On St.-Venant's Problem for Thin-walled Tubes

J. L. ERICKSEN

To Clifford Truesdell on his sixtieth Birthday

I. Introduction

With most physical theories, we face the difficulty that, in practice, we rarely know the detailed boundary data, initial conditions, etc. which are needed to make predictions unique. A reasonable approach is to attempt to devise methods of constructing solutions, each of which can serve as the representative of some set, to give criteria for characterizing the set associated with one representative, and to clarify the meaning of "representative". For linearly elastic prisms, in equilibrium under loads applied only near the ends, St.-Venant set himself the problem of accomplishing this task. St.-Venant's Principle gave a rule of thumb for dividing all solutions into equivalence classes; one solution serves as a representative for the set which has the same resultant force and moment. Also, it suggested what is to be meant by "representative", in rough terms. St.-Venant's semi-inverse methods provide a scheme for constructing solutions for which these resultants have any desired value. Nominally, then, a six parameter family of solutions serves to represent all. This resolution has its faults. Later studies of St.-Venant's Principle, such as are covered by TOUPIN [1], indicate that solutions having the same resultants need not be close together, away from the ends. Said differently, solutions produced by semi- inverse methods do not provide a complete list of representatives. MAIS- SONEUVE [2] has given a simple rule for generating another set of solutions which can reproduce any desired values of the resultants. On each end, the displacement is to have the form of an infinitesimal rigid motion. Without real loss of generality, one can require that it vanish on one end, so this also involves a six-parameter family. St.-Venant's flexure solutions, for example, seem not to spring from any simple concept so, on aesthetic grounds, we might prefer Maissoneuve's solutions. In a well-defined sense, they can also be characterized as the solutions of minimum energy. This vaguely suggests a high degree of stability, but it is not clear which version of the energy criterion is most appropriate. Considering the difficulty with St.-Venant's Principle, it is not obvious that the St.-Venant solutions and MAISSONEUVE solutions fairly repre- sent the same collection of solutions. If we permit finite deformations, it seems clear that we will need additional representatives to cover such phenomena as Euler buckling, which are outside the range of linear theory. We thus have the

8 J.L. ERICKSEN

problem, only partially solved, of characterizing a complete set of representative solutions for elastic prisms, in equilibrium under loads applied only near the ends. Roughly, this explains what I mean by St.-Venant's Problem.

Using nonlinear elasticity theory, I showed in [3] that there is another average over cross-sections which, like the resultant force, has the same value on all cross-sections, so there is similar reason to match it. Roughly, it is an analog of the "energy integral" for the elastica. Near solutions of the St.-Venant type for bending, or combined tension and torsion, it seems to behave peculiarly. This is an aspect of St.-Venant's Problem which seems to deserve more thought, so I probe it in a different way. For variety, I use a different theory, appropriate for those prisms which we call thin-walled tubes. It is not hard to adapt the analysis to some theories of rods. To some degree, the cruder theories summa- rize our estimates of what is needed to describe the main features of repre- sentative solutions, so it seems unwise to ignore them completely. Also, sim- plified versions of St.-Venant's Problem there arise, and we might need to cut our teeth on simpler versions.

2. Basic Equations

Kinematically, the formal equilibrium theory of Cosserat surfaces involves a surface S, given parametrically by

x=x(X~), e = l , 2 , (2.1)

and a director or "thickness vector" field, defined on S,

n = n (X~), (2.2)

where the X" are surface coordinates, x representing the triplet of rectangular Cartesian coordinates in space, n similarly representing Cartesian components. The energy U of a part of S is given by

U =6 WdX1 dX2, (2.3)

where W is given by a constitutive equation of the form

W= W(x, ~,; n; n,p; X~). (2.4)

Throughout, commas denote partial derivatives. Of course, W is required to be Galilean invariant,

W(x,~; n; n,~,XV) = W(Rx,~; Rn; Rn, t~; X ~) (2.5)

for all rotation matrices R. This gives rise to the differential identity,

~W (~W cgW n^-~n +n,~^ ~3n,~ +x,~^ ~x,~ = 0 . (2.6)

Roughly speaking, S and the director serve to represent configurations of a three-dimensional body, of the form brought to mind by the phrase "thin shell".

