Journal of Elasticity, Vol. 6, No. I, January 1976 Noordhoff International Publishing - Leyden Printed in The Netherlands

Location of extreme stresses

H. L. L A N G H A A R and M. C. STIPPES

Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, Illinois 61801, USA

(Received April, 1975)

Introduction

A casual survey of particular solutions in elasticity theory suggests that, when a bounded Hookean body is loaded only by boundary forces that are in equilibrium, the absolute maximum tensile stress occurs at a boundary point. Using potential theory, G. Polya [1] investigated this question in 1930. He demonstrated that extremes of the cubical dilatation and the rotation necessarily occur on the boundary, and he proved several other theorems, but his conclusions concerning extreme principal stresses appear to be inferential. This question is discussed in this note.

Normal stresses are compared algebraically; e.g., a tensile stress is considered to be greater than a compression stress. A stress will be said to have a proper (improper) relative maximum at a point P if, throughout a deleted neighborhood of P, its value is less than (less than or equal to) its value at P. Similarly, proper and improper relative minima are defined. It is not possible for all three principal stresses (o-1, o-2, °3) to have relative maxima (or minima) at an interior point P of a body unless all the extremes are improper, since otherwise the sum o-1 +a2+o-3 would have a proper relative maximum (or minimum) at P. However, this is impossible, since o-1 + a2 + a3 is a harmonic function, and it is known [2] that a harmonic function cannot have a proper relative extreme at an interior point of the region of regularity.

1. Three-dimensional examples

Axially symmetric loading of a solid sphere is considered. Spherical coordinates are denoted by (r, 0, ~b), in which 0 is colatitude and ~b is longitude. One principal stress is a~ = a3. The other principal stresses are

o-,, o-2 = ~(ar + a0) + [~-(a, - a0) 2 + r~0] ~ (1)

in which Timoshenko's stress notations are used [3]. For brevity, cos O=p and sin 0 =/3. Legendre polynomials are denoted by P,(p). Two basic solutions of the equations of

Journal of Elasticity 6 (1976) 83-87

84 H. L. Langhaar and M. C. Stippes

elasticity for axially symmetric loading of a solid sphere by boundary forces are [4]:

(7~ =(n+ l)(n+ 2)r"P.+ 2

(7o =[P' ,+t-(n+ l)(n+ Z)P.+2]r" Solution [A.] . , ( 2 )

(7~ = - r Pn+l

Z,o = - ( n + l)/sr"P'.+ 2

ar = - ( n + 1)[(n+ l ) ( n - 2 ) - 2 v ] r " P , Solution [B.]

(70 = [(n+ 1)(n2+ 5n+ V - 2 v ) P . - ( n + 5-4v)P'.+t]r" (3) try, = [ - ( 1 - 2 v ) ( n + l)(Zn+ 3)P.+(n+ 5-4v)P'+t]r"

z, o = (nZ + 2 n - 1 + 2v) p r" P'.

in which v is Poisson's ratio, n is any non-negative integer, and P',=dP,/dp. Solution

[Ao] is

a~=3p2-1 , 60=3/52-1, a ~ = - l , Zro=-3p/5 (4)

Solution [B0] is such that at = o2 = os = constant. If v=¼, the solution [Ao] - [A2]+ [B2] yields, when expanded in powers of r,

at = 2 - ~ ( 3 - - p 2 ) r 2 + ...

(7 2 = (7 3 = - - 1 - ~ ( 1 - 7 p E ) r 2 + . . . . (5)

Since (3 _ p 2 ) > 0 in - l ~ < p <~ 1, (Tt attains a proper relative maximum at the center r=0 . Since ( 1 - 7 p 2) takes both positive and negative values, (72 and (Ta do not have extremes at r=0 . For a sphere of sufficiently small radius, the stress (Tt = 2 at r = 0 is an absolute maximum; i.e., the greatest tensile stress in the sphere occurs at the center. A reversal of the loading on the surface of the sphere changes the signs of all the stresses. Then the absolute maximum compressive stress occurs at the center of the sphere. The maximum shearing stress at any point in the sphere is

z = ½((Tt - (72) = -a2 - ~ - ( 1 + 3p2) r2 + . - . (6)

Consequently, for a sphere of sufficiently small radius, the shearing stress has an absolute maximum at the center r = 0. The same conclusion applies for the octahedral shearing stress Zo, since, in this example, % = (2½/3)(al- (Tz).

