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A fundamental solution for transversely isotropic and nonhomogeneous media
Int. J. Engng Sci. Vol. 25, No. 1, pp. 117 122, 1987 Printed in Great Britain
0020 7725/87 $3.00 + .00 Pergamon Journals Ltd
A FUNDAMENTAL SOLUTION FOR TRANSVERSELY ISOTROPIC AND NONHOMOGENEOUS MEDIA M. E. ERGUVEN* Technical University of Istanbul, Faculty of Civil Engineering, Maslak-Istanbul, Turkey Abstract--The purpose of this paper is to consider the concept of a ring of sources or forces using the integral transform techniques to derive the axisymmetric fundamental solution for nonhomogeneous transversely isotropic elastic media. Firstly, the formulation of the problem in homogeneous media to derive the fundamental solutions is shown. In the case ofa nonhomogeneous medium, the shear modulus of the material varies with the z-coordinate exponentially.
1. I N T R O D U C T I O N IN MANY industrial applications, the engineers are faced with the stress analysis of 3dimensional bodies in which the geometry and the loading involved are axisymmetric. In the solution of such one can be taken advantage of the axial symmetry. In axisymmetric problems, the point loads at the point P(x) are generalized to ring loads. Two techniques can be applied to determine the elastostatic fundamental solutions for these loads. The first is to take the 3-dimensional point load solution, in [1-3], and integrate the previously established fundamental solution along a circular path around the axis of symmetry. The second technique, adopted by Kermanidis [4], Mayr [5] and Cruse et al. [6], is to formulate a singular body force representation of the ring loads in a direct fashion. These papers employ the concept of a ring sources or forces using the Galerkin vector solution in homogeneous and elastic media. The purpose of this paper is to employ the concept of a ring of sources or forces using the Hankel integral transform to derive the axisymmetric fundamental solution for nonhomogeneous transversely isotropic media.
2. T H E
FORMULATION
OF THE PROBLEM ISOTROPIC MEDIA
IN HOMOGENEOUS
Following Cruse et al. [6],
_-
V2V2Gz = - L / ~ c~2
1
0
02
V 2 = T r ~ + - + r ~ + - - Oz2
(1)
where (G,,Go, Gz) are the Galerkin vector components, (fr,fo,fz) are the body force
* Present address: Visiting Research Scientist, MIT, Dept. of Mech. Engng. Cambridge, U.S.A. 117
118
M.E. ERGUVEN
components and/~ is the material shear modulus. The fundamental body force term can be represented through the use of a Dirac delta function, 6(x - ~), as follows:
=
2rtr
(2)
In (2), p and ~ represent the components of the load point; r and z, the field point. The body force loadings as given in (2) satisfy the following conditions
=
p,~#r,z
and
f i d V = 1,
( i = r , 0,z)
(3)
The solution of body force loading in the r- and z-directions, obtained in terms of Legendre functions of the second king using the integral transform methods [6], are as follows:
(4) In the axisymmetric problem corresponding to body force loading in the 0-direction, the governing equation is
(1j
v ~ - 7~ ~, = 0
where
(5)
Go= ~(r,z). The only nonzero displacement is
,v(,)
u0=~-
V2- 7
(6)
¢,
The nonzero stresses are
=
efuo)
azo = U T zz = (l -
v)
') )
V2 - 7
~
(7)
Integral transform methods can be also used to obtain the solution and the result is given by
uo(r,z)= (- ~--~g)L~e-S¢'--¢)J,(sp)Jl(sr)ds
(8)
A fundamental solution for transverselyisotropic and nonhomogeneousmedia
119
This is equivalent to the Laplace transform of the product of Bessel functions, which can be expressed in terms of a Legendre function as follows:
u°=
(')'
--~5n2 ~ p Q l / 2 ( 7 )
(9)
where the parameter 7 is given by
y= 1+
(z - ~)z + ( r - - p)2
(10)
2rp
For computational purposes, the Legendre function in equation (9) is written in terms of elliptical functions of the first and second kinds [7] as follows: 2
2
(11)
Stresses are calculated using equation (7): 1 "~ dQ1/2(T)(z - ~)
aro =
( - - ~ '){#
(12)
,
2pl/Zr3/2 Q1/z(T)
1 dQ1/2Fr_2 - p2 _ (z - ~)2] + (rp) 11~ d? L -2pfi A
0.1/2 .'~
pl/2r3/ZJ
(13)
{ K (d~)2 _ __~E(dl~T)t y
(14)
where dQx/2(?) _
1
d?
