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Some mixed boundary value problems for the semi-infinite elastic solid subjected to tangential surface tractions
Int. J. me~h. 8c/. Pergamon Press. 1974. Vol. 16, pp. 727-734. Printed in Great Britain
SOME M I X E D B O U N D A R Y V A L U E PROBLEMS FOR THE S E M I - I N F I N I T E ELASTIC SOLID S U B J E C T E D TO T A N G E N T I A L S U R F A C E TRACTIONS J. R. BARBER University Department of Mechanical Engineering, Newcastle upon Tyne NE 1 7RU, England
(Received 24 July 1973, and in revised form 18 March 1974) Summ~ry--A method is developed for representing the tangential displacements and tractions at the surface of the semi-infinite solid in terms of potential functions. In this form, a mathematical analogy is revealed between corresponding mixed boundary value problems involving tangential and normal surface displacements respectively. This analogy enables a general solution to be obtained to the problem in which the surface tangential displacements are specified axisymmetrie functions inside the circle a i> r>~ 0 and the tangential surface traction is zero outside this circle. The method can also be used for certain non-axisymmetric problems, but it fails if the indentation analogue has a stress singularity at the boundary of the stressed are~.
NOTATION X, y , z a
E e
P .P~.9 .~0~, e t c . Ur9 "t.g,z~U 0
V~
~,~
cylindrical polar co-ordinates rectangular Cartesian co-ordinates radius of a circular area on the ~urface of the semi-infinite solid Young's modulus surface dilatation concentrated force acting at the surface stress components in double suffix notation components of surface displacement - (~21~x~) + (021~y~)
Poisson's ratio stress functions displacement functions
= ~+iv
surface rotation
I. I N T R O D U C T I O N THE PROBLEMS t o be considered in this p a p e r are t h e t a n g e n t i a l analogues of the a x i s y m m e t r i c i n d e n t a t i o n p r o b l e m associated with the n a m e o f Boussinesq a n d o f certain related n o n - a x i s y m m e t r i c problems. W e shall find the distribution o f t a n g e n t i a l t r a c t i o n within the circle a t> r I> 0 on the surface of the semi-infinite elastic solid which is necessary to p r o d u c e prescribed t a n g e n t i a l surface displacements within this circle, t h e rest of the surface being stress free. T h e semi-infinite solid is considered t o o c c u p y the space z > 0 in t h e cylindrical polar co-ordinate s y s t e m r, 0, z. 727
728
J . R . BARBER W i t h this notation, a formal s t a t e m e n t of the a b o v e b o u n d a r y conditions is : n o r m a l surface t r a c t i o n : Pzz = 0,
z -- 0,
all r, 0,
t a n g e n t i a l surface t r a c t i o n s : Prz = P0z -- 0, t a n g e n t i a l surface displacements: ur, u 0 prescribed functions of r, 8,
z = 0,
z -- 0,
r > a,
a >/r >/0.
Various p a r t i c u l a r solutions of this problem are known, n o t a b l y those involving s y m m e t r i c torsion o f the semi-infinite solid a b o u t the z axis. Weinstein 1 derived the solution for a solid containing a p e n n y - s h a p e d crack in torsion a b o u t its axis o f s y m m e t r y a n d P a y n e 2 showed t h a t this a n d other solutions can be o b t a i n e d from existing results b y v i r t u e o f a m a t h e m a t i c a l a n a l o g y between a x i s y m m e t r i c torsion a n d a x i s y m m e t r i c indentation. I n this paper, we shall d e m o n s t r a t e the existence of another, more general, a n a l o g y between these classes of problems, which is n o t restricted either to torsional or to a x i s y m m e t r i c systems.
2. R E P R E S E N T A T I O N OF T A N G E N T I A L S U R F A C E D I S P L A C E M E N T AND T R A C T I O N BY P O T E N T I A L F U N C T I O N S The tangential displacement at the surface of the semi-infinite solid is a vector with two components (e.g. ur, ue). It is convenient to define this vector in terms of the gradient of a potential function in order to facilitate the transformation of co-ordinate systems. Following the methods of two-dimensional elasticity, we find that the most general definition of this type requires two independent scalar functions ~:,~7, in terms of which u~=~.
~y
u,=~y.~.
(1)
I t is easily verified that ff n, t are two orthogonal directions, inclined to x, y respectively at some angle 0, we have
u.=0n
; ut=~*~
(2)
(see, for example, Dugdale, s Section 1:6). A more compact statement of equation (2) is a~ • . ~
u. + iu t = -~ * ~'~,
(3)
where = ~+iv.
(4)
The tangential surface traction (components p~, P0,) also lends itseff to this form of representation. We shall use the two stress functions 4, ~b, where P"" = H - g /
p,, = ~ - . ~ .
