On generalized saint-venant's problems

ht. 3. Engng Sci. Vol. 24, No. 5, pp. 849-858, Printed in Great Britain.

OOZO-7225/86 $3.00 + .W 0 1986 Pergamon Pm39 Ltd.

1986

ON GENERALIZED

SAINT-VENANT’S

PROBLEMS

DORIN IE$AN Department of Mathematics, University of Ia$, Ia& Romania Abstract-In this paper is presented the solution of a problem proposed by C.Truesdell [2-41, for the torsion of inhomogeneous and anisotropic cylinders. I. INTRODUCTION

IN [ 11, W. A. DAY has solved the following problem proposed by C. Truesdell [2-41: For an isotropic linearly elastic cylinder subject to end tractions equipollent to a torque A4, define a “twist” 7 in such a way that

where pR, the torsional rigidity of the cylinder, is resolved into the factors CL,the shear modulus, and R, a geometric quantity depending only on the cross section. As remarked by Truesdell, such a twist would generalize Saint-Venant’s notion of twist so as to apply also to solutions of the torsion problem corresponding to distributions of end tractions different from the one assumed by Saint-Venant. In [5], P. Podio-Guidugli has solved Truesdell’s problem rephrased for extension and pure bending. Thus, for an isotropic and homogeneous linearly elastic cylinder subject to end tractions equipollent to an axial force N, he defines a global axial strain E corresponding to any solution of the extension problem in such a way that N = 2p( 1 + Y)AE, where v is the Poisson ratio and A is the area of the cross section. For the pure bending problem, P. Podio-Guidugli defines a global axial curvature x corresponding to any solution of the problem so that

where Z, is the moment of inertia of the cross section with respect to the y axis and M, is the bending moment. Let us note that these results are established for homogeneous and isotropic bodies, where each of the three problems (extension, pure bending, torsion) can be studied independently. It is known (see [6, 71) that the torsion of an anisotropic linearly elastic cylinder is accompanied by extension and pure bending. The extension and the pure bending are coupled problems even within the theory of inhomogeneous and isotropic solids [8]. In order to extend the results of [ 1, 51 to such situations, we rephrase Truesdell’s problem in the following manner: For a linearly elastic cylinder subject to end tractions equipollent to a force F(0, 0, F3) and a moment M(M, , M2, M3) define the constants c, x1, ~2, T in such a way that -Ml MI

= D,rx, + DIzx2 + D,3~ + D,47, = Dz~x~ +

D22~2 +

D23~ +

F3 = D3,x, + D32x2 + D33~+ M3 = D4,x,

+ D4#2

0247, D34r,

+ D43e + 0447,

(1.1)

where Dij (i, j = 1, 4) are quantities depending only on the cross section and the elasticity tensor. The aim of this paper is to solve this problem. 849

850

D. IESAN 2.

BASIC

EQUATIONS

Throughout this paper B denotes a bounded regular region of three-dimensional Euclidean space. We call aB the boundary of B, and designate by n the outward unit normal of aB. Throughout this paper a rectangular Cartesian coordinate system Oxk (k = 1, 2, 3) is used. Letters in boldface stand for tensors of an order p 3 1, and if v has the order p, we write Vi,. . . k, (p subscripts) for the components of v in the rectangular Cartesian coordinate frame. We shall employ the usual summation and differentiation conventions: Greek subscripts are understood to range over the integers (1, 2), where Latin subscripts-unless otherwise specified-are confined to the range ( 1,2,3); summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate. Let B be occupied by a linearly elastic material. If u is a displacement field over B, then the strain field associated with u is E(u) = sym Vu. The stress-displacement Section 27)

(2.1)

relation appropriate to the linearized theory of elasticity is (cf. [lo], S(u) = C[Vu],

(2.2)

where S(u) is the stress field associated with u, and C is the elasticity field. We assume that C is positive-definite, symmetric, and smooth on I?. For the particular case of the isotropic elastic medium the tensor field C admits the representation cijki

= Afiijakl

+ d&kaj/

+ bdjkh

where X and CLare the Lame moduli, and 6, is the Kronecker delta. In the special case of the homogeneous medium C is constant on B. We call a vector field u an equilibrium displacement field for B if and only if u E C’(B> rl C*(B) and div S(u) = 0,

(2.3)

holds on B. Let s(u) be the traction vector at regular points of dB belonging to a stress field S(u) defined on 8. i.e. s(u) = S(u)n. The strain energy U(u) corresponding to a smooth displacement field u on B is (cf. [lo], Section 32) U(u) = ;

sB

Vu. C[Vu]dv.

