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On stress singularities at plane bi- and tri-material junctions -
A way to derive some closed-form analytical solutions
C. Satora, W. Beckera
aFachgebiet Strukturmechanik, Technische Universität Darmstadt, Hochschulstr. 1,
D-64298 Darmstadt, Germany
Abstract
Stress singularities at 2D bi- and tri-material junctions, consisting of dissimilar, homogeneous, isotropic and linear-
elastic wedges under a plane strain state are considered. The stresses formed at the vertex of this composite situation
are analyzed by the complex variable method, based on an appropriate choice of the Kolosov-potentials which are
applicable in the vicinity of the vertex. In doing so, the identification of the singularity exponent is performed. With
the help of a novel approach it is demonstrated how to derive some solutions for the orders of the stress singularities
at bi- and tri-material combinations in a closed-form analytical manner.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of ICM11
Keywords: stress singularity order, bi-material-junction, tri-material-junction, complex variable method
1. Introduction
In various exact solutions of boundary value problems in linear elasticity the stress field is found to
have singularities, e.g. when discontinuities are present in the geometry or mechanical properties of the
material. In many technical areas dissimilar materials have to be joined together, and the stress state
around points, where several materials meet can be expected to be singular. Although it might in general
not be sufficient to consider only the orders of the singularities, the type of singularity affects the
structural strength of elastic materials in a decisive manner, because these singularities correspond to
locations of high stress from which the initiation of fracture is prone to occur. For an assessment in the
scope of linear elasticity the knowledge of the orders of the stress singularities is of particular interest.
There is a multitude of contributions related to stress singularities in linear elastic media and a review
article on notch problems and crack stress-singularities has been given by Paggi and Carpinteri [1]. Some
1877-7058 © 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of ICM11
doi:10.1016/j.proeng.2011.04.026
Procedia Engineering 10 (2011) 141–146
142 C. Sator and W. Becker / Procedia Engineering 10 (2011) 141–146
early works related to stress singularities in linear elastic media are from Williams [2], Bogy [3,4], Hein
and Erdogan [5], England [6] and Theocaris [7]. Dempsey and Sinclair reported and studied power-
logarithmic singularities in a series of papers [8,9,10]. Recently, there is a renewed interest in singularity
problems of linear elasticity, because solutions of such problems can e.g. be useful for checking the
effectiveness of new numerical approaches for anisotropic multi-material junctions by comparison with
limit solutions.
In all works dealing with singularity analyses, to the best knowledge of the authors of this contribution,
the results are at last obtained numerically. In this work it is demonstrated with the help of a novel
approach how to derive some solutions for the orders of the stress singularities at bi- and tri-material
combinations in a closed-form analytical manner. From different techniques available for singularity
analyses, the complex variable method has been chosen to analyze the stresses formed at the vertex of a
plane multi-material composite situation. Based on an appropriate choice of the Kolosov-potentials
(Theocaris [7]) which are applicable in the vicinity of the vertex, the orders of the stress singularities are
determined.
2. Theoretical Setting
As shown in Fig. 1a, the general geometry consisting of dissimilar, homogeneous, isotropic and linear
elastic (Young's moduli k and Poisson's ratios k
E
Q
) material sectors under a plane strain state is
considered. The interfaces between the respective sectors are perfectly bonded and the stresses and
displacements are expressed in terms of two complex potentials and consistent with the complex
potential method given by Muskhelishvili [11].
