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An elastic contact problem for a half-space indented by a flat annular rigid stamp
ht. J. EngngSci., 1974,Vol. 12, pp. 759-771. Pergamon Press. Printed in Great Britain
AN ELASTIC CONTACT PROBLEM FOR A HALF-SPACE INDENTED BY A FLAT ANNULAR RIGID STAMP TOSHIKAZU
SHIBUYA, TAKASHI
KOIZUMI and ICHIRO NAKAHARA
Tokyo Institute of Technology,
Tokyo,Japan
(Communicated by I. N. SNEDDON) Abstract-We consider an elastic contact problem for a half-space indented by a flat annular rigid stamp, which is a three-part mixed boundary value problem. If the distribution of pressure on the contact region is assumed, the contact problem will be reduced to the solution of an infinite set of simultaneous equations. Numerical results are also illustrated for the distributions of the displacements and stresses in the half-space.
1. INTRODUCTION the simplest three-part mixed boundary value problems in elasticity is the contact problem for a flat annular rigid stamp and, at present, there seems to be no method available for obtaining the solution in a closed form. Williams[l], Cooke[2], Noble[3], Collins[4] and Jain and Kanwel[Sl showed that such a problem can be reduced to the solution of a Fredholm integral equation, and tried to solve the equation by an iterative procedure. Gubenko and Mossakovskii [6] showed that the problem can be reduced to the solution of a pair of simultaneous Fredholm integral equations, but that the iterative solution of these equations is very complicated. The purpose of this paper is to propose a simple technique for the solution of the same problem. The normal pressure on the contact region is continuous at all points except the inner and outer edges of the stamp. In consideration of the fact, assuming the contact pressure in an appropriate series, we will reduce the problem to the solution of an infinite system of simultaneous linear algebraic equations. ONE OF
2. FORMULATION
OF THE PROBLEM
In terms of cylindrical coordinates (4,8, z), let the half-space occupy the region z 2 0, and let the annular stamp be attached to the portion z = 0, r; 5 r 5 rO,for all 8, where I; and r, are the inner and outer radii of the stamp, as shown in Fig. 1. In this coordinate P
--i----
4--_-__
Annular
1
stamp
-b-
Fig. 1.Geometry of the problem. 759
T.
160
system we component components is indented frictionless, ditions are
SHIBUYA,
T. KOIZUMI
and I. NAKAHARA
take the components of the displacement vector to be (u,, 0, w,)-the 8being identically zero because of the symmetry-and the non-vanishing of the stress tensor to be a,, a,, oZ and r,,. We assume that the rigid stamp to a depth E” and the interface between the stamp and the half-space is while the rest of the surface is free from traction. Thus, the boundary con-
(a) (~7~)~=~=0, (OSr
(1)
(T,~)~_~ = 0, (0 d r);
(d) all displacements
and stresses vanish at infinity.
In the absence of body forces, the general solution of the equations of elastic equilibrium in the case of axisymmetry without torsion may be represented by the displacement field 2Gu
r
=?%!+z&k?? ar
ar'
(2)
where
In these equations, G and v denote respectively the shear modulus and the Poisson’s ratio. In order to satisfy the boundary conditions (c) and (d) of equation (l), the stress functions cpoand (p3 are expressed by the cylindrical harmonics as follows: cp”= - (1 - 2V)
I
*
0
A~‘B(A)J,,(Ar)e~“‘dh,
and B(A) is an arbitrary function of A. The displacements pond to the stress functions (3) are given as follows:
2Gu,
=
I
2Gw, = -
e
o ((1 -
2~) - Az}B(A)J,(Ar)e-“‘dh,
Z{2(1 - V) + Az}B(A)Jo(Ar)e~“‘dA,
(3)
and stresses which corres-
An elastic contact problem
a =II 0; =
[{(I + 2~) - hz&(hr)
T{(l -
761
2~) - hz}fi(hr)lhB(h)e-“‘dh, (4)
=
(1 t hz)hBfh)Jo(hr)e-“‘dh,
Using the boundary conditions (a) and (b) of equation (l), we can find that the function B(A) must satisfy the triple integral equations hB(h)Jo(hr)dh
