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Goursat functions for an infinite plate with a generalized curvilinear hole in ζ -plane
Applied Mathematics and Computation 212 (2009) 23–36
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Goursat functions for an infinite plate with a generalized curvilinear hole in f-plane M.A. Abdou a,*, S.A. Aseeri b a b
Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt Department of Mathematics, Faculty of Education, Umm Al-Qura University, P.O. Box 6425, Postal Code 21955, Makkah, Saudi Arabia
a r t i c l e
i n f o
Keywords: Conformal mapping Plane elasticity Complex variable method Curvilinear hole
a b s t r a c t We used a rational mapping function with complex constants to derive exact and close expressions of Goursat functions for the first and second fundamental problems (plane elasticity problems) of an infinite plate weakened by a hole having arbitrary shape. Notable, the area outside the hole is conformally mapped outside a unit circle by means of the rational mapping. The complex variable method has been applied in the procedure and it transforms the fundamental problems to the integro-differential equations that can be solved by using Cauchy type integrals. The hole takes different famous shapes which can be found throughout the nature like tunnels, caves, excavation in soil or rock, etc. Many previous works have been considered as special cases of this work. Also many new cases can be derived from the problem. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction For many years, contact and mixed problems in the theory of elasticity has been recognized as a rich and challenging subject for study, see Popov [8] and Atkin and Fox [9]. These problems can be established from the initial value problems, or from the boundary value problems, or from the mixed problems, see Colton and Kress [5] and Abdou [19]. Also, many different methods are established for solving the contact and mixed problems in elastic and thermoelastic problems, the books written by Noda et al. [20], Hetnarski and Ignaczak [21], Parkus [3] and Popov [8] contain many different methods to solve the problems in the theory of elasticity in one, two and three dimensions. In Muskhelishvili [1], it is known that the first and second fundamental problems in the plane theory of elasticity are equivalent to finding two analytic functions /1 ðzÞ and w1 ðzÞ of one complex argument z ¼ x þ iy. These functions, Goursat functions, satisfy the boundary conditions,
k/1 ðtÞ t/01 ðtÞ w1 ðtÞ ¼ f ðtÞ;
ð1:1Þ
where t denotes the affix of a point on the boundary. In terms of z ¼ cxðfÞ; c > 0; x ðfÞ does not vanish or become infinite for j f j> 1, the infinite region outside a closed contour is conformally mapped outside a unit circle c. For k ¼ 1 and f ðtÞ is a given function of stress in (1.1), we have the boundary condition for the first fundamental problem (or in other words the stress boundary value problem); 0
/1 ðtÞ þ t/01 ðtÞ þ w1 ðtÞ ¼ f1 ðtÞ;
f 1 ðtÞ ¼ f ðtÞ:
* Corresponding author. E-mail addresses: [email protected] (M.A. Abdou), [email protected] (S.A. Aseeri). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.01.079
ð1:2Þ
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M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
l While for k ¼ v ¼ kþ3 > 1; k; l are the Lamé constants and f ðtÞ ¼ 2lgðtÞ is a given function of displacement, we have the kþl principal formula for the second fundamental problem (or the displacement boundary value problem);
v/1 ðtÞ t/01 ðtÞ w1 ðtÞ ¼ 2lgðtÞ:
ð1:3Þ
The stress problem is ordered the first because any displacement for a body is resulted after a stress effect on this body. In the absence of body forces, it is known from Muskhelishvili [1] that stress components, in the plane theory of elasticity, take the form
rxx þ ryy ¼ 4Ref/0 ðzÞg; ryy rxx þ 2irxy ¼ 2½z/00 ðzÞ þ w0 ðzÞ;
ð1:4Þ
where the complex functions of potentials /1 ðzÞ and w1 ðzÞ, take the form
/1 ðzÞ ¼ 2pXþiY ln f þ cCf þ /ðfÞ; ð1þvÞ
ð1:5Þ
ln f þ cC f þ wðfÞ; w1 ðzÞ ¼ 2vpðXiYÞ ð1þvÞ
where X; Y are the components of the resultant vector of all external forces acting on the boundary and C; C are complex constants. Generally, the two complex potential functions /ðfÞ; wðfÞ are single-valued analytic functions within the region outside the unit circle and /ð1Þ ¼ 0; wð1Þ ¼ 0. Muskhelishvili [1] used the transformation
z ¼ cðf þ mf1 Þ;
ðc > 0; m; real numbersÞ;
ð1:6Þ
for solving the problem of stretching of an infinite plate weakened by an elliptic hole. Sokolnikoff [7] used the same rational mapping of Eq. (1.6) to solve the problem of the elliptical ring, where the Laurant’s theorem is used. This transformation of (1.6), conformally maps the infinite domain bounded internally by an ellipse into the domain outside the unit circle j f j¼ 1 in the f-plane. The application of the Hilbert problem is used by Muskhelishvili [1] to discuss the case of a stretched infinite plate weakened by a circular cut. Kalandiya [2] used the transformation mapping (1.6) to solve the torsion problem of an elastic beam, in two dimensions, in the theory of elasticity. The same author used the same rational mapping (1.6) to solve the first fundamental problem in the theory of elasticity by using Laurant’s method. England [4] considered an infinite plate which is weakened by a hypotrochoid hole, conformally mapped into a unit circle jfj ¼ 1 by the transformation mapping
