non-planar interface crack under general loading ii. … · rev. ivoir. sci. technol., 30 (2017) 1...
TRANSCRIPT
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 1 ISSN 1813-3290, http://www.revist.ci
P. N. B. ANONGBA
NON-PLANAR INTERFACE CRACK UNDER GENERAL LOADING
II. CLIMB-TYPE SINUSOIDAL EDGE DISLOCATION
P. N. B. ANONGBA
U.F.R. Sciences des Structures de la Matière et de Technologie, Université
F.H.B. de Cocody, 22 BP 582 Abidjan 22, Côte d’Ivoire
_______________________
*Correspondance, e-mail : [email protected]
ABSTRACT
This study's objective is to analyse the conditions of propagation of an
oscillatory front crack along a non-planar interface, under mixed mode I + II +
III loading. The crack model consists of a continuous distribution of three
families of non-straight dislocations having the shape of the crack front :
families 1 and 2 are edges (on average) and family 3 is screw. The associated
Burgers vectors j
b
(j=I, II, III) are directed along the applied tension and shears
2x ,
1x and
3x directions, respectively. The dislocations are aligned along the
3
x direction and spread in 32
xx planes in a small oscillating form ),(31
xx
at an average elevation )(1
xh . It is sufficient to investigate a simpler model
where the interface is in the form of a corrugated sheet and the dislocations
have a sinusoidal form. The first part I of this study has provided expressions
for the displacement and stress fields of sinusoidal dislocations with I
b
(glide-type edges) and III
b
(screws). Those
)( mu
and )(
)(m
of dislocation family
2 with II
b
(climb-type edges) are the subject of this part II. It is shown that in
order to guarantee conditions of continuity of the elastic fields at the crossing
of the interface, the relative variation of volume must be continuous at the
crossing of the interface with well-defined non-zero values which depend on
Poisson’s ratios. This results in an estimated mismatch, far from the interface,
between the displacements )( m
u
and )( m
u
; the latter corresponding to the field
due to the same dislocation in a totally homogeneous medium.
Keywords : linear elasticity, interface dislocations, Galerkin vector, three-
dimensional biharmonic functions, Fourier forms, linear systems
of equations.
2 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
RÉSUMÉ
Fissure d'interface non plane sous sollicitation extérieure arbitraire
II. Dislocation coin de type montée
La présente étude se fixe pour objectif d'analyser les conditions de propagation
d'une fissure de front oscillatoire le long d'une interface non plane sous
sollicitation en mode mixte I+II+III. Le modèle de fissure adopté, est une
distribution continue de trois familles de dislocations non rectilignes ayant la
forme du front de fissure : les familles 1 et 2 sont des coins (en moyenne) et la
famille 3 est vis. Les vecteurs de Burgers infinitésimaux associés j
b
(j= I, II, III) sont suivant les directions2
x , 1
x et 3
x , correspondant à la tension
et aux cisaillements appliqués, respectivement. Les dislocations sont suivant la
direction 3
x et s'étalent dans les plans 32
xx dans la forme ),(31
xx à la hauteur )(1
xh
. Il suffit d’analyser un modèle de fissure plus simple dans lequel l’interface a la forme
d’une tôle ondulée et les dislocations sont sinusoïdales. La première partie de cette
étude a fourni les expressions des champs de déplacement et contrainte des
dislocations sinusoïdales avec I
b
(coins de type "glissile") et
IIIb
(vis). Ceux de la
famille 2 avec II
b
(coins de type "montée") sont l’objet de cette partie II. On montre
que, dans le but de garantir la continuité des champs élastiques à la traversée de
l’interface, la variation relative de volume doit être continue au passage de l’interface
avec une valeur bien définie, non nulle, laquelle dépend des rapports de Poisson des
milieux environnants. Il en résulte un écart estimé, loin de l’interface, entre les
déplacements )( m
u
et )( m
u
; le dernier correspondant aux champs de la même
dislocation dans un milieu totalement homogène.
Mots-clés : élasticité linéaire, dislocations d'interface, Vecteur de Galerkin,
fonctions biharmoniques à trois dimensions, expansions en séries
de Fourier, systèmes d'équations linéaires.
I - INTRODUCTION
This study is involved with the determination of the elastic fields of a
non-planar interface crack model in equilibrium under an externally applied
general loading. The model of crack and its pertinence to represent large real
cracks have been introduced in part I of this work [1] ; it consists of two
infinitely extended elastic mediums R1 and R2 firmly welded along a
non-planar interface R (Figure 1). With respect to a Cartesian coordinate
systemi
x , the crack extends from ax 1
to a, fluctuates on average about
31xOx -plane; its front runs indefinitely in the
3x - direction and spreads in
32
xx planes in the Fourier series form f, written as
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 3
P. N. B. ANONGBA
hhxxf
n
nnnn
33cossin (1)
where n is a positive integer; h, n
, n
and n
are real numbers that depend
on position 1
x along the crack length. The system is subjected to remote applied
Figure 1 : Schematic illustration of the crack front in two elastic solids (1)
and (2) welded along a non-planar wavy surface that contains an
interface crack. The crack fronts lie in 32
xx planes in the form f
(1); in this geometry, the system is subjected to mixed mode I+II+III
loading with the applied tension in the 2
x direction. The average
fracture surface (dashed) is shown perpendicular to that direction
general loading, corresponding to a tension along2
x , and two shears
(parallel to 31
xx ) directed along1
x and3
x , respectively. Our method of analysis
consists in representing the crack by a continuous distribution of three families
of non-straight infinitesimal dislocations having the form f. Families 1 and 2
are edges (on average) and family 3 is screw. The associated Burgers vectors
jb
(j=I, II, III) are directed along the applied tension and shears
2x ,
1x and
3x
directions, respectively. It is sufficient to investigate a simpler model where the
interface is in the form of a corrugated surface S and the dislocations have a
sinusoidal form n
A given by
3sin xA
nnn . (2)
The applied loadings and dislocation Burgers vectors are unchanged. The case
of a general shape, with crack front f (1), is achieved by superposition [1]. From
the elastic fields of the dislocations, the crack-tip stress and crack extension
force can be evaluated. Such analyses, with variable complexity of the crack
front, exist in the case of an infinitely extended isotropic medium ([2 - 9],
4 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
among others). We plan to extend the analysis to non-planar interface cracks
under general loading (mixed mode I+II+III). The first part I of this study [1]
has provided expressions for the displacement and stress fields of sinusoidal
dislocations with I
b
(glide-type edges) and
IIIb
(screws). Those of dislocation
family 2 with II
b
are the subject of this part II. For the various types of
dislocation, the general geometry is that of Figure 2; because )0,0,(bbII
is
out of the 32
xOx –plane of location of the dislocation, this is a climb-type edge.
In what follows, the methodology and elastic field expressions form Section 2
and 3, respectively. Discussion and conclusion form Section 4 and 5
respectively. The third part III of the work will deal with crack-tip stresses and
crack extension force when the non-planar interface crack is loaded in mixed
mode I+II+III.
II - METHODOLOGY
We seek the elastic fields of a dislocation ( )0,0,(bbII
) lying on a non-planar
interface S having the form of a corrugated sheet that separates two firmly
welded elastic solids S1 and S2 of infinite sizes. S is defined by the point
),sin,(3321
xxxxPnnS
and S1 and S2 occupy the regions 32
sin xxnn
and32
sin xxnn
, respectively. The situation is shown in Figure 2 where S1
and S2 are confined for illustration purpose in a parallelepiped of finite sizes.
The dislocation is located at the origin, runs indefinitely in the 3
x direction
and spreads in the 32
xx plane in the formn
A (2).
Figure 2 : Two elastic mediums (1) and (2) welded along a non-planar
sinusoidal surface and containing an interface sinusoidal
dislocation at the origin. The dislocation lies in the 32
xOx
plane in the formn
A and runs indefinitely in the 3
x direction
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 5
P. N. B. ANONGBA
The methodology has been detailed in different places [1, 10, 11]. The elastic
fields ( )( mu
, )()(
m ) are assumed to be the difference between two quantities
( )( mu
, )()(
m ) and ( Wm
u)(
, Wm )()( ):
Wmmm
Wmmmuuu
)()()(
)()()(
)()()(
. (3)
The former with corresponds to the fields of a sinusoidal dislocation
(edge or screw) in an infinitely extended homogeneous solid (m); the latter with
W satisfies the equations of equilibrium and is constructed in such a way that :
(a) ( )( mu
, )()(
m ) are continuous at the crossing of the interface, implying that
)()()1()2()1()2(
S
WWW
SPuuuuuPu
)()()()()()()()()1()2()1()2(
S
WWW
SPP
; (4)
(b) ( )( mu
, )()(
m ) tends to ( )( m
u
, )()(
m ) when one moves far away from the
interface in the 2
x direction. This means that
0)(
0
)(
)(
Wm
Wmu
(5)
when 2
x . ( )( mu
, )()(
m ) may be taken from [6, 7]; they are given to linear
expressions with respect ton
. The associated terms ( ))(0( mu
, ))(0()(
m ) of zero
order correspond to the fields of a straight dislocation and second terms ( )( mAnu
,
)()(
mAn ) are proportional to either
nA or its spatial derivative
3/ xA
n . Hence, we
have
n
n
A
S
A
S
P
uuPu
)()()()(
)(
)0(
)0(
(6)
on the interface point ),sin,(3321
xxxxPnnS
; Appendix A below gives the
complete list, component by component, for the climb-type sinusoidal edge dislocation; the corresponding values for the glide-type edge and screw sinusoidal
dislocation have been given in [1, 10, 11]. ( Wmu
)(, Wm )(
)( ) are obtained with the
help of Galerkin vectors; these are available for the glide-type edge and screw
(see [1, 10, 11]). For the climb-type sinusoidal edge, we arrive at Galerkin vectors
V
with only one non-zero
1x component, arranged in the form
xkixkiexkekxV
.