= 0, (0 6 Y< rj, ro < r),
(3
The present problem is equivalent to the next cases: (A) the circular stamp of the radius r. is indented on the elastic surface, and then the circular portion of the radius G( < rO) is removed; (B) the elastic surface is pressed by the infinite rigid plate with a circular hole of the radius rj, and then the infinite portion with the hole of r, is released from pressing. In the case (A), the singuIarity of (o~)~=,,at r = r. takes together the form (r i - 9)~“* before and after removing the circular portion. Similarly, in the case (B), the singularity of (aZLZO at r = ri takes together the form (r2 - r :)--I” before and after releasing the infinite portion with the hole. Therefore, the singui~ities of (c~)~+, in this problem are considered to have the forms (ri- rZ)-“2 at r = r. and (r’ - r:))‘” at r = ri:,.Then, we assume (o~),_~ in the region on the annular stamp as follows: (uz)z=o=-
Eof ($1
v?r4- r2)(r2- If)’
(ri < r -=Lro),
(6)
where f (r) is an unknown function which is assumed to be continuous in I;:~5r 5 rot and is non-zero at r = r. and r = ri. Putting r, = (fi + r&2, b = (r, - &)/2, 2r,bcoscjb=rZ+b*-r2,
(7)
the variable r in I; s r 5 r. will be replaced to the new one # in 0 s # s W,when r = rj corresponds to 4 = 0, and when r = r. to &,= n: Then, the function f(r) in ri 5 r s r, can be expressed by the Fourier series with respect to 4 in 0 ~5(f, 5 r, Considering f(6) + 0 and f(r,) + 0, we may take the representation: cc
fW=r,n=o
DES.-Vol. 12. No. 9.4
a:,c0sn#,(fi5$~$&
(8)
162
T. SHIHUYA.
where
a:, are unknown
T. KOIZUMI
coefficients.
and I. NAKAHARA
Using the relations
r’--ri=2r,b(l-cosb). we can rewrite
equation
(6) as
(a:),
Using equation Hankel
0 =
-
,C;, uL=,
+2r,b
(9)
(9) and the fact of (a, )_ ,, = 0 in 0 5 r < r, and r,, < r. we find from the
inversion
theorem
that
cos
Recalling
(r, < r < r,,).
+ h’-
n@,,(AdrT
2r,b cos
4) d&.
(IO)
the formula
-I T
I
n
II
cosn~#J,,(A~rT+
b’-2r,bcosd)d~+
=J,(Ar,)J,(Ab),
(11)
we obtain B(A)
Substituting
equation
= -i
TTTE,, i aX(Ar,
)J,,(Ab).
” -,I
(12)
(12) into (MJ,)~-~~and using the second condition
of equation
(5),
we get
I = i
,1 0
a,,
‘ J,,(Ar)J,(Ar, I0
dA, (r, d r 2 rd.
)J.(Ab)
(13)
where 1-v a. = 2G In consideration
?ru:,, (n = 0, I, 2,-----).
of the formula
J,,(Ar) = .J,,(Ar, )J,,(Ab) + 2 c J,(Ar, WI I equation
(13) can be rewritten
)J,(Ab)
(14)
(r, 5 r 5 rd,
as follows: z
I
1 = 2 ~“1 J.(Ar.).I”(Ab)(~,,(Ar~)~C,(Ab) ,I 0 0 Since equation
cos m$,
(15) must hold for an arbitrary
+2
c .L(Ar,.).L(Ab)cosm4 m I
dA.
(15)
value of 4, we find that the equation
is
An elastic contact problem
reduced to the following system of simultaneous coefficients a. :
equations for the determination
J,,,(arc )A, (Ab ).L,(Ar<1.L(Ab ) dA = km
763
of the
(16)
(m = 0, I, 2,3,-----), where So,,, is the Kronecker’s delta. Consequently, the present mixed boundary problem is reduced to the solution of the simultaneous equation (16). Using a,, we rewrite B(A) as
value
(17) In particular,
the displacements
and stresses on the surface are:
(18)
where lo =
I
I, =
=J,(Ar).L(Ar, ).I, (Ah) dh. I0
=J,,(Ar)J, (At-
To evaluate the infinite integrals I,, and I,, we make use of the following formulae 171; I-(n + l/2) 1 b ” F f, n ++; n + 1; sin’ Q lyn + l)r(1/2) r, 0r, (
(19)
T. SHIBUYA,
764
T. KOIZUMI
and I. NAKAHARA
and
r < I;),
+lnr, (ri 5 r S ro, n = O), ={ sin nqblnw, (r, S r 5 r,, n = I), Ur, (r > r(,, n = 01,
cm)
,O, (r > ro, n 2 I),
where W(x), T(x) and F(a, p ; y ; x) are in this order the step function, function and the Gauss hypergeometric series. Also, cp and I#Iare:
the gamma
when 0 S r 5 ri, and
when r Zzr,. Moreover,
# is
u, and all stresses in 0 I r < ri on the surface are always zero, while the displacements and stresses in r > r. on the surface decrease in order of r-’ and r--’ respectively. The total load P on the punch is
2arfo, 1;=,) dr
p=-
3. NUMERICAL
27rGElJ =
-
l-v
a,.
