[ 492 ]

BOUSSLNESQ'S PROBLEM FOR A RIGID CONEB Y IAN N. SNEDDON

Received 17 April 1947

1. The problem of determining the distribution of stress in a semi-infinite solidmedium when its plane boundary is deformed by the pressure against it of a perfectlyrigid cone is of considerable importance in various branches of applied mechanics. I tarises in soil mechanics where the cone is the base of a conical-headed cylindrical pillarand the semi-infinite medium is the soil upon which it rests (l). In this instance thedistribution of stress in the soil is known to be more or less similar to that calculatedon the assumption that the soil is a perfectly elastic, isotropic and homogeneous medium,at least if the factor of safety of a mass of soil with respect to failure by plastic flowexceeds a value of about three (2). The same problem occurs in the theory of indentationtests in which a ductile material is indo .ted by cylindrical punches with conicalheads (3).

A complete solution of the problem would depend on the following factors:(a) the stress-strain relationship of the medium being deformed;(6) the velocity with which the cone penetrates the medium;(c) the coefficient of friction between the cone and the material under it.I t has so far proved impossible to develop the mathematical theory of plasticity to

the point where it could solve a three-dimensional problem of this type. The completedetermination of the stresses set up would involve a discussion of the wave propagationof stress in a medium which is part elastic and part plastic, and the difficulties thisgeneral problem presents are so great that it was only recently that the simplestprocess of this type (the propagation of plastic waves in thin wires) was investigatedby Sir Geoffrey Taylor, von Karman and others (see, for example, (4)). In the staticapproximation the distributions of stress at any given instant during the penetrationwere computed several years ago (5) for various two-dimensional indenters, but theprogress of the deformation with increasing penetration was not followed untilrecently (6), when a detailed account of the history of the motion was given in the caseof a wedge, allowance being made for the continually altering position of the surfaceand for the spreading of the plastic flow.

As a first step towards the solution of the more complex three-dimensional problemof indentation by a cone an account is given here of the distribution of stress in anelastic medium when it is deformed by a rigid cone. The problem of the indentation ofthe plane surface of a semi-infinite elastic solid by a rigid body was first considered byBoussinesq (7), and for that reason problems of this type are usually referred to as' Boussinesq' problems. More recently, Love (8) has treated the case where the surfaceof the rigid punch is a right circular cone whose axis is normal to the indented plane.The analysis given by Love is successful because of his skill in guessing a combinationof potentials which satisfies the boundary conditions, and it is difficult to see how the

Boussinesq's problem for a rigid cone 493method could be extended to cover other shapes of punch. Furthermore, Love's solutionis in such a form that the actual computation of the distribution of stress in the interiorof the elastic solid would be a matter of some difficultyand was not in fact attemptedin Love's original paper. A solution of the indentation problem for a rigid punchpossessing axial symmetry but otherwise of arbitrary shape was given recently (9), andit was shown that the method of integral transforms leads to Love's solution in astraightforward manner without the necessity of guessing combinations of simplepotential functions to satisfy the boundary conditions. Besides being easily extendedto more complicated^problems this method has the advantage that it leads to expressionsfor the components of stress which lend themselves easily to computation. In thepaper (9) the solutions for the special cases in which the free surface was indented bya cone, a flat-ended cylindrical punch and a Brinel ball were derived, but no attemptwas made to describe fully the distribution of stress in the interior of the medium inany of these cases. Such an account for the case of a flat-ended cylindrical punch wasgiven later (10).