With v = ½, the solution -4 [A2] + [B2] yields, with Eq. (1),

(Tt, (72 = ( - 8.5 + 1 lp 2) r 2 +_ r E ( 110.25 + 877p 2 + 1365p4) ~ . (7)

It is easily seen that the values (Tt = (72 = 0 at r = 0 provide a proper absolute minimum to at and a proper absolute maximum to a 2. The stress (73 does not have an extreme at r=O.

If v=¼, the solution [ A o ] - 3 [A2] - [B2] yields, when expanded in powers of r,

(72 = - 1 -~ (17 - 7p2)r 2 + . . . (8)

as = - 1 - ¼ ( l + 9 p E ) r 2 + . . . .

Location o f extreme stresses 85

Since 1 7 - 7 p 2 > 0 and 1 +9p2>0 in - 1 ~<p~< 1, these solutions ensure that o2 and a3 both have proper relative maxima at r = 0. A reversal of the loading provides a case in which two principal stresses have proper relative minima at r = 0. In this example, o 1 does not have an extreme at r=0 .

In view of the remarks in the Introduction, all three principal stresses cannot have proper relative maxima or minima at r--0. Conceivably, however, all three principal stresses might have proper extremes at r = 0, such that two are maxima and one is a minimum, or vice versa. We have been unable to find such an example. Lengthy linear combinations of solutions [A,] and [B,] seem to be unpromising because of rapid fluctuations of P,(p) if n is large. Nonsymmetric solutions of the equations of elasticity, expressible in terms of solid harmonics [5], might provide an example, but they look formidable, particularly since the principal stresses are roots of a cubic equation.

2. Plane elasticity

If z is a complex position variable in the plane of a two-dimensional body, there exist [6] holomorphic functions ~b(z) and ~,(z), such that the stresses in polar coordinates are determined by

~rr+ a o = 2 [q~'(z) + ~(-~-]

a 0 - a t+ 2izr0 = 2 [eq~"(z) + O'(z)] e z'° , (9)

in which bars denote complex conjugates. The principal stresses are

a , , o.2 = 1(o., + o.0) + ½ [ (a0- a,) 2 + 42201 ~ •

It is convenient to introduce the notations (a ' ( z )= f ( z ) and O'(z )=9(z) . Then,

t~,, o.z = / ( z ) + f ( z ) + I~ f ' ( z )+g(z) l (10)

in which the sign _+ is taken to be + for %. Hence, o.1 >az . I f f ( z ) = z 2, g(z)= - l + z 2, and z = p e i°, Eq. (10) yields

t71 = 2p 2 cos 20 + [2p 2 - 1 +p2 cos 20+ ip 2 sin 20[ • (11)

Accordingly, o.1 = 1 at p=0 . On the boundary of the circle, p=0.10, Eq. (11) yields

a~ = 0.02 cos 20+(0.9605-0.0196 cos 20) 4, p =0.10. (12)

The maximum occurs at 0=0 ; its value is o.1=0.990000. The minimum occurs at 0=~/2; its value is o.~ =0.970000. In the interval 0<p~<0.10, the value of o.1 is every- where less than 1. Accordingly, in the circle p=0.10, the absolute maximum tensile stress occurs at the center. A reversal of the loading provides an example in which the absolute maximum compressive stress occurs at the center. In this example, the second principal stress o.2 does not have an extreme at z= 0.

I f f ( z ) = 0 and 9 ( z ) = z 2, Eq. (10) yields o . l = p 2 and a2= _p2. In this case, o.1 has an absolute minimum and ~2 has an absolute maximum at z = 0.

