Some estimates for the anisotropic Navier-Stokes equations and for the hydrostatic approximation RAIRO-Modélisation mathématique et analyse numérique, tome 26, n o 7 (1992), p. 855-865. © AFCET, 1992, tous droits réservés. L'accès aux archives de la revue « RAIRO-Modélisation mathématique et analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

We study the yield conditions of phase transformation initiation for shape memory alloys exhibiting asymmetry between tension and compression. An extension of the choice of the classical invariant parameters such as those of Lode is proposed. A necessary and sufficient condition of convexity of these surfaces representing the elastic domain of austenite in the stress space, is established. Moreover the transport of these surfaces in the space of effective transformations strains of martensite is done. Hence, the duality between these two spaces is built. Some applications involving Cu-Al-Be and NiTi shape memory alloys end the purpose.

Shape Memory Alloys (SMAs) such as NiTi exhibit stress induced martensitic phase transformation. The purpose of this paper is to provide a better understanding of SMA (such as NiTi) fracture behavior, by considering the vicinity of the crack tip where the transformation occurs. This analysis integrates the asymmetry between tension and compression in an analytical prediction of the surface of phase transformation around the crack tip for loading modes 1, 2, 3 and mixed 1+2. The influence of the asymmetry between tension-compression is more important in plane stress conditions than in plane strain conditions, particularly for mode 1 loading. In order to validate this model, we are currently setting up an experimental investigation to observe strain localization during crack propagation (transformation and martensitic saturation regions) on NiTi thin sheets.

One important feature in the description of the shape memory alloys is the determination of the yield surfaces of phase transformation. They can be presented as surfaces in the phase transformation martensitic strain space. In this paper the transition from this stress to the classical stress space is performed. Two application cases concerning bi-axial loading (bi-tension and tension-torsion) are discussed in details.

The present study is an extension of a recent paper of Freed et al. (J Mech Phys Solids 56:3003-3020, 2008). The final aim is to describe the transformation toughening behavior of a static crack along an interface between a shape memory alloy (SMA) and a linear elastic isotropic material. With an SMA as an equivalent Huber-Von Mises stress model (hypothesis of symmetric behavior between tension and compression), Freed et al. determine the initiation (ending) phase transformation yield surfaces in terms of the local phase angle introduced by Rice et al. (Metal ceramic interfaces, Pergamon Press, New York, pp 269-294, 1990). In this paper we give the general framework to determine this angle for a model integrating the asymmetry between tension and compression (experimentally measured: Vacher and Lexcellent in Proc ICM 6:231-236, 1991; Orgéas and Favier in Acta Mater 46(15): [5579][5580][5581][5582][5583][5584][5585][5586][5587][5588][5589][5590][5591] 2000), the Huber-Von Mises model being only a particular case. We demonstrate the local phase angle existence in an appropriate framing domain and give a sufficient hypothesis for its uniqueness and an algorithm to obtain it. Estimates are obtained in terms of physical quantities such as the Young modulus ratio, the bimaterial Poisson modulus values and also the choice of the yield loading functions. Finally, we illustrate this theoretical study by an application linking the asymmetry intensity on the width and the shape on predicted phase transformation surfaces and by a comparison with the symmetric case.

We study the convergence of general finite volume schemes for the diphasic flow problem in porous media ut − div(u∇p) = 0 and ∆p = 0 in a bounded domain. A general formula for the numerical flux on a triangular mesh is given. The stability and an estimate of the total variation of the approximate solutions are obtained by means of a variational method; the convergence follows easily for general data.Résumé. Onétudie la convergence de schémas aux volumes finis pour le problème d'écoulement diphasique en milieu poreux ut − div(u∇p) = 0 et ∆p = 0 dans un domaine borné. Une formule générale pour le flux numérique est présentée sur un maillage triangulaire. Le fluxà deux points tel que celuià décentrage amont ou celui de Lax-Friedrichs en sont des cas particuliers. La stabilité et une estimation de la variation totale des solutions approchées sont obtenues par une méthode variationnelle; la convergence en découle facilement pour des données générales.

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: In shape memory alloys (SMAs), a precise determination of phase transformation surfaces around the crack tip, is very important for the prediction of fracture parameters. In this work, the size of phase transformation region surrounding the tip of a edge crack is evaluated analytically. The purpose is restricted to tip without and with curvature loaded in mode I.