
8 9 8 9 < xx >
> = >  sinð=2Þ½2 þ cosð=2Þ cosð3=2Þ >
> cosð=2Þ½1  sinð=2Þ sinð3=2Þ > KII < =
< xx > = KI
>
< = yy ¼ pffiffiffiffiffiffiffiffi sinð=2Þ cosð=2Þ cosð3=2Þ ,
>
: >
; 2p > : >
;
yy ¼ pffiffiffiffiffiffiffiffi cosð=2Þ½1 þ sinð=2Þ sinð3=2Þ , txy cosð=2Þ½1  sinð=2Þ sinð3=2Þ
>
: >
; 2p > : >
;
txy sinð=2Þ cosð=2Þ cosð3=2Þ
a ð14Þ
a ð7Þ
where
where
pffiffiffiffiffiffi
pffiffiffiffiffiffi KII ¼ t0 pa ð15Þ
KI ¼ 0 pa ð8Þ
For completeness the local displacement field is also Again, for completeness the local displacement field is
recorded, which is given by also recorded as
  rffiffiffiffiffiffi    rffiffiffiffiffiffi  
ux KI cosð=2Þð  cos Þ ux KII 2 þ  þ cos 
¼ ð9Þ ¼ ð16Þ
uy 2 2p sinð=2Þð  cos Þ uy 2 2p 2    cos 

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556 D A HILLS, D DINI, A MAGADU AND A M KORSUNSKY

As before, this solution is of more general relevance, as it bounded, but the application of a remote load parallel
applies whenever the remote loading tends simply to with the crack faces, T , is, to a first-order approxima-
shear the crack ( yy ¼ 0, txy > 0 on  ¼ 0, > 0), and tion, unaffected by the presence of the crack. It therefore
is known in the jargon as mode II loading. Again, the gives rise to a contribution to the local stress state that is
analysis for a wedge or notch will show this as a special less significant than the first-order singular terms
case. derived, but may be more important than the second-
There is a third form of loading that may arise, known order bounded terms, at small but finite distances from
as ‘anti-plane’ deformation. In the notation of Fig. 1 this the crack tip. It is known, in fracture mechanics jargon,
occurs when the remote loading is in the form of a shear as the ‘T-stress’.
into or out of the plane, rather like tearing a book, and Monotonic fracture of a highly brittle material subject
the remote boundary condition is now tyz ¼ tz0 . The to remote tension applied perpendicular to the faces of
corresponding solution is given by the crack is governed by the celebrated Griffith criterion
[4, 10]. This simply states that a crack cannot extend until
tz0 z the strain energy which is released, dU, under remote
ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi ð17Þ
z 2  a2 controlled-displacement conditions, when the crack
extends by an amount da, is at least as great as that
where, in this kind of loading, the non-zero stresses are needed to form the new crack surfaces,  per unit area,
given by assumed to be a material property. A quantity G is
introduced, the strain energy release rate or generalized
tzx ¼ Im½ 0 ðzÞ crack extension force, defined by G ¼ dU=da, and
tyz ¼ Re½ 0 ðzÞ crack extension occurs when
ð18Þ
G ¼ 2 ð23Þ
while the transverse direction (normal to the xy plane) It may be shown that G is related to the mode I stress
displacement, w, is given by intensity factor by evaluating the energy released when
the crack extends by a small amount. This is found by
1
w¼ Im½ ðzÞ ð19Þ evaluating the work done in closing the crack, over an
infinitesimal distance, , giving
The asymptotic solution then takes the form  ð 
1
G ¼ 2 Limit ð  Þvð Þ dr ð24Þ
    2 0 yy
txz KIII  sinð=2Þ
¼ pffiffiffiffiffiffiffiffi ð20Þ
tyz 2p cosð=2Þ where, because the distance  is small compared with
the crack length so that the integral is evaluated
where within the region dominated byp the singular ffi region, the
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffi traction is given by
pffiffiffiffiffiffiffiffiffiffi yy ¼ KI = 2pð  Þ and vðrÞ ¼
KIII ¼ tz0 pa ð21Þ KI ½ð þ 1Þ=2 =2p. This is one part of the so-called
Irwin K  G relation, and is
and the displacement is given by
rffiffiffiffiffiffi ð þ 1ÞKI2
G¼ ð25Þ
2KIII 8
w¼ sinð=2Þ ð22Þ
2p
It demonstrates that the scaling factor for the asymptotic
crack tip stress field, the stress intensity factor, controls
2.1 Physical relevance of the solution fracture for a brittle material.
The asymptotic field has greater relevance than this,
The solutions obtained above have been discovered by however. Firstly, in the presence of a very modest
several analysts and by different routes. Each gives an amount of plasticity, the assumption that the stress
independent contribution to the singular elastic stress intensity factor controls fracture is adopted as a hypoth-
field adjacent to a crack tip. As the solutions are elastic esis, simply because the state of stress in the fracture zone
they may be applied together, from the principle of is itself proportional to KI . Thus, it is argued that there is
superposition, to give a ‘mixed mode’ crack tip field. a material property that represents the critical value of
Note that, in terms of a series expansion of the state of the stress intensity factor at which fracture occurs. This
stress, the next term in the series is one where the stress ‘material property’ is dependent on the degree of trans-
pffiffiffi
state varies as for all three components of loading, verse constraint but is substantially constant under
i.e. next in sequence to equations (7), (14) and (20). plane strain conditions, when it is known as the fracture
The contributions from the second terms are therefore toughness and given the symbol KIC . This hypothesis has
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 A REVIEW OF ASYMPTOTIC PROCEDURES IN STRESS ANALYSIS 557