21
.
11)()()( (7)
6 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
under the condition 02
3
2
2
2
1
2 kkkk
that ensures the biharmonicity of
1V .
For 1
V to cancel far from the interface, we write
2
3
2
1
1)(
22)1( kkikk
mm
(8)
with 1m when 32
sin xxnn
(half-space 1) and 2m when
32sin xx
nn (half-space 2). We use the notations
),,(3
)(
21
)(kkkk
mm
, )(
)(
1
)(
1
mmk
, )(
)(
1
)(
1
mmk
;
hence for half-space 1 (32
sin xxnn
), solid (1)
xkixkiexexVxV
.
2
)1(
1
.)1(
1
)1(
11
)1()1(
)()(
and for half-space 2 (32
sin xxnn
), solid (2)
xkixkiexexVxV
.
2
)2(
1
.)2(
1
)2(
11
)2()2(
)()( .
The elastic fields corresponding to 1
V (7) may be first calculated; then, more
general forms Vmu
)( and Vm )(
)( are constructed from the previous ones by
superposition over 1
k and 3
k ; we may write
BAmBmAmVm
BAmBmAmVmuuuu
)()()()(
)()()()(
)()()()(
(9)
where terms with A and B refer to 1
and 1
in (7) respectively. For Vmu
)( and
Vm )()( to conform with )( m
u
and )()(
m , the summation over
1k is
continuous and that over 3
k is discrete. 3
k takes three values: n
, 0, n
. The
fields corresponding to 03k are denoted Vm
u))(0(
and Vm ))(0()( , and terms
associated with n
k 3
and n
are merged to form expressions denoted byVmA
nu)(
and VmAn
)()( ; this is made possible by requiring that
)()()(
1
)(
1 n
m
n
m , )()(
)(
1
)(
1 n
m
n
m . (10)
In (10), )()(
1 n
m stands for ),,(
)(
211 n
mkk . We write
VmAVmVm
VmAVmVm
n
nuuu
)())(0()(
)())(0()(
)()()(
(11)
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 7
P. N. B. ANONGBA
introducing subsequently the notation Amu
))(0( , AmA
nu)(
, Bmu
))(0( , BmA
nu)(
and
even for the stress. Vmu
))(0( and Vm ))(0(
)( are 3
x independent; VmAnu
)( and
VmAn
)()( are proportional to the sinusoid )(
3xA
n or to its spatial derivative
3/ xA
n . Here also, for points
SP on the interface, V
u
and V)( are
expanded up to terms of first order with respect to 2
x in a similar manner
as in (A.2) (see Appendix A) for u
and
)( . Requiring uu
V and
)()(
V lead to the following equations, writing first the conditions
corresponding to 03k (i.e.
)0()0(
i
V
iuu and
)0()0(
ij
V
ij ).
)0(
1
)0(
1uu
V
0)1()1(
4
1
)1(
11
2
)2(
12
1
)1(
1
2
)2(
1
1
k (a)
2
1
1
1
)1(
11
2
)2(
12
1
)1(
1
2
)2(
1
1
)sgn(
4
)45()45(
k
kibCk
(b)
)0(
2
)0(
2uu
V
2
1
1
1
)1(
1
2
)2(
1
1
)1(
1
2
)2(
1
1
)sgn(
4 k
kibCk
(c)
02
1
)1(
1
2
)2(
1
1
)1(
1
2
)2(
1
1
k (d)
)0(
11
)0(
11
V
0)2(2)2(2)1(
11
)2(
12
)1(
1
)2(
11 k (e)
2
1
1)1(
11
)2(
12
)1(
1
)2(
11
)sgn(6)25()25(
k
kQk
b
(f)
)0(
22
)0(
22
V
0)1(2)1(2)1(
11
)2(
12
)1(
1
)2(
11 k (g)
2
1
1)1(
11
)2(
12
)1(
1
)2(
11
)sgn(2)23()23(
k
kQk
b
(h)
8 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
)0(
33
)0(
33
V
0)1(
11
)2(
12 (i)
2
1
1)1(
11
)2(
12
)sgn(2
k
kQ
c (j)
)0(
12
)0(
12
V (h) and (e) above (12)
where )1(2/mmm
bC , 4/)(12
CCiQb
, 4/)(1122
CCiQc
;
111/)sgn( kkk ;
m and
m are shear modulus and Poisson's ratio. In equations
(12 a to j) above, )(
1
m and )(
1
m stand for )0,,(
3
)(
211kkk
m and
)0,,(3
)(
211kkk
m , respectively. Other elastic field components are zero. The
conditions corresponding to
nnA
i
VA
iuu and
nnA
ij
VA
ij are now listed as :
nnAVA
uu11
1
)1(
11
2
)2(
1222
1
1
)1(
1
2
)2(
12
1
)1()1(4
n
kk
2/3
22
1
22
112
8n
nn
k
kkbC
(a)
2
2
2
12
2
)2(
1
1
)1(
1
2
)2(
122
1
2
11445
nnkkk
014452
1
2
11
1
)1(
1
nk
(b)
nnAVA
uu22
0
1
)1(
1
2
)2(
1
1
)1(
1
2
)2(
122
1
nk (c)
22
11
22
1
1
)1(
1
2
)2(
1
1
)1(
1
2
)2(
122
1
23
82
n
nn
n
kk
kbCk
(d)
nnAVA
uu33
(c) above and
2/3
22
11
22
1
1
)1(
1
2
)2(
12
8n
nn
kk
kbC
(e)
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 9
P. N. B. ANONGBA
nnAVA
1111
)1(
11
)2(
12
22
1
)1(
1
)2(
1
2
1)2()2(2
nkk
2/3
22
1
22
11)43(
n
nnb
k
kkiQ
(f)
)2(
1
2
2
2
12
)1(
1
)2(
1
22
1
2
1)2(2)25(
nnkkk
0)2(2)25()1(
1
2
1
2
11
nk (g)
nnAVA
2222
)1(
11
)2(
12
)1(
1
)2(
1
22
1)1()1(2
nk
22
11
22
1)2(
n
ncbn
kk
QkQi
(h)
0)23()23()1(
11
)2(
12
)1(
1
)2(
1
22
1
nk (i)
nnAVA
3333
)1(
11
)2(
12
22
1
)1(
1
)2(
1
22
nnk
2/3
22
11
422
1
2
12)4(2
n
ncnbccn
kk
QQQkQki
(j)
)2(
1
2
2
2
12
)1(
1
)2(
1
22
1
2)21(2
nnnkk
0)21(2)1(
1
2
1
2
11
nk (k)
nnAVA
1212
)2(
1
2
2
2
12
)1(
1
)2(
1
22
1
2
1)1(2)23(
nnkkk
0)1(2)23()1(
1
2
1
2
11
nk (l)
)2(
1
2
2
2
12
)1(
1
)2(
1
22
1
2
1)1()2(2
nnkkk
22
1
22
11)1(
1
2
1
2
11
)23()1()2(2
n
nnb
n
k
kkiQk
(m)
nnAVA
1313 (l) above and
10 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
)1(
11
)2(
12
22
1
)1(
1
)2(
1
2
1)1()1(2
nkk
2/3
22
1
22
11)2(
n
nnb
k
kkiQ
(n)
nnAVA
2323
0)1(
1
)2(
1
)1(
1
)2(
1
22
1
nk
(o)
)1(
1
)2(
1
)1(
1
)2(
1
22
12
nk
22
11
22
12)2(
n
ncbcn
kk
QkQQi
(p) (13)
Next, we are concerned with satisfying boundary conditions : (12) leads to the
displacement and stress fields due to an interface straight edge dislocation
( )0,0,(bbII
) parallel to the
3x direction at the origin; the interface is the
31
xOx plane. (13) provides the complementary terms (to first order in n
) in
the elastic fields of an interfacial sinusoidal climb-type edge dislocation.
