Modelling of inelastic bending of a metal sheet with thermal coupling

Abstract

The modelling and numerical analysis of inelastic bending of a metal sheet are presented. The distribution strains and stresses are found for slow and fast bending while taking into account the geometrical and material non-linearities. The constitutive relation of Klepaczko, (Int. J. Plast. 17 (2001) 87, Comput. Methods Appl. Mech. Eng. 175 (1999) 19) was used which includes strain hardening, strain rate sensitivity and temperature effect in material behaviour. The large strains are assumed under plane strain conditions. The results with complete thermal coupling are compared with solutions obtained in the isothermal conditions of bending.

Introduction

The theory of cylindrical bending of rigid, perfectly plastic plates in plane strain was developed by Hill [1], also by Lubahn and Sachs [2]. Thereafter, Wolter [3] has considered work hardening materials and has developed solutions based entirely on the analysis of strain field. Hill's concept of fibre movement and his suggestions with regard of displacement equation were used by Proksa [4] to solve plane strain bending of rigid plates with linear strain hardening. Movement of the neutral axis in symmetrical pure bending of plastic beams was discussed later by Phillips [5]. The problem was studied further on by Phillips and Donath [6] for a non-symmetrical pure bending of beams in creep. Malinin and Chirchov [7] studied the problem of the cylindrical plastic bending of a plate by the kinematics approach with the assumption of plane strain. Crafoord [8] has extended Proksa's theory to bending of rigid plates with work hardening and included Bauschinger effect into his analysis. Dadras and Majlessi [9] have proposed a numerical solution based on linear approximation between equivalent stress and strain for fibres in reversed loading, and made an extension of Proksa's analysis to materials described by Ludwik's equation.

Klepaczko et al. [10], [11] presented experimental results on bending of sheet metals with large curvatures. Klepaczko [12] tested the effect of the width of metal strips in the plastic state, subjected to cylindrical bending up to large curvatures, on the bending moments. An analytical solution was obtained describing the bending process of a strip for the plane strain assuming the power law of strain hardening.

The paper by Verguts and Sowerby [13] reported study on pure bending of bonded, laminated metals under plane strain conditions. Each laminate is classified by its initial yield strength and work hardening characteristics. The calculations were performed for rigid, perfectly plastic bi- and tri-metals. Majlessi and Dadras [14] investigated pure plastic bending of two- and three-ply laminates in plane strain conditions. They assumed a rigid with strain hardening material behaviour.

The work by Bell [15] may also be mentioned. The study was focussed on cantilever beams submitted to large deflection, rotation and plastic strain. The complete description of the detailed finite plastic strain response of beams with very large deflections, compatible with general finite strain continuum theory, is given.

In this work, the modelling and numerical computation of inelastic bending process of a sheet metal is considered. The plane strain bending with strain hardening, strain rate sensitivity and thermal effects is analysed. The elasto-viscoplastic temperature-dependent constitutive relations based on earlier works of one of the authors [16], [17] and formulated recently to be applied in numerical codes [18], [19] were introduced in the Marc software package which permits analysis of thermo-mechanical stress fields taking into account geometrical and material non-linearities. It was assumed that the inelastic deformations generate the heat and influence the material properties. The numerical test examples were based on the scheme of a simply supported strip of metal sheet (ES steel-product of SOLLAC®) submitted to displacements in three-point bending by a rigid cylinder. The results of complete, thermally coupled analysis, are compared with solutions obtained in the isothermal conditions. The main motivation of this study stems from the problem of vehicle safety during a crash. During the vehicle crash many components are submitted to fast bending and it is not clear whether the complete thermo-viscoplastic analysis may improve the overall crashworthiness analysis, or the isothermal approach is sufficient (much cheaper solution).