Boundary element formulation for elastoplastic analysis of axisymmetric bodies

Boundary element formulation for

elastoplastic analysis of axisymmetric

bodies

M. Doblare

Catedra de Estructuras, E.T.S.I. Industriales, Polytechnic University of Madrid, Madrid, Spain

J. C. F. Telles and C. A. Brebbia

Department of Civil Engineering, University of Southampton, Southampton S09 5NH, UK

E. Alarcon

Catedra de Estructuras. E.T.S.I. Industriales, Polytechnic University of Madrid, Madrid. Spain

The complete formulation of B.E.M. applied to the analysis of

axisymmetric bodies acting in the plastic range is presented in this paper. The concept of derivative of a singular integral given by Mikhlin has been used in order to calculate the stresses in internal points.

Also a semianalytical approach is proposed to compute the matrix coefficients, presenting the way in which it can be done and the results obtained.

Key words: mathematical model, boundary element formulation

Introduction

The analysis of bodies of revolution has many practical applications in engineering, such as pressure vessels, certain pipes, rotating disks, and many different types of con-tainers, including nuclear pressure vessels. Their importance justifies applying more efficient and accurate methods of analysis, such as boundary elements to them. This paper deals with the elastoplastic analysis of bodies of revolution subject to axisymmetric loads.

Several authors have discussed the application of elasto-plasticity using boundary elements. The more complete formulations for three and two dimensional plane stress-plane strain are found elsewhere,1-4 and the first

applica-tion was due to Ricardella.5 Cruse was the first to present

a formulation for axisymmetric elasticity6 based on the

fundamental solution as presented by Massonet.7 Other

authors8 have solved the axisymmetric potential problem

using boundary elements. In a recent paper9 the

axi-symmetric elastoplastic case was analysed but the complete formulation was not presented.

In addition, internal stresses were calculated using an approximate formulation based on a simple finite difference procedure. By contrast Telles and Brebbia3 have calculated

the stresses at internal points using the proper integral equation. This correct formulation is extended in the present paper to deal with axisymmetric bodies.

The paper starts by formulating the integral equations of elastoplasticity and the corresponding stress tensor

components for axisymmetric cases in the way previously presented for three and two dimensions (plane stress-plane strain) in reference 1. This paper also applies the technique developed by Wrobel and Brebbia8 to integrate the Legendre

function around the singularity for potential axisymmetric problems.

The analytical expressions required to complete the formulation are presented in the Appendix.

General formulation

The Navier equation for the elastoplastic case can be written as:

-

( * '

7 T S « " ) - 6

where iiy- are the components of the displacement rates, G

and v are the shear modulus and Poisson's ratio of the material bj are the body forces rate components and ep

indi-cates the plastic strain rates. The comma indiindi-cates deriva-tives.

•tku,(pc) = ju?kb,(y) dfi - f p&u/OO d r n r

• j « ^ / d r + f s/ / f ce g d n (2)

n

where u* and p* indicate the fundamental solution for displacements and surface tractions. The cik components

given by the Kronecker delta 6,y for internal points are all zero for external points and they have different values for points on the surface. For smooth boundaries for instance they are equal to the Kronecker delta divided by 2, i.e. \b{j.

The stresses are given by:

2Gv

\-2v UiAj+G(u u+uhi)

2Gv

- 2Gef: e£fc5,y

" i _ 2v kk '

The tensor components £//* are as follows:

(3)

y =7G

1 - V dUrl

\r.

^zrr ^rzr

2v dr l-2v\r dUrr bU2

\ dz

J ^ ^ z r

-2v\r

)

Urr + -

_d

~t)}

r l - v duzr v / 1 hu„ \]

2ZZr = 2G\ + \—U„ +——

L l - 2 i > 3z \-2v\r dr !\

\ \-v U„ v (dUrr W„\[

^>6rr ~ ^>r0r ~ ^Bzr ~ ^z6r ~ 0 (4)

\\-vdUrz v I dUzz\]

T,rrz = 2G + \-U„ +

L l - 2 v dr \-7v\r dz /J /dU„ ydUzz\

\ dz dr I

ll-2v dz \-2v \r dr Jl

Y\-2v r \-2v\ dr dz ) \ ^6rz ~ ^rdz : 2fl?j_{zz ~ ^zBz} = 2 0

Finding the derivatives of Somigliana's equation (2) for internal points we have:

§ « " i » = \u?k,mPi d r - \p*k,m"i d r

+ r ^ - {uTkbi dn + — f 2Vkef} da (5) dxm J dxm J

n n

where the derivatives due to the singularity have to be taken into consideration in the way proposed by Mikhlin.10

As indicated by Telles and Brebbia1 the above integrals

can be taken to the limit which for axisymmetric cases will give:

5fc/",-,m = J ufumPi dT - J P*k,mUi d r

+ juT^b, dn + j ^iiKmePij dO.