CALCULATIONS
To solve the simultaneous equation (16), we calculate the infinite integrals for the product of four Bessel functions by the following method. The integral of m-line and n-column in equation (16), A,, is A,, = A,,,,, = where A’rnn=AL=
I
ho
0
~,,_,(hr,)J,,.. l(Ab)J,-,(hr,)J,-I(Ab)
* J,_,(Ar,)J,-,(Ab)J,-,(XT,-)J,-l(hb) _Iho
dA + A A,
dA,
(22)
(231
165
An elastic contact problem
and ho is taken as a certain large value. The first term of the right hand of equation (22) is integrated numerically by means of the Simpson’s second rule. Here, we choose A,,= 500 and AA = 0.2, where AA is the interval in numerical integrals. The second term is integrated by using the approximation of Bessel function. For a large value of A, the product of Bessel functions is J,-,(Ar,)J,_,(Ab) Substituting
+ db{cos
hri + (- I)‘“-’ sin Arc}.
E
equation (24) into equation (23), we find that
+ (- l),+” sin* Ar,] dh. Integrating
(25)
equation (25) by parts and using the sine- and cosine-integral si(x)=
functions
x sin t * cos t t dt, ci (x) = tdt, Cz I_
I
we obtain the equation =- 1 A’ IFan v2r,b
COS’ A,G ----+ ri si (2Aon). A0
-I-{(- I)‘“-’ + (- I)“-‘){ sin Aor~fos ‘Ori- r, ci (2AorC)- b ci (ZA,b)} + (_ ,),+,
r. si (2Aoro) .
(26)
II
si (x) and ci (x) are computed with good accuracy by using approximate formulae [S]. The approximate solutions of the set of simultaneous equation (16) are caiculated and are shown in Table 1. It is known that the solution is independent of the material properties of the half-space. On the surface, the displacements and stresses are calcuTable 1. Values of a. n
r, = 0.25 r,= I.0
:
0.635 380 5 173 38 5 -0.382 265 -0.061
3
-0.019
636 71
:
-0+03 -0+07
464 870 64 63
76 8 9
736 86 16 -O#XJ -0.001 688 -0.000 720 44 083 78 O~OOO
0
r, = 0.5 r,= 1.0 0.624 670 -0.283 978 -0-028 467 -o@Os 414 - 0,001 278 -O%IO 272 -0.000 213 O-000 214 -oGOt? 530 O+IOO648
r, =0.75 r” = 1.0 2 8 31 23 55 37 39 33 07 25
04.586 628 -0.153 899 -0.007 093 -0*000 593 -0dXtO 136 OGOO 126 -0.000 573 0.000 764 -0dIO2 230 0+002 221
3 1 49 64 75 79 99 29 14 99
766
T. SHIBUYA,
T. KOIZUMI
and I. NAKAHARA
lated by equations (18)-(20), and, in the half-space, they are integrated numerically by equation (4). Here, the Poisson’s ratio is taken as 0.3. In Figs. 2-7 respectively the radial distributions of the displacements and stresses are shown for the case of I;, = 1 and r, = O-5. In the figures, the dashed-curves denote the results of a circular stamp with radius r,,. In particular, we are interested in the results on the surface. (u,), (, is always zero in 0 ~5r 5 ri and is negative in r > I;. The
0
-0.2
c
Fig. 2. The distribution of IA,for ril = 1 and ri = 0.5.
0
0.5 I
I.0
I.5
I
Fig. 3. The distribution of w, for r, = 1 and vi = 0.5.
2.0
An elastic
0
0.5
contact
167
problem
1.C
_A
1; \
--I
Fig. 4. The distribution of a, for r, =
1and r, = 0.5.
slopes of (u,), =0tend to infinity as r + ri + 0 and r + r,, - 0. Since (w,), __”is considerably larger than (u,)~~~, (w,),=, shows approximately the surface deformation. The surface deforms approximately elliptically in 0 5 r 5 ri,and decreases in order of r-’ in r 2 r,. The slopes of the deformation, also, tend to infinity as r + ri- 0 and r --f r0 + 0. The stresses (a,),=O, (u,)~=~ and (a,),=,, are always compressive in the contact region and tend to infinity as r + r,+ 0 and r-~ r.- 0, and are finite or zero in the rest of the surface. In particular, ((T,), =0coincides with the contact pressure of the annular stamp. In Figs. 8 and 9 and Figs. 10 and 11, respectively, the radial distributions of w, and a, in the vicinity of the surface are shown for the cases of r,,= 1 and r,= 0.25 and 0.75. In each case, the results are qualitatively similar to the case of r. = 1 and r,= 0.5.