III - CALCULATION RESULTS
III-1. Displacement and stress fields of an interface straight edge
dislocation
Four distinct couples of values for ),()(
1
)(
1
mm are extracted from (12). These
are (j= a to d) :
)(
13
1
)(
)(
1
m
j
m
jm
k
)(
12
1
1)()(
1
)sgn( m
j
m
j
m
k
k (14)
where
4
)(
CiDmm
a , 0
)(
m
a ;
4
)65()( bmm
b
Q
,
4
3)1(
1)( bmm
b
Q ;
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 11
P. N. B. ANONGBA
m
cm
m
c
m
cmm
c
QQQDiC
)89(
22
1
1
22
1
2
11
2)( , m
cmm
c
Q
1)()1(
;
m
c
m
c
m
cmm
d
QQQDiC
3
22
1
1
22
1
2
11
2)( , )()( m
c
m
d (15)
whereij
is the Kronecker delta and )1(2/mmmm
CbD . The couples
),()(
1
)(
1
m
j
m
j , j=a to d, are obtained from (12a to d) associated with the
displacement, (12 e to h) associated with stresses, (12 a, b, i, j) and (12 c, d, i, j), respectively. None of these values satisfies the entire (12). The associated
elastic fields denoted Vm
dau
))(0(
to
and
Vm
da
))(0(
to)( are displayed below. A superposition
of these partial fields will provide the complete form of the solution. We have
at position ),,(321
xxxx
( Vm
j
Vmuu
))(0())(0( , Vm
j
Vm ))(0())(0()()( ; j=a to d):
2
21
)(
2
11
)()(
))(0(
1tan
)1(4)1(
r
xxi
x
xiu
m
m
j
m
m
jm
mm
jVm
j
,
2
2
2
)(
21
)()(
))(0(
2
)1()(ln
)1(
r
xixx
iu
m
m
j
m
A
m
m
j
m
j
m
Vm
j
,
)()()2(2)1(2
212
2)()())(0(
11xx
r
xi
A
m
jm
mm
j
Vm
j
4
2
2
2
12
)()(2
r
xxxim
j
,
)()()1(2)1(2
212
2)()())(0(
22xx
r
xi
A
m
jm
mm
j
Vm
j
4
2
2
2
12
)()(2
r
xxxim
j
,
)()(4)1(
212
2)())(0(
33xx
r
xi
A
m
jm
mVm
j ,
)()()23()1(2
22
1
12
1)()())(0(
12x
r
xJ
r
xi
A
m
jm
m
j
mVm
j
4
2
212
)()sgn(4)1(
r
xxxim
j
m
; (16)
0
1111sin dkxkJ .
Here,iii
xxx /)sgn( , )(1
x is the Dirac delta function andA
has the
12 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
following definition: 0)(2
xA
when 02x and 1)(
2x
A when 0
2x ;
2
2
2
1
2xxr . Constant terms are omitted in the displacement. The other elastic
fields are zero. We define the elastic fields )())(0(
xum
and )()())(0(
xm
of an
interface straight edge dislocation as
Wmmm
uuu))(0())(0())(0(
Wmmm ))(0())(0())(0()()()(
(17)
with
daj
Vm
j
m
j
Wmuu
to
))(0()())(0(
daj
Vm
j
m
j
Wm
to
))(0()())(0()()( . (18)
Again ))(0( mu
and ))(0()(
m are due to a straight edge )0,0,(bb
II
parallel to
the 3
x direction at the origin in an infinite medium (see [7] for example);
Vm
dau
))(0(
to
and Vm
da
))(0(
to)( are given in (16). )(
to
m
da are real, to be determined by the
condition that the elastic fields satisfy the following relations :
)())(0(
xum
and )()())(0(
xm
are continuous when crossing the 31
xOx plane.
bdum
))(0(
1 for a closed contour in
21xx encircling the dislocation.
Wmu
))(0(vanishes far from the interface (i.e. when
2x ).
It is easy to show that the second condition above correspond to ))(0(
1
mu constant
with m on the interface. All the stresses involved in ))(0(
)(m
and Vm
da
))(0(
to)(
vanish at infinity. Under such conditions, )())(0(
xum
and )()())(0(
xm
correspond
to an interface straight edge dislocation. Next, we express the quantities
involved in the above-mentioned requirements and proceed to satisfy these.
),0,(),0,(321
)2)(0(
1321
)1)(0(
1xxxuxxxu
17
))(0()()()1(4
1eme
m
j
m
j
m
j
m
;
)2)(0(
2
)1)(0(
2uu
27
))(0()()()1(2)12(
1emeiC
j
m
j
m
j
m
mm
m
;
)2)(0(
11
)1)(0(
11
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 13
P. N. B. ANONGBA
37
))(0()()()2(2 eme
m
j
m
j
m
j ;
)2)(0(
22
)1)(0(
22
47
))(0()()()1(2 eme
m
j
m
j
m
j ;
)2)(0(
33
)1)(0(
33
57)( eme
m ;
)2)(0(
12
)1)(0(
12
67
))(0()()()23()1(2 emeiC
m
j
m
j
m
j
m
m
(19)
where
)()1(7
))(0()(me
j
m
j
m
j
m .
Here, the subscript j takes the values a to d. The continuity of an elastic field
requires associated i
e constant with m at the crossing of the interface. The
additional condition, ))(0( mu
tends to ))(0( mu
when 2
x , corresponds to
vanishing )())(0(
xuWm
far from the interface. This gives
)())(0(
xuWm
vanishes when 2
x
0/)(7
))(0(
2
m
Wmmieu . (20)
This condition is due to the 2
x component of the displacement )())(0(
xuWm
only; the other 1
x component tends to zero. Because )(7
me is a constant in
medium (m), imposing its value to be identically zero is not mandatory; elastic
displacements, differing only by a constant, are admissible in elasticity.
Instead, we shall choose a value to )(7
me that ensures strongest continuity of
the elastic fields at the crossing of the interface. We have checked that
0)(7
me leads to poor continuity (i.e. only a restricted number of elastic field
components, ))(0( m
iu and
))(0( m
ij , are continuous when crossing the interface). A
best continuity behaviour of the elastic fields at the interface is presented as
follows. The relative volume variation))(0(
)/(m
VV at arbitrary position
),0,(31
xx is obtained to be
14 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
)0(
131
))(0(
)(2),0,(R
m
iVxxxV
V
(21)
with
),,(21)(21
31
))(0(
27
)0(xxiumeV
Wm
m
m
m
R
.
It has a singularity given by the Dirac delta function )(1
x . We demand )0(
RV to
be constant with m = 1 and 2; this gives a value to )(7
me that is m dependent.
We have
2
1
7
7
21
21
)1(
)2(
e
e (22)
where 12
/ . The expressions for )(1
me and )(6
me in (19)
are merged to read
)()()1(26
)0(
1meVmeiC
Rm
m
m . (23)
We seek continuity of the elastic fields on the interface, hence we impose
)(1
me and )(6
me to be constant with m; this leads, with the help of (23), to
21
1221
6
CCe ,
)0(
21
12
1
2
)(
RV
CCie
. (24)
These expressions are unchanged by inverting the elastic constants. We isolate
j
m
j
m
j
))(0()( with the expression for
1e in (19) and obtain
)0(
12
12))(0()(
21
)23(
2
)(
R
m
mmm
j
m
j
m
jV
CCi
. (25)
Introducing (25) in the expression for )(2
me in (19) and imposing )(2
me to be
constant with m, provide a value to )0(
RV in the form
)1)(1(8
)21)(21(
21
21)0(
ibV
R. (26)
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 15
P. N. B. ANONGBA
This relation is unchanged by inverting the elastic constants. The other elastic
parameters are obtained as
)0(
1221
122211
2
))(21)(21(
)21)(1()21)(1(4
RV
ibe
,
)0(
12
12
3
212
)()(
R
m
mmV
CCime
,
)0(
12
12
4
212
)()(
R
m
mmV
CCime
,
)0(
5
21)(
R
m
mmVme
. (27)
The continuities at the interface of ))(0(
1
mu ,
))(0(
2
mu , ))(0(
12
m and ))(0(
)/(m
VV are
achieved. The elastic fields, at position ),,(321
xxxx
, are :
)0(1
2
21
2
11
1
1
21))(0(
1
21
1)1(
2tantan
2R
m
m
m
mmiV
C
r
xx
x
xie
x
xbu
,
)0(
1
1
12
12
21
1
11)1)(0(
2
21
1
4)(ln2ln
2
)21(
RAiV
CCxxr
Cu
)0(
11
1
2
2
2
21
1
2R
iVC
r
x
,
)(2)()()3(
3212
2
4
2
1
2
22))(0(
11miexx
r
x
r
xxxC
Am
m
)0(
4
2
2
2
122
21)1(
)(
R
m
mmiV
r
xxx
,
)(2)()(21
2)1(
)(
4212
2
)0(
4
2
2
2
12))(0(
22miexx
r
xiVC
r
xxx
A
m
Rmm
m
m
,
)0(
212
2
2
2))(0(
334
21)()(2
R
m
mm
Amm
miVxx
r
x
r
xC
,
12
121
22
1
12
1
4
2
2
2
11
1
)1)(0(
12
)()()(
)(
CCx
r
xJ
r
x
r
xxxC
A
)0(
1
1
4
2
21
24
21)sgn(
RiV
r
xxx
,
16 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
))(0(
33
))(0(
22
))(0(
11
))(0(
)1(2
21mmm
mm
m
m
V
V
m
Rm
RA
iV
r
xxxiVxx
r
x
1
2)1(
)(2)()(
)0(
4
2
2
2
12)0(
212
2
2
2)21(
r
xC
m
mm
. (28)
Corresponding expressions for half-space 2m are obtained by inverting the
elastic constants. The relations between )(mei
and )0(
RV are taken from relations
(21) and (27); the value of )0(
RV is given by (26). Other elastic fields are zero.
Dirac delta function )(1
x is present in ),0,(31
))(0(xx
m
ii (i=1 to 3) and
))(0()/(
mVV , on the interface. The only stress component with a singularity
1/1 x on the interface is
))(0(
12
m .