St.-Venant's Problem 9

If the shell is loaded only at its edge ~3S, the equilibrium equations are

,28,

Whenever these are satisfied, we have

~W c~W ) O W _ W 6 ~ - x , ~ . �9 n , , , (2.9) ~X ~ 0x,~ ~3n,a -, ~

where n, n,, and x, z are held constant in calculating the derivatives occuring on the left side. Also, with (2.6), we have

0W OW ] ( x - a ) , , - - + n =0

Ox ,~ ^ ~ n . . . . (2.1o)

where a is any constant. In differential form, (2.7) represents balance of forces, while (2.10) respresents balance of moments.

Consider a body which, when it is not loaded, has the usual prismatic form, generated by translating one plane cross-section along its normals, to get a family of physically indistinguishable cross-sections. If the body is of tubular form, not necessarily circular, it will be generated by a simple closed curve C, and some director field, which need not concern us. Let C be described by a parameter X ~,

X~e[0,A], (2.11)

where A is some constant, X ~ =0 and X 1 =A >0 describing the same point on C. For X 2 we use the normal distance from one cross-section,

XZe[0, L], (2.12)

where L > 0 is the length in this reference configuration. This fixes the domain of the functions occurring in (2.1) and (2.2). To compensate for the defect in the X 1- coordinate, I follow common practice; these functions together with W should be the restrictions to [0, A] of functions which are periodic in X 1, with period A, and smooth, so that, for example,

n(o, x ~) = n(A, X~).

Also, the discussion serves to motivate the constitutive restriction

0W =0. (2.13) •X 2

If A and the constitutive function W are fixed, (2.7) and (2.8) will admit some set of solutions s The interpretation leaves open the possibility of consider- ing like prisms of different lengths, and we all know that St.-Venant's Principle

10 J. L, ERICKSEN

is of little or no use for very short prisms. Some properties of solutions are easily to establish. First,

0 < L 1 < L =~ S(L)cZ(L1). (2.14)

That is, if we have a solution for one length, restrictions of it will be solutions for prisms of shorter length. If we integrate (2.7) and (2.10) with respect to X 1, we get, with the assumed periodicity,

A

F(X 2) = ~ p d X 1 = F(0), (2.15) 0

and

where

A

M(X2, a) = ~ [(x - a) ^ p + n ^ q) d X 1 = M(O, a), (2.16) 0

OW OW ' q = ~?n 2' (2.17) P = ~X,2

and a is any constant. Of course, St.-Venant accounted for such balances of forces and moments in formulating his Principle. Another such integral follows from using (2.13) and similarly integrating (2.9), viz.

A

E(X 2) = ~ r d X a =E(0), (2.18) 0

where r = W - x , 2 . p - n , 2"q. (2.19)

Here, this is the analog of the "energy integral" for the elastica. Clearly, the Principle should be amended to include matching of E, as well as F and M. The question then arises as to what freedom we have in specifying these values. Said differently, what is the range R(L) of these functionals, with domain 2:(L)? It is convenient to note that, by using translations and rotations to normalize solutions, we can arrange that, with a = 0, both F and M can be made parallel to a preassigned unit vector e:

F = F e, M = M e . (2.20)

I presume this done so that R(L), represented by (E, F, M), is some subset of R 3. Of course, (2.4) implies that R ( L ) c R ( L 1 ) . On the face of it, it does not seem implausible that R(L) is, in general, a three-dimensional submanifold, perhaps with complicated topology, and one can give some plausibility arguments favoring this speculation. Granted this, there is rather strong evidence that it has a boundary, OR(L), and that solutions of the St.-Venant type correspond to some of the boundary points. I now turn to an analysis which has led me to this rather surprising conclusion.