-efllim f Xiikf,mdr

c-i-0 J

re

where f represents the distance between the field and singular points.

Taking this into consideration expression (3) for stresses now becomes:

6km = J DkmiPi d r - Skmi(ii d r + J Dkmibi dO,

/

n

+ I Sto.^dn-e^Iim j

r 2Gv i

Sffirfikm + G(2I / f crm + 2,/ mr k) d r

ll-2u J

+ 2Cefm +

2 G i >

l - 2 » »

e?,5 _{7;° ton} (6)

In the limit:

akm = JDkmiPi dr - j Sfcm/«, dr + J£>fcm/6,- dft

r r n

JE^eP.dfi

4 ( 1 - » 0 n

x [ ^ L + « * w ( e g - e f ) ] (7)

Tensors components such as Dkmi and Sfcmi- have the same

expressions as for elasticity.6 The 2fcm,;- components are

given by:

AG1 _{( , 3 tf«}

( I - * )2

I 3r3Z + v(l - v)

(l-2«0:

•1W„ d2UZ2 b2Urr 1W,

xl + + T\dUr2 ^d2U22 | 32t/„ t 1 3 ^ 1 I r 3Z 3z3Z 3r 9/? i? 9r J

v2\ Urr + + +

iRr R dz r dR dzdRl) ₍₈₎

2G2 ( [dUn bUzr\ 1

LdR \ dz dr I dZ\ dz dr 4G2

-\{\-v)2 -V bzdfi

1 V. w - + K ( 1 - » 0 ( ) - 2 v )2 I 3z3/2

"13f/rr ldUzr d2Urr d2Uzz dz dr dR dz dZ

]

x + —

ir dR R

r 1 rr \dUrrldUrz d2Urz]\ + v2\— Urr+ + +

^rzzz ^zrzz'

2G2

\-2v (1

..[

d2U„ b2Ur.

+ •

dz bZ bz bR _

lr\ bZ bR I

b2U„ b2Urz dR I brbZ brbR

, Wrz | 9 ^ z ,

I A T I Z " R dz J drdZ bzbR!

J 1 1 3£/„. 1 bUrr b2Urrl\

v2\— Urr+ - + - +

IRr R br r dR brbR})

AG2

-'88rr

1 buri

-, U - " )a - + v{\-v)

(1 - 7v)2 V R br

r 1 1 bUzr b2Urr b2U,

x — Urr + - + - + '

IRr R bz brbR brbZ\

+ v

lr bR

b2U7r 1 bUr. bU,.. bzbR r bZ bzbZ.

2G2 f 9 lbUrr bUzr\

|

r i tbu

^u

\ b /bu

^bUzXH

"iR \ bz br J bZ\ bz br )\)

y = y = y = y

'-'rzrz "zm ^rzzr '-'zrzr

IbZ \bz br I bR \ bz br l\

' |

(1

_„±(B.

t

»=)

V bZ\ bz br I

^zzrz ^Jzzzr

\-2v

r i /bu

bu

\ b ibu

^u

rY\

IR \ bz br 1 bR\ bz br /J

AG2

O-2.0'

( ,3

A^^Yr

b2Ur,

bR

+ K1-")

JUrr ^ \bUzr | \bUrz ^2Uzz~\\

V VrR R bz r bZ bzbzW

. = £ , 2G

2

\-2v l LdrdZ brbR]

L r \ bR bZ 1

b2Uzr b2Uzz

AG2

->88zz '

V-2v)2

x — Ur r + IrR

{\-u)2

-bz bZ -bz bR

1 bUz

R bz

1 bU„ b2U:

+ v(\ - v)

•r b2U„

}

AlbUrr [ b2Urr | \bUrz b2Urzl\ V lr bR brbR r bZ brbZl)

R br 2

bz bR bz bZ 2

-'rr88

(1

AG2 ( 1 bUrr -2vf \ r dR

Urr | aat/«1

Rr bz bZ\ b2U„ d2U„ Urr b2Uz .brbR bzbR

+ v2 1 bU

rr 1 W„ 1 9Ur

-•rzee = 2

R br 2G2

•zr88

\-2v

+

-R bz r

(

( I

-4T

3r3Z

1 3£/«

az r a/?