(5)
We define the surface dilatation, e, by the equation
0u.
e = ~+W' = V" ~
from equation (1).
0u,
(8) (7)
Tangential displacements a n d tractions on semi-infinite solid
729
Similarly, we define the surface rotation, o~, b y =
Ou,
Ou~
(8)
= V2~.
(9)
Equations (7) a n d (9) provide a convenient means of finding ~, ~/, for a given displacem e n t field.
3. T H E
CONCENTRATED
TANGENTIAL
FORCE
I f a concentrated tangential force, P , acts at the origin on the b o u n d a r y of the semiinfinite solid z > 0, in the direction of the x axis, the surface displacements will be P ( 1 + v) I
vx2\ [l-,,+-;r),
--
<1ol
P(1 +v) vxy uv =
lrEr s
,
P(I+v) (1-2v)x us
(11) (12)
2~rErS
=
(see Love d, Art. 166), where E, v, are Young's modulus and Poisson's ratio respectively, for the material. F r o m equations (7), (9), (10) and (11) we have
V2~=
P(1-v2) x IrE r a
( 13 )
and V2~/ = P ( l + v ) y ~rEra
(14)
giving P
x
(15)
4. D I S P L A C E M E N T POTENTIAL OF TANGENTIAL
DUE TO A DISTRIBUTION TRACTION
We suppose that the semi&n_finite solid is subjected to tangential tractions defined b y equation (5), where ~b,~bare single-valued functions. This condition will ensure that there are no concentrated forces acting a t the surface. We further suppose t h a t there is some finite closed region outside which the surface is stress free. I n this unstressed region we shall take ~b,~b to be zero. These restrictions require t h a t ~b,~b should approach zero at the b o u n d a r y of the stressed region. Consider the displacement potential at an arbitrary point (A) due to the traction on a n element of surface at another point (B). We choose A as the origin of a polar coordinate system, relative to which B has co-ordinates (r, 0). The area of the surface element is (r d0 dr). F r o m equation (15) we have
d~ = (--~--~ (I--v 2)
ipa,(1 +u)~ ] rdOdr
(16)
and hence
ds~
1--v ~Ir a~'~ dSdr; -.----~~ 04 -.~--.~! 1 +v I
d~ = --~
a~.
0~
~r ~ + D } dO dr.
(17) (18)
730
J . R . BARBER
We now integrate over the st~rface, noting that -~d0 =
~-~d0 = 0,
(19)
since 4, ~b have been assumed to be single valued. Thus, 1 -
v ~ /'2~
= -
?o~ 04
Jo 3o r
dr dO,
(20)
The conditions which we have imposed on the function ~ ensure that the integrand of the first term is zero for all 0, provided t h a t ~ is bounded at A. Hence, ~a = -- -- ~ ( 1 - r ' ) ~ d r d O (22) J0 ao ~rE By a similar argument, we also have "qa = 5. A N A L O G Y
WITH
THE
('~ (l + v) ~bdr dO J0 j0 ~rE NORMAL
(23)
INDENTATION
PROBLEM
Equations (22) a n d (23) reveal a n important analogy between tangential and normal loading of the semi-infinite solid. The normal surface displacement (uz) of the semiinfinite solid at the origin due to a distribution of normal pressure p (r, 0) is us = f2"( °°(1-r2)pdOdr JO JO ~E
(24)
(see Timoskenko and Goodier, s Art. 139). which has the same form as equations (22) and (23). Note that in all these results the choice of origin of co-ordinates is arbitrary a n d hence equations (22)-(24) apply throughout the surface plane, provided only that 4, ~b,p are suitably defined relative to the appropriate origin. Various methods a-8 have been developed for finding the pressure distribution p, necessary to produce a proscribed displacement us, over a closed (usually circular) region a n d we can use these methods directly to find corresponding solutions to the problem defined in Section 1. As a n illustrative example, it is well known that the pressure distribution, P=c~j(a2-r2), -~ 09
a>~r>~O,
)
(25)
J
r>a,
produces a paraboloidal depression defined by us = vrc(1--v~)4E(2a2-r~) ,
a>~r>~O
(26)
(Timoshenko a n d Goodier6, Art. 140). By virtue of the analogy between equations (22) and (24), it follows that a stress potential 9~ = c~/(a~--r~), a~r>>-O, t (27) = 0, r > a , corresponding to a tangential stress distribution cr
Prz ---- ~/(a~_ru ) = 0
r>a,
P0~ = 0
allr,
a>>-r>~O, (28)
T a n g e n t i a l displacements a n d t r a c t i o n s on semi-infinite solid
731
f r o m e q u a t i o n s (2), (27), will produce a d i s p l a c e m e n t p o t e n t i a l
= rrc(1--v 2) (2a2-- r s)
4E
, a >~r >~O,
(29)
and hence displacements
Irc(1-v~)r u, =
a~r>~O }
2E
u0 = 0
(30)
all r.
