A note on stationary stress distributions during nonlinear creep

MECHANICS RESEARCH COMMUNICATIONS 0093-6413/83 $3.00 + .00

Vol.10(6), 379-384, 1983. Printed in the USA. Copyright (c) Pergamon Press Ltd.

A NOTE ON STATIONARY STRESS DISTRIBUTIONS

J. Hult Division of Solid Mechanics, Chalmers S-412 96 Gothenburg, Sweden

DURING N O N L I N E A R CREEP

University

of Technology,

(Received 17 August 1983; accepted for print 31 October 1983) Introduction

When a load system (surface tractions and/or body forces) is applied to a statically indeterminate structure under nonlinear creep conditions, and then kept constant, the ensuing stress state will be transient in time. The initially arising stresses will redistribute, and the stress state will approach a stationary one, constant in time [i]. The stress profile in such states depends upon the n o n l i n e a r i t y of the creep law, often expressed by means of the creep exponent n in the N o r t o n - B a i l e y creep law. In several cases the important stationary stress quantity, e.g. the effective stress, is remarkably insensitive to n at certain regions in the structure, cf. [2] - [5]. This is i l l u s t r a t e d in Fig. la, which shows the uniaxial stress field in a beam subject to pure bending, and in Fig. ib, which shows the von Mises effective stress field in a thickwalled tube with closed ends subject to internal pressure. The two extreme cases n=l and n=~ correspond to linearly elastic and rigid, ideally plastic behaviour, respectively. Intermediate stress profiles are seen to pass very close to the point where those two stress profiles intersect. It is the purpose of this note to examine that wellknown circumstance in some detail.

,/o 2

n

M

neutral plane inner

surface

a) Beam under bending Fig.

I. Stationary creep stress creep exponent n.

outer

surface

b) Tube under internal pressure fields 379

for different

values of

380

J. HULT

Analysis

As

shown

may

in

[6],

be w r i t t e n

elements under

eq.

in

(2.10),

the

steady

following

form

(e.g.

bisymmetric

beam

torsion,

cylindrical

and

s(r,8)

= N

state

creep

for

under

stress

a number

bending,

spherical

of

profiles

structural

circular

pressure

cylinder

vessels)

fS(r)

b 8

I f

(I)

(p)q(p)dp

a

Here

s denotes

a location be

left

creep

r, w h e r e

out

in

0 =

(integration

following). here

the

The

stress

frequently

effective limits also

stress,

a and

depends

appears

in

b will on

the

the

form

i/n,

(2) follows

which

limiting

0 < ~ < i.

corresponds

parameter,

s (r,l)

at

notation

1 < n < ~

two

a < r < b

e.g.

1/n

a load

The

quantity,

n, w h i c h

the

3 < n < i0, is

the

exponent

justifying

With

a stress

and

stress

f(r)

to and

In m o s t 0.i0

engineering

< O < 0.33.

q(r)

are

shape

The

applications quantity

functions.

profiles

f(r)

= N

N

(3)

If (p)q (p)dp s (r,0)

1

= N

(4)

Sq (p)dp correspond

to

linearly

elastic

(8=0)

behaviour,

respectively.

These

two

profiles

i.e.

c is

stress defined

s(c,l) With

(3) f(c)

It

is of

which

and

(@=i) Note

intersect

and

rigid,

ideally

that

s(r,0)

is

at

r=c

(point

plastic

a constant. P in Fig.

by

= s(c,0) (4)

(5)

follows

from

(5)

= $f(p)q(p)dp Sq (p) dp interest

qhows

how

2b),

to

the

(6)

consider stress

at

the

stress

P depends

ratio on

8,

s(c,8)/s(c,0), i.e.

on

the

creep

STATIONARY STRESS IN CREEP exponent.

With c given by

(6) there follows

381 from

(i)

s(c,8) = [/f(o)~(p)d0]8.[f~(p)dp] I-8 s (c,0) if8 (p)q (p)dp H61der's

inequality

negative

functions

/fI(P)f2(P)dP

[7] states that, fl(r)

and f2(r)

(7)

for sufficiently

smooth non-

in [a,b]

p i/Pl P2 i/P2 ~ [/fl I(p)dp] "[If2 (p)dp]

(8)

where Pl > i, P2 > i, and (9)

i/Pl + i/P2 = 1 Equality holds if a non-zero constant Pl P2 fl (r) ~ A'f2 (r)

in

With fl(r) = fS(r).qS(r), P2 = 1/(1-8),

A exists,

such that

[a,b]

(i0)

f2(r ) = ql-8(r),

the inequality

Pl = 1/8, and

(8) takes the form

Ife (p)q (p)dp < [/q (p)d~] 1-e •[If(0)q(p)dp] 8 and hence

(7) yields

s(c,8)/s(c,O) for all 0<8
> 1

(12)

In the limiting case 8=1, i.e. n=l,

s(c,8)/s(c,0) =i. occurs in

(7) yields

This is the case of the Maxwell material,

no stress redistribution

occurs.

