(BQ) Part 2 book "Mechanics of sheet metal forming" has contents: Simplified analysis of circular shells, cylindrical deep drawing, stretching circular shells, hydroforming. | 7 Simplified analysis of circular shells Introduction Many sheet forming operations can be modelled as the deformation of circular (axisymmetric) shells. Examples are stretching a sheet over a domed punch, drawing a circular disc to form a cylindrical cup and tube forming processes such as flaring and necking. Detailed analyses usually require numerical methods, but by making certain simplifying assumptions, useful information about forming loads and strains can obtained from approximate analytical models. This chapter introduces the basic simplifying assumptions and some applications to typical forming processes. The shell element An element of an axisymmetric shell is shown in Figure . The surface element is at a radius r and subtends an angle dθ. The thickness is t and the principal stresses are σθ in the hoop direction and σφ along the meridian; the radial stress perpendicular to the element is considered small so that the element is assumed to deform in plane stress. Also acting on the element are the principal tensions, Tθ = σθ t and Tφ = σφ t. If the element is deforming plastically, the principal stresses will satisfy the yield condition and here we select the Tresca criterion. It follows that in a region in which the thickness is uniform, the tensions will also satisfy a similar condition, and this is illustrated for plane stress, in Figure . This may be considered as a tension yield locus and following an approach similar to that in Section , we identify an effective or representative tension function T . The major simplifying assumption employed here is that the yielding tension T in Figure will remain constant throughout the process. This implies that strain-hardening will balance material thinning, . σf t = T , is constant. This happens to be a better assumption than neglecting strain-hardening. The reason can be seen by reference to Figures (a) and . In drawing processes (along the left-hand diagonal) the material does