k
)k
<
Fig. 1a) n-material-junction; b) n-material-junction with notch; c) a special bi-material configuration
It is convenient to formulate stress continuity and stress boundary conditions with the help of stress
resultants. Omitting further details, the continuity conditions at a perfectly bonded interface )( kk
M
M
*
become
>@>@
kk zz
z
kkk
zz
z
kkk zzzzzzzz
<)c
) <)c
) 0
111
0)()()()()()( , (1)
>@>@
)()()(
1
)()()(
1
1111
1
kkkkkkkk
k
kkkkkkkk
k
zzzzzzzz
<)c
) <)c
)
N
P
N
P
, (2)
where the subscripts k and k+1 define quantities or functions referred to the sectors k and k+1,
respectively. Furthermore and hold,
M
i
rez k
i
krez
M
))1(2/(
Q
P
Edenotes the shear modulus, and
if
Q
is Poisson’s ratio, then the Kolosov-constant
N
takes on the value
Q
43 for plane strain and
)1/()(3
Q
Q
for plane stress. Equation (1) enforces stress continuity along the interface between the
k
*
C. Sator and W. Becker / Procedia Engineering 10 (2011) 141–146 143
two materials k and k+1 and equation (2) enforces displacement continuity along the same interface. If an
n-material-junction with notch is considered, as shown in Fig. 1b, the stress-free surfaces and
require:
0
*n
*
>@
0)()()(
0
0
111 <)c
)
i
erz
z
zzzz ,
>
0)()()( 0 <)c
)
n
i
erz
z
nnn zzzz
M
@
. (3) and (4)
In case of an n-material-junction as in Fig. 1a, the possibility of a branch cut must be taken into
account. This can be done without loss of generality by defining 0
0
M
for material 1 and
S
M
n2 for
material n. In order to investigate the asymptotic behaviour of the stress distribution in the vicinity of the
vertex of the composite, complex functions of the form
OO
zazaz kkk 21
)( ) and
OO
zbzbz kkk 21
)( < with
ikik ba ,,
O
Ю (5)
are assumed, following an idea of Theocaris [7]. Again, the subscript k defines quantities referred to the
sector k. Introducing the potentials (5) into any of the conditions (1) - (4) leads to equations that must
hold for every value of the variable r and thus the coefficients for
O
r
and
O
r
must be equal to zero. In
this manner two equations can be gained and after conjugating the second equation, two additional
homogeneous equations for the unknown constants k
k
k
kbbaa 2
1
2
1,,, will result ([7] gives further details).
Finally there will be 4n homogeneous equations for the unknowns }...,,3,2,1{,,, 22 nkbb kk ,aa 11 kk and
for a non-trivial solution of this system of equations 0a
B
the determinant of the coefficient matrix
must be equal to zero. yields the characteristic equation, from which the roots (or
"eigenvalues")
B0B det
O
can be gained. The general expression of the stress-field is then given by the
asymptotic expansion:
>@
)()ln)sin(Im()()ln)cos(Im(),( sincos
1
1)Re(
MOMOMV
O
mm
mijmijm
m
mij frfrrKr ¦
, (6)
with the generalized stress-intensity factors m
Kъ. Note, that the orders of the stress singularities are
defined by the exponents m
O
and an imaginary part of m
O
gives an oscillatory term superimposed on the
singular term. Singular stresses result, when 1Re
O
and requiring finite strain energy in any region of
the body leads to 1Re0
O
. Another form of singularity that may appear is the power-logarithmic
stress singularity, i.e. a singularity of the form ъ, which shall not be taken into account
in this study.
O
O
1),ln r(rO
3. Examples
As an example the configuration of Fig. 1c is analyzed. This bi-material configuration has already
been investigated by Hein and Erdogan [5] using the Mellin transformation technique applied to an Airy
stress function formulation. Therefore an in-depth discussion of the configuration can be omitted. All
results of [5] were obtained numerically, whereas in the following a novel approach is presented, that
allows to derive closed-form analytical solutions in many cases. It is assumed that there are two traction-
free boundaries and one perfectly bonded interface. The elastic constants are given by k
E and k
Q
(k=1,2) and no branch cut has to be taken into account. The traction-free boundary 0
M
, the perfectly
bonded interface )0( !