In the last example, it is the larger principal stress that has the minimum. The question arises whether it is possible for the larger principal stress tr, to have a proper maximum

86 H. L. Lanohaar and M. C. Stippes

and the smaller principal stress o" 2 to have a proper minimum at the same interior point. It can be shown that such a condition is impossible if f (z) and g(z) are second- degree polynomials. However, o"x has an improper maximum and o"2 has an improper minimum at z=0 , if f ( z )= - a z 2 and g (z )= 1, in which a>0 . We can strengthen the extremes by adding a term of higher degree. An example is provided by f (z) = - z 2 - z 6, 9(z) = 1. Then Eq. (10) yields

o"1, °2 = - 2P 2 cos 28 - 2p 6 cos 68 ___ [1 - 4p z + 4p 4 + 36p x 2 _ 12p6( 1 - 2p 2) cos 40] ~ .

Expanding the square root by the binomial theorem as far as sixth-degree terms in p, we get

a 1 = 1 -2p2(1 +cos 20 ) -2p6(3 cos 40+cos 68)+ ... (12)

o"2 = - 1 +2p2( 1 - c o s 28) +2p6(3 cos 4 0 - c o s 68)+ . . . .

Equation (12) shows that 0" 1 has a proper maximum and a2 has a proper minimum at p = 0, provided that there is a positive number e such that

l + c o s 28+p4(3 cos 48+cos 68 )>0 )

and / , 0 < p < e .

1 - c o s 28+p4(3 cos 4 8 - c o s 68 )>0

The only places where these inequalities might break down are at 0 = 0 and 8= n/2, since otherwise 1 _ cos 28 > 0. It is clear that the values 0 = 0 and 8 = re/2 do not vitiate the inequalities. Consequently, o"1 has a proper relative maximum and a2 has a proper relative minimum at z = 0. For a disk of sufficiently small radius, the stresses o1 = 1 and o-2 = - 1 at z = 0 are absolute extremes. Also, since

z = ½(al-o-2) = 1 - 2 p 2 - 6 p 6 c o s 48+ ... ,

the shearing stress attains an absolute maximum at z = 0. For a plane body loaded by boundary forces, both principal stresses cannot have

proper relative maxima or minima at the same interior point. The proof is the same as that given in the Introduction for the three-dimensional analogue of this theorem.

3. Conclusions

In three-dimensional linear isotropic elasticity theory, there are continuous boundary loadings of a bounded body that furnish absolute maxima to the tensile stress, the compressive stress, the shearing stress, and the octahedral shearing stress at interior points of the body. Also, it is possible for two of the principal stresses to have proper relative extremes at the same interior p o i n t - both of them being maxima, minima, or one a maximum and the other a minimum. It is not possible for all three principal stresses to have proper relative maxima or proper relative minima at the same interior point. Whether all three principal stresses can have proper relative extremes at the same interior point (with two maxima and one minimum, or vice versa) has not been decided.

Location of extreme stresses 87

Analogous results are found in plane elasticity. There, one principal stress can have a proper maximum and the other a proper minimum at the same interior point. Either the larger or the smaller principal stress may have the maximum.

REFERENCES

[1] Polya, G., Liegt die Stelle der gr6ssten Beanspruchung an der Oberfl~iche? Zeit. fiir angew. Math. und Mech., 10 (1930) 353-360.

[2] Kellogg, O. D., Foundations of Potential Theory, 1st ed., Dover Pubs., New York. [3] Timoshenko, S., Theory of Elasticity, 1st ed., McGraw-Hill, New York 1934. [4] Sternberg, E., R. A. Eubanks and M. A. Sadowsky, On the Axisymmetric Problem of Elasticity Theory

for a Reoion Bounded by Two Concentric Spheres, Proc. First U.S. Nat. Congr. Appl. Mech., pp~ 209-215, A.S.M.E., New York, 1952.

[5] Love, A. E. H., The Mathematical Theory of Elasticity, 4th ed., Chap. XI, Cambridge University Press, 1934.

[6] Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, English Translation by J. R. M. Radok, 4th ed., Noordhoff Int. Publ., Leyden 1975.