A transient creep solution for uniaxial tension rectangular thin plates with central circular holes

NUCLEAR ENGINEERING AND DESIGN 32 (1975) 325-336. © NORTH-HOLLAND PUBLISHING COMPANY

A TRANSIENT

CREEP SOLUTION FOR UNIAXIAL TENSION RECTANGULAR

THIN

PLATES WITH CENTRAL CIRCULAR HOLES T. HATA

Faculty of Education, Shizuoka University, Shizuoka City. Japan Received 16 January 1975 A general method for the solution of non-linear creep problems is presented. This method reduces the creep problem to a sequence of elastic problems with initial strains. The solution of the elastic problem with initial strains is determined using four displacement functions together with the point-matching method. The method is based on the extension of Goodier's concept of reducing a thermoelastic problem to one at constant temperature with no body force. Using this method, creep behaviour in the vicinity of a circular hole in a uniaxial tension thin rectangular plate has been investigated for a time-hardening material law. The results show how stress concentration factors vary with time for various combinations of geometrical shape and stress index.

1. Introduction Recently, creep analysis has undergone marked development in connection with important problems arising during the design of steam and gas turbines, jet motors, nuclear reactors and so on. Some high-temperature materials have also been developed. Thus, it has become very important to analyse creep problems in equipment under nonstationary creep conditions. To determine the creep stresses, the following consideration of Reissner [1 ] may be effective. Imagine a body subdivided into small elements and suppose that each of these elements undergoes a certain permanent plastic strain or change in shape produced by metallurgical transformation. Let this deformation be defined by the permanent strain components e~ ( i , / = 1, 2, 3) which do not satisfy the compatibility conditions. It is assumed that these strain components are small and represented by continuous functions of the coordinates. If the elements into which the body is subdivided fit each other after the permanent t! set ei/, there will be initial stresses produced. Reissner [1] showed that the problem of determining these initial stresses'is reduced to the usual system o f equations of the theory of elasticity in which the magnitudes of the fictitious body and surface forces are completely determined. Recently, the author [2] discussed this stress problem and reduced it to a boundary-value problem similar to the well-known thermoelastic problem. Then, adopting Goodier's concept [3] of reducing a thermoelastic problem to one at constant temperature with no body force, the inelastic stresses are determined by using four displacement functions similar to the Neuber-Papkovitch stress functions [4]. The creep solution of large thin plates with central circular holes subjected to edge tractions can be found in refs [5] and [6]. In these studies, the finite difference and finite element methods were employed. But the studies were not concerned with the creep solution in the rectangular plate with a large hole. In this paper, using the method of the four displacement functions together with the point-matching method, a transient creep solution for uniaxial tension rectangular thin plates with large centre circular holes is obtained. The first purpose of this analysis is to show the direct application of the theory of elasticity to the creep problem; the second purpose is to elucidate the effect of various combinations of geometric shape and stress index on the variation of stress concentration through creep.

326

T. Hata, Transient creep solution

2. Formulation of problem The constitutive and field equations for a solid with inelastic deformation are presented in the rectangular Cartesian coordinate xi as follows. Solution of the problem requires the determination of the stress components oii (i, / = 1,2, 3) and strain components eli in region D with boundary/L The stress-strain relations are expressed as ¢

(1)

oq = ?~Siie' + 2Geii, r

t

where eii, ~ and G are the elastic strain components and Lam6's constants respectively; e' = eii , where the repetition of subscripts in the term denotes the summation from 1 to 3; and 6ii is the Kronecker delta. At any time, the total strains eii consist of a sum of two components: t

tP

(2)

eij = eij + eij, t!

where eli are the inelastic strain components. The equilibrium conditions are written as oii,i = 0.

(3)

Using the following strain-displacement relations and eqs (1)-(3), (4)

ei i = l(ui, i + ui,i),

the displacement equations of equilibrium are 1

Aui + 1 -- 2u e'i

--

2v tt tr 2eii'i - 1 - 2u e'i = 0,

(5)

where u is Poisson's ratio. The particular displacement ui may be represented by up. Here we introduce the displacement functions ~boP and ~I,F in the forms uP = 4(1 - v ) 4 p - (~P + xicbP), i.

(6)

Substituting from eqs (6) into eqs (5), we know that the system of equations (5) is satisfied if AqbP° -

1

,,

1 ,,.

2(1 -- u) xi(eij'i - ~e'i)

1 l+v

,,

3 1 -- u e ,

A~bP - 2(1

1

,,

1 ,,.