According

to

where

(I0) equality also

(ii) if

f(r).q(r) which requires

E l.q(r)

terminate

(13)

the shape

except at one point.

function q(r)

This corresponds

to be zero everywhere e.g. to the statically

cases of an idealized H cross section in bending,

thinwalled

tube in torsion,

The general conclusion profiles

(ii)

dea

or thinwalled pressure vessels.

is that the intersection

for linearly elastic and rigid,

underes#ima#e8 the stress for stationary

of the stress

ideally plastic behaviour creep as governed by

(i).

382

J. HULT

Numerical cross

calculations,

section

closed

ends

and

for

under

e.g.

for b e n d i n g

a thickwalled

internal

of

tube

pressure,

a beam

with

(diameter

show

the

rectangular

ratio

deviation

2:1)

with

to be

quite

small:

Beam

0 = 1

0.333

0.200

0.143

0.iii

0

_ 1

1.019

1.014

1.011

1.009

1

7

9

- 1

1.018

1.013

1.009

1.008

1

(c,e)

e s (c,0) e interval

3 < n < i0 the

a thinwalled

ablout

5

s(c,0)

s

For

3

s(c,8)

Tube

In the

n = i

tube

deviation

in b e n d i n g

the

is

less

maximum

than

2 percent.

deviation

is

less,

1 percent.

Discussion

r

r

8=I

/P

s a) T r a n s i e n t for f i x e d Fig.

2. C r e e p

In t r a n s i e n t initial They

states,

stress creep

profile

intersect s(d,0,8)

the

r=d

in

stress to

(point

Stationary state, for v a r y i n g @

statically field

indeterminate

s(r,t,@)

a limiting Q in Fig.

changes

stationary 2a),

i.e.

t =~ structure.

from

one

an

s(r,~,8).

d is d e f i n e d

= s(d,~,0)

Numerical

analyses

transient

stress

the

that

sense

fields

s(r,0,8) at

S b)

t = 0 +

(14)

[8],

[9],

profiles

s(d,t,@)

by

have

shown

s(r,t,8)

varies

only

pass

that,

for m a n y

cases,

very

near

the

point

slightly

with

t for

the

Q in

0
STATIONARY STRESS IN CREEP

The expression usefulness

'skeletal

in approximate

existence

[6], eq:s

was coined

analyses

of true skeletal

0 < t, has been proven

From

point'

points,

383

[9] for Q, and its

has been noted where

[i0]. The non-

s(d,t,8) ~ 0 for all

[6].

(2.5)-(2.7),

follows,

with A denoting

a positive

constant

(d,O e) {[Sq(P)do]l-e"[sfl/e ,

(P)dple} I/e

=

-

1

(15)

$f(p)q(p)dp With

fl(r)

= f(r)qS(r),

P2 = i/(1-8), ~(d,0,8) If s(d,t,8)

f2(r)

then follows

is non-oscillating,

The stationary

the

(8) that

to

(17) (12).

cross over point P in Fig.

to those of Q in Fig.

stress

'transient

then

> 1

creep stress

similar

a 'stationary

and

(16)

for all 0 < t in analogy

with

from

Pl = 1/8,

> 0

s(d,t,e)/s(d,~,8)

properties

= ql-%(r),

skeletal stress

may be used in simplified,

point',

skeletal

2a.

It might be termed

marking

point'

approximate

2b has

the difference

Q. The stress

analyses

at P may

of stationary

creep states.

Re fe rence s i. J.Hult, Proc. IUTAM Symposium on Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, Haifa, Israel 1962 (eds. M.Reiner & D.Abir), 352. McMillan, New York (1964) 2. F.K.G.Odqvist & J.Hult, Kriechfestigkeit stoffe. Springer, Berlin (1962) 3. R.K.Penny & D.L.Marriott, London (1971) 4. H.Kraus,

Creep Analysis.

5. J.T.Boyle & J.Spence, London (1983) 6. W.S.Edelstein

Design

for Creep.

Wiley-Interscience,

Stress

& J.Hult,

metallischer

Analysis

Int.J.Solids

Werk-

McGraw-Hill New York

for Creep. Structures,

7. I.S.Gradshteyn & I.M.Ryzhik, Table of integrals, products, 1099. Academic Press, New York (1980)

(UK), (1980)

Butterworths, to be published series

and

384

J. HULT

8. W.J.Goodey, 9. D.L.Marriott 1963-64, Vol i0.

Aircraft

Engineering

30,

170

(1958)

& F.A.Leckie, Inst Mech Engrs London, Proc. 178, Part 3L, 115-125; discussion 169-182

D.L.Marriott, Proc. IUTAM Symposium on Creep in Structures, Gothenburg, Sweden 1970 (ed. J.Hult), 137. Springer, Berlin (1972).