Lagrangian analysis. Modern tool of the dynamics of solids

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Abstract
Explosive metal-working, material synthesis under shock loading, terminal ballistics, and explosive rock-blasting, are some of the civil and military fields of activity that call for a wider knowledge about the behavior of materials subjected to strong dynamic pressures. It is in these fields that Lagrangian analysis methods, the subject of this work, prove to be a useful investigative tool for the physicist. Lagrangian analysis was developed around 1970 by Fowles and Williams. The idea is based on the integration of the conservation equations of mechanics using stress or particle velocity records obtained by means of transducers placed in the path of a stress wave. In this way, all the kinematical and mechanical quantities contained in the conservation equations are obtained. In the first chapter the authors introduce the mathematical tools used to analyze plane and spherical one-dimensional motions. For plane motion, they describe the mathematical analysis methods pertinent to the three regimes of wave propagation encountered : the non-attenuating unsteady wave, the simple wave, and the attenuating unsteady wave. In each of these regimes, cases are treated for which either stress or particle velocity records are initially available. The authors insist that one or the other groups of data (stress and particle velocity) are sufficient to integrate the conservation equations in the case of the plane motion when both groups of data are necessary in the case of the spherical motion. However, in spite of this additional difficulty, Lagrangian analysis of the spherical motion remains particularly interesting for the physicist because it allows access to the behavior of the material under deformation processes other than that imposed by plane one-dimensional motion. The methods expounded in the first chapter are based on Lagrangian measurement of particle velocity and stress in relation to time in a material compressed by a plane or spherical dilatational wave. The Lagrangian specificity of the required measurements is assured by the fact that a transducer enclosed within a solid material is necessarily linked in motion to the particles of the material which surround it. This Lagrangian instrumentation is described in the second chapter. The authors are concerned with the techniques considered today to be the most effective. These are, for stress : piezoresistive gauges (50 Ω and low impedance) and piezoelectric techniques (PVF2 gauges, quartz transducers) ; and for particle velocity : electromagnetic gauges, VISAR and IDL Doppler laser interferometers. In each case both the physical principles as well as techniques of use are set out in detail. For the most part, the authors use their own experience to describe the calibration of these instrumentation systems and to compare their characteristics : measurement range, response time, accuracy, useful recording time, detection area... These characteristics should be taken into account by the physicist when he has to choose the instrumentation systems best adapted to the Lagrangian analysis he intends to apply to any given material. The discussion at the end of chapter 2 should guide his choice both for plane and spherical one-dimensional motions. The third chapter examines to what extent the accuracy of Lagrangian analysis is affected by the accuracies of the numerical analysis methods and experimental techniques. By means of a discussion of different cases of analysis, the authors want to make the reader aware of the different kinds of sources of errors that may be encountered. This work brings up to date the state of studies on Lagrangian analysis methods based on a wide review of bibliographical sources together with the contribution made to research in this field by the four authors themselves in the course of the last ten years. Le formage des métaux par explosif, la consolidation dynamique des poudres, la balistique terminale, l'abattage des roches par explosif, sont autant d'applications, dans les domaines civil et militaire, qui exigent d'approfondir les connaissances que l'on a des comportements des matériaux chargés par des ondes de contrainte de forte intensité. C'est dans ce domaine que les méthodes d'analyse lagrangienne, sujets de cet ouvrage, seront un outil d'investigation intéressant pour le mécanicien du solide. L'analyse lagrangienne a été développée autour des années 1970 par Fowles et Williams. L'idée de base repose sur l'intégration des équations de la mécanique en utilisant l'histoire des contraintes ou des vitesses matérielles obtenues au moyen de capteurs minces sur le trajet d'une onde de contrainte. Sont ainsi obtenues toutes les grandeurs cinématiques et mécaniques contenues dans les équations de conservation. Dans le premier chapitre, les auteurs introduisent les outils mathématiques destinés à analyser les mouvements monodimensionnels plan et sphérique. Pour le premier de ces mouvements, sont décrites les méthodes mathématiques d'analyse propres aux trois régimes de propagation d'onde rencontrés: l'onde instationnaire non amortie, l'onde simple et l'onde instationnaire amortie. Par ailleurs, pour chacun de ces régimes sont traités les cas où l'on dispose initialement des profils des contraintes ou des vitesses matérielles. Les auteurs insistent sur le fait que l'une ou l'autre des deux collections de données (contraintes et vitesses matérielles) sont suffisantes pour intégrer les équations de conservation dans le cas du mouvement plan, alors que les deux collections de données sont indispensables dans le cas de mouvement sphérique. Cependant, malgré cette difficulté supplémentaire, l'analyse lagrangienne du mouvement sphérique reste particulièrement intéressante pour le mécanicien...

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