(7)

v) (eq'l - ~e'i)"

Then, the associated stresses may be given by a~=ZG[2vSiie~,k

+ 2(1-v)(¢~,i+q~i)-(,l,~

+xk,l,P),ii ] - 2 G

( e i"i + - ~2 l ~ v

hiie") .

(8)

For a bounded body with inelastic properties, it will, in general, be found that the solutions qbP o and qbP give nonzero boundary tractions on the surface of the solid. The effects of removing these are found by solving an ordinary stress problem, using the following homogeneous equations: G A u i + (X

+ G ) U k , ki

= O.

(9)

327

T. Hata, Transient creep solution

In cylindrical coordinates (r, 0, z) in a state of plane stress eqs (7) with e" = 0 are represented as AsP=

l¢

t¢

l +v 1 /3(er - e ~ ) +_l /oero e r - - e o 2 r 2 ~or r /30 + r

tt

]

1 + v .... (er + eo), -- ~

¢¢

AS p

t¢]

/ s P + 2 / 3 S p] l + v [ 1 / 3 ( e r ' - e ; ) + l /oe;o+ er r eo , -k7 ~-~-]=--2--[ 2 /3r 7 /30

AS0P_(S0P k r2

(lO)

'. . . . 2 ] 2/OSrP1 = l + v [ 1 /3(e~'--er)+~_rO+rerO r 2 - ~ ] ---2-[ 2 r/30 ]'

where /32 1 /3 1 /32 A=/3r --~ + -+- - - r / or2/302 r ,

4

/3

4

u = --~1v SrP - ~rr (Sop + rSrP),

/3

v = 1 + v S~ - ~

(Sop + rSrP).

(11)

The associated stresses in cylindrical coordinates are 2G [ 4 (/OSrP vsrp+v /3s_PI_[/32 v /3 v /32~ ]-~(er aPr(r'O)=~--v -f~v\--~-r + r 7 /30 ] k~--r2+r~+~o-~J(s°P+rSD 2G [ 4 o~a(r, 0) = -1- - ~ [ 1--~v

( /os

1/3s 1 (

~-T/r +7 +r - - ~ - ] -

[ 2 {±as,. /30 + ~or

rPr°(r'O)=2G[l + v k r

1/3 + -~1 ~/3]2 t (sP + rSPr) ] -

~ ~ + r-/3-;

+

veo), "

2G (eg + Ver)'

1- v

(12)

(1_°_!r ~or/301.l(sp + r S P)I - 2Gero."

SPr + ~ /30

3. Elastic stress

Consider a rectangular plate containing a circular hole as shown in fig. 1. The boundary conditions for the loading are given by OrE = 0 ,

rE =0

on r =a,

oE = p ,

TEy--0

onx=b,

oE=0,

rEy=0

o n y =¢.

(13) (14)

To obtain the elastic stress we initially use the Airy stress function in solving v4× = 0.

(15)

The function X is the series form as X = ao log r + b 0r2 + ~

(a2m r 2m + b2m r 2ra ÷2 + a'2rar -2m + b'2,n r-2m +2) cos (2toO).

m=l

t

I

The constants a o, bo, a2m, b2m, a2m and b2m are now determined from the boundary conditions (13) and (14).

T. Hata, Transient creep solution

328

P

P

2c

2b

Fig. 1. Representation of a hole contained in a rectangular plate in uniaxial tension.

The corresponding stresses are

oEr = ao r-2 + 2bo +

[ - 2 m ( 2 m - 1)a2mr 2 m - 2 -

~

(2m + 1)(2m - 2)b2m r 2 m - 2 m ( 2 m + 1)a'2m r - 2 m - 2

m=I

- (2m - 1)(2m + 2)b'2m r-2m] c o s ( 2 m 0 ) ,

o E = - a o r-2 + 2b o +

~

[2m(2m - 1)a2mr 2m- 2 + (2m + 1)(2m

+

2)b2m r2m + 2 m ( 2 m +

1)a'2mr - 2 m - 2

m=l

+ (2m - 2 ) ( 2 m - 1)b'2m r-2rn ] c o s ( 2 m 0 ) , ~'E0 =

~

(16)

