Michel Potier-Ferry

A numerical continuation method based on Padé approximants

A continuation algorithm is presented with a new predictor, which is based on a rational representation of the solution path. This algorithm belongs to the class of asymptotic numerical methods that connect perturbation techniques with a discretization principle without the use of a correction process. Several examples from shell buckling and from contact mechanics are analyzed, to assess the efficiency and the reliability of the method.

Iterative algorithms for non-linear eigenvalue problems. Application to vibrations of viscoelastic shells

In this paper, two numerical iterative algorithms are developed for the vibrations of damped sandwich structures. These methods associate homotopy, asymptotic numerical techniques and Pade approximants. The first one is a sort of high order Newton method and the second one uses a more or less arbitrary matrix. So one can determine the natural frequencies and the loss factors of viscoelastically damped sandwich structures. To assess their efficiency, a few sandwich beams and plates have been considered. The techniques can be applied to large scale structures, to large damping and to strongly non-linear viscoelastic modulus.

A critical review of asymptotic numerical methods

SummaryVarious sorts of asymptotic-numerical methods have been propsed in the literature: the reduced basis technique, direct computation of series or the use of Padé approximants. The efficiency of the method may also depend on the chosen path parameter, on the order of truncature and on alternative parameters. In this paper, we compare the three classes of asymptotic-numerical method, with a view to define the “best” numerical strategy.

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.

Lateral post-buckling analysis of thin-walled open section beams

Abstract Thin-walled beams with open sections are studied using a nonlinear model. This model is developed in the context of large displacements and small deformations, by accounting for bending-bending and bending-torsion couplings. The warping and shortening effects are considered in the torsion equilibrium equation. The governing coupled equilibrium equations obtained from Galerkin’s method are solved by a Newton–Raphson iterative process. It is established that the buckling loads are highly dependent on the pre-buckling deformations of the beam. The bifurcated branches are unstable and strongly influenced by shortening effects. Some comparisons are presented with the solutions commonly used in linear stability, like in the standard European steel code (Eurocode 3). The regular solutions appear to be very conservative, especially for I sections with large flanges.

An amplitude equation for the non-linear vibration of viscoelastically damped sandwich beams

Abstract An elementary theory for non-linear vibrations of viscoelastic sandwich beams is presented. The harmonic balance method is coupled with a one mode Galerkin analysis. This results in a scalar complex frequency–response relationship. So the non-linear free vibration response is governed by only two complex numbers. This permits one to recover first the concept of linear loss factor, second a parabolic approximation of the backbone curve that accounts for the amplitude dependence of the frequency. A new amplitude–loss factor relationship is also established in this way. The forced vibration analysis leads to resonance curves that are classical within non-linear vibration theory. They are extended here to any viscoelastic constitutive behaviour. This elementary approach could be extended to a large class of structures and in a finite element framework. The amplitude equation is obtained in closed form for a class of sandwich beams. The effects of the boundary conditions and of the temperature on the response are discussed.

The asymptotic-numerical method: an efficient perturbation technique for nonlinear structural mechanics

ABSTRACT Perturbation techniques (asymptotic expansions) have been widely used in many engineering fields for solving nonlinear problems. However, the solution is often represented by the first few terms of a perturbation expansion, which leads to a qualitative approximation rather than a quantitative one. Our aim is to show that a perturbation technique can also lead to a powerfull numerical method for some classes of structural problems, provided that it is combined with a finite element method to account for complex geometries, and that a large number of terms of expansions are determined.

Flexural-torsional post-buckling analysis of thin-walled elements with open sections

Abstract The post-buckling analysis of thin-walled elements under compression is investigated. A nonlinear model is developed by using nonlinear relationships between curvatures and bending moments. Warping and shortening effects are considered in the torsion equilibrium equation. Based on Galerkins method, a nonlinear algebraic system is obtained for simply supported boundary conditions. The three resulting equations in bending and torsion are highly coupled and the Newton–Raphson algorithm with displacement control is adopted for the solution. The post-buckling equilibrium curves are obtained for various sections shapes, such as bisymmetric and monosymmetric sections. The importance of the shortening effect is outlined.

A structural approach of plastic microbuckling in long fibre composites: comparison with theoretical and experimental results

Abstract The aim of this paper is to compare some predictions obtained from a structural plastic microbuckling model presented in detail by Drapier et al. (1999) , with theoretical and experimental results from the literature. After a short presentation of this model, it is established that with our approach it is possible to find the elastic modes determined by Drapier et al. (1996) on the composite microstructure. The plastic instability mechanism is then investigated and its understanding is refined. Some simulations are carried out varying the fibre initial imperfection, and the results are detailed and compared with predictions from a kink-band model (Budiansky and Fleck, 1993) . Compared to the present knowledge, the understanding of the influence of the imperfection shape and of its distribution across the ply thickness is improved and new results are exposed. Validation of the present approach is completed by comparing the influence of both matrix and fibre behaviours as predicted by Budiansky and Fleck (1993) with the ones obtained from our numerical tool. Results demonstrate the influence of the change in the matrix tangent stiffness. Secondly, we have quantified the effects of the applied loading, thickness and stacking sequence on the compressive strength of laminates. Numerical predictions provide new results that yield a proper justification of the very high compressive strength measured with bending tests. These predictions also fit well experimental measurements from the literature showing the effects of the thickness (Wisnom, 1992) and of the stacking sequence (Grandsire-Vincon, 1993) on the compressive strength. For the first time, the effect of the gradient of loading across the laminate thickness is predicted. Results are shown to correlate well with experimental results from Wisnom et al. (1997) .

Microbuckling and strength in long-fiber composites: theory and experiments

Abstract Failure of laminated composites in compression is often ascribed to a mechanism of fiber microbuckling. Following this interpretation, a model is presented where the fiber is schematized as an elastic beam and the matrix as an elastic foundation. In comparison with previous models, however, the strain in the matrix is distributed through the thickness of a ply and lower failure stresses are obtained. An original flexion compression device designed for investigating the elastic behavior in compression and the effect of the loading mode on the characteristics of failure is then described. Results on glass/epoxy resin unidirectional composites and laminates are discussed.