been experimentally tested and been shown to be true, key parameters at the crack tip may be thought of as
provided that the plasticity which is present at the functions of both Kmax and Kmin . These are not the two
crack tip is modest. best choices of characteristic parameter, however: it is
The relationship between the strain energy released better to use the range of stress intensity experienced
when the crack extends and the stress intensity factors (K ¼ Kmax  Kmin ) and something quantifying the
present may be extended to modes II and III loading, average value of the local loads. The R ratio is normally
using, again, an infinitesimal increase in crack length as chosen for this, which may be defined either classically
the vehicle. The following complete form for the relation- using the nominal stress present in the crack’s absence
ship between G and KI , KII , KIII emerges or, for consistency, in terms of the stress intensity factors
as R ¼ Kmin =Kmax : Note that this definition requires
ð þ 1ÞðKI2 þ KII2 Þ KIII
2
that Kmin 5 0, and hence R 5 0. The crack growth rate
G¼ þ ð26Þ
8 2 is a strong function of K and a weak function of
R. It is emphasized again here that the actual size of
It is postulated that the crack will extend when the value the plastic zone is necessarily very small for the principles
of G reaches a critical value, with contributions from all behind elastic fracture mechanics to hold. The plastic
three forms of remote load, although this extension to zone must be much smaller than the region over
the original concept is not universally accepted (see Sih which the singular term solutions dominate the state of
[11, 12]). The point is made, however, that the elastic stress.
singular solution contributions must again control the Note that the fracture of heterogeneous materials may
complete fracture process. not introduce any fundamentally new principles, pro-
A consideration of the relevance of the crack tip stress viding that the size of each characteristic phase or com-
intensity factors to the problem of crack propagation ponent is large compared with the process zone size,
under cyclic loading will now be considered. Crack pro- and the fracture principles cited may be applied on a
pagation occurs by the local exhaustion of plasticity in a pointwise basis. The presence of interfaces between
small region (the ‘process zone’) at the tip of the crack. It those phases may, however, introduce further problems,
is not known what attributes of the local stress or strain because the singularities induced there may be different.
field actually control the exhaustion process, but it is This issue will be addressed in section 5.
clear that they must be associated with the local plastic
stress and strain. These quantities are themselves difficult
to quantify, because they require the solution of a fully- 3 NOTCH PROBLEM
fledged elastic–plastic problem: indeed, even if it were
possible to find the true local values of stress and The results reviewed above, abstracted from standard
strain, that would still not represent a complete solution fracture mechanics theory, may be arrived at in another
to the problem, because there is no universal criterion for way, involving not the asymptotic expansion of the local
the exhaustion of ductility. However, the argument is stress field around the tip of a crack in a particular geo-
made that, if the process zone is small compared with metry but from a solution constructed in an infinite
an elastic hinterland in which the singular solution domain. The best way to do this is to consider a semi-
itself dominates, all the relevant quantities controlling infinite notch or wedge (Fig. 2) of internal angle 2’
the inner process zone are themselves determined by a and to deduce the stress field in the neighbourhood of
single scaling variable, viz. the stress intensity factor. the notch tip. This is a well-posed problem, first solved
The concept of ‘small-scale yielding’, which must be by Williams [14], and the general results obtained are
satisfied for crack growth to be controlled by the stress of practical relevance, as they may be used to define
intensity factor, is precisely this. In fact, there are stron- characteristic singular local fields. As a special case, by
ger requirements on the nature of the stress state that the making the internal notch half-angle equal to p, the
crack tip experiences: (a) if the T-stress is significant in characteristic stress fields derived above may be deduced
proportion to the singular field at the position of the directly. The details of the algebra needed may be found
process zone front, that may have an influence on the in the books by Barber [15] and by Aliabadi and Rooke
fatigue performance; (b) if the crack tip is experiencing [8], and here only an outline of the method will be repro-
combined modes loading, each non-zero mode contri- duced. The nomenclature used in fracture mechanics
butes to the local stress field and, if the ratio between relating to the three independent modes of crack tip
the stress intensity factors varies during the loading loading is adopted for notches. In-plane loading will
cycle (the so-called ‘non-proportional loading’ regime), therefore be considered first. Polar coordinates ðr, Þ
the crack may propagate in a way that is difficult to centred on the notch apex (Fig. 2) are appropriate and
predict; the latter is the subject of current research [13]. the biharmonic equation is then given by
Lastly, it is known that it is the range of plastic strain  2 
(rather than its maximum value) which is principally @ 1 @ 1 @2 2
þ þ  ¼0 ð27Þ
responsible for the rate of crack extension. Thus, the @r2 r @r r2 @2