[2m(2m -- 1)a2mr2m-2+ 2 m ( 2 m + 1 ) b 2 m r 2 m - 2 m ( 2 m + 1)a'2mr - 2 m - 2

rn=l

- 2 m ( 2 m - 1)b'2m r-2m] sin (2m0). Since eqs (13) must hold for all values o f 0, the coefficients satisfy the conditions ao = - 2bo a2,

t

a2m = [b2ma 4m+2 - (2m - 1)b'2ma2]/2m,

a2m = [ - ( 2 m + 1)b2ma 2 - b'2ma-4m+2]/2m. The stresses crE, o E and ~'xEyare obtained f r o m eqs (16) as follows:

o E = ao r-2 cos 20 + 2b o +

~

[cos 2(m - 1)O(-2m(2m - 1)a2mr 2rn-2 - 2 m ( 2 m + 1)b2mr 2m}

m=l

+ cos 2(m + 1)O{-2m(2m + 1)a'2m r-2m - 2 _ 2 m ( 2 m - 1)b'2m r-2m) + cos 2too (2(2m + 1)b2m r2m - 2(2m - 1)b'2mr-2m}],

(17)

T. Hata, Transient creep solution OEy = - a o r-2 cos 20 + 2b o + ~

329

[cos 2(rn - l)O(2m(2m - 1)a2mr 2m-2 + 2m(2m + 1)b2mr 2m}

(18)

m=l

+ cos 2(m + 1)O(2m(2m + 1)a'2m r-2m-2 + 2m(2m -- 1)b'2m r-2m} + cos 2mO{2(2m + 1)b2mr 2m - 2(2m - 1)b'2mr-2m}], [sin 2(m + 1)0 (--2m(2m + 1)a'2m r ' 2 m - 2 -- 2m(2m -- 1)b'2mr -2m)

r E =ao r-2 sin 20 + ~ m=l

+ sin 2(m - 1)O(2m(2m - 1)a2m r2m-2 + 2m(2m + 1)b2mr2m)]. Finally, the remaining coefficients must be determined by the requirement that the stresses of eqs (18) satisfy the boundary conditions (14)• The evaluation of the remaining coefficients is determined by the point-matching method. We replace the summation Z m : l by the summation y M= ~ and put points on the external boundary so that we may determine all the coefficients.

4. Solution of creep-stress problem In this study the following time hardening uniaxial creep law will be used: • tt

e e = (oe/oc)nf(t),

(19)

where o c and n are material constants, f(t) is a time function and d~' is the effective creep strain-rate. If we extend the resulting relations into the multiaxial stress state by means of creep theory of the yon Mises type, the following relations are obtained:

\oo

Or--2

'

Aeo =

--

\oe/

O0

-

-

Y '

3/Aee~r ~ Ae'r'O : -~ ~ o--~] m,

(2o) ,, [Aeet{°r 2) Aez = - ~-~e ] ~ ~ + ,

where Aee- = (Oe/oc)nf(t) At,

~ 2 11/2 0e = [02 -- OrO0 + 0 2 + JrrO 1 .

(21)

It will be recalled from section 2 that the determination of the desired creep deformations and stresses necessitates the construction of the particular solutions q~o,Pq~rPand 4~ which satisfy eqs (10). These solutions may be obtained by expanding the strains in Fourier series. As the stress distribution in the problem is symmetric about the x and y axes, creep strains can be expanded in the following forms:

e; =fro(r) + ~ m=l

frm(r)

cos

2mO,

e~ =foo(r) + ~ m=l

fore(r) cos 2mO,

e'r'o= ~ m=l

from(r) sin 2mO,

(22)

T. Hata, Transient creep solution

330

where M2

~12

rrl2

e r dO,

= --

4

frm (r) = ~- f er cos 2mO dO,

eo dO,

foo(r) = --

0

0

0

(23)

"rr/2 4~r f fore(r) = --

eo. cos 2mO dO,

f,,

from (r) = 4

ero sin 2mO dO.

0

0

Substituting eqs (22) into eqs (10) the particular solutions of eqs (10) may be obtained as follows: .P = +Pro(r)+ ~ di'Prm(r)cos 2mO,

rbPo= ~ ~i'~m(r)sin 2mO,

m=l

m=l

*Po =+go( r) + Z

(24)

dPPm(r)cos 2mO.

m=l

Finally, the substitution of eqs (24) into eqs (12) yields the creep stresses: o[(r,O)=ZG[Ao(r)+ ~ A2m(r) cos2mO], rPro(,, O) = 2G

~ m=l

B2m(r ) cos 2mO],

o~(r,O)=ZG[Bo(r)+

m=l

m =1

(25)

Czm(r) sin 2mO.