T h i s represents a u n i f o r m a x i s y m m e t r i c d i l a t a t i o n of t h e surface w i t h i n t h e circle a >t r I> 0. The d i l a t a t i o n ~c(1 - v ~) 2E '
e =
(31)
f r o m e q u a t i o n s (7) a n d (29). T h e s a m e solution used in c o n j u n c t i o n w i t h e q u a t i o n (23) shows t h a t a stress distribution cr
poz=-- /(a~_r2), a~r~O, = O,
(32)
r>a;
p,~ = O,
allr,
produces a displacement u r = 0, uo
=
~rc(l+v) r 2E
,
a~r~>0.
)
(33)
This solution represents a c o n s t a n t r o t a t i o n of t h e circle a/> r I> 0 a b o u t t h e origin and corresponds to t h e case in which a rigid circular cylinder is j o i n e d to t h e semi-infinite solid o v e r t h e circle a n d is t h e n t w i s t e d a b o u t its axis. T h e r o t a t i o n ¢o ----
6. T H E
GENERAL
Ire(1 +v) 2E
AXISYMMETRIC
(34)
SOLUTION
T h e general solution of t h e a x i s y m m e t r i c t a n g e n t i a l displacement p r o b l e m p r e s e n t s no n e w difficulties. W e first express t h e r e q u i r e d d i s p l a c e m e n t w i t h i n t h e circle a/> r >/0 in t e r m s o f t h e d i s p l a c e m e n t p o t e n t i a l s ~, 7, using e q u a t i o n s {7) a n d (9). W e t h e n solve s e p a r a t e l y for t h e d i s t r i b u t i o n of t h e stress functions ¢, ~b, respectively, necessary to produce these potentials, m a k i n g use of t h e analogies b e t w e e n e q u a t i o n s (22) a n d (24) and one of t h e k n o w n m e t h o d s for solving e q u a t i o n (24). Since we require t h e c o n t a c t pressure p o n l y in e q u a t i o n (24), t h e m e t h o d s of Green e a n d Segedin 7 p r o v e to be t h e m o s t s t r a i g h t f o r w a r d . F i n a l l y , we f i n d t h e corresponding t a n g e n t i a l stresses in t h e region a I> r I> 0 b y s u b s t i t u t i n g for ~b, ~b, in e q u a t i o n (2). The solution of t h e corresponding i n d e n t a t i o n p r o b l e m has been v e r y e x t e n s i v e l y discussed in t h e cited references a n d elsewhere a n d will n o t be considered here. H o w e v e r , we n o t e t h a t t h e c o m p l e t e solution o u t l i n e d a b o v e c o m p a r e s f a v o u r a b l y in m a t h e m a t i c a l s i m p l i c i t y w i t h o t h e r m e t h o d s (for e x a m p l e , t h e use of W e s t m a n n ' s ~ solution for t h e elastic half-space in shear}. 7. R E S T R I C T I O N S AND
ON THE STRESS SINGULARITIES
POTENTIAL
I n Section 4, we laid d o w n c e r t a i n conditions on t h e v a l u e s of ¢, ~b, s o m e of which can n o w be relaxed.
732
J.R.
BARBER
(a) S t e p changes i n stress potential Suppose there exists a step change ~b0 in the stress function ~ across some line S. We could regard such a step as the limiting case of a ramp of slope ~0/8 and width 8. From equation (2), the stress normal to S will be
P.= = ¢0/8
(35)
a n d hence the ramp corresponds to a force ~0 per u n i t length of line, which remains constant as we approach a step by allowing ~ to tend to zero. The force acts normal to the line of the step in the direction of increasing ~. Similarly, a step change, ~b0, in the stress function ~b corresponds to a force ~b0 per unit length, tangential to the line of the step. (b) " P o i n t f o r c e " singularities Analogues of the point force normal to the surface can be found by supposing that ~, ~b are zero everywhere on the surface except over a small circle of radius 8 with its centre a t the origin. I f ~ = C0 over this circle, there will be a radial force C0 per u n i t length acting around the circumference towards the origin. I f we allow ~ to approach zero whilst (Ir8~0) remains constant and equal to ~0, we generate a singularity which we can describe as a centre of dilatation b y analogy with the corresponding singularity in two-dimensional elasticity. The displacement potential due to this singularity is ~)°(1 -v~) =
(36)
~Er
The corresponding singularity in ~b is the "centre of rotation", ~b0, which represents a concentrated m o m e n t acting about a n axis normal to the surface and which produces the displacement potential ~Fo(1 +v) =
(37)
wet
The formal analogy between these singularities and the normal displacement P( 1 - v~) uz =
lrEr
(38)
'
due to a normal point force P, provides an alternative method of deriving the results of Section 4.