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558 D A HILLS, D DINI, A MAGADU AND A M KORSUNSKY

Fig. 2 Semi-infinite wedge of notch

where the stress components are related to the stress homogeneous equations:
function, , by 2
cosð  1Þ’ cosð þ 1Þ’
6  sinð  1Þ’ sinð þ 1Þ’
1 @ 1 @ 2  6
¼ þ 6
rr
r @r r2 @2 4 0 0
@2 0 0
 ¼ 38 9
@r2 0 0 > A1 >
> >
> >
1 @ 1 @ 2  0 0 7< A =
7 2
tr ¼  7
r2 @ r @r @ sinð  1Þ’ > A3 >
sinð þ 1Þ’ 5>
> >
ð28Þ : > ;
 cosð  1Þ’ cosð þ 1Þ’ A4
8 9
If it is assumed, pro tem, that there are tractions varying > 0>
> >
> >
in a power series along the faces of the notch, the form of < 0=
the above equations would suggest an investigation of a ¼ ð31Þ
>0>
> >
variables-separable solution of the form : >
> ;
0
þ1
 ¼r FðÞ ð29Þ where  ¼ ð  1Þ=ð þ 1Þ. The first two equations
correspond to a symmetric solution while the second
If this is substituted into the biharmonic equation the r two are antisymmetric. The solution of these equations
variation is seen to be satisfied, and FðÞ emerges as requires the determinants of the separate pairs of equa-
tions to vanish, leading to the following eigenequations:
FðÞ ¼ A1 cosð  1Þ þ A2 cosð þ 1Þ
sin ’  sin 2’ ¼ 0 ð32Þ
þ A3 sinð  1Þ þ A4 sinð þ 1Þ ð30Þ
where the symmetric solution is associated with the þ
This solution is now substituted into equation (28) to sign and the antisymmetric solution with the  sign.
obtain the stresses and the traction components are The corresponding eigenvalues are plotted in Fig. 3. It
set on the faces  ¼ ’ to 0. This then yields the should be recalled that the stress state varies as r  1 ,
following set of two pairs of uncoupled simultaneous so the stress state is bounded when  1 > 0 and is
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 A REVIEW OF ASYMPTOTIC PROCEDURES IN STRESS ANALYSIS 559

Fig. 3 Order of singularity,  1, at a notch apex

singular when  1 < 0. The figure displays the results in [16]. For example, under pure mode I loading a plot is
the practically important singular region and indicates made of  ðr, 0Þr1  I against r. As r ! 0,  ! 1 but
that the symmetric solution is singular if ’ > p=2 radians, their product is finite and equal to KI .
while the antisymmetric solution is singular only if Note that if ’ is set to p in equation (32) above and the
’ > 128:78. Further, the symmetric solution is always resulting eigenvalue ( ¼ 0:5) is back-substituted into
more strongly singular than the antisymmetric one, for equation (31) for the case of symmetric loading, equation
any given notch, so that, unless special precautions are (33) becomes identical to equation (14).
taken to suppress the symmetric mode (in practice this The singular solutions for notches are less well known
rarely happens; indeed, it is very hard to contrive), the than their specialized counterparts, the singular crack-tip
symmetric solution will dominate the solution for suffi- solutions, described in section 2. It must be emphasized
ciently small r:† If the eigenvalue is back-substituted into that the mode I and II terms do not necessarily contri-
the relevant pair of equations in equation (31) the ratio bute equally to the magnitude of the local stress and
A1 =A2 (symmetric solution) or A3 =A4 (antisymmetric the former dominates. Potentially, these solutions take
solution) may be found. These then serve to fix the spatial on a function analogous to the crack tip stress fields,
distribution of stress with the polar angle , allowing the but they have found rather less practical application so
overall solutions to be written in the form far. This is partly because sharp notches are always
avoided, if at all possible, and partly because, even
ij ðr, Þ ¼ KI r I 1
FI ðÞ þ KII r II  1
FII ðÞ ð33Þ when the geometry does arise, there is very often at
least some notional radius present at the notch root
where the multiplicative factors Kn take on the rôle of that may invalidate the solution. This point will be
generalized stress intensity factors. Their value is, in addressed in a later section. There remains significant
practice, found by collocating the asymptotic solution practical objections to the use of the generalized stress
with the local stress state given by the full solution, nor- intensity factors as controllers of the notch monotonic
mally found by the finite element method of Tur et al. fracture strength, not least because there is no release
of energy to propel the subsequent crack when the
†
The asymptotic solution is usually used to infer properties of the notch forms a crack [17]. However, there is less objection
process zone. As this is of finite extent the contribution from the
antisymmetric term may be significant, especially if the multiplicative to their use as controllers of the size of the local process
factors in the solution are comparable. zone and hence as the basis of a crack initiation criterion.
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560 D A HILLS, D DINI, A MAGADU AND A M KORSUNSKY