Here, the functions A o(r), Bo(r), A 2re(r), B2m(r) and C2m(r) are given by Ao(r) = -l -+7u-f l ~ [fro(~) - foo(~)] d~ -

2

a

-- [.fro(~) +foo(~)] d~ - ~ - -

1 + l~u

a

× [go(a) - foo(a)], r

Bo(r) = -l -+~u f l -~ [f~o(~) - foo(~)] d~ +--2- j ~ [fro(~) +foo(~)] d~ - - ~ -

1

l +v \ r] J

a

a

x [3').o(a)-foo(a)] - (1 + v)foo(r),

A2m(') --~'j

r

(1 +v)(1 - m )

+ {2 +m(1 +v)}

[~2,n(~) +32m(~)] d~

a

+~-

(1 +v)(l +m)~r )

+ { 2 - m ( 1 + v)}~-)

][~zm(~)-32m(~)] d~

a

_.~ i [(2m [r\2m-2 1 -1)[~-)---(2m+l)(r) a

~

2m+2

] 72~(~-~)d~

_

[ f r m (r) -- fore ( r ) ]

2

(26)

331

T. Hata, Transient creep s o l u t i o n

B2m(r) = -~l

2m_ (2 + (1 +v)m}t r J

(1 + v)(m + 1)

J [a2m(~) +/32m(~)] d~

a

+-~

(1 +v)(1 - m )

-~

(1 - 2m) ~ - )

Cz,n(r) = 1

1 f [

4 a

(1 + v)m

1 r

(.~r) 2m

7~--~(~--~)d~-½[~frm(r)+(v+2)fom(r)],

+(1 +2m)

(1 +v)m

a

+ --

(2-(1 +

+ {2 +m(, + v ) ) t r J

- (2 - m(1 + v))

[ r~2m-2

(~)2m-2]

J [a2m(~)+l~m(~)]

[ot2rn(~) -/32m(~)] d~

2m+2

a where

1 a

1)

O~2m(r) = 5 ~r + --r [frm(r) --fore(r)] + 2mfr°m(r)'r

~2m(r) = _ m [fore(r) -- frm(r)]

r

(27) l+v 72re(r) = -- ~ [rOt2m(r) + frm(r) + fore(r)].

+

Since the creep stresses of eqs (25) do not satisfy the boundary conditions of zero boundary tractions on the surface, we employ the biharmonic stress function in plane stress state. The boundary conditions for the stress function are Ors = _ o r

p,

rSrO =-rrPo

onr=a,

{Ts = _ ffp,

TxyS

on x = b,

Oys

py T sx y = --7" x

=_oyP

,

= - - TPxy

o n y = C.

(28) (29)

Considering the boundary conditions, the stress function may be given by Xs = a~ log r + bSor2 + ~

m=l

Constants

a~, bo, s a2m, b~m, a2m s' s

(aS2mr2m+ u2mr~.S 2m+2-1.a2mrS' -2m + bS2~r-2m+2) cos (2m0). s' and b2m may be determined from eqs (28) and (29).

(30)

T. Hata, Transient creep s o l u t i o n

332

The corresponding stresses are

o s = aSor-2 + 2bSo + ~

[ - 2 m ( Z m - 1)aS2mr2m-2 _ (2m + 1)(2m - 2)bS2m r2m - 2m(Zm + 1)a~'mr -2m-2

rn=l

- ( 2 m - 1)(2m + 2)bS2'mr-2m ] cos(2m0),

oSo =-aSo r-2 + 2bSo + ~

[2m(2m - 1)aSzmr2 m - 2 + (2m + 1)(2m + 2)bS2mr 2m + 2m(2m + l)aS2'mr- 2 m - 2

rn=l

+ (2m - 2)(2m ~r0

~

=

(31)

s, -2m ] cos(2mO), 1)b2,nr

-

[2rn(2m

1)a s2 r n r2m-2 + 2m(2m + 1)bS2mr2 m - 2m(2m + 1~ lja2mr s p - 2 m - 2 _ 2m(2m -- l)b2mrS' -2~n]

--

m=l

x sin (2toO). Since eqs (28) must hold for all values of O, the coefficients must satisfy the conditions

aSo = -2bSoa 2 - 2Ga2A o(a), s, _bS2ma4m+2-( 2m a2m -

p b S ' a2 -

) 2m_+2 G

[A2m(a)+C2m(a)] azm+2

2m

-(2m+

a~rn -

4m(2m + 1)