3. A Control Problem

With W,, L, etc. fixed let us pick some particular solution, letting bars denote quantities associated with it. This gives a point (E, F, M ) e R . In R, pick a smooth

St.-Venant's Problem 11

curve

(/~(~), if(z), M(z)), (3.1)

defined near ~ = 0, with

/~(0) =/~, F(0) = F, M(0) = ~t. (3.2)

The problem is then to find a family of solutions, depending smoothly on z, with the corresponding functionals satisfying

E(z) = s F = if(T) e, M(~) = M(z) e (3.3)

in some neighborhood of z=0. Suppose that some such family exists. Then, differentiating with respect to z and evaluating at T=0, I find, after a little calculation, that

A 6E = [E 2" 6x + q, 2" 6 n - X , 2" 6p- , 2" 6q] d X 1 = E ' ( 0 ) ,

0 A

6 F = ! 6 p d X 1 = ff'(O) e, \ (3 .3)

A 6M= ~ [6x ^p+~^ 6~+6n ^ q+~ ^ 6q] dX' = ~t'(0)e,

0

where the 3's have the interpretation which is traditional for Gateaux differen- tials. I presume enough smoothness so that these hold, with the functions involved in the integrands being at least square integrable. Roughly, we might expect the problem to be soluble for a curve starting out in any direction, from an interior point of R, if the starting solution does not have some unusual pathology. If we start at a smooth part of dR, we similarly expect existence for curves in OR; thus

A, (0), F (0), M'(0)) (3.4)

will be restricted to lie in the tangent plane. In neither case would we expect uniqueness of solutions. Now suppose that the starting solution has the property that there exist constants ~o and b such that

1~, 2 =toe^ ~, q, 2 =toe^ t], ~ , 2 = a ~ e ^ ~ + b e ' ~, 2 = o ) e ^ ti" (3.5)

It then follows that, no matter what be the functions 6p etc., may be,

3E +be. 3 F + me. 3M =0. (3.6)

Consequently, (3.3) demands

/~' (0) + b tO' (0) + o~_M' (0) = 0, (3.7)

suggesting that we are at a boundary point of R where the normal to 0R has direction numbers

(1, b, co). (3.8)

12 J.L. ERICKSEN

Formally, such solutions can be generated by making semi-inverse assumptions analogous to those which produce the St.-Venant solutions for torsion or bending, viz.

x=R(X2)y(Xl )+bX2e , n=R(X2)z(X1), (3.9)

where R is the rotation matrix given by

R = exp (12 X2), (3.10)

and f2 is the skew matrix such that

~v=coe^ v for all v. (3.11)

As is discussed by Ericksen [4], y and z must satisfy a system of ordinary differential equations, depending on co and b, so it is likely to be feasible to analyze their existence, smooth dependence on co and b, etc., with some restrictions on the constitutive function W. Topological considerations suggest that some such are likely to be acceptable only if L is sufficiently small; the inclusion represented by (2.14) can be proper. Such analyses would give us a firmer grasp of the corresponding part of R, independent of my speculations concerning its structure. A complete characterization of R would hardly give us a solution of St.-Venant's Problem, but it is an important piece of the puzzle. Analyses like those given by ERICKSEN [3] indicate that it is not easy to construct other solutions corresponding to other points on 8R, if such exist.

Acknowledgement. This work was supported by the National Science Foundation and Grant ENG 7614765-A01.

References

1. TOUPIN, R.A., Saint-Venant's principle, Arch. Rational Mech. Analysis, 18, 83 96 (1955)

2. MAISSONEUVE, O., Sur le Principe de Saint-Venant, Th6se, Poitiers, 1971. 3. ERICKSEN, J.L., On the Foundations of St.-Venants Problem, Heriot-Watt Univer-

sity: Symposium on Nonlinear Analysis and Mechanics, vol. 1, pp. 158 186. London, San Francisco, Melbourne: Pitman Pub. Corp., 1977.

4. ERICKSEN, J.L., Simpler problems for elastic Cosserat surfaces, J. Elasticity, 7, 1-11 (1977).

Department of Mechanics The Johns Hopkins University

Baltimore, Maryland

(Received November 4, 1977)