0
0.5
I.0
I.5
,5
0
1 a L
b”
0
B
-“O
J
-2.0
Fig. 5. The distribution
of me for r,, =
1and r, = 0.5.
T. SHIBUYA,
T. KOIZUMI
Fig. 6. The distribution
Fig. 7. The distribution
and
I. NAKAHARA
of vz for r, =
1and r, = 0.5.
of 7,: for r(, =
1and r, = 0.5.
769
An elastic contact problem
I3
I.5
0.5
2.0 I
--l--TI
I 21
y”
0.: ,-
-.--.-.-.~--
1
:
!-.-
IX )-
Fig. 8. The distribution of w, for r, =
I and r, = 0.25.
Fig. 9. The distribution of o, for r, =
1and r, = 0.25.
770
T. SHIBUYA,
T. KOIZUMI
and
I. NAKAHARA
r 0
I.0
0.5
I.5
2.0
I
Fig. 10. The distribution
of W, for r,, =
1and r, = 0.75.
C
n
2.0 A
\ \ ._______
F=
Fig.
a’P
11.The distribution
I.0
-
k
;F
of V, for r. = 1 and r, = 0.75.
---___
0.5. :: f 4: 0
0.5
I.0
r; I,
Fig. 12. The variations
of (w,),_
z=”and the total load P with the ratio r, IT,,.
An elastic contact
problem
771
Figure 12 shows the variations of (w,),=~+, and the total load P with the ratio n/r+ The former is denoted the solid curve and the Iatter the dashed-curve. (w~)~=~=~decreases linearly in ri /rO< 0.7 and very rapidly in ri fro> O-7. The total load is nearly equal to that for a circular stamp if ri /r,, < O-5. REFERENCES [I] W. E. WILLIAMS, Proc. Edinburgh Math. Sot. 13, 317 (1963). 121 .J. C. COOKE, Q. J. Mech. appl. Math. 16, 193 (1963). [3] [4] (51 [61 I71 [81
B. NOBLE,
Proc. Camb. Phi/. Sot. 59, 351 (1%3).
W. D. COLLINS, Proc. Edinburgh Math. Sot. 13, 235 (1%3). D. L. JAIN and R. P. KANWEL, SIAMJ. appl. M&h. 20, 642 (1971). V. S. GUBENKO and V. I. MOSSAKOVSKII, FMM 24, 334 (l%O). A. ERDELYI (Editor), Tables of Integral Transforms. McGraw-Hill (1954). C. HASTINGS, Approximations for Digitat Computers. Princeton University (Received
19 September
Press
(1955).
1973)
R&um&Nous considerons un probleme de contact Clastique pour un demi plan entail@ par un poincon rigide annulaire plat, ce qui represente un problime mixte de valeurs aux limites h trois parties. Si la dist~bution de la pression sur la rCgion de contact est supposee connue, le probf~me de contact sera ramene a la resolution d’un systeme infini d’equations simultan~es. Des resultats num~riques sont egalement don&s en illustration des dist~butions des deplacements et des contraintes dans le demi espace. Zusa~nfa~n~Wir untersuchen ein elastisches Kontaktproblem ftir einen Halbraum, der durch einen flachen ringfiirmigen starren Stempel eingedrtickt ist, was ein dreiteiliges gemischtes Grenzwertproblem darstellt. Falls die Druekverteilung auf die Kontaktzone angenommen ist, wird das Kontaktproblem auf die Losung eines unendlichen Satzes simultaner Gteichungen reduziert. Es werden such numerische Resultate fur die Verteilungen der Verdrangungen und Spannungen im Halbraum illustriert. Sommario-Si considera it problema de! contatto elastic0 per un semispazio depress0 da uno stamp0 rigid0 anulare piatto, the i: un problema a tre parti con valori limite misti. Assumendo la distribuzione uniforme delle pressioni sulla regione di contatto, il problema viene ridotto alla soluzione di un gruppo infinite di equazioni simultanee. Vengono inoltre illustrati i risultati numerici per la distribuzione degli spostamenti e delle sollecitazioni
nel semispazio.