3. T H E F O R M U L A T I O N OF T H E P R O B L E M IN N O N H O M O G E N E O U S AND TRANSVERSELY ISOTROPIC MEDIA In this section, the axisymmetric problem corresponding to the body force loading in the 0-direction will be considered. The only nonzero components of the stress tensor are the shearing stresses
""z
r
: '-'44- ~Z'
_ /0u0
O.r0 : ( , 6 6 t ~
;0
)
(15)
where C44 and C66 denote the shear moduli of the material which vary with the z-coordinate as follows:
C44(z ) = p e 2kz,
C66(Z ) = ~'e 2kz
(16)
in which/a and #' are positive quantities, and k is the nonhomogeneity parameter. Two of the equilibrium equations are identically satisfied and the third reduces to
M. E. ERGUVEN
120
Or2
r\~r
+ a k--~z2 + 2k
= 0
(17)
where a z = #/#' > 0
(18)
It is assumed
Uo ~ O,
x / ~ + z2) -~ ov
(19)
Using integral transform techniques, the general solution of the equation (17) under condition (19) can be written as
uo(r, z) =
;o
sA(s) e x p ( - k z + x/kZa 2 + s 2
z)
Jl(rs)ds
(20)
To calculate A(s), the ring of potential source or force at (r = P, z = 0) is represented, using equation (2), as follows: (21)
#eZk~z0 z= ° = # 6(2~rP) where 6(r - p) has the following property:
fo °~6(r - p)F(r) dr = F(p)
(22)
in which F is an arbitrary function. Therefore, the Hankel transform of equation (21) can be written as
HlIC'4(z)~fz z=ol = #Jl(xp)2~
(23)
and the value of A(s) can be determined from equations (20) and (23) as
A(s) =
aJl(sp) 2zc(ak + x/kZa2 + s 2)
(24)
Using this expression A(s) in equation (20), the fundamental solution is given by
s 2 + S2) e - ~ k + ~ ~ ) z J a ( p s ) J a ( r s ) d s uo(r, z) = - ~ a fo ~ (ak q- x/a2k
(25)
or changing the integrand
uo(r,z) =
~
2
Z (-1)"(ak)" / s -
Jl (pS)Jl(rS)ds
(26)
In order to obtain a more convenient form of equation (26), we introduce the following functions:
A fundamental solution for transversely isotropic and nonhomogeneous media Fo(r,z ) = fo ~ ( a2k2 "~s $2) 1 / 2 -e _~Z a 2 k2 +~2~1/~Jx(pS)Jl(rS)ds
F.(r, z) =
f0 t3
121 (27)
(a2k2 +Ss2). + 1/2 e _ ~(a2k 2 + S2) 1/2jl(ps)Jl(rs)ds
(28)
Taking Fo and F. into account, we write the following relation: a" d"F"(r' z) _ ( _ 1).Fo(r, z) dz"
(29)
Integrating equation (29), we have
a"F,(r, z) = -
f
~ (z n!. O" Fo(r, Od(
(30)
z
Making use of equations (27), (28) and (30), Uo(r, z) can be written as ae_kZ
{
f
n=O
) Fo(r, 0 d (
0o(z
,a0
}
(31)
or simply
uo(r,z)-
ae_kZ _ o~ ~ {Fo(r,z) kfo Fo(r, Oek(~-°
d~'}
(32)
Once Fo(r, z) has been found, uo(r, z) can be calculated by using equation (32). The stress components can also be calculated by using equation (15). In the homogeneous isotropic case, i.e. when k = 1 and a = 1, uo(r, z) takes the following form:
uo(r, z) = -
Fo(r, z) = - 2n
e - ~z jl(ps)Jl(rS)ds
(33)
Evaluating this integral, we find that
uo(r,z) =
- ~
1)
1
~Q,/2(~
)
(34)
i
x~ rp
where the parameter 7' is given by y'=l+
z 2 + (r -- p)Z 2rp
(35)
As can be seen, these results coincide with equations (9) and (10). Let us modify the integrand as follows in order to calculate Fo(r, z). Fo(r,z) =
us l
_ a e-~(a2k2 +~2),/~ Jl(sp)Jl(Sr)ds z )
(36)
122
M.E. ERGUVEN
=
ae-~a2k2 +s2)'/2
[Jl(sp)Jl(sr)]ds
(37)