z ¼ cðf þ mfn Þ;
1 ; c > 0; 0 6 m < n
ð1:7Þ
where z0 ðfÞ does not vanish or become infinite outside the unit circle c and he solved the boundary value problems of the first fundamental problem. In El-Sirafy and Abdou [6], the complex variable method has been applied to solve the first and second fundamental problems for the same domain of the infinite plate with general curvilinear hole C conformally mapped on the domain outside a unit circle by using the rational mapping function
z¼c
f þ mf1 1 nf
1
;
j n j< 1; > 0; c > 0; 0 6 m <
1 are real parameters : n
ð1:8Þ
In the papers, Abdou and Khar-El din [10–12], Abdou and Badr [13], Abdou and Khamis [14], Abdou [17] and Abdou et al. [15], many rational mappings are used to solve the first and second fundamental problems of an infinite plate with a curvilinear hole, using Cauchy complex variable method, where the two complex Goursat functions are obtained. Moreover, in all previous works the coefficients of the rational mappings were real. In this paper the complex variable method will be applied to solve the first and second fundamental problems for the same previous domain of the infinite plate with a general curvilinear hole C conformally mapped on the domain outside a unit circle c by the rational function
z¼
lf þ mf1 1 nf1
;
j n j< 1;
ð1:9Þ
where l ¼ l1 þ il2; m ¼ m1 þ im2 ; n ¼ n1 þ in2 ; and j ml j is a parameter restricted such that z0 ðfÞ does not vanish or become infinite outside the unit circle c. Moreover, the results of Goursat functions for, the transformation mapping (1.9) when j f j< 1, are discussed. Also, many applications for the first and second fundamental problems are considered. 2. The rational mapping The parametric equations of the hole C, by assuming that h is the parameter of the curve, are obtained from (1.9) as
x ¼ SU þ TV;
ð2:1Þ
y ¼ SV TU;
ð2:2Þ
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
25
where
S¼
1 2n1 cos h þ n21 n22 ; d
ð2:3aÞ
T¼
2ðn1 n2 n2 cos hÞ ; d
ð2:3bÞ
U ¼ ðl1 cos h l2 sin h þ m1 cos h þ m2 sin h ðl1 n1 l2 n2 Þ cos 2h þ ðn1 l2 þ n2 l1 Þ sin 2h m1 n1 þ m2 n2 Þ;
ð2:3cÞ
V ¼ ðl1 sin h þ l2 cos h m1 sin h þ m2 cos h ðl1 n1 l2 n2 Þ sin 2h ðn1 l2 þ n2 l1 Þ cos 2h m2 n1 m1 n2 Þ;
ð2:3dÞ
d ¼ ð1 2n1 cos h þ n21 n22 Þ2 þ 4ðn1 n2 n2 cos hÞ2 :
ð2:3eÞ
and
The present mapping function deals with famous shapes of tunnels, therefore it is useful to use it in studying stresses and strains around tunnels. In underground engineering the tunnel is assumed to be driven in a homogeneous, isotropic, linear elastic and pre-stressed geometrical situation. Also, the tunnel is considered to be deep enough such that the stress distribution before excavation is homogeneous. Excavating underground openings in soils and rocks is done for several purposes and in multi-sizes. At least, excavation of the opening will cause the soil or rock to deform elastically. The excavation in soil or rock is a complicated, dangerous, and expensive process. The mechanics of this can be very complex. However, the use of conformal mapping that allows us to study stresses and strains around a unit circle makes it useful for engineers and easier for mathematicians. The physical interest of the mapping (1.9) comes from its different shapes of holes it treats where we find from Figs. 1–9 the following notations;
Fig. 1.
Fig. 2.
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M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
Fig. 3.
Fig. 4.
Fig. 5.
(i) The number of the holes corners is subjected to n’s values. They are given by n þ 1. (ii) The complex constant m works on circling the shape from the symmetry situation and the circling angle is given by 2 ðm ¼ m1 þ im2 Þ. Positive values of h means that the circling will be in the positive direction i.e. in the antih ¼ tan1 m m1 clockwise direction and for negative values the circling will be in the negative direction i.e. in clockwise direction. (iii) The complex constant l works on expanding the corners of the hole shape.
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
27
Fig. 6.
Fig. 7.
Fig. 8.
3. Method of solution In this section, we will use the transformation mapping (1.9) in the boundary condition (1.1), then we will apply the complex variable method with the residue theorems to obtain a closed expression for Goursat functions. Therefore, the expression xðfÞ=x0 ðf1 Þ will be written in the form
28
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
Fig. 9.
xðfÞ ¼ aðfÞ þ bðfÞ; x0 ðf1 Þ