In the present paper a detailed account is given of the distribution of stress in asemi-infinite elastic medium with a Poisson ratio 0-25 when its boundary is indentedby a rigid right circular cone whose axis is normal to the original surface of the medium.The value

494 IAN N. SNEDDONWe assume that the cone penetrates a distance b + e into the medium, and that at

that depth of penetration the strained surface of the elastic medium fits the coneover the area between the vertex of the cone and a circular section of radius a (cf.Fig. 1). Since it is obvious on physical grounds that the normal components of stress,

Boussinesq's problem for a rigid cone 495where A and fi are Lame's elastic constants,

p = rja, i = z\a, 1% = I pnf(p)e~P^rm(pp)dp,Jo

and/(p) is the solution of a pair of dual integral equations which depend on the surfacevalue of the normal component of the displacement vector. In the case of a conicalpunchcorresponding to the boundary condition (2)f(p) is given by the relation

Substituting from (6) into the expressions for the components of stress, and trans-forming from the Lame constants to the Young's modulus, E, and the Poisson ratio, a,we obtain the expressions

for the complete determination of the stress in the elastic solid. In these expressions

496 IAN N. SNEDDONDifferentiating this expression with respect to we find that dr/d^ is negative for allpositive values of so that the principal shearing stress, T, decreases steadily as zincreases along the axis r = 0. I t is obvious from (14) that r is in fact infinite at theorigin of coordinates r = 2 = 0. Physically this means that a certain volume of materialunder the apex of the cone will flow plastically and hence reduce the high concentrationof stress in that neighbourhood. The problem is in reality not a purely elastic problembut a mixed plastic-elastic problem in which the shape of the boundary of the plasticregion is unknown. If, however, the depth of penetration of the apex of the cone issmall in comparison with the radius of the circle round which it fits the elastic medium,plastic flow will be confined to a very small region and the stresses given accurately bythe elastic solution at all points of the medium not in the immediate vicinity of theapex of the cone. I t would also appear that the elastic solution has a wider field ofapplication in the case of cones of large semi-vertical angle than in the case of cones ofsmall semi-vertical angle.

4. In a similar way we can derive simple expressions for the components of stressand displacement on the boundary plane z = 0. The integrals occurring in this con-nexion are . , , , ., , . .. ,.

and

(16)

Putting = 0 in equation (7) and substituting for JJ(p, 0) from equation (15), weconfirm that the normal component of stress 1. To determine the other components of stress we put = 0 in equations (8)and (9) to obtain

a(Te

*e, (19)where } (2)

Boussinesq's problem for a rigid cone 497Similarly, when 2 = 0 the radial component of the displacement vector assumes thevalues f l a r f p l V ( l p ) ) ..H ^ 1 *l_2

498 IAN N. SNEDDON

Similarly from the result

Jo P Pwe obtain

-1

Putting M> = +*, equating real parts and substituting from equations (25) and (28)we obtain the relation

Jl(P, o = HpJUp, o + si(P, o - tJUp, 0], (29)where S{(p, Q = P ^ ^ e^ J1(pp)dp = -(l-R sin

Boussinesq's problem for a rigid cone 499

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Boussinesq's problem for a rigid cone 501

0-6

0-5

0-4

C 0-3

0-2

0-1.

-0-1

5=05 =0-2 \

Fig. 2. The variation of the radial stress

502 IAN N. SNEDDON

0-6

0-4

0-2

Fig. 4. The variation of the normal stress crz with p and .

Fig. 5. The variation of the shearing stress Tzr with p and .

Boussinesq's problem for a rigid cone 503the isochromatic lines of photoelasticity. The semi-vertical angle of the cone is shownin this figure to be nearly a right angle, merely to give a large value of a to show thedetail of the stress distribution under the punch. The principal shearing stress becomesinfinite at the apex of the cone, indicating that a certain amount of plastic flow will

Table 5. Variation of arjeE with p,

p \ \

0-00-20-40-60-81-01-21-41-61-82-0

0-0

00

0-186501393011030-08790-06670-04630-0340002610-02060-0167

0-2

0-49320-38900-28910-2161014720-07990-04900-02860-02280019500173

0-4

0-41080-37850-31540-25630-19220-12810-08930-05200-03060-01860-0139

0-6

0-33840-32200-28740-24240-19760-14660-10310-06470-04640031400232

0-8

0-27530-26760-24410-21150-17520-13820-10420-07630-05510-03990-0291

1-0

0-22310-21990-20270-179501519012530-09930-07670-06570-04440-0338

Fig. 6. The variation of the principal shearing stress 7 with p and .