III-2. Elastic fields of an interface climb-type sinusoidal edge dislocation
Five values for ( )()(
1 n
m , )(
)(
1 n
m ) are extracted from (13); these are
a) )(
122
1
2
22
11
)(
1)(
11
mA
a
n
n
n
m
anm n
kkk
l
, )(
1
)(
10
mA
a
m n ;
b) )(
12
1
2
1
)(
4
2
22
1
)(
3
2
)(
222
11
)(
1)(
1 ~mA
b
nb
m
bn
n
m
bnm
b
n
m
bnm n
k
l
k
ll
kk
l
,
)(
12
1
2
1
1
22
1
1
)(
1
)(
1 ~
~1 mA
b
nb
b
n
m
bn
m n
kkkt
;
c) )(
122
1
2
22
11
)(
1)(
121
mA
c
n
n
m
n
m
cnm n
kkk
l
nA
c
n
bnm
k
kiQ
122
1
1)(
1
2
independent of m;
d) )(
122
1
)(
3
2
)(
222
11
)(
1)(
1
mA
d
n
m
dnm
d
n
m
dnm n
k
ll
kk
l
,
)(
122
1
1)(
1
)(
1
mA
d
n
m
dn
m n
k
kt
;
e)
22
1
)(
4
2
1
2
1
)(
3
2
2
2
1
)(
2
22
1
1
)(
1)(
1
nm
m
e
n
m
e
n
m
e
n
m
enm
rk
l
rk
l
rk
l
k
kl
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 17
P. N. B. ANONGBA
)(
12
1
22
1
42 mA
e
c
n
b n
k
Q
k
Q
)(
122
1
22
1
)(
11)(
1
2
mA
e
n
b
nm
m
enm n
k
Q
rk
tki
(29)
where
16/)1(1)(
1bCl
m
mm
a
, )1/(1)1/(1
21
C ,
mmmr /)1(
,
1221 mmm , )21)(1()21)(1(/)1)(1(4
~
1221211
b;
)~
1)(1)(1(4/)1(121
)(
1
1)(
1 b
m
a
mm
bll
,
)~
21)(1)(1(4)1(~
)()~
1(121
1
1211
)(
2 b
m
bb
m
bl
,
bb
mm
bl
121121
1)(
3
~)()
~1)(1)(1(4)1(
,
bb
m
b
m
bl
121121
12
1
)(
4
~)()
~1)(1)(1(8)1(
~
,
)(
11
)(
1)1(
~2
m
bmb
m
blt ;
2/)1()(
1 b
mm
ciQl ;
)1()1(2/)1(2112
)(
1 il
mm
d,
)1()1()1()1(23421122112
)(
2
bmmc
m
dQQl ,
)1()1(2342112
)(
3
bmmc
m
dQQl ,
)(
1
1)(
1)1(2)1(
m
dmc
mm
dlQt
;
4/)1(1)(
1il
mm
e
, )1/()/(
221
)(
2
rQQl
bcm
m
e ,
)1/()/(112
)(
3
rQQl
bcm
m
e , )1/()/5(
)(
4
mbcm
m
erQQl ,
)1/()/1()(
1
mbcm
m
erQQt . (30)
None of these couples satisfies the entire conditions (13). For each couple, we
give in Appendix B the associated oscillating elastic fields BmAAmAVmA
nnn uuu)()()(
and BmAAmAVmAnnn
)()()()()()( defined in (11).
A superposition of these partial fields will provide the complete form of
solution (to first order inn
). The elastic fields )()(
xum
and )()()(
xm
of an
interface sinusoidal edge dislocation may be written as
)())(0()( mAmm nuuu
)())(0()(
)()()(mAmm n . (31)
))(0( m
u
and ))(0()(
m (28) correspond to the fields of a straight edge dislocation
18 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
lying on a planar interface; )( mAnu
and )(
)(mA
n are oscillating expressions
proportional to either the sinusoid )(3
xAn
or its spatial derivative 3
/ xAn
in
the forms
WmAmAmA
WmAmAmA
nnn
nnn uuu
)()()(
)()()(
)()()(
(32)
with
eaj
VmA
j
mA
j
WmAnnn uu
to
)()()(
eaj
VmA
j
mA
j
WmAnnn
to
)()()()()( . (33)
VmA
ea
nu)(
to
and
VmA
ea
n)(
to)( are given in Appendix B (for )( mA
nu
and )()(
mAn , see [7])
; )(
to
mA
ea
n are real to be determined by the requirement that the elastic fields be
continuous when crossing the interface. It is sufficient to write this condition
for points on the average interface plane. Before displaying the corresponding
equations, we stress what follows.
Both ),0,(31
)(
1xxu
mAn and ),0,(
31
)(
13xx
mAn contain terms with singularity
1/1 x only; the associated coefficients n
Ae
1and n
Ae
8, respectively, have
been taken constant.
),0,(31
)(
2xxu
mAn , ),0,(
31
)(
12xx
mAn and ),0,(
31
)(
23xx
mAn are bounded
functions. Under such conditions, we have considered their linear forms
with respect to 1
x and posed the terms proportional to 1
x constant with
m=1 and 2.
),0,(31
)(
11xx
mAn , ),0,(
31
)(
22xx
mAn and ),0,(
31
)(
33xx
mAn have terms with
singularities 2
1/1 x and
1ln x ; the associated coefficients ( n
Ae
41, n
Ae
51and
nA
e61
) and ( nA
e42
, nA
e52
and nA
e62
), respectively, have been set constant.
),0,(31
)(
3xxu
mAn contains a term with the singularity
1ln x only. The
associated coefficients nA
e3
is taken constant.
We may write :
),0,(),0,(31
)2(
131
)1(
1xxuxxu nn
AA
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 19
P. N. B. ANONGBA
)~
1()1(4)1(44)23(1
1
)(
1
)(
2
)(
1
)()(
1
)(
b
m
bm
mm
b
m
b
mA
b
m
a
mA
amm
m
tlllC nn
)1(4)1(4)23(4)(
1
)(
2
)(
1
)()(
1
)(
m
mm
d
m
d
m
d
mA
dm
m
c
mA
ctlll nn
nnA
b
m
em
m
cb
m
e
m
e
m
e
m
e
mA
eeQtiQQllll
1
)(
1
1)(
4
)(
3
)(
2
)(
1
)())(1(2)1(]42[4
;
)2(
2
)1(
2
nnAA
uu
)1~
()()1(22)1(1 2
1
)(
1
)(
4
)(
3
)(
1
)()(
1
)(
b
m
b
m
b
m
b
m
b
mmA
b
m
a
mmA
a
m
tllll nn
)(
1
)(
3
)(
1
1)()1(2
m
d
m
d
m
d
mmA
dtlln
nnA
m
m
ebbm
m
e
m
e
m
e
m
e
mmA
eertQiQrlrlrll
2
)(
1
)(
41
)(
32
)(
2
)(
1
)()(]2[2)1(
;
)2(
3
)1(
3
nnAA
uu
)21(444)21(1 )(
1
)()(
2
)(
1
)()(
1
)(
m
m
c
mA
c
m
b
m
b
mA
b
m
a
mA
amm
m
llllC nnn
nnnA
cb
m
e
m
e
m
e
m
e
mA
e
m
d
m
d
mA
deQQllllll
3
)(
4
)(
3
)(
2
)(
1
)()(
2
)(
1
)(]42[44 ;
)2(
11
)1(
11
nnAA
)~
1)(2(2)1(4431
)(
1
)(
2
)(
1
)()(
1
)(
bm
mm
b
m
b
m
b
mA
b
m
a
mA
amtlllC nn
)2(2)1(412)(
1
)(
2
)(
1
)()(
1
)(
m
mm
d
m
d
m
d
mA
d
m
c
mA
ctlll nn
nnA
b
m
em
m
cb
m
e
m
e
m
e
m
e
mA
eeQtiQQllll
41
)(
1
1)(
4
)(
3
)(
2
)(
1
)())(2()1(]42[4
,
)~
1)(2(2)1()(22
1
)(
1
)(
4
)(
3
)(
2
)(
1
)(
bm
m
b
mm
b
m
b
m
b
m
b
mA
bmtllllC n
)2(2)1()(24)(
1
)(
3
)(
2
)(
1
)()(
1
)(
m
mm
d
m
d
m
d
m
d
mA
d
m
c
mA
ctllll nn
)1()1([21
)(
32
)(
2
)(
1
)( rlrll
m
e
m
e
m
e
mA
e
n
nA
m
m
ebm
m
cbm
m
eertQiQQrl
42
)(
1
1)(
4))(2()1(]44)1(
;
)2(
22
)1(
22
nnAA
)~
1)(1(2)1(441
)(
1
)(
2
)(
1
)()(
1
)(
bm
mm
b
m
b
m
b
mA
b
m
a
mA
amtlllC nn
)1(2)1(44)(
1
)(
2
)(
1
)()(
1
)(
m
mm
d
m
d
m
d
mA
d
m
c
mA
ctlll nn
nnA
b
m
em
m
cb
m
e
m
e
m
e
m
e
mA
eeQtiQQllll
51
)(
1
1)(
4
)(
3
)(
2
)(
1
)())(1()1(]42[4
,
)1~
)(1(2)1()(44)21(2
1
)(
1
)(
4
)(
3
)(
1
)()(
1
)(
bm
m
b
mm
b
m
b
m
b
mA
b
m
a
mA
ammtllllC nn
)1(2)1(4)21(4)(
1
)(
3
)(
1
)()(
1
)(
m
mm
d
m
d
m
d
mA
dm
m
c
mA
ctlll nn
nnA
m
m
ebm
m
bm
m
e
m
e
m
e
m
e
mA
eertQiQrlrlrll
52
)(
1
)(
41
)(
32
)(
2
)(
1
)())(1()1(]2[4
;
20 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
)2(
33
)1(
33
nnAA
)(
1
)(
1
1)(
1
)(4)
~1()1(4
m
cm
mA
cbm
mm
b
mA
bmmltC nn
nnnA
b
m
em
mmA
em
mm
d
mA
deQtit
61
)(
1
)(1)(
1
)()()1(2)1(4
,
)1~
(2)1(442
