Numerical Methods in . Laminar and Turbulent Flow I

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Numerical Methods in . Laminar and Turbulent Flow Volume VII, Part 2 Editors: C. TAYLOR J. H. CHIN G. M. HOMSY

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Proceedings of lhe Seventh lnternational Conference held in Stanford, 15th-19th July, 1991

First Published 1991 by Pineridge Press Limited 54, Newton Road, Mumbles, Swansea, U.K. ISBN 0-906674-,77-8 Copyright

© 1991

Pineridge Press

British Library Cataloguing

in Publication

Data

Numerical Methods in Laminar and Turbulent Flow: Proceedings of the Seventh International Conference 1. F!uids - Flow - Mathematics - Numerical Methods I. Taylor, C. (Cedric, 1936- ) 535'051'01511 ISBN 0-906674-77-8

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Ali rights reserved. Na part of this publication may be repraduced, stored in a retrieval system ar transmitted, in any form ar by any means, electronic, mechanical photocopying ar otherwise, without prior permission of lhe publishers Pineridge Press Ltd,

SOLID-FLUID INTERACTION ANALYSIS FOR STRUCTURAL FAILURE PREDICTION OF INELASTIC PIPELI1\"'ES F. B. F. Rachid and H. S. da Costa Mattos Mechanical Engineering Departrnent Pontifícia Universidade Católica do Rio de Janeiro Rio de Janeiro, RJ, 22453, Braail SUMMARY In this paper it is presented a solid-fluid interaction analysis for structural failure prediction of damageable elasto-viscoplastic pipes. To solve the coupled non-linear equations describing the dynamics of fluid and pipe wall motions, a numerical procedure based on the method of characteristics is used. To illustrate the applicability of tbe model, a numerical simuJation of a hypothetical core disruptive accident in a simple nuclear pipeline is presented and analyz ed,' 1. 1NTRODUCTION Industrial pipelines used for liquid transmission may have critical periods of operation during unexpected unsteady flow conditions. In such events, severe pressure peaks may arise into the pipeline as a consequence of fluid flow disturbances (such as valve slam or ant accident, for instance) and associated interaction phenomenon between liquid and pipe motion. Depending on temperature operating conditions, piping stiffness and pressure surge magnitudes metalic pipelines are subjected, significant inelastic deformations of tbe pipe wall may occur. On the other hand it is well-known that excessive inelastic strain is one of the mechanisms responsible for degradation of metalic materials. Thus, a reliable design of pipings conveying liquids must take into account a structural failure analysis. In general, studies [1,2,3J about solid-fluid interaction analysis in compliant pipings has not been concerned with the possibility of treating different rnech anical behaviours of the pipe material. In a recent paper [4], the theory has been extended in arder to include the inelastic viscous behaviour of the pipe wall. Nevertheless, it has not taken into account any continuum damage theory capable to predict structural failure. This article presents a coupled solid-fluid interaction analysis in complaint inelastic pipelines together with a continuum damage mechanics theory to predict structural integrity of the pipe walls. The constitutive equations used to describe the inelastic behaviour of the pipe material as well as the damage model are derived from an internal variable constitutive theory with strong thermodynamical basis. The set of non-linear differential equations describing the dynamics of the liquid and the inelastic pipe walI motion are coupled and solved by using a simpie numerical technique based on the method of characteristics. The damage model allows the calculation of the damage evolution with time in each section of the piping as a function of the stress and inelastic strain fields.

1387

1386 Equations

(1) to (4), along with expression (5), constitute the basic confor the solid-fluid problem and are based on the model described by Walker and Phillips [5J. ln addition to these equations, one must add the constitutive relations which wil! describe the mechanical behaviour of the pipe material.