1)bS2ma2 - O2ma - s ' -4m+2 2m

+ 2G

(32)

'

[A2,n(a) - C2m(a)]a -2m+2 4m (2m - 1)

The equations for the remaining coefficients are given from eqs (29) and (3 l ) by

aSor-2 cos 20 + 2b~ +

~

[cos 2(m - 1)0 {-2rn (2m - 1)a~mr 2rn-2 _ 2m(2m + lab s ) 2mr 2rna;

rn=l

, s' - 2 m - 2 - 2m(2m - 1")o2,nr ' s ' -2m, + c o s 2 m O { 2 ( 2 m + 1)b~rnr 2m + cos 2(m + 1)O{-2m(2m + •l)a2mr - 2(2m - 1 ) b 2 m s, r -2m }] = - a x p

-aSor -2 cos 20 + 2b~ + ~

onx=b,

[cos 2(m - 1)O{2m(2m - 1)aS2mr2m-2 + 2m(2m + 1)bS2mrzm}

m=l

+ cos 2(m + 1)O(2m(2rn + 1)aS2'mr- 2 m - 2 + 2m(2m - 1)bS2mr-2m} + cos 2mO{2(2m + 1)bS2mr2m - 2(2m - 1)bS2mr-2m}] = - O yp

aSor-2

sin

20 +

~

ony =c,

[sin 2(m + 1)O(-2m(2m + 1)aS2~r - 2 m - 2

(33)

--

2m(2rn -- la~s' . I w 2 m r -2rn~

rn=l

+sin2(m-1)0{2m(2m-

1)aS2mr2 m - 2 + 2 m ( 2 m + l ) b ~ m r 2 m } ]

=-rxPy

onx=b

and

y=c.

We use the point-matching method to satisfy eqs (33) at a selected finite set of outer boundary points of rectangular region. Replacing £~n= 1 in eqs (33) by £m-M 1 approximately and solving the sets of simultaneous linear equations, we can determine the remaining coefficients. Finally, the stress field may be determined as follows. At zero time, the values of elastic stresses are known from eqs (16) with the coefficients of eqs (14), (17) and (18). If the stresses are allowed to redistribute for an

T. Hata, Transient creep solution

4.0 -

r'ELAST I C ~

-t-lo = - ~

a.o

333

f

-1"-, o' -~ ~ ' - . V ~ / ~ ' ' ' "

2.0

1.0

! /

,

,O

-I°0

-2.0

Fig. 2. Variation of a 0 at the edge of a hole for a plate (b/a = 3, n = 5).

2.0

1.0

Y/a

01-0

1.5

2.0

2.5

3.0

I:ig. 3. Stress distributions along the y axis of a hole contained in a plate (b/a = 3, n = 5). i n c r e m e n t o f t i m e A t at t h e rate calculated f r o m eqs ( 2 0 ) , using eqs ( 2 5 ) a n d ( 3 1 ) w i t h eqs ( 3 2 ) a n d ( 3 3 ) , t h e stress d i s t r i b u t i o n a f t e r this t i m e interval is

(oii)t=At = (o~)t=o + (o~I + os/)At.

(34)

A t t = A t t h e stresses are n o w k n o w n so t h a t n e w stress rates can b e d e t e r m i n e d . This p e r m i t s n e w stresses to be calculated a f t e r a n o t h e r t i m e interval has elapsed. In general, we have (o'i/)t + A , =

(aii)t + (o~. + osj)At.

(35)

T. Hata, Transient creep solution

334

4"0

3.5

n-5

~/P 3-0 b~ -6 / n .5

b/a= oo

n-5

2.5

2.0

1.5

I

I

I

I 02

o

I

I 03

04

0s

f~ Fig. 4. Variation o f stress

oo(r = a,

0 = 90 °) w i t h time.

4.0

2.0

/

/

/

3.0

/.

/ ~j,S

J

s

,s

I.o

i r 1 I

0

0"2

0"6

0"4

0"8

I'0

±

rl

Fig. 5. Variation o f stationary state stress plate (b/a = ~,) f r o m ref. [5].