ð3:1Þ
where
aðfÞ ¼
h ; fn
h¼
2 nÞ2 ðln þ mÞð1 n ; l 2ln n mn 2
ð3:2Þ
and bðfÞ is a regular function for j f j> 1. Using (3.1) in the boundary condition (1.1) and on f ¼ r, for generality, we get
k/ðrÞ aðrÞ/0 ðrÞ w ðrÞ ¼ f ðrÞ;
ð3:3Þ
where
w ðrÞ ¼ wðrÞ þ bðrÞ/0 ðrÞ; f ðrÞ ¼ FðrÞ lkCr þ N ¼ lC
lC
r
ð3:4aÞ hðX iYÞ X iY r bðrÞ; þ NaðrÞ þ lC 2pð1 þ vÞ 2pð1 þ vÞ
X iY n; 2pð1 þ vÞ
ð3:4bÞ
ð3:4cÞ
and
FðrÞ ¼ f ðtÞ:
ð3:4dÞ
The function FðrÞ with its derivatives must satisfy the Hölder condition. Multiplying both sides of (3.3) by ð1=2piÞ1=ðr fÞ and integrating with respect to r on c, one has
k/ðfÞ þ
1 2p i
Z
aðrÞ/0 ðrÞdr lC hN AðfÞ; ¼ þ fn f rf
c
ð3:5Þ
where
AðfÞ ¼
1 2pi
Z c
FðrÞdr : rf
Using (3.2), we have
1 2pi
Z c
aðrÞ/0 ðrÞdr hb ; ¼ nf rf
ð3:6Þ
where b is a complex constant to be determined. Substituting from (3.6) into (3.5), we have
k/ðfÞ ¼ AðfÞ
lC h ðb þ NÞ: þ nf f
ð3:7Þ
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29
Differentiating (3.7) with respect to f, and using the result in (3.6), yields
mðb þ NÞ þ ln2 C þ A0 ðnÞ ¼ 0; kb þ h where
m¼
n2 . n1Þ2 ðn
ð3:8Þ
Hence, from (3.8), we get
kE mhE b¼ 2 ; 2 k m2 h
ð3:9Þ
2 where E ¼ ½A0 ðnÞ þ ln C þ mhN. Also from (3.3), wðfÞ can be determined in the form
wðfÞ ¼
klC xðf1 Þ hf Þ þ BðfÞ B; / ðn 0 / ðfÞ þ f 1n f x ðfÞ
ð3:10Þ
where
/ ðfÞ ¼ /0 ðfÞ þ lC BðfÞ ¼
1 2pi
Z
X þ iY 1 f ; 2pð1 þ vÞ
FðrÞ dr; rf
c
ð3:11aÞ ð3:11bÞ
and
B¼
1 2pi
Z c
FðrÞ
r
dr:
ð3:11cÞ
4. Special cases
1. For n ¼ 0, 0 6 m 6 1, we get the function z ¼ l f þ mf1 . Considering the reality of the constants l and m in (3.7) and (3.10) besides the imposed restrictions, gives
lC mC ; f f ! klC lf1 þ mf / ðfÞ þ mCf þ Bðf1 Þ B: wðfÞ ¼ f l mf2 k/ðfÞ ¼ Aðf1 Þ
ð4:1Þ ð4:2Þ
These two forms agree’s with those obtained by Muskhelishvili [1] for an elliptic hole if we let l ¼ c and m ¼ cm. 2. By considering the reality of the constants in the rational mapping (1.9), we get
k/ðfÞ ¼ AðfÞ
l C h ðb þ NÞ; þ nf f
! klC lf1 þ mf ð1 nf1 Þ2 hf / ðfÞ þ wðfÞ ¼ / ðnÞ þ BðfÞ B: 1 nf ð1 nf1 Þðl mf2 Þ ðlf þ mf1 Þðnf2 Þ 1 nf f 2
2 2
ð4:3Þ
ð4:4Þ
2
þmÞð1n Þ where h ¼ ðlnl2ln and m ¼ ðn2n1Þ2 in b. These two forms agree’s with those obtained by El-Sirafy and Abdou [6] if we let 2 mn2 l ¼ c and m ¼ cm. 3. The rational mapping (1.9) can be written in the form
2
z ¼ nl þ lf þ ðln þ mÞ
1 X
na1 fa :
ð4:5Þ
a¼1
So by approximating this form to the first term only i.e. a ¼ 1 and then letting n ¼ 0, we get
z ¼ lf þ mf1 :
ð4:6Þ
The corresponding forms of Goursat functions are
k/ðfÞ ¼ AðfÞ
wðfÞ ¼
lC hN ; f f
klC lf1 þ mf þ BðfÞ B: / ðfÞ þ lChf 2 f l mf
ð4:7Þ
ð4:8Þ
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M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
These forms are approximately equivalent to those obtained by Aseeri [23] when n ¼ 1. P j 4. The form (4.5) is approximately equivalent to the rational mapping, z ¼ lf þ M j¼1 mj f , which was used in Abdou and Aseeri [22] and its resultant forms of Goursat functions are equivalent to the forms in (3.7) and (3.10) by omitting constant terms. 5. An important case appears when we replace f by 1f in the conformal mapping (1.9), as
z¼
1 X lf1 þ mf 2 na1 fa ; ¼ nl þ lf1 þ ðln þ mÞ 1 nf a¼1
j n j< 1:
ð4:9Þ
This mapping, when z0 ðfÞ – 0; 1 inside the j f j< 1, transforms the points in the z-plane inside the unit circle c in f-plane. And the Goursat functions, in view of (4.9), becomes
k/ðfÞ ¼ Aðf1 Þ lC f þ
h n f1
ðb þ NÞ;
ð4:10Þ
1 xðfÞ hf 1 Þ þ Bðf1 Þ B: / ðf Þ þ / ðn wðfÞ ¼ klCf f1 x0 ðf1 Þ 1n
ð4:11Þ
6. In the mapping function (4.9), which maps the hole into the interior of a unit circle, if we put m ¼ 0 and then ignore the P constant term with assuming the constants na1 ¼ 0 for a > 3, we have the mapping z ¼ l 1f þ 3a¼1 naþ1 fa . The Goursat functions according to these substitutions are equivalent to the forms found by Exadaktylos and Stavropoulou [16]. 7. Also in the mapping function (4.9), if we put m ¼ 0 and then ignore the constant term with assuming the constants P n2a1 ¼ 0 for a ¼ 1; 2; . . . ; r where r < 1, we have the mapping z ¼ l 1f þ ra¼1 n2aþ1 f2a1 by omitting the terms after r. The Goursat functions according to these substitutions are equivalent to the forms found by Exadaktylos [18].
5. Applications 5.1. Curvilinear hole for an infinite plate subjected to a uniform tensile stress For k ¼ 1; C ¼ 4p ; C ¼ 12 P expð2ihÞ and X ¼ Y ¼ f ¼ 0, we have an infinite plate stretched at infinity by the application of a uniform tensile stress of intensity P making an angle h with x-axis. The plate is weakened by a curvilinear hole C which is free from stress. The functions (3.7) and (3.10) take the form
/ðfÞ ¼
! lPf 2ln2 Pf2 þ mh½ m hlP 2ln 2 Pf2 h ½mhlP þ lP ; þ 2 4ðn fÞ 2 1 m2 h
wðfÞ ¼
lP xðf1 Þ hf lP lP Þ þ 0 /0 ðn þ : /0 ðfÞ þ f 4f 1n 4 4 x ðfÞ
ð5:1Þ
ð5:2Þ
And for l ¼ i; m ¼ 1 þ i; n ¼ 2 þ i; P ¼ 0:25; h ¼ p4, the stress components rxx ; ryy and rxy are obtained in large forms calculated by computer and illustrated in Figs. 10–14;
Fig. 10.