A third asymptotic solution applies when the notch Now set the traction component of stress ðtz Þ to zero on
root is loaded in anti-plane shear (the mode III solution the faces  ¼ ’, to give the following characteristic
in fracture mechanics nomenclature), although the equation:
authors have not been able to find a reference to it in
sin 2 ’ ¼ 0 ð39Þ
the literature. Its derivation is straightforward and a
beginning is made by writing the governing differential from which the relevant eigenvalue is given by
equation for this class of problem, viz.
p
¼ ð40Þ
Fðr, Þ ¼ 0 ð34Þ 2’
where Back-substituting, as before, yields the eigenfunction
@F and hence the spatial distribution of the non-zero shear
trz ¼  stresses, which take on a particularly simple form. The
@r
solution (see Fig. 3) is very similar in behaviour to the
1 @F symmetrical solution for the in-plane problem. It shows
tz ¼
r @ that all re-entrant wedges are singular, with, as expected,
ð35Þ the strongest singularity (square root) when the wedge is
It is assumed that Fðr, Þ may be written in the variables- folded to form a crack, i.e. 2’ ¼ 2p radians. For any
separable form Fðr, Þ ¼ r TðÞ, so that given wedge angle the mode III singularity is weaker
than the mode I solution, but stronger than the mode
2 2 @2T II solution.
Fðr, Þ ¼ r Tþ ð36Þ
@2
from which 4 BONDED COMPONENTS

Fðr, Þ ¼ r ðc1 cos  þ c2 sin Þ ð37Þ

When two elastically similar bodies are bonded together,
and the only non-zero stresses are given by there may be an abrupt change in stiffness across the bond
line. Where the bond line meets the free surface a singular
@F 1
trz ¼  ¼ r ðc1 cos  þ c2 sin Þ state of stress may arise, depending on the local contact
@r geometry. Some simple cases are shown in Fig. 4. They
1 @F 1 include the butt joint, where the interface line meets the
tz ¼ ¼ r ðc1 sin  þ c2 cos Þ
r @ free surface normally (Fig. 4a), and this is seen to be a
ð38Þ special case of the scarf joint (Fig. 4b). In these cases

Fig. 4 Examples of bonded components for idealized geometries

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 A REVIEW OF ASYMPTOTIC PROCEDURES IN STRESS ANALYSIS 561

Fig. 5 (a) Bonded elastically dissimilar wedges. (b) Dundurs’ parallelogram for 1 ¼ p and 2 ¼ p=2

the free surface is locally straight, whereas locally attach- appeared in the solution. Subsequently, Dundurs [19]
ing a block also produces a geometric discontinuity (Fig. showed that the key features of the solution depended
4c). If it is found that this gives rise to a very severe stress on only two quantities, which have now entered the litera-
intensity it is possible to modify the local geometry to ture as the ‘Dundurs’ constants’. These are
produce a pedestal mounting (Fig. 4d), and in the case
of a soldered or brazed joint the weld/braze material ð 2 = 1 Þð1 þ 1Þ  ð2 þ 1Þ
¼
may produce a local feature such as that shown in Fig. ð 2 = 1 Þð1 þ 1Þ þ ð2 þ 1Þ
4(e). Problems of this kind were studied by Bogy [18], ð 2 = 1 Þð1  1Þ  ð2  1Þ
by representing the feature as two bonded elastic semi- ¼
ð 2 = 1 Þð1 þ 1Þ þ ð2 þ 1Þ
infinite wedges (Fig. 5a), so that the salient features of ð41Þ
the geometry and interface may be included. The under-
lying problem that has to be studied is therefore that of where the subscripts 1 and 2 refer to the bodies shown in
two elastically dissimilar bonded wedges, as shown in Fig. 5a. Bogy subsequently re-examined his results [20]
Fig. 5a. The solutions to such problems clearly depend and published them in a more comprehensive form,
on a total of four elastic constants (two for each material) making use of the ‘Dundurs’ parallelogram diagram’
and, indeed, in the original paper all four constants (Fig. 5b). If Poisson’s ratio for both materials lies in the
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562 D A HILLS, D DINI, A MAGADU AND A M KORSUNSKY