In the following, two displacement fields differing by an (infinitesimal) rigid displacement will be regarded identical. The functional U( * ) generates the bilinear functional U(u, v) = f

s

Vu - C[Vv]dv.

B

The set of smooth fields over B can be made into a real vector space with the inner product (u, v) = 2U(u, v).

(2.4)

This inner product generates the energy norm (see, e.g. [9], p. 93)

llullf = (4 u).

(2.5)

On generalized Saint-Venant’s problems

851

For any equilibrium displacement fields u and v, we have (cf. [lo], Section 30)

(u, v) =

s,,

s(u).v da,

(2.6)

which implies the well-known relation s(u) - v da =

s JB

s JB

s(v) - u da.

(2.7)

Following [ 1, 51, for any given equilibrium displacement fields u, v(l), v(*),vc3),vc4)we define the real function of the variables [i, t2, t3, t4 by

.fl4l

9 t;2 5 t3r

44)

=

2w

-

i

.w’).

i=l

In the following section the vector field u will be a solution of a certain boundary-value problem and the vectors vu’)(s = 1, 4) will be prescribed. Obviously,

where A, = ($0, ,ti)),

(i,j = 1, 4).

We have assumed that C is positive definite so that the matrix (A,) is positive definite, and j‘will be a minimum at & = ai (i = 1, 4) if and only if aI, (~2, ~1x3, a4 is the solution of the following system of equations: 4 C

Aijaj =

(i=

(II, di’),

1,4).

653)

j=l

The vector field i aivci)provides the best {v(l), v(*),vt3),v(4)}-type approximation energy norm.

i=

to u in the

I

3. GENERALIZED

EXTENSION-BENDING-TORSION

PROBLEM

Assume that the region B from here on refers to the interior of a right cylinder of length h with the open cross section Z and the lateral boundary II. We denote by L the boundary of the generic cross section 2. The rectangular Cartesian coordinate frame is chosen such that the x3 axis is parallel to the generators of B and the x,0x2 plane contains one of the terminal cross sections. We denote by 2, and Z2, respectively, the cross section located at x3 = 0 and x3 = h. In view of the foregoing agreements we have B = XI

=

{x1(x,, x2) E 2, 0 <

{XKX,,x2)E 2, x3

x3

<

h),

= O},

n 22

=

=

{x1(x,

{4(x,,

) x2)

x2)

E

where x = (x, , x2, x3). By a solution of the generalized extension-bending-torsion librium displacement field u that satisfies the conditions s(u) = 0

on II,

G3aW(x,

9 x2,0)

=

(S,,(U))(X, , x2,

E 2,

L, x3

0 G x3 =s h}, = h},

problem we mean an equi-

h),

(Xl,

x2)

E

2,

(3.1) (3.2)

852

D. IESAN

We denote by F(u) and M(u), respectively, the resultant force and the resultant moment of the traction associated with II, acting on ZZ, i.e. F(u) =

M(u) =

s(u)du, s \‘2

s 22

x X s(u)da.

(3.3)

The condition (3.2) requires that F,(u) = 0. The principal aim of this problem is to solve Truesdell’s problem for anisotropic bodies. In order to best emphasize the method, we study first Truesdell’s problem for the case of inhomogeneous and isotropic solids. Then, we consider the case of anisotropic elastic bodies. (i) Inhomogeneous und isotropic elastic bodies. We assume that the elastic coefficients are independent of the axial coordinate

x = X(x,

x.2),

7

CL =

AXI

(XI

> x2),

9 x2)

E

2.

We define the vector field v(l) by (1’ -

-

Vl

-4x: + lp(x, x2),

v’:’ = u’:‘(x,

x2),

)

)

v:I)

=

x1x3,

(3.4)

XEB,

where ulf) are the components of the plane displacement vector u(l) in the plane strain problem (see [lo], Section 45) associated with the cross section Z, the body forces (Xx,),, and the tractions -A_x,n, on L. If we denote by e(” and t(l), respectively, the strain field and the stress field associated with II(‘),then t$J$= Ae$yi& + 2pe$, tjj& + (Ax.,),, = 0

tf2na = -Ax,n,

in Z,

on L.

(3.5)

It is easy to prove that the necessary and sufficient conditions for the existence ([ 1 I], Section 12) of the solution of the plane strain problem (3.5) are satisfied. In the case of homogeneous and isotropic bodies, the solution of the problem (3.5) is #)

I

x

= _

4(X + P)

x

l,(I) = -

cd - x9,

2

2(X + /A)x’x2.