1
)(
1
)(
2
)(
1
)()(
1
)(
bm
m
b
mm
b
m
b
mA
b
m
a
mA
amtlllC nn
m
mm
d
m
d
m
d
mA
d
m
c
mA
ctlll nn 2)1(44
1)(
1
)(
2
)(
1
)()(
1
)(
nn
A
m
m
ebm
m
cb
m
e
m
e
m
e
m
e
mA
eertQiQQllll
62
)(
1
)(
4
)(
3
)(
2
)(
1
)()()1(]42[4
;
)2(
12
)1(
12
nnAA
)~
()1(22)1(2/1
1
)(
4
)(
3
)(
1
)()(
1
)(
b
m
b
m
b
m
b
mmA
b
m
a
mmA
allll nn
]~
)23(~
)1(21[2/5
1
2/3
1
)(
1 bmbm
m
bt
nnnAm
e
m
b
mA
e
m
d
m
d
m
d
mmA
deiliQtll
7
)(
1
1)()(
1
)(
3
)(
1
1)(4)1(1)1(2
;
)2(
13
)1(
13
nnAA
)~
1)(1(2)1(441
)(
1
)(
2
)(
1
)()(
1
)(
bm
mm
b
m
b
m
b
mA
b
m
a
mA
amtlllC nn
)1(2)1(44)(
1
)(
2
)(
1
)()(
1
)(
m
mm
d
m
d
m
d
mA
d
m
c
mA
ctlll nn
nnA
b
m
em
m
cb
m
e
m
e
m
e
m
e
mA
eeQtiQQllll
8
)(
1
1)(
4
)(
3
)(
2
)(
1
)())(1()1(]42[4
;
)2(
23
)1(
23
nnAA
)1~
()()1(22)1(2
1
)(
1
)(
4
)(
3
)(
1
)()(
1
)(
b
m
b
m
b
m
b
m
b
mmA
b
m
a
mmA
atllll nn
)(
1
)(
3
)(
1
1)()1(2
m
d
m
d
m
d
mmA
dtlln
nnA
m
m
ebbm
m
e
m
e
m
e
m
e
mmA
eertQiQrlrlrll
9
)(
1
)(
41
)(
32
)(
2
)(
1
)()(]2[2)1(
. (34)
All )( me nA
i are constant with m=1 and 2 (i.e. )2()1( nn
A
i
A
iee ).We seek values
for )(
to
mA
ea
n that provide continuity at the interface for the largest number of the
elastic field components )( mA
i
nu and )( mA
ij
n . We shall proceed in a manner like
that for ))(0( m
iu and
))(0( m
ij in Section 3.1. Restricting ourselves to terms that are
singular, the trace )()(
mA
r
nT of the stress matrix )()(
mAn takes the form
2
)(
2
2
2
0
131
)()()(),0,()(
xmt
x
ImtAxxT
zA
Rn
A
Rn
mA
r
nnn (35)
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 21
P. N. B. ANONGBA
With
)(2)()(2
1)(
6151411memememt nnnn
AAAA
R ,
)()()(22
1)(
6252422memememt nnnn
AAAA
R .
The associated oscillating part of the relative volume variation)(
)/(mA
nVV at
position ),0,(31
xx has the same form (35) with similar coefficients
)()1(2
)21()(
11mtmV nn
A
R
mm
mA
R
,
)()1(2
)21()(
22mtmV nn
A
R
mm
mA
R
. (36)
Several relations, interconnecting the )( me nA
i and )(
to
mA
ea
n , are involved in the
procedure under way; these are (by using (34))
)()()1(8
)21()()(
4
)21()(
311411mememememV nnnnn
AA
m
mA
m
A
mm
mA
R
; (a)
)(21
2)(
161mVme nn
A
R
m
mmA
; (b)
)(21
4)()(
14151mVmeme nnn
A
R
m
mAA
; (c)
)2(4)1()()(1)(
851 bc
mmA
e
AAQQimeme nnn
; (d)
)()()()()()()()(
7279mmmemememe
b
mA
be
mA
e
AA
m
AAnnnnnn ,
mmmmcmmmbe
rrrQrrrQi
m /3//32
)(1122212112
,
)~
2)(1)(1(4~
)~
1(2)(2/1
121
2
1
2/1
1
)(
1 bbb
m
bblm
b12121
~4)(32 ; (e)
)(
1
1
11
)(
21)1)(
~1(
~4
2
21)()(
m
b
m
bb
mA
b
m
mA
R
A
RtmVmV nnn
m
m
e
mmA
etin /)1(2
)(
1
1)( ; (f)
22 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
)(
1
)(
1
)(
1
)(
14)
~1(4)1(
2
21)(
m
c
mA
cb
m
b
mmA
bm
m
mA
RltCmV nnn
)()1(2)1(4)(
1
)()(
1
)(
b
m
e
mmA
e
m
d
mmA
dQtit nn . (g) (37)
We impose the relative volume variation to be continuous at the crossing of the
interface implying that nA
RV
2and1 (36) are constant with m=1 and 2. This provides
values to )(2and1
mt nA
R that are m-dependent. Using (36) and (37 a, b, c) under n
Ae
1
also constant with m, we obtain successively
)(
)1()21()2()21(
2121
4112241211
1
nn
n
AA
A eee ,
)(4
)2()1()21)(21(
2121
14124121
1
nn
n
AA
A
R
eeV ,
nnA
R
mm
mmA
RVmt
1
1122
111222
1
)21()21(
)1()1(2)(
,
nn
A
R
mmm
mmmAVme
1
1122
111222
61
)21()21(1
)1()1(2)(
,
nnn
A
R
mmm
mmAAVmeme
1
1122
111222
4151
)21()21(1
)1()1(4)()(
. (38)
These expressions depend on )1(41
nA
e and )2(41
nA
e ; nA
e1
and nA
RV
1are unchanged by
inverting the elastic constants, as required. The only oscillating stress that has
a singularity 1
/1 x at ),0,(21
xx is nA
13 ; hence, this stress contributes a non-zero
value to the crack extension force when a non-planar interface crack (see Figure 1) is
subjected to general loading (this will be the subject of part III of our study).
Consequently, we demand nA
13 to be continuous at the interface; this requires the
associated coefficient nA
e8
(34) to be constant with m=1 and 2. Inspection of (37 d)
tells us that taking 0)(
mA
e
n leads to )()(518
meme nnAA
, an expression of which is
displayed in (38). Imposing )()(518
meme nnAA
constant opens the possibility to find
a relation between )1(41
nA
e and )2(41
nA
e . In this way, we find
)21()1(
)21()1()1()2(
212121
122112
4141
nn
AAee . (39)
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 23
P. N. B. ANONGBA
Further progression comes from (37 e). It tells us that first )()(29
meme nnA
m
A
and second )()(79
meme nnAA
; to see this, it is sufficient to write down three
equations corresponding successively to m=1 and 2, and (third) the one that
results from the inversion of the elastic constants in equation for m=1;
summing, member by member, the two last equations yields )2()2(79
nnAA
ee
and in like manner )1()1(79
nnAA
ee . Furthermore, we work with the value zero
for nA
e , this implies that n
A
b must be zero. A link between n
A
RV
1 and n
A
RV
2
emerges from (37 f). Hence, we write
)()()(279
mememe nnnA
m
AA ,
00)()(
mA
b
mA
e
nn ,
nnA
R
A
RVV
12 . (40)
Under nA
e and n
A
b equal to zero, we have the following relation :
)1(2)1()(4)21/()(2)(
1
)(
3
)(
2
)(
1
)(
1 m
mm
d
m
d
m
d
m
d
mA
dm
A
RmtlllmV nn
)()1(2)(7
1
8meme nn
AmA . (41)
It remains to find values to )1(41
nA
e and )(7
me nA . This can be done by providing
first a value to )( mA
d
n ; for this purpose, we use (37 g) in which 0)(
mA
c
n ( nA
e
and nA
b are equally zero from (40)) and obtain
nnA
R
m
m
m
m
m
mA
dVC
CC1
1
2211
1221)(
21
2)1(
)1(
)1()1(
. (42)
Introducing this value in (41) and imposing )(7
me nA
(as also nA
e8
) to be constant
with m=1 and 2 yields a value to )1(41
nA
e in the form
)(
)()1(
41
41
41n
n
n
A
A
A
eDe
eNue (43)
with
)()1)(1(2)1()21(2)(212121221141
CCQeNuc
An
)1()1()1()1(22111221
CCQb
,
)21()21()1)(1()()(1221212141
c
AQeDe n
24 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
)21)(1()21)(1()1()1(1222111221
b
Q .