The p r o d u c t of the Bessel function in equation (37) can be written as follows [7]:
Jl(sp)Jl(sr) - (sp/2)(sr/2) ~, ( - 1)~(sp/2)2~F(- m, - 1 - m; 2; r2/p 2) 2 m!F(2 + m)
(38)
M a k i n g use of equation (38), Fo(r, z) m a y be expressed as
Fo(r,z)=
~
(-1)rp2'~+lF(-m'-l-rn;2;r2/p2)f°~ae-~a2k2+~2)lnIds2+2"tds
(39)
or, by evaluating the integrals: ( - 1)mrp 2m+1 Fo(r,z) = m=O L ~ / ~ 2 m m ! (ak)'n+l/2F(-m' - 1 - m ,. 2. , r /2p 2)K_,n_3/2 (kz)
(40)
where F(a, b; c; x) is a hypergeometric function and K is the modified Bessel function of order v. REFERENCES [1] D. J. SHIPPY, F. J. RIZZO and A. K. GUPTA, Proc. lOth Southwestern Conf. on Theoretical and Applied Mech., Tennessee (1980). I2] D. J. SHIPPY, F. J. RIZZO and R. K. NIGAM, Proc. 2nd Int. Symp. Innovative Numerical Analysis in Applied Engng Sci., Montreal (1980). 1'3] L. C. WROBEL and C. A. BREBBIA, Proc. 2nd Int. Semin. Recent Advances in Boundary Element Methods, Southampton (1980). [4] T. KERMANIDIS, Int. J. Solids Struct. 11, 493-500 (1975). I-5] M. MAYR, Mech. Res. Commun. 3, 393-398 (1976). [6] T. A. CRUSE, D. W. SNOW and R. B. WILSON, Computer Struct. 7, 445-459 (1977). I-7] M. ABRAMOWlTZ and I. A. STEGUN, Handbook of Mathematical Functions. Dover (1972). 1-8] I. S. GRADSHTEYN and I. M. RYZHIK, Table of Integrals Series and Products. Academic Press, New York (1965).
(Received 13 July 1985)
0020 7725/87 $3.00 + .00 Pergamon Journals Ltd
A FUNDAMENTAL SOLUTION FOR TRANSVERSELY ISOTROPIC AND NONHOMOGENEOUS MEDIA M. E. ERGUVEN* Technical University of Istanbul, Faculty of Civil Engineering, Maslak-Istanbul, Turkey Abstract--The purpose of this paper is to consider the concept of a ring of sources or forces using the integral transform techniques to derive the axisymmetric fundamental solution for nonhomogeneous transversely isotropic elastic media. Firstly, the formulation of the problem in homogeneous media to derive the fundamental solutions is shown. In the case ofa nonhomogeneous medium, the shear modulus of the material varies with the z-coordinate exponentially.
1. I N T R O D U C T I O N IN MANY industrial applications, the engineers are faced with the stress analysis of 3dimensional bodies in which the geometry and the loading involved are axisymmetric. In the solution of such one can be taken advantage of the axial symmetry. In axisymmetric problems, the point loads at the point P(x) are generalized to ring loads. Two techniques can be applied to determine the elastostatic fundamental solutions for these loads. The first is to take the 3-dimensional point load solution, in [1-3], and integrate the previously established fundamental solution along a circular path around the axis of symmetry. The second technique, adopted by Kermanidis [4], Mayr [5] and Cruse et al. [6], is to formulate a singular body force representation of the ring loads in a direct fashion. These papers employ the concept of a ring sources or forces using the Galerkin vector solution in homogeneous and elastic media. The purpose of this paper is to employ the concept of a ring of sources or forces using the Hankel integral transform to derive the axisymmetric fundamental solution for nonhomogeneous transversely isotropic media.
