This article focuses on computational modelling of extrusion for viscoelastic and viscoelastoplastic materials. A hybrid parent finite element-subcell finite volume algorithm is utilised with dynamic free-surface location, which draws upon a fractional staged, predictor-corrector, semi-implicit time-stepping procedure. The viscoelastic dimension is introduced by appealing to the exponential Phan-Thien Tanner network class of models, suitable for typical polymer melt response, with properties of shear-thinning, strain-hardening/softening, and moderate-high Trouton ratios. The analysis is then extended into viscoelastoplasticity through Papanastasiou regularisation (viscous-limiting), by combining the viscoplastic Papanastasiou-Bingham model with the viscoelastic Phan-Thien Tanner-(Eptt) model. A systematic study is undertaken on swelling ratio, exit pressure loss and flow response, as a consequence of viscous, plastic and viscoelastic material behaviour. The investigation follows parametric variation in solvent fraction, strain-hardening and yield stress adjustment. Though analysis of normal tensile stresses, root causes are identified for differences in swell effects (and exit pressure losses) due to viscoplastic and viscoelastic response, with elevation in plasticity suppressing swell whilst rise in elasticity stimulates swell. The extent and shape of yielded and unyielded regions is also determined, from which the various yield fronts may be distinguished as yield stress rises. (C) 2012 Elsevier B.V. All rights reserved.