This step-by-step

method

I(Oe)at/(Oe)t+At

oo(r = a,

0 = 90 °) with stress e x p o n e n t . - -

is u s e d in t h e p a p e r . T h e t i m e i n c r e m e n t

Imax < e2,

w h e r e w e set e 2 = 0 . 0 5 in t h i s p a p e r .

is for a plate

(b/a =

3); and . . . . . . . is for a

was selected from

(36)

T. Hata, Transient creep solution

335

5. Numerical results and discussion A numerical example was evaluated for a square plate, b --- c, with a hole. The non-dimensional time variable t* =

1 +r,

Ar =--E

t)At

x 10 s,

(37)

P was used in the presentation of results. Poisson's ratio was selected as v = 0.30. An upper index o f m = 8 was used in the Fourier series of equations (25). In the evaluation of coefficients by the point-matching method in eqs (18), (14) and (33), we replaced Nm= 1 by Nsm=I. The numerical integrations were carried out using Simpson's formula. (The computations described in the foregoing sections were performed with the aid of a FACOM 230-60 electronic computer at Nagoya University.) The results of the numerical evaluation of stress variation are illustrated in figs 2 - 5 . The variation of stress o0 at the edges of the circular holes with time is shown in fig. 2 for the case n = 5, b/a ; 3. The spatial variations of the tangential stresses along the y axis are shown in fig. 3 for n ; 5, b/a = 3. In these figures the stresses are significantly changed from the elastic stress distributions (t* = 0) with time. Fig. 4 shows how the tangential stress at the edge o f the circular hole and on the y axis diminishes with time. The greater the creep exponent n the faster the stress redistribution process in achieving a stationary state of stress for the case b/a = 3. Fig. 4 also shows the shape effect for the creep exponent n = 5. In fig. 4 it can be seen that the stress for b/a ; 3 decreases faster to the final value than do the stresses for b/a = 5 and b/a = ~. Stationary state values for o0 at the edge of the hole and at t h e y axis for the case b/a = 3 verify the approximately linear relationship with 1/n shown in fig. 5. Stationary stress for b/a -- ~ obtained by Hayhurst [5] is shown in the figure.

6. Conclusions This paper is concerned with a method for calculating stress distribution in a medium which undergoes elastic and inelastic deformation. The particular solution of the inelastic strain problem has been determined using four displacement functions. For a bounded body, the effect of removing the tractions on the surface will be found, in general, by solving an ordinary stress problem by means of the point-matching method. The method is adequate for obtaining the stress distribution in any shape of plate with a hole. For the particular structure element under consideration, the stationary values of ao at the edge of the hole and at t h e y axis is almost a linear function of 1In and is sensitive to the shape of a plate.

Acknowledgement The author would like to thank Professor A. Atsumi of Tohoku University for his helpful suggestions in the preparation of this paper.

Notation O~ b, C

o,¢

eli ee, Oe

= geometric constants in fig. 1 = strain and stress tensors = elastic and inelastic strain tensors = effective creep strain rate and stress

T. Ham, Transient creep solution

336

r~ O~z t*

= cylindrical coordinates = non-dimensional time variable = displacement tensor ui ll~ 1) = radial and tangential displacements = Cartesian coordinates xi x,y = rectilinear coordinates 5q = K r o n e c k e r delta X,G = Lam6's constants V = Poisson's ratio Oc, ,7, f ( t ) = creep constants and time hardening f u n c t i o n Or~ 00, "rro = stresses in cylindrical coordinates Ox, Oy, rxy = stresses in rectilinear coordinates 4~Po, ~/P = displacement functions in Cartesian coordinates qbPo, 4~rp, 4%p = displacement functions in cylindrical coordinates × = stress function

Superscripts E p

= refers to elastic c o m p o n e n t = refers to creep c o m p o n e n t .

References [ 1] [2] [3] [4] [5 ]

H. Reissner, Eigenspannungen und Eigenspannungsquellen, ZAMM 11 (1931) 1. T. Hata, Application of Goodier's concept to the inelastic problem, ZAMM 52 (1972) 245. J.N. Goodier, Integration of thermoelastic equations, Phil. Mag. 23 (1937) 1017. H. Neuber, Kerbspannungslehre, 2nd edn, Springer (1958). D.R. Hayhurst, A time hardening transient creep solution for steadily loaded uniaxial tension panels containing circular and elliptical holes under conditions of plane stress, Int. J. Mech. Sci. 14 (1972) 888. [6] I.W. Goodall and E.J. Chubb, Creep of large thin plates with central circular holes subjected to biaxial edge tractions, Nucl. Eng. Des. 12 (1970) 89.