(c) B e h a v i o u r of ~, ~b at large values of r I n Section 4, we assumed t h a t there is some finite closed region outside which ¢, ~b are identically zero, b u t it is clear from equation (21) that the argument would still be valid provided t h a t r~, r~b -~ 0 as r -~ ~ . I f this condition is n o t satisfied, it will usually be possible to meet it b y "subtracting out" a uniform state of stress, as in the solution of crack problems.
8. N O N - A X I S Y M M E T R I C
PROBLEMS
Although every normal identation solution has a tangential displacement analogue and vice versa, not all these solutions represent physically possible situations. An axisymmetric example is provided by the fiat-ended punch problem, the tangential analogue of which has a n unacceptable singularity, involving an infinite concentrated force, at the outer radius r = a. This is not a serious difficulty in the axisymmetric case since all physically possible axisymmetric tangential displacement problems prove to have wellbehaved indentation analogues. However, when we t r y to extend the method to non-axisymmetric systems, using the results of Green, 6 we find t h a t this is no longer the case. The simplest example is provided
Tangential displacements and tractions on semi-lnfinite solid
733
by the tilted punch "solution". A normal pressure distribution
2cEx P =~'(1--v ~)~/(az-r~) ' = O,
} a>~r>~O,
(39)
r>a,
is known to produce a normal displacement
u, = cx,
a~r~O.
(40)
I f we write ~ for u~ and ~ for p, following the analogy between equations (22) and (24) and substitute in equation (1) we obtain
uz = c,
u v = O, a>~r>~O,
(41)
which corresponds to the "rigid body" displacement of the circle a>~r>>-O in the x direction, relative to the extremities of the solid. However, the corresponding stress distribution from equations (5) and (39) has a physically unacceptable singularity at r = a. The actual stress distribution needed to produce the displacements defined by equation (41) can be shown by integration of the point tangential force solution (equations (10) and (11)) to be
2cE Pxz ----~(1 +v) ( 2 - v ) ~](a~--r~) '
a>~r>~O,
= O, r > a , Pvz---- 0, a l l r
(42)
which corresponds to the stress potentials
2c E ~](a 2 -- r~)x ¢ = -- ~r(1 + v ) ( 2 - - v ) r ~' a>~r>~O, 1 2cE ~/(a~ -- r2)y a >~r >~O, ~r(1 +v) (2--v) r ~' ¢=tb=O, r>a. ~=+
(43)
These potentials in turn have no physically possible indentation analogue because of the singularity at r -- 0, which is self-cancelling in the tangential displacement problem. The results of equation (42) have been derived using a different method by Mindlin. le
9. C O N C L U S I O N T h e a n a l o g y b e t w e e n n o r m a l a n d t a n g e n t i a l d i s p l a c e m e n t o f t h e surface of t h e semi-infinite solid, d e v e l o p e d in Section 4, enables us to give a general solution to t h e m i x e d b o u n d a r y v a l u e p r o b l e m in which t h e t a n g e n t i a l surface d i s p l a c e m e n t is specified as a n a x i s y m m e t r i c f u n c t i o n inside t h e circle a >/r 1>0, a n d t h e t a n g e n t i a l surface t r a c t i o n is zero outside this circle. T h e m e t h o d finds its p r i m a r y a p p l i c a t i o n in p r o b l e m s for which o n l y surface values o f stress a n d d i s p l a c e m e n t are required, since these can b e f o u n d d i r e c t l y w i t h o u t a n a l y s i n g t h e s t a t e o f stress w i t h i n t h e solid. A similar a p p r o a c h can be used for corresponding n o n - a x i s y m m e t r i c p r o b l e m s , b u t it fails ff t h e i n d e n t a t i o n a n a l o g u e h a s a stress s i n g u l a r i t y a t t h e b o u n d a r y o f t h e c o n t a c t area.
734
J . R . BARBER
10. R E F E R E N C E S 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
A. WEI~STEIN, Quart. appl. Math. IO, 77 (1952). L. E. PAYEE, S I A M J. 1, 53 (1953). D. S. DUGDALE,Elements of Elasticity. Pergamon Press, Oxford (1968). A. E. H. LOVE, A Treatise on the Mathematical Theory of Elasticity, 4th edn. University Press, Cambridge (1927). S. P. TIMOSHE:NKOand J. N. GOODIER, Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970). A. E. GREEN, Proc. Cambridge Phil. Soc. 45, 251 (1949). C. M. SEG~.DIN, Mathematika 4, 156 (1957). J. W. HARDING and I. N. SNEDDON, Proc. Cambridge Phil. Soc. 41, 16 (1945). R. A. WEST~XN~, J. appl. Mech. 82, 411 (1965). R. D. MINDLIN, J. appl. Mech. 16, 259 (1949).