range 0 4 i 4 12 , all physically acceptable combinations where

of materials must lie within this parallelogram. In
particular, Fig. 5b shows the values of to be found Að1 , 2 ; Þ ¼ 4Kð , 1 ÞKð , 2 Þ
below, relating to Fig. 5a, for the special case 1 ¼ p
2
and 2 ¼ p=2. Although the solution method used by Bð1 , 2 ; Þ ¼ 2 sin2 ð1 ÞKð , 2 Þ
Bogy was slightly different, a direct method of attack, 2
based on a variation of the technique employed by Wil- þ2 sin2 ð2 ÞKð , 1 Þ
liams, would seem to be the most efficient way to derive
the results. For each wedge, a variables-separable solu- Cð1 , 2 ; Þ ¼ 4 2 ð 2
 1Þ sin2 ð1 Þ sin2 ð2 Þ
tion of the kind employed in section 3 is written down. þ K½ , ð1  2 Þ
Subscripts 1 and 2 are added for the two bodies and it
is supposed that the included angles of each are Dð1 , 2 ; Þ ¼ 2 2 ½sin2 ð1 Þ sin2 ð 2 Þ
1 ð< 0Þ and 2 ð> 0Þ, with the line  ¼ 0 denoting the
interface (Fig. 5). In equation (28) a start was made by  sin2 ð2 Þ sin2 ð 1 Þ
displaying the stress–stress function relationships, in
order to be able to write down the traction components. Eð1 , 2 ; Þ ¼ Dð1 , 2 ; Þ þ Kð , 2 Þ  Kð , 1 Þ
The displacement to stress function relationships are
Fð1 , 2 ; Þ ¼ K½ , ð1 þ 2 Þ
now additionally required, in order to be able to develop
ð46Þ
the continuity conditions across the interface. These are
  and
@ur 1 1 @ 1 @ 2  3
¼ þ 2 2  r2 
@r 2 r @r r @r 4
Kð , sÞ ¼ sin2 ð sÞ  2
sin2 ðsÞ ð47Þ
 
@u u 1 @ur 1 1 @2 1 @
 þ ¼  þ The roots of this determinant govern the behaviour of
@r r r @r r @r @ r2 @
ð42Þ the state of stress in the following way. If 1 is the root
of  having the smallest real part in the strip
As before, the free outer faces are traction free, so that, 0 < Reð Þ 4 1, then
on  ¼ 1 , 2 , the condition  ¼ tr ¼ 0 is required,
On the interface  ¼ 0 the following continuity condi- ij ¼
tions apply: 8 1
>
> Oðr 1
Þ if 1 real
>
>
¼ >
> 1 1
1 2 >
> O½r 1
cosð 1 log rÞ or O½r 1
sinð 1 log rÞ
>
>
tr1 ¼ tr2 >
< if 1 ¼ 1 þ i1 is complex
u1 ¼ u2 >
>
>
> Oðlog rÞ if 1 ¼ 1 and @=@ ¼ 0 at 1 ¼1
ð43Þ >
>
>
>
>
> Oð1Þ if no zeros of  in the strip
>
:
and, in addition, there is no relative radial displacement,
and @=@ 6¼ 0 at 1 ¼1
so that
ð48Þ
ur1  ur2 ¼ 0 ð44Þ
The conditions for bounded/singular behaviour are
These eight conditions give rise to a homogeneous set more complicated than before and are found in the
of equations for the pair of wedges and the correspond- following manner. From equation (48) this condition is
ing eigenvalue problem permits the nature of the defined by the locus of points along which the logarith-
exponent in the expressions for the stresses to be mic singularities occur (as long as no other root occurs
determined. This is very much more complicated than in the strip). Therefore, @=@ is found and 1 ¼ 1 is
the monolithic case and the eigenequation will not be substituted into the resulting expression which, when
derived here. Instead, it will merely be quoted; the evaluated to zero, gives the curve delineating the
determinant that is required to vanish is bounded/singular regions. As an illustration consider
the special case of 1 ¼ p and 2 ¼ p=2 (Fig. 5a). Fol-
ð1 , 2 , , ; Þ
lowing the above procedure, 2ð1 þ Þ2 ¼ 0 is obtained
¼ Að1 , 2 ; Þ2 as the locus and note that the point  ¼ 1 satisfies this
condition. Results for this example case are given in
þ 2Bð1 , 2 ; Þ þ Cð1 , 2 ; Þ2 Fig. 5b. As there are now four independent physical vari-
þ 2Dð1 , 2 ; Þ þ 2Eð1 , 2 ; Þ ables in the problem (1 , 2 ,  and ) a comprehensive
display of the characteristic solution to the problem is
þ Fð1 , 2 ; Þ ð45Þ not feasible.
J. Strain Analysis Vol. 39 No. 6 S00903 # IMechE 2004
 A REVIEW OF ASYMPTOTIC PROCEDURES IN STRESS ANALYSIS 563