']

- H

VbrbZ

b2U„ b2U„ b2UT br bZ bz 3Z dr dR bz bR AG2

^zzee (1 - 2vf

, 1 3£/rz

( l - « 02 rJ.+ v(l-v)

<[•

/• 3Z

/?r r 3/? J

a2t/„ 32t/.

3r 3/? 3z 3/?.

^eeee •

r 1 bU„ 1 3f/zr

+ v2\ +

Li? br R bz

AG2 l , U„ ( l - 2 « 02r Rr

r 1 bUrr 1 bUzr 1 3f/rr 1 bUrzl

XIR br R dz r bR r bZ\

Xd2Urr b2Uzr \bUrz b2Urzl\ + v2\ + + +

IbrbR dzdR r bz drdZll

The axisymmetric elastoplastic boundary problem does not imply any additional compilation over the elastic case and in both cases the integrals are treated as shown in Cruse et al.6

Discretization

Let us assume that the boundary is discretized into Af constant elements and the domain into M internal cells. This produces, as shown in reference 11, the following system of equations:

a = G'P-H'U + B' + Q'ep

• • ( 9 )

HU = GP + B + Qep

where G' and H' are matrices of dimensions 4Mx 2N, Q' is

AM x AM, H and G are IN x 2N, Q is 27V x AM and the

others are vectors of dimension 2N, except for the aP vector which is AM.

For elements without the singularity one can use the standard Gaussian quadrature formula.12 When the element

to be integrated presents a singularity these formulae are not efficient and a large number of integration points are needed to obtain an accurate result. For this case the semianalytical integration can be used.8 Another possibility is to develop

more accurate integration formulae for the axisymmetric case, such as the logarithmic presented in reference 12 or the special ones of reference 13.

Consider a boundary element as shown in Figure 1 and let us assume that the integration will be carried out over a length as indicated in the figure.

Similarly, for domain integrations the two situations indicated in Figures 2 and 3 can be present. Figure 2 repre-sents the integration of terms in the matrix Q (i.e. from a point on the boundary) and Figure 3 the integration of terms in Q' (i.e. for a singularity in the domain). The dark area in the figures corresponds to the area where analytical integration is used. The analytical integration refers only to the variable S (see Figure 4) as the integration with respect to 6 can be accurately done numerically. Hence for Figure 4 we have:

e2 S(e)

\fdv = 2n \rsfdA = 2n f dd lim f rsfds

fdv=2n f dOI

x / dv = 2n

v e,

S(8)

'= Urn f

e^-0 J

rsfds

(10)

(11)

i

--L/2

Figure 1

- S / 2 S / 2 L/2

Figure 2

We need to consider how to calculate the /integral for each different case. The/functions to be integrated corre-spond to the first and second derivatives of the displace-ments and are the ones that appear in the tensors S,/fc and

2,/fc/. In what follows we present the particular case of the

integrals in the Q matrix. The same development applies for those integrals in Q'.

Matrix Q

For the calculation of the terms in this matrix it is necessary to consider terms in the first derivatives of the displace-ments in the £1 domain, see Figure 6. These integrals take the following expression:

\"-7

- 1 87rG(l - v)s/R

8,

3,-Av

J dr

dv-• /1+ ( 6 - 8 i > + r2)/2

9,

dd (12)

8TTG(1 - v)y/R

J

9,

xtz # ? /4- fr 2/6+ — h trR

4 ( 4 r l - l ) r . 3+r;

I i - 's - 8Jhide 3

I

w„

•dv-dz 8nG(l - p)y/R ,

v e,

°

l - I O V ^ J

X [(1 - 2t\) trI„ + \{\ + t\)h + J / J d6

S ( 6 )

Figure 4

References

JV

8TTG(1 - V)y/R J

x?z Rt2zh-t2rI6 + (t2z-l)trI2

rr.R 3 + t \ , 1

—fr^t—--/s + l/s \de

02

f

8TTG(1 - V)y/R J

CWrr

J

br

xtz[y3 + Ri4+U2i5-i6]de

e2

dv- 1

8TTG(1 - i/)Vfl ' 0,

J

d0

{^-l^)/.^-

3

^)

x U + [(3 - 4v) + (3 - 2*0 r | - tt] I2 + K i }

1

1 Telles, J. C. F. and Brebbia, C. A. /Ipp/. Mar/;. Modelling 1979, 3,466

2 Bui, H. D. Int. J. Solids Struct. 1978, 14, 935

3 Telles, J. C. F. and Brebbia, C. A. In Proc. 2nd int. seminar on

recent advances in boundary element methods (ed. Brebbia,

C. A.), Southampton, 1980

4 Cathie, D. N. In 'New developments in boundary element methods (ed. Brebbia, C. A.), CML Publications, Southampton, 1980