SOME M I X E D B O U N D A R Y V A L U E PROBLEMS FOR THE S E M I - I N F I N I T E ELASTIC SOLID S U B J E C T E D TO T A N G E N T I A L S U R F A C E TRACTIONS J. R. BARBER University Department of Mechanical Engineering, Newcastle upon Tyne NE 1 7RU, England
(Received 24 July 1973, and in revised form 18 March 1974) Summ~ry--A method is developed for representing the tangential displacements and tractions at the surface of the semi-infinite solid in terms of potential functions. In this form, a mathematical analogy is revealed between corresponding mixed boundary value problems involving tangential and normal surface displacements respectively. This analogy enables a general solution to be obtained to the problem in which the surface tangential displacements are specified axisymmetrie functions inside the circle a i> r>~ 0 and the tangential surface traction is zero outside this circle. The method can also be used for certain non-axisymmetric problems, but it fails if the indentation analogue has a stress singularity at the boundary of the stressed are~.
NOTATION X, y , z a
E e
P .P~.9 .~0~, e t c . Ur9 "t.g,z~U 0
V~
~,~
cylindrical polar co-ordinates rectangular Cartesian co-ordinates radius of a circular area on the ~urface of the semi-infinite solid Young's modulus surface dilatation concentrated force acting at the surface stress components in double suffix notation components of surface displacement - (~21~x~) + (021~y~)
Poisson's ratio stress functions displacement functions
= ~+iv
surface rotation
I. I N T R O D U C T I O N THE PROBLEMS t o be considered in this p a p e r are t h e t a n g e n t i a l analogues of the a x i s y m m e t r i c i n d e n t a t i o n p r o b l e m associated with the n a m e o f Boussinesq a n d o f certain related n o n - a x i s y m m e t r i c problems. W e shall find the distribution o f t a n g e n t i a l t r a c t i o n within the circle a t> r I> 0 on the surface of the semi-infinite elastic solid which is necessary to p r o d u c e prescribed t a n g e n t i a l surface displacements within this circle, t h e rest of the surface being stress free. T h e semi-infinite solid is considered t o o c c u p y the space z > 0 in t h e cylindrical polar co-ordinate s y s t e m r, 0, z. 727
728
J . R . BARBER W i t h this notation, a formal s t a t e m e n t of the a b o v e b o u n d a r y conditions is : n o r m a l surface t r a c t i o n : Pzz = 0,
z -- 0,
all r, 0,
t a n g e n t i a l surface t r a c t i o n s : Prz = P0z -- 0, t a n g e n t i a l surface displacements: ur, u 0 prescribed functions of r, 8,
z = 0,
z -- 0,
r > a,
a >/r >/0.
Various p a r t i c u l a r solutions of this problem are known, n o t a b l y those involving s y m m e t r i c torsion o f the semi-infinite solid a b o u t the z axis. Weinstein 1 derived the solution for a solid containing a p e n n y - s h a p e d crack in torsion a b o u t its axis o f s y m m e t r y a n d P a y n e 2 showed t h a t this a n d other solutions can be o b t a i n e d from existing results b y v i r t u e o f a m a t h e m a t i c a l a n a l o g y between a x i s y m m e t r i c torsion a n d a x i s y m m e t r i c indentation. I n this paper, we shall d e m o n s t r a t e the existence of another, more general, a n a l o g y between these classes of problems, which is n o t restricted either to torsional or to a x i s y m m e t r i c systems.
2. R E P R E S E N T A T I O N OF T A N G E N T I A L S U R F A C E D I S P L A C E M E N T AND T R A C T I O N BY P O T E N T I A L F U N C T I O N S The tangential displacement at the surface of the semi-infinite solid is a vector with two components (e.g. ur, ue). It is convenient to define this vector in terms of the gradient of a potential function in order to facilitate the transformation of co-ordinate systems. Following the methods of two-dimensional elasticity, we find that the most general definition of this type requires two independent scalar functions ~:,~7, in terms of which u~=~.
~y
u,=~y.~.
(1)
I t is easily verified that ff n, t are two orthogonal directions, inclined to x, y respectively at some angle 0, we have
u.=0n
; ut=~*~
(2)
(see, for example, Dugdale, s Section 1:6). A more compact statement of equation (2) is a~ • . ~
u. + iu t = -~ * ~'~,
(3)
where = ~+iv.
(4)
The tangential surface traction (components p~, P0,) also lends itseff to this form of representation. We shall use the two stress functions 4, ~b, where P"" = H - g /
p,, = ~ - . ~ .
(5)
We define the surface dilatation, e, by the equation
0u.
e = ~+W' = V" ~
from equation (1).
