Hamid Zahrouni

Computing finite rotations of shells by an asymptotic-numerical method

Abstract We present an asymptotic numerical algorithm for the computation of elastic shells with large rotations. The theoretical formulation involves a three-field Hu—Washizu functional, which allows us to put the problem into a quadratic framework. The spatial discretization is based on geometrically exact element, recently presented by Buchter et al. [Buchter et al., Three dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept, Int. J. Numer. Methods Engrg. 37 (1994) 2551-1568] Several classical benchmarks are discussed to define the best strategy and to assess the validity and the efficiency of the present method, as compared to more classical iterative algorithms.

Asymptotic numerical method for problems coupling several nonlinearities

Abstract The objective of this paper is to assess the efficiency of the asymptotic numerical method to solve problems coupling various nonlinearities. The 3D hemispherical stretching of a circular sheet, that involves geometrical, material and red unilateral contact nonlinearities is chosen as an example. An elastoplastic model based on the plasticity deformation theory is adopted. The structural discretization is performed by a shell finite element well adapted for problems involving large displacements and large rotations. The unilateral contact problem is identified to boundary conditions which are replaced by force–displacement relations and solved using a special algorithm. Comparisons with results obtained by the help of an industrial code establish the interest and the performance of the present method.

A semi-analytical buckling analysis of imperfect cylindrical shells under axial compression

Abstract In the framework of the cellular bifurcation theory, we investigate the effect of distributed and/or localized imperfections on the buckling of long cylindrical shells under axial compression. Using a double scale perturbative approach including modes interaction, we establish that the evolution of amplitudes of instability patterns is governed by a non-homogeneous second order system of three non-linear complex equations. The localized imperfections are included by employing jump conditions for their amplitude and permitting discontinuous derivatives. By solving these amplitude equations, we show the influence of distributed and/or localized imperfections on the reduction of the critical load. To assess the validity of the present method, our results are compared to those given by two finite element codes.

Asymptotic Numerical Method for Nonlinear Constitutive Laws

ABSTRACT This paper deals with the application of the asymptomatic numerical method (ANM) to problems involving nonlinear constitutive laws. We are interested in the deformation theory of plasticity which does not take into account the elastic unloading. We show how to obtain a quadratic form of the problem, what allows us to apply easily the perturbation techniques and to obtain the fastest algorithm. Three constitutive behaviors will be analyzed and some examples will be presented to assess the interest of the proposed algorithm as compared with the classical iterative method of Newton-Raphson.

Strip flatness modelling including buckling phenomena during thin strip cold rolling

Abstract This paper describes the application of a steady state elastic–viscoplastic finite element model to the prediction of strip flatness and out of bite buckling during cold rolling of thin strips. This model is applied here to the last stand of a tinplate tandem cold mill. Results show that strip buckling reshuffles the non-homogeneous strip velocity profile just at the roll bite exit; strip stresses inside the roll bite are little impacted, so that roll stack deformation and roll force profile are not modified. The paper focuses on latent strip flatness, i.e. the profile in the transverse direction of the stress in the rolling direction. The profile predicted by the model far downstream the roll bite is in good agreement with shapemeter measurements only if strip buckling (i.e. manifest flatness) just at the roll bite exit and its interaction with the roll bite are considered, using a strong coupling of bite model and post-bite buckling model.

Asymptotic Numerical Method for strong nonlinearities

Plastic constitutive laws and frictional contact conditions induce strong nonlinearities that one has to take into account in the numerical simulation of material forming processes. In this work, we present a review of the different techniques which permit the asymptotic numerical method (ANM) to be adapted to these nonlinearities. ANM needs regular relations and quadratic equations if possible. Several examples show the effectiveness of the proposed method.

An implicit algorithm based on continuous moving least square to simulate material mixing in friction stir welding process

An implicit iterative algorithm, based on the continuous moving least square (CMLS), is developed to simulate material mixing in Friction Stir Welding (FSW) process. Strong formulation is chosen for the modeling of the mechanical problem in Lagrangian framework to avoid the drawback of numerical integration. This algorithm is well adapted to large deformations in the mixing zone in the neighborhood of the welding tool. We limit ourselves to bidimensional viscoplastic problem to show the performance of the proposed implicit algorithm. The results show that the proposed algorithm can be employed to simulate FSW.

A 2D mechanical–thermal coupled model to simulate material mixing observed in friction stir welding process

The aim of our work is the numerical modeling of two dimensional mechanical–thermal material mixing observed in stir welding process using a high order algorithm. This algorithm is based on coupling a meshless approach, a time discretization, a homotopy transformation, a development in Taylor series and a continuation method. The performance of the proposed model is the consideration of large deformations in the formulation of the posed problem. For the spatial treatment, we use the moving least squares approximation which will be applied directly to the strong form formulation of conservation equations. Each collocation point holds mechanical–thermal variables. The high order algorithm and the homotopy transformation allow reducing the number of tangent matrices to decompose and to avoid iterative procedure. Comparisons with the classical iterative solver (Jamal et al. in J Comput Mech 28:375–380, 2002) are performed. Numerical results reveal that a few number of matrix factorization is needed with the proposed approach, decreasing the computation time.

Computing Flatness Defects in Sheet Rolling by Arlequin and Asymptotic Numerical Methods

Rolling of thin sheets generally induces flatness defects due to thermo-elastic deformation of rolls. This leads to heterogeneous plastic deformations throughout the strip width and then to out of plane displacements to relax residual stresses. In this work we present a new numerical technique to model the buckling phenomena under residual stresses induced by rolling process. This technique consists in coupling two finite element models: the first one consists in a three dimensional model based on 8-node tri-linear hexahedron which is used to model the three dimensional behaviour of the sheet in the roll bite; we introduce in this model, residual stresses from a full simulation of rolling (a plane-strain elastoplastic finite element model) or from an analytical profile. The second model is based on a shell formulation well adapted to large displacements and rotations; it will be used to compute buckling of the strip out of the roll bite. We propose to couple these two models by using Arlequin method. The originality of the proposed algorithm is that in the context of Arlequin method, the coupling area varies during the rolling process. Furthermore we use the asymptotic numerical method (ANM) to perform the buckling computations taking into account geometrical nonlinearities in the shell model. This technique allows one to solve nonlinear problems using high order algorithms well adapted to problems in the presence of instabilities. The proposed algorithm is applied to some rolling cases where “edges-waves” and “center-waves” defects of the sheet are observed.

Multiscale analysis of instabilities in heterogeneous materials using ANM and multilevel FEM

In this study, we propose a numerical technique which combines a perturbation approach (asymptotic numerical method) and a multilevel finite element analysis. This procedure allows dealing with instability phenomena in the context of heterogeneous materials where buckling may occur at both macroscopic and/or microscopic scales. Different constitutive relations are applied and geometrical non-linearity is taken into account at both scales. Numerical examples involving instabilities at both micro and macro levels are presented.