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
31
Fig. 11.
Fig. 12.
Fig. 13.
5.2. When the edge of the hole is subject to a uniform pressure P For k ¼ 1; X ¼ y ¼ C ¼ C ¼ 0 and f ¼ Pt, where P is a real constant, the formulae (3.7) and (3.10) becomes
/ðfÞ ¼
þn Þ hPðmhn 2 Þ ; ðn fÞð1 m2 h
ð5:3Þ
32
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
"
wðfÞ ¼
#
"
#
þn þn Þ Þ hf xðf1 Þ hPðmhn hPðmhn þ : 2 2 0 x ðfÞ ðn fÞ ð1 m2 h2 Þ 1 nf ðn nÞ ð1 m2 h2 Þ
And for l ¼ i; m ¼ 1 þ i; n ¼ 2 þ i; P ¼ 0:25, the stress components computer and illustrated in Figs. 15–19;
Fig. 14.
Fig. 15.
Fig. 16.
ð5:4Þ
rxx ; ryy and rxy are obtained in large forms calculated by
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
33
Fig. 17.
Fig. 18.
Fig. 19.
5.3. When the external force acts on the center of the curvilinear For C ¼ C ¼ f ¼ 0 and k ¼ v, we have the second fundamental problem when the force acts on the curvilinear kernel. It is assumed that the stresses vanish at infinity and it is easily seen that the kernel does not rotate.
34
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
In general the kernel remains in the original position. The Goursat’s functions are
/ðfÞ ¼
h X iY b n ; ðn fÞv 2pð1 þ vÞ
wðfÞ ¼
ð5:5Þ
xðf1 Þ 0 X þ iY 1 X þ iY hf Þ þ /0 ðn / ðfÞ f n : 0 f 2pð1 þ vÞ 2pð1 þ vÞ 1n x ðfÞ
Fig. 20.
Fig. 21.
Fig. 22.
ð5:6Þ
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
35
Fig. 23.
Fig. 24.
And for l ¼ i; m ¼ 1 þ i; n ¼ 2 þ i; X ¼ Y ¼ 10; , ¼ 2, the stress components rxx ; ryy and rxy are obtained in large forms calculated by computer and illustrated in Figs. 20–24; 6. Conclusions From the above results and discussions the following may be concluded: (1) In the theory of two dimensional linear elasticity one of the most useful techniques for the solutions of boundary value problem for a region weakened by a curvilinear hole is to transform the region into a simpler shape to get solutions without difficulties. (2) The transformation mapping z ¼ cxðsÞ; s > 0; s ¼ r þ is, transforms the domain of the infinite plate with a curvilinear hole into the domain of the right half-plane. While the mapping z ¼ cxðfÞ; j f j> 0; f ¼ qeih , transforms the domain of the infinite plate with a curvilinear hole into the domain outside a unit circle. (3) The physical interest of the mapping (1.9) comes from its different shapes of holes it treats and different directions it takes as shown in Figs. 1–9. (4) The complex variable method (Cauchy method) is considered one of the best methods for solving the integro-differential Eq. (1.1), and obtaining the two complex potential functions /ðzÞ and wðzÞ directly. (5) Stress is an internal force whereas positive values of it mean that stress is in the positive direction, i.e. stress acts as a tension force. On the other side, negative values of stress mean that stress is in the negative direction, i.e. stress acts as a press force. (6) The most important issue deduced from (Figs. 10,11 and 15,16 and 20,21) is that max rxx ¼ min ryy and vice versa (min rxx ¼ max ryy ).
36
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
r
xx xx (7) By following the intendance of rryy and ryy at (Figs. 13,14 and 18,19 and 23,24), we find that rryy ! 0 at the same points xx r ryy rxx ! 1 and vice versa ( ! 1 at the same points where ! 0). where ryy ryy rxx xx xx ! 0 the perpendicular stress on y-axis is the maximum value and presents the body interior resistance of (8) When rryy treatment (like rocks for example). Whereas, the perpendicular stress on x-axis is small according to y-axis. Thereon, xx . it is better to treat the problem at points determined by angles that gives minimum values of rryy
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
N.I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, Holland, 1953. A.I. Kalandiya, Mathematical Method of Two Dimensional Elasticity, Moscow, 1975. H. Parkus, Thermoelasticity, Spring-Verlag, 1976. A.H. England, Complex Variable Methods in Elasticity, third ed., London, New York, 1980. D. Colton, R. Kress, Integral Equation Methods in Scattering Theory, John Wiley, New York, 1983. I.H. El-Sirafy, M.A. Abdou, First and second fundamental problems of infinite plate with a curvilinear hole, J. Math. Phys. Sci. 18 (1984). I.S. Sokolnikoff, Mathematical Theory of Elasticity, fifth ed., New York, 1985. G. Ya. Popov, Contact Problems for a Linearly Deformable Foundation, Kiev, Odessa, 1988. R.J. Atkin, N. Fox, An Introduction to the Theory of Elasticity, Longman, 1990. M.A. Abdou, E.A. Khar-El din, An infinite plate weakened by a hole having arbitrary shape, J. Comp. Appl. Math. 56 (1994) 341–361. M.A. Abdou, E.A. Khar-El din, On the problems of stretched infinite plate weakened by a curvilinear hole, J. Pure Appl. Math. Sci. 14 (1994) 106–113. M.A. Abdou, E.A. Khar-El din, Stretched infinite plate weakened by arbitrary curvilinear hole, J. Pure. Appl. Math. Sci. 14 (1994) 114–122. M.A. Abdou, A.A. Badr, Boundary value problems of an infinite plate weakened by arbitrary shape hole, J. India Acad. Sci. 27 (1999) 215–227. M.A. Abdou, A.K. Khamis, On a problem of an infinite plate with a curvilinear hole having three poles and arbitrary shape, Bull. Cal. Math. Soc. 9 (4) (2000) 13–326. A.S. Sabbah, M.A. Abdou, A.S. Ismail, An infinite plate with a curvilinear hole and flowing heat, Proc. Math. Phys. Soc. Egypt (2002). G.E. Exadaktylos, M.C. Stavropoulou, A closed form elastic solution for stress and displacement around tunnels, Inter. J. Rock Mech. Mining Sci. 39 (2002) 905–916. M.A. Abdou, Fundamental problems for infinite plate with a curvilinear hole having finite poles, Appl. Math. Comput. 