5 INTERFACE CRACK This solution implies an oscillatory displacement field

with points of interpenetration, which cannot be physi-
A special case of the above problem is when two bodies cally correct. The exact solution implies that, over a
are bonded along a common, straight interface, save over short distance attached to the crack ends, the crack
a small region where the bond fails, leaving an interface faces are pressed together, so that there is no mode I
crack (Fig. 4f ). This may be studied using the Bogy pro- singularity. There is, however, a mode II singularity
cedure by specializing each of the wedges to a half-plane, ahead of the crack tip.
which gives rise to some interesting and surprising The most important feature of this solution is the
results. In particular, it is found that, with external load- nature of the ‘physically incorrect’ bilateral solution
ing, which would normally cause opening of the crack (the one ignoring the possibility of crack face contact).
tips, in mode I, they actually remain closed. If an attempt Although this predicts contact at a number of points
is made to model the crack as if it were open to the tips, it within the crack, the interpenetration zones are widely
is found that there is local implied interpenetration, as separated. If a new angle, i , is defined by the relation
found by England [21]. In order to get round this prob- i ¼ argðuy þ iux Þ ð52Þ
lem a complete solution to the problem of a finite
crack in an infinite plane was first provided by Comni- then equation (51) gives
nou [22], who assumed that the crack faces adjacent to
r
the crack tip were pressed together, giving rise to a i ¼ argðKÞ þ " log  tan1 ð2"Þ ð53Þ
local contact pressure. The solution she derived implies, a
in fact, a singular distribution of contact pressure, so that Interpenetration zones are defined by cosði Þ < 0, i.e.
there is a mode I compression field in (a, b), whereas
ahead of the crack tip the material sees mode II loading. ð2n  32Þp < i < ð2n  12Þp ð54Þ
This solution was subsequently refined by Gautesen and
Dundurs [23]. It seems counterintuitive, even though it is where n is any integer. This shows that the relative dis-
mathematically rigorous. The contact length adjacent to tance between any adjacent pair of contact regions is in
the crack tip is very small indeed (although it may be the ratio ½expð2"=pÞ 4 . This is a very large number
quite large if there is also remote mode II loading [24]) indeed: its minimum possible value is about 3:9  1015 ,
and a little further out from the crack tip the nominal so that this implies that, if there is a point of inter-
stress fields look very much like the conventional ones penetration of order of the crack length, the next one
present at an open crack tip. If the process (plastic) down in size is truly minute, so minute as to be swamped
zone extends to a region in which the local stress field by a plastic region at the crack tip for any practically
looks like that of an open crack tip, the refined solution important load. This illustrates that, while the classical
may not be needed. This question will be addressed in bilaterally defined solution is formally incorrect, in prac-
more detail later, where the application of nested asymp- tice it is of good quality, with physical inconsistencies
totes to help address this problem is described. occurring only within the region where, in any case, the
The classical ‘bilateral’ (traction-free faces) solution to material is not in an elastic state and therefore the solu-
the interface crack problem gives rise to stresses present tion is invalid. Details of the argument, and its extension
along the line of the interface of the form [25] to configurations loaded remotely in shear, are given in
reference [25].
i"
K r
yy þ itxy ¼ pffiffi ð49Þ
r a
6 FRICTIONAL CONTACTS
where r is the distance from the crack tip, a the crack
half-width, K a complex quantity analogous to the The problem of interfaces between components in rela-
stress intensity factor and " is a dimensionless bimaterial tive motion is now discussed. The intended application
parameter defined by of this class of solution is to components in contact but
suffering relative tangential motion in the presence of
1þ friction. In particular, the results to be derived may be
2p" ¼ log ð50Þ
1 applied to components suffering fretting damage, pro-
viding that the relative motion is not sufficiently dama-
and  is Dundurs’ parameter, defined above. The ging for wear to change the local profile, which
corresponding discontinuity in displacement across the remains unchanged. If the coefficient of friction is suffi-
crack faces is given by ciently high for the bodies to adhere, the problem
pffiffi becomes that of a monolithic contact, with a solution
½ð1 þ 1Þ= 1 þ ð2 þ 1Þ= 2 Kðr=aÞi" r
v þ iu ¼ given either by the Williams solution if the bodies
2ð1 þ 2i"Þ coshðp"Þ have the same elastic constants or the Bogy solution if
ð51Þ they are dissimilar. If the coefficient of friction is not
S00903 # IMechE 2004 J. Strain Analysis Vol. 39 No. 6
564 D A HILLS, D DINI, A MAGADU AND A M KORSUNSKY