From (3.4) we obtain

s&q

= Ax&j

s&q

+ t’bB,

= 0,

S33(v(‘))

=

(A

+

2p)x, + x24(‘) n,u’

(3.6)

On the basis of (3.5) and (3.6), we conclude that v(l) is an equilibrium displacement field which satisfies the conditions (3.1), (3.2). Let us consider now the vector field v(‘), defined by (2) -

u(i2) = 11i2’(x,, x2),

v2

where II(,) are the components

-

-4x:

+

lC(zz’(X,

) x2),

(2) u3

-

x2x3,

E

l?,

(3.7)

of the displacement vector in the plane strain problem t$$ = AeLf&, + 2pe(,j, t$-$ + (Xx,),, = 0

@no = -Ax2n, In the case of homogeneous

x

on L.

isotropic bodies, we obtain

in Z, (3.8)

On generalized Saint-Venant’s problems

853

The stress tensor associated with vc2)is

s,&‘*‘) = Ax&3 +

s,3(v(*‘)= 0,

t’aj,

S&(2)) = (A + 2/A)x2+ M2’a.a*

(3.9)

With the help of (3.8) and (3.9) we conclude that v(‘) 1s . an equilibrium displacement field which satisfies the conditions (3. I), (3.2). We define the field vC3)by V’l”= z4(13)(x, ) .x2), where uf’ are the components

21i3)=

up) = u$3’(xl ) x2),

X3)

x E

B,

(3.10)

of the displacement vector in the plane strain problem

The solution of the plane strain problem (3.1 I), in the case of homogeneous bodies, is t((3) = o(

x

-

2(X + /J) x0.

From (3.10) we obtain

Sas(d3’)=

X6,, + tg,

S,,(v(39

= 0

s33(v’3’)

3

=

x +

2p + XUC3) a,01 9

(3.12)

so that, by (3.11) and (3.12) we conclude that vc3)is an equilibrium displacement field which satisfies the conditions (3. I), (3.2). Finally, we define the field vt4)on B by (4) VI

(4) *2

-x2x3,

XIX39

UP

where cp is the solution of the boundary-value

(W.J., = Q3WsL

= dx,

9 x2),

B,

(3.13)

on L.

(3.14)

x E

problem

in Z,

P,cA = C@x+?%

Here crrgis the 2-dimensional alternating symbol. It is easy to see that

sap(v(4)) = 0,

&3(v(49

= P(cp,,

-

Q3%),

S33(v’4’)

= 0.

(3.15)

By using (3.14) and (3.15) it follows that vC4)is an equilibrium displacement field which satisfies (3. l), (3.2). The necessary and sufficient conditions for the existence of the solution are satisfied for each plane boundary-value problem (3.8), (3.1 I), (3.14). In what follows we assume that u!$ (s = n) and cpare known. Let us note that the vectors vu) depend only on the domain Z and the elasticity tensor. Let u be any equilibrium displacement field which satisfies (3.1) and (3.2). Let us establish the form of the system (2.8). First, by (2.4) and (2.6), we obtain

(u, v(I)) = s

aB

s(u) - v”‘da = h s

x,S’33(u)da = -hM2(u). 22

(3.16)

Then, by (2.7) we find (II’ d’)) =

s as

s(v(‘)). uda = E,(u)

”

(3.17)

854

D. IESAN

where

E,(u) =

s 22

[(A + 2~)x, + Xz&!&jda -

s SI

[(A + 21*)x1+ x,ubf,b]tl,da.

(3.18)

In a similar manner we get

(u, v(Z))= hM,(u),

(u,

v(3))

=

(u,

@3(u),

v(4))

=

hM3(u).

(3.19)

On the other hand, we have (u, v@)) = Es(u),

(3.20)

(s = 1,),

where

E,(u) =

s

[(A + 2j~)x, + Aug;]u3da -

s ZI

x2

E3(u)

=

2~ + X$;)u3da

(A

+

+

u2@,2

-

s ZI

s z2

E4(u)

PL[u,((P,,

=

-

x2)

+

s 22

xdlda -

s 21

[(A + 2/~)x, + XuK;]u,da, (A + 2,.~+ Xu(3))usda, P,P

duds,, - x2) + u2k2 + dida.

(3.21)

By the same procedure we obtain (,(O,

,W)

s

=

S( V(‘))

*

v”‘da = hH;j)

(i,j=

1,4)

(3.22)

JB

where Haa =

Ha3

sZ

= s

H33 =

HM =

Z

sZ s

+ 2p)xa + Xu$]du,

x&I

xa[X

+

2~ + X@;]da,

[A + 2~ + Xu’3’]da P.P 7

H4i = 0, (3.23)

z (Q-W,B + x,xJda.