Using (43) and (39), (38) provides values
n
n
n
nA
RA
R
A
RA
RV
VDe
VNuV
2
1
1
1
)(
)( (44)
with
)()1)(1(2)21)(21()(2121211
CCQVNuc
A
R
n
)1()1()1()1(22111221
CCQb
,
)21()21()1)(1(2)(1221211
c
A
RQVDe n
)21)(1()21)(1()1()1(1222111221
b
Q ;
)(
)(
8
8
518n
n
nn
A
A
AA
eDe
eNuee (45)
With
)()1)(1(2)1()21)(21(2)(2121
2
2121218CCQeNu
c
An
)1()1()1()1(22111221
CCQb
,
)21()21()1)(1())(()(12212121218
c
AQeDe n
)21)(1()21)(1()1()1(1222111221
b
Q .
These values are unchanged by inverting the elastic constants, as expected. The
associated nA
e7
and )(2
me nA
are
))(21)(1(
)1()2(
22)(
112212
112
2
22112118
297
CC
CCVemeee
nn
nnn
A
R
A
A
m
AA
)1(4
)21(
))(21)(1(
)1()2(
2
1221
1122
12
221121
221
2
121212
CCC
CCCC
CC
)1(4
)21(
1
2112
CCC. (46)
Beginning with five parameters )(
to
mA
ea
n , only survive )( mA
a
n and )( mA
d
n ; the
others are zero. Using (34), we have
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 25
P. N. B. ANONGBA
)(
1
)(
3
)(
1
1)(
7)(
1
)()1(2
2
1)1(
m
d
m
d
m
d
mmA
d
A
m
a
mmA
atlle
l
nnn
. (47)
Here,
)( mA
d
n and nA
e7
are given by (42) and (46). With )( mA
a
n (47) and )( mA
d
n
(42), the elastic fields )( mAnu
and )(
)(mA
n (32) are completely determined, thus
providing the elastic fields )()(
xum
and )()()(
xm
(31) of an interface
sinusoidal climb-type edge dislocation.
IV - DISCUSSION
In the present study, we considered two welded infinitely extended elastic
media S1 and S2 (Figure 2), with an interface S having the form of a corrugated
sheet. We have considered a sinusoidal dislocation running indefinitely on this
surface and have studied the elastic displacement and stress fields ( )()(
xum
and
)()()(
xm
) produced in the surrounding media. The elastic fields are
decomposed individually, in the linear approximation with respect to the
amplitude of the sinusoid, into two terms. The first terms correspond to the
elastic fields ( )())(0(
xum
and )()())(0(
xm
) of a straight dislocation on a flat
interface and the second ones ( )( mAnu
and )(
)(mA
n ) constitute an oscillating
part, proportional to the sinusoid or to its spatial derivative with respect to the
direction where the dislocation runs. We left with an initial vision that the
elastic fields are continuous at the crossing of the interface and join those of a
sinusoidal dislocation in a homogeneous infinite medium, far from the
interface. In Section 3.1, expressions (28) for )())(0(
xum
and )()())(0(
xm
have
been obtained. The result is that in order to ensure the continuity of the elastic
fields at the crossing of the interface, the relative variation of volume at the
interface ))(0(
)/(m
VV exhibits a spatial dependence proportional to the delta
function of Dirac )(1
x , with a coefficient of proportionality )2()0(
RiV (26),
well-defined, dependent on the Poisson ratios and which remains unchanged at
the crossing of the interface. In addition, the following result holds
)1)(1)(21(8
)21)(21(),,(),,(
21
21
31
))(0(
231
))(0(
2
m
mm bxxuxxu . (48)
In other words, under conditions of continuity at the interface, the displacement
component))(0(
2
mu (far from the interface) differs from that of a straight
dislocation ))(0(
2
mu in an infinitely extended homogeneous medium. The
difference observed is constant, well defined, depends on the Poisson ratios of
26 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
the two media and changes from medium S1 to medium S2. Moreover, this
difference is related to the relative variation of volume at the interface, in the
vicinity of dislocation. The properties which we have just described apply only
when the conditions of continuity are observed. In the contrary case, for
example if 0)0(
RV , the expression of ))(0(
2
mu (28) is not constant with m=1 and
2 on the interface; as a result, ))(0(
2
mu and ))(0(
2
mu meet far from the interface.
The elastic fields )( mAnu
and )(
)(mA
n are completely determined in Section 3.2,
thus providing the elastic fields )()(
xum
and )()()(
xm
of an interface
sinusoidal climb-type edge dislocation. One may wonder if the proposed solution
is the best under condition of continuity of the elastic fields at the interface. The
answer is yes because it is not possible to increase the number of elastic field
components constant with m=1 and 2 at the interface. This is because relations
interconnecting the parameters )( me nA
i exist. The relation (37a) shows that if n
A
RV
1
and nA
e1
are constant with m, then )(41
me nA
and )(3
me nA
will depend on m; similarly,
)(61
me nA
depends on m if nA
RV
1is constant with m, by the relation (37b). It will also
be noted that the relation )()(29
meme nnA
m
A (40) indicates that if n
Ae
9 is constant
with m then )(2
me nA
depends on m. In our study, the constant oscillating elastic
fields with m at the interface are : )(
1
mAnu ,
)()/(
mAnVV ,
)(
22
mAn ,
)(
12
mAn ,
)(
13
mAn
and )(
23
mAn ; the others
)(
2
mAnu ,
)(
3
mAnu ,
)(
11
mAn and
)(
33
mAn are not.
V - CONCLUSION
This study considers two elastic solids S1 and S2, of infinite sizes, firmly
welded along a non-planar interface S having the form of a corrugated sheet
and provides expressions for the elastic fields (displacement and stress) of a
dislocation lying indefinitely on that interface in the sinusoidal direction with
a Burgers vector II
b
(Figure 2) perpendicular to its shape (climb-type
sinusoidal edge dislocation). The proposed solutions ()( m
u
, )(
)(m
: Section 3)
are sums of terms ())(0( m
u
, ))(0(
)(m
) corresponding to a straight interfacial edge
dislocation and ()( mA
nu
, )(
)(mA
n ) proportional to the sinusoid n
A (2) or its
spatial derivative 3
/ xAn
. It is shown that optimum continuity of the elastic
fields at the crossing of the interface is achieved when the relative variation of
volume is continuous at the interface and expressions of the elastic fields are
written under such conditions. The forthcoming work will deal with the crack-
tip stress and the crack extension force when a non-planar crack defined in Section 1 (Figure 1) is loaded in mixed mode I +II +III (work under way, part III).
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 27
P. N. B. ANONGBA
REFERENCES
APPENDIX A : DIFFERENCE OF THE VALUES OF )( mu
AND )()(
m
(CLIMB-TYPE EDGE) WHEN CROSSING THE INTERFACE
Our purpose here is to write down the differences u
and
)( (4) on
crossing the interface at arbitrary point ),sin,(3321
xxxxPnnS
. We use the
notation 2
x (3
sin xnn
small) and take the MacLaurin series
expansions of the elastic fields up to terms of first order with respect to ; this
means that
),0,(),0,(),,(31
2
31321xx
x
uxxuxxxu
,
),0,()(
),0,()(),,()(31
2
31321xx
xxxxxx
. (A.1)
)( m
u
and )()(
m are taken from our previous works [7]. We obtain (
iu is the i-
component of vector u
and ij
the ij-element of the stress matrix )( ; i, j= 1 to 3)
n
A
iiiuuxxxu
)0(
321),,(
n
A
ijijijxxx
)0(
321),,(
as
11
)0(
1
11)sgn(8
dkekibC
uxik
,
[1] - P. N. B. ANONGBA, Rev. Ivoir. Sci. Technol., 28 (2016) 24 - 59
[2] - B. A. BILBY and J. D. ESHELBY, In : ‟Fracture”, Ed. Academic Press
(H. Liebowitz), New York, Vol. 1, (1968) 99 - 182
[3] - P. N. B. ANONGBA, Physica Stat. Sol. B, 194 (1996) 443 - 452
[4] - P. N. B. ANONGBA and V. VITEK, Int. J. Fract., 124 (2003) 1 - 15
[5] - P. N. B. ANONGBA, Int. J. Fract., 124 (2003) 17 - 31
[6] - P. N. B. ANONGBA, Rev. Ivoir. Sci. Technol., 14 (2009) 55 - 86
[7] - P. N. B. ANONGBA, Rev. Ivoir. Sci. Technol., 16 (2010) 11 - 50
[8] - P. N. B. ANONGBA, J. BONNEVILLE and A. JOULAIN, Rev. Ivoir.
Sci. Technol., 17 (2011) 37 - 53
[9] - P. N. B. ANONGBA, Rev. Ivoir. Sci. Technol., 26 (2015) 76 - 90
[10] - P. N. B. ANONGBA, Rev. Ivoir. Sci. Technol., 25 (2015) 34 - 55
[11] - P. N. B. ANONGBA, Rev. Ivoir. Sci. Technol., 26 (2015) 36 - 75
28 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
12/322
1
22
11
1
112
8dke
k
kkibCAu
xik
n
n
n
An
;
1
1
)0(
2
111
8dke
k
bCu
xik
,
122
1
22
1
2
1123
8dke
k
kbCAu
xik
n
n
n
An
;
0)0(
3
u ,
12/322
1
22
1
3
3
112
8dke
k
kbC
x
Au
xik
n
nnAn
;
11
)0(
11
116 dkekiQxik
b
,
12/322
1
22
1
2
1
11
11)43(
2 dke
k
kkAiQ
xik
n
n
nb
An
;
11
)0(
22
112 dkekiQxik
b
,
122
1
22
1
22
112
2 dke
k
QkQiA
xik
n
ncb
n
An
;
11
)0(
33
114 dkekiQxik
c
,
12/322
1
422
1
2
1
33
11
2
2)4(24 dke
k
QQQkQkiA
xik
n
nccbnc
n
An
;
11
)0(
12
11)sgn(2 dkekQxik
b
,
122
1
22
1
112
1123
2 dke
k
kkQA
xik
n
n
bn
An
;
0)0(
13
,
12/322
1
22
1
1
3
13
112
2 dke
k
kkQ
x
A xik
n
n
b
nAn
;
0)0(
23
,
12/122
1
22
1
3
23
112)2(
2 dke
k
QkQQi
x
A xik
n
ncbcnAn
; (A.2)
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 29
P. N. B. ANONGBA
where
)]1/(1)1/(1[21
C , 4/)(12
CCiQb
,
4/)(1122
CCiQc
, )1(2/mmm
bC ;
111/)sgn( kkk ;
m and
m are shear modulus and Poisson's ratio.