2. T H E
FORMULATION
OF THE PROBLEM ISOTROPIC MEDIA
IN HOMOGENEOUS
Following Cruse et al. [6],
_-
V2V2Gz = - L / ~ c~2
1
0
02
V 2 = T r ~ + - + r ~ + - - Oz2
(1)
where (G,,Go, Gz) are the Galerkin vector components, (fr,fo,fz) are the body force
* Present address: Visiting Research Scientist, MIT, Dept. of Mech. Engng. Cambridge, U.S.A. 117
118
M.E. ERGUVEN
components and/~ is the material shear modulus. The fundamental body force term can be represented through the use of a Dirac delta function, 6(x - ~), as follows:
=
2rtr
(2)
In (2), p and ~ represent the components of the load point; r and z, the field point. The body force loadings as given in (2) satisfy the following conditions
=
p,~#r,z
and
f i d V = 1,
( i = r , 0,z)
(3)
The solution of body force loading in the r- and z-directions, obtained in terms of Legendre functions of the second king using the integral transform methods [6], are as follows:
(4) In the axisymmetric problem corresponding to body force loading in the 0-direction, the governing equation is
(1j
v ~ - 7~ ~, = 0
where
(5)
Go= ~(r,z). The only nonzero displacement is
,v(,)
u0=~-
V2- 7
(6)
¢,
The nonzero stresses are
=
efuo)
azo = U T zz = (l -
v)
') )
V2 - 7
~
(7)
Integral transform methods can be also used to obtain the solution and the result is given by
uo(r,z)= (- ~--~g)L~e-S¢'--¢)J,(sp)Jl(sr)ds
(8)
A fundamental solution for transverselyisotropic and nonhomogeneousmedia
119
This is equivalent to the Laplace transform of the product of Bessel functions, which can be expressed in terms of a Legendre function as follows:
u°=
(')'
--~5n2 ~ p Q l / 2 ( 7 )
(9)
where the parameter 7 is given by
y= 1+
(z - ~)z + ( r - - p)2
(10)
2rp
For computational purposes, the Legendre function in equation (9) is written in terms of elliptical functions of the first and second kinds [7] as follows: 2
2
(11)
Stresses are calculated using equation (7): 1 "~ dQ1/2(T)(z - ~)
aro =
( - - ~ '){#
(12)
,
2pl/Zr3/2 Q1/z(T)
1 dQ1/2Fr_2 - p2 _ (z - ~)2] + (rp) 11~ d? L -2pfi A
0.1/2 .'~
pl/2r3/ZJ
(13)
{ K (d~)2 _ __~E(dl~T)t y
(14)
where dQx/2(?) _
1
d?
3. T H E F O R M U L A T I O N OF T H E P R O B L E M IN N O N H O M O G E N E O U S AND TRANSVERSELY ISOTROPIC MEDIA In this section, the axisymmetric problem corresponding to the body force loading in the 0-direction will be considered. The only nonzero components of the stress tensor are the shearing stresses
""z
r
: '-'44- ~Z'
_ /0u0
O.r0 : ( , 6 6 t ~
;0
)
(15)
where C44 and C66 denote the shear moduli of the material which vary with the z-coordinate as follows:
C44(z ) = p e 2kz,
C66(Z ) = ~'e 2kz
(16)
in which/a and #' are positive quantities, and k is the nonhomogeneity parameter. Two of the equilibrium equations are identically satisfied and the third reduces to
M. E. ERGUVEN
120
Or2
r\~r
+ a k--~z2 + 2k
= 0
(17)
where a z = #/#' > 0
(18)
It is assumed
Uo ~ O,
x / ~ + z2) -~ ov
(19)
Using integral transform techniques, the general solution of the equation (17) under condition (19) can be written as
uo(r, z) =
;o
sA(s) e x p ( - k z + x/kZa 2 + s 2
z)
Jl(rs)ds
(20)
To calculate A(s), the ring of potential source or force at (r = P, z = 0) is represented, using equation (2), as follows: (21)
#eZk~z0 z= ° = # 6(2~rP) where 6(r - p) has the following property:
fo °~6(r - p)F(r) dr = F(p)
