Numerical Simulation of Thermoforming

Numerical simulations of the large constrained deformation of polymeric structures frequently involves, at some stage, geometrical situations which are diicult to model. These are in addition to any diiculties encountered in the description and in the implementation of complicated con-stitutive relations. In the case of thermoforming the dii-cult geometrical situations occur as thin polymeric sheets are deformed into complicated mould shapes to form container structures. In fact in the simplest implementation of thermoforming which involves pressure only forcing action the deformation is essentially`geometry driven' and as a consequence the geometrical diiculties are the main dii-culties in the problem. Geometrical quantities, speciically the thickness distribution, are also among the main quantities of interest. To construct a model of thermoforming the set up is usually as follows. The thin sheet is modelled as a membrane which has the eeect of reducing a 3D problem to a more manageable 2D problem. The membrane is assumed to stick on contact with the mould which corresponds to the warm sheet eeectively freezing to the cool mould surface. With e 1 , e 2 and e 3 as the usual Cartesian base vectors the deformation of the mid-surface is described by F denoting the deformation gradient, F = RU = V R being the polar decomposition, C = F T F and B = F F T = R T CR denoting the right and left Cauchy Green deformation tensors and with denoting the Cauchy Stress tensor the membrane assumption implies that for an incompress-ible material C = 0 @ 21) = 1. With 1 , 2 and 3 = denoting the principal values (eigenvalues) of C and with 1 and 2 denoting the non-zero principal stresses, the stress-stretch relation for an isotropic hyperelastic material with strain energy function W can be expressed in the form Although the deformation in thermoforming occurs quickly, the situation is still essentially quasi-static. At any given stage, when the applied pressure is P , the equilibrium of the part of the sheet fr (P) not yet stuck to the mould is described by ZZ fr h 0 (T : rv) ? P g(u; v) dx 1 dx 2 = 0; 8appropriate v, where = F ?1 is the nominal stress ten-sor and where 3g = v (@w @x 1 @w @x 2) + w (@v @x 1 @w @x 2) + w (@w @x 1 @v @x 2) : In …