125 (2002) 177–193. G.E. Exadaktylos, P.A. Liolios, M.C. Stavropoulou, A semi-analytical elastic stress–displacement solution for notched circular openings in rocks, Int. J. Solid Struct. 40 (2003) 1165–1187. M.A. Abdou, On asymptotic method for Fredholm–Volterra integral equation of the second kind in contact problems, J. Comp. Appl. Math. 154 (2003) 431–446. N. Noda, R.B. Hetnarski, Y. Tanigawa, Thermal Stress, Taylor and Francis, 2003. R.B. Hetnarski, J. Ignaczak, Mathematical Theory of Elasticity, Taylor and Francis, 2004. M.A. Abdou, S.A. Aseeri, Closed forms of Goursat functions in presence of heat for curvilinear holes, in: Proc. Sixth Int. Elast. Conf. USA, 2007. S.A. Aseeri, Goursat functions for a problem of an isotropic plate with a curvilinear hole, in: Proc. Seventh Saudi Eng. Conf., 2007.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Goursat functions for an infinite plate with a generalized curvilinear hole in f-plane M.A. Abdou a,*, S.A. Aseeri b a b
Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt Department of Mathematics, Faculty of Education, Umm Al-Qura University, P.O. Box 6425, Postal Code 21955, Makkah, Saudi Arabia
a r t i c l e
i n f o
Keywords: Conformal mapping Plane elasticity Complex variable method Curvilinear hole
a b s t r a c t We used a rational mapping function with complex constants to derive exact and close expressions of Goursat functions for the first and second fundamental problems (plane elasticity problems) of an infinite plate weakened by a hole having arbitrary shape. Notable, the area outside the hole is conformally mapped outside a unit circle by means of the rational mapping. The complex variable method has been applied in the procedure and it transforms the fundamental problems to the integro-differential equations that can be solved by using Cauchy type integrals. The hole takes different famous shapes which can be found throughout the nature like tunnels, caves, excavation in soil or rock, etc. Many previous works have been considered as special cases of this work. Also many new cases can be derived from the problem. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction For many years, contact and mixed problems in the theory of elasticity has been recognized as a rich and challenging subject for study, see Popov [8] and Atkin and Fox [9]. These problems can be established from the initial value problems, or from the boundary value problems, or from the mixed problems, see Colton and Kress [5] and Abdou [19]. Also, many different methods are established for solving the contact and mixed problems in elastic and thermoelastic problems, the books written by Noda et al. [20], Hetnarski and Ignaczak [21], Parkus [3] and Popov [8] contain many different methods to solve the problems in the theory of elasticity in one, two and three dimensions. In Muskhelishvili [1], it is known that the first and second fundamental problems in the plane theory of elasticity are equivalent to finding two analytic functions /1 ðzÞ and w1 ðzÞ of one complex argument z ¼ x þ iy. These functions, Goursat functions, satisfy the boundary conditions,
k/1 ðtÞ t/01 ðtÞ w1 ðtÞ ¼ f ðtÞ;
ð1:1Þ
where t denotes the affix of a point on the boundary. In terms of z ¼ cxðfÞ; c > 0; x ðfÞ does not vanish or become infinite for j f j> 1, the infinite region outside a closed contour is conformally mapped outside a unit circle c. For k ¼ 1 and f ðtÞ is a given function of stress in (1.1), we have the boundary condition for the first fundamental problem (or in other words the stress boundary value problem); 0
/1 ðtÞ þ t/01 ðtÞ þ w1 ðtÞ ¼ f1 ðtÞ;
f 1 ðtÞ ¼ f ðtÞ:
* Corresponding author. E-mail addresses: [email protected] (M.A. Abdou), [email protected] (S.A. Aseeri). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.01.079
ð1:2Þ
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l While for k ¼ v ¼ kþ3 > 1; k; l are the Lamé constants and f ðtÞ ¼ 2lgðtÞ is a given function of displacement, we have the kþl principal formula for the second fundamental problem (or the displacement boundary value problem);
v/1 ðtÞ t/01 ðtÞ w1 ðtÞ ¼ 2lgðtÞ:
ð1:3Þ
The stress problem is ordered the first because any displacement for a body is resulted after a stress effect on this body. In the absence of body forces, it is known from Muskhelishvili [1] that stress components, in the plane theory of elasticity, take the form
rxx þ ryy ¼ 4Ref/0 ðzÞg; ryy rxx þ 2irxy ¼ 2½z/00 ðzÞ þ w0 ðzÞ;
ð1:4Þ
where the complex functions of potentials /1 ðzÞ and w1 ðzÞ, take the form
/1 ðzÞ ¼ 2pXþiY ln f þ cCf þ /ðfÞ; ð1þvÞ
ð1:5Þ
ln f þ cC f þ wðfÞ; w1 ðzÞ ¼ 2vpðXiYÞ ð1þvÞ
where X; Y are the components of the resultant vector of all external forces acting on the boundary and C; C are complex constants. Generally, the two complex potential functions /ðfÞ; wðfÞ are single-valued analytic functions within the region outside the unit circle and /ð1Þ ¼ 0; wð1Þ ¼ 0. Muskhelishvili [1] used the transformation