(22)
in which F is an arbitrary function. Therefore, the Hankel transform of equation (21) can be written as
HlIC'4(z)~fz z=ol = #Jl(xp)2~
(23)
and the value of A(s) can be determined from equations (20) and (23) as
A(s) =
aJl(sp) 2zc(ak + x/kZa2 + s 2)
(24)
Using this expression A(s) in equation (20), the fundamental solution is given by
s 2 + S2) e - ~ k + ~ ~ ) z J a ( p s ) J a ( r s ) d s uo(r, z) = - ~ a fo ~ (ak q- x/a2k
(25)
or changing the integrand
uo(r,z) =
~
2
Z (-1)"(ak)" / s -
Jl (pS)Jl(rS)ds
(26)
In order to obtain a more convenient form of equation (26), we introduce the following functions:
A fundamental solution for transversely isotropic and nonhomogeneous media Fo(r,z ) = fo ~ ( a2k2 "~s $2) 1 / 2 -e _~Z a 2 k2 +~2~1/~Jx(pS)Jl(rS)ds
F.(r, z) =
f0 t3
121 (27)
(a2k2 +Ss2). + 1/2 e _ ~(a2k 2 + S2) 1/2jl(ps)Jl(rs)ds
(28)
Taking Fo and F. into account, we write the following relation: a" d"F"(r' z) _ ( _ 1).Fo(r, z) dz"
(29)
Integrating equation (29), we have
a"F,(r, z) = -
f
~ (z n!. O" Fo(r, Od(
(30)
z
Making use of equations (27), (28) and (30), Uo(r, z) can be written as ae_kZ
{
f
n=O
) Fo(r, 0 d (
0o(z
,a0
}
(31)
or simply
uo(r,z)-
ae_kZ _ o~ ~ {Fo(r,z) kfo Fo(r, Oek(~-°
d~'}
(32)
Once Fo(r, z) has been found, uo(r, z) can be calculated by using equation (32). The stress components can also be calculated by using equation (15). In the homogeneous isotropic case, i.e. when k = 1 and a = 1, uo(r, z) takes the following form:
uo(r, z) = -
Fo(r, z) = - 2n
e - ~z jl(ps)Jl(rS)ds
(33)
Evaluating this integral, we find that
uo(r,z) =
- ~
1)
1
~Q,/2(~
)
(34)
i
x~ rp
where the parameter 7' is given by y'=l+
z 2 + (r -- p)Z 2rp
(35)
As can be seen, these results coincide with equations (9) and (10). Let us modify the integrand as follows in order to calculate Fo(r, z). Fo(r,z) =
us l
_ a e-~(a2k2 +~2),/~ Jl(sp)Jl(Sr)ds z )
(36)
122
M.E. ERGUVEN
=
ae-~a2k2 +s2)'/2
[Jl(sp)Jl(sr)]ds
(37)
The p r o d u c t of the Bessel function in equation (37) can be written as follows [7]:
Jl(sp)Jl(sr) - (sp/2)(sr/2) ~, ( - 1)~(sp/2)2~F(- m, - 1 - m; 2; r2/p 2) 2 m!F(2 + m)
(38)
M a k i n g use of equation (38), Fo(r, z) m a y be expressed as
Fo(r,z)=
~
(-1)rp2'~+lF(-m'-l-rn;2;r2/p2)f°~ae-~a2k2+~2)lnIds2+2"tds
(39)
or, by evaluating the integrals: ( - 1)mrp 2m+1 Fo(r,z) = m=O L ~ / ~ 2 m m ! (ak)'n+l/2F(-m' - 1 - m ,. 2. , r /2p 2)K_,n_3/2 (kz)
(40)
where F(a, b; c; x) is a hypergeometric function and K is the modified Bessel function of order v. REFERENCES [1] D. J. SHIPPY, F. J. RIZZO and A. K. GUPTA, Proc. lOth Southwestern Conf. on Theoretical and Applied Mech., Tennessee (1980). I2] D. J. SHIPPY, F. J. RIZZO and R. K. NIGAM, Proc. 2nd Int. Symp. Innovative Numerical Analysis in Applied Engng Sci., Montreal (1980). 1'3] L. C. WROBEL and C. A. BREBBIA, Proc. 2nd Int. Semin. Recent Advances in Boundary Element Methods, Southampton (1980). [4] T. KERMANIDIS, Int. J. Solids Struct. 11, 493-500 (1975). I-5] M. MAYR, Mech. Res. Commun. 3, 393-398 (1976). [6] T. A. CRUSE, D. W. SNOW and R. B. WILSON, Computer Struct. 7, 445-459 (1977). I-7] M. ABRAMOWlTZ and I. A. STEGUN, Handbook of Mathematical Functions. Dover (1972). 1-8] I. S. GRADSHTEYN and I. M. RYZHIK, Table of Integrals Series and Products. Academic Press, New York (1965).
(Received 13 July 1985)