APPENDIX B : PARTIAL OSCILLATING ELASTIC FIELDS
(CLIMB-TYPE EDGE)
The couple ),()(
3
)(
3
mA
a
mA
a
nn is obtained from (13 a, b, c and e) associated with
the displacement. We have at position ),,(321
xxxx
( VmA
a
VmAnn uu
)()( ,
VmA
a
VmAnn
)()()()( ):
aniiniii
m
m
aVmA
iaI
xA
xx
lu n
1
2
2
1
31
3
32
1
1
)(
1)()(
1
2
0
1
2)1(
n
m
iI ,
2
12
231
4
31
2
0
21
)(
1
)()()()(2
xI
x
IlA
niianiiii
m
an
VmA
iia
n ,
1
2
0
1
)(
1
)(
12)1(2
n
mm
an
VmA
aI
xlAn ,
1
2
021
2
2
1
1
1
)(
1
3
)(
3)1(2
n
m
janj
m
a
nVmA
ajII
xxl
x
An ; (B.1)
122
1
122
1
)1(
11
2
22
1
11
2
22
1
dkezk
edke
zk
e xik
n
xk
xik
n
xk
z
nn
m
,
1
2
11
)1(
0
2112
22
1112
22
1
Kr
xdkeedkeeI
nxikxkxikxknn
m
,
12/3
22
1
12/322
1
)1(
1
11
2
22
1
11
2
22
1
dke
k
edke
k
eI
xik
n
xk
xik
n
xk
a
nn
m
.
Terms in brackets are operators acting on n
A and others, separately; 1
is
the value of z
for z=1, 2
2
2
1
2xxr and subscripts i=1 to 3 and j=1 and 2;
][ xKn
is the nth-order modified Bessel function usually so denoted and ij
is
30 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
the Kronecker delta. We stress that the various integrations (such as in a
I1
and
z ) performed in the present study are given for spatial positions satisfying
the condition 0)1(2 x
m (i.e. )sgn()1(2
1x
m
) with 1m when
32sin xx
nn (half-space 1) and 2m when
32sin xx
nn (half-space 2).
However, this makes no difference in the elastic fields to first order in n
. The
pair ),()(
3
)(
3
mA
b
mA
b
nn is obtained using (13 a to d) associated with the
displacement. We have (using the similar notations)
b
m
b
m
b
m
bb
a
m
bn
m
m
b
n
AmA
b
bn
lll
xIl
x
lAu
1
~)(
41
)(
4
)(
21
2
1
)(
3
2
1
)(
1)(
1 ~1
)~
1(1
,
b
n
bm
m
m
m
b
n
BmA
b
xx
tAu
1
~11
21
)(
1)(
1
~)1(4)1(
b
bnnbIx
1
~2
1
2
1
2
012
~)1
~( ;
b
n m
bn
m
bn
m
b
m
m
m
bnAmA
bllIl
lAu
1
~)(
4
2
1
)(
3
2
0
)(
2
)(
1)(
2)1(
,
b
n
bnnb
m
m
m
bnBmA
bI
xx
tAu
1
~2
1
2
1
2
01
2
2
1
)(
1)(
2
~)
~1()1(1
;
b
m
b
m
b
m
bb
a
m
bn
m
m
bnAmA
b
bn
lll
xIl
l
x
Au
1
~)(
41
)(
4
)(
21
2
1
)(
3
2
)(
1
3
)(
3 ~1
)~
1(1
,
b
n
bnnb
m
m
bnBmA
bIx
t
x
Au
1
~2
1
2
1
2
012
)(
1
3
)(
3
~)
~1(
;
1
1
)(
4)(
3
)(
2
2
0
)(
2
2
1
)(
3
4)(
1
)(
11 ~1
2
b
m
bm
b
m
bn
m
ba
m
bn
m
bn
AmA
b
lllIl
xIllAn
b
b
b
m
bnl
1
~
1
1
)(
4
2
~1
~
,
b
n
bnnbm
mm
bn
BmA
bI
xtA
1
~2
1
2
1
2
01
2
)(
1
)(
11
~)
~1()2(2)1(2
b
bnnbnbI
xx
1
~3
1
4
1
4
02
1
2
1
2
12
~)
~1()1
~( ;
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 31
P. N. B. ANONGBA
b
n m
bn
m
bn
m
b
m
bn
AmA
bllIl
xlA
1
~)(
4
2
1
)(
3
2
0
)(
2
2
)(
1
)(
222
,
b
n
bnnbm
mm
bn
BmA
bI
xtA
1
~2
1
2
1
2
01
2
1)(
1
)(
22
~)
~1()1(2)1(2
b
bnbnbI
x
x1
~2
1
4
02
2
2
1
2
12
~)
~1()
~1( ;
a
m
bn
m
b
m
bb
m
b
b
n
m
bn
AmA
bIllll
xlA
b
n
1
)(
3
2~
)(
41
)(
41
)(
2
21
2)(
1
)(
331
)~
1(~1
12 ,
b
n
bnnbm
m
n
m
bn
BmA
bI
xxtA
1
~2
1
2
1
2
01
2
2
2)(
1
)(
33
~)
~1(2)1(2
;
b
n m
bn
m
bn
m
b
mm
bn
AmA
bllIl
xlA
1
~)(
4
2
1
)(
3
2
0
)(
2
1
)(
1
)(
12)1(2
,
b
n
bmn
m
bn
BmA
b
xtA
1
~11
2
1
)(
1
)(
12
~)1(22
bbnnb
m
mI
xx
1
~2
1
2
1
2
01
2
2
1 ~)
~1()1(23 ;
a
m
bn
b
m
b
m
bb
m
bm
b
nAmA
bIl
lll
xxl
x
Abn
1
)(
3
2
1
~)(
41
)(
41
)(
2
21
)(
1
3
)(
13
1~
)~
1(
21
,
b
n
bnnb
m
b
nBmA
bIx
xt
x
A
1
~2
1
2
1
2
012
1
)(
1
3
)(
13
~)
~1(2
bbm
m
x 1
~11
2
1 ~)1(2)1( ;
b
n m
bn
m
bn
m
b
mm
b
nAmA
bllIll
x
A
1
~)(
4
2
1
)(
3
2
0
)(
2
)(
1
3
)(
23)1(2
,
b
n
bnnb
mm
b
nBmA
bI
xxt
x
A
1
~2
1
2
1
2
01
2
2
1)(
1
3
)(
23
~)
~1()1(12
; (B.2)
The couple ),()(
1
)(
1
mA
c
mA
c
nn is obtained using (13 f, g, l and m) associated with
stresses. We obtain
32 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
1
2
1
2
1
)(
1)(
1)21(
xI
x
lAu
man
m
m
c
n
AmA
c
n
,
01
2
21
21
)(
1)1(4)1(
2Ix
xx
iQAu
nm
m
m
b
n
BmA
c
n
;
1
2
0
)(
1)(
2)21()1(
nm
m
m
m
c
n
AmA
cI
lAu n
,
01
2
2
2
1)(
2)1(1
2I
xx
iQAu
n
m
m
b
n
BmA
c
n
;
1
2
1
2
)(
1
3
)(
3)21(
xI
l
x
Au
man
m
m
cnAmA
c
n
,
01
2
2
3
)(
3
2Ix
iQ
x
Au
n
m
bnBmA
c
n
;
01
2
2
1
4)(
1
)(
11)21()1(22 I
xIlA
mnman
m
cn
AmA
c
n ,
01
2
2
)(
11)2(2)1( I
xiQA
nm
m
bn
BmA
c
n
1
2
0
2
02
1
2
2 nnII
xx ;
01
2
2
)(
1
)(
22)21(2 I
xlA
mn
m
cn
AmA
c
n
,
02
1
2
21
2
0
2
)(
22)1(2)1( I
xxI
xiQA
nm
m
bn
BmA
c
n ;
1
2
1
22)(
1
)(
33)21(2
xIlA
mann
m
cn
AmA
c
n ,
1
2
0
2
1
2
2)(
332)1(
nm
m
nbn
BmA
cI
xxiQAn ;
1
2
0
1
)(
1
)(
12)21()1(2
nm
mm
cn
AmA
cI
xlAn ,
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 33
P. N. B. ANONGBA
1
2
0
2
2