z ¼ cðf þ mf1 Þ;
ðc > 0; m; real numbersÞ;
ð1:6Þ
for solving the problem of stretching of an infinite plate weakened by an elliptic hole. Sokolnikoff [7] used the same rational mapping of Eq. (1.6) to solve the problem of the elliptical ring, where the Laurant’s theorem is used. This transformation of (1.6), conformally maps the infinite domain bounded internally by an ellipse into the domain outside the unit circle j f j¼ 1 in the f-plane. The application of the Hilbert problem is used by Muskhelishvili [1] to discuss the case of a stretched infinite plate weakened by a circular cut. Kalandiya [2] used the transformation mapping (1.6) to solve the torsion problem of an elastic beam, in two dimensions, in the theory of elasticity. The same author used the same rational mapping (1.6) to solve the first fundamental problem in the theory of elasticity by using Laurant’s method. England [4] considered an infinite plate which is weakened by a hypotrochoid hole, conformally mapped into a unit circle jfj ¼ 1 by the transformation mapping
z ¼ cðf þ mfn Þ;
1 ; c > 0; 0 6 m < n
ð1:7Þ
where z0 ðfÞ does not vanish or become infinite outside the unit circle c and he solved the boundary value problems of the first fundamental problem. In El-Sirafy and Abdou [6], the complex variable method has been applied to solve the first and second fundamental problems for the same domain of the infinite plate with general curvilinear hole C conformally mapped on the domain outside a unit circle by using the rational mapping function
z¼c
f þ mf1 1 nf
1
;
j n j< 1; > 0; c > 0; 0 6 m <
1 are real parameters : n
ð1:8Þ
In the papers, Abdou and Khar-El din [10–12], Abdou and Badr [13], Abdou and Khamis [14], Abdou [17] and Abdou et al. [15], many rational mappings are used to solve the first and second fundamental problems of an infinite plate with a curvilinear hole, using Cauchy complex variable method, where the two complex Goursat functions are obtained. Moreover, in all previous works the coefficients of the rational mappings were real. In this paper the complex variable method will be applied to solve the first and second fundamental problems for the same previous domain of the infinite plate with a general curvilinear hole C conformally mapped on the domain outside a unit circle c by the rational function
z¼
lf þ mf1 1 nf1
;
j n j< 1;
ð1:9Þ
where l ¼ l1 þ il2; m ¼ m1 þ im2 ; n ¼ n1 þ in2 ; and j ml j is a parameter restricted such that z0 ðfÞ does not vanish or become infinite outside the unit circle c. Moreover, the results of Goursat functions for, the transformation mapping (1.9) when j f j< 1, are discussed. Also, many applications for the first and second fundamental problems are considered. 2. The rational mapping The parametric equations of the hole C, by assuming that h is the parameter of the curve, are obtained from (1.9) as
x ¼ SU þ TV;
ð2:1Þ
y ¼ SV TU;
ð2:2Þ
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
25
where
S¼
1 2n1 cos h þ n21 n22 ; d
ð2:3aÞ
T¼
2ðn1 n2 n2 cos hÞ ; d
ð2:3bÞ
U ¼ ðl1 cos h l2 sin h þ m1 cos h þ m2 sin h ðl1 n1 l2 n2 Þ cos 2h þ ðn1 l2 þ n2 l1 Þ sin 2h m1 n1 þ m2 n2 Þ;
ð2:3cÞ
V ¼ ðl1 sin h þ l2 cos h m1 sin h þ m2 cos h ðl1 n1 l2 n2 Þ sin 2h ðn1 l2 þ n2 l1 Þ cos 2h m2 n1 m1 n2 Þ;
ð2:3dÞ
d ¼ ð1 2n1 cos h þ n21 n22 Þ2 þ 4ðn1 n2 n2 cos hÞ2 :
ð2:3eÞ
and
The present mapping function deals with famous shapes of tunnels, therefore it is useful to use it in studying stresses and strains around tunnels. In underground engineering the tunnel is assumed to be driven in a homogeneous, isotropic, linear elastic and pre-stressed geometrical situation. Also, the tunnel is considered to be deep enough such that the stress distribution before excavation is homogeneous. Excavating underground openings in soils and rocks is done for several purposes and in multi-sizes. At least, excavation of the opening will cause the soil or rock to deform elastically. The excavation in soil or rock is a complicated, dangerous, and expensive process. The mechanics of this can be very complex. However, the use of conformal mapping that allows us to study stresses and strains around a unit circle makes it useful for engineers and easier for mathematicians. The physical interest of the mapping (1.9) comes from its different shapes of holes it treats where we find from Figs. 1–9 the following notations;
Fig. 1.
Fig. 2.
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M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
Fig. 3.
Fig. 4.
Fig. 5.
(i) The number of the holes corners is subjected to n’s values. They are given by n þ 1. (ii) The complex constant m works on circling the shape from the symmetry situation and the circling angle is given by 2 ðm ¼ m1 þ im2 Þ. Positive values of h means that the circling will be in the positive direction i.e. in the antih ¼ tan1 m m1 clockwise direction and for negative values the circling will be in the negative direction i.e. in clockwise direction. (iii) The complex constant l works on expanding the corners of the hole shape.
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27
Fig. 6.
Fig. 7.
Fig. 8.