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A6CTpaKT
AN ELASTIC CONTACT PROBLEM FOR A HALF-SPACE INDENTED BY A FLAT ANNULAR RIGID STAMP TOSHIKAZU
SHIBUYA, TAKASHI
KOIZUMI and ICHIRO NAKAHARA
Tokyo Institute of Technology,
Tokyo,Japan
(Communicated by I. N. SNEDDON) Abstract-We consider an elastic contact problem for a half-space indented by a flat annular rigid stamp, which is a three-part mixed boundary value problem. If the distribution of pressure on the contact region is assumed, the contact problem will be reduced to the solution of an infinite set of simultaneous equations. Numerical results are also illustrated for the distributions of the displacements and stresses in the half-space.
1. INTRODUCTION the simplest three-part mixed boundary value problems in elasticity is the contact problem for a flat annular rigid stamp and, at present, there seems to be no method available for obtaining the solution in a closed form. Williams[l], Cooke[2], Noble[3], Collins[4] and Jain and Kanwel[Sl showed that such a problem can be reduced to the solution of a Fredholm integral equation, and tried to solve the equation by an iterative procedure. Gubenko and Mossakovskii [6] showed that the problem can be reduced to the solution of a pair of simultaneous Fredholm integral equations, but that the iterative solution of these equations is very complicated. The purpose of this paper is to propose a simple technique for the solution of the same problem. The normal pressure on the contact region is continuous at all points except the inner and outer edges of the stamp. In consideration of the fact, assuming the contact pressure in an appropriate series, we will reduce the problem to the solution of an infinite system of simultaneous linear algebraic equations. ONE OF
2. FORMULATION
OF THE PROBLEM
In terms of cylindrical coordinates (4,8, z), let the half-space occupy the region z 2 0, and let the annular stamp be attached to the portion z = 0, r; 5 r 5 rO,for all 8, where I; and r, are the inner and outer radii of the stamp, as shown in Fig. 1. In this coordinate P
--i----
4--_-__
Annular
1
stamp
-b-
Fig. 1.Geometry of the problem. 759
T.
160
system we component components is indented frictionless, ditions are
SHIBUYA,
T. KOIZUMI
and I. NAKAHARA
take the components of the displacement vector to be (u,, 0, w,)-the 8being identically zero because of the symmetry-and the non-vanishing of the stress tensor to be a,, a,, oZ and r,,. We assume that the rigid stamp to a depth E” and the interface between the stamp and the half-space is while the rest of the surface is free from traction. Thus, the boundary con-
(a) (~7~)~=~=0, (OSr
(1)
(T,~)~_~ = 0, (0 d r);
(d) all displacements
and stresses vanish at infinity.
In the absence of body forces, the general solution of the equations of elastic equilibrium in the case of axisymmetry without torsion may be represented by the displacement field 2Gu
r
=?%!+z&k?? ar
ar'
(2)
where
In these equations, G and v denote respectively the shear modulus and the Poisson’s ratio. In order to satisfy the boundary conditions (c) and (d) of equation (l), the stress functions cpoand (p3 are expressed by the cylindrical harmonics as follows: cp”= - (1 - 2V)
I
*
0
A~‘B(A)J,,(Ar)e~“‘dh,
and B(A) is an arbitrary function of A. The displacements pond to the stress functions (3) are given as follows:
2Gu,
=
I
2Gw, = -
e
o ((1 -
2~) - Az}B(A)J,(Ar)e-“‘dh,
Z{2(1 - V) + Az}B(A)Jo(Ar)e~“‘dA,
(3)
and stresses which corres-
An elastic contact problem
a =II 0; =
[{(I + 2~) - hz&(hr)
T{(l -
761
2~) - hz}fi(hr)lhB(h)e-“‘dh, (4)
=
(1 t hz)hBfh)Jo(hr)e-“‘dh,
Using the boundary conditions (a) and (b) of equation (l), we can find that the function B(A) must satisfy the triple integral equations hB(h)Jo(hr)dh
= 0, (0 6 Y< rj, ro < r),
(3
The present problem is equivalent to the next cases: (A) the circular stamp of the radius r. is indented on the elastic surface, and then the circular portion of the radius G( < rO) is removed; (B) the elastic surface is pressed by the infinite rigid plate with a circular hole of the radius rj, and then the infinite portion with the hole of r, is released from pressing. In the case (A), the singuIarity of (o~)~=,,at r = r. takes together the form (r i - 9)~“* before and after removing the circular portion. Similarly, in the case (B), the singularity of (aZLZO at r = ri takes together the form (r2 - r :)--I” before and after releasing the infinite portion with the hole. Therefore, the singui~ities of (c~)~+, in this problem are considered to have the forms (ri- rZ)-“2 at r = r. and (r’ - r:))‘” at r = ri:,.Then, we assume (o~),_~ in the region on the annular stamp as follows: (uz)z=o=-
Eof ($1
v?r4- r2)(r2- If)’
(ri < r -=Lro),
(6)
where f (r) is an unknown function which is assumed to be continuous in I;:~5r 5 rot and is non-zero at r = r. and r = ri. Putting r, = (fi + r&2, b = (r, - &)/2, 2r,bcoscjb=rZ+b*-r2,
(7)
the variable r in I; s r 5 r. will be replaced to the new one # in 0 s # s W,when r = rj corresponds to 4 = 0, and when r = r. to &,= n: Then, the function f(r) in ri 5 r s r, can be expressed by the Fourier series with respect to 4 in 0 ~5(f, 5 r, Considering f(6) + 0 and f(r,) + 0, we may take the representation: cc
fW=r,n=o
DES.-Vol. 12. No. 9.4
a:,c0sn#,(fi5$~$&
(8)
