Unsteady Flow and Fluid Transients, Bettess & Watts (eds) © 1992 Balkema, Rotterdam.ISBN 90 5410 046 X Water hammer in inelastic pipes: An approach via an internal variable constitutive theory EB.ERachid Universidade Federal Fluminense, Brazil H.C.Mattos Pontifícia Universidade Católica do Rio de Janeiro, 8razi! S.Stuckenbruck Brazil ABSTRACT: This paper presents an extension of the classical water hammer theory capable to handle several kinds of inelastic pipe wall behaviours within a same framework. The inelastic behaviour is described by means of an internal variable constitutive theory. To solve the governing equations, a numerical method based on the operator splitting technique is proposed. Examples of viscoelastic, elasto-viscoplastic and elasto-plastic - behaviours are given to illustrate the versatility of the theory. 1 INTRODUCTION There has been a long time since there exists a great interest in the modelling of hydraulic transients in compliant inelastic pipes. Due to severe pressure loadings generated by unsteady fluid flow conditions, the pipe is likely subjected to inelastic deformations. Such a problem arises in many practical engineering situations ranging from nuclear and thermohydraulics installations to modern water supply transmission lines. Yet, several analytical models have been proposed to describe the unsteady fluid flow phenomenon in in- elastic pipes. For viscoelastic pipes, the models pro- posed by Gally et al. (1979), Rieutord (1982) and Ghi- lardi & Paoletti (1986) are the most representatives. AIso, pipe materiaIs which exhibt plastic behaviour have been considered by Fox & Stepnewski (1974), Youngdahl & Kot (1975) and recently by Rachid et aI. (1991) in a fluid-structure interaction contexto How- _ ever, since inelastic behaviour encompasses a great number of different mechanical responses, these mod- eIs are in genéral restrict to a particular pipe wall me- chanical behaviour. Moreover, for each specific model - it has been proposed an intrinsic numerical technique for obtaining numerical solutions of the governing equations. As a result, it becomes an impracticable task to handle such a variety of pipe wall mechanical respones within a same computational programo Aiming to overcome this difficulty, it is presented in this work an extension of the classical water ham- mer theory in order to incorporate a broad class of inelastic pipe responses within a same mathematical framework. For this purpose, an internal variable con- stitutive theory with strong thermodynamical support is used to describe the inelastic behaviour of the pipe material. To solve the governing equations describing the unsteady fluid flow and pipe wall deformation, the operator splitting technique together with the method of characteristics is proposed. One of the main features of the proposed theory (mechanical model and method of solution) is that it can be easily implemented on an existing computer program that performs the classical method of char- acteristics water hammer analysis. To do so, it suffices to include a single subroutine to take into account the evolution equations which describe the inelastic be- haviour of the pipe material. Examples of viscoelastic, elasto-plastic and elasto viscoplastic behaviours are given to illustrate the ver- satility of the modeI. For some of these examples, numerical results are shown to be in good agreement when compared to experimental data available in the literature. 2 MODELLING The water hammer model for inelastic pipes presented here is derived in three main steps. Fisrtly, the ba- sic water hammer equations are presented and then a constitutive theory capable to handle several pipe material behaviours within a same structure. In a fi- nal step, the model is obtained by combining the basic and constitutive equations. To attain a general formulation able to include several pipe wall mechanical responses within a same mathematical framework, the equations of the model, together with the method of solution proposed, are presented in an abstract formo Afterwards some spe- cific forms of the constitutive equations are given to illustrate the versatility of the theory. 63