occur and hence that there is in reality no purely elastic problem. In the normal way,for low loading, the elastic stresses will predominate everywhere in the medium exceptin the immediate vicinity of the apex of the cone, and the distribution of stress derivedabove will approximate closely to the true result. It is also of interest to observe that

504 IAN N. SNEDDONthe contours of equal principal shearing stress are roughly of the same shape as theisochromatic lines for a point force acting normally to the surface of a two-dimensionalelastic mediumcircles with centres on the 2-axis touching 2 = 0.

Fig. 7. The curves of intersection by the plane 9 = constant of the surfaces of equal principalshearing stress. The numbers associated -with the curves denote the values of ar/eE alongthe curves.

6. We now consider the application of these results to the normal penetration ofa solid medium by a rigid conical-headed punch of the type used in certain hardnesstests (3). In applying the analysis of the preceding sections it must always be kept inmind that we have considered only an idealized statical problem in the theory ofelasticity, neglecting dynamical effects and the effects due to the occurrence of largestrains or of regions of plastic flow in the solid being indented.

In fact, a wave motion, complicated even in the case where there is no failure dueto plastic flow, is created, and it is obvious that the difficulties in the way of exactcalculation are enormous. An estimate of how far the relatively simple stress systemfound above approximates to the actual distribution during penetration can be basedonly on an exact solution of the dynamical equations. No solution of these equationsappropriate to our boundary conditions is available, but much attention has beengiven to problems of a similar nature in Hertz's theory of the collision of elastic spheres.In this connexion a paper of the late Lord Rayleigh's (13) is of some interest, for althoughthe problem considered in it differs from that considered here the general conclusionsgive us a criterion for the reliability of the results reached by the static approximation.It is that we are justified in using the equations of elastic equilibrium provided that thevelocity with which the cone strikes the surface of the elastic medium is small comparedwith the velocity of elastic waves in the1 medium.

For similar reasons it is difficult to estimate the error due to neglecting the possibilityof regions of plastic flow existing in the medium'. A set of equations governing a plasticmass in its equilibrium state has been obtained by Hencky(l4) by minimizing the

Boussinesq's problem for a rigid cone 505integral of the elastic energy density expressed in terms of the stress components whichare supposed to obey the plasticity condition of von Mises. When the variation iscarried out we find that the components of stress and strain satisfy equations analogousto those of elastic theory except that the Young's modulus E is replaced by a functionof the coordinates. Thus, though the solution of the Hencky equations presents aformidable problem we may make use of the concept they provide of a medium inwhich the Young's modulus varies from point to point. In the parts of the mediumremote from the apex of the cone the function E has its usual constant value, Eo say,as in the elastic theory. In the vicinity of the apex of the cone E will have a modifiedvalue E' varying from point to point. As a first approximation we might suppose thatE' is constant but different from Eo in a certain region of the medium, i.e. we mightpostulate the existence of a region R such that

(E'

E(P) denoting the value of the Young's modulus of the material at the point P of themedium being deformed. Estimates of the value of E' and of the extent of R might bederived from the analysis of experimental results. For example, in the photographsof the experiments performed with wedges by Dr J. F. Allen and Mr J. M. Speight (6)the extent of the region of plastic flow can be clearly seen, and hence an estimate of theshape of the curve bounding R can be obtained. The modified value E' of the Young'smodulus must be chosen to fit the experimental results. Calculations based on asimple model of this type are facilitated by the fact that the tables above give thevalues of (Tr/2?, etc., so that changes in the values of the Young's modulus are easilyincorporated in the calculations.