3. Method of solution In this section, we will use the transformation mapping (1.9) in the boundary condition (1.1), then we will apply the complex variable method with the residue theorems to obtain a closed expression for Goursat functions. Therefore, the expression xðfÞ=x0 ðf1 Þ will be written in the form
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M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
Fig. 9.
xðfÞ ¼ aðfÞ þ bðfÞ; x0 ðf1 Þ
ð3:1Þ
where
aðfÞ ¼
h ; fn
h¼
2 nÞ2 ðln þ mÞð1 n ; l 2ln n mn 2
ð3:2Þ
and bðfÞ is a regular function for j f j> 1. Using (3.1) in the boundary condition (1.1) and on f ¼ r, for generality, we get
k/ðrÞ aðrÞ/0 ðrÞ w ðrÞ ¼ f ðrÞ;
ð3:3Þ
where
w ðrÞ ¼ wðrÞ þ bðrÞ/0 ðrÞ; f ðrÞ ¼ FðrÞ lkCr þ N ¼ lC
lC
r
ð3:4aÞ hðX iYÞ X iY r bðrÞ; þ NaðrÞ þ lC 2pð1 þ vÞ 2pð1 þ vÞ
X iY n; 2pð1 þ vÞ
ð3:4bÞ
ð3:4cÞ
and
FðrÞ ¼ f ðtÞ:
ð3:4dÞ
The function FðrÞ with its derivatives must satisfy the Hölder condition. Multiplying both sides of (3.3) by ð1=2piÞ1=ðr fÞ and integrating with respect to r on c, one has
k/ðfÞ þ
1 2p i
Z
aðrÞ/0 ðrÞdr lC hN AðfÞ; ¼ þ fn f rf
c
ð3:5Þ
where
AðfÞ ¼
1 2pi
Z c
FðrÞdr : rf
Using (3.2), we have
1 2pi
Z c
aðrÞ/0 ðrÞdr hb ; ¼ nf rf
ð3:6Þ
where b is a complex constant to be determined. Substituting from (3.6) into (3.5), we have
k/ðfÞ ¼ AðfÞ
lC h ðb þ NÞ: þ nf f
ð3:7Þ
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29
Differentiating (3.7) with respect to f, and using the result in (3.6), yields
mðb þ NÞ þ ln2 C þ A0 ðnÞ ¼ 0; kb þ h where
m¼
n2 . n1Þ2 ðn
ð3:8Þ
Hence, from (3.8), we get
kE mhE b¼ 2 ; 2 k m2 h
ð3:9Þ
2 where E ¼ ½A0 ðnÞ þ ln C þ mhN. Also from (3.3), wðfÞ can be determined in the form
wðfÞ ¼
klC xðf1 Þ hf Þ þ BðfÞ B; / ðn 0 / ðfÞ þ f 1n f x ðfÞ
ð3:10Þ
where
/ ðfÞ ¼ /0 ðfÞ þ lC BðfÞ ¼
1 2pi
Z
X þ iY 1 f ; 2pð1 þ vÞ
FðrÞ dr; rf
c
ð3:11aÞ ð3:11bÞ
and
B¼
1 2pi
Z c
FðrÞ
r
dr:
ð3:11cÞ
4. Special cases
1. For n ¼ 0, 0 6 m 6 1, we get the function z ¼ l f þ mf1 . Considering the reality of the constants l and m in (3.7) and (3.10) besides the imposed restrictions, gives
lC mC ; f f ! klC lf1 þ mf / ðfÞ þ mCf þ Bðf1 Þ B: wðfÞ ¼ f l mf2 k/ðfÞ ¼ Aðf1 Þ
ð4:1Þ ð4:2Þ
These two forms agree’s with those obtained by Muskhelishvili [1] for an elliptic hole if we let l ¼ c and m ¼ cm. 2. By considering the reality of the constants in the rational mapping (1.9), we get
k/ðfÞ ¼ AðfÞ
l C h ðb þ NÞ; þ nf f
! klC lf1 þ mf ð1 nf1 Þ2 hf / ðfÞ þ wðfÞ ¼ / ðnÞ þ BðfÞ B: 1 nf ð1 nf1 Þðl mf2 Þ ðlf þ mf1 Þðnf2 Þ 1 nf f 2
2 2
ð4:3Þ
ð4:4Þ
2
þmÞð1n Þ where h ¼ ðlnl2ln and m ¼ ðn2n1Þ2 in b. These two forms agree’s with those obtained by El-Sirafy and Abdou [6] if we let 2 mn2 l ¼ c and m ¼ cm. 3. The rational mapping (1.9) can be written in the form
2
z ¼ nl þ lf þ ðln þ mÞ
1 X
na1 fa :
ð4:5Þ
a¼1
So by approximating this form to the first term only i.e. a ¼ 1 and then letting n ¼ 0, we get
z ¼ lf þ mf1 :
ð4:6Þ
The corresponding forms of Goursat functions are
k/ðfÞ ¼ AðfÞ
wðfÞ ¼
lC hN ; f f
klC lf1 þ mf þ BðfÞ B: / ðfÞ þ lChf 2 f l mf
ð4:7Þ
ð4:8Þ
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M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
These forms are approximately equivalent to those obtained by Aseeri [23] when n ¼ 1. P j 4. The form (4.5) is approximately equivalent to the rational mapping, z ¼ lf þ M j¼1 mj f , which was used in Abdou and Aseeri [22] and its resultant forms of Goursat functions are equivalent to the forms in (3.7) and (3.10) by omitting constant terms. 5. An important case appears when we replace f by 1f in the conformal mapping (1.9), as
z¼
1 X lf1 þ mf 2 na1 fa ; ¼ nl þ lf1 þ ðln þ mÞ 1 nf a¼1
j n j< 1:
ð4:9Þ
This mapping, when z0 ðfÞ – 0; 1 inside the j f j< 1, transforms the points in the z-plane inside the unit circle c in f-plane. And the Goursat functions, in view of (4.9), becomes
k/ðfÞ ¼ Aðf1 Þ lC f þ
h n f1
ðb þ NÞ;
ð4:10Þ
1 xðfÞ hf 1 Þ þ Bðf1 Þ B: / ðf Þ þ / ðn wðfÞ ¼ klCf f1 x0 ðf1 Þ 1n
ð4:11Þ
6. In the mapping function (4.9), which maps the hole into the interior of a unit circle, if we put m ¼ 0 and then ignore the P constant term with assuming the constants na1 ¼ 0 for a > 3, we have the mapping z ¼ l 1f þ 3a¼1 naþ1 fa . The Goursat functions according to these substitutions are equivalent to the forms found by Exadaktylos and Stavropoulou [16]. 7. Also in the mapping function (4.9), if we put m ¼ 0 and then ignore the constant term with assuming the constants P n2a1 ¼ 0 for a ¼ 1; 2; . . . ; r where r < 1, we have the mapping z ¼ l 1f þ ra¼1 n2aþ1 f2a1 by omitting the terms after r. The Goursat functions according to these substitutions are equivalent to the forms found by Exadaktylos [18].