5 Riccardella, P. C. Rep. No. SM73 - 1° Dept. Mech. Eng. Carnegie Mellon University, Pittsburg, 1973

6 Cruse, T. A. et al. Comput. Struct. 1977, 7, 445

7 Massonet, C. E. In 'Stress Analysis' (ed. Zienkiewicz, O. C ) , Wiley, London, 1965

8 Wrobel, L. C. and Brebbia, C. A. In 'New developments in boundary element methods' (ed. Brebbia, C. A.), CML Publica-tions, Southampton, 1980

9 Cathie, D. N. and Banerjee, P. K. In 'Innovative numerical analysis in the applied engineering sciences' (ed. R. P. Shaw

etai), 1980

10 Mikhlin, S. G. 'Integral equations' (Pergamon, Oxford), 1957 11 Telles, J. and Brebbia, C. A. 'Progress in boundary elements',

Vol. 1, Pentech Press, London, Wiley, USA, 1981

12 Brebbia, C. A. 'The boundary element method for engineers', Pentech Press, 1978

13 Pina, H. L. Appl. Math. Modelling 1981, 5, 209 14 Doblare, M.PhD thesis, Univ. Madrid, 1981

CbUrr - l r

— - d v = Iz(3+2t2-4p)It

J bz 8nG(\-v)RJ v

P^L*,- !—=f

J bz 8TTG(T - i/K/fi J

Appendix

bz 8TTG(1 - v)s/R

v e

x [rr(l - It1)h - i ( l + tl)h - U.] d0

r s

s

2 r ,

/ i = - p i n ds = — Ub-2RyjR+b

J y/r 64Rr 3t2 L 'v

r^

d0

=—>—(

J br SnG(l-v)y/RJ

x In

x [tlifl - 2v) I2 - tr{3 -4v + 2t\)\2

x(h-Rh) + \h]de

-4Ril2)n

64R(R + b)

y/RTb + y/R

- 4/3(7? + bf12 + 4R y/R+b

4R3l2(\n2tr + \)

\ W2Z

dv= —

bz 8irG(l~u)

v e

W'-

[2,?

2- ( 3 - 4 * > ) ] /6d 0

(12) The expression for each of the / integrals is given in the Appendix. The integrals presented in (12) can then be used to compute the elements of Q. Similarly one can find the integrals necessary for Q', see Doblare ,14

y/RTb-y/R S

h = ! -4= d* = T K* - 2R)y/R~+b + 2R3n] J s/r t2

£

2 RR+b)2-6R(R+b)-3R2

r r r 2 1

J r3'2 64Rr t2 L

,x In

(,4R(R + b)

3y/R+b

2j9(R + b)3n

+ 8l3Rs/R~Tb + 2R2 Conclusions

Following the work by Telles and Brebbia11 and Cruse et al.,6 the complete boundary element formulation for the elastoplastic axisymmetric case is given in this paper. To this end the different tensor components corresponding to elastoplasticity have been calculated together with the principal value term for the plastic deformations. The boundary and domain integrals needed for the constant boundary element case are also given following the semi-analytical approach used by Telles and Brebbia11 and Wrobel

and Brebbia.8

16 In 3

y/R+~b y/R~Tb + y/R y/RTB-s/R

2

8/3/?3/2(21n2rr + 5)l

f 1 2 , _

h=\ - p d s = - ( v 7 T + 7 J - y/R)

J

V

t

e

3/2-i3

/«= fy/Fds=—[(R+b)3/2-R3'2]

J J * f

e

S

r s 2 r 2R + b _1

With 6 = rrS.

For the case fr = 0, one has:

h= \—7-\n ds =

J

y/F

Ow-

1

)

S2

n

2y/R

h =

/£"»

64Rr 3R3n \ 64R2 3 /

r l 5

h= -ds

e V *

-' J ,3/2

s2 S3

e

S

h= \ y/rds = Sy/R

e

, S S2