Fig. 6 Asymptotic frictional contact problem for sliding semi-infinite wedges

sufficiently high to prevent local slip, the situation at the The case is also recorded where the two components have
contact edge may be represented by two wedges with the same elastic constants, which occurs more frequently
mixed interfacial boundary conditions; there must be in practice, so that equation [54] becomes
continuity in the  direction displacement and the shear-
ing traction must be equal in magnitude to the product of Fð f , 2 , 0, 0; Þ ¼ cos pðsin2 2  2
sin2 2 Þ
the coefficient of friction, f , and the direct traction at that
point (Fig. 6a): þ 12 sin pðsin 2 2 þ sin 22 Þ
þ f sin p ð1 þ Þ sin2 2 ð58Þ
u1 ¼ u2
jtr j ¼ f  ,  <0 It is noteworthy that, in the case where one of the
ð55Þ wedges is an elastic half-plane and the other wedge
represents the edge of a finite contacting body, the gradi-
This problem was tackled by Gdoutos and Theocaris [26] ent of the displacement field immediately exterior to the
and also by Comninou [27], which should be consulted contact edge is singular, i.e. it has a gradient normal to
for full details. In order to reduce the number of indepen- the free surface. It follows that the solution described
dent variables, body 1 is assumed to be a half-plane above is strictly valid only when the punch has an
(1 ¼ p). A further point to bear in mind is that the internal angle which is p=2 or less, if additional contact
nature of the solution depends on the direction in exterior to the contact face along the inclined flank
which slippage occurs, so that the following sign conven- (Fig. 6b) is to be avoided. A related problem has been
tion is used: studied by Adams [28] who solves explicitly for the con-
f >0 ) wedge ð2 Þ slipping in the þve x direction tacting length. This is normally quite small and, provid-
ing the plastic zone is larger in size than the contact
(away from the corner) length, the validity of a single asymptotic solution in
f <0 ) wedge ð2 Þ slipping in the ve x direction which its presence is neglected will not be invalidated.
Figure 7 shows a plot of the order of singularity,  1,
(towards the corner) against the wedge angle, 2 , when  ¼  ¼ 0 for
sample coefficients of friction. These contours are
Each of the papers cited deals also with the case where
found by plotting Fð f , 2 , 0, 0; Þ ¼ 0 [from equation
the bodies may be elastically dissimilar, when, in the
(58) above] for various 2 and constant f . Details of
same nomenclature used above (Fig. 5a), the eigen-
the interpretation of this figure can be found in
equation corresponds to the determinant of the bound-
Mugadu et al. [29]. It is clear that, for a given contact
ary value problem given by
angle, 2 , the singularity at the corner is stronger when
Dð f , 2 , , ; Þ ¼ 8ð1 þ Þ sinð pÞFð f , 2 , , ; Þ the frictional traction is directed towards the contact
ð56Þ corner than when it is directed away. Also, it is note-
worthy that, for strong coefficients of friction and
where contact angles around 1008, singularities that are slightly
stronger than the square root can be anticipated.
Fð f , 2 , , ; Þ
¼ ð1 þ Þ cos pðsin2 2  2
sin2 2 Þ 6.1 Mode III frictional contact
þ 12 ð1  Þ sin pðsin 2 2 þ sin 22 Þ
Suppose that two elastically similar wedges are pressed
þ f sin p½ð1  Þ ð1 þ Þ sin2 2 together and slid in the z direction to produce what
may be thought of as mode III loading, i.e. where the
 2ðsin2 2  2
sin2 2 Þ ð57Þ frictional shearing traction is directed parallel with a
J. Strain Analysis Vol. 39 No. 6 S00903 # IMechE 2004
 A REVIEW OF ASYMPTOTIC PROCEDURES IN STRESS ANALYSIS 565

Fig. 7 Plot showing the order of singularity,  1, against the pad angle, 2 , as a function of the coefficient of
friction, f

tangent to the contact edge, e.g. as when a ball is pressed faces of the wedges, the in-plane solution corresponds
into a block and twisted or as shown schematically in exactly to that of frictionless loading and that this part
Fig. 8a. This problem is of practical interest in determin- of the problem may be solved separately. When once
ing the local behaviour of, for example, a shrink-fitted this has been done, the effect of the antiplane traction
shaft subject to torsion, as it describes the stress state may be found by considering a straightforward bound-
when the shaft emerges from the pulley or wheel into ary value problem for each of the wedges, in isolation.
which the shrink has been made (Fig. 8b). This problem The free boundary,  ¼ 2 , is devoid of tractions:
differs fundamentally from the ‘mode II’ frictional con-
 ¼ tr ¼ 0,  ¼ 2 ð59Þ
tact described above, in that slip in the z direction pro-
duces an antiplane problem that is uncoupled from the while on the interface boundary,  ¼ 0, the direct trac-
in-plane component of loading, which here is the contact tion,  ðrÞ, is specified by the eigenfunction for the fric-
pressure,  . It follows that, as tr is zero on the slipping tionless contact problem, and tz ¼ f  .