D
D
M
and the traction-free boundary
S
M
require:
0
2121
11 baa
O
and 0
21
1111 aba
O
, (7) and (8)
>@>@
,
11
22
2
22
2
122
2
21
2
21
2
111
1
beaeabeaea iiii
DDODDO
ON
P
ON
P
(9)
144 C. Sator and W. Becker / Procedia Engineering 10 (2011) 141–146
,
22
2
22
2
12
21
2
21
2
11 beaeabeaea iiii
DDODDO
OO
(10)
>@>
,
11
12
2
12
2
22
2
2
11
2
11
2
21
1
1
beaeabeaea iiii
DDODDO
ON
P
ON
P
@
(11)
,
12
2
22
2
1211
2
21
2
11 beaeabeaea iiii
DODDOD
OO
(12)
0
2222
12
2
baae i
O
SO
and 0
12
22
2
12 baea i
SO
O
. (13) and (14)
Equations (9) and (11) are multiplied with 21
P
P
and this leads to the corresponding homogeneous
system of linear equations 0a
B
with the unknown quantities assembled in
>
@
T
b22
/
11 121211 . There will be nontrivial solutions if and only if is
fulfilled. With the modular ratio 21
baabbaa 222121
a0B det
:
P
P
K
, the characteristic polynomial has been
calculated with the help of the computer algebra system Mathematica which took only a few seconds on a
standard personal computer. After some simplifications, the following expression results:
B det)1/(:D 2
2
P
^
>
`
@
>
^
`
@
)2cos())(2cos(0
)2cos()(sin41)sin())32sin(()sin(2)(sin)2cos(4
)(sin))(2cos(4)sin())32sin((2)(sin2)2cos(
))(2cos()2cos(1)(sin)2cos(3))(2cos(4)(sin162
))(sin2)(sin2()1)(sin2)(
)(2cos(2)2cos(
))2(2cos()(sin2)(sin8)2cos(2)(sin6)2cos()2cos(22D
2
2222
1
222
2
2244
2
22222
2
242222
NDOODS
SODODOODSDODDOON
DODSODOODSDONDO
ODSSODDOODSODOK
NDODODOODSSO
ODSNDODODODDODOK
(17)
.)2cos(2)(sin41))((sin2)(sin22 2
11
22222
NNDODODSODO
The studies showed that changes in Poisson's ratio hardly affect the results, so a variation of k
Q
will not
be taken into account. With 10/2
21
Q
Q
(hence 10/83
21
N
N
for plane strain and 21 /EE
K
)
the characteristic polynomial can be written as ,
with
),(
OD
),(), 2
KODKOD
l
f(),,(DD
KOD
q
f c
f
(18)
^
`
>
@
,)(sin)2cos(2)(sin4321
(19)
(20)
In the following, some configurations with a constant value
D
are investigated. It is obvious, that
closed-form analytical solutions )(
K
O
O
cannot be found, since is a (highly) non-linear
transcendental equation and that is why in all studies known to the authors of this article, the results are at
last obtained numerically. However, there is a possibility to derive solutions in a closed-form analytical
manner by considering the "inverse" problem and identifying solutions
0D
()
O
K
K
. In this case, solutions
can simply be calculated as:
>
@
,)2)32( 22
DODODODSO
cos(73)(sin110)sin()sin((110)cos146
)(sin6073))(2cos()(sin)2cos(15984)(sin200
25
4
),( 222244
OS
DOODSDDOODOOD
l
f
.))((sin)(sin)(sin50)2cos(5573
25
8
),( 22222
ODSDODODOOD
c
f
)2
OO
cos(
)2cos(
5
11
)2cos())2(2cos())(2cos(2
5
11
)1)(sin2(
5
11
2),
2222
22
DDODOD
DOSOODSODSDOOD
q
f(
,
2
2/1
q
cqll
f
K
4
2ffff r
C. Sator and W. Becker / Procedia Engineering 10 (2011) 141–146 145
(21)
with clq given by (18) - (20). It is needless to say, that only real-valued solutions
fff ,,
K
ъ are of
interest. There is however a slight restriction since complex roots
O
cannot be investigated with this
method. In that case all the complex-valued branches were calculated by solving with Newton's
method. Calculating the roots turned out to be a bad conditioned problem, so the analysis had to be done
very carefully. Summing up, it can be stated, that a well-adjusted Mathematica-implementation enables to
do calculations in a robust and highly efficient manner: The pure calculating time for a plot as shown
below is only about 4 seconds on a standard PC. The ordinates of all following plots presented are limited
to the real part of
0D
O
in the range 1Re0
O
and corresponding (positive) imaginary parts of the roots are
given when the roots are complex. Note, that complex roots always occur as a pair of complex
conjugates, but in the plots only the positive imaginary part is displayed. Furthermore, it has to be kept in
mind, that ij
V
~ 1
O
Re
r
holds, thus 5.0Re
O
indicates the "classical" square root - stress singularity. In
Figs. 2-3 the angle
D
of the bi-material configuration takes on the values .