0u,
(8) (7)
Tangential displacements a n d tractions on semi-infinite solid
729
Similarly, we define the surface rotation, o~, b y =
Ou,
Ou~
(8)
= V2~.
(9)
Equations (7) a n d (9) provide a convenient means of finding ~, ~/, for a given displacem e n t field.
3. T H E
CONCENTRATED
TANGENTIAL
FORCE
I f a concentrated tangential force, P , acts at the origin on the b o u n d a r y of the semiinfinite solid z > 0, in the direction of the x axis, the surface displacements will be P ( 1 + v) I
vx2\ [l-,,+-;r),
--
<1ol
P(1 +v) vxy uv =
lrEr s
,
P(I+v) (1-2v)x us
(11) (12)
2~rErS
=
(see Love d, Art. 166), where E, v, are Young's modulus and Poisson's ratio respectively, for the material. F r o m equations (7), (9), (10) and (11) we have
V2~=
P(1-v2) x IrE r a
( 13 )
and V2~/ = P ( l + v ) y ~rEra
(14)
giving P
x
(15)
4. D I S P L A C E M E N T POTENTIAL OF TANGENTIAL
DUE TO A DISTRIBUTION TRACTION
We suppose that the semi&n_finite solid is subjected to tangential tractions defined b y equation (5), where ~b,~bare single-valued functions. This condition will ensure that there are no concentrated forces acting a t the surface. We further suppose t h a t there is some finite closed region outside which the surface is stress free. I n this unstressed region we shall take ~b,~b to be zero. These restrictions require t h a t ~b,~b should approach zero at the b o u n d a r y of the stressed region. Consider the displacement potential at an arbitrary point (A) due to the traction on a n element of surface at another point (B). We choose A as the origin of a polar coordinate system, relative to which B has co-ordinates (r, 0). The area of the surface element is (r d0 dr). F r o m equation (15) we have
d~ = (--~--~ (I--v 2)
ipa,(1 +u)~ ] rdOdr
(16)
and hence
ds~
1--v ~Ir a~'~ dSdr; -.----~~ 04 -.~--.~! 1 +v I
d~ = --~
a~.
0~
~r ~ + D } dO dr.
(17) (18)
730
J . R . BARBER
We now integrate over the st~rface, noting that -~d0 =
~-~d0 = 0,
(19)
since 4, ~b have been assumed to be single valued. Thus, 1 -
v ~ /'2~
= -
?o~ 04
Jo 3o r
dr dO,
(20)
The conditions which we have imposed on the function ~ ensure that the integrand of the first term is zero for all 0, provided t h a t ~ is bounded at A. Hence, ~a = -- -- ~ ( 1 - r ' ) ~ d r d O (22) J0 ao ~rE By a similar argument, we also have "qa = 5. A N A L O G Y
WITH
THE
('~ (l + v) ~bdr dO J0 j0 ~rE NORMAL
(23)
INDENTATION
PROBLEM
Equations (22) a n d (23) reveal a n important analogy between tangential and normal loading of the semi-infinite solid. The normal surface displacement (uz) of the semiinfinite solid at the origin due to a distribution of normal pressure p (r, 0) is us = f2"( °°(1-r2)pdOdr JO JO ~E
(24)
(see Timoskenko and Goodier, s Art. 139). which has the same form as equations (22) and (23). Note that in all these results the choice of origin of co-ordinates is arbitrary a n d hence equations (22)-(24) apply throughout the surface plane, provided only that 4, ~b,p are suitably defined relative to the appropriate origin. Various methods a-8 have been developed for finding the pressure distribution p, necessary to produce a proscribed displacement us, over a closed (usually circular) region a n d we can use these methods directly to find corresponding solutions to the problem defined in Section 1. As a n illustrative example, it is well known that the pressure distribution, P=c~j(a2-r2), -~ 09
a>~r>~O,
)
(25)
J
r>a,
produces a paraboloidal depression defined by us = vrc(1--v~)4E(2a2-r~) ,
a>~r>~O
(26)
(Timoshenko a n d Goodier6, Art. 140). By virtue of the analogy between equations (22) and (24), it follows that a stress potential 9~ = c~/(a~--r~), a~r>>-O, t (27) = 0, r > a , corresponding to a tangential stress distribution cr
Prz ---- ~/(a~_ru ) = 0
r>a,
P0~ = 0
allr,
a>>-r>~O, (28)
T a n g e n t i a l displacements a n d t r a c t i o n s on semi-infinite solid
731
f r o m e q u a t i o n s (2), (27), will produce a d i s p l a c e m e n t p o t e n t i a l
= rrc(1--v 2) (2a2-- r s)
4E
, a >~r >~O,
(29)
and hence displacements
Irc(1-v~)r u, =
a~r>~O }
2E
u0 = 0
(30)
all r.
