A thermodynamic framework has been developed for a class of amorphous
polymers used in fused deposition modeling (FDM), in order to predict the residual
stresses and the accompanying distortion of the geometry of the printed part
(warping). When a polymeric melt is cooled, the inhomogeneous distribution of
temperature causes spatially varying volumetric shrinkage resulting in the generation
of residual stresses. Shrinkage is incorporated into the framework by introducing an
isotropic volumetric expansion/contraction in the kinematics of the body. We show that
the parameter for shrinkage also appears in the systematically derived rate-type
constitutive relation for the stress. The solidification of the melt around the glass
transition temperature is emulated by drastically increasing the viscosity of the melt.
In order to illustrate the usefulness and efficacy of the constitutive relation that
has been developed, we consider four ribbons of polymeric melt stacked on top of
each other such as those extruded using a flat nozzle: each layer laid instantaneously
and allowed to cool for one second before another layer is laid on it. Each layer cools,
shrinks and warps until a new layer is laid, at which time the heat from the newly laid
layer flows into the previous laid layer and heats up the bottom layers. The residual
stresses of the existing and newly laid layers readjust to satisfy equilibrium. Such
mechanical and thermal interactions amongst the layers result in a complex
distribution of residual stresses. The plane strain approximation predicts nearly
equibiaxial tensile stress conditions in the core region of the solidified part, implying
that a preexisting crack in that region is likely to propagate and cause failure of the part during service. The free-end of the interface between the first and the second
layer is subjected to the largest magnitude of combined shear and tension in the plane
with a propensity for delamination.