a b s t r a c tMaterial flow during friction stir welding is very complex and not fully understood. Most of studies in literature used threaded pins since most industrial applications currently use threaded pins. However, initially threaded tools may become unthreaded because of the tool wear when used for high melting point alloys or reinforced aluminium alloys. In this study, FSW experiments were performed using two different pin profiles. Both pins are unthreaded but have or do not have flat faces. The primary goal is to analyse the flow when unthreaded pins are used to weld thin plates. Cross-sections and longitudinal sections of welds were observed with and without the use of material marker (MM) to investigate the material flow. Material flow with unthreaded pin was found to have the same features as material flow using classical threaded pins: material is deposited in the advancing side (AS) in the upper part of the weld and in the retreating side (RS) in the lower part of the weld; a rotating layer appears around the tool. However, the analysis revealed a too low vertical motion towards the bottom of the weld, attributed to the lack of threads. The product of the plunge force and the rotational speed was found to affect the size of the shoulder dominated zone. This effect is reduced using the cylindrical tapered pin with flats.

SUMMARYThe aim of this work is to develop a reliable and fast algorithm to compute bifurcation points and bifurcated branches. It is based upon the asymptotic numerical method (ANM) and Padé approximants. The bifurcation point is detected by analysing the poles of Padé approximants or by evaluating, along the computed solution branch, a bifurcation indicator well adapted to ANM. Several examples are presented to assess the effectiveness of the proposed method, that emanate from buckling problems of thin elastic shells. Especially problems involving large rotations are discussed.

International audienceThis paper presents a numerical technique to deal with instability phenomena in the context of heterogeneous materials where buckling may occur at both macroscopic and/or microscopic scales. We limit ourselves to elastic materials but geometrical nonlinearity is taken into account at both scales. The proposed approach combines the multilevel finite element analysis (FE2) and the asymptotic method (ANM). In that framework, the unknown nonlinear constitutive relationship at the macroscale is found by solving a local finite element problem at the microscale. In contrast with FE2, the use of the asymptotic development allows to transform the nonlinear microscopic problems into a sequence of linear problems. Thus, a direct analogy with classical linear homogenization can be made to construct a localisation tensor at each step of the asymptotic development, and an explicit macroscopic constitutive relationship can be constructed at each step. Furthermore, the salient features of the ANM allow treating instabilities and limit points in a very simple way at both scales. The method is tested and illustrated through numerical examples involving local instabilities which have ignificant influence on the macroscopic behavior

SUMMARYNew predictor-corrector algorithms are presented for the computation of solution paths of non-linear partial di erential equations. The predictors and the correctors are based on perturbation techniques and Padà e approximants. This extends the Asymptotic Numerical Method (ANM), which is an e cient highorder continuation technique without corrector. The e ciency and the reliability of the new technique are assessed by several examples within thin shell theory and Navier-Stokes equations. Many variants have been tested to establish an optimal algorithm.