T h i s represents a u n i f o r m a x i s y m m e t r i c d i l a t a t i o n of t h e surface w i t h i n t h e circle a >t r I> 0. The d i l a t a t i o n ~c(1 - v ~) 2E '
e =
(31)
f r o m e q u a t i o n s (7) a n d (29). T h e s a m e solution used in c o n j u n c t i o n w i t h e q u a t i o n (23) shows t h a t a stress distribution cr
poz=-- /(a~_r2), a~r~O, = O,
(32)
r>a;
p,~ = O,
allr,
produces a displacement u r = 0, uo
=
~rc(l+v) r 2E
,
a~r~>0.
)
(33)
This solution represents a c o n s t a n t r o t a t i o n of t h e circle a/> r I> 0 a b o u t t h e origin and corresponds to t h e case in which a rigid circular cylinder is j o i n e d to t h e semi-infinite solid o v e r t h e circle a n d is t h e n t w i s t e d a b o u t its axis. T h e r o t a t i o n ¢o ----
6. T H E
GENERAL
Ire(1 +v) 2E
AXISYMMETRIC
(34)
SOLUTION
T h e general solution of t h e a x i s y m m e t r i c t a n g e n t i a l displacement p r o b l e m p r e s e n t s no n e w difficulties. W e first express t h e r e q u i r e d d i s p l a c e m e n t w i t h i n t h e circle a/> r >/0 in t e r m s o f t h e d i s p l a c e m e n t p o t e n t i a l s ~, 7, using e q u a t i o n s {7) a n d (9). W e t h e n solve s e p a r a t e l y for t h e d i s t r i b u t i o n of t h e stress functions ¢, ~b, respectively, necessary to produce these potentials, m a k i n g use of t h e analogies b e t w e e n e q u a t i o n s (22) a n d (24) and one of t h e k n o w n m e t h o d s for solving e q u a t i o n (24). Since we require t h e c o n t a c t pressure p o n l y in e q u a t i o n (24), t h e m e t h o d s of Green e a n d Segedin 7 p r o v e to be t h e m o s t s t r a i g h t f o r w a r d . F i n a l l y , we f i n d t h e corresponding t a n g e n t i a l stresses in t h e region a I> r I> 0 b y s u b s t i t u t i n g for ~b, ~b, in e q u a t i o n (2). The solution of t h e corresponding i n d e n t a t i o n p r o b l e m has been v e r y e x t e n s i v e l y discussed in t h e cited references a n d elsewhere a n d will n o t be considered here. H o w e v e r , we n o t e t h a t t h e c o m p l e t e solution o u t l i n e d a b o v e c o m p a r e s f a v o u r a b l y in m a t h e m a t i c a l s i m p l i c i t y w i t h o t h e r m e t h o d s (for e x a m p l e , t h e use of W e s t m a n n ' s ~ solution for t h e elastic half-space in shear}. 7. R E S T R I C T I O N S AND
ON THE STRESS SINGULARITIES
POTENTIAL
I n Section 4, we laid d o w n c e r t a i n conditions on t h e v a l u e s of ¢, ~b, s o m e of which can n o w be relaxed.
732
J.R.
BARBER
(a) S t e p changes i n stress potential Suppose there exists a step change ~b0 in the stress function ~ across some line S. We could regard such a step as the limiting case of a ramp of slope ~0/8 and width 8. From equation (2), the stress normal to S will be
P.= = ¢0/8
(35)
a n d hence the ramp corresponds to a force ~0 per u n i t length of line, which remains constant as we approach a step by allowing ~ to tend to zero. The force acts normal to the line of the step in the direction of increasing ~. Similarly, a step change, ~b0, in the stress function ~b corresponds to a force ~b0 per unit length, tangential to the line of the step. (b) " P o i n t f o r c e " singularities Analogues of the point force normal to the surface can be found by supposing that ~, ~b are zero everywhere on the surface except over a small circle of radius 8 with its centre a t the origin. I f ~ = C0 over this circle, there will be a radial force C0 per u n i t length acting around the circumference towards the origin. I f we allow ~ to approach zero whilst (Ir8~0) remains constant and equal to ~0, we generate a singularity which we can describe as a centre of dilatation b y analogy with the corresponding singularity in two-dimensional elasticity. The displacement potential due to this singularity is ~)°(1 -v~) =
(36)
~Er
The corresponding singularity in ~b is the "centre of rotation", ~b0, which represents a concentrated m o m e n t acting about a n axis normal to the surface and which produces the displacement potential ~Fo(1 +v) =
(37)
wet
The formal analogy between these singularities and the normal displacement P( 1 - v~) uz =
lrEr
(38)
'
due to a normal point force P, provides an alternative method of deriving the results of Section 4.
