| We describe two mathematical models for the plastic deformation of flat sheet metal in order to minimize work due to stretching during deformation. The first model is for constant-thickness sheet metal and the second is for thin sheet metal in which thickness varies. Since thickness is considered very small, in both cases we model the sheet metal as a two-dimensional Riemannian manifold S of disc type, and we describe the deformation as a diffeomorphism F from S to the Euclidean plane, {dollar}{lcub}bf R{rcub}sp2{dollar}.; In the constant-thickness case, incompressibility requires that F be area-preserving. We present a way to construct an area-preserving map from any given diffeomorphism between two regions of equal area. We further present an algorithm to compute the second component g of an area-preserving map F = (f,g) when given the level sets of f and boundary data for g. We state a formula for work, or energy of deformation, as a function of the largest eigenvalue {dollar}lambda{dollar} of {dollar}DFsp*DF{dollar}. If no limits are imposed on {dollar}lambda{dollar}, hence on the amount of stretching, we show that there is no non-isometry minimum for the energy.; In the variable-thickness case, we state a formula for the energy of deformation as a function of both eigenvalues of {dollar}DFsp*DF{dollar}, and impose constraints on these eigenvalues from a forming limit diagram. We use a penalty-function gradient-projection algorithm on the resulting constrained minimization problem to construct a deformation numerically.; Finally we calculate the Euler-Lagrange equation for the constant-thickness variational problem and we describe the present state of that problem. |