The results are strictly valid only for small strains, but the difficulties of taking intoaccount large strains are very great, and it is well known that calculations made on thebasis of small strains often give useful results when applied outside their proper rangeof validity.

With these reservations in mind we now consider the total work done in pushinga conical-headed punch of radius a0 and semi-vertical angle a into a semi-infinitemedium which we assume to behave elastically and to possess a modified Young'smodulus determined experimentally. This work is done in two parts, that done in thepenetration to the edge of the conical head and that done in pushing the conical headdown until part of the cylindrical surface of the punch is embedded in the medium.From equation (3-1-13) of the paper (9) we have for the total pressure on the pressedarea when w < w0 = \naQ cot a

2 E tan a Jr = W .

When the depth of penetration w exceeds w0 it is easily shown from equation (3-2-5)of (9) that the resultant excess pressure on the pressed area is given by

PSP44.4 33

506 IAN N. SNEDDONThus we have for the total pressure

(2Eta.nocj-, 2Tw *or w ^ hnao co^ a>

For the work done in making a penetration w we have

Jo Pdw,

so that W =

, cot a.

w3 for w^^7ra0cota,

" w

2 w(w ^7ra0 cot a) for w>^7ra0cota.

w/o0 * -

Fig. 8. Variation of the total load P with depth of penetration w and cone angle. The numbersassociated with the curves are the values of the semi-vertical angle a of the conical head(cf. inset).

Curves illustrating the variation of the total load P with the depth of penetration wfor various values of the semi-vertical angle of the cone are shown in Fig. 8. A com-parison of these curves with the experimental curves derived by Bishop, Hill andMott (3) shows that the above formula for the total load P gives the correct form forthe variation of P up to penetrations of the order 2a0 cot a, even when the strains areno longer small and a certain amount of plastic flow is occurring. For larger values ofthe penetration the experimental results show that the P-w curve becomes increasinglyflatter until it is almost parallel to the w-axis, but the elastic theory developed abovedoes not account for this phenomenon. We might then expect that the distribution ofstress described in 5 will give a reasonably accurate picture of the state of stress in

Boussinesq's problem for a rigid cone- 507the solid for values of the penetration lying in the range indicated above and forvelocities of penetration small compared with the velocity of elastic waves in themedium.

REFERENCES

(1) KRYNTNE, D. F. Soil mechanics (New York, 1941), Chap. rv.(2) TERZAGHI, K. Theoretical soil mechanics (New York, 1943), p. 367.(3) BISHOP, R. F., HELL, R. and MOTT, N. F. Proc. Phys. Soc. 57 (1945), 147.(4) v. RAHMAN, TH. and DUWEZ, P. Comptes rendus du 6" congres de mecan. appl. (in course of

publication).(5) PRANDTL, L. Z. angew. Math. Mech. 3 (1923), 6, 401.(6) HILL, R., LEE, E. H. and TUPPER, S. J. Proc. Roy. Soc. A, 188 (1947), 273.(7) BOUSSESTBSQ, J. Applications des potentiels (Paris, 1885).(8) LOVE, A. E. H. Quart. J. Math. 10 (1939), 161.(9) HARDING, J. W. and SNEDDON, I. N. Proc. Cambridge Phil. Soc. 41 (1945), 16.

(10) SNEDDON, I. N. Proc. Cambridge Phil. Soc. 42 (1946), 29.(11) WATSON, G. N. The theory of Bessel functions (2nd ed., Cambridge, 1944).(12) SNEDDON, I. N. Proc. Roy. Soc. A, 187 (1946), 229.(13) RAYLEIGH, LORD. Phil. Mag. (6), 11 (1906), 283.(14) HENCKY, H. Z. angew. Math. Mech. 4 (1924), 323.For further references see

NADAI, A. Handbuch der Physik, 6, p. 470.

THE DEPARTMENT OF NATURAL PHILOSOPHYTHE UNIVERSITY OF GLASGOW

33-2