162
T. SHIHUYA.
where
a:, are unknown
T. KOIZUMI
coefficients.
and I. NAKAHARA
Using the relations
r’--ri=2r,b(l-cosb). we can rewrite
equation
(6) as
(a:),
Using equation Hankel
0 =
-
,C;, uL=,
+2r,b
(9)
(9) and the fact of (a, )_ ,, = 0 in 0 5 r < r, and r,, < r. we find from the
inversion
theorem
that
cos
Recalling
(r, < r < r,,).
+ h’-
n@,,(AdrT
2r,b cos
4) d&.
(IO)
the formula
-I T
I
n
II
cosn~#J,,(A~rT+
b’-2r,bcosd)d~+
=J,(Ar,)J,(Ab),
(11)
we obtain B(A)
Substituting
equation
= -i
TTTE,, i aX(Ar,
)J,,(Ab).
” -,I
(12)
(12) into (MJ,)~-~~and using the second condition
of equation
(5),
we get
I = i
,1 0
a,,
‘ J,,(Ar)J,(Ar, I0
dA, (r, d r 2 rd.
)J.(Ab)
(13)
where 1-v a. = 2G In consideration
?ru:,, (n = 0, I, 2,-----).
of the formula
J,,(Ar) = .J,,(Ar, )J,,(Ab) + 2 c J,(Ar, WI I equation
(13) can be rewritten
)J,(Ab)
(14)
(r, 5 r 5 rd,
as follows: z
I
1 = 2 ~“1 J.(Ar.).I”(Ab)(~,,(Ar~)~C,(Ab) ,I 0 0 Since equation
cos m$,
(15) must hold for an arbitrary
+2
c .L(Ar,.).L(Ab)cosm4 m I
dA.
(15)
value of 4, we find that the equation
is
An elastic contact problem
reduced to the following system of simultaneous coefficients a. :
equations for the determination
J,,,(arc )A, (Ab ).L,(Ar<1.L(Ab ) dA = km
763
of the
(16)
(m = 0, I, 2,3,-----), where So,,, is the Kronecker’s delta. Consequently, the present mixed boundary problem is reduced to the solution of the simultaneous equation (16). Using a,, we rewrite B(A) as
value
(17) In particular,
the displacements
and stresses on the surface are:
(18)
where lo =
I
I, =
=J,(Ar).L(Ar, ).I, (Ah) dh. I0
=J,,(Ar)J, (At-
To evaluate the infinite integrals I,, and I,, we make use of the following formulae 171; I-(n + l/2) 1 b ” F f, n ++; n + 1; sin’ Q lyn + l)r(1/2) r, 0r, (
(19)
T. SHIBUYA,
764
T. KOIZUMI
and I. NAKAHARA
and
r < I;),
+lnr, (ri 5 r S ro, n = O), ={ sin nqblnw, (r, S r 5 r,, n = I), Ur, (r > r(,, n = 01,
cm)
,O, (r > ro, n 2 I),
where W(x), T(x) and F(a, p ; y ; x) are in this order the step function, function and the Gauss hypergeometric series. Also, cp and I#Iare:
the gamma
when 0 S r 5 ri, and
when r Zzr,. Moreover,
# is
u, and all stresses in 0 I r < ri on the surface are always zero, while the displacements and stresses in r > r. on the surface decrease in order of r-’ and r--’ respectively. The total load P on the punch is
2arfo, 1;=,) dr
p=-
3. NUMERICAL
27rGElJ =
-
l-v
a,.