Fig. 8 Mode III frictional slip: (a) example of where it arises and (b) idealization of the asymptotic solution

S00903 # IMechE 2004 J. Strain Analysis Vol. 39 No. 6

566 D A HILLS, D DINI, A MAGADU AND A M KORSUNSKY

It is possible to extend this approach to the case be thought of as collocating a semi-infinite crack solution
of mixed mode II and III loadings. It will be assumed into the finite crack problem. Now, moving inwards
that the two semi-infinite wedges are being slid, relative again, the outer solution for the semi-infinite slot solu-
to each other, along a line making an angle with tion is crack-like in behaviour, so this may now be
the r direction, and this solution is currently under embedded within the semi-infinite crack solution, using
investigation. the stress intensity factor as the scaling quantity. Explicit
solutions monitoring the increasing discrepancy between
the crack solution and the slot solution may be found,
7 EMBEDDED ASYMPTOTIC SOLUTIONS and hence these provide a more precise measure of the
minimum plastic (process) zone radius.
A number of situations arise in which the asymptotic The innermost solution under consideration is that for
solutions developed above are insufficient to describe a semi-infinite slot having a radiused end, recently devel-
the behaviour of the local stress state, usually because oped by Filippi et al. [31]. This paper should be consulted
the geometry of the problem in the immediate neighbour- for full details, but here the results found will simply be
hood of the discontinuity is not that implied by the math- recorded. The paper itself deals with a rounded-root,
ematical description. The most obvious case where this straight-sided notch, so the geometry is first specialized
may be so is at the root of a V-notch or crack, where by making the root sides parallel. Note that there are
there will normally be a finite radius present, and it will two solutions to this problem, one that is symmetrical
not be atomically sharp. Clearly, the presence of a in nature, so that direct stresses arise on the line of sym-
finite radius at the crack root will formally invalidate metry but with no shearing tractions (mode I in fracture
the asymptotic solution, but, if the radius is very small, mechanics jargon), while the other is antisymmetrical.
it seems intuitively correct that the asymptotic solution Here only the shear stress is non-zero on the line of
will be recovered, for practical purposes. However, symmetry and this is therefore ‘mode II’. The solutions
how big can the crack tip radius be? A simple answer are, of course, elastic, and hence may be superposed to
to this question is to say that, providing that the radius obtain any mode mixity required.
is small compared with the size of the process (plastic) A generalization of the procedure described above is
zone, then its presence will not be felt, and this is the possible for the case when the basic singular solution
usual criterion employed. Thus the range of loads that refers to a sharp notch. Exactly the same philosophy of
can be supported by a cracked component and for ‘nesting’ the solutions applies and the procedure has
linear elastic fracture mechanics to be valid may be again been made possible by the results found in Filippi
found in the following way: the upper bound to the et al. [31].
load is imposed by the requirement that the boundary
of the process zone must be well within the region in
which the singular (KI ) crack tip field dominates the 7.2 Complete contact solution (finite end radius)
state of stress, and the full field is itself geometry-depen-
The solution for the frictional slipping of wedges
dent. On the other hand, the lower bound is dictated by
described above assumes that the contact corner is
the requirement that the plastic zone must be large com-
extremely sharp and, of course, in practice this is unlikely
pared with the radius of any crack tip radius present. If
to be achieved. In the spirit of the solution described
the crack is extending by fatigue, and hence the compo-
above, the state of stress in the neighbourhood of the
nent is subjected to a fluctuating load, these bounds will
edge of a semi-infinite punch having a rounded profile
set bounds to the upper and lower loads respectively and,
has recently been deduced. Interest is principally on a
if these bounds are not satisfied, predictions based on
punch where the faces of the punch are perpendicular
linear elastic fracture mechanics theory will be unsafe.
to the end face, and therefore an ‘outer asymptote’ is
needed in the form of a semi-infinite square-ended
7.1 Crack tip and notch root solutions (finite root radius) punch. This could be found in one of two ways: either
a start could be made from the general solution for two
Recently, a more sophisticated solution to the question
wedges, specializing one to a half-plane and the other
of the tolerable crack tip radius has been devised [30],
to a quarter-plane, or, if the punch is rigid, the standard
which employs a semi-infinite rounded slot solution to
solution could be employed for a finite rigid, square-
represent the crack end and which is embedded within
ended punch and an asymptotic expansion performed
the singular crack tip solution. The spirit of the pro-
at the punch corner [32].
cedure is that a nested set of asymptotic solutions is
developed. There is an outer ‘full-field’ solution, as pre-
vious described, and, moving in towards the singularity, 7.3 Interface crack
an asymptotic solution (in the case cited, a conventional
crack tip stress intensity solution, which is scaled by the The problem of a crack present at an interface between
stress intensity factor). This element of the analysis may elastically dissimilar materials introduces issues that are
J. Strain Analysis Vol. 39 No. 6 S00903 # IMechE 2004
 A REVIEW OF ASYMPTOTIC PROCEDURES IN STRESS ANALYSIS 567

counterintuitive; it does not seem reasonable that the procedure has been applied to the crack tip, the V-
application of a remote tensile force leaves the crack notch and the slipping asymptotic solutions.
tips closed (and hence giving rise to no mode I loading),
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J. Strain Analysis Vol. 39 No. 6 S00903 # IMechE 2004