New Horizons in Computer-Aided Design of Sheet Metal Stampings

Abstract

<p>The determination of principal finite strains from measurements made on a pair of deformed line elements is discussed. The deformation process is assumed to be either homogeneous or pure homogeneous in nature. Emphasis is placed on the pure homogenous mode since it leads to a simpler finite strain tensor and this technique is applied to determine the strain over the surface of an industrial stamping. The present work draws the distinction between homogeneous and pure homogeneous deformation. In the latter mode, an orthogonal triad can be identified (the principal axes), which remains orogonal throughout the deformation. An appropriate strain measure in such processes is that of logarithmic strain. Furthermore, its material derivative equals the rate of deformation tensor. Such a simple expression does not hold when the deformation gradient tensor is nonsymmetric, as in homogeneous processes.</p> <p>The material derivative of the tensor logarithm is no longer simply related to the rate of deformation tensor, and this is exemplified herein. The resulting expression involves the spin of the triad of the Eulerian and Lagrangian ellipsoids.</p> <p>Stress components vary as a result of material rotation and constitutive equations whereby rotational effect of material has been accounted for must be formulated. In finite deformation various "rotation" tensors can be defined. Consequently a wide choice of objective stress rates is available for adoption in constitutive equations, and a number of objective stress rates are examined herein. The utility of the resulting expressions is demonstrated for the case of a hypoelastic material undergoing finite deformation in simple (rectilinear) shear.</p> <p>Another aspect of this work has been an attempt to establish an approximate computer-aided technique for blank development, referred to as Geometric Modelling, and the investigation of possible strain distributions in forming sheet metal components. The technique is based on the initial assumption (this can be refined at a later stage) that a sheet metal component is transformed from a fiat sheet into a non-developable surface without change in thickness. Although few practical forming process occur in this way, many traditional die design procedures are based on similar notions; either there is no change in surface area or that a line length on the undeformed blank is unchanged during forming. Simple plasticity theory also suggests that the membrane stresses in a sheet would be minimized (in the absence of a normal stress) if no change in thickness occurred, therefore given the opportunity, the deformation is likely to take place in this ideal manner. The method of geometric modelling simulates the traditional manual calculations performed by experienced tool designers. The technique does not aim to replace the skill and experience of the designers but rather to enhance them.</p> <p>The present work describes the formulation of the fundamental theories of the method which comprise of the element-by-element mapping and remapping procedures and techniques of surface adjustment. The basic geometric assumptions employed in the development are also described. Two particular automotive stampings have been considered; one is the corner section of a car seat panel and the other is an inner deck-lid of a mid-size vehicle.</p> <p>A computer-aided design package for tool/die designers has been developed and the detailed analytical procedure is implemented in Fortran code. The analysis has been performed without access to advanced computer graphics. However, it is carried out in a way that future modelling using interactive computer graphics may well be attainable.</p>