CALCULATIONS
To solve the simultaneous equation (16), we calculate the infinite integrals for the product of four Bessel functions by the following method. The integral of m-line and n-column in equation (16), A,, is A,, = A,,,,, = where A’rnn=AL=
I
ho
0
~,,_,(hr,)J,,.. l(Ab)J,-,(hr,)J,-I(Ab)
* J,_,(Ar,)J,-,(Ab)J,-,(XT,-)J,-l(hb) _Iho
dA + A A,
dA,
(22)
(231
165
An elastic contact problem
and ho is taken as a certain large value. The first term of the right hand of equation (22) is integrated numerically by means of the Simpson’s second rule. Here, we choose A,,= 500 and AA = 0.2, where AA is the interval in numerical integrals. The second term is integrated by using the approximation of Bessel function. For a large value of A, the product of Bessel functions is J,-,(Ar,)J,_,(Ab) Substituting
+ db{cos
hri + (- I)‘“-’ sin Arc}.
E
equation (24) into equation (23), we find that
+ (- l),+” sin* Ar,] dh. Integrating
(25)
equation (25) by parts and using the sine- and cosine-integral si(x)=
functions
x sin t * cos t t dt, ci (x) = tdt, Cz I_
I
we obtain the equation =- 1 A’ IFan v2r,b
COS’ A,G ----+ ri si (2Aon). A0
-I-{(- I)‘“-’ + (- I)“-‘){ sin Aor~fos ‘Ori- r, ci (2AorC)- b ci (ZA,b)} + (_ ,),+,
r. si (2Aoro) .
(26)
II
si (x) and ci (x) are computed with good accuracy by using approximate formulae [S]. The approximate solutions of the set of simultaneous equation (16) are caiculated and are shown in Table 1. It is known that the solution is independent of the material properties of the half-space. On the surface, the displacements and stresses are calcuTable 1. Values of a. n
r, = 0.25 r,= I.0
:
0.635 380 5 173 38 5 -0.382 265 -0.061
3
-0.019
636 71
:
-0+03 -0+07
464 870 64 63
76 8 9
736 86 16 -O#XJ -0.001 688 -0.000 720 44 083 78 O~OOO
0
r, = 0.5 r,= 1.0 0.624 670 -0.283 978 -0-028 467 -o@Os 414 - 0,001 278 -O%IO 272 -0.000 213 O-000 214 -oGOt? 530 O+IOO648
r, =0.75 r” = 1.0 2 8 31 23 55 37 39 33 07 25
04.586 628 -0.153 899 -0.007 093 -0*000 593 -0dXtO 136 OGOO 126 -0.000 573 0.000 764 -0dIO2 230 0+002 221
3 1 49 64 75 79 99 29 14 99
766
T. SHIBUYA,
T. KOIZUMI
and I. NAKAHARA
lated by equations (18)-(20), and, in the half-space, they are integrated numerically by equation (4). Here, the Poisson’s ratio is taken as 0.3. In Figs. 2-7 respectively the radial distributions of the displacements and stresses are shown for the case of I;, = 1 and r, = O-5. In the figures, the dashed-curves denote the results of a circular stamp with radius r,,. In particular, we are interested in the results on the surface. (u,), (, is always zero in 0 ~5r 5 ri and is negative in r > I;. The
0
-0.2
c
Fig. 2. The distribution of IA,for ril = 1 and ri = 0.5.
0
0.5 I
I.0
I.5
I
Fig. 3. The distribution of w, for r, = 1 and vi = 0.5.
2.0
An elastic
0
0.5
contact
167
problem
1.C
_A
1; \
--I
Fig. 4. The distribution of a, for r, =
1and r, = 0.5.
slopes of (u,), =0tend to infinity as r + ri + 0 and r + r,, - 0. Since (w,), __”is considerably larger than (u,)~~~, (w,),=, shows approximately the surface deformation. The surface deforms approximately elliptically in 0 5 r 5 ri,and decreases in order of r-’ in r 2 r,. The slopes of the deformation, also, tend to infinity as r + ri- 0 and r --f r0 + 0. The stresses (a,),=O, (u,)~=~ and (a,),=,, are always compressive in the contact region and tend to infinity as r + r,+ 0 and r-~ r.- 0, and are finite or zero in the rest of the surface. In particular, ((T,), =0coincides with the contact pressure of the annular stamp. In Figs. 8 and 9 and Figs. 10 and 11, respectively, the radial distributions of w, and a, in the vicinity of the surface are shown for the cases of r,,= 1 and r,= 0.25 and 0.75. In each case, the results are qualitatively similar to the case of r. = 1 and r,= 0.5.