5. Applications 5.1. Curvilinear hole for an infinite plate subjected to a uniform tensile stress For k ¼ 1; C ¼ 4p ; C ¼ 12 P expð2ihÞ and X ¼ Y ¼ f ¼ 0, we have an infinite plate stretched at infinity by the application of a uniform tensile stress of intensity P making an angle h with x-axis. The plate is weakened by a curvilinear hole C which is free from stress. The functions (3.7) and (3.10) take the form
/ðfÞ ¼
! lPf 2ln2 Pf2 þ mh½ m hlP 2ln 2 Pf2 h ½mhlP þ lP ; þ 2 4ðn fÞ 2 1 m2 h
wðfÞ ¼
lP xðf1 Þ hf lP lP Þ þ 0 /0 ðn þ : /0 ðfÞ þ f 4f 1n 4 4 x ðfÞ
ð5:1Þ
ð5:2Þ
And for l ¼ i; m ¼ 1 þ i; n ¼ 2 þ i; P ¼ 0:25; h ¼ p4, the stress components rxx ; ryy and rxy are obtained in large forms calculated by computer and illustrated in Figs. 10–14;
Fig. 10.
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
31
Fig. 11.
Fig. 12.
Fig. 13.
5.2. When the edge of the hole is subject to a uniform pressure P For k ¼ 1; X ¼ y ¼ C ¼ C ¼ 0 and f ¼ Pt, where P is a real constant, the formulae (3.7) and (3.10) becomes
/ðfÞ ¼
þn Þ hPðmhn 2 Þ ; ðn fÞð1 m2 h
ð5:3Þ
32
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
"
wðfÞ ¼
#
"
#
þn þn Þ Þ hf xðf1 Þ hPðmhn hPðmhn þ : 2 2 0 x ðfÞ ðn fÞ ð1 m2 h2 Þ 1 nf ðn nÞ ð1 m2 h2 Þ
And for l ¼ i; m ¼ 1 þ i; n ¼ 2 þ i; P ¼ 0:25, the stress components computer and illustrated in Figs. 15–19;
Fig. 14.
Fig. 15.
Fig. 16.
ð5:4Þ
rxx ; ryy and rxy are obtained in large forms calculated by
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
33
Fig. 17.
Fig. 18.
Fig. 19.
5.3. When the external force acts on the center of the curvilinear For C ¼ C ¼ f ¼ 0 and k ¼ v, we have the second fundamental problem when the force acts on the curvilinear kernel. It is assumed that the stresses vanish at infinity and it is easily seen that the kernel does not rotate.
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M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
In general the kernel remains in the original position. The Goursat’s functions are
/ðfÞ ¼
h X iY b n ; ðn fÞv 2pð1 þ vÞ
wðfÞ ¼
ð5:5Þ
xðf1 Þ 0 X þ iY 1 X þ iY hf Þ þ /0 ðn / ðfÞ f n : 0 f 2pð1 þ vÞ 2pð1 þ vÞ 1n x ðfÞ
Fig. 20.
Fig. 21.
Fig. 22.
ð5:6Þ
M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
35
Fig. 23.
Fig. 24.
And for l ¼ i; m ¼ 1 þ i; n ¼ 2 þ i; X ¼ Y ¼ 10; , ¼ 2, the stress components rxx ; ryy and rxy are obtained in large forms calculated by computer and illustrated in Figs. 20–24; 6. Conclusions From the above results and discussions the following may be concluded: (1) In the theory of two dimensional linear elasticity one of the most useful techniques for the solutions of boundary value problem for a region weakened by a curvilinear hole is to transform the region into a simpler shape to get solutions without difficulties. (2) The transformation mapping z ¼ cxðsÞ; s > 0; s ¼ r þ is, transforms the domain of the infinite plate with a curvilinear hole into the domain of the right half-plane. While the mapping z ¼ cxðfÞ; j f j> 0; f ¼ qeih , transforms the domain of the infinite plate with a curvilinear hole into the domain outside a unit circle. (3) The physical interest of the mapping (1.9) comes from its different shapes of holes it treats and different directions it takes as shown in Figs. 1–9. (4) The complex variable method (Cauchy method) is considered one of the best methods for solving the integro-differential Eq. (1.1), and obtaining the two complex potential functions /ðzÞ and wðzÞ directly. (5) Stress is an internal force whereas positive values of it mean that stress is in the positive direction, i.e. stress acts as a tension force. On the other side, negative values of stress mean that stress is in the negative direction, i.e. stress acts as a press force. (6) The most important issue deduced from (Figs. 10,11 and 15,16 and 20,21) is that max rxx ¼ min ryy and vice versa (min rxx ¼ max ryy ).
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M.A. Abdou, S.A. Aseeri / Applied Mathematics and Computation 212 (2009) 23–36
r
xx xx (7) By following the intendance of rryy and ryy at (Figs. 13,14 and 18,19 and 23,24), we find that rryy ! 0 at the same points xx r ryy rxx ! 1 and vice versa ( ! 1 at the same points where ! 0). where ryy ryy rxx xx xx ! 0 the perpendicular stress on y-axis is the maximum value and presents the body interior resistance of (8) When rryy treatment (like rocks for example). Whereas, the perpendicular stress on x-axis is small according to y-axis. Thereon, xx . it is better to treat the problem at points determined by angles that gives minimum values of rryy
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