(c) B e h a v i o u r of ~, ~b at large values of r I n Section 4, we assumed t h a t there is some finite closed region outside which ¢, ~b are identically zero, b u t it is clear from equation (21) that the argument would still be valid provided t h a t r~, r~b -~ 0 as r -~ ~ . I f this condition is n o t satisfied, it will usually be possible to meet it b y "subtracting out" a uniform state of stress, as in the solution of crack problems.
8. N O N - A X I S Y M M E T R I C
PROBLEMS
Although every normal identation solution has a tangential displacement analogue and vice versa, not all these solutions represent physically possible situations. An axisymmetric example is provided by the fiat-ended punch problem, the tangential analogue of which has a n unacceptable singularity, involving an infinite concentrated force, at the outer radius r = a. This is not a serious difficulty in the axisymmetric case since all physically possible axisymmetric tangential displacement problems prove to have wellbehaved indentation analogues. However, when we t r y to extend the method to non-axisymmetric systems, using the results of Green, 6 we find t h a t this is no longer the case. The simplest example is provided
Tangential displacements and tractions on semi-lnfinite solid
733
by the tilted punch "solution". A normal pressure distribution
2cEx P =~'(1--v ~)~/(az-r~) ' = O,
} a>~r>~O,
(39)
r>a,
is known to produce a normal displacement
u, = cx,
a~r~O.
(40)
I f we write ~ for u~ and ~ for p, following the analogy between equations (22) and (24) and substitute in equation (1) we obtain
uz = c,
u v = O, a>~r>~O,
(41)
which corresponds to the "rigid body" displacement of the circle a>~r>>-O in the x direction, relative to the extremities of the solid. However, the corresponding stress distribution from equations (5) and (39) has a physically unacceptable singularity at r = a. The actual stress distribution needed to produce the displacements defined by equation (41) can be shown by integration of the point tangential force solution (equations (10) and (11)) to be
2cE Pxz ----~(1 +v) ( 2 - v ) ~](a~--r~) '
a>~r>~O,
= O, r > a , Pvz---- 0, a l l r
(42)
which corresponds to the stress potentials
2c E ~](a 2 -- r~)x ¢ = -- ~r(1 + v ) ( 2 - - v ) r ~' a>~r>~O, 1 2cE ~/(a~ -- r2)y a >~r >~O, ~r(1 +v) (2--v) r ~' ¢=tb=O, r>a. ~=+
(43)
These potentials in turn have no physically possible indentation analogue because of the singularity at r -- 0, which is self-cancelling in the tangential displacement problem. The results of equation (42) have been derived using a different method by Mindlin. le
9. C O N C L U S I O N T h e a n a l o g y b e t w e e n n o r m a l a n d t a n g e n t i a l d i s p l a c e m e n t o f t h e surface of t h e semi-infinite solid, d e v e l o p e d in Section 4, enables us to give a general solution to t h e m i x e d b o u n d a r y v a l u e p r o b l e m in which t h e t a n g e n t i a l surface d i s p l a c e m e n t is specified as a n a x i s y m m e t r i c f u n c t i o n inside t h e circle a >/r 1>0, a n d t h e t a n g e n t i a l surface t r a c t i o n is zero outside this circle. T h e m e t h o d finds its p r i m a r y a p p l i c a t i o n in p r o b l e m s for which o n l y surface values o f stress a n d d i s p l a c e m e n t are required, since these can b e f o u n d d i r e c t l y w i t h o u t a n a l y s i n g t h e s t a t e o f stress w i t h i n t h e solid. A similar a p p r o a c h can be used for corresponding n o n - a x i s y m m e t r i c p r o b l e m s , b u t it fails ff t h e i n d e n t a t i o n a n a l o g u e h a s a stress s i n g u l a r i t y a t t h e b o u n d a r y o f t h e c o n t a c t area.
734
J . R . BARBER
10. R E F E R E N C E S 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
A. WEI~STEIN, Quart. appl. Math. IO, 77 (1952). L. E. PAYEE, S I A M J. 1, 53 (1953). D. S. DUGDALE,Elements of Elasticity. Pergamon Press, Oxford (1968). A. E. H. LOVE, A Treatise on the Mathematical Theory of Elasticity, 4th edn. University Press, Cambridge (1927). S. P. TIMOSHE:NKOand J. N. GOODIER, Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970). A. E. GREEN, Proc. Cambridge Phil. Soc. 45, 251 (1949). C. M. SEG~.DIN, Mathematika 4, 156 (1957). J. W. HARDING and I. N. SNEDDON, Proc. Cambridge Phil. Soc. 41, 16 (1945). R. A. WEST~XN~, J. appl. Mech. 82, 411 (1965). R. D. MINDLIN, J. appl. Mech. 16, 259 (1949).