Taking into account (3.16, 3.19, 3.22 and 3.23) the system (2.8) becomes

H,wi = w%(u), H3iai

=

F3(u),

H44a4

=

M3(u).

(3.24)

Let us note that ‘Hij (i, j = I, 4) are quantities depending only on the cross section and the elasticity field. Thus, the formulae ( 1.1) apply to the generalized extension-bending-torsion problem. In the case of homogeneous bodies we have H,,

= E

sz

X&&I

= EI,,,

Ho3 = EAxo,,

H33 = EA,

where E is Young’s modulus, A is the area of the cross section and (x?, xs, h) are the coordinates of the centroid of Z2. In this case the system (3.24) becomes

On generalized Saint-Venant’s problems

855

mq9q.3+ ‘eb3) = q%&?m, EA(a,xy + (Yzxq+ cq) = FJ(U), H44a4

=

(3.25)

M3(u).

Comparing (3.24) with (I. I) it follows that for inhomogeneous and isotropic solids we find D,= H,(i,j= 1,4)andXg=ag,c=a3,~=a4. If we use for (u, v(I)) (i = 1, 4), the expressions from (3.17) (3.21), then the system (2.8) becomes H~~xB+ Hi3&= k Ei(u),

(3.26a)

Hu7 = ; E4(u).

(3.26b)

In this case the extension and the bending are coupled problems. The torsion problem can be studied independently. The relation (3.26b) defines the generalized twist associated with any solution of the torsion problem as the global measure of strain appropriate to torsion. If the body is homogeneous then Day’s solution is recovered. From (3.26a) we obtain xl, x2 and c corresponding to any solution of the extension and bending problem. If the body is homogeneous and the X, axes are principal centroidal axes of the terminal cross section X2, then Podio-Guidugli’s solutions are recovered. In this case (vu), v”‘) = 0 if i f j. The classical Saint-Venant problem for inhomogeneous and isotropic elastic bodies has been solved in [8]. (ii) Anisotropic elastic bodies. We consider an inhomogeneous and anisotropic medium for which c = C(x,,

x2),

(Xl, x2) E

z.

will have occasion to use some results concerning the generalized plane strain problem [6, 71. We define the state of generalized plane strain of the considered cylinder to be that state in which the components of the displacement vector depend only on xl and x2, i.e. We

From the stress-displacement

relation (2.2) we have (3.27)

The equations of equilibrium, with the body forcesfi‘ take the form

amla

+.A = 0,

(3.28)

from which it follows that the components of body force be independent of x3. Let us assume that on the lateral surface of the cylinder we have the boundary conditions (3.29) Obviously, the surface traction must be independent of x3. The generalized plane strain problem consists in the solving of equations (3.27), (3.28) in Z with the boundary conditions (3.29) on L. The stress S3, can be calculated after the displacement components are determined. We assume now that the functions and pi belong to C”. It is known [7, 1 l] that the boundary-value problem (3.27)-(3.29) has solutions belonging to ?(Z) if and only if the following conditions hold Cijk,,fi

(3.30)

D. IESAN

856

We introduce four special problems PCS)(S = 1, 4) of generalized plane strain. In what follows we denote by z$) and t0’ the components of the displacement vector and the comThe problems PCs) are characterized by ponents of the stress tensor from the problem P ('). the equations t!“’ ,a = C.rak@u(s) k,@, &!a

+

(G,33Xp),a t’3’ + u,,a

t!ZL

-

(P =

= 0, c.

1, a,

rp33,p = 0,

&pp(Cia&Q,a

(3.3 1)

in 2,

= 0

and the boundary conditions

t’,4’n,= -Cia3&#Za t’!‘n 011

a

&h

=

-C.

=

q3Gup3-yb

ru33

n

(P = 1, 3,

00

(3.32)

on L.

It is easy to show that the necessary and sufficient conditions (3.30) for the existence of the solution of the problem P(') are satisfied for each boundary-value problem. In what follows we assume that the functions z$’ are known. We define the vector fields vcS)(S = 1, 4) on I? by 1)(O)= _L-&

a

a

where?’

Vi3’

= X36i3

+

vp

= &paX~X3

B + U(B) 01

(8) -

V3

7

x3xj3

+

uy)

(P

=

1,2),

Uf’, +

(4) = V3

up,

are the components

ucp,

(3.33)

of the displacement

vector

from the problem

PCs)

(s= 1,4). From (2. I), (2.2) and (3.33) we obtain t”)lJ

LS;j(V’“‘)

=

Clj33Xa

s..(v'3') U

=

c,,,

S;j(d4))

=

t!J”’ - Cija3&&X#.