$$ 45,15
D
$$ 90,60,
Fig. 2) Orders of singularities for depicted configurations
Fig. 3) Orders of singularities for depicted configurations
146 C. Sator and W. Becker / Procedia Engineering 10 (2011) 141–146
With the method proposed in this work it is possible to derive closed-form analytical solutions for any
other bi-material configuration and in addition for tri-material configurations, too. This possibility to the
best of the authors' knowledge is a novelty. It should be noted, that analyzing a tri-material configuration
will lead to a quartic equation with the variable
K
. It is a well-known fact, that quartic equations can be
solved algebraically in terms of a finite number of operations such as addition, subtraction, multiplication,
division and root extraction.
4. Summary and Conclusion
In this contribution, stress singularities at two-dimensional multi-material-junctions, consisting of
dissimilar, homogeneous, isotropic and linear-elastic wedges under a plane strain state have been
considered. The stresses formed at the vertex of this multi-material composite situation were analyzed by
the complex variable method, based on an appropriate choice of the Kolosov-potentials which are
applicable in the vicinity of the vertex. For the case of a bi-material configuration it has been
demonstrated how to derive some closed-form analytical solutions for the orders of the stress singularities
O
ъ from the characteristic equation. Complex-valued roots
O
were calculated by solving the
characteristic equation with Newton's method. With the help of a well-adjusted Mathematica-
implementation calculations were carried out in a robust and highly efficient manner. With the method
proposed in this work it is possible to calculate all real-valued eigenvalues
O
of any bi-material- and any
tri-material configuration in a closed-form analytical manner. This possibility is a valuable novelty, since
in all works dealing with singularity analyses, to the best knowledge of the authors of this article, the
results are at last obtained numerically.
References
[1] Paggi M, Carpinteri A. On the stress-singularities at multi-material interfaces and related analogies with fluid dynamics and
diffusion. Applied Mechanics Reviews 2008; 61(2), p. 1-22.
[2] Williams ML. Stress singularities resulting from various boundary conditions in angular corners of plates in extension.
Journal of Applied Mechanics 1952; 19, p. 526-528.
[3] Bogy DB. Two edge bonded elastic wedges of different materials and wedge angles under surface tractions. Journal of
Applied Mechanics 1971; 38, p. 377-386.
[4] Bogy DB. On the plane elastostatic problem of a loaded crack terminating at a material interface. Journal of Applied
Mechanics 1971; 38, p. 911-918.
[5] Hein VL, Erdogan F. Stress singularities in a two-material wedge. International Journal of Fracture Mechanics 1971; 7, p.
317-330.
[6] England AH. On stress singularities in linear elasticity. International Journal of Engineering Science 1971; 9, p. 571-585.
[7] Theocaris PS. The order of singularity at a multi-wedge-corner of a composite plate. International Journal of Engineering
Science 1974; 12, p. 107-120.
[8] Dempsey JP, Sinclair GB. On the stress singularities in the plane elasticity of the composite wedge. Journal of Elasticity
1979; 9, p. 373-391.
[9] Dempsey JP, Sinclair GB. On the singular behaviour at the vertex of a bi-material wedge. Journal of Elasticity 1981; 11, p.
317-327.
[10] Dempsey JP. Power-logarithmic stress singularities at bi-material corners and interface cracks. Journal of Adhesion Science
Technology 1995; 9, p. 253-265.
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