A thermo-rheological model for predicting extrusion force in fused filament fabrication

Abstract

We present a thermo-rheological model to predict the extrusion force required for polymer flow in fused filament fabrication hot ends. To capture the complex rheology of polymers, the model accounts for their yield-stress-dominated, solid-like behavior below the glass transition temperature and their shear-thinning behavior above it, using a hybrid rheological formulation to smoothly describe the transition. Thermal boundary conditions of the problem are derived from energy balance principles, eliminating the need for empirical calibration. This approach allows for the dynamic evolution of these conditions, similar to the wall boundary condition, which changes from free-slip to no-slip depending on the local thermal state. For a prescribed volumetric flow rate, the axial pressure gradient is obtained sequentially through a scalar root-finding procedure assuming a quasi-steady, locally fully developed velocity. This axial-marching formulation reduces the computational cost of solving the coupled flow and temperature fields by two orders of magnitude compared to conventional computational methods. Validation against measured extrusion forces for three amorphous and crystalline polymers across a wide range of wall temperatures demonstrates that the model accurately captures the nonlinear increase in force with print speed. Parametric sensitivity analysis reveals a clear hierarchy: thermal properties govern the temperature field, melt rheology influences the force magnitude due to sensitivity to temperature, and solid-state rheological parameters have negligible direct impact on force predictions, while influencing the solid-to-melt transition. The framework provides quantitative guidance for designing 3D printers and identifies progressive thermal under-development at high Péclet numbers as the physical origin of the observed force–velocity transition.