+

)

+ &!) U’

(3.34)

On the basis of (3.31) and (3.32) we find that v(‘), (S = 1, 4), are equilibrium displacement fields which satisfy the conditions (3.1) and (3.2) (see also [7]). Let II be any equilibrium displacement field which satisfies (3.1), (3.2). By (2.4), (2.6) and (3.33) we obtain (u, v’“‘) =

s

s(u) . v’O”da= hc,~I4~(u) 3

(II, v(3)) = hdL4(4,

(u, vy

=

(3.35)

hM,(u).

On the other hand, by (2.7) we find (II,

Vu))

=

S,,S(V')). uda

=

Ej(U),

(j

=

1,

where

E3(u)

=

s

z2

[C/333+ tl:‘]u;da -

s ZI

[C,,,, + t;;‘]uida,

4),

(3.36)

On generalized Saint-Venant’s problems

857

(3.37) In a similar manner we get (Vci),

V(j))

=

(3.38)

(i, j = 1,),

hLij,

where

L3

La4

=

xJC3333

+

s

2

s

z xa[C33,3q3px,3

=

~533

=

sr

LC3333

L34

=

sz

[C33o13qs~p

L44

=

z

s

+

@Ida, +

d‘!iW

$ild~, +

&kh

~,~x,[C,~~~~~AJA

+

$%a.

In view of (3.35) and (3.38) the system (2.8) becomes 4 c

L&f,

=

%BMp(U),

L3s%

=

F3(u),

L4sas

=

M3(u).

_V=l 4 c s=l

ii

(3.39)

s=l Note that L,s (r, s = 1, 4) are quantities depending only on the cross section and the elasticity tensor. Thus, the formulae ( 1.1) apply to the generalized torsion problem for anisotropic bodies. In the case of homogeneous and isotropic bodies the system (3.39) reduces to (3.25). Comparing (3.39) with (1.1) we find D, = L,s, and x, = (Y,,E = (Ye,7 = (Ye. If we use for (u, v(j)) (j = 1, 4) the expressions from (3.36), then the system (2.8) becomes Lip%, + Lij& + Lid7 = i E;(U),

(i =

The system (3.40) defines x,, E, T associated with any solution torsion as global measures of strain. Thus Truesdell’s problem tropic bodies. The classical Saint-Venant’s problem for inhomogeneous been solved in [7]. The method given in [7] leads to a new problem even for homogeneous elastic bodies.

1, 4).

(3.40)

u of the extension-bendingcan be solved also for anisoand anisotropic bodies has treatment of Saint-Venant’s

REFERENCES

[ 11 W. A. DAY, Generalized torsion: The solution of a problem of Truesdell’s. Arch. Ration. Mech. Anal. 76, 283 (1981). [2] C. TRUESDELL, The rational mechanics of materials-past,

present, future. Appl. Mech. Rev. 12, 75 (1959).

D. IESAN [3] C. TRUESDELL, The rational mechanics of materials-past, present, future (corrected and modified reprint of [2]), pp. 225-236 of Applied Mechanics Surveys. Spartan Books, Washington, DC (1966). [4] C. TRUESDELL, Some challenges offered to ana&sis by rational thermomechanics, in Contemporary Davelopments in Continuum Mechanics and Partial D@erential Equations (Edited by G. M. de la Penha and L. A. Medeiros), pp. 495-603. Nosh-Holland, New York ( 1978). [5] P. ~DIO-GUIDUGL~, St. Venant formulae for generalized St. Venant problems. Arc/r. Ration. Me& Anal. 81, 13 (1983). [6] S. G. LEKHNITSKII, Theory @Elasticity ofan Anisotropic Elustic Body. Holden-Day, San Francisco (1963). [7] D. IESAN, Saint-Venant’s problem for inhomogeneous and anisotropic elastic bodies. J. Elusticit!: 6, 277 (1976). [S] D. IESAN, Saint-Venant’s problem for inhomogeneous bodies. Inr. J. Engng Sci. 14, 353 (1976). [9] S. G. MIKHLIN, The Problem ojthe Minimum ofa Quadratic Functional. Holden-Day, San Francisco (1965). [ 101 M. E. GURTIN, The linear Theory o~~~a.~t~~ity,.in Vol. Vi a/2 ~andbuctz der Ph~~sik,(Edited by C. Truesdell), pp. l-295. Springer-Verlag, Berlin (1972). [ 111G. FICHERA, Existence Theorems in Elasticityain Vol. VI a/2 of Handbuch der Physik (Edited by C. Truesdell). Springer-Verlag, Berlin (1972). (Received

I1 March 1985)