1
1
2
1
)(
12)1(23)1(2
n
m
mmnbn
BmA
cI
xx
xiQAn ;
anm
m
c
nAmA
cI
xxl
x
An
1
2
2
1
1
)(
1
3
)(
13)21(2 ,
1
2
02
2
1
13
)(
13)1(2)1(
nm
m
b
nBmA
cIx
xxiQ
x
An ;
1
2
0
)(
1
3
)(
23)21()1(2
nm
mm
c
nAmA
cIl
x
An ,
1
2
0
2
2
1
3
)(
23)1(1
n
m
b
nBmA
cI
xxiQ
x
An ; (B.3)
The pair ),()(
1
)(
1
mA
d
mA
d
nn is calculated from (13 h, i and o, p) associated with
stresses. We have
a
m
dn
m
d
m
m
d
n
AmA
dIl
xl
x
lAu n
1
)(
3
2
2
1)(
2
1
)(
1)(
1
,
1
2
02
2
11
1
)(
1)(
1)1(4)1(
nm
m
m
m
d
n
BmA
dIx
xx
tAu n
;
1
)(
3
2
0
)(
2
)(
1)(
2)1(
m
dn
m
d
m
m
m
d
n
AmA
dlIl
lAu n
,
1
2
0
2
2
1
)(
1)(
2)1(1
n
m
m
m
d
n
BmA
dI
xx
tAu n
;
a
m
dn
m
d
m
m
dnAmA
dIl
xl
l
x
Au n
1
)(
3
2
2
1)(
2
)(
1
3
)(
3
,
1
2
02
)(
1
3
)(
3
n
m
m
dnBmA
dIx
t
x
Au n
;
a
m
dn
m
d
m
dn
m
d
m
dn
AmA
dIlllIl
xlAn
1
)(
3
4
1
)(
3
)(
2
2
0
)(
2
2
)(
1
)(
11)(2 ,
1
4
02
1
2
2
21
2
0
2
)(
1
)(
11)2(2)1(2
nnnm
mm
dn
BmA
dI
xxI
xtAn ;
34 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
1
)(
3
2
0
)(
2
2
)(
1
)(
222
m
dn
m
d
m
dn
AmA
dlIl
xlAn ,
2
1
0
2
21
2
0
2
1)(
1
)(
22)1(2)1(2
x
IxI
xtA
nm
mm
dn
BmA
d
n ;
a
m
dn
m
dn
m
dn
AmA
dIl
xllAn
1
)(
3
2
2
1)(
2
2)(
1
)(
332 ,
1
2
0
2
1
2
2)(
1
)(
332)1(2
nm
m
n
m
dn
BmA
dI
xxtAn ;
1
)(
3
2
0
)(
2
1
)(
1
)(
12)1(2
m
dn
m
d
mm
dn
AmA
dlIl
xlAn ,
1
2
0
2
2
1
1
2
1
)(
1
)(
12)1(23)1(22
n
m
mmn
m
dn
BmA
dI
xx
xtAn ;
a
m
dn
m
d
m
d
nAmA
dIl
xl
xl
x
An
1
)(
3
2
2
1)(
2
1
)(
1
3
)(
132 ,
1
2
02
2
11
1
)(
1
3
)(
13)1(2)1(2
nm
mm
d
nBmA
dIx
xxt
x
An ;
1
)(
3
2
0
)(
2
)(
1
3
)(
23)1(2
m
dn
m
d
mm
d
nAmA
dlIll
x
An ,
1
2
0
2
2
1)(
1
3
)(
23)1(12
n
mm
d
nBmA
dI
xxt
x
An ; (B.4)
The couple ),()(
3
)(
3
mA
e
mA
e
nn is obtained from (13 j, k, and n, l ) associated with
stresses. We have (here, we write m
r for
mr , m=1 and 2)
2
)(
)(
4
2
)(
1
1
)(
3
2
)(
2
2
)(
2
1
)(
1)(
1
111
12
xr
rl
xr
rl
xr
rl
x
lAu
mnr
m
m
m
er
m
er
m
e
m
m
e
n
AmA
e
abncb
m
m
e
m
e
m
eIQ
xQQ
r
l
r
l
r
l
1
2
2
1
)(
4
1
)(
3
2
)(
2242
111 ,
)(
2
1
2
2)(
1
1
)(
1)1)(1(4
2m
n
r
m
mmn
m
e
m
nBmA
e
xxrt
x
iAu
Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36 35
P. N. B. ANONGBA
1
2
2
2)(
20
)(
12)1)(1(4)(
xxtIQtx
m
mn
m
eb
m
e ;
)(
)(
4)(1
)(
3)(2
)(
21
2
)(
1)(
212
2)1(m
n
rm
m
er
m
er
m
ebn
m
m
m
enAmA
erlrlrlQ
lAu
0
)(
4
)(
3
)(
242 IQQlll
cb
m
e
m
e
m
e ,
1)(
)(
1
2
0
)(
1
2
2
1)(
2)()1(1
2
brm
m
enb
m
e
m
m
nBmA
eQrtIQt
xx
iAu
m
n
;
2
)(
)(
4
2
)(
1
1
)(
3
2
)(
2
2
)(
2
)(
1
3
)(
3
111
12
xr
rl
xr
rl
xr
rll
x
Au
mnr
m
m
m
er
m
er
m
e
m
m
enAmA
e
abncb
m
m
e
m
e
m
eIQ
xQQ
r
l
r
l
r
l
1
2
2
1
)(
4
1
)(
3
2
)(
2242
111 ,
1)(
)(
1
2
0
)(
12
3
)(
3)(
2
brm
m
enb
m
e
m
nBmA
eQrtIQtx
i
x
Au
m
n
;
2
)(
2)(
4
2
)(
1
2
1
)(
3
2
)(
2
2
2
)(
22)(
1
)(
11
1112
12
xr
rl
xr
rl
xr
rllA
mnr
m
m
m
er
m
er
m
e
n
m
en
AmA
e
abnb
m
m
m
e
m
e
m
e
nIQ
xQ
r
rl
r
rl
r
rl
1
4
2
1
)(
4
1
1
)(
3
2
2
)(
2222
111
1
2
0
2
)(
4
)(
3
)(
242
ncb
m
e
m
e
m
eI
xQQlll ,
2
1
2
2
2
1)(
11)2(2)1(
xx
xiA
m
m
n
BmA
e
n
1)(
)(
1
2
0
)(
1)(
brm
m
enb
m
eQrtIQt
m
;
)(
)(
4)(1
)(
3)(2
)(
21
2
2
)(
1
)(
2212
22m
n
rm
m
er
m
er
m
ebn
m
en
AmA
erlrlrlQ
xlA
0
)(
4
)(
3
)(
242 IQQlll
cb
m
e
m
e
m
e ,
2
2
2
2
2
)(
22)1(2)1(
x
xx
iAm
m
n
BmA
e
n
1)(
)(
1
2
0
)(
1)(
brm
m
enb
m
eQrtIQt
m
;
36 Rev. Ivoir. Sci. Technol., 30 (2017) 1 - 36
P. N. B. ANONGBA
2
)(
)(
4
2
)(
1
1
)(
3
2
)(
2
2
)(
22)(
1
)(
33
1112
12
xr
rl
xr
rl
xr
rllA
mnr
m
m
m
er
m
er
m
e
n
m
en
AmA
e
abncb
m
m
e
m
e
m
eIQ
xQQ
r
l
r
l
r
l
1
2
2
1
)(
4
1
)(
3
2
)(
2242
111 ,
2
2
2
1)(
332)1(
nm
m
n
BmA
ex
xiAn
1)(
)(
1
2
0
)(
1)(
brm
m
enb
m
eQrtIQt
m
;
)(
)(
4)(1
)(
3)(2
)(
21
2
1
)(
1
)(
1212
2)1(2m
n
rm
m
er
m
er
m
ebn
mm
en
AmA
erlrlrlQ
xlA
0
)(
4
)(
3
)(
242 IQQlll
cb
m
e
m
e
m
e ,
2
1
21)(
)(
1
2
1
)(
12)1(23)1(2
xxQt
xiA
m
mbr
m
emnn
BmA
em
n
1)(
)(
1
2
0
)(
1)(
brm
m
enb
m
eQrtIQt
m
;
2
)(
)(
4
2
)(
1
1
)(
3
2
)(
2
2
)(
2
1
)(
1
3
)(
13
1112
12
xr
rl
xr
rl
xr
rl
xl
x
Amn
r
m
m
m
er
m
er
m
em
e
nAmA
e
abncb
m
m
e
m
e
m
eIQ
xQQ
r
l
r
l
r
l
1
2
2
1
)(
4
1
)(
3
2
)(
2242
111 ,
1)(
)(
1
213
)(
13)1(2)1(
br
m
em
mnBmA
eQt
xxi
x
A
m
n
1)(
)(
1
2
0
)(
12)(
brm
m
enb
m
eQrtIQtx
m
;
)(
)(
4)(1
)(
3)(2
)(
21
2)(
1
1
)(
2312
2)1(2m
n
rm
m
er
m
er
m
ebn
mm
e
nAmA
erlrlrlQl
x
A
0
)(
4
)(
3
)(
242 IQQlll
cb
m
e
m
e
m
e ,
1)(
)(
1
2
0
)(
1
2
2
3
)(
23)(1)1(
brm
m
enb
m
e
mnBmA
eQrtIQt
xxi
x
A
m
n ; (B.5)
122
1
)(
11
2
22
1
dkezk
e xik
n
xk
z
n
.