The ge.nerality of th e proposed medel presented herein is illustrated by a nurnerical example concerning the damage evolution caused by pressure tr ansients in st.ainless steel pipe networks of nuclear power plants. StainJess steels are ofen ernployed in heat transport systems of breeder reactors and present an elasto-viscoplastic behaviour at the reactor operating temperatures, which is about 600° C. Numerical results of tbe plastic damage evoJution induced by a hypothetical core disruptive accident in a simple nuclear pipeline are presented and discussed. 2. SOLID-FLUID

MODEL A~D CONSTITUTIVE

servat ion equations

2.2 Constitutive

The set of elasto-viscoplastic constitutive equations used in this work describes the mechanical behaviour of metalic materiaIs submitted to nonmonotonic Ioadings. Such kind of constitutive theory has a strong thermodynamic basis and a careful! presentation can be found in [6,7J. Here, due to the limited space, we shall only surnrnarize the main features and restrict the study to isothermal transformations. Within the general frarnework of Thermodynamics of Irreversible Processes, it is assumed that the mechanical state of an elasto-viscoplastic damageable body at a given material point and at a given time is completely defined by the set of state variables: (e,ga,p,c,D). e is the total strain; e" is the anelastic strain; p is a scalar variable associated with the isotropic hardening induced by the anelastic deformation; c is a tensorial variable associated with the kinematic hardening induced by the anelastic deformation and the scalar' variable D ElO, lJ is a macroscopic quantity which can be interpreted as a local mesure of the degradation (darnage) of the material ind uced by the anelastic deformation. If D = O, the rnaterial.Is virgin and if D = 1 the material is locally "broken". In general, for the sake of security, tbe material is considered locally "broken" when the variable D reaches a critical value Dcr such that O < Dcr < 1. t·-The reversible behaviour of the material is obtained by the knowledge of the state laws which relate the state variables (e,g",p,c,D) with the thermodynamical forces (CT,BI',BC,BD): . _ .' _ ...

EQUATIONS

The solid-fluid model presented in tbe next paragraphs is intended to describe the fluid-structure interactíon phenomenon in damazeable elastoviscoplastic pipes. It is a wave equation formulation which accounts for the existence ofaxial stress waves in the pipe wall and pressure waves in the liquido To begin with, the basic conservation equations are presented and then the constitutive relations for the pipe material. Throughout this article, boldface letters are use~ to designate vectors and tensors and the following notations are adopted: ( ) = a( )/at; (x) = max{O,x}. 2.1 Basic Equations Consider an inviscid transient axisymmetrical compressible fiow of an acoustic liquid contained in a thin-walled pipe (inside radius R and wall thickness e) for which both liquid and pipe walJ motion are relevant. Under the assumptions of long pulse approximation, constant liquid bulk modulus K, sm.aIl ?eforma~ions an? axisymmetrical plane-stress distribution in the pipe wall, liquid and pipe rnotions are governed by the following set of equations.:

1 ap 2tÚ --+-+-=0 K õt R

ap as

av

prãt + aü

"

OCT.

as

Ptat atÚ

av as

-

+

(1)

.

Eguations

. CT= C(e - ga)

o

=

=

J ..

(2)

o

(3)

.~ ~ . l' .' BD = -- [C(g - g")] . (e - g")] 2

BI' = -b[l- exp(-d

=o

(6.1) (6.2) (6.3)

p)]

(4)

(6.4)

In the above equations P, V, u and w represent the pressure, liquid axial velocity and pipe wall displacements in the axial and radial directions while CT. and CTe designate the only non-vanishing stress components in the axial and circurnferential directions, respectively. AlI of these variables are functions of the axial coordinate s along the pipe and the time t. p! and Pt stand for fluid and pipe densities. '. ', , Equations (1) and (2) are the continuity' and axial momentum equations for the liquid whereas equations (3) and (4) are the axial and radial momentum equations for the pipe. Pipe wall displacements are related to the axial and circumferential str ains through the strain-displacement relationships which are given by: -' , _.' ,

where a, b and d are material parametres, CT is the stress tensor and C is the classical symrnetric fourth-order tensor of elasticity. To complete the constitutive equations, evolution laws are needed for the internal variables (g",p,c,D). The irreversible behaviour of the material can be described by the introduction of an yield function F(CT, BI' ,BC) such that ia = O, 'fi = O, C O, b = O if F(CT,BI',BC) < O for alI (CT,BI',BC). ln this case, the yield function F has the following forro: ~ :.. , -' - - .

PtRe-;:;ot

RP

, -o· :

:

~.

••

.J

.) ~.:-

.:,'

ou as

e. =-

,'

CTee

'.

".

z,

o.",) ,. .r .•...

__

(5)

- .~.

=

(7)

.. ~ ,