0
0.5
I.0
I.5
,5
0
1 a L
b”
0
B
-“O
J
-2.0
Fig. 5. The distribution
of me for r,, =
1and r, = 0.5.
T. SHIBUYA,
T. KOIZUMI
Fig. 6. The distribution
Fig. 7. The distribution
and
I. NAKAHARA
of vz for r, =
1and r, = 0.5.
of 7,: for r(, =
1and r, = 0.5.
769
An elastic contact problem
I3
I.5
0.5
2.0 I
--l--TI
I 21
y”
0.: ,-
-.--.-.-.~--
1
:
!-.-
IX )-
Fig. 8. The distribution of w, for r, =
I and r, = 0.25.
Fig. 9. The distribution of o, for r, =
1and r, = 0.25.
770
T. SHIBUYA,
T. KOIZUMI
and
I. NAKAHARA
r 0
I.0
0.5
I.5
2.0
I
Fig. 10. The distribution
of W, for r,, =
1and r, = 0.75.
C
n
2.0 A
\ \ ._______
F=
Fig.
a’P
11.The distribution
I.0
-
k
;F
of V, for r. = 1 and r, = 0.75.
---___
0.5. :: f 4: 0
0.5
I.0
r; I,
Fig. 12. The variations
of (w,),_
z=”and the total load P with the ratio r, IT,,.
An elastic contact
problem
771
Figure 12 shows the variations of (w,),=~+, and the total load P with the ratio n/r+ The former is denoted the solid curve and the Iatter the dashed-curve. (w~)~=~=~decreases linearly in ri /rO< 0.7 and very rapidly in ri fro> O-7. The total load is nearly equal to that for a circular stamp if ri /r,, < O-5. REFERENCES [I] W. E. WILLIAMS, Proc. Edinburgh Math. Sot. 13, 317 (1963). 121 .J. C. COOKE, Q. J. Mech. appl. Math. 16, 193 (1963). [3] [4] (51 [61 I71 [81
B. NOBLE,
Proc. Camb. Phi/. Sot. 59, 351 (1%3).
W. D. COLLINS, Proc. Edinburgh Math. Sot. 13, 235 (1%3). D. L. JAIN and R. P. KANWEL, SIAMJ. appl. M&h. 20, 642 (1971). V. S. GUBENKO and V. I. MOSSAKOVSKII, FMM 24, 334 (l%O). A. ERDELYI (Editor), Tables of Integral Transforms. McGraw-Hill (1954). C. HASTINGS, Approximations for Digitat Computers. Princeton University (Received
19 September
Press
(1955).
1973)
R&um&Nous considerons un probleme de contact Clastique pour un demi plan entail@ par un poincon rigide annulaire plat, ce qui represente un problime mixte de valeurs aux limites h trois parties. Si la dist~bution de la pression sur la rCgion de contact est supposee connue, le probf~me de contact sera ramene a la resolution d’un systeme infini d’equations simultan~es. Des resultats num~riques sont egalement don&s en illustration des dist~butions des deplacements et des contraintes dans le demi espace. Zusa~nfa~n~Wir untersuchen ein elastisches Kontaktproblem ftir einen Halbraum, der durch einen flachen ringfiirmigen starren Stempel eingedrtickt ist, was ein dreiteiliges gemischtes Grenzwertproblem darstellt. Falls die Druekverteilung auf die Kontaktzone angenommen ist, wird das Kontaktproblem auf die Losung eines unendlichen Satzes simultaner Gteichungen reduziert. Es werden such numerische Resultate fur die Verteilungen der Verdrangungen und Spannungen im Halbraum illustriert. Sommario-Si considera it problema de! contatto elastic0 per un semispazio depress0 da uno stamp0 rigid0 anulare piatto, the i: un problema a tre parti con valori limite misti. Assumendo la distribuzione uniforme delle pressioni sulla regione di contatto, il problema viene ridotto alla soluzione di un gruppo infinite di equazioni simultanee. Vengono inoltre illustrati i risultati numerici per la distribuzione degli spostamenti e delle sollecitazioni
nel semispazio.
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A6CTpaKT